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Study of exotic hadrons in s-wave chiral dynamics Tetsuo Hyodo1,∗), Daisuke Jido1 and Atsushi Hosaka2 1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502, Japan 2Research Center for Nuclear Physics (RCNP), Ibaraki, 567-0047 Japan We study the exotic hadrons in s-wave scattering of the Nambu-Goldstone boson with a target hadron based on chiral dynamics. Utilizing the low energy theorem of chiral symme- try, we show that the s-wave interaction is not strong enough to generate bound states in exotic channels in flavor SU(3) symmetric limit, although the interaction is responsible for generating some nonexotic hadron resonances dynamically. We discuss the renormalization condition adopted in this analysis. §1. Introduction One of the nontrivial issues in hadron physics is almost complete absence of flavor exotic hadrons. Experimentally, we have been observing more than hundred of hadrons,1) whose flavor quantum numbers can be expressed by minimal valence quark contents of q̄q or qqq. The only one exception is the exotic baryon Θ+ with S = +1,2) which is composed of at lease five valence quarks. In this way, the exotic hadrons are indeed “exotic” as an experimental fact. On the other hand, there is no clear theoretical explanation for the nonobservation of the exotic hadrons. Our current knowledge does not forbid to construct four or five quark states in QCD and in effective models. Moreover, the multiquark components in nonexotic hadrons are evident, as seen in the antiquark distribution (or pion cloud) in nucleon and successful descriptions of some excited hadrons as resonances in two-hadron scatterings. In view of these facts, it is fair to say that the nonobservation of the exotic hadrons is not fully understood theoretically. §2. Exotic hadrons in s-wave chiral dynamics In chiral coupled-channel dynamics, some hadron resonances have been success- fully described in s-wave scattering of a hadron and the Nambu-Goldstone (NG) boson,3), 4), 5), 6) along the same line with the old studies with phenomenological vec- tor meson exchange interaction.7), 8) It was found that the generated resonances turned into bound states in flavor SU(3) symmetric limit.9) We therefore conjecture that the bound states in the SU(3) limit are the origin of a certain class of physical resonances, and we examine the possible existence of exotic hadrons as hadron-NG boson bound states.10), 11) The low energy interaction of the NG boson (Ad) with a target hadron (T ) in ∗) e-mail address: hyodo@yukawa.kyoto-u.ac.jp typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.1527v1 2 T. Hyodo, D. Jido and A. Hosaka s-wave is given by Vα = − Cα,T , (2.1) with the decay constant (f) and the energy (ω) of the NG boson. The factor Cα,T is determined by specifying the flavor representations of the target T and the scattering system α ∈ T⊗Ad: Cα,T = −〈2FT · FAd〉α = C2(T )− C2(α) + 3, (2.2) where C2(R) is the quadratic Casimir of SU(3) for the representation R. Eq. (2.1) is the model-independent consequence of chiral symmetry, known as the Weinberg- Tomozawa theorem.12), 13) We have written down the general expression of the coupling strengths (2.2) for arbitrary representations of target hadrons in SU(3). In order to specify the exotic channels, we introduced the exoticness quantum number, as the number of valence antiquarks to construct the given flavor multiplet for the states with positive baryon number. Then we find that the Weinberg-Tomozawa interaction in the exotic channels is repulsive in most cases, and that possible strength of the attractive interaction is given by a universal value Cexotic = 1, (2.3) with α = [p− 1, 2] for T = [p, 0] and p ≥ 3B.10), 11) Next we construct the scattering amplitude with unitarity condition using the N/D method.5) The unitarized amplitude is given by 1− Vα( as a function of the center-of-mass energy s. The loop function G( s) is regularized by the once subtraction as s) = −ã(s0)− ρ(s′) s′ − s − ρ(s s′ − s0 , (2.4) where the phase space integrand is ρ(s) = 2MT (s− s+)(s− s−)/(8πs), s± = (m±MT )2, and m and MT are the masses of the target hadron and the NG boson. In order to determine the subtraction constant ã(s0) and the subtraction point s0, we adopt the renormalization condition given in Refs. 14), 6), G(µ) = 0, µ = MT , (2.5) which is equivalent to tα(µ) = Vα(µ) at this scale. We will discuss the implication of this prescription in section 3. With the condition (2.5), we show that the bound state can be obtained if the coupling strength (2.2) is larger than the critical value Ccrit = −G(MT +m) Exotic hadrons in s-wave chiral dynamics 3 Varying the parameters m, MT , and f in the physically allowed region, we show that the attraction in the exotic channels (2.3) is always smaller than the critical value Ccrit. Thus, it is not possible to generate bound states in exotic channels in the SU(3) symmetric limit. We would like to emphasize that this conclusion is model independent in the SU(3) limit, as far as we respect chiral symmetry. In this study, we only consider the exotic hadrons composed of the NG boson and a hadron, so the existence of exotic states generated by quark dynamics or rotational excitations of chiral solitons is not excluded. In practice, one should bear in mind that the SU(3) breaking effect and higher order terms in the chiral Lagrangian would play a substantial role. Nevertheless, the study of exotic hadrons in a simple extension of a successful model of hadron resonances as we have done in the present work can partly explain difficulty of observation of the exotic hadrons. §3. Interpretation of the renormalization condition Here we discuss the renormalization condition (2.5) in this analysis. The inter- action kernel Vα( s) is constructed from chiral perturbation theory so as to satisfy the low energy theorem. The low energy theorem also constrains the behavior of the full unitarized amplitude tα( s) at a scale s = µm where the chiral expansion is valid. Therefore we can match the unitarized amplitude tα( s) with the tree level one Vα( s) at the scale µm: tα(µm) = Vα(µm) + Vα(µm)G(µm)Vα(µm) + · · · = Vα(µm), (3.1) This condition determines the subtraction constant such that the loop function G(µm) vanishes. This is only possible within the region MT −m ≤ µm ≤ MT +m, (3.2) since the loop function has an imaginary part outside this region and the subtraction constant is a real number. We consider that by employing this renormalization condition, a natural unitarization of the kernel interaction based on chiral symmetry is realized. Interestingly, if we apply this prescription for the case of the octet baryon target, the subtraction constant turns out to be “natural size” which was found in the comparison with three-momentum cutoff,5) and the experimental observables in the S = −1 meson-baryon channel are successfully reproduced. We take µm = MT in the present study. The dependence on µm within the region (3.2) is found that the binding energy of the bound state increases if we shift the matching scale µm to the lower energy region. This is discussed also in Refs. 17),18) by varying the subtraction constant. The region µm ≤ MT corresponds to the u-channel scattering. Thus µm = MT is the most favorable to generate a bound state within the s-channel regime. The unitarized amplitudes in the above prescription do not always reproduce ex- perimental data. In such a case, the subtraction constants ã(s0) should be adjusted in order to satisfy experimental data. The subtraction constants determined in this 4 T. Hyodo, D. Jido and A. Hosaka way supplement the role of the higher order chiral Lagrangians, which is lacking in the kernel interaction. As shown for the ρ meson effect in the meson-meson scatter- ing,16) the higher order terms may contain the effect of the resonances. Therefore, if the natural condition (3.1) is badly violated, one may speculate that the seeds of resonances in the higher order Lagrangian, which are possibly the genuine quark states, appear in the unitarized amplitude, as in the study of Ref. 19). In summary, we have argued the following issues. • In order for the unitarized amplitude tα( s) to satisfy the low energy theorem, the loop function should vanish in the region (3.2). • This requirement can be regarded as a natural unitarization, without introduc- ing the effect of resonances in the higher order Lagrangian. • Lower matching scale µm is more favorable to generate a bound state. Turning to the problem of exotic hadrons, what we have shown is the nonexistence of the exotic bound states with the most favorable condition to generate a bound state, without introducing the seed of genuine quark state. Acknowledgements T. H. thanks the Japan Society for the Promotion of Science (JSPS) for finan- cial support. This work is supported in part by the Grant for Scientific Research (No. 17959600, No. 18042001, and No. 16540252) and by Grant-in-Aid for the 21st Century COE ”Center for Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. References 1) Particle Data Group, W.M. Yao et al., J. of Phys. G33 (2006), 1. 2) T. Nakano et al., (LEPS Collaboration), Phys. Rev. Lett. 91 (2003), 012002 3) N. Kaiser, P.B. Siegel, and W. Weise, Nucl. Phys. A 594 (1995), 325 4) E. Oset and A. Ramos, Nucl. Phys. A 635 (1998), 99 5) J.A. Oller and U.G. Meissner, Phys. Lett. B 500 (2001), 263 6) M.F.M. Lutz and E.E. Kolomeitsev, Nucl. Phys. A 700 (2002), 193 7) R.H. Dalitz and S.F. Tuan, Ann. of Phys. 10 (1960), 307 8) J.H.W. Wyld, Phys. Rev. 155 (1967), 1649 9) D. Jido, J.A. Oller, E. Oset, A. Ramos, and U.G. Meissner, Nucl. Phys. A 725 (2003), 181 10) T. Hyodo, D. Jido, and A. Hosaka, Phys. Rev. Lett. 97 (2006), 192002 11) T. Hyodo, D. Jido, and A. Hosaka, Phys. Rev. D 75 (2007), 034002 12) S. Weinberg, Phys. Rev. Lett. 17 (2966), 616 13) Y. Tomozawa, Nuovo Cim. 46A (1966), 707 14) K. Igi and K.-i. Hikasa, Phys. Rev. D 59 (1999), 034005 15) T. Hyodo, D. Jido and A. Hosaka, hep-ph/0612333 16) G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B 321 (1989), 311 17) T. Hyodo, S.I. Nam, D. Jido and A. Hosaka, Phys. Rev. C 68 (2003), 018201 18) T. Hyodo, S.I. Nam, D. Jido, and A. Hosaka, Prog. Theor. Phys. 112 (2004), 73 19) J.A. Oller, E. Oset and J.R. Pelaez, Phys. Rev. D 59 (1999), 074001, Erratum-ibid. D 60 (1999) 099906. http://arxiv.org/abs/hep-ph/0612333 Introduction Exotic hadrons in s-wave chiral dynamics Interpretation of the renormalization condition

We study the exotic hadrons in s-wave scattering of the Nambu-Goldstone boson with a target hadron based on chiral dynamics. Utilizing the low energy theorem of chiral symmetry, we show that the s-wave interaction is not strong enough to generate bound states in exotic channels in flavor SU(3) symmetric limit, although the interaction is responsible for generating some nonexotic hadron resonances dynamically. We discuss the renormalization condition adopted in this analysis.

Introduction One of the nontrivial issues in hadron physics is almost complete absence of flavor exotic hadrons. Experimentally, we have been observing more than hundred of hadrons,1) whose flavor quantum numbers can be expressed by minimal valence quark contents of q̄q or qqq. The only one exception is the exotic baryon Θ+ with S = +1,2) which is composed of at lease five valence quarks. In this way, the exotic hadrons are indeed “exotic” as an experimental fact. On the other hand, there is no clear theoretical explanation for the nonobservation of the exotic hadrons. Our current knowledge does not forbid to construct four or five quark states in QCD and in effective models. Moreover, the multiquark components in nonexotic hadrons are evident, as seen in the antiquark distribution (or pion cloud) in nucleon and successful descriptions of some excited hadrons as resonances in two-hadron scatterings. In view of these facts, it is fair to say that the nonobservation of the exotic hadrons is not fully understood theoretically. §2. Exotic hadrons in s-wave chiral dynamics In chiral coupled-channel dynamics, some hadron resonances have been success- fully described in s-wave scattering of a hadron and the Nambu-Goldstone (NG) boson,3), 4), 5), 6) along the same line with the old studies with phenomenological vec- tor meson exchange interaction.7), 8) It was found that the generated resonances turned into bound states in flavor SU(3) symmetric limit.9) We therefore conjecture that the bound states in the SU(3) limit are the origin of a certain class of physical resonances, and we examine the possible existence of exotic hadrons as hadron-NG boson bound states.10), 11) The low energy interaction of the NG boson (Ad) with a target hadron (T ) in ∗) e-mail address: hyodo@yukawa.kyoto-u.ac.jp typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.1527v1 2 T. Hyodo, D. Jido and A. Hosaka s-wave is given by Vα = − Cα,T , (2.1) with the decay constant (f) and the energy (ω) of the NG boson. The factor Cα,T is determined by specifying the flavor representations of the target T and the scattering system α ∈ T⊗Ad: Cα,T = −〈2FT · FAd〉α = C2(T )− C2(α) + 3, (2.2) where C2(R) is the quadratic Casimir of SU(3) for the representation R. Eq. (2.1) is the model-independent consequence of chiral symmetry, known as the Weinberg- Tomozawa theorem.12), 13) We have written down the general expression of the coupling strengths (2.2) for arbitrary representations of target hadrons in SU(3). In order to specify the exotic channels, we introduced the exoticness quantum number, as the number of valence antiquarks to construct the given flavor multiplet for the states with positive baryon number. Then we find that the Weinberg-Tomozawa interaction in the exotic channels is repulsive in most cases, and that possible strength of the attractive interaction is given by a universal value Cexotic = 1, (2.3) with α = [p− 1, 2] for T = [p, 0] and p ≥ 3B.10), 11) Next we construct the scattering amplitude with unitarity condition using the N/D method.5) The unitarized amplitude is given by 1− Vα( as a function of the center-of-mass energy s. The loop function G( s) is regularized by the once subtraction as s) = −ã(s0)− ρ(s′) s′ − s − ρ(s s′ − s0 , (2.4) where the phase space integrand is ρ(s) = 2MT (s− s+)(s− s−)/(8πs), s± = (m±MT )2, and m and MT are the masses of the target hadron and the NG boson. In order to determine the subtraction constant ã(s0) and the subtraction point s0, we adopt the renormalization condition given in Refs. 14), 6), G(µ) = 0, µ = MT , (2.5) which is equivalent to tα(µ) = Vα(µ) at this scale. We will discuss the implication of this prescription in section 3. With the condition (2.5), we show that the bound state can be obtained if the coupling strength (2.2) is larger than the critical value Ccrit = −G(MT +m) Exotic hadrons in s-wave chiral dynamics 3 Varying the parameters m, MT , and f in the physically allowed region, we show that the attraction in the exotic channels (2.3) is always smaller than the critical value Ccrit. Thus, it is not possible to generate bound states in exotic channels in the SU(3) symmetric limit. We would like to emphasize that this conclusion is model independent in the SU(3) limit, as far as we respect chiral symmetry. In this study, we only consider the exotic hadrons composed of the NG boson and a hadron, so the existence of exotic states generated by quark dynamics or rotational excitations of chiral solitons is not excluded. In practice, one should bear in mind that the SU(3) breaking effect and higher order terms in the chiral Lagrangian would play a substantial role. Nevertheless, the study of exotic hadrons in a simple extension of a successful model of hadron resonances as we have done in the present work can partly explain difficulty of observation of the exotic hadrons. §3. Interpretation of the renormalization condition Here we discuss the renormalization condition (2.5) in this analysis. The inter- action kernel Vα( s) is constructed from chiral perturbation theory so as to satisfy the low energy theorem. The low energy theorem also constrains the behavior of the full unitarized amplitude tα( s) at a scale s = µm where the chiral expansion is valid. Therefore we can match the unitarized amplitude tα( s) with the tree level one Vα( s) at the scale µm: tα(µm) = Vα(µm) + Vα(µm)G(µm)Vα(µm) + · · · = Vα(µm), (3.1) This condition determines the subtraction constant such that the loop function G(µm) vanishes. This is only possible within the region MT −m ≤ µm ≤ MT +m, (3.2) since the loop function has an imaginary part outside this region and the subtraction constant is a real number. We consider that by employing this renormalization condition, a natural unitarization of the kernel interaction based on chiral symmetry is realized. Interestingly, if we apply this prescription for the case of the octet baryon target, the subtraction constant turns out to be “natural size” which was found in the comparison with three-momentum cutoff,5) and the experimental observables in the S = −1 meson-baryon channel are successfully reproduced. We take µm = MT in the present study. The dependence on µm within the region (3.2) is found that the binding energy of the bound state increases if we shift the matching scale µm to the lower energy region. This is discussed also in Refs. 17),18) by varying the subtraction constant. The region µm ≤ MT corresponds to the u-channel scattering. Thus µm = MT is the most favorable to generate a bound state within the s-channel regime. The unitarized amplitudes in the above prescription do not always reproduce ex- perimental data. In such a case, the subtraction constants ã(s0) should be adjusted in order to satisfy experimental data. The subtraction constants determined in this 4 T. Hyodo, D. Jido and A. Hosaka way supplement the role of the higher order chiral Lagrangians, which is lacking in the kernel interaction. As shown for the ρ meson effect in the meson-meson scatter- ing,16) the higher order terms may contain the effect of the resonances. Therefore, if the natural condition (3.1) is badly violated, one may speculate that the seeds of resonances in the higher order Lagrangian, which are possibly the genuine quark states, appear in the unitarized amplitude, as in the study of Ref. 19). In summary, we have argued the following issues. • In order for the unitarized amplitude tα( s) to satisfy the low energy theorem, the loop function should vanish in the region (3.2). • This requirement can be regarded as a natural unitarization, without introduc- ing the effect of resonances in the higher order Lagrangian. • Lower matching scale µm is more favorable to generate a bound state. Turning to the problem of exotic hadrons, what we have shown is the nonexistence of the exotic bound states with the most favorable condition to generate a bound state, without introducing the seed of genuine quark state. Acknowledgements T. H. thanks the Japan Society for the Promotion of Science (JSPS) for finan- cial support. This work is supported in part by the Grant for Scientific Research (No. 17959600, No. 18042001, and No. 16540252) and by Grant-in-Aid for the 21st Century COE ”Center for Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. References 1) Particle Data Group, W.M. Yao et al., J. of Phys. G33 (2006), 1. 2) T. Nakano et al., (LEPS Collaboration), Phys. Rev. Lett. 91 (2003), 012002 3) N. Kaiser, P.B. Siegel, and W. Weise, Nucl. Phys. A 594 (1995), 325 4) E. Oset and A. Ramos, Nucl. Phys. A 635 (1998), 99 5) J.A. Oller and U.G. Meissner, Phys. Lett. B 500 (2001), 263 6) M.F.M. Lutz and E.E. Kolomeitsev, Nucl. Phys. A 700 (2002), 193 7) R.H. Dalitz and S.F. Tuan, Ann. of Phys. 10 (1960), 307 8) J.H.W. Wyld, Phys. Rev. 155 (1967), 1649 9) D. Jido, J.A. Oller, E. Oset, A. Ramos, and U.G. Meissner, Nucl. Phys. A 725 (2003), 181 10) T. Hyodo, D. Jido, and A. Hosaka, Phys. Rev. Lett. 97 (2006), 192002 11) T. Hyodo, D. Jido, and A. Hosaka, Phys. Rev. D 75 (2007), 034002 12) S. Weinberg, Phys. Rev. Lett. 17 (2966), 616 13) Y. Tomozawa, Nuovo Cim. 46A (1966), 707 14) K. Igi and K.-i. Hikasa, Phys. Rev. D 59 (1999), 034005 15) T. Hyodo, D. Jido and A. Hosaka, hep-ph/0612333 16) G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B 321 (1989), 311 17) T. Hyodo, S.I. Nam, D. Jido and A. Hosaka, Phys. Rev. C 68 (2003), 018201 18) T. Hyodo, S.I. Nam, D. Jido, and A. Hosaka, Prog. Theor. Phys. 112 (2004), 73 19) J.A. Oller, E. Oset and J.R. Pelaez, Phys. Rev. D 59 (1999), 074001, Erratum-ibid. D 60 (1999) 099906. http://arxiv.org/abs/hep-ph/0612333 Introduction Exotic hadrons in s-wave chiral dynamics Interpretation of the renormalization condition

Microsoft Word - K-SC6.doc April 2, 2007 I. INTRODUCTION The study of non-Cu-based oxide superconductors has been extended during the last decade, aiming at understanding the role of electron correlations in the mechanism of superconductivity or searching for a novel pairing mechanism, hopefully to reach a higher Tc. An interesting example recently found is a family of pyrochlore oxide superconductors. The first discovered is α-pyrochlore rhenate Cd2Re2O7 with Tc = 1.0 K 1-3 and the second β-pyrochlore osmate AOs2O6 with Tc = 3.3, 6.3, and 9.6 K for A = Cs,4 Rb,5-7 and K,8 respectively. They crystallize in the cubic pyrochlore structure of space group Fd-3m and commonly possess a 3D skeleton made of ReO6 or OsO6 octahedra. 9 The "pyrochlore" sublattice occupied by the transition metal ions is comprised of corner-sharing tetrahedra that are known to be highly frustrating for a localized spin system with antiferromagnetic nearest-neighbor interactions. A unique structural feature for the β-pyrochlores is that a relatively small A ion is located in an oversized atomic cage made of OsO6 octahedra, Fig. 1. Due to this large size mismatch, the A atom can rattle in the cage.10 The rattling has been recognized recently as an interesting phenomenon for a class of compounds like filled skutterudites11 and Ge/Si clathrates12 and attracted many researchers, because it may suppress thermal conductivity leading to an enhanced thermoelectric efficiency. On the other hand, the rattling is also intriguing from the viewpoint of lattice dynamics: it gives an almost localized mode even in a crystalline material and often exhibits unusual anharmonicity.11 Hence, it is considered that the rattling is a new type of low-lying excitations that potentially affects various properties in a crystal at low temperature. In the β pyrochlores, specific heat experiments found low-energy contributions that could be described approximately by the Einstein model and determined the Einstein temperature TE to be 70 K, 60 K, and 40 or 31 K for A = Cs, Rb, and K, respectively.13, 14 This tendency illustrates uniqueness of the rattling, because, to the contrary, one expects a higher frequency for a lighter atom in the case of conventional phonons. Moreover, it was demonstrated that the specific heat shows an unusual T5 dependence at low temperature below 7 K for A = Cs and Rb, instead of a usual T3 dependence from a Debye-type phonon.13 On one hand, structural refinements revealed large atomic displacement parameters at room temperature of 100Uiso = 2.48, 4.26, and 7.35 Å2 for A = Cs, Rb, and K, respectively,10 and 3.41 Å2 for Rb.15 Particularly, the value for K is enormous and may be the largest among rattlers so far known in related compounds. This trend over the β-pyrochlore series is ascribed to the fact that an available space for the A ion to move in a rather rigid cage increases with decreasing the ionic radius of the A ion.10 Kuneš et al. calculated an energy potential for each A ion and found in fact a large anharmonicity, that is, a deviation from a quadratic form expected for the harmonic oscillator approximation.16 Especially for the smallest K ion, they found 4 shallow potential minima locating away from the center (8b site) along the <111> directions pointing to the nearest K ions, as schematically shown in Fig. 1. The potential minima are so shallow that the K ion may not stop at one of them even at very low temperature. The electronic structures of α-Cd2Re2O7 and β-AOs2O6 have been calculated by first-principle density-functional methods, which reveal that a metallic conduction occurs in the (Re, Os)-O network:16-20 electronic states near the Fermi level originate from transition metal 5d and O 2p orbitals. Although the overall shape of the density of states (DOS) is similar for the two pyrochlores, a difference in band filling may result in different properties; Re5+ for α-Cd2Re2O7 has two 5d electrons, while Os 5.5+ for β-AOs2O6 has two and a half. Moreover, a related α-pyrochlore Cd2Os2O7 with Os5+ (5d3) exhibits a metal-to-insulator transition at 230 K.21, Various experiments have been carried out on the pyrochlore oxide superconductors to elucidate the mechanism of the superconductivity. Most of the results obtained for α-Cd2Re2O7 indicate that it is a weak-coupling BCS-type superconductor.23, 24 In contrast, results on the β-pyrochlores are somewhat controversial. Although the Tc increases smoothly from Cs to K, the jump in specific heat at Tc, the upper critical field, and the Sommerfeld coefficient all exhibit a large enhancement toward K.25 Thus, the K compound is always distinguished from the others. Pressure dependence of Tc was also studied for the two pyrochlores, showing a common feature: as pressure increases, Tc first increases, exhibits a broad maximum and goes to zero above a critical pressure that depends on the system, for example, about 6 GPa for the K compound.26-29 On the symmetry of the superconducting gap for the β-pyrochlores, Rb-NMR and µSR experiments gave evidence for s-wave superconductivity for RbOs2O6. 30-32 In contrast, Arai et al. carried out K-NMR experiments and found no coherence peak in the relaxation rate below Tc for KOs2O6, which seemed to indicate unconventional superconductivity.32 However, their recent interpretation is that the absence of a coherence peak does not necessarily mean non-s pairing, because the relaxation rate probed by the K nuclei can be affected dominantly by strongly overdamped phonons.33 Kasahara et al. measured thermal conductivity using a KOs2O6 single crystal and concluded a full Extremely strong-coupling superconductivity and anomalous lattice properties in the β-pyrochlore oxide KOs2O6 Z. Hiroi, S. Yonezawa, Y. Nagao and J. Yamaura Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Superconducting and normal-state properties of the β-pyrochlore oxide KOs2O6 are studied by means of thermodynamic and transport measurements. It is shown that the superconductivity is of conventional s-wave type and lies in the extremely strong-coupling regime. Specific heat and resistivity measurements reveal that there are characteristic low-energy phonons that give rise to unusual scattering of carriers due to strong electron-phonon interactions. The entity of the low-energy phonons is ascribed to the heavy rattling of the K ion confined in an oversized cage made of OsO6 octahedra. It is suggested that this electron-rattler coupling mediates the Cooper pairing, resulting in the extremely strong-coupling superconductivity. gap from the insensitivity of thermal conductivity to magnetic fields.34 Moreover, very recent photoemission spectroscopy (PES) experiments revealed the opening of a large isotropic gap below Tc. 35 On one hand, a µSR experiment claimed that the gap of KOs2O6 is anisotropic, or otherwise, there are two gaps. Therefore, the pairing symmetry of the β-AOs2O6 superconductors may be of the conventional s wave, aside from minor aspects such as anisotropy or multi gaps, which means that the fundamental pairing mechanism is ascribed to phonons. Then, an important question is what kind of phonons are relevant for the occurrence. To find out the reason of the observed singular appearance toward K in the series must be the key to understand interesting physics involved in this system. Previous studies have suffered poor quality of samples, because only polycrystalline samples were available. The Sommerfeld coefficient γ was estimated from specific heat by extracting contributions from Os metal impurity to be 40 mJ K-2 mol-1 for both Cs and Rb,13 and 34 mJ K-2 mol-1 for Rb.7 Recently, Brühwiler et al. obtained large values of γ = 76-110 mJ K-2 mol-1 for KOs2O6 by collecting five dozen of tiny crystals. 14 However, there is an ambiguity in their values, because of uncertainty in their extrapolation method. They also reported strong-coupling superconductivity with a coupling constant λep = 1.0-1.6. Recently, we successfully prepared a large single crystal of 1 mm size for KOs2O6 and reported two intriguing phenomena: one is a sharp and huge peak in specific heat at Tp = 7.5-7.6 K below Tc, indicative of a first-order structural transition,37, 38 and the other is anisotropic flux pinning at low magnetic fields around 2 T.39 It was suggested that the former is associated with the rattling freedom of the K ion. Moreover, anomalous concave-downward resistivity was observed down to Tc, suggesting a peculiar scattering mechanism of carriers. In this paper, we present specific heat, magnetization and resistivity measurements on the same high-quality single crystal of KOs2O6. Reliable data on the superconducting and normal-state properties are obtained, which provide evidence for an extremely strong-coupling superconductivity realized in this compound. We discuss the role of rattling vibrations of the K ion on the mechanism of the superconductivity. II. EXPERIMENTAL A. Sample preparation A high-quality single crystal was prepared and used for all the measurements in the present study, which was named KOs-729 after the date of July 29, 2005 when the first experiment was performed on this crystal. It was grown from a pellet containing an equimolar mixture of KOsO4 and Os metal in a sealed quartz tube at 723 K for 24 h. Additional oxygen was supplied by using the thermal decomposition of AgO placed away from the pellet in the tube. The KOsO4 powder had been prepared in advance from KO2 and Os metal in the presence of excess oxygen. It was necessary to pay attention to avoid the formation of OsO4 in the course of preparation, which is volatile even at room temperature and highly toxic to eyes or nose. After the reaction, several tiny crystals had grown on the surface of the pellet. Although the mechanism of the crystal growth has not yet been understood clearly, probably it occurs through partial melting and the following reaction with a vapor phase. The KOs-729 crystal possesses a truncated octahedral shape with a large (111) facet as shown in Fig. 2 and is approximately 1.0 × 0.7 × 0.3 mm3 in size and 1.302 mg in weight. The high quality of the crystal has been demonstrated by a sharp peak at Tp in specific heat,38 which was absent in the previous polycrystalline samples or appeared as broad humps in our previous aggregate of tiny crystals37 or in five dozen of tiny crystals by Brühwiler et al.14 Moreover, a dramatic angle dependence of flux-flow resistance was observed on this crystal, indicating that flux pinning is enhanced in magnetic fields along certain crystallographic directions such as [110], [001], and [112].39 This evidences the absence of domains in this relatively large crystal. A special care has been taken to keep the crystal always in a dry atmosphere, because it readily undergoes hydration in air, as reported in isostructural compounds such as KNbWO6. 40, 41 Once partial hydration takes place in KOs2O6, the second sharp anomaly in specific heat tends to collapse. In contrast, the superconducting transition was robust, just slightly broadened after hydration. The hydration must be relatively slow in a single crystal compared with the case of polycrystalline samples, which may be the reason for the broad anomaly or the absence in previous samples. B. Physical-property measurements Both specific heat and electrical resistivity were measured in a temperature range between 300 K and 0.4 K and in magnetic fields of up to 14 T in a Quantum Design Physical Property Measurement System (PPMS) equipped with a 3He refrigerator. The magnetic fields had been calibrated by measuring the magnetization of a standard Pd specimen and also by measuring the voltage of a Hall device (F.W. BELL, BHA-921). Specific heat measurements FIG. 1. (Color online) Crystal structure of the β-pyrochlore oxide KOs2O6. The K ion (big ball) is located in an oversized atomic cage made of OsO6 octahedra and can move along the 4 <111> directions pointing to the neighboring K ions in adjacent cages. FIG. 2. (Color online) Photograph of the KOs-729 crystal used in the present study. It possesses a truncated octahedral shape with 111 facets. The approximate size is 1.0 × 0.7 × 0.3 mm3. Resistivity measurements were carried out with a current flow along the [-110] direction. were performed by the heat-relaxation method. The KOs-729 crystal was attached to an alumina platform by a small amount of Apiezon N grease. In each measurement, heat capacity was obtained by fitting a heat relaxation curve recorded after a heat pulse giving a temperature rise of approximately 2%. The heat capacity of an addendum had been measured in a separate run without a sample, and was subtracted from the data. The measurements were done three times at each temperature with a scatter less than 0.3% at most. Resistivity measurements were carried out by the four-probe method with a current flow along the [-110] direction and magnetic fields along the [111], [110], [001] or [112] direction of the cubic crystal structure. All the measurements were done at a current density of 1.5 A cm-2. Magnetization was measured in magnetic fields up to 7 T in a Quantum Design magnetic property measurement system and also up to 14 T in PPMS. The magnetic fields were applied approximately along the [111] or [-110] direction. III. RESULTS A. Superconducting properties 1. Specific heat First of all, we analyze specific heat data in order to obtain a reliable value of the Sommerfeld coefficient γ. There are two obstacles: one is the large upper critical field Hc2 that is approximately 2 times greater than our experimental limit of 14 T. Brühwiler et al. reported γ = 76 (110) mJ K-2 mol-1 assuming µ0Hc2 = 24 (35) T by an extrapolation method.14 Their values should be modified to γ ~ 100 mJ K-2 mol-1, because recent high magnetic field experiments revealed µ0Hc2 = 30.6 T or 33 T. 42, 43 Nevertheless, there are still large ambiguity in their extrapolation method using specific heat data obtained only at H/Hc2 < 0.5. The other difficulty comes from unusual lattice contributions in specific heat at low temperature and the existence of a sharp peak at Tp. Thus, it is not easy to extract the lattice contribution in a standard way used so far. Here we carefully analyze specific heat data and reasonably divide them into electronic and lattice parts, from which a reliable value of γ is determined, and information on the superconducting gap is attained. Figure 3 shows the temperature dependence of specific heat of the KOs-729 crystal measured on cooling at zero field and in a magnetic field of 14 T applied along the [111] direction. A superconducting transition at zero field takes place with a large jump, followed by a huge peak due to the second phase transition at Tp = 7.5 K. The entropy-conserving construction shown in the inset gives Tc = 9.60 K, ΔC/Tc = 201.2 mJ K -2 mol-1, which is close to the values previously reported,14, 37 and a transition width (ΔTc) of 0.3 K. Tc is reduced to 5.2 K at 14 T, which is evident as a bump in the 14-T data shown in Fig. 3. In contrast, Brühwiler et al. reported Tc = 6.2 K at 14 T, though the transition was not clearly observed in their specific heat data. The C/T at zero field rapidly decreases to zero as T approaches absolute zero. The absence of a residual T-linear contribution in specific heat indicates the high quality of the crystal. The specific heat of a crystal (C) is the sum of an electronic contribution (Ce) and an H-independent lattice contribution (Cl). The former becomes Cen for the normal state above Tc, which is taken as γT, and Ces for the superconducting state below Tc. The γ is assumed to be T-independent, though it can not be the case for compounds with strong electron-phonon couplings.44 In the case of KOs2O6, Cl is large relative to Ce: for example, Cl is as large as ~90% of the total C at just above Tc, as shown later. In order to determine the value of γ, it is crucial to know the low-temperature form of Cl. Since the minimum Tc attained at 14 T is 5.2 K, one has to estimate the Cl from the T dependence of the total C above ~5.5 K. Two terms in the harmonic-lattice approximation are often required for an adequate fit; Cl = β3T 3 + β5T 5. The first term comes from a Debye-type acoustic phonon, and thus is dominant at low temperature, while the second term expresses a deviation at high temperature. Actually, this approximation is valid for α-Cd2Re2O7, where β3 = 0.222 mJ K -4 mol-1 and β5 = 2.70 × 10 mJ K-6 mol-1 are obtained by a fit to the data below 10 K.1 The Debye temperature ΘD is 458 K from the β3 value. In strong contrast, it was found for two members of β-AOs2O6 that the T term prevails in a wide temperature range; β5 = 14.2 × 10 -3 mJ K-6 mol-1 below 5 K for CsOs2O6 and β5 = 30.2 × 10 -3 mJ K-6 mol-1 below 7 K for RbOs2O6. The C/T at H = 0 below 7 K shown in Fig. 3 is again plotted in two ways as functions of T2 and T4 in Fig. 4. It is apparent from the T2 plot that possible T3 terms expected for ΘD = 458 K and 300 K are negligibly small compared with the whole magnitude of specific heat, just as observed in other members. On the other hand, in the T4 plot, there is distinct linear behavior at low temperature below 4 K, indicating that the C approaches asymptotically to T5 behavior as T → 0 with a large slope of 0.3481(6) mJ K-6 mol-1. It is reasonable to ascribe this T5 contribution to the lattice, because Ces should decrease quickly as T → 0. Note that the value of the β5 for KOs2O6 is more than one order larger than those in other members. At high temperatures above 4 K, a downward deviation from the initial T5 behavior is observed in Fig. 4b. The temperature dependence of the 14-T data above 5.5 K, which is taken as γT + Cl, is also close to T 5, but with a smaller slope, which means that a single T5 term is not appropriate to describe the Cl in such a wide temperature range and also that an inclusion of higher order term of Tn is not helpful. Therefore, we adopt expediently an alternative empirical form to express this strange lattice contribution; Cl = β5T 5f(T), where f(T) = [1 + exp(1 - pT-q)]-1. Since the f(T) is almost unity below a certain temperature and decreases gradually with increasing T, this Cl can reproduce T5 behavior at low temperatures and a weaker T FIG. 3. (Color online) Specific heat divided by temperature measured at zero field (circle) and a magnetic field of 14 T (triangle) applied along the [111] direction. The inset shows an enlargement of the superconducting transition with an entropy-conserving construction. dependence at high temperatures. As shown in Fig. 4b, the 14-T data in the 5.5-7 K range can be fitted well by the function for a value of β5 fixed to the initial slope of 0.3481 mJ K -6 mol-1 and a given value of γ; for example, p = 6.96(3) and q = 1.09(1) for γ = 70 mJ K-2 mol-1. In order to determine the value of γ univocally, the entropy conservation is taken into account for the 14-T data, as shown in Fig. 5: since the normal-state specific heat expected for the case of Tc = 0 is given by (Cen + Cl) (dotted line in Fig. 5), the integration of [Cen + Cl - C(14 T)]/T should become zero due to entropy balance. It is shown in the inset to Fig. 5 that the integrated value changes almost linearly with γ and vanishes around γ = 70 mJ K-2 mol-1. Hence, one can determine the value of γ unambiguously. A certain ambiguity may arise from the assumed lattice function. However, since the temperature dependence of Cl is substantially weak in the T range of interest, a possible correction on the γ value must be minimal, say, less than 1 mJ K-2 mol-1. Next we determine Ces at zero field by subtracting the Cl estimated above. The temperature dependence of Ces does have the BCS form, aexp(-Δ/kBTc), as shown in Fig. 6. The energy gap Δ obtained by fitting is 22.5 K, which corresponds to 2Δ/kBTc = 4.69, much larger than the BCS value of 3.53. The above Ces at low temperature below 7 K is again plotted in Fig. 7 together with high-temperature Ces above 5.5 K, which is obtained by subtracting the 14-T data from the zero field data as Ces = C(0) - C(14 T). The two data sets obtained independently overlap well in the 5.5-7 K range, assuring the validity of the above analyses. Because of the existence of the second peak and its small shifts under magnetic fields, the data between 7 K and 8.3 K is to be excluded in the following discussion. Taking γ = 70 mJ K-2 mol-1, the jump in specific heat at Tc, ΔC/γTc, reaches 2.87, much larger than 1.43 expected for a weak-coupling superconductor, indicating that KOs2O6 lies in the strong-coupling regime. Comparisons to other typical strong-coupling superconductors are made in section IV-B. Here we analyze the data based on the α model that was developed to provide a semi-empirical approximation to the thermodynamic properties of strong-coupled superconductors in a wide range of coupling strengths with a single adjustable parameter, α = Δ0/kBTc. 45 Recently, it was generalized to a multi-gap superconductor and successfully applied to the analyses on MgB2 or Nb3Sn. 46, 47 Using the α model, the data in the vicinity of Tc is well reproduced, as shown in Fig. 7, and we obtain α = 2.50 (2Δ0/kBTc = 5.00), slightly larger than the value obtained above from the temperature dependence of Ces. One interesting point to be noted is that there is a significant deviation between the data and the fitting curve at intermediate temperatures, suggesting the existence of an additional structure in the gap. Presumably, this enhancement would be explained if one assumes the coexistence of another smaller gap (not so small as in MgB2, but intermediate). This possibility has been already pointed out in the previous µSR experiment.36 However, ambiguity associated with the second peak in the present data prevents us from further analyzing the data. This important issue will be revisited in future work, where the FIG. 6. (Color online) Temperature dependence of electronic specific heat measured at H = 0 for the superconducting state showing an exponential decrease at low temperature. A magnitude of the gap obtained is 2Δ/kBTc = 4.69. FIG. 4. (Color online) Low-temperature specific heat below 7 K plotted as functions of T2 (a) and T4 (b). The broken and dotted lines in (a) show calculated contributions from Debye T3 phonons of ΘD = 460 K and 300 K, respectively, which are much smaller than the experimental values. The broken line in (b) is a linear fit to the zero-field data as T → 0, which gives a coefficient of the T5 term, β5 = 0.3481(6) mJ K-6 mol-1. The solid and dotted line is a fit to Cl = β5T5f(T). See text for detail. FIG. 5. (Color online) Specific heat data same as shown in Fig. 3. The dotted line shows the estimated contribution of Cen + Cl, and the broken line represents Cl in the case of γ = 70 mJ K-2 mol-1. The inset shows a change of entropy balance as a function of γ, from which the value of γ is decided to be 70 mJ K-2 mol-1.

Superconducting and normal-state properties of the beta-pyrochlore oxide KOs2O6 are studied by means of thermodynamic and transport measurements. It is shown that the superconductivity is of conventional s-wave type and lies in the extremely strong-coupling regime. Specific heat and resistivity measurements reveal that there are characteristic low-energy phonons that give rise to unusual scattering of carriers due to strong electron-phonon interactions. The entity of the low-energy phonons is ascribed to the heavy rattling of the K ion confined in an oversized cage made of OsO6 octahedra. It is suggested that this electron-rattler coupling mediates the Cooper pairing, resulting in the extremely strong-coupling superconductivity.

Microsoft Word - K-SC6.doc April 2, 2007 I. INTRODUCTION The study of non-Cu-based oxide superconductors has been extended during the last decade, aiming at understanding the role of electron correlations in the mechanism of superconductivity or searching for a novel pairing mechanism, hopefully to reach a higher Tc. An interesting example recently found is a family of pyrochlore oxide superconductors. The first discovered is α-pyrochlore rhenate Cd2Re2O7 with Tc = 1.0 K 1-3 and the second β-pyrochlore osmate AOs2O6 with Tc = 3.3, 6.3, and 9.6 K for A = Cs,4 Rb,5-7 and K,8 respectively. They crystallize in the cubic pyrochlore structure of space group Fd-3m and commonly possess a 3D skeleton made of ReO6 or OsO6 octahedra. 9 The "pyrochlore" sublattice occupied by the transition metal ions is comprised of corner-sharing tetrahedra that are known to be highly frustrating for a localized spin system with antiferromagnetic nearest-neighbor interactions. A unique structural feature for the β-pyrochlores is that a relatively small A ion is located in an oversized atomic cage made of OsO6 octahedra, Fig. 1. Due to this large size mismatch, the A atom can rattle in the cage.10 The rattling has been recognized recently as an interesting phenomenon for a class of compounds like filled skutterudites11 and Ge/Si clathrates12 and attracted many researchers, because it may suppress thermal conductivity leading to an enhanced thermoelectric efficiency. On the other hand, the rattling is also intriguing from the viewpoint of lattice dynamics: it gives an almost localized mode even in a crystalline material and often exhibits unusual anharmonicity.11 Hence, it is considered that the rattling is a new type of low-lying excitations that potentially affects various properties in a crystal at low temperature. In the β pyrochlores, specific heat experiments found low-energy contributions that could be described approximately by the Einstein model and determined the Einstein temperature TE to be 70 K, 60 K, and 40 or 31 K for A = Cs, Rb, and K, respectively.13, 14 This tendency illustrates uniqueness of the rattling, because, to the contrary, one expects a higher frequency for a lighter atom in the case of conventional phonons. Moreover, it was demonstrated that the specific heat shows an unusual T5 dependence at low temperature below 7 K for A = Cs and Rb, instead of a usual T3 dependence from a Debye-type phonon.13 On one hand, structural refinements revealed large atomic displacement parameters at room temperature of 100Uiso = 2.48, 4.26, and 7.35 Å2 for A = Cs, Rb, and K, respectively,10 and 3.41 Å2 for Rb.15 Particularly, the value for K is enormous and may be the largest among rattlers so far known in related compounds. This trend over the β-pyrochlore series is ascribed to the fact that an available space for the A ion to move in a rather rigid cage increases with decreasing the ionic radius of the A ion.10 Kuneš et al. calculated an energy potential for each A ion and found in fact a large anharmonicity, that is, a deviation from a quadratic form expected for the harmonic oscillator approximation.16 Especially for the smallest K ion, they found 4 shallow potential minima locating away from the center (8b site) along the <111> directions pointing to the nearest K ions, as schematically shown in Fig. 1. The potential minima are so shallow that the K ion may not stop at one of them even at very low temperature. The electronic structures of α-Cd2Re2O7 and β-AOs2O6 have been calculated by first-principle density-functional methods, which reveal that a metallic conduction occurs in the (Re, Os)-O network:16-20 electronic states near the Fermi level originate from transition metal 5d and O 2p orbitals. Although the overall shape of the density of states (DOS) is similar for the two pyrochlores, a difference in band filling may result in different properties; Re5+ for α-Cd2Re2O7 has two 5d electrons, while Os 5.5+ for β-AOs2O6 has two and a half. Moreover, a related α-pyrochlore Cd2Os2O7 with Os5+ (5d3) exhibits a metal-to-insulator transition at 230 K.21, Various experiments have been carried out on the pyrochlore oxide superconductors to elucidate the mechanism of the superconductivity. Most of the results obtained for α-Cd2Re2O7 indicate that it is a weak-coupling BCS-type superconductor.23, 24 In contrast, results on the β-pyrochlores are somewhat controversial. Although the Tc increases smoothly from Cs to K, the jump in specific heat at Tc, the upper critical field, and the Sommerfeld coefficient all exhibit a large enhancement toward K.25 Thus, the K compound is always distinguished from the others. Pressure dependence of Tc was also studied for the two pyrochlores, showing a common feature: as pressure increases, Tc first increases, exhibits a broad maximum and goes to zero above a critical pressure that depends on the system, for example, about 6 GPa for the K compound.26-29 On the symmetry of the superconducting gap for the β-pyrochlores, Rb-NMR and µSR experiments gave evidence for s-wave superconductivity for RbOs2O6. 30-32 In contrast, Arai et al. carried out K-NMR experiments and found no coherence peak in the relaxation rate below Tc for KOs2O6, which seemed to indicate unconventional superconductivity.32 However, their recent interpretation is that the absence of a coherence peak does not necessarily mean non-s pairing, because the relaxation rate probed by the K nuclei can be affected dominantly by strongly overdamped phonons.33 Kasahara et al. measured thermal conductivity using a KOs2O6 single crystal and concluded a full Extremely strong-coupling superconductivity and anomalous lattice properties in the β-pyrochlore oxide KOs2O6 Z. Hiroi, S. Yonezawa, Y. Nagao and J. Yamaura Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Superconducting and normal-state properties of the β-pyrochlore oxide KOs2O6 are studied by means of thermodynamic and transport measurements. It is shown that the superconductivity is of conventional s-wave type and lies in the extremely strong-coupling regime. Specific heat and resistivity measurements reveal that there are characteristic low-energy phonons that give rise to unusual scattering of carriers due to strong electron-phonon interactions. The entity of the low-energy phonons is ascribed to the heavy rattling of the K ion confined in an oversized cage made of OsO6 octahedra. It is suggested that this electron-rattler coupling mediates the Cooper pairing, resulting in the extremely strong-coupling superconductivity. gap from the insensitivity of thermal conductivity to magnetic fields.34 Moreover, very recent photoemission spectroscopy (PES) experiments revealed the opening of a large isotropic gap below Tc. 35 On one hand, a µSR experiment claimed that the gap of KOs2O6 is anisotropic, or otherwise, there are two gaps. Therefore, the pairing symmetry of the β-AOs2O6 superconductors may be of the conventional s wave, aside from minor aspects such as anisotropy or multi gaps, which means that the fundamental pairing mechanism is ascribed to phonons. Then, an important question is what kind of phonons are relevant for the occurrence. To find out the reason of the observed singular appearance toward K in the series must be the key to understand interesting physics involved in this system. Previous studies have suffered poor quality of samples, because only polycrystalline samples were available. The Sommerfeld coefficient γ was estimated from specific heat by extracting contributions from Os metal impurity to be 40 mJ K-2 mol-1 for both Cs and Rb,13 and 34 mJ K-2 mol-1 for Rb.7 Recently, Brühwiler et al. obtained large values of γ = 76-110 mJ K-2 mol-1 for KOs2O6 by collecting five dozen of tiny crystals. 14 However, there is an ambiguity in their values, because of uncertainty in their extrapolation method. They also reported strong-coupling superconductivity with a coupling constant λep = 1.0-1.6. Recently, we successfully prepared a large single crystal of 1 mm size for KOs2O6 and reported two intriguing phenomena: one is a sharp and huge peak in specific heat at Tp = 7.5-7.6 K below Tc, indicative of a first-order structural transition,37, 38 and the other is anisotropic flux pinning at low magnetic fields around 2 T.39 It was suggested that the former is associated with the rattling freedom of the K ion. Moreover, anomalous concave-downward resistivity was observed down to Tc, suggesting a peculiar scattering mechanism of carriers. In this paper, we present specific heat, magnetization and resistivity measurements on the same high-quality single crystal of KOs2O6. Reliable data on the superconducting and normal-state properties are obtained, which provide evidence for an extremely strong-coupling superconductivity realized in this compound. We discuss the role of rattling vibrations of the K ion on the mechanism of the superconductivity. II. EXPERIMENTAL A. Sample preparation A high-quality single crystal was prepared and used for all the measurements in the present study, which was named KOs-729 after the date of July 29, 2005 when the first experiment was performed on this crystal. It was grown from a pellet containing an equimolar mixture of KOsO4 and Os metal in a sealed quartz tube at 723 K for 24 h. Additional oxygen was supplied by using the thermal decomposition of AgO placed away from the pellet in the tube. The KOsO4 powder had been prepared in advance from KO2 and Os metal in the presence of excess oxygen. It was necessary to pay attention to avoid the formation of OsO4 in the course of preparation, which is volatile even at room temperature and highly toxic to eyes or nose. After the reaction, several tiny crystals had grown on the surface of the pellet. Although the mechanism of the crystal growth has not yet been understood clearly, probably it occurs through partial melting and the following reaction with a vapor phase. The KOs-729 crystal possesses a truncated octahedral shape with a large (111) facet as shown in Fig. 2 and is approximately 1.0 × 0.7 × 0.3 mm3 in size and 1.302 mg in weight. The high quality of the crystal has been demonstrated by a sharp peak at Tp in specific heat,38 which was absent in the previous polycrystalline samples or appeared as broad humps in our previous aggregate of tiny crystals37 or in five dozen of tiny crystals by Brühwiler et al.14 Moreover, a dramatic angle dependence of flux-flow resistance was observed on this crystal, indicating that flux pinning is enhanced in magnetic fields along certain crystallographic directions such as [110], [001], and [112].39 This evidences the absence of domains in this relatively large crystal. A special care has been taken to keep the crystal always in a dry atmosphere, because it readily undergoes hydration in air, as reported in isostructural compounds such as KNbWO6. 40, 41 Once partial hydration takes place in KOs2O6, the second sharp anomaly in specific heat tends to collapse. In contrast, the superconducting transition was robust, just slightly broadened after hydration. The hydration must be relatively slow in a single crystal compared with the case of polycrystalline samples, which may be the reason for the broad anomaly or the absence in previous samples. B. Physical-property measurements Both specific heat and electrical resistivity were measured in a temperature range between 300 K and 0.4 K and in magnetic fields of up to 14 T in a Quantum Design Physical Property Measurement System (PPMS) equipped with a 3He refrigerator. The magnetic fields had been calibrated by measuring the magnetization of a standard Pd specimen and also by measuring the voltage of a Hall device (F.W. BELL, BHA-921). Specific heat measurements FIG. 1. (Color online) Crystal structure of the β-pyrochlore oxide KOs2O6. The K ion (big ball) is located in an oversized atomic cage made of OsO6 octahedra and can move along the 4 <111> directions pointing to the neighboring K ions in adjacent cages. FIG. 2. (Color online) Photograph of the KOs-729 crystal used in the present study. It possesses a truncated octahedral shape with 111 facets. The approximate size is 1.0 × 0.7 × 0.3 mm3. Resistivity measurements were carried out with a current flow along the [-110] direction. were performed by the heat-relaxation method. The KOs-729 crystal was attached to an alumina platform by a small amount of Apiezon N grease. In each measurement, heat capacity was obtained by fitting a heat relaxation curve recorded after a heat pulse giving a temperature rise of approximately 2%. The heat capacity of an addendum had been measured in a separate run without a sample, and was subtracted from the data. The measurements were done three times at each temperature with a scatter less than 0.3% at most. Resistivity measurements were carried out by the four-probe method with a current flow along the [-110] direction and magnetic fields along the [111], [110], [001] or [112] direction of the cubic crystal structure. All the measurements were done at a current density of 1.5 A cm-2. Magnetization was measured in magnetic fields up to 7 T in a Quantum Design magnetic property measurement system and also up to 14 T in PPMS. The magnetic fields were applied approximately along the [111] or [-110] direction. III. RESULTS A. Superconducting properties 1. Specific heat First of all, we analyze specific heat data in order to obtain a reliable value of the Sommerfeld coefficient γ. There are two obstacles: one is the large upper critical field Hc2 that is approximately 2 times greater than our experimental limit of 14 T. Brühwiler et al. reported γ = 76 (110) mJ K-2 mol-1 assuming µ0Hc2 = 24 (35) T by an extrapolation method.14 Their values should be modified to γ ~ 100 mJ K-2 mol-1, because recent high magnetic field experiments revealed µ0Hc2 = 30.6 T or 33 T. 42, 43 Nevertheless, there are still large ambiguity in their extrapolation method using specific heat data obtained only at H/Hc2 < 0.5. The other difficulty comes from unusual lattice contributions in specific heat at low temperature and the existence of a sharp peak at Tp. Thus, it is not easy to extract the lattice contribution in a standard way used so far. Here we carefully analyze specific heat data and reasonably divide them into electronic and lattice parts, from which a reliable value of γ is determined, and information on the superconducting gap is attained. Figure 3 shows the temperature dependence of specific heat of the KOs-729 crystal measured on cooling at zero field and in a magnetic field of 14 T applied along the [111] direction. A superconducting transition at zero field takes place with a large jump, followed by a huge peak due to the second phase transition at Tp = 7.5 K. The entropy-conserving construction shown in the inset gives Tc = 9.60 K, ΔC/Tc = 201.2 mJ K -2 mol-1, which is close to the values previously reported,14, 37 and a transition width (ΔTc) of 0.3 K. Tc is reduced to 5.2 K at 14 T, which is evident as a bump in the 14-T data shown in Fig. 3. In contrast, Brühwiler et al. reported Tc = 6.2 K at 14 T, though the transition was not clearly observed in their specific heat data. The C/T at zero field rapidly decreases to zero as T approaches absolute zero. The absence of a residual T-linear contribution in specific heat indicates the high quality of the crystal. The specific heat of a crystal (C) is the sum of an electronic contribution (Ce) and an H-independent lattice contribution (Cl). The former becomes Cen for the normal state above Tc, which is taken as γT, and Ces for the superconducting state below Tc. The γ is assumed to be T-independent, though it can not be the case for compounds with strong electron-phonon couplings.44 In the case of KOs2O6, Cl is large relative to Ce: for example, Cl is as large as ~90% of the total C at just above Tc, as shown later. In order to determine the value of γ, it is crucial to know the low-temperature form of Cl. Since the minimum Tc attained at 14 T is 5.2 K, one has to estimate the Cl from the T dependence of the total C above ~5.5 K. Two terms in the harmonic-lattice approximation are often required for an adequate fit; Cl = β3T 3 + β5T 5. The first term comes from a Debye-type acoustic phonon, and thus is dominant at low temperature, while the second term expresses a deviation at high temperature. Actually, this approximation is valid for α-Cd2Re2O7, where β3 = 0.222 mJ K -4 mol-1 and β5 = 2.70 × 10 mJ K-6 mol-1 are obtained by a fit to the data below 10 K.1 The Debye temperature ΘD is 458 K from the β3 value. In strong contrast, it was found for two members of β-AOs2O6 that the T term prevails in a wide temperature range; β5 = 14.2 × 10 -3 mJ K-6 mol-1 below 5 K for CsOs2O6 and β5 = 30.2 × 10 -3 mJ K-6 mol-1 below 7 K for RbOs2O6. The C/T at H = 0 below 7 K shown in Fig. 3 is again plotted in two ways as functions of T2 and T4 in Fig. 4. It is apparent from the T2 plot that possible T3 terms expected for ΘD = 458 K and 300 K are negligibly small compared with the whole magnitude of specific heat, just as observed in other members. On the other hand, in the T4 plot, there is distinct linear behavior at low temperature below 4 K, indicating that the C approaches asymptotically to T5 behavior as T → 0 with a large slope of 0.3481(6) mJ K-6 mol-1. It is reasonable to ascribe this T5 contribution to the lattice, because Ces should decrease quickly as T → 0. Note that the value of the β5 for KOs2O6 is more than one order larger than those in other members. At high temperatures above 4 K, a downward deviation from the initial T5 behavior is observed in Fig. 4b. The temperature dependence of the 14-T data above 5.5 K, which is taken as γT + Cl, is also close to T 5, but with a smaller slope, which means that a single T5 term is not appropriate to describe the Cl in such a wide temperature range and also that an inclusion of higher order term of Tn is not helpful. Therefore, we adopt expediently an alternative empirical form to express this strange lattice contribution; Cl = β5T 5f(T), where f(T) = [1 + exp(1 - pT-q)]-1. Since the f(T) is almost unity below a certain temperature and decreases gradually with increasing T, this Cl can reproduce T5 behavior at low temperatures and a weaker T FIG. 3. (Color online) Specific heat divided by temperature measured at zero field (circle) and a magnetic field of 14 T (triangle) applied along the [111] direction. The inset shows an enlargement of the superconducting transition with an entropy-conserving construction. dependence at high temperatures. As shown in Fig. 4b, the 14-T data in the 5.5-7 K range can be fitted well by the function for a value of β5 fixed to the initial slope of 0.3481 mJ K -6 mol-1 and a given value of γ; for example, p = 6.96(3) and q = 1.09(1) for γ = 70 mJ K-2 mol-1. In order to determine the value of γ univocally, the entropy conservation is taken into account for the 14-T data, as shown in Fig. 5: since the normal-state specific heat expected for the case of Tc = 0 is given by (Cen + Cl) (dotted line in Fig. 5), the integration of [Cen + Cl - C(14 T)]/T should become zero due to entropy balance. It is shown in the inset to Fig. 5 that the integrated value changes almost linearly with γ and vanishes around γ = 70 mJ K-2 mol-1. Hence, one can determine the value of γ unambiguously. A certain ambiguity may arise from the assumed lattice function. However, since the temperature dependence of Cl is substantially weak in the T range of interest, a possible correction on the γ value must be minimal, say, less than 1 mJ K-2 mol-1. Next we determine Ces at zero field by subtracting the Cl estimated above. The temperature dependence of Ces does have the BCS form, aexp(-Δ/kBTc), as shown in Fig. 6. The energy gap Δ obtained by fitting is 22.5 K, which corresponds to 2Δ/kBTc = 4.69, much larger than the BCS value of 3.53. The above Ces at low temperature below 7 K is again plotted in Fig. 7 together with high-temperature Ces above 5.5 K, which is obtained by subtracting the 14-T data from the zero field data as Ces = C(0) - C(14 T). The two data sets obtained independently overlap well in the 5.5-7 K range, assuring the validity of the above analyses. Because of the existence of the second peak and its small shifts under magnetic fields, the data between 7 K and 8.3 K is to be excluded in the following discussion. Taking γ = 70 mJ K-2 mol-1, the jump in specific heat at Tc, ΔC/γTc, reaches 2.87, much larger than 1.43 expected for a weak-coupling superconductor, indicating that KOs2O6 lies in the strong-coupling regime. Comparisons to other typical strong-coupling superconductors are made in section IV-B. Here we analyze the data based on the α model that was developed to provide a semi-empirical approximation to the thermodynamic properties of strong-coupled superconductors in a wide range of coupling strengths with a single adjustable parameter, α = Δ0/kBTc. 45 Recently, it was generalized to a multi-gap superconductor and successfully applied to the analyses on MgB2 or Nb3Sn. 46, 47 Using the α model, the data in the vicinity of Tc is well reproduced, as shown in Fig. 7, and we obtain α = 2.50 (2Δ0/kBTc = 5.00), slightly larger than the value obtained above from the temperature dependence of Ces. One interesting point to be noted is that there is a significant deviation between the data and the fitting curve at intermediate temperatures, suggesting the existence of an additional structure in the gap. Presumably, this enhancement would be explained if one assumes the coexistence of another smaller gap (not so small as in MgB2, but intermediate). This possibility has been already pointed out in the previous µSR experiment.36 However, ambiguity associated with the second peak in the present data prevents us from further analyzing the data. This important issue will be revisited in future work, where the FIG. 6. (Color online) Temperature dependence of electronic specific heat measured at H = 0 for the superconducting state showing an exponential decrease at low temperature. A magnitude of the gap obtained is 2Δ/kBTc = 4.69. FIG. 4. (Color online) Low-temperature specific heat below 7 K plotted as functions of T2 (a) and T4 (b). The broken and dotted lines in (a) show calculated contributions from Debye T3 phonons of ΘD = 460 K and 300 K, respectively, which are much smaller than the experimental values. The broken line in (b) is a linear fit to the zero-field data as T → 0, which gives a coefficient of the T5 term, β5 = 0.3481(6) mJ K-6 mol-1. The solid and dotted line is a fit to Cl = β5T5f(T). See text for detail. FIG. 5. (Color online) Specific heat data same as shown in Fig. 3. The dotted line shows the estimated contribution of Cen + Cl, and the broken line represents Cl in the case of γ = 70 mJ K-2 mol-1. The inset shows a change of entropy balance as a function of γ, from which the value of γ is decided to be 70 mJ K-2 mol-1.

Analysis of Low Energy Pion Spectra Suk Choi and Kang Seog Lee∗ Department of Physics, Chonnam National University, Gwangju 500-757, Korea (Dated: Apr. 11, 2007) The transverse mass spectra and the rapidity distributions of π+ and π− in Au-Au collisions at 2, 4, 6, and 8 GeV·A by E895 collaboration are fitted using an elliptically expanding fireball model with the contribution from the resonance decays and the final state Coulomb interaction. The ratio of the total number of produced π− and π+ is used to fit the data. The resulting freeze-out temperature is rather low(Tf < 60 MeV) with large transverse flow and thus resonance contribution is very small. The difference in the shape of mt spectra of the oppositely charged pions are found to be due to the Coulomb interaction of the pions. PACS numbers: 24.10.Pa,25.75.-q Pion production just above the threshold energy is quite different from that at very high energies such as RHIC energy since the ratio of π− to π+ at very high en- ergies is one which is not the case at low energies. At just above the threshold energy, pions are produced through the production of ∆ resonances and counting all the pos- sible channels of ∆ decay the difference in the compo- sition of isospins in the colliding nuclei appears as the difference in the numbers of the two oppositely charged pions[1, 2, 3], whereas at high energies many channels producing pions are open and small asymmetry in the initial isospin does not matter. Other features of the pion spectra at low energies are[1, 2, 3, 4, 5, 6, 7]: (1)The transverse momentum spectra both of the π− and π+ seem to have two temper- atures. Usually the low temperature component in the low momentum region is attributed to the pions decayed from resonances, especially the delta resonance, while the higher temperature component in the mid-momentum re- gion is the thermal ones. (2) Transverse momentum spec- tra of π− and π+ at very small momentum are different in the sense that the π+ spectra is convex in its shape while the π− spectra does not show this behavoir. This difference in low momentum region is due to the Coulomb effect. The hadronic matter formed during the collision has charge which comes from the initially colliding two nuclei and thus the thermal pions escaping from the sys- tem experience the Coulomb interaction. The Coulomb interaction of π− may bend the spectrum in the low mo- mentum region upward and thus it is hard to disentan- gle the contribution from the delta resonance and the Coulomb interaction in the low momentum region. (3) Width of the rapidity spectra of π− and π+ are much wider than those from the isotropically expanding ther- mal model. The wide width may either come from partly transparent nature of the collision dynamics or the ellip- soidal expansion geometry. In order to fit large rapidity width using expanding fireball model one usually needs large longitudinal expansion velocity. ∗kslee@chonnam.ac.kr Even though all those features mentioned above are not new, calculations with all those features put in together comparing each contributions in detail is hard to find. There are claims that the properties of ∆ resonance are modified inside the hadronic matter formed even at this low energy[5, 6]. In order to draw any conclusion, one should have a model which can explain all of the features above mentioned. Lacking one or two features in the calculations, the result may not be conclusive. In this paper, we analyze the pion spectra in Au+Au collisions at 2, 4, 6, and 8 A·GeV measured by the E895 collaboration[4] using the expanding fireball[8, 9, 10] with the resonance contribution[10] and the final state Coulomb interaction[1, 11, 12, 13]. The geometry of the expansion used is ellipsoidal[8] and can be varied to sphere and cylinder, by taking the transverse size R as a function of the longitudinal coordinate, z. At just above the threshold energy, pions are produced through the production of ∆ resonances and their sub- sequent decays. The ratio of π− and π+ is given from the initial isospin conservation as π 5Z2+NZ ∼ 1.94 for Au+Au collisions[1, 2, 3] . Hence it is expected that at low beam energies near 2 GeV·A in Au+Au collisions, the ratio is near 1.94 and then as the beam energy is in- creased the ratio will decrease eventually to one. In the present calculation, the ratio of normalization constants for π− and π+, R is taken as a fit parameter in order to investigate the beam energy dependence of R. We assume that once the pions are produced they rescatter among themselves and thermalize before they decouple from the system. Hence we assume thermaliza- tion of the pionic matter. However, the total number of negatively and positively charged pions are not in chem- ical equilibrium and the ratio is governed by the isospin asymmetry of the initially colliding nuclei. We keep the ratio as a fitting parameter and want to compare with the expected value of 1.94 near threshold of the pion produc- tion. After the formation of a pionic fireball, it expands and cools down until freeze-out when the particle production is described from the formalism of Cooper-Frye[14]. For the equilibrium distribution function we use the Lorentz- boosted Boltzmann distribution function, where uµ is the http://arxiv.org/abs/0704.1529v1 TABLE I: Fitted values for each parameters. Ebeam V ηm ρ0 T Pc π −/π+ χ2 /n (GeV) (×105) GeV GeV/c 2 1.41 1.12 0.88 46 25 1.96 1.3 4 0.93 1.32 0.92 57 24 1.95 2.9 6 1.44 1.50 1.11 54 18 1.40 2.4 8 1.62 1.58 1.12 55 15 1.38 1.8 expansion velocity the space-time of the system. (2π)3 pµdσµ(x)f(x, p) (1) where f(x, p) = exp(− pνuν(x)− µ ) (2) For the geometry of the expanding fireball[8], we as- sume azimuthal symmetry and further assume boost- invariance collective dynamics along the longitudinal direction[15]. In this case it is convenient to use as the coordinates the longitudinal proper time τ = t2 − z2, space-time rapidity η = tanh−1(z/t) and the trans- verse coordinate r⊥. Then the 4-velocity of expansion can be expressed as uµ(x) = γ(1, v⊥(x)er , vz(x)) where vz(τ, r⊥, η) = tanh η, which is the result of the longitudi- nal boost-invariance. In the transverse direction we take a linear flow rapidity profile, tanh−1 v⊥ = ρ(η)(r⊥/R0) where R0 is the transverse radius at midrapidity. Here one takes ρ(η) = ρ0 1− (η2/η2max) for the elliptic ge- ometry and a constant value of ρ, i.e. ρ(η) = ρ0 for the cylindrical geometry. As is the same for the SPS energy by Dobbler et. al., the elliptic case fits the pion spectra a little better. As pions escape from the system at freeze-out, they ex- perience the Coulomb interaction with the charge of the system which are mainly due to the initial protons in the colliding nuclei. The Coulomb effect on the particle spec- tra are studied in detail in refs.[11, 12, 13] for the static and dynamical cases. Here due to the low beam energy we restrict ourselves only to the static case. The system is expanding rapidly in the longitudinal direction and the change in the longitudinal momentum is negligible. Only the transverse momentum of the charged particles will be shifted by an average amount[11] pc = ∆p⊥ ∼ 2e2 dN ch , (3) where Rf is the transverse radius of the system at freeze- out. Due to the lack of detailed knowledge, pc is taken as a fit parameter in the present calculation. And in this way the beam energy dependence of pc can be studied. Since the transverse momentum of the escaping ther- mal pions are shifted by the amount pc, i.e. pt = pt,0±pc. the invariant cross section can be written in terms of the unshifted momentum (pt,0, y0). -1.5 -1 -0.5 0 0.5 1 1.5 E895, Au+Au at 2A·GeV π- fit π+ fit FIG. 1: Rapidity distribution of π− and π+ in Au+Au col- lision at 2 A·GeV by E895 collaboration[4]. ). (4) where (E d )0 is the unshifted invariant cross section. Integrating over the rapidity y one gets the equation for the transverse momentum spectra, ptdpt pt,0dpt,0dy . (5) and the rapidity spectra is obtained by integrating over the transverse mass. mt,0dmt,0( pt,0dpt,0 . (6) Finally one has to add the contribution from resonance decay[10] to both the transverse spectrum and rapidity distribution. Here we assume that the resonances decay far outside the system and the Coulomb interaction of the pions decayed from the resonances is neglected. The fitted values for the parameters are tabulated in the Tab. 1 and the results of the fitting are shown in Figs.1-2 for 2 A·GeV. The fitted value for the ratio R = π−/π+ is close to 1.94 at 2 and 4 A·GeV as expected and decreases to 1.38 at 8 A·GeV, which eventually becomes 1 at higher energies such as RHIC energies. In other words, at this very low energy pion isospin is not in chemical equilibrium. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mt - m0 E895, Au+Au at 2A·GeV 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mt - m0 E895, Au+Au at 2A·GeV FIG. 2: π− transverse mass spectrum for each rapidity bin(∆y = 1.0) measured by E895 collaboration[4]in Au+Au collisions at 2 A·GeV. Data on the top line is for the ra- pidity bin −0.65 < y < −0.55 and the next one is for −0.55 < y < −0.45 scaled by 0.1, etc. The freeze-out temperature is rather small, Tf < 60 MeV and the expansion velocities in both the longitudi- nal and transverse direction are quite large. The large longitudinal expansion velocity(> 0.8c) is needed to fit the large width of the rapidity distribution. The low freeze-out temperature together with the large transverse expansion fits the transverse spectra of pions quite nicely. If not for the large expansion velocity, one usually gets much larger freeze-out temperature(Tf > 80 MeV. Since the freeze-out temperature is small, there are very few resonances at freeze-out, especially ∆, and thus the resonance contribution is negligible. This is reason- able since at this low energy near the pion threshold en- ergy, production of particles with mass larger than pions is rare and their contribution to the pion spectrum is negligible. The pion transverse momentum spectra looks like that there are two slopes; one for small momentum region and another for the higher momentum region near the pion mass. The smaller slope at lower transverse momentum is usually attributed to the pions from the resonance de- cay, especially from the ∆ decay[3, 6]. However, present calculation shows that this is not the case at low beam energies. The shape of the transverse mass spectra of π− and π+ are different especially in the small mass region. As the transverse mass mt decreases, the mt spectrum of π increases sharply while that of π+ saturates showing the convex shape. This difference is due to the Coulomb in- teraction of pions leaving the system which has the charge from the initially bombarding nucleons. The change in the transverse momentum due to the Coulomb interac- tion decreases from 25 GeV/c at 2 A·GeV to 15 GeV/c at 8 A·GeV. This behavior can be understood from the in- crease of the screening effect since the number of charged pions increase at higher energies. At very high energies such as RHIC energies, the momentum change from the Coulomb interaction will be small. The emerging picture of pion production at low en- ergy is that the pions are produced through the inter- mediate ∆ formation and thus they are not in chemical equilibrium in isospin. They make collisions and ther- malize to form a fireball which expands and cools until the freeze-out. Since the fireball has charge which is from the initially colliding nucleus, the pions leaving the sys- tem experiences the Coulomb interaction which makes the difference of the mt spectra of the two oppositely charged pions. Acknowledgments This work is financially supported by Chonnam Na- tional University and the post-BK21 program. We wish to acknowledge U.W. Heinz for providing the program and useful discussions. [1] A. Wagner et. al., Phys. Lett. B420, 20(1998). [2] B.J. VerWest and R.A. Arndt, Phys. Rev. C25, 1979(1982) [3] Phys. Rep. 135, 259(1986). [4] J.L. Klay, et. al., Phys. Rev. C68, 054905(2003). [5] B. Pin-zhen and J. Rafelski, nucl-th/0507037. [6] B.S. Hong, J. Korean Phys. Soc. 46, 1083(2005). [7] B. Hong et. al., Phys. Lett. B407, 115(1999); ibid., J. Korean Phys. Soc. 46, 1083(2005). [8] H. Dobbler, J. Sollfrank and U. Heinz, Phys. Lett. B457,353(1999) . [9] Kang S. Lee, U. Heinz, and E. Schnedermann, Z. Phys. C 48,525 (1990). [10] E. Schnedermann, J. Sollfrank, and U. Heinz, Phys. Rev. C 48, 2462 (1993). [11] H.W. Barz, J.P. Bondorf,J.J. Gaardhoje and H. Heisel- berg, Phys. Rev. C57, 2536(1998). [12] M. Gyulassy and S. K. Kaufmann, Nucl. Phys. A362, 503(1981). [13] A. Ayala and J. Kapusta, Phys. Rev. C56, 407(1997). [14] F. Cooper and G. Frye, Phys. Rev. D10, 186(1974) [15] J.D. Bjorken, Phys. Rev. D27, 140(1983) http://arxiv.org/abs/nucl-th/0507037

The transverse mass spectra and the rapidity distributions of $\pi^+$ and $\pi^-$ in Au-Au collisions at 2, 4, 6, and 8 GeV$\cdot$A by E895 collaboration are fitted using an elliptically expanding fireball model with the contribution from the resonance decays and the final state Coulomb interaction. The ratio of the total number of produced $\pi^-$ and $\pi^+$ is used to fit the data. The resulting freeze-out temperature is rather low($T_f < 60$ MeV) with large transverse flow and thus resonance contribution is very small. The difference in the shape of $m_t$ spectra of the oppositely charged pions are found to be due to the Coulomb interaction of the pions.

Analysis of Low Energy Pion Spectra Suk Choi and Kang Seog Lee∗ Department of Physics, Chonnam National University, Gwangju 500-757, Korea (Dated: Apr. 11, 2007) The transverse mass spectra and the rapidity distributions of π+ and π− in Au-Au collisions at 2, 4, 6, and 8 GeV·A by E895 collaboration are fitted using an elliptically expanding fireball model with the contribution from the resonance decays and the final state Coulomb interaction. The ratio of the total number of produced π− and π+ is used to fit the data. The resulting freeze-out temperature is rather low(Tf < 60 MeV) with large transverse flow and thus resonance contribution is very small. The difference in the shape of mt spectra of the oppositely charged pions are found to be due to the Coulomb interaction of the pions. PACS numbers: 24.10.Pa,25.75.-q Pion production just above the threshold energy is quite different from that at very high energies such as RHIC energy since the ratio of π− to π+ at very high en- ergies is one which is not the case at low energies. At just above the threshold energy, pions are produced through the production of ∆ resonances and counting all the pos- sible channels of ∆ decay the difference in the compo- sition of isospins in the colliding nuclei appears as the difference in the numbers of the two oppositely charged pions[1, 2, 3], whereas at high energies many channels producing pions are open and small asymmetry in the initial isospin does not matter. Other features of the pion spectra at low energies are[1, 2, 3, 4, 5, 6, 7]: (1)The transverse momentum spectra both of the π− and π+ seem to have two temper- atures. Usually the low temperature component in the low momentum region is attributed to the pions decayed from resonances, especially the delta resonance, while the higher temperature component in the mid-momentum re- gion is the thermal ones. (2) Transverse momentum spec- tra of π− and π+ at very small momentum are different in the sense that the π+ spectra is convex in its shape while the π− spectra does not show this behavoir. This difference in low momentum region is due to the Coulomb effect. The hadronic matter formed during the collision has charge which comes from the initially colliding two nuclei and thus the thermal pions escaping from the sys- tem experience the Coulomb interaction. The Coulomb interaction of π− may bend the spectrum in the low mo- mentum region upward and thus it is hard to disentan- gle the contribution from the delta resonance and the Coulomb interaction in the low momentum region. (3) Width of the rapidity spectra of π− and π+ are much wider than those from the isotropically expanding ther- mal model. The wide width may either come from partly transparent nature of the collision dynamics or the ellip- soidal expansion geometry. In order to fit large rapidity width using expanding fireball model one usually needs large longitudinal expansion velocity. ∗kslee@chonnam.ac.kr Even though all those features mentioned above are not new, calculations with all those features put in together comparing each contributions in detail is hard to find. There are claims that the properties of ∆ resonance are modified inside the hadronic matter formed even at this low energy[5, 6]. In order to draw any conclusion, one should have a model which can explain all of the features above mentioned. Lacking one or two features in the calculations, the result may not be conclusive. In this paper, we analyze the pion spectra in Au+Au collisions at 2, 4, 6, and 8 A·GeV measured by the E895 collaboration[4] using the expanding fireball[8, 9, 10] with the resonance contribution[10] and the final state Coulomb interaction[1, 11, 12, 13]. The geometry of the expansion used is ellipsoidal[8] and can be varied to sphere and cylinder, by taking the transverse size R as a function of the longitudinal coordinate, z. At just above the threshold energy, pions are produced through the production of ∆ resonances and their sub- sequent decays. The ratio of π− and π+ is given from the initial isospin conservation as π 5Z2+NZ ∼ 1.94 for Au+Au collisions[1, 2, 3] . Hence it is expected that at low beam energies near 2 GeV·A in Au+Au collisions, the ratio is near 1.94 and then as the beam energy is in- creased the ratio will decrease eventually to one. In the present calculation, the ratio of normalization constants for π− and π+, R is taken as a fit parameter in order to investigate the beam energy dependence of R. We assume that once the pions are produced they rescatter among themselves and thermalize before they decouple from the system. Hence we assume thermaliza- tion of the pionic matter. However, the total number of negatively and positively charged pions are not in chem- ical equilibrium and the ratio is governed by the isospin asymmetry of the initially colliding nuclei. We keep the ratio as a fitting parameter and want to compare with the expected value of 1.94 near threshold of the pion produc- tion. After the formation of a pionic fireball, it expands and cools down until freeze-out when the particle production is described from the formalism of Cooper-Frye[14]. For the equilibrium distribution function we use the Lorentz- boosted Boltzmann distribution function, where uµ is the http://arxiv.org/abs/0704.1529v1 TABLE I: Fitted values for each parameters. Ebeam V ηm ρ0 T Pc π −/π+ χ2 /n (GeV) (×105) GeV GeV/c 2 1.41 1.12 0.88 46 25 1.96 1.3 4 0.93 1.32 0.92 57 24 1.95 2.9 6 1.44 1.50 1.11 54 18 1.40 2.4 8 1.62 1.58 1.12 55 15 1.38 1.8 expansion velocity the space-time of the system. (2π)3 pµdσµ(x)f(x, p) (1) where f(x, p) = exp(− pνuν(x)− µ ) (2) For the geometry of the expanding fireball[8], we as- sume azimuthal symmetry and further assume boost- invariance collective dynamics along the longitudinal direction[15]. In this case it is convenient to use as the coordinates the longitudinal proper time τ = t2 − z2, space-time rapidity η = tanh−1(z/t) and the trans- verse coordinate r⊥. Then the 4-velocity of expansion can be expressed as uµ(x) = γ(1, v⊥(x)er , vz(x)) where vz(τ, r⊥, η) = tanh η, which is the result of the longitudi- nal boost-invariance. In the transverse direction we take a linear flow rapidity profile, tanh−1 v⊥ = ρ(η)(r⊥/R0) where R0 is the transverse radius at midrapidity. Here one takes ρ(η) = ρ0 1− (η2/η2max) for the elliptic ge- ometry and a constant value of ρ, i.e. ρ(η) = ρ0 for the cylindrical geometry. As is the same for the SPS energy by Dobbler et. al., the elliptic case fits the pion spectra a little better. As pions escape from the system at freeze-out, they ex- perience the Coulomb interaction with the charge of the system which are mainly due to the initial protons in the colliding nuclei. The Coulomb effect on the particle spec- tra are studied in detail in refs.[11, 12, 13] for the static and dynamical cases. Here due to the low beam energy we restrict ourselves only to the static case. The system is expanding rapidly in the longitudinal direction and the change in the longitudinal momentum is negligible. Only the transverse momentum of the charged particles will be shifted by an average amount[11] pc = ∆p⊥ ∼ 2e2 dN ch , (3) where Rf is the transverse radius of the system at freeze- out. Due to the lack of detailed knowledge, pc is taken as a fit parameter in the present calculation. And in this way the beam energy dependence of pc can be studied. Since the transverse momentum of the escaping ther- mal pions are shifted by the amount pc, i.e. pt = pt,0±pc. the invariant cross section can be written in terms of the unshifted momentum (pt,0, y0). -1.5 -1 -0.5 0 0.5 1 1.5 E895, Au+Au at 2A·GeV π- fit π+ fit FIG. 1: Rapidity distribution of π− and π+ in Au+Au col- lision at 2 A·GeV by E895 collaboration[4]. ). (4) where (E d )0 is the unshifted invariant cross section. Integrating over the rapidity y one gets the equation for the transverse momentum spectra, ptdpt pt,0dpt,0dy . (5) and the rapidity spectra is obtained by integrating over the transverse mass. mt,0dmt,0( pt,0dpt,0 . (6) Finally one has to add the contribution from resonance decay[10] to both the transverse spectrum and rapidity distribution. Here we assume that the resonances decay far outside the system and the Coulomb interaction of the pions decayed from the resonances is neglected. The fitted values for the parameters are tabulated in the Tab. 1 and the results of the fitting are shown in Figs.1-2 for 2 A·GeV. The fitted value for the ratio R = π−/π+ is close to 1.94 at 2 and 4 A·GeV as expected and decreases to 1.38 at 8 A·GeV, which eventually becomes 1 at higher energies such as RHIC energies. In other words, at this very low energy pion isospin is not in chemical equilibrium. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mt - m0 E895, Au+Au at 2A·GeV 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mt - m0 E895, Au+Au at 2A·GeV FIG. 2: π− transverse mass spectrum for each rapidity bin(∆y = 1.0) measured by E895 collaboration[4]in Au+Au collisions at 2 A·GeV. Data on the top line is for the ra- pidity bin −0.65 < y < −0.55 and the next one is for −0.55 < y < −0.45 scaled by 0.1, etc. The freeze-out temperature is rather small, Tf < 60 MeV and the expansion velocities in both the longitudi- nal and transverse direction are quite large. The large longitudinal expansion velocity(> 0.8c) is needed to fit the large width of the rapidity distribution. The low freeze-out temperature together with the large transverse expansion fits the transverse spectra of pions quite nicely. If not for the large expansion velocity, one usually gets much larger freeze-out temperature(Tf > 80 MeV. Since the freeze-out temperature is small, there are very few resonances at freeze-out, especially ∆, and thus the resonance contribution is negligible. This is reason- able since at this low energy near the pion threshold en- ergy, production of particles with mass larger than pions is rare and their contribution to the pion spectrum is negligible. The pion transverse momentum spectra looks like that there are two slopes; one for small momentum region and another for the higher momentum region near the pion mass. The smaller slope at lower transverse momentum is usually attributed to the pions from the resonance de- cay, especially from the ∆ decay[3, 6]. However, present calculation shows that this is not the case at low beam energies. The shape of the transverse mass spectra of π− and π+ are different especially in the small mass region. As the transverse mass mt decreases, the mt spectrum of π increases sharply while that of π+ saturates showing the convex shape. This difference is due to the Coulomb in- teraction of pions leaving the system which has the charge from the initially bombarding nucleons. The change in the transverse momentum due to the Coulomb interac- tion decreases from 25 GeV/c at 2 A·GeV to 15 GeV/c at 8 A·GeV. This behavior can be understood from the in- crease of the screening effect since the number of charged pions increase at higher energies. At very high energies such as RHIC energies, the momentum change from the Coulomb interaction will be small. The emerging picture of pion production at low en- ergy is that the pions are produced through the inter- mediate ∆ formation and thus they are not in chemical equilibrium in isospin. They make collisions and ther- malize to form a fireball which expands and cools until the freeze-out. Since the fireball has charge which is from the initially colliding nucleus, the pions leaving the sys- tem experiences the Coulomb interaction which makes the difference of the mt spectra of the two oppositely charged pions. Acknowledgments This work is financially supported by Chonnam Na- tional University and the post-BK21 program. We wish to acknowledge U.W. Heinz for providing the program and useful discussions. [1] A. Wagner et. al., Phys. Lett. B420, 20(1998). [2] B.J. VerWest and R.A. Arndt, Phys. Rev. C25, 1979(1982) [3] Phys. Rep. 135, 259(1986). [4] J.L. Klay, et. al., Phys. Rev. C68, 054905(2003). [5] B. Pin-zhen and J. Rafelski, nucl-th/0507037. [6] B.S. Hong, J. Korean Phys. Soc. 46, 1083(2005). [7] B. Hong et. al., Phys. Lett. B407, 115(1999); ibid., J. Korean Phys. Soc. 46, 1083(2005). [8] H. Dobbler, J. Sollfrank and U. Heinz, Phys. Lett. B457,353(1999) . [9] Kang S. Lee, U. Heinz, and E. Schnedermann, Z. Phys. C 48,525 (1990). [10] E. Schnedermann, J. Sollfrank, and U. Heinz, Phys. Rev. C 48, 2462 (1993). [11] H.W. Barz, J.P. Bondorf,J.J. Gaardhoje and H. Heisel- berg, Phys. Rev. C57, 2536(1998). [12] M. Gyulassy and S. K. Kaufmann, Nucl. Phys. A362, 503(1981). [13] A. Ayala and J. Kapusta, Phys. Rev. C56, 407(1997). [14] F. Cooper and G. Frye, Phys. Rev. D10, 186(1974) [15] J.D. Bjorken, Phys. Rev. D27, 140(1983) http://arxiv.org/abs/nucl-th/0507037

A study of the p d → p d η reaction N.J.Upadhyay,1, ∗ K.P.Khemchandani,1, 2, † B.K.Jain,1, ‡ and N.G.Kelkar3, § 1Department of Physics, University of Mumbai, Vidyanagari, Mumbai - 400 098, India 2Departamento de F́ısica Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de Paterna, Aptd. 22085, 46071 Valencia, Spain 3Departamento de Fisica, Universidad de los Andes, Cra.1E No. 18A-10, Santafe de Bogota, Colombia (Dated: October 25, 2018) http://arxiv.org/abs/0704.1530v1 Abstract A study of the p d → p d η reaction in the energy range where the recent data from Uppsala are available, is done in the two-step model of η production including the final state interaction. The η − d final state interaction is incorporated through the solution of the Lippmann Schwinger equation using an elastic scattering matrix element, Tη d→ η d, which is required to be half off-shell. It is written in a factorized form, with an off-shell form factor multiplying an on-shell part given by an effective range expansion up to the fourth power in momentum. The parameters of this expansion have been taken from an existing recent relativistic Faddeev equation solution for the ηNN system corresponding to different η −N scattering amplitudes. Calculations have also been done using few body equations within a finite rank approximation (FRA) to generate Tη d→ η d. The p − d final state interaction is included in the spirit of the Watson-Migdal prescription by multiplying the matrix element by the inverse of the Jost function. The η − d interaction is found to be dominant in the region of small invariant η − d mass, Mηd. The p − d interaction enhances the cross section in the whole region of Mηd, but is larger for large Mηd. We find nearly isotropic angular distributions of the proton and the deuteron in the final state. All the above observations are in agreement with data. The production mechanism for the entire range of the existing data on the p d → p d η reaction seems to be dominated by the two-step model of η production. PACS numbers: 25.10.+s, 25.40.Ve, 24.10.Eq ∗Electronic address: neelam@physbu.mu.ac.in †Electronic address: kanchan@ific.uv.es ‡Electronic address: brajeshk@gmail.com §Electronic address: nkelkar@uniandes.edu.co mailto:neelam@physbu.mu.ac.in mailto:kanchan@ific.uv.es mailto:brajeshk@gmail.com mailto:nkelkar@uniandes.edu.co I. INTRODUCTION The current great interest in the η-nucleus interaction exists because of the attractive nature of the η − N interaction in the s-wave [1], and the consequent possibility of the existence of quasi-bound, virtual or resonant η-nucleus states [2]. The exact nature of these states, of course, depends upon the precise knowledge of the η −N scattering matrix at low energies. As the η is a highly unstable meson (lifetime ∼ 10−18s), this precise information is difficult to obtain directly. It can only be obtained from the eta producing reactions through the final state interaction. With this motivation, starting with the early experiments near threshold at Saclay on the p d → 3He η and the p d → p d η reactions, measurements have been carried out near threshold and beyond at Jülich and Uppsala using the COSY and Celsius rings respectively. In this series of experiments, the recent data on the p d → p d η reaction using the Wasa/Promice setup at the Celsius storage ring of the Svedberg laboratory, Uppsala are thematically complete and cover the excess energy, Q, (Q = s − mη − mp − md) ranging from around threshold to 107 MeV. The data [3] (integrated over other variables) include the invariant mass distribution over the whole excess energy range for the η− d, η− p and p− d systems and angular distributions for the proton, deuteron and the eta meson. Like the p d → 3He η reaction, the (inclusive) η − d invariant mass distribution exhibits a large enhancement near threshold and hence appears promising to study the η − d interaction. The η − p and p− d invariant mass distributions do not show any such enhancement. All observed angular distributions are nearly isotropic. Like in our earlier studies on the p d → 3He η reaction our primary aim in this paper is to investigate the above mentioned data on the η − d invariant mass distribution to obtain a better understanding of the η − N as well as the η − d interaction. We speculate, from our experience on the study of the p d → 3He η reaction [4], that in the region of low η− d relative energy this set of data will be mainly determined by the η − d interaction, though the three-body nature of the final state may introduce some uncertainty in this conclusion. We present a study of the p d → p d η reaction which includes the effect of the final state interaction. We have investigated two possible diagrams for the production mechanism: the direct mechanism and the two step process of η production. The direct mechanism proceeds via an intermediate p n → d η reaction with one of the nucleons in the deuteron as a spectator. The η meson in the two step model is produced in two steps, namely, p p → d π+ and π+N → η N , hence involving the participation and sharing of the transferred momentum by three nucleons. The two step model for η production was first used in [5] and the data on the p d → 3He η reaction was well explained. The vertices at the two steps have been described by the corresponding off-shell T -matrices. The T -matrix for π+N → η N is taken from a coupled channel calculation [1], and that for p p → d π+ is obtained from the SAID program provided by the authors of Ref. [6]. The final state interaction between the η and the deuteron is explicitly incorporated through an η − d T -matrix, Tηd. This T -matrix, which is required to be half-off-shell, is described in two ways. One choice involves taking a “factorized form” which is given by an off-shell form factor multiplied by an on-shell part given by an effective range expansion up to the fourth power in momentum. The parameters of this expansion have been taken from an existing recent relativistic Faddeev equation solution for the ηNN system [7] corresponding to different η − N scattering amplitudes. The off-shell form factor will be described in the next sections and is chosen to have a form without any adjustable parameters. The second prescription involves solving few body equations within the finite rank approximation (FRA) to obtain Tηd. This approach has been used in literature for the η − d, −3He and −4He systems [8]. We perform calculations for both the prescriptions using different models of the elementary coupled channel η-nucleon T -matrix which characterize them. The interaction between the η meson and the proton in the final state, to a certain extent is contained implicitly in our calculations. This is due to the fact that we describe the π+N → η N vertex by a T -matrix, which has been modeled to include the η − N interaction. This off-shell T -matrix treats the π N , η N and π∆ channels in a coupled channel formalism [1] and reproduces the experimental data on this reaction very well. The effect of p−d final state interaction (FSI) is incorporated in the spirit of the Watson- Migdal FSI prescription [9], in which our model p d → p d η production amplitude is mul- tiplied by a factor which incorporates the FSI between the proton and the deuteron. This factor is taken to be the frequently used [10, 11, 12, 13] inverse Jost function, [J(p)]−1, where p is the relative p− d momentum. The assumption implicit in this approximation that the mechanism for the primary reaction be short ranged is very well fulfilled in the η-production reactions. The momentum transfer in these reactions near threshold is around 700 MeV/c. We include FSI for both doublet (2S1/2) and quadruplet ( 4S3/2) p− d states. The η-nucleon T -matrix, which characterizes our calculations, is not precisely known. Recent theoretical works on the n p → d η reaction [14] conclude that the data on this reaction can be reproduced with the strength of the real part of the η-nucleon scattering length ranging between 0.42 and 0.72 fm. In our earlier work on the p d → 3He η reaction [4], we found a good agreement with data, with the real part of the scattering length taken to be around 0.75 fm. This value was also found to be in agreement with the n p → d η data in a K-matrix calculation of the final state η− d interaction in [15]. The same authors as in [15], recently performed a fit to a wide variety of data which includes the π N → π N , π N → η N , γ N → π N and γ N → η N reactions and gave their best fit value of the η-nucleon scattering length, aηN to be (0.91, 0.27) fm [16]. The η − d effective range parameters are given in [7] for aηN up to (1.07 , 0.26) fm. Hence, in the present work we perform calculations with different models of the η − N interaction, which correspond to three different values of the η − N scattering length, ranging from aηN = (0.42 , 0.34) fm to (1.07 , 0.26) fm. We find that the cross sections calculated using the two-step model and the above in- puts for the final state interaction reproduce most of the features of the experimental data reasonably well. A theoretical effort to understand the Uppsala data [3] was made earlier by Tengblad et. al. [17]. In [17] the contribution of three different diagrams, namely, the pick-up (a direct one-step mechanism of η production), the impulse approximation and the two-step mechanism (here the η meson is produced in two steps via the p p → π+ d and π+N → η N reactions) to the cross section for the p d → p d η reaction is determined. The authors in [17] conclude that the impulse approximation is in general negligible as compared to the other two diagrams, the two-step mechanism is dominant in the near threshold region and the contribution of the pick-up diagram (referred to as the direct mechanism in the present work) increases with energy and matches the two-step contribution at an excess energy of Q = 95 MeV. The latter conclusions regarding the contributions of the two step and pick up diagrams are in contrast to the findings of the present work as well as to existing literature on similar kind of reactions. We note here that the authors in [17] do not include the final state interaction in their calculations in any way. They treat the kinematics and the dependence of the pion propagator (appearing in the two step model) on the Fermi momenta in an approximate way. The T -matrices which enter as an input to the two step model are simply extracted from experimental cross sections and are hence not proper off-shell T -matrices. As a result of the above approximations, the authors in [17] do not reproduce the observed enhancement in the η−d invariant mass distribution near threshold, and unlike the observed isotropic distributions, find anisotropy in their calculated angular distributions. The contribution from the direct mechanism (or the so-called pick-up diagram of [17]) to the total cross sections is found to be about four orders of magnitude smaller than the two- step contribution at threshold in the present work. The one-step contribution does increase with energy (as also found in [17]), however, even at the highest energy for which data is available (Tp = 1096 MeV) it remains two orders of magnitude smaller than that due to the two-step model. This is in contrast to the observations in [17], where the two processes give comparable contributions at high energies. The difference of orders of magnitude between the two processes can be understood as a result of the large momentum transfer, q, in the one-step process. This q, which is very large in the threshold region (∼ 840 MeV/c) continues to be large even at high energies. For example, it is ∼ 600 MeV/c even at the highest beam energy of 1096 MeV. This finding of ours is very similar to the previous studies of the reactions involving high momentum transfer. For example, as mentioned above too, in [5], for the pd →3 Heη reaction up to 2.5 GeV beam energy, the authors comment that the one-step cross sections underestimate the data by more than two orders of magnitude. In yet another calculation [18] of the cross section for the pd →3He X reaction (where X = η, η′, ω, φ) the two-step model was found to describe the data on these reactions up to 3 GeV quite well. In [19], in connection with the p d → 3HΛK+ reaction, the authors claim that for a beam energy of 1 - 3 GeV, the one-step mechanism predicts 2 to 3 orders of magnitude smaller cross sections as compared to the two-step mechanism. The cross sections obtained from the one-step model, in Ref. [17] are, however, reported to be only one order of magnitude less than those due to the two-step model at threshold and comparable to the two-step ones at high energies. In the next section, we describe the details of the formalism. In the subsequent sections we present and discuss the results and finally the conclusions. II. THE FORMALISM The differential cross section for the p d → p d η reaction, in the center of mass, can be written as, 2 (2 π)5 s |~kp| dΩp′ | ~kp′| dMη d |~kη d| dΩη d 〈 |T |2 〉 (1) where s is the total energy in the center of mass and ~kp and ~kp′ are the proton momenta in the initial and final states respectively. Mη d denotes the invariant mass of the η − d system and ~kη d and Ωη d denote, in the η − d center of mass, the η momentum and its solid angle, respectively. Ωp′ represents the solid angle of the outgoing proton. Angular brackets around |T |2 in Eq. (1) represent the sum over the final and initial spins. The T -matrix, which includes the interaction between the η and the deuteron is given by T = 〈ψηd( ~kηd), ~kp′; mp′, md′ | Tpd→pdη | ~kp, ~kd (= − ~kp); mp, md 〉 (2) where the spin projections for the proton and the deuteron in the initial and final states have been labeled as mp, md, mp′, and md′ respectively. Tp d→ p d η is the production operator. The wave function of the interacting η − d in the final state has been represented as ψηd( ~kηd). In terms of the elastic η − d scattering T -matrix, Tηd, it is written as 〈ψ−ηd | = 〈 ~kηd |+ (2π)3 〈 ~kηd | Tηd | ~q 〉 E(kηd) − E(q) + iǫ 〈 ~q | (3) The second term here represents the scattered wave. It has two parts originating from the principal-value and the delta-function part of the propagator in the intermediate state. Physically they represent the off-shell and the on-shell scattering between the η and the deuteron. The on-shell part can be shown to be roughly proportional to the η−d momentum and hence dominant at higher energies. The relative contribution of these terms in our case would be determined after we substitute the above expression for ψηd( ~kηd) in Eq. (2). We then get T = 〈 ~kηd, ~kp′; mp′, md′ | Tpd→pdη | ~kp, ~kd(= −~kp); mp, md 〉 (4) (2π)3 〈 ~kηd;md′ | Tηd | ~q;m2′ 〉 E(kηd)−E(q) + iǫ 〈 ~q , ~kp′;m2′ , mp′ | Tpd→pdη | ~kp, ~kd; mp, md〉 It can be seen that the Tηd here appears as a half-off-shell T -matrix. = - k p kd/ 2 - P FIG. 1: The two step process production mechanism for the p d → p d η reaction. A. The production mechanism For evaluating the η production T -matrix, 〈 | Tpd→pdη | 〉, we assume a two-step mechanism as shown in Fig. 1. In this model, the incident proton produces a pion in the first step on interacting with one of the nucleons of the target deuteron. In the second step this pion produces an η meson on interacting with the other nucleon. Both these nucleons are off-shell and have a momentum distribution given by the deuteron bound state wave function. To write the production matrix, we resort to certain standard approximations used in literature [20] (in particular for the triangle diagram appearing in Fig. 1). The amplitude for the pN → πd process, which in principal is off-shell, is taken at an on-shell energy. Considering the high proton beam energy, off-shell effects are not expected to be significant. The π N → η N process is included via an off-shell T -matrix. The production matrix is written as [4, 5], 〈 | Tpd→ p d η | 〉 = (2π)3 〈 p n | d 〉 〈 | Tpp→ π+ d | 〉 k2π − m2π + iǫ 〈 | Tπ+ n→ η p | 〉 (5) where, the squared four momentum of the intermediate pion, k2π = E π−~k2π, with the energy, Eπ, calculated at zero fermi momentum and ~kπ = ~kη + ~kp′ − ~kd/2 + ~P . The summation is over internal spin projections and the matrix element 〈 p n | d 〉 represents the deuteron wave function in momentum space, which has been written using the Paris parametrization [21]. The factor 3/2 is a result of summing the diagrams with an intermediate π0 and π+. The integral over the pion momentum in above includes the contribution from the pole as well as the principal value term. For the pion propagator itself, as we see, we have taken = - k | = - ( kη + kd| ) -1/2 k FIG. 2: The direct process production mechanism for the p d → p d η reaction. the plane wave propagator. This thus excludes any effect in our results due to medium mod- ification of this propagator due to other nucleons. This aspect may be worth investigating in future. The T -matrix for the intermediate p p → π+ d process has been taken from an energy dependent partial wave analysis of the π+ d → p p reaction from threshold to 500 MeV [6]. The various observables in [6] are given in terms of amplitudes which are parametrized to fit the existing database. We refer the reader to [6] and the references therein for the relevant expressions of the helicity and partial wave amplitudes and the notation followed by the authors in [6]. For the π+ n → η p sub-process, different forms of T -matrices are available. We use the T -matrix from [1] which treats the π N , η N , and π∆ channels in a coupled channel formalism. This T -matrix consists of the meson - N∗ vertices and the N∗ propagator as given below: Tπ+ n→ η p(k ′ , k ; z) = gN∗ β (k′ 2 + β2) τN∗(z) gN∗ β (k2 + β2) with, τN∗(z) = ( z − M0 − Σπ(z) − Ση(z) + i ǫ)−1 where Σα(z) (α = π , η) are the self energies from the πN and η N loops. The parameters of this model are, gN∗ = 0.616, β = 2.36 fm −1 and M0 = 1608.1 MeV. This T -matrix reproduces the data on the π+n→ ηp reaction very well. Although the contribution of the direct mechanism (Fig. 2) is known to be small (owing to the large momentum transfer involved in the process) [5, 18, 19], for completeness, we calculate its contribution to the total cross section. The T -matrix for this mechanism can be written as 〈|Tp d→ p d η|2〉 = 〈|Tpn→ d η( sη d)|2〉 × | φd(q)|2, (7) where φd represents the deuteron wave function in the initial state. The spin summed 〈| Tpn→ d η |2〉 is given in terms of the total cross section for the p n → d η reaction by σT (p n → d η) = 2mpmnmd | ~pf | |~pi| 〈| Tpn→ d η|2 〉, (8) where ~pi and ~pf are the initial and final momenta c.m. system. The momentum transfer ~q, as shown in Fig. 2, is defined as, ~kp + ~kp′ . (9) The total cross section, σT , for p n → d η reaction is taken from the experiments [22]. B. Final state interaction 1. η − d interaction This is incorporated through a half-off-shell η− d T -matrix. We construct this T -matrix using the following two prescriptions: 1. Factorized form of Tηd In one ansatz we obtain it by multiplying the on-shell η − d T -matrix by an off-shell extrapolation factor g(k′ , k). Requiring that this T -matrix goes to its on-shell value in the case of on-shell momenta, we write Tη−d(k , E(k0) , k ′) = g(k , k0) Tη−d(E(k0)) g(k ′ , k0), (10) with g(p , q) → 1 as p → q. For a half-off-shell case, this obviously is the ratio of the half-off-shell to the on-shell scattering amplitude. For the on-shell η − d T -matrix we use the effective range expansion of the scattering amplitude up to the fourth power in momentum, F (k) = Rk2 + Sk4 − ik , (11) where F is related to T by Tηd(k, k ′) = − 1 (2π)2µηd Fηd(k, E(k), k ′). (12) The effective range expansion parameters (A,R,S) are taken from a recent relativistic Faddeev equation (RFE) calculation of [7]. This calculation uses the relativistic version of the Faddeev equations for a three particle mNN system, where m is a meson and it can be an η, π or a σ meson. These particles interact pairwise, and these interactions are represented with separable potentials. The parameters of the ηN − πN − σN potentials are fitted to the S11 resonant amplitude and the π −p → ηn cross sections. The η − d effective range parameters obtained from these calculations are listed in [7] for different sets of the meson-nucleon potentials. Each of these sets gives a specific value of the η −N scattering length, which is also listed in [7]. Since the half-off-shell extrapolation factor g(k′ , k0) is not known with any certainty, we choose the following two forms for it. (i) Following the method in [15] for the final state interaction in the η− d system, we express the off-shell form factor in terms of the deuteron form factor, g(k′ , k0) = d~rj0(rk ′/2)φ2d(r)j0(rk0/2) (13) where for the deuteron wave function, φd(r), we take the Paris parametrization. (ii) As a second choice, the form factor is taken to be the ratio of the off-shell η − d T -matrix to its on-shell value, where both of them are calculated using the three body equations within FRA. The input to these calculations is the elementary η −N scattering matrix, the details of which are given in the next Section. 2. Few body equations within the finite rank approximation The other prescription of η−d FSI involves the use of the half-off-shell η−d T -matrix obtained by solving few body equations within the finite rank approximation (FRA). For the details of this formalism and the expression for the η-nucleus T -matrix, we refer the reader to our earlier works [4]. To mention briefly, the FRA involves restricting the spectral decomposition of the nuclear Hamiltonian in the intermediate state to the ground state, neglecting thereby all excited and break-up channels of the nucleus. This is justified in the η − 4He and possibly in the η − 3He case, but in η−deuteron collisions, where the break-up energy is just 2.225 MeV, the applicability of the FRA may be limited. However, it should be noted that a comparative study [23] of the η−d scattering lengths calculated using the FRA and the exact Alt-Grassberger-Sandhas (AGS) [24] equations (which include these intermediate excitations) shows that they are not very different if the real part of the η−N scattering length is restricted up to about 0.5 fm. 2. p− d interaction We incorporate the p− d FSI in our calculations by multiplying our model T -matrix by the inverse Jost function, [J(p)]−1. We include the FSI in both the 1/2 and 3/2 spin states of p − d and restrict it to the s-wave. Since the p and d are charged we also include the Coulomb effects. Following standard procedure, we write the Jost function in terms of phase shifts and use the effective range expansion for the later. The complete expression for the s-wave inverse Jost function squared is written as, [Jo (kpd)] −2 = [Jo (kpd)] Q + [(1 + )Jo (kpd)] D . (14) Here, to include the effect of the existence of one bound state, namely, the spin 1/2 state (3He), the doublet Jost function is multiplied by a factor (1 + ), where |EB| is the separation energy of 3He into p− d. Its value is taken to be 5.48 MeV. The expressions for spin quadruplet (Q) and doublet (D) [Jo (kpd)] −2 are given by [Jo (kpd)] (k2pd + α 2)2 (bcQ) 3C2o k 1 + cot2 δQ [Jo (kpd)] (k2pd + α 2)2 (bcD) 3C2o k 1 + cot2 δD where, 2 bcµ and acµ and b µ are defined as − 2 γ kpdHγ bcµ = where µ stands for either Q or D. The factor C2o in above has its origin in the Coulomb interaction. The phase shifts δQ ,D are obtained from an effective-range expansion [25, 26], C2o kpd cot δµ = − pd − 2 γ kpdHγ (20) αmred ~ kpd C2o = 2 π γ e2 π γ − 1 n(n2 + γ2) − ln (γ) − 0.57722 (23) Here mred is the reduced mass in the p− d system, γ the Coulomb parameter and α is the usual electromagnetic coupling constant. The values of the expansion coefficients aµ, bµ in Eq. (20) are taken as aQ = 11.88 fm, bQ = 2.63 fm, aD = 2.73 fm, and bD = 2.27 fm. They have been determined from a fit to the p − d elastic-scattering phase shifts in the relative p− d momentum range up to around 200 MeV/c [27]. The above expression for the Jost function has the required property that for large p, J0(p) → 1. III. RESULTS AND DISCUSSION Before we discuss the results of the present work, in order to highlight the FSI effects in the experimental η − d invariant mass distribution we remove the phase space from the experimental dσ/dMηd and plot in Fig. 3 the |f |2, which is then given by, |f |2 = phase space , (24) where, phase space = 12 (2 π)5 s |~kp| dΩp′ | ~kp′| |~kη d| dΩη d (25) as a function of the excess energy, Qηd = Mηd − mη − md, where Mηd is the invariant mass of the η − d system. In this figure we also show the plane wave result (i.e. Tp d→ p d η does not include any FSI). The cross section, dσ/dMη d in Eq. (24), is evaluated for each Mηd by performing an integral over the p − d centre of mass momenta, kpd. The range of the allowed values of kpd at each Mηd is shown by the hashed region. One clearly sees a large 0 10 20 30 40 50 60 70 Qηd ( = Mηd − mη − md ) (MeV) = 1032 MeV Bilger et. al. Plane wave FIG. 3: The ratio of experimental differential cross sections [3] to the phase space (Eq. (25)) as a function of the excess energy, Qηd, along with range of p−d relative momenta, kpd (hashed region), contributing to |f |2 at each Qηd. enhancement in the experimental |f |2 near small values of Qη d, which, most likely is due to the η − d FSI. We also observe a rise at large values of Qηd. Examining the range of p− d relative momenta which contribute to |f |2 at each Qηd, one can see that this rise occurs at small values of kpd, indicating thereby the possibility of a large effect of p − d FSI in this region. In Fig. 4, we show two sets of the calculated |f |2 along with the experimental results for a beam energy of 1032 MeV. These results include only η − d FSI. We limit the range of Qηd up to about 10 MeV, where, this effect is large. In Fig. 4(a) we show results for the factorized prescription with the off-shell factor generated from the deuteron form factor and the on-shell part arising from the relativistic Faddeev equation (RFE) calculation of [7]. The results are shown for three different sets of interaction parameters in the RFE. Since these sets give uniquely different values of the η − N scattering lengths aηN , we identify them by their corresponding aηN values. For the results presented here, these values are 0.42 + i0.34 fm, 0.75 + i0.27 fm and 1.07 + i0.26 fm. We see that our results reproduce the enhancement seen in the experimental |f |2 at small values of Qηd. The absolute magnitude depends upon the choice of the RFE parameters. It increases with aηN , which designate these parameter sets. The set corresponding to aηN = 1.07 + i0.26 fm, gives results closest 0 2 4 6 8 10 12 14 16 Qηd ( = Mηd − mη − md ) (MeV) Bilger et. al. Plane wave ηd FSI (aηN = 1.07 + i0.26 fm) ηd FSI (aηN = 0.75 + i0.27 fm) ηd FSI (aηN = 0.42 + i0.34 fm) 0 2 4 6 8 10 12 14 16 Qηd ( = Mηd − mη − md ) (MeV) Bilger et. al. Plane wave ηd FSI (aηN = 0.92 + i0.27 fm) ηd FSI (aηN = 0.77 + i0.25 fm) ηd FSI (aηN = 0.4 + i0.3 fm) FIG. 4: The calculated |f |2 along with the experimental results for a beam energy of 1032 MeV. (a) The results correspond to the factorized form of Tηd with the off-shell factor generated from the deuteron form factor. (b) The results correspond to Tηd obtained from few body equations within the FRA. The data is the same as in Fig. 3. to the experimental values. In Fig. 4(b) we show |f |2 calculated using few body equations within the FRA, for η− d FSI. These results are shown for three different inputs of the η − N T -matrix taken from [16]. The choice of these T -matrices is such that their scattering length values are close to those used in Fig. 4(a). Though this model has the limitation of retaining the intermediate nucleus in its ground state in the η− nucleus elastic scattering, the off-shell re-scattering effects have been properly included. If we compare Fig. 4(a) and 4(b), the two sets of results are similar. In order to check the sensitivity of the results to the off-shell form factor used in the factorized η − d T -matrix, in Fig. 5(a), we show the |f |2 calculated using two different off-shell form factors. The on-shell Tηd is obtained from RFE and the off-shell part is either treated with a deuteron form factor (solid line) or a few body FRA form factor (dash dotted line) as explained in section II B. The elementary η −N T -matrix parameters required for the calculation of the FRA form factor are taken from the parametrization of Green and Wycech [16]. Even though the results (as shown in Fig. 4(a)) corresponding to the aηN = 1.07 + i0.26 fm seem to be the closest to the data, to compare the effect of using different 0 2 4 6 8 10 12 14 16 Qηd ( = Mηd − mη − md ) (MeV) = 1032 MeV Bilger et. al. Plane wave Deuteron form factor FRA form factor 2 4 6 8 10 12 14 k (fm = 51 MeV/c Deuteron form factor FRA form factor FIG. 5: Comparison of the two form factors for aη N = 0.75 + i0.27 fm. (a) Effect of using two different off-shell extrapolation factors for η − d FSI on |f |2. (b) Two form factors as function of off-shell momentum (k′). off-shell form factor, we choose the results corresponding to aη N = 0.75 + i0.27 fm. We make this choice such that we can compare the two calculations for the inputs corresponding to a similar η − N scattering length. It should be expected then, that the off-shell form factors obtained from two different methods should not differ much. This is seen explicitly in Fig. 5(b) where the two form factors are shown as a function of off-shell momentum (k′) for an on-shell value, k0 near the low energy peak in the η − d invariant mass distribution (to be discussed in Fig. 9 later). Next, we include in our calculations the effect of the p−d FSI. This is done by multiplying the pd → pdη squared T -matrix (Eq. (4)) used above by the inverse Jost function squared in Eq. (15) and Eq. (16), and integrating it over the allowed range (as shown in Fig. 3) of p − d momenta, kpd for each Qη d. We show these results in Fig. 6 for the RFE (with deuteron form factor) model of η − d FSI, for the parameter set corresponding to aηN = 1.07 + i0.26 fm. We find, that the p− d FSI affects the results in the whole region of Qη d, while the effect of η − d FSI is confined to small value of Qηd. The large effect of p− d FSI in the region of small Qηd, however, may not be taken with confidence as the value of kpd in this region is large (as shown in Fig. 3), where the s-wave effective range expansion (Eq. (20)) for the calculation of Jost function might not be sufficient. In any case, it appears 0 10 20 30 40 50 60 70 Qηd ( = Mηd − mη − md ) (MeV) = 1032 MeV Bilger et. al. Plane wave ηd FSI ηd & pd FSI FIG. 6: The proton-deuteron final state interaction effects on the p d → p d η reaction at the beam energy of 1032 MeV. The dashed line shows the plane wave results and the dashed dot (solid) line shows the effect of η − d (η − d & p− d) FSI for aηN = 1.07 + i0.26 fm. that the effects of both the η− d and the p− d FSI on the η− d invariant mass distribution are significant. If we disregard the calculated p− d effect for small Qηd, the η− d and p− d FSI dominate in regions well separated from each other. Apart from the FSI, another important ingredient of our calculations is the two-step description of the production vertex. Because of the large momentum transfer, we believe, as has also been stressed in Ref. [17], that the angular distribution of the outgoing particles is probably more sensitive to the description of the production vertex. Inclusive angular distributions have been measured for all the three outgoing particles in the p d → p d η reaction. In Fig. 7, we show the calculated angular distributions for all the three outgoing particles along with the measured distributions. We show results without any FSI, with η− d FSI and with η− d and p− d FSI both included. As each angle has contribution from a range of Qη d as well as kpd, the calculated results include integration of the cross section over these variables. We find that the observed nearly isotropic nature of the experimental angular distributions for the proton, deuteron and eta already gets reproduced by the plane wave calculations. The effect of both η − d and p − d FSI is large and persists over all the angles. Their inclusion brings the magnitudes of the proton and deuteron angular distributions near to experiments. The magnitude of the eta distribution, however, does not seem to be affected much with the FSI. Experimental data also exist on the total cross section. In Fig. 8, we compare the total Bilger et al Plane wave ηd FSI ηd & pd FSI -0.8 -0.4 0 0.4 0.8 cos θ =1032 MeV FIG. 7: The calculated angular distributions of (a) the deuteron, (b) the proton and (c) the η, along with the measured cross sections for aηN = 1.07 + i0.26 fm [3]. 950 1000 1050 1100 Beam Energy (MeV) Bilger et al. (PRC 69, 014003 (2004)) Hibou et al. (EPJA 7, 537 (2002)) Two step mechanism Two step planewave The direct mechanism 0 20 40 60 80 100 Excess Energy (MeV) FIG. 8: A comparison of the total cross section for the p d → p d η reaction calculated with the description of the production vertex as a two step mechanism and direct mechanism, along with the measured cross sections for aη N = 1.07 + i0.26 fm [3, 28]. 2430 2440 2450 2460 2470 2480 2490 dη (MeV) Total contribution Off shell contribution On shell contribution Plane wave FIG. 9: Contributions from the off-shell and the on-shell η − d scattering in the final state. The results are for aη N = 1.07 + i0.26 fm with the inclusion of only the η − d FSI. cross sections calculated including both the η − d and p − d FSI with the measured cross sections. The results are shown with the factorized form of η − d FSI with deuteron form factor for the set corresponding to η −N scattering length equal to 1.07 + i0.26 fm. As we see, the calculated cross sections are in good agreement with the experimental data. In Fig. 8 we also give the cross sections calculated for the one-step direct mechanism (Fig. 2) mentioned in the previous section. Near threshold, these cross sections are about four orders of magnitude below those obtained from the two-step model and two orders of magnitude smaller in the high energy range. As mentioned in the Introduction, this observation is similar to that in other works involving large momentum transfer reactions [5, 18, 19], and is understandable because the momentum transfer continues to be large (∼ 600 MeV/c) in the p d → p d η reaction even at an excess energy as large as 100 MeV. Now we make an observation about the importance of off-shell scattering in treating η − d FSI near threshold. The scattering part of the η − d wave function (Eq. (3)), gets contributions from the off-shell as well as the on-shell scattering in the nucleus. To see quantitatively the relative importance of these two contributions to the cross section for the p d → p d η reaction, in Fig. 9 we show their contributions separately in the η− d invariant mass distribution. These results include only the η − d FSI generated from the factorized prescription using RFE and the deuteron form factor for the η − d T -matrix. We find that near threshold the off-shell scattering completely dominates the threshold enhancement. At 2430 2440 2450 2460 2470 2480 2490 dη (MeV) Bilger et. al. Total contribution Plane wave FIG. 10: A comparison of the calculated results including both the η − d and p − d FSI with the experimental results. The results are for aηN = 1.07 + i0.26 fm. higher excess energy, as expected, the on-shell contribution takes over. Finally we show the nature of agreement of our calculated results with the invariant η−d mass distribution. In Fig. 10, we compare the calculated results including both the η − d and p − d FSI with the experimental results. The results are for aηN = 1.07 + i0.26 fm calculated with the factorized prescription using RFE and the off-shell factor generated from the deuteron form factor. As we see the overall agreement is reasonably good. IV. SUMMARY The invariant η − d mass distribution in the p d → p d η reaction has been studied by describing the production mechanism in terms of a two step model with a pion being produced in the intermediate state. The η−d final state interaction (FSI) has been included in (a) a factorized form involving an on-shell Tηd and two types of off-shell form factors and (b) by solving few body equations within the FRA. The p−d FSI is included through a Jost function. The conclusions of this investigation can be summarized as: 1. Experimentally observed large enhancement in the cross section near small η−d excess energy, Qηd is reproduced by the η − d FSI. The rise in the cross section at large Qηd (which corresponds to a range of small momenta, kpd) can be accounted for by the p− d FSI. 2. Quantitative reproduction of the large enhancement requires η− d FSI corresponding to large values of aηN . In the present calculation it is around 1.07 + i0.26 fm. 3. The calculations successfully reproduce the observed isotropic angular distribution of the proton and the deuteron in the final state. The total cross sections for the pd→ pdη reaction are also well reproduced. 4. The off-shell part of the η − d scattering dominates near threshold. 5. The results for two different choices of the off-shell extrapolation factor in the factorized form of the η − d FSI are similar. V. ACKNOWLEDGMENTS The authors wish to thank R. A. Arndt for providing the computer codes for evaluating the p p → π+ d T -matrix. This work is done under a research grant by the Department of Science and Technology, Government of India. The authors (NJU, KPK and BKJ) gratefully acknowledge the same. [1] R. S. Bhalerao and L. C. Liu, Phys. Rev. Lett. 54 (1985) 865. [2] Q. Haider and L. C. Liu, Phys. Lett. B 172, 257 (1986); ibid 174, 465(E) (1986); ibid, Phys. Rev. C 66, 045208 (2002); N. G. Kelkar, K. P. Khemchandani and B. K. Jain, J. Phys. G:Nucl. Part. Phys. 32, L19 (2006); arXiv:nucl-th/0601080; for experimental claims, see M. Pfeiffer et. al., Phys. Rev. Lett. 92, 252001 (2004) and references therein. [3] R. Bilger et. al., Phys. Rev. C69 (2004) 014003. [4] K. P. Khemchandani, N. G. Kelkar and B. K. Jain, Nucl. Phys. A708 (2002) 312; ibid, Phys. Rev. C 68 (2003) 064610. [5] J. M. Laget and J. F. LeColley, Phys. Rev. Lett. 61 (1988) 2069. [6] R. A. Arndt, I. Strakovsky, R. L. Workman and D. A. Bugg, Phys. Rev. C48 (1993) 1926. The amplitudes can be obtained from the SAID program available at (http://gwdac.phys.gwu.edu). [7] H. Garcilazo, Phys. Rev. C71 (2005) 048201. http://arxiv.org/abs/nucl-th/0601080 http://gwdac.phys.gwu.edu [8] S.A. Sofianos and S.A. Rakityansky, nucl-th/9707044, S. A. Rakityansky et. al., Phys. Rev. C53 (1996) 2043. [9] K. M. Watson, Phys. Rev. 88, 1163 (1952); A. B. Migdal, Sov. Phys.-JETP 1, 2 (1955). [10] J. Gillespie, Final State Interactions, Holden-Day, Inc., San Francisco (1964). [11] Goldberger and Watson, Collision Theory, John Wiley & Sons, Inc., New York, (1964). [12] R. Shyam, Phys. Rev. C 60, 055213 (1999). [13] J. Dubach, W. M. Kloet and R. R. Silbar, Phys. Rev. C 33, 373 (1986). [14] H. Garcilazo and M. T. Peña, Phys. Rev. C72 (2005) 014003 ; ibid, C66 (2002) 034606. [15] S. Wycech and A. M. Green, Phys. Rev. C64 (2001) 045206. [16] A. M. Green and S. Wycech, Phys. Rev. C71 (2005) 014001; see (also the erratum in, Phys. Rev. C72 (2005) 029902(E). [17] U. Tengblad, G. Fäldt and C. Wilkin, Eur. Phys. J. A25 (2005) 267; arXiv:nucl-th/0506024. [18] L. A. Kondratyuk and Yu. N. Uzikov, Phys. Atom. Nucl. 60 468 (1997); arXiv:nucl-th/9510010. [19] V. I. Komarov, A. V. Lado and Yu. N. Uzikov, J. Phys. G:Nucl. Part. Phys. 21, L69 (1995); Phys. Atom. Nucl. 59, (1996) 804; arXiv:nucl-th/9804050. [20] L.A. Kondratyuk and M.G. Sapozhnikov, Phys. Lett. B220, (1989) 333; L.A. Kondratyuk, A.V. Lado and Yu.N. Uzikov, Phys. Atom. Nucl. 58, (1995) 473, Yad. Fiz. 58, 1995 (524); A. Nakamura and L. Satta, Nucl. Phys. A445, (1985) 706; V.M. Kolybasov and N.Ya. Smorodin- skaya, Phys. Lett. B37, (1971) 272. [21] M. Lacombe et. al., Phys. Lett. B101 (1981) 139. [22] H. Calèn Phys. Rev. Lett. 80 (1998) 2069. [23] N. V. Shevchenko et. al., Phys. Rev. C58 (1998) 3055(R). [24] E. O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. B2 (1967) 167. [25] H. A. Bethe, Phys. Rev. 76, 38 (1949). [26] H. O. Meyer and J.A. Niskanen, Phys. Rev. C 47, 2474 (1993). [27] J. Arvieux, Nucl. Phys. A221, 253 (1974). [28] F. Hibou et. al., Eur. Phys. J. A7 (2000) 537. http://arxiv.org/abs/nucl-th/9707044 http://arxiv.org/abs/nucl-th/0506024 http://arxiv.org/abs/nucl-th/9510010 http://arxiv.org/abs/nucl-th/9804050 Introduction The Formalism The production mechanism Final state interaction -d interaction p-d interaction Results and Discussion Summary Acknowledgments References

A study of the $p d \to p d \eta$ reaction in the energy range where the recent data from Uppsala are available, is done in the two-step model of $\eta$ production including the final state interaction. The $\eta -d$ final state interaction is incorporated through the solution of the Lippmann Schwinger equation using an elastic scattering matrix element, $T_{\eta d \to \eta d}$, which is required to be half off-shell. It is written in a factorized form, with an off-shell form factor multiplying an on-shell part given by an effective range expansion up to the fourth power in momentum. The parameters of this expansion have been taken from an existing recent relativistic Faddeev equation solution for the $\eta NN$ system corresponding to different $\eta-N$ scattering amplitudes. Calculations have also been done using few body equations within a finite rank approximation (FRA) to generate $T_{\eta d \to \eta d}$. The $p-d$ final state interaction is included in the spirit of the Watson-Migdal prescription by multiplying the matrix element by the inverse of the Jost function. The $\eta-d$ interaction is found to be dominant in the region of small invariant $\eta -d$ mass, $M_{\eta d}$. The $p-d$ interaction enhances the cross section in the whole region of $M_{\eta d}$, but is larger for large $M_{\eta d}$. We find nearly isotropic angular distributions of the proton and the deuteron in the final state. All the above observations are in agreement with data. The production mechanism for the entire range of the existing data on the $p d \to p d \eta$ reaction seems to be dominated by the two-step model of $\eta$ production.

Introduction, this observation is similar to that in other works involving large momentum transfer reactions [5, 18, 19], and is understandable because the momentum transfer continues to be large (∼ 600 MeV/c) in the p d → p d η reaction even at an excess energy as large as 100 MeV. Now we make an observation about the importance of off-shell scattering in treating η − d FSI near threshold. The scattering part of the η − d wave function (Eq. (3)), gets contributions from the off-shell as well as the on-shell scattering in the nucleus. To see quantitatively the relative importance of these two contributions to the cross section for the p d → p d η reaction, in Fig. 9 we show their contributions separately in the η− d invariant mass distribution. These results include only the η − d FSI generated from the factorized prescription using RFE and the deuteron form factor for the η − d T -matrix. We find that near threshold the off-shell scattering completely dominates the threshold enhancement. At 2430 2440 2450 2460 2470 2480 2490 dη (MeV) Bilger et. al. Total contribution Plane wave FIG. 10: A comparison of the calculated results including both the η − d and p − d FSI with the experimental results. The results are for aηN = 1.07 + i0.26 fm. higher excess energy, as expected, the on-shell contribution takes over. Finally we show the nature of agreement of our calculated results with the invariant η−d mass distribution. In Fig. 10, we compare the calculated results including both the η − d and p − d FSI with the experimental results. The results are for aηN = 1.07 + i0.26 fm calculated with the factorized prescription using RFE and the off-shell factor generated from the deuteron form factor. As we see the overall agreement is reasonably good. IV. SUMMARY The invariant η − d mass distribution in the p d → p d η reaction has been studied by describing the production mechanism in terms of a two step model with a pion being produced in the intermediate state. The η−d final state interaction (FSI) has been included in (a) a factorized form involving an on-shell Tηd and two types of off-shell form factors and (b) by solving few body equations within the FRA. The p−d FSI is included through a Jost function. The conclusions of this investigation can be summarized as: 1. Experimentally observed large enhancement in the cross section near small η−d excess energy, Qηd is reproduced by the η − d FSI. The rise in the cross section at large Qηd (which corresponds to a range of small momenta, kpd) can be accounted for by the p− d FSI. 2. Quantitative reproduction of the large enhancement requires η− d FSI corresponding to large values of aηN . In the present calculation it is around 1.07 + i0.26 fm. 3. The calculations successfully reproduce the observed isotropic angular distribution of the proton and the deuteron in the final state. The total cross sections for the pd→ pdη reaction are also well reproduced. 4. The off-shell part of the η − d scattering dominates near threshold. 5. The results for two different choices of the off-shell extrapolation factor in the factorized form of the η − d FSI are similar. V. ACKNOWLEDGMENTS The authors wish to thank R. A. Arndt for providing the computer codes for evaluating the p p → π+ d T -matrix. This work is done under a research grant by the Department of Science and Technology, Government of India. The authors (NJU, KPK and BKJ) gratefully acknowledge the same. [1] R. S. Bhalerao and L. C. Liu, Phys. Rev. Lett. 54 (1985) 865. [2] Q. Haider and L. C. Liu, Phys. Lett. B 172, 257 (1986); ibid 174, 465(E) (1986); ibid, Phys. Rev. C 66, 045208 (2002); N. G. Kelkar, K. P. Khemchandani and B. K. Jain, J. Phys. G:Nucl. Part. Phys. 32, L19 (2006); arXiv:nucl-th/0601080; for experimental claims, see M. Pfeiffer et. al., Phys. Rev. Lett. 92, 252001 (2004) and references therein. [3] R. Bilger et. al., Phys. Rev. C69 (2004) 014003. [4] K. P. Khemchandani, N. G. Kelkar and B. K. Jain, Nucl. Phys. A708 (2002) 312; ibid, Phys. Rev. C 68 (2003) 064610. [5] J. M. Laget and J. F. LeColley, Phys. Rev. Lett. 61 (1988) 2069. [6] R. A. Arndt, I. Strakovsky, R. L. Workman and D. A. Bugg, Phys. Rev. C48 (1993) 1926. The amplitudes can be obtained from the SAID program available at (http://gwdac.phys.gwu.edu). [7] H. Garcilazo, Phys. Rev. C71 (2005) 048201. http://arxiv.org/abs/nucl-th/0601080 http://gwdac.phys.gwu.edu [8] S.A. Sofianos and S.A. Rakityansky, nucl-th/9707044, S. A. Rakityansky et. al., Phys. Rev. C53 (1996) 2043. [9] K. M. Watson, Phys. Rev. 88, 1163 (1952); A. B. Migdal, Sov. Phys.-JETP 1, 2 (1955). [10] J. Gillespie, Final State Interactions, Holden-Day, Inc., San Francisco (1964). [11] Goldberger and Watson, Collision Theory, John Wiley & Sons, Inc., New York, (1964). [12] R. Shyam, Phys. Rev. C 60, 055213 (1999). [13] J. Dubach, W. M. Kloet and R. R. Silbar, Phys. Rev. C 33, 373 (1986). [14] H. Garcilazo and M. T. Peña, Phys. Rev. C72 (2005) 014003 ; ibid, C66 (2002) 034606. [15] S. Wycech and A. M. Green, Phys. Rev. C64 (2001) 045206. [16] A. M. Green and S. Wycech, Phys. Rev. C71 (2005) 014001; see (also the erratum in, Phys. Rev. C72 (2005) 029902(E). [17] U. Tengblad, G. Fäldt and C. Wilkin, Eur. Phys. J. A25 (2005) 267; arXiv:nucl-th/0506024. [18] L. A. Kondratyuk and Yu. N. Uzikov, Phys. Atom. Nucl. 60 468 (1997); arXiv:nucl-th/9510010. [19] V. I. Komarov, A. V. Lado and Yu. N. Uzikov, J. Phys. G:Nucl. Part. Phys. 21, L69 (1995); Phys. Atom. Nucl. 59, (1996) 804; arXiv:nucl-th/9804050. [20] L.A. Kondratyuk and M.G. Sapozhnikov, Phys. Lett. B220, (1989) 333; L.A. Kondratyuk, A.V. Lado and Yu.N. Uzikov, Phys. Atom. Nucl. 58, (1995) 473, Yad. Fiz. 58, 1995 (524); A. Nakamura and L. Satta, Nucl. Phys. A445, (1985) 706; V.M. Kolybasov and N.Ya. Smorodin- skaya, Phys. Lett. B37, (1971) 272. [21] M. Lacombe et. al., Phys. Lett. B101 (1981) 139. [22] H. Calèn Phys. Rev. Lett. 80 (1998) 2069. [23] N. V. Shevchenko et. al., Phys. Rev. C58 (1998) 3055(R). [24] E. O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. B2 (1967) 167. [25] H. A. Bethe, Phys. Rev. 76, 38 (1949). [26] H. O. Meyer and J.A. Niskanen, Phys. Rev. C 47, 2474 (1993). [27] J. Arvieux, Nucl. Phys. A221, 253 (1974). [28] F. Hibou et. al., Eur. Phys. J. A7 (2000) 537. http://arxiv.org/abs/nucl-th/9707044 http://arxiv.org/abs/nucl-th/0506024 http://arxiv.org/abs/nucl-th/9510010 http://arxiv.org/abs/nucl-th/9804050 Introduction The Formalism The production mechanism Final state interaction -d interaction p-d interaction Results and Discussion Summary Acknowledgments References

On the exact formula for neutrino oscillation probability by Kimura, Takamura and Yokomakura Osamu Yasuda∗ Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan Abstract The exact formula for the neutrino oscillation probability in matter with constant density, which was discovered by Kimura, Takamura and Yokomakura, has been applied mostly to the standard case with three flavor neutrino so far. In this paper applications of their formula to more general cases are discussed. It is shown that this formalism can be generalized to various cases where the matter potential have off-diagonal components, and the two non-trivial examples are given: the case with magnetic moments and a magnetic field and the case with non-standard interactions. It is pointed out that their formalism can be applied also to the case in the long baseline limit with matter whose density varies adiabatically as in the case of solar neutrino. PACS numbers: 14.60.Pq, 14.60.St ∗Electronic address: yasuda˙at˙phys.metro-u.ac.jp Typeset by REVTEX 1 http://arxiv.org/abs/0704.1531v2 mailto:yasuda_at_phys.metro-u.ac.jp I. INTRODUCTION Neutrino oscillations in matter (See, e.g., Ref. [1] for review.) have been discussed by many people in the past because the oscillation probability has non-trivial behaviors in matter and due to the matter effect it may exhibit non-trivial enhancement which could be physically important. Unfortunately, it is not easy to get an analytical formula for the oscillation probability in the three flavor neutrino scheme in matter, and investigation of its behaviors has been a difficult but important problem in the phenomenology of neutrino oscillations. In 2002 Kimura, Takamura and Yokomakura derived a nice compact formula [2, 3] for the neutrino oscillation probability in matter with constant density. Basically what they showed is that the quantity Ũ∗αjŨβj , which is a factor crucial to express the oscillation probability analytically, can be expressed as a linear combination of U∗αjUβj, where Ũαj and Uαj stand for the matrix element of the MNS matrix in matter and in vacuum, respectively. However, their formula is only applicable to the standard three flavor case. In this pa- per we show that their result can be generalized to various cases. We also show that their formalism can be applied also to the case with slowly varying matter density in the limit of the long neutrino path. In Sect. II, we review briefly some aspects of the oscillation probabilities, including a simple derivation for the formula by Kimura, Takamura and Yoko- makura which was given in Ref. [4], because these are used in the following sections. Their formalism is generalized to the various cases where the matter potential has off-diagonal components, and we will discuss the case with large magnetic moments and a magnetic field (Sect. III) and the case with non-standard interactions (Sect. IV). In Sect. V we summarize our conclusions. II. GENERALITIES ABOUT OSCILLATION PROBABILITIES A. The case of constant density It has been known [5] (See also earlier works [6, 7, 8].) that after eliminating the nega- tive energy states by a Tani-Foldy-Wouthusen-type transformation, the Dirac equation for neutrinos propagating in matter is reduced to the familiar form: UEU−1 +A(t) Ψ, (1) where E ≡ diag (E1, E2, E3) , A(t) ≡ 2GFdiag (Ne(t)−Nn(t)/2,−Nn(t)/2,−Nn(t)/2) , ΨT ≡ (νe, νµ, ντ ) is the flavor eigenstate, U is the Maki-Nakagawa-Sakata (MNS) matrix, m2j + ~p (j = 1, 2, 3) is the energy eigenvalue of each mass eigenstate, and the matter effect A(t) at time (or position ) t is characterized by the density Ne(t) of electrons and the one Nn(t) of neutrons, respectively. Throughout this paper we assume for simplicity that the density of matter is either constant or slowly varying so that its derivative is negligible. The 3× 3 matrix on the right hand side of Eq. (1) can be formally diagonalized as: UEU−1 +A(t) = Ũ(t)Ẽ(t)Ũ−1(t), (2) where Ẽ(t) ≡ diag Ẽ1(t), Ẽ2(t), Ẽ3(t) is a diagonal matrix with the energy eigenvalues Ẽj(t) in the presence of the matter effect. First of all, let us assume that the matter density A(t) is constant. Then all the t dependence disappears and Eq. (1) can be easily solved, resulting the flavor eigenstate at the distance L: Ψ(L) = Ũ exp −iẼL Ũ−1Ψ(0). (3) Thus the oscillation probability P (να → νβ) is given by P (να → νβ) = Ũ exp (−iEL) Ũ−1 = δαβ − 4 ∆ẼjkL ∆ẼjkL , (4) where we have defined j ≡ ŨαjŨ∗βj , ∆Ẽjk ≡ Ẽj − Ẽk, and throughout this paper the indices α, β = (e, µ, τ) and j, k = (1, 2, 3) stand for those of the flavor and mass eigenstates, respectively. Once we know the eigenvalues Ẽj and the quantity X̃ j , the oscillation probability can be expressed analytically. B. The case of adiabatically varying density Secondly, let us consider the case where the density of the matter varies adiabatically as in the case of the solar neutrino deficit phenomena. In this case, instead of Eq. (3), we get Ψ(L) = Ũ(L) exp Ẽ(t) dt Ũ(0)−1Ψ(0), where Ũ(0) and Ũ(L) stand for the effective mixing matrices at the origin t = 0 and at the end point t = L. The oscillation probability is given by P (να → νβ) = Ũ(L) exp Ẽ(t) dt Ũ(0)−1 Ũ(L)βjŨ(L) βkŨ(0) αjŨ(0)αk exp ∆Ẽ(t)jk dt . (5) 1 In the standard case with three flavors of neutrinos in matter, the energy eigenvalues Ẽj can be analytically obtained by the root formula for a cubic equation [9]. So the only non-trivial problem in the standard case is to obtain the expression for X̃ j , and this was done by Kimura, Takamura and Yokomakura [2, 3]. In general cases, however, the analytic expression for Ẽj is very difficult or impossible to obtain, and we will discuss below only examples in which the analytic expression for Ẽj is known. Eq. (5) requires in general the quantity like Ũ(t)βjŨ ∗(t)βk which has the same flavor index β but different mass eigenstate indices j, k, and it turns out that the analytical expression for Ũ(t)βjŨ ∗(t)βk is hard to obtain. However, if the length L of the neutrino path is very large and if | 0 ∆Ẽ(t)jk dt| ≫ 1 is satisfied for j 6= k, as in the case of the solar neutrino deficit phenomena, after averaging over rapid oscillations Eq. (5) is reduced to P (να → νβ) = j (L)X̃ j (0), where we have defined X̃ααj (t) ≡ ∣Ũ(t)αj In the case of the solar neutrinos deficit process νe → νe during the daylight, X̃ββj (L) at the end point t = L and X̃ααj (0) at the origin t = 0 correspond to X j in vacuum and [X̃ at the center of the Sun, respectively, where j ≡ UαjU∗βj ŨαjŨ are bilinear products of the elements of the mixing matrices in vacuum and at the center of the Sun, respectively. Thus we obtain P (νe → νe) = X̃eej Hence we see that evaluation of the quantity X̃ααj in the presence of the matter effect is important not only in the case of constant matter density but also in the case of adiabatically varying density. C. Another derivation of the formula by Kimura, Takamura and Yokomakura In this subsection a systematic derivation of their formula is given because such a deriva- tion will be crucial for the generalizations in the following sections.2 The arguments are based on the trivial identities. From the unitarity condition of the matrix Ũ , we have δαβ = Ũ Ũ−1 ŨαjŨ j . (6) Next we take the (α, β) component of the both hand sides in Eq. (2): UEU−1 +A Ũ ẼŨ−1 ŨαjẼjŨ ẼjX̃ j (7) 2 The argument here is the same as that in Ref. [4]. Since this derivation does not seem to be widely known, it is reviewed here. Furthermore, we take the (α, β) component of the square of Eq. (2): UEU−1 +A Ũ Ẽ2Ũ−1 ŨαjẼ Ẽ2j X̃ j (8) Putting Eqs. (6)–(8) together, we have 1 1 1 Ẽ1 Ẽ2 Ẽ3 Ẽ21 Ẽ [UEU−1 +A]αβ (UEU−1 +A)2 which can be easily solved by inverting the Vandermonde matrix: ∆Ẽ21∆Ẽ31 (Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), 1) ∆Ẽ21∆Ẽ32 (Ẽ3Ẽ1, −(Ẽ3 + Ẽ1), 1) ∆Ẽ31∆Ẽ32 (Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), 1) [UEU−1 +A]αβ (UEU−1 +A)2 . (9) [(UEU−1 +A)j]αβ (j = 1, 2) on the right hand side are given by the known quantities: UEU−1 +A j + Aδαeδβe UEU−1 +A j + A j + δβeX + A2 δαeδβe. It can be shown that Eq. (9) coincides with the original results by Kimura, Takamura and Yokomakura [2, 3]. A remark is in order on Eq. (9). Addition of a matrix c1 to Eq. (2) where c is a constant and 1 is the identity matrix, or in other words, the shift Ej → Ej + c (j = 1, 2, 3), (10) should give the same result for X̃ j (j = 1, 2, 3), since Eq. (10) only affects the overall phase of the oscillation amplitude and the phase has to disappear in the probability. It is easy to show that the shift (10) indeed gives the same result as Eq. (9). The proof is given in Appendix A. In practical calculations below, we will always put c = −E1, i.e., we will consider the mass matrix U(E −E11)U−1 +A instead of the original one UEU−1 +A, since all the diagonal elements (E − E11)jj = ∆Ej1 = ∆m2j1/2E are expressed in terms of the relevant variables ∆m2j1, and therefore calculations become simpler. To save space, however, we will use the matrix UEU−1 +A in most of the following discussions. D. The case with arbitrary number of neutrinos It is straightforward to generalize the discussions in sect. II C to the case with arbitrary number of neutrinos where the matter potential is diagonal in the flavor eigenstate. The scheme with number of sterile neutrinos is one of the example of these cases [4, 10]. The time evolution of such a scheme with N neutrino flavors is described by UNENU−1N +AN where ΨTN ≡ (να1 , να2 , · · · , ναN ) is the flavor eigenstate, EN ≡ diag (E1, E2, · · · , EN) (11) is the energy matrix of the mass eigenstate, AN ≡ diag (A1, A2, · · · , AN) , is the potential matrix for the flavor eigenstate, and UN is the N × N MNS matrix. As in the previous sect., by taking the α, β components, we get Ẽmj X̃ UNENU−1N +AN for m = 0, · · · , N − 1, which leads to the simultaneous equation 1 1 · · · 1 Ẽ1 Ẽ2 · · · ẼN ẼN−11 Ẽ 2 · · · ẼN−1N UNEU−1N +AN UNEU−1N +AN . (12) Eq. (12) can be solved by inverting the N ×N Vandermonde matrix VN : = V −1N UNENU−1N +AN UNENU−1N +AN . (13) The determinant of VN is the Vandermonde determinant j<k ∆Ẽjk, and therefore V can be analytically obtained as long as we know the value of Ẽj. The factors [(UNENU−1N + AN)j]αβ on the right hand side of Eq. (13) can be expressed as functions of the energy Ej , the quantity X j in vacuum and the matter potential Aγ , since the matrix (UNENU−1N +AN)j is a sum of products of the matrices [(UNENU−1N )ℓ]γδ = k (0 ≤ ℓ ≤ j) and [(AN)m]ǫη = mδǫη (0 ≤ m ≤ j). From Eq. (13) it is clear that enhancement of the oscillation probability due to the matter effect occurs only when some of ∆Ẽjk becomes small. III. THE CASE WITH LARGE MAGNETIC MOMENTS AND A MAGNETIC FIELD So far we have assumed that the potential term is diagonal in the flavor basis. We can generalize the present result to the cases where we have off-diagonal potential terms. One of such examples is the case where there are only three active neutrinos with magnetic moments and the magnetic field (See, e.g., Ref. [1] for review.). The hermitian matrix3 UEU−1 B B† U∗E(U∗)−1 B ≡ B µαβ is the mass matrix for neutrinos and anti-neutrinos without the matter effect where neutrinos have the magnetic moments µαβ in the magnetic field B. Here we assume the magnetic interaction of Majorana type µαβ ν̄α Fλκσ λκ νcβ + h.c., (15) and in this case the magnetic moments µαβ are real and anti-symmetric in flavor indices: µαβ = −µβα. If the magnetic field is constant, then the oscillation probability can be written as P (νA → νB) = δAB − 4 X̃ABJ X̃ ∆ẼJKL X̃ABJ X̃ ∆ẼJKL , (16) where A,B run e, µ, τ, ē, µ̄, τ̄ , and J,K run 1, · · ·, 6, respectively, and X̃ABJ ≡ UAJU∗BJ . ẼJ (J = 1, · · · , 6) are the eigenvalues of the 6×6 matrix M. On the other hand, if the magnetic field varies very slowly and if the length L of the baseline is so long that |∆ẼJKL| ≫ 1 is satisfied for J 6= K, then the oscillation probability is given by P (νA → νB) = X̃BBJ (L)X̃ J (0). (17) Following the same arguments as before, the quantity X̃ABJ is given by inverting the 6 × 6 Vandermonde matrix V6: X̃AB1 X̃AB2 X̃AB6 = V −16 [M]AB . (18) 3 See [5] for derivation of Eq. (14) from the Dirac Eq. As in the previous sections, [(M)J ]AB (J = 0, · · · , 5) on the right hand side of Eq. (18) can be expressed in terms of the known quantities XABK and BCD, and Eqs. (16) and (18) are useful only when we know the eigenvalues ẼJ . To demonstrate the usefulness of these formulae, let us consider the case where the magnetic field is large at origin but is zero at the end point and the magnetic field varies adiabatically. For simplicity we assume that θ13 and all the CP phases vanish. 4 In this case the 6 × 6 matrix M in Eq. (14) becomes real, and we obtain the following oscillation probabilities: P (να → νβ) = P (ν̄α → ν̄β) = (Uβj) 2[Re Ũ(0)αj] P (να → ν̄β) = P (ν̄α → νβ) = (Uβj) 2[Im Ũ(0)αj] 2, (19) where Ũ(0) the 3× 3 unitary matrix which diagonalizes the 3× 3 matrix UEU−1 + iB(0) at the origin: UEU−1 + iB(0) = Ũ(0)Ẽ(0)Ũ−1(0). In this example the energy eigenvalues are degenerate, i.e., the 6 × 6 energy matrix be- comes diag(Ẽ , Ẽ), and the oscillation probability differs from Eq. (17) because the condition |∆ẼJKL| ≫ 1(J 6= K) is not satisfied (e.g., ∆ẼJK = 0 not only for J = K = 1 but also for J = 1, K = 4). Each probability in Eqs. (19) itself is not expressed in terms of X̃ααj (0), but we find that the following relation holds: P (να → νβ) + P (ν̄α → νβ) = (Uβj) 2|Ũ(0)αj|2 = j (0). (20) Eq. (20) is a new result and without the present formalism it would be hard to derive it. The details of derivation of Eq. (19) and explicit forms of X̃ααj (0) are given in Appendix B. Eq. (20) may be applicable to the case where high energy astrophysical neutrinos, which are produced in a relatively large magnetic field, are observed on the Earth, on the assumption that the fluxes of neutrinos and anti-neutrinos are almost equal. IV. THE CASE WITH NON-STANDARD INTERACTIONS Another interesting application is the oscillation probability in the presence of new physics in propagation [11, 12]. In this case the mass matrix is given by UEU−1 +ANP (21) 4 In the presence of the magnetic interaction (15) of Majorana type, the two CP phases, which are absorbed by redefinition of the charged lepton fields in the standard case, cannot be absorbed and therefore become physical. Here, however, we will assume for simplicity that these CP phases vanish. where ANP ≡ 2GFNe 1 + ǫee ǫeµ ǫeτ ǫ∗eµ ǫµµ ǫµτ ǫ∗eτ ǫ µτ ǫττ The dimensionless quantities ǫαβ stand for possible deviation from the standard matter effect. Also in this case the oscillation probability is given by Eqs. (4) and (9), where the standard potential matrix A has to be replaced by ANP . The extra complication compared to the standard case is calculations of the eigenvalues Ẽj and the elements [(UEU−1 + ANP )m]αβ (m = 1, 2). Again to demonstrate the usefulness of the formalism, here we will discuss for simplicity the case in which the eigenvalues are the roots of a quadratic equation. It is known [13] that the constraints on the three parameters ǫee, ǫeτ , ǫττ from various experimental data are weak and they could be as large as O(1). In Ref. [14] it was found that large values (∼ O(1)) of the parameters ǫee, ǫeτ , ǫττ are consistent with all the experimental data including those of the atmospheric neutrino data, provided that one of the eigenvalues of the matrix (21) at high energy limit, i.e., ANP , becomes zero. Simplifying even further, here we will neglect the parameters ǫeµ, ǫµµ, ǫµτ which are smaller than O(10−2) and we will consider the potential matrix ANP = A 1 + ǫee 0 ǫeτ 0 0 0 ǫ∗eτ 0 ǫττ , (22) where A ≡ 2GFNe, the three parameters ǫee, ǫeτ , ǫττ are constrained in such a way that two of the three eigenvalues become zero. We will assume that Ne is constant, and we will take the limit ∆m221 → 0. The oscillation probability P (νµ → νe) in this case can be analytically expressed and is given by P (νµ → νe) = −4Re − 4Re (Λ+ − Λ−L)L 8A(∆E31) Λ+Λ−(Λ+ − Λ−) |ǫeτXeµ3 X 3 | sin(arg(ǫeµ) + δ) × sin (Λ+ − Λ−)L . (23) Eq. (23) is another new result and it would be difficult to obtain it without using the present formalism. The details of derivation of Eq. (23), explanation of the notations and the explicit forms of all the variables in Eq. (23) are described in Appendix C. V. CONCLUSIONS The essence of the exact formula for the neutrino oscillation probability in constant matter which was discovered by Kimura, Takamura and Yokomakura lies in the fact that the combination X̃ j ≡ ŨαjŨβj∗ of the mixing matrix elements in matter can be expressed as polynomials in the same quantity X j ≡ UαjUβj∗ in vacuum. In this paper we have discussed applications of their formalism to more general cases. We have pointed out that their formalism can be useful for the cases in matter not only with constant density but also with density which varies adiabatically as in the case of the solar neutrino problem, after taking the limit of the long neutrino path. We have shown that their formalism can be generalized to the cases where the matter potential has off-diagonal components. As concrete non-trivial examples, we discussed the case with magnetic moments and a magnetic field, and the case with non-standard interactions. The application of the present formalism to the case with unitarity violation has been discussed elsewhere [15]. The formalism by Kimura, Takamura and Yokomakura is quite general and can be applicable to many problems in neutrino oscillation phenomenology. APPENDIX A: PROOF THAT EQ. (10) GIVES THE SAME (9) In this appendix we show that Eq. (10) gives the same result for X̃ j (j = 1, 2, 3). The value of X̃ j (j = 1, 2, 3) for Ẽ + c1 Ũ−1 = UEU−1 +A+ c1 becomes at most quadratic5 in c, and all one has to do is to show that the coefficients of the terms linear and quadratic in c vanish. Let us introduce the notation 1 1 1 Ẽ1 + c Ẽ2 + c Ẽ3 + c (Ẽ1 + c) 2 (Ẽ2 + c) 2 (Ẽ3 + c) ≡ (V −1)(0) + c(V −1)(1) + c2(V −1)(2) [UEU−1 +A+ c1]αβ (UEU−1 +A+ c1)2 ≡ ~B(0) + c ~B(1) + c2 ~B(2), where V (k) is the coefficient of the inverted Vandermonde matrix which is k-th order in c, and B j is the coefficient of the vector (UEU−1 +A+ c1) which is k-th order in c. Then the terms linear in c are given by (V −1)(1) ~B(0) + (V −1)(0) ~B(1) ∆Ẽ21∆Ẽ31 (Ẽ2 + Ẽ3, −2, 0) ∆Ẽ21∆Ẽ32 (−(Ẽ3 + Ẽ1), +2, 0) ∆Ẽ31∆Ẽ32 (Ẽ1 + Ẽ2, −2, 0) [UEU−1 +A]αβ (UEU−1 +A)2 5 Notice that all the factors ∆Ẽjk are invariant under the shift (10), and the only change by this shift comes either from the terms ẼjẼk or from Ẽj + Ẽk in the inverse of the Vandermonde matrix (cf. Eq. (9)). Hence the difference by Eq. (10) is at most quadratic in c. ∆Ẽ21∆Ẽ31 (+Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), +1) ∆Ẽ21∆Ẽ32 (−Ẽ3Ẽ1, +(Ẽ3 + Ẽ1), −1) ∆Ẽ31∆Ẽ32 (+Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), +1) 2 [UEU−1 +A]αβ  = 0, and the terms quadratic in c are given by (V −1)(2) ~B(0) + (V −1)(1) ~B(1) + (V −1)(0) ~B(2) ∆Ẽ21∆Ẽ31 (+1, 0, 0) ∆Ẽ21∆Ẽ32 (−1, 0, 0) ∆Ẽ31∆Ẽ32 (+1, 0, 0) [UEU−1 +A]αβ (UEU−1 +A)2 ∆Ẽ21∆Ẽ31 (Ẽ2 + Ẽ3, −2, 0) ∆Ẽ21∆Ẽ32 (−(Ẽ3 + Ẽ1), +2, 0) ∆Ẽ31∆Ẽ32 (Ẽ1 + Ẽ2, −2, 0) 2 [UEU−1 +A]αβ ∆Ẽ21∆Ẽ31 (+Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), +1) ∆Ẽ21∆Ẽ32 (−Ẽ3Ẽ1, +(Ẽ3 + Ẽ1), −1) ∆Ẽ31∆Ẽ32 (+Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), +1)  = 0. Thus X̃ j (j = 1, 2, 3) is independent of c, as is claimed. APPENDIX B: DERIVATION OF EQ. (19) The matrix (14) can be rewritten as M = 1 UEU−1 + iB 0 0 UEU−1 − iB 1 −i1 −i1 1 so the problem of diagonalizing the 6 × 6 matrix (14) is reduced to diagonalizing the 3× 3 matrices UEU−1 ± iB. Since we are assuming that θ13 and all the CP phases vanish, all the matrix elements Uαj and Bαβ = −Bβα are real, UEU−1 ± iB can be diagonalized by a unitary matrix and its complex conjugate: UEU−1 + iB = Ũ ẼŨ−1 UEU−1 − iB = Ũ∗Ẽ(Ũ∗)−1. Therefore, we can diagonalize M by a 6× 6 unitary matrix Ũ as M = Ũ Ũ−1, where Ũ = 1√ 1 −i1 −i1 1 0 Ũ∗ Ũ − iŨ∗ −iŨ Ũ∗ We note in passing that the reason why diagonalization of the 6 × 6 matrix is reduced to that of the 3× 3 matrix is because the two matrices UEU−1 and B are real. On the other hand, without a magnetic field the 6× 6 unitary matrix U is given by where the CP phase δ has dropped out because θ13 = 0. From these we can integrate the equation of motion and we get the fields at the end point: Ψc(L) = Ũ(L) e−iΦ 0 0 e−iΦ Ũ(0)−1 Ψc(0) Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1 −i(Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1) i(Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1) Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1 Ψc(0) where Ẽ(t) dt, and we have assumed that a large magnetic field exists at the origin whereas there is no magnetic field at the end point. Thus the oscillation probabilities for the adiabatic transition are give by: P (να → νβ) = P (ν̄α → ν̄β) = lim Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1 |Uβj|2 Re(Ũαj) P (ν̄α → νβ) = P (να → ν̄β) = lim Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1 |Uβj|2 Im(Ũαj) Hence we obtain the following relation: P (να → νβ) + P (ν̄α → νβ) = P (να → νβ) + P (να → ν̄β) = |Uβj |2|Ũαj |2. To get |Ũαj |2, we need the explicit expression for the eigenvalues and the quantity X̃ααj in the presence of a magnetic field. In the following we will subtract E11 from the energy matrix E because it will only change the phase of the oscillation amplitude. For simplicity we will put θ13 = 0, θ23 = π/4, and we will consider the limit ∆m 21 → 0. Defining ∆Ejk ≡ ∆m2jk/2E Bαβ = Bµαβ ≡ 0 −p −q p 0 −r q r 0 we have the eigenvalue equation 0 = |λ1− U(E − E11)U−1 − iB| = λ3 −∆E31λ2 − (p2 + q2 + r2)λ+ (p− q)2. (B1) The three roots of the cubic equation (B1) are given by λ1 = 2R cosϕ+ , λ2 = 2R cos(ϕ+ , λ3 = 2R cos(ϕ− where R ≡ [(∆E31/3)2 + (p2 + q2 + r2)/3]3/2, ϕ ≡ (1/3) cos−1 {(∆E31/3)3 +∆E31(p2 + q2 + r2)/6−∆E31(p− q)2/4}/R The quantity X̃ααj in the presence of a magnetic field is given by X̃αα1 X̃αα2 X̃αα3 ∆λ21∆λ31 (λ2λ3, −(λ2 + λ3), 1) ∆λ21∆λ32 (λ3λ1, −(λ3 + λ1), 1) ∆λ31∆λ32 (λ1λ2, −(λ1 + λ2), 1) Y αα2 Y αα3  , (B2) where Y αα2 = U(E −E11)U−1 + iB = ∆E31X 0 (α = e) ∆E31/2 (α = µ, τ) Y αα3 = U(E −E11)U−1 + iB = (∆E31) 2Xαα3 − (B2)αα q2 + r2 (α = e) r2 + p2 + (∆E31) 2/2 (α = µ) p2 + q2 + (∆E31) 2/2 (α = τ) . (B4) In evaluating Y ααj , we have used the facts θ13 = 0, θ23 = π/4, ∆E21 = 0, Bαβ = −Bβα, and that U(E −E11)U−1 is a symmetric matrix. Using all these results, it is straightforward to obtain the explicit form for P (να → νβ) +P (ν̄α → νβ) by plugging the results of Eqs. (B2), (B3), (B4) into the following (although calculations are tedious): P (να → νe) + P (ν̄α → νe) = c212X̃αα1 + s212X̃αα2 P (να → νβ) + P (ν̄α → νβ) = X̃αα1 + X̃αα2 + X̃αα3 (β = µ, τ), where s12 ≡ sin θ12, c12 ≡ cos θ12. APPENDIX C: DERIVATION OF EQ. (23) The oscillation probability (23) is obtained in two steps. First we will obtain the eigenval- ues of the matrix (21) with Eq. (22) and then we will plug the expressions for the eigenvalues into Eq. (9) with A replaced by ANP given in Eq. (22). Let us introduce notations for 3× 3 hermitian matrices: 0 −i 0 i 0 0 0 0 0  , λ5 ≡ 0 0 −i 0 0 0 i 0 0  , λ7 ≡ 0 0 0 0 0 −i 0 i 0 1 0 0 0 0 0 0 0 1  , λ9 ≡ 1 0 0 0 0 0 0 0 −1 where λ2, λ5 and λ7 are the standard Gell-Mann matrices whereas λ0 and λ9 are the notations which are defined only in this paper. Simple calculations show that the matrix ANP in Eq. (22) can be rewritten as ANP = Aeiγλ9e−iβλ5 1 + ǫee + ǫττ 1 + ǫee − ǫττ + |ǫµτ |2  eiβλ5e−iγλ9 , (C1) where β ≡ 1 tan−1 2|ǫeτ |2 1 + ǫee − ǫττ γ ≡ 1 arg (ǫeµ). From Eq. (C1) we see that the two potentially non-zero eigenvalues λe′ and λτ ′ of the matrix (22) are given by 1 + ǫee + ǫττ 1 + ǫee − ǫττ + |ǫµτ |2 In order for this scheme to be consistent with the atmospheric neutrino data particularly at high energy, which are perfectly described by vacuum oscillations, λτ ′ has to vanish [14]. In this case, we have tanβ = |ǫeτ | 1 + ǫee ǫττ = |ǫeτ |2 1 + ǫee λe′ = A(1 + ǫee) |ǫeτ |2 (1 + ǫee)2 A(1 + ǫee) cos2 β Thus we have ANP = Aeiγλ9e−iβλ5diag (λe′ , 0, 0) eiβλ5e−iγλ9 . (C2) If we did not have β and γ, Eq. (C2) would be the same as the standard three flavor scheme in matter, which was analytically worked out in Ref. [16] in the limit of ∆m221 → 0. It turns out that, by redefining the parametrization of the MNS matrix Eq. (C2) can be also treated analytically in the limit of ∆m221 → 0 as was done in Ref. [16]. The mass matrix can be written as UEU−1 +ANP = eiγλ9e−iβλ5 eiβλ5e−iγλ9UEU−1eiγλ9e−iβλ5 + diag (λe′, 0, 0) eiβλ5e−iγλ9 . Here we introduce the following two unitary matrices: U ′ ≡ eiβλ5e−iγλ9 U ≡ diag(1, 1, eiargU ′τ3)U ′′ diag(eiargU ′e1 , eiargU ′e2 , 1), where U is the 3× 3 MNS matrix in the standard parametrization [17] and U ′′ was defined in the second line in such a way that the elements U ′′e1, U e2, U τ3 be real to be consistent with the standard parametrization in Ref. [17] 6. Then we have UEU−1 +ANP = eiγλ9e−iβλ5diag(1, 1, eiargU U ′′EU ′′−1 + diag (λe′ , 0, 0) ×diag(1, 1, e−iargU ′τ3) eiβλ5e−iγλ9 . (C3) Before proceeding further, let us obtain the expression for the three mixing angles θ′′jk and the Dirac phase δ′′ in U ′′. Since U ′ = −iγUe1 + sβe iγUτ1 cβe −iγUe2 + sβe iγUτ2 cβe −iγUe3 + sβe iγUτ3 Uµ1 Uµ2 Uµ3 −iγUτ1 − sβeiγUe1 cβe−iγUτ2 − sβeiγUe2 cβe−iγUτ3 − sβeiγUe3 where cβ ≡ cos β, sβ ≡ sin β, we get θ′′13 = sin −1 |U ′′e3| = sin−1 |cβe−iγUe3 + sβeiγUτ3| θ′′12 = tan −1(U ′′e2/U e1) = tan |cβe−iγUe2 + sβeiγUτ2|/|cβe−iγUe1 + sβeiγUτ1| θ′′23 = tan −1(U ′′µ3/U τ3) = tan Uµ3/|cβe−iγUτ3 − sβeiγUe3| δ′′ = −argU ′′e3 = −arg (cβe−iγUe3 + sβeiγUτ3). As was shown in Ref. [16], in the limit ∆m221 → 0, the matrix on the right hand side of Eq. (C3) can be diagonalized as follows: U ′′EU ′′−1 + diag (λe′ , 0, 0)−E11 = eiθ λ7Γδ′′e λ5Γ−1δ′′ e λ2diag (0, 0,∆E31) e −iθ′′ λ2Γδ′′e −iθ′′ λ5Γ−1δ′′ e −iθ′′ λ7 + diag (λe′, 0, 0) = eiθ λ7Γδ′′ λ5diag (0, 0,∆E31) + diag (λe′ , 0, 0) Γ−1δ′′ e −iθ′′ = eiθ λ7Γδ′′e iθ̃′′ λ5diag (Λ−, 0,Λ+) e −iθ̃′′ λ5Γ−1δ′′ e where Γδ′′ ≡ diag(1, 1, e−iδ ), ∆E31 ≡ ∆m231/2E, we have used the standard parametriza- tion [17] U ′′ ≡ eiθ′′23λ7Γδ′′eiθ λ5Γ−1δ′′ e λ2 , and the eigenvalues Λ± are defined by (∆E31 + λe′)± (∆E31 cos 2θ 13 − λe′) + (∆E31 sin 2θ 6 The element U ′′τ2 has to be also real, but it is already satisfied because U τ2 = Uτ2. Having obtained the eigenvalues, by plugging these into Eq. (9) with A → ANP , Ẽ1 → Λ−, Ẽ2 → 0, Ẽ3 → Λ+, we obtain X̃µe: Λ−(Λ+ − Λ−) (0, −Λ+, 1) (−Λ+Λ−, −(Λ+ + Λ−), 1) Λ+(Λ+ − Λ−) (0, −Λ−, 1) −Y µe3 + Λ+Y Λ−(Λ+ − Λ−) 3 − (Λ+ + Λ−)Y 3 − Λ−Y Λ+(Λ+ − Λ−) where Y j are defined by UEU−1 +ANP and are given by 2 = ∆E31 X 3 = [(∆E31) 2 + A(1 + ǫee)∆E31]X 3 + A∆E31ǫ Furthermore, by introducing the notations ξ ≡ [(∆E31)2 + A(1 + ǫee)∆E31]Uµ3|Ue3| η ≡ A∆E31|ǫeτ |Uµ3Uτ3 ζ ≡ ∆E31Uµ3|Ue3|, we can rewrite Y 2 = ζe iδ and Y 3 = ξe iδ + ηe−2iγ , where δ is the Dirac CP phase of the MNS matrix U , so we have Λ−(Λ+ − Λ−) [ξ + ηe−i(2γ+δ) − Λ+ζ ] [ξ + ηe−i(2γ+δ) − (Λ+ + Λ−)ζ ] Λ+(Λ+ − Λ−) [ξ + ηe−i(2γ+δ) − Λ−ζ ]. Notice that the phase factor eiδ in front of each X̃ j drops out in the oscillation probability P (νµ → νe) because P (νµ → νe) is expressed in terms of X̃µej X̃ k , and the oscillation probability (23) depends only on the combination 2γ + δ = arg (ǫeµ) + δ. In the present case, the matrix Ũ is unitary and because of this three flavor unitarity all the T violating terms are proportional to one factor: ∆ẼjkL = 2 Im ∆Ẽ12L − sin ∆Ẽ13L + sin ∆Ẽ23L = −8 Im ∆Ẽ21L ∆Ẽ31L ∆Ẽ32L This modified Jarlskog factor Im(X̃ 2 ) in matter can be rewritten as Im(X̃ 2 ) = Λ+Λ−(Λ+ − Λ−) 2 ) = − ηζ sin(2γ + δ) Λ+Λ−(Λ+ − Λ−) = − A(∆E31) Λ+Λ−(Λ+ − Λ−) |ǫeτXeµ3 X 3 | sin(arg(ǫeµ) + δ). This completes derivation of Eq. (23). ACKNOWLEDGMENTS The author would like to thank Alexei Smirnov for bringing my attention to Refs. [5, 6, 7, 8]. He would also like to thank He Zhang for calling my attention to Refs. [4, 10] which were missed in the first version of this paper. This research was supported in part by a Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture, #19340062. [1] J. N. Bahcall, R. Davis, P. Parker, A. Smirnov and R. Ulrich, Reading, USA: Addison-Wesley (1995) 440 p. (Frontiers in physics. 92) [2] K. Kimura, A. Takamura and H. Yokomakura, Phys. Lett. B 537, 86 (2002) [arXiv:hep-ph/0203099]. [3] K. Kimura, A. Takamura and H. Yokomakura, Phys. Rev. D 66, 073005 (2002) [arXiv:hep-ph/0205295]. [4] Z. z. Xing and H. Zhang, Phys. Lett. B 618, 131 (2005) [arXiv:hep-ph/0503118]. [5] W. Grimus and T. Scharnagl, Mod. Phys. Lett. A 8, 1943 (1993). [6] A. Halprin, Phys. Rev. D 34, 3462 (1986). [7] P. D. Mannheim, Phys. Rev. D 37, 1935 (1988). [8] R. F. Sawyer, Phys. Rev. D 42, 3908 (1990). [9] V. D. Barger, K. Whisnant, S. Pakvasa and R. J. N. Phillips, Phys. Rev. D 22, 2718 (1980). [10] H. Zhang, arXiv:hep-ph/0606040. [11] M. M. Guzzo, A. Masiero and S. T. Petcov, Phys. Lett. B 260, 154 (1991); [12] E. Roulet, Phys. Rev. D 44, 935 (1991). [13] S. Davidson, C. Pena-Garay, N. Rius and A. Santamaria, JHEP 0303, 011 (2003) [arXiv:hep-ph/0302093]. [14] A. Friedland and C. Lunardini, Phys. Rev. D 72, 053009 (2005) [arXiv:hep-ph/0506143]. [15] E. Fernandez-Martinez, M. B. Gavela, J. Lopez-Pavon and O. Yasuda, arXiv:hep-ph/0703098. [16] O. Yasuda, Proceedings of Symposium on New Era in Neutrino Physics (Universal Academy Press, Inc., Tokyo, eds. H. Minakata and O. Yasuda), p 165 – 177 (1999) [arXiv:hep-ph/9809205]. [17] S. Eidelman et al. [Particle Data Group], Phys. Lett. B 592, 1 (2004). http://arxiv.org/abs/hep-ph/0203099 http://arxiv.org/abs/hep-ph/0205295 http://arxiv.org/abs/hep-ph/0503118 http://arxiv.org/abs/hep-ph/0606040 http://arxiv.org/abs/hep-ph/0302093 http://arxiv.org/abs/hep-ph/0506143 http://arxiv.org/abs/hep-ph/0703098 http://arxiv.org/abs/hep-ph/9809205 introduction generalities about oscillation probabilities The case of constant density The case of adiabatically varying density Another derivation of the formula by Kimura, Takamura and Yokomakura The case with arbitrary number of neutrinos the case with large magnetic moments and a magnetic field the case with non-standard interactions conclusions proof that Eq. (??) gives the same (??) Derivation of Eq. (??) Derivation of Eq. (??) Acknowledgments References

The exact formula for the neutrino oscillation probability in matter with constant density, which was discovered by Kimura, Takamura and Yokomakura, has been applied mostly to the standard case with three flavor neutrino so far. In this paper applications of their formula to more general cases are discussed. It is shown that this formalism can be generalized to various cases where the matter potential have off-diagonal components, and the two non-trivial examples are given: the case with magnetic moments and a magnetic field and the case with non-standard interactions. It is pointed out that their formalism can be applied also to the case in the long baseline limit with matter whose density varies adiabatically as in the case of solar neutrino.

On the exact formula for neutrino oscillation probability by Kimura, Takamura and Yokomakura Osamu Yasuda∗ Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan Abstract The exact formula for the neutrino oscillation probability in matter with constant density, which was discovered by Kimura, Takamura and Yokomakura, has been applied mostly to the standard case with three flavor neutrino so far. In this paper applications of their formula to more general cases are discussed. It is shown that this formalism can be generalized to various cases where the matter potential have off-diagonal components, and the two non-trivial examples are given: the case with magnetic moments and a magnetic field and the case with non-standard interactions. It is pointed out that their formalism can be applied also to the case in the long baseline limit with matter whose density varies adiabatically as in the case of solar neutrino. PACS numbers: 14.60.Pq, 14.60.St ∗Electronic address: yasuda˙at˙phys.metro-u.ac.jp Typeset by REVTEX 1 http://arxiv.org/abs/0704.1531v2 mailto:yasuda_at_phys.metro-u.ac.jp I. INTRODUCTION Neutrino oscillations in matter (See, e.g., Ref. [1] for review.) have been discussed by many people in the past because the oscillation probability has non-trivial behaviors in matter and due to the matter effect it may exhibit non-trivial enhancement which could be physically important. Unfortunately, it is not easy to get an analytical formula for the oscillation probability in the three flavor neutrino scheme in matter, and investigation of its behaviors has been a difficult but important problem in the phenomenology of neutrino oscillations. In 2002 Kimura, Takamura and Yokomakura derived a nice compact formula [2, 3] for the neutrino oscillation probability in matter with constant density. Basically what they showed is that the quantity Ũ∗αjŨβj , which is a factor crucial to express the oscillation probability analytically, can be expressed as a linear combination of U∗αjUβj, where Ũαj and Uαj stand for the matrix element of the MNS matrix in matter and in vacuum, respectively. However, their formula is only applicable to the standard three flavor case. In this pa- per we show that their result can be generalized to various cases. We also show that their formalism can be applied also to the case with slowly varying matter density in the limit of the long neutrino path. In Sect. II, we review briefly some aspects of the oscillation probabilities, including a simple derivation for the formula by Kimura, Takamura and Yoko- makura which was given in Ref. [4], because these are used in the following sections. Their formalism is generalized to the various cases where the matter potential has off-diagonal components, and we will discuss the case with large magnetic moments and a magnetic field (Sect. III) and the case with non-standard interactions (Sect. IV). In Sect. V we summarize our conclusions. II. GENERALITIES ABOUT OSCILLATION PROBABILITIES A. The case of constant density It has been known [5] (See also earlier works [6, 7, 8].) that after eliminating the nega- tive energy states by a Tani-Foldy-Wouthusen-type transformation, the Dirac equation for neutrinos propagating in matter is reduced to the familiar form: UEU−1 +A(t) Ψ, (1) where E ≡ diag (E1, E2, E3) , A(t) ≡ 2GFdiag (Ne(t)−Nn(t)/2,−Nn(t)/2,−Nn(t)/2) , ΨT ≡ (νe, νµ, ντ ) is the flavor eigenstate, U is the Maki-Nakagawa-Sakata (MNS) matrix, m2j + ~p (j = 1, 2, 3) is the energy eigenvalue of each mass eigenstate, and the matter effect A(t) at time (or position ) t is characterized by the density Ne(t) of electrons and the one Nn(t) of neutrons, respectively. Throughout this paper we assume for simplicity that the density of matter is either constant or slowly varying so that its derivative is negligible. The 3× 3 matrix on the right hand side of Eq. (1) can be formally diagonalized as: UEU−1 +A(t) = Ũ(t)Ẽ(t)Ũ−1(t), (2) where Ẽ(t) ≡ diag Ẽ1(t), Ẽ2(t), Ẽ3(t) is a diagonal matrix with the energy eigenvalues Ẽj(t) in the presence of the matter effect. First of all, let us assume that the matter density A(t) is constant. Then all the t dependence disappears and Eq. (1) can be easily solved, resulting the flavor eigenstate at the distance L: Ψ(L) = Ũ exp −iẼL Ũ−1Ψ(0). (3) Thus the oscillation probability P (να → νβ) is given by P (να → νβ) = Ũ exp (−iEL) Ũ−1 = δαβ − 4 ∆ẼjkL ∆ẼjkL , (4) where we have defined j ≡ ŨαjŨ∗βj , ∆Ẽjk ≡ Ẽj − Ẽk, and throughout this paper the indices α, β = (e, µ, τ) and j, k = (1, 2, 3) stand for those of the flavor and mass eigenstates, respectively. Once we know the eigenvalues Ẽj and the quantity X̃ j , the oscillation probability can be expressed analytically. B. The case of adiabatically varying density Secondly, let us consider the case where the density of the matter varies adiabatically as in the case of the solar neutrino deficit phenomena. In this case, instead of Eq. (3), we get Ψ(L) = Ũ(L) exp Ẽ(t) dt Ũ(0)−1Ψ(0), where Ũ(0) and Ũ(L) stand for the effective mixing matrices at the origin t = 0 and at the end point t = L. The oscillation probability is given by P (να → νβ) = Ũ(L) exp Ẽ(t) dt Ũ(0)−1 Ũ(L)βjŨ(L) βkŨ(0) αjŨ(0)αk exp ∆Ẽ(t)jk dt . (5) 1 In the standard case with three flavors of neutrinos in matter, the energy eigenvalues Ẽj can be analytically obtained by the root formula for a cubic equation [9]. So the only non-trivial problem in the standard case is to obtain the expression for X̃ j , and this was done by Kimura, Takamura and Yokomakura [2, 3]. In general cases, however, the analytic expression for Ẽj is very difficult or impossible to obtain, and we will discuss below only examples in which the analytic expression for Ẽj is known. Eq. (5) requires in general the quantity like Ũ(t)βjŨ ∗(t)βk which has the same flavor index β but different mass eigenstate indices j, k, and it turns out that the analytical expression for Ũ(t)βjŨ ∗(t)βk is hard to obtain. However, if the length L of the neutrino path is very large and if | 0 ∆Ẽ(t)jk dt| ≫ 1 is satisfied for j 6= k, as in the case of the solar neutrino deficit phenomena, after averaging over rapid oscillations Eq. (5) is reduced to P (να → νβ) = j (L)X̃ j (0), where we have defined X̃ααj (t) ≡ ∣Ũ(t)αj In the case of the solar neutrinos deficit process νe → νe during the daylight, X̃ββj (L) at the end point t = L and X̃ααj (0) at the origin t = 0 correspond to X j in vacuum and [X̃ at the center of the Sun, respectively, where j ≡ UαjU∗βj ŨαjŨ are bilinear products of the elements of the mixing matrices in vacuum and at the center of the Sun, respectively. Thus we obtain P (νe → νe) = X̃eej Hence we see that evaluation of the quantity X̃ααj in the presence of the matter effect is important not only in the case of constant matter density but also in the case of adiabatically varying density. C. Another derivation of the formula by Kimura, Takamura and Yokomakura In this subsection a systematic derivation of their formula is given because such a deriva- tion will be crucial for the generalizations in the following sections.2 The arguments are based on the trivial identities. From the unitarity condition of the matrix Ũ , we have δαβ = Ũ Ũ−1 ŨαjŨ j . (6) Next we take the (α, β) component of the both hand sides in Eq. (2): UEU−1 +A Ũ ẼŨ−1 ŨαjẼjŨ ẼjX̃ j (7) 2 The argument here is the same as that in Ref. [4]. Since this derivation does not seem to be widely known, it is reviewed here. Furthermore, we take the (α, β) component of the square of Eq. (2): UEU−1 +A Ũ Ẽ2Ũ−1 ŨαjẼ Ẽ2j X̃ j (8) Putting Eqs. (6)–(8) together, we have 1 1 1 Ẽ1 Ẽ2 Ẽ3 Ẽ21 Ẽ [UEU−1 +A]αβ (UEU−1 +A)2 which can be easily solved by inverting the Vandermonde matrix: ∆Ẽ21∆Ẽ31 (Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), 1) ∆Ẽ21∆Ẽ32 (Ẽ3Ẽ1, −(Ẽ3 + Ẽ1), 1) ∆Ẽ31∆Ẽ32 (Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), 1) [UEU−1 +A]αβ (UEU−1 +A)2 . (9) [(UEU−1 +A)j]αβ (j = 1, 2) on the right hand side are given by the known quantities: UEU−1 +A j + Aδαeδβe UEU−1 +A j + A j + δβeX + A2 δαeδβe. It can be shown that Eq. (9) coincides with the original results by Kimura, Takamura and Yokomakura [2, 3]. A remark is in order on Eq. (9). Addition of a matrix c1 to Eq. (2) where c is a constant and 1 is the identity matrix, or in other words, the shift Ej → Ej + c (j = 1, 2, 3), (10) should give the same result for X̃ j (j = 1, 2, 3), since Eq. (10) only affects the overall phase of the oscillation amplitude and the phase has to disappear in the probability. It is easy to show that the shift (10) indeed gives the same result as Eq. (9). The proof is given in Appendix A. In practical calculations below, we will always put c = −E1, i.e., we will consider the mass matrix U(E −E11)U−1 +A instead of the original one UEU−1 +A, since all the diagonal elements (E − E11)jj = ∆Ej1 = ∆m2j1/2E are expressed in terms of the relevant variables ∆m2j1, and therefore calculations become simpler. To save space, however, we will use the matrix UEU−1 +A in most of the following discussions. D. The case with arbitrary number of neutrinos It is straightforward to generalize the discussions in sect. II C to the case with arbitrary number of neutrinos where the matter potential is diagonal in the flavor eigenstate. The scheme with number of sterile neutrinos is one of the example of these cases [4, 10]. The time evolution of such a scheme with N neutrino flavors is described by UNENU−1N +AN where ΨTN ≡ (να1 , να2 , · · · , ναN ) is the flavor eigenstate, EN ≡ diag (E1, E2, · · · , EN) (11) is the energy matrix of the mass eigenstate, AN ≡ diag (A1, A2, · · · , AN) , is the potential matrix for the flavor eigenstate, and UN is the N × N MNS matrix. As in the previous sect., by taking the α, β components, we get Ẽmj X̃ UNENU−1N +AN for m = 0, · · · , N − 1, which leads to the simultaneous equation 1 1 · · · 1 Ẽ1 Ẽ2 · · · ẼN ẼN−11 Ẽ 2 · · · ẼN−1N UNEU−1N +AN UNEU−1N +AN . (12) Eq. (12) can be solved by inverting the N ×N Vandermonde matrix VN : = V −1N UNENU−1N +AN UNENU−1N +AN . (13) The determinant of VN is the Vandermonde determinant j<k ∆Ẽjk, and therefore V can be analytically obtained as long as we know the value of Ẽj. The factors [(UNENU−1N + AN)j]αβ on the right hand side of Eq. (13) can be expressed as functions of the energy Ej , the quantity X j in vacuum and the matter potential Aγ , since the matrix (UNENU−1N +AN)j is a sum of products of the matrices [(UNENU−1N )ℓ]γδ = k (0 ≤ ℓ ≤ j) and [(AN)m]ǫη = mδǫη (0 ≤ m ≤ j). From Eq. (13) it is clear that enhancement of the oscillation probability due to the matter effect occurs only when some of ∆Ẽjk becomes small. III. THE CASE WITH LARGE MAGNETIC MOMENTS AND A MAGNETIC FIELD So far we have assumed that the potential term is diagonal in the flavor basis. We can generalize the present result to the cases where we have off-diagonal potential terms. One of such examples is the case where there are only three active neutrinos with magnetic moments and the magnetic field (See, e.g., Ref. [1] for review.). The hermitian matrix3 UEU−1 B B† U∗E(U∗)−1 B ≡ B µαβ is the mass matrix for neutrinos and anti-neutrinos without the matter effect where neutrinos have the magnetic moments µαβ in the magnetic field B. Here we assume the magnetic interaction of Majorana type µαβ ν̄α Fλκσ λκ νcβ + h.c., (15) and in this case the magnetic moments µαβ are real and anti-symmetric in flavor indices: µαβ = −µβα. If the magnetic field is constant, then the oscillation probability can be written as P (νA → νB) = δAB − 4 X̃ABJ X̃ ∆ẼJKL X̃ABJ X̃ ∆ẼJKL , (16) where A,B run e, µ, τ, ē, µ̄, τ̄ , and J,K run 1, · · ·, 6, respectively, and X̃ABJ ≡ UAJU∗BJ . ẼJ (J = 1, · · · , 6) are the eigenvalues of the 6×6 matrix M. On the other hand, if the magnetic field varies very slowly and if the length L of the baseline is so long that |∆ẼJKL| ≫ 1 is satisfied for J 6= K, then the oscillation probability is given by P (νA → νB) = X̃BBJ (L)X̃ J (0). (17) Following the same arguments as before, the quantity X̃ABJ is given by inverting the 6 × 6 Vandermonde matrix V6: X̃AB1 X̃AB2 X̃AB6 = V −16 [M]AB . (18) 3 See [5] for derivation of Eq. (14) from the Dirac Eq. As in the previous sections, [(M)J ]AB (J = 0, · · · , 5) on the right hand side of Eq. (18) can be expressed in terms of the known quantities XABK and BCD, and Eqs. (16) and (18) are useful only when we know the eigenvalues ẼJ . To demonstrate the usefulness of these formulae, let us consider the case where the magnetic field is large at origin but is zero at the end point and the magnetic field varies adiabatically. For simplicity we assume that θ13 and all the CP phases vanish. 4 In this case the 6 × 6 matrix M in Eq. (14) becomes real, and we obtain the following oscillation probabilities: P (να → νβ) = P (ν̄α → ν̄β) = (Uβj) 2[Re Ũ(0)αj] P (να → ν̄β) = P (ν̄α → νβ) = (Uβj) 2[Im Ũ(0)αj] 2, (19) where Ũ(0) the 3× 3 unitary matrix which diagonalizes the 3× 3 matrix UEU−1 + iB(0) at the origin: UEU−1 + iB(0) = Ũ(0)Ẽ(0)Ũ−1(0). In this example the energy eigenvalues are degenerate, i.e., the 6 × 6 energy matrix be- comes diag(Ẽ , Ẽ), and the oscillation probability differs from Eq. (17) because the condition |∆ẼJKL| ≫ 1(J 6= K) is not satisfied (e.g., ∆ẼJK = 0 not only for J = K = 1 but also for J = 1, K = 4). Each probability in Eqs. (19) itself is not expressed in terms of X̃ααj (0), but we find that the following relation holds: P (να → νβ) + P (ν̄α → νβ) = (Uβj) 2|Ũ(0)αj|2 = j (0). (20) Eq. (20) is a new result and without the present formalism it would be hard to derive it. The details of derivation of Eq. (19) and explicit forms of X̃ααj (0) are given in Appendix B. Eq. (20) may be applicable to the case where high energy astrophysical neutrinos, which are produced in a relatively large magnetic field, are observed on the Earth, on the assumption that the fluxes of neutrinos and anti-neutrinos are almost equal. IV. THE CASE WITH NON-STANDARD INTERACTIONS Another interesting application is the oscillation probability in the presence of new physics in propagation [11, 12]. In this case the mass matrix is given by UEU−1 +ANP (21) 4 In the presence of the magnetic interaction (15) of Majorana type, the two CP phases, which are absorbed by redefinition of the charged lepton fields in the standard case, cannot be absorbed and therefore become physical. Here, however, we will assume for simplicity that these CP phases vanish. where ANP ≡ 2GFNe 1 + ǫee ǫeµ ǫeτ ǫ∗eµ ǫµµ ǫµτ ǫ∗eτ ǫ µτ ǫττ The dimensionless quantities ǫαβ stand for possible deviation from the standard matter effect. Also in this case the oscillation probability is given by Eqs. (4) and (9), where the standard potential matrix A has to be replaced by ANP . The extra complication compared to the standard case is calculations of the eigenvalues Ẽj and the elements [(UEU−1 + ANP )m]αβ (m = 1, 2). Again to demonstrate the usefulness of the formalism, here we will discuss for simplicity the case in which the eigenvalues are the roots of a quadratic equation. It is known [13] that the constraints on the three parameters ǫee, ǫeτ , ǫττ from various experimental data are weak and they could be as large as O(1). In Ref. [14] it was found that large values (∼ O(1)) of the parameters ǫee, ǫeτ , ǫττ are consistent with all the experimental data including those of the atmospheric neutrino data, provided that one of the eigenvalues of the matrix (21) at high energy limit, i.e., ANP , becomes zero. Simplifying even further, here we will neglect the parameters ǫeµ, ǫµµ, ǫµτ which are smaller than O(10−2) and we will consider the potential matrix ANP = A 1 + ǫee 0 ǫeτ 0 0 0 ǫ∗eτ 0 ǫττ , (22) where A ≡ 2GFNe, the three parameters ǫee, ǫeτ , ǫττ are constrained in such a way that two of the three eigenvalues become zero. We will assume that Ne is constant, and we will take the limit ∆m221 → 0. The oscillation probability P (νµ → νe) in this case can be analytically expressed and is given by P (νµ → νe) = −4Re − 4Re (Λ+ − Λ−L)L 8A(∆E31) Λ+Λ−(Λ+ − Λ−) |ǫeτXeµ3 X 3 | sin(arg(ǫeµ) + δ) × sin (Λ+ − Λ−)L . (23) Eq. (23) is another new result and it would be difficult to obtain it without using the present formalism. The details of derivation of Eq. (23), explanation of the notations and the explicit forms of all the variables in Eq. (23) are described in Appendix C. V. CONCLUSIONS The essence of the exact formula for the neutrino oscillation probability in constant matter which was discovered by Kimura, Takamura and Yokomakura lies in the fact that the combination X̃ j ≡ ŨαjŨβj∗ of the mixing matrix elements in matter can be expressed as polynomials in the same quantity X j ≡ UαjUβj∗ in vacuum. In this paper we have discussed applications of their formalism to more general cases. We have pointed out that their formalism can be useful for the cases in matter not only with constant density but also with density which varies adiabatically as in the case of the solar neutrino problem, after taking the limit of the long neutrino path. We have shown that their formalism can be generalized to the cases where the matter potential has off-diagonal components. As concrete non-trivial examples, we discussed the case with magnetic moments and a magnetic field, and the case with non-standard interactions. The application of the present formalism to the case with unitarity violation has been discussed elsewhere [15]. The formalism by Kimura, Takamura and Yokomakura is quite general and can be applicable to many problems in neutrino oscillation phenomenology. APPENDIX A: PROOF THAT EQ. (10) GIVES THE SAME (9) In this appendix we show that Eq. (10) gives the same result for X̃ j (j = 1, 2, 3). The value of X̃ j (j = 1, 2, 3) for Ẽ + c1 Ũ−1 = UEU−1 +A+ c1 becomes at most quadratic5 in c, and all one has to do is to show that the coefficients of the terms linear and quadratic in c vanish. Let us introduce the notation 1 1 1 Ẽ1 + c Ẽ2 + c Ẽ3 + c (Ẽ1 + c) 2 (Ẽ2 + c) 2 (Ẽ3 + c) ≡ (V −1)(0) + c(V −1)(1) + c2(V −1)(2) [UEU−1 +A+ c1]αβ (UEU−1 +A+ c1)2 ≡ ~B(0) + c ~B(1) + c2 ~B(2), where V (k) is the coefficient of the inverted Vandermonde matrix which is k-th order in c, and B j is the coefficient of the vector (UEU−1 +A+ c1) which is k-th order in c. Then the terms linear in c are given by (V −1)(1) ~B(0) + (V −1)(0) ~B(1) ∆Ẽ21∆Ẽ31 (Ẽ2 + Ẽ3, −2, 0) ∆Ẽ21∆Ẽ32 (−(Ẽ3 + Ẽ1), +2, 0) ∆Ẽ31∆Ẽ32 (Ẽ1 + Ẽ2, −2, 0) [UEU−1 +A]αβ (UEU−1 +A)2 5 Notice that all the factors ∆Ẽjk are invariant under the shift (10), and the only change by this shift comes either from the terms ẼjẼk or from Ẽj + Ẽk in the inverse of the Vandermonde matrix (cf. Eq. (9)). Hence the difference by Eq. (10) is at most quadratic in c. ∆Ẽ21∆Ẽ31 (+Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), +1) ∆Ẽ21∆Ẽ32 (−Ẽ3Ẽ1, +(Ẽ3 + Ẽ1), −1) ∆Ẽ31∆Ẽ32 (+Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), +1) 2 [UEU−1 +A]αβ  = 0, and the terms quadratic in c are given by (V −1)(2) ~B(0) + (V −1)(1) ~B(1) + (V −1)(0) ~B(2) ∆Ẽ21∆Ẽ31 (+1, 0, 0) ∆Ẽ21∆Ẽ32 (−1, 0, 0) ∆Ẽ31∆Ẽ32 (+1, 0, 0) [UEU−1 +A]αβ (UEU−1 +A)2 ∆Ẽ21∆Ẽ31 (Ẽ2 + Ẽ3, −2, 0) ∆Ẽ21∆Ẽ32 (−(Ẽ3 + Ẽ1), +2, 0) ∆Ẽ31∆Ẽ32 (Ẽ1 + Ẽ2, −2, 0) 2 [UEU−1 +A]αβ ∆Ẽ21∆Ẽ31 (+Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), +1) ∆Ẽ21∆Ẽ32 (−Ẽ3Ẽ1, +(Ẽ3 + Ẽ1), −1) ∆Ẽ31∆Ẽ32 (+Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), +1)  = 0. Thus X̃ j (j = 1, 2, 3) is independent of c, as is claimed. APPENDIX B: DERIVATION OF EQ. (19) The matrix (14) can be rewritten as M = 1 UEU−1 + iB 0 0 UEU−1 − iB 1 −i1 −i1 1 so the problem of diagonalizing the 6 × 6 matrix (14) is reduced to diagonalizing the 3× 3 matrices UEU−1 ± iB. Since we are assuming that θ13 and all the CP phases vanish, all the matrix elements Uαj and Bαβ = −Bβα are real, UEU−1 ± iB can be diagonalized by a unitary matrix and its complex conjugate: UEU−1 + iB = Ũ ẼŨ−1 UEU−1 − iB = Ũ∗Ẽ(Ũ∗)−1. Therefore, we can diagonalize M by a 6× 6 unitary matrix Ũ as M = Ũ Ũ−1, where Ũ = 1√ 1 −i1 −i1 1 0 Ũ∗ Ũ − iŨ∗ −iŨ Ũ∗ We note in passing that the reason why diagonalization of the 6 × 6 matrix is reduced to that of the 3× 3 matrix is because the two matrices UEU−1 and B are real. On the other hand, without a magnetic field the 6× 6 unitary matrix U is given by where the CP phase δ has dropped out because θ13 = 0. From these we can integrate the equation of motion and we get the fields at the end point: Ψc(L) = Ũ(L) e−iΦ 0 0 e−iΦ Ũ(0)−1 Ψc(0) Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1 −i(Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1) i(Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1) Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1 Ψc(0) where Ẽ(t) dt, and we have assumed that a large magnetic field exists at the origin whereas there is no magnetic field at the end point. Thus the oscillation probabilities for the adiabatic transition are give by: P (να → νβ) = P (ν̄α → ν̄β) = lim Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1 |Uβj|2 Re(Ũαj) P (ν̄α → νβ) = P (να → ν̄β) = lim Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1 |Uβj|2 Im(Ũαj) Hence we obtain the following relation: P (να → νβ) + P (ν̄α → νβ) = P (να → νβ) + P (να → ν̄β) = |Uβj |2|Ũαj |2. To get |Ũαj |2, we need the explicit expression for the eigenvalues and the quantity X̃ααj in the presence of a magnetic field. In the following we will subtract E11 from the energy matrix E because it will only change the phase of the oscillation amplitude. For simplicity we will put θ13 = 0, θ23 = π/4, and we will consider the limit ∆m 21 → 0. Defining ∆Ejk ≡ ∆m2jk/2E Bαβ = Bµαβ ≡ 0 −p −q p 0 −r q r 0 we have the eigenvalue equation 0 = |λ1− U(E − E11)U−1 − iB| = λ3 −∆E31λ2 − (p2 + q2 + r2)λ+ (p− q)2. (B1) The three roots of the cubic equation (B1) are given by λ1 = 2R cosϕ+ , λ2 = 2R cos(ϕ+ , λ3 = 2R cos(ϕ− where R ≡ [(∆E31/3)2 + (p2 + q2 + r2)/3]3/2, ϕ ≡ (1/3) cos−1 {(∆E31/3)3 +∆E31(p2 + q2 + r2)/6−∆E31(p− q)2/4}/R The quantity X̃ααj in the presence of a magnetic field is given by X̃αα1 X̃αα2 X̃αα3 ∆λ21∆λ31 (λ2λ3, −(λ2 + λ3), 1) ∆λ21∆λ32 (λ3λ1, −(λ3 + λ1), 1) ∆λ31∆λ32 (λ1λ2, −(λ1 + λ2), 1) Y αα2 Y αα3  , (B2) where Y αα2 = U(E −E11)U−1 + iB = ∆E31X 0 (α = e) ∆E31/2 (α = µ, τ) Y αα3 = U(E −E11)U−1 + iB = (∆E31) 2Xαα3 − (B2)αα q2 + r2 (α = e) r2 + p2 + (∆E31) 2/2 (α = µ) p2 + q2 + (∆E31) 2/2 (α = τ) . (B4) In evaluating Y ααj , we have used the facts θ13 = 0, θ23 = π/4, ∆E21 = 0, Bαβ = −Bβα, and that U(E −E11)U−1 is a symmetric matrix. Using all these results, it is straightforward to obtain the explicit form for P (να → νβ) +P (ν̄α → νβ) by plugging the results of Eqs. (B2), (B3), (B4) into the following (although calculations are tedious): P (να → νe) + P (ν̄α → νe) = c212X̃αα1 + s212X̃αα2 P (να → νβ) + P (ν̄α → νβ) = X̃αα1 + X̃αα2 + X̃αα3 (β = µ, τ), where s12 ≡ sin θ12, c12 ≡ cos θ12. APPENDIX C: DERIVATION OF EQ. (23) The oscillation probability (23) is obtained in two steps. First we will obtain the eigenval- ues of the matrix (21) with Eq. (22) and then we will plug the expressions for the eigenvalues into Eq. (9) with A replaced by ANP given in Eq. (22). Let us introduce notations for 3× 3 hermitian matrices: 0 −i 0 i 0 0 0 0 0  , λ5 ≡ 0 0 −i 0 0 0 i 0 0  , λ7 ≡ 0 0 0 0 0 −i 0 i 0 1 0 0 0 0 0 0 0 1  , λ9 ≡ 1 0 0 0 0 0 0 0 −1 where λ2, λ5 and λ7 are the standard Gell-Mann matrices whereas λ0 and λ9 are the notations which are defined only in this paper. Simple calculations show that the matrix ANP in Eq. (22) can be rewritten as ANP = Aeiγλ9e−iβλ5 1 + ǫee + ǫττ 1 + ǫee − ǫττ + |ǫµτ |2  eiβλ5e−iγλ9 , (C1) where β ≡ 1 tan−1 2|ǫeτ |2 1 + ǫee − ǫττ γ ≡ 1 arg (ǫeµ). From Eq. (C1) we see that the two potentially non-zero eigenvalues λe′ and λτ ′ of the matrix (22) are given by 1 + ǫee + ǫττ 1 + ǫee − ǫττ + |ǫµτ |2 In order for this scheme to be consistent with the atmospheric neutrino data particularly at high energy, which are perfectly described by vacuum oscillations, λτ ′ has to vanish [14]. In this case, we have tanβ = |ǫeτ | 1 + ǫee ǫττ = |ǫeτ |2 1 + ǫee λe′ = A(1 + ǫee) |ǫeτ |2 (1 + ǫee)2 A(1 + ǫee) cos2 β Thus we have ANP = Aeiγλ9e−iβλ5diag (λe′ , 0, 0) eiβλ5e−iγλ9 . (C2) If we did not have β and γ, Eq. (C2) would be the same as the standard three flavor scheme in matter, which was analytically worked out in Ref. [16] in the limit of ∆m221 → 0. It turns out that, by redefining the parametrization of the MNS matrix Eq. (C2) can be also treated analytically in the limit of ∆m221 → 0 as was done in Ref. [16]. The mass matrix can be written as UEU−1 +ANP = eiγλ9e−iβλ5 eiβλ5e−iγλ9UEU−1eiγλ9e−iβλ5 + diag (λe′, 0, 0) eiβλ5e−iγλ9 . Here we introduce the following two unitary matrices: U ′ ≡ eiβλ5e−iγλ9 U ≡ diag(1, 1, eiargU ′τ3)U ′′ diag(eiargU ′e1 , eiargU ′e2 , 1), where U is the 3× 3 MNS matrix in the standard parametrization [17] and U ′′ was defined in the second line in such a way that the elements U ′′e1, U e2, U τ3 be real to be consistent with the standard parametrization in Ref. [17] 6. Then we have UEU−1 +ANP = eiγλ9e−iβλ5diag(1, 1, eiargU U ′′EU ′′−1 + diag (λe′ , 0, 0) ×diag(1, 1, e−iargU ′τ3) eiβλ5e−iγλ9 . (C3) Before proceeding further, let us obtain the expression for the three mixing angles θ′′jk and the Dirac phase δ′′ in U ′′. Since U ′ = −iγUe1 + sβe iγUτ1 cβe −iγUe2 + sβe iγUτ2 cβe −iγUe3 + sβe iγUτ3 Uµ1 Uµ2 Uµ3 −iγUτ1 − sβeiγUe1 cβe−iγUτ2 − sβeiγUe2 cβe−iγUτ3 − sβeiγUe3 where cβ ≡ cos β, sβ ≡ sin β, we get θ′′13 = sin −1 |U ′′e3| = sin−1 |cβe−iγUe3 + sβeiγUτ3| θ′′12 = tan −1(U ′′e2/U e1) = tan |cβe−iγUe2 + sβeiγUτ2|/|cβe−iγUe1 + sβeiγUτ1| θ′′23 = tan −1(U ′′µ3/U τ3) = tan Uµ3/|cβe−iγUτ3 − sβeiγUe3| δ′′ = −argU ′′e3 = −arg (cβe−iγUe3 + sβeiγUτ3). As was shown in Ref. [16], in the limit ∆m221 → 0, the matrix on the right hand side of Eq. (C3) can be diagonalized as follows: U ′′EU ′′−1 + diag (λe′ , 0, 0)−E11 = eiθ λ7Γδ′′e λ5Γ−1δ′′ e λ2diag (0, 0,∆E31) e −iθ′′ λ2Γδ′′e −iθ′′ λ5Γ−1δ′′ e −iθ′′ λ7 + diag (λe′, 0, 0) = eiθ λ7Γδ′′ λ5diag (0, 0,∆E31) + diag (λe′ , 0, 0) Γ−1δ′′ e −iθ′′ = eiθ λ7Γδ′′e iθ̃′′ λ5diag (Λ−, 0,Λ+) e −iθ̃′′ λ5Γ−1δ′′ e where Γδ′′ ≡ diag(1, 1, e−iδ ), ∆E31 ≡ ∆m231/2E, we have used the standard parametriza- tion [17] U ′′ ≡ eiθ′′23λ7Γδ′′eiθ λ5Γ−1δ′′ e λ2 , and the eigenvalues Λ± are defined by (∆E31 + λe′)± (∆E31 cos 2θ 13 − λe′) + (∆E31 sin 2θ 6 The element U ′′τ2 has to be also real, but it is already satisfied because U τ2 = Uτ2. Having obtained the eigenvalues, by plugging these into Eq. (9) with A → ANP , Ẽ1 → Λ−, Ẽ2 → 0, Ẽ3 → Λ+, we obtain X̃µe: Λ−(Λ+ − Λ−) (0, −Λ+, 1) (−Λ+Λ−, −(Λ+ + Λ−), 1) Λ+(Λ+ − Λ−) (0, −Λ−, 1) −Y µe3 + Λ+Y Λ−(Λ+ − Λ−) 3 − (Λ+ + Λ−)Y 3 − Λ−Y Λ+(Λ+ − Λ−) where Y j are defined by UEU−1 +ANP and are given by 2 = ∆E31 X 3 = [(∆E31) 2 + A(1 + ǫee)∆E31]X 3 + A∆E31ǫ Furthermore, by introducing the notations ξ ≡ [(∆E31)2 + A(1 + ǫee)∆E31]Uµ3|Ue3| η ≡ A∆E31|ǫeτ |Uµ3Uτ3 ζ ≡ ∆E31Uµ3|Ue3|, we can rewrite Y 2 = ζe iδ and Y 3 = ξe iδ + ηe−2iγ , where δ is the Dirac CP phase of the MNS matrix U , so we have Λ−(Λ+ − Λ−) [ξ + ηe−i(2γ+δ) − Λ+ζ ] [ξ + ηe−i(2γ+δ) − (Λ+ + Λ−)ζ ] Λ+(Λ+ − Λ−) [ξ + ηe−i(2γ+δ) − Λ−ζ ]. Notice that the phase factor eiδ in front of each X̃ j drops out in the oscillation probability P (νµ → νe) because P (νµ → νe) is expressed in terms of X̃µej X̃ k , and the oscillation probability (23) depends only on the combination 2γ + δ = arg (ǫeµ) + δ. In the present case, the matrix Ũ is unitary and because of this three flavor unitarity all the T violating terms are proportional to one factor: ∆ẼjkL = 2 Im ∆Ẽ12L − sin ∆Ẽ13L + sin ∆Ẽ23L = −8 Im ∆Ẽ21L ∆Ẽ31L ∆Ẽ32L This modified Jarlskog factor Im(X̃ 2 ) in matter can be rewritten as Im(X̃ 2 ) = Λ+Λ−(Λ+ − Λ−) 2 ) = − ηζ sin(2γ + δ) Λ+Λ−(Λ+ − Λ−) = − A(∆E31) Λ+Λ−(Λ+ − Λ−) |ǫeτXeµ3 X 3 | sin(arg(ǫeµ) + δ). This completes derivation of Eq. (23). ACKNOWLEDGMENTS The author would like to thank Alexei Smirnov for bringing my attention to Refs. [5, 6, 7, 8]. He would also like to thank He Zhang for calling my attention to Refs. [4, 10] which were missed in the first version of this paper. This research was supported in part by a Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture, #19340062. [1] J. N. Bahcall, R. Davis, P. Parker, A. Smirnov and R. Ulrich, Reading, USA: Addison-Wesley (1995) 440 p. (Frontiers in physics. 92) [2] K. Kimura, A. Takamura and H. Yokomakura, Phys. Lett. B 537, 86 (2002) [arXiv:hep-ph/0203099]. [3] K. Kimura, A. Takamura and H. Yokomakura, Phys. Rev. D 66, 073005 (2002) [arXiv:hep-ph/0205295]. [4] Z. z. Xing and H. Zhang, Phys. Lett. B 618, 131 (2005) [arXiv:hep-ph/0503118]. [5] W. Grimus and T. Scharnagl, Mod. Phys. Lett. A 8, 1943 (1993). [6] A. Halprin, Phys. Rev. D 34, 3462 (1986). [7] P. D. Mannheim, Phys. Rev. D 37, 1935 (1988). [8] R. F. Sawyer, Phys. Rev. D 42, 3908 (1990). [9] V. D. Barger, K. Whisnant, S. Pakvasa and R. J. N. Phillips, Phys. Rev. D 22, 2718 (1980). [10] H. Zhang, arXiv:hep-ph/0606040. [11] M. M. Guzzo, A. Masiero and S. T. Petcov, Phys. Lett. B 260, 154 (1991); [12] E. Roulet, Phys. Rev. D 44, 935 (1991). [13] S. Davidson, C. Pena-Garay, N. Rius and A. Santamaria, JHEP 0303, 011 (2003) [arXiv:hep-ph/0302093]. [14] A. Friedland and C. Lunardini, Phys. Rev. D 72, 053009 (2005) [arXiv:hep-ph/0506143]. [15] E. Fernandez-Martinez, M. B. Gavela, J. Lopez-Pavon and O. Yasuda, arXiv:hep-ph/0703098. [16] O. Yasuda, Proceedings of Symposium on New Era in Neutrino Physics (Universal Academy Press, Inc., Tokyo, eds. H. Minakata and O. Yasuda), p 165 – 177 (1999) [arXiv:hep-ph/9809205]. [17] S. Eidelman et al. [Particle Data Group], Phys. Lett. B 592, 1 (2004). http://arxiv.org/abs/hep-ph/0203099 http://arxiv.org/abs/hep-ph/0205295 http://arxiv.org/abs/hep-ph/0503118 http://arxiv.org/abs/hep-ph/0606040 http://arxiv.org/abs/hep-ph/0302093 http://arxiv.org/abs/hep-ph/0506143 http://arxiv.org/abs/hep-ph/0703098 http://arxiv.org/abs/hep-ph/9809205 introduction generalities about oscillation probabilities The case of constant density The case of adiabatically varying density Another derivation of the formula by Kimura, Takamura and Yokomakura The case with arbitrary number of neutrinos the case with large magnetic moments and a magnetic field the case with non-standard interactions conclusions proof that Eq. (??) gives the same (??) Derivation of Eq. (??) Derivation of Eq. (??) Acknowledgments References

Microsoft Word - fluo_manuscript_rev_S.Lantz.doc Absolute measurement of the nitrogen fluorescence yield in air between 300 and 430 nm. G. Lefeuvrea1, P. Gorodetzkya2, J. Dolbeaua, T. Patzaka, P. Salina a APC - AstroParticule et Cosmologie, CNRS : UMR7164 – CEA – IN2P3 – Observatoire de Paris, Université Denis Diderot - Paris VII, 10, rue Alice Domon et Léonie Duquet 75205 Paris Cedex 13, France Abstract The nitrogen fluorescence induced in air is used to detect ultra-high energy cosmic rays and to measure their energy. The precise knowledge of the absolute fluorescence yield is the key quantity to improve the accuracy on the cosmic ray energy. The total yield has been measured in dry air using a 90Sr source and a [300-430 nm] filter. The fluorescence yield in air is 4.23 ± 0.20 photons per meter when normalized to 760 mmHg, 15°C and with an electron energy of 0.85 MeV. This result is consistent with previous experiments made at various energies, but with an accuracy improved by a factor of about 3. For the first time, the absolute continuous spectrum of nitrogen excited by 90Sr electrons has also been measured with a spectrometer. Details of this experiment are given in one of the author's PhD thesis [32]. Keywords: nitrogen fluorescence, air fluorescence, extensive air showers, ultra-high energy cosmic rays. PACS: 29.30.Dn ; 98.70.Sa 1 Actual address: Department of Physics, Syracuse University, Syracuse, NY 13244, USA 2 Corresponding author: E-mail address: philippe.gorodetzky@cern.ch Introduction One of the current challenges in high energy particle physics is to find the origin and nature of Ultra-High Energy Cosmic Rays (UHECR, E > 1018 eV). The measurement of their energy spectrum could confirm their interaction with the cosmological background (GZK theory). But this is a difficult task: the UHECR are not detectable themselves, but only the air shower they induce while going through the atmosphere. Experiments which have tried to solve this problem still have non consistent results for E > 5•1019 eV. They use different detecting techniques. On one hand, AGASA [1, 2] has an array of ground detectors separated by about 0.7 km. Muons and electrons of the showers arriving at the ground are sampled. The energy reconstruction is difficult because of the lack of knowledge on hadronic cross sections. This method relays on Monte-Carlo particle distributions in the shower approximation to find the energy of the incoming particle. On the other hand, HiRes [3] uses fluorescence telescopes to detect the continuous development of the shower. The accuracy of this method can still be improved if the uncertainties on the fluorescence itself are lowered. Today, the Auger experiment uses both methods but still calibrates energy extracted from the ground detectors by the fluorescence (also on the ground, [4]). In the future, space-based experiments like JEM-EUSO will look for the fluorescence signal from above [5]. Since 1964, several authors [6, 7, 8] have measured the fluorescence yield of each emission band of nitrogen (see the spectrum in fig. 5). The fluorescence yield is defined as the number of photons produced when an electron goes through one meter of air. In 2002, the cosmic ray community [9, 10, 11] decided to start an active campaign of fluorescence measurements for cosmic ray physics. Some were made at high energy in electron accelerators, the energy ranging from 80 MeV to 28 GeV [12, 13]. Others were made at low energy, around 1 MeV with a 90Sr radioactive source, or much lower with electron guns ([14, 15, 18, 21]). These experiments confirmed the hypothesis that the fluorescence yield, up to some tens of MeV, is proportional to the energy deposit of the particle, dE/dX. The difference between energy loss and energy deposit is of no importance from 0.1 to 10 MeV [16, 17], which includes our range. Therefore, the absolute scale of the yield can be set with a measurement performed with electrons from a 90Sr source. Now, the relative variation of the fluorescence yield with altitude is generally considered as quite well known [18]. The parameterization of the yield as a function of altitude is deduced from the relationship between pressure and altitude in an atmospheric model [19, 20]. These parameterizations are fairly consistent with one another, but can only give the variation of the yield with altitude and not the absolute value of the yield. Moreover, measurements of the absolute fluorescence yield have never reached a precision better than 13 to 20 % [13, 21]. The experiment presented in this paper has been realized with a double purpose: • to improve significantly the precision on the fluorescence yield: it is now 5 % ; • to achieve a first continuous measurement of the fluorescence spectrum in the range from 300 to 430 nm, which had never been done before with a radioactive source. 1. Experiment 1.1. Principle Two measurements have been carried out, named “integral” and “spectral” measurements. Electrons from a strontium source are used to excite nitrogen. Three detectors are needed, all of them based on Photonis photomultiplier tubes (PMT). One detects the electrons (electron-PMT), the other two, the photons (photon-PMTs). On one hand, the nitrogen emits fluorescence light through bands emitting lines from 300 to 430 nm. Thus cosmic ray experiments like HiRes and Auger use band pass filters of that range in front of their detectors. This is what is done here in the integral measurement. On the other hand, fluorescence yield experiments generally use narrow filters to separate the lines. But the overlap between some spectral filter bandwidth and the close proximity of some nitrogen bands make the separation difficult in reality. For this reason, the spectral measurement has been performed with an optical grating spectrometer, allowing a continuous check and a good separation of the lines. 1.2. Description of the setup This setup has been conceived with the aim of improving the measurement accuracy. The fluorescence chamber is a cross-shaped stainless steel chamber (a 6-ports accelerator pipe cross). It is schematically shown in fig. 1. Electrons are emitted by a strong 90Sr source (370 MBq) which is placed on the top part of the cross. On the bottom part, a plastic scintillator (truncated cone, 20 and 28 mm diameters, 20 mm thick and its angle is the same that the one made by the electron trajectories at the cone periphery). The electrons path is hence defined as the inside of a cone (100 mm height), with the source at its top. To homogenize the light between the scintillator and the PMT, a kaleidoscopic hexagonal light guide is used to carry the light. The phototube is a Photonis XP2262 [22] equipped with an active voltage divider (the rate is high, of the order of 2.5 MHz, and very stable). Figure 1. Schematic view of the fluorescence chamber. The strontium source and the scintillator are inside the lead shield. The internal structure of the lead is shown in fig. 2 (side view). The consequence of the high counting rate is a high X-ray noise level, due to the interacting electrons with the surrounding matter, including the PMT. To shield to a maximum these X-rays, a 10 cm diameter lead cylinder has been placed inside the chamber with the 90Sr source at its centre. A vertical cone is dug in the lead so the electrons can reach the scintillator. This cone is 30 mm diameter at its base, slightly wider than the electrons cone defined by the scintillator. Such a cone minimises the number of interactions between electrons and the lead: either they are totally stopped (around the source), or they touch in a grazing way the cone in their way to the scintillator. A GEANT Monte- Carlo simulation shows that half of the useful electrons do reach the scintillator without an interaction with the lead, the other half touching only once. Figure 2. Internal structure of the lead shield. Inside the hollow cone is the smaller cone of the path of the useful electrons defined by the source and the scintillator (thick dotted line). An horizontal cylinder (40 mm diameter) is also dug in the lead, centred on the optical axis, to let the photons go to the photon-PMTs. This structure is represented in fig. 2. The effective fluorescence volume is thus, on a first approximation, the intersection of the vertical cone with the horizontal cylinder. In a more precise way, all geometrical efficiencies have been determined with Monte-Carlo simulations. They take into account all possible interactions between electrons and their surroundings and give the exact photon solid angles as well as the exact useful electron path length (the length used in the yield expressed in photons per meter). Two kinds of interactions can occur: with the gas and with the lead. First, part of the energy of the source electrons is lost by ionization, i.e. creation of secondary electrons. If the secondaries go too far away from the useful fluorescence volume, they can either reach the lead or enter the horizontal cylinder. Both modify the detection efficiencies (geometry wise) of the photon-PMTs calculated. But 99 % of these secondaries have an energy smaller than 5 keV. Their range in air at atmospheric pressure is less than 2 mm. This has been checked through a geometrical simulation not to be enough to have an influence on the solid angles. The effect of these δ rays on the yield expressed in photons per deposited energy is explained later in paragraph 3.1 The second type of interaction concerns the electrons from the source that do not reach directly the scintillator. Some of them are scattered by the lead and then bounce back to the scintillator. This has three consequences, studied by a GEANT Monte-Carlo simulation. The first is the increase in the electron counting rate, due to the larger solid angle available for the electrons: it is doubled by this effect. The second is a modification in the electron energy spectrum, which contains more low energy electrons. This has been taken into account in the calculation of the electrons average energies. The third is a slight change in the useful electron path length, also taken into account. Some electrons can also bounce from the scintillator back into the fluorescence volume. This has been simulated to be a 10-9 effect on the photon detection, hence totally negligible. Figure 3. Schematic view of the optical components for the spectral measurement. For the integral measurement, fluorescence photons are detected and counted by a Photonis XP2020Q (2 in. diameter) placed on the left hand-side of the setup [23]. This PMT will be named integral-PMT and it is equipped with a fused silica window. As the fluorescence efficiency is known to be low (around 4 photons per meter), it has to work in single photoelectron mode, meaning at a very high gain. It is polarized positively (photocathode at ground) in order to reduce its own noise level to around 300 Hz (instead of 3 kHz with a negative polarity). This effectively removes the tiny discharges in the silica between the photocathode and the outside world. The optical filter on the entrance window is a Schott-Desag BG3 (2 mm thick, 34 mm diameter) [24]. It is the same as JEM- EUSO intends to use, and very similar to those used by Auger. It is glued with Epotec 301-2 [25] which has the same refraction index than both the window and the filter. Figure 4. Absolute efficiency of the integral-PMT (dashed line) and spectral-PMT (bold line) as a function of the size of the effective detecting surface size. A 20 mm diameter diaphragm is used to limit the effective detecting area of the integral-PMT to the flattest region of the photocathode. Using a diaphragm of 20 mm diameter (314 mm2), the efficiency is 18.7%. The spectral-PMT is used in its central region (~ 10×18 mm) where the efficiency including the filter is 19.3%. The uncertainty is 1.7% of the efficiency values. Note that the zero suppression on the vertical axis. For the spectral measurement, photons are analyzed by an optical grating spectrometer (Jobin- Yvon H25) and counted by another Photonis XP2020Q, attached to the spectrometer. This PMT will be named spectral-PMT. The light rays incoming the spectrometer have to be inside its numerical aperture (f = 250 mm, NA = f/4), so a silica converging lens of 150 mm focal length and 46 mm diameter [26] is inserted at the right place between the fluorescence volume and the entrance slit of the spectrometer. The optical image of this volume is thus on the latter. The optical magnification is 1/7. Entrance and exit slits of the spectrometer have the same dimensions: 2 x 7 mm, the maximum size available, inducing a resolution (measured with lasers) of 6 nm FWHM. This part of the setup is shown in fig. 3. The main uncertainty in this kind of experiments, where very few photons are counted, arises from the absolute efficiency of the photon-PMTs themselves. The manufacturers provides this value with an uncertainty of 15 to 20 % (at 1σ), which is not precise enough. For this reason, we designed a new way to measure the absolute efficiency of the photon-PMTs in the single photoelectron mode very accurately, based on the comparison with a NIST photodiode. The relative efficiency map of the photocathodes of the PMTs working in single photoelectron mode was measured with a 377 nm LED every 3 mm in X and Y with a relative precision better than 0.5%. This precision was needed to be better than the point to point variations in efficiency (about 2%), in order to control these variations. A measurement of the absolute efficiency of one of the map points with an accuracy of 1.7%, transforms that relative efficiency map in an absolute efficiency map. A dedicated paper, following a patent, will present this measurement. This result was used to determine the flattest region of the photocathode (see fig. 4). The effective detecting area of the integral-PMT was limited by a 20 mm diameter diaphragm where the efficiency stays roughly constant when the diaphragm size is increased. That way, a small error on the diaphragm diameter has a negligible consequence on the PMT efficiency (it has however on the solid angle, but this is easy to control). Figure 5. Fluorescence spectrum (vertical lines) with relative intensities and absolute response of the photon detector (PMT + filter + diaphragm, dashed curve) (source of the fluorescence spectrum : [14]). The integral- PMT efficiency for this spectrum is 17.8%. The absolute spectral efficiency of the detector {PMT + filter + diaphragm} is the dashed line superimposed on the fluorescence spectrum in fig. 5. Its value for photons at 377 nm is (18.9 ± 0.3)%, and, when convoluted with the fluorescence bands [14] and the relative variations given by Photonis [22], is (17.8± 0.4)%. The spectral-PMT is illuminated on a 10 x 18 mm area (smaller area than the 20 mm circle for the integral PMT, hence all photons at the exit slit of the spectrometer reaching different parts of this area would here also have roughly the same probability to produce photoelectrons). On their way from the fluorescence chamber to any of the photon-PMTs, integral or spectral-PMT, the 20 cm long tubes are baffled to prevent reflected photons to reach the photocathode. 1.3. Gas The gas used is either nitrogen for the adjustments or Messer dry air for the measurements. Fig. 6 shows the gas setup. It is possible to mix gases and to introduce controlled quantities of different impurities and water vapour. Those measurements including pressure variations are planned for the future. The gas circulates at a rate of 1 L.h-1 through the chamber to avoid impurities build-up and degassing of the walls. This circulation is controlled by a precise flow meter [27], followed by a pressure controller [28] and an ultra-clean primary pump with an ultimate vacuum of 50 mbar [29]. Internal and external pressures and temperatures are regularly monitored during data taking. Figure 6. The gas circulation scheme. 1.4. Data acquisition Hardware data acquisition is basically the same that for other authors, but had to be adapted to high counting rates. Fig. 7 shows its principle. The lifetimes of excited levels of nitrogen molecules at atmospheric pressure are of the order of a few nanoseconds [8, 14, 15, 18, 21]. A measurement based on the time coincidence between an electron and a photon is made to discriminate fluorescence from background photons. Figure 7. The data acquisition principle Basically, an electron in the scintillator is the trigger for the photon measurements. An electron produces about 0.16 photon in the 4 cm long fluorescence volume. Geometrical efficiencies toward each photon-PMT are very small: 3.69·10-4 for the integral-PMT and 7.48·10-6 for the spectral- PMT. As a consequence, only 5.9·10-5 and 1.2·10-6 photon respectively reach the PMTs. So, both photon-PMTs work in single photoelectron mode. Their spectra are very “pure” in “1 photo-electron”, with a negligible amount of “2”. So a discriminator is enough to select the “1 photoelectron” peak. This can be seen in fig. 8. ADCs (LRS 2249A, 0.25 pC per channel) record the electron and photon spectra. They are used only to check the gain stability of all PMTs during data taking. The “detection inefficiency”, D, defined as the proportion of lost fluorescence counts due to the mandatory discriminator threshold on the single photoelectron spectrum (see fig. 8), has also been measured and been found to be 3.76 % of the measured single photoelectron peak. Figure 8. Single photoelectron spectrum, with and without the pedestal. Time to Digital Converters (TDCs) (LRS 2228A, 0.3 ps per channel, monohit) are used to record the time difference spectra between electrons and photons. Both ADCs and TDCs could be triggered by the electron signal, but this would produce a very high random rate. We cleaned up the trigger by replacing it by an electron-photon coincidence. This coincidence is performed by a fast NIM coincidence unit. The photon pulse is 10 ns wide. The electron pulse is 100 ns wide to take into account the delay in photon emission (lifetimes of the levels). This was set to prepare acquisitions at low pressure, where the lifetimes are longer than at atmospheric pressure. Even at very low pressure, 100 ns ensures the loss due to that effect will be negligible. In other terms, a coincidence occurs when the photon arrives with less than 100 ns delay after the electron. The TDC is stopped by the same electron (see fig. 7), this time as a 10 ns wide pulse and delayed by 150 ns. Therefore, the start has a rate much lower than the stop and this inverts the time axis. Results can be seen in fig. 9. The useful signal is the peak in the spectrum. It can be easily separated from the flat background. The effect on the peak left slope due to the lifetimes of the nitrogen levels cannot be seen on this figure because it has been measured in air at atmospheric pressure. The narrow peak on the far right-hand side is due to random photons arriving some nanoseconds before electrons. These spectra suffer from an important dead time. But the ratio of the peak to total spectrum is not affected. The real number of counts is extracted from scalers measuring the total spectrum with a small and well corrected dead time. These fast scalers (CAMAC CERNSPEC NE003, 25 MHz, and VME V560E, 100 MHz) count all electron, photon and coincidence pulses. Two thresholds, creating two triggers, are set on the 90Sr spectrum to study the possible dependence of the fluorescence yield with the electron energy even if it is difficult to imagine an influence other than energy deposit. One is set at ~ 600 keV and another at ~ 1.2 MeV. The corresponding average energies are 1.1 and 1.5 MeV. There are around 1 000 photon pulses per second counted by the integral-PMT (with a background of 300 Hz) and 40 by the spectral-PMT on at the top of the 337 nm line (with a background of 30 Hz). Moreover, the dead times, TM, of both electron scalers have been specifically measured and found to be 1.2 % (scaler rate 2·106 Hz) and 0.7 % (scaler rate 6·105 Hz). 2. Results 2.1. Integral measurement Figure 9. Example of TDC spectrum for the integral measurement The photon yield per unit length, Y, is derived from the signal portion of the TDC spectrum through the following formula: where: • Π is the number of signal counts in the TDC spectrum (see fig. 9) ; • H is the integral of this spectrum ; • C is the number of coincidences sent to the TDC measured by a scaler. Only H coincidences have effectively started the TDC due to its dead time ; • 1 + D is the correction to the detection inefficiency ; • Ne is the number of electrons according to the scaler ; • 1 + TM is the correction to the dead time of the scaler itself ; • Lmoy is the mean length of the electrons path in the fluorescence volume ; • εPMT is the efficiency of the integral-PMT ; • Tw is the transmission of the fused silica window closing the fluorescence chamber ; • εgeo is the geometrical efficiency. Systematic errors are presented in table 1. The main uncertainty of this experiment is due to the high counting rate of the electrons, leading to a non-linear dead time dependence in the TDC module. This effect, which varies from channel to channel, depends on the internal TDC time constants and is not fully understood. It has been evaluated by using all the module’s channels, and another TDC module (CAEN V1290N, multihit) to compare their results. All values are found equal within 4% (at 1σ) and this uncertainty has been chosen in a conservative way. In the future, to further reduce this uncertainty, fast flash ADCs will be used, allowing to discriminate pulses with and without pile-up on an event per event basis. On the contrary, the uncertainty on the efficiency of the integral-PMT is very low. This is due to the photomultiplier efficiency measurement made especially for the fluorescence measurement by the patented method. Table 1. Systematic uncertainties of the experiment. An example of typical data sample is presented in tab. 2. The low (1.1 MeV) and high (1.5 MeV) energy measurements give respectively 3.95 and 4.34 photons per meter. Statistical uncertainties are 0.2 % and 0.8 %. The energy normalization is made at 0.85 MeV using the dE/dX ratios with values interpolated from NIST data [30] and yields respectively 4.05 and 4.41 photons per meter. Table 2. Data sample for the low energy measurement. The pressure and temperature normalizations are then applied. Each excited level has its own lifetime, and the yield can be written with respect to pressure and temperature using the kinetic theory [6]. Thus this normalization has been made for each band using previous parameterizations (which give the same variations for the first kilometers above the ground [20]). Here, Nagano's model [21], who uses the different bands yields is used. The normalized pressure and temperature are those of the US- Standard 1976 model [19] at sea level: 760 mmHg and 15°C. In this model, for an electron energy of 0.85 MeV, one finds that the ratio of the yield at 753.8 mmHg and 295.95° K (the 1.1 MeV conditions) to the yield at 760.0 mmHg and 288.15° K (the US Standard conditions), is 0.9863, and the similar ratio for the 1.5 MeV conditions which are 751.8 mmHg and 296.05° K is 0.9860. These ratios are then used to normalize the measured values. Finally, the two normalized values are 4.05 and 4.42 ph/m and have, due to this normalization, an added uncertainty of 0.6 %, setting the total relative uncertainty to 5.0 %. They are separated by 8.5%, inside an error bar (± 1σ). Hence, these are averaged to get the fluorescence yield, at 760 mmHg, 15°C and for 0.85 MeV electrons: 4.23 ± 0.21 photons / m. This yield per meter is that of the primary particle, therefore it takes into account its production of δ rays and their fluorescence. This number can also be written in units of photons per deposited energy. For a given energy, these two quantities are strictly proportional, since: where Φν is the fluorescence efficiency at the wavelength corresponding to the frequency ν, with the yield per deposited energy being [31]: The energy of the δ rays is naturally included in the dE/dX of the initial electron. Nevertheless, in the context of this experiment, a special care must be taken for the δ rays. A Monte- Carlo simulation has been performed and shows that here, with the geometry described earlier and at atmospheric pressure, only 1 % of the δ rays have more than 5 keV and some of these could be lost in the lead before having produced any fluorescence. The effective dE/dX should thus be reduced. This simulation shows then that almost 67 % of these δ rays with 5keV or more will produce detectable fluorescence. Therefore only 33 % of them are effectively lost and contribute to the reduction of the dE/dX. Finally, this reduction, hence the reduction in yield, is 0.4 %. The central value is shifted from 0.4 % (we measured 4.21 photons/m before applying this correction). The uncertainty stays the same at 5 %. This would be very different at lower pressures, where a more extensive Monte-Carlo simulation would be required and the correction would be increased. The energy deposited by a 0.85 MeV electron in a meter of the US Standard air is 0.2059 MeV [30]. The fluorescence yield per deposited energy is thus 20.38 ± 0.98 photons per MeV. The nitrogen yield to air yield ratio has also been measured, (at one electron energy only: 1.1 MeV), and found to be 4.90 ± 0.01, where the low uncertainty is only statistical: all systematic errors compensate, the gas being the only element changed from one measurement to the other. This ratio is compatible with what has been measured at different energies : 6 ± 2 at 28.5 GeV [13], 5.5 ± 0.3 at 1 MeV [21] 2.2. Spectral measurement Spectral measurements are interesting for many purposes. The gas kinetic theory is valid for individual bands. Is the sum of the different bands yield equal to the integrated yield used by the majority of cosmic ray experiments using fluorescence? How do the different bands change yield when temperature and pressure are modified? Up to now, the different bands have been observed either with a grating spectrometer [6, 7, 14] in a first method, the electrons being produced by an electron gun. Their energy (around 10 keV) is so small that the electron scattering in the gas prevents any measurement of path length, hence cannot give a yield in photons per meter. The second method [8,15, 18, 21] uses narrow interference filters to analyze the bands. Here, the electrons, as in this work, come from a 90Sr source. Fig. 1 of [21] illustrates the method complexity. It is difficult to separate overlapping bands in some filters. The only solution to have a good resolution on the bands is to use a grating spectrometer, and the easiest way to have electrons with an energy large enough to know their path is to use a 90Sr source. The acceptance of a spectrometer is roughly an order of magnitude smaller than that of a filter. So the very first step is to prove the feasibility of such a method. This is what is done here. Hence, in this experiment, a 90Sr source 10 times stronger than in [8,15, 18, 21] was used, The counterpart of the high number of incident electrons is that a very special attention has to be given to pile-up effects and to the lead shield geometry to avoid the introduction of a large X-ray background. Results of the spectral measurements are represented in fig. 10. The whole fluorescence spectrum has been measured, from 300 to 436 nm, in dry air at room temperature and normal pressure, with electrons of average energy of 1.1 MeV. The time allowed to measure this spectrum was short and the spectrometer equipped with an output slit can measure only one wavelength at a time. So it was decided to open the slits at their maximum, setting the 1σ resolution to 3 nm. Thus measurements were made with steps of 3 nm. But even with 2·106 electrons detected every second, only 0.16 fluorescence photons per second are recorded in the signal part of the TDC spectrum for the most intense line (337 nm). This is the reason why this spectrum has only been made for the “low” energy threshold, as defined above. Fluorescence lines are much narrower than the resolution set for the spectrometer as is seen in [14]. The absolute yield is thus given by the height of the curve at a given wavelength, and not by the area integrated over a bandwidth dλ. The sum of the yields of all lines indicated in fig. 10 is 3.9 ± 0.8 photons per meter. This value is in close agreement with the result of the integral measurement. The 20% uncertainty is due most entirely to the bad knowledge of the spectrometer absolute efficiency, especially in the UV. Superimposed on this spectrum is Ulrich’s spectrum [14], convoluted with a 3 nm resolution. As it is a relative yield spectrum, it is shown here normalized to the 337 nm line. Our spectrum is well compatible with what would measure [14] with such a resolution. Figure 10. Spectrum of the absolute fluorescence yield of nitrogen in air between 300 and 436 nm. The bold line is the result of the present experiment, and the dashed line is Ulrich's spectrum [14] as if analyzed by our spectrometer. Discrepancies could be explained by the ageing of our spectrometer. The first observation is the confirmation of the feasibility of this measurement: all the main lines are indeed observed. Evidence is given that an absolute spectrum can be measured with a basic apparatus, and taking care of reducing the PMT background as explained earlier. The sum of the lines yield is consistent with the previous integral measurement. Moreover, its uncertainty is 20 %, when previous experiments do not give better than 15 %. The second observation concerns the discrepancies, around 316 nm and beyond 375 nm. The extraction of the absolute yield involves the efficiency of the spectrometer. Its spectral efficiency curve is provided by the manufacturer with a low accuracy in the UV (which is true also for Ulrich's results). Furthermore two effects induce a very important loss in our spectrometer: • the ageing of each optical element (mirrors and grating). The spectrometer efficiency can be reduced to a value around 15 % after ten years of use ; • the extensive use with intense UV light before this experiment. At 400 nm, the absolute efficiency of the spectrometer was measured with an accuracy of about 2% to be only 15 %, instead of the 61 % given by the manufacturer (who confirms in a private communication such a low value compatible with ageing). The method used is very similar to that taken to determine the absolute efficiency of the photon-PMTs (comparison to a NIST photodiode). This value of 15 % has been used to calculate the absolute yields in the entire spectral range. But there is no reason that this loss is constant with respect to wavelength. The 20% uncertainty on the yield arises from this unknown but limited variation. The question of getting the absolute efficiency of spectrometers in the UV is quite challenging and is the object of specific attentions by the community of atomic / spectral physics. It is unfortunate that this spectrometer method does not yield yet an accuracy better than the "integrated yield" method. It will if the spectrometer resolution can be made high enough to totally separate the bands, which is possible if it is equipped with a CCD readout to minimize the experiment duration. Then, the absolute efficiency of the spectrometer will have to be determined with a high accuracy in the UV, not an easy task according to Ulrich, but possible through our patented method. In the future, two measurements will be done: • calibrate in an absolute way the old Jobin-Yvon; • use the new spectrometer able to measure the full spectrum at once with a 0.1 nm resolution and equipped with a light intensifier to measure the fluorescence yield. 3. Conclusion The absolute fluorescence yield of nitrogen in dry air at atmospheric pressure has been measured. The precision of the measurement is improved by a factor of three, which has an immediate impact on the cosmic ray energies found by HiRes which uses Bunner's [6] yield. Their energies are increased by 22%, hence their spectrum is much closer to AGASA’s (which incidentally have been recently lowered the energy of their points by 10% [33]). The first continuous fluorescence spectrum of nitrogen excited by electrons from a 90Sr source was also measured. Next steps are to introduce impurities in the gas, such as argon, water vapour and pollutants. A pressure study of the total yield will be made. On another hand, the spectral measurement will be improved thanks to a new spectrometer. Papers will follow to account for these measurements, which will provide an overall and realistic view of the fluorescence phenomenon. Acknowledgements We would like to thank Bernard LEFIEVRE for his help with GEANT simulation of the setup, and François LELONG and Jean-Paul RENY for their help in building the bench. REFERENCES [1] M. Takeda et al., Astropart. Phys. 19 (2003) 447 [2] M. Takeda et al., Phys. Rev. Lett. 81 (1998) 1163 [3] R.U. Abbasi et al., Phys. Rev. Lett. 92 (2004) 151101. [4] P. Sommers, C. R. Physique 5 (2004) 463 [5] http://www.euso-mission.org/docs/RedBookEUSO_21apr04.pdf [6] A.N. Bunner, Ph.D. thesis, Cornell University, 1967. [7] G. Davidson, R. O'Neil, J. Chem. Phys. 41 (1964) 3946. [8] F. Kakimoto et al., NIM A 372 (1996) 527. [9] http://www.auger.de/events/air-light-03/ [10] http://lappweb.in2p3.fr/IWFM05/index.html [11] http://www.particle.cz/conferences/floret2006/ [12] F. Arciprete et al., Nucl. Phys. B Proc. Supp. 150 (2006) 186-189 [13] J.W. Belz et al., Astropart. Phys. 25 (2006) 129. [14] A. Ulrich, private communication, and in [9], [10] and [11] [15] T. Waldenmaier, PhD thesis, Forschunzentrum Karlsruhe, 2006 [16] F. Arqueros et al., Astropart. Phys. 26 (2006) 231 [17] F. Blanco and F. Arqueros, Phys. Lett. A 345 (2005) 355 [18] M. Nagano, K. Kobayakawa, N. Sakaki, K. Ando, Astropart. Phys. 20 (2003) 293. [19] http://www.pdas.com/atmos.htm [20] B. Keilhauer et al., Astropart. Phys. 25 (2006) 259. [21] M. Nagano, K. Kobayakawa, N. Sakaki, K. Ando, Astropart. Phys. 22 (2004) 235. [22] http://www.photonis.com/data/cms-resources/File/Photomultiplier_tubes/spec/XP2262.pdf [23] http://www.photonis.com/data/cms-resources/File/Photomultiplier_tubes/spec/XP2020Q.PDF [24] Optical filters/Band Pass filter/BG3 in http://www.schott.com/optics_devices/english/download/index.html [25] http://www.epotek.com/SSCDocs/datasheets/301-2.PDF [26] Edmund Optics C46278 [27] http://www.bronkhorst.fr/fr/produits/débitmètres_et_régulateurs_gaz/lowdpflow [28] http://www.laa.fr/upload/44-4700-Fr-TESCOM.pdf?PHPSESSID=9ae88e24346e03c9765b5350ac7930af [29] http://www.knf.fr/images_messages/image1/61.dat [30] http://physics.nist.gov/PhysRefData/Star/Text/contents.html [31] B. Keilhauer, Ph. D. thesis, Universität Karlsruhe, 2003 [32] G. Lefeuvre, Ph. D. thesis, Paris 7 University, 2006 (ref : APC-26-06). [33] M. Teshima, Ultra High Energy Cosmic Rays Observed by AGASA, XXXIII International Conf. On High Energy Phys (ICHEP'06), July 26, 2006, Moscow, Russia.

The nitrogen fluorescence induced in air is used to detect ultra-high energy cosmic rays and to measure their energy. The precise knowledge of the absolute fluorescence yield is the key quantity to improve the accuracy on the cosmic ray energy. The total yield has been measured in dry air using a 90Sr source and a [300-430 nm] filter. The fluorescence yield in air is 4.23 $\pm$ 0.20 photons per meter when normalized to 760 mmHg, 15 degrees C and with an electron energy of 0.85 MeV. This result is consistent with previous experiments made at various energies, but with an accuracy improved by a factor of about 3. For the first time, the absolute continuous spectrum of nitrogen excited by 90Sr electrons has also been measured with a spectrometer. Details of this experiment are given in one of the author's PhD thesis [32].

Introduction One of the current challenges in high energy particle physics is to find the origin and nature of Ultra-High Energy Cosmic Rays (UHECR, E > 1018 eV). The measurement of their energy spectrum could confirm their interaction with the cosmological background (GZK theory). But this is a difficult task: the UHECR are not detectable themselves, but only the air shower they induce while going through the atmosphere. Experiments which have tried to solve this problem still have non consistent results for E > 5•1019 eV. They use different detecting techniques. On one hand, AGASA [1, 2] has an array of ground detectors separated by about 0.7 km. Muons and electrons of the showers arriving at the ground are sampled. The energy reconstruction is difficult because of the lack of knowledge on hadronic cross sections. This method relays on Monte-Carlo particle distributions in the shower approximation to find the energy of the incoming particle. On the other hand, HiRes [3] uses fluorescence telescopes to detect the continuous development of the shower. The accuracy of this method can still be improved if the uncertainties on the fluorescence itself are lowered. Today, the Auger experiment uses both methods but still calibrates energy extracted from the ground detectors by the fluorescence (also on the ground, [4]). In the future, space-based experiments like JEM-EUSO will look for the fluorescence signal from above [5]. Since 1964, several authors [6, 7, 8] have measured the fluorescence yield of each emission band of nitrogen (see the spectrum in fig. 5). The fluorescence yield is defined as the number of photons produced when an electron goes through one meter of air. In 2002, the cosmic ray community [9, 10, 11] decided to start an active campaign of fluorescence measurements for cosmic ray physics. Some were made at high energy in electron accelerators, the energy ranging from 80 MeV to 28 GeV [12, 13]. Others were made at low energy, around 1 MeV with a 90Sr radioactive source, or much lower with electron guns ([14, 15, 18, 21]). These experiments confirmed the hypothesis that the fluorescence yield, up to some tens of MeV, is proportional to the energy deposit of the particle, dE/dX. The difference between energy loss and energy deposit is of no importance from 0.1 to 10 MeV [16, 17], which includes our range. Therefore, the absolute scale of the yield can be set with a measurement performed with electrons from a 90Sr source. Now, the relative variation of the fluorescence yield with altitude is generally considered as quite well known [18]. The parameterization of the yield as a function of altitude is deduced from the relationship between pressure and altitude in an atmospheric model [19, 20]. These parameterizations are fairly consistent with one another, but can only give the variation of the yield with altitude and not the absolute value of the yield. Moreover, measurements of the absolute fluorescence yield have never reached a precision better than 13 to 20 % [13, 21]. The experiment presented in this paper has been realized with a double purpose: • to improve significantly the precision on the fluorescence yield: it is now 5 % ; • to achieve a first continuous measurement of the fluorescence spectrum in the range from 300 to 430 nm, which had never been done before with a radioactive source. 1. Experiment 1.1. Principle Two measurements have been carried out, named “integral” and “spectral” measurements. Electrons from a strontium source are used to excite nitrogen. Three detectors are needed, all of them based on Photonis photomultiplier tubes (PMT). One detects the electrons (electron-PMT), the other two, the photons (photon-PMTs). On one hand, the nitrogen emits fluorescence light through bands emitting lines from 300 to 430 nm. Thus cosmic ray experiments like HiRes and Auger use band pass filters of that range in front of their detectors. This is what is done here in the integral measurement. On the other hand, fluorescence yield experiments generally use narrow filters to separate the lines. But the overlap between some spectral filter bandwidth and the close proximity of some nitrogen bands make the separation difficult in reality. For this reason, the spectral measurement has been performed with an optical grating spectrometer, allowing a continuous check and a good separation of the lines. 1.2. Description of the setup This setup has been conceived with the aim of improving the measurement accuracy. The fluorescence chamber is a cross-shaped stainless steel chamber (a 6-ports accelerator pipe cross). It is schematically shown in fig. 1. Electrons are emitted by a strong 90Sr source (370 MBq) which is placed on the top part of the cross. On the bottom part, a plastic scintillator (truncated cone, 20 and 28 mm diameters, 20 mm thick and its angle is the same that the one made by the electron trajectories at the cone periphery). The electrons path is hence defined as the inside of a cone (100 mm height), with the source at its top. To homogenize the light between the scintillator and the PMT, a kaleidoscopic hexagonal light guide is used to carry the light. The phototube is a Photonis XP2262 [22] equipped with an active voltage divider (the rate is high, of the order of 2.5 MHz, and very stable). Figure 1. Schematic view of the fluorescence chamber. The strontium source and the scintillator are inside the lead shield. The internal structure of the lead is shown in fig. 2 (side view). The consequence of the high counting rate is a high X-ray noise level, due to the interacting electrons with the surrounding matter, including the PMT. To shield to a maximum these X-rays, a 10 cm diameter lead cylinder has been placed inside the chamber with the 90Sr source at its centre. A vertical cone is dug in the lead so the electrons can reach the scintillator. This cone is 30 mm diameter at its base, slightly wider than the electrons cone defined by the scintillator. Such a cone minimises the number of interactions between electrons and the lead: either they are totally stopped (around the source), or they touch in a grazing way the cone in their way to the scintillator. A GEANT Monte- Carlo simulation shows that half of the useful electrons do reach the scintillator without an interaction with the lead, the other half touching only once. Figure 2. Internal structure of the lead shield. Inside the hollow cone is the smaller cone of the path of the useful electrons defined by the source and the scintillator (thick dotted line). An horizontal cylinder (40 mm diameter) is also dug in the lead, centred on the optical axis, to let the photons go to the photon-PMTs. This structure is represented in fig. 2. The effective fluorescence volume is thus, on a first approximation, the intersection of the vertical cone with the horizontal cylinder. In a more precise way, all geometrical efficiencies have been determined with Monte-Carlo simulations. They take into account all possible interactions between electrons and their surroundings and give the exact photon solid angles as well as the exact useful electron path length (the length used in the yield expressed in photons per meter). Two kinds of interactions can occur: with the gas and with the lead. First, part of the energy of the source electrons is lost by ionization, i.e. creation of secondary electrons. If the secondaries go too far away from the useful fluorescence volume, they can either reach the lead or enter the horizontal cylinder. Both modify the detection efficiencies (geometry wise) of the photon-PMTs calculated. But 99 % of these secondaries have an energy smaller than 5 keV. Their range in air at atmospheric pressure is less than 2 mm. This has been checked through a geometrical simulation not to be enough to have an influence on the solid angles. The effect of these δ rays on the yield expressed in photons per deposited energy is explained later in paragraph 3.1 The second type of interaction concerns the electrons from the source that do not reach directly the scintillator. Some of them are scattered by the lead and then bounce back to the scintillator. This has three consequences, studied by a GEANT Monte-Carlo simulation. The first is the increase in the electron counting rate, due to the larger solid angle available for the electrons: it is doubled by this effect. The second is a modification in the electron energy spectrum, which contains more low energy electrons. This has been taken into account in the calculation of the electrons average energies. The third is a slight change in the useful electron path length, also taken into account. Some electrons can also bounce from the scintillator back into the fluorescence volume. This has been simulated to be a 10-9 effect on the photon detection, hence totally negligible. Figure 3. Schematic view of the optical components for the spectral measurement. For the integral measurement, fluorescence photons are detected and counted by a Photonis XP2020Q (2 in. diameter) placed on the left hand-side of the setup [23]. This PMT will be named integral-PMT and it is equipped with a fused silica window. As the fluorescence efficiency is known to be low (around 4 photons per meter), it has to work in single photoelectron mode, meaning at a very high gain. It is polarized positively (photocathode at ground) in order to reduce its own noise level to around 300 Hz (instead of 3 kHz with a negative polarity). This effectively removes the tiny discharges in the silica between the photocathode and the outside world. The optical filter on the entrance window is a Schott-Desag BG3 (2 mm thick, 34 mm diameter) [24]. It is the same as JEM- EUSO intends to use, and very similar to those used by Auger. It is glued with Epotec 301-2 [25] which has the same refraction index than both the window and the filter. Figure 4. Absolute efficiency of the integral-PMT (dashed line) and spectral-PMT (bold line) as a function of the size of the effective detecting surface size. A 20 mm diameter diaphragm is used to limit the effective detecting area of the integral-PMT to the flattest region of the photocathode. Using a diaphragm of 20 mm diameter (314 mm2), the efficiency is 18.7%. The spectral-PMT is used in its central region (~ 10×18 mm) where the efficiency including the filter is 19.3%. The uncertainty is 1.7% of the efficiency values. Note that the zero suppression on the vertical axis. For the spectral measurement, photons are analyzed by an optical grating spectrometer (Jobin- Yvon H25) and counted by another Photonis XP2020Q, attached to the spectrometer. This PMT will be named spectral-PMT. The light rays incoming the spectrometer have to be inside its numerical aperture (f = 250 mm, NA = f/4), so a silica converging lens of 150 mm focal length and 46 mm diameter [26] is inserted at the right place between the fluorescence volume and the entrance slit of the spectrometer. The optical image of this volume is thus on the latter. The optical magnification is 1/7. Entrance and exit slits of the spectrometer have the same dimensions: 2 x 7 mm, the maximum size available, inducing a resolution (measured with lasers) of 6 nm FWHM. This part of the setup is shown in fig. 3. The main uncertainty in this kind of experiments, where very few photons are counted, arises from the absolute efficiency of the photon-PMTs themselves. The manufacturers provides this value with an uncertainty of 15 to 20 % (at 1σ), which is not precise enough. For this reason, we designed a new way to measure the absolute efficiency of the photon-PMTs in the single photoelectron mode very accurately, based on the comparison with a NIST photodiode. The relative efficiency map of the photocathodes of the PMTs working in single photoelectron mode was measured with a 377 nm LED every 3 mm in X and Y with a relative precision better than 0.5%. This precision was needed to be better than the point to point variations in efficiency (about 2%), in order to control these variations. A measurement of the absolute efficiency of one of the map points with an accuracy of 1.7%, transforms that relative efficiency map in an absolute efficiency map. A dedicated paper, following a patent, will present this measurement. This result was used to determine the flattest region of the photocathode (see fig. 4). The effective detecting area of the integral-PMT was limited by a 20 mm diameter diaphragm where the efficiency stays roughly constant when the diaphragm size is increased. That way, a small error on the diaphragm diameter has a negligible consequence on the PMT efficiency (it has however on the solid angle, but this is easy to control). Figure 5. Fluorescence spectrum (vertical lines) with relative intensities and absolute response of the photon detector (PMT + filter + diaphragm, dashed curve) (source of the fluorescence spectrum : [14]). The integral- PMT efficiency for this spectrum is 17.8%. The absolute spectral efficiency of the detector {PMT + filter + diaphragm} is the dashed line superimposed on the fluorescence spectrum in fig. 5. Its value for photons at 377 nm is (18.9 ± 0.3)%, and, when convoluted with the fluorescence bands [14] and the relative variations given by Photonis [22], is (17.8± 0.4)%. The spectral-PMT is illuminated on a 10 x 18 mm area (smaller area than the 20 mm circle for the integral PMT, hence all photons at the exit slit of the spectrometer reaching different parts of this area would here also have roughly the same probability to produce photoelectrons). On their way from the fluorescence chamber to any of the photon-PMTs, integral or spectral-PMT, the 20 cm long tubes are baffled to prevent reflected photons to reach the photocathode. 1.3. Gas The gas used is either nitrogen for the adjustments or Messer dry air for the measurements. Fig. 6 shows the gas setup. It is possible to mix gases and to introduce controlled quantities of different impurities and water vapour. Those measurements including pressure variations are planned for the future. The gas circulates at a rate of 1 L.h-1 through the chamber to avoid impurities build-up and degassing of the walls. This circulation is controlled by a precise flow meter [27], followed by a pressure controller [28] and an ultra-clean primary pump with an ultimate vacuum of 50 mbar [29]. Internal and external pressures and temperatures are regularly monitored during data taking. Figure 6. The gas circulation scheme. 1.4. Data acquisition Hardware data acquisition is basically the same that for other authors, but had to be adapted to high counting rates. Fig. 7 shows its principle. The lifetimes of excited levels of nitrogen molecules at atmospheric pressure are of the order of a few nanoseconds [8, 14, 15, 18, 21]. A measurement based on the time coincidence between an electron and a photon is made to discriminate fluorescence from background photons. Figure 7. The data acquisition principle Basically, an electron in the scintillator is the trigger for the photon measurements. An electron produces about 0.16 photon in the 4 cm long fluorescence volume. Geometrical efficiencies toward each photon-PMT are very small: 3.69·10-4 for the integral-PMT and 7.48·10-6 for the spectral- PMT. As a consequence, only 5.9·10-5 and 1.2·10-6 photon respectively reach the PMTs. So, both photon-PMTs work in single photoelectron mode. Their spectra are very “pure” in “1 photo-electron”, with a negligible amount of “2”. So a discriminator is enough to select the “1 photoelectron” peak. This can be seen in fig. 8. ADCs (LRS 2249A, 0.25 pC per channel) record the electron and photon spectra. They are used only to check the gain stability of all PMTs during data taking. The “detection inefficiency”, D, defined as the proportion of lost fluorescence counts due to the mandatory discriminator threshold on the single photoelectron spectrum (see fig. 8), has also been measured and been found to be 3.76 % of the measured single photoelectron peak. Figure 8. Single photoelectron spectrum, with and without the pedestal. Time to Digital Converters (TDCs) (LRS 2228A, 0.3 ps per channel, monohit) are used to record the time difference spectra between electrons and photons. Both ADCs and TDCs could be triggered by the electron signal, but this would produce a very high random rate. We cleaned up the trigger by replacing it by an electron-photon coincidence. This coincidence is performed by a fast NIM coincidence unit. The photon pulse is 10 ns wide. The electron pulse is 100 ns wide to take into account the delay in photon emission (lifetimes of the levels). This was set to prepare acquisitions at low pressure, where the lifetimes are longer than at atmospheric pressure. Even at very low pressure, 100 ns ensures the loss due to that effect will be negligible. In other terms, a coincidence occurs when the photon arrives with less than 100 ns delay after the electron. The TDC is stopped by the same electron (see fig. 7), this time as a 10 ns wide pulse and delayed by 150 ns. Therefore, the start has a rate much lower than the stop and this inverts the time axis. Results can be seen in fig. 9. The useful signal is the peak in the spectrum. It can be easily separated from the flat background. The effect on the peak left slope due to the lifetimes of the nitrogen levels cannot be seen on this figure because it has been measured in air at atmospheric pressure. The narrow peak on the far right-hand side is due to random photons arriving some nanoseconds before electrons. These spectra suffer from an important dead time. But the ratio of the peak to total spectrum is not affected. The real number of counts is extracted from scalers measuring the total spectrum with a small and well corrected dead time. These fast scalers (CAMAC CERNSPEC NE003, 25 MHz, and VME V560E, 100 MHz) count all electron, photon and coincidence pulses. Two thresholds, creating two triggers, are set on the 90Sr spectrum to study the possible dependence of the fluorescence yield with the electron energy even if it is difficult to imagine an influence other than energy deposit. One is set at ~ 600 keV and another at ~ 1.2 MeV. The corresponding average energies are 1.1 and 1.5 MeV. There are around 1 000 photon pulses per second counted by the integral-PMT (with a background of 300 Hz) and 40 by the spectral-PMT on at the top of the 337 nm line (with a background of 30 Hz). Moreover, the dead times, TM, of both electron scalers have been specifically measured and found to be 1.2 % (scaler rate 2·106 Hz) and 0.7 % (scaler rate 6·105 Hz). 2. Results 2.1. Integral measurement Figure 9. Example of TDC spectrum for the integral measurement The photon yield per unit length, Y, is derived from the signal portion of the TDC spectrum through the following formula: where: • Π is the number of signal counts in the TDC spectrum (see fig. 9) ; • H is the integral of this spectrum ; • C is the number of coincidences sent to the TDC measured by a scaler. Only H coincidences have effectively started the TDC due to its dead time ; • 1 + D is the correction to the detection inefficiency ; • Ne is the number of electrons according to the scaler ; • 1 + TM is the correction to the dead time of the scaler itself ; • Lmoy is the mean length of the electrons path in the fluorescence volume ; • εPMT is the efficiency of the integral-PMT ; • Tw is the transmission of the fused silica window closing the fluorescence chamber ; • εgeo is the geometrical efficiency. Systematic errors are presented in table 1. The main uncertainty of this experiment is due to the high counting rate of the electrons, leading to a non-linear dead time dependence in the TDC module. This effect, which varies from channel to channel, depends on the internal TDC time constants and is not fully understood. It has been evaluated by using all the module’s channels, and another TDC module (CAEN V1290N, multihit) to compare their results. All values are found equal within 4% (at 1σ) and this uncertainty has been chosen in a conservative way. In the future, to further reduce this uncertainty, fast flash ADCs will be used, allowing to discriminate pulses with and without pile-up on an event per event basis. On the contrary, the uncertainty on the efficiency of the integral-PMT is very low. This is due to the photomultiplier efficiency measurement made especially for the fluorescence measurement by the patented method. Table 1. Systematic uncertainties of the experiment. An example of typical data sample is presented in tab. 2. The low (1.1 MeV) and high (1.5 MeV) energy measurements give respectively 3.95 and 4.34 photons per meter. Statistical uncertainties are 0.2 % and 0.8 %. The energy normalization is made at 0.85 MeV using the dE/dX ratios with values interpolated from NIST data [30] and yields respectively 4.05 and 4.41 photons per meter. Table 2. Data sample for the low energy measurement. The pressure and temperature normalizations are then applied. Each excited level has its own lifetime, and the yield can be written with respect to pressure and temperature using the kinetic theory [6]. Thus this normalization has been made for each band using previous parameterizations (which give the same variations for the first kilometers above the ground [20]). Here, Nagano's model [21], who uses the different bands yields is used. The normalized pressure and temperature are those of the US- Standard 1976 model [19] at sea level: 760 mmHg and 15°C. In this model, for an electron energy of 0.85 MeV, one finds that the ratio of the yield at 753.8 mmHg and 295.95° K (the 1.1 MeV conditions) to the yield at 760.0 mmHg and 288.15° K (the US Standard conditions), is 0.9863, and the similar ratio for the 1.5 MeV conditions which are 751.8 mmHg and 296.05° K is 0.9860. These ratios are then used to normalize the measured values. Finally, the two normalized values are 4.05 and 4.42 ph/m and have, due to this normalization, an added uncertainty of 0.6 %, setting the total relative uncertainty to 5.0 %. They are separated by 8.5%, inside an error bar (± 1σ). Hence, these are averaged to get the fluorescence yield, at 760 mmHg, 15°C and for 0.85 MeV electrons: 4.23 ± 0.21 photons / m. This yield per meter is that of the primary particle, therefore it takes into account its production of δ rays and their fluorescence. This number can also be written in units of photons per deposited energy. For a given energy, these two quantities are strictly proportional, since: where Φν is the fluorescence efficiency at the wavelength corresponding to the frequency ν, with the yield per deposited energy being [31]: The energy of the δ rays is naturally included in the dE/dX of the initial electron. Nevertheless, in the context of this experiment, a special care must be taken for the δ rays. A Monte- Carlo simulation has been performed and shows that here, with the geometry described earlier and at atmospheric pressure, only 1 % of the δ rays have more than 5 keV and some of these could be lost in the lead before having produced any fluorescence. The effective dE/dX should thus be reduced. This simulation shows then that almost 67 % of these δ rays with 5keV or more will produce detectable fluorescence. Therefore only 33 % of them are effectively lost and contribute to the reduction of the dE/dX. Finally, this reduction, hence the reduction in yield, is 0.4 %. The central value is shifted from 0.4 % (we measured 4.21 photons/m before applying this correction). The uncertainty stays the same at 5 %. This would be very different at lower pressures, where a more extensive Monte-Carlo simulation would be required and the correction would be increased. The energy deposited by a 0.85 MeV electron in a meter of the US Standard air is 0.2059 MeV [30]. The fluorescence yield per deposited energy is thus 20.38 ± 0.98 photons per MeV. The nitrogen yield to air yield ratio has also been measured, (at one electron energy only: 1.1 MeV), and found to be 4.90 ± 0.01, where the low uncertainty is only statistical: all systematic errors compensate, the gas being the only element changed from one measurement to the other. This ratio is compatible with what has been measured at different energies : 6 ± 2 at 28.5 GeV [13], 5.5 ± 0.3 at 1 MeV [21] 2.2. Spectral measurement Spectral measurements are interesting for many purposes. The gas kinetic theory is valid for individual bands. Is the sum of the different bands yield equal to the integrated yield used by the majority of cosmic ray experiments using fluorescence? How do the different bands change yield when temperature and pressure are modified? Up to now, the different bands have been observed either with a grating spectrometer [6, 7, 14] in a first method, the electrons being produced by an electron gun. Their energy (around 10 keV) is so small that the electron scattering in the gas prevents any measurement of path length, hence cannot give a yield in photons per meter. The second method [8,15, 18, 21] uses narrow interference filters to analyze the bands. Here, the electrons, as in this work, come from a 90Sr source. Fig. 1 of [21] illustrates the method complexity. It is difficult to separate overlapping bands in some filters. The only solution to have a good resolution on the bands is to use a grating spectrometer, and the easiest way to have electrons with an energy large enough to know their path is to use a 90Sr source. The acceptance of a spectrometer is roughly an order of magnitude smaller than that of a filter. So the very first step is to prove the feasibility of such a method. This is what is done here. Hence, in this experiment, a 90Sr source 10 times stronger than in [8,15, 18, 21] was used, The counterpart of the high number of incident electrons is that a very special attention has to be given to pile-up effects and to the lead shield geometry to avoid the introduction of a large X-ray background. Results of the spectral measurements are represented in fig. 10. The whole fluorescence spectrum has been measured, from 300 to 436 nm, in dry air at room temperature and normal pressure, with electrons of average energy of 1.1 MeV. The time allowed to measure this spectrum was short and the spectrometer equipped with an output slit can measure only one wavelength at a time. So it was decided to open the slits at their maximum, setting the 1σ resolution to 3 nm. Thus measurements were made with steps of 3 nm. But even with 2·106 electrons detected every second, only 0.16 fluorescence photons per second are recorded in the signal part of the TDC spectrum for the most intense line (337 nm). This is the reason why this spectrum has only been made for the “low” energy threshold, as defined above. Fluorescence lines are much narrower than the resolution set for the spectrometer as is seen in [14]. The absolute yield is thus given by the height of the curve at a given wavelength, and not by the area integrated over a bandwidth dλ. The sum of the yields of all lines indicated in fig. 10 is 3.9 ± 0.8 photons per meter. This value is in close agreement with the result of the integral measurement. The 20% uncertainty is due most entirely to the bad knowledge of the spectrometer absolute efficiency, especially in the UV. Superimposed on this spectrum is Ulrich’s spectrum [14], convoluted with a 3 nm resolution. As it is a relative yield spectrum, it is shown here normalized to the 337 nm line. Our spectrum is well compatible with what would measure [14] with such a resolution. Figure 10. Spectrum of the absolute fluorescence yield of nitrogen in air between 300 and 436 nm. The bold line is the result of the present experiment, and the dashed line is Ulrich's spectrum [14] as if analyzed by our spectrometer. Discrepancies could be explained by the ageing of our spectrometer. The first observation is the confirmation of the feasibility of this measurement: all the main lines are indeed observed. Evidence is given that an absolute spectrum can be measured with a basic apparatus, and taking care of reducing the PMT background as explained earlier. The sum of the lines yield is consistent with the previous integral measurement. Moreover, its uncertainty is 20 %, when previous experiments do not give better than 15 %. The second observation concerns the discrepancies, around 316 nm and beyond 375 nm. The extraction of the absolute yield involves the efficiency of the spectrometer. Its spectral efficiency curve is provided by the manufacturer with a low accuracy in the UV (which is true also for Ulrich's results). Furthermore two effects induce a very important loss in our spectrometer: • the ageing of each optical element (mirrors and grating). The spectrometer efficiency can be reduced to a value around 15 % after ten years of use ; • the extensive use with intense UV light before this experiment. At 400 nm, the absolute efficiency of the spectrometer was measured with an accuracy of about 2% to be only 15 %, instead of the 61 % given by the manufacturer (who confirms in a private communication such a low value compatible with ageing). The method used is very similar to that taken to determine the absolute efficiency of the photon-PMTs (comparison to a NIST photodiode). This value of 15 % has been used to calculate the absolute yields in the entire spectral range. But there is no reason that this loss is constant with respect to wavelength. The 20% uncertainty on the yield arises from this unknown but limited variation. The question of getting the absolute efficiency of spectrometers in the UV is quite challenging and is the object of specific attentions by the community of atomic / spectral physics. It is unfortunate that this spectrometer method does not yield yet an accuracy better than the "integrated yield" method. It will if the spectrometer resolution can be made high enough to totally separate the bands, which is possible if it is equipped with a CCD readout to minimize the experiment duration. Then, the absolute efficiency of the spectrometer will have to be determined with a high accuracy in the UV, not an easy task according to Ulrich, but possible through our patented method. In the future, two measurements will be done: • calibrate in an absolute way the old Jobin-Yvon; • use the new spectrometer able to measure the full spectrum at once with a 0.1 nm resolution and equipped with a light intensifier to measure the fluorescence yield. 3. Conclusion The absolute fluorescence yield of nitrogen in dry air at atmospheric pressure has been measured. The precision of the measurement is improved by a factor of three, which has an immediate impact on the cosmic ray energies found by HiRes which uses Bunner's [6] yield. Their energies are increased by 22%, hence their spectrum is much closer to AGASA’s (which incidentally have been recently lowered the energy of their points by 10% [33]). The first continuous fluorescence spectrum of nitrogen excited by electrons from a 90Sr source was also measured. Next steps are to introduce impurities in the gas, such as argon, water vapour and pollutants. A pressure study of the total yield will be made. On another hand, the spectral measurement will be improved thanks to a new spectrometer. Papers will follow to account for these measurements, which will provide an overall and realistic view of the fluorescence phenomenon. Acknowledgements We would like to thank Bernard LEFIEVRE for his help with GEANT simulation of the setup, and François LELONG and Jean-Paul RENY for their help in building the bench. REFERENCES [1] M. Takeda et al., Astropart. Phys. 19 (2003) 447 [2] M. Takeda et al., Phys. Rev. Lett. 81 (1998) 1163 [3] R.U. Abbasi et al., Phys. Rev. Lett. 92 (2004) 151101. [4] P. Sommers, C. R. Physique 5 (2004) 463 [5] http://www.euso-mission.org/docs/RedBookEUSO_21apr04.pdf [6] A.N. Bunner, Ph.D. thesis, Cornell University, 1967. [7] G. Davidson, R. O'Neil, J. Chem. Phys. 41 (1964) 3946. [8] F. Kakimoto et al., NIM A 372 (1996) 527. [9] http://www.auger.de/events/air-light-03/ [10] http://lappweb.in2p3.fr/IWFM05/index.html [11] http://www.particle.cz/conferences/floret2006/ [12] F. Arciprete et al., Nucl. Phys. B Proc. Supp. 150 (2006) 186-189 [13] J.W. Belz et al., Astropart. Phys. 25 (2006) 129. [14] A. Ulrich, private communication, and in [9], [10] and [11] [15] T. Waldenmaier, PhD thesis, Forschunzentrum Karlsruhe, 2006 [16] F. Arqueros et al., Astropart. Phys. 26 (2006) 231 [17] F. Blanco and F. Arqueros, Phys. Lett. A 345 (2005) 355 [18] M. Nagano, K. Kobayakawa, N. Sakaki, K. Ando, Astropart. Phys. 20 (2003) 293. [19] http://www.pdas.com/atmos.htm [20] B. Keilhauer et al., Astropart. Phys. 25 (2006) 259. [21] M. 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CONCRETE CLASSIFICATION AND CENTRALIZERS OF CERTAIN Z2 ⋊ SL(2,Z)-ACTIONS HIROKI SAKO Abstract. We introduce a new class of actions of the group Z2 ⋊ SL(2,Z) on finite von Neumann algebras and call them twisted Bernoulli shift actions. We classify these actions up to conjugacy and give an explicit description of their centralizers. We also distinguish many of those actions on the AFD II1 factor in view of outer conjugacy. 1. Introduction We consider the classification of Z2 ⋊ SL(2,Z)-actions on finite von Neumann algebras in this paper. Mainly, we concentrate on the case that the finite von Neumann algebra is the AFD factor of type II1 or non-atomic abelian. There are two difficulties for analyzing discrete group actions on operator algebras. The first is that we do not have various ways to construct actions. The second is that we can not analyze them by concrete calculation in most cases. To give many examples of actions which admit concrete analysis, we introduce a class of trace preserving Z2 ⋊ SL(2,Z)-actions on finite von Neumann algebras and call them twisted Bernoulli shift actions. We classify those actions up to conjugacy and study them up to outer conjugacy. An action β(H, µ, χ) in the class is defined for a triplet (H, µ, χ), where H is an abelian countable discrete group, µ is a normalized scalar 2-cocycle of H and χ is a character of H . We obtain the action by restricting the so-called generalized Bernoulli shift action to a subalgebra N(H, µ) and “twisting” it by the character χ. The process of restriction has a vital role in concrete analysis of these actions. A ∗-isomorphism which gives conjugacy between two twisted Bernoulli shift ac- tions β(Ha, µa, χa) and β(Hb, µb, χb) must be induced from an isomorphism between the two abelian groups Ha and Hb. We prove this by concrete calculation (Section 4). It turns out that there exist continuously many, non-conjugate Z2 ⋊ SL(2,Z)- actions on the AFD factor of type II1 (Section 5). By using the same technique, we describe the centralizers of all twisted Bernoulli shift actions. Here we should men- tion that the present work was motivated by the previous ones [Ch], [NPS], where similar studies were carried out in the case of SL(n,Z). In Section 6, we distinguish many twisted Bernoulli shift actions in view of outer conjugacy. The classification for actions of discrete amenable groups on the AFD factor of type II1 was given by Ocneanu [Oc]. Outer actions of countable amenable groups are outer conjugate. In the contrast to this, V. F. R. Jones [Jon] proved that any discrete non-amenable group has at least two non outer conjugate actions on 2000 Mathematics Subject Classification. Primary 46L40; Secondary 46L10. Key words and phrases. von Neumann algebras; automorphisms. http://arxiv.org/abs/0704.1533v3 2 HIROKI SAKO the AFD factor of type II1. S. Popa ([Po3], [Po4], [PoSa], etc.) used the malleabil- ity/deformation arguments for the Bernoulli shift actions to study (weak) 1-cocycles for the actions. For some of twisted Bernoulli shift actions, which we introduce in this paper, it is shown that (weak) 1-cocycles are represented in simple forms under some assumption on the (weak) 1-cocycles. We prove that there exist continuously many twisted Bernoulli shift actions which are mutually non outer conjugate. This strengthens the above mentioned result due to Jones in the Z2 ⋊ SL(2,Z) cases. 2. Preparations 2.1. Functions det and gcd. For the definition of twisted Bernoulli shift actions in Section 3, we define two Z-valued functions det and gcd. The function det is given by the following equation: = qr0 − rq0, ∈ Z2. The value of the function gcd at k ∈ Z2 is the greatest common divisor of the two entries. For 0 ∈ Z2, let the value of gcd be 0. Lemma 2.1. (1) The action of SL(2,Z) on Z2 preserves the functions det and gcd, that is, det(k, k0) = det(γ · k, γ · k0), gcd(k) = gcd(γ · k), k, k0 ∈ Z2, γ ∈ SL(2,Z). (2) The following equation holds true: det(k, k0) = gcd(k) + gcd(k0)− gcd(k + k0) mod 2, k, k0 ∈ Z2. Proof. The claim (1) is a well-known fact, so we prove the claim (2). For the function gcd, we get 1 mod 2, (either q or r is odd), 0 mod 2, (both q and r are even). Since the action of SL(2,Z) on Z2 preserves the functions det and gcd, it suffices to show the desired equation against the following four pairs: (k, k0) = mod 2. 2.2. Scalar 2-cocycles for abelian groups. We fix some notations for countable abelian groups and their scalar 2-cocycles. For the rest of this paper, let H be an abelian countable discrete group and suppose that any scalar 2-cocycle µ : H×H → T = {z ∈ C | |z| = 1} is normalized, that is, µ(g, 0) = 1 = µ(0, g) for g ∈ H . We denote by µ∗ the 2-cocycle for H given by µ∗(g, h) = µ(h, g), g, h ∈ H . Let µ∗µ be the function on H ×H defined by µ∗µ(g, h) = µ(h, g)µ(g, h), g, h ∈ H. CLASSIFICATION AND CENTRALIZERS 3 This is a bi-character, that is, µ∗µ(g, ·) and µ∗µ(·, h) are characters of H . By using this function, we can describe the cohomology class of µ. See [OPT] for the proof of the following Proposition: Proposition 2.2. Two scalar 2-cocycles µ1 and µ2 of H are cohomologous if and only if µ∗1µ1 = µ Let Cµ(H) be the twisted group algebra of H with respect to the 2-cocycle µ. We denote by {uh | h ∈ H} the standard basis for Cµ(H) as C-linear space. We recall that the C-algebra Cµ(H) has a structure of ∗-algebra defined by ug uh = µ(g, h) ug+h, u g = µ(g,−g)u−g, g, h ∈ H. Let µ̃ be the T-valued function on ⊕Z2H ×⊕Z2H defined by µ̃(λ1, λ2) = µ(λ1(k), λ2(k)), λ1, λ2 ∈ ⊕Z2H.(Eq1) The function µ̃ is a normalized scalar 2-cocycle for ⊕Z2H . Let Λ(H) be the abelian group defined by Λ(H) = λ : Z2 → H ∣∣∣∣∣ finitely supported and λ(k) = 0 Its additive rule is defined by pointwise addition. 2.3. Definition of a Z2 ⋊ SL(2,Z)-action on Λ(H). The group SL(2,Z) acts on Z2 as matrix-multiplication and the group Z2 also does on Z2 by addition. These two actions define the action of Z2 ⋊ SL(2,Z) on Z2 which is explicitly described as q + xq0 + yr0 r + zq0 + wr0 for all (( ∈ Z2 ⋊ SL(2,Z), ∈ Z2. We define an action of Z2 ⋊ SL(2,Z) on ⊕Z2H as (γ · λ)(k) = λ(γ−1 · k), k ∈ Z2, for γ ∈ Z2 ⋊ SL(2,Z) and λ ∈ ⊕Z2H . 2.4. On the relative property (T) of Kazhdan. We give the definition of the relative property (T) of Kazhdan for a pair of discrete groups. Definition 2.3. Let G ⊂ Γ be an inclusion of discrete groups. We say that the pair (Γ, G) has the relative property (T) if the following condition holds: There exist a finite subset F of Γ and δ > 0 such that if π : Γ → U(H) is a unitary representation of Γ on a Hilbert space H with a unit vector ξ ∈ H satisfying ‖π(g)ξ − ξ‖ < δ for g ∈ F , then there exists a non-zero vector η ∈ H such that π(h)η = η for h ∈ G. Instead of this original definition, we use the following condition. 4 HIROKI SAKO Proposition 2.4. ([Jol]) Let G ⊂ Γ be an inclusion of discrete groups. The pair (Γ, G) has the relative property (T) if and only if the following condition holds: For any ǫ > 0, there exist a finite subset F of Γ and δ > 0 such that if π : Γ → U(H) is a unitary representation of Γ on a Hilbert space H with a unit vector ξ ∈ H satisfying ‖π(g)ξ − ξ‖ < δ for g ∈ F , then ‖π(h)ξ − ξ‖ < ǫ for h ∈ G. The pair (Z2⋊SL(2,Z),Z2) is a typical example of group with the relative property (T). See [Bu] or [Sh] for the proof. 2.5. Weakly mixing actions. An action of a countable discrete group G on a von Neumann algebra N is said to be ergodic if any G-invariant element of N is a scalar multiple of 1. The weak mixing property is a stronger notion of ergodicity. Definition 2.5. Let N be a von Neumann algebra with a faithful normal state φ. A state preserving action (ρg)g∈G of a countable discrete group G on N is said to be weakly mixing if for every finite subset {a1, a2, . . . , an} ⊂ N and ǫ > 0, there exists g ∈ G such that |φ(aiρg(aj))− φ(ai)φ(aj)| < ǫ, i, j = 1, . . . , n. The following is a basic characterization of the weak mixing property. Between two von Neumann algebra N and M , N ⊗ M stands for the tensor product von Neumann algebra. Proposition 2.6 (Proposition D.2 in [Vaes]). Let a countable discrete group G act on a finite von Neumann algebra (N, tr) by trace preserving automorphisms (ρg)g∈G. The following statements are equivalent: (1) The action (ρg) is weakly mixing. (2) The only finite-dimensional invariant subspace of N is C1. (3) For any action (αg) of G on a finite von Neumann algebra (M, τ), we have (N ⊗ M)ρ⊗α = 1 ⊗ Mα, where (N ⊗ M)ρ⊗α and Mα are the fixed point subalgebras. 2.6. A remark on group von Neumann algebras. Let Γ be a discrete group and let µ be a scalar 2-cocycle of a countable group Γ. A group Γ acts on the Hilbert space ℓ2Γ by the following two ways; uγ(δg) = µ(γ, g)δγg, ργ(δg) = µ(g, γ −1)δgγ−1 , γ, g ∈ Γ. These two representations commute with each other. The von Neumann algebra Lµ(Γ) generated by the image of u is called the group von Neumann algebra of Γ twisted by µ. The normal state 〈·δe, δe〉 is a trace on Lµ(Γ). The vector δe is separating for Lµ(Γ). For any element a ∈ Lµ(Γ), we define the square summable function a(·) on Γ by aδe = a(g)δg. The function a(·) is called the Fourier coefficient of a. We write a = g∈Γ a(g)ug and call this the Fourier expansion of a. The Fourier expansion of a∗ is given by a∗ = g∈Γ µ(g, g −1)a(g−1)ug, since the Fourier coefficient a∗(g) = 〈a∗δe, δg〉 is described as 〈δe, aρg−1δe〉 = ρ∗g−1δe, a(g)δg = µ(g, g−1)a(g−1). CLASSIFICATION AND CENTRALIZERS 5 Here we used the equation ρ∗ = µ(g, g−1)ρg, which is verified by direct computa- tion. For two elements a, b, the Fourier coefficient of ab is given by ab(γ) = 〈bδe, ργ−1a∗δe〉 = a∗(g)µ(g, γ)b(gγ) = µ(g−1, gγ)a(g−1)b(gγ). This equation allows us to calculate the Fourier coefficient algebraically, that is, µ(g−1, gγ)a(g−1)b(gγ) a(g)b(h) For a subgroup Λ ⊂ Γ, the subalgebra {uλ | λ ∈ Λ}′′ ⊂ Lµ(Γ) is isomorphic to Lµ(Λ). We sometimes identify them. An element a ∈ Lµ(Γ) is in the subalgebra Lµ(Λ) if and only if the Fourier expansion a(·) : Γ → C is supported on Λ, since the trace 〈·δe, δe〉 preserving conditional expectation E from Lµ(Γ) onto Lµ(Λ) is described as E(a) = λ∈Λ a(λ)uλ. 3. Definition of twisted Bernoulli shift actions In this section, we introduce twisted Bernoulli shift actions of Z2 ⋊ SL(2,Z) on finite von Neumann algebras. The action is defined for a triplet i = (H, µ, χ), where H 6= {0} is an abelian countable discrete group, µ is a normalized scalar 2-cocycle of H and χ is a character of H . The finite von Neumann algebra, on which the group Z2 ⋊ SL(2,Z) acts, is defined by the pair (H, µ). We introduce a group structure on the set Γ0 = Ĥ × Z2 × SL(2,Z) as (c1, k, γ1)(c2, l, γ2) = c1c2χ det(k,γ1·l), k + γ1 · l, γ1γ2 for any c1, c2 ∈ Ĥ, k, l ∈ Z2, γ1, γ2 ∈ SL(2,Z). The associativity is verified by Lemma 2.1. It turns out that the subsets Ĥ = Ĥ × {0} × {e} and G0 = Ĥ × Z2 × {e} are subgroups in Γ0. It is easy to see that G0 is a normal subgroup of Γ0 and that Ĥ is a normal subgroup of G0 and Γ0. We get a normal inclusion of groups G0/Ĥ ⊂ Γ0/Ĥ and this is isomorphic to Z2 ⊂ Z2 ⋊ SL(2,Z). Before stating the definition of the twisted Bernoulli shift action, we define a Γ0- action ρ on the von Neumann algebra Lµ̃(⊕Z2H). We denote by u(λ) ∈ Lµ̃(⊕Z2H) the unitary corresponding to λ ∈ ⊕Z2H . We define a faithful normal trace tr of Lµ̃(⊕Z2H) in the usual way. For c ∈ Ĥ, k ∈ Z2, γ ∈ SL(2,Z), let ρ(c), ρ(k), ρ(γ) be the linear transformations on Cµ̃(⊕Z2H) given by, ρ(c)(u(λ)) = c(λ(l)) u(λ), ρ(k)(u(λ)) = χ(λ(m))det(k,m) u(k · λ), ρ(γ)(u(λ)) = u(γ · λ), λ ∈ Λ(H), These maps are compatible with the multiplication rule and the ∗-operation of Cµ̃(⊕Z2H) . Since these maps preserve the trace, they extend to ∗-automorphisms 6 HIROKI SAKO on Lµ̃(⊕Z2H). It is immediate to see that ρ(c) commutes with ρ(k) and ρ(γ). For k, l ∈ Z2, we have the following relation: ρ(k) ◦ ρ(l)(u(λ)) = χ(λ(m))det(l,m)ρ(k)(u(l · λ)) χ(λ(m))det(l,m) χ((l · λ)(m))det(k,m)u(k · (l · λ)) χ(λ(m))det(l,m)χ(λ(m))det(k,m+l)u((k + l) · λ). By det(l, m) + det(k,m+ l) = det(k, l) + det(k + l, m), this equals to χ(λ(m)) )det(k,l) ∏ χ(λ(m))det(k+l,m)u((k + l) · λ) = ρ(χdet(k,l)) ◦ ρ(k + l)(u(λ)). Since det is SL(2,Z)-invariant (Lemma 2.1), for k ∈ Z2, γ ∈ SL(2,Z), we get ρ(γ · k) ◦ ρ(γ)(u(λ)) = ρ(γ · k)(u(γ · λ)) χ((γ · λ)(l))det(γ·k,l)u((γ · k) · (γ · λ)) χ(λ(l))det(γ·k,γ·l)u(γ · (k · λ)) χ(λ(l))det(k,l)ρ(γ)(u(k · λ)) = ρ(γ) ◦ ρ(k)(u(λ)), λ ∈ Λ(H). By using the above two equations, ρ satisfies the following formula: (ρ(c1) ◦ ρ(k) ◦ ρ(γ1)) ◦ (ρ(c2) ◦ ρ(l) ◦ ρ(γ2)) = ρ(c1) ◦ ρ(c2) ◦ ρ(k) ◦ ρ(γ1) ◦ ρ(l) ◦ ρ(γ2) = ρ(c1) ◦ ρ(c2) ◦ ρ(k) ◦ ρ(γ1 · l) ◦ ρ(γ1) ◦ ρ(γ2) = ρ(c1) ◦ ρ(c2) ◦ ρ(χdet(k,γ1·l)) ◦ ρ(k + (γ1 · l)) ◦ ρ(γ1γ2) = ρ(c1c2χ det(k,γ1·l)) ◦ ρ(k + (γ1 · l)) ◦ ρ(γ1γ2). With ρ(c, k, γ) = ρ(c) ◦ ρ(k) ◦ ρ(γ), ρ gives a Γ0-action on Lµ̃(⊕Z2H). We define the finite von Neumann algebra N(H, µ) as the group von Neumann algebra Lµ̃(Λ(H)). By using Fourier coefficients, we can prove that N(H, µ) is the fixed point algebra under the Ĥ-action ρ(Ĥ, 0, e) on Lµ̃(⊕Z2H). We get a Z2 ⋊ SL(2,Z)-action on N(H, µ) by β(k, γ)(x) = ρ(1, k, γ)(x), k ∈ Z2, γ ∈ SL(2,Z), x ∈ N(H, µ). This is the definition of the twisted Bernoulli shift action β = β(H, µ, χ) on N(H, µ). We obtained the actions β(H, µ, χ) not only by twisting generalized Bernoulli shift actions but also restricting to subalgebras N(H, µ) ⊂ Lµ̂(⊕Z2H) = Lµ(H). This restriction allows us to classify the actions up to conjugacy in the next section. CLASSIFICATION AND CENTRALIZERS 7 In order to give a variety of the actions, we twisted the shift actions by the character χ of the abelian group H . Remark 3.1. The action β|Z2 = β(H, µ, χ)|Z2 has the weak mixing property. In definition 2.5, we may assume that the Fourier coefficients of ai (i = 1, 2, · · · , n) are finitely supported, by approximating in the L2-norm. Then for appropriate k ∈ Z2, we get tr(aiβ(k)(aj)) = tr(ai)tr(aj), i, j = 1, 2, · · · , n. 4. Classification up to conjugacy In this section, we classify the twisted Bernoulli shift actions {β(H, µ, χ)} up to conjugacy (Theorem 4.1). We prove that an isomorphism which gives conjugacy between two twisted Bernoulli shift actions is of a very special form. In fact it comes from an isomorphism in the level of base groups H . We also determine the centralizer of the Z2 ⋊ SL(2,Z)-action β(H, µ, χ) on N(H, µ) (Theorem 4.4). We fix some notations for the proofs. We define 0, e1, e2 ∈ Z2 as , e1 = , e2 = Let ξ be the element of Z2 ⋊ SL(2,Z) satisfying ξ · 0 = e1, ξ · e1 = e2, ξ · e2 = 0. The elements ξ and ξ2 are explicitly described as −1 −1 −1 −1 The order of ξ is 3. Let η, δ ∈ SL(2,Z) be given by η = , δ = Let D be the subset of all elements of Z2 fixed under the action of δ, that is, n ∈ Z Then we get ξ ·D = n ∈ Z , ξ2 ·D = n ∈ Z We define the subgroup ΛD(H) of Λ(H) by ΛD(H) = {λ ∈ Λ(H) | λ : Z2 → H is supported on D}. Let (Ha, µa, χa) and (Hb, µb, χb) be triplets of countable abelian groups, their normalized 2-cocycles and characters. For h ∈ Ha, we define λh ∈ ΛD(Ha) as λh(k) = h (k = e1), −h (k = 0), 0 (k 6= e1, 0). 8 HIROKI SAKO For g ∈ Hb, we define σg ∈ ΛD(Hb) as σg(k) = g (k = e1), −g (k = 0), 0 (k 6= e1, 0). We denote by v(σ) ∈ N(Hb, µb) the unitary corresponding to σ ∈ Λ(Hb). Theorem 4.1. If π : N(Ha, µa) → N(Hb, µb) is a ∗-isomorphism giving conjugacy between βa = β(Ha, µa, χa) and βb = β(Hb, µb, χb), then there exists a group isomor- phism φ = φπ : Ha → Hb satisfying (1) π(u(λ)) = v(φ ◦ λ) mod T for λ ∈ Λ(Ha), (2) the 2-cocycles µa(·, ·) and µb(φ(·), φ(·)) of Ha are cohomologous, (3) χ2a = (χb ◦ φ)2. Conversely, given a group isomorphism φ : Ha → Hb satisfying (2) and (3), there exists a ∗-isomorphism π = πφ : N(Ha, µa) → N(Hb, µb) which satisfies condition (1) and gives conjugacy between βa, βb. We note that by Proposition 2.2 condition (2) for φ is equivalent to (2)′ µ∗aµa(g, h) = µ bµb(φ(g), φ(h)), g, h ∈ Ha. Proof for the first half of Theorem 4.1. Suppose that there exists a (not necessarily trace preserving) ∗-isomorphism π from N(Ha, µa) onto N(Hb, µb) such that π ◦ βa(γ) = βb(γ) ◦ π, γ ∈ Z2 ⋊ SL(2,Z). We prove that for every h ∈ Ha there exists φ(h) ∈ Hb satisfying π(u(λh)) = v(σφ(h)) mod T. Let Uh denote the unitary in N(Hb, µb) Uh = π µa(h,−h) u(λh) , h ∈ Ha. We identify N(H, µ) with the subalgebra of the infinite tensor product Lµ(H), which is canonically isomorphic to Lµ̃ (⊕Z2H). The preimage π−1(Uh) can be written as u∗h ⊗ uh. Here uh is the unitary corresponding to h ∈ Ha and placed on 1 ∈ Z2 and the unitary u∗h is placed on 0 ∈ Z2. We describe Uh as the Fourier expansion σ∈Λ(Hb) c(σ)v(σ). Since e1 and 0 are fixed under the action of δ, one has βb(δ) n(Uh) = π ◦ βa(δ)n(π−1(Uh)) = Uh. It follows that the Fourier expansion Uh = σ∈Λ(Hb) c(σ)v(σ) must satisfy that c(σ) = c(δ−n · σ) for every σ ∈ Λ(Hb) and n ∈ Z. For σ ∈ Λ(Hb) \ ΛD(Hb), the orbit of σ under the action of δ−1 is an infinite set, since the support supp(σ) ⊂ Z2 is not included in D. It turns out that c(σ) = 0 for all σ ∈ Λ(Hb) \ ΛD(Hb) due to Σ|c(σ)|2 = 1 < +∞, so that Uh = σ∈ΛD(Hb) c(σ)v(σ). The unitary χa(h)U h is also fixed under the action of δ and can be written as χa(h)U h = π χa(h)µa(h,−h)u(−λh) χa(h)µa(h,−h)u(ξ · λh) µa(h,−h)u(ξ2 · λh) = βb(ξ)(Uh) βb(ξ 2)(Uh). CLASSIFICATION AND CENTRALIZERS 9 Letting ne1 = (n, 0) T ∈ Z2, we get βb(ξ)(Uh) = βb(ξ) c(σ) v(σ) σ∈ΛD(Hb) c(σ) v(ξ · σ) χb(σ(ne1)) 2)(Uh) = βb(ξ c(σ) v(σ) σ∈ΛD(Hb) c(σ) v(ξ2 · σ). Since Fourier expansion admits algebraical calculation as in subsection 2.6, the ex- pansion of χa(h)U χa(h)U h = βb(ξ)(Uh) βb(ξ 2)(Uh) σ1,σ2∈ΛD(Hb) c(σ1) c(σ2) v(ξ · σ1) v(ξ2 · σ2) χb(σ1(ne1)) σ1,σ2∈ΛD(Hb) c(σ1) c(σ2) µ̃b(ξ · σ1, ξ2 · σ2) χb(σ1(ne1)) n v(ξ · σ1 + ξ2 · σ2). The map ΛD(Hb)×ΛD(Hb) ∋ (σ1, σ2) 7→ ξ ·σ1+ξ2·σ2 ∈ Λ(Hb) is injective. Indeed, σ1 is uniquely determined by ξ ·σ1+ξ2·σ2, since σ1(k) = (ξ ·σ1+ξ2·σ2)(ξ ·k), k ∈ D\{e1} and σ1(e1) = − k∈D\{e1} σ1(k). Here we used the condition σ1(k) = 0. The element σ2 is also determined by ξ · σ1 + ξ2 · σ2. Thus the index (σ1, σ2) uniquely determines ξ · σ1 + ξ2 · σ2. We take arbitrary elements σ1, σ2 ∈ ΛD(Hb) and suppose that c(σ1) 6= 0, c(σ2) 6= 0. Since the unitary χa(h)U h is invariant under the action of δ and the coefficient of ξ ·σ1+ξ2 ·σ2 is not zero, ξ ·σ1+ξ2 ·σ2 is supported on D. It follows that the elements σ1 and σ2 can be written as σ1 = σφ(h) = σ2, by some φ(h) ∈ Hb. Indeed, since the subsets D \ {0, e1}, ξD \ {e1, e2} and ξ2D \ {e2, 0} are mutually disjoint, the element ξ · σ1 must be supported on {e1, e2} and the element ξ2 · σ2 must be supported on {e2, 0}. By the assumption k∈Z2 σi(k) = 0 (i = 1, 2), σi can be written as σφ(hi). Then using the fact that (ξ · σ1 + ξ2 · σ2)(e2) = σ1(e1) + σ2(0) = 0, we get that σ1 = σφ(h) = σ2 for some h ∈ Hb. This means that there exists only one σ ∈ Λ(Hb) such that c(σ) 6= 0 and that it is of the form σ = σφ(h). Then the unitary Uh satisfies Uh = π(u(λh)) = v(σφ(h)) mod T. We claim that the map φ = φπ : Ha → Hb is a group isomorphism. For all h1, h2 ∈ Ha, we get π(u(λh1+h2)) = π(u(λh1)) π(u(λh2)) = v(σφ(h1)) v(σφ(h2)) = v(σφ(h1) + σφ(h2)) = v(σφ(h1)+φ(h2)) mod T. On the other hand, we get π(u(λh1+h2)) = v(σφ(h1+h2)) mod T. Since {v(σ)} are linearly independent, we get σφ(h1+h2) = σφ(h1)+φ(h2), and hence φ(h1 + h2) = φ(h1) + φ(h2). This means that the map φ is a group homomorphism. The bijectivity of the ∗- isomorphism π leads to that of the group homomorphism φ = φπ. Since {γ ·λh | γ ∈ Z2⋊SL(2,Z), h ∈ Ha} ⊂ Λ(Ha) generates Λ(Ha), we get π(u(λ)) = v(φ◦λ) mod T for λ ∈ Λ(Ha). 10 HIROKI SAKO We prove that the group isomorphism φ = φπ satisfies conditions (2) and (3) in the theorem. For all h ∈ Ha, there exists c(h) ∈ T satisfying Uh = π µa(h,−h) u(λh) = c(h)µb(φ(h),−φ(h)) v(σφ(h)). Since (e1, η) ∈ Z2 ⋊ SL(2,Z) acts on Z2 as (e1, η) · e1 = 0, (e1, η) · 0 = e1, we get Uh βb(e1, η)(Uh) = π µa(h,−h) u(λh)µa(h,−h) u(−λh) = µa(h,−h) µ̃a(λh,−λh) = 1. The following equation also holds: Uh βb(e1, η)(Uh) = c(h)µb(φ(h),−φ(h)) v(σφ(h)) c(h)µb(φ(h),−φ(h)) v(σ−φ(h)) = c(h)2 µb(φ(h),−φ(h)) µ̃b(σφ(h),−σφ(h)) = c(h)2. Thus we have c(h) ∈ {1,−1} for h ∈ Ha. Since ξ · e1 = e2, ξ · e2 = 0 and ξ · 0 = e1, we have Uh βb(ξ)(Uh) βb(ξ 2)(Uh) µa(h,−h) u(λh) χa(h)µa(h,−h) u(ξ · λh) µa(h,−h) u(ξ2 · λh) = χa(h). On the other hand, we have the following: Uh βb(ξ)(Uh) βb(ξ 2)(Uh) = c(h)µb(φ(h),−φ(h)) v(σφ(h)) c(h)χb(φ(h))µb(φ(h),−φ(h)) v(ξ · σφ(h)) c(h)µb(φ(h),−φ(h)) v(ξ2 · σφ(h)) = c(h)3 χb(φ(h)) = c(h)χb(φ(h)). It follows that c(h) = χb(φ(h))χa(h)(Eq2) and χb(φ(h)) 2 = χa(h) 2, for all h ∈ Ha. We recall that the algebra Lµ̃a(⊕Z2Ha) is canonically identified wit the infinite tensor product Lµa(Ha). The unitary π −1(Uh) ∈ N(Ha, µa) ⊂ Lµa(Ha) can be written as 1⊗ u∗h ⊗ uh, where 1 is placed on −e1 ∈ Z2, u∗h is placed on 0 and uh is placed on e1. Since η ∈ Z2 ⋊ SL(2,Z) acts on Z2 as η · e1 = −e1, η · 0 = 0, the unitary π−1(βb(η)(Ug)) can be written as ug ⊗ u∗g ⊗ 1. We have the following equation: Ug βb(η)(Uh)U g βb(η)(Uh) = π((1⊗ u∗g ⊗ ug)(uh ⊗ u∗h ⊗ 1)(1⊗ u∗g ⊗ ug)∗(uh ⊗ u∗h ⊗ 1)∗) = µ∗aµa(g, h). The unitary Uh can be written as c(h)(1⊗ v∗φ(h) ⊗ vφ(h)) ∈ N(Hb, µb) ⊂ Lµb(Hb). Here we write vφ(h) for the unitary in Lµb(Hb) corresponding to φ(h). The unitary βb(η)(Ug) can be written as c(g)(vφ(g) ⊗ v∗φ(g) ⊗ 1). Then we get Ug βb(η)(Uh)U g βb(η)(U = (1⊗ v∗φ(g) ⊗ vφ(g))(vφ(h) ⊗ v∗φ(h) ⊗ 1)(1⊗ v∗φ(g) ⊗ vφ(g))∗(vφ(h) ⊗ v∗φ(h) ⊗ 1)∗ = µ∗bµb(φ(g), φ(h)). CLASSIFICATION AND CENTRALIZERS 11 Thus we get µ∗aµa(g, h) = µ bµb(φ(g), φ(h)), for all g, h ∈ Ha. We proved that the group isomorphism φ = φπ satisfies conditions (1), (2) and (3). � From a group homomorphism which satisfies conditions (2) and (3), we construct a ∗-homomorphism from N(Ha, µa) to N(Hb, µb) with condition (1). In the con- struction, the function µ̂ on Λ(H) given below is useful. We fix an index for Z2 as Z2 = {k0, k1, k2, · · · } throughout the rest of this section. For a scalar 2-cocycle µ of H , we define the function µ̂ by µ̂(λ) = λ(ki), λ(kj) , λ ∈ Λ(H), where λ is supported on {k0, k1, k2, · · · , kn}. This definition depends on the choice of an order on Z2. Since i λ(ki) = 0, the function µ̂ is also given by the following relation in Cµ(H): µ̂(λ)1 = uλ(k0)uλ(k1)uλ(k2) · · ·uλ(kn), λ ∈ Λ(H). If µ is a coboundary, then the definition of µ̂ does not depend on the order on Z2, since Cµ(H) is commutative. Lemma 4.2. Let µ0 be another normalized scalar 2-cocycle for H. Let µ̃0 be the scalar 2-cocycle on Λ(H)×Λ(H) given in the same way as equation (Eq1) in subsec- tion 2.2 and let µ̂0 be the function on Λ(H) constructed from µ0 in the above manner. If the scalar 2-cocycles µ and µ0 are cohomologous, then for all λ1, λ2 ∈ Λ(H), we have the equation µ̃(λ1, λ2) µ̂(λ1) µ̂(λ2) µ̂(λ1 + λ2) = µ̃0(λ1, λ2) µ̂0(λ1) µ̂0(λ2) µ̂0(λ1 + λ2). Proof. We denote by {ν(g, h)} the scalar 2-cocycle {µ0(g, h)µ(g, h)} of H . Since ν is a 2-coboundary, there exists {c(g)}g∈H ⊂ T satisfying ν(g, h) = b(g)b(h)b(g + h). Then the map ν̂ becomes ν̂(λ) = b(λ(ki)). Since ν̂(λ1) ν̂(λ2) = b(λ1(ki)) b(λ2(ki)), ν̃(λ1, λ2) = b(λ1(ki)) b(λ2(ki)) b(λ1(ki) + λ2(ki)), ν̂(λ1 + λ2) = b(λ1(ki) + λ2(ki)), we get ν̂(λ1) ν̂(λ2) = ν̃(λ1, λ2) ν̂(λ1 + λ2). By the definitions of µ̃, µ̃0, µ̂ and µ̂0, the maps ν̂ and ν̃ are given by ν̂(λ) = µ̂(λ) µ̂0(λ), ν̃(λ1, λ2) = µ̃(λ1, λ2) µ̃0(λ1, λ2), Thus the desired equality immediately follows. � Proof for the second half of Theorem 4.1. Suppose that there exists a group isomorphism φ satisfying conditions (2) and (3) in the theorem. We prove that there exists a ∗-isomorphism π = πφ from N(Ha, µa) onto N(Hb, µb) preserving the Z 2 ⋊ SL(2,Z)-actions with condition (1). We define a group homomorphism cφ from Ha to {1,−1} ⊂ T by cφ(h) = χb(φ(h))χa(h), h ∈ Ha. 12 HIROKI SAKO Let c̃φ be the group homomorphism from Λ(Ha) to {1,−1} ⊂ T given by c̃φ(λ) = cφ(λ(k)) gcd(k) = χa(λ(k)) gcd(k)χb(φ(λ(k))) gcd(k) , λ ∈ Λ(Ha). We define a linear map π from the group algebra Cµ̃a(Λ(Ha)) onto Cµ̃b(Λ(Hb)) by µ̂a(λ) u(λ) = c̃φ(λ) µ̂b(φ ◦ λ) v(φ ◦ λ), λ ∈ Λ(Ha). By direct computations, for all λ1, λ2 ∈ Λ(Ha), we get µ̂a(λ1) u(λ1) µ̂a(λ2)u(λ2) = c̃φ(λ1) c̃φ(λ2) µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) v(φ ◦ λ1) v(φ ◦ λ2) = c̃φ(λ1 + λ2) µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) µ̃b(φ ◦ λ1, φ ◦ λ2) v(φ ◦ (λ1 + λ2)). On the other hand, we have the following equation: µ̂a(λ1) µ̂a(λ2)u(λ1) u(λ2) µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) u(λ1 + λ2) = µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) π µ̂a(λ1 + λ2) u(λ1 + λ2) = c̃φ(λ1 + λ2) µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) µ̂b(φ ◦ (λ1 + λ2)) v(φ ◦ (λ1 + λ2)). By Lemma 4.2 and condition (2) for the group isomorphism φ in the theorem, we have that µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) µ̃b(φ ◦ λ1, φ ◦ λ2) = µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) µ̂b(φ ◦ (λ1) + φ ◦ (λ2)). Therefore we get π(u(λ1)) π(u(λ2)) = π(u(λ1) u(λ2)). The linear map π also pre- serves the ∗-operation. As a consequence, π is a ∗-isomorphism from Cµ̃a(Λ(Ha)) onto Cµ̃b(Λ(Hb)) and this preserves the trace. The map π = πφ is extended to a normal ∗-isomorphism from N(Ha, µa) onto N(Hb, µb). We next prove that this π preserves the Z2 ⋊ SL(2,Z)-actions. The group ho- momorphism c̃φ from Λ(Ha) to {1,−1} is invariant under the action of SL(2,Z), by Lemma 2.1 (1). The scalar 2-cocycle ν(g, h) = µa(g, h)µb(φ(h), φ(g)) satisfies ν(g, h) = ν(h, g) by condition (2), so the function ν̂(·) = µ̂a(·)µ̂b(φ ◦ ·) on Λ(Ha) does not depend on the order on Z2 chosen before. Since π ◦ βa(γ)(u(λ)) = π(u(γ · λ)) = c̃φ(γ · λ)µ̂a(γ · λ)µ̂b(φ ◦ (γ · λ))v(φ ◦ (γ · λ)) = c̃φ(λ)µ̂a(λ)µ̂b(φ ◦ λ)v(γ · (φ ◦ λ)) = βb(γ) c̃φ(λ)µ̂a(λ)µ̂b(φ ◦ λ)v(φ ◦ λ) = βb(γ) ◦ π(u(λ)), γ ∈ SL(2,Z), λ ∈ Λ(Ha), it turns out that the ∗-isomorphism π preserves the SL(2,Z)-action. CLASSIFICATION AND CENTRALIZERS 13 For all λ ∈ Λ(Ha) and k ∈ Z2, we have π ◦ βa(k)(u(λ)) = π χa(λ(l)) det(k,l)u(k · λ) cφ((k · λ)(l))gcd(l) χa(λ(l)) det(k,l)µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(φ ◦ (k · λ)). Since cφ(h) det(k,l)cφ(h) gcd(k+l) = cφ(h) gcd(k)cφ(h) gcd(l), by Lemma 2.1 (2), the unitary π ◦ βa(k)(u(λ)) equals to cφ(λ(l)) gcd(k+l) cφ(λ(l)) det(k,l)χb(φ ◦ λ(l))det(k,l) µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(φ ◦ (k · λ)) cφ(λ(l)) gcd(k) cφ(λ(l)) gcd(l) χb(φ ◦ λ(l))det(k,l) µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(k · (φ ◦ λ)) = c̃φ(λ) µ̂a(k · λ) µ̂b(k · (φ ◦ λ))βb(k)(v(φ · λ)) = βb(k) ◦ π(u(λ)). This means that the ∗-isomorphism π preserves the Z2-actions. We get the ∗-isomorphism π = πφ fromN(Ha, µa) ontoN(Hb, µb) giving conjugacy between βa and βb. � Remark 4.3. The proof of the first half of Theorem 4.1 shows that any isomorphism π giving conjugacy between βa and βb is of the form πφ. This means that an isomorphism which gives conjugacy between two twisted Bernoulli shift actions must be trace preserving. This proof shows that an isomorphism giving conjugacy between the two actions β(Ha, µa, χa), β(Hb, µb, χb) is of a very special form derived from a group isomor- phism between Ha and Hb. Taking notice of this fact, we can describe the centralizer of a twisted Bernoulli shift action. We define two topological groups before we state Theorem 4.4. Let β be a trace preserving action of some group Γ on a separable finite von Neumann algebra (N, tr). We denote by Aut(N, β) the group of all automorphisms which commute with the action β, that is, {α ∈ Aut(N) | β(γ) ◦ α = α ◦ β(γ), γ ∈ Γ}. We regard the group Aut(N, β) as a topological group equipped with the pointwise- strong topology. When β is a twisted Bernoulli shift action on N , an automorphism α commuting with β is necessarily trace preserving by Remark 4.3. We consider that Aut(N, β) is equipped with the pointwise-2-norm topology. Let Aut(H, µ, χ) be the group of all automorphisms of an abelian group H which preserve its 2-cocycle µ and character χ, that is, {φ ∈ Aut(H) | µ(g, h) = µ(φ(g), φ(h)), χ(g) = χ(φ(g)), g, h ∈ H}. We define the topology of Aut(H, µ, χ) by pointwise convergence. 14 HIROKI SAKO Theorem 4.4. For π ∈ Aut(N(H, µ), β(H, µ, χ)), there exists a unique element φ = φπ ∈ Aut(H, µ∗µ, χ2) satisfying π(u(λ)) = u(φ ◦ λ) mod T for λ ∈ Λ(H). The map π 7→ φπ gives an isomorphism between two topological groups Aut(N(H, µ), β(H, µ, χ)) ∼= Aut(H, µ∗µ, χ2). Proof. We use the notations in the proof of the previous theorem letting Ha = Hb = H , µa = µb = µ and χa = χb = χ. Denote N = N(H, µ) and β = β(H, µ, χ). We have already have shown the first claim. Let Aut(H, µ∗µ, χ2) ∋ φ 7→ πφ ∈ Aut(N, β), be the map given as in the proof of Theorem 4.1, that is, µ̂(λ)u(λ) = c̃φ(λ) µ̂(φ ◦ λ)u(φ ◦ λ), λ ∈ Λ(H), where c̃φ(λ) = χ(λ(k))gcd(k) χ(φ ◦ λ(k)) gcd(k) It is easy to prove that φπφ = φ by the definition. Thus the map π 7→ φπ is surjective. This map is also injective. Let φ be an element of Aut(H, µ∗µ, χ2). Suppose that π is an arbitrary element of Aut(N, β) satisfying φ = φπ. The set {β(γ)(u(λh)) | h ∈ H, γ ∈ Z2 ⋊ SL(2,Z)} generates N , so we have only to prove the uniqueness of c(h) ∈ T satisfying µ(h,−h) u(λh) = c(h)µ(φ(h),−φ(h))u(λφ(h)), for all h ∈ H . In the proof of the first half of the previous theorem (equation (Eq2)), we have already shown that c(h) = χ(h)χ(φ(h)). Thus the ∗-isomorphism π is uniquely determined and the map π 7→ φπ is injective. We prove the two maps φ 7→ πφ and π 7→ φπ are continuous. Let (φi) be a net in Aut(H, µ∗µ, χ2) converging to φ. For all h ∈ H , we have χ(h)µ(h,−h) u(λh) = χ(φi(h))µ(φi(h),−φi(h))u(λφi(h)). The right side of the equation converges to χ(φ(h))µ(φ(h),−φ(h))u(λφ(h)) = πφ χ(h)µ(h,−h) u(λh) This proves that πφi converges to πφ in pointwise 2-norm topology on the generating set {β(γ)(u(λh)) | h ∈ H, γ ∈ Z2 ⋊SL(2,Z)} of N . Thus πφi converges to πφ on N . Conversely, let (πi) be a net in Aut(N, β) converging to π. For all h ∈ H , we get πi(u(λh)) = u(λφπi(h)) mod T. The left side of the equation converges to π(u(λh)) = u(λφπ(h)). If φπi(h) 6= φπ(h), then the distance between Tu(λφπ(h)) and Tu(λφπi (h)) is 2 in the 2-norm. Thus φπi(h) = φπ(h) for large enough i. This means that (φπi) converges to φπ. As a consequence, the two maps φ 7→ πφ and π 7→ φπ are continuous group homomorphisms and inverse maps of each other. � CLASSIFICATION AND CENTRALIZERS 15 5. Examples 5.1. Twisted Bernoulli shift actions on L∞(X). In this subsection, we consider the case of µ = 1 and H 6= {0}. Then the algebra N(H, 1) is abelian and has a faithful normal state, so it is isomorphic to L∞(X), whereX is a standard probability space. The measure of X is determined by the trace on N(H, 1). Furthermore, X is non-atomic, since N(H, 1) is infinite dimensional and the action β(H, 1, χ) is ergodic. As corollaries of Theorems 4.1 and 4.4, we get trace preserving Z2⋊SL(2,Z)-actions on L∞(X) whose centralizers are isomorphic to some prescribed groups. Remark 5.1. The Z2 ⋊ SL(2,Z)-action on X defined by β = β(H, 1, χ) is free. An automorphism α ∈ Aut(L∞(X), β) is free or the identity map for any twisted Bernoulli shift action β on L∞(X). This is proved as follows. We identify β(γ) (γ ∈ Z2 ⋊ SL(2,Z)) and α with measure preserving Borel isomorphisms on X here. Suppose that there exists a non-null Borel subset Y ⊂ X whose elements are fixed under α. All elements in Ỹ = ∪{β(γ)(Y ) | γ ∈ Z2⋊SL(2,Z)} are fixed under α. By the ergodicity of β, the measure of Ỹ is 1. Then α is the identity map of L∞(X). Corollary 5.2. For any abelian countable discrete group H 6= {0}, there exists a trace preserving essentially free ergodic action β of Z2⋊SL(2,Z) on L∞(X) satisfying Aut(L∞(X), β) ∼= Aut(H). Proof. When we define β = β(H, 1, 1), we have the above relation by Theorem 4.4. � In the next corollary we use the effect of twisting by a character χ. Corollary 5.3. For every abelian countable discrete group H 6= {0}, there exist continuously many trace preserving essentially free ergodic actions {βc} of Z2 ⋊ SL(2,Z) on L∞(X) which are mutually non-conjugate and satisfy Aut(L∞(X), βc) ∼= H ⋊Aut(H). Here the topology of H ⋊ Aut(H) is the product of the discrete topology on H and the pointwise convergence topology on Aut(H). Proof. Let c ∈ {eiπt | t ∈ (0, 1/2) \Q}. We put βc = β(H ⊕ Z, 1, 1× χc), where the character χc of Z is defined as χc(n) = c n. By Theorem 4.4, we get Aut(L∞(X), βc) ∼= Aut(H ⊕ Z, 1, 1× χ2c). Since the character χ2c is injective, a group automorphism α ∈ Aut(H⊕Z, 1, 1×χ2c) preserves the second entry. For all α ∈ Aut(H⊕Z, 1, 1×χ2c), there exist φα ∈ Aut(H) and hα ∈ H satisfying α(h, n) = (φα(h) + nhα, n), (h, n) ∈ H ⊕ Z. The map Aut(H ⊕ Z, 1, 1× χ2c) ∋ α 7→ (hα, φα) ∈ H ⋊ Aut(H) is a homeomorphic group isomorphism. If c1, c2 ∈ {eiπt | t ∈ (0, 1/2) \ Q} and c1 6= c2, then there exists no isomorphism from H ⊕ Z to H ⊕ Z whose pull back of the character 1 × χ2c2 is equal to 1 × χ The two actions βc1 and βc2 are not conjugate by Theorem 4.1. � 16 HIROKI SAKO Corollary 5.4. There exist continuously many trace preserving essentially free er- godic actions {βc} of Z2 ⋊ SL(2,Z) on L∞(X) which are mutually non-conjugate and have the trivial centralizer Aut(L∞(X), βc) = {idL∞X}. Proof. Let {χc | c = eiπt, t ∈ (0, 1/2)} be characters of Z such that χc(m) = cm. Since χ2c(1) is in the upper half plane, the identity map is the only automorphism of Z preserving χ2c . By Theorem 4.4, we get Aut(β(Z, 1, χc)) = {id}. If β(Z, 1, χc1), β(Z, 1, χc2) are conjugate, then there exists a group isomorphism on Z whose pull back of χ2c2 is χ by Theorem 4.1. This means c1 = c2. Thus the actions {β(Z, 1, χc)} are mutually non-conjugate. � 5.2. Twisted Bernoulli shift actions on the AFD factor of type II1. Firstly, we find a condition that the finite von Neumann algebra N(H, µ) is the AFD factor of type II1. Lemma 5.5. For an abelian countable discrete group H 6= {0} and its normalized scalar 2-cocycle µ, the following statements are equivalent: (1) The algebra N(H, µ) is the AFD factor of type II1. (2) The group von Neumann algebra Lµ(H) twisted by the scalar 2-cocycle µ is a factor (of type II1 or In). (3) For all g ∈ H \ {0}, there exists h ∈ H such that µ(g, h) 6= µ(h, g). Proof. The amenability of the group Λ(H) leads the injectivity for N(H, µ). The injectivity for N(H, µ) implies that N(H, µ) is approximately finite dimensional ([Co]). We have only to show the equivalence of conditions (2), (3) and (1)′ The algebra N(H, µ) is a factor. By using Fourier expansion it is easy to see that condition (2) holds true if and only if for any g ∈ H \ {0} there exists h ∈ H satisfying uguh 6= uhug. This is equivalent to condition (3). Similarly, condition (1)′ is equivalent to (1)′′ For any λ1 ∈ Λ(H) \ {0}, there exists λ2 ∈ Λ(H) satisfying µ̃(λ2, λ1)µ̃(λ1, λ2) 6= 1. Suppose condition (3). For any λ1, choose element k, l ∈ Z2 so that k ∈ supp(λ1) and l /∈ supp(λ1). By condition (3), there exists h ∈ H satisfying µ∗µ(λ1(k), h) 6= 1. Let λ2 be the element in Λ(H) which takes h at k, −h at l and 0 for the other places. The element λ2 satisfies µ̃(λ2, λ1)µ̃(λ1, λ2) = µ ∗µ(λ1(k), h) 6= 1. Here we get condition (1)′′. The implication from (1)′′ to (3) is easily shown. � Remark 5.6. The twisted Bernoulli shift action β = β(H, µ, χ) is an outer action of Z2 ⋊ SL(2,Z). Any non-trivial automorphism in Aut(R, β(H, µ, χ)) is also outer. This is proved by the weak mixing property of the action β(H, µ, χ) as follows. If α ∈ Aut(R, β(H, µ, χ)) is an inner automorphism Ad(u), then we have Ad(β(γ)(u))(x) = β(γ)(uβ(γ)−1(x)u∗) = β(γ) ◦ α ◦ β(γ)−1(x) = α(x) = Ad(u)(x), for all x ∈ R and γ ∈ Z2 ⋊ SL(2,Z). Since Ad(β(γ)(u)u∗) = id, Cu ⊂ R is an invariant subspace of the action β. The only subspace invariant under the weakly mixing action β is C1 (Proposition 2.6), thus we get α = id. CLASSIFICATION AND CENTRALIZERS 17 Using Theorems 4.1 and 4.4, we give continuously many actions of Z2 ⋊ SL(2,Z) on R such that there exists no commuting automorphism except for trivial one. Corollary 5.7. There exist continuously many ergodic outer actions {βc} of Z2 ⋊ SL(2,Z) on the AFD factor R of type II1 which are mutually non-conjugate and have the trivial centralizer Aut(R, βc) = {idR}. Proof. We can choose and fix a character χ on Z2 such that χ2 is injective. Let {µc | c = eiπt, t ∈ (0, 1/2) \Q} be scalar 2-cocycles for Z2 defined by = cs1t2−t1s2, s1, t1, s2, t2 ∈ Z. We put βc = β(Z 2, µc, χ). The 2-cocycle µc satisfies condition (3) in Lemma 5.5. Thus βc defines a Z 2 ⋊ SL(2,Z)-action on R. By Theorem 4.4, we get the following isomorphism between topological groups: Aut(R, βc) ∼= Aut(Z2, µ∗cµc, χ2) = Aut(Z2, µc2, χ2). Since the character χ2 of Z2 is injective, so the group of the right side is {id|Z2}. This means that the action βc has trivial centralizers. Finally, we prove that the actions {βc | c = eiπt, t ∈ (0, 1/2) \ Q} are mutually non-conjugate. Suppose that actions βc1 and βc2 are conjugate. By Theorem 4.1, there exists a group isomorphism φ of Z2 satisfying (g, h) = µc2 (φ(g), φ(h)), g, h ∈ Z2. A group isomorphism of Z2 is given by an element of GL(2,Z). If the automorphism φ is given by an element of SL(2,Z), we get c22 = c 1. If φ is given by an element of GL(2,Z) \ SL(2,Z), then we get c22 = −c21. Since both c1 and c2 have the form eiπt, t ∈ (0, 1/2), we get c1 = c2. � Any cyclic group of an odd order can be realized as the centralizer of a twisted Bernoulli shift actions on R. Corollary 5.8. Let q be an odd natural number ≥ 3 and denote by Hq the abelian group (Z/qZ)2. We define the 2-cocycle µq and the character χq on Hq as = exp (2πis1t2/q), χq = exp (2πis1/q). Then the algebra N(Hq, µq) is the AFD factor R of type II1 and the centralizer of the twisted Bernoulli shift action βa = β(Hq, µq, χq) is isomorphic to Z/qZ. Proof. By Lemma 5.5, it is shown that the algebra N(HQ, µQ) is the AFD factor of type II1. Using Theorem 4.4, we have only to prove that Aut(Hq, µ qµq, χ ∼= Z/qZ. Let φ be in Aut(Hq, µ qµq, χ q). The automorphism φ of Hq is given by a 2 × 2 matrix A of Z/qZ. Since φ preserves µ∗qµq, the determinant of A must be 1. Since q is odd, the value of χ2q determines the first entry of (Z/qZ) 2 and φ preserves χ2q . The matrix A is of the form( , tφ ∈ Z/qZ. The map φ 7→ tφ is an isomorphism. In turn, if the matrix A is of this form, it defines an element in Aut(Hq, µ qµq, χ q). � 18 HIROKI SAKO Corollary 5.9. For a set Q consisting of odd prime numbers, let βQ be the tensor product q∈Q βq of the actions βq on the AFD factor of type II1. The centralizer of βQ is isomorphic to q∈Q Z/qZ. Proof. The action βQ is the twisted Bernoulli shift action β(HQ, µQ, χQ), where HQ is the abelian group ⊕q∈QHq and the scalar 2-cocycle µq and a character χQ on HQ are given by µQ((sq), (tq)) = µq(sq, tq), χQ((sq)) = χq(sq), (sq), (tq) ∈ HQ, sq, tq ∈ Hq. Using Theorem 4.4, we have only to prove Aut(HQ, µ QµQ, χ Z/qZ. A group automorphism φ of HQ = ⊕q∈QHq has a form φ((kq)) = (φq(kq)), for some {φq ∈ Aut(Hq)}. Thus we get Aut(HQ, µ QµQ, χ Aut(Hq, µ qµq, χ Together with the previous corollary, we get the conclusion. � Remark 5.10. If Q1 6= Q2, then the two groups Z/qZ and Z/qZ are not isomorphic. The continuously many outer actions {βQ} are distinguished in view of conjugacy only by using the centralizers {Aut(R, βQ)}. 6. Malleability and rigidity arguments In this section, we give malleability and rigidity type arguments invented by S. Popa, in order to examine weak 1-cocycles for actions. See Popa [Po2], [Po3], [Po4] and Popa–Sasyk [PoSa] for the references. S. Popa in [Po3] showed that every 1-cocycle for a Connes-Størmer Bernoulli shift by property (T) group (or w-rigid group like Z2 ⋊ SL(2,Z)) vanishes modulo scalars. As a consequence, two such actions are cocycle conjugate if and only if they are conjugate. In our case, 1- cocycles do not vanish modulo scalars but they are still in the situation that cocycle (outer) conjugacy implies conjugacy. We need the following notion to examine outer conjugacy of two group actions. Definition 6.1. Let α be an action of discrete group Γ on a von Neumann algebra M. A weak 1-cocycle for α is a map w : Γ → U(M) satisfying wgh = wgαg(wh) mod T, g, h ∈ Γ. The weak 1-cocycle w is called a weak 1-coboundary if there exists a unitary v ∈ U(M) satisfying wg = vαg(v)∗ mod T. Two weak 1-cocycles w and w′ are said to be equivalent when w′g = vwgαg(v) ∗ mod T for some v ∈ U(M). Let N be a finite von Neumann algebra with a faithful normal trace. The following is directly obtained by combining Lemmas 2.4 and 2.5 in [PoSa], although these Lemmas were proved for Bernoulli shift actions on standard probability space. The CLASSIFICATION AND CENTRALIZERS 19 following can be also regarded as a weak 1-cocycle version of Proposition 3.2 in [Po4]. Proposition 6.2. Let G be a countable discrete group. Let β be a trace preserving weakly mixing action of G on N . A weak 1-cocycle {wg}g∈G ⊂ N for β is a weak 1-coboundary if only if there exists a non-zero element x̃0 ∈ N ⊗N satisfying (wg ⊗ 1)(βg ⊗ βg)(x̃0)(1⊗ w∗g) = x̃0, g ∈ G. The following is a weak 1-cocycle version of Proposition 3.6.3◦ in [Po4]. Proposition 6.3. Let Γ be a countable discrete group and G be a normal subgroup of Γ. The group Γ acts on a finite von Neumann algebra N in a trace-preserving way by β. Suppose that the restriction of β to G is weakly mixing. Let {wγ}γ∈Γ be a weak 1-cocycle for β. If w|G is a weak 1-coboundary, then w is a weak 1-coboundary for the Γ-action. Proof. Suppose that w|G is a weak 1-coboundary, that is, there exists a unitary element v in N such that wg = vβg(v ∗) mod T for g ∈ G. It suffices to show that {w′γ} = {v∗wγβγ(v)} is in T for all γ ∈ Γ. Take arbitrary γ ∈ Γ, g ∈ G. Write h = γ−1gγ ∈ G. Let πγ be the unitary on L2(N) induced from βγ . Since w′h, w′g ∈ T, we get w′γπgw = (w′γπγ)(w hπh)(w ∗ = w′γhγ−1πγhγ−1 = πg mod T, By applying these operators to 1̂ ∈ N̂ ⊂ L2(N), it follows that w′γβg(w′γ ) ∈ T. Since the G-action is weakly mixing, we have w′γ ∈ T. � By using the above propositions, we will “untwist” some weak 1-cocycles later. We require some ergodicity assumption on the weak 1-cocycles. Definition 6.4. Let Γ be a discrete group and G be a subgroup of Γ. Suppose that its restriction to G is ergodic. Let β be a trace preserving action of Γ on N . A weak 1-cocycle w = {wg}g∈Γ for β is said to be ergodic on G, if the action βw of G is still ergodic, where βw is defined by βwg = Adwg ◦ βg, g ∈ G. Let β be a Γ-action onN . Suppose that the diagonal action β⊗β on (N⊗N, tr⊗tr) has an extension β̃ on a finite von Neumann algebra (Ñ , τ). The algebra Ñ is not necessarily identical with N⊗N . When the action β̃ is ergodic on a normal subgroup G ⊂ Γ, we get the following: Proposition 6.5. Let {wγ}γ∈Γ ⊂ N be a weak 1-cocycle for β. Let α be a trace preserving continuous action of R on Ñ satisfying the following properties: • α1(x⊗ 1) = 1⊗ x, for all x ∈ N . • αt ◦ β̃(γ)(x̃) = β̃(γ) ◦ αt(x̃), for all t ∈ R, γ ∈ Γ and x̃ ∈ Ñ . Suppose that the weak 1-cocycle {wγ⊗1} ⊂ Ñ is ergodic for the G-action β̃|G. If the group inclusion G ⊂ Γ has the relative property (T) of Kazhdan, then there exists a non-zero element x̃0 ∈ Ñ so that (wg ⊗ 1)β̃g(x̃0)(1⊗ w∗g) = x̃0, g ∈ G. This is proved in the same way for Bernoulli shift actions on the infinite tensor product of abelian von Neumann algebras ([PoSa], Lemma 3.5). Since we are in- terested in actions on the AFD II1 factor, we require the ergodicity assumption on 20 HIROKI SAKO weak 1-cocycle {wγ ⊗ 1}. For the self-containedness and in order to make it clear where the ergodicity assumption works, we write down a complete proof. Proof. For t ∈ (0, 1], let Kt be the convex weak closure of {(wg ⊗ 1)αt(w∗g ⊗ 1) | g ∈ G} ⊂ Ñ and x̃t ∈ Kt be the unique element whose 2-norm is minimum in Kt. Since (wg ⊗ 1)β̃g((wg1 ⊗ 1)αt(w∗g1 ⊗ 1))αt(w g ⊗ 1) = (wgβg(wg1)⊗ 1)αt(βg(w∗g1)w g ⊗ 1) = (wgg1 ⊗ 1)αt(w∗gg1 ⊗ 1), g, g1 ∈ G, we have (wg ⊗ 1)β̃g(Kt)αt(w∗g ⊗ 1) = Kt, for g ∈ G. By the uniqueness of x̃t, we get (wg ⊗ 1)β̃g(x̃t)αt(w∗g ⊗ 1) = x̃t, g ∈ G.(Eq3) By the assumption, the action (Ad(wg ⊗ 1) ◦ β̃g)g∈G is ergodic on Ñ . By the calcu- lation (wg ⊗ 1)β̃g(x̃tx̃t∗)(w∗g ⊗ 1) = (wg ⊗ 1)β̃g(x̃t)αt(w∗g ⊗ 1)αt(wg ⊗ 1)β̃g(x̃t )(w∗g ⊗ 1) = x̃tx̃t , g ∈ G, we get x̃tx̃t ∗ ∈ C1. The element x̃t is a scalar multiple of a unitary in Ñ . We shall next prove that x̃1/n is not zero for some positive integer n. The pair (Γ, G) has the relative property (T) of Kazhdan. By proposition 2.4, we can find a positive number δ and a finite subset F ⊂ Γ satisfying the following condition: If a unitary representation (π,H) of Γ and a unit vector ξ of H satisfy ‖π(γ)ξ − ξ‖ ≤ δ (γ ∈ F ), then ‖π(g)ξ− ξ‖ ≤ 1/2 (g ∈ G). By the continuity of the action α, there exists n such that ‖(wγ ⊗ 1)α1/n(w∗γ ⊗ 1)− 1‖tr,2 ≤ δ, γ ∈ F, The actions β and (αl1/n)l∈Z on Ñ give a Γ× Z action on Ñ . Let P be the crossed product von Neumann algebra P = Ñ ⋊ (Γ × Z). Let (Uγ)γ∈Γ and W be the implementing unitaries in P for Γ and 1 ∈ Z respectively. We put Vγ = (wγ ⊗ 1)Uγ , γ ∈ Γ. We regard AdV· as a unitary representation of Γ on L2(P ). Since ‖AdVγ(W )−W‖L2(P ) = ‖(wγ ⊗ 1)W (w∗γ ⊗ 1)W ∗ − 1‖L2(P ) = ‖(wγ ⊗ 1)α1/n(w∗γ ⊗ 1)− 1‖L2(Ñ) ≤ δ, γ ∈ F, we have the following inequality: 1/2 ≥ ‖AdVg(W )−W‖L2(P ) = ‖(wg ⊗ 1)α1/n(w∗g ⊗ 1)− 1‖L2(Ñ), g ∈ G. We get 1/2 ≥ ‖x̃1/n − 1‖L2(Ñ) and x̃1/n 6= 0. Let ũ1/n be the unitary of Ñ given by a scalar multiple of x̃1/n. By equation (Eq3), the unitary satisfies (wg ⊗ 1)β̃g(ũ1/n)α1/n(w∗g ⊗ 1) = ũ1/n, g ∈ G. CLASSIFICATION AND CENTRALIZERS 21 Let x̃0 be the unitary defined by x̃0 = ũ1/nα1/n(ũ1/n)α2/n(ũ1/n) . . . α(n−1)/n(ũ1/n). By direct computations, we have the following desired equality: (wg ⊗ 1)β̃g(x̃0)(1⊗ w∗g) = (wg ⊗ 1)β̃g(x̃0)α1(w∗g ⊗ 1) = x̃0, g ∈ G. Theorem 6.6. Let β = β(H, µ, χ) be a twisted Bernoulli shift action on N(H, µ). Suppose that N(H, µ) is the AFD factor of type II1 and that there exists a continuous R-action (α t )t∈R on Lµ(H)⊗ Lµ(H) satisfying the following properties: • For any x ∈ Lµ(H), α(0)1 (x⊗ 1) = 1⊗ x, • The automorphism α(0)t commutes with the diagonal action of Ĥ. Let β(1) be another action of Z2 ⋊ SL(2,Z) on the AFD factor N (1) of type II1 and suppose that its restriction to Z2 is ergodic. The action β(1) is outer conjugate to β, if and only if β(1) is conjugate to β. Proof. We deduce from outer conjugacy to conjugacy in the above situation. Let θ be a ∗-isomorphism from N (1) onto N(H, µ) which gives the outer conjugacy of the action β(1) and β = β(H, µ, χ). There exists a weak 1-cocycle {wγ}γ∈Z2⋊SL(2,Z) for β satisfying θ ◦ β(1)(γ) = Adwγ ◦ β(γ) ◦ θ, γ ∈ Z2 ⋊ SL(2,Z). Since the action β(1) is ergodic on Z2, the weak 1-cocycle w is ergodic on Z2. We use the notations Γ0, G0 given in Section 3. Let ρ̃ be the diagonal action ρ⊗ρ of Γ0 on the tensor product algebra M̃ = Lµ̃(⊕Z2H)⊗ Lµ̃(⊕Z2H): ρ̃(γ0)(a⊗ b) = ρ(γ0)(a)⊗ ρ(γ0)(b). The fixed point algebra Ñ ⊂ M̃ of the diagonal Ĥ-action contains N(H, µ) ⊗ N(H, µ). Since Z2⋊SL(2,Z) = Γ0/Ĥ, the action ρ̃ gives a Z 2⋊SL(2,Z)-action β̃ on Ñ . The action β̃ is the extension of the diagonal action β⊗β on N(H, µ)⊗N(H, µ). We denote by αt the action on M̃ ∼= (Lµ(H)⊗Lµ(H)) given by the infinite tensor product of the R-action α t . By the assumption on α t , the R-action αt commutes with the action ρ̃. It follows that the subalgebra Ñ is globally invariant under αt. The set of unitary {Wγ = wγ ⊗ 1}γ∈Z2⋊SL(2,Z) ⊂ Ñ is a weak 1-cocycle for β̃. We shall prove that this weak 1-cocycle is ergodic on Z2. Let a be an element in Ñ fixed under β̃|Z2 . The element a can be written as a = H aλ⊗u(λ) in L2M̃ , where aλ ⊗ 1 = EM⊗C(a(1⊗ u(λ))∗). Since a is fixed under the action of Z2, we have a = β̃W (k)(a) = Adwk ◦ ρ(1, k)(aλ)⊗ ρ(1, k)(u(λ)) Adwk ◦ ρ(1, k)(aλ)⊗ χ(λ(l))det(k,l)u(k · λ). Since Adwk ◦ ρ(1, k) preserves the 2-norm, we get ‖aλ‖2 = ‖ak−1·λ‖2. Since ‖a‖22 =∑ ‖aλ‖22 < ∞ and the set {ak−1·λ | k ∈ Z2} is infinite for λ 6= 0, it turns out that aλ = 0 for λ 6= 0 and thus a ∈ Ñ ∩ (M ⊗ C) = N(H, µ) ⊗ C. By the ergodicity of 22 HIROKI SAKO the Z2-action {Adwk ◦ β(k)}, we get a ∈ C. We conclude that the weak 1-cocycle {Wγ} ⊂ Ñ is ergodic on Z2. By the relative property (T) for the inclusion Z2 ⊂ Z2⋊SL(2,Z) and Proposition 6.5, there exists a non-zero element x̃0 ∈ Ñ satisfying (wk ⊗ 1)β̃(k)(x̃0)(1⊗ w∗k) = x̃0, k ∈ Z2. The element x̃0 can be written as the following Fourier expansion: x̃0 = c(λ1, λ2)u(λ1)⊗ u(λ2) ∈ Ñ ⊂ Lµ̃(⊕Z2H)⊗ Lµ̃(⊕Z2H). Here c(λ1, λ2) is a complex number and (λ1, λ2) ∈ (⊕Z2H)2 runs through all pairs satisfying k∈Z2(λ1(k) + λ2(k)) = 0. Choose and fix a pair (λ1, λ2) satisfying λ1(k) = h = λ2(k), c(λ1, λ2) 6= 0. Let v′h ∈ M be the unitary written as v′h = uh ⊗ 1 ⊗ 1 ⊗ · · · , where uh ∈ Lµ(H) is placed on 0 ∈ Z2. The following unitaries {w′γ} ⊂ N(H, µ) give a weak 1-cocycle for β: w′(k,γ0) = v hw(k,γ0)ρ(1, k, γ0)(v ), (k, γ0) ∈ Z2 ⋊ SL(2,Z). Letting ỹ = (v′h ⊗ 1)x̃0(1⊗ v′h)∗ ∈ M̃ , we get ỹ = (w′k ⊗ 1)β̃(k)(ỹ)(1⊗ w′k ), k ∈ Z2. Applying the trace preserving conditional expectation E = EN(H,µ)⊗N(H,µ), we get E(ỹ) = (w′k ⊗ 1)E(β̃(k)(ỹ))(1⊗ w′k = (w′k ⊗ 1)β̃(k)(E(ỹ))(1⊗ w′k ), k ∈ Z2. Since the Fourier coefficient of x̃0 at (λ1, λ2) ∈ (⊕Z2H)2 is not zero, that of E(ỹ) at (λ1 + δh,0, λ2 − δh,0) ∈ Λ(H)2 is also non-zero, where δh,0 ∈ ⊕Z2H is zero on Z2 \ {0} and is h on 0 ∈ Z2. By Proposition 6.2, it follows that the weak 1-cocycle {w′(k,e)}k∈Z2 ⊂ N(H, µ) is a weak 1-coboundary of β|Z2. Since the Z2-action β|Z2 is weakly mixing, w′ is a weak 1-coboundary on Z2 ⋊ SL(2,Z), by Proposition 6.3. In other words, there exists v ∈ N(H, µ) satisfying w′γ = vβ(γ)(v ∗) mod T, wγ = v vρ(1, γ)(v∗v′h) mod T, γ ∈ Z2 ⋊ SL(2,Z). Noting that u = v∗v′h ∈ M is a normalizer of N(H, µ), we get (Ad(u) ◦ θ) ◦ β(1)(γ) = Ad(u) ◦ Ad(wγ) ◦ β(γ) ◦ θ = Ad(ρ(1, γ)(u)) ◦ β(γ) ◦ θ = ρ(1, γ) ◦ Ad(u) ◦ θ = β(γ) ◦ (Ad(u) ◦ θ), γ ∈ Z2 ⋊ SL(2,Z). Thus we get the conjugacy of two Z2 ⋊ SL(2,Z)-actions β(0) and β. � We can always apply Theorem 6.6 if H is finite. CLASSIFICATION AND CENTRALIZERS 23 Corollary 6.7. Let H be a finite abelian group and let β = β(H, µ, χ) be a twisted Bernoulli shift action on N(H, µ). Suppose that N(H, µ) is the AFD factor of type II1. Let β (1) be an action of Z2 ⋊ SL(2,Z) on the AFD factor N (1) of type II1 and suppose that its restriction to Z2 is ergodic. The action β(1) is outer conjugate to β, if and only if β(1) is conjugate to β. Proof. We have only to construct an R-action on Lµ(H) ⊗ Lµ(H) satisfying the properties in Theorem 6.6. Let U be an element of Lµ(H)⊗ Lµ(H) defined by |H|1/2 uh ⊗ u∗h. We note that µ∗µ(g, ·) is a character of H and that it is not identically 1 provided g 6= 0 by Lemma 5.5. The element U is self-adjoint and unitary, since |H|1/2 u∗h ⊗ uh = |H|1/2 µ(h,−h)u−h ⊗ µ(h,−h)u∗−h = U, g,h∈H uguh ⊗ u∗gu∗h = g,h∈H µ∗µ(g, h)ug+h ⊗ u∗g+h µ∗µ(g, h− g) ug ⊗ u∗g = 1. The operator U is a fixed point under the action of Ĥ , so the projections P1 = (1 + U)/2 and P−1 = (1 − U)/2 are also fixed points. Thus the R-action α(0)t = Ad(P1+exp (iπt)P−1) commutes with the Ĥ-action. The automorphism α 1 satisfies 1 (ug ⊗ 1) = U(ug ⊗ 1)U∗ = (1⊗ ug)UU∗ = 1⊗ ug, g ∈ H. This verifies the first condition for α(0). � Corollary 6.8. Let Q be a set consisting of odd prime numbers and βQ be the twisted Bernoulli shift action defined in Corollary 5.9. Let β be a Z2 ⋊ SL(2,Z)-action on the AFD factor of type II1 whose restriction to Z 2 is ergodic. The actions βQ and β are outer conjugate if and only if they are conjugate. In particular, {βQ} is an uncountable family of Z2⋊SL(2,Z)-actions which are mutually non outer conjugate. Proof. We will use the notation given in Corollary 5.8 and 5.9. Let α t be the R- action on Lµq (Hq)⊗Lµq(Hq) constructed as in the previous corollary. We define the R-action α(Q) on LµQ(HQ)⊗LµQ(HQ) by α t (⊗q∈Qxq) = ⊗q∈Qα t (xq), where xq ∈ Lµq (Hq) ⊗ Lµq (Hq) and xq 6= 1 only for finitely many q. The R-action satisfies the conditions in Theorem 6.6. By Corollary 5.9, {βQ} are mutually non conjugate and their restriction to Z2 is ergodic. Thus they are mutually non outer conjugate. � Acknowledgment . The author would like to thank Professor Yasuyuki Kawahigashi for helpful conversations. 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Shalom, Bounded generation and Kazhdan’s Property (T). Publ. Math. IHES, 90 (1999), 145–168. [Vaes] S. Vaes, Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa), Sminaire Bourbaki, exp. no. 961, Astrisque 311 (2007) 237–294. Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan E-mail address : hiroki@ms.u-tokyo.ac.jp 1. Introduction 2. Preparations 2.1. Functions det and gcd 2.2. Scalar 2-cocycles for abelian groups 2.3. Definition of a Z2 SL(2,Z)-action on (H) 2.4. On the relative property (T) of Kazhdan 2.5. Weakly mixing actions 2.6. A remark on group von Neumann algebras 3. Definition of twisted Bernoulli shift actions 4. Classification up to conjugacy 5. Examples 5.1. Twisted Bernoulli shift actions on L(X) 5.2. Twisted Bernoulli shift actions on the AFD factor of type II1 6. Malleability and rigidity arguments References

We introduce a new class of actions of the group $\G$ on finite von Neumann algebras and call them twisted Bernoulli shift actions. We classify these actions up to conjugacy and give an explicit description of their centralizers. We also distinguish many of those actions on the AFD $\mathrm{II}_1$ factor in view of outer conjugacy.

Introduction We consider the classification of Z2 ⋊ SL(2,Z)-actions on finite von Neumann algebras in this paper. Mainly, we concentrate on the case that the finite von Neumann algebra is the AFD factor of type II1 or non-atomic abelian. There are two difficulties for analyzing discrete group actions on operator algebras. The first is that we do not have various ways to construct actions. The second is that we can not analyze them by concrete calculation in most cases. To give many examples of actions which admit concrete analysis, we introduce a class of trace preserving Z2 ⋊ SL(2,Z)-actions on finite von Neumann algebras and call them twisted Bernoulli shift actions. We classify those actions up to conjugacy and study them up to outer conjugacy. An action β(H, µ, χ) in the class is defined for a triplet (H, µ, χ), where H is an abelian countable discrete group, µ is a normalized scalar 2-cocycle of H and χ is a character of H . We obtain the action by restricting the so-called generalized Bernoulli shift action to a subalgebra N(H, µ) and “twisting” it by the character χ. The process of restriction has a vital role in concrete analysis of these actions. A ∗-isomorphism which gives conjugacy between two twisted Bernoulli shift ac- tions β(Ha, µa, χa) and β(Hb, µb, χb) must be induced from an isomorphism between the two abelian groups Ha and Hb. We prove this by concrete calculation (Section 4). It turns out that there exist continuously many, non-conjugate Z2 ⋊ SL(2,Z)- actions on the AFD factor of type II1 (Section 5). By using the same technique, we describe the centralizers of all twisted Bernoulli shift actions. Here we should men- tion that the present work was motivated by the previous ones [Ch], [NPS], where similar studies were carried out in the case of SL(n,Z). In Section 6, we distinguish many twisted Bernoulli shift actions in view of outer conjugacy. The classification for actions of discrete amenable groups on the AFD factor of type II1 was given by Ocneanu [Oc]. Outer actions of countable amenable groups are outer conjugate. In the contrast to this, V. F. R. Jones [Jon] proved that any discrete non-amenable group has at least two non outer conjugate actions on 2000 Mathematics Subject Classification. Primary 46L40; Secondary 46L10. Key words and phrases. von Neumann algebras; automorphisms. http://arxiv.org/abs/0704.1533v3 2 HIROKI SAKO the AFD factor of type II1. S. Popa ([Po3], [Po4], [PoSa], etc.) used the malleabil- ity/deformation arguments for the Bernoulli shift actions to study (weak) 1-cocycles for the actions. For some of twisted Bernoulli shift actions, which we introduce in this paper, it is shown that (weak) 1-cocycles are represented in simple forms under some assumption on the (weak) 1-cocycles. We prove that there exist continuously many twisted Bernoulli shift actions which are mutually non outer conjugate. This strengthens the above mentioned result due to Jones in the Z2 ⋊ SL(2,Z) cases. 2. Preparations 2.1. Functions det and gcd. For the definition of twisted Bernoulli shift actions in Section 3, we define two Z-valued functions det and gcd. The function det is given by the following equation: = qr0 − rq0, ∈ Z2. The value of the function gcd at k ∈ Z2 is the greatest common divisor of the two entries. For 0 ∈ Z2, let the value of gcd be 0. Lemma 2.1. (1) The action of SL(2,Z) on Z2 preserves the functions det and gcd, that is, det(k, k0) = det(γ · k, γ · k0), gcd(k) = gcd(γ · k), k, k0 ∈ Z2, γ ∈ SL(2,Z). (2) The following equation holds true: det(k, k0) = gcd(k) + gcd(k0)− gcd(k + k0) mod 2, k, k0 ∈ Z2. Proof. The claim (1) is a well-known fact, so we prove the claim (2). For the function gcd, we get 1 mod 2, (either q or r is odd), 0 mod 2, (both q and r are even). Since the action of SL(2,Z) on Z2 preserves the functions det and gcd, it suffices to show the desired equation against the following four pairs: (k, k0) = mod 2. 2.2. Scalar 2-cocycles for abelian groups. We fix some notations for countable abelian groups and their scalar 2-cocycles. For the rest of this paper, let H be an abelian countable discrete group and suppose that any scalar 2-cocycle µ : H×H → T = {z ∈ C | |z| = 1} is normalized, that is, µ(g, 0) = 1 = µ(0, g) for g ∈ H . We denote by µ∗ the 2-cocycle for H given by µ∗(g, h) = µ(h, g), g, h ∈ H . Let µ∗µ be the function on H ×H defined by µ∗µ(g, h) = µ(h, g)µ(g, h), g, h ∈ H. CLASSIFICATION AND CENTRALIZERS 3 This is a bi-character, that is, µ∗µ(g, ·) and µ∗µ(·, h) are characters of H . By using this function, we can describe the cohomology class of µ. See [OPT] for the proof of the following Proposition: Proposition 2.2. Two scalar 2-cocycles µ1 and µ2 of H are cohomologous if and only if µ∗1µ1 = µ Let Cµ(H) be the twisted group algebra of H with respect to the 2-cocycle µ. We denote by {uh | h ∈ H} the standard basis for Cµ(H) as C-linear space. We recall that the C-algebra Cµ(H) has a structure of ∗-algebra defined by ug uh = µ(g, h) ug+h, u g = µ(g,−g)u−g, g, h ∈ H. Let µ̃ be the T-valued function on ⊕Z2H ×⊕Z2H defined by µ̃(λ1, λ2) = µ(λ1(k), λ2(k)), λ1, λ2 ∈ ⊕Z2H.(Eq1) The function µ̃ is a normalized scalar 2-cocycle for ⊕Z2H . Let Λ(H) be the abelian group defined by Λ(H) = λ : Z2 → H ∣∣∣∣∣ finitely supported and λ(k) = 0 Its additive rule is defined by pointwise addition. 2.3. Definition of a Z2 ⋊ SL(2,Z)-action on Λ(H). The group SL(2,Z) acts on Z2 as matrix-multiplication and the group Z2 also does on Z2 by addition. These two actions define the action of Z2 ⋊ SL(2,Z) on Z2 which is explicitly described as q + xq0 + yr0 r + zq0 + wr0 for all (( ∈ Z2 ⋊ SL(2,Z), ∈ Z2. We define an action of Z2 ⋊ SL(2,Z) on ⊕Z2H as (γ · λ)(k) = λ(γ−1 · k), k ∈ Z2, for γ ∈ Z2 ⋊ SL(2,Z) and λ ∈ ⊕Z2H . 2.4. On the relative property (T) of Kazhdan. We give the definition of the relative property (T) of Kazhdan for a pair of discrete groups. Definition 2.3. Let G ⊂ Γ be an inclusion of discrete groups. We say that the pair (Γ, G) has the relative property (T) if the following condition holds: There exist a finite subset F of Γ and δ > 0 such that if π : Γ → U(H) is a unitary representation of Γ on a Hilbert space H with a unit vector ξ ∈ H satisfying ‖π(g)ξ − ξ‖ < δ for g ∈ F , then there exists a non-zero vector η ∈ H such that π(h)η = η for h ∈ G. Instead of this original definition, we use the following condition. 4 HIROKI SAKO Proposition 2.4. ([Jol]) Let G ⊂ Γ be an inclusion of discrete groups. The pair (Γ, G) has the relative property (T) if and only if the following condition holds: For any ǫ > 0, there exist a finite subset F of Γ and δ > 0 such that if π : Γ → U(H) is a unitary representation of Γ on a Hilbert space H with a unit vector ξ ∈ H satisfying ‖π(g)ξ − ξ‖ < δ for g ∈ F , then ‖π(h)ξ − ξ‖ < ǫ for h ∈ G. The pair (Z2⋊SL(2,Z),Z2) is a typical example of group with the relative property (T). See [Bu] or [Sh] for the proof. 2.5. Weakly mixing actions. An action of a countable discrete group G on a von Neumann algebra N is said to be ergodic if any G-invariant element of N is a scalar multiple of 1. The weak mixing property is a stronger notion of ergodicity. Definition 2.5. Let N be a von Neumann algebra with a faithful normal state φ. A state preserving action (ρg)g∈G of a countable discrete group G on N is said to be weakly mixing if for every finite subset {a1, a2, . . . , an} ⊂ N and ǫ > 0, there exists g ∈ G such that |φ(aiρg(aj))− φ(ai)φ(aj)| < ǫ, i, j = 1, . . . , n. The following is a basic characterization of the weak mixing property. Between two von Neumann algebra N and M , N ⊗ M stands for the tensor product von Neumann algebra. Proposition 2.6 (Proposition D.2 in [Vaes]). Let a countable discrete group G act on a finite von Neumann algebra (N, tr) by trace preserving automorphisms (ρg)g∈G. The following statements are equivalent: (1) The action (ρg) is weakly mixing. (2) The only finite-dimensional invariant subspace of N is C1. (3) For any action (αg) of G on a finite von Neumann algebra (M, τ), we have (N ⊗ M)ρ⊗α = 1 ⊗ Mα, where (N ⊗ M)ρ⊗α and Mα are the fixed point subalgebras. 2.6. A remark on group von Neumann algebras. Let Γ be a discrete group and let µ be a scalar 2-cocycle of a countable group Γ. A group Γ acts on the Hilbert space ℓ2Γ by the following two ways; uγ(δg) = µ(γ, g)δγg, ργ(δg) = µ(g, γ −1)δgγ−1 , γ, g ∈ Γ. These two representations commute with each other. The von Neumann algebra Lµ(Γ) generated by the image of u is called the group von Neumann algebra of Γ twisted by µ. The normal state 〈·δe, δe〉 is a trace on Lµ(Γ). The vector δe is separating for Lµ(Γ). For any element a ∈ Lµ(Γ), we define the square summable function a(·) on Γ by aδe = a(g)δg. The function a(·) is called the Fourier coefficient of a. We write a = g∈Γ a(g)ug and call this the Fourier expansion of a. The Fourier expansion of a∗ is given by a∗ = g∈Γ µ(g, g −1)a(g−1)ug, since the Fourier coefficient a∗(g) = 〈a∗δe, δg〉 is described as 〈δe, aρg−1δe〉 = ρ∗g−1δe, a(g)δg = µ(g, g−1)a(g−1). CLASSIFICATION AND CENTRALIZERS 5 Here we used the equation ρ∗ = µ(g, g−1)ρg, which is verified by direct computa- tion. For two elements a, b, the Fourier coefficient of ab is given by ab(γ) = 〈bδe, ργ−1a∗δe〉 = a∗(g)µ(g, γ)b(gγ) = µ(g−1, gγ)a(g−1)b(gγ). This equation allows us to calculate the Fourier coefficient algebraically, that is, µ(g−1, gγ)a(g−1)b(gγ) a(g)b(h) For a subgroup Λ ⊂ Γ, the subalgebra {uλ | λ ∈ Λ}′′ ⊂ Lµ(Γ) is isomorphic to Lµ(Λ). We sometimes identify them. An element a ∈ Lµ(Γ) is in the subalgebra Lµ(Λ) if and only if the Fourier expansion a(·) : Γ → C is supported on Λ, since the trace 〈·δe, δe〉 preserving conditional expectation E from Lµ(Γ) onto Lµ(Λ) is described as E(a) = λ∈Λ a(λ)uλ. 3. Definition of twisted Bernoulli shift actions In this section, we introduce twisted Bernoulli shift actions of Z2 ⋊ SL(2,Z) on finite von Neumann algebras. The action is defined for a triplet i = (H, µ, χ), where H 6= {0} is an abelian countable discrete group, µ is a normalized scalar 2-cocycle of H and χ is a character of H . The finite von Neumann algebra, on which the group Z2 ⋊ SL(2,Z) acts, is defined by the pair (H, µ). We introduce a group structure on the set Γ0 = Ĥ × Z2 × SL(2,Z) as (c1, k, γ1)(c2, l, γ2) = c1c2χ det(k,γ1·l), k + γ1 · l, γ1γ2 for any c1, c2 ∈ Ĥ, k, l ∈ Z2, γ1, γ2 ∈ SL(2,Z). The associativity is verified by Lemma 2.1. It turns out that the subsets Ĥ = Ĥ × {0} × {e} and G0 = Ĥ × Z2 × {e} are subgroups in Γ0. It is easy to see that G0 is a normal subgroup of Γ0 and that Ĥ is a normal subgroup of G0 and Γ0. We get a normal inclusion of groups G0/Ĥ ⊂ Γ0/Ĥ and this is isomorphic to Z2 ⊂ Z2 ⋊ SL(2,Z). Before stating the definition of the twisted Bernoulli shift action, we define a Γ0- action ρ on the von Neumann algebra Lµ̃(⊕Z2H). We denote by u(λ) ∈ Lµ̃(⊕Z2H) the unitary corresponding to λ ∈ ⊕Z2H . We define a faithful normal trace tr of Lµ̃(⊕Z2H) in the usual way. For c ∈ Ĥ, k ∈ Z2, γ ∈ SL(2,Z), let ρ(c), ρ(k), ρ(γ) be the linear transformations on Cµ̃(⊕Z2H) given by, ρ(c)(u(λ)) = c(λ(l)) u(λ), ρ(k)(u(λ)) = χ(λ(m))det(k,m) u(k · λ), ρ(γ)(u(λ)) = u(γ · λ), λ ∈ Λ(H), These maps are compatible with the multiplication rule and the ∗-operation of Cµ̃(⊕Z2H) . Since these maps preserve the trace, they extend to ∗-automorphisms 6 HIROKI SAKO on Lµ̃(⊕Z2H). It is immediate to see that ρ(c) commutes with ρ(k) and ρ(γ). For k, l ∈ Z2, we have the following relation: ρ(k) ◦ ρ(l)(u(λ)) = χ(λ(m))det(l,m)ρ(k)(u(l · λ)) χ(λ(m))det(l,m) χ((l · λ)(m))det(k,m)u(k · (l · λ)) χ(λ(m))det(l,m)χ(λ(m))det(k,m+l)u((k + l) · λ). By det(l, m) + det(k,m+ l) = det(k, l) + det(k + l, m), this equals to χ(λ(m)) )det(k,l) ∏ χ(λ(m))det(k+l,m)u((k + l) · λ) = ρ(χdet(k,l)) ◦ ρ(k + l)(u(λ)). Since det is SL(2,Z)-invariant (Lemma 2.1), for k ∈ Z2, γ ∈ SL(2,Z), we get ρ(γ · k) ◦ ρ(γ)(u(λ)) = ρ(γ · k)(u(γ · λ)) χ((γ · λ)(l))det(γ·k,l)u((γ · k) · (γ · λ)) χ(λ(l))det(γ·k,γ·l)u(γ · (k · λ)) χ(λ(l))det(k,l)ρ(γ)(u(k · λ)) = ρ(γ) ◦ ρ(k)(u(λ)), λ ∈ Λ(H). By using the above two equations, ρ satisfies the following formula: (ρ(c1) ◦ ρ(k) ◦ ρ(γ1)) ◦ (ρ(c2) ◦ ρ(l) ◦ ρ(γ2)) = ρ(c1) ◦ ρ(c2) ◦ ρ(k) ◦ ρ(γ1) ◦ ρ(l) ◦ ρ(γ2) = ρ(c1) ◦ ρ(c2) ◦ ρ(k) ◦ ρ(γ1 · l) ◦ ρ(γ1) ◦ ρ(γ2) = ρ(c1) ◦ ρ(c2) ◦ ρ(χdet(k,γ1·l)) ◦ ρ(k + (γ1 · l)) ◦ ρ(γ1γ2) = ρ(c1c2χ det(k,γ1·l)) ◦ ρ(k + (γ1 · l)) ◦ ρ(γ1γ2). With ρ(c, k, γ) = ρ(c) ◦ ρ(k) ◦ ρ(γ), ρ gives a Γ0-action on Lµ̃(⊕Z2H). We define the finite von Neumann algebra N(H, µ) as the group von Neumann algebra Lµ̃(Λ(H)). By using Fourier coefficients, we can prove that N(H, µ) is the fixed point algebra under the Ĥ-action ρ(Ĥ, 0, e) on Lµ̃(⊕Z2H). We get a Z2 ⋊ SL(2,Z)-action on N(H, µ) by β(k, γ)(x) = ρ(1, k, γ)(x), k ∈ Z2, γ ∈ SL(2,Z), x ∈ N(H, µ). This is the definition of the twisted Bernoulli shift action β = β(H, µ, χ) on N(H, µ). We obtained the actions β(H, µ, χ) not only by twisting generalized Bernoulli shift actions but also restricting to subalgebras N(H, µ) ⊂ Lµ̂(⊕Z2H) = Lµ(H). This restriction allows us to classify the actions up to conjugacy in the next section. CLASSIFICATION AND CENTRALIZERS 7 In order to give a variety of the actions, we twisted the shift actions by the character χ of the abelian group H . Remark 3.1. The action β|Z2 = β(H, µ, χ)|Z2 has the weak mixing property. In definition 2.5, we may assume that the Fourier coefficients of ai (i = 1, 2, · · · , n) are finitely supported, by approximating in the L2-norm. Then for appropriate k ∈ Z2, we get tr(aiβ(k)(aj)) = tr(ai)tr(aj), i, j = 1, 2, · · · , n. 4. Classification up to conjugacy In this section, we classify the twisted Bernoulli shift actions {β(H, µ, χ)} up to conjugacy (Theorem 4.1). We prove that an isomorphism which gives conjugacy between two twisted Bernoulli shift actions is of a very special form. In fact it comes from an isomorphism in the level of base groups H . We also determine the centralizer of the Z2 ⋊ SL(2,Z)-action β(H, µ, χ) on N(H, µ) (Theorem 4.4). We fix some notations for the proofs. We define 0, e1, e2 ∈ Z2 as , e1 = , e2 = Let ξ be the element of Z2 ⋊ SL(2,Z) satisfying ξ · 0 = e1, ξ · e1 = e2, ξ · e2 = 0. The elements ξ and ξ2 are explicitly described as −1 −1 −1 −1 The order of ξ is 3. Let η, δ ∈ SL(2,Z) be given by η = , δ = Let D be the subset of all elements of Z2 fixed under the action of δ, that is, n ∈ Z Then we get ξ ·D = n ∈ Z , ξ2 ·D = n ∈ Z We define the subgroup ΛD(H) of Λ(H) by ΛD(H) = {λ ∈ Λ(H) | λ : Z2 → H is supported on D}. Let (Ha, µa, χa) and (Hb, µb, χb) be triplets of countable abelian groups, their normalized 2-cocycles and characters. For h ∈ Ha, we define λh ∈ ΛD(Ha) as λh(k) = h (k = e1), −h (k = 0), 0 (k 6= e1, 0). 8 HIROKI SAKO For g ∈ Hb, we define σg ∈ ΛD(Hb) as σg(k) = g (k = e1), −g (k = 0), 0 (k 6= e1, 0). We denote by v(σ) ∈ N(Hb, µb) the unitary corresponding to σ ∈ Λ(Hb). Theorem 4.1. If π : N(Ha, µa) → N(Hb, µb) is a ∗-isomorphism giving conjugacy between βa = β(Ha, µa, χa) and βb = β(Hb, µb, χb), then there exists a group isomor- phism φ = φπ : Ha → Hb satisfying (1) π(u(λ)) = v(φ ◦ λ) mod T for λ ∈ Λ(Ha), (2) the 2-cocycles µa(·, ·) and µb(φ(·), φ(·)) of Ha are cohomologous, (3) χ2a = (χb ◦ φ)2. Conversely, given a group isomorphism φ : Ha → Hb satisfying (2) and (3), there exists a ∗-isomorphism π = πφ : N(Ha, µa) → N(Hb, µb) which satisfies condition (1) and gives conjugacy between βa, βb. We note that by Proposition 2.2 condition (2) for φ is equivalent to (2)′ µ∗aµa(g, h) = µ bµb(φ(g), φ(h)), g, h ∈ Ha. Proof for the first half of Theorem 4.1. Suppose that there exists a (not necessarily trace preserving) ∗-isomorphism π from N(Ha, µa) onto N(Hb, µb) such that π ◦ βa(γ) = βb(γ) ◦ π, γ ∈ Z2 ⋊ SL(2,Z). We prove that for every h ∈ Ha there exists φ(h) ∈ Hb satisfying π(u(λh)) = v(σφ(h)) mod T. Let Uh denote the unitary in N(Hb, µb) Uh = π µa(h,−h) u(λh) , h ∈ Ha. We identify N(H, µ) with the subalgebra of the infinite tensor product Lµ(H), which is canonically isomorphic to Lµ̃ (⊕Z2H). The preimage π−1(Uh) can be written as u∗h ⊗ uh. Here uh is the unitary corresponding to h ∈ Ha and placed on 1 ∈ Z2 and the unitary u∗h is placed on 0 ∈ Z2. We describe Uh as the Fourier expansion σ∈Λ(Hb) c(σ)v(σ). Since e1 and 0 are fixed under the action of δ, one has βb(δ) n(Uh) = π ◦ βa(δ)n(π−1(Uh)) = Uh. It follows that the Fourier expansion Uh = σ∈Λ(Hb) c(σ)v(σ) must satisfy that c(σ) = c(δ−n · σ) for every σ ∈ Λ(Hb) and n ∈ Z. For σ ∈ Λ(Hb) \ ΛD(Hb), the orbit of σ under the action of δ−1 is an infinite set, since the support supp(σ) ⊂ Z2 is not included in D. It turns out that c(σ) = 0 for all σ ∈ Λ(Hb) \ ΛD(Hb) due to Σ|c(σ)|2 = 1 < +∞, so that Uh = σ∈ΛD(Hb) c(σ)v(σ). The unitary χa(h)U h is also fixed under the action of δ and can be written as χa(h)U h = π χa(h)µa(h,−h)u(−λh) χa(h)µa(h,−h)u(ξ · λh) µa(h,−h)u(ξ2 · λh) = βb(ξ)(Uh) βb(ξ 2)(Uh). CLASSIFICATION AND CENTRALIZERS 9 Letting ne1 = (n, 0) T ∈ Z2, we get βb(ξ)(Uh) = βb(ξ) c(σ) v(σ) σ∈ΛD(Hb) c(σ) v(ξ · σ) χb(σ(ne1)) 2)(Uh) = βb(ξ c(σ) v(σ) σ∈ΛD(Hb) c(σ) v(ξ2 · σ). Since Fourier expansion admits algebraical calculation as in subsection 2.6, the ex- pansion of χa(h)U χa(h)U h = βb(ξ)(Uh) βb(ξ 2)(Uh) σ1,σ2∈ΛD(Hb) c(σ1) c(σ2) v(ξ · σ1) v(ξ2 · σ2) χb(σ1(ne1)) σ1,σ2∈ΛD(Hb) c(σ1) c(σ2) µ̃b(ξ · σ1, ξ2 · σ2) χb(σ1(ne1)) n v(ξ · σ1 + ξ2 · σ2). The map ΛD(Hb)×ΛD(Hb) ∋ (σ1, σ2) 7→ ξ ·σ1+ξ2·σ2 ∈ Λ(Hb) is injective. Indeed, σ1 is uniquely determined by ξ ·σ1+ξ2·σ2, since σ1(k) = (ξ ·σ1+ξ2·σ2)(ξ ·k), k ∈ D\{e1} and σ1(e1) = − k∈D\{e1} σ1(k). Here we used the condition σ1(k) = 0. The element σ2 is also determined by ξ · σ1 + ξ2 · σ2. Thus the index (σ1, σ2) uniquely determines ξ · σ1 + ξ2 · σ2. We take arbitrary elements σ1, σ2 ∈ ΛD(Hb) and suppose that c(σ1) 6= 0, c(σ2) 6= 0. Since the unitary χa(h)U h is invariant under the action of δ and the coefficient of ξ ·σ1+ξ2 ·σ2 is not zero, ξ ·σ1+ξ2 ·σ2 is supported on D. It follows that the elements σ1 and σ2 can be written as σ1 = σφ(h) = σ2, by some φ(h) ∈ Hb. Indeed, since the subsets D \ {0, e1}, ξD \ {e1, e2} and ξ2D \ {e2, 0} are mutually disjoint, the element ξ · σ1 must be supported on {e1, e2} and the element ξ2 · σ2 must be supported on {e2, 0}. By the assumption k∈Z2 σi(k) = 0 (i = 1, 2), σi can be written as σφ(hi). Then using the fact that (ξ · σ1 + ξ2 · σ2)(e2) = σ1(e1) + σ2(0) = 0, we get that σ1 = σφ(h) = σ2 for some h ∈ Hb. This means that there exists only one σ ∈ Λ(Hb) such that c(σ) 6= 0 and that it is of the form σ = σφ(h). Then the unitary Uh satisfies Uh = π(u(λh)) = v(σφ(h)) mod T. We claim that the map φ = φπ : Ha → Hb is a group isomorphism. For all h1, h2 ∈ Ha, we get π(u(λh1+h2)) = π(u(λh1)) π(u(λh2)) = v(σφ(h1)) v(σφ(h2)) = v(σφ(h1) + σφ(h2)) = v(σφ(h1)+φ(h2)) mod T. On the other hand, we get π(u(λh1+h2)) = v(σφ(h1+h2)) mod T. Since {v(σ)} are linearly independent, we get σφ(h1+h2) = σφ(h1)+φ(h2), and hence φ(h1 + h2) = φ(h1) + φ(h2). This means that the map φ is a group homomorphism. The bijectivity of the ∗- isomorphism π leads to that of the group homomorphism φ = φπ. Since {γ ·λh | γ ∈ Z2⋊SL(2,Z), h ∈ Ha} ⊂ Λ(Ha) generates Λ(Ha), we get π(u(λ)) = v(φ◦λ) mod T for λ ∈ Λ(Ha). 10 HIROKI SAKO We prove that the group isomorphism φ = φπ satisfies conditions (2) and (3) in the theorem. For all h ∈ Ha, there exists c(h) ∈ T satisfying Uh = π µa(h,−h) u(λh) = c(h)µb(φ(h),−φ(h)) v(σφ(h)). Since (e1, η) ∈ Z2 ⋊ SL(2,Z) acts on Z2 as (e1, η) · e1 = 0, (e1, η) · 0 = e1, we get Uh βb(e1, η)(Uh) = π µa(h,−h) u(λh)µa(h,−h) u(−λh) = µa(h,−h) µ̃a(λh,−λh) = 1. The following equation also holds: Uh βb(e1, η)(Uh) = c(h)µb(φ(h),−φ(h)) v(σφ(h)) c(h)µb(φ(h),−φ(h)) v(σ−φ(h)) = c(h)2 µb(φ(h),−φ(h)) µ̃b(σφ(h),−σφ(h)) = c(h)2. Thus we have c(h) ∈ {1,−1} for h ∈ Ha. Since ξ · e1 = e2, ξ · e2 = 0 and ξ · 0 = e1, we have Uh βb(ξ)(Uh) βb(ξ 2)(Uh) µa(h,−h) u(λh) χa(h)µa(h,−h) u(ξ · λh) µa(h,−h) u(ξ2 · λh) = χa(h). On the other hand, we have the following: Uh βb(ξ)(Uh) βb(ξ 2)(Uh) = c(h)µb(φ(h),−φ(h)) v(σφ(h)) c(h)χb(φ(h))µb(φ(h),−φ(h)) v(ξ · σφ(h)) c(h)µb(φ(h),−φ(h)) v(ξ2 · σφ(h)) = c(h)3 χb(φ(h)) = c(h)χb(φ(h)). It follows that c(h) = χb(φ(h))χa(h)(Eq2) and χb(φ(h)) 2 = χa(h) 2, for all h ∈ Ha. We recall that the algebra Lµ̃a(⊕Z2Ha) is canonically identified wit the infinite tensor product Lµa(Ha). The unitary π −1(Uh) ∈ N(Ha, µa) ⊂ Lµa(Ha) can be written as 1⊗ u∗h ⊗ uh, where 1 is placed on −e1 ∈ Z2, u∗h is placed on 0 and uh is placed on e1. Since η ∈ Z2 ⋊ SL(2,Z) acts on Z2 as η · e1 = −e1, η · 0 = 0, the unitary π−1(βb(η)(Ug)) can be written as ug ⊗ u∗g ⊗ 1. We have the following equation: Ug βb(η)(Uh)U g βb(η)(Uh) = π((1⊗ u∗g ⊗ ug)(uh ⊗ u∗h ⊗ 1)(1⊗ u∗g ⊗ ug)∗(uh ⊗ u∗h ⊗ 1)∗) = µ∗aµa(g, h). The unitary Uh can be written as c(h)(1⊗ v∗φ(h) ⊗ vφ(h)) ∈ N(Hb, µb) ⊂ Lµb(Hb). Here we write vφ(h) for the unitary in Lµb(Hb) corresponding to φ(h). The unitary βb(η)(Ug) can be written as c(g)(vφ(g) ⊗ v∗φ(g) ⊗ 1). Then we get Ug βb(η)(Uh)U g βb(η)(U = (1⊗ v∗φ(g) ⊗ vφ(g))(vφ(h) ⊗ v∗φ(h) ⊗ 1)(1⊗ v∗φ(g) ⊗ vφ(g))∗(vφ(h) ⊗ v∗φ(h) ⊗ 1)∗ = µ∗bµb(φ(g), φ(h)). CLASSIFICATION AND CENTRALIZERS 11 Thus we get µ∗aµa(g, h) = µ bµb(φ(g), φ(h)), for all g, h ∈ Ha. We proved that the group isomorphism φ = φπ satisfies conditions (1), (2) and (3). � From a group homomorphism which satisfies conditions (2) and (3), we construct a ∗-homomorphism from N(Ha, µa) to N(Hb, µb) with condition (1). In the con- struction, the function µ̂ on Λ(H) given below is useful. We fix an index for Z2 as Z2 = {k0, k1, k2, · · · } throughout the rest of this section. For a scalar 2-cocycle µ of H , we define the function µ̂ by µ̂(λ) = λ(ki), λ(kj) , λ ∈ Λ(H), where λ is supported on {k0, k1, k2, · · · , kn}. This definition depends on the choice of an order on Z2. Since i λ(ki) = 0, the function µ̂ is also given by the following relation in Cµ(H): µ̂(λ)1 = uλ(k0)uλ(k1)uλ(k2) · · ·uλ(kn), λ ∈ Λ(H). If µ is a coboundary, then the definition of µ̂ does not depend on the order on Z2, since Cµ(H) is commutative. Lemma 4.2. Let µ0 be another normalized scalar 2-cocycle for H. Let µ̃0 be the scalar 2-cocycle on Λ(H)×Λ(H) given in the same way as equation (Eq1) in subsec- tion 2.2 and let µ̂0 be the function on Λ(H) constructed from µ0 in the above manner. If the scalar 2-cocycles µ and µ0 are cohomologous, then for all λ1, λ2 ∈ Λ(H), we have the equation µ̃(λ1, λ2) µ̂(λ1) µ̂(λ2) µ̂(λ1 + λ2) = µ̃0(λ1, λ2) µ̂0(λ1) µ̂0(λ2) µ̂0(λ1 + λ2). Proof. We denote by {ν(g, h)} the scalar 2-cocycle {µ0(g, h)µ(g, h)} of H . Since ν is a 2-coboundary, there exists {c(g)}g∈H ⊂ T satisfying ν(g, h) = b(g)b(h)b(g + h). Then the map ν̂ becomes ν̂(λ) = b(λ(ki)). Since ν̂(λ1) ν̂(λ2) = b(λ1(ki)) b(λ2(ki)), ν̃(λ1, λ2) = b(λ1(ki)) b(λ2(ki)) b(λ1(ki) + λ2(ki)), ν̂(λ1 + λ2) = b(λ1(ki) + λ2(ki)), we get ν̂(λ1) ν̂(λ2) = ν̃(λ1, λ2) ν̂(λ1 + λ2). By the definitions of µ̃, µ̃0, µ̂ and µ̂0, the maps ν̂ and ν̃ are given by ν̂(λ) = µ̂(λ) µ̂0(λ), ν̃(λ1, λ2) = µ̃(λ1, λ2) µ̃0(λ1, λ2), Thus the desired equality immediately follows. � Proof for the second half of Theorem 4.1. Suppose that there exists a group isomorphism φ satisfying conditions (2) and (3) in the theorem. We prove that there exists a ∗-isomorphism π = πφ from N(Ha, µa) onto N(Hb, µb) preserving the Z 2 ⋊ SL(2,Z)-actions with condition (1). We define a group homomorphism cφ from Ha to {1,−1} ⊂ T by cφ(h) = χb(φ(h))χa(h), h ∈ Ha. 12 HIROKI SAKO Let c̃φ be the group homomorphism from Λ(Ha) to {1,−1} ⊂ T given by c̃φ(λ) = cφ(λ(k)) gcd(k) = χa(λ(k)) gcd(k)χb(φ(λ(k))) gcd(k) , λ ∈ Λ(Ha). We define a linear map π from the group algebra Cµ̃a(Λ(Ha)) onto Cµ̃b(Λ(Hb)) by µ̂a(λ) u(λ) = c̃φ(λ) µ̂b(φ ◦ λ) v(φ ◦ λ), λ ∈ Λ(Ha). By direct computations, for all λ1, λ2 ∈ Λ(Ha), we get µ̂a(λ1) u(λ1) µ̂a(λ2)u(λ2) = c̃φ(λ1) c̃φ(λ2) µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) v(φ ◦ λ1) v(φ ◦ λ2) = c̃φ(λ1 + λ2) µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) µ̃b(φ ◦ λ1, φ ◦ λ2) v(φ ◦ (λ1 + λ2)). On the other hand, we have the following equation: µ̂a(λ1) µ̂a(λ2)u(λ1) u(λ2) µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) u(λ1 + λ2) = µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) π µ̂a(λ1 + λ2) u(λ1 + λ2) = c̃φ(λ1 + λ2) µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) µ̂b(φ ◦ (λ1 + λ2)) v(φ ◦ (λ1 + λ2)). By Lemma 4.2 and condition (2) for the group isomorphism φ in the theorem, we have that µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) µ̃b(φ ◦ λ1, φ ◦ λ2) = µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) µ̂b(φ ◦ (λ1) + φ ◦ (λ2)). Therefore we get π(u(λ1)) π(u(λ2)) = π(u(λ1) u(λ2)). The linear map π also pre- serves the ∗-operation. As a consequence, π is a ∗-isomorphism from Cµ̃a(Λ(Ha)) onto Cµ̃b(Λ(Hb)) and this preserves the trace. The map π = πφ is extended to a normal ∗-isomorphism from N(Ha, µa) onto N(Hb, µb). We next prove that this π preserves the Z2 ⋊ SL(2,Z)-actions. The group ho- momorphism c̃φ from Λ(Ha) to {1,−1} is invariant under the action of SL(2,Z), by Lemma 2.1 (1). The scalar 2-cocycle ν(g, h) = µa(g, h)µb(φ(h), φ(g)) satisfies ν(g, h) = ν(h, g) by condition (2), so the function ν̂(·) = µ̂a(·)µ̂b(φ ◦ ·) on Λ(Ha) does not depend on the order on Z2 chosen before. Since π ◦ βa(γ)(u(λ)) = π(u(γ · λ)) = c̃φ(γ · λ)µ̂a(γ · λ)µ̂b(φ ◦ (γ · λ))v(φ ◦ (γ · λ)) = c̃φ(λ)µ̂a(λ)µ̂b(φ ◦ λ)v(γ · (φ ◦ λ)) = βb(γ) c̃φ(λ)µ̂a(λ)µ̂b(φ ◦ λ)v(φ ◦ λ) = βb(γ) ◦ π(u(λ)), γ ∈ SL(2,Z), λ ∈ Λ(Ha), it turns out that the ∗-isomorphism π preserves the SL(2,Z)-action. CLASSIFICATION AND CENTRALIZERS 13 For all λ ∈ Λ(Ha) and k ∈ Z2, we have π ◦ βa(k)(u(λ)) = π χa(λ(l)) det(k,l)u(k · λ) cφ((k · λ)(l))gcd(l) χa(λ(l)) det(k,l)µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(φ ◦ (k · λ)). Since cφ(h) det(k,l)cφ(h) gcd(k+l) = cφ(h) gcd(k)cφ(h) gcd(l), by Lemma 2.1 (2), the unitary π ◦ βa(k)(u(λ)) equals to cφ(λ(l)) gcd(k+l) cφ(λ(l)) det(k,l)χb(φ ◦ λ(l))det(k,l) µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(φ ◦ (k · λ)) cφ(λ(l)) gcd(k) cφ(λ(l)) gcd(l) χb(φ ◦ λ(l))det(k,l) µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(k · (φ ◦ λ)) = c̃φ(λ) µ̂a(k · λ) µ̂b(k · (φ ◦ λ))βb(k)(v(φ · λ)) = βb(k) ◦ π(u(λ)). This means that the ∗-isomorphism π preserves the Z2-actions. We get the ∗-isomorphism π = πφ fromN(Ha, µa) ontoN(Hb, µb) giving conjugacy between βa and βb. � Remark 4.3. The proof of the first half of Theorem 4.1 shows that any isomorphism π giving conjugacy between βa and βb is of the form πφ. This means that an isomorphism which gives conjugacy between two twisted Bernoulli shift actions must be trace preserving. This proof shows that an isomorphism giving conjugacy between the two actions β(Ha, µa, χa), β(Hb, µb, χb) is of a very special form derived from a group isomor- phism between Ha and Hb. Taking notice of this fact, we can describe the centralizer of a twisted Bernoulli shift action. We define two topological groups before we state Theorem 4.4. Let β be a trace preserving action of some group Γ on a separable finite von Neumann algebra (N, tr). We denote by Aut(N, β) the group of all automorphisms which commute with the action β, that is, {α ∈ Aut(N) | β(γ) ◦ α = α ◦ β(γ), γ ∈ Γ}. We regard the group Aut(N, β) as a topological group equipped with the pointwise- strong topology. When β is a twisted Bernoulli shift action on N , an automorphism α commuting with β is necessarily trace preserving by Remark 4.3. We consider that Aut(N, β) is equipped with the pointwise-2-norm topology. Let Aut(H, µ, χ) be the group of all automorphisms of an abelian group H which preserve its 2-cocycle µ and character χ, that is, {φ ∈ Aut(H) | µ(g, h) = µ(φ(g), φ(h)), χ(g) = χ(φ(g)), g, h ∈ H}. We define the topology of Aut(H, µ, χ) by pointwise convergence. 14 HIROKI SAKO Theorem 4.4. For π ∈ Aut(N(H, µ), β(H, µ, χ)), there exists a unique element φ = φπ ∈ Aut(H, µ∗µ, χ2) satisfying π(u(λ)) = u(φ ◦ λ) mod T for λ ∈ Λ(H). The map π 7→ φπ gives an isomorphism between two topological groups Aut(N(H, µ), β(H, µ, χ)) ∼= Aut(H, µ∗µ, χ2). Proof. We use the notations in the proof of the previous theorem letting Ha = Hb = H , µa = µb = µ and χa = χb = χ. Denote N = N(H, µ) and β = β(H, µ, χ). We have already have shown the first claim. Let Aut(H, µ∗µ, χ2) ∋ φ 7→ πφ ∈ Aut(N, β), be the map given as in the proof of Theorem 4.1, that is, µ̂(λ)u(λ) = c̃φ(λ) µ̂(φ ◦ λ)u(φ ◦ λ), λ ∈ Λ(H), where c̃φ(λ) = χ(λ(k))gcd(k) χ(φ ◦ λ(k)) gcd(k) It is easy to prove that φπφ = φ by the definition. Thus the map π 7→ φπ is surjective. This map is also injective. Let φ be an element of Aut(H, µ∗µ, χ2). Suppose that π is an arbitrary element of Aut(N, β) satisfying φ = φπ. The set {β(γ)(u(λh)) | h ∈ H, γ ∈ Z2 ⋊ SL(2,Z)} generates N , so we have only to prove the uniqueness of c(h) ∈ T satisfying µ(h,−h) u(λh) = c(h)µ(φ(h),−φ(h))u(λφ(h)), for all h ∈ H . In the proof of the first half of the previous theorem (equation (Eq2)), we have already shown that c(h) = χ(h)χ(φ(h)). Thus the ∗-isomorphism π is uniquely determined and the map π 7→ φπ is injective. We prove the two maps φ 7→ πφ and π 7→ φπ are continuous. Let (φi) be a net in Aut(H, µ∗µ, χ2) converging to φ. For all h ∈ H , we have χ(h)µ(h,−h) u(λh) = χ(φi(h))µ(φi(h),−φi(h))u(λφi(h)). The right side of the equation converges to χ(φ(h))µ(φ(h),−φ(h))u(λφ(h)) = πφ χ(h)µ(h,−h) u(λh) This proves that πφi converges to πφ in pointwise 2-norm topology on the generating set {β(γ)(u(λh)) | h ∈ H, γ ∈ Z2 ⋊SL(2,Z)} of N . Thus πφi converges to πφ on N . Conversely, let (πi) be a net in Aut(N, β) converging to π. For all h ∈ H , we get πi(u(λh)) = u(λφπi(h)) mod T. The left side of the equation converges to π(u(λh)) = u(λφπ(h)). If φπi(h) 6= φπ(h), then the distance between Tu(λφπ(h)) and Tu(λφπi (h)) is 2 in the 2-norm. Thus φπi(h) = φπ(h) for large enough i. This means that (φπi) converges to φπ. As a consequence, the two maps φ 7→ πφ and π 7→ φπ are continuous group homomorphisms and inverse maps of each other. � CLASSIFICATION AND CENTRALIZERS 15 5. Examples 5.1. Twisted Bernoulli shift actions on L∞(X). In this subsection, we consider the case of µ = 1 and H 6= {0}. Then the algebra N(H, 1) is abelian and has a faithful normal state, so it is isomorphic to L∞(X), whereX is a standard probability space. The measure of X is determined by the trace on N(H, 1). Furthermore, X is non-atomic, since N(H, 1) is infinite dimensional and the action β(H, 1, χ) is ergodic. As corollaries of Theorems 4.1 and 4.4, we get trace preserving Z2⋊SL(2,Z)-actions on L∞(X) whose centralizers are isomorphic to some prescribed groups. Remark 5.1. The Z2 ⋊ SL(2,Z)-action on X defined by β = β(H, 1, χ) is free. An automorphism α ∈ Aut(L∞(X), β) is free or the identity map for any twisted Bernoulli shift action β on L∞(X). This is proved as follows. We identify β(γ) (γ ∈ Z2 ⋊ SL(2,Z)) and α with measure preserving Borel isomorphisms on X here. Suppose that there exists a non-null Borel subset Y ⊂ X whose elements are fixed under α. All elements in Ỹ = ∪{β(γ)(Y ) | γ ∈ Z2⋊SL(2,Z)} are fixed under α. By the ergodicity of β, the measure of Ỹ is 1. Then α is the identity map of L∞(X). Corollary 5.2. For any abelian countable discrete group H 6= {0}, there exists a trace preserving essentially free ergodic action β of Z2⋊SL(2,Z) on L∞(X) satisfying Aut(L∞(X), β) ∼= Aut(H). Proof. When we define β = β(H, 1, 1), we have the above relation by Theorem 4.4. � In the next corollary we use the effect of twisting by a character χ. Corollary 5.3. For every abelian countable discrete group H 6= {0}, there exist continuously many trace preserving essentially free ergodic actions {βc} of Z2 ⋊ SL(2,Z) on L∞(X) which are mutually non-conjugate and satisfy Aut(L∞(X), βc) ∼= H ⋊Aut(H). Here the topology of H ⋊ Aut(H) is the product of the discrete topology on H and the pointwise convergence topology on Aut(H). Proof. Let c ∈ {eiπt | t ∈ (0, 1/2) \Q}. We put βc = β(H ⊕ Z, 1, 1× χc), where the character χc of Z is defined as χc(n) = c n. By Theorem 4.4, we get Aut(L∞(X), βc) ∼= Aut(H ⊕ Z, 1, 1× χ2c). Since the character χ2c is injective, a group automorphism α ∈ Aut(H⊕Z, 1, 1×χ2c) preserves the second entry. For all α ∈ Aut(H⊕Z, 1, 1×χ2c), there exist φα ∈ Aut(H) and hα ∈ H satisfying α(h, n) = (φα(h) + nhα, n), (h, n) ∈ H ⊕ Z. The map Aut(H ⊕ Z, 1, 1× χ2c) ∋ α 7→ (hα, φα) ∈ H ⋊ Aut(H) is a homeomorphic group isomorphism. If c1, c2 ∈ {eiπt | t ∈ (0, 1/2) \ Q} and c1 6= c2, then there exists no isomorphism from H ⊕ Z to H ⊕ Z whose pull back of the character 1 × χ2c2 is equal to 1 × χ The two actions βc1 and βc2 are not conjugate by Theorem 4.1. � 16 HIROKI SAKO Corollary 5.4. There exist continuously many trace preserving essentially free er- godic actions {βc} of Z2 ⋊ SL(2,Z) on L∞(X) which are mutually non-conjugate and have the trivial centralizer Aut(L∞(X), βc) = {idL∞X}. Proof. Let {χc | c = eiπt, t ∈ (0, 1/2)} be characters of Z such that χc(m) = cm. Since χ2c(1) is in the upper half plane, the identity map is the only automorphism of Z preserving χ2c . By Theorem 4.4, we get Aut(β(Z, 1, χc)) = {id}. If β(Z, 1, χc1), β(Z, 1, χc2) are conjugate, then there exists a group isomorphism on Z whose pull back of χ2c2 is χ by Theorem 4.1. This means c1 = c2. Thus the actions {β(Z, 1, χc)} are mutually non-conjugate. � 5.2. Twisted Bernoulli shift actions on the AFD factor of type II1. Firstly, we find a condition that the finite von Neumann algebra N(H, µ) is the AFD factor of type II1. Lemma 5.5. For an abelian countable discrete group H 6= {0} and its normalized scalar 2-cocycle µ, the following statements are equivalent: (1) The algebra N(H, µ) is the AFD factor of type II1. (2) The group von Neumann algebra Lµ(H) twisted by the scalar 2-cocycle µ is a factor (of type II1 or In). (3) For all g ∈ H \ {0}, there exists h ∈ H such that µ(g, h) 6= µ(h, g). Proof. The amenability of the group Λ(H) leads the injectivity for N(H, µ). The injectivity for N(H, µ) implies that N(H, µ) is approximately finite dimensional ([Co]). We have only to show the equivalence of conditions (2), (3) and (1)′ The algebra N(H, µ) is a factor. By using Fourier expansion it is easy to see that condition (2) holds true if and only if for any g ∈ H \ {0} there exists h ∈ H satisfying uguh 6= uhug. This is equivalent to condition (3). Similarly, condition (1)′ is equivalent to (1)′′ For any λ1 ∈ Λ(H) \ {0}, there exists λ2 ∈ Λ(H) satisfying µ̃(λ2, λ1)µ̃(λ1, λ2) 6= 1. Suppose condition (3). For any λ1, choose element k, l ∈ Z2 so that k ∈ supp(λ1) and l /∈ supp(λ1). By condition (3), there exists h ∈ H satisfying µ∗µ(λ1(k), h) 6= 1. Let λ2 be the element in Λ(H) which takes h at k, −h at l and 0 for the other places. The element λ2 satisfies µ̃(λ2, λ1)µ̃(λ1, λ2) = µ ∗µ(λ1(k), h) 6= 1. Here we get condition (1)′′. The implication from (1)′′ to (3) is easily shown. � Remark 5.6. The twisted Bernoulli shift action β = β(H, µ, χ) is an outer action of Z2 ⋊ SL(2,Z). Any non-trivial automorphism in Aut(R, β(H, µ, χ)) is also outer. This is proved by the weak mixing property of the action β(H, µ, χ) as follows. If α ∈ Aut(R, β(H, µ, χ)) is an inner automorphism Ad(u), then we have Ad(β(γ)(u))(x) = β(γ)(uβ(γ)−1(x)u∗) = β(γ) ◦ α ◦ β(γ)−1(x) = α(x) = Ad(u)(x), for all x ∈ R and γ ∈ Z2 ⋊ SL(2,Z). Since Ad(β(γ)(u)u∗) = id, Cu ⊂ R is an invariant subspace of the action β. The only subspace invariant under the weakly mixing action β is C1 (Proposition 2.6), thus we get α = id. CLASSIFICATION AND CENTRALIZERS 17 Using Theorems 4.1 and 4.4, we give continuously many actions of Z2 ⋊ SL(2,Z) on R such that there exists no commuting automorphism except for trivial one. Corollary 5.7. There exist continuously many ergodic outer actions {βc} of Z2 ⋊ SL(2,Z) on the AFD factor R of type II1 which are mutually non-conjugate and have the trivial centralizer Aut(R, βc) = {idR}. Proof. We can choose and fix a character χ on Z2 such that χ2 is injective. Let {µc | c = eiπt, t ∈ (0, 1/2) \Q} be scalar 2-cocycles for Z2 defined by = cs1t2−t1s2, s1, t1, s2, t2 ∈ Z. We put βc = β(Z 2, µc, χ). The 2-cocycle µc satisfies condition (3) in Lemma 5.5. Thus βc defines a Z 2 ⋊ SL(2,Z)-action on R. By Theorem 4.4, we get the following isomorphism between topological groups: Aut(R, βc) ∼= Aut(Z2, µ∗cµc, χ2) = Aut(Z2, µc2, χ2). Since the character χ2 of Z2 is injective, so the group of the right side is {id|Z2}. This means that the action βc has trivial centralizers. Finally, we prove that the actions {βc | c = eiπt, t ∈ (0, 1/2) \ Q} are mutually non-conjugate. Suppose that actions βc1 and βc2 are conjugate. By Theorem 4.1, there exists a group isomorphism φ of Z2 satisfying (g, h) = µc2 (φ(g), φ(h)), g, h ∈ Z2. A group isomorphism of Z2 is given by an element of GL(2,Z). If the automorphism φ is given by an element of SL(2,Z), we get c22 = c 1. If φ is given by an element of GL(2,Z) \ SL(2,Z), then we get c22 = −c21. Since both c1 and c2 have the form eiπt, t ∈ (0, 1/2), we get c1 = c2. � Any cyclic group of an odd order can be realized as the centralizer of a twisted Bernoulli shift actions on R. Corollary 5.8. Let q be an odd natural number ≥ 3 and denote by Hq the abelian group (Z/qZ)2. We define the 2-cocycle µq and the character χq on Hq as = exp (2πis1t2/q), χq = exp (2πis1/q). Then the algebra N(Hq, µq) is the AFD factor R of type II1 and the centralizer of the twisted Bernoulli shift action βa = β(Hq, µq, χq) is isomorphic to Z/qZ. Proof. By Lemma 5.5, it is shown that the algebra N(HQ, µQ) is the AFD factor of type II1. Using Theorem 4.4, we have only to prove that Aut(Hq, µ qµq, χ ∼= Z/qZ. Let φ be in Aut(Hq, µ qµq, χ q). The automorphism φ of Hq is given by a 2 × 2 matrix A of Z/qZ. Since φ preserves µ∗qµq, the determinant of A must be 1. Since q is odd, the value of χ2q determines the first entry of (Z/qZ) 2 and φ preserves χ2q . The matrix A is of the form( , tφ ∈ Z/qZ. The map φ 7→ tφ is an isomorphism. In turn, if the matrix A is of this form, it defines an element in Aut(Hq, µ qµq, χ q). � 18 HIROKI SAKO Corollary 5.9. For a set Q consisting of odd prime numbers, let βQ be the tensor product q∈Q βq of the actions βq on the AFD factor of type II1. The centralizer of βQ is isomorphic to q∈Q Z/qZ. Proof. The action βQ is the twisted Bernoulli shift action β(HQ, µQ, χQ), where HQ is the abelian group ⊕q∈QHq and the scalar 2-cocycle µq and a character χQ on HQ are given by µQ((sq), (tq)) = µq(sq, tq), χQ((sq)) = χq(sq), (sq), (tq) ∈ HQ, sq, tq ∈ Hq. Using Theorem 4.4, we have only to prove Aut(HQ, µ QµQ, χ Z/qZ. A group automorphism φ of HQ = ⊕q∈QHq has a form φ((kq)) = (φq(kq)), for some {φq ∈ Aut(Hq)}. Thus we get Aut(HQ, µ QµQ, χ Aut(Hq, µ qµq, χ Together with the previous corollary, we get the conclusion. � Remark 5.10. If Q1 6= Q2, then the two groups Z/qZ and Z/qZ are not isomorphic. The continuously many outer actions {βQ} are distinguished in view of conjugacy only by using the centralizers {Aut(R, βQ)}. 6. Malleability and rigidity arguments In this section, we give malleability and rigidity type arguments invented by S. Popa, in order to examine weak 1-cocycles for actions. See Popa [Po2], [Po3], [Po4] and Popa–Sasyk [PoSa] for the references. S. Popa in [Po3] showed that every 1-cocycle for a Connes-Størmer Bernoulli shift by property (T) group (or w-rigid group like Z2 ⋊ SL(2,Z)) vanishes modulo scalars. As a consequence, two such actions are cocycle conjugate if and only if they are conjugate. In our case, 1- cocycles do not vanish modulo scalars but they are still in the situation that cocycle (outer) conjugacy implies conjugacy. We need the following notion to examine outer conjugacy of two group actions. Definition 6.1. Let α be an action of discrete group Γ on a von Neumann algebra M. A weak 1-cocycle for α is a map w : Γ → U(M) satisfying wgh = wgαg(wh) mod T, g, h ∈ Γ. The weak 1-cocycle w is called a weak 1-coboundary if there exists a unitary v ∈ U(M) satisfying wg = vαg(v)∗ mod T. Two weak 1-cocycles w and w′ are said to be equivalent when w′g = vwgαg(v) ∗ mod T for some v ∈ U(M). Let N be a finite von Neumann algebra with a faithful normal trace. The following is directly obtained by combining Lemmas 2.4 and 2.5 in [PoSa], although these Lemmas were proved for Bernoulli shift actions on standard probability space. The CLASSIFICATION AND CENTRALIZERS 19 following can be also regarded as a weak 1-cocycle version of Proposition 3.2 in [Po4]. Proposition 6.2. Let G be a countable discrete group. Let β be a trace preserving weakly mixing action of G on N . A weak 1-cocycle {wg}g∈G ⊂ N for β is a weak 1-coboundary if only if there exists a non-zero element x̃0 ∈ N ⊗N satisfying (wg ⊗ 1)(βg ⊗ βg)(x̃0)(1⊗ w∗g) = x̃0, g ∈ G. The following is a weak 1-cocycle version of Proposition 3.6.3◦ in [Po4]. Proposition 6.3. Let Γ be a countable discrete group and G be a normal subgroup of Γ. The group Γ acts on a finite von Neumann algebra N in a trace-preserving way by β. Suppose that the restriction of β to G is weakly mixing. Let {wγ}γ∈Γ be a weak 1-cocycle for β. If w|G is a weak 1-coboundary, then w is a weak 1-coboundary for the Γ-action. Proof. Suppose that w|G is a weak 1-coboundary, that is, there exists a unitary element v in N such that wg = vβg(v ∗) mod T for g ∈ G. It suffices to show that {w′γ} = {v∗wγβγ(v)} is in T for all γ ∈ Γ. Take arbitrary γ ∈ Γ, g ∈ G. Write h = γ−1gγ ∈ G. Let πγ be the unitary on L2(N) induced from βγ . Since w′h, w′g ∈ T, we get w′γπgw = (w′γπγ)(w hπh)(w ∗ = w′γhγ−1πγhγ−1 = πg mod T, By applying these operators to 1̂ ∈ N̂ ⊂ L2(N), it follows that w′γβg(w′γ ) ∈ T. Since the G-action is weakly mixing, we have w′γ ∈ T. � By using the above propositions, we will “untwist” some weak 1-cocycles later. We require some ergodicity assumption on the weak 1-cocycles. Definition 6.4. Let Γ be a discrete group and G be a subgroup of Γ. Suppose that its restriction to G is ergodic. Let β be a trace preserving action of Γ on N . A weak 1-cocycle w = {wg}g∈Γ for β is said to be ergodic on G, if the action βw of G is still ergodic, where βw is defined by βwg = Adwg ◦ βg, g ∈ G. Let β be a Γ-action onN . Suppose that the diagonal action β⊗β on (N⊗N, tr⊗tr) has an extension β̃ on a finite von Neumann algebra (Ñ , τ). The algebra Ñ is not necessarily identical with N⊗N . When the action β̃ is ergodic on a normal subgroup G ⊂ Γ, we get the following: Proposition 6.5. Let {wγ}γ∈Γ ⊂ N be a weak 1-cocycle for β. Let α be a trace preserving continuous action of R on Ñ satisfying the following properties: • α1(x⊗ 1) = 1⊗ x, for all x ∈ N . • αt ◦ β̃(γ)(x̃) = β̃(γ) ◦ αt(x̃), for all t ∈ R, γ ∈ Γ and x̃ ∈ Ñ . Suppose that the weak 1-cocycle {wγ⊗1} ⊂ Ñ is ergodic for the G-action β̃|G. If the group inclusion G ⊂ Γ has the relative property (T) of Kazhdan, then there exists a non-zero element x̃0 ∈ Ñ so that (wg ⊗ 1)β̃g(x̃0)(1⊗ w∗g) = x̃0, g ∈ G. This is proved in the same way for Bernoulli shift actions on the infinite tensor product of abelian von Neumann algebras ([PoSa], Lemma 3.5). Since we are in- terested in actions on the AFD II1 factor, we require the ergodicity assumption on 20 HIROKI SAKO weak 1-cocycle {wγ ⊗ 1}. For the self-containedness and in order to make it clear where the ergodicity assumption works, we write down a complete proof. Proof. For t ∈ (0, 1], let Kt be the convex weak closure of {(wg ⊗ 1)αt(w∗g ⊗ 1) | g ∈ G} ⊂ Ñ and x̃t ∈ Kt be the unique element whose 2-norm is minimum in Kt. Since (wg ⊗ 1)β̃g((wg1 ⊗ 1)αt(w∗g1 ⊗ 1))αt(w g ⊗ 1) = (wgβg(wg1)⊗ 1)αt(βg(w∗g1)w g ⊗ 1) = (wgg1 ⊗ 1)αt(w∗gg1 ⊗ 1), g, g1 ∈ G, we have (wg ⊗ 1)β̃g(Kt)αt(w∗g ⊗ 1) = Kt, for g ∈ G. By the uniqueness of x̃t, we get (wg ⊗ 1)β̃g(x̃t)αt(w∗g ⊗ 1) = x̃t, g ∈ G.(Eq3) By the assumption, the action (Ad(wg ⊗ 1) ◦ β̃g)g∈G is ergodic on Ñ . By the calcu- lation (wg ⊗ 1)β̃g(x̃tx̃t∗)(w∗g ⊗ 1) = (wg ⊗ 1)β̃g(x̃t)αt(w∗g ⊗ 1)αt(wg ⊗ 1)β̃g(x̃t )(w∗g ⊗ 1) = x̃tx̃t , g ∈ G, we get x̃tx̃t ∗ ∈ C1. The element x̃t is a scalar multiple of a unitary in Ñ . We shall next prove that x̃1/n is not zero for some positive integer n. The pair (Γ, G) has the relative property (T) of Kazhdan. By proposition 2.4, we can find a positive number δ and a finite subset F ⊂ Γ satisfying the following condition: If a unitary representation (π,H) of Γ and a unit vector ξ of H satisfy ‖π(γ)ξ − ξ‖ ≤ δ (γ ∈ F ), then ‖π(g)ξ− ξ‖ ≤ 1/2 (g ∈ G). By the continuity of the action α, there exists n such that ‖(wγ ⊗ 1)α1/n(w∗γ ⊗ 1)− 1‖tr,2 ≤ δ, γ ∈ F, The actions β and (αl1/n)l∈Z on Ñ give a Γ× Z action on Ñ . Let P be the crossed product von Neumann algebra P = Ñ ⋊ (Γ × Z). Let (Uγ)γ∈Γ and W be the implementing unitaries in P for Γ and 1 ∈ Z respectively. We put Vγ = (wγ ⊗ 1)Uγ , γ ∈ Γ. We regard AdV· as a unitary representation of Γ on L2(P ). Since ‖AdVγ(W )−W‖L2(P ) = ‖(wγ ⊗ 1)W (w∗γ ⊗ 1)W ∗ − 1‖L2(P ) = ‖(wγ ⊗ 1)α1/n(w∗γ ⊗ 1)− 1‖L2(Ñ) ≤ δ, γ ∈ F, we have the following inequality: 1/2 ≥ ‖AdVg(W )−W‖L2(P ) = ‖(wg ⊗ 1)α1/n(w∗g ⊗ 1)− 1‖L2(Ñ), g ∈ G. We get 1/2 ≥ ‖x̃1/n − 1‖L2(Ñ) and x̃1/n 6= 0. Let ũ1/n be the unitary of Ñ given by a scalar multiple of x̃1/n. By equation (Eq3), the unitary satisfies (wg ⊗ 1)β̃g(ũ1/n)α1/n(w∗g ⊗ 1) = ũ1/n, g ∈ G. CLASSIFICATION AND CENTRALIZERS 21 Let x̃0 be the unitary defined by x̃0 = ũ1/nα1/n(ũ1/n)α2/n(ũ1/n) . . . α(n−1)/n(ũ1/n). By direct computations, we have the following desired equality: (wg ⊗ 1)β̃g(x̃0)(1⊗ w∗g) = (wg ⊗ 1)β̃g(x̃0)α1(w∗g ⊗ 1) = x̃0, g ∈ G. Theorem 6.6. Let β = β(H, µ, χ) be a twisted Bernoulli shift action on N(H, µ). Suppose that N(H, µ) is the AFD factor of type II1 and that there exists a continuous R-action (α t )t∈R on Lµ(H)⊗ Lµ(H) satisfying the following properties: • For any x ∈ Lµ(H), α(0)1 (x⊗ 1) = 1⊗ x, • The automorphism α(0)t commutes with the diagonal action of Ĥ. Let β(1) be another action of Z2 ⋊ SL(2,Z) on the AFD factor N (1) of type II1 and suppose that its restriction to Z2 is ergodic. The action β(1) is outer conjugate to β, if and only if β(1) is conjugate to β. Proof. We deduce from outer conjugacy to conjugacy in the above situation. Let θ be a ∗-isomorphism from N (1) onto N(H, µ) which gives the outer conjugacy of the action β(1) and β = β(H, µ, χ). There exists a weak 1-cocycle {wγ}γ∈Z2⋊SL(2,Z) for β satisfying θ ◦ β(1)(γ) = Adwγ ◦ β(γ) ◦ θ, γ ∈ Z2 ⋊ SL(2,Z). Since the action β(1) is ergodic on Z2, the weak 1-cocycle w is ergodic on Z2. We use the notations Γ0, G0 given in Section 3. Let ρ̃ be the diagonal action ρ⊗ρ of Γ0 on the tensor product algebra M̃ = Lµ̃(⊕Z2H)⊗ Lµ̃(⊕Z2H): ρ̃(γ0)(a⊗ b) = ρ(γ0)(a)⊗ ρ(γ0)(b). The fixed point algebra Ñ ⊂ M̃ of the diagonal Ĥ-action contains N(H, µ) ⊗ N(H, µ). Since Z2⋊SL(2,Z) = Γ0/Ĥ, the action ρ̃ gives a Z 2⋊SL(2,Z)-action β̃ on Ñ . The action β̃ is the extension of the diagonal action β⊗β on N(H, µ)⊗N(H, µ). We denote by αt the action on M̃ ∼= (Lµ(H)⊗Lµ(H)) given by the infinite tensor product of the R-action α t . By the assumption on α t , the R-action αt commutes with the action ρ̃. It follows that the subalgebra Ñ is globally invariant under αt. The set of unitary {Wγ = wγ ⊗ 1}γ∈Z2⋊SL(2,Z) ⊂ Ñ is a weak 1-cocycle for β̃. We shall prove that this weak 1-cocycle is ergodic on Z2. Let a be an element in Ñ fixed under β̃|Z2 . The element a can be written as a = H aλ⊗u(λ) in L2M̃ , where aλ ⊗ 1 = EM⊗C(a(1⊗ u(λ))∗). Since a is fixed under the action of Z2, we have a = β̃W (k)(a) = Adwk ◦ ρ(1, k)(aλ)⊗ ρ(1, k)(u(λ)) Adwk ◦ ρ(1, k)(aλ)⊗ χ(λ(l))det(k,l)u(k · λ). Since Adwk ◦ ρ(1, k) preserves the 2-norm, we get ‖aλ‖2 = ‖ak−1·λ‖2. Since ‖a‖22 =∑ ‖aλ‖22 < ∞ and the set {ak−1·λ | k ∈ Z2} is infinite for λ 6= 0, it turns out that aλ = 0 for λ 6= 0 and thus a ∈ Ñ ∩ (M ⊗ C) = N(H, µ) ⊗ C. By the ergodicity of 22 HIROKI SAKO the Z2-action {Adwk ◦ β(k)}, we get a ∈ C. We conclude that the weak 1-cocycle {Wγ} ⊂ Ñ is ergodic on Z2. By the relative property (T) for the inclusion Z2 ⊂ Z2⋊SL(2,Z) and Proposition 6.5, there exists a non-zero element x̃0 ∈ Ñ satisfying (wk ⊗ 1)β̃(k)(x̃0)(1⊗ w∗k) = x̃0, k ∈ Z2. The element x̃0 can be written as the following Fourier expansion: x̃0 = c(λ1, λ2)u(λ1)⊗ u(λ2) ∈ Ñ ⊂ Lµ̃(⊕Z2H)⊗ Lµ̃(⊕Z2H). Here c(λ1, λ2) is a complex number and (λ1, λ2) ∈ (⊕Z2H)2 runs through all pairs satisfying k∈Z2(λ1(k) + λ2(k)) = 0. Choose and fix a pair (λ1, λ2) satisfying λ1(k) = h = λ2(k), c(λ1, λ2) 6= 0. Let v′h ∈ M be the unitary written as v′h = uh ⊗ 1 ⊗ 1 ⊗ · · · , where uh ∈ Lµ(H) is placed on 0 ∈ Z2. The following unitaries {w′γ} ⊂ N(H, µ) give a weak 1-cocycle for β: w′(k,γ0) = v hw(k,γ0)ρ(1, k, γ0)(v ), (k, γ0) ∈ Z2 ⋊ SL(2,Z). Letting ỹ = (v′h ⊗ 1)x̃0(1⊗ v′h)∗ ∈ M̃ , we get ỹ = (w′k ⊗ 1)β̃(k)(ỹ)(1⊗ w′k ), k ∈ Z2. Applying the trace preserving conditional expectation E = EN(H,µ)⊗N(H,µ), we get E(ỹ) = (w′k ⊗ 1)E(β̃(k)(ỹ))(1⊗ w′k = (w′k ⊗ 1)β̃(k)(E(ỹ))(1⊗ w′k ), k ∈ Z2. Since the Fourier coefficient of x̃0 at (λ1, λ2) ∈ (⊕Z2H)2 is not zero, that of E(ỹ) at (λ1 + δh,0, λ2 − δh,0) ∈ Λ(H)2 is also non-zero, where δh,0 ∈ ⊕Z2H is zero on Z2 \ {0} and is h on 0 ∈ Z2. By Proposition 6.2, it follows that the weak 1-cocycle {w′(k,e)}k∈Z2 ⊂ N(H, µ) is a weak 1-coboundary of β|Z2. Since the Z2-action β|Z2 is weakly mixing, w′ is a weak 1-coboundary on Z2 ⋊ SL(2,Z), by Proposition 6.3. In other words, there exists v ∈ N(H, µ) satisfying w′γ = vβ(γ)(v ∗) mod T, wγ = v vρ(1, γ)(v∗v′h) mod T, γ ∈ Z2 ⋊ SL(2,Z). Noting that u = v∗v′h ∈ M is a normalizer of N(H, µ), we get (Ad(u) ◦ θ) ◦ β(1)(γ) = Ad(u) ◦ Ad(wγ) ◦ β(γ) ◦ θ = Ad(ρ(1, γ)(u)) ◦ β(γ) ◦ θ = ρ(1, γ) ◦ Ad(u) ◦ θ = β(γ) ◦ (Ad(u) ◦ θ), γ ∈ Z2 ⋊ SL(2,Z). Thus we get the conjugacy of two Z2 ⋊ SL(2,Z)-actions β(0) and β. � We can always apply Theorem 6.6 if H is finite. CLASSIFICATION AND CENTRALIZERS 23 Corollary 6.7. Let H be a finite abelian group and let β = β(H, µ, χ) be a twisted Bernoulli shift action on N(H, µ). Suppose that N(H, µ) is the AFD factor of type II1. Let β (1) be an action of Z2 ⋊ SL(2,Z) on the AFD factor N (1) of type II1 and suppose that its restriction to Z2 is ergodic. The action β(1) is outer conjugate to β, if and only if β(1) is conjugate to β. Proof. We have only to construct an R-action on Lµ(H) ⊗ Lµ(H) satisfying the properties in Theorem 6.6. Let U be an element of Lµ(H)⊗ Lµ(H) defined by |H|1/2 uh ⊗ u∗h. We note that µ∗µ(g, ·) is a character of H and that it is not identically 1 provided g 6= 0 by Lemma 5.5. The element U is self-adjoint and unitary, since |H|1/2 u∗h ⊗ uh = |H|1/2 µ(h,−h)u−h ⊗ µ(h,−h)u∗−h = U, g,h∈H uguh ⊗ u∗gu∗h = g,h∈H µ∗µ(g, h)ug+h ⊗ u∗g+h µ∗µ(g, h− g) ug ⊗ u∗g = 1. The operator U is a fixed point under the action of Ĥ , so the projections P1 = (1 + U)/2 and P−1 = (1 − U)/2 are also fixed points. Thus the R-action α(0)t = Ad(P1+exp (iπt)P−1) commutes with the Ĥ-action. The automorphism α 1 satisfies 1 (ug ⊗ 1) = U(ug ⊗ 1)U∗ = (1⊗ ug)UU∗ = 1⊗ ug, g ∈ H. This verifies the first condition for α(0). � Corollary 6.8. Let Q be a set consisting of odd prime numbers and βQ be the twisted Bernoulli shift action defined in Corollary 5.9. Let β be a Z2 ⋊ SL(2,Z)-action on the AFD factor of type II1 whose restriction to Z 2 is ergodic. The actions βQ and β are outer conjugate if and only if they are conjugate. In particular, {βQ} is an uncountable family of Z2⋊SL(2,Z)-actions which are mutually non outer conjugate. Proof. We will use the notation given in Corollary 5.8 and 5.9. Let α t be the R- action on Lµq (Hq)⊗Lµq(Hq) constructed as in the previous corollary. We define the R-action α(Q) on LµQ(HQ)⊗LµQ(HQ) by α t (⊗q∈Qxq) = ⊗q∈Qα t (xq), where xq ∈ Lµq (Hq) ⊗ Lµq (Hq) and xq 6= 1 only for finitely many q. The R-action satisfies the conditions in Theorem 6.6. By Corollary 5.9, {βQ} are mutually non conjugate and their restriction to Z2 is ergodic. Thus they are mutually non outer conjugate. � Acknowledgment . The author would like to thank Professor Yasuyuki Kawahigashi for helpful conversations. He thanks the referee for careful reading and numerous detailed comments. He is supported by JSPS Research Fellowships for Young Sci- entists. 24 HIROKI SAKO References [Bu] M. Burger, Kazhdan constants for SL3(Z), J. Reine Angew. Math., 413 (1991), 36–67. [Ch] M. 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Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan E-mail address : hiroki@ms.u-tokyo.ac.jp 1. Introduction 2. Preparations 2.1. Functions det and gcd 2.2. Scalar 2-cocycles for abelian groups 2.3. Definition of a Z2 SL(2,Z)-action on (H) 2.4. On the relative property (T) of Kazhdan 2.5. Weakly mixing actions 2.6. A remark on group von Neumann algebras 3. Definition of twisted Bernoulli shift actions 4. Classification up to conjugacy 5. Examples 5.1. Twisted Bernoulli shift actions on L(X) 5.2. Twisted Bernoulli shift actions on the AFD factor of type II1 6. Malleability and rigidity arguments References

The Phase-resolved High Energy Spectrum of the Crab Pulsar J.J. Jia a, Anisia P.S. Tang a, J. Takata b, H.K. Chang c, K.S.Cheng a aDepartment of Physics, The University of Hong Kong, Hong Kong, China, hrspksc@hkucc.hku.hk bASIAA/National Tsing Hua University - TIARA, PO Box 23-141, Taipei, Taiwan cDepartment of Physics and Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan Abstract We present a modified outer gap model to study the phase-resolved spectra of the Crab pulsar. A theoretical double peak profile of the light curve containing the whole phase is shown to be consistent with the observed light curve of the Crab pulsar by shifting the inner boundary of the outer gap inwardly to ∼ 10 stellar radii above the neutron star surface. In this model, the radial distances of the photons corresponding to different phases can be determined in the numerical calculation. Also the local electrodynamics, such as the accelerating electric field, the curvature radius of the magnetic field line and the soft photon energy, are sensitive to the radial distances to the neutron star. Using a synchrotron self-Compton mechanism, the phase-resolved spectra with the energy range from 100 eV to 3 GeV of the Crab pulsar can also be explained. Key words: neutron stars, pulsars, radiation mechanism, radiation processes PACS: 97.60.Jd, 97.60.Gb, 95.30.Gv, 94.05.Dd 1 Introduction There are eight pulsars have been detected in gamma-ray energy range (cf. Thompson 2006 for a recent review) with period ranging from 0.033s to 0.237s and age younger than million years old. Theoretically, it is suggested that high-energy photons are produced by the radiation of charged particles that are accelerated in the pulsar magnetosphere. There are two kinds of theoretical Preprint submitted to Elsevier 18 November 2018 http://arxiv.org/abs/0704.1534v1 models: one is the polar gap model (e.g., Harding 1981, Daugherty & Hard- ing 1996, for more detail review of polar cap model cf. Harding 2006), and another is the outer gap model (e.g. Cheng, Ho, & Ruderman 1986a, 1986b; Chiang & Romani 1994). Both models predict that electrons and positrons are accelerated in a charge depletion region called a gap by the electric field along the magnetic field lines and assume that charged particles lose their energies via curvature radiation in both polar and outer gaps. The key differences are polar gap are located near stellar surface and the outer gap are located near the null charge surface, where are at least several tens stellar radii away the star. The continuous observations of powerful young pulsars, including the Crab, the Vela and the Geminga, have collected large number of high energy photons, which allow us to carry out much more detailed analysis. Fierro et al. (1998) divided the whole phase into eight phase intervals, i.e. leading wing 1, peak 1, trailing 1, bridge, leading wing 2, peak 2, trailing 2 and off-pulse. They showed that the data in each of these phases can be roughly fitted with a simple power law. However, the photon indices of these phases are very different, they range from 1.6 to 2.6. Massaro et al. (2000) have shown that X-ray pulse profile is energy dependent and the X-ray spectral index also depends on the phase of the rotation. The Crab pulsar has been extensively studied from the radio to the extremely high energy ranges, and the phase of the double-pulse with separation of 144◦ is found to be consistent over all wavelengths. Recently Kuiper et al. (2001) have combined the X-ray and gamma-ray data of the Crab pulsar, they showed that the phase-dependent spectra exhibit a double-peak structure, i.e. one very broad peak in soft gamma-rays and another broad peak in higher energy gamma-rays. The position of these peaks depend on the phase. Although the double-peak structure is a signature of synchrotron self-Compton mechanism, it is impossible to fit the phase dependent spectrum by a simple particle energy spectrum. Actually it is not surprised that the spectrum is phase dependent because photons are emitted from different regions of the magnetosphere. The local properties, e.g. electric field E(r), magnetic field B(r), particles density and energy distribution are very much different for different regions. Therefore these phase dependent data provide very important information for emission region. Consequently, the phase-resolved properties provide very important clues and constraints for the theoretical models. So far, the three-dimensional outer gap model seems to be the most successful model in explaining both the double-peak pulse profile and the phase-resolved spectra of the Crab pulsar (e.g. Chiang & Romani 1992, 1994; Dyks & Rudak 2003; Cheng, Ruderman & Zhang, 2000, hereafter CRZ). However, the leading-edge and trailing-edge of the light curve cannot be given out, since the inner boundary of the outer gap is located at the null charge surface in this model. Recently, the electro- dynamics of the pulsar magnetosphere has been studied carefully by solving the Poisson equation for electrostatic potential and the Boltzmann equations for electrons/positrons (Hirotani & Shibata, 1999a,b,c; Takata et al. 2004, 2006; Hirotani 2005), and the inner boundary of the gap is shown to be located near the stellar surface. We will organize the paper as follows. We describe the modified outer gap model in §2. In §3, we calculate the phase-resolved spectra and present the fitting result of the spectra in different phase intervals. Finally, we discuss our results and draw conclusions in §4. 2 A modified outer gap model Originally proposed by Holloway (1973) that vacuum gaps may form in the outer regions of pulsar’s magnetosphere, Cheng, Ho and Ruderman (1986a, 1986b; hereafter CHR) developed the idea of outer magnetosphere gaps and explained the radiation mechanisms of the γ-rays from the Crab and Vela pulsars. CHR argued that a global current flowing through the null surface of a rapidly spinning neutron star would result in large regions of charge depletion, which form the gaps in the magnetosphere. They assume the outer gap should begin at the null charge surface and extend to the light cylinder. In the gaps, a large electric field parallel to the magnetic field lines is induced ( ~E · ~B 6= 0), and it can accelerate the electrons or positrons to extremely relativistic speed. Thus, those charges can emit high energy photons through various mechanisms, and further produce copious e+e− pairs to sustain the gaps and the currents. Based on the CHR model, Chiang and Romani (1992, 1994) generated gamma ray light curves for various magnetosphere geometries by assuming that gap- type regions could be supported along all field lines which define the boundary between the closed region and open field line region rather than just on the bundle of field lines lying in the plane containing the rotation and magnetic dipole axes. In their model, photons are generated and travel tangential to the local magnetic field lines and there are beams in both the outward and inward directions. They suggested that a single pole would produce a double-peak emission profile when the line of sight crosses the enhanced regions of the γ-ray beam, while the inner region of the beam results in the bridge emission between these two pulses. The peak phase separation can be accommodated by choosing a proper observer viewing angle. Because the location of emission of each point in phase along a given line of sight can be mapped approximately in this model, the outer gap is thus divided into small subzones. As the curvature radius, photon densities and the local electrodynamics in different subzones are not the same, the spectral variation of the high energy radiation in different phase intervals varies. Later, Romani and Yadigaroglu (1995) developed the single gap model by involving the effects of aberration, retarded potential and time of flight across the magnetosphere. The light curve profiles in this modified model is simply determined by only two parameters, which are magnetic inclination angle α and the viewing angle ζ . They argued that the γ-ray emission can only be observed from pulsars with large viewing angle (ζ ≥ 45◦), and we cannot receive the γ photons but radio emissions from the aligned pulsars (α ≤ 35◦). Furthermore, they showed the gap would grow larger as the pulsar slows down, and more open field lines can occupy the outer gap, which means the older pulsar are more efficient for producing GeV γ-ray photons (Yadigaroglu & Romani, 1995). However, the assumptions of the model proposed by Romani’s group are not self-consistent. Why is there only a single pole and only outgoing current in the magnetosphere? Cheng, Ruderman and Zhang (2000) proposed another version of three dimensional outer gap model for high energy pulsars based on the pioneering work of Romani, and made it more natural in physics. In the CRZ model, two outer gaps and both outward and inward currents are allowed (though it turns out that outgoing currents dominate the emitted ra- diation intensities), and the azimuthal extension of the outer gap is restricted on a bundle of fields instead of the whole lines. Like the previous work by Yadigaroglu and Romani (1995), the CRZ model also contains the same two parameters, but more self-consistent in gap geometry and radiation morphol- ogy by using the pair production conditions. The electric field parallel to the magnetic field lines is ΩB(r)f 2(r)R2L cs(r) , (1) where f(r) ∝ r3/2 and s(r) ∝ r1/2 are the fractional size of the outer gap and the curvature radius at the distance r. The characteristic fractional size of the outer gap evaluated at r ∼ rL, where rL is the light cylinder radius, can be estimated by the condition of pair creation (Zhang & Cheng 1997; CRZ) and is given by f0 = 5.5P 26/21B 12 ∆Φ 1/7 , (2) where ∆Φ is the azimuthal extension of the outer gap. CRZ estimates its value by considering the local pair production condition and give ∆Φ ∼ 160◦ for the Crab pular. It has been pointed out that if the inclination angle is small, f0 can be changed by a factor of several (Zhang et al. 2004). We want to remark that equation (1) is the solution of vacuum solution, for regions near null surface and the inward extension of the gap the electric field is shown to be deviated from the vacuum solution (e.g Muslimov & Harding 2004; Hirotani 2006). Nevertheless for simplicity we shall assume the vacuum solution for the entire gap. In the numerical calculation, the outer gap should be divided into several layers in space. The shape of each layer at the stellar surface is similar to that of the polar cap, but smaller in size. Thus, for a thin gap, the calculation of only one representative layer is enough; while for a thick one (e.g. Geminga), several different layers should be added in the calculation (Zhang & Cheng, 2001). The coordinate of the footprint of the last closed field lines on the stellar surface is determined as (x0, y0, z0), then the coordinates values (x 0) of the inner layers can be defined by x′0 = a1x0, y 0 = a1y0, and z 1− x′02 − y′02, where a1 corresponds to the various layers in the open volume. Inside the light cylinder, high energy photons will be emitted nearly tangent to the magnetic field lines in the corotating frame because of the relativistic 1/γ beaming inherent in high energy processes unless |E×B| ∼ B2. Then the propagation direction of each emitted photons by relativistic charged particles can be expressed as (ζ ,Φ), where ζ is the polar angle from the rotation axis and Φ is the phase of rotation of the star. Effects of the time of flight and aberration are taken into account. A photon with velocity u = (ux, uy, uz) along a magnetic field line with a relativistic addition of velocity along the azimuthal angle gives an aberrated emission direction u′ = (u′x, u z). The time of flight gives a change of the phase of the rotation of the star. Combining these two effects, and choosing Φ = 0 for radiation in the (x,z) plane from the center of the star, ζ and Φ are given by cos ζ = u′z and Φ = −φu′ −~r · û′, where φu′ is the azimuthal angle of û′ and ~r is the emitting location in units of RL. In panel A of Fig. 1, the emission morphology in the (ζ , Φ) plane is shown. For a given observer with a fixed viewing angle ζ , a double-pulsed structure is observed because photons are clustered near two edges of the emission pattern due to the relativistic effects (cf. panel B of Fig. 1). In Fig. 1, we can see that this model can only produce radiation between two peaks. However, the observed data of the Crab, Vela and Geminga indicate that the leading wing 1 and the trailing wing 2 are quite strong, and even the intensity in off-pulse cannot be ignored. Hirotani and his co-workers (Hirotani & Shibata 2001; Hirotani, Harding & Shibata 2003) have pointed out that the large current in the outer gap can change the boundary of the outer gap. They solve the set of Maxwell and Boltzmann equations in pulsar magnetospheres and demonstrate the existence of outer-gap accelerators, whose inner bound- ary position depends the detail of the current flow and it is not necessarily located at the null charge surface. For the gap current lower than 25% of the Goldreich-Julian current, the inner boundary of the outer gap is very close to the null surface (Hirotani 2005). On the other hand if the current is close to the Goldreich-Julian current, the inner boundary can be as close as 10 stellar radii. In Fig. 2, we show the light curve by assuming the inner boundary is extended inward from the null charge surface to 10 stellar radii (cf. panel A Fig. 1. Emission projection onto the (ζ,Φ) plane and pulse profile for the single pole outer gap. The photons are emitted outwards from the outer gap. (a) The emission projection (a1 = 0.9) and (b) the corresponding pulse profile (∆a1 = 0.03), for Crab parameters α = 65◦ and ζ = 82◦. of Fig. 2). In panel B of Fig. 2, the solid line represents emission trajectory of outgoing radiation of one gap from the null surface to the light cylinder with α = 50◦ and ζ = 75◦ and the dashed line represents the outgoing radiation from another gap from the inner boundary to the null surface. In the presence of the extended emission region from the near the stellar surface to the null charge surface, leading wing 1, trailing wing 2 and the off-pulse components can also be produced. 0 60 120 180 240 300 360 Phase Fig. 2. Upper panel: the simulated pulse profile of the Crab pulsar; lower panel: variation of radial distance with pulse phase for the Crab pulsar in units of RL, where the bold line represents those in the outer magnetosphere, and the dashed line represents those in the inner magnetosphere. The inclination angle is 50◦ and the viewing angle is 75◦. 3 The phase-resolved spectra 3.1 radiation spectrum The Crab pulsar has enough photons for its spectra to be analyzed, and the phase-resolved spectra are useful for study of the local properties of the mag- netosphere. Here, we summarize the calculation procedure of the radiation spectrum given in CRZ, which is used to calculate the spectrum in different phases. The electric field of a thin outer gap (CHR) is given by E‖(r) = ΩB(r)a2(r) cs(r) ΩB(r)f2(r)R2 cs(r) , where a(r) is the thickness of the outer gap at position r, and f(r) = a(r)/RL is the local fractional size of the outer gap. Assuming that the magnetic flux subtended in the outer gap is constant in the steady state, we get the local size factor f(r) ∼ f(RL)( rRL ) 3/2, where f(RL) is estimated by using the pair creation condition (cf. Zhang & Cheng 1997, CRZ). As the equilibrium between the energy loss in radiation and gain from accelerating electric field, the local Lorentz factor of the electrons/positrons in the outer gap is γe(r) = ( eE‖(r)c) For a volume element ∆V in the outer gap, the number of primary charged particles can be roughly written as dN = nGJ∆A∆l, where nGJ = is the local Goldreich-Julian number density, B∆A is the magnetic flux through the accelerator and ∆l is the path length along its magnetic field lines. (Here, We would like to remark that this could overestimate the primary charge num- ber density near the null surface, where the positronic charge density dom- inates the Goldreich-Julian charge density. However, the observed radiation comes from the wide range of magnetospheric region, an slight overestima- tion of a small region should not cause a qualitative difference.) Thus, the total number of the charged particles in the outer gap is N ∼ ΩΦ RL, where Φ ∼ f(RL)B(RL)R2L∆φ is the typical angular width of the magnetic flux tube subtend in the outer gap. The primary e± pairs radiate curvature pho- tons with a characteristic energy Ecur(r) = ~γ3e (r) , and the power into curvature radiation for dN e± pairs in a unit volume is dLcur ≈ lcurnGJ(r), where lcur = eE‖c is the local power into the curvature radiation from a single electron/positron. The spectrum of primary photons from a unit volume is dV dEγ ≈ lcurnGJ , Eγ ≤ Ecur. (3) These primary curvature photons collide with the soft photons produced by synchrotron radiation of the secondary e± pairs, and produce the secondary e± pairs by photon-photon production. In a steady state, the distribution of secondary electrons/positrons in a unit volume is dV dEe ∫ d2Ṅ(E ′γ = 2E dV dEγ dE ′e ≈ lcurnGJ ), (4) with Ėe the electron energy loss into synchrotron radiation, which is Ėe = e4B2(r) sin2 β(r) )2, where B(r) is the local magnetic field and β(r) the local pitch angle, sin β(r) ∼ sin β(RL)( rRL ) 1/2, sin β(RL) is the pitch angle at the light cylinder. Therefore, the energy distribution of the secondary elec- trons/positrons in volume ∆V (r) can be written as dN(r) dV dEe ∆V (r) ∼ lcurnGJ∆V (r) ). (5) The corresponding photon spectrum of the synchrotron radiation is Fsyn(Eγ , r) = 3e3B(r) sinβ mec2h dN(r) F (x)dEe, (6) where x = Eγ/Esyn, and Esyn(r) = heB(r) sinβ(r) is the typical photon energy, and F (x) = x x K5/3(y)dy, where K5/3(y) is the modified Bessel function of order 5/3. Also, the spectrum of the inverse Compton scattered photons in the volume ∆V (r) is FICS(Eγ, r) = dN(r) d2NICS(r) dEγdt dEe, (7) where d2NICS(r) dEγdt nsyn(ǫ, r)F (%epsilon, Eγ , Ee)dǫ, and F (ǫ, Eγ , Ee) = 3σT c 4(Ee/mc2)2 [2q ln q+ (1 + 2q)(1− q) + (Γq) 2(1−q) 2(1+Γq) ], where Γ = 4ǫ(Ee/mec 2)/mec 2, q = E1/Γ(1− E1) with E1 = Eγ/Ee and 1/4(Ee/mec 2) < q < 1. The number density of the synchrotron photons with energy ǫ is nsyn(ǫ, r) = Fsyn(ǫ) cr2∆Ω , where Fsyn is the calculated synchrotron radiation flux, and ∆Ω is the usual beam solid angle. Fig. 3 shows the observed data of the phase-resolved spectra from 100 eV to 3 GeV of the Crab pulsar, and the theoretical fitting results calculated by using the synchrotron self-Compton mechanism. The phase intervals are defined by division given by Fierro (1998), and the amplitude of the spectrum in each phase interval is proportional to the number of photons counted in it. In this fitting, f(RL) = 0.21, and B = 3.0× 1012Gauss are used, which give a consistent fitting of the phase-resolved spectra of the seven phase intervals. In order to obtain a better fit, we treat the pitch angle (β) and the beam solid angle (∆Ω) near the light cylinder as free parameters and vary from phase to phase in the calculation. sin β(RL) = 0.06 and ∆Ω = 5.0 are chosen for trailing wing 1, bridge and leading wing 2; sin β(RL) = 0.02,∆Ω = 1.0 for leading wing 1; sin β(RL) = 0.04,∆Ω = 3.5 for peak 1, sin β(RL) = 0.07,∆Ω = 3.0 for peak 2, and sin β(RL) = 0.03,∆Ω = 6.0 for trailing wing 2. Additionally, the phase- averaged spectrum of the total pulse of the Crab pulsar is shown in Fig. 4, where the parameters are chosen as sin β(RL) = 0.05 and ∆Ω = 5.0. 3.2 Analysis of the Phase-Resolved Spectra The high energy spectra of the Crab pulsar is explained by using the syn- chrotron self-Compton mechanism, which involves both the synchrotron radia- tion and the Inverse Compton-Scattering (ICS) caused by the ultra-relativistic electron/positron pairs created by the extremely high-energy curvature pho- tons. The secondary e± pairs gyrate in the strong magnetic field and radiate F(E) (MeV cm Fig. 3. Phase resolved spectra of the Crab pulsar from 100 eV to 3 GeV in the 7 narrow pulse-phase intervals. Two spectra (for the TW1 and LW2) are displayed twice. The curved line is calculated by the theoretical model, and the observed data are taken from Kuiper et al. (2001). synchrotron photons. While in the far regions of the magnetosphere where the magnetic field decays rapidly, the relativistic pairs collide with the soft syn- chrotron photons through the ICS process. Thus, the spectra of the radiation contain two main components: one is the synchrotron radiation from the soft X-ray to ∼10 MeV, and the other is the ICS component in the even higher energy range. Usually, the synchrotron spectrum has stronger amplitude than 1x10 1x10 2 F s - E (MeV) Fig. 4. Phase-averaged spectrum of the Crab pulsar. The observed data are taken from Kuiper et al. (2001). that of ICS, and there is obvious turning frequency between these two com- ponents, e.g. about 3MeV for peak 1. As we know, the power of synchrotron radiation and ICS can be compared by the ratio of the local magnetic energy density and the photon energy density, i.e. ∝ B(r) ǫsyn(r)nsyn(ǫsyn, r) , (8) where ǫsyn(r) is the synchrotron photon energy in location r. In Fig. 3, the spectra in trailing wing 1, bridge and leading wing 2 have broad synchrotron spectra, which cover from 100 eV to ∼30 MeV. In Fig. 2, it is demonstrated that the radiation of these three phase intervals are dominated by the photons generated in the near surface region, where the magnetic field is so strong that synchrotron radiation takes up the most emission. However, the radiation of peak 1 and 2 are from the far regions near the light cylinder, where the magnetic field decays rapidly (B ∝ r−3), thus, the ICS radiation becomes more important above 3 MeV. The peak of the synchrotron spectrum is determined by the characteristic synchrotron photon energy. Since Esyn ∝ γ2eB sin β, where γe is the Lorentz factor of the secondary pairs and β is the pitch angle of the electron/positron to the magnetic field, the peak of the synchrotron spectrum can shift if the β varies. Since the outward radiation direction covers a wider range than that of the inward radiation, so the solid angle (∆Ω) is no longer the unity as assumed in CRZ model. The solid angle can effect the amplitude of the ICS spectrum because the number density of the synchrotron photons is proportional to 1 Therefore it is reasonable for us to choose β(RL) and ∆Ω as a set of parameters in fitting the phase-resolved spectra of the Crab pulsar. 4 Conclusion and Discussion We have tried to explain the high energy light curve and the phase-resolved spectra in the energy range from 100 eV to 3 GeV of the Crab pulsar by mod- ifying the three dimensional outer magnetosphere gap model. Compared to the classical outer gap with the inner boundary at the null charge surface, the modified model allows the outer gap to start at the region about several stel- lar radii above the neutron star surface, and the ”inwardly-extended” part of the outer gap contributes to the outer wings and off-pulse of the light curve. Such modified outer gap geometry also plays a vital role in explaining the optical polarization properties of the Crab pulsar (Takata et al. 2006). Two adjustable parameters are used to simulate the light curve: one is the inclina- tion angle of the magnetic axis to the rotational axis α, and the other is the viewing angle also to the rotational axis ζ . As constrained by the phase sepa- ration of the double peaks, we choose the values for these two parameters that α = 50◦ and ζ = 75◦. So far, these two parameters have not been determined from the observations. From radio observations, Rankin (1993) estimated that α ≈ 84◦ and ζ is not known. Moffett and Hankins (1999) gave that α ≈ 56◦ and ζ = 117◦ by using the polarimetric observations at frequencies between 1.4 and 8.4 GHz. Of course, our values cannot be the true ones, and require further observations to give strong restrictions of them. In fitting the phase-resolved spectra of the Crab pulsar, our model performs well from 100 eV to 1 GeV, but fails beyond 1 GeV. The inverse Compton scattering spectrum of our results falls down quickly when the energy is over 1 GeV, but the observation data indicates that the spectrum still increases, es- pecially in the first trailing wing, the bridge and the second leading wing phase intervals. We have assumed that the curvature photons are all absorbed by the magnetic field lines, however, some of these multi-GeV photons produced near the light cylinder should be easily escaped from the photon-photon pair creation process. In the spectrum fitting of peak 1, our result has a frequency shift below 1 MeV, and we found that in order to well fit the spectrum we should reduce the curvature photon energy by a quarter. The energy of the curvature photon Ecur ∝ s−1(r), where s(r) is the local curvature radius. As the high energy photons are produced in the far regions of the magnetosphere, where s(r) maybe not follow the dipole form, we can change the photon energy slightly. Moreover, the stellar radius of a neutron star is usually treated as 106 cm when calculating the strength of the surface magnetic field. However, the equation of state inside the neutron star of the current theoretical models cannot give a convincing value of the neutron star size. Thus, we can only determine the magnetic moment, i.e. BpR 0, of the pulsar from the energy loss rate. Therefore, we can rewrite the magnetic field of the Crab pulsar as B12R 6 = 3.8. 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We present a modified outer gap model to study the phase-resolved spectra of the Crab pulsar. A theoretical double peak profile of the light curve containing the whole phase is shown to be consistent with the observed light curve of the Crab pulsar by shifting the inner boundary of the outer gap inwardly to $\sim 10$ stellar radii above the neutron star surface. In this model, the radial distances of the photons corresponding to different phases can be determined in the numerical calculation. Also the local electrodynamics, such as the accelerating electric field, the curvature radius of the magnetic field line and the soft photon energy, are sensitive to the radial distances to the neutron star. Using a synchrotron self-Compton mechanism, the phase-resolved spectra with the energy range from 100 eV to 3 GeV of the Crab pulsar can also be explained.

Introduction There are eight pulsars have been detected in gamma-ray energy range (cf. Thompson 2006 for a recent review) with period ranging from 0.033s to 0.237s and age younger than million years old. Theoretically, it is suggested that high-energy photons are produced by the radiation of charged particles that are accelerated in the pulsar magnetosphere. There are two kinds of theoretical Preprint submitted to Elsevier 18 November 2018 http://arxiv.org/abs/0704.1534v1 models: one is the polar gap model (e.g., Harding 1981, Daugherty & Hard- ing 1996, for more detail review of polar cap model cf. Harding 2006), and another is the outer gap model (e.g. Cheng, Ho, & Ruderman 1986a, 1986b; Chiang & Romani 1994). Both models predict that electrons and positrons are accelerated in a charge depletion region called a gap by the electric field along the magnetic field lines and assume that charged particles lose their energies via curvature radiation in both polar and outer gaps. The key differences are polar gap are located near stellar surface and the outer gap are located near the null charge surface, where are at least several tens stellar radii away the star. The continuous observations of powerful young pulsars, including the Crab, the Vela and the Geminga, have collected large number of high energy photons, which allow us to carry out much more detailed analysis. Fierro et al. (1998) divided the whole phase into eight phase intervals, i.e. leading wing 1, peak 1, trailing 1, bridge, leading wing 2, peak 2, trailing 2 and off-pulse. They showed that the data in each of these phases can be roughly fitted with a simple power law. However, the photon indices of these phases are very different, they range from 1.6 to 2.6. Massaro et al. (2000) have shown that X-ray pulse profile is energy dependent and the X-ray spectral index also depends on the phase of the rotation. The Crab pulsar has been extensively studied from the radio to the extremely high energy ranges, and the phase of the double-pulse with separation of 144◦ is found to be consistent over all wavelengths. Recently Kuiper et al. (2001) have combined the X-ray and gamma-ray data of the Crab pulsar, they showed that the phase-dependent spectra exhibit a double-peak structure, i.e. one very broad peak in soft gamma-rays and another broad peak in higher energy gamma-rays. The position of these peaks depend on the phase. Although the double-peak structure is a signature of synchrotron self-Compton mechanism, it is impossible to fit the phase dependent spectrum by a simple particle energy spectrum. Actually it is not surprised that the spectrum is phase dependent because photons are emitted from different regions of the magnetosphere. The local properties, e.g. electric field E(r), magnetic field B(r), particles density and energy distribution are very much different for different regions. Therefore these phase dependent data provide very important information for emission region. Consequently, the phase-resolved properties provide very important clues and constraints for the theoretical models. So far, the three-dimensional outer gap model seems to be the most successful model in explaining both the double-peak pulse profile and the phase-resolved spectra of the Crab pulsar (e.g. Chiang & Romani 1992, 1994; Dyks & Rudak 2003; Cheng, Ruderman & Zhang, 2000, hereafter CRZ). However, the leading-edge and trailing-edge of the light curve cannot be given out, since the inner boundary of the outer gap is located at the null charge surface in this model. Recently, the electro- dynamics of the pulsar magnetosphere has been studied carefully by solving the Poisson equation for electrostatic potential and the Boltzmann equations for electrons/positrons (Hirotani & Shibata, 1999a,b,c; Takata et al. 2004, 2006; Hirotani 2005), and the inner boundary of the gap is shown to be located near the stellar surface. We will organize the paper as follows. We describe the modified outer gap model in §2. In §3, we calculate the phase-resolved spectra and present the fitting result of the spectra in different phase intervals. Finally, we discuss our results and draw conclusions in §4. 2 A modified outer gap model Originally proposed by Holloway (1973) that vacuum gaps may form in the outer regions of pulsar’s magnetosphere, Cheng, Ho and Ruderman (1986a, 1986b; hereafter CHR) developed the idea of outer magnetosphere gaps and explained the radiation mechanisms of the γ-rays from the Crab and Vela pulsars. CHR argued that a global current flowing through the null surface of a rapidly spinning neutron star would result in large regions of charge depletion, which form the gaps in the magnetosphere. They assume the outer gap should begin at the null charge surface and extend to the light cylinder. In the gaps, a large electric field parallel to the magnetic field lines is induced ( ~E · ~B 6= 0), and it can accelerate the electrons or positrons to extremely relativistic speed. Thus, those charges can emit high energy photons through various mechanisms, and further produce copious e+e− pairs to sustain the gaps and the currents. Based on the CHR model, Chiang and Romani (1992, 1994) generated gamma ray light curves for various magnetosphere geometries by assuming that gap- type regions could be supported along all field lines which define the boundary between the closed region and open field line region rather than just on the bundle of field lines lying in the plane containing the rotation and magnetic dipole axes. In their model, photons are generated and travel tangential to the local magnetic field lines and there are beams in both the outward and inward directions. They suggested that a single pole would produce a double-peak emission profile when the line of sight crosses the enhanced regions of the γ-ray beam, while the inner region of the beam results in the bridge emission between these two pulses. The peak phase separation can be accommodated by choosing a proper observer viewing angle. Because the location of emission of each point in phase along a given line of sight can be mapped approximately in this model, the outer gap is thus divided into small subzones. As the curvature radius, photon densities and the local electrodynamics in different subzones are not the same, the spectral variation of the high energy radiation in different phase intervals varies. Later, Romani and Yadigaroglu (1995) developed the single gap model by involving the effects of aberration, retarded potential and time of flight across the magnetosphere. The light curve profiles in this modified model is simply determined by only two parameters, which are magnetic inclination angle α and the viewing angle ζ . They argued that the γ-ray emission can only be observed from pulsars with large viewing angle (ζ ≥ 45◦), and we cannot receive the γ photons but radio emissions from the aligned pulsars (α ≤ 35◦). Furthermore, they showed the gap would grow larger as the pulsar slows down, and more open field lines can occupy the outer gap, which means the older pulsar are more efficient for producing GeV γ-ray photons (Yadigaroglu & Romani, 1995). However, the assumptions of the model proposed by Romani’s group are not self-consistent. Why is there only a single pole and only outgoing current in the magnetosphere? Cheng, Ruderman and Zhang (2000) proposed another version of three dimensional outer gap model for high energy pulsars based on the pioneering work of Romani, and made it more natural in physics. In the CRZ model, two outer gaps and both outward and inward currents are allowed (though it turns out that outgoing currents dominate the emitted ra- diation intensities), and the azimuthal extension of the outer gap is restricted on a bundle of fields instead of the whole lines. Like the previous work by Yadigaroglu and Romani (1995), the CRZ model also contains the same two parameters, but more self-consistent in gap geometry and radiation morphol- ogy by using the pair production conditions. The electric field parallel to the magnetic field lines is ΩB(r)f 2(r)R2L cs(r) , (1) where f(r) ∝ r3/2 and s(r) ∝ r1/2 are the fractional size of the outer gap and the curvature radius at the distance r. The characteristic fractional size of the outer gap evaluated at r ∼ rL, where rL is the light cylinder radius, can be estimated by the condition of pair creation (Zhang & Cheng 1997; CRZ) and is given by f0 = 5.5P 26/21B 12 ∆Φ 1/7 , (2) where ∆Φ is the azimuthal extension of the outer gap. CRZ estimates its value by considering the local pair production condition and give ∆Φ ∼ 160◦ for the Crab pular. It has been pointed out that if the inclination angle is small, f0 can be changed by a factor of several (Zhang et al. 2004). We want to remark that equation (1) is the solution of vacuum solution, for regions near null surface and the inward extension of the gap the electric field is shown to be deviated from the vacuum solution (e.g Muslimov & Harding 2004; Hirotani 2006). Nevertheless for simplicity we shall assume the vacuum solution for the entire gap. In the numerical calculation, the outer gap should be divided into several layers in space. The shape of each layer at the stellar surface is similar to that of the polar cap, but smaller in size. Thus, for a thin gap, the calculation of only one representative layer is enough; while for a thick one (e.g. Geminga), several different layers should be added in the calculation (Zhang & Cheng, 2001). The coordinate of the footprint of the last closed field lines on the stellar surface is determined as (x0, y0, z0), then the coordinates values (x 0) of the inner layers can be defined by x′0 = a1x0, y 0 = a1y0, and z 1− x′02 − y′02, where a1 corresponds to the various layers in the open volume. Inside the light cylinder, high energy photons will be emitted nearly tangent to the magnetic field lines in the corotating frame because of the relativistic 1/γ beaming inherent in high energy processes unless |E×B| ∼ B2. Then the propagation direction of each emitted photons by relativistic charged particles can be expressed as (ζ ,Φ), where ζ is the polar angle from the rotation axis and Φ is the phase of rotation of the star. Effects of the time of flight and aberration are taken into account. A photon with velocity u = (ux, uy, uz) along a magnetic field line with a relativistic addition of velocity along the azimuthal angle gives an aberrated emission direction u′ = (u′x, u z). The time of flight gives a change of the phase of the rotation of the star. Combining these two effects, and choosing Φ = 0 for radiation in the (x,z) plane from the center of the star, ζ and Φ are given by cos ζ = u′z and Φ = −φu′ −~r · û′, where φu′ is the azimuthal angle of û′ and ~r is the emitting location in units of RL. In panel A of Fig. 1, the emission morphology in the (ζ , Φ) plane is shown. For a given observer with a fixed viewing angle ζ , a double-pulsed structure is observed because photons are clustered near two edges of the emission pattern due to the relativistic effects (cf. panel B of Fig. 1). In Fig. 1, we can see that this model can only produce radiation between two peaks. However, the observed data of the Crab, Vela and Geminga indicate that the leading wing 1 and the trailing wing 2 are quite strong, and even the intensity in off-pulse cannot be ignored. Hirotani and his co-workers (Hirotani & Shibata 2001; Hirotani, Harding & Shibata 2003) have pointed out that the large current in the outer gap can change the boundary of the outer gap. They solve the set of Maxwell and Boltzmann equations in pulsar magnetospheres and demonstrate the existence of outer-gap accelerators, whose inner bound- ary position depends the detail of the current flow and it is not necessarily located at the null charge surface. For the gap current lower than 25% of the Goldreich-Julian current, the inner boundary of the outer gap is very close to the null surface (Hirotani 2005). On the other hand if the current is close to the Goldreich-Julian current, the inner boundary can be as close as 10 stellar radii. In Fig. 2, we show the light curve by assuming the inner boundary is extended inward from the null charge surface to 10 stellar radii (cf. panel A Fig. 1. Emission projection onto the (ζ,Φ) plane and pulse profile for the single pole outer gap. The photons are emitted outwards from the outer gap. (a) The emission projection (a1 = 0.9) and (b) the corresponding pulse profile (∆a1 = 0.03), for Crab parameters α = 65◦ and ζ = 82◦. of Fig. 2). In panel B of Fig. 2, the solid line represents emission trajectory of outgoing radiation of one gap from the null surface to the light cylinder with α = 50◦ and ζ = 75◦ and the dashed line represents the outgoing radiation from another gap from the inner boundary to the null surface. In the presence of the extended emission region from the near the stellar surface to the null charge surface, leading wing 1, trailing wing 2 and the off-pulse components can also be produced. 0 60 120 180 240 300 360 Phase Fig. 2. Upper panel: the simulated pulse profile of the Crab pulsar; lower panel: variation of radial distance with pulse phase for the Crab pulsar in units of RL, where the bold line represents those in the outer magnetosphere, and the dashed line represents those in the inner magnetosphere. The inclination angle is 50◦ and the viewing angle is 75◦. 3 The phase-resolved spectra 3.1 radiation spectrum The Crab pulsar has enough photons for its spectra to be analyzed, and the phase-resolved spectra are useful for study of the local properties of the mag- netosphere. Here, we summarize the calculation procedure of the radiation spectrum given in CRZ, which is used to calculate the spectrum in different phases. The electric field of a thin outer gap (CHR) is given by E‖(r) = ΩB(r)a2(r) cs(r) ΩB(r)f2(r)R2 cs(r) , where a(r) is the thickness of the outer gap at position r, and f(r) = a(r)/RL is the local fractional size of the outer gap. Assuming that the magnetic flux subtended in the outer gap is constant in the steady state, we get the local size factor f(r) ∼ f(RL)( rRL ) 3/2, where f(RL) is estimated by using the pair creation condition (cf. Zhang & Cheng 1997, CRZ). As the equilibrium between the energy loss in radiation and gain from accelerating electric field, the local Lorentz factor of the electrons/positrons in the outer gap is γe(r) = ( eE‖(r)c) For a volume element ∆V in the outer gap, the number of primary charged particles can be roughly written as dN = nGJ∆A∆l, where nGJ = is the local Goldreich-Julian number density, B∆A is the magnetic flux through the accelerator and ∆l is the path length along its magnetic field lines. (Here, We would like to remark that this could overestimate the primary charge num- ber density near the null surface, where the positronic charge density dom- inates the Goldreich-Julian charge density. However, the observed radiation comes from the wide range of magnetospheric region, an slight overestima- tion of a small region should not cause a qualitative difference.) Thus, the total number of the charged particles in the outer gap is N ∼ ΩΦ RL, where Φ ∼ f(RL)B(RL)R2L∆φ is the typical angular width of the magnetic flux tube subtend in the outer gap. The primary e± pairs radiate curvature pho- tons with a characteristic energy Ecur(r) = ~γ3e (r) , and the power into curvature radiation for dN e± pairs in a unit volume is dLcur ≈ lcurnGJ(r), where lcur = eE‖c is the local power into the curvature radiation from a single electron/positron. The spectrum of primary photons from a unit volume is dV dEγ ≈ lcurnGJ , Eγ ≤ Ecur. (3) These primary curvature photons collide with the soft photons produced by synchrotron radiation of the secondary e± pairs, and produce the secondary e± pairs by photon-photon production. In a steady state, the distribution of secondary electrons/positrons in a unit volume is dV dEe ∫ d2Ṅ(E ′γ = 2E dV dEγ dE ′e ≈ lcurnGJ ), (4) with Ėe the electron energy loss into synchrotron radiation, which is Ėe = e4B2(r) sin2 β(r) )2, where B(r) is the local magnetic field and β(r) the local pitch angle, sin β(r) ∼ sin β(RL)( rRL ) 1/2, sin β(RL) is the pitch angle at the light cylinder. Therefore, the energy distribution of the secondary elec- trons/positrons in volume ∆V (r) can be written as dN(r) dV dEe ∆V (r) ∼ lcurnGJ∆V (r) ). (5) The corresponding photon spectrum of the synchrotron radiation is Fsyn(Eγ , r) = 3e3B(r) sinβ mec2h dN(r) F (x)dEe, (6) where x = Eγ/Esyn, and Esyn(r) = heB(r) sinβ(r) is the typical photon energy, and F (x) = x x K5/3(y)dy, where K5/3(y) is the modified Bessel function of order 5/3. Also, the spectrum of the inverse Compton scattered photons in the volume ∆V (r) is FICS(Eγ, r) = dN(r) d2NICS(r) dEγdt dEe, (7) where d2NICS(r) dEγdt nsyn(ǫ, r)F (%epsilon, Eγ , Ee)dǫ, and F (ǫ, Eγ , Ee) = 3σT c 4(Ee/mc2)2 [2q ln q+ (1 + 2q)(1− q) + (Γq) 2(1−q) 2(1+Γq) ], where Γ = 4ǫ(Ee/mec 2)/mec 2, q = E1/Γ(1− E1) with E1 = Eγ/Ee and 1/4(Ee/mec 2) < q < 1. The number density of the synchrotron photons with energy ǫ is nsyn(ǫ, r) = Fsyn(ǫ) cr2∆Ω , where Fsyn is the calculated synchrotron radiation flux, and ∆Ω is the usual beam solid angle. Fig. 3 shows the observed data of the phase-resolved spectra from 100 eV to 3 GeV of the Crab pulsar, and the theoretical fitting results calculated by using the synchrotron self-Compton mechanism. The phase intervals are defined by division given by Fierro (1998), and the amplitude of the spectrum in each phase interval is proportional to the number of photons counted in it. In this fitting, f(RL) = 0.21, and B = 3.0× 1012Gauss are used, which give a consistent fitting of the phase-resolved spectra of the seven phase intervals. In order to obtain a better fit, we treat the pitch angle (β) and the beam solid angle (∆Ω) near the light cylinder as free parameters and vary from phase to phase in the calculation. sin β(RL) = 0.06 and ∆Ω = 5.0 are chosen for trailing wing 1, bridge and leading wing 2; sin β(RL) = 0.02,∆Ω = 1.0 for leading wing 1; sin β(RL) = 0.04,∆Ω = 3.5 for peak 1, sin β(RL) = 0.07,∆Ω = 3.0 for peak 2, and sin β(RL) = 0.03,∆Ω = 6.0 for trailing wing 2. Additionally, the phase- averaged spectrum of the total pulse of the Crab pulsar is shown in Fig. 4, where the parameters are chosen as sin β(RL) = 0.05 and ∆Ω = 5.0. 3.2 Analysis of the Phase-Resolved Spectra The high energy spectra of the Crab pulsar is explained by using the syn- chrotron self-Compton mechanism, which involves both the synchrotron radia- tion and the Inverse Compton-Scattering (ICS) caused by the ultra-relativistic electron/positron pairs created by the extremely high-energy curvature pho- tons. The secondary e± pairs gyrate in the strong magnetic field and radiate F(E) (MeV cm Fig. 3. Phase resolved spectra of the Crab pulsar from 100 eV to 3 GeV in the 7 narrow pulse-phase intervals. Two spectra (for the TW1 and LW2) are displayed twice. The curved line is calculated by the theoretical model, and the observed data are taken from Kuiper et al. (2001). synchrotron photons. While in the far regions of the magnetosphere where the magnetic field decays rapidly, the relativistic pairs collide with the soft syn- chrotron photons through the ICS process. Thus, the spectra of the radiation contain two main components: one is the synchrotron radiation from the soft X-ray to ∼10 MeV, and the other is the ICS component in the even higher energy range. Usually, the synchrotron spectrum has stronger amplitude than 1x10 1x10 2 F s - E (MeV) Fig. 4. Phase-averaged spectrum of the Crab pulsar. The observed data are taken from Kuiper et al. (2001). that of ICS, and there is obvious turning frequency between these two com- ponents, e.g. about 3MeV for peak 1. As we know, the power of synchrotron radiation and ICS can be compared by the ratio of the local magnetic energy density and the photon energy density, i.e. ∝ B(r) ǫsyn(r)nsyn(ǫsyn, r) , (8) where ǫsyn(r) is the synchrotron photon energy in location r. In Fig. 3, the spectra in trailing wing 1, bridge and leading wing 2 have broad synchrotron spectra, which cover from 100 eV to ∼30 MeV. In Fig. 2, it is demonstrated that the radiation of these three phase intervals are dominated by the photons generated in the near surface region, where the magnetic field is so strong that synchrotron radiation takes up the most emission. However, the radiation of peak 1 and 2 are from the far regions near the light cylinder, where the magnetic field decays rapidly (B ∝ r−3), thus, the ICS radiation becomes more important above 3 MeV. The peak of the synchrotron spectrum is determined by the characteristic synchrotron photon energy. Since Esyn ∝ γ2eB sin β, where γe is the Lorentz factor of the secondary pairs and β is the pitch angle of the electron/positron to the magnetic field, the peak of the synchrotron spectrum can shift if the β varies. Since the outward radiation direction covers a wider range than that of the inward radiation, so the solid angle (∆Ω) is no longer the unity as assumed in CRZ model. The solid angle can effect the amplitude of the ICS spectrum because the number density of the synchrotron photons is proportional to 1 Therefore it is reasonable for us to choose β(RL) and ∆Ω as a set of parameters in fitting the phase-resolved spectra of the Crab pulsar. 4 Conclusion and Discussion We have tried to explain the high energy light curve and the phase-resolved spectra in the energy range from 100 eV to 3 GeV of the Crab pulsar by mod- ifying the three dimensional outer magnetosphere gap model. Compared to the classical outer gap with the inner boundary at the null charge surface, the modified model allows the outer gap to start at the region about several stel- lar radii above the neutron star surface, and the ”inwardly-extended” part of the outer gap contributes to the outer wings and off-pulse of the light curve. Such modified outer gap geometry also plays a vital role in explaining the optical polarization properties of the Crab pulsar (Takata et al. 2006). Two adjustable parameters are used to simulate the light curve: one is the inclina- tion angle of the magnetic axis to the rotational axis α, and the other is the viewing angle also to the rotational axis ζ . As constrained by the phase sepa- ration of the double peaks, we choose the values for these two parameters that α = 50◦ and ζ = 75◦. So far, these two parameters have not been determined from the observations. From radio observations, Rankin (1993) estimated that α ≈ 84◦ and ζ is not known. Moffett and Hankins (1999) gave that α ≈ 56◦ and ζ = 117◦ by using the polarimetric observations at frequencies between 1.4 and 8.4 GHz. Of course, our values cannot be the true ones, and require further observations to give strong restrictions of them. In fitting the phase-resolved spectra of the Crab pulsar, our model performs well from 100 eV to 1 GeV, but fails beyond 1 GeV. The inverse Compton scattering spectrum of our results falls down quickly when the energy is over 1 GeV, but the observation data indicates that the spectrum still increases, es- pecially in the first trailing wing, the bridge and the second leading wing phase intervals. We have assumed that the curvature photons are all absorbed by the magnetic field lines, however, some of these multi-GeV photons produced near the light cylinder should be easily escaped from the photon-photon pair creation process. In the spectrum fitting of peak 1, our result has a frequency shift below 1 MeV, and we found that in order to well fit the spectrum we should reduce the curvature photon energy by a quarter. The energy of the curvature photon Ecur ∝ s−1(r), where s(r) is the local curvature radius. As the high energy photons are produced in the far regions of the magnetosphere, where s(r) maybe not follow the dipole form, we can change the photon energy slightly. Moreover, the stellar radius of a neutron star is usually treated as 106 cm when calculating the strength of the surface magnetic field. However, the equation of state inside the neutron star of the current theoretical models cannot give a convincing value of the neutron star size. Thus, we can only determine the magnetic moment, i.e. BpR 0, of the pulsar from the energy loss rate. Therefore, we can rewrite the magnetic field of the Crab pulsar as B12R 6 = 3.8. 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Massive N = 1 supermultiplets with arbitrary superspins Yu. M. Zinoviev ∗ Institute for High Energy Physics Protvino, Moscow Region, 142280, Russia Abstract In this paper we give explicit construction of massive N = 1 supermultiplets in flat d = 4 Minkowski space-time. We work in a component on-shell formalism based on gauge invariant description of massive integer and half-integer spin particles where massive supermultiplets are constructed out of appropriate set of massless ones. ∗E-mail address: yurii.zinoviev@ihep.ru http://arxiv.org/abs/0704.1535v1 Introduction In a flat space-time massive spin s particles in a massless limit decompose into massless spin s, s− 1, . . . ones. This, in particular, leads to the possibility of gauge invariant description of massive spin s particles, e.g. [1]-[13]. In this, two different approaches could be used. From one hand, one can start with usual non gauge invariant description of massive particle and achieve gauge invariance through the introduction of additional fields (thus promoting second class constraints into the first class ones). From the other hand, one can start with the appropriate set of massless particles having gauge invariance from the very beginning and obtain massive particle description as a deformation of massless theory. This last approach closely mimic situation in spontaneous gauge symmetry breaking where gauge field has to eat some Goldstone field(s) to become massive. In the supersymmetric theories all particles must belong to some supermultiplet, massive or massless. Till now most of investigations in supersymmetric theories where bounded to massless supermultiplets. Quite a few results on massive supermultiplets mainly devoted to superspins 1 and 3/2 exist [14]-[21]. The aim of this paper is to extend these results to include massive N = 1 supermultiplets with arbitrary superspins. Certainly, it would be nice to have superfield off-shell description of such supermultiplets, but as previous results clearly show it is a highly non-trivial task. So in this paper we restrict ourselves with component on-shell formalism in terms of physical fields. The same reasoning on the massless limit means that massive supermultiplets could (should) be constructed out of the massless ones in the same way as massive particles out of the massless ones. So our approach will be supersymmetric generalization of the second approach to massive particle description mentioned above. Namely, we will start with appropriate set of massless supermultiplets and obtain massive one as a smooth deformation. The paper is organized as follows. Though our previous examples on massive superspin 1 [21] and superspin 3/2 [15] supermultiplets already give important hints on how general case of arbitrary superspin could looks like, due to peculiarities of lower spin fields they are not enough to achieve such generalization. Thus, in the first two sections we give two more concrete examples, namely massive supermultiplets with superspin 2 and 5/2, correspond- ingly. All these and subsequent results heavily depend on the gauge invariant description of massive particles with integer [2, 3] and half-integer [10] spins as well as on the known form of massless supermultiplets [22]. For reader convenience and to make paper self-contained, in the next two sections we give all necessary formulas in compact condensed notations. One of the lessons from previous investigations is that the structures of massive supermultiplets with integer and half-integer superspins are different, so in the last two sections we consider these two cases separately. We will see that, in spite of large number of fields, all calculations are pretty straightforward and mainly combinatorical. 1 Superspin 2 Massive superspin 2 supermultiplet contains four massive particles with spins 5/2, 2, 2’ and 3/2, correspondingly. In the massless limit massive supermultilets must decompose into the appropriate set of massless ones in the same way as massive spin s particles — into massless spin s, s-1, ... ones. Simple counting of physical degrees of freedom immediately gives: 0, 0′ So we will start with five massless supermultiplets (Φµν , hµν), (fµν ,Ψµ), (Ωµ, Aµ), (Bµ, ψ) and (χ, z). From our previous experience with massive superspin 1 and superspin 3/2 su- permultiplets we know that it is crucial for the construction of massive supermultiplets to make dual rotation of vector Aµ and axial-vector Bµ fields mixing massless supermultiplets containing these fields. But now we have two tensor fields hµν and fµν as well, moreover they necessarily must be tensor and pseudo-tensor ones. Thus we have to consider the possibility to mix massless supermultiplets with these fields as well and the real structure of massless supermultiplets we are going to work with looks like: hµν fµν Aµ Bµ Then, introducing a sum of the massless Lagrangians for bosonic fields: ∂αhµν∂αhµν − (∂h)µ(∂h)µ + (∂h)µ∂µh− ∂µh∂µh+ ∂αfµν∂αfµν − (∂f)µ(∂f)µ + (∂f)µ∂µf − ∂µf∂µf − (∂µϕ) (∂µπ) 2 (1) as well as sum of the massless Lagrangians for fermionic fields: Φ̄µν ∂̂Φµν − 2i(Φ̄γ)µ(∂Φ)µ + i(Φ̄γ)µ∂̂(γΦ)µ + i(Φ̄γ∂)Φ− Φ̄∂̂Φ− Ψ̄µ∂̂Ψµ + i(Ψ̄γ)(∂Ψ)− (Ψ̄γ)∂̂(γΨ) + ψ̄∂̂ψ − Ω̄µ∂̂Ωµ + i(Ω̄γ)(∂Ω)− (Ω̄γ)∂̂(γΩ) + χ̄∂̂χ (2) it is not hard to check that the most general supertransformations leaving sum of massless Lagrangians invariant have the form (round brackets denote symmetrization): δΦµν = − σαβ∂α(cos(θ2)hβ(µγν) − sin(θ2)γ5fβ(µγν))η δhµν = 2 cos(θ2)(Φ̄µνη) + i sin(θ2)(Ψ̄(µγν)η) (3) δfµν = 2 sin(θ2)(Φ̄µνγ5η) + i cos(θ2)(Ψ̄(µγν)γ5η) δΨµ = −σαβ∂α(sin(θ2)hβµ + cos(θ2)fβµγ5)η for the supermultiplets containing spin 2 fields and δΩµ = − σαβ(cos(θ1)Aαβ − sin(θ1)γ5Bαβ)γµη δAµ = 2 cos(θ1)(Ω̄µη) + i sin(θ1)(ψ̄γµη) (4) δBµ = 2 sin(θ1)(Ω̄µγ5η) + i cos(θ1)(ψ̄γµγ5η) δψ = − σαβ(sin(θ1)Aαβ + cos(θ1)γ5Bαβ)η for those with (axial)vector ones. The last supermultiplets is simple: δχ = −iγµ∂µ(ϕ+ γ5π)η, δϕ = (χ̄η), δπ = (χ̄γ5η) To construct massive supermultiplet we have to add mass terms for all fields as well as appropriate corrections to fermionic supertransformations. In this, the most important ques- tion is which lower spin fields play the roles of Goldstone ones and have to be eaten to make main gauge fields massive. For the bosonic fields (taking into account parity conservation) the choice is unambiguous: vector Aµ and scalar ϕ fields for tensor field hµν and axial-vector Bµ and pseudo-scalar π — for pseudo-tensor fµν . Thus bosonic mass terms will be: 2[hµν∂µAν − h(∂A)] − 3Aµ∂µϕ+ 2[fµν∂µBν − f(∂B)]− 3Bµ∂µπ L2 = − (hµνhµν − h2)− hϕ+ ϕ2 − (fµνhµν − f 2)− fπ + π2 (5) But for fermions we have two spin 3/2 and two spin 1/2 fields and there is no evident choice. Thus we introduce the most general mass terms for the fermions: Lm = − Φ̄µνΦµν + (Φ̄γ) µ(γΦ)µ + −iα1[Φ̄µνγµΨν − Φ̄(γΨ)]− iα2[Φ̄µνγµΩν − Φ̄(γΩ)] + +a1[Ψ̄ µΨµ − (Ψ̄γ)(γΨ)] + a2[Ω̄µΩµ − (Ω̄γ)(γΩ)] + a3[Ψ̄µΩµ − (Ψ̄γ)(γΩ)] + +ia4(Ψ̄γ)ψ + ia5(Ψ̄γ)χ+ ia6(Ω̄γ)ψ + ia7(Ω̄γ)χ+ +a8ψ̄ψ + a9ψ̄χ+ a10χ̄χ (6) and proceed with calculations. Cancellation of variations with one derivative gives: sin(θ2) = cos(θ2) = sin(θ1) = cos(θ1) = , α2 = a1 = − , a3 = 1, a4 = 2, a5 = 0 a6 = a2 2, a7 = 3, a8 = 0, a9 = − while variations without derivatives give: , a10 = − Resulting fermionic mass terms: Lm = − Φ̄µνΦµν + (Φ̄γ) µ(γΦ)µ + Φ̄µνγµΨν + Φ̄(γΨ)− i 2Φ̄µνγµΩν + Φ̄(γΩ) + Ψ̄µΨµ + (Ψ̄γ)(γΨ) + Ω̄µΩµ − (Ω̄γ)(γΩ) + Ψ̄µΩµ − (Ψ̄γ)(γΩ) + 2(Ψ̄γ)ψ + (Ω̄γ)ψ + i 3(Ω̄γ)χ− 6ψ̄χ− χ̄χ (7) correspond to invariance of the Lagrangian (besides global supertransformations) under three local spinor gauge transformations: δΦµν = ∂(µξν) + γ(µξν) + gµνξ1 + gµνξ2 δΨµ = ∂µξ1 + γµξ1 + δΩµ = ∂µξ2 +m 2ξµ + γµξ1 + γµξ2 (8) δψ = m 2ξ1 + ξ2 δχ = m From these formula one can easily determine which combination of spin 3/2 fields Ψµ and Ωµ plays the role of Goldstone field for spin 5/2 field Φµν . Indeed, if one introduces two orthogonal combinations: Ψ̃µ = Ωµ, Ω̃µ = − then after diagonalization of mass terms one finds that Ψ̃µ is a Goldstone field, while Ω̃µ — physical field with the same mass as Φµν . Similar to the case of massive supermultiplet with superspin 1, both mixing angles have been fixed: θ1 = θ2 = π/4 and this, in turn, means that all bosonic fields enter through the complex combinations only: Hµν = hµν + γ5fµν , Cµ = Aµ + γ5Bµ, z = ϕ+ γ5π Introducing gauge covariant derivatives: ∇µHαβ = ∂µHαβ − Cµgαβ, ∇µz = ∂µz −m we can write final form of fermionic supertransformations as: δΦµν = [− σαβ∇αH̄β(µγν) −mHµν + γ(µ(γH)ν) + gµνz]η δΨµ = [− σαβ∇αHβµ − (γH)µ + ∇µz + γµz]η δΩµ = [− σαβC̄αβγµ + (γH)µ + ∇µz − γµz]η (9) δψ = − σαβCαβη δχ = −iγµ∇µzη Note also that due to complexification of bosonic fields the Lagrangian and supertrans- formations are invariant under global axial U(1)A symmetry, axial charges for all fields being: field η Φµν , Ψµ, Ωµ, ψ, χ Hµν , Cµ, z qA +1 0 –1 2 Superspin 5/2 Our next example — massive supermultiplet with superpin 5/2. It also contains four massive fields: with spin 3, 5/2, 5/2’ and 2 and in the massless limit it should reduce to six massless supermultiplets: 5/2 5/2 0, 0′ By analogy with all previous cases we will take into account possible mixing for bosonic tensor and vector fields, so we will start with the following structure of massless supermultiplets: hµν fµν Aµ Bµ So we introduce sum of the massless Lagrangians for bosonic fields: L0 = − ∂ρΦµνλ∂ρΦµνλ + (∂Φ)µν(∂Φ)µν − 3(∂Φ)µν∂µΦν + ∂µΦν∂µΦν + (∂Φ)2 ∂αhµν∂αhµν − (∂h)µ(∂h)µ + (∂h)µ∂µh− ∂µh∂µh− (∂µϕ) ∂αfµν∂αfµν − (∂f)µ(∂f)µ + (∂f)µ∂µf − ∂µf∂µf − (∂µπ) 2 (10) as well as sum of the massless Lagrangians for fermionic fields: Ψ̄µν ∂̂Ψµν − 2i(Ψ̄γ)µ(∂Ψ)µ + i(Ψ̄γ)µ∂̂(γΨ)µ + i(Ψ̄γ∂)Ψ− Ψ̄∂̂Ψ+ Ω̄µν ∂̂Ωµν − 2i(Ω̄γ)µ(∂Ω)µ + i(Ω̄γ)µ∂̂(γΩ)µ + i(Ω̄γ∂)Ω− Ω̄∂̂Ω− Ψ̄µ∂̂Ψµ + i(Ψ̄γ)(∂Ψ)− (Ψ̄γ)∂̂(γΨ) + ψ̄∂̂ψ − Ω̄µ∂̂Ωµ + i(Ω̄γ)(∂Ω) − (Ω̄γ)∂̂(γΩ) + χ̄∂̂χ (11) and start with the following global supertransformations: δΦµνλ = i(Ψ̄(µνγλ)η) δΨµν = [−σαβ∂αΦβµν + ∂(µγν)(γΦ)]η (12) for the supermultiplet (3, 5/2), δΩµν = − σαβ∂α(cos(θ2)hβ(µγν) − sin(θ2)γ5fβ(µγν))η δhµν = 2 cos(θ2)(Ω̄µνη) + i sin(θ2)(Ψ̄(µγν)η) (13) δfµν = 2 sin(θ2)(Ω̄µνγ5η) + i cos(θ2)(Ψ̄(µγν)γ5η) δΨµ = −σαβ∂α(sin(θ2)hβµ + cos(θ2)fβµγ5)η for the mixed (5/2, 2) and (2, 3/2) supermultiplets, δΩµ = − σαβ(cos(θ1)Aαβ − sin(θ1)γ5Bαβ)γµη δAµ = 2 cos(θ1)(Ω̄µη) + i sin(θ1)(ψ̄γµη) (14) δBµ = 2 sin(θ1)(Ω̄µγ5η) + i cos(θ1)(ψ̄γµγ5η) δψ = − σαβ(sin(θ1)Aαβ + cos(θ1)γ5Bαβ)η for the mixed (3/2, 1) and (1, 1/2) supermultiplets and δχ = −iγµ∂µ(ϕ+ γ5π)η, δϕ = (χ̄η), δπ = (χ̄γ5η) for the last one. By analogy with the superspin 3/2 case, we will assume that fermionic mass terms are Dirac ones: Lm = −Ψ̄µνΩµν + 2(Ψ̄γ)µ(γΩ)µ + Ψ̄Ω + [−Ψ̄µνγµΨν + Ψ̄(γΨ)− Ω̄µνγµΩν + Ω̄(γΩ)] + Ψ̄µΩµ − (Ψ̄γ)(γΩ) + 2i(Ψ̄γ)ψ + 2i(Ω̄γ)χ− 3ψ̄χ (15) where all coefficients are completely fixed by the requirement that the Lagrangian has to be invariant not only under the global supertransformations, but under four (by the number of fermionic gauge fields) spinor gauge transformations: δΨµν = ∂(µξν) + γ(µην) + gµνξ1, δΩµν = ∂(µην) + γ(µξν) + gµνξ2, δΨµ = ∂µξ1 +m ξµ + im γµξ2, δΩµ = ∂µξ2 +m ηµ + im γµξ1, δψ = 2mξ1, δχ = 2mξ2 As for the bosonic fields, here the roles of the fields are evident (again taking into account parity conservation): we need tensor hµν , vector Aµ and scalar ϕ fields to make spin 3 field Φµνλ massive, while pseudo-tensor fµν field needs to eat axial-vector Bµ and pseudo-scalar π fields. So the bosonic mass terms are also completely fixed: 3[−Φµνλ∂µhνλ + 2Φµ(∂h)µ − Φµ∂µh] + 5[hµν∂µAν − h(∂A)]− 6Aµ∂µϕ + 2[fµν∂µBν − f(∂B)]− 3Bµ∂µπ (16) ΦµνλΦµνλ − ΦµΦµ + ΦµAµ − (fµνfµν − f 2)− fπ + π2 (17) Now we require that the whole Lagrangian be invariant under global supertransformations with appropriate corrections to fermionic transformations. This fixes both mixing angles: sin(θ2) = , cos(θ2) = , sin(θ1) = , cos(θ1) = and gives the following form of additional terms for fermionic supertransformations: δΨµν = [− γ(µ(γh)ν) − fµνγ5 + γ(µ(γf)ν)γ5]η δΩµν = i[(γΦ)µν + gµν(γΦ)− γ(µAν) + gµνÂ− γ(µBν)γ5 + gµνB̂γ5]η δΨµ = [ γµ(γΦ)− γµÂ− Bµγ5 − γµB̂γ5]η (18) δΩµ = i[ (γh)µ + (γf)µγ5 − γµϕ− γµγ5π]η δψ = [−ϕ− 2γ5π]η δχ = i[ 6 + 3B̂γ5]η The complete supertransformations for fermionic fields could be simplified by introduction of gauge invariant derivatives; ∇µhαβ = ∂µhαβ − Aµgαβ, ∇µϕ = ∂µϕ−m ∇µfαβ = ∂µfαβ − Bµgαβ, ∇µπ = ∂µπ −m This time bosonic fields do not combine into complex combinations, but due to the fact that fermionic mass terms are Dirac ones the Lagrangian and supertransformations are invariant under global axial U(1)A transformations, provided axial charges of all fields are assigned as follows: field η, Ψµν , Ψµ, ψ hµν , fµν , Aµ, Bµ, ϕ, π Ωµν , Ωµ, χ qA +1 0 –1 3 Massive particles All our previous and subsequent calculations heavily depend on the gauge invariant descrip- tion of massive high spin particles. For reader convenience and to make paper self-contained we will give here gauge invariant formulations for massive particles with arbitrary integer [2, 3] and half-integer [10] spins. We restrict ourselves to flat d = 4 Minkowski space but all results could be easily generalized to the case of (A)dS space with arbitrary dimension d. 3.1 Integer spin The simplest way to describe massless bosonic field with arbitrary spin s is to use completely symmetric rank s tensor Φ(α1α2...αs) which is double traceless. In what follows we will use condensed notations where index denotes just number of free indices and not the indices themselves. For example, the tensor field itself will be denoted as Φs, it’s contraction with derivative as (∂Φ)s−1, it’s trace as Φ̃s−2 and so on. As we will see this does not lead to any ambiguities then working with free Lagrangians quadratic in fields. In these notations the Lagrangian for massless particles of arbitrary spin s could be written as: L0 = (−1)s[ ∂µΦs∂µΦ (∂Φ)s−1(∂Φ)s−1 − s(s− 1) ∂µΦ̃s−2∂µΦ̃ s−2 + s(s− 1) (∂Φ)s−1∂(1Φ̃s−2) − s(s− 1)(s− 2) (∂Φ̃)s−3(∂Φ̃)s−3] (19) where Φ = 0. This Lagrangian is invariant under the following gauge transformations: δ0Φs = ∂(1ξs−1), ξ̃s−3 = 0 where parameter ξs−1 is completely symmetric traceless tensor of rank s− 1. To construct gauge invariant Lagrangian for massive particle which has correct (i.e. with right number of physical degrees of freedom) massless limit, we start with the sum of massless Lagrangians with 0 ≤ k ≤ s: (−1)k[ ∂µΦk∂µΦ (∂Φ)k−1(∂Φ)k−1 − k(k − 1) ∂µΦ̃k−2∂µΦ̃ k(k − 1) (∂Φ)k−1∂(1Φ̃k−2) − k(k − 1)(k − 2) (∂Φ̃)k−3(∂Φ̃)k−3] (20) Then we add the following cross terms with one derivative as well as mass terms without derivatives: (−1)kak[(∂Φ)k−1Φk−1 + (k − 1)Φ̃k−2(∂Φ)k−2 + (k − 1)(k − 2) (∂Φ̃)k−3Φ̃k−3] (−1)k[dkΦkΦk + ekΦ̃k−2Φ̃k−2 + fkΦ̃k−2Φk−2] (21) and try to achieve gauge invariance with the help of appropriate corrections to gauge trans- formations: δΦk = αkξk + βkg(2ξk−2) Straightforward but lengthy calculations give a number of algebraic equations on the un- known coefficients which could be solved (and this is non-trivial because we obtain overde- termined system of equations) and give us: (s− k)(s+ k + 1) 2(k + 1)2 , βk+1 = k + 1 αk, 0 ≤ k ≤ s− 1 ak = − (s− k + 1)(s+ k) , dk = (s− k − 1)(s+ k + 2) 4(k + 1) k(k − 1) 16(k + 1) [(s− k + 2)(s+ k − 1) + 6] fk = − (s− k + 2)(s+ k − 1)(s− k + 1)(s+ k) 3.2 Half-integer spin For the description of massless spin s+1/2 particles we will use completely symmetric rank s tensor-spinor Ψs such that (γΨ̃)s−3 = 0 (in the same condensed notations as before). Then Lagrangain for such field could be written as: L0 = i(−1)s[ Ψ̄s∂̂Ψs − s(Ψ̄γ)s−1(∂Ψ)s−1 + (Ψ̄γ)s−1∂̂(γΨ)s−1 + s(s− 1) (Ψ̄γ∂)s−2Ψ̃s−2 − s(s− 1) ∂̂Ψ̃s−2] (22) and is invariant under the following gauge transformations: δ0Ψs = ∂(1ξs−1), (γξ) = 0, where gauge parameter ξs−1 is a γ-traceless tensor-spinor of rank s− 1. Once again we start with the sum of massless Lagrangians with 0 ≤ k ≤ s: i(−1)k[ Ψ̄k∂̂Ψk − k(Ψ̄γ)k−1(∂Ψ)k−1 + (Ψ̄γ)k−1∂̂(γΨ)k−1 + k(k − 1) (Ψ̄γ∂)k−2Ψ̃k−2 − k(k − 1) ∂̂Ψ̃k−2] (23) To combine all these massless fields into one massive particle we have to add the following mass terms: (−1)k 2(k + 1) [Ψ̄kΨk − k(Ψ̄γ)k−1(γΨ)k−1 − k(k − 1) Ψ̃k−2]+ −ick[(Ψ̄γ)k−1Ψk−1 − k − 1 (γΨ)k−2] and corresponding corrections to gauge transformations: δΨk = αkξk + iβkγ(1ξk−1) + ρkg(2ξk−2) Then total Lagrangian will be gauge invariant provided: (s+ 1)2 − k2 , αk = k + 1 , βk = 2k(k + 1) , ρk = 4 Massless supermultiplets It is not easy to find in the recent literature the explicit component form of massless su- permultiplets with arbitrary superspin [22], so for completeness we will give their short description here. As we have already seen on the lower superspin cases, supermultiplets with integer and half-integer superspins have different structure and have to be considered separately. (s, s+1/2). Supermultiplet with integer superspin s contains bosonic spin s field and fermionic spin s+1/2 one. In this and in two subsequent sections we will use the same con- densed notations as in the previous one. By analogy with superspins 1 and 2 supermultiplets, we start with the following ansatz for the supertransformations: δΨs = iα1σ µν∂µΦν(s−1γ1)η δΦs = β(Ψ̄sη) Indeed, calculating variations of the sum of two massless Lagrangians one can see that most of variations cancel, provided one set α1 = −β2 . The residue: δL = −(−1)sβ (s− 1)(s− 2) [2(Ψ̄∂∂)s−2Φ̃s−2 − ˜̄Ψ ∂2Φ̃s−2 − (s− 2)( ˜̄Ψ∂)s−3(∂Φ̃)s−3 − −2(Ψ̄γ∂)s−2∂̂Φ̃s−2 + 2(Ψ̄γ∂∂)s−3(γΦ̃)s−3 − ( ˜̄Ψ∂)s−3∂̂(γΦ̃)s−3] contains terms with Φ̃s−2 only, so we proceed by adding to fermionic supertransformations one more term: δ′Ψs = iα2∂(1γ1Φ̃s−2) Then the choice α2 = (s−1)(s−2)β leaves us with: δL = −(−1)sβ (s− 1)(s− 2) [2(Ψ̄γ∂∂)s−3(γΦ̃)s−3 − ( ˜̄Ψ∂)s−3∂̂(γΦ̃)s−3] where the only terms are whose with (γΦ̃)s−3. So we make one more (last) correction to supertransformations: δ′′Ψs = iα3g(2∂1(γΦ̃)s−3) and obtain full invariance with α3 = − (s−1)(s−2)β4s . To fix concrete normalization we will use closure of the superalgebra. Calculating the commutator of two supertransformations we obtain: [δ1, δ2]Φs = −iβ2(η̄2γνη1)∂µΦs + . . . where dots mean “up to gauge transformation”. So we set β = 2 and our final result looks like: δΨs = − σµν∂µΦ̄ν(s−1γ1)η + i(s− 1)(s− 2) [∂(1γ1Φ̃s−2) − g(2∂1(γΦ̃)s−3)]η δΦs = 2(Ψ̄sη) (25) (s+1/2, s+1). Half-integer superspin multiplet contains fermionic spin s + 1/2 fields and bosonic spin s + 1 one. Again by analogy with lower superspin case we will make the following ansatz for supertransformations: δΨs = α1σ µν∂µΦν(s)η, δΦs+1 = iβ(Ψ̄(sγ1)η) This time most of the variations cancel if one set α1 = −β leaving us with: δL = i(−1)sβ s(s− 1) [2(Ψ̄∂∂)s−2(γΦ̃)s−2 − ˜̄Ψ ∂2(γΦ̃)s−2 − −2(Ψ̄γ∂)s−2∂̂(γΦ̃)s−2 − (s− 2)( ˜̄Ψ∂)s−3(γ∂Φ̃)s−3] Then the full invariance could be achieved with the following correction to supertransforma- tions: δ′Ψs = α2∂(1γ1(γΦ̃)s−2) provided α2 = . To check the closure of superalgebra and to choose normalization we calculate commutator of two supertransformations: [δ1, δ2]Φs+1 = −2iβ2(η̄2γµη1)∂µΦs+1 + . . . Then our choice will be β = 1 and our final form: δΨs = −σµν∂µΦν(s)η + ∂(1γ1(γΦ̃)s−2)η, δΦs+1 = i(Ψ̄(sγ1)η) Note that starting with superspin 2 the structure of supertransformations are defined up to possible field dependent gauge transformations and our choice differs from that of [22]. It makes no difference for massless theories but for massive case the structure of corrections for fermionic supertransformations depends on the choice made. 5 Integer superspin Now, having in our disposal gauge invariant description of massive particles with arbitrary (half-)integer spins, known form of supertransformations for massless arbitrary superspin supermultiplets and concrete examples of massive supermultiplets with lower superspins, we are ready to construct massive arbitrary superspin supermultiplets. As we have seen, integer and half-integer cases have different structures and have to be considered separately. In this section we consider massive supermultiplet with integer superspin. Such super- multiplet also contains four massive fields: two bosonic spin s fields (with opposite parity) and fermionic spin (s+1/2) and (s-1/2) ones. Calculating total number of physical degrees of freedom and taking into account possible mixing of supermultiplets containing bosonic fields with equal spins and opposite parity, we start with the following structure of massless supermultiplets: As Bs Ak Bk By analogy with superspin 1 and 2 cases, we will assume that all bosonic fields enter through the complex combinations Ck = Ak + iBk only (so that all possible mixing angles are fixed and equal π/4). Thus we choose the following form of supertransformations for massless supermultiplets with 1 ≤ k ≤ s: δΦk = − σµν∂µC̄ν(k−1γ1)η + i(k − 1)(k − 2) [∂(1γ1C̃k−2) − g(2∂1(γC̃)k−3)]η δC̄k = 2(Φ̄kη) + i 2(Ψ̄(k−1γ1)η) (26) δΨk−1 = − σµν∂µCν(k−1)η + k − 2 ∂(1γ1(γC̃)k−3)η and also δΦ0 = −i∂̂zη, δz̄ = 2(Φ̄0η) As a result of our assumption mass terms for bosonic fields are completely fixed: (−1)kck[C̄k∂(1Ck−1) − (k − 1) ˜̄C (∂C)k−2 + (k − 1)(k − 2) ( ˜̄C ∂(1C̃k−3) + h.c.)] (−1)k[dkC̄kCk + ek ˜̄C C̃k−2 + fk( Ck−2 + h.c.)] (27) where (s+ k)(s− k + 1) , dk = (s− k − 1)(s+ k + 2) 4(k + 1) k(k − 1) 16(k + 1) [(s− k + 2)(s+ k − 1) + 6] fk = − (s− k + 2)(s+ k − 1)(s− k + 1)(s+ k) As for the fermionic mass terms, apriori we don’t have any restrictions on them so we have to consider the most general possible form: (−1)k a1k[Φ̄ kΦk − k(Φ̄γ)k−1(γΦ)k−1 − k(k − 1) Φ̃k−2]+ +a2k[Φ̄ kΨk − k(Φ̄γ)k−1(γΨ)k−1 − k(k − 1) Ψ̃k−2] + +a3k[Ψ̄ kΨk − k(Ψ̄γ)k−1(γΨ)k−1 − k(k − 1) Ψ̃k−2] + +ib1k[(Φ̄γ) k−1Φk−1 − k − 1 (γΦ)k−2] + +ib2k[(Φ̄γ) k−1Ψk−1 − k − 1 (γΨ)k−2] + +ib3k[(Ψ̄γ) k−1Φk−1 − k − 1 (γΦ)k−2] + + ib4k[(Ψ̄γ) k−1Ψk−1 − k − 1 (γΨ)k−2] Where: a1s = − , a2s = a3s = b3s = b4s = 0 The requirement that total Lagrangian be invariant under (appropriately corrected) super- transformations gives: a1k = − , a2k = − k + 1 ck+1, a3k = 2(k + 1) b1k = −2ck, b2k = − , b3k = 0, b4k = −2ck+1 In this, additional terms for fermionic supertransformations look like: δ′Φk = 2ick+1 k + 1 [(γC)k + k(k − 1) 4(k + 1) γ(1C̃k−1) − (k − 1)2(2k + 1) 8k(k + 1) g(2(γC̃)k−2)]− −Ck + k − 1 γ(1(γC)k−1 − (k − 1)(k − 2) g(2C̃k−2) − [γ(1Ck−1) − g(2(γC)k−2)] (29) δ′Ψk = kck+2 2(k + 1) γ(1(γC̃)k−1) − 2(k + 1) [(γC)k − k(k − 1) 4(k + 1) γ(1C̃k−1) + (k − 1)(3k + 1) 8k(k + 1) g(2(γC̃)k−2)]− 2ck+1 k + 1 [Ck + γ(1(γC)k−1) + (k − 1)(k − 2) g(2C̃k−2)] (30) Here the supertransformations for Φk field contain terms with Ck+1, Ck and Ck−1 fields in the first, second and third lines correspondingly, while that of Ψk contain terms with Ck+2, Ck+1 and Ck fields. 6 Half-integer superspin Next we turn to the half-integer superspin case. This time we have two fermionic spin (s+1/2) fields and bosonic ones with spins (s+1) and s. Usual reasoning on physical degrees of freedom and possible mixings leads us to the following structure of massless supermultiplets we will start with: Φs Ψs Ak Bk We see that this structure is rather similar to that of integer superspin case. The main difference (besides the presence of As+1, Ψs supermultiplet) comes from the mixing of bosonic fields. We have no reasons to suggest that all mixing angles could be fixed from the very beginning so we have to consider the most general possibility here. Let us denote: Ck = cos(θk)Ak + γ5 sin(θk)Bk, Dk = sin(θk)Ak + γ5 cos(θk)Bk In these notations supertransformations for massless supermultiplets could be written as follows. Highest supermultiplet: δAs+1 = i(Ψ̄(sγ1)η) δΨs = −σµν∂µAν(s)η + ∂(1γ1(γÃ)s−2)η Main set (1 ≤ k ≤ s): δΦk = − σµν∂µC̄ν(k−1γ1)η + i(k − 1)(k − 2) [∂(1γ1C̃k−2) − g(2∂1(γC̃)k−3)]η δAk = 2 cos(θk)(Φ̄kη) + i sin(θk)(Ψ̄(k−1γ1)η) (31) δBk = 2 sin(θk)(Φ̄kγ5η) + i cos(θk)(Ψ̄(k−1γ1)γ5η) δΨk−1 = −σµν∂µDν(k−1)η + k − 2 ∂(1γ1(γD̃)k−3η and the last supermultiplet: δΦ0 = −i∂̂zη, δz̄ = 2(Φ̄0η) By analogy with superspins 3/2 and 5/2 cases we will assume that fermionic mass terms are Dirac ones. This immediately gives: (−1)k s + 1 k + 1 [Ψ̄kΦk − k(Ψ̄γ)k−1(γΦ)k−1 − k(k − 1) Φ̃k−2]− − ick[(Ψ̄γ)k−1Ψk−1 − k − 1 (γΨ)k−2 + (Ψ → Φ)] where: (s+ k + 1)(s− k + 1) The choice for the bosonic mass terms (taking into account parity) is also unambiguous: (−1)kak[Ak∂(1Ak−1) − (k − 1)Ãk−2(∂A)k−2 + (k − 1)(k − 2) Ãk−2∂(1Ãk−3)] + (−1)kbk[Bk∂(1Bk−1) − (k − 1)B̃k−2(∂B)k−2 + (k − 1)(k − 2) B̃k−2∂(1B̃k−3)] (33) for the terms with one derivative, where: (s+ k + 1)(s− k + 2) , bk = (s+ k)(s− k + 1) and the following terms without derivatives: (−1)k[d̂kAkAk + êkÃk−2Ãk−2 + f̂kÃk−2Ak−2] + (−1)k[dkBkBk + ekB̃k−2B̃k−2 + fkB̃k−2Bk−2] (34) Here: d̂k = (s− k)(s+ k + 3) 4(k + 1) , êk = k(k − 1) 16(k + 1) [(s− k + 3)(s+ k) + 6] f̂k = − (s− k + 3)(s+ k)(s− k + 2)(s+ k + 1) (s− k − 1)(s+ k + 2) 4(k + 1) , ek = k(k − 1) 16(k + 1) [(s− k + 2)(s+ k − 1) + 6] fk = − (s− k + 2)(s+ k − 1)(s− k + 1)(s+ k) Note that hatted coefficients differ from the unhatted ones by replacement s→ s+ 1. Now we require that total Lagrangian be invariant under the supertransformations. First of all this fixes all mixing angles: sin(θk) = s+ k + 1 2(s+ 1) , cos(θk) = s− k + 1 2(s+ 1) and gives us additional terms for fermionic supertransformations: δ′Ψk = α1Ak + α2γ(1(γA)k−1) + α3g(2Ãk−2) + +β1Bk + β2γ(1(γB)k−1) + β3g(2B̃k−2) + kck+1 4(k + 1) [sin(θk+2)γ(1(γÃ)k−1) + cos(θk+1)γ(1(γB̃)k−1)] δ′Φk = α4(γA)k + α5γ(1Ãk−1) + α6g(2(γÃ)k−2) + +β4(γB)k + β5γ(1B̃k−1) + β6g(2(γB̃)k−2) − cos(θk)[γ(1Ak−1) − g(2(γA)k−2]− sin(θk)[γ(1Bk−1) − g(2(γB)k−2)] Where: α1 = − s− k√ 2(k + 1) cos(θk), α2 = − k2 + s+ k + 1√ 2k(k + 1) cos(θk) α3 = − (s + 1)(k − 1)(k − 2) 2k2(k + 1) cos(θk) β1 = − s+ k + 2√ 2(k + 1) sin(θk), β2 = k2 − s+ k − 1√ 2k(k + 1) sin(θk) β3 = − (s + 1)(k − 1)(k − 2) 2k2(k + 1) sin(θk) k + 1 sin(θk+1), α5 = k(k − 1)(s− k) 4(k + 1)2 sin(θk+1) (k − 1)[(k + 1)(s+ 1)− 2k2(s− k)] 8k(k + 1)2 sin(θk+1) k + 1 cos(θk+1), β5 = k(k − 1)(s+ k + 2) 4(k + 1)2 cos(θk+1) (k − 1)[(k + 1)(s+ 1)− 2k2(s+ k + 2)] 8k(k + 1)2 cos(θk+1) We have explicitely checked that (rather complicated) formulas from this and previous sections correctly reproduce all lower superspins results. 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Lett.B85 (1979) http://arxiv.org/abs/hep-th/0608005 http://arxiv.org/abs/hep-th/0609029 http://arxiv.org/abs/hep-th/0609170 http://arxiv.org/abs/hep-th/0612279 http://arxiv.org/abs/hep-th/0703049 http://arxiv.org/abs/hep-th/0201096 http://arxiv.org/abs/hep-th/0206209 http://arxiv.org/abs/hep-th/0207243 http://arxiv.org/abs/hep-th/0403224 http://arxiv.org/abs/hep-th/0506255 http://arxiv.org/abs/hep-th/0501199 http://arxiv.org/abs/hep-th/0610333 http://arxiv.org/abs/hep-th/0703118 Superspin 2 Superspin 5/2 Massive particles Integer spin Half-integer spin Massless supermultiplets Integer superspin Half-integer superspin

In this paper we give explicit construction of massive N=1 supermultiplets in flat d=4 Minkowski space-time. We work in a component on-shell formalism based on gauge invariant description of massive integer and half-integer spin particles where massive supermultiplets are constructed out of appropriate set of massless ones.

Introduction In a flat space-time massive spin s particles in a massless limit decompose into massless spin s, s− 1, . . . ones. This, in particular, leads to the possibility of gauge invariant description of massive spin s particles, e.g. [1]-[13]. In this, two different approaches could be used. From one hand, one can start with usual non gauge invariant description of massive particle and achieve gauge invariance through the introduction of additional fields (thus promoting second class constraints into the first class ones). From the other hand, one can start with the appropriate set of massless particles having gauge invariance from the very beginning and obtain massive particle description as a deformation of massless theory. This last approach closely mimic situation in spontaneous gauge symmetry breaking where gauge field has to eat some Goldstone field(s) to become massive. In the supersymmetric theories all particles must belong to some supermultiplet, massive or massless. Till now most of investigations in supersymmetric theories where bounded to massless supermultiplets. Quite a few results on massive supermultiplets mainly devoted to superspins 1 and 3/2 exist [14]-[21]. The aim of this paper is to extend these results to include massive N = 1 supermultiplets with arbitrary superspins. Certainly, it would be nice to have superfield off-shell description of such supermultiplets, but as previous results clearly show it is a highly non-trivial task. So in this paper we restrict ourselves with component on-shell formalism in terms of physical fields. The same reasoning on the massless limit means that massive supermultiplets could (should) be constructed out of the massless ones in the same way as massive particles out of the massless ones. So our approach will be supersymmetric generalization of the second approach to massive particle description mentioned above. Namely, we will start with appropriate set of massless supermultiplets and obtain massive one as a smooth deformation. The paper is organized as follows. Though our previous examples on massive superspin 1 [21] and superspin 3/2 [15] supermultiplets already give important hints on how general case of arbitrary superspin could looks like, due to peculiarities of lower spin fields they are not enough to achieve such generalization. Thus, in the first two sections we give two more concrete examples, namely massive supermultiplets with superspin 2 and 5/2, correspond- ingly. All these and subsequent results heavily depend on the gauge invariant description of massive particles with integer [2, 3] and half-integer [10] spins as well as on the known form of massless supermultiplets [22]. For reader convenience and to make paper self-contained, in the next two sections we give all necessary formulas in compact condensed notations. One of the lessons from previous investigations is that the structures of massive supermultiplets with integer and half-integer superspins are different, so in the last two sections we consider these two cases separately. We will see that, in spite of large number of fields, all calculations are pretty straightforward and mainly combinatorical. 1 Superspin 2 Massive superspin 2 supermultiplet contains four massive particles with spins 5/2, 2, 2’ and 3/2, correspondingly. In the massless limit massive supermultilets must decompose into the appropriate set of massless ones in the same way as massive spin s particles — into massless spin s, s-1, ... ones. Simple counting of physical degrees of freedom immediately gives: 0, 0′ So we will start with five massless supermultiplets (Φµν , hµν), (fµν ,Ψµ), (Ωµ, Aµ), (Bµ, ψ) and (χ, z). From our previous experience with massive superspin 1 and superspin 3/2 su- permultiplets we know that it is crucial for the construction of massive supermultiplets to make dual rotation of vector Aµ and axial-vector Bµ fields mixing massless supermultiplets containing these fields. But now we have two tensor fields hµν and fµν as well, moreover they necessarily must be tensor and pseudo-tensor ones. Thus we have to consider the possibility to mix massless supermultiplets with these fields as well and the real structure of massless supermultiplets we are going to work with looks like: hµν fµν Aµ Bµ Then, introducing a sum of the massless Lagrangians for bosonic fields: ∂αhµν∂αhµν − (∂h)µ(∂h)µ + (∂h)µ∂µh− ∂µh∂µh+ ∂αfµν∂αfµν − (∂f)µ(∂f)µ + (∂f)µ∂µf − ∂µf∂µf − (∂µϕ) (∂µπ) 2 (1) as well as sum of the massless Lagrangians for fermionic fields: Φ̄µν ∂̂Φµν − 2i(Φ̄γ)µ(∂Φ)µ + i(Φ̄γ)µ∂̂(γΦ)µ + i(Φ̄γ∂)Φ− Φ̄∂̂Φ− Ψ̄µ∂̂Ψµ + i(Ψ̄γ)(∂Ψ)− (Ψ̄γ)∂̂(γΨ) + ψ̄∂̂ψ − Ω̄µ∂̂Ωµ + i(Ω̄γ)(∂Ω)− (Ω̄γ)∂̂(γΩ) + χ̄∂̂χ (2) it is not hard to check that the most general supertransformations leaving sum of massless Lagrangians invariant have the form (round brackets denote symmetrization): δΦµν = − σαβ∂α(cos(θ2)hβ(µγν) − sin(θ2)γ5fβ(µγν))η δhµν = 2 cos(θ2)(Φ̄µνη) + i sin(θ2)(Ψ̄(µγν)η) (3) δfµν = 2 sin(θ2)(Φ̄µνγ5η) + i cos(θ2)(Ψ̄(µγν)γ5η) δΨµ = −σαβ∂α(sin(θ2)hβµ + cos(θ2)fβµγ5)η for the supermultiplets containing spin 2 fields and δΩµ = − σαβ(cos(θ1)Aαβ − sin(θ1)γ5Bαβ)γµη δAµ = 2 cos(θ1)(Ω̄µη) + i sin(θ1)(ψ̄γµη) (4) δBµ = 2 sin(θ1)(Ω̄µγ5η) + i cos(θ1)(ψ̄γµγ5η) δψ = − σαβ(sin(θ1)Aαβ + cos(θ1)γ5Bαβ)η for those with (axial)vector ones. The last supermultiplets is simple: δχ = −iγµ∂µ(ϕ+ γ5π)η, δϕ = (χ̄η), δπ = (χ̄γ5η) To construct massive supermultiplet we have to add mass terms for all fields as well as appropriate corrections to fermionic supertransformations. In this, the most important ques- tion is which lower spin fields play the roles of Goldstone ones and have to be eaten to make main gauge fields massive. For the bosonic fields (taking into account parity conservation) the choice is unambiguous: vector Aµ and scalar ϕ fields for tensor field hµν and axial-vector Bµ and pseudo-scalar π — for pseudo-tensor fµν . Thus bosonic mass terms will be: 2[hµν∂µAν − h(∂A)] − 3Aµ∂µϕ+ 2[fµν∂µBν − f(∂B)]− 3Bµ∂µπ L2 = − (hµνhµν − h2)− hϕ+ ϕ2 − (fµνhµν − f 2)− fπ + π2 (5) But for fermions we have two spin 3/2 and two spin 1/2 fields and there is no evident choice. Thus we introduce the most general mass terms for the fermions: Lm = − Φ̄µνΦµν + (Φ̄γ) µ(γΦ)µ + −iα1[Φ̄µνγµΨν − Φ̄(γΨ)]− iα2[Φ̄µνγµΩν − Φ̄(γΩ)] + +a1[Ψ̄ µΨµ − (Ψ̄γ)(γΨ)] + a2[Ω̄µΩµ − (Ω̄γ)(γΩ)] + a3[Ψ̄µΩµ − (Ψ̄γ)(γΩ)] + +ia4(Ψ̄γ)ψ + ia5(Ψ̄γ)χ+ ia6(Ω̄γ)ψ + ia7(Ω̄γ)χ+ +a8ψ̄ψ + a9ψ̄χ+ a10χ̄χ (6) and proceed with calculations. Cancellation of variations with one derivative gives: sin(θ2) = cos(θ2) = sin(θ1) = cos(θ1) = , α2 = a1 = − , a3 = 1, a4 = 2, a5 = 0 a6 = a2 2, a7 = 3, a8 = 0, a9 = − while variations without derivatives give: , a10 = − Resulting fermionic mass terms: Lm = − Φ̄µνΦµν + (Φ̄γ) µ(γΦ)µ + Φ̄µνγµΨν + Φ̄(γΨ)− i 2Φ̄µνγµΩν + Φ̄(γΩ) + Ψ̄µΨµ + (Ψ̄γ)(γΨ) + Ω̄µΩµ − (Ω̄γ)(γΩ) + Ψ̄µΩµ − (Ψ̄γ)(γΩ) + 2(Ψ̄γ)ψ + (Ω̄γ)ψ + i 3(Ω̄γ)χ− 6ψ̄χ− χ̄χ (7) correspond to invariance of the Lagrangian (besides global supertransformations) under three local spinor gauge transformations: δΦµν = ∂(µξν) + γ(µξν) + gµνξ1 + gµνξ2 δΨµ = ∂µξ1 + γµξ1 + δΩµ = ∂µξ2 +m 2ξµ + γµξ1 + γµξ2 (8) δψ = m 2ξ1 + ξ2 δχ = m From these formula one can easily determine which combination of spin 3/2 fields Ψµ and Ωµ plays the role of Goldstone field for spin 5/2 field Φµν . Indeed, if one introduces two orthogonal combinations: Ψ̃µ = Ωµ, Ω̃µ = − then after diagonalization of mass terms one finds that Ψ̃µ is a Goldstone field, while Ω̃µ — physical field with the same mass as Φµν . Similar to the case of massive supermultiplet with superspin 1, both mixing angles have been fixed: θ1 = θ2 = π/4 and this, in turn, means that all bosonic fields enter through the complex combinations only: Hµν = hµν + γ5fµν , Cµ = Aµ + γ5Bµ, z = ϕ+ γ5π Introducing gauge covariant derivatives: ∇µHαβ = ∂µHαβ − Cµgαβ, ∇µz = ∂µz −m we can write final form of fermionic supertransformations as: δΦµν = [− σαβ∇αH̄β(µγν) −mHµν + γ(µ(γH)ν) + gµνz]η δΨµ = [− σαβ∇αHβµ − (γH)µ + ∇µz + γµz]η δΩµ = [− σαβC̄αβγµ + (γH)µ + ∇µz − γµz]η (9) δψ = − σαβCαβη δχ = −iγµ∇µzη Note also that due to complexification of bosonic fields the Lagrangian and supertrans- formations are invariant under global axial U(1)A symmetry, axial charges for all fields being: field η Φµν , Ψµ, Ωµ, ψ, χ Hµν , Cµ, z qA +1 0 –1 2 Superspin 5/2 Our next example — massive supermultiplet with superpin 5/2. It also contains four massive fields: with spin 3, 5/2, 5/2’ and 2 and in the massless limit it should reduce to six massless supermultiplets: 5/2 5/2 0, 0′ By analogy with all previous cases we will take into account possible mixing for bosonic tensor and vector fields, so we will start with the following structure of massless supermultiplets: hµν fµν Aµ Bµ So we introduce sum of the massless Lagrangians for bosonic fields: L0 = − ∂ρΦµνλ∂ρΦµνλ + (∂Φ)µν(∂Φ)µν − 3(∂Φ)µν∂µΦν + ∂µΦν∂µΦν + (∂Φ)2 ∂αhµν∂αhµν − (∂h)µ(∂h)µ + (∂h)µ∂µh− ∂µh∂µh− (∂µϕ) ∂αfµν∂αfµν − (∂f)µ(∂f)µ + (∂f)µ∂µf − ∂µf∂µf − (∂µπ) 2 (10) as well as sum of the massless Lagrangians for fermionic fields: Ψ̄µν ∂̂Ψµν − 2i(Ψ̄γ)µ(∂Ψ)µ + i(Ψ̄γ)µ∂̂(γΨ)µ + i(Ψ̄γ∂)Ψ− Ψ̄∂̂Ψ+ Ω̄µν ∂̂Ωµν − 2i(Ω̄γ)µ(∂Ω)µ + i(Ω̄γ)µ∂̂(γΩ)µ + i(Ω̄γ∂)Ω− Ω̄∂̂Ω− Ψ̄µ∂̂Ψµ + i(Ψ̄γ)(∂Ψ)− (Ψ̄γ)∂̂(γΨ) + ψ̄∂̂ψ − Ω̄µ∂̂Ωµ + i(Ω̄γ)(∂Ω) − (Ω̄γ)∂̂(γΩ) + χ̄∂̂χ (11) and start with the following global supertransformations: δΦµνλ = i(Ψ̄(µνγλ)η) δΨµν = [−σαβ∂αΦβµν + ∂(µγν)(γΦ)]η (12) for the supermultiplet (3, 5/2), δΩµν = − σαβ∂α(cos(θ2)hβ(µγν) − sin(θ2)γ5fβ(µγν))η δhµν = 2 cos(θ2)(Ω̄µνη) + i sin(θ2)(Ψ̄(µγν)η) (13) δfµν = 2 sin(θ2)(Ω̄µνγ5η) + i cos(θ2)(Ψ̄(µγν)γ5η) δΨµ = −σαβ∂α(sin(θ2)hβµ + cos(θ2)fβµγ5)η for the mixed (5/2, 2) and (2, 3/2) supermultiplets, δΩµ = − σαβ(cos(θ1)Aαβ − sin(θ1)γ5Bαβ)γµη δAµ = 2 cos(θ1)(Ω̄µη) + i sin(θ1)(ψ̄γµη) (14) δBµ = 2 sin(θ1)(Ω̄µγ5η) + i cos(θ1)(ψ̄γµγ5η) δψ = − σαβ(sin(θ1)Aαβ + cos(θ1)γ5Bαβ)η for the mixed (3/2, 1) and (1, 1/2) supermultiplets and δχ = −iγµ∂µ(ϕ+ γ5π)η, δϕ = (χ̄η), δπ = (χ̄γ5η) for the last one. By analogy with the superspin 3/2 case, we will assume that fermionic mass terms are Dirac ones: Lm = −Ψ̄µνΩµν + 2(Ψ̄γ)µ(γΩ)µ + Ψ̄Ω + [−Ψ̄µνγµΨν + Ψ̄(γΨ)− Ω̄µνγµΩν + Ω̄(γΩ)] + Ψ̄µΩµ − (Ψ̄γ)(γΩ) + 2i(Ψ̄γ)ψ + 2i(Ω̄γ)χ− 3ψ̄χ (15) where all coefficients are completely fixed by the requirement that the Lagrangian has to be invariant not only under the global supertransformations, but under four (by the number of fermionic gauge fields) spinor gauge transformations: δΨµν = ∂(µξν) + γ(µην) + gµνξ1, δΩµν = ∂(µην) + γ(µξν) + gµνξ2, δΨµ = ∂µξ1 +m ξµ + im γµξ2, δΩµ = ∂µξ2 +m ηµ + im γµξ1, δψ = 2mξ1, δχ = 2mξ2 As for the bosonic fields, here the roles of the fields are evident (again taking into account parity conservation): we need tensor hµν , vector Aµ and scalar ϕ fields to make spin 3 field Φµνλ massive, while pseudo-tensor fµν field needs to eat axial-vector Bµ and pseudo-scalar π fields. So the bosonic mass terms are also completely fixed: 3[−Φµνλ∂µhνλ + 2Φµ(∂h)µ − Φµ∂µh] + 5[hµν∂µAν − h(∂A)]− 6Aµ∂µϕ + 2[fµν∂µBν − f(∂B)]− 3Bµ∂µπ (16) ΦµνλΦµνλ − ΦµΦµ + ΦµAµ − (fµνfµν − f 2)− fπ + π2 (17) Now we require that the whole Lagrangian be invariant under global supertransformations with appropriate corrections to fermionic transformations. This fixes both mixing angles: sin(θ2) = , cos(θ2) = , sin(θ1) = , cos(θ1) = and gives the following form of additional terms for fermionic supertransformations: δΨµν = [− γ(µ(γh)ν) − fµνγ5 + γ(µ(γf)ν)γ5]η δΩµν = i[(γΦ)µν + gµν(γΦ)− γ(µAν) + gµνÂ− γ(µBν)γ5 + gµνB̂γ5]η δΨµ = [ γµ(γΦ)− γµÂ− Bµγ5 − γµB̂γ5]η (18) δΩµ = i[ (γh)µ + (γf)µγ5 − γµϕ− γµγ5π]η δψ = [−ϕ− 2γ5π]η δχ = i[ 6 + 3B̂γ5]η The complete supertransformations for fermionic fields could be simplified by introduction of gauge invariant derivatives; ∇µhαβ = ∂µhαβ − Aµgαβ, ∇µϕ = ∂µϕ−m ∇µfαβ = ∂µfαβ − Bµgαβ, ∇µπ = ∂µπ −m This time bosonic fields do not combine into complex combinations, but due to the fact that fermionic mass terms are Dirac ones the Lagrangian and supertransformations are invariant under global axial U(1)A transformations, provided axial charges of all fields are assigned as follows: field η, Ψµν , Ψµ, ψ hµν , fµν , Aµ, Bµ, ϕ, π Ωµν , Ωµ, χ qA +1 0 –1 3 Massive particles All our previous and subsequent calculations heavily depend on the gauge invariant descrip- tion of massive high spin particles. For reader convenience and to make paper self-contained we will give here gauge invariant formulations for massive particles with arbitrary integer [2, 3] and half-integer [10] spins. We restrict ourselves to flat d = 4 Minkowski space but all results could be easily generalized to the case of (A)dS space with arbitrary dimension d. 3.1 Integer spin The simplest way to describe massless bosonic field with arbitrary spin s is to use completely symmetric rank s tensor Φ(α1α2...αs) which is double traceless. In what follows we will use condensed notations where index denotes just number of free indices and not the indices themselves. For example, the tensor field itself will be denoted as Φs, it’s contraction with derivative as (∂Φ)s−1, it’s trace as Φ̃s−2 and so on. As we will see this does not lead to any ambiguities then working with free Lagrangians quadratic in fields. In these notations the Lagrangian for massless particles of arbitrary spin s could be written as: L0 = (−1)s[ ∂µΦs∂µΦ (∂Φ)s−1(∂Φ)s−1 − s(s− 1) ∂µΦ̃s−2∂µΦ̃ s−2 + s(s− 1) (∂Φ)s−1∂(1Φ̃s−2) − s(s− 1)(s− 2) (∂Φ̃)s−3(∂Φ̃)s−3] (19) where Φ = 0. This Lagrangian is invariant under the following gauge transformations: δ0Φs = ∂(1ξs−1), ξ̃s−3 = 0 where parameter ξs−1 is completely symmetric traceless tensor of rank s− 1. To construct gauge invariant Lagrangian for massive particle which has correct (i.e. with right number of physical degrees of freedom) massless limit, we start with the sum of massless Lagrangians with 0 ≤ k ≤ s: (−1)k[ ∂µΦk∂µΦ (∂Φ)k−1(∂Φ)k−1 − k(k − 1) ∂µΦ̃k−2∂µΦ̃ k(k − 1) (∂Φ)k−1∂(1Φ̃k−2) − k(k − 1)(k − 2) (∂Φ̃)k−3(∂Φ̃)k−3] (20) Then we add the following cross terms with one derivative as well as mass terms without derivatives: (−1)kak[(∂Φ)k−1Φk−1 + (k − 1)Φ̃k−2(∂Φ)k−2 + (k − 1)(k − 2) (∂Φ̃)k−3Φ̃k−3] (−1)k[dkΦkΦk + ekΦ̃k−2Φ̃k−2 + fkΦ̃k−2Φk−2] (21) and try to achieve gauge invariance with the help of appropriate corrections to gauge trans- formations: δΦk = αkξk + βkg(2ξk−2) Straightforward but lengthy calculations give a number of algebraic equations on the un- known coefficients which could be solved (and this is non-trivial because we obtain overde- termined system of equations) and give us: (s− k)(s+ k + 1) 2(k + 1)2 , βk+1 = k + 1 αk, 0 ≤ k ≤ s− 1 ak = − (s− k + 1)(s+ k) , dk = (s− k − 1)(s+ k + 2) 4(k + 1) k(k − 1) 16(k + 1) [(s− k + 2)(s+ k − 1) + 6] fk = − (s− k + 2)(s+ k − 1)(s− k + 1)(s+ k) 3.2 Half-integer spin For the description of massless spin s+1/2 particles we will use completely symmetric rank s tensor-spinor Ψs such that (γΨ̃)s−3 = 0 (in the same condensed notations as before). Then Lagrangain for such field could be written as: L0 = i(−1)s[ Ψ̄s∂̂Ψs − s(Ψ̄γ)s−1(∂Ψ)s−1 + (Ψ̄γ)s−1∂̂(γΨ)s−1 + s(s− 1) (Ψ̄γ∂)s−2Ψ̃s−2 − s(s− 1) ∂̂Ψ̃s−2] (22) and is invariant under the following gauge transformations: δ0Ψs = ∂(1ξs−1), (γξ) = 0, where gauge parameter ξs−1 is a γ-traceless tensor-spinor of rank s− 1. Once again we start with the sum of massless Lagrangians with 0 ≤ k ≤ s: i(−1)k[ Ψ̄k∂̂Ψk − k(Ψ̄γ)k−1(∂Ψ)k−1 + (Ψ̄γ)k−1∂̂(γΨ)k−1 + k(k − 1) (Ψ̄γ∂)k−2Ψ̃k−2 − k(k − 1) ∂̂Ψ̃k−2] (23) To combine all these massless fields into one massive particle we have to add the following mass terms: (−1)k 2(k + 1) [Ψ̄kΨk − k(Ψ̄γ)k−1(γΨ)k−1 − k(k − 1) Ψ̃k−2]+ −ick[(Ψ̄γ)k−1Ψk−1 − k − 1 (γΨ)k−2] and corresponding corrections to gauge transformations: δΨk = αkξk + iβkγ(1ξk−1) + ρkg(2ξk−2) Then total Lagrangian will be gauge invariant provided: (s+ 1)2 − k2 , αk = k + 1 , βk = 2k(k + 1) , ρk = 4 Massless supermultiplets It is not easy to find in the recent literature the explicit component form of massless su- permultiplets with arbitrary superspin [22], so for completeness we will give their short description here. As we have already seen on the lower superspin cases, supermultiplets with integer and half-integer superspins have different structure and have to be considered separately. (s, s+1/2). Supermultiplet with integer superspin s contains bosonic spin s field and fermionic spin s+1/2 one. In this and in two subsequent sections we will use the same con- densed notations as in the previous one. By analogy with superspins 1 and 2 supermultiplets, we start with the following ansatz for the supertransformations: δΨs = iα1σ µν∂µΦν(s−1γ1)η δΦs = β(Ψ̄sη) Indeed, calculating variations of the sum of two massless Lagrangians one can see that most of variations cancel, provided one set α1 = −β2 . The residue: δL = −(−1)sβ (s− 1)(s− 2) [2(Ψ̄∂∂)s−2Φ̃s−2 − ˜̄Ψ ∂2Φ̃s−2 − (s− 2)( ˜̄Ψ∂)s−3(∂Φ̃)s−3 − −2(Ψ̄γ∂)s−2∂̂Φ̃s−2 + 2(Ψ̄γ∂∂)s−3(γΦ̃)s−3 − ( ˜̄Ψ∂)s−3∂̂(γΦ̃)s−3] contains terms with Φ̃s−2 only, so we proceed by adding to fermionic supertransformations one more term: δ′Ψs = iα2∂(1γ1Φ̃s−2) Then the choice α2 = (s−1)(s−2)β leaves us with: δL = −(−1)sβ (s− 1)(s− 2) [2(Ψ̄γ∂∂)s−3(γΦ̃)s−3 − ( ˜̄Ψ∂)s−3∂̂(γΦ̃)s−3] where the only terms are whose with (γΦ̃)s−3. So we make one more (last) correction to supertransformations: δ′′Ψs = iα3g(2∂1(γΦ̃)s−3) and obtain full invariance with α3 = − (s−1)(s−2)β4s . To fix concrete normalization we will use closure of the superalgebra. Calculating the commutator of two supertransformations we obtain: [δ1, δ2]Φs = −iβ2(η̄2γνη1)∂µΦs + . . . where dots mean “up to gauge transformation”. So we set β = 2 and our final result looks like: δΨs = − σµν∂µΦ̄ν(s−1γ1)η + i(s− 1)(s− 2) [∂(1γ1Φ̃s−2) − g(2∂1(γΦ̃)s−3)]η δΦs = 2(Ψ̄sη) (25) (s+1/2, s+1). Half-integer superspin multiplet contains fermionic spin s + 1/2 fields and bosonic spin s + 1 one. Again by analogy with lower superspin case we will make the following ansatz for supertransformations: δΨs = α1σ µν∂µΦν(s)η, δΦs+1 = iβ(Ψ̄(sγ1)η) This time most of the variations cancel if one set α1 = −β leaving us with: δL = i(−1)sβ s(s− 1) [2(Ψ̄∂∂)s−2(γΦ̃)s−2 − ˜̄Ψ ∂2(γΦ̃)s−2 − −2(Ψ̄γ∂)s−2∂̂(γΦ̃)s−2 − (s− 2)( ˜̄Ψ∂)s−3(γ∂Φ̃)s−3] Then the full invariance could be achieved with the following correction to supertransforma- tions: δ′Ψs = α2∂(1γ1(γΦ̃)s−2) provided α2 = . To check the closure of superalgebra and to choose normalization we calculate commutator of two supertransformations: [δ1, δ2]Φs+1 = −2iβ2(η̄2γµη1)∂µΦs+1 + . . . Then our choice will be β = 1 and our final form: δΨs = −σµν∂µΦν(s)η + ∂(1γ1(γΦ̃)s−2)η, δΦs+1 = i(Ψ̄(sγ1)η) Note that starting with superspin 2 the structure of supertransformations are defined up to possible field dependent gauge transformations and our choice differs from that of [22]. It makes no difference for massless theories but for massive case the structure of corrections for fermionic supertransformations depends on the choice made. 5 Integer superspin Now, having in our disposal gauge invariant description of massive particles with arbitrary (half-)integer spins, known form of supertransformations for massless arbitrary superspin supermultiplets and concrete examples of massive supermultiplets with lower superspins, we are ready to construct massive arbitrary superspin supermultiplets. As we have seen, integer and half-integer cases have different structures and have to be considered separately. In this section we consider massive supermultiplet with integer superspin. Such super- multiplet also contains four massive fields: two bosonic spin s fields (with opposite parity) and fermionic spin (s+1/2) and (s-1/2) ones. Calculating total number of physical degrees of freedom and taking into account possible mixing of supermultiplets containing bosonic fields with equal spins and opposite parity, we start with the following structure of massless supermultiplets: As Bs Ak Bk By analogy with superspin 1 and 2 cases, we will assume that all bosonic fields enter through the complex combinations Ck = Ak + iBk only (so that all possible mixing angles are fixed and equal π/4). Thus we choose the following form of supertransformations for massless supermultiplets with 1 ≤ k ≤ s: δΦk = − σµν∂µC̄ν(k−1γ1)η + i(k − 1)(k − 2) [∂(1γ1C̃k−2) − g(2∂1(γC̃)k−3)]η δC̄k = 2(Φ̄kη) + i 2(Ψ̄(k−1γ1)η) (26) δΨk−1 = − σµν∂µCν(k−1)η + k − 2 ∂(1γ1(γC̃)k−3)η and also δΦ0 = −i∂̂zη, δz̄ = 2(Φ̄0η) As a result of our assumption mass terms for bosonic fields are completely fixed: (−1)kck[C̄k∂(1Ck−1) − (k − 1) ˜̄C (∂C)k−2 + (k − 1)(k − 2) ( ˜̄C ∂(1C̃k−3) + h.c.)] (−1)k[dkC̄kCk + ek ˜̄C C̃k−2 + fk( Ck−2 + h.c.)] (27) where (s+ k)(s− k + 1) , dk = (s− k − 1)(s+ k + 2) 4(k + 1) k(k − 1) 16(k + 1) [(s− k + 2)(s+ k − 1) + 6] fk = − (s− k + 2)(s+ k − 1)(s− k + 1)(s+ k) As for the fermionic mass terms, apriori we don’t have any restrictions on them so we have to consider the most general possible form: (−1)k a1k[Φ̄ kΦk − k(Φ̄γ)k−1(γΦ)k−1 − k(k − 1) Φ̃k−2]+ +a2k[Φ̄ kΨk − k(Φ̄γ)k−1(γΨ)k−1 − k(k − 1) Ψ̃k−2] + +a3k[Ψ̄ kΨk − k(Ψ̄γ)k−1(γΨ)k−1 − k(k − 1) Ψ̃k−2] + +ib1k[(Φ̄γ) k−1Φk−1 − k − 1 (γΦ)k−2] + +ib2k[(Φ̄γ) k−1Ψk−1 − k − 1 (γΨ)k−2] + +ib3k[(Ψ̄γ) k−1Φk−1 − k − 1 (γΦ)k−2] + + ib4k[(Ψ̄γ) k−1Ψk−1 − k − 1 (γΨ)k−2] Where: a1s = − , a2s = a3s = b3s = b4s = 0 The requirement that total Lagrangian be invariant under (appropriately corrected) super- transformations gives: a1k = − , a2k = − k + 1 ck+1, a3k = 2(k + 1) b1k = −2ck, b2k = − , b3k = 0, b4k = −2ck+1 In this, additional terms for fermionic supertransformations look like: δ′Φk = 2ick+1 k + 1 [(γC)k + k(k − 1) 4(k + 1) γ(1C̃k−1) − (k − 1)2(2k + 1) 8k(k + 1) g(2(γC̃)k−2)]− −Ck + k − 1 γ(1(γC)k−1 − (k − 1)(k − 2) g(2C̃k−2) − [γ(1Ck−1) − g(2(γC)k−2)] (29) δ′Ψk = kck+2 2(k + 1) γ(1(γC̃)k−1) − 2(k + 1) [(γC)k − k(k − 1) 4(k + 1) γ(1C̃k−1) + (k − 1)(3k + 1) 8k(k + 1) g(2(γC̃)k−2)]− 2ck+1 k + 1 [Ck + γ(1(γC)k−1) + (k − 1)(k − 2) g(2C̃k−2)] (30) Here the supertransformations for Φk field contain terms with Ck+1, Ck and Ck−1 fields in the first, second and third lines correspondingly, while that of Ψk contain terms with Ck+2, Ck+1 and Ck fields. 6 Half-integer superspin Next we turn to the half-integer superspin case. This time we have two fermionic spin (s+1/2) fields and bosonic ones with spins (s+1) and s. Usual reasoning on physical degrees of freedom and possible mixings leads us to the following structure of massless supermultiplets we will start with: Φs Ψs Ak Bk We see that this structure is rather similar to that of integer superspin case. The main difference (besides the presence of As+1, Ψs supermultiplet) comes from the mixing of bosonic fields. We have no reasons to suggest that all mixing angles could be fixed from the very beginning so we have to consider the most general possibility here. Let us denote: Ck = cos(θk)Ak + γ5 sin(θk)Bk, Dk = sin(θk)Ak + γ5 cos(θk)Bk In these notations supertransformations for massless supermultiplets could be written as follows. Highest supermultiplet: δAs+1 = i(Ψ̄(sγ1)η) δΨs = −σµν∂µAν(s)η + ∂(1γ1(γÃ)s−2)η Main set (1 ≤ k ≤ s): δΦk = − σµν∂µC̄ν(k−1γ1)η + i(k − 1)(k − 2) [∂(1γ1C̃k−2) − g(2∂1(γC̃)k−3)]η δAk = 2 cos(θk)(Φ̄kη) + i sin(θk)(Ψ̄(k−1γ1)η) (31) δBk = 2 sin(θk)(Φ̄kγ5η) + i cos(θk)(Ψ̄(k−1γ1)γ5η) δΨk−1 = −σµν∂µDν(k−1)η + k − 2 ∂(1γ1(γD̃)k−3η and the last supermultiplet: δΦ0 = −i∂̂zη, δz̄ = 2(Φ̄0η) By analogy with superspins 3/2 and 5/2 cases we will assume that fermionic mass terms are Dirac ones. This immediately gives: (−1)k s + 1 k + 1 [Ψ̄kΦk − k(Ψ̄γ)k−1(γΦ)k−1 − k(k − 1) Φ̃k−2]− − ick[(Ψ̄γ)k−1Ψk−1 − k − 1 (γΨ)k−2 + (Ψ → Φ)] where: (s+ k + 1)(s− k + 1) The choice for the bosonic mass terms (taking into account parity) is also unambiguous: (−1)kak[Ak∂(1Ak−1) − (k − 1)Ãk−2(∂A)k−2 + (k − 1)(k − 2) Ãk−2∂(1Ãk−3)] + (−1)kbk[Bk∂(1Bk−1) − (k − 1)B̃k−2(∂B)k−2 + (k − 1)(k − 2) B̃k−2∂(1B̃k−3)] (33) for the terms with one derivative, where: (s+ k + 1)(s− k + 2) , bk = (s+ k)(s− k + 1) and the following terms without derivatives: (−1)k[d̂kAkAk + êkÃk−2Ãk−2 + f̂kÃk−2Ak−2] + (−1)k[dkBkBk + ekB̃k−2B̃k−2 + fkB̃k−2Bk−2] (34) Here: d̂k = (s− k)(s+ k + 3) 4(k + 1) , êk = k(k − 1) 16(k + 1) [(s− k + 3)(s+ k) + 6] f̂k = − (s− k + 3)(s+ k)(s− k + 2)(s+ k + 1) (s− k − 1)(s+ k + 2) 4(k + 1) , ek = k(k − 1) 16(k + 1) [(s− k + 2)(s+ k − 1) + 6] fk = − (s− k + 2)(s+ k − 1)(s− k + 1)(s+ k) Note that hatted coefficients differ from the unhatted ones by replacement s→ s+ 1. Now we require that total Lagrangian be invariant under the supertransformations. First of all this fixes all mixing angles: sin(θk) = s+ k + 1 2(s+ 1) , cos(θk) = s− k + 1 2(s+ 1) and gives us additional terms for fermionic supertransformations: δ′Ψk = α1Ak + α2γ(1(γA)k−1) + α3g(2Ãk−2) + +β1Bk + β2γ(1(γB)k−1) + β3g(2B̃k−2) + kck+1 4(k + 1) [sin(θk+2)γ(1(γÃ)k−1) + cos(θk+1)γ(1(γB̃)k−1)] δ′Φk = α4(γA)k + α5γ(1Ãk−1) + α6g(2(γÃ)k−2) + +β4(γB)k + β5γ(1B̃k−1) + β6g(2(γB̃)k−2) − cos(θk)[γ(1Ak−1) − g(2(γA)k−2]− sin(θk)[γ(1Bk−1) − g(2(γB)k−2)] Where: α1 = − s− k√ 2(k + 1) cos(θk), α2 = − k2 + s+ k + 1√ 2k(k + 1) cos(θk) α3 = − (s + 1)(k − 1)(k − 2) 2k2(k + 1) cos(θk) β1 = − s+ k + 2√ 2(k + 1) sin(θk), β2 = k2 − s+ k − 1√ 2k(k + 1) sin(θk) β3 = − (s + 1)(k − 1)(k − 2) 2k2(k + 1) sin(θk) k + 1 sin(θk+1), α5 = k(k − 1)(s− k) 4(k + 1)2 sin(θk+1) (k − 1)[(k + 1)(s+ 1)− 2k2(s− k)] 8k(k + 1)2 sin(θk+1) k + 1 cos(θk+1), β5 = k(k − 1)(s+ k + 2) 4(k + 1)2 cos(θk+1) (k − 1)[(k + 1)(s+ 1)− 2k2(s+ k + 2)] 8k(k + 1)2 cos(θk+1) We have explicitely checked that (rather complicated) formulas from this and previous sections correctly reproduce all lower superspins results. 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Critical Current of Type-II Superconductors in a Broken Bose Glass State J. P. Rodriguez1 1Department of Physics and Astronomy, California State University, Los Angeles, California 90032 (Dated: November 1, 2018) Abstract The tilt modulus of a defective Abrikosov vortex lattice pinned by material line defects is com- puted using the boson analogy. It tends to infinity at long wavelength, which yields a Bose glass state that is robust to the addition of weak point-pinning centers, and which implies a restoring force per vortex line for rigid translations about mechanical equilibrium that is independent of magnetic field. It also indicates that the Bose glass state breaks into pieces along the direction of the correlated pinning centers if the latter have finite length. The critical current is predicted to crossover from two dimensional to three dimensional behavior as a function of sample thickness along the correlated pinning centers in such case. That crossover notably can occur at a film thickness that is much larger than that expected from point pins of comparable strength. The above is compared to the dependence on thickness shown by the critical current in certain films of high-temperature superconductors currently being developed for wire technology. http://arxiv.org/abs/0704.1536v6 INTRODUCTION It well known that thin films of high-temperature superconductors exhibit larger critical currents than their single-crystal counterparts. Thin films of the high-temperature supercon- ductor YBa2Cu3O7−δ (YBCO) grown by pulsed laser deposition (PLD), which are actively being developed for wire technology, achieve critical currents that are a significant fraction of the maximum depairing current, for example[1]. Evidence exists that lines of dislocations that run parallel to the crystalline c axis in PLD-YBCO act as correlated pinning centers for vortex lines inside of the superconducting state[2], and thereby give rise to such high critical currents. This is confirmed by the peak observed in the critical current of PLD-YBCO at orientations of the c axis aligned parallel to an applied magnetic field, as well as by the dra- matic enhancement of the former peak after more material defects in the form of nano-rods aligned parallel to the c axis are added[3][4]. The microstructure described above for PLD-YBCO films immediately suggests that the vortex lattice that emerges from the superconducting state in applied magnetic field aligned parallel to the c axis is some form of Bose glass characterized by a divergent tilt modulus[5]. In the limit of a rigid vortex lattice, to be expected at large magnetic fields, two-dimensional (2D) collective pinning of vortex lines by the material line defects then determines the critical current [6][7][8][9]. Recent theoretical calculations that follow this line of reasoning find moderate quantitative agreement with the critical current measured in films of PLD-YBCO in c-axis magnetic fields of a few to many kG, at liquid nitrogen temperature[10]. A potential problem with the Bose glass hypothesis, however, is that the material line defects in PLD-YBCO films can meander, or they can be of relatively short length[3][4]. That question is addressed in this paper, where we find that the Bose glass breaks up into pieces along the direction of the correlated pinning centers when the effective length of the latter is less than the film thickness. In particular, the profile of the critical current versus film thickness is predicted to reflect a crossover from two-dimensional to three- dimensional (3D) collective pinning of the vortex lines[11]. This cross-over can occur at a length scale that is notably much larger than that expected from point pins of comparable strength[12]. We also find that the unbroken Bose glass state is robust to the addition of weak point pins. Good agreement is achieved between the dependence on thickness shown by the critical current in certain films of PLD-YBCO at applied magnetic field[12] and that predicted for the true Bose glass state[10]. Last, the effective restoring force per vortex line due to a rigid translation of the Bose glass about mechanical equilibrium, which is gauged by the Labusch parameter[13], is found to depend only weakly on applied magnetic field. This prediction agrees with recent measurements of the microwave surface resistance on PLD-YBCO films with nano-rod inclusions[14]. TILT MODULUS OF BOSE GLASS Material line defects in thin enough films of PLD-YBCO can be considered to be per- fectly parallel to the c-axis. They notably arrange themselves in a manner that resembles a snapshot of a 2D liquid[2], as opposed to a gas[15][16]. In particular, such correlated pin- ning centers do not show clusters or voids. A defective vortex lattice that assumes a hexatic Bose glass state will then occur for external magnetic fields aligned in parallel to such a microstructure, in the limit of weak correlated pinning. It is characterized by parallel lines of edge dislocation defects that are injected into the pristine vortex lattice in order to relieve shear stress due to the correlated pins. The former do not show any intrinsic tendency to arrange themselves into grain boundaries, however, due to the absence of clusters and voids in the “liquid” arrangement of linear pinning centers. (Cf. refs. [15] and [16].) This results in a vortex lattice whose translational order is destroyed at long range by the isolated lines of edge dislocations, but which retains long-range orientational order. Collective pinning of the dislocation defects by the correlated pins then results both in an elastic response to shear and in a net superfluid density. Theoretical calculations[17] and numerical Monte Carlo simulations (see fig. 1 and caption) of the corresponding 2D Coulomb gas ensemble[18] confirm this picture. The correlation length Lc(|) for order along the magnetic field direction is infinite for a Bose glass state. In the limit of weak correlated pins, the density of dislocation defects that thread the corresponding vortex lattice can then be obtained by applying the theory of 2D collective pinning[6][7][8]. Each line of unbound edge dislocations is in one-to-one correspondence with a well-ordered bundle of vortex lattice, or Larkin domain, that has dimensions Rc(|) × Rc(|) in the directions transverse to the magnetic field. The injection of the dislocation lines into the pristine vortex lattice then results in plastic creep of each Larkin domain by a Burgers vector[19]. The transverse Larkin scale is hence obtained by minimizing the sum of the elastic energy cost due to the edge dislocations with the gain in pinning energy due to the translation of a Larkin domain by an elementary Burgers vector of the triangular vortex lattice, b = a△. This yields the estimate[6][7][8] Rc(|)−2 = C20np(fp/c66b)2, (1) for the density of Larkin domains, which coincides with the density of lines of unbound dislocations. Here np denotes the density of pinned vortex lines, fp denotes the maximum pinning force along a material line defect per unit length, and c66 denotes the elastic shear modulus of the pristine vortex lattice. The prefactor above is of order[8] C0 ∼= π/ln(Rc/a′df)2, where a′df is the core diameter of a dislocation in the vortex lattice. Consider now the limit of weak pinning centers that do not crowd together: fp → 0 and πr2p · nφ ≪ 1, respectively, where nφ denotes the density of material line defects, and where rp denotes their range. Simple considerations of probability then yield the identity np/nφ = nB · πr2p between the fraction of occupied pinning centers and the product of the density of vortex lines, nB, with the effective area of each pinning center. Substituting it plus the estimate c66 = (Φ0/8πλL) 2nB for the shear modulus of the pristine vortex lattice[21] into Eq. (1) then yields the result Rc(|)−2 ∼= ( 3π/2)C20(4fprp/ε0) 2nφ for the density of Larkin domains, which depends only weakly on magnetic field. Here, λL denotes the London penetration depth and ε0 = (Φ0/4πλL) 2 is the maximum tension of a flux line in the superconductor. All of the above is valid in the 2D collective pinning regime that exists at perpendicular magnetic fields beyond the threshold Bcp = C 3/2)(4fp/ε0) 2Φ0, in which case many vortex lines are pinned by material line defects within a Larkin domain of transverse dimensions Rc(|)×Rc(|) [10]. We will now exploit the boson analogy for vortex matter in order to compute the uniform tilt modulus of the hexatic Bose glass in the absence of point pinning centers[5]. It amounts to a London model set by the free-energy density gB({r}, z) = V0(ri, rj) + Vp(ri) (2) for vortex lines located at transverse positions {ri(z)}, at a coordinate z along the field direction. Here ε̃l denotes the tension of an isolated vortex line, while the pair potential V0(r, r ′) describes the interaction between vortex lines at the same longitudinal coordinate z. The energy landscape for the correlated pinning centers is described by the potential energy Vp(r), which is independent of the coordinate z along the field direction. The ther- modynamics of this system in the presence of an external tilt stress nBa is then set by the partition function ZB[a] = (Πi D[ri(z)])exp[−(kBT )−1 dz[gB({r}, z)− d2r jB · a]], (3) which under periodic boundary conditions, ri(z+Lz) = ri(z), is equivalent to a system of 2D bosons. Above, jB(r, z) = (2)[r − ri(z)](dri/dz) is the current density within the boson analogy. Observe now that the kernel Πµ,ν(ω) of the uniform electromagnetic response for alternating current (AC) is connected to this partition function through the proportionality relationship ZB[a] ∝ exp (kBT ) a ·Π · a in the limit that the corresponding uniform tilt stress vanishes, a → 0. Above, the Matsubara frequencies iωn are given by the allowed wavenumbers along the magnetic field, qz = 2πn/Lz, and V = LxLyLz is the volume of the system. The fact that the tilt stress is given by nBa then yields the identity C44(qz) = n B/Π⊥(ωn) (5) between the uniform tilt modulus and the uniform AC electromagnetic response of the 2D Bose glass. The subscript “⊥” above is a tag for the pure shear component of the electromagnetic kernel Πµ,ν (Cf. ref. [5]). The hexatic Bose glass state is clearly a 2D dielectric insulator within the boson analogy. Its electromagnetic response can therefore be modeled by the kernel Π⊥(ω) = (nB/ε̃l)ω 2/(ω2 − ω20), (6) which is dielectric in the low-frequency limit, and which conserves charge by satisfying the oscillator f-sum rule. Above, ω0 is the natural frequency of the electric dipole degrees of free- dom in the Bose glass. The latter correspond to the 2D Larkin domains in reality, since they represent the smallest units of well-ordered vortex lattice that can respond independently to an applied force. We then have that the above natural frequency is of order the resonant frequency for transverse sound inside a Larkin domain of the 2D Bose glass: ω0 ∼ γ/Rc(|), where γ = (c66/c44) 1/2 is the effective mass anisotropy parameter equal to the transverse sound speed within the boson analogy. Here c44 = nBε̃l is the tilt modulus due to isolated flux lines. After substitution into Eqs. (6) for the AC response, the identity (5) then yields a divergent tilt modulus for the Bose glass at long wavelength C44(qz) = nBε̃l[1 + (qzL∗) −2], (7) with a longitudinal scale L∗ = ω 0 that is related to the transverse Larkin length by the anisotropic scale transformation L∗ ∼ Rc(|)/γ. Expression (7) for the uniform tilt modulus of a Bose glass is the central result of the paper. We shall first extract the Labusch parameter[13] from the singular behavior that it shows at long wavelength. In particular, observe that the elastic energy density for a periodic tilt of the Bose glass by a displacement u0 at long wavelength, C44(qz)q 0, acquires a contribution of the form 1 0 from this divergence, with k0 = c66/Rc(|)2. The latter is simply the spring constant per unit volume of the restoring force for a rigid translation of the Bose glass state about mechanical equilibrium. Using the estimate c66 = ε0nB for the shear modulus of the vortex lattice[21] yields an effective spring constant per vortex line due to 2D collective pinning limited by plastic creep, k0/nB, that is given by kp = ε0/Rc(|)2. Notice that kp depends only weakly on magnetic field. The Labusch parameter extracted from the microwave surface resistance on PLD-YBCO films with nano-rod inclusions also exhibits only a weak dependence on external magnetic field aligned parallel to the nano-rods (or c-axis)[14]! The above should be compared to the corresponding Labusch parameter due to point pins[13], which depends strongly on magnetic field. Indeed, given the conjecture kp = (c66/R c + c44/L c)/nB for the Labusch parameter due to 3D collective-pinning implies that it decays with increasing magnetic field instead as 1/B2 in such case. Here we have assumed anisotropic scaling, and we have used[9] Rc ∝ B. We shall next use Eq. (7) for the uniform tilt modulus to test how robust the hexatic Bose glass is to the addition of point pinning centers. The hexatic Bose glass shows long-range orientational order in all directions (see fig. 1). The addition of point pins will therefore break it up into Larkin domains, of dimensions R′c × R′c × Lc transverse and parallel to the magnetic induction, that tilt in the transverse direction by a distance of order the size of the vortex core, ξ. Such a break-up then has an elastic energy cost per unit volume and a pinning energy gain per unit volume that sum to[6][9] R′2c Lc f ′0ξ. (8) Here C66 denotes the shear modulus of the hexatic Bose glass, which can be approximated by c66 in the limit of weak correlated pinning, while the corresponding tilt modulus C44 is given by expression (7) evaluated at wavenumber qz = 1/Lc. Also, n 0 denotes the density of point pins, while f ′0 denotes the magnitude of their characteristic force. Minimizing δu with respect to the dimensions of the Larkin domains then yields standard results for these[9]: Lc = Lc(·)[1 + (qzL∗)−2] and R′c = Rc(·)[1 + (qzL∗)−2]1/2, where Lc(·) = 2c44c66ξ2/n′0f ′20 and Rc(·) = 21/2c1/244 c 2/n′0f 0 are respectively the longitudinal Larkin scale and the transverse Larkin scale in the absence of correlated pinning centers. The first equation is quadratic in terms of the variable L−1c , and it has a formal solution L c = (2Lc(·))−1+[(2Lc(·))−2−L−2∗ ]1/2. The hexatic Bose glass is therefore robust to the addition of weak point pins. In particular, the longitudinal Larkin scale Lc remains divergent for Lc(·) > L∗/2, which is equivalent to the inequality 23/2Rc(·) > Rc(|). 2D-3D CROSSOVER IN CRITICAL CURRENT The critical current density of the above hexatic Bose glass, which is robust to the addition of point pins, can be obtained by applying 2D collective pinning theory[10]. All vortex lines can be considered to be rigid rods. Balancing the Lorentz force against the collective pinning force over a Larkin domain[6][9] yields the identity JcB/c = [(npf p + n p )/R 1/2 for the product of the critical current density along the film, Jc, with the perpendicular magnetic field, B, aligned parallel to the material line defects[10]. Here np and n p denote the density of vortex lines pinned by material line defects and the density of interstitial vortex lines pinned by material point defects, while fp and f p are the maximum force exerted on the respective vortex lines per unit length. Again, it is important to observe that the critical current is limited by plastic creep of the vortex lattice due to slip of the quenched-in lines of edge dislocations along their respective glide planes[19]. Minimization of the sum of the elastic and pinning energy densities then yields a higher density of Larkin domains in the hexatic Bose glass with material point defects added by comparison (1): R−2c = C 0 (npf p )/(c66b) This reflects the injection of extra lines of edge dislocations that relieve shear stress caused by point pins. Substitution in turn yields a critical current density, Jc = jc + j c, that has a component due to correlated pins set by the identity jcB/c = C0npf p/c66b, (9) and that has a component due to point pins set by the ratio j′c/jc = n p /npf p . The critical current density notably varies as Jc ∝ B−1/2 with magnetic field in the limit of weak pinning[10]. Consider now a film geometry of thickness τ along the axis of the material line defects. The forces due to point pins add up statistically along a rigid interstitial vortex line. The effective pinning force per unit length experienced by an interstitial vortex line is then given by[7] f ′p = f 1/2 at film thicknesses τ that are much greater than the average separation τ ′p between such pins along the field direction. Again, f 0 denotes the maximum force exerted by a point pin. The relative contribution by point pins to the critical current density is then predicted to show an inverse dependence on film thickness, j′c/jc = τ0/τ , that is set by the scale τ0 = (n p/np)(f p). We thereby obtain the linear dependence on film thickness I(2D)c = (τ0 + τ)jc for the net critical current per unit width, τJc. Last, comparison of Eq. (1) with Eq. (9) yields the useful expression L∗ = γ −1[(35/4/27/2C0)(ξ · avx)(j0/jc)]1/2 (10) for the longitudinal scale characteristic of the Bose glass as a function of the bulk critical current density, jc. Here avx = n B is the average distance between vortex lines and j0 is the depairing current density (see ref. [9]). Figure 2 shows a fit to data for the critical current versus thickness obtained from a thin film of PLD-YBCO at liquid-nitrogen temperature in 1T magnetic field aligned parallel to the c axis[12]. A bulk critical current density jc = 0.22MA/cm is extracted from it. Using a value of j0 = 36MA/cm for the depairing current (ref. [12]), of ξ = 11 nm for the coherence length, of γ = 7 for the mass anisotropy parameter, and setting C0 = 1 yields a longitudinal scale L∗ = 24 nm. Finally, consider again an arrangement of material line defects that are aligned along the c axis, that show no voids or clusters in the transverse directions, but that are broken up into relatively long rods of length L0 ≫ L∗. It can be realized by meandering material line defects[1][2], where L0 is the correlation length for alignment along the c-axis, or by artificial nano-rod defects[3][4]. Expression (7) for the divergent tilt modulus indicates proximity to the regime of 2D collective pinning, where both Lc and L0 are divergent. It hence indicates that Lc is also large compared to L∗ here. Second, observe that L∗ ∼ Rc(|)/γ coincides with the longitudinal Larkin scale if the rods are considered to be point defects. The previous ultimately implies the chain of inequalities L0 ≥ Lc ≫ L∗. They are consistent with a broken Bose glass state for the vortex lattice, which is threaded by isolated lines of edge dislocations of length Lc along the c-axis that are connected together by lines of screw-dislocations[19] along the transverse directions[11]. As in the limiting case of the true Bose glass state (fig. 1), the lines of edge dislocations are injected into the pristine vortex lattice in order to relieve shear stress due to the correlated pins. Larkin domains hence are finite volumes[6][9]. In contrast to their transverse dimensions, however, Larkin domains exhibit well defined boundaries along the direction parallel to the correlated pinning centers, across which the vortex lattice slips by a Burger’s vector due to the presence of the screw dislocations[19]. The critical current density expected from this peculiar example of 3D collective pinning therefore coincides with that of a Bose glass of thickness Lc: Jc = jc + j with a bulk component due to interstitial vortex lines j′c/jc = τ0/Lc. The critical current per unit width then varies with film thickness as I(3D)c = τJc, showing no offset. Figure 2 depicts the predicted dependence on film thickness for the critical current of such a broken Bose glass. It notably exhibits 2D-3D cross-over at film thicknesses in the vicinity of Lc [11]. CONCLUDING REMARKS The dependence of the critical current on thickness τ shown by certain films of PLD- YBCO is in fact consistent with dimensional cross-over at τ ∼ 1µm. A previous attempt by Gurevich to account for such behavior by collective pinning of individual vortex lines at point defects yields a longitudinal Larkin scale Lc ∼ 10 nm that is too small, however[12]. We find here, on the other hand, that pinning due to correlated material defects of length L0 yields a Larkin scale Lc that is much longer than that [L∗ = 24 nm, see Eq. (10)] expected from point pins of comparable strength if L0 is much longer than that scale as well. Indeed, we predict here that the film thickness at which the critical current crosses over from 2D to 3D behavior is of order the effective length of the correlated pinning centers when that length satisfies L0 ≫ L∗. The author thanks Leonardo Civale, Chandan Das Gupta and Sang-il Kim for discussions. This work was supported in part by the US Air Force Office of Scientific Research under grant no. FA9550-06-1-0479. [1] B. Dam, J.M. Huijbregtse, F.C. Klaassen, R.C.F. van der Geest, G. Doornbos, J.H. Rector, A.M. Testa, S. Freisem, J.C. Martinez, B. Stauble-Pumpin and R. Griessen, Nature 399, 439 (1999). [2] F.C. Klaassen, G. Doornbos, J.M. Huijbregtse, R.C.F. van der Geest, B. Dam and R. Griessen, Phys. Rev. B 64, 184523 (2001). [3] J.L. MacManus-Driscoll, S.R. Foltyn, Q.X. Jia, H. Wang, A. Serquis, L. Civale, B. Maiorov, M.E. Hawley, M.P. Maley and D.E. Peterson, Nature Materials 3, 439 (2004). [4] A. Goyal, S. Kang, K.J. Leonard, P.M. Martin, A.A. Gapud, M. Varela, M. Paranthaman, A.O. Ijaduola, E.D. Specht, J.R. Thompson, D.K. Christen, S.J. Pennycook and F.A List, Supercond. Sci. Technol. 18, 1533 (2005). [5] D.R. Nelson and V.M. Vinokur, Phys. Rev. B 48, 13060 (1993). [6] A.I. Larkin and Yu V. Ovchinnikov, J. Low Temp. Phys. 34, 409 (1979). [7] P.H. Kes and C.C. Tsuei, Phys. Rev. B 28, 5126 (1983). [8] S.J. Mullock and J.E. Evetts, J. Appl. Phys. 57, 2588 (1985). [9] M. Tinkham, Introduction to Superconductivity (McGraw-Hill, New York, 1996) 2nd ed.. [10] J.P. Rodriguez and M.P. Maley, Phys. Rev. B 73, 094502 (2006). [11] R. Wördenweber and P. H. Kes, Phys. Rev. B 34, 494 (1986). [12] Chang-Beom Eom, Alex Gurevich and David Larbalestier, Superconductivity for Electric Systems 2004 Annual (DOE) Peer Review (http://www.energetics.com/supercon04.html). [13] G. Blatter, V.B. Geshkenbein, and J.A.G. Koopmann, Phys. Rev. Lett. 92, 067009 (2004). [14] N. Pompeo, R. Rogai, E. Silva, A. Augieri, V. Galluzzi and G. Celentano, Appl. Phys. Lett. 91, 182507 (2007). (arXiv:0710.1754). [15] M. Menghini, Y. Fasano, F. de la Cruz, S.S. Banerjee, Y. Myasoedov, E. Zeldov, C.J. van der Beek, M. Konczykowski, and T. Tamegai, Phys. Rev. Lett. 90, 147001 (2003). [16] C. Dasgupta and O.T. Valls, Phys. Rev. Lett. 91, 127002 (2003). [17] J.P. Rodriguez, Phys. Rev. B 72, 214503 (2005). [18] C.E. Creffield and J.P. Rodriguez, Phys. Rev. B 67, 144510 (2003). [19] D. Hull and D.J. Bacon, Introduction to Dislocations, 3rd ed. (Pergamon, Oxford, 1984). [20] J.P. Rodriguez, Phys. Rev. B 70, 224507 (2004). http://www.energetics.com/supercon04.html http://arxiv.org/abs/0710.1754 [21] E.H. Brandt, J. Low Temp. Phys. 26, 735 (1977). FIG. 1: Shown is a Delaunay triangulation of a low-temperature groundstate made up of 2016 vortices that interact logarithmically over a 336 × 336 grid with periodic boundary conditions, and that experience an equal number of identically weak δ-function pinning centers arranged in a “liquid” fashion. (See ref. [18].) The state shows macroscopic phase coherence and long-range hexatic order, with a supefluid density and a hexatic order parameter, respectively, that are 29% and 58% of the maximum possible values attained by the perfect triangular vortex lattice. It was obtained after simulated annealing from the liquid state down to low temperature, which resulted in 371 pinned vortices. A red and green pair of disclinations forms a dislocation (see ref. [19]). The above Monte Carlo simulation results are consistent with theoretical predictions of a hexatic vortex glass state in two dimensions, in the zero-temperature limit, for pinning arrangements that do not show any clusters or voids (ref. [17]). Josephson coupling between layers then produces a Bose glass transition above zero temperature as long as the 2D glass transtion is second order (ref. [20]). -0.1 0 0.1 0.2 0.3 0.4 0.5 film thickness (microns) PLD-YBCO in liq. N @ H=1T (ref. 12) 2D COLLECTIVE PINNING 3D COLLECTIVE PINNING 2D-3D CROSSOVER FIG. 2: Shown is the dependence on film thickness predicted for the critical current of a defective vortex lattice found in a broken Bose glass state (solid line). The dashed and dotted lines are extrapolated from the 2D and 3D behaviors, respectively. Measurements made by Sang Kim (circles, ref. [12]) of the critical current on a thin film of PLD-YBCO at liquid nitrogen temperature subject to 1T magnetic field aligned parallel to the c-axis are fit to the straight dashed line predicted by 2D collective pinning (ref. [10]). This yields an intercept −τ0 = −69 nm and a slope jc = 0.22MA/cm . Although the value of Lc shown here is indeed larger than the lower bound L∗ = 24nm [see Eq. (10)] and is consistent with the fit to 2D collective pinning, it is only hypothetical. Introduction Tilt modulus of Bose glass 2D-3D crossover in critical current Concluding remarks Acknowledgments References

The tilt modulus of a defective Abrikosov vortex lattice pinned by material line defects is computed using the boson analogy. It tends to infinity at long wavelength, which yields a Bose glass state that is robust to the addition of weak point-pinning centers, and which implies a restoring force per vortex line for rigid translations about mechanical equilibrium that is independent of magnetic field. It also indicates that the Bose glass state breaks into pieces along the direction of the correlated pinning centers if the latter have finite length. The critical current is predicted to crossover from two dimensional to three dimensional behavior as a function of sample thickness along the correlated pinning centers in such case. That crossover notably can occur at a film thickness that is much larger than that expected from point pins of comparable strength. The above is compared to the dependence on thickness shown by the critical current in certain films of high-temperature superconductors currently being developed for wire technology.

Introduction to Superconductivity (McGraw-Hill, New York, 1996) 2nd ed.. [10] J.P. Rodriguez and M.P. Maley, Phys. Rev. B 73, 094502 (2006). [11] R. Wördenweber and P. H. Kes, Phys. Rev. B 34, 494 (1986). [12] Chang-Beom Eom, Alex Gurevich and David Larbalestier, Superconductivity for Electric Systems 2004 Annual (DOE) Peer Review (http://www.energetics.com/supercon04.html). [13] G. Blatter, V.B. Geshkenbein, and J.A.G. Koopmann, Phys. Rev. Lett. 92, 067009 (2004). [14] N. Pompeo, R. Rogai, E. Silva, A. Augieri, V. Galluzzi and G. Celentano, Appl. Phys. Lett. 91, 182507 (2007). (arXiv:0710.1754). [15] M. Menghini, Y. Fasano, F. de la Cruz, S.S. Banerjee, Y. Myasoedov, E. Zeldov, C.J. van der Beek, M. Konczykowski, and T. Tamegai, Phys. Rev. Lett. 90, 147001 (2003). [16] C. Dasgupta and O.T. Valls, Phys. Rev. Lett. 91, 127002 (2003). [17] J.P. Rodriguez, Phys. Rev. B 72, 214503 (2005). [18] C.E. Creffield and J.P. Rodriguez, Phys. Rev. B 67, 144510 (2003). [19] D. Hull and D.J. Bacon, Introduction to Dislocations, 3rd ed. (Pergamon, Oxford, 1984). [20] J.P. Rodriguez, Phys. Rev. B 70, 224507 (2004). http://www.energetics.com/supercon04.html http://arxiv.org/abs/0710.1754 [21] E.H. Brandt, J. Low Temp. Phys. 26, 735 (1977). FIG. 1: Shown is a Delaunay triangulation of a low-temperature groundstate made up of 2016 vortices that interact logarithmically over a 336 × 336 grid with periodic boundary conditions, and that experience an equal number of identically weak δ-function pinning centers arranged in a “liquid” fashion. (See ref. [18].) The state shows macroscopic phase coherence and long-range hexatic order, with a supefluid density and a hexatic order parameter, respectively, that are 29% and 58% of the maximum possible values attained by the perfect triangular vortex lattice. It was obtained after simulated annealing from the liquid state down to low temperature, which resulted in 371 pinned vortices. A red and green pair of disclinations forms a dislocation (see ref. [19]). The above Monte Carlo simulation results are consistent with theoretical predictions of a hexatic vortex glass state in two dimensions, in the zero-temperature limit, for pinning arrangements that do not show any clusters or voids (ref. [17]). Josephson coupling between layers then produces a Bose glass transition above zero temperature as long as the 2D glass transtion is second order (ref. [20]). -0.1 0 0.1 0.2 0.3 0.4 0.5 film thickness (microns) PLD-YBCO in liq. N @ H=1T (ref. 12) 2D COLLECTIVE PINNING 3D COLLECTIVE PINNING 2D-3D CROSSOVER FIG. 2: Shown is the dependence on film thickness predicted for the critical current of a defective vortex lattice found in a broken Bose glass state (solid line). The dashed and dotted lines are extrapolated from the 2D and 3D behaviors, respectively. Measurements made by Sang Kim (circles, ref. [12]) of the critical current on a thin film of PLD-YBCO at liquid nitrogen temperature subject to 1T magnetic field aligned parallel to the c-axis are fit to the straight dashed line predicted by 2D collective pinning (ref. [10]). This yields an intercept −τ0 = −69 nm and a slope jc = 0.22MA/cm . Although the value of Lc shown here is indeed larger than the lower bound L∗ = 24nm [see Eq. (10)] and is consistent with the fit to 2D collective pinning, it is only hypothetical. Introduction Tilt modulus of Bose glass 2D-3D crossover in critical current Concluding remarks Acknowledgments References

arXiv:0704.1537v1 [math.CA] 12 Apr 2007 ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION Shuichi Sato Abstract. In this note, we study singular integrals with rough kernels, which belong to a class of singular Radon transforms. We prove certain estimates for the singular integrals that are useful in an extrapolation argument. As an application, we prove Lp boundedness of the singular integrals under a certain sharp size condition on their kernels. 1. Introduction Let Ω be a function in L1(Sn−1) satisfying (1.1) Ω(θ) dσ(θ) = 0, where dσ denotes the Lebesgue surface measure on the unit sphere Sn−1 in Rn. In this note we assume n ≥ 2. For s ≥ 1, let ∆s denote the collection of measurable functions h on R+ = {t ∈ R : t > 0} satisfying ‖h‖∆s = sup ∫ 2j+1 |h(t)|s dt/t where Z denotes the set of integers. We note that ∆s ⊂ ∆t if s > t. In this note we always assume h ∈ ∆1. Let P (y) = (P1(y), P2(y), . . . , Pd(y)) be a polynomial mapping, where each Pj is a real-valued polynomial on R n. We consider a singular integral operator of the form: (1.2) T (f)(x) = p. v. f(x− P (y))K(y) dy = lim |y|>ǫ f(x− P (y))K(y) dy, for an appropriate function f on Rd, where K(y) = h(|y|)Ω(y′)|y|−n, y′ = |y|−1y. Then, T (f) belongs to a class of singular Radon transforms. See Stein [17], Fan-Pan [8] and Al-Salman-Pan [1] for this singular integral. When h = 1 (a constant function), n = d and P (y) = y, we also write T (f) = S(f). Let f̂(ξ) = f(x)e−2πi〈x,ξ〉 dx be the Fourier transform of f , where 〈·, ·〉 Key words and phrases. Singular integrals, singular Radon transforms, maximal functions, extrapolation. 2000 Mathematics Subject Classification. Primary 42B20, 42B25 Typeset by AMS-TEX http://arxiv.org/abs/0704.1537v1 2 SHUICHI SATO denotes the inner product in Rd. Then it is known that (Sf )̂ (ξ) = m(ξ′)f̂(ξ), where m(ξ′) = − sgn(〈ξ′, θ〉) + log |〈ξ′, θ〉| dσ(θ). Using this, we can show that S extends to a bounded operator on L2 if Ω ∈ L logL(Sn−1), where L logL(Sn−1) denotes the Zygmund class of all those func- tions Ω on Sn−1 which satisfy |Ω(θ)| log(2 + |Ω(θ)|) dσ(θ) <∞. Furthermore, if Ω ∈ L logL(Sn−1), by the method of rotations of Calderón-Zygmund (see [2]) it can be shown that S extends to a bounded operator on Lp for all p ∈ (1,∞). When n = d and P (y) = y, R. Fefferman [10] proved that if h is bounded and Ω satisfies a Lipschitz condition of positive order on Sn−1, then the singular integral operator T in (1.2) is bounded on Lp for 1 < p < ∞. Namazi [13] improved this result by replacing the Lipschitz condition by the condition that Ω ∈ Lq(Sn−1) for some q > 1. In [7], Duoandikoetxea and Rubio de Francia developed methods which can be used to study mapping properties of several kinds of operators in harmonic analysis including the singular integrals considered in [13]. Also, see [6, 22] for weighted Lp boundedness of singular integrals, and [18, 19] for background materials. For the rest of this note we assume that the polynomial mapping P in (1.2) satisfies P (−y) = −P (y) and P 6= 0. We shall prove the following: Theorem 1. Let Ω ∈ Lq(Sn−1), q ∈ (1, 2] and h ∈ ∆s, s ∈ (1, 2]. Suppose Ω satisfies (1.1). Let T be as in (1.2). Then we have ‖T (f)‖Lp(Rd) ≤ Cp(q − 1) −1(s− 1)−1‖Ω‖Lq(Sn−1)‖h‖∆s‖f‖Lp(Rd) for all p ∈ (1,∞), where the constant Cp is independent of q, s,Ω and h. Also, the constant Cp is independent of polynomials Pj if we fix deg(Pj) (j = 1, 2, . . . , d). In Al-Salman-Pan [1], the Lp boundedness of T was proved under the condition that Ω is a function in L logL(Sn−1) satisfying (1.1) and h ∈ ∆s for some s > 1 ([1, Theorem 1.3]). Also it is noted there that estimates like those in Theorem 1 (with s being fixed) can be used to prove the same result by applying an extrapolation method, but such estimates are yet to be proved (see [1, p. 156]). In [1], the authors also considered singular integrals defined by certain polynomial mappings P which do not satisfy the condition P (−y) = −P (y). As a consequence of Theorem 1 we can give a different proof of [1, Theorem 1.3] via an extrapolation method; in fact, we can get an improved result. For a positive number a and a function h on R+, let La(h) = sup ∫ 2j+1 |h(r)| (log(2 + |h(r)|)) dr/r. We define a class La to be the space of all those measurable functions h on R+ which satisfy La(h) <∞. Also, let Na(h) = ma2mdm(h), ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 3 where dm(h) = supk∈Z 2 −k|E(k,m)| with E(k,m) = {r ∈ (2k, 2k+1] : 2m−1 < |h(r)| ≤ 2m} for m ≥ 2, E(k, 1) = {r ∈ (2k, 2k+1] : |h(r)| ≤ 2}. We denote by Na the class of all those measurable functions h on R+ such that Na(h) < ∞. Then we readily see that Na(h) < ∞ implies La(h) < ∞. Conversely, if La+b(h) < ∞ for some b > 1, then Na(h) <∞. To see this, note that 2mma+b2−k|E(k,m)| ≤ C E(k,m) |h(r)| (log(2 + |h(r)|)) dr/r ≤ CLa+b(h) for m ≥ 2; thus Na(h) ≤ 2d1(h) + CLa+b(h) −b < ∞. By Theorem 1 and an extrapolation method we have the following: Theorem 2. Suppose Ω is a function in L logL(Sn−1) satisfying (1.1) and h ∈ N1. Let T be as in (1.2). Then ‖T (f)‖Lp(Rd) ≤ Cp‖f‖Lp(Rd) for all p ∈ (1,∞), where Cp is independent of polynomials Pj if the polynomials are of fixed degree. By Theorem 2 and the remark preceding it we see that T is bounded on Lp for all p ∈ (1,∞) if Ω is as in Theorem 2 and h ∈ La for some a > 2. When n = d, P (y) = y, Ω is as in Theorem 2 and h is a constant function, it is known that T is of weak type (1, 1); see [5, 15]. Also, see [4, 9, 11, 12, 16, 20, 21] for related results. In Section 2, we shall prove Theorem 1. Applying the methods of [7] involving the Littlewood-Paley theory and using results of [8, 14], we shall prove Lp estimates for certain maximal and singular integral operators related to the operator T in Theorem 1 (Lemmas 1 and 2). Lemma 1 is used to prove Lemma 2. By Lemma 2 we can easily prove Theorem 1. A key idea of the proof of Theorem 1 is to apply a Littlewood-Paley decomposition adapted to a suitable lacunary sequence depending on q and s for which Ω ∈ Lq(Sn−1) and h ∈ ∆s. The method of appropriately choosing the lacunary sequence was inspired by [1], where, in a somewhat different way from ours, a similar method was used to study several classes of singular integrals. We shall prove Theorem 2 in Section 3. Finally, in Section 4, we consider the maximal operator (1.3) T ∗(f)(x) = sup N,ǫ>0 ǫ<|y|<N f(x− P (y))K(y) dy where P and K are as in (1.2). We shall prove analogs of Theorems 1 and 2 for the operator T ∗. Throughout this note, the letter C will be used to denote non-negative constants which may be different in different occurrences. 2. Proof of Theorem 1 Let Ω, h be as in Theorem 1. We consider the singular integral T (f) defined in (1.2). Let ρ ≥ 2 and Ek = {x ∈ R n : ρk < |x| ≤ ρk+1}. Then T (f)(x) = −∞ σk ∗ f(x), where {σk} is a sequence of Borel measures on R d such that (2.1) σk ∗ f(x) = f(x− P (y))K(y) dy. 4 SHUICHI SATO We note that (σk ∗ f )̂ (ξ) = f̂(ξ) e−2πi〈P (y),ξ〉K(y) dy. We write P (y) = Qj(y), Qj(y) = |γ|=N(j) γ (aγ ∈ R where Qj 6= 0, 1 ≤ N(1) < N(2) < · · · < N(ℓ), γ = (γ1, . . . , γn) is a multi- index, yγ = y 1 . . . y n and |γ| = γ1 + · · · + γn. Let βm = ρ N(m) and αm = (q − 1)(s− 1)/(2qsN(m)) for 1 ≤ m ≤ ℓ. Put P (m)(y) = j=1Qj(y) and define a sequence µ(m) = {µ k } of positive measures on R k ∗ f(x) = x− P (m)(y) |K(y)| dy for m = 1, 2, . . . , ℓ. Also, define µ(0) = {µ k } by µ k = ( |K(y)| dy)δ, where δ is Dirac’s delta function on Rd. For a sequence ν = {νk} of finite Borel measures on Rd, we define the maximal operator ν∗ by ν∗(f)(x) = supk ||νk| ∗ f(x)|, where |νk| denotes the total variation. We consider the maximal operators m ≤ ℓ). We also write = µ∗ρ. Lj(ξ) = (〈aγ(j,1), ξ〉, 〈aγ(j,2), ξ〉, . . . , 〈aγ(j,rj), ξ〉), where {γ(j, k)} k=1 is an enumeration of {γ}|γ|=N(j) for 1 ≤ j ≤ ℓ. Then Lj is a linear mapping from Rd to Rrj . Let sj = rankLj. There exist non-singular linear transformations Rj : R d → Rd and Hj : R sj → Rsj such that Rj(ξ)| ≤ |Lj(ξ)| ≤ C|Hjπ Rj(ξ)|, where πdsj (ξ) = (ξ1, . . . , ξsj ) is the projection and C depends only on rj (a proof can be found in [8]). Let {σ k } (0 ≤ m ≤ ℓ) be a sequence of Borel measures on d such that k ∗ f(x) = x− P (m)(y) K(y) dy for m = 1, 2, . . . , ℓ, while σ k = 0. Let ϕ ∈ C 0 (R) be supported in {|r| ≤ 1} and ϕ(r) = 1 for |r| < 1/2. Define a sequence τ (m) = {τ k } of Borel measures by (2.2) τ̂ k (ξ) = σ̂ k (ξ)Φk,m(ξ)− σ̂ (m−1) k (ξ)Φk,m−1(ξ) for m = 1, 2, . . . , ℓ, where Φk,m(ξ) = j=m+1 βkj |Hjπ Rj(ξ)| if 0 ≤ m ≤ ℓ− 1 and Φk,ℓ = 1. Then σk = σ m=1 τ k . We note that Φk,m(ξ)ϕ βkm|Hmπ Rm(ξ)| = Φk,m−1(ξ) (1 ≤ m ≤ ℓ). For 1 ≤ m ≤ ℓ, let T ρ (f) = k ∗ f . Then T = m=1 T For p ∈ (1,∞) we put p′ = p/(p − 1) and δ(p) = |1/p − 1/p′|. Let θ ∈ (0, 1). Then we have the following Lp estimates for (µ(m))∗ and T ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 5 Lemma 1. For p > 1 + θ and 0 ≤ j ≤ ℓ, we have (2.3) (µ(j))∗(f) Lp(Rd) ≤ C(log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s 1− ρ−θ/(2q )−2/p ‖f‖Lp(Rd). Lemma 2. For p ∈ (1 + θ, (1 + θ)/θ) and 1 ≤ m ≤ ℓ, we have ‖T (m)ρ (f)‖Lp(Rd) ≤ C(log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s 1− ρ−θ/(2q )−1−δ(p) ‖f‖Lp(Rd). The constants C in Lemmas 1 and 2 are independent of q, s ∈ (1, 2], Ω ∈ Lq(Sn−1), h ∈ ∆s, ρ and the coefficients of the polynomials Pk (1 ≤ k ≤ d). We prove Lemma 2 first, taking Lemma 1 for granted for the moment. Let A = (log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s and B = 1− β−θαmm 1− ρ−θ/(2q . Then, we have the following estimates: (2.4) ‖τ k ‖ ≤ c1A (‖τ k ‖ = |τ k |(R (2.5) |τ̂ k (ξ)| ≤ c2A βkm|Lm(ξ)| (2.6) |τ̂ k (ξ)| ≤ c3A βk+1m |Lm(ξ)| (2.7) (τ (m))∗(f) ≤ CpAB 2/p‖f‖p for p > 1 + θ, for some constants ci (1 ≤ i ≤ 3) and Cp, where we simply write ‖f‖Lp(Rd) = ‖f‖p. Now we prove the estimates (2.4)–(2.7). First we see that k ‖ ≤ C k ‖+ ‖σ (m−1) (2.8) ≤ C‖Ω‖1 ∫ ρk+1 |h(r)| dr/r ≤ C(log ρ)‖Ω‖1‖h‖∆1. From this (2.4) follows. To prove (2.5), define F (r, ξ) = Ω(θ) exp(−2πi〈ξ, P (m)(rθ)〉) dσ(θ). Then, via Hölder’s inequality, for s ∈ (1, 2] we see that k (ξ)| = ∫ ρk+1 h(r)F (r, ξ) dr/r (2.9) ∫ ρk+1 |h(r)|s dr/r )1/s( ∫ ρk+1 |F (r, ξ)| )1/s′ ≤ C(log ρ)1/s‖h‖∆s‖Ω‖ (s′−2)/s′ ∫ ρk+1 |F (r, ξ)| )1/s′ We need the following estimates for the last integral: 6 SHUICHI SATO Lemma 3. Let 1 < q ≤ 2 and Ω ∈ Lq(Sn−1). Then there exists a constant C > 0 independent of q, ρ,Ω and the coefficients of the polynomial components of P (m) such that ∫ ρk+1 |F (r, ξ)| dr/r ≤ C(log ρ) βkm|Lm(ξ)| )−1/(2q′N(m)) ‖Ω‖2q. Proof. Take an integer ν such that 2ν < ρ ≤ 2ν+1. By the proof of Proposition 5.1 of [8] we have ∫ ρk+1 |F (r, ξ)| dr/r = ∣F (ρkr, ξ) dr/r ≤ ∫ 2j+1 ∣F (ρkr, ξ) ∣F (2jρkr, ξ) )2/q′ 2jN(m)ρkN(m)|Lm(ξ)| )−1/(2N(m)q′) ‖Ω‖2q ≤ C(log ρ) ρkN(m)|Lm(ξ)| )−1/(2N(m)q′) ‖Ω‖2q. This completes the proof of Lemma 3. By (2.9) and Lemma 3 we have |σ̂ k (ξ)| ≤ CA βkm|Lm(ξ)| . Also, we have (m−1) k ‖ ≤ CA by (2.8). We can prove the estimate (2.5) by using these estimates in the definition of τ k in (2.2) and by noting that ϕ is compactly supported. Next, to prove (2.6), using (1.1) when m = 1, we see that k (ξ)| ≤ k (ξ)− σ̂ (m−1) k (ξ) Φk,m(ξ) (Φk,m(ξ) − Φk,m−1(ξ)) σ̂ (m−1) k (ξ) ≤ C‖Ω‖1β m |Lm(ξ)| ∫ ρk+1 |h(r)| dr/r + C‖σ (m−1) m|Lm(ξ)| ≤ C(log ρ)‖Ω‖1‖h‖∆1β m |Lm(ξ)|, where to get the last inequality we have used (2.8). By this and (2.8), we have k (ξ)| ≤ C(log ρ)‖Ω‖1‖h‖∆1 βk+1m |Lm(ξ)| for all c ∈ (0, 1], which implies (2.6). Finally, the estimate (2.7) follows from Lemma 1 since (τ (m))∗(f) (µ(m))∗(|f |) (µ(m−1))∗(|f |) ≤ CAB2/p‖f‖p for p > 1 + θ, where the first inequality can be seen by change of variables and a well-known result on maximal functions (see [8]). Let {ψk} −∞ be a sequence of functions in C ∞((0,∞)) such that supp(ψk) ⊂ [β m , β ψk(t) 2 = 1, |(d/dt)jψk(t)| ≤ cj/t j (j = 1, 2, . . . ), ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 7 where the constants cj are independent of βm. Define an operator Sk by (Sk(f)) (̂ξ) = Rm(ξ)| f̂(ξ) and let j (f) = k ∗ Sj+k(f) Then by Plancherel’s theorem and the estimates (2.4)–(2.6) we have j (f) D(j+k) k (ξ)| 2|f̂(ξ)|2 dξ (2.10) ≤ CA2 min 1, β−2αm(|j|−2)m D(j+k) |f̂(ξ)|2 dξ ≤ CA2 min 1, β−2αm(|j|−2)m ‖f‖22, where D(k) = {β−k−1m ≤ |Hmπ Rm(ξ)| ≤ β Applying the proof of Lemma in [7, p. 544] and using the estimates (2.4) and (2.7), we can prove the following. Lemma 4. Let u ∈ (1 + θ, 2]. Define a number v by 1/v− 1/2 = 1/(2u). Then we have the vector valued inequality (2.11) k ∗ gk| ≤ (c1Cu) 1/2AB1/u where the constants c1 and Cu are as in (2.4) and (2.7), respectively. By the Littlewood-Paley theory we have j (f)‖p ≤ cp k ∗ Sj+k(f)| ,(2.12) |Sk(f)| ≤ cp‖f‖p,(2.13) where 1 < p < ∞ and cp is independent of βm and the linear transformations Rm, Hm. Suppose that 1 + θ < p ≤ 4/(3− θ). Then we can find u ∈ (1 + θ, 2] such that 1/p = 1/2 + (1 − θ)/(2u). Let v be defined by u as in Lemma 4. Then by (2.11)–(2.13) we have (2.14) ‖V j (f)‖v ≤ CAB 1/u‖f‖v. Since 1/p = θ/2 + (1− θ)/v, interpolating between (2.10) and (2.14), we have j (f)‖p ≤ CAB (1−θ)/u min 1, β−θαm(|j|−2)m ‖f‖p. 8 SHUICHI SATO It follows that ‖T (m)ρ (f)‖p ≤ j (f)‖p ≤ CAB (1−θ)/u(1− β−θαmm ) −1‖f‖p (2.15) ≤ CAB2/p‖f‖p, where we have used the inequality −θαm(|j|−2) 1− β−θαmm We also have ‖T ρ (f)‖2 ≤ j (f)‖2 ≤ CAB‖f‖2 by (2.10), since B ≥ (1− β−αmm ) . By duality and interpolation, we can now get the conclusion of Lemma 2. Next, we give a proof of Lemma 1. We prove Lemma 1 by induction on j. Now we assume (2.3) for j = m − 1, 1 ≤ m ≤ ℓ, and prove (2.3) for j = m. Let ϕ ∈ C∞0 (R) be as above. Define a sequence η (m) = {η k } of Borel measures on R k (ξ) = ϕ βkm|Hmπ Rm(ξ)| (m−1) k (ξ). Then by (2.3) with j = m− 1, we have (2.16) (η(m))∗(f) (µ(m−1))∗(f) ≤ CAB2/p‖f‖p for p > 1 + θ. Furthermore, we have the following: k ‖+ ‖µ k ‖ ≤ C‖µ (m−1) k ‖+ ‖µ k ‖ ≤ C‖Ω‖1 ∫ ρk+1 |h(r)| dr/r (2.17) ≤ C(log ρ)‖Ω‖1‖h‖∆1 ≤ CA, k (ξ)− η̂ k (ξ)| ≤ C(log ρ)‖Ω‖1‖h‖∆1 βk+1m |Lm(ξ)| (2.18) βk+1m |Lm(ξ)| (2.19) |µ̂ k (ξ)| ≤ CA βkm|Lm(ξ)| (2.20) |η̂ k (ξ)| ≤ C(log ρ)‖h‖∆1‖Ω‖1 βkm|Lm(ξ)| βkm|Lm(ξ)| To see (2.18) we note that k (ξ)−η̂ k (ξ)| ≤ |µ̂ k (ξ)−µ̂ (m−1) k (ξ)|+ βkm|Hmπ Rm(ξ)| (m−1) k (ξ) Thus arguing as in the proof of (2.6), we have the first inequality of (2.18). The estimate (2.19) follows from the arguments used to prove (2.5). Also, we can see the first inequality of (2.20) by the definition of η k and (2.17). ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 9 Since ‖(µ(m))∗(f)‖∞ ≤ CA‖f‖∞, by taking into account an interpolation, it suffices to prove (2.3) with j = m for p ∈ (1+θ, 2]. Define a sequence ν(m) = {ν of Borel measures by ν k = µ k − η k . Let gm(f)(x) = k ∗ f(x) (2.21) (µ(m))∗(f) ≤ gm(f) + (η (m))∗(|f |). Thus, by (2.16), to get (2.3) with j = m it suffices to prove ‖gm(f)‖p ≤ CAB 2/p‖f‖p for p ∈ (1 + θ, 2] with an appropriate constant C. By a well-known property of Rademacher’s functions, this follows from (2.22) U (m)ǫ (f) ≤ CAB2/p‖f‖p for p ∈ (1 + θ, 2], where U ǫ (f) = k ǫkν k ∗ f with ǫ = {ǫk}, ǫk = 1 or −1, and the constant C is independent of ǫ. The estimate (2.22) is a consequence of the following: Lemma 5. We define a sequence {pj} 1 by p1 = 2 and 1/pj+1 = 1/2+(1−θ)/(2pj) for j ≥ 1. (We note that 1/pj = (1 − a j)/(1 + θ), where a = (1 − θ)/2, so {pj} is decreasing and converges to 1 + θ.) Then, for j ≥ 1 we have U (m)ǫ (f) ≤ CjAB 2/pj ‖f‖pj . Proof. Let j (f) = ǫkSj+k k ∗ Sj+k(f) Then by Plancherel’s theorem and the estimates (2.17)–(2.20), as in (2.10) we have (2.23) j (f) ≤ CAmin 1, β−αm(|j|−2)m ‖f‖2. It follows that ǫ (f) j (f)‖2 ≤ CAB‖f‖2. If we denote by A(s) the assertion of Lemma 5 for j = s, this proves A(1). Now we derive A(s+1) from A(s) assuming that A(s) holds, which will complete the proof of Lemma 5 by induction. Using (2.21), we see that (ν(m))∗(f) ≤ (µ(m))∗(|f |) + (η(m))∗(|f |) ≤ gm(|f |) + 2(η (m))∗(|f |). Note that A(s) implies ‖gm(f)‖ps ≤ CAB 2/ps‖f‖ps . By this and (2.16) we have (2.24) (ν(m))∗(f) ≤ ‖gm(|f |)‖ps + 2 (η(m))∗(|f |) ≤ CAB2/ps‖f‖ps . 10 SHUICHI SATO By (2.17), (2.23) and (2.24), we can now apply the arguments used in the proof of (2.15) to get A(s+ 1). This completes the proof of Lemma 5. Now we prove (2.22) for p ∈ (1 + θ, 2]. Let {pj} 1 be as in Lemma 5. Then we have pN+1 < p ≤ pN for some N . Thus, interpolating between the estimates of Lemma 5 for j = N and j = N + 1, we have (2.22). This proves (2.3) for j = m. Finally, we can easily see that (µ(0))∗(f) ≤ C(log ρ)‖Ω‖1‖h‖∆1|f | (see (2.17)), which implies the estimate (2.3) for j = 0. Therefore, by induction we have (2.3) for all 0 ≤ j ≤ ℓ. This completes the proof of Lemma 1. Now we can prove Theorem 1. Since θ ∈ (0, 1) is arbitrary, by taking ρ = 2q in Lemma 2 we have (f)‖p ≤ Cp(q − 1) −1(s− 1)−1‖Ω‖q‖h‖∆s‖f‖p for all p ∈ (1,∞). This completes the proof of Theorem 1, since T = m=1 T 3. Proof of Theorem 2 Theorem 2 can be proved by Theorem 1 and an extrapolation argument. Let T (f) be the singular integral in (1.2). We also write T (f) = Th,Ω(f). We fix q ∈ (1, 2], Ω ∈ Lq(Sn−1), p ∈ (1,∞) and a function f with ‖f‖p ≤ 1 and put S(h) = ‖Th,Ω(f)‖p. Then we have the following subadditivity: (3.1) S(h+ k) ≤ S(h) + S(k). Set E1 = {r ∈ R+ : |h(r)| ≤ 2} and Em = {r ∈ R+ : 2 m−1 < |h(r)| ≤ 2m} for m ≥ 2. Then, applying Theorem 1, we see that (3.2) S (hχEm) ≤ C(q − 1) −1(s− 1)−1‖Ω‖q‖hχEm‖∆s for s ∈ (1, 2], where χE denotes the characteristic function of a set E. Now we follow the extrapolation argument of Zygmund [23, Chap. XII, pp. 119–120]. First, note ‖hχEm‖∆1+1/m ≤ 2 mdm/(m+1)m (h) for m ≥ 1, where dm(h) is as in Section 1. Using this and (3.2) we see that S (hχEm) ≤ C(q − 1) −1‖Ω‖q m‖hχEm‖∆1+1/m ≤ C(q − 1)−1‖Ω‖q m2mdm/(m+1)m (h). Recalling the definition of Na(h), we have m2mdm/(m+1)m (h) = dm(h)<3−m m2mdm/(m+1)m (h) + dm(h)≥3−m m2mdm/(m+1)m (h) m2m3−m 2/(m+1) + m2mdm(h)3 m/(m+1) ≤ C(1 +N1(h)). ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 11 Therefore, by (3.1) we see that (3.3) S(h) ≤ S (hχEm) ≤ C(q − 1) −1‖Ω‖q (1 +N1(h)) . Next, fix h ∈ N1, p ∈ (1,∞) and f with ‖f‖p ≤ 1 and let R(Ω) = ‖Th,Ω(f)‖p. Put em = σ(Fm) for m ≥ 1, where Fm = {θ ∈ S n−1 : 2m−1 < |Ω(θ)| ≤ 2m} for m ≥ 2 and F1 = {θ ∈ S n−1 : |Ω(θ)| ≤ 2}. We decompose Ω as Ω = m=1 Ωm, where Ωm = ΩχFm − σ(S n−1)−1 Ω dσ. We note that Ωm dσ = 0, ‖Ωm‖r ≤ m for 1 < r <∞. Now, by (3.3) and the subadditivity of R(Ω) we see that R(Ω) ≤ R(Ωm) ≤ C (1 +N1(h)) m‖Ωm‖1+1/m ≤ C (1 +N1(h)) m2mem/(m+1)m = C (1 +N1(h)) em<3−m em≥3−m ≤ C (1 +N1(h)) m2m3−m 2/(m+1) + m2mem3 m/(m+1) ≤ C (1 +N1(h))) |Ω(θ)| log(2 + |Ω(θ)|) dσ(θ) This completes the proof of Theorem 2. 4. Estimates for maximal functions For the maximal operator T ∗ in (1.3) we have a result similar to Theorem 1. Theorem 3. Let q ∈ (1, 2], s ∈ (1, 2] and Ω ∈ Lq(Sn−1), h ∈ ∆s. Suppose Ω satisfies (1.1). Then we have ‖T ∗(f)‖Lp(Rd) ≤ Cp(q − 1) −1(s− 1)−1‖Ω‖Lq(Sn−1)‖h‖∆s‖f‖Lp(Rd) for all p ∈ (1,∞), where Cp is independent of q, s, Ω and h. As Theorem 1 implies Theorem 2, we have the following as a consequence of Theorem 3. Theorem 4. Let Ω be a function in L logL(Sn−1) satisfying (1.1) and h ∈ N1. ‖T ∗(f)‖Lp(Rd) ≤ Cp‖f‖Lp(Rd) for all p ∈ (1,∞). As in the cases of Theorems 1 and 2, the constants Cp of Theorems 3 and 4 are also independent of polynomials Pj if we fix deg(Pj) (j = 1, 2, . . . , d). When Ω is as in Theorem 4 and h ∈ ∆s for some s > 1, the L p boundedness of T ∗ was proved in [1]. When n = d, P (y) = y, Ω ∈ Lq for some q > 1 and h is bounded, the Lp boundedness of T ∗ is due to [3]. We use the following to prove Theorem 3. 12 SHUICHI SATO Lemma 6. Let τ (m) = {τ k } (1 ≤ m ≤ ℓ), where the measures τ k are as in (2.2). Let θ ∈ (0, 1) and let positive numbers A = (log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s, 1− β−θαmm be as above. We define (4.1) T ∗ρ,m(f)(x) = sup j ∗ f(x) Then, for p ∈ (2(1 + θ)/(θ2 − θ + 2), (1 + θ)/θ) =: Iθ we have ‖T ∗ρ,m(f)‖p ≤ CA B1+δ(p) +B2/p+1−θ/2 ‖f‖p, where C is independent of q, s ∈ (1, 2], Ω ∈ Lq(Sn−1), h ∈ ∆s, ρ and the coefficients of the polynomials Pj (1 ≤ j ≤ d). Proof. Let T ρ (f) = k ∗ f be as in Lemma 2. Let a function ϕ be as in the definition of τ k in (2.2). Define ϕk by ϕ̂k(ξ) = ϕ βkm|Hmπ Rm(ξ)| . Let δ be the delta function as above. Following [8], we decompose j ∗ f = ϕk ∗ T ρ (f)− ϕk ∗ j ∗ f + (δ − ϕk) ∗ j ∗ f It follows that (4.2) T ∗ρ,m(f) ≤ sup ϕk ∗ T ρ (f) j (f), whereN j (f) = supk k−j−1 ∗ f +supk (δ − ϕk) ∗ j+k ∗ f . By Lemma 2 we have (4.3) ϕk ∗ T ρ (f) ≤ CAB1+δ(p)‖f‖p for p ∈ (1 + θ, (1 + θ)/θ). Also, by (2.7) we see that (4.4) ‖N j (f)‖r ≤ CAB 2/r‖f‖r for r > 1 + θ. On the other hand, we have j (f) ≤ (δ − ϕk) ∗ j+k ∗ f k−j−1 ∗ f Therefore, by the estimates (2.5), (2.6) and Plancherel’s theorem, as in [8, p. 820] we see that (4.5) ‖N j (f)‖2 ≤ CAβ 1− β−2αmm )−1/2 ‖f‖2. ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 13 For p ∈ Iθ we can find r ∈ (1 + θ, 2(1 + θ)/θ) such that 1/p = (1 − θ)/r + θ/2, so an interpolation between (4.4) and (4.5) implies that (4.6) ‖N j (f)‖p ≤ CAB 2(1−θ)/r 1− β−2αmm )−θ/2 β−αmθjm ‖f‖p. Therefore, by (4.2), (4.3) and (4.6), for p ∈ Iθ we have ‖T ∗ρ,m(f)‖p ≤ CA B1+δ(p) +B2(1−θ)/r+1 1− β−2αmm )−θ/2 ‖f‖p. This implies the conclusion of Lemma 6, since 1− β−2αmm ≤ B and 2(1−θ)/r+ θ/2 + 1 = 2/p+ 1− θ/2. Proof of Theorem 3. Note that T ∗(f) ≤ 2T ∗0 (f) + 2µ ρ(|f |), where T 0 (f) is defined by the formula in (4.1) with {τ j } replaced by the sequence {σj} of measures in (2.1) and µ∗ρ = (µ (ℓ))∗ is as in Lemma 1. We note that T ∗0 (f) ≤ m=1 T ρ,m(f). Now, Lemma 6 implies that ‖T ∗ρ,m(f)‖p ≤ C(log ρ) 1− ρ−θ/(2q ‖Ω‖q‖h‖∆s‖f‖p for p ∈ Iθ. By using this with ρ = 2 q′s′ , since θ ∈ (0, 1) is arbitrary, we can conclude ′s′ ,m (f)‖p ≤ Cp(q − 1) −1(s− 1)−1‖Ω‖q‖h‖∆s‖f‖p for p ∈ (1,∞). Also, by Lemma 1 µ∗ρ satisfies a similar estimate when ρ = 2 q′s′ . Collecting results, we have Theorem 3. Remark. Let M(f)(x) = sup |y|<t |f(x− P (y))||Ω(y′)||h(|y|)| dy. It is easy to see that M(f) ≤ Cµ∗ρ(f), where C is independent of ρ ≥ 2. Therefore, by Lemma 1 we can prove results similar to Theorems 1 and 2 for the maximal operator M . In [1], Lp boundedness of M was proved under the condition that Ω ∈ L logL(Sn−1) and h ∈ ∆s for some s > 1. When n = d, P (y) = y, it is known that M is of weak type (1, 1) if Ω ∈ L logL(Sn−1) and h is bounded (see [5, 4]). References 1. A. Al-Salman and Y. Pan, Singular integrals with rough kernels in L logL(Sn−1), J. London Math. Soc. (2) 66 (2002), 153–174. 2. A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309. 3. L. K. Chen, On a singular integral, Studia Math. 85 (1987), 61–72. 4. M. Christ, Weak type (1, 1) bounds for rough operators, Ann. of Math. 128 (1988), 19–42. 5. M. Christ and J. L. Rubio de Francia, Weak type (1, 1) bounds for rough operators, II, Invent. Math. 93 (1988), 225–237. 6. J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), 869–880. 7. J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541–561. 14 SHUICHI SATO 8. D. Fan and Y. Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math. 119 (1997), 799–839. 9. D. Fan and S. Sato, Weighted weak type (1, 1) estimates for singular integrals and Littlewood- Paley functions, Studia Math. 163 (2004), 119-136. 10. R. Fefferman, A note on singular integrals, Proc. Amer. Math. Soc. 74 (1979), 266–270. 11. S. Hofmann, Weak (1, 1) boundedness of singular integrals with nonsmooth kernel, Proc. Amer. Math. Soc. 103 (1988), 260–264. 12. S. Hofmann, Weighted weak-type (1, 1) inequalities for rough operators, Proc. Amer. Math. Soc. 107 (1989), 423–435. 13. J. Namazi, On a singular integral, Proc. Amer. Math. Soc. 96 (1986), 421–424. 14. F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals, I, J. Func. Anal. 73 (1987), 179–194. 15. A. Seeger, Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (1996), 95–105. 16. A. Seeger and T. Tao, Sharp Lorentz space estimates for rough operators, Math. Ann. 320 (2001), 381–415. 17. E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proceedings of International Congress of Mathematicians, Berkeley (1986), 196–221. 18. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality And Oscillatory In- tegrals, Princeton University Press, Princeton, NJ, 1993. 19. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1971. 20. T. Tao, The weak-type (1, 1) of L logL homogeneous convolution operator, Indiana Univ. Math. J. 48 (1999), 1547–1584. 21. A. Vargas, Weighted weak type (1, 1) bounds for rough operators, J. London Math. Soc. (2) 54 (1996), 297–310. 22. D. Watson, Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J. 60 (1990), 389-399. 23. A. Zygmund, Trigonometric series 2nd ed., Cambridge Univ. Press, Cambridge, London, New York and Melbourne, 1977. Department of Mathematics Faculty of Education Kanazawa University Kanazawa, 920-1192 Japan E-mail address: shuichi@kenroku.kanazawa-u.ac.jp

We prove a sharp Lp estimate for a singular Radon transform according to a size condition of its kernel, which is useful for extrapolation.

Introduction Let Ω be a function in L1(Sn−1) satisfying (1.1) Ω(θ) dσ(θ) = 0, where dσ denotes the Lebesgue surface measure on the unit sphere Sn−1 in Rn. In this note we assume n ≥ 2. For s ≥ 1, let ∆s denote the collection of measurable functions h on R+ = {t ∈ R : t > 0} satisfying ‖h‖∆s = sup ∫ 2j+1 |h(t)|s dt/t where Z denotes the set of integers. We note that ∆s ⊂ ∆t if s > t. In this note we always assume h ∈ ∆1. Let P (y) = (P1(y), P2(y), . . . , Pd(y)) be a polynomial mapping, where each Pj is a real-valued polynomial on R n. We consider a singular integral operator of the form: (1.2) T (f)(x) = p. v. f(x− P (y))K(y) dy = lim |y|>ǫ f(x− P (y))K(y) dy, for an appropriate function f on Rd, where K(y) = h(|y|)Ω(y′)|y|−n, y′ = |y|−1y. Then, T (f) belongs to a class of singular Radon transforms. See Stein [17], Fan-Pan [8] and Al-Salman-Pan [1] for this singular integral. When h = 1 (a constant function), n = d and P (y) = y, we also write T (f) = S(f). Let f̂(ξ) = f(x)e−2πi〈x,ξ〉 dx be the Fourier transform of f , where 〈·, ·〉 Key words and phrases. Singular integrals, singular Radon transforms, maximal functions, extrapolation. 2000 Mathematics Subject Classification. Primary 42B20, 42B25 Typeset by AMS-TEX http://arxiv.org/abs/0704.1537v1 2 SHUICHI SATO denotes the inner product in Rd. Then it is known that (Sf )̂ (ξ) = m(ξ′)f̂(ξ), where m(ξ′) = − sgn(〈ξ′, θ〉) + log |〈ξ′, θ〉| dσ(θ). Using this, we can show that S extends to a bounded operator on L2 if Ω ∈ L logL(Sn−1), where L logL(Sn−1) denotes the Zygmund class of all those func- tions Ω on Sn−1 which satisfy |Ω(θ)| log(2 + |Ω(θ)|) dσ(θ) <∞. Furthermore, if Ω ∈ L logL(Sn−1), by the method of rotations of Calderón-Zygmund (see [2]) it can be shown that S extends to a bounded operator on Lp for all p ∈ (1,∞). When n = d and P (y) = y, R. Fefferman [10] proved that if h is bounded and Ω satisfies a Lipschitz condition of positive order on Sn−1, then the singular integral operator T in (1.2) is bounded on Lp for 1 < p < ∞. Namazi [13] improved this result by replacing the Lipschitz condition by the condition that Ω ∈ Lq(Sn−1) for some q > 1. In [7], Duoandikoetxea and Rubio de Francia developed methods which can be used to study mapping properties of several kinds of operators in harmonic analysis including the singular integrals considered in [13]. Also, see [6, 22] for weighted Lp boundedness of singular integrals, and [18, 19] for background materials. For the rest of this note we assume that the polynomial mapping P in (1.2) satisfies P (−y) = −P (y) and P 6= 0. We shall prove the following: Theorem 1. Let Ω ∈ Lq(Sn−1), q ∈ (1, 2] and h ∈ ∆s, s ∈ (1, 2]. Suppose Ω satisfies (1.1). Let T be as in (1.2). Then we have ‖T (f)‖Lp(Rd) ≤ Cp(q − 1) −1(s− 1)−1‖Ω‖Lq(Sn−1)‖h‖∆s‖f‖Lp(Rd) for all p ∈ (1,∞), where the constant Cp is independent of q, s,Ω and h. Also, the constant Cp is independent of polynomials Pj if we fix deg(Pj) (j = 1, 2, . . . , d). In Al-Salman-Pan [1], the Lp boundedness of T was proved under the condition that Ω is a function in L logL(Sn−1) satisfying (1.1) and h ∈ ∆s for some s > 1 ([1, Theorem 1.3]). Also it is noted there that estimates like those in Theorem 1 (with s being fixed) can be used to prove the same result by applying an extrapolation method, but such estimates are yet to be proved (see [1, p. 156]). In [1], the authors also considered singular integrals defined by certain polynomial mappings P which do not satisfy the condition P (−y) = −P (y). As a consequence of Theorem 1 we can give a different proof of [1, Theorem 1.3] via an extrapolation method; in fact, we can get an improved result. For a positive number a and a function h on R+, let La(h) = sup ∫ 2j+1 |h(r)| (log(2 + |h(r)|)) dr/r. We define a class La to be the space of all those measurable functions h on R+ which satisfy La(h) <∞. Also, let Na(h) = ma2mdm(h), ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 3 where dm(h) = supk∈Z 2 −k|E(k,m)| with E(k,m) = {r ∈ (2k, 2k+1] : 2m−1 < |h(r)| ≤ 2m} for m ≥ 2, E(k, 1) = {r ∈ (2k, 2k+1] : |h(r)| ≤ 2}. We denote by Na the class of all those measurable functions h on R+ such that Na(h) < ∞. Then we readily see that Na(h) < ∞ implies La(h) < ∞. Conversely, if La+b(h) < ∞ for some b > 1, then Na(h) <∞. To see this, note that 2mma+b2−k|E(k,m)| ≤ C E(k,m) |h(r)| (log(2 + |h(r)|)) dr/r ≤ CLa+b(h) for m ≥ 2; thus Na(h) ≤ 2d1(h) + CLa+b(h) −b < ∞. By Theorem 1 and an extrapolation method we have the following: Theorem 2. Suppose Ω is a function in L logL(Sn−1) satisfying (1.1) and h ∈ N1. Let T be as in (1.2). Then ‖T (f)‖Lp(Rd) ≤ Cp‖f‖Lp(Rd) for all p ∈ (1,∞), where Cp is independent of polynomials Pj if the polynomials are of fixed degree. By Theorem 2 and the remark preceding it we see that T is bounded on Lp for all p ∈ (1,∞) if Ω is as in Theorem 2 and h ∈ La for some a > 2. When n = d, P (y) = y, Ω is as in Theorem 2 and h is a constant function, it is known that T is of weak type (1, 1); see [5, 15]. Also, see [4, 9, 11, 12, 16, 20, 21] for related results. In Section 2, we shall prove Theorem 1. Applying the methods of [7] involving the Littlewood-Paley theory and using results of [8, 14], we shall prove Lp estimates for certain maximal and singular integral operators related to the operator T in Theorem 1 (Lemmas 1 and 2). Lemma 1 is used to prove Lemma 2. By Lemma 2 we can easily prove Theorem 1. A key idea of the proof of Theorem 1 is to apply a Littlewood-Paley decomposition adapted to a suitable lacunary sequence depending on q and s for which Ω ∈ Lq(Sn−1) and h ∈ ∆s. The method of appropriately choosing the lacunary sequence was inspired by [1], where, in a somewhat different way from ours, a similar method was used to study several classes of singular integrals. We shall prove Theorem 2 in Section 3. Finally, in Section 4, we consider the maximal operator (1.3) T ∗(f)(x) = sup N,ǫ>0 ǫ<|y|<N f(x− P (y))K(y) dy where P and K are as in (1.2). We shall prove analogs of Theorems 1 and 2 for the operator T ∗. Throughout this note, the letter C will be used to denote non-negative constants which may be different in different occurrences. 2. Proof of Theorem 1 Let Ω, h be as in Theorem 1. We consider the singular integral T (f) defined in (1.2). Let ρ ≥ 2 and Ek = {x ∈ R n : ρk < |x| ≤ ρk+1}. Then T (f)(x) = −∞ σk ∗ f(x), where {σk} is a sequence of Borel measures on R d such that (2.1) σk ∗ f(x) = f(x− P (y))K(y) dy. 4 SHUICHI SATO We note that (σk ∗ f )̂ (ξ) = f̂(ξ) e−2πi〈P (y),ξ〉K(y) dy. We write P (y) = Qj(y), Qj(y) = |γ|=N(j) γ (aγ ∈ R where Qj 6= 0, 1 ≤ N(1) < N(2) < · · · < N(ℓ), γ = (γ1, . . . , γn) is a multi- index, yγ = y 1 . . . y n and |γ| = γ1 + · · · + γn. Let βm = ρ N(m) and αm = (q − 1)(s− 1)/(2qsN(m)) for 1 ≤ m ≤ ℓ. Put P (m)(y) = j=1Qj(y) and define a sequence µ(m) = {µ k } of positive measures on R k ∗ f(x) = x− P (m)(y) |K(y)| dy for m = 1, 2, . . . , ℓ. Also, define µ(0) = {µ k } by µ k = ( |K(y)| dy)δ, where δ is Dirac’s delta function on Rd. For a sequence ν = {νk} of finite Borel measures on Rd, we define the maximal operator ν∗ by ν∗(f)(x) = supk ||νk| ∗ f(x)|, where |νk| denotes the total variation. We consider the maximal operators m ≤ ℓ). We also write = µ∗ρ. Lj(ξ) = (〈aγ(j,1), ξ〉, 〈aγ(j,2), ξ〉, . . . , 〈aγ(j,rj), ξ〉), where {γ(j, k)} k=1 is an enumeration of {γ}|γ|=N(j) for 1 ≤ j ≤ ℓ. Then Lj is a linear mapping from Rd to Rrj . Let sj = rankLj. There exist non-singular linear transformations Rj : R d → Rd and Hj : R sj → Rsj such that Rj(ξ)| ≤ |Lj(ξ)| ≤ C|Hjπ Rj(ξ)|, where πdsj (ξ) = (ξ1, . . . , ξsj ) is the projection and C depends only on rj (a proof can be found in [8]). Let {σ k } (0 ≤ m ≤ ℓ) be a sequence of Borel measures on d such that k ∗ f(x) = x− P (m)(y) K(y) dy for m = 1, 2, . . . , ℓ, while σ k = 0. Let ϕ ∈ C 0 (R) be supported in {|r| ≤ 1} and ϕ(r) = 1 for |r| < 1/2. Define a sequence τ (m) = {τ k } of Borel measures by (2.2) τ̂ k (ξ) = σ̂ k (ξ)Φk,m(ξ)− σ̂ (m−1) k (ξ)Φk,m−1(ξ) for m = 1, 2, . . . , ℓ, where Φk,m(ξ) = j=m+1 βkj |Hjπ Rj(ξ)| if 0 ≤ m ≤ ℓ− 1 and Φk,ℓ = 1. Then σk = σ m=1 τ k . We note that Φk,m(ξ)ϕ βkm|Hmπ Rm(ξ)| = Φk,m−1(ξ) (1 ≤ m ≤ ℓ). For 1 ≤ m ≤ ℓ, let T ρ (f) = k ∗ f . Then T = m=1 T For p ∈ (1,∞) we put p′ = p/(p − 1) and δ(p) = |1/p − 1/p′|. Let θ ∈ (0, 1). Then we have the following Lp estimates for (µ(m))∗ and T ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 5 Lemma 1. For p > 1 + θ and 0 ≤ j ≤ ℓ, we have (2.3) (µ(j))∗(f) Lp(Rd) ≤ C(log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s 1− ρ−θ/(2q )−2/p ‖f‖Lp(Rd). Lemma 2. For p ∈ (1 + θ, (1 + θ)/θ) and 1 ≤ m ≤ ℓ, we have ‖T (m)ρ (f)‖Lp(Rd) ≤ C(log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s 1− ρ−θ/(2q )−1−δ(p) ‖f‖Lp(Rd). The constants C in Lemmas 1 and 2 are independent of q, s ∈ (1, 2], Ω ∈ Lq(Sn−1), h ∈ ∆s, ρ and the coefficients of the polynomials Pk (1 ≤ k ≤ d). We prove Lemma 2 first, taking Lemma 1 for granted for the moment. Let A = (log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s and B = 1− β−θαmm 1− ρ−θ/(2q . Then, we have the following estimates: (2.4) ‖τ k ‖ ≤ c1A (‖τ k ‖ = |τ k |(R (2.5) |τ̂ k (ξ)| ≤ c2A βkm|Lm(ξ)| (2.6) |τ̂ k (ξ)| ≤ c3A βk+1m |Lm(ξ)| (2.7) (τ (m))∗(f) ≤ CpAB 2/p‖f‖p for p > 1 + θ, for some constants ci (1 ≤ i ≤ 3) and Cp, where we simply write ‖f‖Lp(Rd) = ‖f‖p. Now we prove the estimates (2.4)–(2.7). First we see that k ‖ ≤ C k ‖+ ‖σ (m−1) (2.8) ≤ C‖Ω‖1 ∫ ρk+1 |h(r)| dr/r ≤ C(log ρ)‖Ω‖1‖h‖∆1. From this (2.4) follows. To prove (2.5), define F (r, ξ) = Ω(θ) exp(−2πi〈ξ, P (m)(rθ)〉) dσ(θ). Then, via Hölder’s inequality, for s ∈ (1, 2] we see that k (ξ)| = ∫ ρk+1 h(r)F (r, ξ) dr/r (2.9) ∫ ρk+1 |h(r)|s dr/r )1/s( ∫ ρk+1 |F (r, ξ)| )1/s′ ≤ C(log ρ)1/s‖h‖∆s‖Ω‖ (s′−2)/s′ ∫ ρk+1 |F (r, ξ)| )1/s′ We need the following estimates for the last integral: 6 SHUICHI SATO Lemma 3. Let 1 < q ≤ 2 and Ω ∈ Lq(Sn−1). Then there exists a constant C > 0 independent of q, ρ,Ω and the coefficients of the polynomial components of P (m) such that ∫ ρk+1 |F (r, ξ)| dr/r ≤ C(log ρ) βkm|Lm(ξ)| )−1/(2q′N(m)) ‖Ω‖2q. Proof. Take an integer ν such that 2ν < ρ ≤ 2ν+1. By the proof of Proposition 5.1 of [8] we have ∫ ρk+1 |F (r, ξ)| dr/r = ∣F (ρkr, ξ) dr/r ≤ ∫ 2j+1 ∣F (ρkr, ξ) ∣F (2jρkr, ξ) )2/q′ 2jN(m)ρkN(m)|Lm(ξ)| )−1/(2N(m)q′) ‖Ω‖2q ≤ C(log ρ) ρkN(m)|Lm(ξ)| )−1/(2N(m)q′) ‖Ω‖2q. This completes the proof of Lemma 3. By (2.9) and Lemma 3 we have |σ̂ k (ξ)| ≤ CA βkm|Lm(ξ)| . Also, we have (m−1) k ‖ ≤ CA by (2.8). We can prove the estimate (2.5) by using these estimates in the definition of τ k in (2.2) and by noting that ϕ is compactly supported. Next, to prove (2.6), using (1.1) when m = 1, we see that k (ξ)| ≤ k (ξ)− σ̂ (m−1) k (ξ) Φk,m(ξ) (Φk,m(ξ) − Φk,m−1(ξ)) σ̂ (m−1) k (ξ) ≤ C‖Ω‖1β m |Lm(ξ)| ∫ ρk+1 |h(r)| dr/r + C‖σ (m−1) m|Lm(ξ)| ≤ C(log ρ)‖Ω‖1‖h‖∆1β m |Lm(ξ)|, where to get the last inequality we have used (2.8). By this and (2.8), we have k (ξ)| ≤ C(log ρ)‖Ω‖1‖h‖∆1 βk+1m |Lm(ξ)| for all c ∈ (0, 1], which implies (2.6). Finally, the estimate (2.7) follows from Lemma 1 since (τ (m))∗(f) (µ(m))∗(|f |) (µ(m−1))∗(|f |) ≤ CAB2/p‖f‖p for p > 1 + θ, where the first inequality can be seen by change of variables and a well-known result on maximal functions (see [8]). Let {ψk} −∞ be a sequence of functions in C ∞((0,∞)) such that supp(ψk) ⊂ [β m , β ψk(t) 2 = 1, |(d/dt)jψk(t)| ≤ cj/t j (j = 1, 2, . . . ), ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 7 where the constants cj are independent of βm. Define an operator Sk by (Sk(f)) (̂ξ) = Rm(ξ)| f̂(ξ) and let j (f) = k ∗ Sj+k(f) Then by Plancherel’s theorem and the estimates (2.4)–(2.6) we have j (f) D(j+k) k (ξ)| 2|f̂(ξ)|2 dξ (2.10) ≤ CA2 min 1, β−2αm(|j|−2)m D(j+k) |f̂(ξ)|2 dξ ≤ CA2 min 1, β−2αm(|j|−2)m ‖f‖22, where D(k) = {β−k−1m ≤ |Hmπ Rm(ξ)| ≤ β Applying the proof of Lemma in [7, p. 544] and using the estimates (2.4) and (2.7), we can prove the following. Lemma 4. Let u ∈ (1 + θ, 2]. Define a number v by 1/v− 1/2 = 1/(2u). Then we have the vector valued inequality (2.11) k ∗ gk| ≤ (c1Cu) 1/2AB1/u where the constants c1 and Cu are as in (2.4) and (2.7), respectively. By the Littlewood-Paley theory we have j (f)‖p ≤ cp k ∗ Sj+k(f)| ,(2.12) |Sk(f)| ≤ cp‖f‖p,(2.13) where 1 < p < ∞ and cp is independent of βm and the linear transformations Rm, Hm. Suppose that 1 + θ < p ≤ 4/(3− θ). Then we can find u ∈ (1 + θ, 2] such that 1/p = 1/2 + (1 − θ)/(2u). Let v be defined by u as in Lemma 4. Then by (2.11)–(2.13) we have (2.14) ‖V j (f)‖v ≤ CAB 1/u‖f‖v. Since 1/p = θ/2 + (1− θ)/v, interpolating between (2.10) and (2.14), we have j (f)‖p ≤ CAB (1−θ)/u min 1, β−θαm(|j|−2)m ‖f‖p. 8 SHUICHI SATO It follows that ‖T (m)ρ (f)‖p ≤ j (f)‖p ≤ CAB (1−θ)/u(1− β−θαmm ) −1‖f‖p (2.15) ≤ CAB2/p‖f‖p, where we have used the inequality −θαm(|j|−2) 1− β−θαmm We also have ‖T ρ (f)‖2 ≤ j (f)‖2 ≤ CAB‖f‖2 by (2.10), since B ≥ (1− β−αmm ) . By duality and interpolation, we can now get the conclusion of Lemma 2. Next, we give a proof of Lemma 1. We prove Lemma 1 by induction on j. Now we assume (2.3) for j = m − 1, 1 ≤ m ≤ ℓ, and prove (2.3) for j = m. Let ϕ ∈ C∞0 (R) be as above. Define a sequence η (m) = {η k } of Borel measures on R k (ξ) = ϕ βkm|Hmπ Rm(ξ)| (m−1) k (ξ). Then by (2.3) with j = m− 1, we have (2.16) (η(m))∗(f) (µ(m−1))∗(f) ≤ CAB2/p‖f‖p for p > 1 + θ. Furthermore, we have the following: k ‖+ ‖µ k ‖ ≤ C‖µ (m−1) k ‖+ ‖µ k ‖ ≤ C‖Ω‖1 ∫ ρk+1 |h(r)| dr/r (2.17) ≤ C(log ρ)‖Ω‖1‖h‖∆1 ≤ CA, k (ξ)− η̂ k (ξ)| ≤ C(log ρ)‖Ω‖1‖h‖∆1 βk+1m |Lm(ξ)| (2.18) βk+1m |Lm(ξ)| (2.19) |µ̂ k (ξ)| ≤ CA βkm|Lm(ξ)| (2.20) |η̂ k (ξ)| ≤ C(log ρ)‖h‖∆1‖Ω‖1 βkm|Lm(ξ)| βkm|Lm(ξ)| To see (2.18) we note that k (ξ)−η̂ k (ξ)| ≤ |µ̂ k (ξ)−µ̂ (m−1) k (ξ)|+ βkm|Hmπ Rm(ξ)| (m−1) k (ξ) Thus arguing as in the proof of (2.6), we have the first inequality of (2.18). The estimate (2.19) follows from the arguments used to prove (2.5). Also, we can see the first inequality of (2.20) by the definition of η k and (2.17). ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 9 Since ‖(µ(m))∗(f)‖∞ ≤ CA‖f‖∞, by taking into account an interpolation, it suffices to prove (2.3) with j = m for p ∈ (1+θ, 2]. Define a sequence ν(m) = {ν of Borel measures by ν k = µ k − η k . Let gm(f)(x) = k ∗ f(x) (2.21) (µ(m))∗(f) ≤ gm(f) + (η (m))∗(|f |). Thus, by (2.16), to get (2.3) with j = m it suffices to prove ‖gm(f)‖p ≤ CAB 2/p‖f‖p for p ∈ (1 + θ, 2] with an appropriate constant C. By a well-known property of Rademacher’s functions, this follows from (2.22) U (m)ǫ (f) ≤ CAB2/p‖f‖p for p ∈ (1 + θ, 2], where U ǫ (f) = k ǫkν k ∗ f with ǫ = {ǫk}, ǫk = 1 or −1, and the constant C is independent of ǫ. The estimate (2.22) is a consequence of the following: Lemma 5. We define a sequence {pj} 1 by p1 = 2 and 1/pj+1 = 1/2+(1−θ)/(2pj) for j ≥ 1. (We note that 1/pj = (1 − a j)/(1 + θ), where a = (1 − θ)/2, so {pj} is decreasing and converges to 1 + θ.) Then, for j ≥ 1 we have U (m)ǫ (f) ≤ CjAB 2/pj ‖f‖pj . Proof. Let j (f) = ǫkSj+k k ∗ Sj+k(f) Then by Plancherel’s theorem and the estimates (2.17)–(2.20), as in (2.10) we have (2.23) j (f) ≤ CAmin 1, β−αm(|j|−2)m ‖f‖2. It follows that ǫ (f) j (f)‖2 ≤ CAB‖f‖2. If we denote by A(s) the assertion of Lemma 5 for j = s, this proves A(1). Now we derive A(s+1) from A(s) assuming that A(s) holds, which will complete the proof of Lemma 5 by induction. Using (2.21), we see that (ν(m))∗(f) ≤ (µ(m))∗(|f |) + (η(m))∗(|f |) ≤ gm(|f |) + 2(η (m))∗(|f |). Note that A(s) implies ‖gm(f)‖ps ≤ CAB 2/ps‖f‖ps . By this and (2.16) we have (2.24) (ν(m))∗(f) ≤ ‖gm(|f |)‖ps + 2 (η(m))∗(|f |) ≤ CAB2/ps‖f‖ps . 10 SHUICHI SATO By (2.17), (2.23) and (2.24), we can now apply the arguments used in the proof of (2.15) to get A(s+ 1). This completes the proof of Lemma 5. Now we prove (2.22) for p ∈ (1 + θ, 2]. Let {pj} 1 be as in Lemma 5. Then we have pN+1 < p ≤ pN for some N . Thus, interpolating between the estimates of Lemma 5 for j = N and j = N + 1, we have (2.22). This proves (2.3) for j = m. Finally, we can easily see that (µ(0))∗(f) ≤ C(log ρ)‖Ω‖1‖h‖∆1|f | (see (2.17)), which implies the estimate (2.3) for j = 0. Therefore, by induction we have (2.3) for all 0 ≤ j ≤ ℓ. This completes the proof of Lemma 1. Now we can prove Theorem 1. Since θ ∈ (0, 1) is arbitrary, by taking ρ = 2q in Lemma 2 we have (f)‖p ≤ Cp(q − 1) −1(s− 1)−1‖Ω‖q‖h‖∆s‖f‖p for all p ∈ (1,∞). This completes the proof of Theorem 1, since T = m=1 T 3. Proof of Theorem 2 Theorem 2 can be proved by Theorem 1 and an extrapolation argument. Let T (f) be the singular integral in (1.2). We also write T (f) = Th,Ω(f). We fix q ∈ (1, 2], Ω ∈ Lq(Sn−1), p ∈ (1,∞) and a function f with ‖f‖p ≤ 1 and put S(h) = ‖Th,Ω(f)‖p. Then we have the following subadditivity: (3.1) S(h+ k) ≤ S(h) + S(k). Set E1 = {r ∈ R+ : |h(r)| ≤ 2} and Em = {r ∈ R+ : 2 m−1 < |h(r)| ≤ 2m} for m ≥ 2. Then, applying Theorem 1, we see that (3.2) S (hχEm) ≤ C(q − 1) −1(s− 1)−1‖Ω‖q‖hχEm‖∆s for s ∈ (1, 2], where χE denotes the characteristic function of a set E. Now we follow the extrapolation argument of Zygmund [23, Chap. XII, pp. 119–120]. First, note ‖hχEm‖∆1+1/m ≤ 2 mdm/(m+1)m (h) for m ≥ 1, where dm(h) is as in Section 1. Using this and (3.2) we see that S (hχEm) ≤ C(q − 1) −1‖Ω‖q m‖hχEm‖∆1+1/m ≤ C(q − 1)−1‖Ω‖q m2mdm/(m+1)m (h). Recalling the definition of Na(h), we have m2mdm/(m+1)m (h) = dm(h)<3−m m2mdm/(m+1)m (h) + dm(h)≥3−m m2mdm/(m+1)m (h) m2m3−m 2/(m+1) + m2mdm(h)3 m/(m+1) ≤ C(1 +N1(h)). ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 11 Therefore, by (3.1) we see that (3.3) S(h) ≤ S (hχEm) ≤ C(q − 1) −1‖Ω‖q (1 +N1(h)) . Next, fix h ∈ N1, p ∈ (1,∞) and f with ‖f‖p ≤ 1 and let R(Ω) = ‖Th,Ω(f)‖p. Put em = σ(Fm) for m ≥ 1, where Fm = {θ ∈ S n−1 : 2m−1 < |Ω(θ)| ≤ 2m} for m ≥ 2 and F1 = {θ ∈ S n−1 : |Ω(θ)| ≤ 2}. We decompose Ω as Ω = m=1 Ωm, where Ωm = ΩχFm − σ(S n−1)−1 Ω dσ. We note that Ωm dσ = 0, ‖Ωm‖r ≤ m for 1 < r <∞. Now, by (3.3) and the subadditivity of R(Ω) we see that R(Ω) ≤ R(Ωm) ≤ C (1 +N1(h)) m‖Ωm‖1+1/m ≤ C (1 +N1(h)) m2mem/(m+1)m = C (1 +N1(h)) em<3−m em≥3−m ≤ C (1 +N1(h)) m2m3−m 2/(m+1) + m2mem3 m/(m+1) ≤ C (1 +N1(h))) |Ω(θ)| log(2 + |Ω(θ)|) dσ(θ) This completes the proof of Theorem 2. 4. Estimates for maximal functions For the maximal operator T ∗ in (1.3) we have a result similar to Theorem 1. Theorem 3. Let q ∈ (1, 2], s ∈ (1, 2] and Ω ∈ Lq(Sn−1), h ∈ ∆s. Suppose Ω satisfies (1.1). Then we have ‖T ∗(f)‖Lp(Rd) ≤ Cp(q − 1) −1(s− 1)−1‖Ω‖Lq(Sn−1)‖h‖∆s‖f‖Lp(Rd) for all p ∈ (1,∞), where Cp is independent of q, s, Ω and h. As Theorem 1 implies Theorem 2, we have the following as a consequence of Theorem 3. Theorem 4. Let Ω be a function in L logL(Sn−1) satisfying (1.1) and h ∈ N1. ‖T ∗(f)‖Lp(Rd) ≤ Cp‖f‖Lp(Rd) for all p ∈ (1,∞). As in the cases of Theorems 1 and 2, the constants Cp of Theorems 3 and 4 are also independent of polynomials Pj if we fix deg(Pj) (j = 1, 2, . . . , d). When Ω is as in Theorem 4 and h ∈ ∆s for some s > 1, the L p boundedness of T ∗ was proved in [1]. When n = d, P (y) = y, Ω ∈ Lq for some q > 1 and h is bounded, the Lp boundedness of T ∗ is due to [3]. We use the following to prove Theorem 3. 12 SHUICHI SATO Lemma 6. Let τ (m) = {τ k } (1 ≤ m ≤ ℓ), where the measures τ k are as in (2.2). Let θ ∈ (0, 1) and let positive numbers A = (log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s, 1− β−θαmm be as above. We define (4.1) T ∗ρ,m(f)(x) = sup j ∗ f(x) Then, for p ∈ (2(1 + θ)/(θ2 − θ + 2), (1 + θ)/θ) =: Iθ we have ‖T ∗ρ,m(f)‖p ≤ CA B1+δ(p) +B2/p+1−θ/2 ‖f‖p, where C is independent of q, s ∈ (1, 2], Ω ∈ Lq(Sn−1), h ∈ ∆s, ρ and the coefficients of the polynomials Pj (1 ≤ j ≤ d). Proof. Let T ρ (f) = k ∗ f be as in Lemma 2. Let a function ϕ be as in the definition of τ k in (2.2). Define ϕk by ϕ̂k(ξ) = ϕ βkm|Hmπ Rm(ξ)| . Let δ be the delta function as above. Following [8], we decompose j ∗ f = ϕk ∗ T ρ (f)− ϕk ∗ j ∗ f + (δ − ϕk) ∗ j ∗ f It follows that (4.2) T ∗ρ,m(f) ≤ sup ϕk ∗ T ρ (f) j (f), whereN j (f) = supk k−j−1 ∗ f +supk (δ − ϕk) ∗ j+k ∗ f . By Lemma 2 we have (4.3) ϕk ∗ T ρ (f) ≤ CAB1+δ(p)‖f‖p for p ∈ (1 + θ, (1 + θ)/θ). Also, by (2.7) we see that (4.4) ‖N j (f)‖r ≤ CAB 2/r‖f‖r for r > 1 + θ. On the other hand, we have j (f) ≤ (δ − ϕk) ∗ j+k ∗ f k−j−1 ∗ f Therefore, by the estimates (2.5), (2.6) and Plancherel’s theorem, as in [8, p. 820] we see that (4.5) ‖N j (f)‖2 ≤ CAβ 1− β−2αmm )−1/2 ‖f‖2. ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 13 For p ∈ Iθ we can find r ∈ (1 + θ, 2(1 + θ)/θ) such that 1/p = (1 − θ)/r + θ/2, so an interpolation between (4.4) and (4.5) implies that (4.6) ‖N j (f)‖p ≤ CAB 2(1−θ)/r 1− β−2αmm )−θ/2 β−αmθjm ‖f‖p. Therefore, by (4.2), (4.3) and (4.6), for p ∈ Iθ we have ‖T ∗ρ,m(f)‖p ≤ CA B1+δ(p) +B2(1−θ)/r+1 1− β−2αmm )−θ/2 ‖f‖p. This implies the conclusion of Lemma 6, since 1− β−2αmm ≤ B and 2(1−θ)/r+ θ/2 + 1 = 2/p+ 1− θ/2. Proof of Theorem 3. Note that T ∗(f) ≤ 2T ∗0 (f) + 2µ ρ(|f |), where T 0 (f) is defined by the formula in (4.1) with {τ j } replaced by the sequence {σj} of measures in (2.1) and µ∗ρ = (µ (ℓ))∗ is as in Lemma 1. We note that T ∗0 (f) ≤ m=1 T ρ,m(f). Now, Lemma 6 implies that ‖T ∗ρ,m(f)‖p ≤ C(log ρ) 1− ρ−θ/(2q ‖Ω‖q‖h‖∆s‖f‖p for p ∈ Iθ. By using this with ρ = 2 q′s′ , since θ ∈ (0, 1) is arbitrary, we can conclude ′s′ ,m (f)‖p ≤ Cp(q − 1) −1(s− 1)−1‖Ω‖q‖h‖∆s‖f‖p for p ∈ (1,∞). Also, by Lemma 1 µ∗ρ satisfies a similar estimate when ρ = 2 q′s′ . Collecting results, we have Theorem 3. Remark. Let M(f)(x) = sup |y|<t |f(x− P (y))||Ω(y′)||h(|y|)| dy. It is easy to see that M(f) ≤ Cµ∗ρ(f), where C is independent of ρ ≥ 2. Therefore, by Lemma 1 we can prove results similar to Theorems 1 and 2 for the maximal operator M . In [1], Lp boundedness of M was proved under the condition that Ω ∈ L logL(Sn−1) and h ∈ ∆s for some s > 1. When n = d, P (y) = y, it is known that M is of weak type (1, 1) if Ω ∈ L logL(Sn−1) and h is bounded (see [5, 4]). References 1. A. Al-Salman and Y. Pan, Singular integrals with rough kernels in L logL(Sn−1), J. London Math. Soc. (2) 66 (2002), 153–174. 2. A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309. 3. L. K. Chen, On a singular integral, Studia Math. 85 (1987), 61–72. 4. M. Christ, Weak type (1, 1) bounds for rough operators, Ann. of Math. 128 (1988), 19–42. 5. M. Christ and J. L. Rubio de Francia, Weak type (1, 1) bounds for rough operators, II, Invent. Math. 93 (1988), 225–237. 6. J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), 869–880. 7. J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541–561. 14 SHUICHI SATO 8. D. Fan and Y. Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math. 119 (1997), 799–839. 9. D. Fan and S. Sato, Weighted weak type (1, 1) estimates for singular integrals and Littlewood- Paley functions, Studia Math. 163 (2004), 119-136. 10. R. Fefferman, A note on singular integrals, Proc. Amer. Math. Soc. 74 (1979), 266–270. 11. S. Hofmann, Weak (1, 1) boundedness of singular integrals with nonsmooth kernel, Proc. Amer. Math. Soc. 103 (1988), 260–264. 12. S. Hofmann, Weighted weak-type (1, 1) inequalities for rough operators, Proc. Amer. Math. Soc. 107 (1989), 423–435. 13. J. Namazi, On a singular integral, Proc. Amer. Math. Soc. 96 (1986), 421–424. 14. F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals, I, J. Func. Anal. 73 (1987), 179–194. 15. A. Seeger, Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (1996), 95–105. 16. A. Seeger and T. Tao, Sharp Lorentz space estimates for rough operators, Math. Ann. 320 (2001), 381–415. 17. E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proceedings of International Congress of Mathematicians, Berkeley (1986), 196–221. 18. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality And Oscillatory In- tegrals, Princeton University Press, Princeton, NJ, 1993. 19. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1971. 20. T. Tao, The weak-type (1, 1) of L logL homogeneous convolution operator, Indiana Univ. Math. J. 48 (1999), 1547–1584. 21. A. Vargas, Weighted weak type (1, 1) bounds for rough operators, J. London Math. Soc. (2) 54 (1996), 297–310. 22. D. Watson, Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J. 60 (1990), 389-399. 23. A. Zygmund, Trigonometric series 2nd ed., Cambridge Univ. Press, Cambridge, London, New York and Melbourne, 1977. Department of Mathematics Faculty of Education Kanazawa University Kanazawa, 920-1192 Japan E-mail address: shuichi@kenroku.kanazawa-u.ac.jp

Rounding of first-order phase transitions and optimal cooperation in scale-free networks M. Karsai,1,2 J-Ch. Anglès d’Auriac,2 and F. Iglói3, 1 Institute of Theoretical Physics, Szeged University, H-6720 Szeged, Hungary Centre de Recherches sur les Trés Basses Températures, B. P. 166, F-38042 Grenoble, France Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O.Box 49, Hungary (Dated: November 20, 2018) We consider the ferromagnetic large-q state Potts model in complex evolving networks, which is equivalent to an optimal cooperation problem, in which the agents try to optimize the total sum of pair cooperation benefits and the supports of independent projects. The agents are found to be typically of two kinds: a fraction of m (being the magnetization of the Potts model) belongs to a large cooperating cluster, whereas the others are isolated one man’s projects. It is shown rigorously that the homogeneous model has a strongly first-order phase transition, which turns to second-order for random interactions (benefits), the properties of which are studied numerically on the Barabási- Albert network. The distribution of finite-size transition points is characterized by a shift exponent, 1/ν̃′ = .26(1), and by a different width exponent, 1/ν′ = .18(1), whereas the magnetization at the transition point scales with the size of the network, N , as: m ∼ N−x, with x = .66(1). I. INTRODUCTION Complex networks have been used to describe the structure and topology of a large class of systems in dif- ferent fields of science, technics, transport, social and political life, etc, see Refs.1,2,3,4 for recent reviews. A complex network is represented by a graph5, in which the nodes stand for the agents and the edges denote the possi- ble interactions. Realistic networks generally have three basic properties. The average distance between the nodes is small, which is called the small-world effect6. There is a tendency of clustering and the degree-distribution of the edges, P (k), has a power-law tail, PD(k) ≃ Ak −γ , k ≫ 1. Thus the edge distribution is scale free7, which is usually attributed to growth and preferential attachment during the evaluation of the network. In reality there is some sort of interaction between the agents of a network which leads to some kind of coop- erative behavior in macroscopic scales. One throughly studied question in this field is the spread of infections and epidemics in networks8,9,10, which problem is closely related to another non-equilibrium processes, such as percolation11, diffusion12, the contact process13 or the zero-range process14, etc. In another investigations one considers simple magnetic models15,16,17,18,19, in which the agents are represented by classical (Ising or Potts) spin variables, the interactions are described by ferro- magnetic couplings, whereas the temperature plays the rôle of a disordering field. In theoretical investigations of the cooperative behav- ior one usually resort on some kind of approximations. For example the sites of the networks are often consid- ered uncorrelated, which is generally not true for evolv- ing networks, such as the Barabási-Albert (BA) network. However this effect is expected to be irrelevant, as far as the singularities in the system are considered. Also the simple mean-field approach could lead to exact results due to long-range interactions in the networks, which has been checked by numerical simulations15 and by another, more accurate theoretical methods19 (Bethe-lattice ap- proach, replica method, etc.). In these calculations the critical behavior of the network is found to depend on the value of the degree exponent, γ. For sufficiently weakly connected networks with γ > γu (γu = 5 for the Ising model) there are conventional mean-field singularities. In the intermediate or unconventional mean-field regime, for γu > γ > γc, the critical exponents are γ dependent. Fi- nally, for γ ≤ γc, when the average of k 2, defined by 〈k2〉 = PD(k)k 2dk, as well as the strength of the aver- age interaction becomes divergent the scale-free network remains in the ordered state at any finite temperature. Since γc = 3, in realistic networks with homogeneous interactions always this type of phenomena is expected to occur. In weighted networks, however, in which the strengths of the interaction is appropriately rescaled with the degrees of the connected vertices, γc is shifted to larger values and therefore the complete phase-transition scenario can be tested13,19. We note that the properties of the phase transitions are generally different for undi- rected (as we consider here) and directed networks20. In several models the phase transition in regular lat- tices is of first order, such as for the q-state Potts model for sufficiently large value of q. Putting these models on a complex network the inhomogeneities of the lattice play the rôle of some kind of disorder and it is expected that the value of the latent heat is reduced or even the tran- sition is smoothened to a continuous one. This type of scenario is indeed found in a mean-field treatment17, in which the transition is of first-order for γ > γ(q) and be- comes continuous for γc < γ < γ(q), where 3 < γ(q) < 4. On the other hand in an effective medium Bethe lattice approach one has obtained γ(q) = 3, thus the unconven- tional mean-field regime is absent in this treatment18. The interactions considered so far were homogeneous, however, in realistic situations the disorder is inevitable, which has a strong influence on the properties of the http://arxiv.org/abs/0704.1538v1 phase transition. In regular lattices and for a second- order transition Harris-type relevance-irrelevance crite- rion can be used to decide about the stability of the pure system’s fixed point in the presence of weak disorder. On the contrary for a first-order transition such type of cri- terion does not exist. In this case rigorous results asserts that in two dimensions (2d) for any type of continuous disorder the originally first order transition softens into a second order one21. In three dimensions there are numer- ical investigations which have shown22,23,24,25,26 that this kind of softening takes place only for sufficiently strong disorder. In this paper we consider interacting models with ran- dom interactions on complex networks and in this way we study the combined effect of network topology and bond disorder. The particular model we consider is the ran- dom bond ferromagnetic Potts model (RBPM) for large value of q. This model besides its relevance in ordering- disordering phenomena and phase transitions has an exact relation with an optimal cooperation problem27. This mapping is based on the observation that in the large-q limit the thermodynamic properties of the sys- tem are dominated by one single diagram28 of the high- temperature expansion29 and its calculation is equiva- lent to the solution of an optimization problem. This optimization problem can be interpreted in terms of co- operating agents which try to maximize the total sum of benefits received for pair cooperations plus a unit sup- port which is paid for each independent projects. For a given realization of the interactions the optimal state is calculated exactly by a combinatorial optimization algo- rithm which works in strongly polynomial time27. The optimal graph of this problem consists of connected com- ponents (representing sets of cooperating agents) and isolated sites and its temperature (support) dependent topology contains information about the collective be- havior of the agents. In the thermodynamic limit one expects to have a sharp phase transition in the system, which separates the ordered (cooperating) phase with a giant clusters from a disordered (non-cooperating) phase, having only clusters of finite extent. The structure of the paper is the following. The model and the optimization method used in the study for large q is presented in Sec. II. The solution for homogeneous non-random evolving networks can be found in Sec. III, whereas numerical study of the random model on the Barabási-Albert network is presented in Sec. IV. Our results are discussed in Sec. V. II. THE MODEL AND ITS RELATION WITH OPTIMAL COOPERATION The q-state Potts model30 is defined by the Hamilto- nian: H = − 〈i,j〉 Jijδ(σi, σj) (1) in terms of the Potts-spin variables, σi = 0, 1, · · · , q − 1. Here i and j are sites of a lattice, which is represented by a complex network in our case and the summation runs over nearest neighbors, i.e. pairs of connected sites. The couplings, Jij > 0, are ferromagnetic and they are either identical, Jij = J , which is the case of homo- geneous networks, or they are identically and indepen- dently distributed random variables. In this paper we use a quasi-continuous distribution: P (Jij) = 1 + ∆ 2i− l − 1 − Jij which consists of large number of l equally spaced discrete values within the range J(1 ± ∆/2) and 0 ≤ ∆ ≤ 2 measures the strength of disorder. For a given set of couplings the partition function of the system is convenient to write in the random cluster representation29 as: qc(G) qβJij − 1 where the sum runs over all subset of bonds, G and c(G) stands for the number of connected components of G. In Eq. (3) we use the reduced temperature, T → T ln q and its inverse β → β/ ln q, which are of O(1) even in the large-q limit31. In this limit we have qβJij ≫ 1 and the partition function can be written as qφ(G), φ(G) = c(G) + β Jij (4) which is dominated by the largest term, φ∗ = maxG φ(G). Note that this graph, which is called the optimal set, generally depends on the temperature. The free-energy per site is proportional to φ∗ and given by −βf = φ∗/N where N stands for the number of sites of the lattice. As already mentioned in the introduction the optimiza- tion in Eq. (4) can be interpreted as an optimal coopera- tion problem27 in which the agents, which cooperate with each other in some projects, form connected components. Each cooperating pair receives a benefit represented by the weight of the connecting edge (which is proportional to the inverse temperature) and also there is a unit sup- port to each component, i.e. for each projects. Thus by uniting two projects the support will be reduced but at the same time the edge benefits will be enhanced. Finally one is interested in the optimal form of cooperation when the total value of the project grants is maximal. In a mathematical point of view the cost-function in Eq. (4), −φ(G), is sub-modular32 and there is an effi- cient combinatorial optimization algorithm which calcu- lates the optimal set (i.e. set of bonds which minimizes the cost-function) exactly at any temperature in strongly polynomial time27. In the algorithm the optimal set is calculated iteratively and at each step one new vertex of the lattice is taken into account. Having the optimal set at a given step, say with n vertices, its connected compo- nents have the property to contain all the edges between their sites. Due to the submodularity of −φ(G) each con- nected component is contracted into a new vertex with effective weights being the sum of individual weights in the original representation. Now adding a new vertex one should solve the optimization problem in terms of the ef- fective vertices, which needs the application of a standard maximum flow algorithm, since any contractions should include the new vertex. After making the possible new contractions one repeats the previous steps until all the vertices are taken into account and the optimal set of the problem is found. This method has already been applied for 2d31,33 and 3d25,26 regular lattices with short range random inter- actions. As a general result the optimal graph at low temperatures is compact and the largest connected sub- graph contains a finite fraction of the sites, m(T ), which is identified by the orderparameter of the system. In the other limit, for high temperature, most of the sites in the optimal set are isolated and the connected clusters have a finite extent, the typical size of which is used to define the correlation length, ξ. Between the two phases there is a sharp phase transition in the thermodynamic limit, the order of which depends on the dimension of the lattice and the strength of disorder, ∆. In the following the optimization problem is solved ex- actly for homogeneous evolving networks in Sec.III and studied numerically in random Barabási-Albert networks in Sec.IV. III. EXACT SOLUTION FOR HOMOGENEOUS EVOLVING NETWORKS In regular d-dimensional lattices the solution of the optimization problem in Eq. (4) is simple, since there are only two distinct optimal sets, which correspond to the T = 0 and T → ∞ solutions, respectively. For T < Tc(0) it is the fully connected diagram, E, with a free- energy: −βNf = 1 + NβJd and for T > Tc(0) it is the empty diagram, Ø, with −βNf = N . In the proof we make use of the fact, that any edge of a regular lattice, e1, can be transformed to any another edge, e2, through operations of the automorphy group of the lattice. Thus if e1 belongs to some optimal set, then e2 belongs to an optimal set, too. Furthermore, due to submodularity the union of optimal sets is also an optimal set, from which follows that at any temperature the optimal set is either Ø or E. By equating the free energies in the two phases we obtain for the position of the transition point: Tc(0) = Jd/(1 − 1/N) whereas the latent heat is maximal: ∆e/Tc(0) = 1− 1/N . In the following we consider the optimization problem in homogeneous evolving networks which are generated by the following rules: • we start with a complete graph with 2µ vertices • at each timestep we add a new vertex • which is connected to µ existing vertices. In definition of these networks there is no restric- tion in which way the µ existing vertices are selected. These could be chosen randomly, as in the Erdős-Rényi model34, or one can follow some defined rule, like the preferential attachment in the BA network7. In the following we show that for such networks the phase- transition point is located at Tc(0) = Jµ and for T < Tc(0) (T > Tc(0)) the optimal set is the fully connected diagram (empty diagram), as for the regular lattices. Furthermore, the latent heat is maximal: ∆e/Tc(0) = 1. In the proof we follow the optimal cooperation algorithm27 outlined in Sec.II, and in application of the algorithm we add the vertices one by one in the same or- der as in the construction of the network. First we note, that the statement is true for the initial graph, which is a complete graph, thus the optimal set can be either fully connected, having a free-energy: −β2µf = 1 + µ(2µ − 1)βJd, or empty, having a free-energy: −β2µf = 2µ, thus the transition point is indeed at T = Tc(0). We suppose then that the property is satisfied after n steps and add a new vertex, v0. Here we investigate the two cases, T ≤ Tc(0) and T ≥ Tc(0) separately. • If T ≤ Tc(0), then according to our statement all vertices of the original graph are contracted into a single vertex, s, which has an effective weight, µ × J/T > µJ/Tc(0) = 1, to the new vertex v0. Consequently in the optimal set s and v0 are con- nected, in accordance with our statement. • If T ≥ Tc(0), then all vertices of the original graph are disconnected, which means that for any subset, S, having ns ≤ n vertices and es edges one has: ns ≥ esJT + 1. Let us denote by µs ≤ µ the number of edges between v0 and the vertices of S. One has µsJ/T ≤ µJ/T ≤ µJ/Tc(0) = 1, so that for the composite S+ v0 we have: ns+1 ≥ esJT + 1+µsJ/T , which proves that the vertex v0 will not be connected to any subset S and thus will not be contracted to any vertex. This result, i.e. a maximally first-order transition of the large-q state Potts model holds for a wide class of evolving networks, which satisfy the construction rules presented above. This is true, among others, for ran- domly selected sites, for the BA evolving network which has a degree exponent γ = 3 and for several generaliza- tions of the BA network1 including nonlinear preferential attachment, initial attractiveness, etc. In these latter network models the degree exponent can vary in a range of 2 < γ < ∞. It is interesting to note that for uncorre- lated random networks with a given degree distribution the q-state Potts model is in the ordered phase17,18 for any γ ≤ 3. This is in contrast to evolving networks in which correlations in the network sites results in the ex- istence of a disordered phase for T > Tc(0), at least for large q. 3.12.92.72.5 Tc(∞) N=256 N=512 N=1024 N=2048 N=4096 0 0.5 1 1.5 2 2.5 3 ∆=0.2 ∆=0.8 ∆=1.4 ∆=2.0 FIG. 1: (Color online) Temperature dependence of the av- erage magnetization in a BA network of N = 1024 sites for different strength of the disorder, ∆. At T = Tc(0) = 2 the magnetization is independent of ∆ > 0 and its value is in- dicated by an arrow. Inset: The average magnetization for uniform disorder, ∆ = 2, close to the transition point for dif- ferent finite sizes. The arrow indicates the critical point of the infinite system. IV. NUMERICAL STUDY OF RANDOM BARABÁSI-ALBERT NETWORKS In this section we study the large-q state Potts model in the BA network with a given value of the connectiv- ity, µ = 2, and the size of the network varies between N = 26 to N = 212. The interactions are independent random variables taken from the quasi-continuous distri- bution in Eq.(2) having l = 1024 discrete peaks and we fix J = 1. The advantage of using quasi-continuous distri- butions is that in this way we avoid extra, non-physical singularities, which could appear for discrete (e.q. bi- modal) distributions31. For a given size we have gener- ated 100 independent networks and for each we have 100 independent realizations of the disordered couplings. A. Magnetization and structure of the optimal set In Fig.1 the temperature dependence of the average magnetization is shown for various strength of disorder, ∆, for a BA network ofN = 1024 sites. It is seen that the sharp first-order phase transition of the pure system with ∆ = 0 is rounded and the magnetization has considerable variation within a temperature range of ∼ ∆. The phase transition seems to be continuous even for weak disorder. Close to the transition point the magnetization curves for uniform disorder (∆ = 2) are presented in the inset of Fig.1, which are calculated for different finite systems. Some features of the magnetization curves and the properties of the phase transition can be understood by analysing the structure of the optimal set. For low enough temperature this optimal set is fully connected, i.e. the magnetization is m = 1, which happens for T < Tc(0)−∆. Indeed, the first sites with k = µ = 2 (i.e. those which have only outgoing edges) are removed from the fully connected diagram, if the sum of the connected bonds is i=1 Ji < T , which happens within the tem- perature range indicated above. From a similar analysis follows that the optimal set is empty for any finite system for T > Tc(0) + ∆. The magnetization can be estimated for t = T − (Tc(0)−∆) ≪ 1, and the correction is given by: 1−m ∼ tµ. For the numerically studied model with µ = 2 and ∆ = 2, we have m(T ) ≈ 1 − T 2/8, which is indeed a good approximation for T < 1. In the tempera- ture range Tc(0)−∆ < T < Tc(0)+∆ typically the sites are either isolated or belong to the largest cluster. There are also some clusters with an intermediate size, which are dominantly two-site clusters for T < Tc(0) and their fraction is less then 1%, as shown in Fig.2. The fraction of two-site clusters for ∆ = 2 and T < Tc(0) = 2 can be estimated as follows. First, we note that since they are not part of the biggest cluster they can be taken out of a fraction of p1 = 1 − m(T ) sites. Before being dis- connected a two-site cluster has typically three bonds to the biggest cluster, denoted by J1, J2 and J3. When it becomes disconnected we have J1 + J2 + J3 < T , which happens with a probability p2 = T 3/48. At the same time the coupling within the two-site cluster should be J4 > T , which happens with probability p3 = (2− T )/2. Thus the fraction of two-site clusters is approximately: n2 ≈ p1 × p2 × p3 ≈ T 5(2− T )/768, which describes well the general behavior of the distribution in Fig.2. In the temperature range T > Tc(0) the intermedi- ate clusters have at least three sites and their fraction is negligible, which is seen in Fig.2. Consequently the in- termediate size clusters do not influence the properties of the phase transition in the system. In the ordered phase, T < Tc, the largest connected cluster contains a finite fraction of m(T ) < 1 of the sites. We have analyzed the degree distribution of this connected giant cluster in Fig. 3, which has scale-free behavior and for any tempera- ture T < Tc there is the same degree exponent, γ = 3, as for the original BA network. We note an interesting feature of the magnetization curves in Fig.1 that cross each other at the transition point of the pure system, at Tc(0) = 2, having a value of m(Tc(0)) = 0.58, for any strength of disorder. This property follows from the fact that for a given realization of the disorder the optimal set at T = Tc(0) only depends on the sign of the sum of fluctuations of given couplings (c.f. some set of sites is connected (disconnected) to the giant cluster only for positive (negative) accumulated fluctuations) and does not depend on the actual value of ∆ > 0. We can thus conclude the following picture about the evaluation of the optimal set. This is basically one large connected cluster with N sites, immersed in the see of isolated vertices. With increasing temperature more and more loosely connected sites are dissolved from the clus- ter, but for T < Tc we have N/N = m(T ) > 0 and the N=128 N=256 N=512 N=1024 FIG. 2: (Color online) Fraction of intermediate size clusters as a function of the temperature. 0 1 2 3 4 5 6 ln(k) T=1.957 T=2.464 T=2.738 T=2.816 T=2.894 T=2.972 T=3.030 T=3.089 T=3.128 FIG. 3: (Color online) Degree distribution of the largest cluster at different temperatures in a finite network with N = 2048. The dashed straight line indicates the range of the critical temperature. cluster has the same type of scale-free character as the underlaying network. On the contrary above the phase- transition point, Tc(0) + ∆ > T > Tc, the large clus- ter has only a finite extent, N < ∞. The order of the transition depends on the way how N behaves close to Tc. A first-order transition, i.e. phase-cooexistence at Tc does not fit to the above scenario. Indeed, as long as N ∼ N the same type of continuous erosion of the large cluster should take place, i.e. the transition is of second order for any strength, ∆ > 0 of the disorder. Approaching the critical point one expects the following singularities: m(T ) ∼ (Tc − T ) β and N ∼ (T − Tc) −ν′ . Finally, at T = Tc the large cluster has N ∼ N 1−x sites, with x = β/ν′. B. Distribution of the finite-size transition temperatures The first step in the study of the critical singularities is to locate the position of the phase-transition point. In this respect it is not convenient to use the magne- tization, which approaches zero very smoothly, see the inset of Fig.1, so that there is a relatively large error by calculating Tc in this way. One might have, however, a better estimate by defining for each given sample, say α, a finite-size transition temperature Tc(α,N), as has been made for regular lattices25,26,31. For a network we use a condition for the size of the connected component: N (T ) ≃ AN1−x , in which x is the magnetization critical exponent and A = O(1) is a free parameter, from which the scaling form of the distribution is expected to be in- dependent. The calculation is made self-consistently: for a fixed A and a starting value of xs = x1 we have deter- mined the distribution of the finite-size transition tem- peratures and at their average value we have obtained an estimate for the exponent, x = x2. Then the whole procedure is repeated with xs = x2, etc. until a good convergence is obtained. Fortunatelly the distribution function, p(Tc, N), has only a weak x-dependence thus it was enough to make only two iterations. We have started with a logarithmic initial condition, N (T ) ≃ A lnN , which means formally x1 = 1 and we have obtained x2 = .69. Then in the next step the critical exponents are converged within the error of the calculation and they are found to be independent of the value of A, which has been set to be A = 1, 2 and 3. The distribution of the finite-size critical temperatures calculated with x2 = 0.69 and A = 3. are shown in Fig.4 for different sizes of the network. One can observe a shift of the position of the maxima as well as a shrinking of the width of the distribution with increasing size of the network. The shift of the average value, T avc (N), is asymptotically given by: T avc (N)− Tc(∞) ∼ N −1/ν̃′ , (5) whereas the width, characterized by the mean standard deviation, ∆Tc(N), scales with another exponent, ν ′, as: ∆Tc(N) ∼ N −1/ν′ . (6) Using Eq.(5) from a three-point fit we have obtained ν̃′ = 3.8(2) and Tc(∞) = 3.03(2). We have determined the position of the transition point in the infinite sys- tem, Tc(∞), in another way by plotting the difference T avc (N) − Tc(∞) vs. N in a log-log scale for different values of Tc(∞), see Fig.5. At the true transition point according to Eq.(5) there is an asymptotic linear depen- dence, which is indeed the case around Tc(∞) = 3.03(2) and the slope of the line is compatible with 1/ν̃′ = .27(1). For the width exponent, ν′, we obtained from Eq.(6) with two-point fit the estimate: ν′ = 5.6(2). With these parameters the data in Fig.4 can be collapsed to a master curve as shown in the inset of Fig.4. This 1.8 2 2.2 2.4 2.6 2.8 3 3.2 N=128 N=256 N=512 N=1024 N=2048 N=4096 -3 -2 -1 0 1 2 3 4 5 FIG. 4: (Color online) Distribution of the finite-size transi- tion temperatures for different sizes of the BA network. In- set: scaling collapse of the data in terms of t = (Tc(N) − (N))/∆Tc(N), using the scaling form in Eqs.(5) and (6) with ν′ = 3.8(2), and ν̃′ = 5.6(2). 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 ln(N) Tc=3.17 Tc=3.14 Tc=3.11 Tc=3.08 Tc=3.05 Tc=3.03 Tc=3.01 Tc=2.98 Tc=2.96 FIG. 5: (Color online) Shift of the average finite-size transi- tion temperatures, T av (N) − Tc(∞), vs. N in a log-log scale plotted for different values of Tc(∞). The lines connecting the points at the same Tc(∞) are guide for the eye. At the true transition point the asymptotic behavior is linear which is indicated by a dotted straight line. master curve looks not symmetric, at least for the fi- nite sizes used in the present calculation, and can be well fitted by a modified Gumbel distribution, Gω(−y) = ωω/Γ(ω)(exp(−y−e−y))ω , with a parameter ω = 4.2. We note that the same type of fitting curve has already been used in Ref.35. For another values of the initial parame- ter, A = 1 and 2 the estimates of the critical exponents as well as the position of the transition point are found to be stable and stand in the range indicated by the error bars. The equations in Eqs.(5) and (6) are generalizations of the relations obtained in regular d-dimensional lat- tices36,37,38,39,40 in which N is replaced by Ld, L being the linear size of the system and therefore instead of ν′ and ν̃′ we have ν = ν′/d and ν̃ = ν̃′/d, respectively. Gen- erally at a random fixed point the two characteristic ex- ponents are equal and satisfy the relation41 ν′ = ν̃′ ≥ 2. This has indeed been observed for the 2d31 and 3d25,26 random bond Potts models for large q at disorder induced critical points. On the other hand if the transition stays first-order there are two distinct exponents35,42 ν̃′ = 1 and ν′ = 2. Interestingly our results on the distribution of the finite-size transition temperatures in networks are differ- ent of those found in regular lattices. Here the transition is of second order but still there are two distinct critical exponents, which are completely different of that at a disordered first-order transition. For our system ν′ > ν̃′, which means that disorder fluctuations in the critical point are dominant over deterministic shift of the tran- sition point. Similar trend is observed about the finite- size transition parameters in the random transverse-field Ising model43, the critical behavior of which is controlled by an infinite disorder fixed point. In this respect the RBPM in scale-free networks can be considered as a new realization of an infinite disorder fixed point. C. Size of the critical cluster Having the distribution of the finite-size transition temperatures we have calculated the size of the largest cluster at T avc (N), which is expected to scale as N [N, T avc (N)] ∼ N 1−x. Then from two-point fit we have obtained an estimate for the magnetization expo- nent: x = .66(1). We have also plotted N [N, T avc (N)] vs. N1−x in Fig.6 for different initial parameters, A. Here we have obtained an asymptotic linear dependence with an exponent, x = .65(1), which agrees with the previous value within the error of the calculation. V. DISCUSSION IN TERMS OF OPTIMAL COOPERATION In this paper we have studied the properties of the Potts model for large value of q on scale-free evolving complex networks, such as the BA network, both for ho- mogeneous and random ferromagnetic couplings. This problem is equivalent to an optimal cooperation prob- lem in which the agents try to optimize the total sum of the benefits coming from pair cooperations (represented by the Potts couplings) and the total sum of the support which is the same for each cooperating projects (given by the temperature of the Potts model). The homogeneous problem is shown exactly to have two distinct states: ei- ther all the agents cooperate with each other or there is no cooperation between any agents. There is a strongly first-order phase transition: by increasing the support the agents stop cooperating at a critical value. 0 10 20 30 40 50 [0 , 0] 0 10 20 30 40 50 [0 , 0] FIG. 6: (Color online) Size dependence of the critical cluster at the average finite-size critical temperature as a function of N1−x with x = .65. The date points for different initial parameters, A, are well described by straight lines, which are guides to the eye. In the random problem, in which the benefits are ran- dom and depend on the pairs of the cooprerating agents, the structure of the optimal set depends on the value of the support. Typically the agents are of two kinds: a frac- tion of m belongs to a large cooperating cluster whereas the others are isolated, representing one man’s projects. With increasing support more and more agents are split off the cluster, thus its size, as well as m is decreasing, but the cluster keeps its scale-free topology. For a critical value of the support m goes to zero continuously and the corresponding singularity is characterized by non-trivial critical exponents. This transition, as shown by the nu- merically calculated critical exponents for the BA net- work, belongs to a new universality class. 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We consider the ferromagnetic large-$q$ state Potts model in complex evolving networks, which is equivalent to an optimal cooperation problem, in which the agents try to optimize the total sum of pair cooperation benefits and the supports of independent projects. The agents are found to be typically of two kinds: a fraction of $m$ (being the magnetization of the Potts model) belongs to a large cooperating cluster, whereas the others are isolated one man's projects. It is shown rigorously that the homogeneous model has a strongly first-order phase transition, which turns to second-order for random interactions (benefits), the properties of which are studied numerically on the Barab\'asi-Albert network. The distribution of finite-size transition points is characterized by a shift exponent, $1/\tilde{\nu}'=.26(1)$, and by a different width exponent, $1/\nu'=.18(1)$, whereas the magnetization at the transition point scales with the size of the network, $N$, as: $m\sim N^{-x}$, with $x=.66(1)$.

Rounding of first-order phase transitions and optimal cooperation in scale-free networks M. Karsai,1,2 J-Ch. Anglès d’Auriac,2 and F. Iglói3, 1 Institute of Theoretical Physics, Szeged University, H-6720 Szeged, Hungary Centre de Recherches sur les Trés Basses Températures, B. P. 166, F-38042 Grenoble, France Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O.Box 49, Hungary (Dated: November 20, 2018) We consider the ferromagnetic large-q state Potts model in complex evolving networks, which is equivalent to an optimal cooperation problem, in which the agents try to optimize the total sum of pair cooperation benefits and the supports of independent projects. The agents are found to be typically of two kinds: a fraction of m (being the magnetization of the Potts model) belongs to a large cooperating cluster, whereas the others are isolated one man’s projects. It is shown rigorously that the homogeneous model has a strongly first-order phase transition, which turns to second-order for random interactions (benefits), the properties of which are studied numerically on the Barabási- Albert network. The distribution of finite-size transition points is characterized by a shift exponent, 1/ν̃′ = .26(1), and by a different width exponent, 1/ν′ = .18(1), whereas the magnetization at the transition point scales with the size of the network, N , as: m ∼ N−x, with x = .66(1). I. INTRODUCTION Complex networks have been used to describe the structure and topology of a large class of systems in dif- ferent fields of science, technics, transport, social and political life, etc, see Refs.1,2,3,4 for recent reviews. A complex network is represented by a graph5, in which the nodes stand for the agents and the edges denote the possi- ble interactions. Realistic networks generally have three basic properties. The average distance between the nodes is small, which is called the small-world effect6. There is a tendency of clustering and the degree-distribution of the edges, P (k), has a power-law tail, PD(k) ≃ Ak −γ , k ≫ 1. Thus the edge distribution is scale free7, which is usually attributed to growth and preferential attachment during the evaluation of the network. In reality there is some sort of interaction between the agents of a network which leads to some kind of coop- erative behavior in macroscopic scales. One throughly studied question in this field is the spread of infections and epidemics in networks8,9,10, which problem is closely related to another non-equilibrium processes, such as percolation11, diffusion12, the contact process13 or the zero-range process14, etc. In another investigations one considers simple magnetic models15,16,17,18,19, in which the agents are represented by classical (Ising or Potts) spin variables, the interactions are described by ferro- magnetic couplings, whereas the temperature plays the rôle of a disordering field. In theoretical investigations of the cooperative behav- ior one usually resort on some kind of approximations. For example the sites of the networks are often consid- ered uncorrelated, which is generally not true for evolv- ing networks, such as the Barabási-Albert (BA) network. However this effect is expected to be irrelevant, as far as the singularities in the system are considered. Also the simple mean-field approach could lead to exact results due to long-range interactions in the networks, which has been checked by numerical simulations15 and by another, more accurate theoretical methods19 (Bethe-lattice ap- proach, replica method, etc.). In these calculations the critical behavior of the network is found to depend on the value of the degree exponent, γ. For sufficiently weakly connected networks with γ > γu (γu = 5 for the Ising model) there are conventional mean-field singularities. In the intermediate or unconventional mean-field regime, for γu > γ > γc, the critical exponents are γ dependent. Fi- nally, for γ ≤ γc, when the average of k 2, defined by 〈k2〉 = PD(k)k 2dk, as well as the strength of the aver- age interaction becomes divergent the scale-free network remains in the ordered state at any finite temperature. Since γc = 3, in realistic networks with homogeneous interactions always this type of phenomena is expected to occur. In weighted networks, however, in which the strengths of the interaction is appropriately rescaled with the degrees of the connected vertices, γc is shifted to larger values and therefore the complete phase-transition scenario can be tested13,19. We note that the properties of the phase transitions are generally different for undi- rected (as we consider here) and directed networks20. In several models the phase transition in regular lat- tices is of first order, such as for the q-state Potts model for sufficiently large value of q. Putting these models on a complex network the inhomogeneities of the lattice play the rôle of some kind of disorder and it is expected that the value of the latent heat is reduced or even the tran- sition is smoothened to a continuous one. This type of scenario is indeed found in a mean-field treatment17, in which the transition is of first-order for γ > γ(q) and be- comes continuous for γc < γ < γ(q), where 3 < γ(q) < 4. On the other hand in an effective medium Bethe lattice approach one has obtained γ(q) = 3, thus the unconven- tional mean-field regime is absent in this treatment18. The interactions considered so far were homogeneous, however, in realistic situations the disorder is inevitable, which has a strong influence on the properties of the http://arxiv.org/abs/0704.1538v1 phase transition. In regular lattices and for a second- order transition Harris-type relevance-irrelevance crite- rion can be used to decide about the stability of the pure system’s fixed point in the presence of weak disorder. On the contrary for a first-order transition such type of cri- terion does not exist. In this case rigorous results asserts that in two dimensions (2d) for any type of continuous disorder the originally first order transition softens into a second order one21. In three dimensions there are numer- ical investigations which have shown22,23,24,25,26 that this kind of softening takes place only for sufficiently strong disorder. In this paper we consider interacting models with ran- dom interactions on complex networks and in this way we study the combined effect of network topology and bond disorder. The particular model we consider is the ran- dom bond ferromagnetic Potts model (RBPM) for large value of q. This model besides its relevance in ordering- disordering phenomena and phase transitions has an exact relation with an optimal cooperation problem27. This mapping is based on the observation that in the large-q limit the thermodynamic properties of the sys- tem are dominated by one single diagram28 of the high- temperature expansion29 and its calculation is equiva- lent to the solution of an optimization problem. This optimization problem can be interpreted in terms of co- operating agents which try to maximize the total sum of benefits received for pair cooperations plus a unit sup- port which is paid for each independent projects. For a given realization of the interactions the optimal state is calculated exactly by a combinatorial optimization algo- rithm which works in strongly polynomial time27. The optimal graph of this problem consists of connected com- ponents (representing sets of cooperating agents) and isolated sites and its temperature (support) dependent topology contains information about the collective be- havior of the agents. In the thermodynamic limit one expects to have a sharp phase transition in the system, which separates the ordered (cooperating) phase with a giant clusters from a disordered (non-cooperating) phase, having only clusters of finite extent. The structure of the paper is the following. The model and the optimization method used in the study for large q is presented in Sec. II. The solution for homogeneous non-random evolving networks can be found in Sec. III, whereas numerical study of the random model on the Barabási-Albert network is presented in Sec. IV. Our results are discussed in Sec. V. II. THE MODEL AND ITS RELATION WITH OPTIMAL COOPERATION The q-state Potts model30 is defined by the Hamilto- nian: H = − 〈i,j〉 Jijδ(σi, σj) (1) in terms of the Potts-spin variables, σi = 0, 1, · · · , q − 1. Here i and j are sites of a lattice, which is represented by a complex network in our case and the summation runs over nearest neighbors, i.e. pairs of connected sites. The couplings, Jij > 0, are ferromagnetic and they are either identical, Jij = J , which is the case of homo- geneous networks, or they are identically and indepen- dently distributed random variables. In this paper we use a quasi-continuous distribution: P (Jij) = 1 + ∆ 2i− l − 1 − Jij which consists of large number of l equally spaced discrete values within the range J(1 ± ∆/2) and 0 ≤ ∆ ≤ 2 measures the strength of disorder. For a given set of couplings the partition function of the system is convenient to write in the random cluster representation29 as: qc(G) qβJij − 1 where the sum runs over all subset of bonds, G and c(G) stands for the number of connected components of G. In Eq. (3) we use the reduced temperature, T → T ln q and its inverse β → β/ ln q, which are of O(1) even in the large-q limit31. In this limit we have qβJij ≫ 1 and the partition function can be written as qφ(G), φ(G) = c(G) + β Jij (4) which is dominated by the largest term, φ∗ = maxG φ(G). Note that this graph, which is called the optimal set, generally depends on the temperature. The free-energy per site is proportional to φ∗ and given by −βf = φ∗/N where N stands for the number of sites of the lattice. As already mentioned in the introduction the optimiza- tion in Eq. (4) can be interpreted as an optimal coopera- tion problem27 in which the agents, which cooperate with each other in some projects, form connected components. Each cooperating pair receives a benefit represented by the weight of the connecting edge (which is proportional to the inverse temperature) and also there is a unit sup- port to each component, i.e. for each projects. Thus by uniting two projects the support will be reduced but at the same time the edge benefits will be enhanced. Finally one is interested in the optimal form of cooperation when the total value of the project grants is maximal. In a mathematical point of view the cost-function in Eq. (4), −φ(G), is sub-modular32 and there is an effi- cient combinatorial optimization algorithm which calcu- lates the optimal set (i.e. set of bonds which minimizes the cost-function) exactly at any temperature in strongly polynomial time27. In the algorithm the optimal set is calculated iteratively and at each step one new vertex of the lattice is taken into account. Having the optimal set at a given step, say with n vertices, its connected compo- nents have the property to contain all the edges between their sites. Due to the submodularity of −φ(G) each con- nected component is contracted into a new vertex with effective weights being the sum of individual weights in the original representation. Now adding a new vertex one should solve the optimization problem in terms of the ef- fective vertices, which needs the application of a standard maximum flow algorithm, since any contractions should include the new vertex. After making the possible new contractions one repeats the previous steps until all the vertices are taken into account and the optimal set of the problem is found. This method has already been applied for 2d31,33 and 3d25,26 regular lattices with short range random inter- actions. As a general result the optimal graph at low temperatures is compact and the largest connected sub- graph contains a finite fraction of the sites, m(T ), which is identified by the orderparameter of the system. In the other limit, for high temperature, most of the sites in the optimal set are isolated and the connected clusters have a finite extent, the typical size of which is used to define the correlation length, ξ. Between the two phases there is a sharp phase transition in the thermodynamic limit, the order of which depends on the dimension of the lattice and the strength of disorder, ∆. In the following the optimization problem is solved ex- actly for homogeneous evolving networks in Sec.III and studied numerically in random Barabási-Albert networks in Sec.IV. III. EXACT SOLUTION FOR HOMOGENEOUS EVOLVING NETWORKS In regular d-dimensional lattices the solution of the optimization problem in Eq. (4) is simple, since there are only two distinct optimal sets, which correspond to the T = 0 and T → ∞ solutions, respectively. For T < Tc(0) it is the fully connected diagram, E, with a free- energy: −βNf = 1 + NβJd and for T > Tc(0) it is the empty diagram, Ø, with −βNf = N . In the proof we make use of the fact, that any edge of a regular lattice, e1, can be transformed to any another edge, e2, through operations of the automorphy group of the lattice. Thus if e1 belongs to some optimal set, then e2 belongs to an optimal set, too. Furthermore, due to submodularity the union of optimal sets is also an optimal set, from which follows that at any temperature the optimal set is either Ø or E. By equating the free energies in the two phases we obtain for the position of the transition point: Tc(0) = Jd/(1 − 1/N) whereas the latent heat is maximal: ∆e/Tc(0) = 1− 1/N . In the following we consider the optimization problem in homogeneous evolving networks which are generated by the following rules: • we start with a complete graph with 2µ vertices • at each timestep we add a new vertex • which is connected to µ existing vertices. In definition of these networks there is no restric- tion in which way the µ existing vertices are selected. These could be chosen randomly, as in the Erdős-Rényi model34, or one can follow some defined rule, like the preferential attachment in the BA network7. In the following we show that for such networks the phase- transition point is located at Tc(0) = Jµ and for T < Tc(0) (T > Tc(0)) the optimal set is the fully connected diagram (empty diagram), as for the regular lattices. Furthermore, the latent heat is maximal: ∆e/Tc(0) = 1. In the proof we follow the optimal cooperation algorithm27 outlined in Sec.II, and in application of the algorithm we add the vertices one by one in the same or- der as in the construction of the network. First we note, that the statement is true for the initial graph, which is a complete graph, thus the optimal set can be either fully connected, having a free-energy: −β2µf = 1 + µ(2µ − 1)βJd, or empty, having a free-energy: −β2µf = 2µ, thus the transition point is indeed at T = Tc(0). We suppose then that the property is satisfied after n steps and add a new vertex, v0. Here we investigate the two cases, T ≤ Tc(0) and T ≥ Tc(0) separately. • If T ≤ Tc(0), then according to our statement all vertices of the original graph are contracted into a single vertex, s, which has an effective weight, µ × J/T > µJ/Tc(0) = 1, to the new vertex v0. Consequently in the optimal set s and v0 are con- nected, in accordance with our statement. • If T ≥ Tc(0), then all vertices of the original graph are disconnected, which means that for any subset, S, having ns ≤ n vertices and es edges one has: ns ≥ esJT + 1. Let us denote by µs ≤ µ the number of edges between v0 and the vertices of S. One has µsJ/T ≤ µJ/T ≤ µJ/Tc(0) = 1, so that for the composite S+ v0 we have: ns+1 ≥ esJT + 1+µsJ/T , which proves that the vertex v0 will not be connected to any subset S and thus will not be contracted to any vertex. This result, i.e. a maximally first-order transition of the large-q state Potts model holds for a wide class of evolving networks, which satisfy the construction rules presented above. This is true, among others, for ran- domly selected sites, for the BA evolving network which has a degree exponent γ = 3 and for several generaliza- tions of the BA network1 including nonlinear preferential attachment, initial attractiveness, etc. In these latter network models the degree exponent can vary in a range of 2 < γ < ∞. It is interesting to note that for uncorre- lated random networks with a given degree distribution the q-state Potts model is in the ordered phase17,18 for any γ ≤ 3. This is in contrast to evolving networks in which correlations in the network sites results in the ex- istence of a disordered phase for T > Tc(0), at least for large q. 3.12.92.72.5 Tc(∞) N=256 N=512 N=1024 N=2048 N=4096 0 0.5 1 1.5 2 2.5 3 ∆=0.2 ∆=0.8 ∆=1.4 ∆=2.0 FIG. 1: (Color online) Temperature dependence of the av- erage magnetization in a BA network of N = 1024 sites for different strength of the disorder, ∆. At T = Tc(0) = 2 the magnetization is independent of ∆ > 0 and its value is in- dicated by an arrow. Inset: The average magnetization for uniform disorder, ∆ = 2, close to the transition point for dif- ferent finite sizes. The arrow indicates the critical point of the infinite system. IV. NUMERICAL STUDY OF RANDOM BARABÁSI-ALBERT NETWORKS In this section we study the large-q state Potts model in the BA network with a given value of the connectiv- ity, µ = 2, and the size of the network varies between N = 26 to N = 212. The interactions are independent random variables taken from the quasi-continuous distri- bution in Eq.(2) having l = 1024 discrete peaks and we fix J = 1. The advantage of using quasi-continuous distri- butions is that in this way we avoid extra, non-physical singularities, which could appear for discrete (e.q. bi- modal) distributions31. For a given size we have gener- ated 100 independent networks and for each we have 100 independent realizations of the disordered couplings. A. Magnetization and structure of the optimal set In Fig.1 the temperature dependence of the average magnetization is shown for various strength of disorder, ∆, for a BA network ofN = 1024 sites. It is seen that the sharp first-order phase transition of the pure system with ∆ = 0 is rounded and the magnetization has considerable variation within a temperature range of ∼ ∆. The phase transition seems to be continuous even for weak disorder. Close to the transition point the magnetization curves for uniform disorder (∆ = 2) are presented in the inset of Fig.1, which are calculated for different finite systems. Some features of the magnetization curves and the properties of the phase transition can be understood by analysing the structure of the optimal set. For low enough temperature this optimal set is fully connected, i.e. the magnetization is m = 1, which happens for T < Tc(0)−∆. Indeed, the first sites with k = µ = 2 (i.e. those which have only outgoing edges) are removed from the fully connected diagram, if the sum of the connected bonds is i=1 Ji < T , which happens within the tem- perature range indicated above. From a similar analysis follows that the optimal set is empty for any finite system for T > Tc(0) + ∆. The magnetization can be estimated for t = T − (Tc(0)−∆) ≪ 1, and the correction is given by: 1−m ∼ tµ. For the numerically studied model with µ = 2 and ∆ = 2, we have m(T ) ≈ 1 − T 2/8, which is indeed a good approximation for T < 1. In the tempera- ture range Tc(0)−∆ < T < Tc(0)+∆ typically the sites are either isolated or belong to the largest cluster. There are also some clusters with an intermediate size, which are dominantly two-site clusters for T < Tc(0) and their fraction is less then 1%, as shown in Fig.2. The fraction of two-site clusters for ∆ = 2 and T < Tc(0) = 2 can be estimated as follows. First, we note that since they are not part of the biggest cluster they can be taken out of a fraction of p1 = 1 − m(T ) sites. Before being dis- connected a two-site cluster has typically three bonds to the biggest cluster, denoted by J1, J2 and J3. When it becomes disconnected we have J1 + J2 + J3 < T , which happens with a probability p2 = T 3/48. At the same time the coupling within the two-site cluster should be J4 > T , which happens with probability p3 = (2− T )/2. Thus the fraction of two-site clusters is approximately: n2 ≈ p1 × p2 × p3 ≈ T 5(2− T )/768, which describes well the general behavior of the distribution in Fig.2. In the temperature range T > Tc(0) the intermedi- ate clusters have at least three sites and their fraction is negligible, which is seen in Fig.2. Consequently the in- termediate size clusters do not influence the properties of the phase transition in the system. In the ordered phase, T < Tc, the largest connected cluster contains a finite fraction of m(T ) < 1 of the sites. We have analyzed the degree distribution of this connected giant cluster in Fig. 3, which has scale-free behavior and for any tempera- ture T < Tc there is the same degree exponent, γ = 3, as for the original BA network. We note an interesting feature of the magnetization curves in Fig.1 that cross each other at the transition point of the pure system, at Tc(0) = 2, having a value of m(Tc(0)) = 0.58, for any strength of disorder. This property follows from the fact that for a given realization of the disorder the optimal set at T = Tc(0) only depends on the sign of the sum of fluctuations of given couplings (c.f. some set of sites is connected (disconnected) to the giant cluster only for positive (negative) accumulated fluctuations) and does not depend on the actual value of ∆ > 0. We can thus conclude the following picture about the evaluation of the optimal set. This is basically one large connected cluster with N sites, immersed in the see of isolated vertices. With increasing temperature more and more loosely connected sites are dissolved from the clus- ter, but for T < Tc we have N/N = m(T ) > 0 and the N=128 N=256 N=512 N=1024 FIG. 2: (Color online) Fraction of intermediate size clusters as a function of the temperature. 0 1 2 3 4 5 6 ln(k) T=1.957 T=2.464 T=2.738 T=2.816 T=2.894 T=2.972 T=3.030 T=3.089 T=3.128 FIG. 3: (Color online) Degree distribution of the largest cluster at different temperatures in a finite network with N = 2048. The dashed straight line indicates the range of the critical temperature. cluster has the same type of scale-free character as the underlaying network. On the contrary above the phase- transition point, Tc(0) + ∆ > T > Tc, the large clus- ter has only a finite extent, N < ∞. The order of the transition depends on the way how N behaves close to Tc. A first-order transition, i.e. phase-cooexistence at Tc does not fit to the above scenario. Indeed, as long as N ∼ N the same type of continuous erosion of the large cluster should take place, i.e. the transition is of second order for any strength, ∆ > 0 of the disorder. Approaching the critical point one expects the following singularities: m(T ) ∼ (Tc − T ) β and N ∼ (T − Tc) −ν′ . Finally, at T = Tc the large cluster has N ∼ N 1−x sites, with x = β/ν′. B. Distribution of the finite-size transition temperatures The first step in the study of the critical singularities is to locate the position of the phase-transition point. In this respect it is not convenient to use the magne- tization, which approaches zero very smoothly, see the inset of Fig.1, so that there is a relatively large error by calculating Tc in this way. One might have, however, a better estimate by defining for each given sample, say α, a finite-size transition temperature Tc(α,N), as has been made for regular lattices25,26,31. For a network we use a condition for the size of the connected component: N (T ) ≃ AN1−x , in which x is the magnetization critical exponent and A = O(1) is a free parameter, from which the scaling form of the distribution is expected to be in- dependent. The calculation is made self-consistently: for a fixed A and a starting value of xs = x1 we have deter- mined the distribution of the finite-size transition tem- peratures and at their average value we have obtained an estimate for the exponent, x = x2. Then the whole procedure is repeated with xs = x2, etc. until a good convergence is obtained. Fortunatelly the distribution function, p(Tc, N), has only a weak x-dependence thus it was enough to make only two iterations. We have started with a logarithmic initial condition, N (T ) ≃ A lnN , which means formally x1 = 1 and we have obtained x2 = .69. Then in the next step the critical exponents are converged within the error of the calculation and they are found to be independent of the value of A, which has been set to be A = 1, 2 and 3. The distribution of the finite-size critical temperatures calculated with x2 = 0.69 and A = 3. are shown in Fig.4 for different sizes of the network. One can observe a shift of the position of the maxima as well as a shrinking of the width of the distribution with increasing size of the network. The shift of the average value, T avc (N), is asymptotically given by: T avc (N)− Tc(∞) ∼ N −1/ν̃′ , (5) whereas the width, characterized by the mean standard deviation, ∆Tc(N), scales with another exponent, ν ′, as: ∆Tc(N) ∼ N −1/ν′ . (6) Using Eq.(5) from a three-point fit we have obtained ν̃′ = 3.8(2) and Tc(∞) = 3.03(2). We have determined the position of the transition point in the infinite sys- tem, Tc(∞), in another way by plotting the difference T avc (N) − Tc(∞) vs. N in a log-log scale for different values of Tc(∞), see Fig.5. At the true transition point according to Eq.(5) there is an asymptotic linear depen- dence, which is indeed the case around Tc(∞) = 3.03(2) and the slope of the line is compatible with 1/ν̃′ = .27(1). For the width exponent, ν′, we obtained from Eq.(6) with two-point fit the estimate: ν′ = 5.6(2). With these parameters the data in Fig.4 can be collapsed to a master curve as shown in the inset of Fig.4. This 1.8 2 2.2 2.4 2.6 2.8 3 3.2 N=128 N=256 N=512 N=1024 N=2048 N=4096 -3 -2 -1 0 1 2 3 4 5 FIG. 4: (Color online) Distribution of the finite-size transi- tion temperatures for different sizes of the BA network. In- set: scaling collapse of the data in terms of t = (Tc(N) − (N))/∆Tc(N), using the scaling form in Eqs.(5) and (6) with ν′ = 3.8(2), and ν̃′ = 5.6(2). 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 ln(N) Tc=3.17 Tc=3.14 Tc=3.11 Tc=3.08 Tc=3.05 Tc=3.03 Tc=3.01 Tc=2.98 Tc=2.96 FIG. 5: (Color online) Shift of the average finite-size transi- tion temperatures, T av (N) − Tc(∞), vs. N in a log-log scale plotted for different values of Tc(∞). The lines connecting the points at the same Tc(∞) are guide for the eye. At the true transition point the asymptotic behavior is linear which is indicated by a dotted straight line. master curve looks not symmetric, at least for the fi- nite sizes used in the present calculation, and can be well fitted by a modified Gumbel distribution, Gω(−y) = ωω/Γ(ω)(exp(−y−e−y))ω , with a parameter ω = 4.2. We note that the same type of fitting curve has already been used in Ref.35. For another values of the initial parame- ter, A = 1 and 2 the estimates of the critical exponents as well as the position of the transition point are found to be stable and stand in the range indicated by the error bars. The equations in Eqs.(5) and (6) are generalizations of the relations obtained in regular d-dimensional lat- tices36,37,38,39,40 in which N is replaced by Ld, L being the linear size of the system and therefore instead of ν′ and ν̃′ we have ν = ν′/d and ν̃ = ν̃′/d, respectively. Gen- erally at a random fixed point the two characteristic ex- ponents are equal and satisfy the relation41 ν′ = ν̃′ ≥ 2. This has indeed been observed for the 2d31 and 3d25,26 random bond Potts models for large q at disorder induced critical points. On the other hand if the transition stays first-order there are two distinct exponents35,42 ν̃′ = 1 and ν′ = 2. Interestingly our results on the distribution of the finite-size transition temperatures in networks are differ- ent of those found in regular lattices. Here the transition is of second order but still there are two distinct critical exponents, which are completely different of that at a disordered first-order transition. For our system ν′ > ν̃′, which means that disorder fluctuations in the critical point are dominant over deterministic shift of the tran- sition point. Similar trend is observed about the finite- size transition parameters in the random transverse-field Ising model43, the critical behavior of which is controlled by an infinite disorder fixed point. In this respect the RBPM in scale-free networks can be considered as a new realization of an infinite disorder fixed point. C. Size of the critical cluster Having the distribution of the finite-size transition temperatures we have calculated the size of the largest cluster at T avc (N), which is expected to scale as N [N, T avc (N)] ∼ N 1−x. Then from two-point fit we have obtained an estimate for the magnetization expo- nent: x = .66(1). We have also plotted N [N, T avc (N)] vs. N1−x in Fig.6 for different initial parameters, A. Here we have obtained an asymptotic linear dependence with an exponent, x = .65(1), which agrees with the previous value within the error of the calculation. V. DISCUSSION IN TERMS OF OPTIMAL COOPERATION In this paper we have studied the properties of the Potts model for large value of q on scale-free evolving complex networks, such as the BA network, both for ho- mogeneous and random ferromagnetic couplings. This problem is equivalent to an optimal cooperation prob- lem in which the agents try to optimize the total sum of the benefits coming from pair cooperations (represented by the Potts couplings) and the total sum of the support which is the same for each cooperating projects (given by the temperature of the Potts model). The homogeneous problem is shown exactly to have two distinct states: ei- ther all the agents cooperate with each other or there is no cooperation between any agents. There is a strongly first-order phase transition: by increasing the support the agents stop cooperating at a critical value. 0 10 20 30 40 50 [0 , 0] 0 10 20 30 40 50 [0 , 0] FIG. 6: (Color online) Size dependence of the critical cluster at the average finite-size critical temperature as a function of N1−x with x = .65. The date points for different initial parameters, A, are well described by straight lines, which are guides to the eye. In the random problem, in which the benefits are ran- dom and depend on the pairs of the cooprerating agents, the structure of the optimal set depends on the value of the support. Typically the agents are of two kinds: a frac- tion of m belongs to a large cooperating cluster whereas the others are isolated, representing one man’s projects. With increasing support more and more agents are split off the cluster, thus its size, as well as m is decreasing, but the cluster keeps its scale-free topology. For a critical value of the support m goes to zero continuously and the corresponding singularity is characterized by non-trivial critical exponents. This transition, as shown by the nu- merically calculated critical exponents for the BA net- work, belongs to a new universality class. 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A New Monte Carlo Method and Its Implications for Generalized Cluster Algorithms C. H. Mak and Arun K. Sharma Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA (Dated: November 13, 2018) We describe a novel switching algorithm based on a “reverse” Monte Carlo method, in which the potential is stochastically modified before the system configuration is moved. This new algorithm facilitates a generalized formulation of cluster-type Monte Carlo methods, and the generalization makes it possible to derive cluster algorithms for systems with both discrete and continuous de- grees of freedom. The roughening transition in the sine-Gordon model has been studied with this method, and high-accuracy simulations for system sizes up to 10242 were carried out to examine the logarithmic divergence of the surface roughness above the transition temperature, revealing clear evidence for universal scaling of the Kosterlitz-Thouless type. PACS numbers: 05.10.Ln, 05.50.+q, 64.60.Ht, 75.40.Mg Large-scale Monte Carlo (MC) simulations are often plagued by slow sampling problems. These problems are especially severe in systems near the critical point or in those with strong correlations. Slow sampling problems manifest themselves as poor scaling of the dynamic relax- ation time with the system size, making large-size sim- ulations extremely slow to converge. The cause of these problems is that most MC simulations are based on lo- cal moves, and when the correlation length of the system grows or as relaxation modes of the system become heav- ily entangled, local moves become increasingly inefficient. But if nonlocal MC moves are used [1], their acceptance ratios are often found to be exceedingly low when system correlations are strong. One way to circumvent these problems was suggested by Swendsen and Wang [2], who devised a clever scheme where large-scale nonlocal MC moves may be constructed to achieve high sampling efficiencies by exploiting certain geometric symmetries in the system. This algorithm led to a marked reduction in the dynamical scaling exponent in the 2-dimensional Ising model near criticality. Since the nonlocal moves in this algorithm update a large num- ber of degrees of freedom at the same time, the Swendsen- Wang method and others inspired by it are also often referred to as “cluster Monte Carlo” methods. Since Swendsen and Wang’s paper in 1987, many cluster-type MC algorithms have appeared [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. But the success of cluster MC has not been universal because the proper cluster moves needed seem to be highly de- pendent on the system, and efficient cluster MC meth- ods have been found for only a small number of models [3, 5, 6, 7, 11, 14, 15, 18, 19] so far. The difficulty of formulating a generalized MC method that could work for any system seems to be associated with the appar- ent geometric nature of the cluster-type MC methods – all existing cluster MC methods have been derived in one way or another by using certain geometric features of the system. For example, in the original Swendsen- Wang formulation a mapping between the Ising model and the percolation model originally described by For- tuin and Kasteleyn [20] was exploited to effect cluster spin flips. In other models, the requisite mapping is not always obvious, making cluster MC methods difficult to implement for general systems. In this letter, we will show that the derivations of clus- ter MC methods do not have to be based on geometric features of the systems. Instead, they may be more con- veniently formulated based on algebraic features of the system potential V (C). We will exploit this algebraic for- mulation and suggest a way to generalize cluster Monte Carlo methods to systems with any potential. We focus on classical systems with partition function Z = Tr e−V (C), where V (C) is the potential in units of the temperature T . Acceptable Monte Carlo methods to sample the system configurations can be constructed using any transition probabilities W (C → C′) as long as the detailed balance condition W (C → C′)e−V (C) = W (C′ → C)e−V (C ′) (1) is satisfied. Conventional MC methods such as Metropo- lis [21] accomplishes this in two steps: a trial move is made from C to C′ with some transition probability, and the move is then accepted or rejected according to an acceptance probability based on V (C′), V (C) or both, so that the composite process satisfies Eqn.(1). This way of constructing the Markov chain – trial moves followed by acceptance/rejection – has been the accepted “stan- dard” method for doing MC since the inception of the MC method [22]. Other MC methods do exist, such as the heat bath algorithm [23], which follow alternative strate- gies, but by far the standard method is conceptually the simplest and most convenient in practice. In the Monte Carlo method we are proposing, we will reverse the order of the two steps in the standard method. That is, we will first determine an acceptable way to mod- ify the potential V and then find a transition C → C′ that is consistent with the new potential. To our knowledge, http://arxiv.org/abs/0704.1539v1 the basic elements of this “reverse MC” idea were first suggested by Kandel et al. [4], who used it to stochasti- cally remove interaction terms from the system’s poten- tial in an Ising model to arrive at an alternative deriva- tion of the Swendsen-Wang method. The formulation of Kandel et al. was limited to discrete systems like the Ising model. In the following, we will show how the re- verse MC idea may be formulated more generally for any system, discrete or continuous, and how it may then be used as a framework to construct generalized cluster al- gorithms. Consider a system with potential V = i vi+Vr, con- sisting of a number of “interaction terms” vi plus a “resid- ual” Vr . These interactions may be bonds between parti- cles, interactions of the particles with a field, or any other additive terms in V . We consider replacing each interac- tion term vi by some pre-selected ṽi with a “switching” probability Si(C) = cie ∆vi(C), (2) where ∆vi = vi− ṽi, ci = e i and ∆v∗i = maxC vi(C). The outcome of the switches defines two complemen- tary sets of interactions – the switched ones σ̃ and the unswitched ones σ̄. Using the outcome of the switches, we define a stochastically modified potential Ṽ as follows: ṽi + v̄j + Vr, (3) with v̄i = vi − ln(1 − Si). An MC pass starts with an attempt to switch every interaction vi to the new ṽi us- ing the Si defined in Eqn.(2). If the switch is successful, the interaction is replaced by ṽi. If not, the interaction is replaced by another interaction v̄i. This is followed by an update in the configuration of the entire system from C to C′ using a transition probability W̃ (C → C′) that satisfies detailed balance on the modified potential Ṽ . This constitutes one pass. The move from C to C′ can of course be carried out using any conventional MC move that satisfies detailed balance on the modified potential. But the reverse formulation of the MC method now offers possibilities that were previously unavailable to conven- tional MC methods — if a simple scheme can be devised to update the configuration of the entire system on the stochastically modified potential, one can envision de- signing global moves for the system to accelerate its sam- pling, and our freedom in choosing the ṽi can be actively exploited to facilitate this. Within this context, the origi- nal formulation of Kandel et al. corresponds to switching vi to ṽi = 0, i.e. simplifying the potential by deleting in- teractions from it. Kandel et al. showed that for the Ising model they could easily construct global moves on this stochastically simplified potential and their formulation regenerates the Swendsen-Wang method. But compared to the deletion formulation of Kandel et al., the switch- ing implementation of the reverse MC method now offers a much wider set of possibilities because the form of the “switch to” interactions is completely arbitrary. Whereas previously there may not be an obvious way to globally update the configuration of the system on the original po- tential, with the proper choices for ṽi large-scale moves may now become possible on the stochastically modified potential. Indeed, we have shown that the switching idea may be used to formulate a cluster MC algorithm for a Lennard-Jones fluid [24]. Equations (2), (3) and the transition probability W̃ define the switching algorithm. To prove detailed bal- ance Eqn.(1) for the switching algorithm, it is suffi- cient to treat a case where there are only two interac- tion terms. Extension to any number of interactions is straightforward. Starting with C, with two interac- tion terms v1 and v2, there are four possible outcomes from the switch: I. both 1 and 2 are switched, which oc- curs with probability PI = S1(C)S2(C), II. 1 is switched and 2 is unswitched, with PII = S1(C)[1 − S2(C)], III. 1 is unswitched and 2 is switched, with PIII = [1 − S1(C)]S2(C), and IV. both 1 and 2 are unswitched, with PIV = [1 − S1(C)][1 − S2(C)]. After the switch, an up- date C → C′ is made with a transition probability W̃ that satisfies detailed balance on the modified potential Ṽ defined in Eqn.(3). Each of the four channels will have a different W̃ : W̃I, W̃II, etc., and W (C → C ′) in Eqn.(1) is the sum PIW̃I + PIIW̃II + PIIIW̃III + PIVW̃IV over all four channels. For the reverse transition, we start with C′ and consider switching v1(C ′) → ṽ1(C ′) and ′) → ṽ2(C ′). Again there are four possible outcomes and we call these scenarios I′, II′, III′ and IV′ as for the forward transition. W (C′ → C) in Eqn.(1) is again the sum PI′W̃I′ +PII′W̃II′ +PIII′W̃III′ +PIV′W̃IV′ . Using the choice of S and Ṽ in Eqs.(2) and (3), it is easy to show that detailed balance is obeyed along each chan- nel, i.e. PIW̃I = PI′W̃I′ , PIIW̃II = PII′W̃II′ , etc. Of course, detailed balance only requires the total W to sat- isfy Eqn.(1), and it is possible to choose alternate forms of S and Ṽ to do that, which may provide further flexi- bilities. In the rest of this letter, we will illustrate the effec- tiveness of the switching implementation of the reverse MC method, and show how it can be used to easily de- rive a cluster MC method in a system with continuous degrees of freedom. Previously, it has been extremely dif- ficult to design cluster MC algorithms for systems with continuous degrees of freedom. The few that have been reported to date [3, 5, 6, 7, 11, 14, 19] were mainly based on embedding discrete degrees of freedom into continu- ous ones. The only exception is the recent discovery of a geometric MC algorithm by Liu and Luitjen [19] where they formulated a rejection-free MC method to sample the Lennard-Jones fluid at its critical point. The switching algorithm we have proposed makes the process of deriving cluster-type MC methods much more straightforward compared to those based on geometric features of the system. We will illustrate this using the sine-Gordon model, which can be used to study the roughening transition on 2-dimensional surfaces. The sine-Gordon (SG) model has the potential VSG = T 〈i,j〉 |φi − φj | cos(φi)  , (4) where φi are continuous variables on a 2-dimensional square lattice, the second sum is over all sites and the first sum is over all nearest-neighbor pairs. The SG model is often considered to be a coarse-grained ver- sion of the discrete Gaussian (DG) model with poten- tial VDG = T 〈i,j〉 |hi − hj | 2, where hi are integers. The DG model can in turn be mapped directly onto the Coulomb gas model [25], and as a result, the SG model should belong in the same universality class as the Kosterlitz-Thouless (KT) transition [26, 27]. Roughening is expected to be a weak transition. The only easily discernible divergence is exhibited in a loga- rithmic dependence of the surface roughness σ2 = 〈|φi − 〈φ〉|2〉 on the system size L at the roughening tempera- ture TR. Below TR, σ 2 is expected to approach a finite value as L → ∞. In addition to this, since the divergence is slow, large lattice sizes are needed to reach the scaling limit. All of these features of the SG model make it hard to accurately study the roughening transition using MC simulations. Previous simulations have been limited to small systems [14, 28, 29, 30, 31, 32]. In order to locate TR and study the scaling behavior at the roughening transition, we make use of the switch- ing algorithm of the reverse MC method proposed above. The essential difficulty in treating the SG model is due to the nonlinear cosine terms in the potential in Eqn.(4). If these nonlinearities could be removed, the residual po- tential becomes a simple Gaussian and we could move the system configuration effectively using uncoupled sur- face modes. With this in mind, we separate the potential into two parts and treat the cosine terms as interactions vi = −T −1 cosφi and the harmonic part as the resid- ual Vr. Each of the interactions is switched to a uni- form potential ṽi = −T −1 with Si = e [1−cosφi]/T . After the switches, a number of φi would have effectively lost their couplings to the cosine potential, while the rest have their interactions with the cosine potential replaced by v̄i = − ln[e cosφi/T − e−1/T ]. In the ensuring MC move, we can update the unswitched φi which are now cou- pled to the replacement interactions v̄i using conventional methods, but try to formulate an update scheme where the rest of the φi, now forming a constrained Gaussian field, may be updated globally. A Gaussian field sub- ject to linear constraints is still Gaussian, and in prin- ciple we can diagonalize the potential to obtain all the normal modes and then move each one independently. This problem is the subject of fracton dynamics and has been studied previously [33]. However, the cost of ob- 64 128 256 512 1024 64 128 256 512 1024 (b)(a) FIG. 1: (a) Surface thickness σ2 as a function of the log of lattice size L for different temperatures T . (b) Expanded view of (a) for several temperatures near TR shifted vertically to coincide at L = 64. Dashed line is the expected KT slope at TR, showing that TR is slightly above T = 25 but below taining all the normal modes of the constrained surface and their frequencies will grow rapidly with the size of the lattice and will only be feasible for small-size simula- tions. Since the scaling limit in the SG model can only be reached with large system sizes, we will need an al- ternative method. The method we have used to update the constrained Gaussian fields is based on the method of Hoffman and Ribak [34]. Since the statistics of the fluctuations of a Gaussian field from its mean is indepen- dent of the value of the mean field, the fluctuations from a free Gaussian field can be transferred to a constrained field with a different mean. Near the roughening tem- perature, the switching procedure produces roughly 5% unswitched field points, and the corresponding mean field with these constraints can easily be determined using a steepest descent molecular dynamics method. To ensure ergodicity, a conventional Metropolis move is also carried out with every reverse MC move. Figure 1(a) shows simulation results for the scaling of the surface roughness σ2 with the length L of the lattice in simulations with different lattice sizes L2 up to 10242 and at several temperatures T from 16 to 30. KT theory [26, 27] predicts a logarithmic divergence for σ2 with a universal slope at TR 2(L) = σ20(TR) + lnL, (5) where a is the lattice constant of the surface, and in the units of Eqn.(4), a = 2π. Therefore, at TR the slope of Fig. 1(a) should be equal to 4. Above TR, the logarith- mic behavior of σ2 continues to hold except the constant σ20 as well as the slope both increase with T . The data in Fig. 1(a) show that for T = 21 and below, σ2 appears to approach a finite value as L → ∞. Therefore, it is 64 128 256 512 1024 10000 Metropolis cluster MC ξ = 1.4 ξ = 2.5 FIG. 2: Dynamic scaling for the relaxation time τ (in units of MC passes) of σ2 as a function of lattice size L in Metropolis versus cluster MC, with their corresponding exponent ξ. clear that TR > 21. The most recent simulation of the SG model by Sanchez et al. [32] (referred to as the “or- dered SG model”, OSGM, in this paper) suggested that TR ≈ 16. Our data show that this is incorrect, and their error is likely due to slow sampling problems. Locating the precise value of TR is more involved, since the data for T > 21 show no obvious tendency toward a finite σ2. There are two possibilities: either these temperatures are above TR or the system size may not be large enough to have reached the scaling limit for these temperatures. To determine which one is the case, we must resort to a comparison between the simulation data with KT the- ory. Figure 1(b) shows an expanded view of Fig. 1(a) for a few temperatures 23 < T < 30, but for each T the curve has been shifted vertically to remove the off- set σ20 so that they all coincide at L = 64. The heavy dashed line indicates the KT slope at TR according to Eqn.(5). The data therefore suggest that TR is slightly larger than 25 but less than 26, which is consistent with the RG prediction for TR = 8π in the continuum model [35, 36]. The apparent lack of an asymptotic σ2 in the data for 21 < T < TR implies that even for L = 1024, these lattice sizes are not yet large enough to be in the scaling limit for those temperatures. Finally, to compare the dynamic scaling behavior of the switching algorithm with Metropolis, Fig. 2 shows the relaxation time in the measurement of σ2 with the lattice size L slightly above TR. Compared with the dynamic exponent ξ ≈ 2.5 in Metropolis, the switching algorithm shows a markedly improved ξ ≈ 1.4. 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We describe a novel switching algorithm based on a ``reverse'' Monte Carlo method, in which the potential is stochastically modified before the system configuration is moved. This new algorithm facilitates a generalized formulation of cluster-type Monte Carlo methods, and the generalization makes it possible to derive cluster algorithms for systems with both discrete and continuous degrees of freedom. The roughening transition in the sine-Gordon model has been studied with this method, and high-accuracy simulations for system sizes up to $1024^2$ were carried out to examine the logarithmic divergence of the surface roughness above the transition temperature, revealing clear evidence for universal scaling of the Kosterlitz-Thouless type.

A New Monte Carlo Method and Its Implications for Generalized Cluster Algorithms C. H. Mak and Arun K. Sharma Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA (Dated: November 13, 2018) We describe a novel switching algorithm based on a “reverse” Monte Carlo method, in which the potential is stochastically modified before the system configuration is moved. This new algorithm facilitates a generalized formulation of cluster-type Monte Carlo methods, and the generalization makes it possible to derive cluster algorithms for systems with both discrete and continuous de- grees of freedom. The roughening transition in the sine-Gordon model has been studied with this method, and high-accuracy simulations for system sizes up to 10242 were carried out to examine the logarithmic divergence of the surface roughness above the transition temperature, revealing clear evidence for universal scaling of the Kosterlitz-Thouless type. PACS numbers: 05.10.Ln, 05.50.+q, 64.60.Ht, 75.40.Mg Large-scale Monte Carlo (MC) simulations are often plagued by slow sampling problems. These problems are especially severe in systems near the critical point or in those with strong correlations. Slow sampling problems manifest themselves as poor scaling of the dynamic relax- ation time with the system size, making large-size sim- ulations extremely slow to converge. The cause of these problems is that most MC simulations are based on lo- cal moves, and when the correlation length of the system grows or as relaxation modes of the system become heav- ily entangled, local moves become increasingly inefficient. But if nonlocal MC moves are used [1], their acceptance ratios are often found to be exceedingly low when system correlations are strong. One way to circumvent these problems was suggested by Swendsen and Wang [2], who devised a clever scheme where large-scale nonlocal MC moves may be constructed to achieve high sampling efficiencies by exploiting certain geometric symmetries in the system. This algorithm led to a marked reduction in the dynamical scaling exponent in the 2-dimensional Ising model near criticality. Since the nonlocal moves in this algorithm update a large num- ber of degrees of freedom at the same time, the Swendsen- Wang method and others inspired by it are also often referred to as “cluster Monte Carlo” methods. Since Swendsen and Wang’s paper in 1987, many cluster-type MC algorithms have appeared [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. But the success of cluster MC has not been universal because the proper cluster moves needed seem to be highly de- pendent on the system, and efficient cluster MC meth- ods have been found for only a small number of models [3, 5, 6, 7, 11, 14, 15, 18, 19] so far. The difficulty of formulating a generalized MC method that could work for any system seems to be associated with the appar- ent geometric nature of the cluster-type MC methods – all existing cluster MC methods have been derived in one way or another by using certain geometric features of the system. For example, in the original Swendsen- Wang formulation a mapping between the Ising model and the percolation model originally described by For- tuin and Kasteleyn [20] was exploited to effect cluster spin flips. In other models, the requisite mapping is not always obvious, making cluster MC methods difficult to implement for general systems. In this letter, we will show that the derivations of clus- ter MC methods do not have to be based on geometric features of the systems. Instead, they may be more con- veniently formulated based on algebraic features of the system potential V (C). We will exploit this algebraic for- mulation and suggest a way to generalize cluster Monte Carlo methods to systems with any potential. We focus on classical systems with partition function Z = Tr e−V (C), where V (C) is the potential in units of the temperature T . Acceptable Monte Carlo methods to sample the system configurations can be constructed using any transition probabilities W (C → C′) as long as the detailed balance condition W (C → C′)e−V (C) = W (C′ → C)e−V (C ′) (1) is satisfied. Conventional MC methods such as Metropo- lis [21] accomplishes this in two steps: a trial move is made from C to C′ with some transition probability, and the move is then accepted or rejected according to an acceptance probability based on V (C′), V (C) or both, so that the composite process satisfies Eqn.(1). This way of constructing the Markov chain – trial moves followed by acceptance/rejection – has been the accepted “stan- dard” method for doing MC since the inception of the MC method [22]. Other MC methods do exist, such as the heat bath algorithm [23], which follow alternative strate- gies, but by far the standard method is conceptually the simplest and most convenient in practice. In the Monte Carlo method we are proposing, we will reverse the order of the two steps in the standard method. That is, we will first determine an acceptable way to mod- ify the potential V and then find a transition C → C′ that is consistent with the new potential. To our knowledge, http://arxiv.org/abs/0704.1539v1 the basic elements of this “reverse MC” idea were first suggested by Kandel et al. [4], who used it to stochasti- cally remove interaction terms from the system’s poten- tial in an Ising model to arrive at an alternative deriva- tion of the Swendsen-Wang method. The formulation of Kandel et al. was limited to discrete systems like the Ising model. In the following, we will show how the re- verse MC idea may be formulated more generally for any system, discrete or continuous, and how it may then be used as a framework to construct generalized cluster al- gorithms. Consider a system with potential V = i vi+Vr, con- sisting of a number of “interaction terms” vi plus a “resid- ual” Vr . These interactions may be bonds between parti- cles, interactions of the particles with a field, or any other additive terms in V . We consider replacing each interac- tion term vi by some pre-selected ṽi with a “switching” probability Si(C) = cie ∆vi(C), (2) where ∆vi = vi− ṽi, ci = e i and ∆v∗i = maxC vi(C). The outcome of the switches defines two complemen- tary sets of interactions – the switched ones σ̃ and the unswitched ones σ̄. Using the outcome of the switches, we define a stochastically modified potential Ṽ as follows: ṽi + v̄j + Vr, (3) with v̄i = vi − ln(1 − Si). An MC pass starts with an attempt to switch every interaction vi to the new ṽi us- ing the Si defined in Eqn.(2). If the switch is successful, the interaction is replaced by ṽi. If not, the interaction is replaced by another interaction v̄i. This is followed by an update in the configuration of the entire system from C to C′ using a transition probability W̃ (C → C′) that satisfies detailed balance on the modified potential Ṽ . This constitutes one pass. The move from C to C′ can of course be carried out using any conventional MC move that satisfies detailed balance on the modified potential. But the reverse formulation of the MC method now offers possibilities that were previously unavailable to conven- tional MC methods — if a simple scheme can be devised to update the configuration of the entire system on the stochastically modified potential, one can envision de- signing global moves for the system to accelerate its sam- pling, and our freedom in choosing the ṽi can be actively exploited to facilitate this. Within this context, the origi- nal formulation of Kandel et al. corresponds to switching vi to ṽi = 0, i.e. simplifying the potential by deleting in- teractions from it. Kandel et al. showed that for the Ising model they could easily construct global moves on this stochastically simplified potential and their formulation regenerates the Swendsen-Wang method. But compared to the deletion formulation of Kandel et al., the switch- ing implementation of the reverse MC method now offers a much wider set of possibilities because the form of the “switch to” interactions is completely arbitrary. Whereas previously there may not be an obvious way to globally update the configuration of the system on the original po- tential, with the proper choices for ṽi large-scale moves may now become possible on the stochastically modified potential. Indeed, we have shown that the switching idea may be used to formulate a cluster MC algorithm for a Lennard-Jones fluid [24]. Equations (2), (3) and the transition probability W̃ define the switching algorithm. To prove detailed bal- ance Eqn.(1) for the switching algorithm, it is suffi- cient to treat a case where there are only two interac- tion terms. Extension to any number of interactions is straightforward. Starting with C, with two interac- tion terms v1 and v2, there are four possible outcomes from the switch: I. both 1 and 2 are switched, which oc- curs with probability PI = S1(C)S2(C), II. 1 is switched and 2 is unswitched, with PII = S1(C)[1 − S2(C)], III. 1 is unswitched and 2 is switched, with PIII = [1 − S1(C)]S2(C), and IV. both 1 and 2 are unswitched, with PIV = [1 − S1(C)][1 − S2(C)]. After the switch, an up- date C → C′ is made with a transition probability W̃ that satisfies detailed balance on the modified potential Ṽ defined in Eqn.(3). Each of the four channels will have a different W̃ : W̃I, W̃II, etc., and W (C → C ′) in Eqn.(1) is the sum PIW̃I + PIIW̃II + PIIIW̃III + PIVW̃IV over all four channels. For the reverse transition, we start with C′ and consider switching v1(C ′) → ṽ1(C ′) and ′) → ṽ2(C ′). Again there are four possible outcomes and we call these scenarios I′, II′, III′ and IV′ as for the forward transition. W (C′ → C) in Eqn.(1) is again the sum PI′W̃I′ +PII′W̃II′ +PIII′W̃III′ +PIV′W̃IV′ . Using the choice of S and Ṽ in Eqs.(2) and (3), it is easy to show that detailed balance is obeyed along each chan- nel, i.e. PIW̃I = PI′W̃I′ , PIIW̃II = PII′W̃II′ , etc. Of course, detailed balance only requires the total W to sat- isfy Eqn.(1), and it is possible to choose alternate forms of S and Ṽ to do that, which may provide further flexi- bilities. In the rest of this letter, we will illustrate the effec- tiveness of the switching implementation of the reverse MC method, and show how it can be used to easily de- rive a cluster MC method in a system with continuous degrees of freedom. Previously, it has been extremely dif- ficult to design cluster MC algorithms for systems with continuous degrees of freedom. The few that have been reported to date [3, 5, 6, 7, 11, 14, 19] were mainly based on embedding discrete degrees of freedom into continu- ous ones. The only exception is the recent discovery of a geometric MC algorithm by Liu and Luitjen [19] where they formulated a rejection-free MC method to sample the Lennard-Jones fluid at its critical point. The switching algorithm we have proposed makes the process of deriving cluster-type MC methods much more straightforward compared to those based on geometric features of the system. We will illustrate this using the sine-Gordon model, which can be used to study the roughening transition on 2-dimensional surfaces. The sine-Gordon (SG) model has the potential VSG = T 〈i,j〉 |φi − φj | cos(φi)  , (4) where φi are continuous variables on a 2-dimensional square lattice, the second sum is over all sites and the first sum is over all nearest-neighbor pairs. The SG model is often considered to be a coarse-grained ver- sion of the discrete Gaussian (DG) model with poten- tial VDG = T 〈i,j〉 |hi − hj | 2, where hi are integers. The DG model can in turn be mapped directly onto the Coulomb gas model [25], and as a result, the SG model should belong in the same universality class as the Kosterlitz-Thouless (KT) transition [26, 27]. Roughening is expected to be a weak transition. The only easily discernible divergence is exhibited in a loga- rithmic dependence of the surface roughness σ2 = 〈|φi − 〈φ〉|2〉 on the system size L at the roughening tempera- ture TR. Below TR, σ 2 is expected to approach a finite value as L → ∞. In addition to this, since the divergence is slow, large lattice sizes are needed to reach the scaling limit. All of these features of the SG model make it hard to accurately study the roughening transition using MC simulations. Previous simulations have been limited to small systems [14, 28, 29, 30, 31, 32]. In order to locate TR and study the scaling behavior at the roughening transition, we make use of the switch- ing algorithm of the reverse MC method proposed above. The essential difficulty in treating the SG model is due to the nonlinear cosine terms in the potential in Eqn.(4). If these nonlinearities could be removed, the residual po- tential becomes a simple Gaussian and we could move the system configuration effectively using uncoupled sur- face modes. With this in mind, we separate the potential into two parts and treat the cosine terms as interactions vi = −T −1 cosφi and the harmonic part as the resid- ual Vr. Each of the interactions is switched to a uni- form potential ṽi = −T −1 with Si = e [1−cosφi]/T . After the switches, a number of φi would have effectively lost their couplings to the cosine potential, while the rest have their interactions with the cosine potential replaced by v̄i = − ln[e cosφi/T − e−1/T ]. In the ensuring MC move, we can update the unswitched φi which are now cou- pled to the replacement interactions v̄i using conventional methods, but try to formulate an update scheme where the rest of the φi, now forming a constrained Gaussian field, may be updated globally. A Gaussian field sub- ject to linear constraints is still Gaussian, and in prin- ciple we can diagonalize the potential to obtain all the normal modes and then move each one independently. This problem is the subject of fracton dynamics and has been studied previously [33]. However, the cost of ob- 64 128 256 512 1024 64 128 256 512 1024 (b)(a) FIG. 1: (a) Surface thickness σ2 as a function of the log of lattice size L for different temperatures T . (b) Expanded view of (a) for several temperatures near TR shifted vertically to coincide at L = 64. Dashed line is the expected KT slope at TR, showing that TR is slightly above T = 25 but below taining all the normal modes of the constrained surface and their frequencies will grow rapidly with the size of the lattice and will only be feasible for small-size simula- tions. Since the scaling limit in the SG model can only be reached with large system sizes, we will need an al- ternative method. The method we have used to update the constrained Gaussian fields is based on the method of Hoffman and Ribak [34]. Since the statistics of the fluctuations of a Gaussian field from its mean is indepen- dent of the value of the mean field, the fluctuations from a free Gaussian field can be transferred to a constrained field with a different mean. Near the roughening tem- perature, the switching procedure produces roughly 5% unswitched field points, and the corresponding mean field with these constraints can easily be determined using a steepest descent molecular dynamics method. To ensure ergodicity, a conventional Metropolis move is also carried out with every reverse MC move. Figure 1(a) shows simulation results for the scaling of the surface roughness σ2 with the length L of the lattice in simulations with different lattice sizes L2 up to 10242 and at several temperatures T from 16 to 30. KT theory [26, 27] predicts a logarithmic divergence for σ2 with a universal slope at TR 2(L) = σ20(TR) + lnL, (5) where a is the lattice constant of the surface, and in the units of Eqn.(4), a = 2π. Therefore, at TR the slope of Fig. 1(a) should be equal to 4. Above TR, the logarith- mic behavior of σ2 continues to hold except the constant σ20 as well as the slope both increase with T . The data in Fig. 1(a) show that for T = 21 and below, σ2 appears to approach a finite value as L → ∞. Therefore, it is 64 128 256 512 1024 10000 Metropolis cluster MC ξ = 1.4 ξ = 2.5 FIG. 2: Dynamic scaling for the relaxation time τ (in units of MC passes) of σ2 as a function of lattice size L in Metropolis versus cluster MC, with their corresponding exponent ξ. clear that TR > 21. The most recent simulation of the SG model by Sanchez et al. [32] (referred to as the “or- dered SG model”, OSGM, in this paper) suggested that TR ≈ 16. Our data show that this is incorrect, and their error is likely due to slow sampling problems. Locating the precise value of TR is more involved, since the data for T > 21 show no obvious tendency toward a finite σ2. There are two possibilities: either these temperatures are above TR or the system size may not be large enough to have reached the scaling limit for these temperatures. To determine which one is the case, we must resort to a comparison between the simulation data with KT the- ory. Figure 1(b) shows an expanded view of Fig. 1(a) for a few temperatures 23 < T < 30, but for each T the curve has been shifted vertically to remove the off- set σ20 so that they all coincide at L = 64. The heavy dashed line indicates the KT slope at TR according to Eqn.(5). The data therefore suggest that TR is slightly larger than 25 but less than 26, which is consistent with the RG prediction for TR = 8π in the continuum model [35, 36]. The apparent lack of an asymptotic σ2 in the data for 21 < T < TR implies that even for L = 1024, these lattice sizes are not yet large enough to be in the scaling limit for those temperatures. Finally, to compare the dynamic scaling behavior of the switching algorithm with Metropolis, Fig. 2 shows the relaxation time in the measurement of σ2 with the lattice size L slightly above TR. Compared with the dynamic exponent ξ ≈ 2.5 in Metropolis, the switching algorithm shows a markedly improved ξ ≈ 1.4. 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The Orbifolds of Permutation-Type as Physical String Systems at Multiples of c = 26 III. The Spectra of ĉ = 52 Strings M.B.Halpern∗ Department of Physics, University of California and Theoretical Physics Group, Lawrence Berkeley National Laboratory University of California, Berkeley CA 94720 USA November 15, 2018 Abstract In the second paper of this series, I obtained the twisted BRST systems and extended physical-state conditions of all twisted open and closed ĉ = 52 strings. In this paper, I supplement the extended physical-state conditions with the explicit form of the extended (twisted) Virasoro generators of all ĉ = 52 strings, which allows us to discuss the physical spectra of these systems. Surprisingly, all the ĉ = 52 spectra admit an equivalent description in terms of generically-unconventional Virasoro generators at c = 26. This description strongly supports our prior conjecture that the ĉ = 52 strings are free of negative-norm states, and moreover shows that the spectra of some of the simpler cases are equivalent to those of ordinary untwisted open and closed c = 26 strings. ∗halpern@physics.berkeley.edu http://arxiv.org/abs/0704.1540v1 Table of Contents 1. Introduction 2. The Extended Virasoro Generators of ĉ = 52 Strings 3. First Discussion of the ĉ = 52 String Spectra 4. Equivalent c = 26 Description of the ĉ = 52 Spectra 5. Conclusions 1 Introduction Opening another chapter in the orbifold program [1-11,12-15], this is the third in a series of papers which considers the critical orbifolds of permutation- type as candidates for new physical string systems at higher central charge. In the first paper [16] of this series, we found that the twisted sectors of these orbifolds are governed by new, extended (permutation-twisted) world- sheet gravities – which indicate that the free-bosonic orbifold-string systems of permutation-type can be free of negative-norm states at critical central charge ĉ = 26K. Correspondingly-extended world-sheet permutation super- gravities are expected in the twisted sectors of the superstring orbifolds of permutation-type, where superconformal matter lives at higher multiples of critical superstring central charges. In the second paper [17] of the series, we found the corresponding twisted BRST systems for all sectors of the free-bosonic orbifolds which couple to the simple case of Z2-twisted permutation gravity, i.e. for all the twisted strings with ĉ = 52 matter. The new BRST systems also implied the following extended physical-state conditions for the physical states {|χ〉} of each of the ĉ = 52 strings: |χ〉 = 0, m ∈ Z, u = 0, 1 (1.1a) , L̂v m− n+ u−v L̂u+v m+ n + u+v (1.1b) δm+n+u+v The algebra in Eq. (1.1b) is called an order-two orbifold Virasoro algebra (or extended, twisted Virasoro algebra) and general orbifold Virasoro algebras [1,18,9,12,16,17] are known to govern all the twisted sectors of the orbifolds of permutation-type at higher multiples of c = 26. The set of all ĉ = 52 orbifold-strings is a very large class of fractional- moded free-bosonic string systems, including e.g. the twisted open-string sectors of the orientation orbifolds, the twisted closed-string sectors of the generalized Z2-permutation orbifolds and many others (see Refs. [16,17] and Sec. 2). Starting from the extended physical-state conditions (1.1) (and a right-mover copy of (1.1) on the same {|χ〉}for the twisted closed-string sectors) this paper begins the concrete study of the physical spectrum of each ĉ = 52 string. As the prerequisite for this analysis, I first provide in Sec. 2 the explicit form – in terms of twisted matter fields – of the extended Virasoro generators , u = 0, 1 of all ĉ = 52 strings. This construction allows us to begin the study of the general ĉ = 52 string spectra in Sec. 3. The same subject is further considered in Sec. 4, where I point out that all the ĉ = 52 spectra admit an equivalent description in terms of generically- unconventional Virasoro generators at c = 26. This description allows us to see clearly a number of spectral regularities which are only glimpsed in Sec. 3, including strong further evidence that the critical orbifolds of permutation- type can be free of negative-norm states. Moreover, although the generic ĉ = 52 spectrum is apparently new, we are able to show that some of the simpler spectra are equivalent to those of ordinary untwisted open and closed critical strings at c = 26. Based on these results, the discussion in Sec. 5 raises some interesting questions about these theories at the interacting level, and speculates on the form of the extended physical-state conditions for more general orbifold- strings of permutation-type. I will return to both of these subjects in suc- ceeding papers of the series. 2 The Twisted Virasoro Generators of ĉ = 52 Strings As emphasized in Ref. [17], the universal form of the twisted BRST systems and the extended physical-state conditions (1.1) are consequences of their origin in Z2-twisted permutation gravity, which governs all twisted ĉ = 52 matter. There are however many distinct ĉ = 52 strings, including the twisted open-string sectors of the orientation orbifolds [12,13,15-17] U(1)26 , H− = Z2(w.s.)×H (2.1) and the twisted closed-string sectors of the generalized Z2-permutation orb- ifolds [15-17] U(1)26 × U(1)26 , H+ = Z2(perm)×H ′ (2.2) as well as the generalized open-string Z2-permutation orbifolds and their T - duals [15-17]. For the orientation orbifolds in Eq. (2.1), I remind that H− is any automorphism group of the untwisted closed string U(1)26 which includes world-sheet orientation-reversing automorphisms. Indeed the twisted open- string orientation-orbifold sectors correspond to the orientation-reversing au- tomorphisms, which have the form τ− × ω, ω ∈ H where the basic automor- phism τ− exchanges the left- and right-movers of the closed string and ω is an extra automorphism which acts uniformly on the left- and right-movers of the closed string. Similarly, the automorphism group H+ of the general- ized Z2-permutation orbifolds in (2.2) is generated by elements of the form τ+×ω, ω ∈ H ′, where the basic automorphism τ+ exchanges the two copies of the closed string and the extra automorphism ω again acts uniformly on the left- and right-movers of each closed string. In both cases, the extra automorphisms ω in τ × ω may or may not form a group (see the examples at the end of this section). The spectra of different ĉ = 52 strings are characterized by their extended (twisted) Virasoro generators, all of which can in fact be written in the following unified form: Gn(r)µ;−n(r)ν(σ) × (2.3a) ×:Ĵn(r)µv Ĵ−n(r),ν,u−v m− p− + u−v +δm+u ,0 ∆̂0(σ) Ĵn(r)µu , Ĵn(s)νv = (2.3b) δn(r)+n(s),0modρ(σ)δm+n+n(r)+n(s) Gn(r),µ;−n(r),ν(σ) , Ĵn(r)µv (2.3c) Ĵn(r)µ,u+v m+ n+ + u+v ∆̂0(σ) = dim[n(r)] (2.3d) dim[n(r)] = 26. (2.3e) Each set of extended Virasoro generators in Eq. (2.3a) satisfies the order-two orbifold Virasoro algebra (1.1b) at ĉ = 52, and the current algebras in Eq. (2.3b) are of the type called doubly-twisted in the orbifold program. For those unfamiliar with the program, I first give a short summary of the standard notation in the result (2.3) – followed by the derivation of the result. As in the extended Virasoro generators themselves, the indices u, v with fundamental range ū, v̄ ∈ {0, 1} describe the twist of the basic permutations τ∓ in each H∓. For each extra automorphism ω(σ) in each τ∓ × ω(σ), the spectral indices {n(r)} and the degeneracy indices {µ ≡ µ(n(r))} of each twisted sector σ are determined by the so-called H-eigenvalue problem [3,5,6] of ω(σ) ω(σ)a U †(σ)b n(r)µ = U †(σ)a n(r)µ −2πin(r) ρ(σ) , ω(σ) ∈ H or H ′ (2.4a) ω(σ)a ω(σ)b Gcd = Gab, Gab = G (2.4b) a, b = 0, 1, . . . , 25, n(r) ∈ (0, 1, . . . , ρ(σ)− 1) (2.4c) where G is the untwisted target-space metric of U(1)26. The quantity ρ(σ) is the order of ω(σ) and all indices {n(r)µ} are periodic modulo ρ(σ), with {n(r)} the pullback to the fundamental region and dim[n(r)] the size of the subspace n(r). The index r is summed once over the fundamental region in Eqs. (2.3a), (2.3d) and (2.3e). The twisted metric G.(σ) and its inverse G are defined in terms of the unitary eigenvectors U(σ) of the H-eigenvalue problem Gn(r)µ;n(s)ν(σ) = χn(r)µχn(s)νU(σ)n(r)µ U(σ)n(s)ν Gab (2.5a) = δn(r)+n(s),0modρ(σ)Gn(r)µ;−n(r),ν(σ) (2.5b) Gn(r)µ;n(s)ν(σ) = χ−1 n(r)µ n(s)ν GabU †(σ)a n(r)µ U †(σ)b n(s)ν (2.5c) = δn(r)+n(s),0modρ(σ)G n(r),µ;−n(r),ν (2.5d) where G is again the untwisted metric and the χ’s are essentially-arbitrary normalization constants. Finally, the standard mode normal-ordering in Eq. (2.3a) is: :Ĵn(r)µu Ĵn(s)νv :M (2.6) Ĵn(s)νv Ĵn(r)µu Ĵn(r)µu Ĵn(s)νv It follows that the quantity ∆0(σ) in Eqs. (2.3a) and (2.3d) Ĵn(r)µu |0〉σ = 0 (2.7a) → L̂u |0〉σ = ∆̂0(σ) δm+u ,0|0〉σ (2.7b) is the conformal weight of the scalar twist-field state |0〉σ of sector σ. I comment briefly on the derivation of the unified form (2.3) of the ĉ = 52 extended Virasoro generators. Essentially this result was given for the twisted open-string sectors of the non-abelian orientation orbifolds in Subsecs. 3.4, 3.5 of Ref. [12], and that result is easily reduced for our abelian case U(1)26/H− in Eq. (2.1). With a right-mover copy of the ex- tended Virasoro generators (and u → ̂ = 0, 1), the result also hold for the twisted closed-string sectors of the generalized Z2-permutation orbifolds (U(1)26 × U(1)26)/H+ in Eq. (2.2). This follows by the substitution G → G, u (2.8) into the known results for the ordinary Z2-permutation orbifolds with trivial H ′ (see Ref. [perm] and Subsec. 4.2 of Ref. [16]). Finally, a single copy of the unified form (2.3) holds as well for each twisted sector of the generalized open-string Z2-permutation orbifolds (U(1) 26 × U(1)26)open/H+ and all pos- sible T -dualizations of each of these sectors. This conclusion follows because the left-mover extended Virasoro generators of the closed-string orbifolds for each H+ are the input data for the construction of the correponding open- string orbifolds [14], and the twisted-current form of each set of extended Virasoro generators is independent of T-dualization [15]. The branes, quasi- canonical algebra and non-commutative geometry of the twisted open strings [13-15,16,17] depend of course on the particular T-dualization, but these will not be needed here. In what follows I will consider each twisted ĉ = 52 string separately, but the reader may find it helpful to bear in mind the complete sector structure of these orbifold-string systems as labelled by the elements of the automorphism groups H∓. Given a particular extra automorphism ωn ∈ H or H ′ of order n, one may list the following low-order examples: (1; τ∓) (2.9a) (1; τ∓ × ω2) (2.9b) (1, ω3, ω 3; τ∓, τ∓ × ω3, τ∓ × ω 3) (2.9c) (1, ω24; τ∓ × ω4, τ∓ × ω 4) (2.9d) (1, ω26, ω 6; τ∓ × ω6, τ∓ × ω 6, τ∓ × ω 6). (2.9e) For the generalized Z2-permutation orbifolds (τ+) all of these sectors are twisted closed strings at ĉ = 52, while all the sectors of the generalized open- string Z2-permutation orbifolds (τ+) and their T-dualizations are twisted open strings at ĉ = 52. For the orientation orbifolds (τ−) the sectors before the semicolon are twisted closed strings at c = 26 (which form an ordinary space-time orbifold) while the sectors after the semicolon are twisted open strings at ĉ = 52. More generally, orientation orbifolds always contain an equal number of twisted open and closed strings. In all cases, the twisting is of course trivial for sectors corresponding to the unit element. 3 First Discussion of the ĉ = 52 String Spectra To frame this discussion, I remind [1] the reader that the Virasoro primary states of our orbifold CFT’s are defined by the integral Virasoro subalgebra (generated by {L̂0(m)}) of the extended Virasoro algebra. Then the extended physical-state conditions (1.1a) tell us that all the physical states {|χ〉} of each ĉ = 52 orbifold-string are Virasoro primary L̂0(m > 0)|χ〉 = 0 (3.1) but only a small subset of these primary states are selected by the rest of the physical-state conditions: L̂0(0)− |χ〉 = L̂1 |χ〉 = 0. (3.2) In what follows, I will refer to the L̂0(0) condition in Eq. (3.2) as the spectral condition, since it will determine the allowed values of momentum-squared for each ĉ = 52 string. The space of physical states of each orbifold-string is then much smaller than the space of states of the underlying orbifold conformal field theory. For the experts, I remark in particular that the extended physical-state condi- tions generically disallow the characteristic sequence [19] of Virasoro primary states known as the principle-primary states [1,9]. This follows first by the spectral condition (which fixes the conformal weight), and second because the physical-state condition {L̂u ≃ 0} is stronger than the principle-primary state condition [1,9] |p.p.s.〉 = 0, u = 0, 1, m > 0 (3.3) which does not extend to m = 0. I turn now to concretize the spectral condition of each twisted ĉ = 52 string, using the explicit form (2.3) of its extended Virasoro generators. For this, recall [12,15] first that these generators contain in general two kinds of commuting zero modes (dimensionless momenta), namely {Ĵ0µ0(0)} and {Ĵρ(σ)/2,µ,1(0)}, where the latter is relevant only when the order ρ(σ) of ω(σ) is even. In what follows, I often refer to these zero modes collectively as {Ĵ(0)}. It is then natural to define the “momentum-squared” operator P̂ 2 as follows: L̂0(0) = −P̂ 2 + R̂(σ) + ∆̂0(σ) (3.4a) P̂ 2 ≡ − G0µ;0ν(σ)Ĵ0µ0(0)Ĵ0ν0(0) + (3.4b) ,µ;− ρ(σ) ,ν(σ)Ĵρ(σ)/2,µ,1(0)Ĵ−ρ(σ)/2,ν,−1(0) R̂(σ) ≡ r,µ,ν Gn(r)µ;−n(r),ν(σ)× (3.4c) ×:Ĵn(r)µu Ĵ−n(r),ν,−u Here the primed sum in the “level-number” operator R̂(σ) indicates omission of the zero modes. With this decomposition, the spectral condition in Eq. (3.2) takes the simple form: P̂ 2|χ〉 = P̂ 2(0) + R̂(σ) |χ〉 (3.5a) P̂ 2(0) ≡ 2 δ̂0(σ)− 1 (3.5b) δ̂0(σ) = dim[n(r)] ≥ 0. (3.5c) Although I will continue the discussion primarily in this form, in fact Eqs. (3.4a) and (3.5a) hold only for the twisted open-string sectors of the orbifolds. For the twisted closed-string sectors, we also have right-mover copies of the extended Virasoro generators (2.3), and a corresponding right-mover copy of the extended physical-state conditions (1.1) on the same {|χ〉}. For simplicity I will limit the discussion of these sectors here to the case of decompactified zero modes, for which it is appropriate to equate the left and right movers ĴR(0) = ĴL(0) = 1√ Ĵ(0) → R̂R(σ) = R̂L(σ) (3.6) where the last equality is level-matching in each twisted sector. Keeping the same definition of the operator P̂ 2 in Eq. (3.4b), the correct closed-string ĉ = 52 spectral condition is then obtained by the substitution P̂ 2 → 1 P̂ 2 (3.7) in both Eqs. (3.4a) and (3.5a). These identifications, and hence P̂ 2 → 2P̂ 2 can be used at any point in the discussion below to obtain the corresponding closed-string results. Returning to the open-string case, one simple solution of the extended physical-state conditions is the ground state |0, Ĵ(0)〉σ of twisted sector σ: R̂(σ)|0, Ĵ(0)〉σ = L̂u |0, Ĵ(σ)〉σ = 0 (3.8a) P̂ 2|0, Ĵ(0)〉σ = P̂ (0)|0, Ĵ(0)〉σ, P̂ (0) = −2 + 2δ̂0(σ) (3.8b) This is the “momentum-boosted” twist-field state (see Eq. (2.7)) of that sector, with ground-state mass-squared P̂ 2 . Moreover Eq. (3.4c) and the commutator (2.3c) give the increments ∆(P̂ 2) = ∆(R̂(σ)) = 4 (3.9) obtained by adding the negatively-moded current Ĵn(r)µu to any previous state. The precise content of these excited levels must of course be determined from the remainder of the extended physical-state con- ditions. I continue this discussion with some specific examples of ĉ = 52 strings, beginning with the simplest twisted open-string orientation-orbifold sectors [12,13,15,16,17]: ω = 1l : ρ = 1, n = 0, U = 1l, G = G, Ĵ0au (3.10a) ∆(P̂ 2) = 4 ∣m+ u ∣ (u = 0 is DD, u = 0 is ND) (3.10b) ω = −1l : ρ = 2, n = 1, U = 1l, G = G, Ĵ1au m+ u+1 (3.11a) ∆(P̂ 2) = 4 ∣m+ u+1 ∣ (u = 0 is DN, u = 0 is NN). (3.11b) In these cases, the extra automorphisms ω act uniformly on the labels a = 0, . . . 25 and G is the untwisted target space metric in Eq. (2.4b). Although both twisted strings have (26+26) = 52 matter degrees of freedom, note that each example has only one of the two types of zero modes {Ĵ(0)}: 26DD zero modes {Ĵ0a0(0)} for ω = 1l and 26NN zero modes {Ĵρ/2,a,1(0)} for ω = −1l. In both cases, the momentum-squared (3.4b) has the schematic form P̂ 2 = ηabĴa(0)Ĵb(0), η = 0 −1l (3.12) where η = −G is the standard (west-coast) 26-dimensional target-space met- ric. Then we compute from Eqs. (3.5b) and (3.5c) that both strings share the same tachyonic ground-state mass-squared δ̂0(σ) = 0, ∆̂0(σ) = , P̂ 2(0) = −2 (3.13) and the first excited state of each is massless: Ĵ0a1 Ĵ1a0 |0, Ĵ(0)〉σ = 0 for ω = (3.14) For this level, I have checked that the L̂1 ≃ 0 gauge eliminates the longitudinal parts of the 26-dimensional “photons”, and moreover the L̂1 and L̂0 (1) gauges together eliminate the negative-norm states at the next level: + βĴ(−1) |0, Ĵ(0)〉σ, P̂ 2 = 2. (3.15) Since the increments ∆(P 2) in Eqs. (3.10b) and (3.11b) are even integers, we are led to suspect that the spectra of these two twisted ĉ = 52 strings are nothing but the spectrum of an ordinary open c = 26 string in disguise 1. I will return to this question in the following section. A larger subset of twisted ĉ = 52 strings is the following. For a particular twisted sector σ, suppose that ω = ±1l acts uniformly on a set of d labels a = 0, 1, . . . d − 1, d ≥ 4 while a non-trivial element ω(perm) of some per- mutation group acts non-trivially on the other 26 − d spatial labels. Then Eqs. (2.4),(2.5) and standard results [3,5-7,9] in the orbifold program give the following explicit form of the extended Virasoro generators (2.3) in this sector: ∆0(σ) + (3.16a) Gab(d) :Ĵǫav p+ v+ǫ Ĵ−ǫ,b,u−v m− p + u−v−ǫ fj(σ) fj(σ)−1 ×:Ĵ̂jv p+ ̂ fj(σ) Ĵ−̂,j,u−v m− p− ̂ fj(σ) + u−v δ̂0(σ) = fj(σ)−1 fj(σ) fj(σ) fj(σ) ≥ 0 (3.16b) fj(σ) = 26− d, 4 ≤ d ≤ 26. (3.16c) Here ǫ = 0 or 1 for ω = 1l or −1l, G(d) is the restriction of the flat target-space metric (2.4b) to the first d labels, fj(σ) is the size of the jth cycle in ω(perm), 1The spectra of these two ĉ = 52 strings look even more familiar in terms of the dimensionful momenta k ≡ Ĵ(0)/ , where α′ is the conventional open-string Regge slope. and the previous cases with δ̂0(σ) = 0 are included when d = 26. The half- integer moded currents in the second term of (3.16) satisfy the twisted current algebra (2.3b) with G → G(d). For the permutation-twisted currents in the last term of (3.16), I have used the standard relation (n(r)/ρ(σ)) = (̂/fj(σ)) and (the inverse of) the twisted metric [3,5-7,9] ̂j;l̂l (σ) = δjlfj(σ)δ̂+l̂,0modfj(σ) (3.17) which also determines the twisted current algebra (2.3b) for these currents. Using Eq. (3.16b), we see that the non-trivial element of Z2 on two labels also gives δ̂0(σ) = 0 and a P̂ = −2 ground state, but a non-trivial element of Z3 on three labels gives a slightly-raised ground state δ̂0(σ) = , ∆̂0(σ) = , P̂ 2(0) = − (3.18) and no photons. Given the cycle-structure {fj(σ)} of any extra automorphism w(perm) (see e.g. Eq. (3.4) of Ref. [16]), it is straightforward to evaluate the sum in Eq. (3.16b). As an illustration, one finds the simple tachyonic ground-state mass-squares 5 ≤ (d = prime) ≤ 23 : P̂ 2(0) = − (d− 2 + 26− d ) (3.19) in twisted sectors which correspond to the action of any non-trivial element of the cyclic group Zλ of prime order on 3 ≤ (λ = 26−d) ≤ 21 spatial labels. The result (3.19) includes Eq. (3.18) when d = 23, but does not extend to the cases d = 26, 24 with P̂ 2(0) = −2 discussed above. I remind that this result applies only to the open orbifold-strings, while twice these values of P̂ 2(0) are obtained for the closed-string versions. Further analysis of the ĉ = 52 strings, including the “larger subset” of examples (3.16), is found in the following section. 4 Equivalent c = 26 Description of the ĉ = 52 Spectra In fact, there exists an entirely equivalent description of all the ĉ = 52 string spectra in terms of generically-unconventional Virasoro generators at c = 26. To obtain the c = 26 description, I first define the relabelled (unhatted) operators Jn(r)µ 2m+ u+ 2n(r) ≡ Ĵn(r)µu , u = 0, 1 (4.1a) L(2m+ u) ≡ 2L̂u (4.1b) in terms of the hatted operators above. This 1−1 map is recognized as a modest generalization of (the inverse of) the order-two orbifold-induction procedure of Borisov, Halpern and Schweigert [1]. SinceM ≡ 2m+u, u = 0, 1 covers the integers once, we then find from (2.3) the explicit form of the c = 26 generators: L(M) = δ̂0(σ)δM,0 + (4.2a) r,µ,ν Gn(r)µ;−n(r),ν(σ) :Jn(r)µ 2n(r) J−n(r),ν M −Q− 2n(r) δ̂0(σ) = dim[n(r)] (4.2b) [L(M), L(N)] = (M −N)L(M +N) + 26 M(M2 − 1)δM+N,0 (4.2c) L(M), Jn(r)µ 2n(r) 2n(r) Jn(r)µ M +N + 2n(r) (4.2d) Jn(r)µ 2n(r) , Jn(s)ν 2n(s) (4.2e) = δn(r)+n(s),0modρ(σ)δM+N+2(n(r)+n(s)ρ(σ) ),0 Gn(r)µ;−n(r),ν(σ). The expression (4.2b) for δ̂0(σ) is the same as above, and the mode-normal ordering in Eq. (4.2a) :Jn(r)µ 2n(r) Jn(s)ν 2n(s) :M (4.3) 2n(r) Jn(s)ν 2n(s) Jn(r)µ 2n(r) 2n(r) Jn(r)µ 2n(r) Jn(s)ν 2n(s) follows from the ĉ = 52 ordering (2.6) because the map (4.1) preserves the sign of all arguments. I emphasize that the c = 26 Virasoro generators in Eq. (4.2) are generically- unconventional because the twisted matter is now summed over the fractions {2n/ρ} instead of the conventional orbifold-fractions {n/ρ}. This distortion of the “extra twist” is the price we must pay in order to unwind the “basic twist” associated to the basic permutations τ∓ of H∓. The map (4.1) also tells us that the ĉ = 52 momenta {Ĵ(0)} and the c = 26 momenta {J(0)} are identical, and we may record J(0) = Ĵ(0) : J0µ(0) = Ĵ0µ0(0), Jρ(σ)/2,µ(0) = Ĵρ(σ)/2,µ,1(0) (4.4a) P 2 = P̂ 2 (4.4b) G0µ:0ν(σ)J0µ(0)J0ν(0) + ,µ;− ρ(σ) ,ν(σ)Jρ(σ)/2,µ(0)J−ρ(σ)/2,ν(0) where the ĉ = 52 form of P̂ 2 was given in Eq. (3.4b). Similarly, the “level- number” operator R(σ) in the decomposition of L(0) is the same L(0) = −1 P 2 +R(σ) + δ̂0(σ) (4.5a) R(σ) = R̂(σ) (4.5b) r,µ,ν Gn(r)µ;−n(r),ν(σ)× ×:Jn(r)µ 2n(r) J−n(r),ν 2n(r) where the ĉ = 52 form of R̂(σ) was given in Eq. (3.4c). By itself, the inverse orbifold-induction procedure (4.1) is only a rela- belling of the operators of the permutation-orbifold CFT’s. The central point of this discussion however is that for the orbifold-string theories – restricted by the extended physical state conditions (1.1) – the map also gives us a completely equivalent c = 26 description of the physical spectrum of each ĉ = 52 orbifold-string. Indeed, it is easily checked that both components ū = 0, 1 of the ĉ = 52 extended physical-state condition (1.1a) map directly onto the simpler and in fact conventional physical-state condition L(M ≥ 0)|χ〉 = δM,0|χ〉 (4.6) in the 26-dimensional description! A right- mover copy of Eq. (4.6) on the same physical states {|χ〉} is similarly obtained in the equivalent c = 26 description of the closed orbifold-strings. I emphasize that the physical states {|χ〉} of the 26-dimensional descrip- tion (4.6) are exactly the original physical states (1.1a) of the ĉ = 52 string. Indeed, each physical state |χ〉 can be regarded as invariant under the map, or each can now be rewritten in 26-dimensional form. In further detail, Eqs. (4.5) and (4.6) give the same spectral condition P 2 ≃ P 20 + R(σ), the same physical ground state 2 |0, J(0)〉σ ≡ |0, Ĵ(0)〉σ, P 0 = P̂ 0 = −2 + 2δ̂0(σ) (4.7) and each negatively-moded hatted current in any physical state can be re- placed according to Eq. (4.1a) by the corresponding unhatted current mode. Note finally that the commutator (4.2d) and the decomposition (4.5a) give the 26-dimensional increment ∆(P̂ 2) = ∆(R(σ)) = 2 2n(r) (4.8) which results from the addition of Jn(r)µ 2n(r) to any previous state. With M = 2m+ n, these are recognized as the same increments (3.9) obtained in the ĉ = 52 description. As simple examples, consider the “larger subset” (3.16) of ĉ = 52 strings – whose equivalent c = 26 physical state condition (4.6) now involves the following subset of the c = 26 Virasoro generators (4.2): L(M) = δM,0δ̂0(σ) + Gab(d) :Jǫa(Q + ǫ)J−ǫ,b(M −Q− ǫ):M + fj(σ) fj(σ)−1 :J̂j Q + 2̂ fj(σ) J−̂,j M −Q− 2̂ fj(σ) :M (4.9a) δ̂0(σ) = fj(σ)−1 fj(σ) fj(σ) − 2̂ fj(σ) (4.9b) 2Although it is not directly relevant in either description of the ĉ = 52 strings, one notes that the conformal weight of the scalar twist-field state |0〉 of sector σ has now shifted from ∆̂0(σ) to δ̂0(σ) in the c = 26 description. a, b = 0, . . . , d− 1, fj(σ) = 26− d, 4 ≤ d ≤ 26. (4.9c) Recall for the larger subset that ǫ = 0, 1 corresponds in the symmetric theory to the action of the extra automorphism ω = ±1l on the first d ≥ 4 labels {a}, while fj(σ) is the length of the j-th cycle of the extra permutation ω(perm) which acts on the remaining 26 − d spatial labels. Shifting the dummy integer Q by the integer ǫ, we note that the second term in Eq. (4.9a) is a set of ordinary Virasoro generators for d untwisted bosons with the ordinary current algebra [Ja(Q), Jb(P )] = G ab QδQ+P,0 (4.10) for both values of ǫ. The currents in the third term satisfy the twisted current algebra (4.2e) with the permutation-twisted metric (3.17), and the value of δ̂0(σ) in Eq. (4.9b) is only a slightly-rewritten form of that given in Eq. (3.16b). We are now in a position to confirm our suspicions in the previous sec- tion about the simplest orbifold-strings, described earlier at ĉ = 52 by the extended Virasoro generators: :Ĵǫav p+ v+ǫ Ĵ−ǫ,b,u−v m− p+ u−v−ǫ ,0, u = 0, 1, ǫ = 0, 1. (4.11) These are now equivalently described by the choice d = 26 in Eq. (4.9), in which case only the second (ordinary) term of Eq. (4.9a) is non-zero – and then the equivalent physical-state condition (4.6) verifies that the physical spectrum of each of these particular twisted ĉ = 52 strings is indeed equiva- lent to that of an ordinary untwisted c = 26 string! These cases include the open-string orientation-orbifold sectors corresponding to τ− × (ω = ±1l) in Eq. (3.10) and their T-duals, as well as the twisted closed-string sectors of the generalized Z2-permutation orbifolds corresponding to τ+ × (ω = ±1l). Additionally, consider the following special cases of the extended Virasoro generators (3.16) at ĉ = 52 = δm+u + (4.12) Gab(24) :Ĵǫav p+ v+ǫ Ĵ−ǫ,b,u−v m− p+ u−v−ǫ :Ĵ̂v p+ ̂+v Ĵ−̂,u−v m− p+ u−v−̂ which result when the extra automorphism in the symmetric theory acts as ω = ±1l on the first d = 24 labels and the non-trivial element of a Z2 on the remaining 2 spatial labels. I have noted in Sec. 3 that δ̂0(σ) = 0 for these cases as well, and indeed the equivalent c = 26 description (4.6) and (4.9) at d = 24 now shows that the open and closed orbifold- strings of this type also have the spectrum of ordinary untwisted c = 26 strings. The common thread for the orbifold-strings in Eqs. (4.10) and (4.12) is that they are at most half-integer moded, so that the shift {n/ρ} → {2n/ρ} gives integer moding in the c = 26 description. Beyond these simple cases, the ĉ = 52 strings are apparently new – with δ̂0(σ) 6= 0, unfamiliar ground-state mass-squares, and fractional modeing (and increments) in either description. 5 Conclusions We have discussed the physical spectrum of the general ĉ = 52 orbifold-string, as well as an equivalent but unconventionally-twisted c = 26 description of the twisted ĉ = 52 matter. The equivalent c = 26 description holds only for the orbifold-string theories – restricted by the extended physical- state conditions (1.1) – and not in the larger Hilbert space of the underlying orbifold conformal field theories. In general we have found that the spectra of these orbifold-string systems are unfamiliar. One simple and unexpected conclusion however is that, as string theories restricted by the extended physical-state conditions, the single twisted ĉ = 52 sector of each of the simplest orbifolds of permutation-type (see Eq. (2.9)) (1; τ∓) (5.1a) (1; τ∓ × ω2), ω 2 = 1 (5.1b) have the same physical spectra as ordinary untwisted c = 26 strings. No such equivalence is found of course in the half-integer moded Hilbert space of the full orbifold CFT’s. The list in Eq. (5.1) includes the simplest orientation orbifolds (with τ−) and their T-duals, as well as the simplest generalized Z2-permutation orbifolds (with τ+). For the simplest orientation orbifolds in particular, the string theories in Eq. (5.1) consist of an ordinary unoriented closed string (the unit element) at c = 26 and a ĉ = 52 twisted open string whose physical spectrum is equivalent to that of an ordinary untwisted c = 26 critical open string. Since both the closed- and open-string spectra of these simple orientation orbifolds are equivalent to those of the archtypal orientifold (without Chan-Paton fac- tors), we are led to suspect that orientation orbifolds include orientifolds. I will return in the next paper of this series to consider this question at the interacting level, where we will also be able to ask about the decoupling of null physical states. Following that, I will consider in a succeeding paper the corresponding situation and modular invariance for the simplest permutation orbifold-string systems. More generally, we have seen that there are many other orientation orb- ifolds, open-string Z2-permutation orbifolds and generalized Z2-permutation orbifolds whose ĉ = 52 spectra show fractional modeing in both the ĉ = 52 and the c = 26 descriptions. These include in particular the orbifolds in Eq. (2.9) when the order n of the extra automorphism is greater than two. There is more to say about no-ghost theorems for the general twisted ĉ = 52 string. The original intuition [16] was that the doubled gauges ū = 0, 1 of the extended physical state condition (1.1) could remove the doubled set of negative-norm states (time-like modes) of the ĉ = 52 strings – which are also associated with ū = 0, 1. For the simplest ĉ = 52 strings in Eq. (5.1), this intuition is certainly born out [20]. More generally, the equivalent c = 26 description of each spectrum shows that both aspects of the doubling are indeed eliminated at the same time, leaving us with the conventional physical state condition (4.6) and only a single set of time-like modes. This is clearly visible in the set of examples (4.9), where the only time-like modes (a = 0) are included in the second term. For the general ĉ = 52 string, the reader should bear in mind that the twisted metric G in Eq. (4.2) is only a unitary transformation (2.5) of the untwisted metric G with a single time-like direction. Although not yet a proof, and illustrated here only for ĉ = 52, I consider this a stronger form of the original arguments [16] that all the critical orbifolds of permutation-type should be free of negative-norm states. The next question I wish to address is the following: I have empha- sized that the equivalent c = 26 Virasoro generators (4.2) are generically- unconventional, being summed over the matter-field fractions {2n/ρ} instead of the conventional orbifold fractions {n/ρ}, but are they actually new Vira- soro generators? I do not know the answer to this question in general, but at least some of them can in fact be re-expressed by further mode-relabeling in terms of more familiar Virasoro generators. As examples, consider the special case of the “larger subset” (4.9) when ω(perm) is one of the elements of order λ of each cyclic group Zλ. (These are the particular, single-cycle elements of Zλ with f0(σ) = λ.) When λ is odd, one finds that the first and third terms of (4.9) can in fact be re-expressed in terms of the conven- tional Virasoro generators associated to a twisted sector of an ordinary cyclic permutation orbifold U(1)λ/Zλ [christ] Lλ(M) = Q+ ̂ M −Q− ̂ :M (5.2) +δM,0 , c = λ = 2l + 1 where I have relabeled the currents J̂ ≡ J̂0. To obtain this result from (4.9), one needs the fact that {2̂/λ} ≃ {̂/λ} modulo the integers when λ is odd. This observation is consistent with the ground-state mass-squares for prime λ in Eq. (3.19). When λ is even, I have also checked that the first and third terms of (4.9) can be re-expressed as the sum of two identical commuting Virasoro generators of this type Lλ(M) = Lλ (M) + L̃λ (M), c = λ = 2l (5.3) each of which is associated to a twisted sector of U(1)λ/2/Zλ/2. This result is also obtained by relabeling the modes modulo the integers, and provides us with another way to understand that the ground-state mass-squared is unshifted when ω(perm) is the non-trivial element of a Z2. My final remark is a conjecture, that the extended physical-state condi- tions for the twisted strings at ĉ = 26λ, λ prime will in fact read m+ ̂ |χ〉 = 0, ̂ = 0, 1, . . . , λ− 1 (5.4a) âλ ≡ 13λ2 − 1 (5.4b) m+ ̂ n+ l̂ m− n+ ̂−l̂ ̂+l̂ m+ n + ̂+l̂ + (5.4c) + 26λ m+ ̂ m+ ̂ ̂+l̂ where Eq. (5.4c) is an orbifold Virasoro algebra [1,18,9] of order λ. This form includes the correct generators {L̂̂} corresponding to the classical extended Polyakov constraints of Ref. [16], and includes the correct value â2 = 17/8 studied here for the ĉ = 52 strings. I obtained the system (5.4) by requiring (as we now know for λ = 2) that it map by the inverse of the order-λ orbifold-induction procedure [1] to the conventional physical-state condition (4.6) with â1 = 1 at c = 26. One way to test this conjecture would be the construction of the corresponding twisted BRST systems [17] for these higher values of ĉ. Extensions to include winding number and twisted B fields at ĉ = 52 are also deferred to another time and place. Acknowledgements For helpful information, discussions and encouragement, I thank L. Alvarez- Gaumé, K. Bardakci, I. Brunner, J. de Boer, D. Fairlie, O. Ganor, E. Gi- mon, C. Helfgott, E. Kiritsis, R. Littlejohn, S. Mandelstam, J. McGreevy, N. Obers, A. Petkou, E. Rabinovici, V. Schomerus, K. Schoutens, C. Schweigert and E. Witten. This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE- AC02-O5CH11231 and in part by the National Science Foundation under grant PHY00-98840. References [1] L. Borisov, M. B. Halpern, and C. Schweigert, “Systematic approach to cyclic orbifolds,” Int. J. Mod. Phys. A13 (1998) 125, hep-th/9701061. http://arxiv.org/abs/hep-th/9701061 [2] J. Evslin, M. B. Halpern, and J. E. Wang, “General Virasoro construc- tion on orbifold affine algebra,” Int. J. Mod. Phys. A14 (1999) 4985, hep-th/9904105. [3] J. de Boer, J. Evslin, M. B. Halpern, and J. E. Wang, “New duality transformations in orbifold theory,” Int. J. Mod. Phys.A15 (2000) 1297, hep-th/9908187. [4] J. Evslin, M. B. Halpern, and J. E. Wang, “Cyclic coset orbifolds,” Int. J. Mod. Phys. A15 (2000) 3829, hep-th/9912084. [5] M. B. Halpern and J. E. Wang, “More about all current-algebraic orb- ifolds,” Int. J. Mod. Phys. A16 (2001) 97, hep-th/0005187. [6] J. de Boer, M. B. Halpern, and N. A. Obers, “The operator algebra and twisted KZ equations of WZW orbifolds,” J. High Energy Phys. 10 (2001) 011, hep-th/0105305. [7] M. B. Halpern and N. A. Obers, “Two large examples in orbifold theory: Abelian orbifolds and the charge conjugation orbifold on su(n),” Int. J. Mod. Phys. A17 (2002) 3897, hep-th/0203056. [8] M. B. Halpern and F. Wagner, “The general coset orbifold action,” Int. J. Mod. Phys. A18 (2003) 19, hep-th/0205143. [9] M. B. Halpern and C. Helfgott, “Extended operator algebra and re- ducibility in the WZW permutation orbifolds,” Int. J. Mod. Phys. A18 (2003) 1773, hep-th/0208087. [10] O. Ganor, M. B. Halpern, C. Helfgott and N. A. Obers, “The outer- automorphic WZW orbifolds on so(2n), including five triality orbifolds on so(8),” J. High Energy Phys. 0212 (2002) 019, hep-th/0211003. [11] J. de Boer, M. B. Halpern and C. Helfgott, “Twisted Einstein ten- sors and orbifold geometry,” Int. J. Mod. Phys. A18 (2003) 3489, hep-th/0212275. [12] M. B. Halpern and C. Helfgott, “Twisted open strings from closed strings: The WZW orientation orbifolds,” Int. J. Mod. Phys.A19 (2004) 2233, hep-th/0306014. http://arxiv.org/abs/hep-th/9904105 http://arxiv.org/abs/hep-th/9908187 http://arxiv.org/abs/hep-th/9912084 http://arxiv.org/abs/hep-th/0005187 http://arxiv.org/abs/hep-th/0105305 http://arxiv.org/abs/hep-th/0203056 http://arxiv.org/abs/hep-th/0205143 http://arxiv.org/abs/hep-th/0208087 http://arxiv.org/abs/hep-th/0211003 http://arxiv.org/abs/hep-th/0212275 http://arxiv.org/abs/hep-th/0306014 [13] M. B. Halpern and C. Helfgott, “On the target-space geometry of the open-string orientation-orbifold sectors,” Ann. of Phys. 310 (2004) 302, hep-th/0309101. [14] M. B. Halpern and C. Helfgott, “A basic class of twisted open WZW strings,” Int. J. Mod. Phys. A19 (2004) 3481, hep-th/0402108. [15] M. B. Halpern and C. Helfgott, “The general twisted openWZW string,” Int. J. Mod. Phys. A20 (2005) 923, hep-th/0406003. [16] M. B. Halpern, “The orbifolds of permutation-type as physical string systems at multiples of c = 26: I. Extended actions and new twisted world-sheet gravities,” hep-th/0703044. [17] M. B.Halpern, “The orbifolds of permutation-type as physical string systems at multiples of c = 26: II. The twisted BRST systems of ĉ = 52 matter,” hep-th/0703208. [18] R. Dijkgraaf, E. Verlinde and H. Verlinde, “Matrix string theory,” Nucl. Phys. B 500 (1997) 43, hep-th/9703030. [19] A. Klemm and M. G. Schmidt, “Orbifolds by cyclic permutations of tensor-product conformal field theories, ” Phys. Lett. B245 (1990) 53. [20] P. Goddard and C. B. Thorn, “Compatibility of the dual pomeron with unitarity and the absence of ghosts in the dual resonance model,” Phys. Lett. B40 (1972) 378. [21] S. Mandelstam “Dual resonance models,” Phys. Rep. 13 (1974) 259. http://arxiv.org/abs/hep-th/0309101 http://arxiv.org/abs/hep-th/0402108 http://arxiv.org/abs/hep-th/0406003 http://arxiv.org/abs/hep-th/0703044 http://arxiv.org/abs/hep-th/0703208 http://arxiv.org/abs/hep-th/9703030 Introduction The Twisted Virasoro Generators of =52 Strings First Discussion of the = 52 String Spectra Equivalent c = 26 Description of the = 52 Spectra Conclusions

In the second paper of this series, I obtained the twisted BRST systems and extended physical-state conditions of all twisted open and closed $\hat{c} = 52$ strings. In this paper, I supplement the extended physical-state conditions with the explicit form of the extended (twisted) Virasoro generators of all $\hat{c} = 52$ strings, which allows us to discuss the physical spectra of these systems. Surprisingly, all the $\hat{c}=52$ spectra admit an equivalent description in terms of generically-unconventional Virasoro generators at $c=26$. This description strongly supports our prior conjecture that the $\hat{c}=52$ strings are free of negative-norm states, and moreover shows that the spectra of some of the simpler cases are equivalent to those of ordinary untwisted open and closed $c=26$ strings.

Introduction 2. The Extended Virasoro Generators of ĉ = 52 Strings 3. First Discussion of the ĉ = 52 String Spectra 4. Equivalent c = 26 Description of the ĉ = 52 Spectra 5. Conclusions 1 Introduction Opening another chapter in the orbifold program [1-11,12-15], this is the third in a series of papers which considers the critical orbifolds of permutation- type as candidates for new physical string systems at higher central charge. In the first paper [16] of this series, we found that the twisted sectors of these orbifolds are governed by new, extended (permutation-twisted) world- sheet gravities – which indicate that the free-bosonic orbifold-string systems of permutation-type can be free of negative-norm states at critical central charge ĉ = 26K. Correspondingly-extended world-sheet permutation super- gravities are expected in the twisted sectors of the superstring orbifolds of permutation-type, where superconformal matter lives at higher multiples of critical superstring central charges. In the second paper [17] of the series, we found the corresponding twisted BRST systems for all sectors of the free-bosonic orbifolds which couple to the simple case of Z2-twisted permutation gravity, i.e. for all the twisted strings with ĉ = 52 matter. The new BRST systems also implied the following extended physical-state conditions for the physical states {|χ〉} of each of the ĉ = 52 strings: |χ〉 = 0, m ∈ Z, u = 0, 1 (1.1a) , L̂v m− n+ u−v L̂u+v m+ n + u+v (1.1b) δm+n+u+v The algebra in Eq. (1.1b) is called an order-two orbifold Virasoro algebra (or extended, twisted Virasoro algebra) and general orbifold Virasoro algebras [1,18,9,12,16,17] are known to govern all the twisted sectors of the orbifolds of permutation-type at higher multiples of c = 26. The set of all ĉ = 52 orbifold-strings is a very large class of fractional- moded free-bosonic string systems, including e.g. the twisted open-string sectors of the orientation orbifolds, the twisted closed-string sectors of the generalized Z2-permutation orbifolds and many others (see Refs. [16,17] and Sec. 2). Starting from the extended physical-state conditions (1.1) (and a right-mover copy of (1.1) on the same {|χ〉}for the twisted closed-string sectors) this paper begins the concrete study of the physical spectrum of each ĉ = 52 string. As the prerequisite for this analysis, I first provide in Sec. 2 the explicit form – in terms of twisted matter fields – of the extended Virasoro generators , u = 0, 1 of all ĉ = 52 strings. This construction allows us to begin the study of the general ĉ = 52 string spectra in Sec. 3. The same subject is further considered in Sec. 4, where I point out that all the ĉ = 52 spectra admit an equivalent description in terms of generically- unconventional Virasoro generators at c = 26. This description allows us to see clearly a number of spectral regularities which are only glimpsed in Sec. 3, including strong further evidence that the critical orbifolds of permutation- type can be free of negative-norm states. Moreover, although the generic ĉ = 52 spectrum is apparently new, we are able to show that some of the simpler spectra are equivalent to those of ordinary untwisted open and closed critical strings at c = 26. Based on these results, the discussion in Sec. 5 raises some interesting questions about these theories at the interacting level, and speculates on the form of the extended physical-state conditions for more general orbifold- strings of permutation-type. I will return to both of these subjects in suc- ceeding papers of the series. 2 The Twisted Virasoro Generators of ĉ = 52 Strings As emphasized in Ref. [17], the universal form of the twisted BRST systems and the extended physical-state conditions (1.1) are consequences of their origin in Z2-twisted permutation gravity, which governs all twisted ĉ = 52 matter. There are however many distinct ĉ = 52 strings, including the twisted open-string sectors of the orientation orbifolds [12,13,15-17] U(1)26 , H− = Z2(w.s.)×H (2.1) and the twisted closed-string sectors of the generalized Z2-permutation orb- ifolds [15-17] U(1)26 × U(1)26 , H+ = Z2(perm)×H ′ (2.2) as well as the generalized open-string Z2-permutation orbifolds and their T - duals [15-17]. For the orientation orbifolds in Eq. (2.1), I remind that H− is any automorphism group of the untwisted closed string U(1)26 which includes world-sheet orientation-reversing automorphisms. Indeed the twisted open- string orientation-orbifold sectors correspond to the orientation-reversing au- tomorphisms, which have the form τ− × ω, ω ∈ H where the basic automor- phism τ− exchanges the left- and right-movers of the closed string and ω is an extra automorphism which acts uniformly on the left- and right-movers of the closed string. Similarly, the automorphism group H+ of the general- ized Z2-permutation orbifolds in (2.2) is generated by elements of the form τ+×ω, ω ∈ H ′, where the basic automorphism τ+ exchanges the two copies of the closed string and the extra automorphism ω again acts uniformly on the left- and right-movers of each closed string. In both cases, the extra automorphisms ω in τ × ω may or may not form a group (see the examples at the end of this section). The spectra of different ĉ = 52 strings are characterized by their extended (twisted) Virasoro generators, all of which can in fact be written in the following unified form: Gn(r)µ;−n(r)ν(σ) × (2.3a) ×:Ĵn(r)µv Ĵ−n(r),ν,u−v m− p− + u−v +δm+u ,0 ∆̂0(σ) Ĵn(r)µu , Ĵn(s)νv = (2.3b) δn(r)+n(s),0modρ(σ)δm+n+n(r)+n(s) Gn(r),µ;−n(r),ν(σ) , Ĵn(r)µv (2.3c) Ĵn(r)µ,u+v m+ n+ + u+v ∆̂0(σ) = dim[n(r)] (2.3d) dim[n(r)] = 26. (2.3e) Each set of extended Virasoro generators in Eq. (2.3a) satisfies the order-two orbifold Virasoro algebra (1.1b) at ĉ = 52, and the current algebras in Eq. (2.3b) are of the type called doubly-twisted in the orbifold program. For those unfamiliar with the program, I first give a short summary of the standard notation in the result (2.3) – followed by the derivation of the result. As in the extended Virasoro generators themselves, the indices u, v with fundamental range ū, v̄ ∈ {0, 1} describe the twist of the basic permutations τ∓ in each H∓. For each extra automorphism ω(σ) in each τ∓ × ω(σ), the spectral indices {n(r)} and the degeneracy indices {µ ≡ µ(n(r))} of each twisted sector σ are determined by the so-called H-eigenvalue problem [3,5,6] of ω(σ) ω(σ)a U †(σ)b n(r)µ = U †(σ)a n(r)µ −2πin(r) ρ(σ) , ω(σ) ∈ H or H ′ (2.4a) ω(σ)a ω(σ)b Gcd = Gab, Gab = G (2.4b) a, b = 0, 1, . . . , 25, n(r) ∈ (0, 1, . . . , ρ(σ)− 1) (2.4c) where G is the untwisted target-space metric of U(1)26. The quantity ρ(σ) is the order of ω(σ) and all indices {n(r)µ} are periodic modulo ρ(σ), with {n(r)} the pullback to the fundamental region and dim[n(r)] the size of the subspace n(r). The index r is summed once over the fundamental region in Eqs. (2.3a), (2.3d) and (2.3e). The twisted metric G.(σ) and its inverse G are defined in terms of the unitary eigenvectors U(σ) of the H-eigenvalue problem Gn(r)µ;n(s)ν(σ) = χn(r)µχn(s)νU(σ)n(r)µ U(σ)n(s)ν Gab (2.5a) = δn(r)+n(s),0modρ(σ)Gn(r)µ;−n(r),ν(σ) (2.5b) Gn(r)µ;n(s)ν(σ) = χ−1 n(r)µ n(s)ν GabU †(σ)a n(r)µ U †(σ)b n(s)ν (2.5c) = δn(r)+n(s),0modρ(σ)G n(r),µ;−n(r),ν (2.5d) where G is again the untwisted metric and the χ’s are essentially-arbitrary normalization constants. Finally, the standard mode normal-ordering in Eq. (2.3a) is: :Ĵn(r)µu Ĵn(s)νv :M (2.6) Ĵn(s)νv Ĵn(r)µu Ĵn(r)µu Ĵn(s)νv It follows that the quantity ∆0(σ) in Eqs. (2.3a) and (2.3d) Ĵn(r)µu |0〉σ = 0 (2.7a) → L̂u |0〉σ = ∆̂0(σ) δm+u ,0|0〉σ (2.7b) is the conformal weight of the scalar twist-field state |0〉σ of sector σ. I comment briefly on the derivation of the unified form (2.3) of the ĉ = 52 extended Virasoro generators. Essentially this result was given for the twisted open-string sectors of the non-abelian orientation orbifolds in Subsecs. 3.4, 3.5 of Ref. [12], and that result is easily reduced for our abelian case U(1)26/H− in Eq. (2.1). With a right-mover copy of the ex- tended Virasoro generators (and u → ̂ = 0, 1), the result also hold for the twisted closed-string sectors of the generalized Z2-permutation orbifolds (U(1)26 × U(1)26)/H+ in Eq. (2.2). This follows by the substitution G → G, u (2.8) into the known results for the ordinary Z2-permutation orbifolds with trivial H ′ (see Ref. [perm] and Subsec. 4.2 of Ref. [16]). Finally, a single copy of the unified form (2.3) holds as well for each twisted sector of the generalized open-string Z2-permutation orbifolds (U(1) 26 × U(1)26)open/H+ and all pos- sible T -dualizations of each of these sectors. This conclusion follows because the left-mover extended Virasoro generators of the closed-string orbifolds for each H+ are the input data for the construction of the correponding open- string orbifolds [14], and the twisted-current form of each set of extended Virasoro generators is independent of T-dualization [15]. The branes, quasi- canonical algebra and non-commutative geometry of the twisted open strings [13-15,16,17] depend of course on the particular T-dualization, but these will not be needed here. In what follows I will consider each twisted ĉ = 52 string separately, but the reader may find it helpful to bear in mind the complete sector structure of these orbifold-string systems as labelled by the elements of the automorphism groups H∓. Given a particular extra automorphism ωn ∈ H or H ′ of order n, one may list the following low-order examples: (1; τ∓) (2.9a) (1; τ∓ × ω2) (2.9b) (1, ω3, ω 3; τ∓, τ∓ × ω3, τ∓ × ω 3) (2.9c) (1, ω24; τ∓ × ω4, τ∓ × ω 4) (2.9d) (1, ω26, ω 6; τ∓ × ω6, τ∓ × ω 6, τ∓ × ω 6). (2.9e) For the generalized Z2-permutation orbifolds (τ+) all of these sectors are twisted closed strings at ĉ = 52, while all the sectors of the generalized open- string Z2-permutation orbifolds (τ+) and their T-dualizations are twisted open strings at ĉ = 52. For the orientation orbifolds (τ−) the sectors before the semicolon are twisted closed strings at c = 26 (which form an ordinary space-time orbifold) while the sectors after the semicolon are twisted open strings at ĉ = 52. More generally, orientation orbifolds always contain an equal number of twisted open and closed strings. In all cases, the twisting is of course trivial for sectors corresponding to the unit element. 3 First Discussion of the ĉ = 52 String Spectra To frame this discussion, I remind [1] the reader that the Virasoro primary states of our orbifold CFT’s are defined by the integral Virasoro subalgebra (generated by {L̂0(m)}) of the extended Virasoro algebra. Then the extended physical-state conditions (1.1a) tell us that all the physical states {|χ〉} of each ĉ = 52 orbifold-string are Virasoro primary L̂0(m > 0)|χ〉 = 0 (3.1) but only a small subset of these primary states are selected by the rest of the physical-state conditions: L̂0(0)− |χ〉 = L̂1 |χ〉 = 0. (3.2) In what follows, I will refer to the L̂0(0) condition in Eq. (3.2) as the spectral condition, since it will determine the allowed values of momentum-squared for each ĉ = 52 string. The space of physical states of each orbifold-string is then much smaller than the space of states of the underlying orbifold conformal field theory. For the experts, I remark in particular that the extended physical-state condi- tions generically disallow the characteristic sequence [19] of Virasoro primary states known as the principle-primary states [1,9]. This follows first by the spectral condition (which fixes the conformal weight), and second because the physical-state condition {L̂u ≃ 0} is stronger than the principle-primary state condition [1,9] |p.p.s.〉 = 0, u = 0, 1, m > 0 (3.3) which does not extend to m = 0. I turn now to concretize the spectral condition of each twisted ĉ = 52 string, using the explicit form (2.3) of its extended Virasoro generators. For this, recall [12,15] first that these generators contain in general two kinds of commuting zero modes (dimensionless momenta), namely {Ĵ0µ0(0)} and {Ĵρ(σ)/2,µ,1(0)}, where the latter is relevant only when the order ρ(σ) of ω(σ) is even. In what follows, I often refer to these zero modes collectively as {Ĵ(0)}. It is then natural to define the “momentum-squared” operator P̂ 2 as follows: L̂0(0) = −P̂ 2 + R̂(σ) + ∆̂0(σ) (3.4a) P̂ 2 ≡ − G0µ;0ν(σ)Ĵ0µ0(0)Ĵ0ν0(0) + (3.4b) ,µ;− ρ(σ) ,ν(σ)Ĵρ(σ)/2,µ,1(0)Ĵ−ρ(σ)/2,ν,−1(0) R̂(σ) ≡ r,µ,ν Gn(r)µ;−n(r),ν(σ)× (3.4c) ×:Ĵn(r)µu Ĵ−n(r),ν,−u Here the primed sum in the “level-number” operator R̂(σ) indicates omission of the zero modes. With this decomposition, the spectral condition in Eq. (3.2) takes the simple form: P̂ 2|χ〉 = P̂ 2(0) + R̂(σ) |χ〉 (3.5a) P̂ 2(0) ≡ 2 δ̂0(σ)− 1 (3.5b) δ̂0(σ) = dim[n(r)] ≥ 0. (3.5c) Although I will continue the discussion primarily in this form, in fact Eqs. (3.4a) and (3.5a) hold only for the twisted open-string sectors of the orbifolds. For the twisted closed-string sectors, we also have right-mover copies of the extended Virasoro generators (2.3), and a corresponding right-mover copy of the extended physical-state conditions (1.1) on the same {|χ〉}. For simplicity I will limit the discussion of these sectors here to the case of decompactified zero modes, for which it is appropriate to equate the left and right movers ĴR(0) = ĴL(0) = 1√ Ĵ(0) → R̂R(σ) = R̂L(σ) (3.6) where the last equality is level-matching in each twisted sector. Keeping the same definition of the operator P̂ 2 in Eq. (3.4b), the correct closed-string ĉ = 52 spectral condition is then obtained by the substitution P̂ 2 → 1 P̂ 2 (3.7) in both Eqs. (3.4a) and (3.5a). These identifications, and hence P̂ 2 → 2P̂ 2 can be used at any point in the discussion below to obtain the corresponding closed-string results. Returning to the open-string case, one simple solution of the extended physical-state conditions is the ground state |0, Ĵ(0)〉σ of twisted sector σ: R̂(σ)|0, Ĵ(0)〉σ = L̂u |0, Ĵ(σ)〉σ = 0 (3.8a) P̂ 2|0, Ĵ(0)〉σ = P̂ (0)|0, Ĵ(0)〉σ, P̂ (0) = −2 + 2δ̂0(σ) (3.8b) This is the “momentum-boosted” twist-field state (see Eq. (2.7)) of that sector, with ground-state mass-squared P̂ 2 . Moreover Eq. (3.4c) and the commutator (2.3c) give the increments ∆(P̂ 2) = ∆(R̂(σ)) = 4 (3.9) obtained by adding the negatively-moded current Ĵn(r)µu to any previous state. The precise content of these excited levels must of course be determined from the remainder of the extended physical-state con- ditions. I continue this discussion with some specific examples of ĉ = 52 strings, beginning with the simplest twisted open-string orientation-orbifold sectors [12,13,15,16,17]: ω = 1l : ρ = 1, n = 0, U = 1l, G = G, Ĵ0au (3.10a) ∆(P̂ 2) = 4 ∣m+ u ∣ (u = 0 is DD, u = 0 is ND) (3.10b) ω = −1l : ρ = 2, n = 1, U = 1l, G = G, Ĵ1au m+ u+1 (3.11a) ∆(P̂ 2) = 4 ∣m+ u+1 ∣ (u = 0 is DN, u = 0 is NN). (3.11b) In these cases, the extra automorphisms ω act uniformly on the labels a = 0, . . . 25 and G is the untwisted target space metric in Eq. (2.4b). Although both twisted strings have (26+26) = 52 matter degrees of freedom, note that each example has only one of the two types of zero modes {Ĵ(0)}: 26DD zero modes {Ĵ0a0(0)} for ω = 1l and 26NN zero modes {Ĵρ/2,a,1(0)} for ω = −1l. In both cases, the momentum-squared (3.4b) has the schematic form P̂ 2 = ηabĴa(0)Ĵb(0), η = 0 −1l (3.12) where η = −G is the standard (west-coast) 26-dimensional target-space met- ric. Then we compute from Eqs. (3.5b) and (3.5c) that both strings share the same tachyonic ground-state mass-squared δ̂0(σ) = 0, ∆̂0(σ) = , P̂ 2(0) = −2 (3.13) and the first excited state of each is massless: Ĵ0a1 Ĵ1a0 |0, Ĵ(0)〉σ = 0 for ω = (3.14) For this level, I have checked that the L̂1 ≃ 0 gauge eliminates the longitudinal parts of the 26-dimensional “photons”, and moreover the L̂1 and L̂0 (1) gauges together eliminate the negative-norm states at the next level: + βĴ(−1) |0, Ĵ(0)〉σ, P̂ 2 = 2. (3.15) Since the increments ∆(P 2) in Eqs. (3.10b) and (3.11b) are even integers, we are led to suspect that the spectra of these two twisted ĉ = 52 strings are nothing but the spectrum of an ordinary open c = 26 string in disguise 1. I will return to this question in the following section. A larger subset of twisted ĉ = 52 strings is the following. For a particular twisted sector σ, suppose that ω = ±1l acts uniformly on a set of d labels a = 0, 1, . . . d − 1, d ≥ 4 while a non-trivial element ω(perm) of some per- mutation group acts non-trivially on the other 26 − d spatial labels. Then Eqs. (2.4),(2.5) and standard results [3,5-7,9] in the orbifold program give the following explicit form of the extended Virasoro generators (2.3) in this sector: ∆0(σ) + (3.16a) Gab(d) :Ĵǫav p+ v+ǫ Ĵ−ǫ,b,u−v m− p + u−v−ǫ fj(σ) fj(σ)−1 ×:Ĵ̂jv p+ ̂ fj(σ) Ĵ−̂,j,u−v m− p− ̂ fj(σ) + u−v δ̂0(σ) = fj(σ)−1 fj(σ) fj(σ) fj(σ) ≥ 0 (3.16b) fj(σ) = 26− d, 4 ≤ d ≤ 26. (3.16c) Here ǫ = 0 or 1 for ω = 1l or −1l, G(d) is the restriction of the flat target-space metric (2.4b) to the first d labels, fj(σ) is the size of the jth cycle in ω(perm), 1The spectra of these two ĉ = 52 strings look even more familiar in terms of the dimensionful momenta k ≡ Ĵ(0)/ , where α′ is the conventional open-string Regge slope. and the previous cases with δ̂0(σ) = 0 are included when d = 26. The half- integer moded currents in the second term of (3.16) satisfy the twisted current algebra (2.3b) with G → G(d). For the permutation-twisted currents in the last term of (3.16), I have used the standard relation (n(r)/ρ(σ)) = (̂/fj(σ)) and (the inverse of) the twisted metric [3,5-7,9] ̂j;l̂l (σ) = δjlfj(σ)δ̂+l̂,0modfj(σ) (3.17) which also determines the twisted current algebra (2.3b) for these currents. Using Eq. (3.16b), we see that the non-trivial element of Z2 on two labels also gives δ̂0(σ) = 0 and a P̂ = −2 ground state, but a non-trivial element of Z3 on three labels gives a slightly-raised ground state δ̂0(σ) = , ∆̂0(σ) = , P̂ 2(0) = − (3.18) and no photons. Given the cycle-structure {fj(σ)} of any extra automorphism w(perm) (see e.g. Eq. (3.4) of Ref. [16]), it is straightforward to evaluate the sum in Eq. (3.16b). As an illustration, one finds the simple tachyonic ground-state mass-squares 5 ≤ (d = prime) ≤ 23 : P̂ 2(0) = − (d− 2 + 26− d ) (3.19) in twisted sectors which correspond to the action of any non-trivial element of the cyclic group Zλ of prime order on 3 ≤ (λ = 26−d) ≤ 21 spatial labels. The result (3.19) includes Eq. (3.18) when d = 23, but does not extend to the cases d = 26, 24 with P̂ 2(0) = −2 discussed above. I remind that this result applies only to the open orbifold-strings, while twice these values of P̂ 2(0) are obtained for the closed-string versions. Further analysis of the ĉ = 52 strings, including the “larger subset” of examples (3.16), is found in the following section. 4 Equivalent c = 26 Description of the ĉ = 52 Spectra In fact, there exists an entirely equivalent description of all the ĉ = 52 string spectra in terms of generically-unconventional Virasoro generators at c = 26. To obtain the c = 26 description, I first define the relabelled (unhatted) operators Jn(r)µ 2m+ u+ 2n(r) ≡ Ĵn(r)µu , u = 0, 1 (4.1a) L(2m+ u) ≡ 2L̂u (4.1b) in terms of the hatted operators above. This 1−1 map is recognized as a modest generalization of (the inverse of) the order-two orbifold-induction procedure of Borisov, Halpern and Schweigert [1]. SinceM ≡ 2m+u, u = 0, 1 covers the integers once, we then find from (2.3) the explicit form of the c = 26 generators: L(M) = δ̂0(σ)δM,0 + (4.2a) r,µ,ν Gn(r)µ;−n(r),ν(σ) :Jn(r)µ 2n(r) J−n(r),ν M −Q− 2n(r) δ̂0(σ) = dim[n(r)] (4.2b) [L(M), L(N)] = (M −N)L(M +N) + 26 M(M2 − 1)δM+N,0 (4.2c) L(M), Jn(r)µ 2n(r) 2n(r) Jn(r)µ M +N + 2n(r) (4.2d) Jn(r)µ 2n(r) , Jn(s)ν 2n(s) (4.2e) = δn(r)+n(s),0modρ(σ)δM+N+2(n(r)+n(s)ρ(σ) ),0 Gn(r)µ;−n(r),ν(σ). The expression (4.2b) for δ̂0(σ) is the same as above, and the mode-normal ordering in Eq. (4.2a) :Jn(r)µ 2n(r) Jn(s)ν 2n(s) :M (4.3) 2n(r) Jn(s)ν 2n(s) Jn(r)µ 2n(r) 2n(r) Jn(r)µ 2n(r) Jn(s)ν 2n(s) follows from the ĉ = 52 ordering (2.6) because the map (4.1) preserves the sign of all arguments. I emphasize that the c = 26 Virasoro generators in Eq. (4.2) are generically- unconventional because the twisted matter is now summed over the fractions {2n/ρ} instead of the conventional orbifold-fractions {n/ρ}. This distortion of the “extra twist” is the price we must pay in order to unwind the “basic twist” associated to the basic permutations τ∓ of H∓. The map (4.1) also tells us that the ĉ = 52 momenta {Ĵ(0)} and the c = 26 momenta {J(0)} are identical, and we may record J(0) = Ĵ(0) : J0µ(0) = Ĵ0µ0(0), Jρ(σ)/2,µ(0) = Ĵρ(σ)/2,µ,1(0) (4.4a) P 2 = P̂ 2 (4.4b) G0µ:0ν(σ)J0µ(0)J0ν(0) + ,µ;− ρ(σ) ,ν(σ)Jρ(σ)/2,µ(0)J−ρ(σ)/2,ν(0) where the ĉ = 52 form of P̂ 2 was given in Eq. (3.4b). Similarly, the “level- number” operator R(σ) in the decomposition of L(0) is the same L(0) = −1 P 2 +R(σ) + δ̂0(σ) (4.5a) R(σ) = R̂(σ) (4.5b) r,µ,ν Gn(r)µ;−n(r),ν(σ)× ×:Jn(r)µ 2n(r) J−n(r),ν 2n(r) where the ĉ = 52 form of R̂(σ) was given in Eq. (3.4c). By itself, the inverse orbifold-induction procedure (4.1) is only a rela- belling of the operators of the permutation-orbifold CFT’s. The central point of this discussion however is that for the orbifold-string theories – restricted by the extended physical state conditions (1.1) – the map also gives us a completely equivalent c = 26 description of the physical spectrum of each ĉ = 52 orbifold-string. Indeed, it is easily checked that both components ū = 0, 1 of the ĉ = 52 extended physical-state condition (1.1a) map directly onto the simpler and in fact conventional physical-state condition L(M ≥ 0)|χ〉 = δM,0|χ〉 (4.6) in the 26-dimensional description! A right- mover copy of Eq. (4.6) on the same physical states {|χ〉} is similarly obtained in the equivalent c = 26 description of the closed orbifold-strings. I emphasize that the physical states {|χ〉} of the 26-dimensional descrip- tion (4.6) are exactly the original physical states (1.1a) of the ĉ = 52 string. Indeed, each physical state |χ〉 can be regarded as invariant under the map, or each can now be rewritten in 26-dimensional form. In further detail, Eqs. (4.5) and (4.6) give the same spectral condition P 2 ≃ P 20 + R(σ), the same physical ground state 2 |0, J(0)〉σ ≡ |0, Ĵ(0)〉σ, P 0 = P̂ 0 = −2 + 2δ̂0(σ) (4.7) and each negatively-moded hatted current in any physical state can be re- placed according to Eq. (4.1a) by the corresponding unhatted current mode. Note finally that the commutator (4.2d) and the decomposition (4.5a) give the 26-dimensional increment ∆(P̂ 2) = ∆(R(σ)) = 2 2n(r) (4.8) which results from the addition of Jn(r)µ 2n(r) to any previous state. With M = 2m+ n, these are recognized as the same increments (3.9) obtained in the ĉ = 52 description. As simple examples, consider the “larger subset” (3.16) of ĉ = 52 strings – whose equivalent c = 26 physical state condition (4.6) now involves the following subset of the c = 26 Virasoro generators (4.2): L(M) = δM,0δ̂0(σ) + Gab(d) :Jǫa(Q + ǫ)J−ǫ,b(M −Q− ǫ):M + fj(σ) fj(σ)−1 :J̂j Q + 2̂ fj(σ) J−̂,j M −Q− 2̂ fj(σ) :M (4.9a) δ̂0(σ) = fj(σ)−1 fj(σ) fj(σ) − 2̂ fj(σ) (4.9b) 2Although it is not directly relevant in either description of the ĉ = 52 strings, one notes that the conformal weight of the scalar twist-field state |0〉 of sector σ has now shifted from ∆̂0(σ) to δ̂0(σ) in the c = 26 description. a, b = 0, . . . , d− 1, fj(σ) = 26− d, 4 ≤ d ≤ 26. (4.9c) Recall for the larger subset that ǫ = 0, 1 corresponds in the symmetric theory to the action of the extra automorphism ω = ±1l on the first d ≥ 4 labels {a}, while fj(σ) is the length of the j-th cycle of the extra permutation ω(perm) which acts on the remaining 26 − d spatial labels. Shifting the dummy integer Q by the integer ǫ, we note that the second term in Eq. (4.9a) is a set of ordinary Virasoro generators for d untwisted bosons with the ordinary current algebra [Ja(Q), Jb(P )] = G ab QδQ+P,0 (4.10) for both values of ǫ. The currents in the third term satisfy the twisted current algebra (4.2e) with the permutation-twisted metric (3.17), and the value of δ̂0(σ) in Eq. (4.9b) is only a slightly-rewritten form of that given in Eq. (3.16b). We are now in a position to confirm our suspicions in the previous sec- tion about the simplest orbifold-strings, described earlier at ĉ = 52 by the extended Virasoro generators: :Ĵǫav p+ v+ǫ Ĵ−ǫ,b,u−v m− p+ u−v−ǫ ,0, u = 0, 1, ǫ = 0, 1. (4.11) These are now equivalently described by the choice d = 26 in Eq. (4.9), in which case only the second (ordinary) term of Eq. (4.9a) is non-zero – and then the equivalent physical-state condition (4.6) verifies that the physical spectrum of each of these particular twisted ĉ = 52 strings is indeed equiva- lent to that of an ordinary untwisted c = 26 string! These cases include the open-string orientation-orbifold sectors corresponding to τ− × (ω = ±1l) in Eq. (3.10) and their T-duals, as well as the twisted closed-string sectors of the generalized Z2-permutation orbifolds corresponding to τ+ × (ω = ±1l). Additionally, consider the following special cases of the extended Virasoro generators (3.16) at ĉ = 52 = δm+u + (4.12) Gab(24) :Ĵǫav p+ v+ǫ Ĵ−ǫ,b,u−v m− p+ u−v−ǫ :Ĵ̂v p+ ̂+v Ĵ−̂,u−v m− p+ u−v−̂ which result when the extra automorphism in the symmetric theory acts as ω = ±1l on the first d = 24 labels and the non-trivial element of a Z2 on the remaining 2 spatial labels. I have noted in Sec. 3 that δ̂0(σ) = 0 for these cases as well, and indeed the equivalent c = 26 description (4.6) and (4.9) at d = 24 now shows that the open and closed orbifold- strings of this type also have the spectrum of ordinary untwisted c = 26 strings. The common thread for the orbifold-strings in Eqs. (4.10) and (4.12) is that they are at most half-integer moded, so that the shift {n/ρ} → {2n/ρ} gives integer moding in the c = 26 description. Beyond these simple cases, the ĉ = 52 strings are apparently new – with δ̂0(σ) 6= 0, unfamiliar ground-state mass-squares, and fractional modeing (and increments) in either description. 5 Conclusions We have discussed the physical spectrum of the general ĉ = 52 orbifold-string, as well as an equivalent but unconventionally-twisted c = 26 description of the twisted ĉ = 52 matter. The equivalent c = 26 description holds only for the orbifold-string theories – restricted by the extended physical- state conditions (1.1) – and not in the larger Hilbert space of the underlying orbifold conformal field theories. In general we have found that the spectra of these orbifold-string systems are unfamiliar. One simple and unexpected conclusion however is that, as string theories restricted by the extended physical-state conditions, the single twisted ĉ = 52 sector of each of the simplest orbifolds of permutation-type (see Eq. (2.9)) (1; τ∓) (5.1a) (1; τ∓ × ω2), ω 2 = 1 (5.1b) have the same physical spectra as ordinary untwisted c = 26 strings. No such equivalence is found of course in the half-integer moded Hilbert space of the full orbifold CFT’s. The list in Eq. (5.1) includes the simplest orientation orbifolds (with τ−) and their T-duals, as well as the simplest generalized Z2-permutation orbifolds (with τ+). For the simplest orientation orbifolds in particular, the string theories in Eq. (5.1) consist of an ordinary unoriented closed string (the unit element) at c = 26 and a ĉ = 52 twisted open string whose physical spectrum is equivalent to that of an ordinary untwisted c = 26 critical open string. Since both the closed- and open-string spectra of these simple orientation orbifolds are equivalent to those of the archtypal orientifold (without Chan-Paton fac- tors), we are led to suspect that orientation orbifolds include orientifolds. I will return in the next paper of this series to consider this question at the interacting level, where we will also be able to ask about the decoupling of null physical states. Following that, I will consider in a succeeding paper the corresponding situation and modular invariance for the simplest permutation orbifold-string systems. More generally, we have seen that there are many other orientation orb- ifolds, open-string Z2-permutation orbifolds and generalized Z2-permutation orbifolds whose ĉ = 52 spectra show fractional modeing in both the ĉ = 52 and the c = 26 descriptions. These include in particular the orbifolds in Eq. (2.9) when the order n of the extra automorphism is greater than two. There is more to say about no-ghost theorems for the general twisted ĉ = 52 string. The original intuition [16] was that the doubled gauges ū = 0, 1 of the extended physical state condition (1.1) could remove the doubled set of negative-norm states (time-like modes) of the ĉ = 52 strings – which are also associated with ū = 0, 1. For the simplest ĉ = 52 strings in Eq. (5.1), this intuition is certainly born out [20]. More generally, the equivalent c = 26 description of each spectrum shows that both aspects of the doubling are indeed eliminated at the same time, leaving us with the conventional physical state condition (4.6) and only a single set of time-like modes. This is clearly visible in the set of examples (4.9), where the only time-like modes (a = 0) are included in the second term. For the general ĉ = 52 string, the reader should bear in mind that the twisted metric G in Eq. (4.2) is only a unitary transformation (2.5) of the untwisted metric G with a single time-like direction. Although not yet a proof, and illustrated here only for ĉ = 52, I consider this a stronger form of the original arguments [16] that all the critical orbifolds of permutation-type should be free of negative-norm states. The next question I wish to address is the following: I have empha- sized that the equivalent c = 26 Virasoro generators (4.2) are generically- unconventional, being summed over the matter-field fractions {2n/ρ} instead of the conventional orbifold fractions {n/ρ}, but are they actually new Vira- soro generators? I do not know the answer to this question in general, but at least some of them can in fact be re-expressed by further mode-relabeling in terms of more familiar Virasoro generators. As examples, consider the special case of the “larger subset” (4.9) when ω(perm) is one of the elements of order λ of each cyclic group Zλ. (These are the particular, single-cycle elements of Zλ with f0(σ) = λ.) When λ is odd, one finds that the first and third terms of (4.9) can in fact be re-expressed in terms of the conven- tional Virasoro generators associated to a twisted sector of an ordinary cyclic permutation orbifold U(1)λ/Zλ [christ] Lλ(M) = Q+ ̂ M −Q− ̂ :M (5.2) +δM,0 , c = λ = 2l + 1 where I have relabeled the currents J̂ ≡ J̂0. To obtain this result from (4.9), one needs the fact that {2̂/λ} ≃ {̂/λ} modulo the integers when λ is odd. This observation is consistent with the ground-state mass-squares for prime λ in Eq. (3.19). When λ is even, I have also checked that the first and third terms of (4.9) can be re-expressed as the sum of two identical commuting Virasoro generators of this type Lλ(M) = Lλ (M) + L̃λ (M), c = λ = 2l (5.3) each of which is associated to a twisted sector of U(1)λ/2/Zλ/2. This result is also obtained by relabeling the modes modulo the integers, and provides us with another way to understand that the ground-state mass-squared is unshifted when ω(perm) is the non-trivial element of a Z2. My final remark is a conjecture, that the extended physical-state condi- tions for the twisted strings at ĉ = 26λ, λ prime will in fact read m+ ̂ |χ〉 = 0, ̂ = 0, 1, . . . , λ− 1 (5.4a) âλ ≡ 13λ2 − 1 (5.4b) m+ ̂ n+ l̂ m− n+ ̂−l̂ ̂+l̂ m+ n + ̂+l̂ + (5.4c) + 26λ m+ ̂ m+ ̂ ̂+l̂ where Eq. (5.4c) is an orbifold Virasoro algebra [1,18,9] of order λ. This form includes the correct generators {L̂̂} corresponding to the classical extended Polyakov constraints of Ref. [16], and includes the correct value â2 = 17/8 studied here for the ĉ = 52 strings. I obtained the system (5.4) by requiring (as we now know for λ = 2) that it map by the inverse of the order-λ orbifold-induction procedure [1] to the conventional physical-state condition (4.6) with â1 = 1 at c = 26. 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Rep. 13 (1974) 259. http://arxiv.org/abs/hep-th/0309101 http://arxiv.org/abs/hep-th/0402108 http://arxiv.org/abs/hep-th/0406003 http://arxiv.org/abs/hep-th/0703044 http://arxiv.org/abs/hep-th/0703208 http://arxiv.org/abs/hep-th/9703030 Introduction The Twisted Virasoro Generators of =52 Strings First Discussion of the = 52 String Spectra Equivalent c = 26 Description of the = 52 Spectra Conclusions

L’espace Riemannien et pseudo-Riemannien non symétrique SO(2m)/Sp(m). Elisabeth REMM ∗- Michel GOZE † Université de Haute Alsace, F.S.T. 4, rue des Frères Lumière - 68093 MULHOUSE - France 2000 Mathematics Subject Classification. Primary 53C30, Secondary 53C20, 53C50, 17Bxx Mots clés. Homogeneous manifolds, Riemannian structures, non symmetric spaces. Abstract In this work, we are interested in a non symmetric homogeneous space, namely SO(2m)/Sp(m). We show that this space admits a structure of (Z2) 2-symmetric space. We describe all the non degenerated metrics and classify the Riemannian and Lorentzian ones. Dans ce travail, nous nous intéressons à un espace homogène non symétrique, à savoir SO(2m)/Sp(m). Nous montrons que cet espace admet une structure d’espace 2-symétrique. Nous décrivons toutes les métriques non dégénérées et classons les métriques riemanniennes et lorentziennes. 1 Rappel. Espaces Γ-symétriques Soit Γ un groupe abélien fini. Un espace Γ-symétrique est un espace homogène M = G/H où G est un groupe de Lie connexe et H un sous-groupe fermé de G tel qu’il existe un homomorphisme injectif de groupes ρ : γ ∈ Γ −→ ρ(γ) ∈ Aut(G), tel que le groupe H vérifie (GΓ)1 ⊂ H ⊂ G Γ où GΓ = {g ∈ G/∀γ ∈ Γ, ρ(γ)(g) = g} et (GΓ)1 sa composante connexe passant par l’élément neutre 1 de G. On a en particulier: ρ(γ1) ◦ ρ(γ2) = ρ(γ1 · γ2), ρ(eΓ) = Id où eΓ est l’élément neutre de Γ, ∀γ ∈ Γ, ρ(γ)(g) = g ⇐⇒ g ∈ H dès que H est connexe. ∗corresponding author: e-mail: E.Remm@uha.fr †M.Goze@uha.fr. http://arxiv.org/abs/0704.1541v1 Si M = G/H est un espace Γ-symétrique, alors l’algèbre de Lie g de G est Γ-graduée: g = ⊕γ∈Γgγ [gγ1 , gγ2 ] ⊂ gγ1·γ2 geΓ = h où h est l’algèbre de Lie de H . On dit que la paire (g, h) est l’espace Γ-symétrique local de G/H . Si G est simplement connexe alors toute algèbre de Lie Γ-graduée g = ⊕γ∈Γgγ définit une paire (g, ge) qui est l’espace Γ-symétrique local d’un espace Γ-symétrique G/H . Soit M = G/H un espace Γ-symétrique. En tout point x de M on peut définir un sous-groupe Γx de Diff(M) isomorphe à Γ tel que x soit le seul point fixe commun à tous les éléments sγ,x de Γx. Les premiers exemples d’espaces Γ-symétriques lorsque Γ n’est pas cyclique correspon- dent au cas où Γ = (Z2) 2. Rappelons que tout espace Z2-symétrique n’est rien d’autre qu’un espace symétrique. Exemple. La sphère S3. Nous savons que la sphère S3 (ou plus généralement Sn) est munie d’une structure d’espace symétrique. Il suffit de considérer un difféomorphisme de Sn sur l’espace homogène SO(n+ 1)/SO(n). L’algèbre de Lie so(n+ 1) admet une Z2-graduation so(n+ 1) = so(n)⊕m où so(n) est l’ensemble des points fixes de l’automorphisme involutif τa : so(n+ 1) → so(n+ 1) donné par τa(A) = SAS avec S = −1 0n 0n In et In est la matrice identité d’ordre n. Nous pouvons munir la sphère S3 d’une structure d’espace (Z2) 2- symétrique en con- sidérant un difféomorphisme de S3 sur l’espace homogène SO(4)/Sp(2). Considérons sur l’algèbre de Lie so(4) les automorphismes τa, τb, τc = τa ◦ τb donnés par τa : M ∈ SO(4) 7→ J a MJa τb : M ∈ SO(4) 7→ J 0 −1 0 0 1 0 0 0 0 0 0 1 0 0 −1 0 et Jb = 0 0 0 1 0 0 −1 0 0 1 0 0 −1 0 0 0 La famille {Id, τa, τb, τc} détermine un sous-groupe de Aut(so(4)) isomorphe à (Z2) 2. Si ga = {M ∈ so(4) / τa(M) = M, τb(M) = −M} , gb = {M ∈ so(4) / τa(M) = −M, τb(M) = M} gc = {M ∈ so(4) / τa(M) = −M, τb(M) = −M} , on a la décomposition (Z2) 2-symétrique de so(4): so(4) = sp(2)⊕ ga ⊕ gb ⊕ gc. En effet l’ensemble de points fixes pour τa, τb et τc est la sous-algèbre de so(4) dont les éléments sont les matrices 0 −a2 −a3 −a4 a2 0 −a4 a3 a3 a4 0 −a2 a4 −a3 a2 0 Cette sous-algèbre est isomorphe à sp(2). Ainsi (so(4), sp(2)) est un espace local (Z2) symétrique déterminant une structure d’espace (Z2) 2-symétrique sur la shpère S3 = SO(4)/Sp(2). La structure d’espace symétrique sur S3 est liée à l’existence en tout point x ∈ S3 d’une symétrie sx ∈ Diff(S 3) vérifiant s2x = Id et x est le seul point tel que sx(x ′) = x′ (seul point fixe). La structure d’espace (Z2) 2-symétrique de S3 donne l’existence d’un sous-groupe Γx de Diff(S3) de ”symétries” de S3: Γx = {id, sa,x, sb,x, sc,x} s2x,a = s = s2x,c = Id sx,a ◦ sx,b = sx,b ◦ sx,a = sx,c sx,b ◦ sx,c = sx,c ◦ sx,b = sx,a sx,a ◦ sx,c = sx,c ◦ sx,a = sx,b et x est le seul point fixe commun à toutes les symétries: (sx,a(x ′) = sx,b(x ′) = sx,c(x ′) = x′) ⇔ x = x′. Notons que chacune des symétries peut avoir un ensemble de points fixes non réduit à {x} et donc chacune des symétries ne déterminent pas une structure d’espace symétrique sur S3. Déterminons ces symétries. Pour tout γ ∈ Γ = (Z2) 2 = {e, a, b, c}, considérons l’automorphisme de SO(4) donné par ρa(A) = J a AJa ρb(A) = J ρc(A) = ρa ◦ ρb(A) ρe(A) = A Si x = [A] désigne la classe dans l’espace homogène SO(4)/Sp(2) de la marice A ∈ SO(4), alors sa,x[A] = [J a AJa], sb,x[A] = [J AJb], sc,x[A] = [J c AJc], où Jc = JaJb. Ainsi chacune des symétries a un grand cercle comme variété de points invariants. 2 Espaces riemanniens Γ-symétriques Soit (M = G/H,Γ) un espace homogène Γ-symétrique. Définition 1 Une métrique riemannienne g sur M est dite adaptée à la structure Γ- symétrique si chacune des symétries sγ,x est une isométrie. Si ▽g est la connexion de Levi-Civita de g, cette connexion ne cöıncide pas en général avec la connexion canonique ▽ de l’espace homogène (Γ-symétrique). Ces deux connexions cöıncident si et seulement si g est naturellement réductive. Par exemple dans le cas de la sphère S3 considérée comme espace (Z2) 2-symétrique, les métriques adaptées à cette structure sont les métriques sur SO(4)/Sp(2) invariantes par SO(4) chacune étant définie par une forme bilinéaire symétrique B sur so(4) qui est ad(sp(2))-invariante. Si so(4) = sp(2) ⊕ ga ⊕ gb ⊕ gc est la décomposition (Z2) 2-graduée correspondante, le fait de dire que sur S3 les symétries sγ,x sont des isométries est équivalent à dire que les espaces ge, ga, gb, gc sont deux à deux orthogonaux pour B. Décrivons en détail cette graduation: sp(2) = 0 −a2 −a3 −a4 a2 0 −a4 a3 a3 a4 0 −a2 a4 −a3 a2 0 , ga = 0 0 0 x 0 0 −x 0 0 x 0 0 −x 0 0 0 0 y 0 0 −y 0 0 0 0 0 0 −y 0 0 y 0 et gc = 0 0 z 0 0 0 0 z −z 0 0 0 0 −z 0 0 Si {A1, A2, A3, X, Y, Z} est une base adaptée à cette graduation et si {α1, α2, α3, ω1, ω2, ω3} en est la base duale alors B |ga⊕gb⊕gc= λ 1 + λ 2 + λ La métrique correspondante sera naturellement réductive si et seulement si λ1 = λ2 = λ3 et dans ce cas-là elle correspond à la restriction de la forme de Killing Cartan. 3 Métriques adaptées à la structure (Z2) 2-symétrique de SO(2m)/Sp(m) 3.1 La graduation (Z2) 2-symétrique Considérons les matrices −In 0 , Xa = , Xb = , Xc = Soit M ∈ so(2m). Les applications τa(M) = J a MJa τb(M) = J τa(M) = J c MJc où Ja = Sm ⊗ Xa, Jb = Sm ⊗ Xb, Jc = Sm ⊗ Xc sont des automorphismes involutifs de so(2m) qui commutent deux à deux. Ainsi {Id, τa, τb, τc} est un sous groupe de Aut(so(2m)) isomorphe à (Z2) 2. Il définit donc une (Z2) 2-graduation so(2m) = ge ⊕ ga ⊕ gb ⊕ gc ge = {M ∈ so(2m) / τa(M) = τb(M) = τc(M) = M} ga = {M ∈ so(2m) / τa(M) = τc(M) = −M, τb(M) = M} gb = {M ∈ so(2m) / τb(M) = τc(M) = −M, τa(M) = M} gc = {M ∈ so(2m) / τa(M) = τb(M) = −M, τc(M) = M} Ainsi A1 B1 A2 B2 −B1 A1 B2 −A2 −tA2 − tB2 A1 B1 tA2 −B1 A1 tA1 = −A1, tB1 = B1 tA2 = A2, tB2 = B2 X1 Y1 Z1 T1 Y1 −X1 −T1 Z1 tT1 −X1 −Y1 −tT1 − tZ1 −Y1 X1 tX1 = −X1, tY1 = −Y1 tZ1 = −Z1, tT1 = T1 X2 Y2 Z2 T2 −Y2 X2 T2 Z2 −tZ2 − tT2 −X2 −Y2 −tT2 − tZ2 Y2 −X2 tX2 = −X2, tY2 = Y2 tZ2 = −Z2, tT2 = −T2 X3 Y3 Z3 T3 Y3 −X3 −T3 Z3 tT3 X3 Y3 −tT3 − tZ3 Y3 −X3 tX3 = −X3, tY3 = −Y3 tZ3 = Z3, tT3 = −T3 Notons que dimge = m(2m+ 1), dimga = dimgb = dimgc = m(2m− 1). Proposition 2 Dans cette graduation ge est isomorphe à sp(m) et toute (Z2) 2-graduation de so(2m) telle que ge soit isomorphe à sp(m) est équivalente à la graduation ci-dessus. En effet ge est simple de rang m et de dimension m(2m+1). La deuxième partie résulte de la classification donnée dans [1] et [2]. Corollaire 3 Il n’existe, à équivalence près, qu’une seule structure d’espace homogène (Z2) symétrique sur l’espace homogène compact SO(2m)/Sp(m). Cette structure est associée à l’existence en tout point x de SO(2m)/Sp(m) d’un sous- groupe de Diff(M) isomorphe à (Z2) 2. Notons Γx ce sous-groupe. Il est entièrement défini dès que l’on connait Γ1̄ où 1̄ est la classe dans SO(2m)/Sp(m) de l’élément neutre 1 de SO(2m). Notons Γ1̄ = se,1̄, sa,1̄, sb,1̄, sc,1̄ les symétries sγ,1̄(x) = π(ργ(A)) où π : SO(2m) → SO(2m)/Sp(m) est la submersion canonique, x = π(A) et ργ est un automorphisme de SO(2m) dont l’application tangente en 1 coincide avec τγ . Ainsi ρa(A) = J a AJa ρb(A) = J ρc(A) = J c AJc Si B ∈ π(A) alors il existe P ∈ Sp(m) tel que B = AP . On a J−1a BJa = J a AJaJ a PJa = J−1a AJa car P est invariante pour tous les automorphismes ρa, ρb, ρc. 3.2 Structure métrique (Z2) 2-symétrique Une métrique non dégénérée g invariante par SO(2m) sur SO(2m)/Sp(m) est adaptée à la 2-structure si les symétries sx,γ sont des isométries c’est-à-dire si les automorphismes ργ induisent des isométries linéaires. Ceci implique que g soit définie par une forme bilinéaire symétrique non dégénérée B sur ga ⊕ gb ⊕ gc telle que les espaces ga, gb, gc soient deux à deux orthogonaux. Déterminons toutes les formes bilinéaires B vérifiant les hypothèses ci-dessus. Une telle forme s’écrit donc B = Ba +Bb +Bc où Ba(resp. Bb, resp. Bc ) est une forme bilinéaire symétrique non dégénérée invariante par ge dont le noyau contient gb ⊕ gc (resp. ga ⊕ gc, resp. ga ⊕ gb). 3.3 Exemples 1) Dans le cas de la sphère SO(4)/Sp(2) la métrique adaptée à la structure (Z2) 2-symétrique est définie par la forme bilinéaire B sur ga⊕gb⊕gc qui est ad(sp(2))-invariante. Nous avons vu qu’une telle forme s’écrivait B = λ21ω 1 + λ 2 + λ Elle est définie positive si et seulement si les coefficients λi sont positifs ou nuls. 2) Considérons l’espace (Z2) 2-symétrique compact SO(8)/Sp(4). Afin de fixer les nota- tions écrivons la (Z2) 2-graduation de so(8) ainsi: X1 Y1 Z1 T1 Y1 −X1 −T1 Z1 tT1 −X1 −Y1 −tT1 − tZ1 −Y1 X1 tX1 = −X1, tY1 = −Y1 tZ1 = −Z1, tT1 = T1 X2 Y2 Z2 T2 −Y2 X2 T2 Z2 −tZ2 − tT2 −X2 −Y2 tZ2 Y2 −X2 tX2 = −X2, tY2 = Y2 tZ2 = −Z2, tT2 = −T2 X3 Y3 Z3 T3 Y3 −X3 −T3 Z3 tT3 X3 Y3 −tT3 − tZ3 Y3 −X3 tX3 = −X3, tY3 = −Y3 tZ3 = Z3, tT3 = −T3 et pour la matrice Xi (resp. Yi, Zi, Ti) on notera Xi = −xi 0 si elle est anti- symétrique ou Xi = x1i x x2i x si elle est symétrique, c’est à dire Xi = Enfin on notera par les lettres αi, βi, γi, δi les formes linéaires duales des vecteurs définis respectivement par les matrices Xi, Yi, Zi, Ti. Ainsi si Xi est antisymétrique, la forme duale correspondante sera notée αi, et si Xi est symétrique, les formes duales α i , α i , α i cor- respondent aux vecteurs . Ceci étant la forme B s’écrit Ba+Bb+Bc où la forme Bγ a pour noyau gγ1 ⊕gγ2 avec gγ 6= gγ1 et gγ 6= gγ2 . Déterminons Ba. Comme elle est invariante par ad(sp(2)) on obtient: Ba(X1, Y1) = Ba(X1, Z1) = Ba(X1, T 1) = 0 Ba(Y1, Z1) = Ba(Y1, T 1) = 0 Ba(Z1, T 1) = Ba(T 1 ) = 0pour i = 1, 3 Ba(X1, X1) = Ba(Y1, Y1) = Ba(Z1, Z1) = Ba(T 1 , T 1 ) = 0 1 , T 1 ) = Ba(T 1 , T Ba(X1, X1) = 2Ba(T 1 , T 1 )− 2Ba(T 1 , T Ainsi la forme quadratique associée s’écrit qga = λ1(α 1 + β 1 + γ 1 + (δ 2) + λ2((δ 2) + (δ31) 2) + (λ2 − )((δ11)(δ qga = λ1(α 1 + β 1 + γ 1 + (δ 2) + ( )(δ11 + δ 2 + ( )(δ11 − δ De même nous aurons qgb = λ3(α 2 + (β 2 + γ22 + δ 2) + ( )(β12 + β 2 + ( )(β12 − β qgc = λ5(α 3 + β 3 + (γ 2 + δ23) + ( )(γ13 + γ 2 + ( )(γ13 − γ Remarques. 1. La forme B définit une métrique riemannienne si et seulement si λ2p > λ2p−1 pour p = 1, 2, 3. Si cette contrainte est relachée, la forme B, supposée non dégénérée, peut définir une métrique pseudo riemannienne sur l’espace (Z2) 2-symétrique. Nous verrons cela dans le dernier paragraphe. 2. Considérons le sous-espace ge ⊕ ga. Comme [ga, ga] ⊂ ge c’est un sous-algèbre de so(8) (ou plus généralement de g) admettant une stucture symétrique. La forme Ba induit donc une structure riemannienne ou pseudo-riemannienne sur l’espace symétrique associé à l’espace symétrique local (ge, ga). Dans l’exemple précédent ge ⊕ ga est la sous-algèbre de so(8) donnée par les matrices: X1 X3 X4 X5 −tX3 X2 X6 − −tX4 − tX6 X2 −tX5 X4 − tX3 X2 tX1 = −X1, tX2 = −X2 tX5 = X5, tX6 = X6 Dans [3], on détermine les espaces réels en étudiant ces structures symétriques ge ⊕ ga données par deux automorphismes commutant de g. En effet si g est simple réelle et si σ est un automorphisme involutif de g, il existe une sous-algèbre compacte maximale g1 de g qui est invariante par σ et l’étude des espaces locaux symétriques (g, ge) se ramène à l’étude des espaces locaux symétriques (g1, g11) où g1 est compacte. Dans ce cas g est définie à partir de g1 par un automorphisme involutif τ commutant avec l’automorphisme σ. Ici notre approche est en partie similaire mais le but est de regarder la structure des espaces non symétrique associés aux paires (g, ge). Dans le cas particulier de l’espace (Z2) 2-symétrique compact SO(8)/Sp(4) l’algèbre de Lie ge⊕ga est isomorphe à so(4)⊕R où R désigne l’algèbre abélienne de dimension 1. Notons également que chacun des espaces symétriques ge⊕ga, ge⊕gb, ge⊕gc est isomorphe à so(4)⊕ R. Mais ceci n’est pas général, les algèbres symétriques peuvent ne pas être isomorphes ni même de même dimension. L’espace symétrique compact connexe associé est l’espace homogène Su(4)/Sp(2) × T où T est le tore à une dimension. C’est un espace riemannien symétrique compact non irréductible. La métrique qga définie précédemment correspond à une métrique riemannienne ou pseudo riemannienne sur cet espace. La restriction au premier facteur correspond à la métrique associée à la forme de Killing Cartan sur su(4). Elle correspond à λ2 = 3.4 Cas général: métriques adaptées sur SO(2m)/Sp(m) Notations. Nous avons écrit une matrice générale de ga sous la forme (3.1). Si on note (X1, Y1, Z1, T1) un élément de ga, on considère la base de ga, {X1,ij , Y1,ij , Z1,ij , T1,ij} cor- respondant aux matrices élémentaires . La base duale sera notée (αa,ij , βa,ij , γa,ij , δa,ij). Rappelons que X1, Y1, Z1 sont antisymétriques alors que T1 est symétrique. Les crochets correspondent aux représentations de so(m ) sur lui-même ou de so(m ) sur l’espace des matrices symétriques. On aura donc qga = λ (α2a,ij + β a,ij + γ a,ij) + δ2a,ij) + λ a,ii) + (λ (δa,iiδa,jj). Les formes qgb et qgc admettent une décomposition analogue, en tenant compte du fait que dans gb ce sont les matrices Y2 qui sont symétriques et pour ga les matrices Z1 (3.1). On note (αb,ij , βb,ij , γb,ij , δb,ij) la base duale de {X2,ij, Y2,ij , Z2,ij, T2,ij} et par (αc,ij , βc,ij, γc,ij , δc,ij) la base duale de {X3,ij , Y3,ij , Z3,ij , T3,ij}. Proposition 4 Toute métrique non dégénéré adaptée à la structure (Z2) 2-symétrique de l’espace homogène SO(2m)/Sp(m) est définie à partir de la forme bilinéaire ad(ge)-invariante sur ga ⊕ gb ⊕ gc B = qga + qgb + qgb avec qga = λ (α2a,ij + β a,ij + γ i6=j δ a,ij) + λ a,ii) + (λ (δa,iiδa,jj) qgb = λ + γ2ij) + δ i6=j β ) + λb2(β ) + (λb2 − (βb,iiβb,jj) qgc = λ (β2c,ij + γ c,ij) + δ c,ij + i6=j α c,ij) + λ c,ii) + (λ (αc,iiαc,jj) 4 Métriques pseudo-riemanniennes (Z2) 2-symétriques sur SO(2m)/Sp(m) 4.1 Signature des formes qgγ Soit γ ∈ {a, b, c}. Les valeurs propres de la forme qgγ sont µ1,γ = λ 1 , µ2,γ = λ 2/2 + λ 1/4, µ3,γ = λ r + 1 r − 1 où r est l’ordre commun des matrices symétriques X4, Y2, Z1. Ces valeurs propres sont respectivement de multiplicité dimgγ − r, r − 1, 1. Le signe des valeurs propres µ2,γ et µ3,γ est donc µ2,γ > 0 ⇐⇒ λ 2 > −λ µ3,γ > 0 ⇐⇒ λ 2 > −λ 2(r+1) On en déduit, si s(q) désigne la signature de la forme quadratique q : s(qgγ ) = (dimgγ , 0) ⇔ (λ 1 > 0, λ 2 > λ 2(r+1) = (dimgγ − 1, 1) ⇔ (λ 1 > 0, −λ 1/2 < λ 2 < λ 2(r+1) = (dimgγ − r, r) ⇔ (λ 1 > 0, λ 2 < −λ = (r, dimgγ − r) ⇔ (λ 1 < 0, λ 2 > −λ = (1, dimgγ − 1) ⇔ (λ 1 < 0, λ 2(r+1) 2 < −λ = (0, dimgγ) ⇔ (λ 1 < 0, λ 2 < λ 2(r+1) Notons que µ2,γ = µ3,γ si et seulement si λ 1 = 2λ 4.2 Classification des métriques riemanniennes adaptées sur SO(2m)/Sp(m) Comme r = m 2+m−2 m2+m+2 on a le résultat suivant Théorème 5 Toute métrique riemannienne sur SO(2m)/Sp(m) adaptée à la structure 2-symétrique est définie à partir de la forme bilinéaire sur gaø plusgb ⊕ gc B = qga(λ 1 , λ 2) + qgb(λ 2) + qgb(λ 1 > 0 2 > λ 2+m−2 2(m2+m+2 pour tout γ ∈ {a, b, c}. Pour une telle métrique, la connexion de Levi-Civita ne cöıncide pas en général avec la connexion canonique associée à la structure (Z2) 2-symétrique ( [2]). Ces deux connexions sont les mêmes si et seulement si la mt́rique riemannienne est naturellement réductive. Elle correspond donc à la restriction de la forme de Killing (au signe près) de SO(2m). Cette métrique correspond à la forme bilinéaire B définie par les paramètres λa1 = λ 1 = λ 1 = 2λ 2 = 2λ 2 = 2λ 4.3 Classification des métriques lorentziennes adaptées sur l’espace SO(2m)/Sp(m) Les métriques lorentziennes adaptées à la structure (Z2) 2-symétrique sont définies par les formes bilinéaires B, définies dans la section précédente, dont la signature est (dim(ga) + dim(gb) + dim(gc)− 1, 1). On a donc Théorème 6 Toute métrique lorentzienne sur SO(2m)/Sp(m) adaptée à la structure (Z2) symétrique est définie par l’une des formes bilinéaires B = qga(λ 1 , λ 2) + qgb(λ 2) + qgb(λ ∀γ ∈ {a, b, c}, λ 1 > 0 ∃γ0 ∈ {a, b, c} tel que − λ 1 /2 < λ 2 < λ 2(r+1) ∀γ 6= γ0, λ 2 > λ 2(r+1) References [1] Bahturin Y., Goze M., Γ-symmetric homogeneous spaces. Preprint Mulhouse (2006). [2] Bouyakoub A., Goze M., Remm E. On Riemannian nonsymmetric spaces and flag manifolds. Preprint Mulhouse (2006). [3] Berger M., Les espaces symétriques non compacts, Ann.E.N.S. 74, 2, (1957), 85-177. Rappel. Espaces -symétriques Espaces riemanniens -symétriques Métriques adaptées à la structure (Z2)2-symétrique de SO(2m)/Sp(m) La graduation (Z2)2-symétrique Structure métrique (Z2)2-symétrique Exemples Cas général: métriques adaptées sur SO(2m)/Sp(m) Métriques pseudo-riemanniennes (Z2)2-symétriques sur SO(2m)/Sp(m) Signature des formes qg Classification des métriques riemanniennes adaptées sur SO(2m)/Sp(m) Classification des métriques lorentziennes adaptées sur l'espace SO(2m)/Sp(m)

In this work, we are interested in a non symmetric homogeneous space, namely $SO(2m)/Sp(m)$. We show that this space admits a structure of $Z_2^2$-symmetric space. We describe all the non degenerated metrics and classify the Riemannian and Lorentzian ones.

L’espace Riemannien et pseudo-Riemannien non symétrique SO(2m)/Sp(m). Elisabeth REMM ∗- Michel GOZE † Université de Haute Alsace, F.S.T. 4, rue des Frères Lumière - 68093 MULHOUSE - France 2000 Mathematics Subject Classification. Primary 53C30, Secondary 53C20, 53C50, 17Bxx Mots clés. Homogeneous manifolds, Riemannian structures, non symmetric spaces. Abstract In this work, we are interested in a non symmetric homogeneous space, namely SO(2m)/Sp(m). We show that this space admits a structure of (Z2) 2-symmetric space. We describe all the non degenerated metrics and classify the Riemannian and Lorentzian ones. Dans ce travail, nous nous intéressons à un espace homogène non symétrique, à savoir SO(2m)/Sp(m). Nous montrons que cet espace admet une structure d’espace 2-symétrique. Nous décrivons toutes les métriques non dégénérées et classons les métriques riemanniennes et lorentziennes. 1 Rappel. Espaces Γ-symétriques Soit Γ un groupe abélien fini. Un espace Γ-symétrique est un espace homogène M = G/H où G est un groupe de Lie connexe et H un sous-groupe fermé de G tel qu’il existe un homomorphisme injectif de groupes ρ : γ ∈ Γ −→ ρ(γ) ∈ Aut(G), tel que le groupe H vérifie (GΓ)1 ⊂ H ⊂ G Γ où GΓ = {g ∈ G/∀γ ∈ Γ, ρ(γ)(g) = g} et (GΓ)1 sa composante connexe passant par l’élément neutre 1 de G. On a en particulier: ρ(γ1) ◦ ρ(γ2) = ρ(γ1 · γ2), ρ(eΓ) = Id où eΓ est l’élément neutre de Γ, ∀γ ∈ Γ, ρ(γ)(g) = g ⇐⇒ g ∈ H dès que H est connexe. ∗corresponding author: e-mail: E.Remm@uha.fr †M.Goze@uha.fr. http://arxiv.org/abs/0704.1541v1 Si M = G/H est un espace Γ-symétrique, alors l’algèbre de Lie g de G est Γ-graduée: g = ⊕γ∈Γgγ [gγ1 , gγ2 ] ⊂ gγ1·γ2 geΓ = h où h est l’algèbre de Lie de H . On dit que la paire (g, h) est l’espace Γ-symétrique local de G/H . Si G est simplement connexe alors toute algèbre de Lie Γ-graduée g = ⊕γ∈Γgγ définit une paire (g, ge) qui est l’espace Γ-symétrique local d’un espace Γ-symétrique G/H . Soit M = G/H un espace Γ-symétrique. En tout point x de M on peut définir un sous-groupe Γx de Diff(M) isomorphe à Γ tel que x soit le seul point fixe commun à tous les éléments sγ,x de Γx. Les premiers exemples d’espaces Γ-symétriques lorsque Γ n’est pas cyclique correspon- dent au cas où Γ = (Z2) 2. Rappelons que tout espace Z2-symétrique n’est rien d’autre qu’un espace symétrique. Exemple. La sphère S3. Nous savons que la sphère S3 (ou plus généralement Sn) est munie d’une structure d’espace symétrique. Il suffit de considérer un difféomorphisme de Sn sur l’espace homogène SO(n+ 1)/SO(n). L’algèbre de Lie so(n+ 1) admet une Z2-graduation so(n+ 1) = so(n)⊕m où so(n) est l’ensemble des points fixes de l’automorphisme involutif τa : so(n+ 1) → so(n+ 1) donné par τa(A) = SAS avec S = −1 0n 0n In et In est la matrice identité d’ordre n. Nous pouvons munir la sphère S3 d’une structure d’espace (Z2) 2- symétrique en con- sidérant un difféomorphisme de S3 sur l’espace homogène SO(4)/Sp(2). Considérons sur l’algèbre de Lie so(4) les automorphismes τa, τb, τc = τa ◦ τb donnés par τa : M ∈ SO(4) 7→ J a MJa τb : M ∈ SO(4) 7→ J 0 −1 0 0 1 0 0 0 0 0 0 1 0 0 −1 0 et Jb = 0 0 0 1 0 0 −1 0 0 1 0 0 −1 0 0 0 La famille {Id, τa, τb, τc} détermine un sous-groupe de Aut(so(4)) isomorphe à (Z2) 2. Si ga = {M ∈ so(4) / τa(M) = M, τb(M) = −M} , gb = {M ∈ so(4) / τa(M) = −M, τb(M) = M} gc = {M ∈ so(4) / τa(M) = −M, τb(M) = −M} , on a la décomposition (Z2) 2-symétrique de so(4): so(4) = sp(2)⊕ ga ⊕ gb ⊕ gc. En effet l’ensemble de points fixes pour τa, τb et τc est la sous-algèbre de so(4) dont les éléments sont les matrices 0 −a2 −a3 −a4 a2 0 −a4 a3 a3 a4 0 −a2 a4 −a3 a2 0 Cette sous-algèbre est isomorphe à sp(2). Ainsi (so(4), sp(2)) est un espace local (Z2) symétrique déterminant une structure d’espace (Z2) 2-symétrique sur la shpère S3 = SO(4)/Sp(2). La structure d’espace symétrique sur S3 est liée à l’existence en tout point x ∈ S3 d’une symétrie sx ∈ Diff(S 3) vérifiant s2x = Id et x est le seul point tel que sx(x ′) = x′ (seul point fixe). La structure d’espace (Z2) 2-symétrique de S3 donne l’existence d’un sous-groupe Γx de Diff(S3) de ”symétries” de S3: Γx = {id, sa,x, sb,x, sc,x} s2x,a = s = s2x,c = Id sx,a ◦ sx,b = sx,b ◦ sx,a = sx,c sx,b ◦ sx,c = sx,c ◦ sx,b = sx,a sx,a ◦ sx,c = sx,c ◦ sx,a = sx,b et x est le seul point fixe commun à toutes les symétries: (sx,a(x ′) = sx,b(x ′) = sx,c(x ′) = x′) ⇔ x = x′. Notons que chacune des symétries peut avoir un ensemble de points fixes non réduit à {x} et donc chacune des symétries ne déterminent pas une structure d’espace symétrique sur S3. Déterminons ces symétries. Pour tout γ ∈ Γ = (Z2) 2 = {e, a, b, c}, considérons l’automorphisme de SO(4) donné par ρa(A) = J a AJa ρb(A) = J ρc(A) = ρa ◦ ρb(A) ρe(A) = A Si x = [A] désigne la classe dans l’espace homogène SO(4)/Sp(2) de la marice A ∈ SO(4), alors sa,x[A] = [J a AJa], sb,x[A] = [J AJb], sc,x[A] = [J c AJc], où Jc = JaJb. Ainsi chacune des symétries a un grand cercle comme variété de points invariants. 2 Espaces riemanniens Γ-symétriques Soit (M = G/H,Γ) un espace homogène Γ-symétrique. Définition 1 Une métrique riemannienne g sur M est dite adaptée à la structure Γ- symétrique si chacune des symétries sγ,x est une isométrie. Si ▽g est la connexion de Levi-Civita de g, cette connexion ne cöıncide pas en général avec la connexion canonique ▽ de l’espace homogène (Γ-symétrique). Ces deux connexions cöıncident si et seulement si g est naturellement réductive. Par exemple dans le cas de la sphère S3 considérée comme espace (Z2) 2-symétrique, les métriques adaptées à cette structure sont les métriques sur SO(4)/Sp(2) invariantes par SO(4) chacune étant définie par une forme bilinéaire symétrique B sur so(4) qui est ad(sp(2))-invariante. Si so(4) = sp(2) ⊕ ga ⊕ gb ⊕ gc est la décomposition (Z2) 2-graduée correspondante, le fait de dire que sur S3 les symétries sγ,x sont des isométries est équivalent à dire que les espaces ge, ga, gb, gc sont deux à deux orthogonaux pour B. Décrivons en détail cette graduation: sp(2) = 0 −a2 −a3 −a4 a2 0 −a4 a3 a3 a4 0 −a2 a4 −a3 a2 0 , ga = 0 0 0 x 0 0 −x 0 0 x 0 0 −x 0 0 0 0 y 0 0 −y 0 0 0 0 0 0 −y 0 0 y 0 et gc = 0 0 z 0 0 0 0 z −z 0 0 0 0 −z 0 0 Si {A1, A2, A3, X, Y, Z} est une base adaptée à cette graduation et si {α1, α2, α3, ω1, ω2, ω3} en est la base duale alors B |ga⊕gb⊕gc= λ 1 + λ 2 + λ La métrique correspondante sera naturellement réductive si et seulement si λ1 = λ2 = λ3 et dans ce cas-là elle correspond à la restriction de la forme de Killing Cartan. 3 Métriques adaptées à la structure (Z2) 2-symétrique de SO(2m)/Sp(m) 3.1 La graduation (Z2) 2-symétrique Considérons les matrices −In 0 , Xa = , Xb = , Xc = Soit M ∈ so(2m). Les applications τa(M) = J a MJa τb(M) = J τa(M) = J c MJc où Ja = Sm ⊗ Xa, Jb = Sm ⊗ Xb, Jc = Sm ⊗ Xc sont des automorphismes involutifs de so(2m) qui commutent deux à deux. Ainsi {Id, τa, τb, τc} est un sous groupe de Aut(so(2m)) isomorphe à (Z2) 2. Il définit donc une (Z2) 2-graduation so(2m) = ge ⊕ ga ⊕ gb ⊕ gc ge = {M ∈ so(2m) / τa(M) = τb(M) = τc(M) = M} ga = {M ∈ so(2m) / τa(M) = τc(M) = −M, τb(M) = M} gb = {M ∈ so(2m) / τb(M) = τc(M) = −M, τa(M) = M} gc = {M ∈ so(2m) / τa(M) = τb(M) = −M, τc(M) = M} Ainsi A1 B1 A2 B2 −B1 A1 B2 −A2 −tA2 − tB2 A1 B1 tA2 −B1 A1 tA1 = −A1, tB1 = B1 tA2 = A2, tB2 = B2 X1 Y1 Z1 T1 Y1 −X1 −T1 Z1 tT1 −X1 −Y1 −tT1 − tZ1 −Y1 X1 tX1 = −X1, tY1 = −Y1 tZ1 = −Z1, tT1 = T1 X2 Y2 Z2 T2 −Y2 X2 T2 Z2 −tZ2 − tT2 −X2 −Y2 −tT2 − tZ2 Y2 −X2 tX2 = −X2, tY2 = Y2 tZ2 = −Z2, tT2 = −T2 X3 Y3 Z3 T3 Y3 −X3 −T3 Z3 tT3 X3 Y3 −tT3 − tZ3 Y3 −X3 tX3 = −X3, tY3 = −Y3 tZ3 = Z3, tT3 = −T3 Notons que dimge = m(2m+ 1), dimga = dimgb = dimgc = m(2m− 1). Proposition 2 Dans cette graduation ge est isomorphe à sp(m) et toute (Z2) 2-graduation de so(2m) telle que ge soit isomorphe à sp(m) est équivalente à la graduation ci-dessus. En effet ge est simple de rang m et de dimension m(2m+1). La deuxième partie résulte de la classification donnée dans [1] et [2]. Corollaire 3 Il n’existe, à équivalence près, qu’une seule structure d’espace homogène (Z2) symétrique sur l’espace homogène compact SO(2m)/Sp(m). Cette structure est associée à l’existence en tout point x de SO(2m)/Sp(m) d’un sous- groupe de Diff(M) isomorphe à (Z2) 2. Notons Γx ce sous-groupe. Il est entièrement défini dès que l’on connait Γ1̄ où 1̄ est la classe dans SO(2m)/Sp(m) de l’élément neutre 1 de SO(2m). Notons Γ1̄ = se,1̄, sa,1̄, sb,1̄, sc,1̄ les symétries sγ,1̄(x) = π(ργ(A)) où π : SO(2m) → SO(2m)/Sp(m) est la submersion canonique, x = π(A) et ργ est un automorphisme de SO(2m) dont l’application tangente en 1 coincide avec τγ . Ainsi ρa(A) = J a AJa ρb(A) = J ρc(A) = J c AJc Si B ∈ π(A) alors il existe P ∈ Sp(m) tel que B = AP . On a J−1a BJa = J a AJaJ a PJa = J−1a AJa car P est invariante pour tous les automorphismes ρa, ρb, ρc. 3.2 Structure métrique (Z2) 2-symétrique Une métrique non dégénérée g invariante par SO(2m) sur SO(2m)/Sp(m) est adaptée à la 2-structure si les symétries sx,γ sont des isométries c’est-à-dire si les automorphismes ργ induisent des isométries linéaires. Ceci implique que g soit définie par une forme bilinéaire symétrique non dégénérée B sur ga ⊕ gb ⊕ gc telle que les espaces ga, gb, gc soient deux à deux orthogonaux. Déterminons toutes les formes bilinéaires B vérifiant les hypothèses ci-dessus. Une telle forme s’écrit donc B = Ba +Bb +Bc où Ba(resp. Bb, resp. Bc ) est une forme bilinéaire symétrique non dégénérée invariante par ge dont le noyau contient gb ⊕ gc (resp. ga ⊕ gc, resp. ga ⊕ gb). 3.3 Exemples 1) Dans le cas de la sphère SO(4)/Sp(2) la métrique adaptée à la structure (Z2) 2-symétrique est définie par la forme bilinéaire B sur ga⊕gb⊕gc qui est ad(sp(2))-invariante. Nous avons vu qu’une telle forme s’écrivait B = λ21ω 1 + λ 2 + λ Elle est définie positive si et seulement si les coefficients λi sont positifs ou nuls. 2) Considérons l’espace (Z2) 2-symétrique compact SO(8)/Sp(4). Afin de fixer les nota- tions écrivons la (Z2) 2-graduation de so(8) ainsi: X1 Y1 Z1 T1 Y1 −X1 −T1 Z1 tT1 −X1 −Y1 −tT1 − tZ1 −Y1 X1 tX1 = −X1, tY1 = −Y1 tZ1 = −Z1, tT1 = T1 X2 Y2 Z2 T2 −Y2 X2 T2 Z2 −tZ2 − tT2 −X2 −Y2 tZ2 Y2 −X2 tX2 = −X2, tY2 = Y2 tZ2 = −Z2, tT2 = −T2 X3 Y3 Z3 T3 Y3 −X3 −T3 Z3 tT3 X3 Y3 −tT3 − tZ3 Y3 −X3 tX3 = −X3, tY3 = −Y3 tZ3 = Z3, tT3 = −T3 et pour la matrice Xi (resp. Yi, Zi, Ti) on notera Xi = −xi 0 si elle est anti- symétrique ou Xi = x1i x x2i x si elle est symétrique, c’est à dire Xi = Enfin on notera par les lettres αi, βi, γi, δi les formes linéaires duales des vecteurs définis respectivement par les matrices Xi, Yi, Zi, Ti. Ainsi si Xi est antisymétrique, la forme duale correspondante sera notée αi, et si Xi est symétrique, les formes duales α i , α i , α i cor- respondent aux vecteurs . Ceci étant la forme B s’écrit Ba+Bb+Bc où la forme Bγ a pour noyau gγ1 ⊕gγ2 avec gγ 6= gγ1 et gγ 6= gγ2 . Déterminons Ba. Comme elle est invariante par ad(sp(2)) on obtient: Ba(X1, Y1) = Ba(X1, Z1) = Ba(X1, T 1) = 0 Ba(Y1, Z1) = Ba(Y1, T 1) = 0 Ba(Z1, T 1) = Ba(T 1 ) = 0pour i = 1, 3 Ba(X1, X1) = Ba(Y1, Y1) = Ba(Z1, Z1) = Ba(T 1 , T 1 ) = 0 1 , T 1 ) = Ba(T 1 , T Ba(X1, X1) = 2Ba(T 1 , T 1 )− 2Ba(T 1 , T Ainsi la forme quadratique associée s’écrit qga = λ1(α 1 + β 1 + γ 1 + (δ 2) + λ2((δ 2) + (δ31) 2) + (λ2 − )((δ11)(δ qga = λ1(α 1 + β 1 + γ 1 + (δ 2) + ( )(δ11 + δ 2 + ( )(δ11 − δ De même nous aurons qgb = λ3(α 2 + (β 2 + γ22 + δ 2) + ( )(β12 + β 2 + ( )(β12 − β qgc = λ5(α 3 + β 3 + (γ 2 + δ23) + ( )(γ13 + γ 2 + ( )(γ13 − γ Remarques. 1. La forme B définit une métrique riemannienne si et seulement si λ2p > λ2p−1 pour p = 1, 2, 3. Si cette contrainte est relachée, la forme B, supposée non dégénérée, peut définir une métrique pseudo riemannienne sur l’espace (Z2) 2-symétrique. Nous verrons cela dans le dernier paragraphe. 2. Considérons le sous-espace ge ⊕ ga. Comme [ga, ga] ⊂ ge c’est un sous-algèbre de so(8) (ou plus généralement de g) admettant une stucture symétrique. La forme Ba induit donc une structure riemannienne ou pseudo-riemannienne sur l’espace symétrique associé à l’espace symétrique local (ge, ga). Dans l’exemple précédent ge ⊕ ga est la sous-algèbre de so(8) donnée par les matrices: X1 X3 X4 X5 −tX3 X2 X6 − −tX4 − tX6 X2 −tX5 X4 − tX3 X2 tX1 = −X1, tX2 = −X2 tX5 = X5, tX6 = X6 Dans [3], on détermine les espaces réels en étudiant ces structures symétriques ge ⊕ ga données par deux automorphismes commutant de g. En effet si g est simple réelle et si σ est un automorphisme involutif de g, il existe une sous-algèbre compacte maximale g1 de g qui est invariante par σ et l’étude des espaces locaux symétriques (g, ge) se ramène à l’étude des espaces locaux symétriques (g1, g11) où g1 est compacte. Dans ce cas g est définie à partir de g1 par un automorphisme involutif τ commutant avec l’automorphisme σ. Ici notre approche est en partie similaire mais le but est de regarder la structure des espaces non symétrique associés aux paires (g, ge). Dans le cas particulier de l’espace (Z2) 2-symétrique compact SO(8)/Sp(4) l’algèbre de Lie ge⊕ga est isomorphe à so(4)⊕R où R désigne l’algèbre abélienne de dimension 1. Notons également que chacun des espaces symétriques ge⊕ga, ge⊕gb, ge⊕gc est isomorphe à so(4)⊕ R. Mais ceci n’est pas général, les algèbres symétriques peuvent ne pas être isomorphes ni même de même dimension. L’espace symétrique compact connexe associé est l’espace homogène Su(4)/Sp(2) × T où T est le tore à une dimension. C’est un espace riemannien symétrique compact non irréductible. La métrique qga définie précédemment correspond à une métrique riemannienne ou pseudo riemannienne sur cet espace. La restriction au premier facteur correspond à la métrique associée à la forme de Killing Cartan sur su(4). Elle correspond à λ2 = 3.4 Cas général: métriques adaptées sur SO(2m)/Sp(m) Notations. Nous avons écrit une matrice générale de ga sous la forme (3.1). Si on note (X1, Y1, Z1, T1) un élément de ga, on considère la base de ga, {X1,ij , Y1,ij , Z1,ij , T1,ij} cor- respondant aux matrices élémentaires . La base duale sera notée (αa,ij , βa,ij , γa,ij , δa,ij). Rappelons que X1, Y1, Z1 sont antisymétriques alors que T1 est symétrique. Les crochets correspondent aux représentations de so(m ) sur lui-même ou de so(m ) sur l’espace des matrices symétriques. On aura donc qga = λ (α2a,ij + β a,ij + γ a,ij) + δ2a,ij) + λ a,ii) + (λ (δa,iiδa,jj). Les formes qgb et qgc admettent une décomposition analogue, en tenant compte du fait que dans gb ce sont les matrices Y2 qui sont symétriques et pour ga les matrices Z1 (3.1). On note (αb,ij , βb,ij , γb,ij , δb,ij) la base duale de {X2,ij, Y2,ij , Z2,ij, T2,ij} et par (αc,ij , βc,ij, γc,ij , δc,ij) la base duale de {X3,ij , Y3,ij , Z3,ij , T3,ij}. Proposition 4 Toute métrique non dégénéré adaptée à la structure (Z2) 2-symétrique de l’espace homogène SO(2m)/Sp(m) est définie à partir de la forme bilinéaire ad(ge)-invariante sur ga ⊕ gb ⊕ gc B = qga + qgb + qgb avec qga = λ (α2a,ij + β a,ij + γ i6=j δ a,ij) + λ a,ii) + (λ (δa,iiδa,jj) qgb = λ + γ2ij) + δ i6=j β ) + λb2(β ) + (λb2 − (βb,iiβb,jj) qgc = λ (β2c,ij + γ c,ij) + δ c,ij + i6=j α c,ij) + λ c,ii) + (λ (αc,iiαc,jj) 4 Métriques pseudo-riemanniennes (Z2) 2-symétriques sur SO(2m)/Sp(m) 4.1 Signature des formes qgγ Soit γ ∈ {a, b, c}. Les valeurs propres de la forme qgγ sont µ1,γ = λ 1 , µ2,γ = λ 2/2 + λ 1/4, µ3,γ = λ r + 1 r − 1 où r est l’ordre commun des matrices symétriques X4, Y2, Z1. Ces valeurs propres sont respectivement de multiplicité dimgγ − r, r − 1, 1. Le signe des valeurs propres µ2,γ et µ3,γ est donc µ2,γ > 0 ⇐⇒ λ 2 > −λ µ3,γ > 0 ⇐⇒ λ 2 > −λ 2(r+1) On en déduit, si s(q) désigne la signature de la forme quadratique q : s(qgγ ) = (dimgγ , 0) ⇔ (λ 1 > 0, λ 2 > λ 2(r+1) = (dimgγ − 1, 1) ⇔ (λ 1 > 0, −λ 1/2 < λ 2 < λ 2(r+1) = (dimgγ − r, r) ⇔ (λ 1 > 0, λ 2 < −λ = (r, dimgγ − r) ⇔ (λ 1 < 0, λ 2 > −λ = (1, dimgγ − 1) ⇔ (λ 1 < 0, λ 2(r+1) 2 < −λ = (0, dimgγ) ⇔ (λ 1 < 0, λ 2 < λ 2(r+1) Notons que µ2,γ = µ3,γ si et seulement si λ 1 = 2λ 4.2 Classification des métriques riemanniennes adaptées sur SO(2m)/Sp(m) Comme r = m 2+m−2 m2+m+2 on a le résultat suivant Théorème 5 Toute métrique riemannienne sur SO(2m)/Sp(m) adaptée à la structure 2-symétrique est définie à partir de la forme bilinéaire sur gaø plusgb ⊕ gc B = qga(λ 1 , λ 2) + qgb(λ 2) + qgb(λ 1 > 0 2 > λ 2+m−2 2(m2+m+2 pour tout γ ∈ {a, b, c}. Pour une telle métrique, la connexion de Levi-Civita ne cöıncide pas en général avec la connexion canonique associée à la structure (Z2) 2-symétrique ( [2]). Ces deux connexions sont les mêmes si et seulement si la mt́rique riemannienne est naturellement réductive. Elle correspond donc à la restriction de la forme de Killing (au signe près) de SO(2m). Cette métrique correspond à la forme bilinéaire B définie par les paramètres λa1 = λ 1 = λ 1 = 2λ 2 = 2λ 2 = 2λ 4.3 Classification des métriques lorentziennes adaptées sur l’espace SO(2m)/Sp(m) Les métriques lorentziennes adaptées à la structure (Z2) 2-symétrique sont définies par les formes bilinéaires B, définies dans la section précédente, dont la signature est (dim(ga) + dim(gb) + dim(gc)− 1, 1). On a donc Théorème 6 Toute métrique lorentzienne sur SO(2m)/Sp(m) adaptée à la structure (Z2) symétrique est définie par l’une des formes bilinéaires B = qga(λ 1 , λ 2) + qgb(λ 2) + qgb(λ ∀γ ∈ {a, b, c}, λ 1 > 0 ∃γ0 ∈ {a, b, c} tel que − λ 1 /2 < λ 2 < λ 2(r+1) ∀γ 6= γ0, λ 2 > λ 2(r+1) References [1] Bahturin Y., Goze M., Γ-symmetric homogeneous spaces. Preprint Mulhouse (2006). [2] Bouyakoub A., Goze M., Remm E. On Riemannian nonsymmetric spaces and flag manifolds. Preprint Mulhouse (2006). [3] Berger M., Les espaces symétriques non compacts, Ann.E.N.S. 74, 2, (1957), 85-177. Rappel. Espaces -symétriques Espaces riemanniens -symétriques Métriques adaptées à la structure (Z2)2-symétrique de SO(2m)/Sp(m) La graduation (Z2)2-symétrique Structure métrique (Z2)2-symétrique Exemples Cas général: métriques adaptées sur SO(2m)/Sp(m) Métriques pseudo-riemanniennes (Z2)2-symétriques sur SO(2m)/Sp(m) Signature des formes qg Classification des métriques riemanniennes adaptées sur SO(2m)/Sp(m) Classification des métriques lorentziennes adaptées sur l'espace SO(2m)/Sp(m)

Quasiparticles in Neon using the Faddeev Random Phase Approximation C. Barbieri Gesellschaft für Schwerionenforschung, Planckstr. 1, D-64291, Darmstadt, Germany D. Van Neck Laboratory of Theoretical Physics, Ghent University, Proeftuinstraat 86, B-9000 Gent, Belgium W.H. Dickhoff Department of Physics, Washington University, St. Louis, MO 63130, USA (Dated: November 2, 2018) The spectral function of the closed-shell Neon atom is computed by expanding the electron self-energy through a set of Faddeev equations. This method describes the coupling of single-particle degrees of free- dom with correlated two-electron, two-hole, and electron-hole pairs. The excitation spectra are obtained using the Random Phase Approximation, rather than the Tamm-Dancoff framework employed in the third-order al- gebraic diagrammatic contruction [ADC(3)] method. The difference between these two approaches is studied, as well as the interplay between ladder and ring diagrams in the self-energy. Satisfactory results are obtained for the ionization energies as well as the energy of the ground state with the Faddeev-RPA scheme that is also appropriate for the high-density electron gas. PACS numbers: 31.10.+z,31.15.Ar I. INTRODUCTION Ab initio treatments of electronic systems become unwork- able for sufficiently complex systems. On the other hand, the Kohn-Sham formulation [1] of density functional the- ory (DFT) [2] incorporates many-body correlations (beyond Hartree-Fock), while only single-particle (sp) equations must be solved. Due to this simplicity DFT is the only feasible approach in some modern applications of electronic structure theory. There is therefore a continuing interest both in devel- oping new and more accurate functionals and in studying con- ceptual improvements and extensions to the DFT framework. In particular it is found that DFT can handle short-range inter- electronic correlations quite well, while there is room for im- provements in the description of long-range (van der Waals) forces and dissociation processes. Microscopic theories offer some guidance in the devel- opment of extensions to DFT. Orbital dependent function- als can be constructed using many-body perturbation theory (MBPT) [3, 4]. More recently, the development of general ab initio DFT [5, 6] addressed the lack of a systematic im- provement in DFT methods. In this approach one considers an expansion of the exact ground-state wave function (e.g., MBPT or coupled cluster) from a chosen reference determi- nant. Requiring that the correction to the density vanishes at a certain level of perturbation theory allows one to construct the corresponding approximation to the Kohn-Sham potential. A different route has been proposed in Ref. [7] by devel- oping a quasi-particle (QP)-DFT formalism. In the QP-DFT approach the full spectral function is decomposed in the con- tribution of the QP excitations, and a remainder or background part. The latter part is complicated, but does not need to be known accurately: it is sufficient to have a functional model for the energy-averaged background part to set up a single- electron selfconsistency problem that generates the QP exci- tations. Such an approach is appealing since it contains the well-developed standard Kohn-Sham formulation of DFT as a special case, while at the same time emphasis is put on the correct description of QPs, in the Landau-Migdal sense [8]. Hence, it can provide an improved description of the dynam- ics at the Fermi surface. Given the close relation between QP- DFT and the Green’s function (GF) formulation of many-body theory [9, 10], it is natural to employ ab initio calculations in the latter formalism to investigate the structure of possible QP-DFT functionals. In this respect it is imperative to iden- tify which classes of diagrams are responsible for the correct description of the QP physics. Some previous calculations, based on GF theory, have fo- cused on a self-consistent treatment of the self-energy at the second order [11, 12, 13] for simple atoms and molecules. For the atomic binding energies it was found that the bulk of corre- lations, beyond Hartree-Fock, are accounted for while signifi- cant disagreement with experiment persists for QP properties like ionization energies and electron affinities. The formal- ism beyond the second-order approximation was taken up in Ref. [14, 15, 16, 17, 18] by employing a self-energy of the GW type [19]. In this approach, the random phase approx- imation (RPA) in the particle-hole (ph) channel is adopted to allow for possible collective effects on the atomic excited states. The latter are coupled to the sp states by means of dia- grams like the last two in Fig. 1(c). Two variants of the G0W0 formalism were employed in Ref. [14] (where the subscript “0” indicates that non-dressed propagators are used). In the first only the direct terms of the interelectron Coulomb po- tential are taken into account. In the second version, also the exchange terms are included when diagonalizing the ph space [generalized RPA (GRPA)] and in constructing the self-energy [generalized GW (GGW)]. Although the exchange terms are known to be crucial in order to reproduce the experimentally observed Rydberg sequence in the excitation spectrum of neu- tral atoms, they were found to worsen the agreement between the theoretical and experimental ionization energies [14]. http://arxiv.org/abs/0704.1542v2 In the GW approach the sp states are directly coupled with the two-particle–one-hole (2p1h) and the two-hole–one- particle (2h1p) spaces. However, only partial diagonalizations (namely, in the ph subspaces) are performed. This procedure unavoidably neglects Pauli correlations with the third particle (or hole) outside the subspace. In the case of the GGW ap- proach, this leads to a double counting of the second order self-energy which must be corrected for explicitly [20, 21]. We note that simply subtracting the double counted diagram is not completely satisfactory here, since it introduces poles with negative residues in the self-energy. More important, the inter- action between electrons in the two-particle (pp) and two-hole (hh) subspaces are neglected altogether in (G)GW. Clearly, it is necessary to identify which contributions, beyond GGW, are needed to correctly reproduce the QP spectrum. In this respect, it is known that highly accurate descrip- tions of the QP properties in finite systems can be obtained with the algebraic diagrammatic construction (ADC) method of Schirmer and co-workers [22]. The most widely used third- order version [ADC(3)] is equivalent to the so-called extended 2p1h Tamm-Dancoff (TDA) method [23] and allows to pre- dict ionization energies with an accuracy of 10-20 mH in atoms and small molecules. Upon inspection of its diagram- matic content, the ADC(3) self-energy is seen to contain all diagrams where TDA excitations are exchanged between the three propagator lines of the intermediate 2p1h or 2h1p prop- agation. The TDA excitations are constructed through a diag- onalization in either 2p1h or 2h1p space, and neglect ground- state correlations. However, it is clear that use of TDA leads to difficulties for extended systems. In the high-density elec- tron gas e.g., the correct plasmon spectrum requires the RPA in the ph channel, rather than TDA. In order to bridge the gap between the QP description in finite and extended systems, it seems therefore necessary to develop a formalism where the intermediate excitations in the 2p1h/2h1p propagator are described at the RPA level. This can be achieved by a formalism based on employing a set of Faddeev equations, as proposed in Ref. [24] and subsequently applied to nuclear structure problems [25, 26, 27]. In this ap- proach the GRPA equations are solved separately in the ph and pp/hh subspaces. The resulting polarization and two-particle propagators are then coupled through an all-order summation that accounts completely for Pauli exchanges in the 2p1h/2h1p spaces. This Faddeev-RPA (F-RPA) formalism is required if one wants to couple propagators at the RPA level or beyond. Apart from correctly incorporating Pauli exchange, F-RPA takes the explicit inclusion of ground-state correlations into account, and can therefore be expected to apply to both finite and extended systems. The ADC(3) formalism is recovered as an approximation by neglecting ground-state correlations in the intermediate excitations (i.e. replacing RPA with TDA phonons). In this work we consider the Neon atom and apply the F- RPA method to a nonrelativistic electronic problem for the first time. The relevant features of the F-RPA formalism (also extensively treated in Ref. [24]), are introduced in Sect. II. The application to the Neon atom is discussed in Sec. III, where we also investigate the separate effects of the ladder and ring series on the self-energy, as well as the differences between including TDA and RPA phonons. Our findings are summarized in Sec. IV. Some more technical aspects are rele- gated to the appendix, where the interested reader can find the derivation of the Faddeev expansion for the 2p1h/2h1p propa- gator, adapted from Ref. [24]. In particular, the approach used to avoid the multiple-frequency dependence of the Green’s functions is discussed in App. A 1, along with its basic as- sumptions. The explicit expressions of the Faddeev kernels are given in App. A 3. Together with Ref. [24], the appendix provides sufficient information for an interested reader to ap- ply the formalism. II. FORMALISM The theoretical framework of the present study is that of propagator theory, where the object of interest is the sp prop- agator, instead of the many-body wave function. In this paper we will employ the convention of summing over repeated in- dices, unless specified otherwise. Given a complete orthonor- mal basis set of sp states, labeled by α,β,..., the sp propagator can be written in its Lehmann representation as [9, 10] gαβ(ω) = ω − ε+n + iη ω − ε−k − iη , (1) where Xnα = 〈Ψ 0 〉 (Y α = 〈Ψ |cα|Ψ 0 〉) are the spectroscopic amplitudes, cα (c ) are the second quantization destruction (creation) operators and ε+n = E n − E EN0 − E ). In these definitions, |ΨN+1n 〉, |Ψ 〉 are the eigenstates, and EN+1n , E k the eigenenergies of the (N ± 1)- electron system. Therefore, the poles of the propagator reflect the electron affinities and ionization energies. The sp propagator solves the Dyson equation gαβ(ω) = g αβ(ω) + g αγ(ω)Σ γδ(ω)gδβ(ω) , (2) which depends on the irreducible self-energy Σ⋆(ω). The lat- ter can be written as the sum of two terms αβ(ω) = Σ Vαλ,µν Rµνλ,γδε(ω) Vγδ,βε , (3) where ΣHF represents the Hartree-Fock diagram for the self- energy. In Eqs. (2) and (3), g0(ω) is the sp propagator for the system of noninteracting electrons, whose Hamiltonian contains only the kinetic energy and the electron-nucleus at- traction. The Vαβ,γδ represent the antisymmetrized matrix el- ements of the interelectron (Coulomb) repulsion. Note that in this work we only consider antisymmetrized elements of the interaction, hence, our result for the ring summation al- ways compare to the generalized GW approach. Equation (3) introduces the 2p1h/2h1p-irreducible propagator R(ω), which carries the information concerning the coupling of sp states to more complex configurations. Both Σ⋆(ω) and R(ω) have a perturbative expansion as a power series in the interelectron FIG. 1: (Color online) a) Diagrammatic expansion of R(ω) in terms of the (antisymmetrized) Coulomb interaction and undressed prop- agators. b) R(ω) is related to the self-energy according to Eq. (3). c) By substituting the diagrams a) in the latter equation, one finds the perturbative expansion of the self-energy. interaction V̂ . Some of the diagrams appearing in the expan- sion of R(ω) are depicted in Fig. 1, together with the corre- sponding contributions to the self-energy. Note that already at zero order in R(ω) (three free lines with no mutual interaction) the second order self-energy is generated. Different approximations to the self-energy can be con- structed by summing particular classes of diagrams. In this work we are interested in the summation of rings and ladders, through the (G)RPA equations. In order to include such effects in R(ω), we first consider the polarization propagator describ- ing excited states in the N-electron system Παβ,γδ(ω) = 〈ΨN0 |c n 〉 〈Ψ γcδ|Ψ ENn − E 〈ΨN0 |c γcδ|Ψ n 〉 〈Ψ ENn − E , (4) and the two-particle propagator, that describes the addi- tion/removal of two electrons gIIαβ,γδ(ω) = 〈ΨN0 |cβcα|Ψ n 〉 〈Ψ |ΨN0 〉 EN+2n − E 〈ΨN0 |c |ΨN−2 〉 〈ΨN−2 |cβcα|Ψ EN0 − E . (5) We note that the expansion of R(ω) arises from applying the equations of motion to the sp propagator (1), which is associ- ated to the ground state |ΨN0 〉. Hence, all the Green’s functions appearing in this expansion will also be ground state based, in- cluding Eqs. (4) and (5). However the latter contain, in their Lehmann representations, all the relevant information regard- ing the excitation of ph and pp/hh collective modes. The ap- proach of Ref. [24] consists in computing these quantities by =g II Π(ph) (pp/hh)II (pp/hh) FIG. 2: (Color online) Diagrammatic equations for the polarization (above) and the two-particle (below) propagators in the (G)RPA ap- proach. Dashed lines are always antisymmetrized Coulomb matrix elements and the full lines represent free (undressed) propagators. solving the ring-GRPA and the ladder-RPA equations [10], which are depicted for propagators in Fig. 2. In the more general case of a self-consistent calculation, a fragmented in- put propagator can be used and the corresponding dressed (G)RPA [D(G)RPA] equations [10, 28] solved [see Eqs. (A2a) and (A2b)]. Since the propagators (4) and (5) reflect two-body correlations, they still have to be coupled to an additional sp propagator in order to obtain the corresponding approximation for the 2p1h and 2h1p components of R(ω). This is achieved by solving two separate sets of Faddeev equations. Taking the 2p1h case as an example, one can split R(2p1h)(ω) in three different components R̄(i)(ω) (i = 1, 2, 3) that differ from each other by the last pair of lines that interact in their diagrammatic expansion, R̄(2p1h) αβγ,µνλ (ω) = αβγ,µνλ(ω) −G βαγ,µνλ(ω) i=1,2,3 R̄(i) αβγ,µνλ (ω) , where G0 (ω) is the 2p1h propagator for three freely propa- gating lines. These components are solutions of the following set Faddeev equations [29] R̄(i) αβγ,µνλ (ω) = G0 αβγ,µ′ν′λ′(ω) Γ µ′ν′λ′,µ′′ν′′λ′′ µ′′ν′′λ′′ ,µνλ (ω) + R̄(k) µ′′ν′′λ′′ ,µνλ (ω) (7) µ′′ν′′λ′′ ,µνλ(ω) −G ν′′µ′′λ′′ ,µνλ(ω) , i = 1, 2, 3 where (i, j, k) are cyclic permutations of (1, 2, 3). The inter- action vertices Γ(i)(ω) contain the couplings of a ph or pp/hh collective excitation and a freely propagating line. These are given in the Appendix in terms of the polarization (4) and two-particle (5) propagators. Equations. (7) include RPA-like phonons and fully describe the resulting energy dependence of R(ω). However, they still neglect energy-independent contributions–even at low order in the interaction–that also correspond to relevant ground-state correlations. The latter can be systematically inserted according to R(2p1h) αβγ,µνλ (ω) = Uµνλ,µ′ν′λ′ R̄ (2p1h) µ′ν′λ′,µ′′ν′′λ′′ (ω) U† µ′′ν′′λ′′ ,µνλ , (8) where R(ω) is the propagator we employ in Eq. (3), R̄(ω) is the one obtained by solving Eqs. (7), U ≡ I + ∆U, and I is the (pp/hh) Π(ph) g II (pp/hh) Π(ph) FIG. 3: (Color online) Example of one of the diagrams that are summed to all orders by means of the Faddeev Eqs. (7) (left). The corresponding contribution to the self-energy, obtained upon inser- tion into Eq. (3), is also shown (right). identity matrix. Following the algebraic diagrammatic con- struction method [22, 23], the energy independent term ∆U was determined by expanding Eq. (8) in terms of the inter- action and imposing that it fulfills perturbation theory up to first order (corresponding to third order in the self-energy). The resulting ∆U, employed in this work, is the same as in Ref. [23] and is reported in App. A 3 for completeness. It has been shown that the additional diagrams introduced by this correction are required to obtain accurate QP properties. Equations. (7) and (8) are valid only in the case in which a mean-field propagator is used to expand R(ω). This is the case of the present work, which employs Hartree-Fock sp propaga- tors as input. The derivation of these equations for the gen- eral case of a fragmented propagator is given in the appendix. More details about the actual implementation of the Faddeev formalism to 2p1h/2h1p propagation have been presented in Ref. [24]. The calculation of the 2h1p component of R(ω) follows completely analogous steps. It is important to note that the present formalism includes the effects of ph and pp/hh motion to be included simulta- neously, while allowing interferences between these modes. These excitations are evaluated here at the RPA level and are then coupled to each other by solving Eqs. (7). This generates diagrams as the one displayed in Fig. 3, with the caveat that two phonons are not allowed to propagate at the same time. Equations. (7) also assure that Pauli correlations are properly taken into account at the 2p1h/2h1p level. In addition, one can in principle employ dressed sp propagators in these equations to generate a self-consistent solution. If we neglect the lad- der propagator gII (ω) (5) in this expansion, we are left with the ring series alone and the analogous physics ingredients as for the generalized GW approach. However, this differs from GGW due to the fact that no double counting of the second- order self-energy occurs, since the Pauli exchanges between the polarization propagator and the third line are properly ac- counted for (see Fig. 3). Alternatively, one can suppress the polarization propagator to investigate the effects of pp/hh lad- ders alone. It is instructive to replace in the above equations all RPA phonons with TDA ones; this amounts to allowing only forward-propagating diagrams in Fig. 2, and is equivalent to separate diagonalisations in the spaces of ph, pp and hh con- l 0 1 2 3 4 5 6 rw 2.0 4.0 0.0 0.0 0.0 0.0 0.0 no 12 21 10 10 5 5 5 TABLE I: Parameters that define the sp basis: radius of the confining wall rw (in atomic units) and number of orbits no used for different partial waves l. The value of cw is always set to 5 a.u.. figurations, relative to the HF ground state. It can be shown that using these TDA phonons to sum all diagrams of the type in Fig. 3 reduces to one single diagonalization in the 2p1h or 2h1p spaces. Therefore, Eqs. (7) and (8) with TDA phonons lead directly to the “extended” 2p1h TDA of Ref. [23], which was later shown to be equivalent to ADC(3) in the general ADC framework [22]. The Faddeev expansion formalism of Ref. [24] creates the possibility to go beyond ADC(3) by in- cluding RPA phonons. This is more satisfactory in the limit of large systems. At the same time, the computational cost remains modest since only diagonalizations in the 2p1h/2h1p spaces are required. Note that complete self-consistency requires the use of fragmented (or dressed) propagators in the evaluation of all in- gredients leading to the self-energy. This is outside the scope of the present paper, but we included partial selfconsistency by taking into account the modifications to the HF diagram by employing the correlated one-body density matrix and it- erating to convergence. This is relatively simple to achieve, since the 2p1h/2h1p propagator is only evaluated once with the input HF propagators. Below we will give results with and without this partial selfconsistency at the HF level. III. RESULTS Calculations have been performed using two different model spaces: (1) a standard quantumchemical Gaussian ba- sis set, aug-cc-pVTZ for Neon [30], with Cartesian repre- sentation of the d and f functions; (2) a numerical basis set based on HF and subsequent discretization of the continuum, to be detailed below. The aug-cc-pVTZ basis set was used primarily to check our formalism with the ADC(3) result in literature (i.e. [31], where this basis was employed). The HF+continuum basis allows to approach, at least for the ion- ization energies, the results for the full sp space (basis set limit). The HF+continuum is the same discrete model space em- ployed previously in Refs. [11, 14]. It consists of: (1) Solv- ing on a radial grid the HF problem for the neutral atom; (2) Adding to this fixed nonlocal HF potential a parabolic poten- tial wall of the type U(r) = θ(r − rw)cw(r − rw) 2, placed at a distance rw of the nucleus. The latter eigenvalue problem has a basis of discrete eigenstates. This basis is truncated by spec- ifying some largest angular momentum lmax and the number of virtual states for each value of l ≤ lmax. (3) Solve the HF problem again, without the potential wall, in this truncated discrete space. The resulting basis set is used for the subse- F-TDA F-RPA F-TDAc F-RPAc Expt. 2p -0.799 -0.791 -0.803 -0.797 (0.94) -0.793 (0.92) 2s -1.796 -1.787 -1.802 -1.793 (0.90) -1.782 (0.85) 1s -32.126 -32.087 -32.140 -32.102 (0.86) -31.70 Etot -128.778 -128.772 -128.836 -128.840 -128.928 TABLE II: Results with the aug-cc-pVTZ basis. The first three rows list the energies of the main sp fragments below the Fermi level, as predicted by different self-energies. F-TDA/F-RPA refers to the Fad- deev summation with TDA/RPA phonons, respectively. In all cases the self-energy was corrected at third order through Eq. (8). The suffix “c” refers to partial selfconsistency, when the static (HF-type) self-energy is consistent with the correlated density matrix. Without “c” the pure HF self-energy was taken. In the F-RPAc column the strength of the fragment is indicated between brackets. The last row is the total electronic binding energy. The experimental values are taken from Refs. [32, 33]. All energies are in atomic units. quent Green’s function calculations. When a sufficiently large number of states is retained after truncation, the final results should approach the basis set limit. In particular the results should not depend on the choice of the auxiliary confining potential. This was verified in Ref. [11] for the second-order, and in Ref. [14] for the G0W0 self-energy; in these cases the self-energy is sufficiently simple that extensive convergence checks can be made for various choices of the auxiliary potential. The parameters of the confining wall and the number of sp states kept in the basis set was optimized in Ref. [11], by requiring that the ionization energy is converged to about 1 mH for the second-order self-energy. In Ref. [14] the same choice of basis set was also seen to bring the ioniza- tion energy for the G0W0 self-energy near convergence. For completeness, the details of this basis are reported in Table I. While the self-energy in the present paper is too complicated to allow similar convergence checks, it seems safe to assume that basis set effects will affect the calculated ionization ener- gies by at most 5 mH. In Table II we compare, for the aug-cc-pVTZ basis, the ion- ization energies of the main single-hole configurations when TDA or RPA phonons are employed in the Faddeev construc- tion (this is labeled F-TDA and F-RPA, respectively, in the table). Note that use of TDA phonons corresponds to the usual ADC(3) self-energy. We find that the replacement of TDA with RPA phonons provides more screening, leading to slightly less bound poles which are shifted towards the exper- imental values. This shift increases with binding energy. As discussed at the end of Sec. I, one can include consistency of the static part of the self-energy. About eight iterations are needed for convergence. This is a nonnegligible correction, providing about 5 mH more binding (i.e. larger ionization en- ergies) for the valence/subvalence 2p and 2s, 15 mH for the deeply bound 1s, and 60 mH to the total binding energy. Our converged result for the Faddeev-TDA self-energy (labeled F- TDAc in Table II) is in good agreement with the ADC(3) value for the 2p ionization energy (-0.804 H) quoted in [31], as it should be. The analogous results obtained with the larger F-TDA F-RPA F-TDAc F-RPAc Expt. 2p -0.807 -0.799 -0.808 -0.801 (0.94) -0.793 (0.92) 2s -1.802 -1.792 -1.804 -1.795 (0.91) -1.782 (0.85) 1s -32.136 -32.097 -32.142 -32.104 (0.81) -31.70 Etot -128.863 -128.857 -128.883 -128.888 -128.928 TABLE III: Results with the HF+continuum basis set from Table I. See also the caption of Table II. -40 -35 -30 [a.u.] F-RPA(ring) F-RPA(ladder) F-RPA FIG. 4: (Color online) Spectral function for the s states in Ne obtained with various self-energy approximations. From the top down: the second-order (Σ(2)) self-energy, the F-RPA(ring), the F- RPA(ladder), and the full F-RPA self-energy. The strength is given relative to the Hartree-Fock occupation of each shell. Only fragments with strength larger than Z > 0.005 are shown. HF+continuum basis are given in Table III, which al- lows to assess overall stability and basis set effects. We find exactly the same trends as for aug-cc-pVTZ. In particular the reduction of ionization energies from the replacement of TDA with RPA phonons is almost independent of the basis set used, while the effect of including partial consistency is roughly halved. Overall, the ionization states are always more bound with the larger basis set; while the basis set limit could be still more bound than the present results with the HF+continuum basis set, it is likely (based on the G0W0 extrapolation in Ref. [14]) that the difference does not exceed 5 mH. As discussed in Sec. I, the the F-RPA self-energy contains RPA excitations of both ph type (ring diagrams) and pp/hh type (ladder diagrams). It is instructive to analyze their sepa- rate contributions to the final ionization energies, in order to 1s 2s 2p HF -32.77 (1.00) -1.931 (1.00) -0.850 (1.00) Σ(2) -31.84 (0.74) -1.736 (0.88) -0.747 (0.91) G0W0 -31.14 (0.85) -1.774 (0.91) -0.801 (0.94) F-RPA (ring) -31.82 (0.73) -1.636 (0.56) -0.730 (0.80) F-RPA (ladder) -32.04 (0.87) -1.802 (0.95) -0.781 (0.96) F-RPA -32.10 (0.81) -1.792 (0.91) -0.799 (0.94) Exp. -31.70 -1.782 (0.85) -0.793 (0.92) TABLE IV: Energy (in a.u.) and strength (bracketed numbers) of the main fragments in the spectral function of Neon, generated by differ- ent self-energies. Results for the HF+continuum basis. Consecutive rows refer to: (1) HF; (2) second-order self-energy; (3) G0W0 results from Ref. [14]; (4) F-RPA self-energy with only ph rings retained; (5) F-RPA self-energy with only pp/hh ladders retained; (6) Com- plete F-RPA self-energy. In all F-RPA results the self-energy was corrected at third order through Eq. (8). The static self-energy was pure HF (no partial self-consistency). The experimental values are taken from Refs. [32, 33]. understand how the F-RPA self-energy is related to the stan- dard (G)GW self-energy. Table IV compares the results for the ionization energies, obtained with the second-order self- energy, to different approximations for including the ring sum- mations. As one can see, the second-order self-energy gener- ates an l=1 sp energy of -0.747 mH, which is 46 mH above the empirical 2p ionization energy. The G0W0 self-energy, which includes the ring summation with only direct Coulomb matrix elements, improves this result and brings it close to ex- periment. The 2s behaves in a similar way. Unfortunately, including the exchange terms of the interelectron repulsion in the GG0W0 method turns out to have the opposite effect (the 2p ionization energy becomes -0.712 H [14] [41]) and the agreement with experiment is lost. Obviously, GG0W0 is too simplistic to account for exchange in the ph channel. With the F-RPA(ring) self-energy one can go one step fur- ther and employ the Faddeev expansion to also force proper Pauli exchange correlations in the 2p1h/2h1p spaces. As shown in Table IV, this enhances the screening due to the exchange interaction terms, leading to even less binding for the 2s and 2p. The corrections relative to the second-order self-energy can be large (100 mH for the 2s state) and in the direction away from the experimental value. We also note that the larger shift, in the 2s orbit, is accompanied by an increase of the fragmentation (see Fig. 4 and Tab. IV). Similar observa- tions were also made in Ref. [14] for other atoms: in general ring summations in the direct channel alone bring the quasi- hole peaks close to the experiment. This agreement is then spoiled as soon as one includes proper exchange terms in the self-energy. On the other hand, exchange in the ph channel is required to reproduce the correct Rydberg sequence in the excitation spectrum of neutral atoms. So further corrections must arise from other diagrams, and obviously the summation of ladder diagrams can play a relevant role, since these con- tribute to the expansion of the self-energy at the same level as that of the ring diagrams. The result when only including ladder-type RPA phonons in the F-RPA self-energy is also shown in Table IV. One can see that pp/hh ladders do actually work in the opposite way as the ph channel ring diagrams, and have the same order of magnitude with, e.g., a shift of 66 mH for the 2s relative to the second-order result. When combined with the ring dia- grams in the full F-RPA self-energy, the agreement with ex- periment is restored again. Note that the final result cannot be obtained by adding the contributions of rings and ladders, but depends nontrivially on the interplay between these classes of diagrams thereby pointing to significant interference effects. With the F-RPA(ring) self-energy, where only the contribu- tions of the ph channel are included, the main peaks listed in Table IV are not only considerably shifted but also strongly depleted, e.g. a strength of only 0.56 for the main 2s peak. The complete spectral function for the l = 0 strength in Fig. 4 shows that the depletion of the main fragment is accompanied by strong fragmentation over several states. While correlation effects are overestimated in F-RPA(ring), they are suppressed in F-RPA(ladder), where only the pp/hh ladders are included in the self-energy. In this case one finds a spectral distribu- tion closer to the HF one, with a main 2s fragment of strength 0.95 and less fragmentation than the the second-order self- energy. The spectral distribution generated by the complete F-RPA self-energy is again a combination of the above ef- fects. The strength of the deeply bound 1s orbital behaves in an analogous way. The strength of the main peak is reduced but several satellite levels appear due to the mixing with 2h1p configurations. In all the calculations reported in Fig. 4 we found a summed l = 0 strength exceeding 0.98 in the interval [-40 H, -30 H] which can be associated with the 1s orbital, and this remains true even in the presence of strong correla- tions using the F-RPA(ring) self-energy. Of course, the mix- ing with 3h2p configurations, not included in this work, may further contribute to the fragmentation pattern in this energy region. IV. CONLUSIONS AND DISCUSSION In conclusion, the electronic self-energy for the Ne atom was computed by the F-RPA method which includes – simultaneously– the effects of both ring and ladder diagrams. This was accomplished by employing an expansion of the self-energy based on a set of Faddeev equations. This tech- nique was originally proposed for nuclear structure applica- tions [24] and is described in the appendix. At the level of the self-energy one sums all diagrams where the three propa- gator lines of the intermediate 2p1h or 2h1p propagation are connected by repeated exchange of RPA excitations in both the ph and the pp/hh channel. This differs from the ADC(3) formalism in the fact that the exchanged excitations are of the RPA type, rather than the TDA type, and therefore take ground-state correlations effects into account. The coupling to the external points of the self-energy uses the same modi- fied vertex as in ADC(3), which must be introduced to include consistently all third-order perturbative contributions. The resulting main ionization energies in the Neon atom are at least of the same quality, and even somewhat improved, compared to the ADC(3) result. Note that, numerically, F- RPA can be implemented as a diagonalization in 2p1h/2h1p space implying about the same cost as ADC(3). The present study also shows that in localized electronic systems subtle cancellations occur between the ring and ladder series. In par- ticular, only a combination of the ring and ladder series leads to sensible results, as the separate ring series tends to correct the second-order result in the wrong direction. Since the limit to extended systems requires an RPA treat- ment of excitations, the F-RPA method holds promise to bridge the gap between an accurate description of quasiparti- cles in both finite and extended systems. In particular, the GW treatment of the electron gas has been shown to yield excellent binding energies, but poor quasiparticle properties [34, 35]. Further progress beyond GW theory requires a consistent in- corporation of exchange in the ph channel. The F-RPA tech- nique may be highly relevant in this respect. A common framework for calculating accurate QP properties in both fi- nite and extended systems, is also important for constrain- ing functionals in quasiparticle density functional theory (QP- DFT) [7]. Finally, complete self-consistency requires sizable compu- tational efforts for bases as large as the HF+continuum basis used here. It would nevertheless represent an important exten- sion of the present work, since it is related to the fulfillment of conservation laws [36, 37]. These issues will be addressed in future work. Acknowledgments This work was supported by the U.S. National Science Foundation under grant PHY-0652900. APPENDIX A: FADDEEV EXPANSION OF THE 2P1H/2H1P PROPAGATOR Although only the one-energy (or two-time) part of the 2p1h/2h1p propagator enters the definition of the self energy, Eq. (3), a full resummation of all its diagrammatic contribu- tions would require to treat explicitly the dependence on three separate frequencies, corresponding to the three final lines in the expansion of R(ω). For example, inserting the RPA ring (ladder) series in R(ω) implies the propagation of a ph (pp/hh) pair of lines both forward and backward in time, while the third line remains unaffected. A way out of this situation is to solve the Bethe-Salpeter-like equations for the polarization and ladder propagators separately and then to couple them to the additional line. If it is assumed that different phonons do not overlap in time, the three lines in between phonon struc- tures will propagate only in one time direction [see figures (3) and (5)]. In this situation the integration over several frequen- cies can be circumvented following the prescription detailed in the next subsection. This approach will be discussed in the following for the general case of a fully fragmented prop- gator, in order to derive a set of Faddeev equations capable FIG. 5: (Color online) Diagrammatic representation of Eq. (A3). Double lines represent fully dressed sp Green’s funcions which, how- ever, are restricted to propagate only in one time direction [i.e., only one of the two terms on the r.h.s. of Eq. (1) is retained]. The Faddeev Eqs. (A9) and (7) allow for both forward and backward propagation of the phonons Γ(π)(ω) and Γ(II)(ω) as long as these do not overlap in time. For the propagators, time ordereing is asumed with forwad propagation in the upward direction. of dressing the sp propagator self-consistently. Since the for- ward (2p1h) and the backward (2h1p) parts of R(ω) decouple in two analogous sets of equations, it is sufficient to focus on the first case alone. 1. Multiple frequencies integrals We start by considering the effective interactions in the ph and pp/hh channels that correspond to Eqs. (4) and (5) stripped of the external legs. In the present work, these are the follow- ing two-time objects: αβ,γδ (ω) = Vαδ,βγ + Vαν,βµ Π µν,ρσ(ω) Vρδ,σγ (A1a) = Vαδ,βγ + ω − επn + iη ω + επn′ − iη αβ,γδ (ω) = Vαβ,γδ + Vαβ,µν g µν,ρσ(ω) Vρσ,γδ (A1b) = Vαδ,βγ + ω − εΓ+n + iη ω − εΓ− where the residues and poles for the ring series are Ωn 〈ΨNn |c µcν|Ψ 0 〉Vµβ,να and ε n = E 0 . For the ladders, ∆ 〈ΨN+2n |c 0 〉Vµν,αβ and ∆ = Vαβ,µν〈Ψ k |cµcν|Ψ 0 〉, with poles εΓ+n = E 0 and ε k = E . Equations. (A1) solve the ring and ladder RPA equations, respectively αβ,γδ (ω) = Vαδ,βγ (A2a) αβ,µν gµρ(ω + ω1)gσν(ω1) Vρδ,σγ , αβ,γδ (ω) = Vαβ,γδ (A2b) αβ,µν gµρ(ω − ω1)gνσ(ω1) Vρσ,γδ . To display how the phonons (A1a) and (A1b) enter the ex- pansion of R(ω), we perform explicitly the frequency integrals for the diagram of Fig. 5. Since it is assumed that the separate propagators lines evolve only in one time direction, only the forwardgoing (g>(ω)) or the backwardgoing (g<(ω)) part of Eq. (1) must be included for particles and holes, respectively. After some algebra, one obtains ∆Rαβγ,µνλ(ω) = g>αα1 (ω −Ω) g (ω1) g (ω1 −Ω) Γ β1γ1,σ1λ1 (Ω) g>σ1σ2 (s + Ω − ω) α1σ2,µ1ν1 (s) g>µ1µ(s − ω2) g (ω2) g (s − ω) ω − (ε+n1 + ε ) + iη Vβ1λ1,γ1σ1 + ω − (ε+n1 + ε nπ) + iη [ω − επ − ε+n1 − ε + ε−k3 − ε+n4 + ε − ε+n2 + ε ][−επ − ε+n4 + ε ω − (ε+n1 + ε ) + iη Vα1σ2,µ1ν1 + +,nII +,nII ω − (εΓ+nII − ε ) + iη [ω + εΓ−kII − ε − ε+n4 − ε − ε+n6 + ε ] ∆−,kIIα1σ2 −,kII − ε+n1 − ε ][εΓ− − ε+n5 − ε ω − (ε+n5 + ε − ε−k7 ) + iη ω − (εΓ− − ε+n4 − ε ) − iη −,kII −,kII − ε+n2 + ε ] [−επ − ε+n4 + ε ] [εΓ− − ε+n1 − ε ] [εΓ− − ε+n5 − ε The last term in this expression contains an energy denom- inator that involves the simultaneous propagation of two phonons. Thus, it will be discarded in accordance with our assumptions. It must be stressed that similar terms, with over- lapping phonons, imply the explicit contribution of at least 3p2h/3h2p. A proper treatment of these would require a non trivial externsion of the present formalism, which is beyond the scope of this paper. The remaining part in Eq. (A3) is the relevant contribution for our purposes. This has the correct energy dependence of a product of denominators that correspond to the intermediate steps of propagation. All of these involve configurations that have at most 2p1h character. Although, ground state corre- lations are implicitely included by having already resummed the RPA series. Still, this term does not factorize in a prod- uct of separate Green’s functions due to the summations over the fragmentation indices ni and ki [labeling the eigenstates of the (N±1)-electron systems]. This is overcome if one defines the matrices G0>(ω), Γ(1,2)(ω) and Γ(3)(ω), with elements (no implicit summation used) αnαβnβγkγ; µnµνnνλkλ (ω) = δnα,nµ δnβ,nν δkγ,kλ ω − (ε+nα + ε − ε−kγ ) + iη , (A4a) αnαβnβγkγ; µnµνnνλkλ (ω) = Γ(2)> βnβαnαγkγ; νnνµnµλkλ (ω) = δα,µ δnα ,nµ Vβλ,γν + ω − (ε+nα + ε nπ) + iη [ω − επ − ε+nα − ε + ε−kγ − ε+nν + ε − ε+nβ + ε ][−επ − ε+nν + ε , (A4b) αnαβnβγkγ; µnµνnνλkλ (ω) = δγ,λ δkγ ,kλ Vαβ,µν + +,nII +,nII ω − (εΓ+nII − ε ) + iη [ω + εΓ−kII − ε − ε+nβ − ε − ε+nν + ε ] ∆−,kII −,kII − ε+nα − ε ][εΓ− − ε+nµ − ε . (A4c) In these definitions, the row and column indices are ordered to represent at first two quasiparticle lines and then a quasihole. The index ‘i’ in Γ(i)> refer to the line that propagates indepen- dently along with the phonon. Using Eqs.(A4), the first term on the r.h.s. of Eq. (A3) can be written as ∆R(2p1h) αβγ,µνλ (ω) = (A5) nα nβ kγ nµ nν kλ G0>(ω)Γ(1)>(ω)G0>(ω)Γ(3)>(ω)G0>(ω) αnαβnβγkγ; µnµνnνλkλ Eq. (A5) generalizes to diagrams involving any number of phonon insertions, as long as the terms involving two or more simultaneous phonons are dropped. Based on this relation, we use the following prescription to avoid performing integrals over frequencies. One extends all the Green’s functions to ob- jects depending not only on the sp basis’ indices (α, β, γ) but also on the indices labeling quasi-particles and holes (ni and ki). Whether a given argument represents a particle or an hole depends on the type of line being propagated. At this point one can perform calculations working with only two-time quanti- ties. The standard propagator is recovered at the end by sum- ming the “extended” one over the quasi-particle/hole indices. 2. Faddeev expansion The 2p1h/2h1p propagator that includes the full resumma- tion of both the ladder and ring diagrams at the (G)RPA level is the solution of the following Bethe-Salpeter-like equation, Rαβγ,µνλ(ω1, ω2, ω3) = (A6) gαµ(ω1)gβν(ω2) − gβµ(ω2)gαν(ω1) gλγ(−ω3) + gββ1(ω2)gγ1γ(−ω3)Vβ1σ,γ1ρ Rαρσ,µνλ(ω1, s, ω2 + ω3 − s) + gαα1 (ω1)gγ1γ(−ω3)Vα1σ,γ1ρ Rρβσ,µνλ(s, ω2, ω1 + ω3 − s) gαα1 (ω1)gββ1(ω2)Vα1β1,ρσ Rρσγ,µνλ(s, ω1 + ω2 − s, ω3) If this equation is solved, a double integration of R(ω1, ω2, ω3) would yield the two-time propagator R(ω) contributing to Eq. (3). However, the numerical solution of Eq. (A6) appears beyond reach of the present day computers and one needs to avoid dealing directly with multiple frequencies integrals. The strategy used is to first solve the RPA equations (A2a) and (A2b) separately. Once this is done it is necessary to re- arrange the series (A6) in such a way that only the resummed phonons appear. Following the formalism introduced by Fad- deev [29, 38], we identify the components R(i)(ω) with the three terms between curly brakets in Eq. (A6). By employ- ing Eqs. (A2a) and (A2b) one is lead to the following set of equations [42], αβγ,µνλ (ω1, ω2, ω3) = gαα1 (ω1)gββ1(ω2)gγ1γ(−ω3) (A7) ds1 ds2 ds3 α1β1γ1,µ1ν1λ1 (ω1, ω2, ω3; s1, s2, s3) gµ1µ(s1)gν1ν(s2) − gν1µ(s2)gµ1ν(s1) gλλ1(−s3) µ1ν1λ1,µνλ (s1, s2, s3) + R µ1ν1λ1,µνλ (s1, s2, s3) , i = 1, 2, 3 , where (i,j,k) are cyclic permutations of (1,2,3) and the inter- action vertices Γ(i)(ω1, ω2, ω3) are given by αβγ,µνλ (ω1, ω2, ω3;ω4, ω5, ω6) = (A8a) βαγ,νµλ (ω2, ω1, ω3;ω5, ω4, ω6) = = δ(ω1 − ω4)δ(ω2 + ω3 − ω5 − ω6)g αµ(ω1)Γ βγ,νλ (ω2 + ω3) , αβγ,µνλ (ω1, ω2, ω3;ω4, ω5, ω6) = (A8b) δ(ω3 − ω6)δ(ω1 + ω2 − ω4 − ω5)g λγ (−ω3)Γ αβ,µν (ω1 + ω2) . Finally, we apply the prescription of Sec. A 1 and substitute R(ω1, ω2, ω3) with its extended but two-time version R(ω). This leads to the following set of Faddeev equations which propagate 2p1h forward in time, R̄(i) αnαβnβγkγ; µnµνnνλkλ (ω) = αnαβnβγkγ; α′n′αβ′n γ′k′γ (ω) Γ(i) α′n′αβ′n γ′k′γ; µ′n ′n′νλ′k αnαβnβγkγ; µnµνnνλkλ (ω) −G0 αnαβnβγkγ; µnµνnνλkλ × R̄( j) µ′n′µν′n ; µnµνnνλkλ (ω) + R̄(k) µ′n′µν′n ; µnµνnνλkλ i = 1, 2, 3 . (A9) Since the full energy dependence is retained in Eq. (A7), the self-energy corresponding to its solution, R(ω1, ω2, ω3), is complete up to third order [see Eq. (3)]. This is no longer the case after the reduction to a two-time propagator. In particu- lar, the approximation that only forward 2p1h propagation is allowed between different phonons implies that all diagrams with different time propagation of their external lines are ne- glected in Eqs. (A9). However, these terms are not energy dependent and can be can be reinserted in a systematic way a posteriori as in Eq. (8). In the general case, Rαβγ,µνλ(ω) = (A10) U (2p1h) αβγ; α′n′αβ′n γ′k′γ R̄(2p1h) α′n′αβ′n γ′k′γ; µ′n ′n′νλ′k (ω) U (2p1h) † µ′n′µν′n ; µνλ U (2p1h) αβγ; µnµνnνλkλ = δαµδβνδγλ + ∆U (2p1h) αβγ; µnµνnνλkλ , (A11) where the correction ∆U can be determined by comparison with perturbation theory. The vertices (A4), that appear in Eqs. (A9), and U (2p1h) are expressed in terms of the fully fragmented propagator. There- fore, this approach allows to obtain self-consistent solutions of the sp Green’s function [25]. Whenever, like in this work, only a mean-field propagator is employed as input there ex- ist a one-to-one correspondence between the fragmentation indices and the sp basis. This is expressed by the relations Xnα = δn,α(1 − δα∈F) and Y α = δk,αδα∈F , where F represents the set of occupied orbits. In this case, it is possible to drop one set of indices so that Eqs. (A9) and (A10) simplify into the form (7) and (8). 3. Faddeev vertices In practical applications, it is worth to note that the poles of the free propagator G0(ω), Eq. (A4a), do not contribute to the kernel of Eqs. (A9). This can be proven by employing the closure relations for the RPA problem, in the form obtained by extracting the free poles in Eqs. (A2). As an example, for the forward poles of the ladder propagator these are ω→ε+n1 +ε+n2 [(ω − ε+n1 − ε ) × (Eq. A2b)] =⇒ (A12) µν,γδ (ω = ε+n1 + ε ) = 0 , ∀n1, n2 , and similarly for other cases. Making use of these relations one can derive the following working expression of the ker- nels of the 2p1h Faddeev equations (no implicit summations used) G0>(ω)Γ(1)>(ω) αnαβnβγkγ; µnµνnνλkλ G0>(ω)Γ(2)>(ω) βnβαnαγkγ; νnνµnµλkλ = (A13a) = δnα,nµ [επnπ − ε ][ω − (ε+nα + ε nπ) + iη] − ε+nβ + ε ][−επ − ε+nν + ε G0>(ω)Γ(3)>(ω) αnαβnβγkγ; µnµνnνλkλ = (A13b) = δkγ,kλ +,nII +,nII [εΓ+nII − ε − ε+nβ][ω − (ε ) + iη] −,kII −,kII − ε+nα − ε ][εΓ− − ε+nµ − ε After substituting Eq. (A10) into (3), one needs the working expression for the matrix product V U(2p1h) (where V is the inter- electron interaction). The minimum correction that guaranties to reproduce all third order self-energy diagrams is V U(2p1h) α; µnµνnνλkλ = Vαλ,µν + Vαλ,γ1δ1 Y Vγ2δ2,µν 2 [ε− − ε+nµ − ε (A14) Vαδ1,µγ1 Y Vγ2λ,δ2ν − ε+nδ − ε Vαδ1,νγ1 Y Vγ2λ,δ2µ − ε+nδ − ε The case of 2h1p is handled in a completely analogous way along the steps of Secs. (A 1) and (A 2). After extending R(ω1, ω2, ω3) to depend on the fragmentation indices (k1,k2,n), the 2h1p equivalent of Eq. (A9) is obtained with the follow- ing definitions of the kernels, G0>(ω)Γ(1)>(ω) αkαβkβγnγ; µkµνkνλnλ G0>(ω)Γ(2)>(ω) βkβαkαγnγ; νkνµkµλnλ = (A15a) = δkα,kµ + ε+nγ ][ω − (ε ) − iη] [επnπ − ε + ε+nγ ][ε nπ − ε + ε+nλ] G0>(ω)Γ(3)>(ω) αkαβkβγnγ; µkµνkνλnλ = (A15b) = δnγ,nλ −,nII −,nII ][ω − (εΓ− − ε+nγ ) − iη] +,kII +,kII [εΓ+nII − ε − ε−kβ ][εΓ+nII − ε − ε−kν and correction to the external legs, V U(2h1p) α; µkµνkνλnλ = Vαλ,µν + Vαλ,γ1δ1 X Vγ2δ2,µν 2 [ε−kµ + ε − ε+nγ − ε (A16) Vαδ1,µγ1 X Vγ2λ,δ2ν [ε−kδ + ε − ε+nγ − ε Vαδ1,νγ1 X Vγ2λ,δ2µ [ε−kδ + ε − ε+nγ − ε It should be pointed out that while the prescription of Sec. A 1 allows sp lines to propagate only in one time di- rection, it allows for backward propagation of the phonons. These contributions translate directly into the energy indepen- dent terms of Eqs. (A13) and (A15) and are a direct conse- quence of the inversion pattern typical of RPA theory. These terms have normally a weaker impact than the direct ones on the solutions of Eqs. (A9). However, it is show in Ref. [24] that they are crucial to guarantee the exact separation of the spurious solutions—always introduced by the Faddeev for- malism [39, 40]—if RPA phonons are used. For the same rea- sons, the last terms in curly brackets of Eqs. (A13) and (A15) should be dropped whenever Tamm-Dancoff (TDA) phonons are propagated. The approach followed in this work for solving Eqs. (A9) is to transform them into a matrix representations [24]. Once this is done, one is left with an eigenvalue prob- lem that depends only on the 2p1h (2h1p) configurations (n, n′, k) [(k, k′, n)]. The spurious states are known ex- actly [24] and can be projected out analytically to reduce the computational load. In any case, they would give vanishing contributions to Eq. (3). [1] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). [2] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). [3] A. 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The definition used here agrees with the standard literature on the subject [29, 38]. http://physics.nist.gov/PhysRefData/ASD/in-dex.html

The spectral function of the closed-shell Neon atom is computed by expanding the electron self-energy through a set of Faddeev equations. This method describes the coupling of single-particle degrees of freedom with correlated two-electron, two-hole, and electron-hole pairs. The excitation spectra are obtained using the Random Phase Approximation, rather than the Tamm-Dancoff framework employed in the third-order algebraic diagrammatic contruction [ADC(3)] method. The difference between these two approaches is studied, as well as the interplay between ladder and ring diagrams in the self-energy. Satisfactory results are obtained for the ionization energies as well as the energy of the ground state with the Faddeev-RPA scheme that is also appropriate for the high-density electron gas.

Quasiparticles in Neon using the Faddeev Random Phase Approximation C. Barbieri Gesellschaft für Schwerionenforschung, Planckstr. 1, D-64291, Darmstadt, Germany D. Van Neck Laboratory of Theoretical Physics, Ghent University, Proeftuinstraat 86, B-9000 Gent, Belgium W.H. Dickhoff Department of Physics, Washington University, St. Louis, MO 63130, USA (Dated: November 2, 2018) The spectral function of the closed-shell Neon atom is computed by expanding the electron self-energy through a set of Faddeev equations. This method describes the coupling of single-particle degrees of free- dom with correlated two-electron, two-hole, and electron-hole pairs. The excitation spectra are obtained using the Random Phase Approximation, rather than the Tamm-Dancoff framework employed in the third-order al- gebraic diagrammatic contruction [ADC(3)] method. The difference between these two approaches is studied, as well as the interplay between ladder and ring diagrams in the self-energy. Satisfactory results are obtained for the ionization energies as well as the energy of the ground state with the Faddeev-RPA scheme that is also appropriate for the high-density electron gas. PACS numbers: 31.10.+z,31.15.Ar I. INTRODUCTION Ab initio treatments of electronic systems become unwork- able for sufficiently complex systems. On the other hand, the Kohn-Sham formulation [1] of density functional the- ory (DFT) [2] incorporates many-body correlations (beyond Hartree-Fock), while only single-particle (sp) equations must be solved. Due to this simplicity DFT is the only feasible approach in some modern applications of electronic structure theory. There is therefore a continuing interest both in devel- oping new and more accurate functionals and in studying con- ceptual improvements and extensions to the DFT framework. In particular it is found that DFT can handle short-range inter- electronic correlations quite well, while there is room for im- provements in the description of long-range (van der Waals) forces and dissociation processes. Microscopic theories offer some guidance in the devel- opment of extensions to DFT. Orbital dependent function- als can be constructed using many-body perturbation theory (MBPT) [3, 4]. More recently, the development of general ab initio DFT [5, 6] addressed the lack of a systematic im- provement in DFT methods. In this approach one considers an expansion of the exact ground-state wave function (e.g., MBPT or coupled cluster) from a chosen reference determi- nant. Requiring that the correction to the density vanishes at a certain level of perturbation theory allows one to construct the corresponding approximation to the Kohn-Sham potential. A different route has been proposed in Ref. [7] by devel- oping a quasi-particle (QP)-DFT formalism. In the QP-DFT approach the full spectral function is decomposed in the con- tribution of the QP excitations, and a remainder or background part. The latter part is complicated, but does not need to be known accurately: it is sufficient to have a functional model for the energy-averaged background part to set up a single- electron selfconsistency problem that generates the QP exci- tations. Such an approach is appealing since it contains the well-developed standard Kohn-Sham formulation of DFT as a special case, while at the same time emphasis is put on the correct description of QPs, in the Landau-Migdal sense [8]. Hence, it can provide an improved description of the dynam- ics at the Fermi surface. Given the close relation between QP- DFT and the Green’s function (GF) formulation of many-body theory [9, 10], it is natural to employ ab initio calculations in the latter formalism to investigate the structure of possible QP-DFT functionals. In this respect it is imperative to iden- tify which classes of diagrams are responsible for the correct description of the QP physics. Some previous calculations, based on GF theory, have fo- cused on a self-consistent treatment of the self-energy at the second order [11, 12, 13] for simple atoms and molecules. For the atomic binding energies it was found that the bulk of corre- lations, beyond Hartree-Fock, are accounted for while signifi- cant disagreement with experiment persists for QP properties like ionization energies and electron affinities. The formal- ism beyond the second-order approximation was taken up in Ref. [14, 15, 16, 17, 18] by employing a self-energy of the GW type [19]. In this approach, the random phase approx- imation (RPA) in the particle-hole (ph) channel is adopted to allow for possible collective effects on the atomic excited states. The latter are coupled to the sp states by means of dia- grams like the last two in Fig. 1(c). Two variants of the G0W0 formalism were employed in Ref. [14] (where the subscript “0” indicates that non-dressed propagators are used). In the first only the direct terms of the interelectron Coulomb po- tential are taken into account. In the second version, also the exchange terms are included when diagonalizing the ph space [generalized RPA (GRPA)] and in constructing the self-energy [generalized GW (GGW)]. Although the exchange terms are known to be crucial in order to reproduce the experimentally observed Rydberg sequence in the excitation spectrum of neu- tral atoms, they were found to worsen the agreement between the theoretical and experimental ionization energies [14]. http://arxiv.org/abs/0704.1542v2 In the GW approach the sp states are directly coupled with the two-particle–one-hole (2p1h) and the two-hole–one- particle (2h1p) spaces. However, only partial diagonalizations (namely, in the ph subspaces) are performed. This procedure unavoidably neglects Pauli correlations with the third particle (or hole) outside the subspace. In the case of the GGW ap- proach, this leads to a double counting of the second order self-energy which must be corrected for explicitly [20, 21]. We note that simply subtracting the double counted diagram is not completely satisfactory here, since it introduces poles with negative residues in the self-energy. More important, the inter- action between electrons in the two-particle (pp) and two-hole (hh) subspaces are neglected altogether in (G)GW. Clearly, it is necessary to identify which contributions, beyond GGW, are needed to correctly reproduce the QP spectrum. In this respect, it is known that highly accurate descrip- tions of the QP properties in finite systems can be obtained with the algebraic diagrammatic construction (ADC) method of Schirmer and co-workers [22]. The most widely used third- order version [ADC(3)] is equivalent to the so-called extended 2p1h Tamm-Dancoff (TDA) method [23] and allows to pre- dict ionization energies with an accuracy of 10-20 mH in atoms and small molecules. Upon inspection of its diagram- matic content, the ADC(3) self-energy is seen to contain all diagrams where TDA excitations are exchanged between the three propagator lines of the intermediate 2p1h or 2h1p prop- agation. The TDA excitations are constructed through a diag- onalization in either 2p1h or 2h1p space, and neglect ground- state correlations. However, it is clear that use of TDA leads to difficulties for extended systems. In the high-density elec- tron gas e.g., the correct plasmon spectrum requires the RPA in the ph channel, rather than TDA. In order to bridge the gap between the QP description in finite and extended systems, it seems therefore necessary to develop a formalism where the intermediate excitations in the 2p1h/2h1p propagator are described at the RPA level. This can be achieved by a formalism based on employing a set of Faddeev equations, as proposed in Ref. [24] and subsequently applied to nuclear structure problems [25, 26, 27]. In this ap- proach the GRPA equations are solved separately in the ph and pp/hh subspaces. The resulting polarization and two-particle propagators are then coupled through an all-order summation that accounts completely for Pauli exchanges in the 2p1h/2h1p spaces. This Faddeev-RPA (F-RPA) formalism is required if one wants to couple propagators at the RPA level or beyond. Apart from correctly incorporating Pauli exchange, F-RPA takes the explicit inclusion of ground-state correlations into account, and can therefore be expected to apply to both finite and extended systems. The ADC(3) formalism is recovered as an approximation by neglecting ground-state correlations in the intermediate excitations (i.e. replacing RPA with TDA phonons). In this work we consider the Neon atom and apply the F- RPA method to a nonrelativistic electronic problem for the first time. The relevant features of the F-RPA formalism (also extensively treated in Ref. [24]), are introduced in Sect. II. The application to the Neon atom is discussed in Sec. III, where we also investigate the separate effects of the ladder and ring series on the self-energy, as well as the differences between including TDA and RPA phonons. Our findings are summarized in Sec. IV. Some more technical aspects are rele- gated to the appendix, where the interested reader can find the derivation of the Faddeev expansion for the 2p1h/2h1p propa- gator, adapted from Ref. [24]. In particular, the approach used to avoid the multiple-frequency dependence of the Green’s functions is discussed in App. A 1, along with its basic as- sumptions. The explicit expressions of the Faddeev kernels are given in App. A 3. Together with Ref. [24], the appendix provides sufficient information for an interested reader to ap- ply the formalism. II. FORMALISM The theoretical framework of the present study is that of propagator theory, where the object of interest is the sp prop- agator, instead of the many-body wave function. In this paper we will employ the convention of summing over repeated in- dices, unless specified otherwise. Given a complete orthonor- mal basis set of sp states, labeled by α,β,..., the sp propagator can be written in its Lehmann representation as [9, 10] gαβ(ω) = ω − ε+n + iη ω − ε−k − iη , (1) where Xnα = 〈Ψ 0 〉 (Y α = 〈Ψ |cα|Ψ 0 〉) are the spectroscopic amplitudes, cα (c ) are the second quantization destruction (creation) operators and ε+n = E n − E EN0 − E ). In these definitions, |ΨN+1n 〉, |Ψ 〉 are the eigenstates, and EN+1n , E k the eigenenergies of the (N ± 1)- electron system. Therefore, the poles of the propagator reflect the electron affinities and ionization energies. The sp propagator solves the Dyson equation gαβ(ω) = g αβ(ω) + g αγ(ω)Σ γδ(ω)gδβ(ω) , (2) which depends on the irreducible self-energy Σ⋆(ω). The lat- ter can be written as the sum of two terms αβ(ω) = Σ Vαλ,µν Rµνλ,γδε(ω) Vγδ,βε , (3) where ΣHF represents the Hartree-Fock diagram for the self- energy. In Eqs. (2) and (3), g0(ω) is the sp propagator for the system of noninteracting electrons, whose Hamiltonian contains only the kinetic energy and the electron-nucleus at- traction. The Vαβ,γδ represent the antisymmetrized matrix el- ements of the interelectron (Coulomb) repulsion. Note that in this work we only consider antisymmetrized elements of the interaction, hence, our result for the ring summation al- ways compare to the generalized GW approach. Equation (3) introduces the 2p1h/2h1p-irreducible propagator R(ω), which carries the information concerning the coupling of sp states to more complex configurations. Both Σ⋆(ω) and R(ω) have a perturbative expansion as a power series in the interelectron FIG. 1: (Color online) a) Diagrammatic expansion of R(ω) in terms of the (antisymmetrized) Coulomb interaction and undressed prop- agators. b) R(ω) is related to the self-energy according to Eq. (3). c) By substituting the diagrams a) in the latter equation, one finds the perturbative expansion of the self-energy. interaction V̂ . Some of the diagrams appearing in the expan- sion of R(ω) are depicted in Fig. 1, together with the corre- sponding contributions to the self-energy. Note that already at zero order in R(ω) (three free lines with no mutual interaction) the second order self-energy is generated. Different approximations to the self-energy can be con- structed by summing particular classes of diagrams. In this work we are interested in the summation of rings and ladders, through the (G)RPA equations. In order to include such effects in R(ω), we first consider the polarization propagator describ- ing excited states in the N-electron system Παβ,γδ(ω) = 〈ΨN0 |c n 〉 〈Ψ γcδ|Ψ ENn − E 〈ΨN0 |c γcδ|Ψ n 〉 〈Ψ ENn − E , (4) and the two-particle propagator, that describes the addi- tion/removal of two electrons gIIαβ,γδ(ω) = 〈ΨN0 |cβcα|Ψ n 〉 〈Ψ |ΨN0 〉 EN+2n − E 〈ΨN0 |c |ΨN−2 〉 〈ΨN−2 |cβcα|Ψ EN0 − E . (5) We note that the expansion of R(ω) arises from applying the equations of motion to the sp propagator (1), which is associ- ated to the ground state |ΨN0 〉. Hence, all the Green’s functions appearing in this expansion will also be ground state based, in- cluding Eqs. (4) and (5). However the latter contain, in their Lehmann representations, all the relevant information regard- ing the excitation of ph and pp/hh collective modes. The ap- proach of Ref. [24] consists in computing these quantities by =g II Π(ph) (pp/hh)II (pp/hh) FIG. 2: (Color online) Diagrammatic equations for the polarization (above) and the two-particle (below) propagators in the (G)RPA ap- proach. Dashed lines are always antisymmetrized Coulomb matrix elements and the full lines represent free (undressed) propagators. solving the ring-GRPA and the ladder-RPA equations [10], which are depicted for propagators in Fig. 2. In the more general case of a self-consistent calculation, a fragmented in- put propagator can be used and the corresponding dressed (G)RPA [D(G)RPA] equations [10, 28] solved [see Eqs. (A2a) and (A2b)]. Since the propagators (4) and (5) reflect two-body correlations, they still have to be coupled to an additional sp propagator in order to obtain the corresponding approximation for the 2p1h and 2h1p components of R(ω). This is achieved by solving two separate sets of Faddeev equations. Taking the 2p1h case as an example, one can split R(2p1h)(ω) in three different components R̄(i)(ω) (i = 1, 2, 3) that differ from each other by the last pair of lines that interact in their diagrammatic expansion, R̄(2p1h) αβγ,µνλ (ω) = αβγ,µνλ(ω) −G βαγ,µνλ(ω) i=1,2,3 R̄(i) αβγ,µνλ (ω) , where G0 (ω) is the 2p1h propagator for three freely propa- gating lines. These components are solutions of the following set Faddeev equations [29] R̄(i) αβγ,µνλ (ω) = G0 αβγ,µ′ν′λ′(ω) Γ µ′ν′λ′,µ′′ν′′λ′′ µ′′ν′′λ′′ ,µνλ (ω) + R̄(k) µ′′ν′′λ′′ ,µνλ (ω) (7) µ′′ν′′λ′′ ,µνλ(ω) −G ν′′µ′′λ′′ ,µνλ(ω) , i = 1, 2, 3 where (i, j, k) are cyclic permutations of (1, 2, 3). The inter- action vertices Γ(i)(ω) contain the couplings of a ph or pp/hh collective excitation and a freely propagating line. These are given in the Appendix in terms of the polarization (4) and two-particle (5) propagators. Equations. (7) include RPA-like phonons and fully describe the resulting energy dependence of R(ω). However, they still neglect energy-independent contributions–even at low order in the interaction–that also correspond to relevant ground-state correlations. The latter can be systematically inserted according to R(2p1h) αβγ,µνλ (ω) = Uµνλ,µ′ν′λ′ R̄ (2p1h) µ′ν′λ′,µ′′ν′′λ′′ (ω) U† µ′′ν′′λ′′ ,µνλ , (8) where R(ω) is the propagator we employ in Eq. (3), R̄(ω) is the one obtained by solving Eqs. (7), U ≡ I + ∆U, and I is the (pp/hh) Π(ph) g II (pp/hh) Π(ph) FIG. 3: (Color online) Example of one of the diagrams that are summed to all orders by means of the Faddeev Eqs. (7) (left). The corresponding contribution to the self-energy, obtained upon inser- tion into Eq. (3), is also shown (right). identity matrix. Following the algebraic diagrammatic con- struction method [22, 23], the energy independent term ∆U was determined by expanding Eq. (8) in terms of the inter- action and imposing that it fulfills perturbation theory up to first order (corresponding to third order in the self-energy). The resulting ∆U, employed in this work, is the same as in Ref. [23] and is reported in App. A 3 for completeness. It has been shown that the additional diagrams introduced by this correction are required to obtain accurate QP properties. Equations. (7) and (8) are valid only in the case in which a mean-field propagator is used to expand R(ω). This is the case of the present work, which employs Hartree-Fock sp propaga- tors as input. The derivation of these equations for the gen- eral case of a fragmented propagator is given in the appendix. More details about the actual implementation of the Faddeev formalism to 2p1h/2h1p propagation have been presented in Ref. [24]. The calculation of the 2h1p component of R(ω) follows completely analogous steps. It is important to note that the present formalism includes the effects of ph and pp/hh motion to be included simulta- neously, while allowing interferences between these modes. These excitations are evaluated here at the RPA level and are then coupled to each other by solving Eqs. (7). This generates diagrams as the one displayed in Fig. 3, with the caveat that two phonons are not allowed to propagate at the same time. Equations. (7) also assure that Pauli correlations are properly taken into account at the 2p1h/2h1p level. In addition, one can in principle employ dressed sp propagators in these equations to generate a self-consistent solution. If we neglect the lad- der propagator gII (ω) (5) in this expansion, we are left with the ring series alone and the analogous physics ingredients as for the generalized GW approach. However, this differs from GGW due to the fact that no double counting of the second- order self-energy occurs, since the Pauli exchanges between the polarization propagator and the third line are properly ac- counted for (see Fig. 3). Alternatively, one can suppress the polarization propagator to investigate the effects of pp/hh lad- ders alone. It is instructive to replace in the above equations all RPA phonons with TDA ones; this amounts to allowing only forward-propagating diagrams in Fig. 2, and is equivalent to separate diagonalisations in the spaces of ph, pp and hh con- l 0 1 2 3 4 5 6 rw 2.0 4.0 0.0 0.0 0.0 0.0 0.0 no 12 21 10 10 5 5 5 TABLE I: Parameters that define the sp basis: radius of the confining wall rw (in atomic units) and number of orbits no used for different partial waves l. The value of cw is always set to 5 a.u.. figurations, relative to the HF ground state. It can be shown that using these TDA phonons to sum all diagrams of the type in Fig. 3 reduces to one single diagonalization in the 2p1h or 2h1p spaces. Therefore, Eqs. (7) and (8) with TDA phonons lead directly to the “extended” 2p1h TDA of Ref. [23], which was later shown to be equivalent to ADC(3) in the general ADC framework [22]. The Faddeev expansion formalism of Ref. [24] creates the possibility to go beyond ADC(3) by in- cluding RPA phonons. This is more satisfactory in the limit of large systems. At the same time, the computational cost remains modest since only diagonalizations in the 2p1h/2h1p spaces are required. Note that complete self-consistency requires the use of fragmented (or dressed) propagators in the evaluation of all in- gredients leading to the self-energy. This is outside the scope of the present paper, but we included partial selfconsistency by taking into account the modifications to the HF diagram by employing the correlated one-body density matrix and it- erating to convergence. This is relatively simple to achieve, since the 2p1h/2h1p propagator is only evaluated once with the input HF propagators. Below we will give results with and without this partial selfconsistency at the HF level. III. RESULTS Calculations have been performed using two different model spaces: (1) a standard quantumchemical Gaussian ba- sis set, aug-cc-pVTZ for Neon [30], with Cartesian repre- sentation of the d and f functions; (2) a numerical basis set based on HF and subsequent discretization of the continuum, to be detailed below. The aug-cc-pVTZ basis set was used primarily to check our formalism with the ADC(3) result in literature (i.e. [31], where this basis was employed). The HF+continuum basis allows to approach, at least for the ion- ization energies, the results for the full sp space (basis set limit). The HF+continuum is the same discrete model space em- ployed previously in Refs. [11, 14]. It consists of: (1) Solv- ing on a radial grid the HF problem for the neutral atom; (2) Adding to this fixed nonlocal HF potential a parabolic poten- tial wall of the type U(r) = θ(r − rw)cw(r − rw) 2, placed at a distance rw of the nucleus. The latter eigenvalue problem has a basis of discrete eigenstates. This basis is truncated by spec- ifying some largest angular momentum lmax and the number of virtual states for each value of l ≤ lmax. (3) Solve the HF problem again, without the potential wall, in this truncated discrete space. The resulting basis set is used for the subse- F-TDA F-RPA F-TDAc F-RPAc Expt. 2p -0.799 -0.791 -0.803 -0.797 (0.94) -0.793 (0.92) 2s -1.796 -1.787 -1.802 -1.793 (0.90) -1.782 (0.85) 1s -32.126 -32.087 -32.140 -32.102 (0.86) -31.70 Etot -128.778 -128.772 -128.836 -128.840 -128.928 TABLE II: Results with the aug-cc-pVTZ basis. The first three rows list the energies of the main sp fragments below the Fermi level, as predicted by different self-energies. F-TDA/F-RPA refers to the Fad- deev summation with TDA/RPA phonons, respectively. In all cases the self-energy was corrected at third order through Eq. (8). The suffix “c” refers to partial selfconsistency, when the static (HF-type) self-energy is consistent with the correlated density matrix. Without “c” the pure HF self-energy was taken. In the F-RPAc column the strength of the fragment is indicated between brackets. The last row is the total electronic binding energy. The experimental values are taken from Refs. [32, 33]. All energies are in atomic units. quent Green’s function calculations. When a sufficiently large number of states is retained after truncation, the final results should approach the basis set limit. In particular the results should not depend on the choice of the auxiliary confining potential. This was verified in Ref. [11] for the second-order, and in Ref. [14] for the G0W0 self-energy; in these cases the self-energy is sufficiently simple that extensive convergence checks can be made for various choices of the auxiliary potential. The parameters of the confining wall and the number of sp states kept in the basis set was optimized in Ref. [11], by requiring that the ionization energy is converged to about 1 mH for the second-order self-energy. In Ref. [14] the same choice of basis set was also seen to bring the ioniza- tion energy for the G0W0 self-energy near convergence. For completeness, the details of this basis are reported in Table I. While the self-energy in the present paper is too complicated to allow similar convergence checks, it seems safe to assume that basis set effects will affect the calculated ionization ener- gies by at most 5 mH. In Table II we compare, for the aug-cc-pVTZ basis, the ion- ization energies of the main single-hole configurations when TDA or RPA phonons are employed in the Faddeev construc- tion (this is labeled F-TDA and F-RPA, respectively, in the table). Note that use of TDA phonons corresponds to the usual ADC(3) self-energy. We find that the replacement of TDA with RPA phonons provides more screening, leading to slightly less bound poles which are shifted towards the exper- imental values. This shift increases with binding energy. As discussed at the end of Sec. I, one can include consistency of the static part of the self-energy. About eight iterations are needed for convergence. This is a nonnegligible correction, providing about 5 mH more binding (i.e. larger ionization en- ergies) for the valence/subvalence 2p and 2s, 15 mH for the deeply bound 1s, and 60 mH to the total binding energy. Our converged result for the Faddeev-TDA self-energy (labeled F- TDAc in Table II) is in good agreement with the ADC(3) value for the 2p ionization energy (-0.804 H) quoted in [31], as it should be. The analogous results obtained with the larger F-TDA F-RPA F-TDAc F-RPAc Expt. 2p -0.807 -0.799 -0.808 -0.801 (0.94) -0.793 (0.92) 2s -1.802 -1.792 -1.804 -1.795 (0.91) -1.782 (0.85) 1s -32.136 -32.097 -32.142 -32.104 (0.81) -31.70 Etot -128.863 -128.857 -128.883 -128.888 -128.928 TABLE III: Results with the HF+continuum basis set from Table I. See also the caption of Table II. -40 -35 -30 [a.u.] F-RPA(ring) F-RPA(ladder) F-RPA FIG. 4: (Color online) Spectral function for the s states in Ne obtained with various self-energy approximations. From the top down: the second-order (Σ(2)) self-energy, the F-RPA(ring), the F- RPA(ladder), and the full F-RPA self-energy. The strength is given relative to the Hartree-Fock occupation of each shell. Only fragments with strength larger than Z > 0.005 are shown. HF+continuum basis are given in Table III, which al- lows to assess overall stability and basis set effects. We find exactly the same trends as for aug-cc-pVTZ. In particular the reduction of ionization energies from the replacement of TDA with RPA phonons is almost independent of the basis set used, while the effect of including partial consistency is roughly halved. Overall, the ionization states are always more bound with the larger basis set; while the basis set limit could be still more bound than the present results with the HF+continuum basis set, it is likely (based on the G0W0 extrapolation in Ref. [14]) that the difference does not exceed 5 mH. As discussed in Sec. I, the the F-RPA self-energy contains RPA excitations of both ph type (ring diagrams) and pp/hh type (ladder diagrams). It is instructive to analyze their sepa- rate contributions to the final ionization energies, in order to 1s 2s 2p HF -32.77 (1.00) -1.931 (1.00) -0.850 (1.00) Σ(2) -31.84 (0.74) -1.736 (0.88) -0.747 (0.91) G0W0 -31.14 (0.85) -1.774 (0.91) -0.801 (0.94) F-RPA (ring) -31.82 (0.73) -1.636 (0.56) -0.730 (0.80) F-RPA (ladder) -32.04 (0.87) -1.802 (0.95) -0.781 (0.96) F-RPA -32.10 (0.81) -1.792 (0.91) -0.799 (0.94) Exp. -31.70 -1.782 (0.85) -0.793 (0.92) TABLE IV: Energy (in a.u.) and strength (bracketed numbers) of the main fragments in the spectral function of Neon, generated by differ- ent self-energies. Results for the HF+continuum basis. Consecutive rows refer to: (1) HF; (2) second-order self-energy; (3) G0W0 results from Ref. [14]; (4) F-RPA self-energy with only ph rings retained; (5) F-RPA self-energy with only pp/hh ladders retained; (6) Com- plete F-RPA self-energy. In all F-RPA results the self-energy was corrected at third order through Eq. (8). The static self-energy was pure HF (no partial self-consistency). The experimental values are taken from Refs. [32, 33]. understand how the F-RPA self-energy is related to the stan- dard (G)GW self-energy. Table IV compares the results for the ionization energies, obtained with the second-order self- energy, to different approximations for including the ring sum- mations. As one can see, the second-order self-energy gener- ates an l=1 sp energy of -0.747 mH, which is 46 mH above the empirical 2p ionization energy. The G0W0 self-energy, which includes the ring summation with only direct Coulomb matrix elements, improves this result and brings it close to ex- periment. The 2s behaves in a similar way. Unfortunately, including the exchange terms of the interelectron repulsion in the GG0W0 method turns out to have the opposite effect (the 2p ionization energy becomes -0.712 H [14] [41]) and the agreement with experiment is lost. Obviously, GG0W0 is too simplistic to account for exchange in the ph channel. With the F-RPA(ring) self-energy one can go one step fur- ther and employ the Faddeev expansion to also force proper Pauli exchange correlations in the 2p1h/2h1p spaces. As shown in Table IV, this enhances the screening due to the exchange interaction terms, leading to even less binding for the 2s and 2p. The corrections relative to the second-order self-energy can be large (100 mH for the 2s state) and in the direction away from the experimental value. We also note that the larger shift, in the 2s orbit, is accompanied by an increase of the fragmentation (see Fig. 4 and Tab. IV). Similar observa- tions were also made in Ref. [14] for other atoms: in general ring summations in the direct channel alone bring the quasi- hole peaks close to the experiment. This agreement is then spoiled as soon as one includes proper exchange terms in the self-energy. On the other hand, exchange in the ph channel is required to reproduce the correct Rydberg sequence in the excitation spectrum of neutral atoms. So further corrections must arise from other diagrams, and obviously the summation of ladder diagrams can play a relevant role, since these con- tribute to the expansion of the self-energy at the same level as that of the ring diagrams. The result when only including ladder-type RPA phonons in the F-RPA self-energy is also shown in Table IV. One can see that pp/hh ladders do actually work in the opposite way as the ph channel ring diagrams, and have the same order of magnitude with, e.g., a shift of 66 mH for the 2s relative to the second-order result. When combined with the ring dia- grams in the full F-RPA self-energy, the agreement with ex- periment is restored again. Note that the final result cannot be obtained by adding the contributions of rings and ladders, but depends nontrivially on the interplay between these classes of diagrams thereby pointing to significant interference effects. With the F-RPA(ring) self-energy, where only the contribu- tions of the ph channel are included, the main peaks listed in Table IV are not only considerably shifted but also strongly depleted, e.g. a strength of only 0.56 for the main 2s peak. The complete spectral function for the l = 0 strength in Fig. 4 shows that the depletion of the main fragment is accompanied by strong fragmentation over several states. While correlation effects are overestimated in F-RPA(ring), they are suppressed in F-RPA(ladder), where only the pp/hh ladders are included in the self-energy. In this case one finds a spectral distribu- tion closer to the HF one, with a main 2s fragment of strength 0.95 and less fragmentation than the the second-order self- energy. The spectral distribution generated by the complete F-RPA self-energy is again a combination of the above ef- fects. The strength of the deeply bound 1s orbital behaves in an analogous way. The strength of the main peak is reduced but several satellite levels appear due to the mixing with 2h1p configurations. In all the calculations reported in Fig. 4 we found a summed l = 0 strength exceeding 0.98 in the interval [-40 H, -30 H] which can be associated with the 1s orbital, and this remains true even in the presence of strong correla- tions using the F-RPA(ring) self-energy. Of course, the mix- ing with 3h2p configurations, not included in this work, may further contribute to the fragmentation pattern in this energy region. IV. CONLUSIONS AND DISCUSSION In conclusion, the electronic self-energy for the Ne atom was computed by the F-RPA method which includes – simultaneously– the effects of both ring and ladder diagrams. This was accomplished by employing an expansion of the self-energy based on a set of Faddeev equations. This tech- nique was originally proposed for nuclear structure applica- tions [24] and is described in the appendix. At the level of the self-energy one sums all diagrams where the three propa- gator lines of the intermediate 2p1h or 2h1p propagation are connected by repeated exchange of RPA excitations in both the ph and the pp/hh channel. This differs from the ADC(3) formalism in the fact that the exchanged excitations are of the RPA type, rather than the TDA type, and therefore take ground-state correlations effects into account. The coupling to the external points of the self-energy uses the same modi- fied vertex as in ADC(3), which must be introduced to include consistently all third-order perturbative contributions. The resulting main ionization energies in the Neon atom are at least of the same quality, and even somewhat improved, compared to the ADC(3) result. Note that, numerically, F- RPA can be implemented as a diagonalization in 2p1h/2h1p space implying about the same cost as ADC(3). The present study also shows that in localized electronic systems subtle cancellations occur between the ring and ladder series. In par- ticular, only a combination of the ring and ladder series leads to sensible results, as the separate ring series tends to correct the second-order result in the wrong direction. Since the limit to extended systems requires an RPA treat- ment of excitations, the F-RPA method holds promise to bridge the gap between an accurate description of quasiparti- cles in both finite and extended systems. In particular, the GW treatment of the electron gas has been shown to yield excellent binding energies, but poor quasiparticle properties [34, 35]. Further progress beyond GW theory requires a consistent in- corporation of exchange in the ph channel. The F-RPA tech- nique may be highly relevant in this respect. A common framework for calculating accurate QP properties in both fi- nite and extended systems, is also important for constrain- ing functionals in quasiparticle density functional theory (QP- DFT) [7]. Finally, complete self-consistency requires sizable compu- tational efforts for bases as large as the HF+continuum basis used here. It would nevertheless represent an important exten- sion of the present work, since it is related to the fulfillment of conservation laws [36, 37]. These issues will be addressed in future work. Acknowledgments This work was supported by the U.S. National Science Foundation under grant PHY-0652900. APPENDIX A: FADDEEV EXPANSION OF THE 2P1H/2H1P PROPAGATOR Although only the one-energy (or two-time) part of the 2p1h/2h1p propagator enters the definition of the self energy, Eq. (3), a full resummation of all its diagrammatic contribu- tions would require to treat explicitly the dependence on three separate frequencies, corresponding to the three final lines in the expansion of R(ω). For example, inserting the RPA ring (ladder) series in R(ω) implies the propagation of a ph (pp/hh) pair of lines both forward and backward in time, while the third line remains unaffected. A way out of this situation is to solve the Bethe-Salpeter-like equations for the polarization and ladder propagators separately and then to couple them to the additional line. If it is assumed that different phonons do not overlap in time, the three lines in between phonon struc- tures will propagate only in one time direction [see figures (3) and (5)]. In this situation the integration over several frequen- cies can be circumvented following the prescription detailed in the next subsection. This approach will be discussed in the following for the general case of a fully fragmented prop- gator, in order to derive a set of Faddeev equations capable FIG. 5: (Color online) Diagrammatic representation of Eq. (A3). Double lines represent fully dressed sp Green’s funcions which, how- ever, are restricted to propagate only in one time direction [i.e., only one of the two terms on the r.h.s. of Eq. (1) is retained]. The Faddeev Eqs. (A9) and (7) allow for both forward and backward propagation of the phonons Γ(π)(ω) and Γ(II)(ω) as long as these do not overlap in time. For the propagators, time ordereing is asumed with forwad propagation in the upward direction. of dressing the sp propagator self-consistently. Since the for- ward (2p1h) and the backward (2h1p) parts of R(ω) decouple in two analogous sets of equations, it is sufficient to focus on the first case alone. 1. Multiple frequencies integrals We start by considering the effective interactions in the ph and pp/hh channels that correspond to Eqs. (4) and (5) stripped of the external legs. In the present work, these are the follow- ing two-time objects: αβ,γδ (ω) = Vαδ,βγ + Vαν,βµ Π µν,ρσ(ω) Vρδ,σγ (A1a) = Vαδ,βγ + ω − επn + iη ω + επn′ − iη αβ,γδ (ω) = Vαβ,γδ + Vαβ,µν g µν,ρσ(ω) Vρσ,γδ (A1b) = Vαδ,βγ + ω − εΓ+n + iη ω − εΓ− where the residues and poles for the ring series are Ωn 〈ΨNn |c µcν|Ψ 0 〉Vµβ,να and ε n = E 0 . For the ladders, ∆ 〈ΨN+2n |c 0 〉Vµν,αβ and ∆ = Vαβ,µν〈Ψ k |cµcν|Ψ 0 〉, with poles εΓ+n = E 0 and ε k = E . Equations. (A1) solve the ring and ladder RPA equations, respectively αβ,γδ (ω) = Vαδ,βγ (A2a) αβ,µν gµρ(ω + ω1)gσν(ω1) Vρδ,σγ , αβ,γδ (ω) = Vαβ,γδ (A2b) αβ,µν gµρ(ω − ω1)gνσ(ω1) Vρσ,γδ . To display how the phonons (A1a) and (A1b) enter the ex- pansion of R(ω), we perform explicitly the frequency integrals for the diagram of Fig. 5. Since it is assumed that the separate propagators lines evolve only in one time direction, only the forwardgoing (g>(ω)) or the backwardgoing (g<(ω)) part of Eq. (1) must be included for particles and holes, respectively. After some algebra, one obtains ∆Rαβγ,µνλ(ω) = g>αα1 (ω −Ω) g (ω1) g (ω1 −Ω) Γ β1γ1,σ1λ1 (Ω) g>σ1σ2 (s + Ω − ω) α1σ2,µ1ν1 (s) g>µ1µ(s − ω2) g (ω2) g (s − ω) ω − (ε+n1 + ε ) + iη Vβ1λ1,γ1σ1 + ω − (ε+n1 + ε nπ) + iη [ω − επ − ε+n1 − ε + ε−k3 − ε+n4 + ε − ε+n2 + ε ][−επ − ε+n4 + ε ω − (ε+n1 + ε ) + iη Vα1σ2,µ1ν1 + +,nII +,nII ω − (εΓ+nII − ε ) + iη [ω + εΓ−kII − ε − ε+n4 − ε − ε+n6 + ε ] ∆−,kIIα1σ2 −,kII − ε+n1 − ε ][εΓ− − ε+n5 − ε ω − (ε+n5 + ε − ε−k7 ) + iη ω − (εΓ− − ε+n4 − ε ) − iη −,kII −,kII − ε+n2 + ε ] [−επ − ε+n4 + ε ] [εΓ− − ε+n1 − ε ] [εΓ− − ε+n5 − ε The last term in this expression contains an energy denom- inator that involves the simultaneous propagation of two phonons. Thus, it will be discarded in accordance with our assumptions. It must be stressed that similar terms, with over- lapping phonons, imply the explicit contribution of at least 3p2h/3h2p. A proper treatment of these would require a non trivial externsion of the present formalism, which is beyond the scope of this paper. The remaining part in Eq. (A3) is the relevant contribution for our purposes. This has the correct energy dependence of a product of denominators that correspond to the intermediate steps of propagation. All of these involve configurations that have at most 2p1h character. Although, ground state corre- lations are implicitely included by having already resummed the RPA series. Still, this term does not factorize in a prod- uct of separate Green’s functions due to the summations over the fragmentation indices ni and ki [labeling the eigenstates of the (N±1)-electron systems]. This is overcome if one defines the matrices G0>(ω), Γ(1,2)(ω) and Γ(3)(ω), with elements (no implicit summation used) αnαβnβγkγ; µnµνnνλkλ (ω) = δnα,nµ δnβ,nν δkγ,kλ ω − (ε+nα + ε − ε−kγ ) + iη , (A4a) αnαβnβγkγ; µnµνnνλkλ (ω) = Γ(2)> βnβαnαγkγ; νnνµnµλkλ (ω) = δα,µ δnα ,nµ Vβλ,γν + ω − (ε+nα + ε nπ) + iη [ω − επ − ε+nα − ε + ε−kγ − ε+nν + ε − ε+nβ + ε ][−επ − ε+nν + ε , (A4b) αnαβnβγkγ; µnµνnνλkλ (ω) = δγ,λ δkγ ,kλ Vαβ,µν + +,nII +,nII ω − (εΓ+nII − ε ) + iη [ω + εΓ−kII − ε − ε+nβ − ε − ε+nν + ε ] ∆−,kII −,kII − ε+nα − ε ][εΓ− − ε+nµ − ε . (A4c) In these definitions, the row and column indices are ordered to represent at first two quasiparticle lines and then a quasihole. The index ‘i’ in Γ(i)> refer to the line that propagates indepen- dently along with the phonon. Using Eqs.(A4), the first term on the r.h.s. of Eq. (A3) can be written as ∆R(2p1h) αβγ,µνλ (ω) = (A5) nα nβ kγ nµ nν kλ G0>(ω)Γ(1)>(ω)G0>(ω)Γ(3)>(ω)G0>(ω) αnαβnβγkγ; µnµνnνλkλ Eq. (A5) generalizes to diagrams involving any number of phonon insertions, as long as the terms involving two or more simultaneous phonons are dropped. Based on this relation, we use the following prescription to avoid performing integrals over frequencies. One extends all the Green’s functions to ob- jects depending not only on the sp basis’ indices (α, β, γ) but also on the indices labeling quasi-particles and holes (ni and ki). Whether a given argument represents a particle or an hole depends on the type of line being propagated. At this point one can perform calculations working with only two-time quanti- ties. The standard propagator is recovered at the end by sum- ming the “extended” one over the quasi-particle/hole indices. 2. Faddeev expansion The 2p1h/2h1p propagator that includes the full resumma- tion of both the ladder and ring diagrams at the (G)RPA level is the solution of the following Bethe-Salpeter-like equation, Rαβγ,µνλ(ω1, ω2, ω3) = (A6) gαµ(ω1)gβν(ω2) − gβµ(ω2)gαν(ω1) gλγ(−ω3) + gββ1(ω2)gγ1γ(−ω3)Vβ1σ,γ1ρ Rαρσ,µνλ(ω1, s, ω2 + ω3 − s) + gαα1 (ω1)gγ1γ(−ω3)Vα1σ,γ1ρ Rρβσ,µνλ(s, ω2, ω1 + ω3 − s) gαα1 (ω1)gββ1(ω2)Vα1β1,ρσ Rρσγ,µνλ(s, ω1 + ω2 − s, ω3) If this equation is solved, a double integration of R(ω1, ω2, ω3) would yield the two-time propagator R(ω) contributing to Eq. (3). However, the numerical solution of Eq. (A6) appears beyond reach of the present day computers and one needs to avoid dealing directly with multiple frequencies integrals. The strategy used is to first solve the RPA equations (A2a) and (A2b) separately. Once this is done it is necessary to re- arrange the series (A6) in such a way that only the resummed phonons appear. Following the formalism introduced by Fad- deev [29, 38], we identify the components R(i)(ω) with the three terms between curly brakets in Eq. (A6). By employ- ing Eqs. (A2a) and (A2b) one is lead to the following set of equations [42], αβγ,µνλ (ω1, ω2, ω3) = gαα1 (ω1)gββ1(ω2)gγ1γ(−ω3) (A7) ds1 ds2 ds3 α1β1γ1,µ1ν1λ1 (ω1, ω2, ω3; s1, s2, s3) gµ1µ(s1)gν1ν(s2) − gν1µ(s2)gµ1ν(s1) gλλ1(−s3) µ1ν1λ1,µνλ (s1, s2, s3) + R µ1ν1λ1,µνλ (s1, s2, s3) , i = 1, 2, 3 , where (i,j,k) are cyclic permutations of (1,2,3) and the inter- action vertices Γ(i)(ω1, ω2, ω3) are given by αβγ,µνλ (ω1, ω2, ω3;ω4, ω5, ω6) = (A8a) βαγ,νµλ (ω2, ω1, ω3;ω5, ω4, ω6) = = δ(ω1 − ω4)δ(ω2 + ω3 − ω5 − ω6)g αµ(ω1)Γ βγ,νλ (ω2 + ω3) , αβγ,µνλ (ω1, ω2, ω3;ω4, ω5, ω6) = (A8b) δ(ω3 − ω6)δ(ω1 + ω2 − ω4 − ω5)g λγ (−ω3)Γ αβ,µν (ω1 + ω2) . Finally, we apply the prescription of Sec. A 1 and substitute R(ω1, ω2, ω3) with its extended but two-time version R(ω). This leads to the following set of Faddeev equations which propagate 2p1h forward in time, R̄(i) αnαβnβγkγ; µnµνnνλkλ (ω) = αnαβnβγkγ; α′n′αβ′n γ′k′γ (ω) Γ(i) α′n′αβ′n γ′k′γ; µ′n ′n′νλ′k αnαβnβγkγ; µnµνnνλkλ (ω) −G0 αnαβnβγkγ; µnµνnνλkλ × R̄( j) µ′n′µν′n ; µnµνnνλkλ (ω) + R̄(k) µ′n′µν′n ; µnµνnνλkλ i = 1, 2, 3 . (A9) Since the full energy dependence is retained in Eq. (A7), the self-energy corresponding to its solution, R(ω1, ω2, ω3), is complete up to third order [see Eq. (3)]. This is no longer the case after the reduction to a two-time propagator. In particu- lar, the approximation that only forward 2p1h propagation is allowed between different phonons implies that all diagrams with different time propagation of their external lines are ne- glected in Eqs. (A9). However, these terms are not energy dependent and can be can be reinserted in a systematic way a posteriori as in Eq. (8). In the general case, Rαβγ,µνλ(ω) = (A10) U (2p1h) αβγ; α′n′αβ′n γ′k′γ R̄(2p1h) α′n′αβ′n γ′k′γ; µ′n ′n′νλ′k (ω) U (2p1h) † µ′n′µν′n ; µνλ U (2p1h) αβγ; µnµνnνλkλ = δαµδβνδγλ + ∆U (2p1h) αβγ; µnµνnνλkλ , (A11) where the correction ∆U can be determined by comparison with perturbation theory. The vertices (A4), that appear in Eqs. (A9), and U (2p1h) are expressed in terms of the fully fragmented propagator. There- fore, this approach allows to obtain self-consistent solutions of the sp Green’s function [25]. Whenever, like in this work, only a mean-field propagator is employed as input there ex- ist a one-to-one correspondence between the fragmentation indices and the sp basis. This is expressed by the relations Xnα = δn,α(1 − δα∈F) and Y α = δk,αδα∈F , where F represents the set of occupied orbits. In this case, it is possible to drop one set of indices so that Eqs. (A9) and (A10) simplify into the form (7) and (8). 3. Faddeev vertices In practical applications, it is worth to note that the poles of the free propagator G0(ω), Eq. (A4a), do not contribute to the kernel of Eqs. (A9). This can be proven by employing the closure relations for the RPA problem, in the form obtained by extracting the free poles in Eqs. (A2). As an example, for the forward poles of the ladder propagator these are ω→ε+n1 +ε+n2 [(ω − ε+n1 − ε ) × (Eq. A2b)] =⇒ (A12) µν,γδ (ω = ε+n1 + ε ) = 0 , ∀n1, n2 , and similarly for other cases. Making use of these relations one can derive the following working expression of the ker- nels of the 2p1h Faddeev equations (no implicit summations used) G0>(ω)Γ(1)>(ω) αnαβnβγkγ; µnµνnνλkλ G0>(ω)Γ(2)>(ω) βnβαnαγkγ; νnνµnµλkλ = (A13a) = δnα,nµ [επnπ − ε ][ω − (ε+nα + ε nπ) + iη] − ε+nβ + ε ][−επ − ε+nν + ε G0>(ω)Γ(3)>(ω) αnαβnβγkγ; µnµνnνλkλ = (A13b) = δkγ,kλ +,nII +,nII [εΓ+nII − ε − ε+nβ][ω − (ε ) + iη] −,kII −,kII − ε+nα − ε ][εΓ− − ε+nµ − ε After substituting Eq. (A10) into (3), one needs the working expression for the matrix product V U(2p1h) (where V is the inter- electron interaction). The minimum correction that guaranties to reproduce all third order self-energy diagrams is V U(2p1h) α; µnµνnνλkλ = Vαλ,µν + Vαλ,γ1δ1 Y Vγ2δ2,µν 2 [ε− − ε+nµ − ε (A14) Vαδ1,µγ1 Y Vγ2λ,δ2ν − ε+nδ − ε Vαδ1,νγ1 Y Vγ2λ,δ2µ − ε+nδ − ε The case of 2h1p is handled in a completely analogous way along the steps of Secs. (A 1) and (A 2). After extending R(ω1, ω2, ω3) to depend on the fragmentation indices (k1,k2,n), the 2h1p equivalent of Eq. (A9) is obtained with the follow- ing definitions of the kernels, G0>(ω)Γ(1)>(ω) αkαβkβγnγ; µkµνkνλnλ G0>(ω)Γ(2)>(ω) βkβαkαγnγ; νkνµkµλnλ = (A15a) = δkα,kµ + ε+nγ ][ω − (ε ) − iη] [επnπ − ε + ε+nγ ][ε nπ − ε + ε+nλ] G0>(ω)Γ(3)>(ω) αkαβkβγnγ; µkµνkνλnλ = (A15b) = δnγ,nλ −,nII −,nII ][ω − (εΓ− − ε+nγ ) − iη] +,kII +,kII [εΓ+nII − ε − ε−kβ ][εΓ+nII − ε − ε−kν and correction to the external legs, V U(2h1p) α; µkµνkνλnλ = Vαλ,µν + Vαλ,γ1δ1 X Vγ2δ2,µν 2 [ε−kµ + ε − ε+nγ − ε (A16) Vαδ1,µγ1 X Vγ2λ,δ2ν [ε−kδ + ε − ε+nγ − ε Vαδ1,νγ1 X Vγ2λ,δ2µ [ε−kδ + ε − ε+nγ − ε It should be pointed out that while the prescription of Sec. A 1 allows sp lines to propagate only in one time di- rection, it allows for backward propagation of the phonons. These contributions translate directly into the energy indepen- dent terms of Eqs. (A13) and (A15) and are a direct conse- quence of the inversion pattern typical of RPA theory. 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DISCRETE NONHOLONOMIC LAGRANGIAN SYSTEMS ON LIE GROUPOIDS DAVID IGLESIAS, JUAN C. MARRERO, DAVID MARTÍN DE DIEGO, AND EDUARDO MARTÍNEZ Abstract. This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for con- tinuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint sub- manifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of non- holonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of the evolution operator and the discrete nonholonomic momentum equation are also consid- ered. Finally, we illustrate with several classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile robot). Contents 1. Introduction 2 2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids 5 2.1. Lie algebroids 5 2.2. Lie groupoids 6 2.3. Discrete Unconstrained Lagrangian Systems 9 3. Discrete Nonholonomic (or constrained) Lagrangian systems on Lie groupoids 11 3.1. Discrete Generalized Hölder’s principle 11 3.2. Discrete Nonholonomic Legendre transformations 14 3.3. Nonholonomic evolution operators and regular discrete nonholonomic Lagrangian systems 20 3.4. Reversible discrete nonholonomic Lagrangian systems 22 3.5. Lie groupoid morphisms and reduction 23 3.6. Discrete nonholonomic Hamiltonian evolution operator 24 3.7. The discrete nonholonomic momentum map 24 4. Examples 26 4.1. Discrete holonomic Lagrangian systems on a Lie groupoid 26 4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid 27 This work has been partially supported by MICYT (Spain) Grants MTM 2006-03322, MTM 2004-7832, MTM 2006-10531 and S-0505/ESP/0158 of the CAM. D. Iglesias thanks MEC for a “Juan de la Cierva” research contract. 2 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ 4.3. Discrete nonholonomic Lagrangian systems on a Lie group 29 4.4. Discrete nonholonomic Lagrangian systems on an action Lie groupoid 32 4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid 35 4.6. Discrete Chaplygin systems 39 5. Conclusions and Future Work 43 References 43 1. Introduction In the paper of Moser and Veselov [40] dedicated to the complete integrability of certain dynamical systems, the authors proposed a discretization of the tangent bundle TQ of a configuration space Q replacing it by the product Q×Q, approx- imating a tangent vector on Q by a pair of ‘close’ points (q0, q1). In this sense, the continuous Lagrangian function L : TQ −→ R is replaced by a discretization Ld : Q×Q −→ R. Then, applying a suitable variational principle, it is possible to derive the discrete equations of motion. In the regular case, one obtains an evolu- tion operator, a map which assigns to each pair (qk−1, qk) a pair (qk, qk+1), sharing many properties with the continuous system, in particular, symplecticity, momen- tum conservation and a good energy behavior. We refer to [32] for an excellent review in discrete Mechanics (on Q×Q) and its numerical implementation. On the other hand, in [40, 44], the authors also considered discrete Lagrangians defined on a Lie group G where the evolution operator is given by a diffeomorphism of G. All the above examples led to A. Weinstein [45] to study discrete mechanics on Lie groupoids. A Lie groupoid is a geometric structure that includes as particular examples the case of cartesian products Q × Q as well as Lie groups and other examples as Atiyah or action Lie groupoids [26]. In a recent paper [27], we studied discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids, deriving from a variational principle the discrete Euler-Lagrange equations. We also introduced a symplectic 2-section (which is preserved by the Lagrange evolution operator) and defined the Hamiltonian evolution operator, in terms of the discrete Legendre transformations, which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. These techniques include as particular cases the classical discrete Euler-Lagrange equations, the discrete Euler-Poincaré equations (see [5, 6, 29, 30]) and the discrete Lagrange-Poincaré equations. In fact, the results in [27] may be applied in the construction of geometric integrators for continuous Lagrangian systems which are invariant under the action of a symmetry Lie group (see also [18] for the particular case when the symmetry Lie group is abelian). From the perspective of geometric integration, there are a great interest in intro- ducing new geometric techniques for developing numerical integrators since stan- dard methods often introduce some spurious effects like dissipation in conservative systems [16, 42]. The case of dynamical systems subjected to constraints is also of considerable interest. In particular, the case of holonomic constraints is well established in the literature of geometric integration, for instance, in simulation of molecular dynamics where the constraints may be molecular bond lengths or angles and also in multibody dynamics (see [16, 20] and references therein). DISCRETE NONHOLONOMIC MECHANICS 3 By contrast, the construction of geometric integrators for the case of nonholo- nomic constraints is less well understood. This type of constraints appears, for instance, in mechanical models of convex rigid bodies rolling without sliding on a surface [41]. The study of systems with nonholonomic constraints goes back to the XIX century. The equations of motion were obtained applying either D’Alembert’s principle of virtual work or Gauss principle of least constraint. Recently, many authors have shown a new interest in that theory and also in its relation to the new developments in control theory and robotics using geometric techniques (see, for instance, [2, 3, 4, 8, 19, 22, 24]). Geometrically, nonholonomic constraints are globally described by a submanifold M of the velocity phase space TQ. If M is a vector subbundle of TQ, we are dealing with the case of linear constraints and, in the case M is an affine subbundle, we are in the case of affine constraints. Lagrange-D’Alembert’s or Chetaev’s principles allow us to determine the set of possible values of the constraint forces only from the set of admissible kinematic states, that is, from the constraint manifold M determined by the vanishing of the nonholonomic constraints φa. Therefore, assuming that the dynamical properties of the system are mathematically described by a Lagrangian function L : TQ −→ R and by a constraint submanifold M, the equations of motion, following Chetaev’s principle, are[ δqi = 0 , where δqi denotes the virtual displacements verifying δqi = 0. By using the Lagrange multiplier rule, we obtain that = λ̄a , (1.1) with the condition q̇(t) ∈ M, λ̄a being the Lagrange multipliers to be determined. Recently, J. Cortés et al [9] (see also [11, 38, 39]) proposed a unified framework for nonholonomic systems in the Lie algebroid setting that we will use along this paper generalizing some previous work for free Lagrangian mechanics on Lie algebroids (see, for instance, [23, 33, 34, 35]). The construction of geometric integrators for Equations (1.1) is very recent. In fact, in [37] appears as an open problem: ...The problem for the more general class of non-holonomic con- straints is still open, as is the question of the correct analogue of symplectic integration for non-holonomically constrained La- grangian systems... Numerical integrators derived from discrete variational principles have proved their adaptability to many situations: collisions, classical field theory, external forces...[28, 32] and it also seems very adequate for nonholonomic systems, since nonholonomic equations of motion come from Hölder’s variational principle which is not a stan- dard variational principle [1], but admits an adequate discretization. This is the procedure introduced by J. Cortés and S. Mart́ınez [8, 10] and followed by other authors [12, 14, 15, 36] extending, moreover, the results to nonholonomic systems defined on Lie groups (see also [25] for a different approach using generating func- tions). In this paper, we tackle the problem from the unifying point of view of Lie groupoids (see [9] for the continuous case). This technique permits to recover all the previous methods in the literature [10, 14, 36] and consider new cases of great 4 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ importance in nonholonomic dynamics. For instance, using action Lie groupoids, we may discretize LR-nonholonomic systems such as the Veselova system or us- ing Atiyah Lie groupoids we find discrete versions for the reduced equations of nonholonomic systems with symmetry. The paper is structured as follows. In section 2 we review some basic results on Lie algebroids and Lie groupoids. In particular, we describe the prolongation of a Lie groupoid [43], which has a double structure of Lie groupoid and Lie algebroid. Then, we briefly expose the geometric structure of discrete unconstrained mechanics on Lie groupoids: Poincaré-Cartan sections, Legendre transformations... The main results of the paper appear in section 3, where the geometric structure of discrete nonholonomic systems on Lie groupoids is considered. In particular, given a discrete Lagrangian Ld : Γ → R on a Lie groupoid Γ, a constraint distribution Dc in the Lie algebroid EΓ of Γ and a discrete constraint submanifold Mc in Γ, we obtain the nonholonomic discrete Euler-Lagrange equations from a discrete Generalized Hölder’s principle (see section 3.1). In addition, we characterize the regularity of the nonholonomic system in terms of the nonholonomic Legendre transformations and decompositions of the prolongation of the Lie groupoid. In the case when the system is regular, we can define the nonholonomic evolution operator. An interesting situation, studied in in Section 3.4, is that of reversible discrete nonholonomic Lagrangian systems, where the Lagrangian and the discrete constraint submanifold are invariants with respect to the inversion of the Lie groupoid. The particular example of reversible systems in the pair groupoid Q×Q was first studied in [36]. We also define the discrete nonholonomic momentum map. In order to give an idea of the breadth and flexibility of the proposed formalism, several examples are discussed, including their regularity and their reversibility: - Discrete holonomic Lagrangian systems on a Lie groupoid, which are a generalization of the Shake algorithm for holonomic systems [16, 20, 32]; - Discrete nonholonomic systems on the pair groupoid, recovering the equa- tions first considered in [10]. An explicit example of this situation is the discrete nonholonomic constrained particle. - Discrete nonholonomic systems on Lie groups, where the equations that are obtained are the so-called discrete Euler-Poincaré-Suslov equations (see [14]). We remark that, although our equations coincide with those in [14], the technique developed in this paper is different to the one in that paper. Two explicit examples which we describe here are the Suslov system and the Chaplygin sleigh. - Discrete nonholonomic Lagrangian systems on an action Lie groupoid. This example is quite interesting since it allows us to discretize a well- known nonholonomic LR-system: the Veselova system (see [44]; see also [13]). For this example, we obtain a discrete system that is not reversible and we show that the system is regular in a neighborhood around the manifold of units. - Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid. With this example, we are able to discretize reduced systems, in particular, we concentrate on the example of the discretization of the equations of motion of a rolling ball without sliding on a rotating table with constant angular velocity. - Discrete Chaplygin systems, which are regular systems (Ld,Mc,Dc) on the Lie groupoid Γ ⇒ M , for which (α, β) ◦ iMc : Mc → M × M is a diffeomorphism and ρ ◦ iDc : Dc → TM is an isomorphism of vector bundles, (α, β) being the source and target of the Lie groupoid Γ and ρ DISCRETE NONHOLONOMIC MECHANICS 5 being the anchor map of the Lie algebroid EΓ. This example includes a discretization of the two wheeled planar mobile robot. We conclude our paper with future lines of work. 2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids 2.1. Lie algebroids. A Lie algebroid E over a manifold M is a real vector bundle τ : E →M together with a Lie bracket [[·, ·]] on the space Sec(τ) of the global cross sections of τ : E → M and a bundle map ρ : E → TM , called the anchor map, such that if we also denote by ρ : Sec(τ) → X(M) the homomorphism of C∞(M)-modules induced by the anchor map then [[X, fY ]] = f [[X,Y ]] + ρ(X)(f)Y, (2.1) for X,Y ∈ Sec(τ) and f ∈ C∞(M) (see [26]). If (E, [[·, ·]], ρ) is a Lie algebroid over M then the anchor map ρ : Sec(τ) → X(M) is a homomorphism between the Lie algebras (Sec(τ), [[·, ·]]) and (X(M), [·, ·]). Moreover, one may define the differential d of E as follows: dµ(X0, . . . , Xk) = (−1)iρ(Xi)(µ(X0, . . . , X̂i, . . . , Xk)) (−1)i+jµ([[Xi, Xj ]], X0, . . . , X̂i, . . . , X̂j , . . . , Xk), (2.2) for µ ∈ Sec(∧kτ∗) and X0, . . . , Xk ∈ Sec(τ). d is a cohomology operator, that is, d2 = 0. In particular, if f : M −→ R is a real smooth function then df(X) = ρ(X)f, for X ∈ Sec(τ). Trivial examples of Lie algebroids are a real Lie algebra of finite dimension (in this case, the base space is a single point) and the tangent bundle of a manifold M. On the other hand, let (E, [[·, ·]], ρ) be a Lie algebroid of rank n over a manifold M of dimension m and π : P →M be a fibration. We consider the subset of E×TP TEP = { (a, v) ∈ E × TP | (Tπ)(v) = ρ(a) } , where Tπ : TP → TM is the tangent map to π. Denote by τπ : TEP → P the map given by τπ(a, v) = τP (v), τP : TP → P being the canonical projection. If dimP = p, one may prove that TEP is a vector bundle over P of rank n + p −m with vector bundle projection τπ : TEP → P . A section X̃ of τπ : TEP → P is said to be projectable if there exists a section X of τ : E →M and a vector field U ∈ X(P ) which is π-projectable to the vector field ρ(X) and such that X̃(p) = (X(π(p)), U(p)), for all p ∈ P . For such a projectable section X̃, we will use the following notation X̃ ≡ (X,U). It is easy to prove that one may choose a local basis of projectable sections of the space Sec(τπ). The vector bundle τπ : TEP → P admits a Lie algebroid structure ([[·, ·]]π, ρπ). Indeed, if (X,U) and (Y, V ) are projectable sections then [[(X,U), (Y, V )]]π = ([[X,Y ]], [U, V ]), ρπ(X,U) = U. (TEP, [[·, ·]]π, ρπ) is the E-tangent bundle to P or the prolongation of E over the fibration π : P →M (for more details, see [23]). Now, let (E, [[·, ·]], ρ) (resp., (E′, [[·, ·]]′, ρ′)) be a Lie algebroid over a manifold M (resp., M ′) and suppose that Ψ : E → E′ is a vector bundle morphism over the map Ψ0 : M →M ′. Then, the pair (Ψ,Ψ0) is said to be a Lie algebroid morphism if d((Ψ,Ψ0) ∗φ′) = (Ψ,Ψ0) ∗(d′φ′), for all φ′ ∈ Sec(∧k(τ ′)∗) and for all k, (2.3) 6 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ where d (resp., d′) is the differential of the Lie algebroid E (resp., E′) (see [23]). In the particular case when M = M ′ and Ψ0 = Id then (2.3) holds if and only if [[Ψ ◦X,Ψ ◦ Y ]]′ = Ψ[[X,Y ]], ρ′(ΨX) = ρ(X), for X,Y ∈ Sec(τ). 2.2. Lie groupoids. A Lie groupoid over a differentiable manifold M is a differ- entiable manifold Γ together with the following structural maps: • A pair of submersions α : Γ → M , the source, and β : Γ → M, the target. The maps α and β define the set of composable pairs Γ2 = { (g, h) ∈ G×G | β(g) = α(h) } . • A multiplication m : Γ2 → Γ, to be denoted simply by m(g, h) = gh, such that – α(gh) = α(g) and β(gh) = β(h). – g(hk) = (gh)k. • An identity section � : M → Γ such that – �(α(g))g = g and g�(β(g)) = g. • An inversion map i : Γ → Γ, to be simply denoted by i(g) = g−1, such – g−1g = �(β(g)) and gg−1 = �(α(g)). A Lie groupoid Γ over a set M will be simply denoted by the symbol Γ ⇒ M . On the other hand, if g ∈ Γ then the left-translation by g and the right- translation by g are the diffeomorphisms lg : α−1(β(g)) −→ α−1(α(g)) ; h −→ lg(h) = gh, rg : β−1(α(g)) −→ β−1(β(g)) ; h −→ rg(h) = hg. Note that l−1g = lg−1 and r g = rg−1 . A vector field X̃ on Γ is said to be left-invariant (resp., right-invariant) if it is tangent to the fibers of α (resp., β) and X̃(gh) = (Thlg)(X̃h) (resp., X̃(gh) = (Tgrh)(X̃(g))), for (g, h) ∈ Γ2. Now, we will recall the definition of the Lie algebroid associated with Γ. We consider the vector bundle τ : EΓ → M , whose fiber at a point x ∈ M is (EΓ)x = V�(x)α = Ker(T�(x)α). It is easy to prove that there exists a bijection between the space Sec(τ) and the set of left-invariant (resp., right-invariant) vector fields on Γ. If X is a section of τ : EΓ →M , the corresponding left-invariant (resp., right-invariant) vector field on Γ will be denoted X (resp., X ), where X (g) = (T�(β(g))lg)(X(β(g))), (2.4) X (g) = −(T�(α(g))rg)((T�(α(g))i)(X(α(g)))), (2.5) for g ∈ Γ. Using the above facts, we may introduce a Lie algebroid structure ([[·, ·]], ρ) on EΓ, which is defined by ←−−−− [[X,Y ]] = [ Y ], ρ(X)(x) = (T�(x)β)(X(x)), (2.6) for X,Y ∈ Sec(τ) and x ∈M . Note that −−−−→ [[X,Y ]] = −[ Y ], [ Y ] = 0, (2.7) (for more details, see [7, 26]). Given two Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′, a morphism of Lie groupoids is a smooth map Φ : Γ→ Γ′ such that (g, h) ∈ Γ2 =⇒ (Φ(g),Φ(h)) ∈ (Γ′)2 DISCRETE NONHOLONOMIC MECHANICS 7 Φ(gh) = Φ(g)Φ(h). A morphism of Lie groupoids Φ : Γ → Γ′ induces a smooth map Φ0 : M → M ′ in such a way that α′ ◦ Φ = Φ0 ◦ α, β′ ◦ Φ = Φ0 ◦ β, Φ ◦ � = �′ ◦ Φ0, α, β and � (resp., α′, β′ and �′) being the source, the target and the identity section of Γ (resp., Γ′). Suppose that (Φ,Φ0) is a morphism between the Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′ and that τ : EΓ → M (resp., τ ′ : EΓ′ → M ′) is the Lie algebroid of Γ (resp., Γ′). Then, if x ∈ M we may consider the linear map Ex(Φ) : (EΓ)x → (EΓ′)Φ0(x) defined by Ex(Φ)(v�(x)) = (T�(x)Φ)(v�(x)), for v�(x) ∈ (EΓ)x. (2.8) In fact, we have that the pair (E(Φ),Φ0) is a morphism between the Lie algebroids τ : EΓ →M and τ ′ : EΓ′ →M ′ (see [26]). Trivial examples of Lie groupoids are Lie groups and the pair or banal groupoid M ×M , M being an arbitrary smooth manifold. The Lie algebroid of a Lie group Γ is just the Lie algebra g of Γ. On the other hand, the Lie algebroid of the pair (or banal) groupoid M ×M is the tangent bundle TM to M . Apart from the Lie algebroid EΓ associated with a Lie groupoid Γ ⇒ M , other interesting Lie algebroids associated with Γ are the following ones: • The EΓ- tangent bundle to E∗Γ: Let TEΓE∗Γ be the EΓ-tangent bundle to E Γ, that is, E∗Γ = (vx, XΥx) ∈ (EΓ)x × TΥxE ∣∣ (TΥxτ∗)(XΥx) = (T�(x)β)(vx)} for Υx ∈ (E∗Γ)x, with x ∈M. As we know, T EΓE∗Γ is a Lie algebroid over E We may introduce the canonical section Θ of the vector bundle (TEΓE∗Γ) ∗ → E∗Γ as follows: Θ(Υx)(ax, XΥx) = Υx(ax), for Υx ∈ (E∗Γ)x and (ax, XΥx) ∈ T E∗Γ. Θ is called the Liouville section as- sociated with EΓ. Moreover, we define the canonical symplectic section Ω associated with EΓ by Ω = −dΘ, where d is the differential on the Lie algebroid TEΓE∗Γ → E Γ. It is easy to prove that Ω is nondegenerate and closed, that is, it is a symplectic section of TEΓE∗Γ (see [23]). Now, if Z is a section of τ : EΓ → M then there is a unique vector field Z∗c on E∗Γ, the complete lift of X to E Γ, satisfying the two following conditions: (i) Z∗c is τ∗-projectable on ρ(Z) and (ii) Z∗c(X̂) = ̂[[Z,X]] for X ∈ Sec(τ) (see [23]). Here, if X is a section of τ : EΓ → M then X̂ is the linear function X̂ ∈ C∞(E∗) defined by X̂(a∗) = a∗(X(τ∗(a∗))), for all a∗ ∈ E∗. Using the vector field Z∗c, one may introduce the complete lift Z∗c of Z as the section of τ τ : TEΓE∗Γ → E Γ defined by Z∗c(a∗) = (Z(τ∗(a∗)), Z∗c(a∗)), for a∗ ∈ E∗. (2.9) 8 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Z∗c is just the Hamiltonian section of Ẑ with respect to the canonical symplectic section Ω associated with EΓ. In other words, iZ∗cΩ = dẐ, (2.10) where d is the differential of the Lie algebroid τ τ : TEΓE∗Γ → E Γ (for more details, see [23]). • The Lie algebroid τ̃Γ : TΓΓ→ Γ : Let TΓΓ be the Whitney sum V β ⊕Γ V α of the vector bundles V β → Γ and V α → Γ, where V β (respectively, V α) is the vertical bundle of β (respectively, α). Then, the vector bundle τ̃Γ : TΓΓ ≡ V β ⊕Γ V α → Γ admits a Lie algebroid structure ([[·, ·]]T ΓΓ, ρT ΓΓ). The anchor map ρT ΓΓ is given by ΓΓ)(Xg, Yg) = Xg + Yg and the Lie bracket bracket [[·, ·]]T ΓΓ on the space Sec(τ̃Γ) is characterized for the following relation Y ), ( Y ′)]]T ΓΓ = (− −−−−−→ [[X,X ′]], ←−−−− [[Y, Y ′]]), for X,Y,X ′, Y ′ ∈ Sec(τ) (for more details, see [27]). On other hand, if X is a section of τ : EΓ → M , one may define the sections X(1,0), X(0,1) (the β and α-lifts) and X(1,1) (the complete lift) of X to τ̃Γ : TΓΓ→ Γ as follows: X(1,0)(g) = ( X (g), 0g), X (0,1)(g) = (0g, X (g)), and X(1,1)(g) = (− X (g), X (g)). We have that [[X(1,0), Y (1,0)]]T ΓΓ = −[[X,Y ]](1,0) [[X(0,1), Y (1,0)]]T ΓΓ = 0, [[X(0,1), Y (0,1)]]T ΓΓ = [[X,Y ]](0,1), and, as a consequence, [[X(1,1), Y (1,0)]]T ΓΓ = [[X,Y ]](1,0), [[X(1,1), Y (0,1)]]T ΓΓ = [[X,Y ]](0,1), [[X(1,1), Y (1,1)]]T ΓΓ = [[X,Y ]](1,1). Now, if g, h ∈ Γ one may introduce the linear monomorphisms (1,0)h : (EΓ) (TΓhΓ) ∗ ≡ V ∗h β ⊕ V h α and (0,1) g : (EΓ)∗β(g) → (T ∗ ≡ V ∗g β ⊕ V ∗g α given by (1,0) h (Xh, Yh) = γ(Th(i ◦ rh−1)(Xh)), (2.11) γ(0,1)g (Xg, Yg) = γ((Tglg−1)(Yg)), (2.12) for (Xg, Yg) ∈ TΓg Γ and (Xh, Yh) ∈ TΓhΓ. Thus, if µ is a section of τ∗ : E∗Γ → M , one may define the corresponding lifts µ(1,0) and µ(0,1) as the sections of τ̃Γ ∗ : (TΓΓ)∗ → Γ given by µ(1,0)(h) = µ(1,0)h , for h ∈ Γ, µ(0,1)(g) = µ(0,1)g , for g ∈ Γ. Note that if g ∈ Γ and {XA} (respectively, {YB}) is a local basis of Sec(τ) on an open subset U (respectively, V ) of M such that α(g) ∈ U (respectively, β(g) ∈ V ) then {X(1,0)A , Y (0,1) B } is a local basis of Sec(τ̃Γ) on the open subset α −1(U)∩β−1(V ). In addition, if {XA} (respectively, {Y B}) is the dual basis of {XA} (respectively, {YB}) then {(XA)(1,0), (Y B)(0,1)} is the dual basis of {X (1,0) A , Y (0,1) DISCRETE NONHOLONOMIC MECHANICS 9 2.3. Discrete Unconstrained Lagrangian Systems. (See [27] for details) A discrete unconstrained Lagrangian system on a Lie groupoid consists of a Lie groupoid Γ ⇒ M (the discrete space) and a discrete Lagrangian Ld : Γ→ 2.3.1. Discrete unconstrained Euler-Lagrange equations. An admissible sequence of order N on the Lie groupoid Γ is an element (g1, . . . , gN ) of ΓN ≡ Γ× · · · ×Γ such that (gk, gk+1) ∈ Γ2, for k = 1, . . . , N − 1. An admissible sequence (g1, . . . , gN ) of order N is a solution of the discrete unconstrained Euler-Lagrange equations for Ld if do[Ld ◦ lgk + Ld ◦ rgk+1 ◦ i](�(xk))|(EΓ)xk = 0 where β(gk) = α(gk+1) = xk and do is the standard differential on Γ, that is, the differential of the Lie algebroid τΓ : TΓ→ Γ (see [27]). The discrete unconstrained Euler-Lagrange operator DDELLd : Γ2 → E∗Γ is given by (DDELLd)(g, h) = d o[Ld ◦ lg + Ld ◦ rh ◦ i](�(x))|(EΓ)x = 0, for (g, h) ∈ Γ2, with β(g) = α(h) = x ∈M (see [27]). Thus, an admissible sequence (g1, . . . , gN ) of order N is a solution of the discrete unconstrained Euler-Lagrange equations if and only if (DDELLd)(gk, gk+1) = 0, for k = 1, . . . , N − 1. 2.3.2. Discrete Poincaré-Cartan sections. Consider the Lie algebroid τ̃Γ : TΓΓ ≡ V β ⊕Γ V α → Γ, and define the Poincaré-Cartan 1-sections Θ−Ld ,Θ Sec((τ̃Γ)∗) as follows Θ−Ld(g)(Xg, Yg) = −Xg(Ld), Θ (g)(Xg, Yg) = Yg(Ld), (2.13) for each g ∈ Γ and (Xg, Yg) ∈ TΓg Γ ≡ Vgβ ⊕ Vgα. Since dLd = Θ −Θ−Ld and so, using d 2 = 0, it follows that dΘ+Ld = dΘ . This means that there exists a unique 2-section ΩLd = −dΘ = −dΘ−Ld , which will be called the Poincaré-Cartan 2-section. This 2-section will be important to study the symplectic character of the discrete unconstrained Euler-Lagrange equations. If g is an element of Γ such that α(g) = x and β(g) = y and {XA} (respectively, {YB}) is a local basis of Sec(τ) on the open subset U (respectively, V ) of M , with x ∈ U (respectively, y ∈ V ), then on α−1(U) ∩ β−1(V ) we have that Θ−Ld = − XA(L)(XA)(1,0), Θ YB(L)(Y B)(0,1), ΩLd = − YB(Ld))(XA)(1,0) ∧ (Y B)(0,1) (2.14) where {XA} (respectively, {Y B}) is the dual basis of {XA} (respectively, {YB}) (for more details, see [27]). 2.3.3. Discrete unconstrained Lagrangian evolution operator. Let Υ : Γ → Γ be a smooth map such that: - graph(Υ) ⊆ Γ2, that is, (g,Υ(g)) ∈ Γ2, for all g ∈ Γ (Υ is a second order operator) and - (g,Υ(g)) is a solution of the discrete unconstrained Euler-Lagrange equa- tions, for all g ∈ Γ, that is, (DDELLd)(g,Υ(g)) = 0, for all g ∈ Γ. 10 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ In such a case ←− X (g)(Ld)− X (Υ(g))(Ld) = 0, (2.15) for every section X of τ : EΓ → M and every g ∈ Γ. The map Υ : Γ→ Γ is called a discrete flow or a discrete unconstrained Lagrangian evolution operator for Ld. Now, let Υ : Γ → Γ be a second order operator. Then, the prolongation TΥ : TΓΓ ≡ V β ⊕Γ V α → TΓΓ ≡ V β ⊕Γ V α of Υ is the Lie algebroid morphism over Υ : Γ→ Γ defined as follows (see [27]): TgΥ(Xg, Yg) = ((Tg(rgΥ(g) ◦ i))(Yg), (TgΥ)(Xg) +(TgΥ)(Yg)− Tg(rgΥ(g) ◦ i)(Yg)), (2.16) for all (Xg, Yg) ∈ TΓg Γ ≡ Vgβ ⊕ Vgα. Moreover, from (2.4), (2.5) and (2.16), we obtain that X (g), Y (g)) = (− Y (Υ(g)), (TgΥ)( X (g) + Y (g)) + Y (Υ(g))), (2.17) for all X,Y ∈ Sec(τ). Using (2.16), one may prove that (see [27]): (i) The map Υ is a discrete unconstrained Lagrangian evolution operator for Ld if and only if (TΥ,Υ)∗Θ = Θ+Ld . (ii) The map Υ is a discrete unconstrained Lagrangian evolution operator for Ld if and only if (TΥ,Υ)∗Θ −Θ−Ld = dLd. (iii) If Υ is discrete unconstrained Lagrangian evolution operator then (TΥ,Υ)∗ΩLd = ΩLd . 2.3.4. Discrete unconstrained Legendre transformations. Given a Lagrangian Ld : Γ → R we define the discrete unconstrained Legendre transformations F−Ld : Γ→ E∗Γ and F +Ld : Γ→ E∗Γ by (see [27]) (F−Ld)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ rh ◦ i), for v�(α(h)) ∈ (EΓ)α(h), (F+Ld)(g)(v�(β(g))) = v�(β(g))(Ld ◦ lg), for v�(β(g)) ∈ (EΓ)β(g). Now, we introduce the prolongations TΓF−Ld : TΓΓ ≡ V β ⊕Γ V α → TEΓE∗Γ and TΓF+Ld : TΓΓ ≡ V β ⊕Γ V α→ TEΓE∗Γ by −Ld(Xh, Yh) = (Th(i ◦ rh−1)(Xh), (ThF−Ld)(Xh) + (ThF−Ld)(Yh)),(2.18) TΓg F +Ld(Xg, Yg) = ((Tglg−1)(Yg), (TgF+Ld)(Xg) + (TgF+Ld)(Yg)), (2.19) for all h, g ∈ Γ and (Xh, Yh) ∈ TΓhΓ ≡ Vhβ ⊕ Vhα and (Xg, Yg) ∈ T g Γ ≡ Vgβ ⊕ Vgα (see [27]). We observe that the discrete Poincaré-Cartan 1-sections and 2- section are related to the canonical Liouville section of (TEΓE∗Γ) ∗ → E∗Γ and the canonical symplectic section of ∧2(TEΓE∗Γ) ∗ → E∗Γ by pull-back under the discrete unconstrained Legendre transformations, that is (see [27]), (TΓF−Ld,F−Ld)∗Θ = Θ−Ld , (T ΓF+Ld,F+Ld)∗Θ = Θ+Ld , (2.20) (TΓF−Ld,F−Ld)∗Ω = ΩLd , (T ΓF+Ld,F+Ld)∗Ω = ΩLd . (2.21) 2.3.5. Discrete regular Lagrangians. A discrete Lagrangian Ld : Γ→ R is said to be regular if the Poincaré-Cartan 2-section ΩLd is nondegenerate on the Lie algebroid τ̃Γ : TΓΓ ≡ V β ⊕Γ V α → Γ (see [27]). In [27], we obtained some necessary and sufficient conditions for a discrete Lagrangian on a Lie groupoid Γ to be regular that we summarize as follows: Ld is regular ⇐⇒ The Legendre transformation F+Ld is a local diffeomorphism ⇐⇒ The Legendre transformation F−Ld is a local diffeomorphism DISCRETE NONHOLONOMIC MECHANICS 11 Locally, we deduce that Ld is regular if and only if for every g ∈ Γ and every local basis {XA} (respectively, {YB}) of Sec(τ) on an open subset U (respectively, V ) of M such that α(g) ∈ U (respectively, β(g) ∈ V ) we have that the matrix Y B(Ld))) is regular on α−1(U) ∩ β−1(V ). Now, let Ld : Γ→ R be a discrete Lagrangian and g be a point of Γ. We define the R-bilinear map GLdg : (EΓ)α(g) ⊕ (EΓ)β(g) → R given by GLdg (a, b) = ΩLd(g)((−T�(α(g))(rg ◦ i)(a), 0), (0, (T�(β(g))lg)(b))). (2.22) Then, using (2.14), we have that Proposition 2.1. The discrete Lagrangian Ld : Γ → R is regular if and only if GLdg is nondegenerate, for all g ∈ Γ, that is, GLdg (a, b) = 0, for all b ∈ (EΓ)β(g) ⇒ a = 0 (respectively, GLdg (a, b) = 0, for all a ∈ (EΓ)α(g) ⇒ b = 0). On the other hand, if Ld : Γ → R is a discrete Lagrangian on a Lie groupoid Γ then we have that τ∗ ◦ F−Ld = α, τ∗ ◦ F+Ld = β, where τ∗ : E∗Γ → M is the vector bundle projection. Using these facts, (2.18) and (2.19), we deduce the following result. Proposition 2.2. Let Ld : Γ → R be a discrete Lagrangian function. Then, the following conditions are equivalent: (i) Ld is regular. (ii) The linear map TΓhF −Ld : Vhβ ⊕ Vhα → TEΓF−Ld(h)E Γ is a linear isomor- phism, for all h ∈ Γ. (iii) The linear map TΓg F +Ld : Vgβ ⊕ Vgα → TEΓF+Ld(g)EΓ ∗ is a linear isomor- phism, for all g ∈ Γ. Finally, let Ld : Γ→ R be a regular discrete Lagrangian function and (g0, h0) ∈ Γ×Γ be a solution of the discrete Euler-Lagrange equations for Ld. Then, one may prove (see [27]) that there exist two open subsets U0 and V0 of Γ, with g0 ∈ U0 and h0 ∈ V0, and there exists a (local) discrete unconstrained Lagrangian evolution operator ΥLd : U0 → V0 such that: (i) ΥLd(g0) = h0, (ii) ΥLd is a diffeomorphism and (iii) ΥLd is unique, that is, if U 0 is an open subset of Γ, with g0 ∈ U ′0, and Υ′Ld : U 0 → Γ is a (local) discrete Lagrangian evolution operator then ΥLd|U0∩U ′0 = Υ Ld|U0∩U ′0 3. Discrete Nonholonomic (or constrained) Lagrangian systems on Lie groupoids 3.1. Discrete Generalized Hölder’s principle. Let Γ be a Lie groupoid with structural maps α, β : Γ→M, � : M → Γ, i : Γ→ Γ, m : Γ2 → Γ. Denote by τ : EΓ → M the Lie algebroid associated to Γ. Suppose that the rank of EΓ is n and that the dimension of M is m. A generalized discrete nonholonomic (or constrained) Lagrangian system on Γ is determined by: 12 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ - a regular discrete Lagrangian Ld : Γ −→ R, - a constraint distribution, Dc, which is a vector subbundle of the bundle EΓ → M of admissible directions. We will denote by τDc : Dc → M the vector bundle projection and by iDc : Dc → EΓ the canonical inclusion. - a discrete constraint embedded submanifold Mc of Γ, such that dimMc = dimDc = m + r, with r ≤ n. We will denote by iMc : Mc → Γ the canonical inclusion. Remark 3.1. Let Ld : Γ→ R be a regular discrete Lagrangian on a Lie groupoid Γ and Mc be a submanifold of Γ such that �(M) ⊆ Mc. Then, dimMc = m + r, with 0 ≤ r ≤ m. Moreover, for every x ∈M , we may introduce the subspace Dc(x) of EΓ(x) given by Dc(x) = T�(x)Mc ∩ EΓ(x). Since the linear map T�(x)α : T�(x)Mc → TxM is an epimorphism, we deduce that dimDc(x) = r. In fact, Dc = x∈M Dc(x) is a vector subbundle of EΓ (over M) of rank r. Thus, we may consider the discrete nonholonomic system (Ld,Mc,Dc) on the Lie groupoid Γ. � For g ∈ Γ fixed, we consider the following set of admissible sequences of order CNg = (g1, . . . , gN ) ∈ ΓN ∣∣ (gk, gk+1) ∈ Γ2, for k = 1, .., N − 1 and g1 . . . gN = g } . Given a tangent vector at (g1, . . . , gN ) to the manifold CNg , we may write it as the tangent vector at t = 0 of a curve in CNg , t ∈ (−ε, ε) ⊆ R −→ c(t) which passes through (g1, . . . , gN ) at t = 0. This type of curves is of the form c(t) = (g1h1(t), h 1 (t)g2h2(t), . . . , h N−2(t)gN−1hN−1(t), h N−1(t)gN ) where hk(t) ∈ α−1(β(gk)), for all t, and hk(0) = �(β(gk)) for k = 1, . . . , N − 1. Therefore, we may identify the tangent space to CNg at (g1, . . . , gN ) with T(g1,g2,..,gN )C g ≡ { (v1, v2, . . . , vN−1) | vk ∈ (EΓ)xk and xk = β(gk), 1 ≤ k ≤ N − 1 } . Observe that each vk is the tangent vector to the curve hk at t = 0. The curve c is called a variation of (g1, . . . , gN ) and (v1, v2, . . . , vN−1) is called an infinitesimal variation of (g1, . . . , gN ). Now, we define the discrete action sum associated to the discrete Lagrangian Ld : Γ −→ R as SLd : CNg −→ R (g1, . . . , gN ) 7−→ Ld(gk). We define the variation δSLd : T(g1,...,gN )C g → R as δSLd(v1, . . . , vN−1) = SLd(c(t)) Ld(g1h1(t)) + Ld(h 1 (t)g2h2(t)) + . . .+ Ld(h N−2(t)gN−1hN−1(t)) + Ld(h N−1(t)gN ) do(Ld ◦ lgk)(�(xk))(vk) + d o(Ld ◦ rgk+1 ◦ i)(�(xk))(vk) where do is the standard differential on Γ, i.e., do is the differential of the Lie algebroid τΓ : TΓ → Γ. It is obvious from the last expression that the definition DISCRETE NONHOLONOMIC MECHANICS 13 of variation δSLd does not depend on the choice of variations c of the sequence g whose infinitesimal variation is (v1, . . . , vN−1). Next, we will introduce the subset (Vc)g of T(g1,...,gN )C g defined by (Vc)g = (v1, . . . , vN−1) ∈ T(g1,...,gN )C ∣∣ ∀k ∈ {1, . . . , N − 1}, vk ∈ Dc } . Then, we will say that a sequence in CNg satisfying the constraints determined by Mc is a Hölder-critical point of the discrete action sum SLd if the restriction of δSLd to (Vc)g vanishes, i.e. (Vc)g Definition 3.2 (Discrete Hölder’s principle). Given g ∈ Γ, a sequence (g1, . . . , gN ) ∈ CNg such that gk ∈ Mc, 1 ≤ k ≤ N , is a solution of the discrete nonholo- nomic Lagrangian system determined by (Ld,Mc,Dc) if and only if (g1, . . . , gN ) is a Hölder-critical point of SLd. If (g1, . . . , gN ) ∈ CNg ∩ (Mc × · · · ×Mc) then (g1, . . . , gN ) is a solution of the nonholonomic discrete Lagrangian system if and only if (do(Ld ◦ lgk) + d o(Ld ◦ rgk+1 ◦ i))(�(xk))|(Dc)xk = 0, where β(gk) = α(gk+1) = xk. For N = 2, we obtain that (g, h) ∈ Γ2 ∩ (Mc ×Mc) (with β(g) = α(h) = x) is a solution if do(Ld ◦ lg + Ld ◦ rh ◦ i)(�(x))|(Dc)x = 0. These equations will be called the discrete nonholonomic Euler-Lagrange equations for the system (Ld,Mc,Dc). Let (g1, . . . , gN ) be an element of CNg . Suppose that β(gk) = α(gk+1) = xk, 1 ≤ k ≤ N − 1, and that {XAk} = {Xak, Xαk} is a local adapted basis of Sec(τ) on an open subset Uk of M , with xk ∈ Uk. Here, {Xak}1≤a≤r is a local basis of Sec(τDc) and, thus, {Xαk}r+1≤α≤n is a local basis of the space of sections of the vector subbundle τD0c : D c →M , where D0c is the annihilator of Dc and {Xak, Xαk} is the dual basis of {Xak, Xαk}. Then, the sequence (g1, . . . , gN ) is a solution of the discrete nonholonomic equations if (g1, . . . , gN ) ∈Mc×· · ·×Mc and it satisfies the following closed system of difference equations gk)(Ld)− gk+1)(Ld) 〈dLd, (Xak)(0,1)〉(gk)− 〈dLd, (Xak)(1,0)〉(gk+1) for 1 ≤ a ≤ r, d being the differential of the Lie algebroid πτ : TΓΓ ≡ V β⊕ΓV α −→ Γ. For N = 2 we obtain that (g, h) ∈ Γ2 ∩ (Mc ×Mc) (with β(g) = α(h) = x) is a solution if Xa(g)(Ld)− Xa(h)(Ld) = 0 where {Xa} is a local basis of Sec(τDc) on an open subset U of M such that x ∈ U . Next, we describe an alternative version of these difference equations. First observe that using the Lagrange multipliers the discrete nonholonomic equations are rewritten as do [Ld ◦ lg + Ld ◦ rh ◦ i] (�(x))(v) = λαXα(x)(v), 14 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ for v ∈ (EΓ)x, with (g, h) ∈ Γ2 ∩ (Mc ×Mc) and β(g) = α(h) = x. Here, {Xα} is a local basis of sections of the annihilator D0c . Thus, the discrete nonholonomic equations are: Y (g)(Ld)− Y (h)(Ld) = λα(X α)(Y )|β(g), (g, h) ∈ Γ2 ∩ (Mc ×Mc), for all Y ∈ Sec(τ) or, alternatively, 〈dLd − λα(Xα)(0,1), Y (0,1)〉(g)− 〈dLd, Y (1,0)〉(h) = 0, (g, h) ∈ Γ2 ∩ (Mc ×Mc), for all Y ∈ Sec(τ). On the other hand, we may define the discrete nonholonomic Euler-Lagrange operator DDEL(Ld,Mc,Dc) : Γ2 ∩ (Mc ×Mc)→ D∗c as follows DDEL(Ld,Mc,Dc)(g, h) = d o [Ld ◦ lg + Ld ◦ rh ◦ i] (�(x))|(Dc)x , for (g, h) ∈ Γ2 ∩ (Mc ×Mc), with β(g) = α(h) = x ∈M . Then, we may characterize the solutions of the discrete nonholonomic equations as the sequences (g1, . . . , gN ), with (gk, gk+1) ∈ Γ2 ∩ (Mc × Mc), for each k ∈ {1, . . . , N − 1}, and DDEL(Ld,Mc,Dc)(gk, gk+1) = 0. Remark 3.3. (i) The set Γ2 ∩ (Mc ×Mc) is not, in general, a submanifold of Mc ×Mc. (ii) Suppose that αMc : Mc → M and βMc : Mc → M are the restrictions to Mc of α : Γ → M and β : Γ → M , respectively. If αMc and βMc are submersions then Γ2∩(Mc×Mc) is a submanifold of Mc×Mc of dimension m+ 2r. 3.2. Discrete Nonholonomic Legendre transformations. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system. We define the discrete nonholo- nomic Legendre transformations F−(Ld,Mc,Dc) : Mc → D∗c and F +(Ld,Mc,Dc) : Mc → D∗c as follows: F−(Ld,Mc,Dc)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ rh ◦ i), for v�(α(h)) ∈ Dc(α(h)),(3.1) F+(Ld,Mc,Dc)(g)(v�(β(g))) = v�(β(g))(Ld ◦ lg), for v�(β(g)) ∈ Dc(β(g)).(3.2) If F−Ld : Γ→ E∗Γ and F +Ld : Γ→ E∗Γ are the standard Legendre transformations associated with the Lagrangian function Ld and i∗Dc : E Γ → D c is the dual map of the canonical inclusion iDc : Dc → EΓ then F−(Ld,Mc,Dc) = i∗Dc ◦ F −Ld ◦ iMc , F +(Ld,Mc,Dc) = i ◦ F+Ld ◦ iMc . (3.3) Remark 3.4. (i) Note that τ∗Dc ◦ F −(Ld,Mc,Dc) = αMc , τ ◦ F+(Ld,Mc,Dc) = βMc . (3.4) (ii) If DDEL(Ld,Mc,Dc) is the discrete nonholonomic Euler-Lagrange opera- tor then DDEL(Ld,Mc,Dc)(g, h) = F+(Ld,Mc,Dc)(g)− F−(Ld,Mc,Dc)(h), for (g, h) ∈ Γ2 ∩ (Mc ×Mc). DISCRETE NONHOLONOMIC MECHANICS 15 On the other hand, since by assumption Ld : Γ → R is a regular discrete La- grangian function, we have that the discrete Poincaré-Cartan 2-section ΩLd is sym- plectic on the Lie algebroid τ̃Γ : TΓΓ → Γ. Moreover, the regularity of L is equiv- alent to the fact that the Legendre transformations F−Ld and F+Ld to be local diffeomorphisms (see Subsection 2.3.5). Next, we will obtain necessary and sufficient conditions for the discrete non- holonomic Legendre transformations associated with the system (Ld,Mc,Dc) to be local diffeomorphisms. Let F be the vector subbundle (over Γ) of τ̃Γ : TΓΓ→ Γ whose fiber at the point h ∈ Γ is (1,0) ∣∣∣ γ ∈ Dc(α(h))0 }0 ⊆ TΓhΓ. In other words, F 0h = (1,0) ∣∣∣ γ ∈ Dc(α(h))0 } . Note that the rank of F is n+ r. We also consider the vector subbundle F̄ (over Γ) of τ̃Γ : TΓΓ→ Γ of rank n+ r whose fiber at the point g ∈ Γ is F̄g = γ(0,1)g ∣∣∣ γ ∈ Dc(β(g))0 }0 ⊆ TΓg Γ. Lemma 3.5. F (respectively, F̄ ) is a coisotropic vector subbundle of the symplectic vector bundle (TΓΓ,ΩLd), that is, F⊥h ⊆ Fh, for every h ∈ Γ (respectively, F̄⊥g ⊆ F̄g, for every g ∈ Γ), where F⊥h (respectively, F̄ g ) is the symplectic orthogonal of Fh (respectively, F̄g) in the symplectic vector space (TΓhΓ, ΩLd(h)) (respectively, (T g Γ,ΩLd(g))). Proof. If h ∈ Γ we have that F⊥h = [ ΩLd (h) (F 0h ), [ΩLd (h) : TΓhΓ→ (T ∗ being the canonical isomorphism induced by the symplectic form ΩLd(h). Thus, using (2.14), we deduce that F⊥h = ΩLd (h) (γ(1,0)h ) ∣∣∣ γ ∈ Dc(α(h))0 } ⊆ {0} ⊕ Vhα ⊆ Fh. The coisotropic character of F̄g is proved in a similar way. � We also have the following result Lemma 3.6. Let TΓF−Ld : TΓΓ→ TEΓE∗Γ (respectively, T ΓF+Ld : TΓΓ→ TEΓE∗Γ) be the prolongation of the Legendre transformation F−Ld : Γ → E∗Γ (respectively, F+Ld : Γ→ E∗Γ). Then, (TΓhF −Ld)(Fh) = T F−Ld(h) E∗Γ = (vα(h), XF−Ld(h)) ∈ T F−Ld(h) ∣∣∣ vα(h) ∈ Dc(α(h))} , for h ∈Mc (respectively, (TΓg F +Ld)(F̄g) = T F+Ld(g) E∗Γ = (vβ(g), XF+Ld(g)) ∈ T F+Ld(g) ∣∣∣ vβ(g) ∈ Dc(β(g))} , for g ∈Mc). Proof. It follows using (2.11), (2.18) (respectively, (2.12), (2.19)) and Proposition 2.2. � Now, we may prove the following theorem. 16 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Theorem 3.7. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system. Then, the following conditions are equivalent: (i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) (re- spectively, F+(Ld,Mc,Dc)) is a local diffeomorphism. (ii) For every h ∈Mc (respectively, g ∈Mc) ΓΓ)−1(ThMc) ∩ F⊥h = {0} (3.5) (respectively, (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}). Proof. (i) ⇒ (ii) If h ∈ Mc and (Xh, Yh) ∈ (ρT ΓΓ)−1(ThMc) ∩ F⊥h then, using the fact that F⊥h ⊆ {0} ⊕ Vhα (see the proof of Lemma 3.5), we have that Xh = 0. Therefore, Yh ∈ Vhα ∩ ThMc. (3.6) Next, we will see that (ThF−(Ld,Mc,Dc))(Yh) = 0. (3.7) From (3.4) and (3.6), it follows that (ThF−(Ld,Mc,Dc))(Yh) is vertical with respect to the projection τ∗Dc : D c →M . Thus, it is sufficient to prove that ((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = 0, for all Z ∈ Sec(τDc). Here, Ẑ : D∗c → R is the linear function on D∗c induced by the section Z. Now, using (3.3), we deduce that ((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = d(Ẑ ◦ i∗Dc)((F −Ld)(h))(0, (ThF−Ld)(Yh)), where d is the differential of the Lie algebroid τ τ : TEΓE∗Γ → E Consequently, if Z∗c : E∗Γ → T EΓE∗Γ is the complete lift of Z ∈ Sec(τ), we have that (see (2.10)), ((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = Ω(F−Ld(h))(Z∗c(F−Ld(h)), (0, (ThF−Ld)(Yh)), (3.8) Ω being the canonical symplectic section associated with the Lie algebroid EΓ. On the other hand, since Z ∈ Sec(τDc), it follows that Z∗c(F−Ld(h)) is in F−Ld(h) E∗Γ and, from Lemma 3.6, we conclude that there exists (X h) ∈ Fh such that (TΓhF −Ld)(X h) = Z ∗c((F−Ld)(h)). (3.9) Moreover, using (2.18), we obtain that (TΓhF −Ld)(0, Yh) = (0, (ThF−Ld)(Yh)). (3.10) Thus, from (2.21), (3.8), (3.9) and (3.10), we deduce that ((ThF−(Ld,M,Dc))(Yh))(Ẑ) = −ΩLd(h)((0, Yh), (X Therefore, since (0, Yh) ∈ F⊥h , it follows that (3.7) holds, which implies that Yh = 0. This proves that (ρT ΓΓ)−1(ThMc) ∩ F⊥h = {0}. If F+(Ld,Mc,Dc) is a local diffeomorphism then, proceeding as above, we have that (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc. (ii) ⇒ (i) Suppose that h ∈Mc and that Yh is a tangent vector to Mc at h such (ThF−(Ld,Mc,Dc))(Yh) = 0. (3.11) DISCRETE NONHOLONOMIC MECHANICS 17 We have that (Thα)(Yh) = 0 and, thus, (0, Yh) ∈ (ρT ΓΓ)−1(ThMc). We will see that (0, Yh) ∈ F⊥h , that is, ΩLd(h)((0, Yh), (X h)) = 0, for (X h) ∈ Fh. (3.12) Now, using (2.18) and (2.21), we deduce that ΩLd(h)((0, Yh), (X h)) = Ω(F −Ld(h))((0, (ThF−Ld)(Yh)), (TΓhF −Ld)(X Therefore, from Lemma 3.6, we obtain that ΩLd(h)((0, Yh), (X h)) = Ω(F −Ld(h))(0, (ThF−Ld)(Yh)), (vα(h), YF−Ld(h))) with (vα(h), YF−Ld(h)) ∈ T F−Ld(h) Next, we take a section Z ∈ Sec(τDc) such that Z(α(h)) = vα(h). Then (see (2.9)), (vα(h), YF−Ld(h)) = Z ∗c(F−Ld(h)) + (0, Y ′F−Ld(h)), where Y ′F−Ld(h) ∈ TF−Ld(h)E Γ and Y F−Ld(h) is vertical with respect to the projection τ∗ : E∗Γ →M . Thus, since (see Eq. (3.7) in [23]) Ω(F−Ld(h))((0, (ThF−Ld)(Yh)), (0, Y ′F−Ld(h))) = 0, we have that ΩLd(h)((0, Yh), (X h)) = −Ω(F −Ld(h))(Z∗c(F−Ld(h)), (0, (ThF−Ld)(Yh))) = −d(Ẑ ◦ i∗Dc)(F −Ld(h))(0, (ThF−Ld)(Yh)) and, from (3.11), we deduce that (3.12) holds. This proves that Yh ∈ (ρT ΓΓ)−1(ThMc) ∩ F⊥h which implies that Yh = 0. Therefore, F−(Ld,Mc,Dc) is a local diffeomorphism. If (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0} for all g ∈ Mc then, proceeding as above, we obtain that F+(Ld,Mc,Dc) is a local diffeomorphism. � Now, let ρT ΓΓ : TΓΓ→ TΓ be the anchor map of the Lie algebroid πτ : TΓΓ→ Γ. Then, we will denote by Hh the subspace of TΓhΓ given by Hh = (ρ TΓΓ)−1(ThMc) ∩ Fh, for h ∈Mc. In a similar way, for every g ∈Mc we will introduce the subspace H̄g of TΓg Γ defined H̄g = (ρ TΓΓ)−1(TgMc) ∩ F̄g. On the other hand, let h be a point of Mc and G h : (EΓ)α(h) ⊕ (EΓ)β(h) → R be the R-bilinear map given by (2.22). We will denote by ( h the subspace of (EΓ)β(h) defined by b ∈ (EΓ)β(h) ∣∣ (T�(β(h))lh)(b) ∈ ThMc } and by GLdch : (Dc)α(h)× ( h → R the restriction to (Dc)α(h)× ( h of the R-bilinear map GLdh . In a similar way, if g is a point of Γ we will consider the subspace ( E Γ)Mcg of (EΓ)α(g) defined by a ∈ (EΓ)α(g) ∣∣ (T�(α(g))(rg ◦ i))(a) ∈ TgMc } 18 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ and the restriction ḠLdcg : ( E Γ)Mcg × (Dc)β(g) → R of GLdg to the space ( E Γ)Mcg × (Dc)β(g). Proposition 3.8. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system. Then, the following conditions are equivalent: (i) For every h ∈Mc (respectively, g ∈Mc) ΓΓ)−1(ThMc) ∩ F⊥h = {0} (respectively, (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}). (ii) For every h ∈ Mc (respectively, g ∈ Mc) the dimension of the vector sub- space Hh (respectively, H̄g) is 2r and the restriction to the vector subbundle H (respectively, H̄) of the Poincaré-Cartan 2-section ΩLd is nondegener- (iii) For every h ∈Mc (respectively, g ∈Mc){ b ∈ ( ∣∣∣ GLdch (a, b) = 0,∀a ∈ (Dc)α(h) } = {0} (respectively, a ∈ ( E Γ)Mcg ∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}). Proof. (i) ⇒ (ii) Assume that h ∈Mc and that ΓΓ)−1(ThMc) ∩ F⊥h = {0}. (3.13) Let U be an open subset of Γ, with h ∈ U , and {φγ}γ=1,...,n−r a set of independent real C∞-functions on U such that Mc ∩ U = {h′ ∈ U | φγ(h′) = 0, for all γ } . If d is the differential of the Lie algebroid τ̃Γ : TΓΓ → Γ then it is easy to prove ΓΓ)−1(ThMc) =< {dφγ(h)} >0 . Thus, dim((ρT ΓΓ)−1(ThMc)) ≥ n+ r. (3.14) On the other hand, dimF⊥h = n−r. Therefore, from (3.13) and (3.14), we obtain dim((ρT ΓΓ)−1(ThMc)) = n+ r TΓhΓ = (ρ TΓΓ)−1(ThMc)⊕ F⊥h . Consequently, using Lemma 3.5, we deduce that Fh = Hh ⊕ F⊥h . (3.15) This implies that dimHh = 2r. Moreover, from (3.15), we also get that Hh ∩H⊥h ⊆ Hh ∩ F and, since Hh ∩ F⊥h = (ρ TΓΓ)−1(ThMc) ∩ F⊥h (see Lemma 3.5), it follows that Hh ∩H⊥h = {0}. Thus, we have proved that Hh is a symplectic subspace of the symplectic vector space (TΓhΓ,ΩLd(h)). If (ρT ΓΓ)−1(TgMc)∩ F̄⊥g = {0}, for all g ∈Mc then, proceeding as above, we ob- tain that H̄g is a symplectic subspace of the symplectic vector space (TΓg Γ,ΩLd(g)), for all g ∈Mc. (ii) ⇒ (i) Suppose that h ∈ Mc and that Hh is a symplectic subspace of the symplectic vector space (TΓhΓ,ΩLd(h)). DISCRETE NONHOLONOMIC MECHANICS 19 If (Xh, Yh) ∈ (ρT ΓΓ)−1(ThMc) ∩ F⊥h then, using Lemma 3.5, we deduce that (Xh, Yh) ∈ Hh. Now, if (X ′h, Y h) ∈ Hh then, since (Xh, Yh) ∈ F h , we conclude that ΩLd(h)((Xh, Yh), (X h)) = 0. This implies that (Xh, Yh) ∈ Hh ∩H⊥h = {0}. Therefore, we have proved that (ρT ΓΓ)−1(ThMc) ∩ F⊥h = {0}. If H̄g ∩ H̄⊥g = {0}, for all g ∈ Mc then, proceeding as above, we obtain that ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc. (i) ⇒ (iii) Assume that ΓΓ)−1(ThMc) ∩ F⊥h = {0} and that b ∈ ( h satisfies the following condition h (a, b) = 0, ∀a ∈ (Dc)α(h). Then, Yh = (T�(β(h))lh)(b) ∈ ThMc ∩ Vhα and (0, Yh) ∈ (ρT ΓΓ)−1(ThMc). Moreover, if (X ′h, Y h) ∈ Fh, we have that X ′h = −(T�(α(h))(rh ◦ i))(a), with a ∈ (Dc)α(h). Thus, using (2.14) and (2.22), we deduce that ΩLd(h)((X h), (0, Yh)) = ΩLd(h)((X h, 0), (0, Yh)) = G h (a, b) = 0. Therefore, (0, Yh) ∈ (ρT ΓΓ)−1(ThMc) ∩ F⊥h = {0}, which implies that b = 0. If (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈ Mc then, proceeding as above, we obtain that{ a ∈ ( ∣∣∣ GLdcg (a, b) = 0, for all b ∈ (Dc)β(g) } = {0}. (iii) ⇒ (i) Suppose that h ∈Mc, that{ b ∈ ( ∣∣∣ GLdh (a, b) = 0, ∀a ∈ (Dc)α(h) } = {0} and let (Xh, Yh) be an element of the set (ρT ΓΓ)−1(ThMc) ∩ F⊥h . Then (see the proof of Lemma 3.5), Xh = 0 and Yh ∈ ThMc∩Vhα. Consequently, Yh = (T�(β(h)lh)(b), with b ∈ ( Now, if a ∈ (Dc)α(h), we have that X ′h = (T�(α(h))(rh ◦ i))(a) ∈ Vhβ and (X h, 0) ∈ Fh. Thus, from (2.22) and since (0, Yh) ∈ F⊥h , it follows that h (a, b) = ΩLd(h)((X h, 0)(0, Yh)) = 0. Therefore, b = 0 which implies that Yh = 0. a ∈ ( E Γ)Mcg ∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}, for all g ∈ Mc, then proceeding as above we obtain that (ρT ΓΓ)−1(TgMc)∩F̄⊥g = {0}, for all g ∈Mc. � Using Theorem 3.7 and Proposition 3.8, we conclude 20 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Theorem 3.9. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system. Then, the following conditions are equivalent: (i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) (re- spectively, F+(Ld,Mc,Dc)) is a local diffeomorphism. (ii) For every h ∈Mc (respectively, g ∈Mc) ΓΓ)−1(ThMc) ∩ F⊥h = {0} (respectively, (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}). (iii) For every h ∈ Mc (respectively, g ∈ Mc) the dimension of the vector sub- space Hh (respectively, H̄g) is 2r and the restriction to the vector subbundle H (respectively, H̄) of the Poincaré-Cartan 2-section ΩLd is nondegener- (iv) For every h ∈Mc (respectively, g ∈Mc){ b ∈ ( ∣∣∣ GLdch (a, b) = 0,∀a ∈ (Dc)α(h) } = {0} (respectively, a ∈ ( E Γ)Mcg ∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}). 3.3. Nonholonomic evolution operators and regular discrete nonholo- nomic Lagrangian systems. First of all, we will introduce the definition of a nonholonomic evolution operator. Definition 3.10. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system and Υnh : Mc →Mc be a differentiable map. Υnh is said to be a discrete nonholo- nomic evolution operator for (Ld,Mc,Dc) if: (i) graph(Υnh) ⊆ Γ2, that is, (g,Υnh(g)) ∈ Γ2, for all g ∈Mc and (ii) (g,Υnh(g)) is a solution of the discrete nonholonomic equations, for all g ∈Mc, that is, do(Ld ◦ lg + Ld ◦ rΥnh(g) ◦ i)(�(β(g)))|Dc(β(g)) = 0, for all g ∈Mc. Remark 3.11. If Υnh : Mc → Mc is a differentiable map then, from (3.1), (3.2) and (3.4), we deduce that Υnh is a discrete nonholonomic evolution operator for (Ld,Mc,Dc) if and only if F−(Ld,Mc,Dc) ◦Υnh = F+(Ld,Mc,Dc). Now, we will introduce the notion of a regular discrete nonholonomic Lagrangian system. Definition 3.12. A discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is said to be regular if the discrete nonholonomic Legendre transformations F−(Ld,Mc,Dc) and F+(Ld,Mc,Dc) are local diffeomorphims. From Theorem 3.9, we deduce Corollary 3.13. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system. Then, the following conditions are equivalent: (i) The system (Ld,Mc,Dc) is regular. (ii) The following relations hold ΓΓ)−1(ThMc) ∩ F⊥h = {0}, for all h ∈Mc, ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc. DISCRETE NONHOLONOMIC MECHANICS 21 (iii) H and H̄ are symplectic subbundles of rank 2r of the symplectic vector bundle (TΓMcΓ,ΩLd). (iv) If g and h are points of Mc then the R-bilinear maps GLdch and Ḡ g are right and left nondegenerate, respectively. The map GLdch (respectively, Ḡ g ) is right nondegenerate (respectively, left non- degenerate) if h (a, b) = 0,∀a ∈ (Dc)α(h) ⇒ b = 0 (respectively, ḠLdcg (a, b) = 0,∀b ∈ (Dc)β(g) ⇒ a = 0). Every solution of the discrete nonholonomic equations for a regular discrete nonholonomic Lagrangian system determines a unique local discrete nonholonomic evolution operator. More precisely, we may prove the following result: Theorem 3.14. Let (Ld,Mc,Dc) be a regular discrete nonholonomic Lagrangian system and (g0, h0) ∈Mc×Mc be a solution of the discrete nonholonomic equations for (Ld,Mc,Dc). Then, there exist two open subsets U0 and V0 of Γ, with g0 ∈ U0 and h0 ∈ V0, and there exists a local discrete nonholonomic evolution operator Υ(Ld,Mc,Dc)nh : U0 ∩Mc → V0 ∩Mc such that: (i) Υ(Ld,Mc,Dc)nh (g0) = h0; (ii) Υ(Ld,Mc,Dc)nh is a diffeomorphism and (iii) Υ(Ld,Mc,Dc)nh is unique, that is, if U 0 is an open subset of Γ, with g0 ∈ U ′0, and Υnh : U ′0 ∩ Mc → Mc is a (local) discrete nonholonomic evolution operator then (Υ(Ld,Mc,Dc)nh )|U0∩U ′0∩Mc = (Υnh)|U0∩U ′0∩Mc . Proof. From remark 3.4, we deduce that F+(Ld,Mc,Dc)(g0) = F−(Ld,Mc,Dc)(h0) = µ0 ∈ D∗c . Thus, we can choose two open subsets U0 and V0 of Γ, with g0 ∈ U0 and h0 ∈ V0, and an open subset W0 of E∗Γ such that µ0 ∈W0 and F+(Ld,Mc,Dc) : U0 ∩Mc →W0 ∩D∗c , F −(Ld,Mc,Dc) : V0 ∩Mc →W0 ∩D∗c are diffeomorphisms. Therefore, from Remark 3.11, we deduce that Υ(Ld,Mc,Dc)nh = (F −(Ld,Mc,Dc) −1 ◦ F+(Ld,Mc,Dc))|U0∩Mc : U0 ∩Mc → V0 ∩Mc is a (local) discrete nonholonomic evolution operator. Moreover, it is clear that Υ(Ld,Mc,Dc)nh (g0) = h0 and it follows that Υ (Ld,Mc,Dc) nh is a diffeomorphism. Finally, if U ′0 is an open subset of Γ, with g0 ∈ U ′0, and Υnh : U ′0 ∩Mc → Mc is another (local) discrete nonholonomic evolution operator then (Υnh)|U0∩U ′0∩Mc is also a (local) discrete nonholonomic evolution operator. Consequently, from Remark 3.11, we conclude that (Υnh)|U0∩U ′0∩Mc = [F −(Ld,Mc,Dc)−1 ◦ F+(Ld,Mc,Dc)]|U0∩U ′0∩Mc = (Υ(Ld,Mc,Dc)nh )|U0∩U ′0∩Mc . 22 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ 3.4. Reversible discrete nonholonomic Lagrangian systems. Let (Ld,Mc, Dc) be a discrete nonholonomic Lagrangian system on a Lie groupoid Γ ⇒ M . Following the terminology used in [36] for the particular case when Γ is the pair groupoid M ×M , we will introduce the following definition Definition 3.15. The discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is said to be reversible if Ld ◦ i = Ld, i(Mc) = Mc, i : Γ→ Γ being the inversion of the Lie groupoid Γ. For a reversible discrete nonholonomic Lagrangian system we have the following result: Proposition 3.16. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian sys- tem on a Lie groupoid Γ. Then, the following conditions are equivalent: (i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) is a local diffeomorphism. (ii) The discrete nonholonomic Legendre transformation F+(Ld,Mc,Dc) is a local diffeomorphism. Proof. If h ∈Mc then, using (3.1) and the fact that Ld ◦ i = Ld, it follows that F−(Ld,Mc,Dc)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ l−1h ) for v�(α(h)) ∈ (Dc)α(h). Thus, from (3.2), we obtain that F−(Ld,Mc,Dc)(h)(v�(α(h))) = −F+(Ld,Mc,Dc)(h−1)(v�(β(h−1))). This implies that F+(Ld,Mc,Dc) = −F−(Ld,Mc,Dc) ◦ i. Therefore, since the inversion is a diffeomorphism (in fact, we have that i2 = id), we deduce the result � Using Theorem 3.9, Definition 3.12 and Proposition 3.16, we prove the following corollaries. Corollary 3.17. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian system on a Lie groupoid Γ. Then, the following conditions are equivalent: (i) The system (Ld,Mc,Dc) is regular. (ii) For all h ∈Mc, ΓΓ)−1(ThMc) ∩ F⊥h = {0}. (iii) H = (ρT ΓΓ)−1(TMc)∩F is a symplectic subbundle of the symplectic vector bundle (TΓMcΓ,ΩLd). (iv) The R-bilinear map GLdch : ( h ×(Dc)α(h) → R is right nondegenerate, for all h ∈Mc. Corollary 3.18. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian system on a Lie groupoid Γ. Then, the following conditions are equivalent: (i) The system (Ld,Mc,Dc) is regular. (ii) For all g ∈Mc, ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}. (iii) H̄ = (ρT ΓΓ)−1(TMc)∩ F̄ is a symplectic subbundle of the symplectic vector bundle (TΓMcΓ,ΩLd). DISCRETE NONHOLONOMIC MECHANICS 23 (iv) The R-bilinear map ḠLdcg : (Dc)β(g) × ( E Γ)Mcg → R is left nondegenerate, for all g ∈Mc. Next, we will prove that a reversible nonholonomic Lagrangian system is dynam- ically reversible. Proposition 3.19. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian sys- tem on a Lie groupoid Γ and (g, h) be a solution of the discrete nonholonomic Euler- Lagrange equations for (Ld,Mc,Dc). Then, (h−1, g−1) is also a solution of these equations. In particular, if the system (Ld,Mc,Dc) is regular and Υ (Ld,Mc,Dc) nh is the (local) discrete nonholonomic evolution operator for (Ld,Mc,Dc) then Υ (Ld,Mc,Dc) is reversible, that is, Υ(Ld,Mc,Dc)nh ◦ i ◦Υ (Ld,Mc,Dc) nh = i. Proof. Using that i(Mc) = Mc, we deduce that (h−1, g−1) ∈ Γ2 ∩ (Mc ×Mc). Now, suppose that β(g) = α(h) = x and that v ∈ (Dc)x. Then, since Ld ◦ i = Ld, it follows that do[Ld ◦ lh−1 + Ld ◦ rg−1 ◦ i](ε(x))(v) = v(Ld ◦ i ◦ rh ◦ i) + v(Ld ◦ i ◦ lg) = v(Ld ◦ lg) + v(Ld ◦ rh ◦ i) = 0. Thus, we conclude that (h−1, g−1) is a solution of the discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc). If the system (Ld,Mc,Dc) is regular and g ∈Mc, we have that (g,Υ (Ld,M,Dc) nh (g)) is a solution of the discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc). Therefore, (i(Υ(Ld,M,Dc)nh (g)), i(g)) is also a solution of the dynamical equations which implies that Υ(Ld,M,Dc)nh (i(Υ (Ld,M,Dc) nh (g))) = i(g). Remark 3.20. Proposition 3.19 was proved in [36] for the particular case when Γ is the pair groupoid. � 3.5. Lie groupoid morphisms and reduction. Let (Φ,Φ0) be a Lie groupoid morphism between the Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′. Denote by (E(Φ),Φ0) the corresponding morphism between the Lie algebroids EΓ and EΓ′ of Γ and Γ′, respectively (see Section 2.2). If Ld : Γ→ R and L′d : Γ ′ → R are discrete Lagrangians on Γ and Γ′ such that Ld = L d ◦ Φ then, using Theorem 4.6 in [27], we have that (DDELLd)(g, h)(v) = (DDELL d)(Φ(g),Φ(h))(Ex(Φ)(v)) for (g, h) ∈ Γ2 and v ∈ (EΓ)x, where x = β(g) = α(h) ∈M. Using this fact, we deduce the following result: Corollary 3.21. Let (Φ,Φ0) be a Lie groupoid morphism between the Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′. Suppose that L′d : Γ ′ → R is a discrete Lagrangian on Γ′, that (Ld = L′d ◦Φ,Mc,Dc) is a discrete nonholonomic Lagrangian system on Γ and that (g, h) ∈ Γ2 ∩ (Mc ×Mc). Then: 24 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ (i) The pair (g, h) is a solution of the discrete nonholonomic problem (Ld,Mc, Dc) if and only if (DDELL′d)(Φ(g),Φ(h)) vanishes over the set (Eβ(g)Φ)((Dc)β(g)). (ii) If (L′d,M c) is a discrete nonholonomic Lagrangian system on Γ ′ such that (Φ(g),Φ(h)) ∈M′c×M′c and (Eβ(g)(Φ))((Dc)β(g)) = (D′c)Φ0(β(g)) then (g, h) is a solution for the discrete nonholonomic problem (Ld,Mc,Dc) if and only if (Φ(g),Φ(h)) is a solution for the discrete nonholonomic problem (L′d,M 3.6. Discrete nonholonomic Hamiltonian evolution operator. Let (Ld,Mc, Dc) a regular discrete nonholonomic system. Assume, without the loss of gener- ality, that the discrete nonholonomic Legendre transformations F−(Ld,Mc,Dc) : Mc −→ D∗c and F+(Ld,Mc,Dc) : Mc −→ D∗c are global diffeomorphisms. Then, (Ld,Mc,Dc) nh = F −(Ld,Mc,Dc)−1◦F+(Ld,Mc,Dc) is the discrete nonholonomic evo- lution operator and one may define the discrete nonholonomic Hamiltonian evolution operator, γ̃nh : D∗c → D∗c , by γ̃nh = F+(Ld,Mc,Dc) ◦ γ (Ld,Mc,Dc) nh ◦ F +(Ld,Mc,Dc) −1 . (3.16) From Remark 3.11, we have the following alternative definitions γ̃nh = F−(Ld,Mc,Dc) ◦ γ (Ld,Mc,Dc) nh ◦ F −(Ld,Mc,Dc) γ̃nh = F+(Ld,Mc,Dc) ◦ F−(Ld,Mc,Dc)−1 of the discrete Hamiltonian evolution operator. The following commutative diagram illustrates the situation Mc Mc (Ld,Mc,Dc) D∗c D F−(Ld, Mc, Dc) F+(Ld, Mc, Dc) F−(Ld, Mc, Dc) F+(Ld, Mc, Dc) γ̃nh γ̃nh Remark 3.22. The discrete nonholonomic evolution operator is an application from D∗c to itself. It is remarkable that D c is also the appropriate nonholonomic momentum space for a continuous nonholonomic system defined by a Lagrangian L : EΓ → R and the constraint distribution Dc. Therefore, in the regular case, the solution of the continuous nonholonomic Lagrangian system also determines a flow from D∗c to itself. We consider that this would be a good starting point to compare the discrete and continuous dynamics and eventually to establish a backward error analysis for nonholonomic systems. � 3.7. The discrete nonholonomic momentum map. Let (Ld,Mc,Dc) be a reg- ular discrete nonholonomic Lagrangian system on a Lie groupoid Γ ⇒ M and τ : EΓ →M be the Lie algebroid of Γ. Suppose that g is a Lie algebra and that Ψ : g→ Sec(τ) is a R-linear map. Then, for each x ∈M, we consider the vector subspace gx of g given by gx = { ξ ∈ g | Ψ(ξ)(x) ∈ (Dc)x } DISCRETE NONHOLONOMIC MECHANICS 25 and the disjoint union of these vector spaces gDc = We will denote by (gDc)∗ the disjoint union of the dual spaces, that is, (gDc)∗ = (gx)∗. Next, we define the discrete nonholonomic momentum map Jnh : Γ → (gDc)∗ as follows: Jnh(g) ∈ (gβ(g))∗ and Jnh(g)(ξ) = Θ+Ld(Ψ(ξ) (1,1))(g) = Ψ(ξ)(g)(Ld), for g ∈ Γ and ξ ∈ gβ(g). If ξ̃ : M → g is a smooth map such that ξ̃(x) ∈ gx, for all x ∈ M, then we may consider the smooth function Jnheξ : Γ→ R defined by Jnheξ (g) = Jnh(g)(ξ̃(β(g))), ∀g ∈ Γ. Definition 3.23. The Lagrangian Ld is said to be g-invariant with respect Ψ if Ψ(ξ)(1,1)(Ld) = Ψ(ξ)(Ld)− Ψ(ξ)(Ld) = 0, ∀ξ ∈ g. Now, we will prove the following result Theorem 3.24. Let Υ(Ld,Mc,Dc)nh : Mc → Mc be the local discrete nonholonomic evolution operator for the regular system (Ld,Mc,Dc). If Ld is g-invariant with respect to Ψ : g→ Sec(τ) and ξ̃ : M → g is a smooth map such that ξ̃(x) ∈ gx, for all x ∈M, then Jnheξ (Υ(Ld,Mc,Dc)nh (g))− Jnheξ (g) = ←−−−−−−−−−−−−−−−−−−−−−−−−−− Ψ(ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))− ξ̃(β(g)))(Υ (Ld,Mc,Dc) nh (g))(Ld) for g ∈Mc. Proof. Using that the Lagrangian Ld is g-invariant with respect to Ψ, we have that −−−−−−−−−−−−−−−−−−→ Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ (Ld,Mc,Dc) nh (g))(Ld) = ←−−−−−−−−−−−−−−−−−− Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ (Ld,Mc,Dc) nh (g))(Ld). (3.17) Also, since (g,Υ(Ld,Mc,Dc)nh (g)) is a solution of the discrete nonholonomic equations: ←−−−−−−− Ψ(ξ̃(β(g)))(g)(Ld) = −−−−−−−−−−−−−−−−−−→ Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ (Ld,Mc,Dc) nh (g))(Ld).(3.18) Thus, from (3.17) and (3.18), we find that ←−−−−−−− Ψ(ξ̃(β(g))(g)(Ld) = ←−−−−−−− Ψ(ξ̃(β(g)))(Υ(Ld,Mc,Dc)nh (g))(Ld). 26 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Therefore, (Υ(Ld,Mc,Dc)nh (g))− J (g) = ←−−−−−−−−−−−−−−−−−−− ξ̃(β(Υ(Ld,Mc,Dc)nh (g))) (Υ(Ld,Mc,Dc)nh (g))(Ld) ←−−−−−−−− ξ̃(β(g)) (g)(Ld) ←−−−−−−−−−−−−−−−−−−− ξ̃(β(Υ(Ld,Mc,Dc)nh (g))) (Υ(Ld,Mc,Dc)nh (g))(Ld) ←−−−−−−− Ψ(ξ̃(β(g))(Υ(Ld,Mc,Dc)nh (g))(Ld) ←−−−−−−−−−−−−−−−−−−−−−−−−−−− ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))− ξ̃(β(g)) (Υ(Ld,Mc,Dc)nh (g))(Ld). Theorem 3.24 suggests us to introduce the following definition Definition 3.25. An element ξ ∈ g is said to be a horizontal symmetry for the discrete nonholonomic system (Ld,Mc,Dc) and the map Ψ : g→ Sec(τ) if Ψ(ξ)(x) ∈ (Dc)x, for all x ∈M. Now, from Theorem 3.24, we conclude that Corollary 3.26. If Ld is g-invariant with respect to Ψ and ξ ∈ g is a horizontal symmetry for (Ld,Mc,Dc) and Ψ : g → Sec(τ) then Jnhξ̃ : Γ → R is a constant of the motion for Υ(Ld,Mc,Dc)nh , that is, ◦Υ(Ld,Mc,Dc)nh = J 4. Examples 4.1. Discrete holonomic Lagrangian systems on a Lie groupoid. Let us examine the case when the system is subjected to holonomic constraints. Let Ld : Γ → R be a discrete Lagrangian on a Lie groupoid Γ ⇒ M . Suppose that Mc ⊆ Γ is a Lie subgroupoid of Γ over M ′ ⊆M , that is, Mc is a Lie groupoid over M ′ with structural maps α|Mc : Mc →M ′, β|Mc : Mc →M ′, �|M ′ : M ′ →Mc, i|Mc : Mc →Mc, the canonical inclusions iMc : Mc −→ Γ and iM ′ : M ′ −→ M are injective immer- sions and the pair (iMc , iM ′) is a Lie groupoid morphism. We may assume, without the loss of generality, that M ′ = M (in other case, we will replace the Lie groupoid Γ by the Lie subgroupoid Γ′ over M ′ defined by Γ′ = α−1(M ′) ∩ β−1(M ′)). Then, if LMc = Ld ◦ iMc and τMc : EMc → M is the Lie algebroid of Mc, we have that the discrete (unconstrained) Euler-Lagrange equations for the Lagrangian function LMc are: X (g)(LMc)− X (h)(LMc) = 0, (g, h) ∈ (Mc)2, (4.1) for X ∈ Sec(τMc). We are interested in writing these equations in terms of the Lagrangian Ld defined on the Lie groupoid Γ. From Corollary 4.7 (iii) in [27], we deduce that (g, h) ∈ (Mc)2 is a solution of Equations 4.1 if and only if DDELLd(g, h) vanishes over Im(Eβ(g)(iMc)). Here, E(iMc) : EMc → EΓ is the Lie algebroid morphism induced between EMc and EΓ by the Lie groupoid morphism (iMc , Id). Therefore, we may consider the discrete holonomic system as the discrete nonholonomic system (Ld,Mc,Dc), where Dc = (E(iMc))(EMc) ∼= EMc . DISCRETE NONHOLONOMIC MECHANICS 27 In the particular case, when the subgroupoid Mc is determined by the vanishing set of n− r independent real C∞-functions φγ : Γ→ R: Mc = { g ∈ Γ | φγ(g) = 0, for all γ } , then the discrete holonomic equations are equivalent to: Y (g)(Ld)− Y (h)(Ld) = λγd oφγ(�(β(g)))(Y (β(g)), φγ(g) = φγ(h) = 0, for all Y ∈ Sec(τ), where do is the standard differential on Γ. This algorithm is a generalization of the Shake algorithm for holonomic systems (see [10, 20, 32, 36] for similar results on the pair groupoid Q×Q). 4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system on the pair group- oid Q×Q ⇒ Q and suppose that (q0, q1) is a point of Mc. Then, using the results of Section 3.1, we deduce that ((q0, q1), (q1, q2)) ∈ (Q × Q)2 is a solution of the discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc) if and only if (D2Ld(q0, q1) +D1Ld(q1, q2))|Dc(q1) = 0, (q1, q2) ∈Mc, or, equivalently, D2Ld(q0, q1) +D1Ld(q1, q2) = j(q1), (q1, q2) ∈Mc, where λj are the Lagrange multipliers and {Aj} is a local basis of the annihilator D0c . These equations were considered in [10] and [36]. Note that if (q1, q2) ∈ Γ = Q×Q then, in this particular case, GLd(q1,q2) : Tq1Q× Tq2Q→ R is just the R-bilinear map (D2D1Ld)(q1, q2). On the other hand, if (q1, q2) ∈Mc we have that (TQ)Mc (q1,q2) vq2 ∈ Tq2Q ∣∣ (0, vq2) ∈ T(q1,q2)Mc } , (TQ)Mc (q1,q2) vq1 ∈ Tq1Q ∣∣ (vq1 , 0) ∈ T(q1,q2)Mc } . Thus, the system (Ld,Mc,Dc) is regular if and only if for every (q1, q2) ∈ Mc the following conditions hold: If vq1 ∈ (TQ)Mc (q1,q2) 〈D2D1Ld(q1, q2)vq1 , vq2〉 = 0, ∀vq2 ∈ Dc(q2)  =⇒ vq1 = 0, If vq2 ∈ (TQ)Mc (q1,q2) 〈D2D1Ld(q1, q2)vq1 , vq2〉 = 0, ∀vq1 ∈ Dc(q1)  =⇒ vq2 = 0. The first condition was obtained in [36] in order to guarantee the existence of a unique local nonholonomic evolution operator Υ(Ld,Mc,Dc)nh for the system (Ld,Mc,Dc). However, in order to assure that Υ (Ld,Mc,Dc) nh is a (local) diffeomor- phism one must assume that the second condition also holds. 28 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Example 4.1 (Discrete Nonholonomically Constrained particle). Consider the discrete nonholonomic system determined by: a) A discrete Lagrangian Ld : R3 × R3 → R: Ld(x0, y0, z0, x1, y1, z1) = x1 − x0 y1 − y0 z1 − z0 b) A constraint distribution of Q = R3, Dc = span ,X2 = c) A discrete constraint submanifold Mc of R3 × R3 determined by the con- straint φ(x0, y0, z0, x1, y1, z1) = z1 − z0 y1 + y0 x1 − x0 The system (Ld,Mc,Dc) is a discretization of a classical continuous nonholonomic system: the nonholonomic free particle (for a discussion on this continuous system see, for instance, [4, 8]). Note that if E(R3×R3) ∼= TR3 is the Lie algebroid of the pair groupoid R3 × R3 ⇒ R3 then T(x1,y1,z1,x1,y1,z1)Mc ∩ E(R3×R3)(x1, y1, z1) = Dc(x1, y1, z1). Since X1 = − X2 = − then, the discrete nonholonomic equations are:( x2 − 2x1 + x0 z2 − 2z1 + z0 = 0, (4.2) y2 − 2y1 + y0 = 0, (4.3) which together with the constraint equation determine a well posed system of dif- ference equations. We have that D2D1Ld = − 1h{dx0 ∧ dx1 + dy0 ∧ dy1 + dz0 ∧ dz1} TR3)Mc (x0,y0,z0,x1,y1,z1) = {a0 ∂∂x0 + b0 + c0 ∂∂z0 ∈ T(x0,y0,z0)R c0 = 12 (a0(y1 + y0)− b0(x1 − x0))}. TR3)Mc (x0,y0,z0,x1,y1,z1) = {a1 ∂∂x1 + b1 + c1 ∂∂z1 ∈ T(x1,y1,z1)R c1 = 12 (a1(y1 + y0) + b1(x1 − x0))}. Thus, if we consider the open subset of Mc defined by{ (x0, y0, z0, x1, y1, z1) ∈Mc ∣∣ 2 + y21 + y1y0 6= 0, 2 + y20 + y0y1 6= 0} then in this subset the discrete nonholonomic system is regular. Let Ψ : g = R2 −→ X(R3) given by Ψ(a, b) = a ∂ + b ∂ . Then gDc = span{Ψ(ξ̃) = X1}, where ξ̃ : R3 → R2 is defined by ξ̃(x, y, z) = (1, y). More- over, the Lagrangian Ld is g-invariant with respect to Ψ. Therefore, (x1, y1, z1, x2, y2, z2)− Jnhξ̃ (x0, y0, z0, x1, y1, z1) ←−−−−−−−−− Ψ(0, y2 − y1)(x1, y1, z1, x2, y2, z2)(Ld), DISCRETE NONHOLONOMIC MECHANICS 29 that is,( x2 − x1 z2 − z1 x1 − x0 z1 − z0 = (y2 − y1) z2 − z1 This equation is precisely Equation (4.2). 4.3. Discrete nonholonomic Lagrangian systems on a Lie group. Let G be a Lie group. G is a Lie groupoid over a single point and the Lie algebra g of G is just the Lie algebroid associated with G. If g, h ∈ G, vh ∈ ThG and αh ∈ T ∗hG we will use the following notation: gvh = (Thlg)(vh) ∈ TghG, vhg = (Thrg)(vh) ∈ ThgG, gαh = (T ∗ghlg−1)(αh) ∈ T ghG, αhg = (T hgrg−1)(αh) ∈ T Now, let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system on the Lie group G, that is, Ld : G → R is a discrete Lagrangian, Mc is a submanifold of G and Dc is a vector subspace of g. If g1 ∈ Mc then (g1, g2) ∈ G × G is a solution of the discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc) if and only if g−11 dLd(g1)− dLd(g2)g gk ∈Mc, k = 1, 2 (4.4) where λj are the Lagrange multipliers and {µj} is a basis of the annihilator D0c of Dc. These equations were obtained in [36] (see Theorem 3 in [36]). Taking pk = dLd(gk)g k , k = 1, 2 then p2 −Ad∗g1p1 = − λjµj , gk ∈Mc, k = 1, 2 (4.5) where Ad : G × g −→ g is the adjoint action of G on g. These equations were obtained in [14] and called discrete Euler-Poincaré-Suslov equations. On the other hand, from (2.14), we have that ΩLd(( −→η ,←−µ ), (−→η ′,←−µ ′)) = −→η ′(←−µ (Ld))−−→η (←−µ ′(Ld)). Thus, if g ∈ G then, using (2.22), it follows that the R-bilinear map GLdg : g×g→ R is given by GLdg (ξ, η) = − ←−η (g)( ξ (Ld)). Therefore, the system (Ld,Mc,Dc) is regular if and only if for every g ∈ Mc the following conditions hold: η ∈ g/←−η (g) ∈ TgMc and ←−η (g)( ξ (Ld)) = 0,∀ξ ∈ Dc =⇒ η = 0, ξ ∈ g/ ξ (g) ∈ TgMc and ←−η (g)( ξ (Ld)) = 0,∀η ∈ Dc =⇒ ξ = 0. We illustrate this situation with two simple examples previously considered in [14]. 30 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ 4.3.1. The discrete Suslov system. (See [14]) The Suslov system studies the motion of a rigid body suspended at its centre of mass under the action of the following nonholonomic constraint: the body angular velocity is orthogonal to some fixed direction. The configuration space is G = SO(3) and the elements of the Lie algebra so(3) may be identified with R3 and represented by coordinates (ωx, ωy, ωz). Without loss of generality, let us choose as fixed direction the third vector of the body frame ē1, ē2, ē3. Then, the nonholonomic constraint is ωz = 0. The discretization of this system is modelled by considering the discrete La- grangian Ld : SO(3) −→ R defined by Ld(Ω) = 12Tr (ΩJ), where J represents the mass matrix (a symmetric positive-definite matrix with components (Jij)1≤i,j≤3). The constraint submanifold Mc is determined by the constraint Tr (ΩE3) = 0 (see [14]) where  0 0 00 0 −1 0 1 0  , E2 =  0 0 10 0 0 −1 0 0  , E3 =  0 −1 01 0 0 0 0 0 is the standard basis of so(3), the Lie algebra of SO(3). The vector subspace Dc = span{E1, E2}. Therefore, D0c = span{E3}. Moreover, the exponential map of SO(3) is a diffeomorphism from an open subset of Dc (which contains the zero vector) to an open subset of Mc (which contains the identity element I). In particular, TIMc = Dc. On the other hand, the discrete Euler-Poincaré-Suslov equations are given Ei(Ω1)(Ld)− Ei(Ω2)(Ld) = 0, Tr (ΩiE3) = 0, i ∈ {1, 2}. After some straightforward operations, we deduce that the above equations are equivalent to: Tr ((EiΩ2 − Ω1Ei)J) = 0, Tr (ΩiE3) = 0, i ∈ {1, 2} or, considering the components Ωk = (Ω ij ) of the elements of SO(3), we have that:( 33 − J33Ω 32 + J22Ω −J23Ω 22 + J12Ω 13 − J13Ω −J23Ω 33 − J22Ω 32 − J12Ω +J33Ω 23 + J23Ω 22 + J13Ω −J13Ω 33 + J33Ω 31 − J12Ω +J23Ω 21 − J11Ω 13 + J13Ω 33 + J12Ω 32 + J11Ω −J33Ω 13 − J23Ω 12 − J13Ω Ω(1)12 = Ω 21 , Ω 12 = Ω Moreover, since the discrete Lagrangian verifies that Ld(Ω) = Tr (ΩJ) = Tr (ΩtJ) = Ld(Ω and also the constraint satisfies Tr (ΩE3) = −Tr (Ω−1E3), then this discretization of the Suslov system is reversible. The regularity condition in Ω ∈ SO(3) is in this particular case: η ∈ so(3) /Tr (E1ΩηJ) = 0, Tr (E2ΩηJ) = 0 and Tr (ΩηE3) = 0 =⇒ η = 0 It is easy to show that the system is regular in a neighborhood of the identity I. DISCRETE NONHOLONOMIC MECHANICS 31 4.3.2. The discrete Chaplygin sleigh. (See [12, 14]) The Chaplygin sleigh system describes the motion of a rigid body sliding on a horizontal plane. The body is supported at three points, two of which slide freely without friction while the third is a knife edge, a constraint that allows no motion orthogonal to this edge (see [41]). The configuration space of this system is the group SE(2) of Euclidean motions of R2. An element Ω ∈ SE(2) is represented by a matrix  cos θ − sin θ xsin θ cos θ y 0 0 1  with θ, x, y ∈ R. Thus, (θ, x, y) are local coordinates on SE(2). A basis of the Lie algebra se(2) ∼= R3 of SE(2) is given by  0 −1 01 0 0 0 0 0  , e1 =  0 0 10 0 0 0 0 0  , e2 =  0 0 00 0 1 0 0 0 and we have that [e, e1] = e2, [e, e2] = −e1, [e1, e2] = 0. An element ξ ∈ se(2) is of the form ξ = ω e+ v1 e1 + v2 e2 and the exponential map exp : se(2) ∼= R3 → SE(2) of SE(2) is given by exp(ω, v1, v2) = (ω, v1 + v2( cosω − 1 ),−v1( cosω − 1 ) + v2 ), if ω 6= 0, exp(0, v1, v2) = (0, v1, v2). Note that the restriction of this map to the open subset U =] − π, π[×R2 ⊆ R3 ∼= se(2) is a diffeomorphism onto the open subset exp(U) of SE(2). A discretization of the Chaplygin sleigh may be constructed as follows: - The discrete Lagrangian Ld : SE(2) −→ R is given by Ld(Ω) = Tr (ΩJΩT )− Tr (ΩJ), where J is the matrix:  (J/2) +ma2 mab mamab (J/2) +mb2 mb ma mb m (see [14]). - The vector subspace Dc of se(2) is Dc = span {e, e1} = { (ω, v1, v2) ∈ se(2) | v2 = 0 } . - The constraint submanifold Mc of SE(2) is Mc = exp(U ∩Dc). (4.6) Thus, we have that Mc = { (θ, x, y) ∈ SE(2) | − π < θ < π, θ 6= 0, (1− cos θ)x− y sin θ = 0 } ∪ { (0, x, 0) ∈ SE(2) | x ∈ R } . 32 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Figure 1. Submanifold Mc From (4.6) it follows that I ∈Mc and TIMc = Dc. In fact, one may prove T(0,x,0)Mc = span { ∂θ |(0,x,0) ∂y |(0,x,0) ∂x |(0,x,0) for x ∈ R. Now, the discrete Euler-Poincaré-Suslov equations are: ←−e (θ1, x1, y1)(Ld)−−→e (θ2, x2, y2)(Ld) = 0, ←−e1(θ1, x1, y1)(Ld)−−→e1(θ2, x2, y2)(Ld) = 0, and the condition (θk, xk, yk) ∈Mc, with k ∈ {1, 2}. We rewrite these equations as the following system of difference equations:( −am cos θ1 − bm sin θ1 + am +mx1 cos θ1 +my1 sin θ1 mx2 + am cos θ2 −bm sin θ2 − am amy1 cos θ1 − amx1 sin θ1 − bmx1 cos θ1 −bmy1 sin θ1 + (a2m+ b2m+ J) sin θ1 amy2 − bmx2 +(a2m+ b2m+ J) sin θ2 together with the condition (θk, xk, yk) ∈Mc, k ∈ {1, 2}. On the other hand, one may prove that the discrete nonholonomic Lagrangian system (Ld,Mc,D) is reversible. Finally, consider a point (0, x, 0) ∈ Mc and an element η ≡ (ω, v1, v2) ∈ se(2) such that ←−η (0, x, 0) ∈ T(0,x,0)Mc, ←−η (0, x, 0)(−→e (Ld)) = 0, ←−η (0, x, 0)(−→e1(Ld)) = 0. Then, if we assume that a2m+ J + amx 6= 0 it follows that η = 0. Thus, the discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is regular in a neighborhood of the identity I. 4.4. Discrete nonholonomic Lagrangian systems on an action Lie group- oid. Let H be a Lie group with identity element e and · : M ×H → M , (x, h) ∈ M × H 7→ xh, a right action of H on M . Thus, we may consider the action Lie groupoid Γ = M ×H over M with structural maps given by α̃(x, h) = x, β̃(x, h) = xh, �̃(x) = (x, e), m̃((x, h), (xh, h′)) = (x, hh′), ĩ(x, h) = (xh, h−1). (4.7) DISCRETE NONHOLONOMIC MECHANICS 33 Now, let h = TeH be the Lie algebra of H and Φ : h→ X(M) the map given by Φ(η) = ηM , for η ∈ h, where ηM is the infinitesimal generator of the action · : M×H →M corresponding to η. Then, Φ is a Lie algebra morphism and the corresponding action Lie algebroid pr1 : M × h→M is just the Lie algebroid of Γ = M ×H. We have that Sec(pr1) ∼= { η̃ : M → h | η̃ is smooth } and that the Lie algebroid structure ([[·, ·]]Φ, ρΦ) on pr1 : M ×H →M is defined by [[η̃, µ̃]]Φ(x) = [η̃(x), µ̃(x)]+(η̃(x))M (x)(µ̃)−(µ̃(x))M (x)(η̃), ρΦ(η̃)(x) = (η̃(x))M (x), for η̃, µ̃ ∈ Sec(pr1) and x ∈M. Here, [·, ·] denotes the Lie bracket of h. If (x, h) ∈ Γ = M ×H then the left-translation l(x,h) : α̃−1(xh) → α̃−1(x) and the right-translation r(x,h) : β̃−1(x)→ β̃−1(xh) are given l(x,h)(xh, h ′) = (x, hh′), r(x,h)(x(h ′)−1, h′) = (x(h′)−1, h′h). (4.8) Now, if η ∈ h then η defines a constant section Cη : M → h of pr1 : M × h→M and, using (2.4), (2.5), (4.7) and (4.8), we have that the left-invariant and the right-invariant vector fields C η and C η, respectively, on M ×H are defined by C η(x, h) = (−ηM (x),−→η (h)), C η(x, h) = (0x, ←−η (h)), (4.9) for (x, h) ∈ Γ = M ×H. Note that if {ηi} is a basis of h then {Cηi} is a global basis of Sec(pr1). On the other hand, we will denote by expΓ : EΓ = M × h → Γ = M × H the map given by expΓ(x, η) = (x, expH(η)), for (x, η) ∈ EΓ = M × h, where expH : h → H is the exponential map of the Lie group H. Note that if Φ(x,e) : R → Γ = M ×H is the integral curve of the left-invariant vector field on Γ = M ×H such that Φ(x,e)(0) = (x, e) then (see (4.9)) expΓ(x, η) = Φ(x,e)(1). Next, suppose that Ld : Γ = M × H → R is a Lagrangian function, Dc is a constraint distribution such that {Xα} is a local basis of sections of the annihilator D0c , and Mc ⊆ Γ is the discrete constraint submanifold. For every h ∈ H (resp., x ∈ M) we will denote by Lh (resp., Lx) the real function on M (resp., on H) given by Lh(y) = Ld(y, h) (resp., Lx(h′) = Ld(x, h′)). A composable pair ((x, hk), (xhk, hk+1)) ∈ Γ2 ∩ (Mc × Mc) is a solution of the discrete nonholonomic Euler-Lagrange equations for the system (Ld,Mc,Dc) if C η(x, hk)(Ld)− C η(xhk, hk+1)(Ld) = λαX α(xhk)(η), for all η ∈ h, or, in other terms (see (4.9)) {(Telhk)(η)}(Lx)− {(Terhk+1)(η)}(Lxhk) + ηM (xhk)(Lhk+1) = λαX α(xhk)(η), for all η ∈ h. 4.4.1. The discrete Veselova system. As a concrete example of a nonholonomic system on a transformation Lie groupoid we consider a discretization of the Veselova system (see [44]). In the continuous theory [9], the configuration manifold is the transformation Lie algebroid pr1 : S2 × so(3)→ S2 with Lagrangian Lc(γ, ω) = ω · Iω −mglγ · e, 34 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ where S2 is the unit sphere in R3, ω ∈ R3 ' so(3) is the angular velocity, γ is the direction opposite to the gravity and e is a unit vector in the direction from the fixed point to the center of mass, all them expressed in a frame fixed to the body. The constants m, g and l are respectively the mass of the body, the strength of the gravitational acceleration and the distance from the fixed point to the center of mass. The matrix I is the inertia tensor of the body. Moreover, the constraint subbundle Dc → S2 is given by γ ∈ S2 7→ Dc(γ) = ω ∈ R3 ' so(3) ∣∣ γ · ω = 0} . Note that the section φ : S2 → S2 × so(3)∗, (x, y, z) 7→ ((x, y, z), xe1 + ye2 + ze3), where {e1, e2, e3} is the canonical basis of R3 and {e1, e2, e3} is the dual basis, is a global basis for D0c . If ω ∈ so(3) and ω̂ is the skew-symmetric matrix of order 3 such that ω̂v = ω×v then the Lagrangian function Lc may be expressed as follows Lc(γ, ω) = Tr(ω̂IIω̂T )−mg l γ · e, where II = 1 Tr(I)I3×3− I. Here, I3×3 is the identity matrix. Thus, we may define a discrete Lagrangian Ld : Γ = S2 × SO(3)→ R for the system by (see [27]) Ld(γ,Ω) = − Tr(IIΩ)− hmg l γ · e. On the other hand, we consider the open subset of SO(3) V = {Ω ∈ SO(3) | Tr Ω 6= ±1 } and the real function ψ : S2 × V → R given by ψ(γ,Ω) = γ · (Ω̂− ΩT ). One may check that the critical points of ψ are (γ,Ω) ∈ S2 × V ∣∣ Ωγ − γ = 0} . Thus, the subset Mc of Γ = S2 × SO(3) defined by (γ,Ω) ∈ (S2 × V )− Cψ ∣∣∣ γ · (Ω̂− ΩT ) = 0} , is a submanifold of Γ of codimension one. Mc is the discrete constraint submanifold. We have that the map expΓ : S 2× so(3)→ S2×SO(3) is a diffeomorphism from an open subset of Dc, which contains the zero section, to an open subset of Mc, which contains the subset of Γ given by �̃(S2) = {(γ, e) ∈ S2 × SO(3)}. So, it follows that (Dc)(γ) = T(γ,e)Mc ∩ EΓ(γ), for γ ∈ S2. Following the computations of [27] we get the nonholonomic discrete Euler-Lagrange equations, for ((γk,Ωk), (γkΩk,Ωk+1)) ∈ Γ2 Mk+1 − ΩTkMkΩk +mglh 2( ̂γk+1 × e) = λγ̂k+1, γk( ̂Ωk − ΩTk ) = 0, γk+1( ̂Ωk+1 − ΩTk+1) = 0, where M = ΩII − IIΩT . Therefore, in terms of the axial vector Π in R3 defined by Π̂ = M , we can write the equations in the form Πk+1 = ΩTk Πk −mglh 2γk+1 × e + λγk+1, γk( ̂Ωk − ΩTk ) = 0, γk+1( ̂Ωk+1 − ΩTk+1) = 0. DISCRETE NONHOLONOMIC MECHANICS 35 Note that, using the expression of an arbitrary element of SO(3) in terms of the Euler angles (see Chapter 15 of [31]), we deduce that the discrete constraint sub- manifold Mc is reversible, that is, i(Mc) = Mc. However, the discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is not reversible. In fact, it is easy to prove that Ld ◦ i 6= Ld. On the other hand, if γ ∈ S2 and ξ, η ∈ R3 ∼= so(3) then it follows that C ξ(γ, I3)( C η(Ld)) = −ξ · Iη. Consequently, the nonholonomic system (Ld,Mc,Dc) is regular in a neighborhood (in Mc) of the submanifold �̃(S2). 4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie group- oid. Let p : Q → M = Q/G be a principal G-bundle and choose a local trivial- ization G× U , where U is an open subset of M . Then, one may identify the open subset (p−1(U) × p−1(U))/G ' ((G × U) × (G × U))/G of the Atiyah groupoid (Q×Q)/G with the product manifold (U ×U)×G. Indeed, it is easy to prove that the map ((G× U)× (G× U))/G→ (U × U)×G, [((g, x), (g′, y))]→ ((x, y), g−1g′)), is bijective. Thus, the restriction to ((G × U) × (G × U))/G of the Lie groupoid structure on (Q × Q)/G induces a Lie groupoid structure in (U × U) × G with source, target and identity section given by α : (U × U)×G→ U ; ((x, y), g)→ x, β : (U × U)×G→ U ; ((x, y), g)→ y, � : U → (U × U)×G; x→ ((x, x), e), and with multiplication m : ((U × U) × G)2 → (U × U) × G and inversion i : (U × U)×G→ (U × U)×G defined by m(((x, y), g), ((y, z), h)) = ((x, z), gh), i((x, y), g) = ((y, x), g−1). (4.10) The Lie algebroid A((U×U)×G) may be identified with the vector bundle TU×g→ U . Thus, the fibre over the point x ∈ U is the vector space TxU × g. Therefore, a section of A((U ×U)×G) is a pair (X, ξ̃), where X is a vector field on U and ξ̃ is a map from U on g. The space Sec(A((U × U)×G)) is generated by sections of the form (X, 0) and (0, Cξ), with X ∈ X(U), ξ ∈ g and Cξ : U → g being the constant map Cξ(x) = ξ, for all x ∈ U (see [27] for more details). Now, suppose that Ld : (U ×U)×G→ R is a Lagrangian function, Dc a vector subbundle of TU×g and Mc a constraint submanifold on (U×U)×G. Take a basis of sections {Y α} of the annihilator Doc . Then, the discrete nonholonomic equations are ←−−−−− (Xα, η̃α)((x, y), gk)(Ld)− −−−−−→ (Xα, η̃α)((y, z), gk+1)(Ld) = 0, with (Xα, η̃α) : U → TU × g a basis of the space Sec(τDc) and (((x, y), gk), ((y, z), gk+1)) ∈ (Mc×Mc)∩ ((U ×U)×G)2. The above equations may be also written as (X, 0)((x, y), gk)(Ld)− (X, 0)((y, z), gk+1)(Ld) = λαY α(y)(X(y)),←−−−− (0, Cξ)((x, y), gk)(Ld)− −−−−→ (0, Cξ)((y, z), gk+1)(Ld) = λαY α(y)(Cξ(y)), with X ∈ X(U), ξ ∈ g and (((x, y), gk), ((y, z), gk+1)) ∈ (Mc×Mc)∩((U×U)×G)2. An equivalent expression of these equations is D2Ld((x, y), gk) +D1Ld((y, z), gk+1) = λαµα(y), pk+1(y, z) = Ad∗gkpk(x, y)− λαη̃ α(y), (4.11) 36 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ where pk(x̄, ȳ) = d(r∗gkL(x̄,ȳ, ))(e) for (x̄, ȳ) ∈ U × U and we write Y α ≡ (µα, η̃α), µα being a 1-form on U and η̃α : U → g∗ a smooth map. 4.5.1. A discretization of the equations of motion of a rolling ball without sliding on a rotating table with constant angular velocity. A (homogeneous) sphere of radius r > 0, mass m and inertia about any axis I rolls without sliding on a horizontal table which rotates with constant angular velocity Ω about a vertical axis through one of its points. Apart from the constant gravitational force, no other external forces are assumed to act on the sphere (see [41]). The configuration space for the continuous system isQ = R2×SO(3) and we shall use the notation (x, y;R) to represent a typical point in Q. Then, the nonholonomic constraints are Tr(ṘRTE2) = −Ωy, Tr(ṘRTE1) = Ωx, where {E1, E2, E3} is the standard basis of so(3). The matrix ṘRT is skew symmetric, therefore we may write ṘRT =  0 −w3 w2w3 0 −w1 −w2 w1 0 where (w1, w2, w3) represents the angular velocity vector of the sphere measured with respect to the inertial frame. Then, we may rewrite the constraints in the usual form: ẋ− rw2 = −Ωy, ẏ + rw1 = Ωx. The Lagrangian for the rolling ball is: Lc(x, y;R, ẋ, ẏ; Ṙ) = m(ẋ2 + ẏ2) + I Tr(ṘRT (ṘRT )T ) m(ẋ2 + ẏ2) + I(ω21 + ω 2 + ω Moreover, it is clear that Q = R2 × SO(3) is the total space of a trivial princi- pal SO(3)-bundle over R2 and the bundle projection φ : Q → M = R2 is just the canonical projection on the first factor. Therefore, we may consider the correspond- ing Atiyah algebroid E′ = TQ/SO(3) over M = R2. We will identify the tangent bundle to SO(3) with so(3)× SO(3) by using right translation. Under this identification between T (SO(3)) and so(3)×SO(3) the tangent action of SO(3) on T (SO(3)) ∼= so(3)× SO(3) is the trivial action (so(3)× SO(3))× SO(3)→ so(3)× SO(3), ((ω,R), S) 7→ (ω,RS). (4.12) Thus, the Atiyah algebroid TQ/SO(3) is isomorphic to the product manifold TR2×so(3) and the vector bundle projection is τR2 ◦pr1, where pr1 : TR2×so(3)→ TR2 and τR2 : TR2 → R2 are the canonical projections. A section of E′ = TQ/SO(3) ∼= TR2 × so(3) → R2 is a pair (X,u), where X is a vector field on R2 and u : R2 → so(3) is a smooth map. Therefore, a global basis of sections of TR2 × so(3)→ R2 is s′1 = ( , 0), s′2 = ( , 0), s′3 = (0, E1), s 4 = (0, E2), s 5 = (0, E3). DISCRETE NONHOLONOMIC MECHANICS 37 The anchor map ρ′ : E′ = TQ/SO(3) ∼= TR2 × so(3) → TR2 is the projection over the first factor and if [[·, ·]]′ is the Lie bracket on the space Sec(E′ = TQ/SO(3)) then the only non-zero fundamental Lie brackets are [[s′3, s ′ = s′5, [[s ′ = s′3, [[s ′ = s′4. Moreover, the Lagrangian function Lc = T and the constraint functions are SO(3)-invariant. Consequently, Lc induces a Lagrangian function L′c on E TQ/SO(3) L′c(x, y, ẋ, ẏ;ω) = m(ẋ2 + ẏ2) + I Tr(ωωT ), m(ẋ2 + ẏ2)− I Tr(ω2), where (x, y, ẋ, ẏ) are the standard coordinates on TR2 and ω ∈ so(3). The con- straint functions defined on E′ = TQ/SO(3) are: ẋ+ r Tr(ωE2) = −Ωy, ẏ − r Tr(ωE1) = Ωx. (4.13) We have a nonholonomic system on the Atiyah algebroid E′ = TQ/SO(3) ∼= TR2× so(3). This kind of systems was recently analyzed by J. Cortés et al [9] (in particular, this example was carefully studied). Eqs. (4.13) define an affine subbundle of the vector bundle E′ ∼= TR2× so(3)→ R2 which is modelled over the vector subbundle D′c generated by the sections D′c = {s 5, rs 1 + s 4, rs 2 − s Our objective is to discretize this example directly on the Atiyah algebroid. The Atiyah groupoid is now identified to R2 × R2 × SO(3) ⇒ R2. We may construct the discrete Lagrangian by L′d(x0, y0, x1, y1;W1) = L c(x0, y0, x1 − x0 y1 − y0 ; (logW1)/h) where log : SO(3) −→ so(3) is the (local)-inverse of the exponential map exp : so(3) −→ SO(3). For simplicity instead of this procedure we use the following approximation: logW1/h ≈ W1 − I3×3 where I3×3 is the identity matrix. L′d(x0, y0, x1, y1;W1) = L c(x0, y0, x1 − x0 y1 − y0 W1 − I3×3 x1 − x0 y1 − y0 (2h)2 Tr(I3×3 −W1) Eliminating constants, we may consider as discrete Lagrangian L′d = x1 − x0 y1 − y0 Tr(W1) The discrete constraint submanifold M′c of R 2 × R2 × SO(3) is determined by the constraints: x1 − x0 Tr(W1E2) = −Ω y1 + y0 y1 − y0 Tr(W1E1) = Ω x1 + x0 38 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ We have that the system (L′d,M c) is not reversible. Note that the Lagrangian function L′d is reversible. However, the constraint submanifold M c is not reversible. The discrete nonholonomic Euler-Lagrange equations for the system (L′d, ,M D′c) are: s′5(x0, y0, x1, y1;W1)(L s′5(x1, y1, x2, y2;W2)(L d) = 0 ←−−−−−− (rs′1 + s 4)(x0, y0, x1, y1;W1)(L −−−−−−→ (rs′1 + s 4)(x1, y1, x2, y2;W2)(L d) = 0 ←−−−−−− (rs′2 − s 3)(x0, y0, x1, y1;W1)(L −−−−−−→ (rs′2 − s 3)(x1, y1, x2, y2;W2)(L d) = 0 with the constraints defining Mc. On the other hand, the vector fields ←−s ′5, −→s ′5, ←−−−−− rs′1 + s −−−−−→ rs′1 + s ←−−−−− rs′2 − s′3 and−−−−−→ rs′2 − s′3 on (R2 × R2)× SO(3) are given by ←−s ′5 = ((0, 0), E 3), −→s ′5 = ((0, 0), E 3),←−−−−− rs′1 + s 4 = ((0, r E 2), −−−−−→ rs′1 + s 4 = ((−r , 0), E 2),←−−−−− rs′2 − s′3 = ((0, r E 1), −−−−−→ rs′2 − s′3 = ((0,−r E 1), where E i (respectively, E i) is the left-invariant (respectively, right-invariant) vec- tor field on SO(3) induced by Ei ∈ so(3), for i ∈ {1, 2, 3}. Thus, we deduce the following system of equations: Tr ((W1 −W2)E3) = 0, x2 − 2x1 + x0 Tr ((W1 −W2)E2) = 0, y2 − 2y1 + y0 Tr ((W1 −W2)E1) = 0, x2 − x1 Tr(W2E2) + Ω y2 + y1 y2 − y1 Tr(W2E1)− Ω x2 + x1 where (x0, x1, y0, y1;W1) are known. Simplifying we obtain the following system of equations: x2 − 2x1 + x0 I +mr2 y2 − y0 = 0 (4.14) y2 − 2y1 + y0 I +mr2 x2 − x0 = 0 (4.15) Tr ((W1 −W2)E3) = 0 (4.16) x2 − x1 Tr(W2E2) + Ω y2 + y1 = 0, (4.17) y2 − y1 Tr(W2E1)− Ω x2 + x1 = 0. (4.18) Now, consider the open subset U of R2 × R2 × SO(3) U = (R2 × R2)× {W ∈ SO(3) | W − Tr(W )I3×3 is regular } . Then, using Corollary 3.13 (iv), we deduce that the discrete nonholonomic La- grangian system (L′d,M c) is regular in the open subset U ′ of M′c given by U ′ = U ∩M′c. DISCRETE NONHOLONOMIC MECHANICS 39 If we denote by uk = (xk+1 − xk)/h and vk = (yk+1 − yk)/h, k ∈ N then from Equations (4.14) and (4.15) we deduce that( 4 + α2h2 4− α2h2 −4αh 4αh 4− α2h2 or in other terms x(k + 2) = 8x(k + 1) + (α2h2 − 4)x(k)− 4αh(y(k + 1)− y(k)) α2h2 + 4 y(k + 2) = 8y(k + 1) + (α2h2 − 4)y(k) + 4α(x(k + 1)− x(k)) α2h2 + 4 where α = IΩ I+mr2 . Since A ∈ SO(2), the discrete nonholonomic model predicts that the point of contact of the ball will sweep out a circle on the table in agreement with the continuous model. Figure 2 shows the excellent behaviour of the proposed numerical method Figure 2. Orbits for the discrete nonholonomic equations of mo- tion (left) and a standard numerical method (right) (initial condi- tions x(0) = 0.99, y(0) = 1, x(1) = 1, y(1) = 0.99 and h = 0.01 after 20000 steps). 4.6. Discrete Chaplygin systems. Now, we present the theory for a particu- lar (but typical) example of discrete nonholonomic systems: discrete Chaplygin systems. This kind of systems was considered in the case of the pair groupoid in [10]. For any groupoid Γ ⇒ M , the map χ : Γ → M × M , g 7→ (α(g), β(g)) is a morphism over M from Γ to the pair groupoid M ×M (usually called the anchor of Γ). The induced morphism of Lie algebroids is precisely the anchor ρ : EΓ → TM of EΓ (the Lie algebroid of Γ). Definition 4.2. A discrete Chaplygin system on the groupoid Γ is a discrete nonholonomic problem (Ld,Mc,Dc) such that - (Ld,Mc,Dc) is a regular discrete nonholonomic Lagrangian system; - χMc = χ ◦ iMc : Mc −→M ×M is a diffeomorphism; - ρ ◦ iDc : Dc −→ TM is an isomorphism of vector bundles. Denote by L̃d : M ×M −→ R the discrete Lagrangian defined by L̃d = Ld ◦ iMc ◦ (χMc)−1. In the following, we want to express the dynamics on M×M , by finding relations between de dynamics defined by the nonholonomic system on Γ and M ×M . 40 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ From our hypothesis, for any vector field Y ∈ X(M) there exists a unique section X ∈ Sec(τDc) such that ρ ◦ iDc ◦X = Y . Now, using (2.4), (2.5) and (2.6), it follows that X (g)) = −Y (α(g)) and Tgβ( X (g)) = Y (β(g)) with some abuse of notation. In other words, Tgχ(X (1,0)(g)) = Y (1,0)(α(g), β(g)) and Tgχ(X (0,1)(g)) = Y (0,1)(α(g), β(g)) for g ∈ Mc, where Tχ : TΓΓ ∼= V β ⊕Γ V α → TM×M (M ×M) ∼= T (M ×M) is the prolongation of the morphism χ given by (Tgχ)(Xg, Yg) = ((Tgα)(Xg), (Tgβ)(Yg)), for g ∈ Γ and (Xg, Yg) ∈ TΓg Γ ∼= Vgβ ⊕ Vgα. Since χMc is a diffeomorphism, there exists a unique X g ∈ TgMc (respectively, X̄ ′g ∈ TgMc) such that (TgχMc)(X g) = Y (1,0)(α(g), β(g)) = (−Y (α(g)), 0β(g)) (respectively, (TgχMc)(X̄ g) = Y (0,1)(α(g), β(g)) = (0α(g), Y (β(g)))) for all g ∈Mc. Thus, X ′g ∈ TgMc ∩ Vgβ, X (g)−X ′g = Z ′g ∈ Vgα ∩ Vgβ, X̄ ′g ∈ TgMc ∩ Vgα, X (g)− X̄ ′g = Z̄ ′g ∈ Vgα ∩ Vgβ, for all g ∈Mc. Now, if (g, h) ∈ Γ2 ∩ (Mc ×Mc) then X (g)(Ld)− X (h)(Ld) = X̄ g(Ld) + Z̄ g(Ld)−X h(Ld)− Z h(Ld) Y (α(g), β(g))(L̃d)− Y (α(h), β(h))(L̃d) +Z̄ ′g(Ld)− Z h(Ld). Therefore, if we use the following notation (α(g), β(g)) = (x, y), (α(h), β(h)) = (y, z) F+Y (x, y) = −Z̄ (x,y) (Ld), F Y (y, z) = Z (y,z) (Ld), X (g)(Ld)− X (h)(Ld) = Y (x, y)(L̃d)− Y (y, z)(L̃d) −F+Y (x, y) + F Y (y, z). In conclusion, we have proved that (g, h) is a solution of the discrete nonholonomic Euler-Lagrange equations for the system (Ld,Mc,Dc) if and only if ((x, y), (y, z)) is a solution of the reduced equations Y (x, y)(L̃d)− Y (y, z)(L̃d) = F Y (x, y)− F Y (y, z), Y ∈ X(M). Note that the above equations are the standard forced discrete Euler-Lagrange equations (see [32]). 4.6.1. The discrete two wheeled planar mobile robot. We now consider a discrete version of the two-wheeled planar mobile robot [8, 9]. The position and orientation of the robot is determined, with respect a fixed cartesian reference, by an element Ω = (θ, x, y) ∈ SE(2), that is, a matrix  cos θ − sin θ xsin θ cos θ y 0 0 1 DISCRETE NONHOLONOMIC MECHANICS 41 Moreover, the different positions of the two wheels are described by elements (φ, ψ) ∈ T2. Therefore, the configuration space is SE(2) × T2. The system is subjected to three nonholonomic constraints: one constraint induced by the condi- tion of no lateral sliding of the robot and the other two by the rolling conditions of both wheels. It is well known that this system is SE(2)-invariant and then the system may be described as a nonholonomic system on the Lie algebroid se(2)×TT2 → T2 (see [9]). In this case, the Lagrangian is Jω2 +m(v1)2 +m(v2)2 + 2m0lωv 2 + J2φ̇ 2 + J2ψ̇ Tr(ξJξT ) + φ̇2 + where ξ = ω e+ v1 e1 + v 2 e2 =  0 −ω v1ω 0 v2 0 0 0  and J =  J/2 0 m0l0 J/2 0 m0l 0 m Here, m = m0 + 2m1, where m0 is the mass of the robot without the two wheels, m1 the mass of each wheel, J its the moment of inertia with respect to the vertical axis, J2 the axial moments of inertia of the wheels and l the distance between the center of mass of the robot and the intersection point of the horizontal symmetry axis of the robot and the horizontal line connecting the centers of the two wheels. The nonholonomic constraints are v1 + R φ̇+ R ψ̇ = 0, v2 = 0, ω + R φ̇− R ψ̇ = 0, (4.19) determining a submanifold M of se(2) × TT2, where R is the radius of the two wheels and 2c the lateral length of the robot. In order to discretize the above nonholonomic system, we consider the Atiyah groupoid Γ = SE(2)×(T2×T2) ⇒ T2. The Lie algebroid of SE(2)×(T2×T2) ⇒ T2 is TT2 × se(2)→ T2. Then: - The discrete Lagrangian Ld : SE(2)× (T2 × T2)→ R is given by: Ld(Ωk, φk, ψk, φk+1, ψk+1) = 12h2 Tr ((Ωk − I3×3)J(Ωk − I3×3) (∆φk) (∆ψk) where I3×3 is the identity matrix, ∆φk = φk+1 − φk, ∆ψk = ψk+1 − ψk  cos θk − sin θk xksin θk cos θk yk 0 0 1 We obtain that mx2k +my k − 2lm0xk(1− cos θk) +2J(1− cos θk) + 2lm0yk sin θk) + (∆φk)2 (∆ψk)2 - The constraint vector subbundle of se(2)×TT2 is generated by the sections:{ , s2 = 42 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ - The continuous constraints of the two-wheeled planar robot are written in matrix form (see 4.19):  0 −ω v1ω 0 v2 0 0 0  0 R2c φ̇− R2c ψ̇ −R2 φ̇− R2 ψ̇− R φ̇+ R ψ̇ 0 0 0 0 0 We discretize the previous constraints using the exponential on SE(2) (see Section 4.3.2) and discretizing the velocities on the right hand side cos( R2c∆φk− R2c∆ψk) sin( R2c∆φk− R2c∆ψk) −c ∆φk+∆ψk ∆φk−∆ψk sin( R2c∆φk− R2c∆ψk) − sin( R2c∆φk− R2c∆ψk) cos( R2c∆φk− R2c∆ψk) c ∆φk+∆ψk ∆φk−∆ψk (1−cos( R2c∆φk− R2c∆ψk)) 0 0 1 if ∆φk 6= ∆ψk and  1 0 −R∆φk0 1 0 0 0 1 if ∆φk = ∆ψk. Therefore, the constraint submanifold Mc is defined as θk = − ∆φk + ∆ψk (4.20) xk = −c ∆φk + ∆ψk ∆φk −∆ψk ∆φk − (4.21) yk = c ∆φk + ∆ψk ∆φk −∆ψk 1− cos ∆φk − (4.22) if ∆φk 6= ∆ψk and θk = 0, xk = −R∆φk and yk = 0 if ∆φk = ∆ψk. We have that the discrete nonholonomic system (Ld,Mc,Dc) is reversible. More- over, if �Γ : T2 → SE(2) × (T2 × T2) is the identity section of the Lie groupoid Γ = SE(2)× (T2 × T2) then it is clear that �Γ(T2) = {I3×3} ×∆T2×T2 ⊆Mc. Here, ∆T2×T2 is the diagonal in T2 × T2. In addition, the system (Ld,Mc,Dc) is regular in a neighborhood U of the submanifold �Γ(T2) = {I3×3} ×∆T2×T2 in Mc. Note that T(I3×3,φ1,ψ1,φ1,ψ1)Mc ∩ EΓ(φ1, ψ1) = Dc(φ1, ψ1), for (φ1, ψ1) ∈ T2, where EΓ = se(2)× TT2 is the Lie algebroid of the Lie groupoid Γ = SE(2)× (T2 × T2). On the other hand, it is easy to show that the system (Ld, U,Dc) is a discrete Chaplygin system. The reduced Lagrangian on T2 × T2 is L̃d =   (mc2( ∆φk + ∆ψk ∆φk −∆ψk )2(1− cos( ∆φk − ∆ψk)) +J(1− cos( ∆φk − ∆ψk))) + (∆φk)2 (∆ψk)2 if ∆φk 6= ∆ψk (J1 + (∆φk)2 , if ∆φk = ∆ψk The discrete nonholonomic equations are: (Ω1,φ1,ψ1,φ2,ψ2) (Ld)−−→s1 (Ω2,φ2,ψ2,φ3,ψ3) (Ld) = 0 (Ω1,φ1,ψ1,φ2,ψ2) (Ld)−−→s2 (Ω2φ2,ψ2,φ3,ψ3) (Ld) = 0 DISCRETE NONHOLONOMIC MECHANICS 43 These equations in coordinates are: 2J1(φ3 − 2φ2 + φ1) = lRm0(cos θ2 + cos θ1) + (sin θ2 − sin θ1) R cos θ1 (lm0y1 + cmx1) + R sin θ1 (lm0x1 − cmy1) (cmx2 + lm0(y2 − 2c)) (4.23) 2J1(ψ3 − 2ψ2 + ψ1) = lRm0(cos θ2 + cos θ1)− (sin θ2 − sin θ1) R cos θ1 (lm0y1 − cmx1)− R sin θ1 (lm0x1 + cmy1) (cmx2 − lm0(y2 + 2c)) (4.24) Substituting constraints (4.20), (4.21) and (4.22) in Equations (4.23) and (4.24) we obtain a set of equations of the type 0 = f1(φ1, φ2, φ3, ψ1, ψ2, ψ3) and 0 = f1(φ1, φ2, φ3, ψ1, ψ2, ψ3) which are the reduced equations of the Chaplygin system. 5. Conclusions and Future Work In this paper we have elucidated the geometrical framework for nonholonomic discrete Mechanics on Lie groupoids. We have proposed discrete nonholonomic equations that are general enough to produce practical integrators for continuous nonholonomic systems (reduced or not). The geometric properties related with these equations have been completely studied and the applicability of these devel- opments has been stated in several interesting examples. Of course, much work remains to be done to clarify the nature of discrete non- holonomic mechanics. Many of this future work was stated in [36] and, in particular, we emphasize: - a complete backward error analysis which explain the very good energy behavior showed in examples or the preservation of a discrete energy (see [14]); - related with the previous question, the construction of a discrete exact model for a continuous nonholonomic system (see [17, 32, 36]); - to study discrete nonholonomic systems which preserve a volume form on the constraint surface mimicking the continuous case (see, for instance, [13, 46] for this last case); - to analyze the discrete hamiltonian framework and the construction of integrators depending on different discretizations; - and the construction of a discrete nonholonomic connection in the case of Atiyah groupoids (see [21, 27]). Related with some of the previous questions, in the conclusions of the paper of R. McLachlan and M. 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Math. 30 (3), (2004), 637–655. [44] Veselov AP and Veselova LE, Integrable nonholonomic systems on Lie groups, Math. Notes 44 (1989), 810–819. [45] Weinstein A, Lagrangian Mechanics and groupoids, Fields Inst. Comm. 7 (1996), 207–231. [46] Zenkov D and Bloch AM, Invariant measures of nonholonomic flows with internal degrees of freedom, Nonlinearity 16 (2003), 1793–1807. D. Iglesias: Instituto de Matemáticas y F́ısica Fundamental, Consejo Superior de Investigaciones Cient́ıficas, Serrano 123, 28006 Madrid, Spain E-mail address: iglesias@imaff.cfmac.csic.es Juan C. Marrero: Departamento de Matemática Fundamental, Facultad de Matemá- ticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain E-mail address: jcmarrer@ull.es D. Mart́ın de Diego: Instituto de Matemáticas y F́ısica Fundamental, Consejo Supe- rior de Investigaciones Cient́ıficas, Serrano 123, 28006 Madrid, Spain E-mail address: d.martin@imaff.cfmac.csic.es Eduardo Mart́ınez: Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain E-mail address: emf@unizar.es 1. Introduction 2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids 2.1. Lie algebroids 2.2. Lie groupoids 2.3. Discrete Unconstrained Lagrangian Systems 3. Discrete Nonholonomic (or constrained) Lagrangian systems on Lie groupoids 3.1. Discrete Generalized Hölder's principle 3.2. Discrete Nonholonomic Legendre transformations 3.3. Nonholonomic evolution operators and regular discrete nonholonomic Lagrangian systems 3.4. Reversible discrete nonholonomic Lagrangian systems 3.5. Lie groupoid morphisms and reduction 3.6. Discrete nonholonomic Hamiltonian evolution operator 3.7. The discrete nonholonomic momentum map 4. Examples 4.1. Discrete holonomic Lagrangian systems on a Lie groupoid 4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid 4.3. Discrete nonholonomic Lagrangian systems on a Lie group 4.4. Discrete nonholonomic Lagrangian systems on an action Lie groupoid 4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid 4.6. Discrete Chaplygin systems 5. Conclusions and Future Work References

This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of the evolution operator and the discrete nonholonomic momentum equation are also considered. Finally, we illustrate with several classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile robot).

Introduction 2 2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids 5 2.1. Lie algebroids 5 2.2. Lie groupoids 6 2.3. Discrete Unconstrained Lagrangian Systems 9 3. Discrete Nonholonomic (or constrained) Lagrangian systems on Lie groupoids 11 3.1. Discrete Generalized Hölder’s principle 11 3.2. Discrete Nonholonomic Legendre transformations 14 3.3. Nonholonomic evolution operators and regular discrete nonholonomic Lagrangian systems 20 3.4. Reversible discrete nonholonomic Lagrangian systems 22 3.5. Lie groupoid morphisms and reduction 23 3.6. Discrete nonholonomic Hamiltonian evolution operator 24 3.7. The discrete nonholonomic momentum map 24 4. Examples 26 4.1. Discrete holonomic Lagrangian systems on a Lie groupoid 26 4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid 27 This work has been partially supported by MICYT (Spain) Grants MTM 2006-03322, MTM 2004-7832, MTM 2006-10531 and S-0505/ESP/0158 of the CAM. D. Iglesias thanks MEC for a “Juan de la Cierva” research contract. 2 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ 4.3. Discrete nonholonomic Lagrangian systems on a Lie group 29 4.4. Discrete nonholonomic Lagrangian systems on an action Lie groupoid 32 4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid 35 4.6. Discrete Chaplygin systems 39 5. Conclusions and Future Work 43 References 43 1. Introduction In the paper of Moser and Veselov [40] dedicated to the complete integrability of certain dynamical systems, the authors proposed a discretization of the tangent bundle TQ of a configuration space Q replacing it by the product Q×Q, approx- imating a tangent vector on Q by a pair of ‘close’ points (q0, q1). In this sense, the continuous Lagrangian function L : TQ −→ R is replaced by a discretization Ld : Q×Q −→ R. Then, applying a suitable variational principle, it is possible to derive the discrete equations of motion. In the regular case, one obtains an evolu- tion operator, a map which assigns to each pair (qk−1, qk) a pair (qk, qk+1), sharing many properties with the continuous system, in particular, symplecticity, momen- tum conservation and a good energy behavior. We refer to [32] for an excellent review in discrete Mechanics (on Q×Q) and its numerical implementation. On the other hand, in [40, 44], the authors also considered discrete Lagrangians defined on a Lie group G where the evolution operator is given by a diffeomorphism of G. All the above examples led to A. Weinstein [45] to study discrete mechanics on Lie groupoids. A Lie groupoid is a geometric structure that includes as particular examples the case of cartesian products Q × Q as well as Lie groups and other examples as Atiyah or action Lie groupoids [26]. In a recent paper [27], we studied discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids, deriving from a variational principle the discrete Euler-Lagrange equations. We also introduced a symplectic 2-section (which is preserved by the Lagrange evolution operator) and defined the Hamiltonian evolution operator, in terms of the discrete Legendre transformations, which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. These techniques include as particular cases the classical discrete Euler-Lagrange equations, the discrete Euler-Poincaré equations (see [5, 6, 29, 30]) and the discrete Lagrange-Poincaré equations. In fact, the results in [27] may be applied in the construction of geometric integrators for continuous Lagrangian systems which are invariant under the action of a symmetry Lie group (see also [18] for the particular case when the symmetry Lie group is abelian). From the perspective of geometric integration, there are a great interest in intro- ducing new geometric techniques for developing numerical integrators since stan- dard methods often introduce some spurious effects like dissipation in conservative systems [16, 42]. The case of dynamical systems subjected to constraints is also of considerable interest. In particular, the case of holonomic constraints is well established in the literature of geometric integration, for instance, in simulation of molecular dynamics where the constraints may be molecular bond lengths or angles and also in multibody dynamics (see [16, 20] and references therein). DISCRETE NONHOLONOMIC MECHANICS 3 By contrast, the construction of geometric integrators for the case of nonholo- nomic constraints is less well understood. This type of constraints appears, for instance, in mechanical models of convex rigid bodies rolling without sliding on a surface [41]. The study of systems with nonholonomic constraints goes back to the XIX century. The equations of motion were obtained applying either D’Alembert’s principle of virtual work or Gauss principle of least constraint. Recently, many authors have shown a new interest in that theory and also in its relation to the new developments in control theory and robotics using geometric techniques (see, for instance, [2, 3, 4, 8, 19, 22, 24]). Geometrically, nonholonomic constraints are globally described by a submanifold M of the velocity phase space TQ. If M is a vector subbundle of TQ, we are dealing with the case of linear constraints and, in the case M is an affine subbundle, we are in the case of affine constraints. Lagrange-D’Alembert’s or Chetaev’s principles allow us to determine the set of possible values of the constraint forces only from the set of admissible kinematic states, that is, from the constraint manifold M determined by the vanishing of the nonholonomic constraints φa. Therefore, assuming that the dynamical properties of the system are mathematically described by a Lagrangian function L : TQ −→ R and by a constraint submanifold M, the equations of motion, following Chetaev’s principle, are[ δqi = 0 , where δqi denotes the virtual displacements verifying δqi = 0. By using the Lagrange multiplier rule, we obtain that = λ̄a , (1.1) with the condition q̇(t) ∈ M, λ̄a being the Lagrange multipliers to be determined. Recently, J. Cortés et al [9] (see also [11, 38, 39]) proposed a unified framework for nonholonomic systems in the Lie algebroid setting that we will use along this paper generalizing some previous work for free Lagrangian mechanics on Lie algebroids (see, for instance, [23, 33, 34, 35]). The construction of geometric integrators for Equations (1.1) is very recent. In fact, in [37] appears as an open problem: ...The problem for the more general class of non-holonomic con- straints is still open, as is the question of the correct analogue of symplectic integration for non-holonomically constrained La- grangian systems... Numerical integrators derived from discrete variational principles have proved their adaptability to many situations: collisions, classical field theory, external forces...[28, 32] and it also seems very adequate for nonholonomic systems, since nonholonomic equations of motion come from Hölder’s variational principle which is not a stan- dard variational principle [1], but admits an adequate discretization. This is the procedure introduced by J. Cortés and S. Mart́ınez [8, 10] and followed by other authors [12, 14, 15, 36] extending, moreover, the results to nonholonomic systems defined on Lie groups (see also [25] for a different approach using generating func- tions). In this paper, we tackle the problem from the unifying point of view of Lie groupoids (see [9] for the continuous case). This technique permits to recover all the previous methods in the literature [10, 14, 36] and consider new cases of great 4 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ importance in nonholonomic dynamics. For instance, using action Lie groupoids, we may discretize LR-nonholonomic systems such as the Veselova system or us- ing Atiyah Lie groupoids we find discrete versions for the reduced equations of nonholonomic systems with symmetry. The paper is structured as follows. In section 2 we review some basic results on Lie algebroids and Lie groupoids. In particular, we describe the prolongation of a Lie groupoid [43], which has a double structure of Lie groupoid and Lie algebroid. Then, we briefly expose the geometric structure of discrete unconstrained mechanics on Lie groupoids: Poincaré-Cartan sections, Legendre transformations... The main results of the paper appear in section 3, where the geometric structure of discrete nonholonomic systems on Lie groupoids is considered. In particular, given a discrete Lagrangian Ld : Γ → R on a Lie groupoid Γ, a constraint distribution Dc in the Lie algebroid EΓ of Γ and a discrete constraint submanifold Mc in Γ, we obtain the nonholonomic discrete Euler-Lagrange equations from a discrete Generalized Hölder’s principle (see section 3.1). In addition, we characterize the regularity of the nonholonomic system in terms of the nonholonomic Legendre transformations and decompositions of the prolongation of the Lie groupoid. In the case when the system is regular, we can define the nonholonomic evolution operator. An interesting situation, studied in in Section 3.4, is that of reversible discrete nonholonomic Lagrangian systems, where the Lagrangian and the discrete constraint submanifold are invariants with respect to the inversion of the Lie groupoid. The particular example of reversible systems in the pair groupoid Q×Q was first studied in [36]. We also define the discrete nonholonomic momentum map. In order to give an idea of the breadth and flexibility of the proposed formalism, several examples are discussed, including their regularity and their reversibility: - Discrete holonomic Lagrangian systems on a Lie groupoid, which are a generalization of the Shake algorithm for holonomic systems [16, 20, 32]; - Discrete nonholonomic systems on the pair groupoid, recovering the equa- tions first considered in [10]. An explicit example of this situation is the discrete nonholonomic constrained particle. - Discrete nonholonomic systems on Lie groups, where the equations that are obtained are the so-called discrete Euler-Poincaré-Suslov equations (see [14]). We remark that, although our equations coincide with those in [14], the technique developed in this paper is different to the one in that paper. Two explicit examples which we describe here are the Suslov system and the Chaplygin sleigh. - Discrete nonholonomic Lagrangian systems on an action Lie groupoid. This example is quite interesting since it allows us to discretize a well- known nonholonomic LR-system: the Veselova system (see [44]; see also [13]). For this example, we obtain a discrete system that is not reversible and we show that the system is regular in a neighborhood around the manifold of units. - Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid. With this example, we are able to discretize reduced systems, in particular, we concentrate on the example of the discretization of the equations of motion of a rolling ball without sliding on a rotating table with constant angular velocity. - Discrete Chaplygin systems, which are regular systems (Ld,Mc,Dc) on the Lie groupoid Γ ⇒ M , for which (α, β) ◦ iMc : Mc → M × M is a diffeomorphism and ρ ◦ iDc : Dc → TM is an isomorphism of vector bundles, (α, β) being the source and target of the Lie groupoid Γ and ρ DISCRETE NONHOLONOMIC MECHANICS 5 being the anchor map of the Lie algebroid EΓ. This example includes a discretization of the two wheeled planar mobile robot. We conclude our paper with future lines of work. 2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids 2.1. Lie algebroids. A Lie algebroid E over a manifold M is a real vector bundle τ : E →M together with a Lie bracket [[·, ·]] on the space Sec(τ) of the global cross sections of τ : E → M and a bundle map ρ : E → TM , called the anchor map, such that if we also denote by ρ : Sec(τ) → X(M) the homomorphism of C∞(M)-modules induced by the anchor map then [[X, fY ]] = f [[X,Y ]] + ρ(X)(f)Y, (2.1) for X,Y ∈ Sec(τ) and f ∈ C∞(M) (see [26]). If (E, [[·, ·]], ρ) is a Lie algebroid over M then the anchor map ρ : Sec(τ) → X(M) is a homomorphism between the Lie algebras (Sec(τ), [[·, ·]]) and (X(M), [·, ·]). Moreover, one may define the differential d of E as follows: dµ(X0, . . . , Xk) = (−1)iρ(Xi)(µ(X0, . . . , X̂i, . . . , Xk)) (−1)i+jµ([[Xi, Xj ]], X0, . . . , X̂i, . . . , X̂j , . . . , Xk), (2.2) for µ ∈ Sec(∧kτ∗) and X0, . . . , Xk ∈ Sec(τ). d is a cohomology operator, that is, d2 = 0. In particular, if f : M −→ R is a real smooth function then df(X) = ρ(X)f, for X ∈ Sec(τ). Trivial examples of Lie algebroids are a real Lie algebra of finite dimension (in this case, the base space is a single point) and the tangent bundle of a manifold M. On the other hand, let (E, [[·, ·]], ρ) be a Lie algebroid of rank n over a manifold M of dimension m and π : P →M be a fibration. We consider the subset of E×TP TEP = { (a, v) ∈ E × TP | (Tπ)(v) = ρ(a) } , where Tπ : TP → TM is the tangent map to π. Denote by τπ : TEP → P the map given by τπ(a, v) = τP (v), τP : TP → P being the canonical projection. If dimP = p, one may prove that TEP is a vector bundle over P of rank n + p −m with vector bundle projection τπ : TEP → P . A section X̃ of τπ : TEP → P is said to be projectable if there exists a section X of τ : E →M and a vector field U ∈ X(P ) which is π-projectable to the vector field ρ(X) and such that X̃(p) = (X(π(p)), U(p)), for all p ∈ P . For such a projectable section X̃, we will use the following notation X̃ ≡ (X,U). It is easy to prove that one may choose a local basis of projectable sections of the space Sec(τπ). The vector bundle τπ : TEP → P admits a Lie algebroid structure ([[·, ·]]π, ρπ). Indeed, if (X,U) and (Y, V ) are projectable sections then [[(X,U), (Y, V )]]π = ([[X,Y ]], [U, V ]), ρπ(X,U) = U. (TEP, [[·, ·]]π, ρπ) is the E-tangent bundle to P or the prolongation of E over the fibration π : P →M (for more details, see [23]). Now, let (E, [[·, ·]], ρ) (resp., (E′, [[·, ·]]′, ρ′)) be a Lie algebroid over a manifold M (resp., M ′) and suppose that Ψ : E → E′ is a vector bundle morphism over the map Ψ0 : M →M ′. Then, the pair (Ψ,Ψ0) is said to be a Lie algebroid morphism if d((Ψ,Ψ0) ∗φ′) = (Ψ,Ψ0) ∗(d′φ′), for all φ′ ∈ Sec(∧k(τ ′)∗) and for all k, (2.3) 6 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ where d (resp., d′) is the differential of the Lie algebroid E (resp., E′) (see [23]). In the particular case when M = M ′ and Ψ0 = Id then (2.3) holds if and only if [[Ψ ◦X,Ψ ◦ Y ]]′ = Ψ[[X,Y ]], ρ′(ΨX) = ρ(X), for X,Y ∈ Sec(τ). 2.2. Lie groupoids. A Lie groupoid over a differentiable manifold M is a differ- entiable manifold Γ together with the following structural maps: • A pair of submersions α : Γ → M , the source, and β : Γ → M, the target. The maps α and β define the set of composable pairs Γ2 = { (g, h) ∈ G×G | β(g) = α(h) } . • A multiplication m : Γ2 → Γ, to be denoted simply by m(g, h) = gh, such that – α(gh) = α(g) and β(gh) = β(h). – g(hk) = (gh)k. • An identity section � : M → Γ such that – �(α(g))g = g and g�(β(g)) = g. • An inversion map i : Γ → Γ, to be simply denoted by i(g) = g−1, such – g−1g = �(β(g)) and gg−1 = �(α(g)). A Lie groupoid Γ over a set M will be simply denoted by the symbol Γ ⇒ M . On the other hand, if g ∈ Γ then the left-translation by g and the right- translation by g are the diffeomorphisms lg : α−1(β(g)) −→ α−1(α(g)) ; h −→ lg(h) = gh, rg : β−1(α(g)) −→ β−1(β(g)) ; h −→ rg(h) = hg. Note that l−1g = lg−1 and r g = rg−1 . A vector field X̃ on Γ is said to be left-invariant (resp., right-invariant) if it is tangent to the fibers of α (resp., β) and X̃(gh) = (Thlg)(X̃h) (resp., X̃(gh) = (Tgrh)(X̃(g))), for (g, h) ∈ Γ2. Now, we will recall the definition of the Lie algebroid associated with Γ. We consider the vector bundle τ : EΓ → M , whose fiber at a point x ∈ M is (EΓ)x = V�(x)α = Ker(T�(x)α). It is easy to prove that there exists a bijection between the space Sec(τ) and the set of left-invariant (resp., right-invariant) vector fields on Γ. If X is a section of τ : EΓ →M , the corresponding left-invariant (resp., right-invariant) vector field on Γ will be denoted X (resp., X ), where X (g) = (T�(β(g))lg)(X(β(g))), (2.4) X (g) = −(T�(α(g))rg)((T�(α(g))i)(X(α(g)))), (2.5) for g ∈ Γ. Using the above facts, we may introduce a Lie algebroid structure ([[·, ·]], ρ) on EΓ, which is defined by ←−−−− [[X,Y ]] = [ Y ], ρ(X)(x) = (T�(x)β)(X(x)), (2.6) for X,Y ∈ Sec(τ) and x ∈M . Note that −−−−→ [[X,Y ]] = −[ Y ], [ Y ] = 0, (2.7) (for more details, see [7, 26]). Given two Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′, a morphism of Lie groupoids is a smooth map Φ : Γ→ Γ′ such that (g, h) ∈ Γ2 =⇒ (Φ(g),Φ(h)) ∈ (Γ′)2 DISCRETE NONHOLONOMIC MECHANICS 7 Φ(gh) = Φ(g)Φ(h). A morphism of Lie groupoids Φ : Γ → Γ′ induces a smooth map Φ0 : M → M ′ in such a way that α′ ◦ Φ = Φ0 ◦ α, β′ ◦ Φ = Φ0 ◦ β, Φ ◦ � = �′ ◦ Φ0, α, β and � (resp., α′, β′ and �′) being the source, the target and the identity section of Γ (resp., Γ′). Suppose that (Φ,Φ0) is a morphism between the Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′ and that τ : EΓ → M (resp., τ ′ : EΓ′ → M ′) is the Lie algebroid of Γ (resp., Γ′). Then, if x ∈ M we may consider the linear map Ex(Φ) : (EΓ)x → (EΓ′)Φ0(x) defined by Ex(Φ)(v�(x)) = (T�(x)Φ)(v�(x)), for v�(x) ∈ (EΓ)x. (2.8) In fact, we have that the pair (E(Φ),Φ0) is a morphism between the Lie algebroids τ : EΓ →M and τ ′ : EΓ′ →M ′ (see [26]). Trivial examples of Lie groupoids are Lie groups and the pair or banal groupoid M ×M , M being an arbitrary smooth manifold. The Lie algebroid of a Lie group Γ is just the Lie algebra g of Γ. On the other hand, the Lie algebroid of the pair (or banal) groupoid M ×M is the tangent bundle TM to M . Apart from the Lie algebroid EΓ associated with a Lie groupoid Γ ⇒ M , other interesting Lie algebroids associated with Γ are the following ones: • The EΓ- tangent bundle to E∗Γ: Let TEΓE∗Γ be the EΓ-tangent bundle to E Γ, that is, E∗Γ = (vx, XΥx) ∈ (EΓ)x × TΥxE ∣∣ (TΥxτ∗)(XΥx) = (T�(x)β)(vx)} for Υx ∈ (E∗Γ)x, with x ∈M. As we know, T EΓE∗Γ is a Lie algebroid over E We may introduce the canonical section Θ of the vector bundle (TEΓE∗Γ) ∗ → E∗Γ as follows: Θ(Υx)(ax, XΥx) = Υx(ax), for Υx ∈ (E∗Γ)x and (ax, XΥx) ∈ T E∗Γ. Θ is called the Liouville section as- sociated with EΓ. Moreover, we define the canonical symplectic section Ω associated with EΓ by Ω = −dΘ, where d is the differential on the Lie algebroid TEΓE∗Γ → E Γ. It is easy to prove that Ω is nondegenerate and closed, that is, it is a symplectic section of TEΓE∗Γ (see [23]). Now, if Z is a section of τ : EΓ → M then there is a unique vector field Z∗c on E∗Γ, the complete lift of X to E Γ, satisfying the two following conditions: (i) Z∗c is τ∗-projectable on ρ(Z) and (ii) Z∗c(X̂) = ̂[[Z,X]] for X ∈ Sec(τ) (see [23]). Here, if X is a section of τ : EΓ → M then X̂ is the linear function X̂ ∈ C∞(E∗) defined by X̂(a∗) = a∗(X(τ∗(a∗))), for all a∗ ∈ E∗. Using the vector field Z∗c, one may introduce the complete lift Z∗c of Z as the section of τ τ : TEΓE∗Γ → E Γ defined by Z∗c(a∗) = (Z(τ∗(a∗)), Z∗c(a∗)), for a∗ ∈ E∗. (2.9) 8 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Z∗c is just the Hamiltonian section of Ẑ with respect to the canonical symplectic section Ω associated with EΓ. In other words, iZ∗cΩ = dẐ, (2.10) where d is the differential of the Lie algebroid τ τ : TEΓE∗Γ → E Γ (for more details, see [23]). • The Lie algebroid τ̃Γ : TΓΓ→ Γ : Let TΓΓ be the Whitney sum V β ⊕Γ V α of the vector bundles V β → Γ and V α → Γ, where V β (respectively, V α) is the vertical bundle of β (respectively, α). Then, the vector bundle τ̃Γ : TΓΓ ≡ V β ⊕Γ V α → Γ admits a Lie algebroid structure ([[·, ·]]T ΓΓ, ρT ΓΓ). The anchor map ρT ΓΓ is given by ΓΓ)(Xg, Yg) = Xg + Yg and the Lie bracket bracket [[·, ·]]T ΓΓ on the space Sec(τ̃Γ) is characterized for the following relation Y ), ( Y ′)]]T ΓΓ = (− −−−−−→ [[X,X ′]], ←−−−− [[Y, Y ′]]), for X,Y,X ′, Y ′ ∈ Sec(τ) (for more details, see [27]). On other hand, if X is a section of τ : EΓ → M , one may define the sections X(1,0), X(0,1) (the β and α-lifts) and X(1,1) (the complete lift) of X to τ̃Γ : TΓΓ→ Γ as follows: X(1,0)(g) = ( X (g), 0g), X (0,1)(g) = (0g, X (g)), and X(1,1)(g) = (− X (g), X (g)). We have that [[X(1,0), Y (1,0)]]T ΓΓ = −[[X,Y ]](1,0) [[X(0,1), Y (1,0)]]T ΓΓ = 0, [[X(0,1), Y (0,1)]]T ΓΓ = [[X,Y ]](0,1), and, as a consequence, [[X(1,1), Y (1,0)]]T ΓΓ = [[X,Y ]](1,0), [[X(1,1), Y (0,1)]]T ΓΓ = [[X,Y ]](0,1), [[X(1,1), Y (1,1)]]T ΓΓ = [[X,Y ]](1,1). Now, if g, h ∈ Γ one may introduce the linear monomorphisms (1,0)h : (EΓ) (TΓhΓ) ∗ ≡ V ∗h β ⊕ V h α and (0,1) g : (EΓ)∗β(g) → (T ∗ ≡ V ∗g β ⊕ V ∗g α given by (1,0) h (Xh, Yh) = γ(Th(i ◦ rh−1)(Xh)), (2.11) γ(0,1)g (Xg, Yg) = γ((Tglg−1)(Yg)), (2.12) for (Xg, Yg) ∈ TΓg Γ and (Xh, Yh) ∈ TΓhΓ. Thus, if µ is a section of τ∗ : E∗Γ → M , one may define the corresponding lifts µ(1,0) and µ(0,1) as the sections of τ̃Γ ∗ : (TΓΓ)∗ → Γ given by µ(1,0)(h) = µ(1,0)h , for h ∈ Γ, µ(0,1)(g) = µ(0,1)g , for g ∈ Γ. Note that if g ∈ Γ and {XA} (respectively, {YB}) is a local basis of Sec(τ) on an open subset U (respectively, V ) of M such that α(g) ∈ U (respectively, β(g) ∈ V ) then {X(1,0)A , Y (0,1) B } is a local basis of Sec(τ̃Γ) on the open subset α −1(U)∩β−1(V ). In addition, if {XA} (respectively, {Y B}) is the dual basis of {XA} (respectively, {YB}) then {(XA)(1,0), (Y B)(0,1)} is the dual basis of {X (1,0) A , Y (0,1) DISCRETE NONHOLONOMIC MECHANICS 9 2.3. Discrete Unconstrained Lagrangian Systems. (See [27] for details) A discrete unconstrained Lagrangian system on a Lie groupoid consists of a Lie groupoid Γ ⇒ M (the discrete space) and a discrete Lagrangian Ld : Γ→ 2.3.1. Discrete unconstrained Euler-Lagrange equations. An admissible sequence of order N on the Lie groupoid Γ is an element (g1, . . . , gN ) of ΓN ≡ Γ× · · · ×Γ such that (gk, gk+1) ∈ Γ2, for k = 1, . . . , N − 1. An admissible sequence (g1, . . . , gN ) of order N is a solution of the discrete unconstrained Euler-Lagrange equations for Ld if do[Ld ◦ lgk + Ld ◦ rgk+1 ◦ i](�(xk))|(EΓ)xk = 0 where β(gk) = α(gk+1) = xk and do is the standard differential on Γ, that is, the differential of the Lie algebroid τΓ : TΓ→ Γ (see [27]). The discrete unconstrained Euler-Lagrange operator DDELLd : Γ2 → E∗Γ is given by (DDELLd)(g, h) = d o[Ld ◦ lg + Ld ◦ rh ◦ i](�(x))|(EΓ)x = 0, for (g, h) ∈ Γ2, with β(g) = α(h) = x ∈M (see [27]). Thus, an admissible sequence (g1, . . . , gN ) of order N is a solution of the discrete unconstrained Euler-Lagrange equations if and only if (DDELLd)(gk, gk+1) = 0, for k = 1, . . . , N − 1. 2.3.2. Discrete Poincaré-Cartan sections. Consider the Lie algebroid τ̃Γ : TΓΓ ≡ V β ⊕Γ V α → Γ, and define the Poincaré-Cartan 1-sections Θ−Ld ,Θ Sec((τ̃Γ)∗) as follows Θ−Ld(g)(Xg, Yg) = −Xg(Ld), Θ (g)(Xg, Yg) = Yg(Ld), (2.13) for each g ∈ Γ and (Xg, Yg) ∈ TΓg Γ ≡ Vgβ ⊕ Vgα. Since dLd = Θ −Θ−Ld and so, using d 2 = 0, it follows that dΘ+Ld = dΘ . This means that there exists a unique 2-section ΩLd = −dΘ = −dΘ−Ld , which will be called the Poincaré-Cartan 2-section. This 2-section will be important to study the symplectic character of the discrete unconstrained Euler-Lagrange equations. If g is an element of Γ such that α(g) = x and β(g) = y and {XA} (respectively, {YB}) is a local basis of Sec(τ) on the open subset U (respectively, V ) of M , with x ∈ U (respectively, y ∈ V ), then on α−1(U) ∩ β−1(V ) we have that Θ−Ld = − XA(L)(XA)(1,0), Θ YB(L)(Y B)(0,1), ΩLd = − YB(Ld))(XA)(1,0) ∧ (Y B)(0,1) (2.14) where {XA} (respectively, {Y B}) is the dual basis of {XA} (respectively, {YB}) (for more details, see [27]). 2.3.3. Discrete unconstrained Lagrangian evolution operator. Let Υ : Γ → Γ be a smooth map such that: - graph(Υ) ⊆ Γ2, that is, (g,Υ(g)) ∈ Γ2, for all g ∈ Γ (Υ is a second order operator) and - (g,Υ(g)) is a solution of the discrete unconstrained Euler-Lagrange equa- tions, for all g ∈ Γ, that is, (DDELLd)(g,Υ(g)) = 0, for all g ∈ Γ. 10 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ In such a case ←− X (g)(Ld)− X (Υ(g))(Ld) = 0, (2.15) for every section X of τ : EΓ → M and every g ∈ Γ. The map Υ : Γ→ Γ is called a discrete flow or a discrete unconstrained Lagrangian evolution operator for Ld. Now, let Υ : Γ → Γ be a second order operator. Then, the prolongation TΥ : TΓΓ ≡ V β ⊕Γ V α → TΓΓ ≡ V β ⊕Γ V α of Υ is the Lie algebroid morphism over Υ : Γ→ Γ defined as follows (see [27]): TgΥ(Xg, Yg) = ((Tg(rgΥ(g) ◦ i))(Yg), (TgΥ)(Xg) +(TgΥ)(Yg)− Tg(rgΥ(g) ◦ i)(Yg)), (2.16) for all (Xg, Yg) ∈ TΓg Γ ≡ Vgβ ⊕ Vgα. Moreover, from (2.4), (2.5) and (2.16), we obtain that X (g), Y (g)) = (− Y (Υ(g)), (TgΥ)( X (g) + Y (g)) + Y (Υ(g))), (2.17) for all X,Y ∈ Sec(τ). Using (2.16), one may prove that (see [27]): (i) The map Υ is a discrete unconstrained Lagrangian evolution operator for Ld if and only if (TΥ,Υ)∗Θ = Θ+Ld . (ii) The map Υ is a discrete unconstrained Lagrangian evolution operator for Ld if and only if (TΥ,Υ)∗Θ −Θ−Ld = dLd. (iii) If Υ is discrete unconstrained Lagrangian evolution operator then (TΥ,Υ)∗ΩLd = ΩLd . 2.3.4. Discrete unconstrained Legendre transformations. Given a Lagrangian Ld : Γ → R we define the discrete unconstrained Legendre transformations F−Ld : Γ→ E∗Γ and F +Ld : Γ→ E∗Γ by (see [27]) (F−Ld)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ rh ◦ i), for v�(α(h)) ∈ (EΓ)α(h), (F+Ld)(g)(v�(β(g))) = v�(β(g))(Ld ◦ lg), for v�(β(g)) ∈ (EΓ)β(g). Now, we introduce the prolongations TΓF−Ld : TΓΓ ≡ V β ⊕Γ V α → TEΓE∗Γ and TΓF+Ld : TΓΓ ≡ V β ⊕Γ V α→ TEΓE∗Γ by −Ld(Xh, Yh) = (Th(i ◦ rh−1)(Xh), (ThF−Ld)(Xh) + (ThF−Ld)(Yh)),(2.18) TΓg F +Ld(Xg, Yg) = ((Tglg−1)(Yg), (TgF+Ld)(Xg) + (TgF+Ld)(Yg)), (2.19) for all h, g ∈ Γ and (Xh, Yh) ∈ TΓhΓ ≡ Vhβ ⊕ Vhα and (Xg, Yg) ∈ T g Γ ≡ Vgβ ⊕ Vgα (see [27]). We observe that the discrete Poincaré-Cartan 1-sections and 2- section are related to the canonical Liouville section of (TEΓE∗Γ) ∗ → E∗Γ and the canonical symplectic section of ∧2(TEΓE∗Γ) ∗ → E∗Γ by pull-back under the discrete unconstrained Legendre transformations, that is (see [27]), (TΓF−Ld,F−Ld)∗Θ = Θ−Ld , (T ΓF+Ld,F+Ld)∗Θ = Θ+Ld , (2.20) (TΓF−Ld,F−Ld)∗Ω = ΩLd , (T ΓF+Ld,F+Ld)∗Ω = ΩLd . (2.21) 2.3.5. Discrete regular Lagrangians. A discrete Lagrangian Ld : Γ→ R is said to be regular if the Poincaré-Cartan 2-section ΩLd is nondegenerate on the Lie algebroid τ̃Γ : TΓΓ ≡ V β ⊕Γ V α → Γ (see [27]). In [27], we obtained some necessary and sufficient conditions for a discrete Lagrangian on a Lie groupoid Γ to be regular that we summarize as follows: Ld is regular ⇐⇒ The Legendre transformation F+Ld is a local diffeomorphism ⇐⇒ The Legendre transformation F−Ld is a local diffeomorphism DISCRETE NONHOLONOMIC MECHANICS 11 Locally, we deduce that Ld is regular if and only if for every g ∈ Γ and every local basis {XA} (respectively, {YB}) of Sec(τ) on an open subset U (respectively, V ) of M such that α(g) ∈ U (respectively, β(g) ∈ V ) we have that the matrix Y B(Ld))) is regular on α−1(U) ∩ β−1(V ). Now, let Ld : Γ→ R be a discrete Lagrangian and g be a point of Γ. We define the R-bilinear map GLdg : (EΓ)α(g) ⊕ (EΓ)β(g) → R given by GLdg (a, b) = ΩLd(g)((−T�(α(g))(rg ◦ i)(a), 0), (0, (T�(β(g))lg)(b))). (2.22) Then, using (2.14), we have that Proposition 2.1. The discrete Lagrangian Ld : Γ → R is regular if and only if GLdg is nondegenerate, for all g ∈ Γ, that is, GLdg (a, b) = 0, for all b ∈ (EΓ)β(g) ⇒ a = 0 (respectively, GLdg (a, b) = 0, for all a ∈ (EΓ)α(g) ⇒ b = 0). On the other hand, if Ld : Γ → R is a discrete Lagrangian on a Lie groupoid Γ then we have that τ∗ ◦ F−Ld = α, τ∗ ◦ F+Ld = β, where τ∗ : E∗Γ → M is the vector bundle projection. Using these facts, (2.18) and (2.19), we deduce the following result. Proposition 2.2. Let Ld : Γ → R be a discrete Lagrangian function. Then, the following conditions are equivalent: (i) Ld is regular. (ii) The linear map TΓhF −Ld : Vhβ ⊕ Vhα → TEΓF−Ld(h)E Γ is a linear isomor- phism, for all h ∈ Γ. (iii) The linear map TΓg F +Ld : Vgβ ⊕ Vgα → TEΓF+Ld(g)EΓ ∗ is a linear isomor- phism, for all g ∈ Γ. Finally, let Ld : Γ→ R be a regular discrete Lagrangian function and (g0, h0) ∈ Γ×Γ be a solution of the discrete Euler-Lagrange equations for Ld. Then, one may prove (see [27]) that there exist two open subsets U0 and V0 of Γ, with g0 ∈ U0 and h0 ∈ V0, and there exists a (local) discrete unconstrained Lagrangian evolution operator ΥLd : U0 → V0 such that: (i) ΥLd(g0) = h0, (ii) ΥLd is a diffeomorphism and (iii) ΥLd is unique, that is, if U 0 is an open subset of Γ, with g0 ∈ U ′0, and Υ′Ld : U 0 → Γ is a (local) discrete Lagrangian evolution operator then ΥLd|U0∩U ′0 = Υ Ld|U0∩U ′0 3. Discrete Nonholonomic (or constrained) Lagrangian systems on Lie groupoids 3.1. Discrete Generalized Hölder’s principle. Let Γ be a Lie groupoid with structural maps α, β : Γ→M, � : M → Γ, i : Γ→ Γ, m : Γ2 → Γ. Denote by τ : EΓ → M the Lie algebroid associated to Γ. Suppose that the rank of EΓ is n and that the dimension of M is m. A generalized discrete nonholonomic (or constrained) Lagrangian system on Γ is determined by: 12 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ - a regular discrete Lagrangian Ld : Γ −→ R, - a constraint distribution, Dc, which is a vector subbundle of the bundle EΓ → M of admissible directions. We will denote by τDc : Dc → M the vector bundle projection and by iDc : Dc → EΓ the canonical inclusion. - a discrete constraint embedded submanifold Mc of Γ, such that dimMc = dimDc = m + r, with r ≤ n. We will denote by iMc : Mc → Γ the canonical inclusion. Remark 3.1. Let Ld : Γ→ R be a regular discrete Lagrangian on a Lie groupoid Γ and Mc be a submanifold of Γ such that �(M) ⊆ Mc. Then, dimMc = m + r, with 0 ≤ r ≤ m. Moreover, for every x ∈M , we may introduce the subspace Dc(x) of EΓ(x) given by Dc(x) = T�(x)Mc ∩ EΓ(x). Since the linear map T�(x)α : T�(x)Mc → TxM is an epimorphism, we deduce that dimDc(x) = r. In fact, Dc = x∈M Dc(x) is a vector subbundle of EΓ (over M) of rank r. Thus, we may consider the discrete nonholonomic system (Ld,Mc,Dc) on the Lie groupoid Γ. � For g ∈ Γ fixed, we consider the following set of admissible sequences of order CNg = (g1, . . . , gN ) ∈ ΓN ∣∣ (gk, gk+1) ∈ Γ2, for k = 1, .., N − 1 and g1 . . . gN = g } . Given a tangent vector at (g1, . . . , gN ) to the manifold CNg , we may write it as the tangent vector at t = 0 of a curve in CNg , t ∈ (−ε, ε) ⊆ R −→ c(t) which passes through (g1, . . . , gN ) at t = 0. This type of curves is of the form c(t) = (g1h1(t), h 1 (t)g2h2(t), . . . , h N−2(t)gN−1hN−1(t), h N−1(t)gN ) where hk(t) ∈ α−1(β(gk)), for all t, and hk(0) = �(β(gk)) for k = 1, . . . , N − 1. Therefore, we may identify the tangent space to CNg at (g1, . . . , gN ) with T(g1,g2,..,gN )C g ≡ { (v1, v2, . . . , vN−1) | vk ∈ (EΓ)xk and xk = β(gk), 1 ≤ k ≤ N − 1 } . Observe that each vk is the tangent vector to the curve hk at t = 0. The curve c is called a variation of (g1, . . . , gN ) and (v1, v2, . . . , vN−1) is called an infinitesimal variation of (g1, . . . , gN ). Now, we define the discrete action sum associated to the discrete Lagrangian Ld : Γ −→ R as SLd : CNg −→ R (g1, . . . , gN ) 7−→ Ld(gk). We define the variation δSLd : T(g1,...,gN )C g → R as δSLd(v1, . . . , vN−1) = SLd(c(t)) Ld(g1h1(t)) + Ld(h 1 (t)g2h2(t)) + . . .+ Ld(h N−2(t)gN−1hN−1(t)) + Ld(h N−1(t)gN ) do(Ld ◦ lgk)(�(xk))(vk) + d o(Ld ◦ rgk+1 ◦ i)(�(xk))(vk) where do is the standard differential on Γ, i.e., do is the differential of the Lie algebroid τΓ : TΓ → Γ. It is obvious from the last expression that the definition DISCRETE NONHOLONOMIC MECHANICS 13 of variation δSLd does not depend on the choice of variations c of the sequence g whose infinitesimal variation is (v1, . . . , vN−1). Next, we will introduce the subset (Vc)g of T(g1,...,gN )C g defined by (Vc)g = (v1, . . . , vN−1) ∈ T(g1,...,gN )C ∣∣ ∀k ∈ {1, . . . , N − 1}, vk ∈ Dc } . Then, we will say that a sequence in CNg satisfying the constraints determined by Mc is a Hölder-critical point of the discrete action sum SLd if the restriction of δSLd to (Vc)g vanishes, i.e. (Vc)g Definition 3.2 (Discrete Hölder’s principle). Given g ∈ Γ, a sequence (g1, . . . , gN ) ∈ CNg such that gk ∈ Mc, 1 ≤ k ≤ N , is a solution of the discrete nonholo- nomic Lagrangian system determined by (Ld,Mc,Dc) if and only if (g1, . . . , gN ) is a Hölder-critical point of SLd. If (g1, . . . , gN ) ∈ CNg ∩ (Mc × · · · ×Mc) then (g1, . . . , gN ) is a solution of the nonholonomic discrete Lagrangian system if and only if (do(Ld ◦ lgk) + d o(Ld ◦ rgk+1 ◦ i))(�(xk))|(Dc)xk = 0, where β(gk) = α(gk+1) = xk. For N = 2, we obtain that (g, h) ∈ Γ2 ∩ (Mc ×Mc) (with β(g) = α(h) = x) is a solution if do(Ld ◦ lg + Ld ◦ rh ◦ i)(�(x))|(Dc)x = 0. These equations will be called the discrete nonholonomic Euler-Lagrange equations for the system (Ld,Mc,Dc). Let (g1, . . . , gN ) be an element of CNg . Suppose that β(gk) = α(gk+1) = xk, 1 ≤ k ≤ N − 1, and that {XAk} = {Xak, Xαk} is a local adapted basis of Sec(τ) on an open subset Uk of M , with xk ∈ Uk. Here, {Xak}1≤a≤r is a local basis of Sec(τDc) and, thus, {Xαk}r+1≤α≤n is a local basis of the space of sections of the vector subbundle τD0c : D c →M , where D0c is the annihilator of Dc and {Xak, Xαk} is the dual basis of {Xak, Xαk}. Then, the sequence (g1, . . . , gN ) is a solution of the discrete nonholonomic equations if (g1, . . . , gN ) ∈Mc×· · ·×Mc and it satisfies the following closed system of difference equations gk)(Ld)− gk+1)(Ld) 〈dLd, (Xak)(0,1)〉(gk)− 〈dLd, (Xak)(1,0)〉(gk+1) for 1 ≤ a ≤ r, d being the differential of the Lie algebroid πτ : TΓΓ ≡ V β⊕ΓV α −→ Γ. For N = 2 we obtain that (g, h) ∈ Γ2 ∩ (Mc ×Mc) (with β(g) = α(h) = x) is a solution if Xa(g)(Ld)− Xa(h)(Ld) = 0 where {Xa} is a local basis of Sec(τDc) on an open subset U of M such that x ∈ U . Next, we describe an alternative version of these difference equations. First observe that using the Lagrange multipliers the discrete nonholonomic equations are rewritten as do [Ld ◦ lg + Ld ◦ rh ◦ i] (�(x))(v) = λαXα(x)(v), 14 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ for v ∈ (EΓ)x, with (g, h) ∈ Γ2 ∩ (Mc ×Mc) and β(g) = α(h) = x. Here, {Xα} is a local basis of sections of the annihilator D0c . Thus, the discrete nonholonomic equations are: Y (g)(Ld)− Y (h)(Ld) = λα(X α)(Y )|β(g), (g, h) ∈ Γ2 ∩ (Mc ×Mc), for all Y ∈ Sec(τ) or, alternatively, 〈dLd − λα(Xα)(0,1), Y (0,1)〉(g)− 〈dLd, Y (1,0)〉(h) = 0, (g, h) ∈ Γ2 ∩ (Mc ×Mc), for all Y ∈ Sec(τ). On the other hand, we may define the discrete nonholonomic Euler-Lagrange operator DDEL(Ld,Mc,Dc) : Γ2 ∩ (Mc ×Mc)→ D∗c as follows DDEL(Ld,Mc,Dc)(g, h) = d o [Ld ◦ lg + Ld ◦ rh ◦ i] (�(x))|(Dc)x , for (g, h) ∈ Γ2 ∩ (Mc ×Mc), with β(g) = α(h) = x ∈M . Then, we may characterize the solutions of the discrete nonholonomic equations as the sequences (g1, . . . , gN ), with (gk, gk+1) ∈ Γ2 ∩ (Mc × Mc), for each k ∈ {1, . . . , N − 1}, and DDEL(Ld,Mc,Dc)(gk, gk+1) = 0. Remark 3.3. (i) The set Γ2 ∩ (Mc ×Mc) is not, in general, a submanifold of Mc ×Mc. (ii) Suppose that αMc : Mc → M and βMc : Mc → M are the restrictions to Mc of α : Γ → M and β : Γ → M , respectively. If αMc and βMc are submersions then Γ2∩(Mc×Mc) is a submanifold of Mc×Mc of dimension m+ 2r. 3.2. Discrete Nonholonomic Legendre transformations. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system. We define the discrete nonholo- nomic Legendre transformations F−(Ld,Mc,Dc) : Mc → D∗c and F +(Ld,Mc,Dc) : Mc → D∗c as follows: F−(Ld,Mc,Dc)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ rh ◦ i), for v�(α(h)) ∈ Dc(α(h)),(3.1) F+(Ld,Mc,Dc)(g)(v�(β(g))) = v�(β(g))(Ld ◦ lg), for v�(β(g)) ∈ Dc(β(g)).(3.2) If F−Ld : Γ→ E∗Γ and F +Ld : Γ→ E∗Γ are the standard Legendre transformations associated with the Lagrangian function Ld and i∗Dc : E Γ → D c is the dual map of the canonical inclusion iDc : Dc → EΓ then F−(Ld,Mc,Dc) = i∗Dc ◦ F −Ld ◦ iMc , F +(Ld,Mc,Dc) = i ◦ F+Ld ◦ iMc . (3.3) Remark 3.4. (i) Note that τ∗Dc ◦ F −(Ld,Mc,Dc) = αMc , τ ◦ F+(Ld,Mc,Dc) = βMc . (3.4) (ii) If DDEL(Ld,Mc,Dc) is the discrete nonholonomic Euler-Lagrange opera- tor then DDEL(Ld,Mc,Dc)(g, h) = F+(Ld,Mc,Dc)(g)− F−(Ld,Mc,Dc)(h), for (g, h) ∈ Γ2 ∩ (Mc ×Mc). DISCRETE NONHOLONOMIC MECHANICS 15 On the other hand, since by assumption Ld : Γ → R is a regular discrete La- grangian function, we have that the discrete Poincaré-Cartan 2-section ΩLd is sym- plectic on the Lie algebroid τ̃Γ : TΓΓ → Γ. Moreover, the regularity of L is equiv- alent to the fact that the Legendre transformations F−Ld and F+Ld to be local diffeomorphisms (see Subsection 2.3.5). Next, we will obtain necessary and sufficient conditions for the discrete non- holonomic Legendre transformations associated with the system (Ld,Mc,Dc) to be local diffeomorphisms. Let F be the vector subbundle (over Γ) of τ̃Γ : TΓΓ→ Γ whose fiber at the point h ∈ Γ is (1,0) ∣∣∣ γ ∈ Dc(α(h))0 }0 ⊆ TΓhΓ. In other words, F 0h = (1,0) ∣∣∣ γ ∈ Dc(α(h))0 } . Note that the rank of F is n+ r. We also consider the vector subbundle F̄ (over Γ) of τ̃Γ : TΓΓ→ Γ of rank n+ r whose fiber at the point g ∈ Γ is F̄g = γ(0,1)g ∣∣∣ γ ∈ Dc(β(g))0 }0 ⊆ TΓg Γ. Lemma 3.5. F (respectively, F̄ ) is a coisotropic vector subbundle of the symplectic vector bundle (TΓΓ,ΩLd), that is, F⊥h ⊆ Fh, for every h ∈ Γ (respectively, F̄⊥g ⊆ F̄g, for every g ∈ Γ), where F⊥h (respectively, F̄ g ) is the symplectic orthogonal of Fh (respectively, F̄g) in the symplectic vector space (TΓhΓ, ΩLd(h)) (respectively, (T g Γ,ΩLd(g))). Proof. If h ∈ Γ we have that F⊥h = [ ΩLd (h) (F 0h ), [ΩLd (h) : TΓhΓ→ (T ∗ being the canonical isomorphism induced by the symplectic form ΩLd(h). Thus, using (2.14), we deduce that F⊥h = ΩLd (h) (γ(1,0)h ) ∣∣∣ γ ∈ Dc(α(h))0 } ⊆ {0} ⊕ Vhα ⊆ Fh. The coisotropic character of F̄g is proved in a similar way. � We also have the following result Lemma 3.6. Let TΓF−Ld : TΓΓ→ TEΓE∗Γ (respectively, T ΓF+Ld : TΓΓ→ TEΓE∗Γ) be the prolongation of the Legendre transformation F−Ld : Γ → E∗Γ (respectively, F+Ld : Γ→ E∗Γ). Then, (TΓhF −Ld)(Fh) = T F−Ld(h) E∗Γ = (vα(h), XF−Ld(h)) ∈ T F−Ld(h) ∣∣∣ vα(h) ∈ Dc(α(h))} , for h ∈Mc (respectively, (TΓg F +Ld)(F̄g) = T F+Ld(g) E∗Γ = (vβ(g), XF+Ld(g)) ∈ T F+Ld(g) ∣∣∣ vβ(g) ∈ Dc(β(g))} , for g ∈Mc). Proof. It follows using (2.11), (2.18) (respectively, (2.12), (2.19)) and Proposition 2.2. � Now, we may prove the following theorem. 16 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Theorem 3.7. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system. Then, the following conditions are equivalent: (i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) (re- spectively, F+(Ld,Mc,Dc)) is a local diffeomorphism. (ii) For every h ∈Mc (respectively, g ∈Mc) ΓΓ)−1(ThMc) ∩ F⊥h = {0} (3.5) (respectively, (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}). Proof. (i) ⇒ (ii) If h ∈ Mc and (Xh, Yh) ∈ (ρT ΓΓ)−1(ThMc) ∩ F⊥h then, using the fact that F⊥h ⊆ {0} ⊕ Vhα (see the proof of Lemma 3.5), we have that Xh = 0. Therefore, Yh ∈ Vhα ∩ ThMc. (3.6) Next, we will see that (ThF−(Ld,Mc,Dc))(Yh) = 0. (3.7) From (3.4) and (3.6), it follows that (ThF−(Ld,Mc,Dc))(Yh) is vertical with respect to the projection τ∗Dc : D c →M . Thus, it is sufficient to prove that ((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = 0, for all Z ∈ Sec(τDc). Here, Ẑ : D∗c → R is the linear function on D∗c induced by the section Z. Now, using (3.3), we deduce that ((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = d(Ẑ ◦ i∗Dc)((F −Ld)(h))(0, (ThF−Ld)(Yh)), where d is the differential of the Lie algebroid τ τ : TEΓE∗Γ → E Consequently, if Z∗c : E∗Γ → T EΓE∗Γ is the complete lift of Z ∈ Sec(τ), we have that (see (2.10)), ((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = Ω(F−Ld(h))(Z∗c(F−Ld(h)), (0, (ThF−Ld)(Yh)), (3.8) Ω being the canonical symplectic section associated with the Lie algebroid EΓ. On the other hand, since Z ∈ Sec(τDc), it follows that Z∗c(F−Ld(h)) is in F−Ld(h) E∗Γ and, from Lemma 3.6, we conclude that there exists (X h) ∈ Fh such that (TΓhF −Ld)(X h) = Z ∗c((F−Ld)(h)). (3.9) Moreover, using (2.18), we obtain that (TΓhF −Ld)(0, Yh) = (0, (ThF−Ld)(Yh)). (3.10) Thus, from (2.21), (3.8), (3.9) and (3.10), we deduce that ((ThF−(Ld,M,Dc))(Yh))(Ẑ) = −ΩLd(h)((0, Yh), (X Therefore, since (0, Yh) ∈ F⊥h , it follows that (3.7) holds, which implies that Yh = 0. This proves that (ρT ΓΓ)−1(ThMc) ∩ F⊥h = {0}. If F+(Ld,Mc,Dc) is a local diffeomorphism then, proceeding as above, we have that (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc. (ii) ⇒ (i) Suppose that h ∈Mc and that Yh is a tangent vector to Mc at h such (ThF−(Ld,Mc,Dc))(Yh) = 0. (3.11) DISCRETE NONHOLONOMIC MECHANICS 17 We have that (Thα)(Yh) = 0 and, thus, (0, Yh) ∈ (ρT ΓΓ)−1(ThMc). We will see that (0, Yh) ∈ F⊥h , that is, ΩLd(h)((0, Yh), (X h)) = 0, for (X h) ∈ Fh. (3.12) Now, using (2.18) and (2.21), we deduce that ΩLd(h)((0, Yh), (X h)) = Ω(F −Ld(h))((0, (ThF−Ld)(Yh)), (TΓhF −Ld)(X Therefore, from Lemma 3.6, we obtain that ΩLd(h)((0, Yh), (X h)) = Ω(F −Ld(h))(0, (ThF−Ld)(Yh)), (vα(h), YF−Ld(h))) with (vα(h), YF−Ld(h)) ∈ T F−Ld(h) Next, we take a section Z ∈ Sec(τDc) such that Z(α(h)) = vα(h). Then (see (2.9)), (vα(h), YF−Ld(h)) = Z ∗c(F−Ld(h)) + (0, Y ′F−Ld(h)), where Y ′F−Ld(h) ∈ TF−Ld(h)E Γ and Y F−Ld(h) is vertical with respect to the projection τ∗ : E∗Γ →M . Thus, since (see Eq. (3.7) in [23]) Ω(F−Ld(h))((0, (ThF−Ld)(Yh)), (0, Y ′F−Ld(h))) = 0, we have that ΩLd(h)((0, Yh), (X h)) = −Ω(F −Ld(h))(Z∗c(F−Ld(h)), (0, (ThF−Ld)(Yh))) = −d(Ẑ ◦ i∗Dc)(F −Ld(h))(0, (ThF−Ld)(Yh)) and, from (3.11), we deduce that (3.12) holds. This proves that Yh ∈ (ρT ΓΓ)−1(ThMc) ∩ F⊥h which implies that Yh = 0. Therefore, F−(Ld,Mc,Dc) is a local diffeomorphism. If (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0} for all g ∈ Mc then, proceeding as above, we obtain that F+(Ld,Mc,Dc) is a local diffeomorphism. � Now, let ρT ΓΓ : TΓΓ→ TΓ be the anchor map of the Lie algebroid πτ : TΓΓ→ Γ. Then, we will denote by Hh the subspace of TΓhΓ given by Hh = (ρ TΓΓ)−1(ThMc) ∩ Fh, for h ∈Mc. In a similar way, for every g ∈Mc we will introduce the subspace H̄g of TΓg Γ defined H̄g = (ρ TΓΓ)−1(TgMc) ∩ F̄g. On the other hand, let h be a point of Mc and G h : (EΓ)α(h) ⊕ (EΓ)β(h) → R be the R-bilinear map given by (2.22). We will denote by ( h the subspace of (EΓ)β(h) defined by b ∈ (EΓ)β(h) ∣∣ (T�(β(h))lh)(b) ∈ ThMc } and by GLdch : (Dc)α(h)× ( h → R the restriction to (Dc)α(h)× ( h of the R-bilinear map GLdh . In a similar way, if g is a point of Γ we will consider the subspace ( E Γ)Mcg of (EΓ)α(g) defined by a ∈ (EΓ)α(g) ∣∣ (T�(α(g))(rg ◦ i))(a) ∈ TgMc } 18 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ and the restriction ḠLdcg : ( E Γ)Mcg × (Dc)β(g) → R of GLdg to the space ( E Γ)Mcg × (Dc)β(g). Proposition 3.8. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system. Then, the following conditions are equivalent: (i) For every h ∈Mc (respectively, g ∈Mc) ΓΓ)−1(ThMc) ∩ F⊥h = {0} (respectively, (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}). (ii) For every h ∈ Mc (respectively, g ∈ Mc) the dimension of the vector sub- space Hh (respectively, H̄g) is 2r and the restriction to the vector subbundle H (respectively, H̄) of the Poincaré-Cartan 2-section ΩLd is nondegener- (iii) For every h ∈Mc (respectively, g ∈Mc){ b ∈ ( ∣∣∣ GLdch (a, b) = 0,∀a ∈ (Dc)α(h) } = {0} (respectively, a ∈ ( E Γ)Mcg ∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}). Proof. (i) ⇒ (ii) Assume that h ∈Mc and that ΓΓ)−1(ThMc) ∩ F⊥h = {0}. (3.13) Let U be an open subset of Γ, with h ∈ U , and {φγ}γ=1,...,n−r a set of independent real C∞-functions on U such that Mc ∩ U = {h′ ∈ U | φγ(h′) = 0, for all γ } . If d is the differential of the Lie algebroid τ̃Γ : TΓΓ → Γ then it is easy to prove ΓΓ)−1(ThMc) =< {dφγ(h)} >0 . Thus, dim((ρT ΓΓ)−1(ThMc)) ≥ n+ r. (3.14) On the other hand, dimF⊥h = n−r. Therefore, from (3.13) and (3.14), we obtain dim((ρT ΓΓ)−1(ThMc)) = n+ r TΓhΓ = (ρ TΓΓ)−1(ThMc)⊕ F⊥h . Consequently, using Lemma 3.5, we deduce that Fh = Hh ⊕ F⊥h . (3.15) This implies that dimHh = 2r. Moreover, from (3.15), we also get that Hh ∩H⊥h ⊆ Hh ∩ F and, since Hh ∩ F⊥h = (ρ TΓΓ)−1(ThMc) ∩ F⊥h (see Lemma 3.5), it follows that Hh ∩H⊥h = {0}. Thus, we have proved that Hh is a symplectic subspace of the symplectic vector space (TΓhΓ,ΩLd(h)). If (ρT ΓΓ)−1(TgMc)∩ F̄⊥g = {0}, for all g ∈Mc then, proceeding as above, we ob- tain that H̄g is a symplectic subspace of the symplectic vector space (TΓg Γ,ΩLd(g)), for all g ∈Mc. (ii) ⇒ (i) Suppose that h ∈ Mc and that Hh is a symplectic subspace of the symplectic vector space (TΓhΓ,ΩLd(h)). DISCRETE NONHOLONOMIC MECHANICS 19 If (Xh, Yh) ∈ (ρT ΓΓ)−1(ThMc) ∩ F⊥h then, using Lemma 3.5, we deduce that (Xh, Yh) ∈ Hh. Now, if (X ′h, Y h) ∈ Hh then, since (Xh, Yh) ∈ F h , we conclude that ΩLd(h)((Xh, Yh), (X h)) = 0. This implies that (Xh, Yh) ∈ Hh ∩H⊥h = {0}. Therefore, we have proved that (ρT ΓΓ)−1(ThMc) ∩ F⊥h = {0}. If H̄g ∩ H̄⊥g = {0}, for all g ∈ Mc then, proceeding as above, we obtain that ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc. (i) ⇒ (iii) Assume that ΓΓ)−1(ThMc) ∩ F⊥h = {0} and that b ∈ ( h satisfies the following condition h (a, b) = 0, ∀a ∈ (Dc)α(h). Then, Yh = (T�(β(h))lh)(b) ∈ ThMc ∩ Vhα and (0, Yh) ∈ (ρT ΓΓ)−1(ThMc). Moreover, if (X ′h, Y h) ∈ Fh, we have that X ′h = −(T�(α(h))(rh ◦ i))(a), with a ∈ (Dc)α(h). Thus, using (2.14) and (2.22), we deduce that ΩLd(h)((X h), (0, Yh)) = ΩLd(h)((X h, 0), (0, Yh)) = G h (a, b) = 0. Therefore, (0, Yh) ∈ (ρT ΓΓ)−1(ThMc) ∩ F⊥h = {0}, which implies that b = 0. If (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈ Mc then, proceeding as above, we obtain that{ a ∈ ( ∣∣∣ GLdcg (a, b) = 0, for all b ∈ (Dc)β(g) } = {0}. (iii) ⇒ (i) Suppose that h ∈Mc, that{ b ∈ ( ∣∣∣ GLdh (a, b) = 0, ∀a ∈ (Dc)α(h) } = {0} and let (Xh, Yh) be an element of the set (ρT ΓΓ)−1(ThMc) ∩ F⊥h . Then (see the proof of Lemma 3.5), Xh = 0 and Yh ∈ ThMc∩Vhα. Consequently, Yh = (T�(β(h)lh)(b), with b ∈ ( Now, if a ∈ (Dc)α(h), we have that X ′h = (T�(α(h))(rh ◦ i))(a) ∈ Vhβ and (X h, 0) ∈ Fh. Thus, from (2.22) and since (0, Yh) ∈ F⊥h , it follows that h (a, b) = ΩLd(h)((X h, 0)(0, Yh)) = 0. Therefore, b = 0 which implies that Yh = 0. a ∈ ( E Γ)Mcg ∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}, for all g ∈ Mc, then proceeding as above we obtain that (ρT ΓΓ)−1(TgMc)∩F̄⊥g = {0}, for all g ∈Mc. � Using Theorem 3.7 and Proposition 3.8, we conclude 20 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Theorem 3.9. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system. Then, the following conditions are equivalent: (i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) (re- spectively, F+(Ld,Mc,Dc)) is a local diffeomorphism. (ii) For every h ∈Mc (respectively, g ∈Mc) ΓΓ)−1(ThMc) ∩ F⊥h = {0} (respectively, (ρT ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}). (iii) For every h ∈ Mc (respectively, g ∈ Mc) the dimension of the vector sub- space Hh (respectively, H̄g) is 2r and the restriction to the vector subbundle H (respectively, H̄) of the Poincaré-Cartan 2-section ΩLd is nondegener- (iv) For every h ∈Mc (respectively, g ∈Mc){ b ∈ ( ∣∣∣ GLdch (a, b) = 0,∀a ∈ (Dc)α(h) } = {0} (respectively, a ∈ ( E Γ)Mcg ∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}). 3.3. Nonholonomic evolution operators and regular discrete nonholo- nomic Lagrangian systems. First of all, we will introduce the definition of a nonholonomic evolution operator. Definition 3.10. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system and Υnh : Mc →Mc be a differentiable map. Υnh is said to be a discrete nonholo- nomic evolution operator for (Ld,Mc,Dc) if: (i) graph(Υnh) ⊆ Γ2, that is, (g,Υnh(g)) ∈ Γ2, for all g ∈Mc and (ii) (g,Υnh(g)) is a solution of the discrete nonholonomic equations, for all g ∈Mc, that is, do(Ld ◦ lg + Ld ◦ rΥnh(g) ◦ i)(�(β(g)))|Dc(β(g)) = 0, for all g ∈Mc. Remark 3.11. If Υnh : Mc → Mc is a differentiable map then, from (3.1), (3.2) and (3.4), we deduce that Υnh is a discrete nonholonomic evolution operator for (Ld,Mc,Dc) if and only if F−(Ld,Mc,Dc) ◦Υnh = F+(Ld,Mc,Dc). Now, we will introduce the notion of a regular discrete nonholonomic Lagrangian system. Definition 3.12. A discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is said to be regular if the discrete nonholonomic Legendre transformations F−(Ld,Mc,Dc) and F+(Ld,Mc,Dc) are local diffeomorphims. From Theorem 3.9, we deduce Corollary 3.13. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system. Then, the following conditions are equivalent: (i) The system (Ld,Mc,Dc) is regular. (ii) The following relations hold ΓΓ)−1(ThMc) ∩ F⊥h = {0}, for all h ∈Mc, ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc. DISCRETE NONHOLONOMIC MECHANICS 21 (iii) H and H̄ are symplectic subbundles of rank 2r of the symplectic vector bundle (TΓMcΓ,ΩLd). (iv) If g and h are points of Mc then the R-bilinear maps GLdch and Ḡ g are right and left nondegenerate, respectively. The map GLdch (respectively, Ḡ g ) is right nondegenerate (respectively, left non- degenerate) if h (a, b) = 0,∀a ∈ (Dc)α(h) ⇒ b = 0 (respectively, ḠLdcg (a, b) = 0,∀b ∈ (Dc)β(g) ⇒ a = 0). Every solution of the discrete nonholonomic equations for a regular discrete nonholonomic Lagrangian system determines a unique local discrete nonholonomic evolution operator. More precisely, we may prove the following result: Theorem 3.14. Let (Ld,Mc,Dc) be a regular discrete nonholonomic Lagrangian system and (g0, h0) ∈Mc×Mc be a solution of the discrete nonholonomic equations for (Ld,Mc,Dc). Then, there exist two open subsets U0 and V0 of Γ, with g0 ∈ U0 and h0 ∈ V0, and there exists a local discrete nonholonomic evolution operator Υ(Ld,Mc,Dc)nh : U0 ∩Mc → V0 ∩Mc such that: (i) Υ(Ld,Mc,Dc)nh (g0) = h0; (ii) Υ(Ld,Mc,Dc)nh is a diffeomorphism and (iii) Υ(Ld,Mc,Dc)nh is unique, that is, if U 0 is an open subset of Γ, with g0 ∈ U ′0, and Υnh : U ′0 ∩ Mc → Mc is a (local) discrete nonholonomic evolution operator then (Υ(Ld,Mc,Dc)nh )|U0∩U ′0∩Mc = (Υnh)|U0∩U ′0∩Mc . Proof. From remark 3.4, we deduce that F+(Ld,Mc,Dc)(g0) = F−(Ld,Mc,Dc)(h0) = µ0 ∈ D∗c . Thus, we can choose two open subsets U0 and V0 of Γ, with g0 ∈ U0 and h0 ∈ V0, and an open subset W0 of E∗Γ such that µ0 ∈W0 and F+(Ld,Mc,Dc) : U0 ∩Mc →W0 ∩D∗c , F −(Ld,Mc,Dc) : V0 ∩Mc →W0 ∩D∗c are diffeomorphisms. Therefore, from Remark 3.11, we deduce that Υ(Ld,Mc,Dc)nh = (F −(Ld,Mc,Dc) −1 ◦ F+(Ld,Mc,Dc))|U0∩Mc : U0 ∩Mc → V0 ∩Mc is a (local) discrete nonholonomic evolution operator. Moreover, it is clear that Υ(Ld,Mc,Dc)nh (g0) = h0 and it follows that Υ (Ld,Mc,Dc) nh is a diffeomorphism. Finally, if U ′0 is an open subset of Γ, with g0 ∈ U ′0, and Υnh : U ′0 ∩Mc → Mc is another (local) discrete nonholonomic evolution operator then (Υnh)|U0∩U ′0∩Mc is also a (local) discrete nonholonomic evolution operator. Consequently, from Remark 3.11, we conclude that (Υnh)|U0∩U ′0∩Mc = [F −(Ld,Mc,Dc)−1 ◦ F+(Ld,Mc,Dc)]|U0∩U ′0∩Mc = (Υ(Ld,Mc,Dc)nh )|U0∩U ′0∩Mc . 22 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ 3.4. Reversible discrete nonholonomic Lagrangian systems. Let (Ld,Mc, Dc) be a discrete nonholonomic Lagrangian system on a Lie groupoid Γ ⇒ M . Following the terminology used in [36] for the particular case when Γ is the pair groupoid M ×M , we will introduce the following definition Definition 3.15. The discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is said to be reversible if Ld ◦ i = Ld, i(Mc) = Mc, i : Γ→ Γ being the inversion of the Lie groupoid Γ. For a reversible discrete nonholonomic Lagrangian system we have the following result: Proposition 3.16. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian sys- tem on a Lie groupoid Γ. Then, the following conditions are equivalent: (i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) is a local diffeomorphism. (ii) The discrete nonholonomic Legendre transformation F+(Ld,Mc,Dc) is a local diffeomorphism. Proof. If h ∈Mc then, using (3.1) and the fact that Ld ◦ i = Ld, it follows that F−(Ld,Mc,Dc)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ l−1h ) for v�(α(h)) ∈ (Dc)α(h). Thus, from (3.2), we obtain that F−(Ld,Mc,Dc)(h)(v�(α(h))) = −F+(Ld,Mc,Dc)(h−1)(v�(β(h−1))). This implies that F+(Ld,Mc,Dc) = −F−(Ld,Mc,Dc) ◦ i. Therefore, since the inversion is a diffeomorphism (in fact, we have that i2 = id), we deduce the result � Using Theorem 3.9, Definition 3.12 and Proposition 3.16, we prove the following corollaries. Corollary 3.17. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian system on a Lie groupoid Γ. Then, the following conditions are equivalent: (i) The system (Ld,Mc,Dc) is regular. (ii) For all h ∈Mc, ΓΓ)−1(ThMc) ∩ F⊥h = {0}. (iii) H = (ρT ΓΓ)−1(TMc)∩F is a symplectic subbundle of the symplectic vector bundle (TΓMcΓ,ΩLd). (iv) The R-bilinear map GLdch : ( h ×(Dc)α(h) → R is right nondegenerate, for all h ∈Mc. Corollary 3.18. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian system on a Lie groupoid Γ. Then, the following conditions are equivalent: (i) The system (Ld,Mc,Dc) is regular. (ii) For all g ∈Mc, ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}. (iii) H̄ = (ρT ΓΓ)−1(TMc)∩ F̄ is a symplectic subbundle of the symplectic vector bundle (TΓMcΓ,ΩLd). DISCRETE NONHOLONOMIC MECHANICS 23 (iv) The R-bilinear map ḠLdcg : (Dc)β(g) × ( E Γ)Mcg → R is left nondegenerate, for all g ∈Mc. Next, we will prove that a reversible nonholonomic Lagrangian system is dynam- ically reversible. Proposition 3.19. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian sys- tem on a Lie groupoid Γ and (g, h) be a solution of the discrete nonholonomic Euler- Lagrange equations for (Ld,Mc,Dc). Then, (h−1, g−1) is also a solution of these equations. In particular, if the system (Ld,Mc,Dc) is regular and Υ (Ld,Mc,Dc) nh is the (local) discrete nonholonomic evolution operator for (Ld,Mc,Dc) then Υ (Ld,Mc,Dc) is reversible, that is, Υ(Ld,Mc,Dc)nh ◦ i ◦Υ (Ld,Mc,Dc) nh = i. Proof. Using that i(Mc) = Mc, we deduce that (h−1, g−1) ∈ Γ2 ∩ (Mc ×Mc). Now, suppose that β(g) = α(h) = x and that v ∈ (Dc)x. Then, since Ld ◦ i = Ld, it follows that do[Ld ◦ lh−1 + Ld ◦ rg−1 ◦ i](ε(x))(v) = v(Ld ◦ i ◦ rh ◦ i) + v(Ld ◦ i ◦ lg) = v(Ld ◦ lg) + v(Ld ◦ rh ◦ i) = 0. Thus, we conclude that (h−1, g−1) is a solution of the discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc). If the system (Ld,Mc,Dc) is regular and g ∈Mc, we have that (g,Υ (Ld,M,Dc) nh (g)) is a solution of the discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc). Therefore, (i(Υ(Ld,M,Dc)nh (g)), i(g)) is also a solution of the dynamical equations which implies that Υ(Ld,M,Dc)nh (i(Υ (Ld,M,Dc) nh (g))) = i(g). Remark 3.20. Proposition 3.19 was proved in [36] for the particular case when Γ is the pair groupoid. � 3.5. Lie groupoid morphisms and reduction. Let (Φ,Φ0) be a Lie groupoid morphism between the Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′. Denote by (E(Φ),Φ0) the corresponding morphism between the Lie algebroids EΓ and EΓ′ of Γ and Γ′, respectively (see Section 2.2). If Ld : Γ→ R and L′d : Γ ′ → R are discrete Lagrangians on Γ and Γ′ such that Ld = L d ◦ Φ then, using Theorem 4.6 in [27], we have that (DDELLd)(g, h)(v) = (DDELL d)(Φ(g),Φ(h))(Ex(Φ)(v)) for (g, h) ∈ Γ2 and v ∈ (EΓ)x, where x = β(g) = α(h) ∈M. Using this fact, we deduce the following result: Corollary 3.21. Let (Φ,Φ0) be a Lie groupoid morphism between the Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′. Suppose that L′d : Γ ′ → R is a discrete Lagrangian on Γ′, that (Ld = L′d ◦Φ,Mc,Dc) is a discrete nonholonomic Lagrangian system on Γ and that (g, h) ∈ Γ2 ∩ (Mc ×Mc). Then: 24 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ (i) The pair (g, h) is a solution of the discrete nonholonomic problem (Ld,Mc, Dc) if and only if (DDELL′d)(Φ(g),Φ(h)) vanishes over the set (Eβ(g)Φ)((Dc)β(g)). (ii) If (L′d,M c) is a discrete nonholonomic Lagrangian system on Γ ′ such that (Φ(g),Φ(h)) ∈M′c×M′c and (Eβ(g)(Φ))((Dc)β(g)) = (D′c)Φ0(β(g)) then (g, h) is a solution for the discrete nonholonomic problem (Ld,Mc,Dc) if and only if (Φ(g),Φ(h)) is a solution for the discrete nonholonomic problem (L′d,M 3.6. Discrete nonholonomic Hamiltonian evolution operator. Let (Ld,Mc, Dc) a regular discrete nonholonomic system. Assume, without the loss of gener- ality, that the discrete nonholonomic Legendre transformations F−(Ld,Mc,Dc) : Mc −→ D∗c and F+(Ld,Mc,Dc) : Mc −→ D∗c are global diffeomorphisms. Then, (Ld,Mc,Dc) nh = F −(Ld,Mc,Dc)−1◦F+(Ld,Mc,Dc) is the discrete nonholonomic evo- lution operator and one may define the discrete nonholonomic Hamiltonian evolution operator, γ̃nh : D∗c → D∗c , by γ̃nh = F+(Ld,Mc,Dc) ◦ γ (Ld,Mc,Dc) nh ◦ F +(Ld,Mc,Dc) −1 . (3.16) From Remark 3.11, we have the following alternative definitions γ̃nh = F−(Ld,Mc,Dc) ◦ γ (Ld,Mc,Dc) nh ◦ F −(Ld,Mc,Dc) γ̃nh = F+(Ld,Mc,Dc) ◦ F−(Ld,Mc,Dc)−1 of the discrete Hamiltonian evolution operator. The following commutative diagram illustrates the situation Mc Mc (Ld,Mc,Dc) D∗c D F−(Ld, Mc, Dc) F+(Ld, Mc, Dc) F−(Ld, Mc, Dc) F+(Ld, Mc, Dc) γ̃nh γ̃nh Remark 3.22. The discrete nonholonomic evolution operator is an application from D∗c to itself. It is remarkable that D c is also the appropriate nonholonomic momentum space for a continuous nonholonomic system defined by a Lagrangian L : EΓ → R and the constraint distribution Dc. Therefore, in the regular case, the solution of the continuous nonholonomic Lagrangian system also determines a flow from D∗c to itself. We consider that this would be a good starting point to compare the discrete and continuous dynamics and eventually to establish a backward error analysis for nonholonomic systems. � 3.7. The discrete nonholonomic momentum map. Let (Ld,Mc,Dc) be a reg- ular discrete nonholonomic Lagrangian system on a Lie groupoid Γ ⇒ M and τ : EΓ →M be the Lie algebroid of Γ. Suppose that g is a Lie algebra and that Ψ : g→ Sec(τ) is a R-linear map. Then, for each x ∈M, we consider the vector subspace gx of g given by gx = { ξ ∈ g | Ψ(ξ)(x) ∈ (Dc)x } DISCRETE NONHOLONOMIC MECHANICS 25 and the disjoint union of these vector spaces gDc = We will denote by (gDc)∗ the disjoint union of the dual spaces, that is, (gDc)∗ = (gx)∗. Next, we define the discrete nonholonomic momentum map Jnh : Γ → (gDc)∗ as follows: Jnh(g) ∈ (gβ(g))∗ and Jnh(g)(ξ) = Θ+Ld(Ψ(ξ) (1,1))(g) = Ψ(ξ)(g)(Ld), for g ∈ Γ and ξ ∈ gβ(g). If ξ̃ : M → g is a smooth map such that ξ̃(x) ∈ gx, for all x ∈ M, then we may consider the smooth function Jnheξ : Γ→ R defined by Jnheξ (g) = Jnh(g)(ξ̃(β(g))), ∀g ∈ Γ. Definition 3.23. The Lagrangian Ld is said to be g-invariant with respect Ψ if Ψ(ξ)(1,1)(Ld) = Ψ(ξ)(Ld)− Ψ(ξ)(Ld) = 0, ∀ξ ∈ g. Now, we will prove the following result Theorem 3.24. Let Υ(Ld,Mc,Dc)nh : Mc → Mc be the local discrete nonholonomic evolution operator for the regular system (Ld,Mc,Dc). If Ld is g-invariant with respect to Ψ : g→ Sec(τ) and ξ̃ : M → g is a smooth map such that ξ̃(x) ∈ gx, for all x ∈M, then Jnheξ (Υ(Ld,Mc,Dc)nh (g))− Jnheξ (g) = ←−−−−−−−−−−−−−−−−−−−−−−−−−− Ψ(ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))− ξ̃(β(g)))(Υ (Ld,Mc,Dc) nh (g))(Ld) for g ∈Mc. Proof. Using that the Lagrangian Ld is g-invariant with respect to Ψ, we have that −−−−−−−−−−−−−−−−−−→ Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ (Ld,Mc,Dc) nh (g))(Ld) = ←−−−−−−−−−−−−−−−−−− Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ (Ld,Mc,Dc) nh (g))(Ld). (3.17) Also, since (g,Υ(Ld,Mc,Dc)nh (g)) is a solution of the discrete nonholonomic equations: ←−−−−−−− Ψ(ξ̃(β(g)))(g)(Ld) = −−−−−−−−−−−−−−−−−−→ Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ (Ld,Mc,Dc) nh (g))(Ld).(3.18) Thus, from (3.17) and (3.18), we find that ←−−−−−−− Ψ(ξ̃(β(g))(g)(Ld) = ←−−−−−−− Ψ(ξ̃(β(g)))(Υ(Ld,Mc,Dc)nh (g))(Ld). 26 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Therefore, (Υ(Ld,Mc,Dc)nh (g))− J (g) = ←−−−−−−−−−−−−−−−−−−− ξ̃(β(Υ(Ld,Mc,Dc)nh (g))) (Υ(Ld,Mc,Dc)nh (g))(Ld) ←−−−−−−−− ξ̃(β(g)) (g)(Ld) ←−−−−−−−−−−−−−−−−−−− ξ̃(β(Υ(Ld,Mc,Dc)nh (g))) (Υ(Ld,Mc,Dc)nh (g))(Ld) ←−−−−−−− Ψ(ξ̃(β(g))(Υ(Ld,Mc,Dc)nh (g))(Ld) ←−−−−−−−−−−−−−−−−−−−−−−−−−−− ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))− ξ̃(β(g)) (Υ(Ld,Mc,Dc)nh (g))(Ld). Theorem 3.24 suggests us to introduce the following definition Definition 3.25. An element ξ ∈ g is said to be a horizontal symmetry for the discrete nonholonomic system (Ld,Mc,Dc) and the map Ψ : g→ Sec(τ) if Ψ(ξ)(x) ∈ (Dc)x, for all x ∈M. Now, from Theorem 3.24, we conclude that Corollary 3.26. If Ld is g-invariant with respect to Ψ and ξ ∈ g is a horizontal symmetry for (Ld,Mc,Dc) and Ψ : g → Sec(τ) then Jnhξ̃ : Γ → R is a constant of the motion for Υ(Ld,Mc,Dc)nh , that is, ◦Υ(Ld,Mc,Dc)nh = J 4. Examples 4.1. Discrete holonomic Lagrangian systems on a Lie groupoid. Let us examine the case when the system is subjected to holonomic constraints. Let Ld : Γ → R be a discrete Lagrangian on a Lie groupoid Γ ⇒ M . Suppose that Mc ⊆ Γ is a Lie subgroupoid of Γ over M ′ ⊆M , that is, Mc is a Lie groupoid over M ′ with structural maps α|Mc : Mc →M ′, β|Mc : Mc →M ′, �|M ′ : M ′ →Mc, i|Mc : Mc →Mc, the canonical inclusions iMc : Mc −→ Γ and iM ′ : M ′ −→ M are injective immer- sions and the pair (iMc , iM ′) is a Lie groupoid morphism. We may assume, without the loss of generality, that M ′ = M (in other case, we will replace the Lie groupoid Γ by the Lie subgroupoid Γ′ over M ′ defined by Γ′ = α−1(M ′) ∩ β−1(M ′)). Then, if LMc = Ld ◦ iMc and τMc : EMc → M is the Lie algebroid of Mc, we have that the discrete (unconstrained) Euler-Lagrange equations for the Lagrangian function LMc are: X (g)(LMc)− X (h)(LMc) = 0, (g, h) ∈ (Mc)2, (4.1) for X ∈ Sec(τMc). We are interested in writing these equations in terms of the Lagrangian Ld defined on the Lie groupoid Γ. From Corollary 4.7 (iii) in [27], we deduce that (g, h) ∈ (Mc)2 is a solution of Equations 4.1 if and only if DDELLd(g, h) vanishes over Im(Eβ(g)(iMc)). Here, E(iMc) : EMc → EΓ is the Lie algebroid morphism induced between EMc and EΓ by the Lie groupoid morphism (iMc , Id). Therefore, we may consider the discrete holonomic system as the discrete nonholonomic system (Ld,Mc,Dc), where Dc = (E(iMc))(EMc) ∼= EMc . DISCRETE NONHOLONOMIC MECHANICS 27 In the particular case, when the subgroupoid Mc is determined by the vanishing set of n− r independent real C∞-functions φγ : Γ→ R: Mc = { g ∈ Γ | φγ(g) = 0, for all γ } , then the discrete holonomic equations are equivalent to: Y (g)(Ld)− Y (h)(Ld) = λγd oφγ(�(β(g)))(Y (β(g)), φγ(g) = φγ(h) = 0, for all Y ∈ Sec(τ), where do is the standard differential on Γ. This algorithm is a generalization of the Shake algorithm for holonomic systems (see [10, 20, 32, 36] for similar results on the pair groupoid Q×Q). 4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system on the pair group- oid Q×Q ⇒ Q and suppose that (q0, q1) is a point of Mc. Then, using the results of Section 3.1, we deduce that ((q0, q1), (q1, q2)) ∈ (Q × Q)2 is a solution of the discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc) if and only if (D2Ld(q0, q1) +D1Ld(q1, q2))|Dc(q1) = 0, (q1, q2) ∈Mc, or, equivalently, D2Ld(q0, q1) +D1Ld(q1, q2) = j(q1), (q1, q2) ∈Mc, where λj are the Lagrange multipliers and {Aj} is a local basis of the annihilator D0c . These equations were considered in [10] and [36]. Note that if (q1, q2) ∈ Γ = Q×Q then, in this particular case, GLd(q1,q2) : Tq1Q× Tq2Q→ R is just the R-bilinear map (D2D1Ld)(q1, q2). On the other hand, if (q1, q2) ∈Mc we have that (TQ)Mc (q1,q2) vq2 ∈ Tq2Q ∣∣ (0, vq2) ∈ T(q1,q2)Mc } , (TQ)Mc (q1,q2) vq1 ∈ Tq1Q ∣∣ (vq1 , 0) ∈ T(q1,q2)Mc } . Thus, the system (Ld,Mc,Dc) is regular if and only if for every (q1, q2) ∈ Mc the following conditions hold: If vq1 ∈ (TQ)Mc (q1,q2) 〈D2D1Ld(q1, q2)vq1 , vq2〉 = 0, ∀vq2 ∈ Dc(q2)  =⇒ vq1 = 0, If vq2 ∈ (TQ)Mc (q1,q2) 〈D2D1Ld(q1, q2)vq1 , vq2〉 = 0, ∀vq1 ∈ Dc(q1)  =⇒ vq2 = 0. The first condition was obtained in [36] in order to guarantee the existence of a unique local nonholonomic evolution operator Υ(Ld,Mc,Dc)nh for the system (Ld,Mc,Dc). However, in order to assure that Υ (Ld,Mc,Dc) nh is a (local) diffeomor- phism one must assume that the second condition also holds. 28 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Example 4.1 (Discrete Nonholonomically Constrained particle). Consider the discrete nonholonomic system determined by: a) A discrete Lagrangian Ld : R3 × R3 → R: Ld(x0, y0, z0, x1, y1, z1) = x1 − x0 y1 − y0 z1 − z0 b) A constraint distribution of Q = R3, Dc = span ,X2 = c) A discrete constraint submanifold Mc of R3 × R3 determined by the con- straint φ(x0, y0, z0, x1, y1, z1) = z1 − z0 y1 + y0 x1 − x0 The system (Ld,Mc,Dc) is a discretization of a classical continuous nonholonomic system: the nonholonomic free particle (for a discussion on this continuous system see, for instance, [4, 8]). Note that if E(R3×R3) ∼= TR3 is the Lie algebroid of the pair groupoid R3 × R3 ⇒ R3 then T(x1,y1,z1,x1,y1,z1)Mc ∩ E(R3×R3)(x1, y1, z1) = Dc(x1, y1, z1). Since X1 = − X2 = − then, the discrete nonholonomic equations are:( x2 − 2x1 + x0 z2 − 2z1 + z0 = 0, (4.2) y2 − 2y1 + y0 = 0, (4.3) which together with the constraint equation determine a well posed system of dif- ference equations. We have that D2D1Ld = − 1h{dx0 ∧ dx1 + dy0 ∧ dy1 + dz0 ∧ dz1} TR3)Mc (x0,y0,z0,x1,y1,z1) = {a0 ∂∂x0 + b0 + c0 ∂∂z0 ∈ T(x0,y0,z0)R c0 = 12 (a0(y1 + y0)− b0(x1 − x0))}. TR3)Mc (x0,y0,z0,x1,y1,z1) = {a1 ∂∂x1 + b1 + c1 ∂∂z1 ∈ T(x1,y1,z1)R c1 = 12 (a1(y1 + y0) + b1(x1 − x0))}. Thus, if we consider the open subset of Mc defined by{ (x0, y0, z0, x1, y1, z1) ∈Mc ∣∣ 2 + y21 + y1y0 6= 0, 2 + y20 + y0y1 6= 0} then in this subset the discrete nonholonomic system is regular. Let Ψ : g = R2 −→ X(R3) given by Ψ(a, b) = a ∂ + b ∂ . Then gDc = span{Ψ(ξ̃) = X1}, where ξ̃ : R3 → R2 is defined by ξ̃(x, y, z) = (1, y). More- over, the Lagrangian Ld is g-invariant with respect to Ψ. Therefore, (x1, y1, z1, x2, y2, z2)− Jnhξ̃ (x0, y0, z0, x1, y1, z1) ←−−−−−−−−− Ψ(0, y2 − y1)(x1, y1, z1, x2, y2, z2)(Ld), DISCRETE NONHOLONOMIC MECHANICS 29 that is,( x2 − x1 z2 − z1 x1 − x0 z1 − z0 = (y2 − y1) z2 − z1 This equation is precisely Equation (4.2). 4.3. Discrete nonholonomic Lagrangian systems on a Lie group. Let G be a Lie group. G is a Lie groupoid over a single point and the Lie algebra g of G is just the Lie algebroid associated with G. If g, h ∈ G, vh ∈ ThG and αh ∈ T ∗hG we will use the following notation: gvh = (Thlg)(vh) ∈ TghG, vhg = (Thrg)(vh) ∈ ThgG, gαh = (T ∗ghlg−1)(αh) ∈ T ghG, αhg = (T hgrg−1)(αh) ∈ T Now, let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system on the Lie group G, that is, Ld : G → R is a discrete Lagrangian, Mc is a submanifold of G and Dc is a vector subspace of g. If g1 ∈ Mc then (g1, g2) ∈ G × G is a solution of the discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc) if and only if g−11 dLd(g1)− dLd(g2)g gk ∈Mc, k = 1, 2 (4.4) where λj are the Lagrange multipliers and {µj} is a basis of the annihilator D0c of Dc. These equations were obtained in [36] (see Theorem 3 in [36]). Taking pk = dLd(gk)g k , k = 1, 2 then p2 −Ad∗g1p1 = − λjµj , gk ∈Mc, k = 1, 2 (4.5) where Ad : G × g −→ g is the adjoint action of G on g. These equations were obtained in [14] and called discrete Euler-Poincaré-Suslov equations. On the other hand, from (2.14), we have that ΩLd(( −→η ,←−µ ), (−→η ′,←−µ ′)) = −→η ′(←−µ (Ld))−−→η (←−µ ′(Ld)). Thus, if g ∈ G then, using (2.22), it follows that the R-bilinear map GLdg : g×g→ R is given by GLdg (ξ, η) = − ←−η (g)( ξ (Ld)). Therefore, the system (Ld,Mc,Dc) is regular if and only if for every g ∈ Mc the following conditions hold: η ∈ g/←−η (g) ∈ TgMc and ←−η (g)( ξ (Ld)) = 0,∀ξ ∈ Dc =⇒ η = 0, ξ ∈ g/ ξ (g) ∈ TgMc and ←−η (g)( ξ (Ld)) = 0,∀η ∈ Dc =⇒ ξ = 0. We illustrate this situation with two simple examples previously considered in [14]. 30 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ 4.3.1. The discrete Suslov system. (See [14]) The Suslov system studies the motion of a rigid body suspended at its centre of mass under the action of the following nonholonomic constraint: the body angular velocity is orthogonal to some fixed direction. The configuration space is G = SO(3) and the elements of the Lie algebra so(3) may be identified with R3 and represented by coordinates (ωx, ωy, ωz). Without loss of generality, let us choose as fixed direction the third vector of the body frame ē1, ē2, ē3. Then, the nonholonomic constraint is ωz = 0. The discretization of this system is modelled by considering the discrete La- grangian Ld : SO(3) −→ R defined by Ld(Ω) = 12Tr (ΩJ), where J represents the mass matrix (a symmetric positive-definite matrix with components (Jij)1≤i,j≤3). The constraint submanifold Mc is determined by the constraint Tr (ΩE3) = 0 (see [14]) where  0 0 00 0 −1 0 1 0  , E2 =  0 0 10 0 0 −1 0 0  , E3 =  0 −1 01 0 0 0 0 0 is the standard basis of so(3), the Lie algebra of SO(3). The vector subspace Dc = span{E1, E2}. Therefore, D0c = span{E3}. Moreover, the exponential map of SO(3) is a diffeomorphism from an open subset of Dc (which contains the zero vector) to an open subset of Mc (which contains the identity element I). In particular, TIMc = Dc. On the other hand, the discrete Euler-Poincaré-Suslov equations are given Ei(Ω1)(Ld)− Ei(Ω2)(Ld) = 0, Tr (ΩiE3) = 0, i ∈ {1, 2}. After some straightforward operations, we deduce that the above equations are equivalent to: Tr ((EiΩ2 − Ω1Ei)J) = 0, Tr (ΩiE3) = 0, i ∈ {1, 2} or, considering the components Ωk = (Ω ij ) of the elements of SO(3), we have that:( 33 − J33Ω 32 + J22Ω −J23Ω 22 + J12Ω 13 − J13Ω −J23Ω 33 − J22Ω 32 − J12Ω +J33Ω 23 + J23Ω 22 + J13Ω −J13Ω 33 + J33Ω 31 − J12Ω +J23Ω 21 − J11Ω 13 + J13Ω 33 + J12Ω 32 + J11Ω −J33Ω 13 − J23Ω 12 − J13Ω Ω(1)12 = Ω 21 , Ω 12 = Ω Moreover, since the discrete Lagrangian verifies that Ld(Ω) = Tr (ΩJ) = Tr (ΩtJ) = Ld(Ω and also the constraint satisfies Tr (ΩE3) = −Tr (Ω−1E3), then this discretization of the Suslov system is reversible. The regularity condition in Ω ∈ SO(3) is in this particular case: η ∈ so(3) /Tr (E1ΩηJ) = 0, Tr (E2ΩηJ) = 0 and Tr (ΩηE3) = 0 =⇒ η = 0 It is easy to show that the system is regular in a neighborhood of the identity I. DISCRETE NONHOLONOMIC MECHANICS 31 4.3.2. The discrete Chaplygin sleigh. (See [12, 14]) The Chaplygin sleigh system describes the motion of a rigid body sliding on a horizontal plane. The body is supported at three points, two of which slide freely without friction while the third is a knife edge, a constraint that allows no motion orthogonal to this edge (see [41]). The configuration space of this system is the group SE(2) of Euclidean motions of R2. An element Ω ∈ SE(2) is represented by a matrix  cos θ − sin θ xsin θ cos θ y 0 0 1  with θ, x, y ∈ R. Thus, (θ, x, y) are local coordinates on SE(2). A basis of the Lie algebra se(2) ∼= R3 of SE(2) is given by  0 −1 01 0 0 0 0 0  , e1 =  0 0 10 0 0 0 0 0  , e2 =  0 0 00 0 1 0 0 0 and we have that [e, e1] = e2, [e, e2] = −e1, [e1, e2] = 0. An element ξ ∈ se(2) is of the form ξ = ω e+ v1 e1 + v2 e2 and the exponential map exp : se(2) ∼= R3 → SE(2) of SE(2) is given by exp(ω, v1, v2) = (ω, v1 + v2( cosω − 1 ),−v1( cosω − 1 ) + v2 ), if ω 6= 0, exp(0, v1, v2) = (0, v1, v2). Note that the restriction of this map to the open subset U =] − π, π[×R2 ⊆ R3 ∼= se(2) is a diffeomorphism onto the open subset exp(U) of SE(2). A discretization of the Chaplygin sleigh may be constructed as follows: - The discrete Lagrangian Ld : SE(2) −→ R is given by Ld(Ω) = Tr (ΩJΩT )− Tr (ΩJ), where J is the matrix:  (J/2) +ma2 mab mamab (J/2) +mb2 mb ma mb m (see [14]). - The vector subspace Dc of se(2) is Dc = span {e, e1} = { (ω, v1, v2) ∈ se(2) | v2 = 0 } . - The constraint submanifold Mc of SE(2) is Mc = exp(U ∩Dc). (4.6) Thus, we have that Mc = { (θ, x, y) ∈ SE(2) | − π < θ < π, θ 6= 0, (1− cos θ)x− y sin θ = 0 } ∪ { (0, x, 0) ∈ SE(2) | x ∈ R } . 32 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ Figure 1. Submanifold Mc From (4.6) it follows that I ∈Mc and TIMc = Dc. In fact, one may prove T(0,x,0)Mc = span { ∂θ |(0,x,0) ∂y |(0,x,0) ∂x |(0,x,0) for x ∈ R. Now, the discrete Euler-Poincaré-Suslov equations are: ←−e (θ1, x1, y1)(Ld)−−→e (θ2, x2, y2)(Ld) = 0, ←−e1(θ1, x1, y1)(Ld)−−→e1(θ2, x2, y2)(Ld) = 0, and the condition (θk, xk, yk) ∈Mc, with k ∈ {1, 2}. We rewrite these equations as the following system of difference equations:( −am cos θ1 − bm sin θ1 + am +mx1 cos θ1 +my1 sin θ1 mx2 + am cos θ2 −bm sin θ2 − am amy1 cos θ1 − amx1 sin θ1 − bmx1 cos θ1 −bmy1 sin θ1 + (a2m+ b2m+ J) sin θ1 amy2 − bmx2 +(a2m+ b2m+ J) sin θ2 together with the condition (θk, xk, yk) ∈Mc, k ∈ {1, 2}. On the other hand, one may prove that the discrete nonholonomic Lagrangian system (Ld,Mc,D) is reversible. Finally, consider a point (0, x, 0) ∈ Mc and an element η ≡ (ω, v1, v2) ∈ se(2) such that ←−η (0, x, 0) ∈ T(0,x,0)Mc, ←−η (0, x, 0)(−→e (Ld)) = 0, ←−η (0, x, 0)(−→e1(Ld)) = 0. Then, if we assume that a2m+ J + amx 6= 0 it follows that η = 0. Thus, the discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is regular in a neighborhood of the identity I. 4.4. Discrete nonholonomic Lagrangian systems on an action Lie group- oid. Let H be a Lie group with identity element e and · : M ×H → M , (x, h) ∈ M × H 7→ xh, a right action of H on M . Thus, we may consider the action Lie groupoid Γ = M ×H over M with structural maps given by α̃(x, h) = x, β̃(x, h) = xh, �̃(x) = (x, e), m̃((x, h), (xh, h′)) = (x, hh′), ĩ(x, h) = (xh, h−1). (4.7) DISCRETE NONHOLONOMIC MECHANICS 33 Now, let h = TeH be the Lie algebra of H and Φ : h→ X(M) the map given by Φ(η) = ηM , for η ∈ h, where ηM is the infinitesimal generator of the action · : M×H →M corresponding to η. Then, Φ is a Lie algebra morphism and the corresponding action Lie algebroid pr1 : M × h→M is just the Lie algebroid of Γ = M ×H. We have that Sec(pr1) ∼= { η̃ : M → h | η̃ is smooth } and that the Lie algebroid structure ([[·, ·]]Φ, ρΦ) on pr1 : M ×H →M is defined by [[η̃, µ̃]]Φ(x) = [η̃(x), µ̃(x)]+(η̃(x))M (x)(µ̃)−(µ̃(x))M (x)(η̃), ρΦ(η̃)(x) = (η̃(x))M (x), for η̃, µ̃ ∈ Sec(pr1) and x ∈M. Here, [·, ·] denotes the Lie bracket of h. If (x, h) ∈ Γ = M ×H then the left-translation l(x,h) : α̃−1(xh) → α̃−1(x) and the right-translation r(x,h) : β̃−1(x)→ β̃−1(xh) are given l(x,h)(xh, h ′) = (x, hh′), r(x,h)(x(h ′)−1, h′) = (x(h′)−1, h′h). (4.8) Now, if η ∈ h then η defines a constant section Cη : M → h of pr1 : M × h→M and, using (2.4), (2.5), (4.7) and (4.8), we have that the left-invariant and the right-invariant vector fields C η and C η, respectively, on M ×H are defined by C η(x, h) = (−ηM (x),−→η (h)), C η(x, h) = (0x, ←−η (h)), (4.9) for (x, h) ∈ Γ = M ×H. Note that if {ηi} is a basis of h then {Cηi} is a global basis of Sec(pr1). On the other hand, we will denote by expΓ : EΓ = M × h → Γ = M × H the map given by expΓ(x, η) = (x, expH(η)), for (x, η) ∈ EΓ = M × h, where expH : h → H is the exponential map of the Lie group H. Note that if Φ(x,e) : R → Γ = M ×H is the integral curve of the left-invariant vector field on Γ = M ×H such that Φ(x,e)(0) = (x, e) then (see (4.9)) expΓ(x, η) = Φ(x,e)(1). Next, suppose that Ld : Γ = M × H → R is a Lagrangian function, Dc is a constraint distribution such that {Xα} is a local basis of sections of the annihilator D0c , and Mc ⊆ Γ is the discrete constraint submanifold. For every h ∈ H (resp., x ∈ M) we will denote by Lh (resp., Lx) the real function on M (resp., on H) given by Lh(y) = Ld(y, h) (resp., Lx(h′) = Ld(x, h′)). A composable pair ((x, hk), (xhk, hk+1)) ∈ Γ2 ∩ (Mc × Mc) is a solution of the discrete nonholonomic Euler-Lagrange equations for the system (Ld,Mc,Dc) if C η(x, hk)(Ld)− C η(xhk, hk+1)(Ld) = λαX α(xhk)(η), for all η ∈ h, or, in other terms (see (4.9)) {(Telhk)(η)}(Lx)− {(Terhk+1)(η)}(Lxhk) + ηM (xhk)(Lhk+1) = λαX α(xhk)(η), for all η ∈ h. 4.4.1. The discrete Veselova system. As a concrete example of a nonholonomic system on a transformation Lie groupoid we consider a discretization of the Veselova system (see [44]). In the continuous theory [9], the configuration manifold is the transformation Lie algebroid pr1 : S2 × so(3)→ S2 with Lagrangian Lc(γ, ω) = ω · Iω −mglγ · e, 34 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ where S2 is the unit sphere in R3, ω ∈ R3 ' so(3) is the angular velocity, γ is the direction opposite to the gravity and e is a unit vector in the direction from the fixed point to the center of mass, all them expressed in a frame fixed to the body. The constants m, g and l are respectively the mass of the body, the strength of the gravitational acceleration and the distance from the fixed point to the center of mass. The matrix I is the inertia tensor of the body. Moreover, the constraint subbundle Dc → S2 is given by γ ∈ S2 7→ Dc(γ) = ω ∈ R3 ' so(3) ∣∣ γ · ω = 0} . Note that the section φ : S2 → S2 × so(3)∗, (x, y, z) 7→ ((x, y, z), xe1 + ye2 + ze3), where {e1, e2, e3} is the canonical basis of R3 and {e1, e2, e3} is the dual basis, is a global basis for D0c . If ω ∈ so(3) and ω̂ is the skew-symmetric matrix of order 3 such that ω̂v = ω×v then the Lagrangian function Lc may be expressed as follows Lc(γ, ω) = Tr(ω̂IIω̂T )−mg l γ · e, where II = 1 Tr(I)I3×3− I. Here, I3×3 is the identity matrix. Thus, we may define a discrete Lagrangian Ld : Γ = S2 × SO(3)→ R for the system by (see [27]) Ld(γ,Ω) = − Tr(IIΩ)− hmg l γ · e. On the other hand, we consider the open subset of SO(3) V = {Ω ∈ SO(3) | Tr Ω 6= ±1 } and the real function ψ : S2 × V → R given by ψ(γ,Ω) = γ · (Ω̂− ΩT ). One may check that the critical points of ψ are (γ,Ω) ∈ S2 × V ∣∣ Ωγ − γ = 0} . Thus, the subset Mc of Γ = S2 × SO(3) defined by (γ,Ω) ∈ (S2 × V )− Cψ ∣∣∣ γ · (Ω̂− ΩT ) = 0} , is a submanifold of Γ of codimension one. Mc is the discrete constraint submanifold. We have that the map expΓ : S 2× so(3)→ S2×SO(3) is a diffeomorphism from an open subset of Dc, which contains the zero section, to an open subset of Mc, which contains the subset of Γ given by �̃(S2) = {(γ, e) ∈ S2 × SO(3)}. So, it follows that (Dc)(γ) = T(γ,e)Mc ∩ EΓ(γ), for γ ∈ S2. Following the computations of [27] we get the nonholonomic discrete Euler-Lagrange equations, for ((γk,Ωk), (γkΩk,Ωk+1)) ∈ Γ2 Mk+1 − ΩTkMkΩk +mglh 2( ̂γk+1 × e) = λγ̂k+1, γk( ̂Ωk − ΩTk ) = 0, γk+1( ̂Ωk+1 − ΩTk+1) = 0, where M = ΩII − IIΩT . Therefore, in terms of the axial vector Π in R3 defined by Π̂ = M , we can write the equations in the form Πk+1 = ΩTk Πk −mglh 2γk+1 × e + λγk+1, γk( ̂Ωk − ΩTk ) = 0, γk+1( ̂Ωk+1 − ΩTk+1) = 0. DISCRETE NONHOLONOMIC MECHANICS 35 Note that, using the expression of an arbitrary element of SO(3) in terms of the Euler angles (see Chapter 15 of [31]), we deduce that the discrete constraint sub- manifold Mc is reversible, that is, i(Mc) = Mc. However, the discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is not reversible. In fact, it is easy to prove that Ld ◦ i 6= Ld. On the other hand, if γ ∈ S2 and ξ, η ∈ R3 ∼= so(3) then it follows that C ξ(γ, I3)( C η(Ld)) = −ξ · Iη. Consequently, the nonholonomic system (Ld,Mc,Dc) is regular in a neighborhood (in Mc) of the submanifold �̃(S2). 4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie group- oid. Let p : Q → M = Q/G be a principal G-bundle and choose a local trivial- ization G× U , where U is an open subset of M . Then, one may identify the open subset (p−1(U) × p−1(U))/G ' ((G × U) × (G × U))/G of the Atiyah groupoid (Q×Q)/G with the product manifold (U ×U)×G. Indeed, it is easy to prove that the map ((G× U)× (G× U))/G→ (U × U)×G, [((g, x), (g′, y))]→ ((x, y), g−1g′)), is bijective. Thus, the restriction to ((G × U) × (G × U))/G of the Lie groupoid structure on (Q × Q)/G induces a Lie groupoid structure in (U × U) × G with source, target and identity section given by α : (U × U)×G→ U ; ((x, y), g)→ x, β : (U × U)×G→ U ; ((x, y), g)→ y, � : U → (U × U)×G; x→ ((x, x), e), and with multiplication m : ((U × U) × G)2 → (U × U) × G and inversion i : (U × U)×G→ (U × U)×G defined by m(((x, y), g), ((y, z), h)) = ((x, z), gh), i((x, y), g) = ((y, x), g−1). (4.10) The Lie algebroid A((U×U)×G) may be identified with the vector bundle TU×g→ U . Thus, the fibre over the point x ∈ U is the vector space TxU × g. Therefore, a section of A((U ×U)×G) is a pair (X, ξ̃), where X is a vector field on U and ξ̃ is a map from U on g. The space Sec(A((U × U)×G)) is generated by sections of the form (X, 0) and (0, Cξ), with X ∈ X(U), ξ ∈ g and Cξ : U → g being the constant map Cξ(x) = ξ, for all x ∈ U (see [27] for more details). Now, suppose that Ld : (U ×U)×G→ R is a Lagrangian function, Dc a vector subbundle of TU×g and Mc a constraint submanifold on (U×U)×G. Take a basis of sections {Y α} of the annihilator Doc . Then, the discrete nonholonomic equations are ←−−−−− (Xα, η̃α)((x, y), gk)(Ld)− −−−−−→ (Xα, η̃α)((y, z), gk+1)(Ld) = 0, with (Xα, η̃α) : U → TU × g a basis of the space Sec(τDc) and (((x, y), gk), ((y, z), gk+1)) ∈ (Mc×Mc)∩ ((U ×U)×G)2. The above equations may be also written as (X, 0)((x, y), gk)(Ld)− (X, 0)((y, z), gk+1)(Ld) = λαY α(y)(X(y)),←−−−− (0, Cξ)((x, y), gk)(Ld)− −−−−→ (0, Cξ)((y, z), gk+1)(Ld) = λαY α(y)(Cξ(y)), with X ∈ X(U), ξ ∈ g and (((x, y), gk), ((y, z), gk+1)) ∈ (Mc×Mc)∩((U×U)×G)2. An equivalent expression of these equations is D2Ld((x, y), gk) +D1Ld((y, z), gk+1) = λαµα(y), pk+1(y, z) = Ad∗gkpk(x, y)− λαη̃ α(y), (4.11) 36 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ where pk(x̄, ȳ) = d(r∗gkL(x̄,ȳ, ))(e) for (x̄, ȳ) ∈ U × U and we write Y α ≡ (µα, η̃α), µα being a 1-form on U and η̃α : U → g∗ a smooth map. 4.5.1. A discretization of the equations of motion of a rolling ball without sliding on a rotating table with constant angular velocity. A (homogeneous) sphere of radius r > 0, mass m and inertia about any axis I rolls without sliding on a horizontal table which rotates with constant angular velocity Ω about a vertical axis through one of its points. Apart from the constant gravitational force, no other external forces are assumed to act on the sphere (see [41]). The configuration space for the continuous system isQ = R2×SO(3) and we shall use the notation (x, y;R) to represent a typical point in Q. Then, the nonholonomic constraints are Tr(ṘRTE2) = −Ωy, Tr(ṘRTE1) = Ωx, where {E1, E2, E3} is the standard basis of so(3). The matrix ṘRT is skew symmetric, therefore we may write ṘRT =  0 −w3 w2w3 0 −w1 −w2 w1 0 where (w1, w2, w3) represents the angular velocity vector of the sphere measured with respect to the inertial frame. Then, we may rewrite the constraints in the usual form: ẋ− rw2 = −Ωy, ẏ + rw1 = Ωx. The Lagrangian for the rolling ball is: Lc(x, y;R, ẋ, ẏ; Ṙ) = m(ẋ2 + ẏ2) + I Tr(ṘRT (ṘRT )T ) m(ẋ2 + ẏ2) + I(ω21 + ω 2 + ω Moreover, it is clear that Q = R2 × SO(3) is the total space of a trivial princi- pal SO(3)-bundle over R2 and the bundle projection φ : Q → M = R2 is just the canonical projection on the first factor. Therefore, we may consider the correspond- ing Atiyah algebroid E′ = TQ/SO(3) over M = R2. We will identify the tangent bundle to SO(3) with so(3)× SO(3) by using right translation. Under this identification between T (SO(3)) and so(3)×SO(3) the tangent action of SO(3) on T (SO(3)) ∼= so(3)× SO(3) is the trivial action (so(3)× SO(3))× SO(3)→ so(3)× SO(3), ((ω,R), S) 7→ (ω,RS). (4.12) Thus, the Atiyah algebroid TQ/SO(3) is isomorphic to the product manifold TR2×so(3) and the vector bundle projection is τR2 ◦pr1, where pr1 : TR2×so(3)→ TR2 and τR2 : TR2 → R2 are the canonical projections. A section of E′ = TQ/SO(3) ∼= TR2 × so(3) → R2 is a pair (X,u), where X is a vector field on R2 and u : R2 → so(3) is a smooth map. Therefore, a global basis of sections of TR2 × so(3)→ R2 is s′1 = ( , 0), s′2 = ( , 0), s′3 = (0, E1), s 4 = (0, E2), s 5 = (0, E3). DISCRETE NONHOLONOMIC MECHANICS 37 The anchor map ρ′ : E′ = TQ/SO(3) ∼= TR2 × so(3) → TR2 is the projection over the first factor and if [[·, ·]]′ is the Lie bracket on the space Sec(E′ = TQ/SO(3)) then the only non-zero fundamental Lie brackets are [[s′3, s ′ = s′5, [[s ′ = s′3, [[s ′ = s′4. Moreover, the Lagrangian function Lc = T and the constraint functions are SO(3)-invariant. Consequently, Lc induces a Lagrangian function L′c on E TQ/SO(3) L′c(x, y, ẋ, ẏ;ω) = m(ẋ2 + ẏ2) + I Tr(ωωT ), m(ẋ2 + ẏ2)− I Tr(ω2), where (x, y, ẋ, ẏ) are the standard coordinates on TR2 and ω ∈ so(3). The con- straint functions defined on E′ = TQ/SO(3) are: ẋ+ r Tr(ωE2) = −Ωy, ẏ − r Tr(ωE1) = Ωx. (4.13) We have a nonholonomic system on the Atiyah algebroid E′ = TQ/SO(3) ∼= TR2× so(3). This kind of systems was recently analyzed by J. Cortés et al [9] (in particular, this example was carefully studied). Eqs. (4.13) define an affine subbundle of the vector bundle E′ ∼= TR2× so(3)→ R2 which is modelled over the vector subbundle D′c generated by the sections D′c = {s 5, rs 1 + s 4, rs 2 − s Our objective is to discretize this example directly on the Atiyah algebroid. The Atiyah groupoid is now identified to R2 × R2 × SO(3) ⇒ R2. We may construct the discrete Lagrangian by L′d(x0, y0, x1, y1;W1) = L c(x0, y0, x1 − x0 y1 − y0 ; (logW1)/h) where log : SO(3) −→ so(3) is the (local)-inverse of the exponential map exp : so(3) −→ SO(3). For simplicity instead of this procedure we use the following approximation: logW1/h ≈ W1 − I3×3 where I3×3 is the identity matrix. L′d(x0, y0, x1, y1;W1) = L c(x0, y0, x1 − x0 y1 − y0 W1 − I3×3 x1 − x0 y1 − y0 (2h)2 Tr(I3×3 −W1) Eliminating constants, we may consider as discrete Lagrangian L′d = x1 − x0 y1 − y0 Tr(W1) The discrete constraint submanifold M′c of R 2 × R2 × SO(3) is determined by the constraints: x1 − x0 Tr(W1E2) = −Ω y1 + y0 y1 − y0 Tr(W1E1) = Ω x1 + x0 38 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ We have that the system (L′d,M c) is not reversible. Note that the Lagrangian function L′d is reversible. However, the constraint submanifold M c is not reversible. The discrete nonholonomic Euler-Lagrange equations for the system (L′d, ,M D′c) are: s′5(x0, y0, x1, y1;W1)(L s′5(x1, y1, x2, y2;W2)(L d) = 0 ←−−−−−− (rs′1 + s 4)(x0, y0, x1, y1;W1)(L −−−−−−→ (rs′1 + s 4)(x1, y1, x2, y2;W2)(L d) = 0 ←−−−−−− (rs′2 − s 3)(x0, y0, x1, y1;W1)(L −−−−−−→ (rs′2 − s 3)(x1, y1, x2, y2;W2)(L d) = 0 with the constraints defining Mc. On the other hand, the vector fields ←−s ′5, −→s ′5, ←−−−−− rs′1 + s −−−−−→ rs′1 + s ←−−−−− rs′2 − s′3 and−−−−−→ rs′2 − s′3 on (R2 × R2)× SO(3) are given by ←−s ′5 = ((0, 0), E 3), −→s ′5 = ((0, 0), E 3),←−−−−− rs′1 + s 4 = ((0, r E 2), −−−−−→ rs′1 + s 4 = ((−r , 0), E 2),←−−−−− rs′2 − s′3 = ((0, r E 1), −−−−−→ rs′2 − s′3 = ((0,−r E 1), where E i (respectively, E i) is the left-invariant (respectively, right-invariant) vec- tor field on SO(3) induced by Ei ∈ so(3), for i ∈ {1, 2, 3}. Thus, we deduce the following system of equations: Tr ((W1 −W2)E3) = 0, x2 − 2x1 + x0 Tr ((W1 −W2)E2) = 0, y2 − 2y1 + y0 Tr ((W1 −W2)E1) = 0, x2 − x1 Tr(W2E2) + Ω y2 + y1 y2 − y1 Tr(W2E1)− Ω x2 + x1 where (x0, x1, y0, y1;W1) are known. Simplifying we obtain the following system of equations: x2 − 2x1 + x0 I +mr2 y2 − y0 = 0 (4.14) y2 − 2y1 + y0 I +mr2 x2 − x0 = 0 (4.15) Tr ((W1 −W2)E3) = 0 (4.16) x2 − x1 Tr(W2E2) + Ω y2 + y1 = 0, (4.17) y2 − y1 Tr(W2E1)− Ω x2 + x1 = 0. (4.18) Now, consider the open subset U of R2 × R2 × SO(3) U = (R2 × R2)× {W ∈ SO(3) | W − Tr(W )I3×3 is regular } . Then, using Corollary 3.13 (iv), we deduce that the discrete nonholonomic La- grangian system (L′d,M c) is regular in the open subset U ′ of M′c given by U ′ = U ∩M′c. DISCRETE NONHOLONOMIC MECHANICS 39 If we denote by uk = (xk+1 − xk)/h and vk = (yk+1 − yk)/h, k ∈ N then from Equations (4.14) and (4.15) we deduce that( 4 + α2h2 4− α2h2 −4αh 4αh 4− α2h2 or in other terms x(k + 2) = 8x(k + 1) + (α2h2 − 4)x(k)− 4αh(y(k + 1)− y(k)) α2h2 + 4 y(k + 2) = 8y(k + 1) + (α2h2 − 4)y(k) + 4α(x(k + 1)− x(k)) α2h2 + 4 where α = IΩ I+mr2 . Since A ∈ SO(2), the discrete nonholonomic model predicts that the point of contact of the ball will sweep out a circle on the table in agreement with the continuous model. Figure 2 shows the excellent behaviour of the proposed numerical method Figure 2. Orbits for the discrete nonholonomic equations of mo- tion (left) and a standard numerical method (right) (initial condi- tions x(0) = 0.99, y(0) = 1, x(1) = 1, y(1) = 0.99 and h = 0.01 after 20000 steps). 4.6. Discrete Chaplygin systems. Now, we present the theory for a particu- lar (but typical) example of discrete nonholonomic systems: discrete Chaplygin systems. This kind of systems was considered in the case of the pair groupoid in [10]. For any groupoid Γ ⇒ M , the map χ : Γ → M × M , g 7→ (α(g), β(g)) is a morphism over M from Γ to the pair groupoid M ×M (usually called the anchor of Γ). The induced morphism of Lie algebroids is precisely the anchor ρ : EΓ → TM of EΓ (the Lie algebroid of Γ). Definition 4.2. A discrete Chaplygin system on the groupoid Γ is a discrete nonholonomic problem (Ld,Mc,Dc) such that - (Ld,Mc,Dc) is a regular discrete nonholonomic Lagrangian system; - χMc = χ ◦ iMc : Mc −→M ×M is a diffeomorphism; - ρ ◦ iDc : Dc −→ TM is an isomorphism of vector bundles. Denote by L̃d : M ×M −→ R the discrete Lagrangian defined by L̃d = Ld ◦ iMc ◦ (χMc)−1. In the following, we want to express the dynamics on M×M , by finding relations between de dynamics defined by the nonholonomic system on Γ and M ×M . 40 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ From our hypothesis, for any vector field Y ∈ X(M) there exists a unique section X ∈ Sec(τDc) such that ρ ◦ iDc ◦X = Y . Now, using (2.4), (2.5) and (2.6), it follows that X (g)) = −Y (α(g)) and Tgβ( X (g)) = Y (β(g)) with some abuse of notation. In other words, Tgχ(X (1,0)(g)) = Y (1,0)(α(g), β(g)) and Tgχ(X (0,1)(g)) = Y (0,1)(α(g), β(g)) for g ∈ Mc, where Tχ : TΓΓ ∼= V β ⊕Γ V α → TM×M (M ×M) ∼= T (M ×M) is the prolongation of the morphism χ given by (Tgχ)(Xg, Yg) = ((Tgα)(Xg), (Tgβ)(Yg)), for g ∈ Γ and (Xg, Yg) ∈ TΓg Γ ∼= Vgβ ⊕ Vgα. Since χMc is a diffeomorphism, there exists a unique X g ∈ TgMc (respectively, X̄ ′g ∈ TgMc) such that (TgχMc)(X g) = Y (1,0)(α(g), β(g)) = (−Y (α(g)), 0β(g)) (respectively, (TgχMc)(X̄ g) = Y (0,1)(α(g), β(g)) = (0α(g), Y (β(g)))) for all g ∈Mc. Thus, X ′g ∈ TgMc ∩ Vgβ, X (g)−X ′g = Z ′g ∈ Vgα ∩ Vgβ, X̄ ′g ∈ TgMc ∩ Vgα, X (g)− X̄ ′g = Z̄ ′g ∈ Vgα ∩ Vgβ, for all g ∈Mc. Now, if (g, h) ∈ Γ2 ∩ (Mc ×Mc) then X (g)(Ld)− X (h)(Ld) = X̄ g(Ld) + Z̄ g(Ld)−X h(Ld)− Z h(Ld) Y (α(g), β(g))(L̃d)− Y (α(h), β(h))(L̃d) +Z̄ ′g(Ld)− Z h(Ld). Therefore, if we use the following notation (α(g), β(g)) = (x, y), (α(h), β(h)) = (y, z) F+Y (x, y) = −Z̄ (x,y) (Ld), F Y (y, z) = Z (y,z) (Ld), X (g)(Ld)− X (h)(Ld) = Y (x, y)(L̃d)− Y (y, z)(L̃d) −F+Y (x, y) + F Y (y, z). In conclusion, we have proved that (g, h) is a solution of the discrete nonholonomic Euler-Lagrange equations for the system (Ld,Mc,Dc) if and only if ((x, y), (y, z)) is a solution of the reduced equations Y (x, y)(L̃d)− Y (y, z)(L̃d) = F Y (x, y)− F Y (y, z), Y ∈ X(M). Note that the above equations are the standard forced discrete Euler-Lagrange equations (see [32]). 4.6.1. The discrete two wheeled planar mobile robot. We now consider a discrete version of the two-wheeled planar mobile robot [8, 9]. The position and orientation of the robot is determined, with respect a fixed cartesian reference, by an element Ω = (θ, x, y) ∈ SE(2), that is, a matrix  cos θ − sin θ xsin θ cos θ y 0 0 1 DISCRETE NONHOLONOMIC MECHANICS 41 Moreover, the different positions of the two wheels are described by elements (φ, ψ) ∈ T2. Therefore, the configuration space is SE(2) × T2. The system is subjected to three nonholonomic constraints: one constraint induced by the condi- tion of no lateral sliding of the robot and the other two by the rolling conditions of both wheels. It is well known that this system is SE(2)-invariant and then the system may be described as a nonholonomic system on the Lie algebroid se(2)×TT2 → T2 (see [9]). In this case, the Lagrangian is Jω2 +m(v1)2 +m(v2)2 + 2m0lωv 2 + J2φ̇ 2 + J2ψ̇ Tr(ξJξT ) + φ̇2 + where ξ = ω e+ v1 e1 + v 2 e2 =  0 −ω v1ω 0 v2 0 0 0  and J =  J/2 0 m0l0 J/2 0 m0l 0 m Here, m = m0 + 2m1, where m0 is the mass of the robot without the two wheels, m1 the mass of each wheel, J its the moment of inertia with respect to the vertical axis, J2 the axial moments of inertia of the wheels and l the distance between the center of mass of the robot and the intersection point of the horizontal symmetry axis of the robot and the horizontal line connecting the centers of the two wheels. The nonholonomic constraints are v1 + R φ̇+ R ψ̇ = 0, v2 = 0, ω + R φ̇− R ψ̇ = 0, (4.19) determining a submanifold M of se(2) × TT2, where R is the radius of the two wheels and 2c the lateral length of the robot. In order to discretize the above nonholonomic system, we consider the Atiyah groupoid Γ = SE(2)×(T2×T2) ⇒ T2. The Lie algebroid of SE(2)×(T2×T2) ⇒ T2 is TT2 × se(2)→ T2. Then: - The discrete Lagrangian Ld : SE(2)× (T2 × T2)→ R is given by: Ld(Ωk, φk, ψk, φk+1, ψk+1) = 12h2 Tr ((Ωk − I3×3)J(Ωk − I3×3) (∆φk) (∆ψk) where I3×3 is the identity matrix, ∆φk = φk+1 − φk, ∆ψk = ψk+1 − ψk  cos θk − sin θk xksin θk cos θk yk 0 0 1 We obtain that mx2k +my k − 2lm0xk(1− cos θk) +2J(1− cos θk) + 2lm0yk sin θk) + (∆φk)2 (∆ψk)2 - The constraint vector subbundle of se(2)×TT2 is generated by the sections:{ , s2 = 42 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ - The continuous constraints of the two-wheeled planar robot are written in matrix form (see 4.19):  0 −ω v1ω 0 v2 0 0 0  0 R2c φ̇− R2c ψ̇ −R2 φ̇− R2 ψ̇− R φ̇+ R ψ̇ 0 0 0 0 0 We discretize the previous constraints using the exponential on SE(2) (see Section 4.3.2) and discretizing the velocities on the right hand side cos( R2c∆φk− R2c∆ψk) sin( R2c∆φk− R2c∆ψk) −c ∆φk+∆ψk ∆φk−∆ψk sin( R2c∆φk− R2c∆ψk) − sin( R2c∆φk− R2c∆ψk) cos( R2c∆φk− R2c∆ψk) c ∆φk+∆ψk ∆φk−∆ψk (1−cos( R2c∆φk− R2c∆ψk)) 0 0 1 if ∆φk 6= ∆ψk and  1 0 −R∆φk0 1 0 0 0 1 if ∆φk = ∆ψk. Therefore, the constraint submanifold Mc is defined as θk = − ∆φk + ∆ψk (4.20) xk = −c ∆φk + ∆ψk ∆φk −∆ψk ∆φk − (4.21) yk = c ∆φk + ∆ψk ∆φk −∆ψk 1− cos ∆φk − (4.22) if ∆φk 6= ∆ψk and θk = 0, xk = −R∆φk and yk = 0 if ∆φk = ∆ψk. We have that the discrete nonholonomic system (Ld,Mc,Dc) is reversible. More- over, if �Γ : T2 → SE(2) × (T2 × T2) is the identity section of the Lie groupoid Γ = SE(2)× (T2 × T2) then it is clear that �Γ(T2) = {I3×3} ×∆T2×T2 ⊆Mc. Here, ∆T2×T2 is the diagonal in T2 × T2. In addition, the system (Ld,Mc,Dc) is regular in a neighborhood U of the submanifold �Γ(T2) = {I3×3} ×∆T2×T2 in Mc. Note that T(I3×3,φ1,ψ1,φ1,ψ1)Mc ∩ EΓ(φ1, ψ1) = Dc(φ1, ψ1), for (φ1, ψ1) ∈ T2, where EΓ = se(2)× TT2 is the Lie algebroid of the Lie groupoid Γ = SE(2)× (T2 × T2). On the other hand, it is easy to show that the system (Ld, U,Dc) is a discrete Chaplygin system. The reduced Lagrangian on T2 × T2 is L̃d =   (mc2( ∆φk + ∆ψk ∆φk −∆ψk )2(1− cos( ∆φk − ∆ψk)) +J(1− cos( ∆φk − ∆ψk))) + (∆φk)2 (∆ψk)2 if ∆φk 6= ∆ψk (J1 + (∆φk)2 , if ∆φk = ∆ψk The discrete nonholonomic equations are: (Ω1,φ1,ψ1,φ2,ψ2) (Ld)−−→s1 (Ω2,φ2,ψ2,φ3,ψ3) (Ld) = 0 (Ω1,φ1,ψ1,φ2,ψ2) (Ld)−−→s2 (Ω2φ2,ψ2,φ3,ψ3) (Ld) = 0 DISCRETE NONHOLONOMIC MECHANICS 43 These equations in coordinates are: 2J1(φ3 − 2φ2 + φ1) = lRm0(cos θ2 + cos θ1) + (sin θ2 − sin θ1) R cos θ1 (lm0y1 + cmx1) + R sin θ1 (lm0x1 − cmy1) (cmx2 + lm0(y2 − 2c)) (4.23) 2J1(ψ3 − 2ψ2 + ψ1) = lRm0(cos θ2 + cos θ1)− (sin θ2 − sin θ1) R cos θ1 (lm0y1 − cmx1)− R sin θ1 (lm0x1 + cmy1) (cmx2 − lm0(y2 + 2c)) (4.24) Substituting constraints (4.20), (4.21) and (4.22) in Equations (4.23) and (4.24) we obtain a set of equations of the type 0 = f1(φ1, φ2, φ3, ψ1, ψ2, ψ3) and 0 = f1(φ1, φ2, φ3, ψ1, ψ2, ψ3) which are the reduced equations of the Chaplygin system. 5. Conclusions and Future Work In this paper we have elucidated the geometrical framework for nonholonomic discrete Mechanics on Lie groupoids. We have proposed discrete nonholonomic equations that are general enough to produce practical integrators for continuous nonholonomic systems (reduced or not). The geometric properties related with these equations have been completely studied and the applicability of these devel- opments has been stated in several interesting examples. Of course, much work remains to be done to clarify the nature of discrete non- holonomic mechanics. Many of this future work was stated in [36] and, in particular, we emphasize: - a complete backward error analysis which explain the very good energy behavior showed in examples or the preservation of a discrete energy (see [14]); - related with the previous question, the construction of a discrete exact model for a continuous nonholonomic system (see [17, 32, 36]); - to study discrete nonholonomic systems which preserve a volume form on the constraint surface mimicking the continuous case (see, for instance, [13, 46] for this last case); - to analyze the discrete hamiltonian framework and the construction of integrators depending on different discretizations; - and the construction of a discrete nonholonomic connection in the case of Atiyah groupoids (see [21, 27]). Related with some of the previous questions, in the conclusions of the paper of R. McLachlan and M. 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[36] McLachlan R and Perlmutter M, Integrators for nonholonomic Mechanical Systems, J. Nonlinear Sci., 16 (2006), 283–328. [37] McLachlan R and Scovel C, Open problems in symplectic integration, Fields Inst. Comm. 10 (1996), 151–180. [38] Mestdag T , Lagrangian reduction by stages for non-holonomic systems in a Lie algebroid framework, J. Phys. A: Math. Gen 38 (2005), 10157–10179. [39] Mestdag T and Langerock B, A Lie algebroid framework for nonholonomic systems, J. Phys. A: Math. Gen 38 (2005) 1097–1111. [40] Moser J and Veselov AP, Discrete versions of some classical integrable systems and fac- torization of matrix polynomials, Comm. Math. Phys. 139 (1991), 217–243. [41] Neimark J and Fufaev N, Dynamics on Nonholonomic systems, Translation of Mathematics Monographs, 33, AMS, Providence, RI, 1972. [42] Sanz-Serna JM and Calvo MP, Numerical Hamiltonian Problems, Chapman& Hall, Lon- don 1994. [43] Saunders D, Prolongations of Lie groupoids and Lie algebroids, Houston J. Math. 30 (3), (2004), 637–655. [44] Veselov AP and Veselova LE, Integrable nonholonomic systems on Lie groups, Math. Notes 44 (1989), 810–819. [45] Weinstein A, Lagrangian Mechanics and groupoids, Fields Inst. Comm. 7 (1996), 207–231. [46] Zenkov D and Bloch AM, Invariant measures of nonholonomic flows with internal degrees of freedom, Nonlinearity 16 (2003), 1793–1807. D. Iglesias: Instituto de Matemáticas y F́ısica Fundamental, Consejo Superior de Investigaciones Cient́ıficas, Serrano 123, 28006 Madrid, Spain E-mail address: iglesias@imaff.cfmac.csic.es Juan C. Marrero: Departamento de Matemática Fundamental, Facultad de Matemá- ticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain E-mail address: jcmarrer@ull.es D. Mart́ın de Diego: Instituto de Matemáticas y F́ısica Fundamental, Consejo Supe- rior de Investigaciones Cient́ıficas, Serrano 123, 28006 Madrid, Spain E-mail address: d.martin@imaff.cfmac.csic.es Eduardo Mart́ınez: Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain E-mail address: emf@unizar.es 1. Introduction 2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids 2.1. Lie algebroids 2.2. Lie groupoids 2.3. Discrete Unconstrained Lagrangian Systems 3. Discrete Nonholonomic (or constrained) Lagrangian systems on Lie groupoids 3.1. Discrete Generalized Hölder's principle 3.2. Discrete Nonholonomic Legendre transformations 3.3. Nonholonomic evolution operators and regular discrete nonholonomic Lagrangian systems 3.4. Reversible discrete nonholonomic Lagrangian systems 3.5. Lie groupoid morphisms and reduction 3.6. Discrete nonholonomic Hamiltonian evolution operator 3.7. The discrete nonholonomic momentum map 4. Examples 4.1. Discrete holonomic Lagrangian systems on a Lie groupoid 4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid 4.3. Discrete nonholonomic Lagrangian systems on a Lie group 4.4. Discrete nonholonomic Lagrangian systems on an action Lie groupoid 4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid 4.6. Discrete Chaplygin systems 5. Conclusions and Future Work References

Pseudogap and charge density waves in two dimensions S. V. Borisenko1, A. A. Kordyuk1,2, A. N. Yaresko3, V. B. Zabolotnyy1, D. S. Inosov1, R. Schuster1, B. Büchner1, R. Weber4, R. Follath4, L. Patthey5, H. Berger6 1Leibniz-Institute for Solid State Research, IFW-Dresden, D-01171, Dresden, Germany 2Institute of Metal Physics, 03142 Kyiv, Ukraine 3Max-Planck-Institute for the Physics of Complex Systems, Dresden, Germany 4BESSY, Berlin, Germany 5Swiss Light Source, Paul Scherrer Institut, CH-5234 Villigen, Switzerland 6Institute of Physics of Complex Matter, EPFL, 1015 Lausanne, Switzerland An interaction between electrons and lattice vibrations (phonons) results in two fundamental quantum phenomena in solids: in three dimensions it can turn a metal into a superconductor whereas in one dimension it can turn a metal into an insulator1, 2, 3. In two dimensions (2D) both superconductivity and charge-density waves (CDW)4, 5 are believed to be anomalous. In superconducting cuprates, critical transition temperatures are unusually high and the energy gap may stay unclosed even above these temperatures (pseudogap). In CDW-bearing dichalcogenides the resistivity below the transition can decrease with temperature even faster than in the normal phase6, 7 and a basic prerequisite for the CDW, the favourable nesting conditions (when some sections of the Fermi surface appear shifted by the same vector), seems to be absent8, 9, 10. Notwithstanding the existence of alternatives11, 12, 13, 14, 15 to conventional theories1, 2, 3, both phenomena in 2D still remain the most fascinating puzzles in condensed matter physics. Using the latest developments in high-resolution angle-resolved photoemission spectroscopy (ARPES) here we show that the normal-state pseudogap also exists in one of the most studied 2D examples, dichalcogenide 2H-TaSe2, and the formation of CDW is driven by a conventional nesting instability, which is masked by the pseudogap. Our findings reconcile and explain a number of unusual, as previously believed, experimental responses as well as disprove many alternative theoretical approaches12, 13, 14, 15. The magnitude, character and anisotropy of the 2D-CDW pseudogap are intriguingly similar to those seen in superconducting cuprates. Variations of the electron density in a metal are highly unfavourable because of the Coulomb repulsion. Though, some low-dimensional systems such as transition-metal dichalcogenides spontaneously develop a static periodic modulation (known as charge-density wave) of the electron gas below a certain temperature. A typical representative of the transition metal dichalcogenides is a 2H (trigonal prismatic) polytype of TaSe2 which exhibits two CDW phase transitions at accessible temperatures: a second-order one at TNIC=122 K from normal to an incommensurate CDW state and a first-order lock-in transition at TICC=90 K from the incommensurate to a 3x3 commensurate CDW phase16. In the one-dimensional case, where all points of the Fermi surface (FS) can be connected by the same vector (perfect nesting), the CDW transition occurs when the energy gain due to opening of a gap at the Fermi level exceeds the energy costs to distort the lattice2, 4, 17. In real 2D materials the energy balance is more delicate since all FS points cannot be connected by the same vector (non-perfect nesting) and thus the FS can be gapped only partially. To understand the CDW mechanism in 2H-TaSe2, a detailed knowledge of the low-energy electronic structure is required. We therefore start with an overview of its temperature evolution in Figs. 1 and 2. The upper panel of Fig. 1 shows the topology of the Fermi surface in the normal state together with the CDW vectors Mqn Γ= 3 defined by other experiments5, 16. Contrary to the earlier band structure calculations8, but in accordance with a recent study10, the FS consists of single hole-like “barrels” centred at Γ and K points, and electron-like “dogbones” around the M-point. The upper row of panels in Fig. 2a shows the corresponding dispersions of the electronic states crossing the Fermi level. FS sheets originate from two bands: one is responsible for the Γ and K barrels with a saddle point in between, the other one supports the dogbone with M being another saddle point. The second row of panels in Fig. 2a showing the data below the first phase transition, suggests that the normal ( 290 K ) and incommensurate CDW state ( TICC < 107 K < TNIC ) dispersions are qualitatively similar, except for the naturally different temperature broadening and weaker crossing #3 (Fig. 2a). This seems to be in agreement with the surprising earlier ARPES results, when virtually no change of the electronic structure has been detected upon entering the incommensurate CDW phase9, 10, 18, thus clearly contradicting the concept of the energy gain that should accompany a CDW transition. In contrast, the lock-in transition to the commensurate CDW state at TICC=90 K is much more pronounced (see lower panel of Fig.1). The new folded FS is schematically shown in Fig. 1 as a set of nearly circles around new Γ’-points and rounded triangles around new K’-points. This topology of the folded FS, though natural, as suggested by the normal state FS (see Supplementary material), has never been detected before. A possible reason could be a rather weak umklapp potential due to small lattice distortion of the order of 0.05 Å (Ref 16, 17), which consequently results in the intensity distribution along the FS that still reminds the one seen in the normal state. Note, that without such an overview of the large portion of the k-space, it is problematic to understand what exactly happens to the electronic structure below 90 K. The energy- momentum intensity distributions in the lower row of panels in Fig. 2a are also considerably modified by the 3x3 folding and show clear signatures of a strong hybridization. Exactly this hybridization, when occurs in the vicinity of the Fermi level, can lower the energy of the system (see right inset to Fig. 3c). This observation of strong changes at TICC is again paradoxical. According to other experimental techniques, the CDW phase transition with the critical energy lowering occurs at TNIC=122 K as is clearly seen in e.g. temperature dependences of the specific heat or resistivity6, 7. The lock-in transition at TICC=90 K, in contrast, is hardly detectable in the mentioned curves6, 7 and appears as a small break in the superlattice strength as seen by neutron scattering16. More detailed data analysis clarifies the situation. ARPES offers a unique opportunity to find out whether the energy gap opens up at the Fermi level anywhere on the FS by tracking the binding energy of the leading edge of an energy distribution curve (EDC) taken at kF. In Fig. 2b we show EDCs corresponding exactly to kF for selected high symmetry cuts in the k-space (Fig. 2a). Both panels unambiguously signal the leading edge shift (~15 meV) of the EDC #6. In both cases a clear suppression (not absence) of the spectral weight at the Fermi level is evident also from the corresponding energy-momentum distributions (rightmost panels in Fig. 2a), which is reinforced by an obviously more diffused appearance of the K-barrels on the normal-state FS map (Fig. 1). In a striking analogy with the superconducting cuprates, we thus conclude the presence of a pseudogap in both, normal and incommensurate CDW states of 2H-TaSe2. In order to understand whether necessary energy gain can come from the pseudogap, we further characterize the pseudogap as a function of temperature and momentum in Fig. 3. The so called maps of gaps19, the plots of binding energies of the EDC leading edges as a function of k (i.e. not only kF), are shown for the normal and incommensurate CDW states in Fig. 3a,b. These maps are a visual demonstration of the energy lowering of the system and are ideal for the determination of the anisotropy of the gap as they make the analysis of the behaviour of the leading edge in the vicinity of the Fermi surface (shown as dotted lines) possible. While the normal state map (Fig. 3a) reveals an isotropic pseudogap detectable only on the K-barrel, in the CDW state a pseudogap opens, in addition, on the dogbone FS and is anisotropic. As a quantitative measure of the pseudogap we take the difference of the leading edge binding energies of the kF-EDCs of M-dogbone and K-barrel which belong to the M-K cut, as is sketched in the left inset to the Fig. 3c. The sharp reproducible increase of the pseudogap magnitude below ~122 K, which escaped the detection before, is distinctly seen in Fig. 3c. It is now clear that it is the NIC transition (TNIC=122K) at which the critical lowering of the electronic energy occurs, and not only because of the larger pseudogap on the K-barrel, but also owing to the opening of the anisotropic pseudogap on the dogbone M-centred FS. In the commensurate CDW phase one can no longer characterize the energy gap by the leading edge position because of the interference with the folded bands --- the rounded corners of the new small triangular FS fall right where the normal state K-barrel was located (see Fig. 1). Instead, we plot in Fig. 3c also the values determined as shown in the right inset. The “band-gap” is a direct consequence of the hybridization, distinctly observed below TICC (Figs. 1, 2). In order to emphasize the existence of a crossover regime where the pseudogap evolves into a band-gap we also plot in a limited T- interval the leading edge positions below TICC and the band-gap above TICC. Both gaps do not exhibit any anomalies at TICC monotonically increasing upon cooling deeper in the CDW state. Presented data already put certain limitations on several alternative theoretical approaches that were stimulated by the earlier experiments. The locations of the saddle points in the momentum-energy space obviously do not support the saddle point nesting scenario13 , proposed as an alternative to the conventional nesting, as both points are far from the Fermi level (280 meV and 330 meV) and are not connected by any of the CDW vectors (the first one is located between Γ- and K-barrels, and the second is the M-point itself) . The pseudogap supported by the K-centred FS stays isotropic (within ~ 2 meV) thus ruling out a six-fold symmetric CDW gap with nodes as suggested in Ref. 14. The value of the band-gap saturates at low temperatures at ~33 meV which is nearly a factor of 5 smaller than the one (~ 150 meV) obtained in the strong-coupling approach12. A recent proposal15 to consider two components of the electronic structure, one of which is gapped and the other one is not, seems to be not supported by the data as well. What is then responsible for the CDW in two dimensions? In the following we demonstrate that the conventional FS nesting scenario, though modified by the presence of the normal-state pseudogap, is perfectly applicable. From our high-resolution measurements, we have analysed the nesting properties of the FS quantitatively, which made it possible to explain the temperature evolution of the electronic structure of TaSe2 step by step. The quantitative measure of the nesting is the charge susceptibility. Here we approximate the charge susceptibility by the autocorrelation of the FS map20. In order to avoid the influence of the matrix elements, for further processing we take the model FS map shown in Fig. 4a, which is an exact copy of the experimental one as far as the locus of kF-points is concerned. In addition, this gives possibility to investigate the FS nesting properties of a hypothetical compound, with the electronic structure that yet have not reacted to the nesting instability (i.e. without gaps). The resulting autocorrelation maps as a function of temperature are shown in Fig. 4b, while the corresponding cuts along the ΓM direction -- in Fig. 4d. The sharp peak, seen in the 290 K curve exactly at 2/3 ΓM in Fig. 4d, is the first clear evidence for the nearly perfect nesting in 2D chalcogenides. This is at variance with the previous calculations17 which have found susceptibility for 2H polytypes to take a broadly humped form. Thus, the Fermi surface shape alone, without taking into account the pseudogap, results in the peak in the charge susceptibility at a wave vector ~2/3ΓM. According to the neutron scattering16, exactly at this wave vector there is a strong Kohn-like anomaly4 of the Σ1 phonon branch already at 300 K, and the matching phonon softens even more as the transition is approached. We consider therefore the formation of both, normal state pseudogap and anomaly of the Σ1 phonon branch as respective reactions of the electronic and lattice subsystems to the instability caused by the strong scattering channel that appears due to the nesting and a presence of the suitable mediating phonon. Such a mutual response can signify a strong electron-phonon interaction in 2H-TaSe2. It is interesting, that despite the favourable conditions, the system does not escape the instability by developing a static commensurate CDW order, presumably because of still too high temperature, which would effectively close not large enough band-gaps. Instead, it opens up a pseudogap. Upon cooling we observe correlated changes of the susceptibility, pseudogap magnitude, and energy of the softened phonon16: the peak in the susceptibility splits into two (middle panel in Fig. 4d), the pseudogap slowly closes (as suggested by the EDC shift at 290 K in Fig. 2b and a weak but detectable trend of the gap to decrease in Fig. 3c) , the phonon energy becomes lower. This worsening of the nesting conditions is fundamentally different from the 1D case, where the susceptibility exhibits a peak, which becomes sharper with lowering the temperature4. We have found out that such anomalous behaviour of the susceptibility, i.e. the splitting and the size of the peak at ~2/3ΓM, is very sensitive to the shape of the Fermi surface itself, namely to the distance between the dogbone and K-centred FSs (distance, marked ‘D’ in Fig. 4a). The data plotted in Fig. 4c clearly suggest that the FS itself is temperature dependent explaining the variation of the charge susceptibility. Moreover, a close correlation with the temperature dependence of the pseudogap magnitude (Fig. 3c) implies that the pseudogap itself is directly related to the charge susceptibility. Nevertheless, the transition finally occurs at TNIC=122 K , but now only into the incommensurate CDW phase as dictated by the split peak in the susceptibility. Further reduction of the temperature results in a reversed modification of the obviously correlated quantities: pseudogap opens up more, peaks in susceptibility move towards each other (right panel in Fig. 4d), which agrees well with the dynamics of the superlattice peaks seen by neutrons, and phonon energy starts to increase again16. In such a manner the system arrives at the transition into the commensurate CDW state at 90 K. Our findings provide a natural explanation not only for the neutron scattering experiments16. Nearly linear in-plane resistivity in the normal state6, 7 is similar to the resistivity of an optimally doped cuprate superconductor21 and shows a typical behaviour of a pseudo-gapped metal. Below 122 K the slope is increasing in close correspondence to the larger pseudogap, and below 90 K the resistivity resembles the one of a normal metal. Optical measurements on the same single crystals7 have indirectly suggested the presence of a pseudogap already at 300 K as well, though with somewhat different energy scale. We also notice an excellent agreement with the Hall coefficient measurements22. According to Ref.22, the Hall coefficient starts to decrease sharply from its positive value below ~120 K, and changes sign at 90 K. The positive value in the normal state is explained by the larger volume of the Γ- and K-centred hole-like barrels in comparison with the electron-like dogbones around the M-points. The sharp increase of the pseudogap at 122 K results in the reduction of the number of charge carriers and finally in the commensurate phase (below 90 K), the area enclosed by the electron-like circular FS around new Γ’-points is clearly larger than the two (per new BZ) hole-like triangular FS, which leads to the negative sign of the Hall coefficient. It was suggested before23 that the HTSCs and 2D chalcogenides have similar phase diagrams and that the pseudogap regime in cuprates is very similar to the CDW regime of chalcogenides. Now, after observation of the pseudogap in TaSe2 we can directly compare both pseudogaps. The energy- momentum distribution of the photoemission intensity near kF of the K-barrel (crossing #6, Fig. 2a) is very similar to the one measured for the bonding barrel in the underdoped Bi2212-cuprate24, 25. In both cases one can still track the dispersion up to the Fermi level, but the spectral weight is significantly suppressed resulting in the shift of the kF-EDC’s leading edge. Furthermore, the detected anisotropy of the pseudogap on the dogbone FS is reminiscent of the famous anisotropic behaviour of the pseudogap in cuprates. Finally, the highly inhomogeneous intensity distribution along some of the small triangular FSs and especially along incomplete circles around new Γ’ points cannot escape a comparison with the famous Fermi surface ‘arcs’. We believe that this series of remarkable similarities to the high-temperature superconducting cuprates calls for more careful comparative studies of the pseudogap phenomenon in these materials. It is interesting, that unlike in the 1D case where a pseudogap has been reported26, 27, 28 to be a consequence of fluctuations (potentially able to suppress the transition temperature up to a quarter of the mean-field value29), the pseudogap in 2D shows unusual non-monotonic behaviour clearly tracking the temperature evolution of both, the bare susceptibility and phonon spectrum, and thus seems to represent a natural response of the system to a nesting instability. 1. Bardeen, J., Cooper, L. N. and Schrieffer, J. R. Theory of Superconductivity. Phys. Rev. 108, 1175 (1957). 2. Peierls, R. E. Quantum Theory of Solids (Oxford: OUP, 1955) p. 108. 3. Frohlich, H. On the Theory of Superconductivity: The One-Dimensional Case. Proc. R. Soc. A 223, 296 (1954). 4. Gruener, G. Density Waves in Solids (Addison-Wesley, Reading, MA, 1994). 5. Wilson, J. A., DiSalvo, F. J. & Mahajan, S. Charge-density waves and superlattices in the metallic layered transition metal diehaleogenides. Adv. Phys. 24, 117 (1975). 6. Craven, R. A. & Meyer, S. F. Specific Heat and Resistivity near the Charge-Density-Wave Phase Transitions in 2H-TaSe2 and 2H-TaS2. Phys. Rev. B 16, 4583 (1977). 7. Vescoli, V. et al. Dynamics of Correlated Two-Dimensional Materials: The 2H-TaSe2 Case. Phys. Rev. Lett. 81, 453 (1998). 8. Wexler, G. & Woolley, A. M. Fermi surfaces and band structures of the 2H metallic transition- metal dichalcogenides. J. Phys. C 9, 1185 (1976). 9. Liu, R. et al. Fermi surface of 2H-TaSe2 and its relation to the charge-density wave. Phys. Rev. B 61, 5212 (2000), Momentum Dependent Spectral Changes Induced by the Charge Density Wave in 2H-TaSe2 and the Implication on the CDW Mechanism. Phys. Lett. 80, 5762 (1998). 10. Rossnagel, K. et al. Fermi surface, charge-density-wave gap, and kinks in 2H-TaSe2. Phys. Rev. B. 72, 121103(R) (2005). 11. Zaanen, J. et al. Towards a complete theory of high Tc. Nature Physics 2, 138 - 143 (2006). 12. McMillan, W. L. Microscopic model of charge-density waves in 2H-TaSe2. Phys. Rev. B 16, 643 (1977). 13. Rice,T. M. & Scott, G. K. New Mechanism for a Charge-Density-Wave Instability. Phys. Rev. Lett. 35, 120 (1975). 14. Castro Neto, A. H. Charge Density Wave, Superconductivity, and Anomalous Metallic Behavior in 2D Transition Metal Dichalcogenides. Phys. Rev. Lett. 86, 4382 (2001). 15. Barnett, R. L. et al. Coexistence of Gapless Excitations and Commensurate Charge-Density Wave in the 2 H Transition Metal Dichalcogenides. Phys. Rev. Lett. 96, 026406 (2006). 16. Moncton, D. E., Axe, J. D. & DiSalvo, F. J. Neutron Scattering Study of the Charge-Density Wave Transitions in 2H-TaSe2 and 2H-NbSe2. Phys. Rev. B 16, 801 (1977). 17. Withers, R. L. & Wilson, J. A. An examination of the formation and characteristics of charge- density waves in inorganic materials with special reference to the two- and one-dimensional transition-metal chalcogenides. J. Phys. C 19, 4809 (1986). 18. Valla, T. et al. Charge-Density-Wave-Induced Modifications to the Quasiparticle Self-Energy in 2H-TaSe2. Phys. Rev. Lett. 85, 4759 (2000). 19. Borisenko, S. V. et al. Superconducting gap in the presence of bilayer splitting in underdoped Bi(Pb)2212. Phys. Rev. B 66, 140509(R) (2002). 20. Kordyuk A. A. et al. From tunneling to photoemission: Correlating two spaces. J. Electron Spectrosc. Rel. Phenom. (2007), in press. 21. Ando, Y. et al. Electronic Phase Diagram of High-Tc Cuprate Superconductors from a Mapping of the In-Plane Resistivity Curvature. Phys. Rev. Lett. 93, 267001 (2004). 22. Lee, H. N. S. et al. The Low-Temperature Electrical and Magnetic Properties of TaSe2 and NbSe2. Journ. Solid State Chem. 1, 190-194 (1970). 23. Klemm, R. A. Striking Similarities Between the Pseudogap Phenomena in Cuprates and in Layered Organic and Dichalcogenide Superconductors Physica C 341-348, 839 (2000). 24. Eckl, T. et al. Change of quasiparticle dispersion in crossing Tc in the underdoped cuprates. Phys. Rev. B 70, 094522 (2004). 25. Borisenko, S. V. et al. Anomalous Enhancement of the Coupling to the Magnetic Resonance Mode in Underdoped Pb-Bi2212. Phys. Rev. Lett. 90, 207001 (2003). 26. Schafer, J. et al. High-Temperature Symmetry Breaking in the Electronic Band Structure of the Quasi-One-Dimensional Solid NbSe3. Phys. Rev. Lett. 87, 196403 (2001). 27. Schafer, J. et al. Unusual Spectral Behavior of Charge-Density Waves with Imperfect Nesting in a Quasi-One-Dimensional Metal. Phys. Rev. Lett. 91, 066401 (2003). 28. Yokoya, T. et al. Role of charge-density-wave fluctuations on the spectral function in a metallic charge-density-wave system. Phys. Rev. B 71, 140504(R) (2005). 29. Lee, P. A., Rice, T. M. & Anderson, P. W. Fluctuation Effects at a Peierls Transition. Phys. Rev. Lett. 31, 462 (1973). Acknowledgements: The project was supported, in part, by the DFG under Grant No. KN393/4. We thank R. Hübel for technical support. This work was partially performed at the Swiss Light Source, Paul Scherrer Institut, Villigen, Switzerland. Correspondence and requests for materials should be addressed to S.V.B. (S. Borisenko@ifw-dresden.de). Figure 1. Fermi surfaces. Momentum distribution of the photoemission intensity at the Fermi level at 180 K and 30 K. Short dashed lines are the BZ boundaries. Figure 2. Electronic structure and leading edge gap. a) Photoemission intensity as a function of energy and momentum in the normal (upper row of panels), incommensurate CDW (middle row) and commensurate CDW (lower row) states. Sketches of the FS on the middle panels show cuts in momentum space along which the data were taken and are valid for all panels in the same column. Hybridization effects in the lower row are seen as “repulsions” of the bands, which occur when a folded band is supposed to cross the original one, as schematically shown in the right inset to Fig. 3b. Since the spectral weight of the folded band is lower, these effects appear as breaks in the intensity of the original bands. Numbers correspond to the different kF. b) kF EDCs from the datasets shown in a). Leading edge gap is clearly seen as a shift of the EDC#6 to the higher binding energies at 290 K and 107 K. Figure 3. Momentum and temperature dependence of the pseudogap. a,b) Binding energies of the leading edges of all EDCs from the momentum range close to the irreducible parts of the M-dogbone and K-barrel FS. Colour scales reproduce the vertical coordinate. Anisotropy of the pseudogap on the M-dogbone FS is seen as a changing colour when going along the dashed line which correspond to FS, i.e. kF points. c) Difference between the binding energies of the leading edges of the EDC#5 and #6 from Fig. 2b as a function of temperature as shown schematically in the left inset (pseudogap) when cycling the temperature (filled symbols). Shift of the EDC maximum (bandgap) which corresponds to the top of the hybridized band as shown in the right inset with respect to its position at TNIC=122 K (open symbols). Note that the given leading edge gap values below 90 K cannot be considered as a measure of the pseudogap because of folding. Figure 4. Nesting properties. a) A model copy of the FS map from Fig. 1 with the homogeneous intensity distribution along the FS. b) Autocorrelation maps of this model at different temperatures. c) Temperature dependence of the momentum distance between the M-dogbone and K-barrel (T > 90 K). The distance below 90 K is just a difference between corresponding maxima at the FS map. d) Intensity of the maps from b) along the cuts shown by dotted lines. Supplementary information. Here we explain in more details the nesting properties of the normal state FS and how to understand the 3x3 folding upon entering the commensurate CDW regime. In panel a we show only Γ- and K-centred barrels and their first order ( 1q ± , 2q ± , 3q ± ) and second order ( 21 qq + , 32 qq + ) replica. One can see that the nesting conditions are nearly perfect for all points of the K-barrel: shifting the Γ-barrel by one of the CDW vectors results in an overlap with the corresponding part of the K-barrel with opposite Fermi velocities. Panel b shows an idealized situation, which is actually very close to the real one, since a more careful examination of Fig. 1a reveals the straight sections of both Γ- and K-barrels. This is, however, not enough to explain the fully gapped K-barrel in the CDW state. In panel a it is seen that 3x3 folding produces a double-walled barrels in the centre of the new BZ which are supposed to interact with the single copies of Γ-barrel. This is not surprising as the original BZ contains only one complete Γ-barrel and two complete K-barrels. It means, that the K- barrels of a system with the FS schematically shown in panel b, will not disappear after a 3x3 folding. Panel c shows the dogbone FS centred around M-points together with its first- and second order replica. Folding of this FS sheet results in the same triangular FSs as in panel a, a rather complicated set of features around the centres of the new BZ and nearly exact copies of the Γ- barrels and its replica. It is these copies of the Γ-barrels which were missing in panel a to interact with double-walled K-barrels. Panel d, where all FSs and their folded replica are shown, summarizes all mentioned above. When hybridization effects are switched on, the former K-barrel completely disappears, doubly degenerate triangular FSs emerge around the corners of the new BZ and the complicated set of features seems to evolve into a four-times degenerate nearly circle FSs around the Γ’- points (at least experimentally we can currently resolve only these semi-circular FSs and not two doubly degenerate hexagons with rounded corners as is suggested by the panel c).

An interaction between electrons and lattice vibrations (phonons) results in two fundamental quantum phenomena in solids: in three dimensions it can turn a metal into a superconductor whereas in one dimension it can turn a metal into an insulator. In two dimensions (2D) both superconductivity and charge-density waves (CDW) are believed to be anomalous. In superconducting cuprates, critical transition temperatures are unusually high and the energy gap may stay unclosed even above these temperatures (pseudogap). In CDW-bearing dichalcogenides the resistivity below the transition can decrease with temperature even faster than in the normal phase and a basic prerequisite for the CDW, the favourable nesting conditions (when some sections of the Fermi surface appear shifted by the same vector), seems to be absent. Notwithstanding the existence of alternatives to conventional theories, both phenomena in 2D still remain the most fascinating puzzles in condensed matter physics. Using the latest developments in high-resolution angle-resolved photoemission spectroscopy (ARPES) here we show that the normal-state pseudogap also exists in one of the most studied 2D examples, dichalcogenide 2H-TaSe2, and the formation of CDW is driven by a conventional nesting instability, which is masked by the pseudogap. Our findings reconcile and explain a number of unusual, as previously believed, experimental responses as well as disprove many alternative theoretical approaches. The magnitude, character and anisotropy of the 2D-CDW pseudogap are intriguingly similar to those seen in superconducting cuprates.

Pseudogap and charge density waves in two dimensions S. V. Borisenko1, A. A. Kordyuk1,2, A. N. Yaresko3, V. B. Zabolotnyy1, D. S. Inosov1, R. Schuster1, B. Büchner1, R. Weber4, R. Follath4, L. Patthey5, H. Berger6 1Leibniz-Institute for Solid State Research, IFW-Dresden, D-01171, Dresden, Germany 2Institute of Metal Physics, 03142 Kyiv, Ukraine 3Max-Planck-Institute for the Physics of Complex Systems, Dresden, Germany 4BESSY, Berlin, Germany 5Swiss Light Source, Paul Scherrer Institut, CH-5234 Villigen, Switzerland 6Institute of Physics of Complex Matter, EPFL, 1015 Lausanne, Switzerland An interaction between electrons and lattice vibrations (phonons) results in two fundamental quantum phenomena in solids: in three dimensions it can turn a metal into a superconductor whereas in one dimension it can turn a metal into an insulator1, 2, 3. In two dimensions (2D) both superconductivity and charge-density waves (CDW)4, 5 are believed to be anomalous. In superconducting cuprates, critical transition temperatures are unusually high and the energy gap may stay unclosed even above these temperatures (pseudogap). In CDW-bearing dichalcogenides the resistivity below the transition can decrease with temperature even faster than in the normal phase6, 7 and a basic prerequisite for the CDW, the favourable nesting conditions (when some sections of the Fermi surface appear shifted by the same vector), seems to be absent8, 9, 10. Notwithstanding the existence of alternatives11, 12, 13, 14, 15 to conventional theories1, 2, 3, both phenomena in 2D still remain the most fascinating puzzles in condensed matter physics. Using the latest developments in high-resolution angle-resolved photoemission spectroscopy (ARPES) here we show that the normal-state pseudogap also exists in one of the most studied 2D examples, dichalcogenide 2H-TaSe2, and the formation of CDW is driven by a conventional nesting instability, which is masked by the pseudogap. Our findings reconcile and explain a number of unusual, as previously believed, experimental responses as well as disprove many alternative theoretical approaches12, 13, 14, 15. The magnitude, character and anisotropy of the 2D-CDW pseudogap are intriguingly similar to those seen in superconducting cuprates. Variations of the electron density in a metal are highly unfavourable because of the Coulomb repulsion. Though, some low-dimensional systems such as transition-metal dichalcogenides spontaneously develop a static periodic modulation (known as charge-density wave) of the electron gas below a certain temperature. A typical representative of the transition metal dichalcogenides is a 2H (trigonal prismatic) polytype of TaSe2 which exhibits two CDW phase transitions at accessible temperatures: a second-order one at TNIC=122 K from normal to an incommensurate CDW state and a first-order lock-in transition at TICC=90 K from the incommensurate to a 3x3 commensurate CDW phase16. In the one-dimensional case, where all points of the Fermi surface (FS) can be connected by the same vector (perfect nesting), the CDW transition occurs when the energy gain due to opening of a gap at the Fermi level exceeds the energy costs to distort the lattice2, 4, 17. In real 2D materials the energy balance is more delicate since all FS points cannot be connected by the same vector (non-perfect nesting) and thus the FS can be gapped only partially. To understand the CDW mechanism in 2H-TaSe2, a detailed knowledge of the low-energy electronic structure is required. We therefore start with an overview of its temperature evolution in Figs. 1 and 2. The upper panel of Fig. 1 shows the topology of the Fermi surface in the normal state together with the CDW vectors Mqn Γ= 3 defined by other experiments5, 16. Contrary to the earlier band structure calculations8, but in accordance with a recent study10, the FS consists of single hole-like “barrels” centred at Γ and K points, and electron-like “dogbones” around the M-point. The upper row of panels in Fig. 2a shows the corresponding dispersions of the electronic states crossing the Fermi level. FS sheets originate from two bands: one is responsible for the Γ and K barrels with a saddle point in between, the other one supports the dogbone with M being another saddle point. The second row of panels in Fig. 2a showing the data below the first phase transition, suggests that the normal ( 290 K ) and incommensurate CDW state ( TICC < 107 K < TNIC ) dispersions are qualitatively similar, except for the naturally different temperature broadening and weaker crossing #3 (Fig. 2a). This seems to be in agreement with the surprising earlier ARPES results, when virtually no change of the electronic structure has been detected upon entering the incommensurate CDW phase9, 10, 18, thus clearly contradicting the concept of the energy gain that should accompany a CDW transition. In contrast, the lock-in transition to the commensurate CDW state at TICC=90 K is much more pronounced (see lower panel of Fig.1). The new folded FS is schematically shown in Fig. 1 as a set of nearly circles around new Γ’-points and rounded triangles around new K’-points. This topology of the folded FS, though natural, as suggested by the normal state FS (see Supplementary material), has never been detected before. A possible reason could be a rather weak umklapp potential due to small lattice distortion of the order of 0.05 Å (Ref 16, 17), which consequently results in the intensity distribution along the FS that still reminds the one seen in the normal state. Note, that without such an overview of the large portion of the k-space, it is problematic to understand what exactly happens to the electronic structure below 90 K. The energy- momentum intensity distributions in the lower row of panels in Fig. 2a are also considerably modified by the 3x3 folding and show clear signatures of a strong hybridization. Exactly this hybridization, when occurs in the vicinity of the Fermi level, can lower the energy of the system (see right inset to Fig. 3c). This observation of strong changes at TICC is again paradoxical. According to other experimental techniques, the CDW phase transition with the critical energy lowering occurs at TNIC=122 K as is clearly seen in e.g. temperature dependences of the specific heat or resistivity6, 7. The lock-in transition at TICC=90 K, in contrast, is hardly detectable in the mentioned curves6, 7 and appears as a small break in the superlattice strength as seen by neutron scattering16. More detailed data analysis clarifies the situation. ARPES offers a unique opportunity to find out whether the energy gap opens up at the Fermi level anywhere on the FS by tracking the binding energy of the leading edge of an energy distribution curve (EDC) taken at kF. In Fig. 2b we show EDCs corresponding exactly to kF for selected high symmetry cuts in the k-space (Fig. 2a). Both panels unambiguously signal the leading edge shift (~15 meV) of the EDC #6. In both cases a clear suppression (not absence) of the spectral weight at the Fermi level is evident also from the corresponding energy-momentum distributions (rightmost panels in Fig. 2a), which is reinforced by an obviously more diffused appearance of the K-barrels on the normal-state FS map (Fig. 1). In a striking analogy with the superconducting cuprates, we thus conclude the presence of a pseudogap in both, normal and incommensurate CDW states of 2H-TaSe2. In order to understand whether necessary energy gain can come from the pseudogap, we further characterize the pseudogap as a function of temperature and momentum in Fig. 3. The so called maps of gaps19, the plots of binding energies of the EDC leading edges as a function of k (i.e. not only kF), are shown for the normal and incommensurate CDW states in Fig. 3a,b. These maps are a visual demonstration of the energy lowering of the system and are ideal for the determination of the anisotropy of the gap as they make the analysis of the behaviour of the leading edge in the vicinity of the Fermi surface (shown as dotted lines) possible. While the normal state map (Fig. 3a) reveals an isotropic pseudogap detectable only on the K-barrel, in the CDW state a pseudogap opens, in addition, on the dogbone FS and is anisotropic. As a quantitative measure of the pseudogap we take the difference of the leading edge binding energies of the kF-EDCs of M-dogbone and K-barrel which belong to the M-K cut, as is sketched in the left inset to the Fig. 3c. The sharp reproducible increase of the pseudogap magnitude below ~122 K, which escaped the detection before, is distinctly seen in Fig. 3c. It is now clear that it is the NIC transition (TNIC=122K) at which the critical lowering of the electronic energy occurs, and not only because of the larger pseudogap on the K-barrel, but also owing to the opening of the anisotropic pseudogap on the dogbone M-centred FS. In the commensurate CDW phase one can no longer characterize the energy gap by the leading edge position because of the interference with the folded bands --- the rounded corners of the new small triangular FS fall right where the normal state K-barrel was located (see Fig. 1). Instead, we plot in Fig. 3c also the values determined as shown in the right inset. The “band-gap” is a direct consequence of the hybridization, distinctly observed below TICC (Figs. 1, 2). In order to emphasize the existence of a crossover regime where the pseudogap evolves into a band-gap we also plot in a limited T- interval the leading edge positions below TICC and the band-gap above TICC. Both gaps do not exhibit any anomalies at TICC monotonically increasing upon cooling deeper in the CDW state. Presented data already put certain limitations on several alternative theoretical approaches that were stimulated by the earlier experiments. The locations of the saddle points in the momentum-energy space obviously do not support the saddle point nesting scenario13 , proposed as an alternative to the conventional nesting, as both points are far from the Fermi level (280 meV and 330 meV) and are not connected by any of the CDW vectors (the first one is located between Γ- and K-barrels, and the second is the M-point itself) . The pseudogap supported by the K-centred FS stays isotropic (within ~ 2 meV) thus ruling out a six-fold symmetric CDW gap with nodes as suggested in Ref. 14. The value of the band-gap saturates at low temperatures at ~33 meV which is nearly a factor of 5 smaller than the one (~ 150 meV) obtained in the strong-coupling approach12. A recent proposal15 to consider two components of the electronic structure, one of which is gapped and the other one is not, seems to be not supported by the data as well. What is then responsible for the CDW in two dimensions? In the following we demonstrate that the conventional FS nesting scenario, though modified by the presence of the normal-state pseudogap, is perfectly applicable. From our high-resolution measurements, we have analysed the nesting properties of the FS quantitatively, which made it possible to explain the temperature evolution of the electronic structure of TaSe2 step by step. The quantitative measure of the nesting is the charge susceptibility. Here we approximate the charge susceptibility by the autocorrelation of the FS map20. In order to avoid the influence of the matrix elements, for further processing we take the model FS map shown in Fig. 4a, which is an exact copy of the experimental one as far as the locus of kF-points is concerned. In addition, this gives possibility to investigate the FS nesting properties of a hypothetical compound, with the electronic structure that yet have not reacted to the nesting instability (i.e. without gaps). The resulting autocorrelation maps as a function of temperature are shown in Fig. 4b, while the corresponding cuts along the ΓM direction -- in Fig. 4d. The sharp peak, seen in the 290 K curve exactly at 2/3 ΓM in Fig. 4d, is the first clear evidence for the nearly perfect nesting in 2D chalcogenides. This is at variance with the previous calculations17 which have found susceptibility for 2H polytypes to take a broadly humped form. Thus, the Fermi surface shape alone, without taking into account the pseudogap, results in the peak in the charge susceptibility at a wave vector ~2/3ΓM. According to the neutron scattering16, exactly at this wave vector there is a strong Kohn-like anomaly4 of the Σ1 phonon branch already at 300 K, and the matching phonon softens even more as the transition is approached. We consider therefore the formation of both, normal state pseudogap and anomaly of the Σ1 phonon branch as respective reactions of the electronic and lattice subsystems to the instability caused by the strong scattering channel that appears due to the nesting and a presence of the suitable mediating phonon. Such a mutual response can signify a strong electron-phonon interaction in 2H-TaSe2. It is interesting, that despite the favourable conditions, the system does not escape the instability by developing a static commensurate CDW order, presumably because of still too high temperature, which would effectively close not large enough band-gaps. Instead, it opens up a pseudogap. Upon cooling we observe correlated changes of the susceptibility, pseudogap magnitude, and energy of the softened phonon16: the peak in the susceptibility splits into two (middle panel in Fig. 4d), the pseudogap slowly closes (as suggested by the EDC shift at 290 K in Fig. 2b and a weak but detectable trend of the gap to decrease in Fig. 3c) , the phonon energy becomes lower. This worsening of the nesting conditions is fundamentally different from the 1D case, where the susceptibility exhibits a peak, which becomes sharper with lowering the temperature4. We have found out that such anomalous behaviour of the susceptibility, i.e. the splitting and the size of the peak at ~2/3ΓM, is very sensitive to the shape of the Fermi surface itself, namely to the distance between the dogbone and K-centred FSs (distance, marked ‘D’ in Fig. 4a). The data plotted in Fig. 4c clearly suggest that the FS itself is temperature dependent explaining the variation of the charge susceptibility. Moreover, a close correlation with the temperature dependence of the pseudogap magnitude (Fig. 3c) implies that the pseudogap itself is directly related to the charge susceptibility. Nevertheless, the transition finally occurs at TNIC=122 K , but now only into the incommensurate CDW phase as dictated by the split peak in the susceptibility. Further reduction of the temperature results in a reversed modification of the obviously correlated quantities: pseudogap opens up more, peaks in susceptibility move towards each other (right panel in Fig. 4d), which agrees well with the dynamics of the superlattice peaks seen by neutrons, and phonon energy starts to increase again16. In such a manner the system arrives at the transition into the commensurate CDW state at 90 K. Our findings provide a natural explanation not only for the neutron scattering experiments16. Nearly linear in-plane resistivity in the normal state6, 7 is similar to the resistivity of an optimally doped cuprate superconductor21 and shows a typical behaviour of a pseudo-gapped metal. Below 122 K the slope is increasing in close correspondence to the larger pseudogap, and below 90 K the resistivity resembles the one of a normal metal. Optical measurements on the same single crystals7 have indirectly suggested the presence of a pseudogap already at 300 K as well, though with somewhat different energy scale. We also notice an excellent agreement with the Hall coefficient measurements22. According to Ref.22, the Hall coefficient starts to decrease sharply from its positive value below ~120 K, and changes sign at 90 K. The positive value in the normal state is explained by the larger volume of the Γ- and K-centred hole-like barrels in comparison with the electron-like dogbones around the M-points. The sharp increase of the pseudogap at 122 K results in the reduction of the number of charge carriers and finally in the commensurate phase (below 90 K), the area enclosed by the electron-like circular FS around new Γ’-points is clearly larger than the two (per new BZ) hole-like triangular FS, which leads to the negative sign of the Hall coefficient. It was suggested before23 that the HTSCs and 2D chalcogenides have similar phase diagrams and that the pseudogap regime in cuprates is very similar to the CDW regime of chalcogenides. Now, after observation of the pseudogap in TaSe2 we can directly compare both pseudogaps. The energy- momentum distribution of the photoemission intensity near kF of the K-barrel (crossing #6, Fig. 2a) is very similar to the one measured for the bonding barrel in the underdoped Bi2212-cuprate24, 25. In both cases one can still track the dispersion up to the Fermi level, but the spectral weight is significantly suppressed resulting in the shift of the kF-EDC’s leading edge. Furthermore, the detected anisotropy of the pseudogap on the dogbone FS is reminiscent of the famous anisotropic behaviour of the pseudogap in cuprates. Finally, the highly inhomogeneous intensity distribution along some of the small triangular FSs and especially along incomplete circles around new Γ’ points cannot escape a comparison with the famous Fermi surface ‘arcs’. We believe that this series of remarkable similarities to the high-temperature superconducting cuprates calls for more careful comparative studies of the pseudogap phenomenon in these materials. It is interesting, that unlike in the 1D case where a pseudogap has been reported26, 27, 28 to be a consequence of fluctuations (potentially able to suppress the transition temperature up to a quarter of the mean-field value29), the pseudogap in 2D shows unusual non-monotonic behaviour clearly tracking the temperature evolution of both, the bare susceptibility and phonon spectrum, and thus seems to represent a natural response of the system to a nesting instability. 1. Bardeen, J., Cooper, L. N. and Schrieffer, J. R. Theory of Superconductivity. Phys. Rev. 108, 1175 (1957). 2. Peierls, R. E. Quantum Theory of Solids (Oxford: OUP, 1955) p. 108. 3. Frohlich, H. On the Theory of Superconductivity: The One-Dimensional Case. Proc. R. Soc. A 223, 296 (1954). 4. Gruener, G. Density Waves in Solids (Addison-Wesley, Reading, MA, 1994). 5. Wilson, J. A., DiSalvo, F. J. & Mahajan, S. Charge-density waves and superlattices in the metallic layered transition metal diehaleogenides. Adv. Phys. 24, 117 (1975). 6. Craven, R. A. & Meyer, S. F. Specific Heat and Resistivity near the Charge-Density-Wave Phase Transitions in 2H-TaSe2 and 2H-TaS2. Phys. Rev. B 16, 4583 (1977). 7. Vescoli, V. et al. Dynamics of Correlated Two-Dimensional Materials: The 2H-TaSe2 Case. Phys. Rev. Lett. 81, 453 (1998). 8. Wexler, G. & Woolley, A. M. Fermi surfaces and band structures of the 2H metallic transition- metal dichalcogenides. J. Phys. C 9, 1185 (1976). 9. Liu, R. et al. Fermi surface of 2H-TaSe2 and its relation to the charge-density wave. Phys. Rev. B 61, 5212 (2000), Momentum Dependent Spectral Changes Induced by the Charge Density Wave in 2H-TaSe2 and the Implication on the CDW Mechanism. Phys. Lett. 80, 5762 (1998). 10. Rossnagel, K. et al. Fermi surface, charge-density-wave gap, and kinks in 2H-TaSe2. Phys. Rev. B. 72, 121103(R) (2005). 11. Zaanen, J. et al. Towards a complete theory of high Tc. Nature Physics 2, 138 - 143 (2006). 12. McMillan, W. L. Microscopic model of charge-density waves in 2H-TaSe2. Phys. Rev. B 16, 643 (1977). 13. Rice,T. M. & Scott, G. K. New Mechanism for a Charge-Density-Wave Instability. Phys. Rev. Lett. 35, 120 (1975). 14. Castro Neto, A. H. Charge Density Wave, Superconductivity, and Anomalous Metallic Behavior in 2D Transition Metal Dichalcogenides. Phys. Rev. Lett. 86, 4382 (2001). 15. Barnett, R. L. et al. Coexistence of Gapless Excitations and Commensurate Charge-Density Wave in the 2 H Transition Metal Dichalcogenides. Phys. Rev. Lett. 96, 026406 (2006). 16. Moncton, D. E., Axe, J. D. & DiSalvo, F. J. Neutron Scattering Study of the Charge-Density Wave Transitions in 2H-TaSe2 and 2H-NbSe2. Phys. Rev. B 16, 801 (1977). 17. Withers, R. L. & Wilson, J. A. An examination of the formation and characteristics of charge- density waves in inorganic materials with special reference to the two- and one-dimensional transition-metal chalcogenides. J. Phys. C 19, 4809 (1986). 18. Valla, T. et al. Charge-Density-Wave-Induced Modifications to the Quasiparticle Self-Energy in 2H-TaSe2. Phys. Rev. Lett. 85, 4759 (2000). 19. Borisenko, S. V. et al. Superconducting gap in the presence of bilayer splitting in underdoped Bi(Pb)2212. Phys. Rev. B 66, 140509(R) (2002). 20. Kordyuk A. A. et al. From tunneling to photoemission: Correlating two spaces. J. Electron Spectrosc. Rel. Phenom. (2007), in press. 21. Ando, Y. et al. Electronic Phase Diagram of High-Tc Cuprate Superconductors from a Mapping of the In-Plane Resistivity Curvature. Phys. Rev. Lett. 93, 267001 (2004). 22. Lee, H. N. S. et al. The Low-Temperature Electrical and Magnetic Properties of TaSe2 and NbSe2. Journ. Solid State Chem. 1, 190-194 (1970). 23. Klemm, R. A. Striking Similarities Between the Pseudogap Phenomena in Cuprates and in Layered Organic and Dichalcogenide Superconductors Physica C 341-348, 839 (2000). 24. Eckl, T. et al. Change of quasiparticle dispersion in crossing Tc in the underdoped cuprates. Phys. Rev. B 70, 094522 (2004). 25. Borisenko, S. V. et al. Anomalous Enhancement of the Coupling to the Magnetic Resonance Mode in Underdoped Pb-Bi2212. Phys. Rev. Lett. 90, 207001 (2003). 26. Schafer, J. et al. High-Temperature Symmetry Breaking in the Electronic Band Structure of the Quasi-One-Dimensional Solid NbSe3. Phys. Rev. Lett. 87, 196403 (2001). 27. Schafer, J. et al. Unusual Spectral Behavior of Charge-Density Waves with Imperfect Nesting in a Quasi-One-Dimensional Metal. Phys. Rev. Lett. 91, 066401 (2003). 28. Yokoya, T. et al. Role of charge-density-wave fluctuations on the spectral function in a metallic charge-density-wave system. Phys. Rev. B 71, 140504(R) (2005). 29. Lee, P. A., Rice, T. M. & Anderson, P. W. Fluctuation Effects at a Peierls Transition. Phys. Rev. Lett. 31, 462 (1973). Acknowledgements: The project was supported, in part, by the DFG under Grant No. KN393/4. We thank R. Hübel for technical support. This work was partially performed at the Swiss Light Source, Paul Scherrer Institut, Villigen, Switzerland. Correspondence and requests for materials should be addressed to S.V.B. (S. Borisenko@ifw-dresden.de). Figure 1. Fermi surfaces. Momentum distribution of the photoemission intensity at the Fermi level at 180 K and 30 K. Short dashed lines are the BZ boundaries. Figure 2. Electronic structure and leading edge gap. a) Photoemission intensity as a function of energy and momentum in the normal (upper row of panels), incommensurate CDW (middle row) and commensurate CDW (lower row) states. Sketches of the FS on the middle panels show cuts in momentum space along which the data were taken and are valid for all panels in the same column. Hybridization effects in the lower row are seen as “repulsions” of the bands, which occur when a folded band is supposed to cross the original one, as schematically shown in the right inset to Fig. 3b. Since the spectral weight of the folded band is lower, these effects appear as breaks in the intensity of the original bands. Numbers correspond to the different kF. b) kF EDCs from the datasets shown in a). Leading edge gap is clearly seen as a shift of the EDC#6 to the higher binding energies at 290 K and 107 K. Figure 3. Momentum and temperature dependence of the pseudogap. a,b) Binding energies of the leading edges of all EDCs from the momentum range close to the irreducible parts of the M-dogbone and K-barrel FS. Colour scales reproduce the vertical coordinate. Anisotropy of the pseudogap on the M-dogbone FS is seen as a changing colour when going along the dashed line which correspond to FS, i.e. kF points. c) Difference between the binding energies of the leading edges of the EDC#5 and #6 from Fig. 2b as a function of temperature as shown schematically in the left inset (pseudogap) when cycling the temperature (filled symbols). Shift of the EDC maximum (bandgap) which corresponds to the top of the hybridized band as shown in the right inset with respect to its position at TNIC=122 K (open symbols). Note that the given leading edge gap values below 90 K cannot be considered as a measure of the pseudogap because of folding. Figure 4. Nesting properties. a) A model copy of the FS map from Fig. 1 with the homogeneous intensity distribution along the FS. b) Autocorrelation maps of this model at different temperatures. c) Temperature dependence of the momentum distance between the M-dogbone and K-barrel (T > 90 K). The distance below 90 K is just a difference between corresponding maxima at the FS map. d) Intensity of the maps from b) along the cuts shown by dotted lines. Supplementary information. Here we explain in more details the nesting properties of the normal state FS and how to understand the 3x3 folding upon entering the commensurate CDW regime. In panel a we show only Γ- and K-centred barrels and their first order ( 1q ± , 2q ± , 3q ± ) and second order ( 21 qq + , 32 qq + ) replica. One can see that the nesting conditions are nearly perfect for all points of the K-barrel: shifting the Γ-barrel by one of the CDW vectors results in an overlap with the corresponding part of the K-barrel with opposite Fermi velocities. Panel b shows an idealized situation, which is actually very close to the real one, since a more careful examination of Fig. 1a reveals the straight sections of both Γ- and K-barrels. This is, however, not enough to explain the fully gapped K-barrel in the CDW state. In panel a it is seen that 3x3 folding produces a double-walled barrels in the centre of the new BZ which are supposed to interact with the single copies of Γ-barrel. This is not surprising as the original BZ contains only one complete Γ-barrel and two complete K-barrels. It means, that the K- barrels of a system with the FS schematically shown in panel b, will not disappear after a 3x3 folding. Panel c shows the dogbone FS centred around M-points together with its first- and second order replica. Folding of this FS sheet results in the same triangular FSs as in panel a, a rather complicated set of features around the centres of the new BZ and nearly exact copies of the Γ- barrels and its replica. It is these copies of the Γ-barrels which were missing in panel a to interact with double-walled K-barrels. Panel d, where all FSs and their folded replica are shown, summarizes all mentioned above. When hybridization effects are switched on, the former K-barrel completely disappears, doubly degenerate triangular FSs emerge around the corners of the new BZ and the complicated set of features seems to evolve into a four-times degenerate nearly circle FSs around the Γ’- points (at least experimentally we can currently resolve only these semi-circular FSs and not two doubly degenerate hexagons with rounded corners as is suggested by the panel c).

RIKEN-TH-96 Supersymmetric Field Theory Based on Generalized Uncertainty Principle SHIBUSA Yuuichirou Theoretical Physics Laboratory, RIKEN (The Institute of Physical and Chemical Research), Wako, 351-0198, Japan Abstract We construct a quantum theory of free fermion field based on the deformed Heisenberg algebra [x̂, p̂] = i~(1+βp̂2) where β is a deformation parameter using supersymmetry as a guiding principle. A supersymmetric field theory with a real scalar field and a Majorana fermion field is given explicitly and we also find that the supersymmetry algebra is deformed from an usual one. PACS numbers: 03.65.Ca, 03.70.+k, 11.10.Ef, 11.30.Pb http://arxiv.org/abs/0704.1545v1 I. INTRODUCTION Physics in extremely high energy regions is particularly of interest to particle physics. In particular, when we discuss gravity, it is expected that there is a minimal length in principle. String theory which has a characteristic scale α′, is one of the most successful theoretical frameworks which overcome the difficulty of ultra-violet divergence in quantum theory of gravity. However, string theory has many difficulties in performing practical computations. Therefore if we construct a field theory which captures some stringy nature and/or includes stringy corrections, it would play a pivotal role in investigating physics in high energy regions even near the Planck scale. Some of the stringy corrections appear as α′ corrections. In other words, it often takes the form as higher derivative corrections i.e. higher order polynomial of momentum. One way to discuss these corrections is deforming the Heisenberg uncertainty principle to a generalized uncertainty principle (GUP): ∆x̂ ≥ ~ ∆p̂, (1.1) where β is a deforming parameter and corresponds to the square of the minimal length scale. If GUP is realized in a certain string theory context, β would take a value of order the string scale (β ∼ α′). This relation comes from various types of studies such as on high energy or short distance behavior of strings [1], [2], gedanken experiment of black hole [3], de Sitter space [4], the symmetry of massless particle [5] and wave packets [6]. There are several canonical commutation algebra which lead to the GUP. Among these algebra we will focus on the algebra; [x̂, p̂] = i~(1 + βp̂2). (1.2) This algebra is investigated in [7]-[10] and an attempt to construct a field theory with minimal length scale is made in [11] by using the Bargmann-Fock representation in 1+1 dimensional spacetime. It has also been used in cosmology, especially in physics at an early universe (see for example, [12]-[15] and references therein). In our previous paper [16], we investigated the quantization of fields based on the de- formed algebra (1.2) in the canonical formalism in 1+1 dimensions and in the path integral formalism as well. Using the path integral formalism we constructed a quantum theory of scalar field in arbitrary spacetime dimensions. This theory has a non-locality which stems from the existence of a minimal length. In this paper, we construct a quantum theory of free fermion field based on the deformed Heisenberg algebra. Where, we respect supersymmetry as a guiding principle. This is be- cause a string theory has this symmetry and we intend to construct a field theory which con- tains the stringy corrections. Moreover, supersymmetry is also an useful tool to understand physics in ultra-violet momentum regions. It manages a behavior of system in extremely high energy regions and eases ultra-violet divergence in quantum theory. Therefore we pro- pose a quantum field theory of fermion to have a supersymmetry for a scalar system which was given in [16]. In two and three-dimensional spacetime, we give a system with one real scalar and one Majorana fermion explicitly. This system has a special symmetry between a boson and a fermion which corresponds to supersymmetry. Although, this symmetry is deformed from ordinary supersymmetry. From the fermionic part of this system, we propose an action of fermionic fields based on GUP in general dimensional spacetime. II. SCALAR FIELD THEORY In the paper [16], we proposed a field theory of scalar based on GUP in the path-integral formalism. We begin with a review of this theory. Our theory is based on the following algebra [7]: x̂i, p̂j = i~(1 + βp̂2)δij . (2.1) This is an extension to higher dimensional spacetime of deformed Heisenberg algebra (1.2). Here i, j run from 1 to d which is the number of spatial coordinates and p̂2 ≡ (p̂i) Hereinafter, we use index i, j for spatial coordinates and a, b for all spacetime coordinates. Jacobi identity determines the full algebra: x̂i, x̂j = −2i~β(1 + βp̂2)L̂ij . (2.2) p̂i, p̂j = 0. (2.3) Here L̂ij are angular momentum like operators L̂ij ≡ 1 2(1+βp̂ (x̂ip̂j − x̂j p̂i + p̂j x̂i − p̂ix̂j). Because operators p̂i commute with each other, we construct a theory in momentum space representation. In momentum space representation, momentum operators are diagonalized simultaneously and we do not distinguish eigenvalues of momentum pi from operators p̂i. In the following, we set Planck constant ~ to be 1 for simplicity. Lagrangian in d+ 1 dimensional spacetime [16] is L = − ddp(1 + βp2)−1φ(−p, t) ∂2t + p 2 +m2 φ(p, t), (2.4) where, p2 ≡ The difference from ordinary quantum field theory is a prefactor (1+βp2)−1 in Lagrangian. Using the Bjorken-Johnson-Low prescription[17], from behavior of T∗-product between φ(p, t) and φ(p′, t′), we obtain canonical commutation relation: [φ(p, t), ∂tφ(p ′, t)] = i(1 + βp2)δd(p+ p′). (2.5) As we can see from this equation, a deforming prefactor (1 + βp̂2) of Heisenberg algebra in the first quantization (2.1) also appears in canonical commutation relation of the second quantized field theory. In a fermion field case, we encounter a difficulty at constructing the second quantized Hilbert space which does not appear in a scalar system. Note that a system of spin 0 particles contains only spin 0 particle. By contrast, a system of spin 1 particles is not closed with only fermions in the sense that it contains bosons as bound states. Therefore algebra of fermion fields must be introduced to be consistent with that of bosons fields. Because the scalar fields in our theory have a different commutation relation (2.5) from ordinary one, we must construct fermion fields so that the composite fields which correspond to scalar particles have the same commutation relations. Or, in two-dimensional ordinary quantum field theory we could use the concepts of bosonization and fermionization which associate fermion fields with boson fields. However, it is obscure which of these principles which relate bosons and fermions remains unchanged in GUP or in extremely high energy regions. Instead of handling this problem directly, we use supersymmetry to construct quantized field theory of fermion. This is because string theory accommodates this symmetry and therefore it is expected that this symmetry is reflected in GUP or in extremely high energy regions. In the next section we construct a quantum field theory of fermions which is consistent with the above scalar theory by using supersymmetry. III. SUPERSYMMETRY IN GUP In two and three-dimensional spacetime, a system with a real scalar and a Majorana fermion has a special symmetry between a boson and a fermion, namely supersymmetry. Thus we construct a quantum field theory of fermion in GUP to have a similar symme- try between bosons and fermions with an above-reviewed scalar system in two and three- dimensional spacetime. Our notation for two and three-dimensional spacetime is as follows: In those dimensional spacetime (with signature −+ or − + +) the Lorentz group has a real (Majorana) two- component spinor representation ψα. In the following, we explain the notation of three- dimensional spacetime. Reduction to two-dimensional spacetime is trivial. We define a representation of Gamma matrices by Pauli matrices1 as follows: {Γa,Γb} = 2ηab = 2diag(−++), (3.1) Γ0 = −iσ2,Γ1 = σ1,Γ2 = −σ3. (3.2) Spinor indices are lowered and raised by charge conjugation matrix Cαβ ≡ Γ0 and its inverse matrix C−1: ψα = ψ βCβα(= ψ̄α), ψ α = ψβ(C −1)βα. (3.3) Because the algebra of scalar field (2.5) is deformed from usual one, it is natural to expect that supersymmetry algebra may also be deformed from ordinary one. We generalize supersymmetry algebra and its actions on a scalar field φ, a Majorana fermion ψ and an auxiliary field F with parameter ǫαas follows: ǭ1Q̂, ǭ2Q̂ = 2∆ǭ1Γ aǫ2P̂a, (3.4) δφ(p, t) = iǭψ(p, t), (3.5) δψα(p, t) = A1F (p, t)ǫ α − A2{(ǭΓ0C−1)α∂t + (ǭΓjC−1)α(ipj)}φ(p, t), (3.6) δF (p, t) = A3iǭ(Γ 0∂t + Γ j(ipj))ψ(p, t). (3.7) Here, we introduce factors ∆, Ai as functions of a deforming parameter β and momentum. These factors should reduce to 1 in the limit of β → 0 and will be determined later by consistency conditions. 1 Pauli matrices are σ1 = , σ2 = , σ3 = From the closeness of algebra on each fields, we obtain conditions A1A3 = A2 = ∆. (3.8) We also generalize a Lagrangian by introducing factorsBi, which are functions of a deforming parameter β and momentum and are to be determined as well: φ(−p, t)(∂2t + p2)φ(p, t)− ψ̄(−p, t)(Γ0∂t + (ipi)Γi +m)ψ(p, t) B1B3mφ(−p, t)F (p, t) + F (−p, t)F (p, t) . (3.9) Here d is the number of spatial coordinates (1 or 2). By integrating out the field F , we obtain Lagrangian with the scalar field and the Majorana field: φ(−p, t)(∂2t + p2 +m2)φ(p, t) − iB2 ψ̄(−p, t)(Γ0∂t + (ipi)Γi +m)ψ(p, t) . (3.10) Invariance of Lagrangian (3.9) under supersymmetry variations (3.5)-(3.7) leads following conditions; A1B2 = B1B3, A1B2 = A3B3, B1 = A1A3B2. (3.11) From conditions (3.8) and (3.11), only B1 and B2 remain to be determined. (Factor A1 can be absorbed into normalization of a field F and we set it to be 1 for a field F to be an auxiliary field.) Noether’s current for supersymmetry can be calculated from Lagrangian (3.10) and supersymmetry charge is found to be dtdpdB1{ − ψα(−p, t)∂tφ(p, t) + (ΓiΓ0ψ(−p, t))α(ipi)φ(p, t) + m(Γ0ψ(−p, t))αφ(p, t)}. (3.12) Then, we obtain Hamiltonian of this system from supersymmetry charge and algebra (3.4), H = P 0 = −1 (CΓ0)αβ{Qα, Qβ}. (3.13) Using the Bjorken-Johnson-Low prescription, from behaviors of T∗-product between fields, we obtain canonical commutation relations as follows, [φ(p, t), ∂tφ(q, t)] = δ(p+ q), (3.14) α(p, t), ψβ(q, t) (Γ0C−1)αβ δ(p+ q). (3.15) Thus we can write the Hamiltonian in the following form; {π(−p, t)π(p, t) + φ(−p, t)(p2 +m2)φ(p, t)} ψ̄(−p, t)((ipi)Γi +m)ψ(p, t). (3.16) Here, we use conjugate momentum π(p, t) = ∂tφ(−p, t) and indices i runs from 1 to d. There is another condition which can be used to determine the factors B1 and B2. It comes from the free energy of supersymmetric vacuum. From algebra (3.4), supersymmetric state has zero energy: TrB ln(B1(E 2 + p2 +m2))− 1 TrF ln(B 2 + p2 +m2)). (3.17) This fact leads to the condition; B1 = B 2 . (3.18) Here TrB and TrF represent trace in bosonic and fermionic Hilbert space respectively. Lastly, we set B1 = (1+βp 2)−1 as we can see from the scalar action (2.4). This determines all of the introduced factors as follows; ∆ = A2 = A3 = B2 = (1 + βp 2 , (3.19) A1 = B3 = 1, (3.20) B1 = (1 + βp 2)−1. (3.21) Thus we construct quantized fields of fermion which is consistent with scalar fields (2.5) as ψα(p, t), ψβ(q, t) = −(1 + βp2) 2 (Γ0C−1)αβδ(p+ q). (3.22) Note that a factor ∆ is not equal to 1 no matter how we set A1. Therefore this super- symmetry algebra is deformed from an usual one as [ǭ1Q, ǭ2Q] = 2(1 + βp 2 ǭ1Γ aǫ2Pa. (3.23) There is no difficulty in generalizing the above quantum fields of fermion to higher d+ 1 dimensions than three dimensions. The action is as follows: (1 + βp2) ψ̄(−p, t)(Γ0∂t + (ipi)Γi +m)ψ(p, t) . (3.24) There appears a universal prefactor (1+βp2)− 2 comparing with usual fermion action regard- less as to whether there were supersymmetry or not. This prefactor ensures that fermion fields are compatible with the scalar fields which had been constructed in our previous paper [16]. From the actions (2.4) and (3.24), we also have supersymmetric field theory in four dimensions with an complex scalar and a Majorana (or Weyl) fermion just as a corresponding ordinary field theory has supersymmetry in four dimensions. IV. CONCLUSION AND DISCUSSIONS In summary, we have constructed a quantum theory of free fermion field based on the deformed Heisenberg algebra. It is consistent with already proposed scalar theory through supersymmetry. We start with a system with an real scalar and a Majorana fermion in two- and three-dimensional spacetime and determine supersymmetric action. We found that supersymmetry algebra is deformed from an usual one. An extension to higher dimensions are trivial and there is also supersymmetric theory in four-dimensional spacetime. We conclude with a brief discussion on Lorentz invariant extension of our theory. Lorentz invariant extension of deformed Heisenberg algebra (2.1) is known as a sort of ‘doubly special relativity’ or ‘κ-deformation’ (for example, see [18], [19] and references therein), [x̂a, p̂b] = i~(1 + βp̂ 2)δab . (4.1) Here a, b run from 0 to d and p̂2 ≡ −(p̂0)2 + i=0(p̂i) 2. Thus we claim that an action where the factor (1+βp2) is replaced with a new factor (1+βp2) describes quantum field theory of doubly special relativity. In such case, time slice is not well-defined because of the existence of minimal time interval. Therefore there is no canonical formalism. Acknowledgments The author is grateful to T. Matsuo, K. Oda, T. Tada, and N. Yokoi for valuable dis- cussions. The author is supported by the Special Postdoctoral Researchers Program at RIKEN. [1] D. J. Gross and P. F. 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Phys. B 716 (2005) 261 [arXiv:gr-qc/0411056]. [15] A. Ashoorioon, J. L. Hovdebo and R. B. Mann, Nucl. Phys. B 727 (2005) 63 [arXiv:gr-qc/0504135]. [16] T. Matsuo and Y. Shibusa, Mod. Phys. Lett. A 21 (2006) 1285 [arXiv:hep-th/0511031]. [17] J. D. Bjorken, Phys. Rev. 148 (1966) 1467. K. Johnson and F. E. Low, Prog. Theor. Phys. Suppl. 37 (1966) 74. [18] J. Lukierski and A. Nowicki, Int. J. Mod. Phys. A 18 (2003) 7 [arXiv:hep-th/0203065]. [19] G. Amelino-Camelia, Int. J. Mod. Phys. D 11 (2002) 1643 [arXiv:gr-qc/0210063]. http://arxiv.org/abs/hep-th/9301067 http://arxiv.org/abs/hep-th/0505183 http://arxiv.org/abs/gr-qc/0610056 http://arxiv.org/abs/hep-th/9412167 http://arxiv.org/abs/hep-th/0305262 http://arxiv.org/abs/hep-ph/0405127 http://arxiv.org/abs/hep-th/0510245 http://arxiv.org/abs/hep-th/9602085 http://arxiv.org/abs/astro-ph/0410139 http://arxiv.org/abs/gr-qc/0410053 http://arxiv.org/abs/gr-qc/0411056 http://arxiv.org/abs/gr-qc/0504135 http://arxiv.org/abs/hep-th/0511031 http://arxiv.org/abs/hep-th/0203065 http://arxiv.org/abs/gr-qc/0210063 Introduction Scalar Field Theory Supersymmetry in GUP Conclusion and Discussions Acknowledgments References

We construct a quantum theory of free fermion field based on the generalized uncertainty principle using supersymmetry as a guiding principle. A supersymmetric field theory with a real scalar field and a Majorana fermion field is given explicitly and we also find that the supersymmetry algebra is deformed from an usual one.

Introduction Scalar Field Theory Supersymmetry in GUP Conclusion and Discussions Acknowledgments References

Optimal flexibility for conformational transitions in macromolecules Richard A. Neher1, Wolfram Möbius1, Erwin Frey1, and Ulrich Gerland1,2 1Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for Nanoscience (CeNS), LMU München, Theresienstraße 37, 80333 München, Germany 2Institute for Theoretical Physics, University of Cologne, Germany (Dated: November 20, 2018) Conformational transitions in macromolecular complexes often involve the reorientation of lever- like structures. Using a simple theoretical model, we show that the rate of such transitions is drastically enhanced if the lever is bendable, e.g. at a localized “hinge”. Surprisingly, the transition is fastest with an intermediate flexibility of the hinge. In this intermediate regime, the transition rate is also least sensitive to the amount of “cargo” attached to the lever arm, which could be exploited by molecular motors. To explain this effect, we generalize the Kramers-Langer theory for multi-dimensional barrier crossing to configuration dependent mobility matrices. Many biological functions depend on transitions in the global conformation of macromolecules, and the associ- ated kinetic rates can be under strong evolutionary pres- sure. For instance, the directed motion of molecular mo- tors is based on power strokes [1], protein binding to DNA can require DNA bending [2] or spontaneous partial unwrapping of DNA from histones [3, 4], and the func- tioning of some ribozymes depends on global transitions in the tertiary structure [5]. These and other examples display two generic features: (i) A long segment within the molecule or complex is turned during the transition, e.g. an RNA stem in a ribozyme, the DNA as it unwraps from histones or bends upon protein binding, or the lever arm of a molecular motor relative to the attached head. (ii) The segment has a certain bending flexibility. Here, we use a minimal physical model to study the coupled dynamics of the transition and the bending fluctuations. Our model, illustrated in Fig. 1, demonstrates explic- itly how even a small bending flexibility can drastically accelerate the transition. Furthermore, if the flexibil- ity arises through a localized “hinge”, e.g. in the protein structure of some molecular motors [6, 7] or an interior loop in an RNA stem, we find that the transition rate is maximal at an intermediate hinge stiffness. Thus, in situations where rapid transition rates are crucial, molec- ular evolution could tune a hinge stiffness to the optimal value. We find that an intermediate stiffness is optimal also from the perspective of robustness, since it renders the transition rate least sensitive to changes in the drag on the lever arm, incurred e.g. by different cargos trans- ported by a molecular motor. Our finding of an optimal rate is reminiscent of a phe- nomenon known as resonant activation [8, 9], where a transition rate displays a peak as a function of the charac- teristic timescale of fluctuations in the potential barrier. However, we will see that the peak in our system has a different origin: a trade-off between the accelerating effect of the bending fluctuations and a decreasing av- erage mobility of the reaction coordinate. The standard Kramers-Langer theory [10] for multi-dimensional transi- tion processes is not sufficient to capture this trade-off. A generalization of the theory to the case of configuration- dependent mobility matrices turns out to be essential to understand the peak at intermediate stiffness. Model.— We model the conformational transition as a thermally activated change in the attachment angle ϕ of a macromolecular lever, see Fig. 1. The lever has two segments connected by a hinge with stiffness ǫ, which ren- ders the lever preferentially straight, but allows thermal fluctuations in the bending angle θ. The energy function V (ϕ, θ) of this ‘Two-Segment Lever’ (TSL) is V (ϕ, θ) = ǫ(1− cos θ)− (aϕ)3 − b(aϕ) , (1) where kBT is the thermal energy unit. The hinge, de- scribed by the first term, serves not only as a sim- ple model for a protein or RNA hinge, but also as a zeroth-order approximation to a more continuously dis- tributed flexibility; see below. The second term is the potential on the attachment angle ϕ, which produces a metastable minimum at (ϕ, θ) = (0, 0). The thermally- assisted escape from this minimum passes through the transition state at (ϕ, θ) = (b/a, 0) with a barrier height ∆V = b3kBT/6 [20]. a) b) FIG. 1: Schematic illustration of the ‘Two-Segment Lever’ (TSL) model for conformational transitions. (a) The two segments of lengths 1 and ρ are connected by a hinge and attached to the origin. The viscous drag acts on the ends of the segments as indicated by the beads. (b) Schematic illustration of the barrier crossing processes. The external meta-stable potential V (ϕ) is indicated by shading (top; dark corresponds to high energy) and is also sketched below. http://arxiv.org/abs/0704.1546v1 In the present context, inertial forces are negligible, i.e. it is sufficient to consider the stochastic dynamics of the TSL in the overdamped limit. We localize the fric- tion forces to the ends of the two segments, as indicated by the beads in Fig. 1(a). The length of the first segment defines our length unit and ρ denotes the relative length of the second segment. Similarly, we choose our time unit such that the friction coefficient of the first bead is unity, and denote the coefficient of the second bead by ξ. To describe the Brownian dynamics of the TSL, we derive the Fokker-Planck equation for the time-evolution of the configurational probability density p(ϕ, θ, t). In general, the derivation of the correct dynamic equations can be a nontrivial task for stochastic systems with con- straints [11, 12]. For instance, implementing fixed seg- ment lengths through the limit of stiff springs, leads to Fokker-Planck equations with equilibrium distributions that depend on the way in which the limit is taken [12]. However, for our overdamped system, we can avoid this problem by imposing the desired equilibrium distribu- tion, i.e. the Boltzmann distribution p = exp(−V/kBT ), which together with the well-defined deterministic equa- tions of motion uniquely determines the Fokker-Planck equation for the TSL. The deterministic equations of motion take the form q̇k = −Mkl ∂V/∂ql with the coordinates (q1, q2) = (ϕ, θ) and a mobility matrix M. We obtain M with a standard Lagrange procedure: Given linear friction, M is the in- verse of the friction matrix, which in turn is the Hessian matrix of the dissipation function [13]. This yields 1 + ξ sin2 θ 1 ρ+cos θ ρ+cos θ ρ+2 cos θ + 1+ξ . (2) The Fokker-Planck equation then follows from the conti- nuity equation ∂tp({qi}, t) = −∂kjk({qi}, t) together with jk({qi}, t) = −Mkl + kBT p({qi}, t) (3) as the probability flux density. Our analytical analysis below is based directly on Eqs. (2) and (3), while we perform all Brownian dynamics simulations with a set of equivalent stochastic differential equations [14]. Transition rate.— To explore the phenomenology of the TSL, we performed simulations to determine its av- erage dwell time τ in the metastable state, for a range of hinge stiffnesses ǫ. The rate for the conformational tran- sition is related to the dwell time by k(ǫ) = 1/τ(ǫ). Fig. 2 shows k(ǫ) (circles) for a barrier ∆V =12 kBT , a distance ∆ϕ=0.4 to the transition state, and ξ= ρ=1 (data for different parameter values behaves qualitatively similar, as long as the process is reaction-limited, i.e. ∆V is suf- ficiently large that τ is much longer than the time for the TSL to freely diffuse over an angle ∆ϕ). We observe a significant flexibility-induced enhancement of the transi- tion rate over a broad range of stiffnesses, compared to 0 10 20 30 40 50 60 70 stiffness ε [kT] Langer rate gen. Langer rate 0 50 100 150 stiff limit FIG. 2: Simulation data of the barrier crossing rate normal- ized by k0 display a prominent peak at finite stiffness (cir- cles, each obtained from 20000 simulation runs initialized at the metastable minimum). The conventional Langer theory fails to describe the non-monotonicity of the rate and over- estimates the rate at small ǫ. The generalized Langer theory captures the non-monotonicity of the rate and describes the simulations data accurately; parameters see main text. the dynamics in the stiff limit (ǫ→∞), see inset. Note that the enhancement persists even at relatively large ǫ, where typical thermal bending fluctuations δϕ ∼ ǫ are significantly smaller than ∆ϕ. Surprisingly, the acceler- ation is strongest at an intermediate stiffness (ǫ ≈ 10). This observation suggests that the stiffness of molecular hinges could be used, by evolution or in synthetic con- structs, to tune and optimize reaction rates. When the friction coefficient ξ of the outer bead is increased, the rate of the conformational transition de- creases; see Fig. 3a. This decrease is most dramatic in the stiff limit (dash-dotted line). In the flexible limit (dia- monds) the decrease is less pronounced. Notably, the rate appears least sensitive to the viscous drag on the outer bead at intermediate ǫ (circles). Indeed, Fig. 3b shows that the ǫ-dependence of this sensitivity (measured as the slope of the curves in Fig. 3a at ξ=1) has a pronounced minimum at ǫ ≈ 20. Hence, intermediate hinge stiffnesses in the TSL lead to maximal robustness, which is an im- portant design constraint for many biomolecular mecha- nisms in the cellular context. For instance, as molecular motors transport various cargos along one-dimensional filaments, it may be advantageous to render their speed insensitive to the cargo size, e.g. to avoid “traffic jams”. In the remainder of this letter, we seek a theoretical understanding of the above phenomenology. First, it is instructive to consider simple bounds on the transition rate. An upper bound is obtained by completely elimi- nating the outer bead. The Kramers rate [15] for the re- maining 1D escape process, k0 = (a 2b/2π) e−∆V/kBT , is used in Figs. 2 and 3 to normalize the transition rates. At the optimal stiffness, the transition rate in Fig. 2 comes within 20% of this upper bound. An obvious lower bound is the stiff limit: For ǫ→∞, the second segment increases 1 10 100 friction ξ 0 20 40 60 80 stiffness ε [kT] a) b) ε=25 stiff FIG. 3: The sensitivity of the rate to the friction coefficient ξ is minimal at intermediate stiffness. (a) Simulation results at ǫ = 0 and ǫ = 25 as well as the theoretical estimates of the rate at ǫ = 0 and in the stiff limit. (b) The derivative of ln k with respect to ln ξ evaluated at ξ = 1, i.e. the slope of the curves in a), is minimal in an intermediate stiffness range. the rotational friction by a factor ζ = 1 + (1 + ρ)2ξ, so that the 1D Kramers rate becomes k∞ = k0/ζ, as shown by the dash-dotted line in Fig. 2 and Fig. 3a. However, to understand how the dynamics of the bending fluctua- tions affects the transition rate, we must consider the full 2D dynamics of the TSL. The multi-dimensional gener- alization of Kramers theory is Langer’s formula for the escape rate over a saddle in a potential landscape [10], kLanger = det e(w) | det e(s)| . (4) Here, e(w) and e(s) denote the Hessian matrix of the po- tential energy, ∂2V/∂qk∂ql, evaluated at the well bot- tom and the saddle point, respectively, whereas λ is the unique negative eigenvalue of the product of the mobility matrix M and e(s). Eq. (4) can be made plausible in sim- ple terms: Given a quasi-equilibrium in the metastable state, the second factor represents the probability of be- ing in the transition region, i.e. the region within ∼ kBT of the saddle. The escape rate is then given by this prob- ability multiplied by the rate λ at which the system re- laxes out of the transition state, analogous to Michaelis- Menten reaction kinetics. For our potential (1), the determinants in (4) can- cel. The eigenvalue can be determined analytically (the dashed line in Fig. 2 shows the resulting kLanger), but for the present purpose it is more instructive to consider the expansions for large and small stiffness. In the stiff limit, the natural small parameter is the stiffness ratio γ/ǫ, where γ = a2b is the absolute curvature or “stiffness” of the external potential at the transition state. The ex- pansion yields kLanger/k∞ = 1+ (ρ 2ξ/ζ) γ/ǫ+O(γ2/ǫ2). As expected, the rate approaches k∞, but the stiff limit is attained only when the bending fluctuations ∼ ǫ are small compared to the width of the barrier ∼ √γ. In the opposite limit, ǫ≪ γ, the rate is given by kLanger/k0 = 1 + ρ−1 ǫ/γ + O(ǫ2/γ). Since the linear term is negative, Langer theory predicts that the transition rate peaks at zero stiffness, with a peak value equal to the Kramers rate k0 for the lever without the second segment. a) b) FIG. 4: The friction opposing rotation of the attachment an- gle ϕ depends on the bending angle θ, since the outer bead is moved by different amounts in different configurations. For an infinitesimal displacement dϕ, the displacement of the outer bead is sin θ dϕ. The projection of the resulting friction force onto the direction of motion adds another factor sin θ, yielding a friction coefficient for ϕ of 1 + ξ sin2 θ. This prediction is clearly at variance with the simulation results. It is interesting to note, however, that the slope of the linear decay is independent of ξ. This is consistent with our observation that the transition rate is insensi- tive to ξ in the intermediate stiffness regime. Indeed, Fig. 2 shows that Langer theory (dashed line) describes the simulation data (circles) reasonably well for interme- diate and large hinge stiffness. To understand the origin of the peak at intermediate stiffness, it is useful to consider the flexible limit (ǫ = 0). In this limit, the transition state is degenerate in θ, and it seems plausible to estimate the transition rate by using a θ-averaged mobility for the reaction coordinate ϕ, k(ǫ = 0) ≈ k0 M11(θ) = 1 + ξ . (5) This estimate agrees well with the simulation data, see the dashed line in Fig. 3a, indicating that the configuration-dependent mobility (2) plays an important role for the transition rate. In contrast, the conventional Langer theory assumes the mobility matrix to be con- stant in the relevant region near the transition state. Fig. 4 illustrates why the mobility M11 of the coordinate ϕ is affected by the bending angle θ and gives a graphical construction for M11. Generalized Langer theory.— To account for the mo- bility effect identified above, we must generalize the Langer theory to configuration-dependent mobility ma- trices. The special case where the mobility varies only along the reaction coordinate has already been studied in [16], however the main effect in our case is due to the vari- ation in the transverse direction. In the following, we out- line the derivation of the central result, while all details will be presented elsewhere. Near the saddle, the mobil- ity matrix takes the form Mij({qi}) = M (s)ij + 12A ij q̂lq̂k, where q̂i are deviations from the saddle and A ij denotes the tensor of second derivatives of the mobility matrix (we assume that the first derivatives of M vanish at the saddle, which is the case for the TSL). The escape rate is given by the probability flux out of the metastable well, divided by the population inside the well. To cal- culate the flux, we construct a steady state solution to the Fokker-Planck equation in the vicinity of the saddle, as described in [15] for the conventional Langer theory. We use the Ansatz p({qi}) = 12peq({qi}) erfc(u), where peq({qi}) = Z−1e−V ({qi})/kBT and erfc(u) is the comple- mentary error function with argument u = Uk q̂k. Insert- ing the Ansatz into the Fokker-Planck equation yields an equation for the vector U, Ui(−Mije(s)jk +Bik)− UiMijUj Uk = 0 , (6) where Bik = kBT ni . Bik q̂k is the noise induced drift, which is absent in the conventional Langer theory. Ignoring higher order terms, this equation determines U to be the left eigenvector of −M(s)e(s)+B to the unique positive eigenvalue λ, and requires U to be normalized such that UiM ij Uj = λ. The directions of the left and right eigenvectors of −M(s)e(s) + B have a physical in- terpretation: U is perpendicular to the stochastic sepa- ratrix, while the corresponding right eigenvector points in the direction of the diffusive flux at the saddle [17]. From p({qi}), the flux density is determined by (3) and the total flux is obtained by integrating the flux density over a plane containing the saddle; a convenient choice is the plane u = 0. Evaluation of the integral is particularly simple in a coordinate system, where the first coordinate is parallel toU, and the remaining coordinates are chosen such that e(s) is diagonal in this subspace, e ij = µiδij for i, j > 1. In this coordinate system, the generalized Langer rate takes the simple form 1 + 1 det e(w) | det e(s)| T , (7) where c = Uie ij Uj + 1 = B1ie i1 /M 11 and e −1 denotes the inverse matrix of e(s). Eq. (7) contains three correc- tions to (4), all of which vanish whenM({qi}) is constant: The most important one is given by l>1 A 11/µl, which changes the mobility M11 in the direction of U to an ef- fective mobility that is averaged over the separatrix with respect to the Boltzmann distribution. In addition, there are two corrections incurred by the noise-induced drift: the factor 1− c and a change due to the fact that λ is now the eigenvalue to M(s)e(s) −B instead of M(s)e(s). The solid line in Fig. 2 shows the application of the generalized Langer formula to the TSL. We observe that it captures the peak in the transition rate and thus the es- sential phenomenology of the TSL. Obviously, the evalu- ation of the Gaussian integral that leads to Eq. (7) is only meaningful, if the harmonic approximation of the mobil- ity matrix is reasonable within the relevant saddle point region. This integral diverges as the saddle point degen- erates, which explains the behavior for ǫ → 0. At high ξ, the very anisotropic friction can also render Langer theory invalid [18, 19]. Conclusion.— We have introduced the “Two-Segment Lever” as a simple model for a class of conformational transitions in biomolecules. The model clearly demon- strates how flexibility can enhance the rate of a confor- mational transition. This remains true, if the hinge in the TSL is replaced by a more continuous bendability. In- terestingly, a discrete hinge has a stiffness regime, where the rate is large and robust against cargo variation, which raises the question, whether these effects are exploited by evolution, for example in the design of molecular motors. To understand these effects theoretically, we derived a generalized Langer theory that takes into account con- figuration dependent mobility matrices. We hope that this theory will find applications also in other fields. We thank the German Excellence Initiative for finan- cial support via the program NIM. RN and UG are grate- ful for the hospitality of the CTBP at UCSD, where part of this work was done, and for financial support by the CeNS in Munich and the DFG. [1] J. Howard, Mechanics of Motor Proteins and the Cy- toskeleton (Sinauer Press: Sunderland, MA, 2001). [2] S. Sugimura and D. M. Crothers, PNAS 103, 18510 (2006). [3] G. Li, M. Levitus, C. Bustamante, and J. Widom, Nat. Struct. Mol. Biol. 12, 46 (2005). [4] W. Möbius, R. A. Neher, and U. Gerland, Phys. Rev. Lett. 97, 208102 (2006). [5] X. Zhuang, H. Kim, M. J. B. Pereira, H. P. Babcock, N. G. Walter, and S. Chu, Science 296, 1473 (2002). [6] M. Terrak, G. Rebowski, R. C. Lu, Z. Grabarek, and R. Dominguez, PNAS 102, 12718 (2005). [7] S. Jeney, E. H. K. Stelzer, H. Grubmüller, and E.-L. Florin, ChemPhysChem 5, 1150 (2004). [8] C. R. Doering and J. C. Gadoua, Phys. Rev. Lett. 69, 2318 (1992). [9] P. Reimann, Phys. Rev. Lett. 74, 4576 (1995). [10] J. S. Langer, Ann. Phys. 54, 258 (1969). [11] E. Helfand, J. Chem. Phys. 71, 5000 (1979). [12] N. van Kampen and J. Lodder, Am. J. Phys. 52, 419 (1984). [13] H. Goldstein, C. P. Poole, and J. L. Safko, Classical Me- chanics (Addison Wesley, 2002). [14] C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 2004). [15] P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990). [16] B. Gavish, Phys. Rev. Lett. 44, 1160 (1980). [17] A. Berezhkovskii and A. Szabo, J. Chem. Phys. 122, 014503 (2005). [18] A. N. Drozdov and P. Talkner, J. Chem. Phys. 105, 4117 (1996). [19] A. M. Berezhkovskii and V. Y. Zitserman, Chemical Physics 157, 141 (1991). [20] With the potential (1) this transition is irreversible, how- ever all of our conclusions equally apply to reversible transitions in a double-well potential.

Conformational transitions in macromolecular complexes often involve the reorientation of lever-like structures. Using a simple theoretical model, we show that the rate of such transitions is drastically enhanced if the lever is bendable, e.g. at a localized "hinge''. Surprisingly, the transition is fastest with an intermediate flexibility of the hinge. In this intermediate regime, the transition rate is also least sensitive to the amount of "cargo'' attached to the lever arm, which could be exploited by molecular motors. To explain this effect, we generalize the Kramers-Langer theory for multi-dimensional barrier crossing to configuration dependent mobility matrices.

Optimal flexibility for conformational transitions in macromolecules Richard A. Neher1, Wolfram Möbius1, Erwin Frey1, and Ulrich Gerland1,2 1Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for Nanoscience (CeNS), LMU München, Theresienstraße 37, 80333 München, Germany 2Institute for Theoretical Physics, University of Cologne, Germany (Dated: November 20, 2018) Conformational transitions in macromolecular complexes often involve the reorientation of lever- like structures. Using a simple theoretical model, we show that the rate of such transitions is drastically enhanced if the lever is bendable, e.g. at a localized “hinge”. Surprisingly, the transition is fastest with an intermediate flexibility of the hinge. In this intermediate regime, the transition rate is also least sensitive to the amount of “cargo” attached to the lever arm, which could be exploited by molecular motors. To explain this effect, we generalize the Kramers-Langer theory for multi-dimensional barrier crossing to configuration dependent mobility matrices. Many biological functions depend on transitions in the global conformation of macromolecules, and the associ- ated kinetic rates can be under strong evolutionary pres- sure. For instance, the directed motion of molecular mo- tors is based on power strokes [1], protein binding to DNA can require DNA bending [2] or spontaneous partial unwrapping of DNA from histones [3, 4], and the func- tioning of some ribozymes depends on global transitions in the tertiary structure [5]. These and other examples display two generic features: (i) A long segment within the molecule or complex is turned during the transition, e.g. an RNA stem in a ribozyme, the DNA as it unwraps from histones or bends upon protein binding, or the lever arm of a molecular motor relative to the attached head. (ii) The segment has a certain bending flexibility. Here, we use a minimal physical model to study the coupled dynamics of the transition and the bending fluctuations. Our model, illustrated in Fig. 1, demonstrates explic- itly how even a small bending flexibility can drastically accelerate the transition. Furthermore, if the flexibil- ity arises through a localized “hinge”, e.g. in the protein structure of some molecular motors [6, 7] or an interior loop in an RNA stem, we find that the transition rate is maximal at an intermediate hinge stiffness. Thus, in situations where rapid transition rates are crucial, molec- ular evolution could tune a hinge stiffness to the optimal value. We find that an intermediate stiffness is optimal also from the perspective of robustness, since it renders the transition rate least sensitive to changes in the drag on the lever arm, incurred e.g. by different cargos trans- ported by a molecular motor. Our finding of an optimal rate is reminiscent of a phe- nomenon known as resonant activation [8, 9], where a transition rate displays a peak as a function of the charac- teristic timescale of fluctuations in the potential barrier. However, we will see that the peak in our system has a different origin: a trade-off between the accelerating effect of the bending fluctuations and a decreasing av- erage mobility of the reaction coordinate. The standard Kramers-Langer theory [10] for multi-dimensional transi- tion processes is not sufficient to capture this trade-off. A generalization of the theory to the case of configuration- dependent mobility matrices turns out to be essential to understand the peak at intermediate stiffness. Model.— We model the conformational transition as a thermally activated change in the attachment angle ϕ of a macromolecular lever, see Fig. 1. The lever has two segments connected by a hinge with stiffness ǫ, which ren- ders the lever preferentially straight, but allows thermal fluctuations in the bending angle θ. The energy function V (ϕ, θ) of this ‘Two-Segment Lever’ (TSL) is V (ϕ, θ) = ǫ(1− cos θ)− (aϕ)3 − b(aϕ) , (1) where kBT is the thermal energy unit. The hinge, de- scribed by the first term, serves not only as a sim- ple model for a protein or RNA hinge, but also as a zeroth-order approximation to a more continuously dis- tributed flexibility; see below. The second term is the potential on the attachment angle ϕ, which produces a metastable minimum at (ϕ, θ) = (0, 0). The thermally- assisted escape from this minimum passes through the transition state at (ϕ, θ) = (b/a, 0) with a barrier height ∆V = b3kBT/6 [20]. a) b) FIG. 1: Schematic illustration of the ‘Two-Segment Lever’ (TSL) model for conformational transitions. (a) The two segments of lengths 1 and ρ are connected by a hinge and attached to the origin. The viscous drag acts on the ends of the segments as indicated by the beads. (b) Schematic illustration of the barrier crossing processes. The external meta-stable potential V (ϕ) is indicated by shading (top; dark corresponds to high energy) and is also sketched below. http://arxiv.org/abs/0704.1546v1 In the present context, inertial forces are negligible, i.e. it is sufficient to consider the stochastic dynamics of the TSL in the overdamped limit. We localize the fric- tion forces to the ends of the two segments, as indicated by the beads in Fig. 1(a). The length of the first segment defines our length unit and ρ denotes the relative length of the second segment. Similarly, we choose our time unit such that the friction coefficient of the first bead is unity, and denote the coefficient of the second bead by ξ. To describe the Brownian dynamics of the TSL, we derive the Fokker-Planck equation for the time-evolution of the configurational probability density p(ϕ, θ, t). In general, the derivation of the correct dynamic equations can be a nontrivial task for stochastic systems with con- straints [11, 12]. For instance, implementing fixed seg- ment lengths through the limit of stiff springs, leads to Fokker-Planck equations with equilibrium distributions that depend on the way in which the limit is taken [12]. However, for our overdamped system, we can avoid this problem by imposing the desired equilibrium distribu- tion, i.e. the Boltzmann distribution p = exp(−V/kBT ), which together with the well-defined deterministic equa- tions of motion uniquely determines the Fokker-Planck equation for the TSL. The deterministic equations of motion take the form q̇k = −Mkl ∂V/∂ql with the coordinates (q1, q2) = (ϕ, θ) and a mobility matrix M. We obtain M with a standard Lagrange procedure: Given linear friction, M is the in- verse of the friction matrix, which in turn is the Hessian matrix of the dissipation function [13]. This yields 1 + ξ sin2 θ 1 ρ+cos θ ρ+cos θ ρ+2 cos θ + 1+ξ . (2) The Fokker-Planck equation then follows from the conti- nuity equation ∂tp({qi}, t) = −∂kjk({qi}, t) together with jk({qi}, t) = −Mkl + kBT p({qi}, t) (3) as the probability flux density. Our analytical analysis below is based directly on Eqs. (2) and (3), while we perform all Brownian dynamics simulations with a set of equivalent stochastic differential equations [14]. Transition rate.— To explore the phenomenology of the TSL, we performed simulations to determine its av- erage dwell time τ in the metastable state, for a range of hinge stiffnesses ǫ. The rate for the conformational tran- sition is related to the dwell time by k(ǫ) = 1/τ(ǫ). Fig. 2 shows k(ǫ) (circles) for a barrier ∆V =12 kBT , a distance ∆ϕ=0.4 to the transition state, and ξ= ρ=1 (data for different parameter values behaves qualitatively similar, as long as the process is reaction-limited, i.e. ∆V is suf- ficiently large that τ is much longer than the time for the TSL to freely diffuse over an angle ∆ϕ). We observe a significant flexibility-induced enhancement of the transi- tion rate over a broad range of stiffnesses, compared to 0 10 20 30 40 50 60 70 stiffness ε [kT] Langer rate gen. Langer rate 0 50 100 150 stiff limit FIG. 2: Simulation data of the barrier crossing rate normal- ized by k0 display a prominent peak at finite stiffness (cir- cles, each obtained from 20000 simulation runs initialized at the metastable minimum). The conventional Langer theory fails to describe the non-monotonicity of the rate and over- estimates the rate at small ǫ. The generalized Langer theory captures the non-monotonicity of the rate and describes the simulations data accurately; parameters see main text. the dynamics in the stiff limit (ǫ→∞), see inset. Note that the enhancement persists even at relatively large ǫ, where typical thermal bending fluctuations δϕ ∼ ǫ are significantly smaller than ∆ϕ. Surprisingly, the acceler- ation is strongest at an intermediate stiffness (ǫ ≈ 10). This observation suggests that the stiffness of molecular hinges could be used, by evolution or in synthetic con- structs, to tune and optimize reaction rates. When the friction coefficient ξ of the outer bead is increased, the rate of the conformational transition de- creases; see Fig. 3a. This decrease is most dramatic in the stiff limit (dash-dotted line). In the flexible limit (dia- monds) the decrease is less pronounced. Notably, the rate appears least sensitive to the viscous drag on the outer bead at intermediate ǫ (circles). Indeed, Fig. 3b shows that the ǫ-dependence of this sensitivity (measured as the slope of the curves in Fig. 3a at ξ=1) has a pronounced minimum at ǫ ≈ 20. Hence, intermediate hinge stiffnesses in the TSL lead to maximal robustness, which is an im- portant design constraint for many biomolecular mecha- nisms in the cellular context. For instance, as molecular motors transport various cargos along one-dimensional filaments, it may be advantageous to render their speed insensitive to the cargo size, e.g. to avoid “traffic jams”. In the remainder of this letter, we seek a theoretical understanding of the above phenomenology. First, it is instructive to consider simple bounds on the transition rate. An upper bound is obtained by completely elimi- nating the outer bead. The Kramers rate [15] for the re- maining 1D escape process, k0 = (a 2b/2π) e−∆V/kBT , is used in Figs. 2 and 3 to normalize the transition rates. At the optimal stiffness, the transition rate in Fig. 2 comes within 20% of this upper bound. An obvious lower bound is the stiff limit: For ǫ→∞, the second segment increases 1 10 100 friction ξ 0 20 40 60 80 stiffness ε [kT] a) b) ε=25 stiff FIG. 3: The sensitivity of the rate to the friction coefficient ξ is minimal at intermediate stiffness. (a) Simulation results at ǫ = 0 and ǫ = 25 as well as the theoretical estimates of the rate at ǫ = 0 and in the stiff limit. (b) The derivative of ln k with respect to ln ξ evaluated at ξ = 1, i.e. the slope of the curves in a), is minimal in an intermediate stiffness range. the rotational friction by a factor ζ = 1 + (1 + ρ)2ξ, so that the 1D Kramers rate becomes k∞ = k0/ζ, as shown by the dash-dotted line in Fig. 2 and Fig. 3a. However, to understand how the dynamics of the bending fluctua- tions affects the transition rate, we must consider the full 2D dynamics of the TSL. The multi-dimensional gener- alization of Kramers theory is Langer’s formula for the escape rate over a saddle in a potential landscape [10], kLanger = det e(w) | det e(s)| . (4) Here, e(w) and e(s) denote the Hessian matrix of the po- tential energy, ∂2V/∂qk∂ql, evaluated at the well bot- tom and the saddle point, respectively, whereas λ is the unique negative eigenvalue of the product of the mobility matrix M and e(s). Eq. (4) can be made plausible in sim- ple terms: Given a quasi-equilibrium in the metastable state, the second factor represents the probability of be- ing in the transition region, i.e. the region within ∼ kBT of the saddle. The escape rate is then given by this prob- ability multiplied by the rate λ at which the system re- laxes out of the transition state, analogous to Michaelis- Menten reaction kinetics. For our potential (1), the determinants in (4) can- cel. The eigenvalue can be determined analytically (the dashed line in Fig. 2 shows the resulting kLanger), but for the present purpose it is more instructive to consider the expansions for large and small stiffness. In the stiff limit, the natural small parameter is the stiffness ratio γ/ǫ, where γ = a2b is the absolute curvature or “stiffness” of the external potential at the transition state. The ex- pansion yields kLanger/k∞ = 1+ (ρ 2ξ/ζ) γ/ǫ+O(γ2/ǫ2). As expected, the rate approaches k∞, but the stiff limit is attained only when the bending fluctuations ∼ ǫ are small compared to the width of the barrier ∼ √γ. In the opposite limit, ǫ≪ γ, the rate is given by kLanger/k0 = 1 + ρ−1 ǫ/γ + O(ǫ2/γ). Since the linear term is negative, Langer theory predicts that the transition rate peaks at zero stiffness, with a peak value equal to the Kramers rate k0 for the lever without the second segment. a) b) FIG. 4: The friction opposing rotation of the attachment an- gle ϕ depends on the bending angle θ, since the outer bead is moved by different amounts in different configurations. For an infinitesimal displacement dϕ, the displacement of the outer bead is sin θ dϕ. The projection of the resulting friction force onto the direction of motion adds another factor sin θ, yielding a friction coefficient for ϕ of 1 + ξ sin2 θ. This prediction is clearly at variance with the simulation results. It is interesting to note, however, that the slope of the linear decay is independent of ξ. This is consistent with our observation that the transition rate is insensi- tive to ξ in the intermediate stiffness regime. Indeed, Fig. 2 shows that Langer theory (dashed line) describes the simulation data (circles) reasonably well for interme- diate and large hinge stiffness. To understand the origin of the peak at intermediate stiffness, it is useful to consider the flexible limit (ǫ = 0). In this limit, the transition state is degenerate in θ, and it seems plausible to estimate the transition rate by using a θ-averaged mobility for the reaction coordinate ϕ, k(ǫ = 0) ≈ k0 M11(θ) = 1 + ξ . (5) This estimate agrees well with the simulation data, see the dashed line in Fig. 3a, indicating that the configuration-dependent mobility (2) plays an important role for the transition rate. In contrast, the conventional Langer theory assumes the mobility matrix to be con- stant in the relevant region near the transition state. Fig. 4 illustrates why the mobility M11 of the coordinate ϕ is affected by the bending angle θ and gives a graphical construction for M11. Generalized Langer theory.— To account for the mo- bility effect identified above, we must generalize the Langer theory to configuration-dependent mobility ma- trices. The special case where the mobility varies only along the reaction coordinate has already been studied in [16], however the main effect in our case is due to the vari- ation in the transverse direction. In the following, we out- line the derivation of the central result, while all details will be presented elsewhere. Near the saddle, the mobil- ity matrix takes the form Mij({qi}) = M (s)ij + 12A ij q̂lq̂k, where q̂i are deviations from the saddle and A ij denotes the tensor of second derivatives of the mobility matrix (we assume that the first derivatives of M vanish at the saddle, which is the case for the TSL). The escape rate is given by the probability flux out of the metastable well, divided by the population inside the well. To cal- culate the flux, we construct a steady state solution to the Fokker-Planck equation in the vicinity of the saddle, as described in [15] for the conventional Langer theory. We use the Ansatz p({qi}) = 12peq({qi}) erfc(u), where peq({qi}) = Z−1e−V ({qi})/kBT and erfc(u) is the comple- mentary error function with argument u = Uk q̂k. Insert- ing the Ansatz into the Fokker-Planck equation yields an equation for the vector U, Ui(−Mije(s)jk +Bik)− UiMijUj Uk = 0 , (6) where Bik = kBT ni . Bik q̂k is the noise induced drift, which is absent in the conventional Langer theory. Ignoring higher order terms, this equation determines U to be the left eigenvector of −M(s)e(s)+B to the unique positive eigenvalue λ, and requires U to be normalized such that UiM ij Uj = λ. The directions of the left and right eigenvectors of −M(s)e(s) + B have a physical in- terpretation: U is perpendicular to the stochastic sepa- ratrix, while the corresponding right eigenvector points in the direction of the diffusive flux at the saddle [17]. From p({qi}), the flux density is determined by (3) and the total flux is obtained by integrating the flux density over a plane containing the saddle; a convenient choice is the plane u = 0. Evaluation of the integral is particularly simple in a coordinate system, where the first coordinate is parallel toU, and the remaining coordinates are chosen such that e(s) is diagonal in this subspace, e ij = µiδij for i, j > 1. In this coordinate system, the generalized Langer rate takes the simple form 1 + 1 det e(w) | det e(s)| T , (7) where c = Uie ij Uj + 1 = B1ie i1 /M 11 and e −1 denotes the inverse matrix of e(s). Eq. (7) contains three correc- tions to (4), all of which vanish whenM({qi}) is constant: The most important one is given by l>1 A 11/µl, which changes the mobility M11 in the direction of U to an ef- fective mobility that is averaged over the separatrix with respect to the Boltzmann distribution. In addition, there are two corrections incurred by the noise-induced drift: the factor 1− c and a change due to the fact that λ is now the eigenvalue to M(s)e(s) −B instead of M(s)e(s). The solid line in Fig. 2 shows the application of the generalized Langer formula to the TSL. We observe that it captures the peak in the transition rate and thus the es- sential phenomenology of the TSL. Obviously, the evalu- ation of the Gaussian integral that leads to Eq. (7) is only meaningful, if the harmonic approximation of the mobil- ity matrix is reasonable within the relevant saddle point region. This integral diverges as the saddle point degen- erates, which explains the behavior for ǫ → 0. At high ξ, the very anisotropic friction can also render Langer theory invalid [18, 19]. Conclusion.— We have introduced the “Two-Segment Lever” as a simple model for a class of conformational transitions in biomolecules. The model clearly demon- strates how flexibility can enhance the rate of a confor- mational transition. This remains true, if the hinge in the TSL is replaced by a more continuous bendability. In- terestingly, a discrete hinge has a stiffness regime, where the rate is large and robust against cargo variation, which raises the question, whether these effects are exploited by evolution, for example in the design of molecular motors. To understand these effects theoretically, we derived a generalized Langer theory that takes into account con- figuration dependent mobility matrices. We hope that this theory will find applications also in other fields. We thank the German Excellence Initiative for finan- cial support via the program NIM. RN and UG are grate- ful for the hospitality of the CTBP at UCSD, where part of this work was done, and for financial support by the CeNS in Munich and the DFG. [1] J. Howard, Mechanics of Motor Proteins and the Cy- toskeleton (Sinauer Press: Sunderland, MA, 2001). [2] S. Sugimura and D. M. Crothers, PNAS 103, 18510 (2006). [3] G. Li, M. Levitus, C. Bustamante, and J. Widom, Nat. Struct. Mol. Biol. 12, 46 (2005). [4] W. Möbius, R. A. Neher, and U. Gerland, Phys. Rev. Lett. 97, 208102 (2006). [5] X. Zhuang, H. Kim, M. J. B. Pereira, H. P. Babcock, N. G. Walter, and S. Chu, Science 296, 1473 (2002). [6] M. Terrak, G. Rebowski, R. C. Lu, Z. Grabarek, and R. Dominguez, PNAS 102, 12718 (2005). [7] S. Jeney, E. H. K. Stelzer, H. Grubmüller, and E.-L. Florin, ChemPhysChem 5, 1150 (2004). [8] C. R. Doering and J. C. Gadoua, Phys. Rev. Lett. 69, 2318 (1992). [9] P. Reimann, Phys. Rev. Lett. 74, 4576 (1995). [10] J. S. Langer, Ann. Phys. 54, 258 (1969). [11] E. Helfand, J. Chem. Phys. 71, 5000 (1979). [12] N. van Kampen and J. Lodder, Am. J. Phys. 52, 419 (1984). [13] H. Goldstein, C. P. Poole, and J. L. Safko, Classical Me- chanics (Addison Wesley, 2002). [14] C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 2004). [15] P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990). [16] B. Gavish, Phys. Rev. Lett. 44, 1160 (1980). [17] A. Berezhkovskii and A. Szabo, J. Chem. Phys. 122, 014503 (2005). [18] A. N. Drozdov and P. Talkner, J. Chem. Phys. 105, 4117 (1996). [19] A. M. Berezhkovskii and V. Y. Zitserman, Chemical Physics 157, 141 (1991). [20] With the potential (1) this transition is irreversible, how- ever all of our conclusions equally apply to reversible transitions in a double-well potential.

Montel_2007_condmat AFM imaging of SWI/SNF action : mapping the nucleosome remodeling and sliding Fabien MONTEL, Emeline FONTAINE, Philippe ST-JEAN, Martin CASTELNOVO and Cendrine FAIVRE-MOSKALENKO Laboratoire Joliot-Curie (CNRS USR 3010) et Laboratoire de Physique (CNRS UMR 5672), Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, 69007 Lyon, France ABSTRACT We propose a combined experimental (Atomic Force Microscopy) and theoretical study of the structural and dynamical properties of nucleosomes. In contrast to biochemical approaches, this method allows to determine simultaneously the DNA complexed length distribution and nucleosome position in various contexts. First, we show that differences in the nucleo-proteic structure observed between conventional H2A and H2A.Bbd variant nucleosomes induce quantitative changes in the in the length distribution of DNA complexed with histones. Then, the sliding action of remodeling complex SWI/SNF is characterized through the evolution of the nucleosome position and wrapped DNA length mapping. Using a linear energetic model for the distribution of DNA complexed length, we extract the net wrapping energy of DNA onto the histone octamer, and compare it to previous studies. Keywords : Atomic Force Microscopy, mono-nucleosome, H2A.Bbd, length distribution of wrapped DNA, nucleosome position distribution, chromatin remodeling factor, histone variant INTRODUCTION DNA is packaged into chromatin in the cell nucleus. The chromatin repeating unit, called the nucleosome, consists of an octamer of the core histones (two each of H2A, H2B, H3 and H4) around which about two superhelical turns of DNA are wrapped (1). The Nucleosome Core Particle (NCP) represents a barrier for the transcription factors binding to their target DNA sequences and interferes with several basic cellular processes (2). Histone modifications, ATP-remodeling machines and the incorporation of histone variants within chromatin are used by the cell to overcome the nucleosomal barrier and modulate DNA accessibility by the control of nucleosome dynamics (3-6). In this work, we use a single molecule technique (Atomic Force Microscopy) to visualize isolated mono-nucleosomes, to quantify the influence of histone octamer composition (H2A-Bbd variant) on the equilibrium nucleosome conformation and to map nucleosome mobility induced by a remodeling complex (SWI/SNF). Chromatin remodeling complexes are used by the cell to overcome the general repression of transcription associated with the DNA organization into chromatin (7-9). In order to destabilize histone-DNA interaction, remodeling factors (like SWI/SNF) consume the energy from ATP hydrolysis to relocate the histone octamer along the DNA sequence (10, 11) and in some cases, the ejection of the octamer from the DNA template is observed (12). The molecular motor SWI/SNF is known to mobilize the histone octamer from a central to an end- position on short DNA templates (13). Nevertheless, the molecular mechanisms involved in the nucleosome remodeling process have not yet been elucidated . Histone variants are nonallelic isoforms of the conventional histones. The function of the different histone variants is far from clear, but the emerging general picture suggests that the incorporation of histone variants (14-19) in the nucleosome has serious impacts on several processes, including transcription and repair, and it may have important epigenetic consequences (20-23). H2A.Bbd (Barr body deficient) is an unusual histone variant whose primary sequence shows only 48% identity compared to its conventional H2A counterpart (24). The current view is that H2A.Bbd is enriched in nucleosomes associated with transcriptionally active regions of the genome (24). In recent studies, the unusual properties of this variant nucleosome were described (16, 25) using a combination of physical methods and molecular biology approaches. Those results were mainly focused on the biological role of the various histone fold domains of H2A.Bbd on the overall structure, stability and dynamics of the nucleosome, whereas we concentrate here on the quantification of the subtle modifications in the nucleosome conformation induced by the presence of this histone variant. Different experimental approaches have been used so far to study the structure and dynamics of the nucleosome, including crystallographic studies by Luger et al. (26) , restriction enzyme accessibility assays (27, 28), and FRET measurements (29, 30). Additionally, physical models (31) and recent computational efforts were developed to describe the nucleosome dynamics and energetics (32-35). Following these numerous contributions, the present study combines experimental (Atomic Force Microscopy) and theoretical tools to bring complementary information regarding the interplay between nucleosome position dynamics and DNA wrapping energetics. Atomic Force Microscopy (36) allows direct visualization of chromatin fibers and isolated nucleosomes (37). Several experimental procedures allow to depose and observe reproducibly, all this without any fixing agent, DNA or chromatin samples (38-44). By scanning the sample with an apex of very high aspect ratio mounted on a flexible lever, the topography of a surface at the nanometric scale can be acquired. Moreover, computer analysis of AFM images enables the extraction of systematic and statistically relevant distributions of structural parameters describing these biological objects (45-47). As nucleosome is a complex and very dynamic structure, it has been observed that, for a given DNA template, the position of the octamer relative to the sequence (13, 48-50) and the length of DNA wrapped around the histone octamer (27-29, 51, 52) both could change drastically in time. This paper is organized as follows. First, we show that mapping the nucleosome position along with the length of DNA complexed with histones within individual nucleosome is a powerful tool to discriminate between conventional and variant nucleosomes. A model is then proposed to explain quantitatively these differences and to calculate the wrapping energy of nucleosomes in each case. Next, we have studied nucleosomes in a more dynamic context by observing the action of chromatin remodeling factor SWI/SNF. To do so, similar mapping of the nucleosome position and DNA complexed length was used to quantify the impact of ATP-activated remodeling and sliding of nucleosomes. The results suggest experimental insights into the processivity of SWI/SNF on mono-nucleosomes. MATERIALS AND METHODS Preparation of DNA fragments The 255 bp and 356 bp DNA fragments, containing the 601 nucleosome positioning sequence(53), were obtained by PCR amplification from plasmid pGem-3Z-601. For the 255 bp template, 147 bp long 601 positioning sequence is flanked by 52 bp on one side and 56 bp on the other side. For the 356 bp template, 147 bp long-601 positioning sequence is flanked by 127 bp on one side and 82 bp on the other side. As both 601 DNA templates are built from the same plasmid, the DNA flanking sequences of the short template are included in the long DNA template. Protein purification, nucleosome reconstitution and remodeling Recombinant Xenopus laevis full-length histone proteins were produced in bacteria and purified as described (54). For the H2A.Bbd protein, the coding sequences for the H2A and for H2A.Bbd were amplified by PCR and introduced in the pET3a vector. Recombinant proteins were purified as previously described (55). Yeast SWI/SNF complex was purified as described previously (56) and its activity was normalized by measuring its effect on the sliding of conventional nucleosomes : 1 unit being defined as the amount of ySWI/SNF required to mobilize 50% of input nucleosomes (~50 ng) at 29°C during 45 minutes. Nucleosome reconstitution was performed by the salt dialysis procedure (57). Nucleosomes reconstituted on a 601 nucleosome positioning sequence (20 ng) were incubated with SWI/SNF as indicated at 29°C and in remodeling buffer (RB) containing 10 mM Tris-HCl, pH = 7.4, 2.5 mM MgCl2, and 1 mM ATP. The reaction was stopped after the time as indicated by diluting about 10 times in TE buffer (Tris- HCl 10 mM, pH = 7.4, EDTA 1 mM) and NaCl 2 mM and deposing the sample onto the functionalized APTES-mica surface. Atomic Force Microscopy and surface preparation For the AFM imaging the conventional and variant nucleosomes were immobilized onto APTES-mica surfaces. The functionalization of freshly cleaved mica disks (muscovite mica, grade V-1, SPI) was obtained by self-assembly of a monolayer of APTES under Argon atmosphere for 2 hours (39). Nucleosomes (DNA concentration ~ 75 ng/µl) were filtered and concentrated using Microcon® centrifugal filters to remove free histones from the solution, and diluted 10 times in TE buffer, just prior to deposition onto APTES-Mica surfaces. A 5 µl droplet of the nucleosome solution is applied on the surface for 1 min, rinsed with 1 mL of milliQ-Ultrapure © water and gently dried by nitrogen flow. The samples were visualized by using a Nanoscope III AFM (Digital Instruments™, Veeco, Santa Barbara, CA). The images were obtained in Tapping Mode in air, with silicon tips (resonant frequency 250-350 kHz) or Diamond Like Carbon Spikes tips (resonant frequency ~150 kHz) at scanning rates of 2 Hz over scan areas of 1 µm wide. This surface functionalization was chosen because it is known to trap 3D conformation of naked DNA molecule on a 2D surface (58, 59). Moreover, under such experimental conditions, rinsing and drying are thought to have little effect on the observed conformation of biomolecule (60). Image analysis We have extracted parameters of interest from the AFM images using a MATLAB© (The Mathworks, Natick, MA) script essentially based on morphological tools such as binary dilatation and erosion (61-64), and height/areas selections . The aim of the first three steps of this algorithm is to select relevant objects : 1. In order to remove the piezoelectric scanner thermal drift, flatten of the image is performed. The use of a height criteria (h>0.5nm where h is the height of the object) allows to avoid the shadow artifact induced by high objects on the image. 2. Building of a binary image using a simple thresholding (h > 0.25 nm where h is the height of the object)) and then selection of the binary objects in the good area range (500 < A < 2000 nm² where A is the area of the object)). 3. Selection of the objects in the good height range using a hysteresis thresholding (65) (hmin1 = 0.25 nm and h 2 = 1.4 nm, where h 1 and h 2 are the height of the two thresholds). These three steps leads to the selection of binary objects whose area is between 500 and 2000 nm² and corresponds in the AFM image to a group of connected pixels whose minimun height is more than 0.25 nm and maximum height is above 1.6 nm. For example a height criterion is used to reject tetrasomes while events with SWI/SNF still complexed with nucleosomes are removed from analysis by a size criterion. The next steps correspond to measurements in itself : 4. Detection of the NCP centroid by shrinking the objects in the binary image. 5. Building of a distance map inside the nucleosome with respect to their NCP centroid using a pseudo-euclidian dilatation based algorithm. 6. Selection of the non-octamer parts of the nucleosomes (d > dc , where d is the constraint distance to the NCP centroid and dc ~ 7.5 nm is the apparent nucleosome radius) and then thinning of the free arm regions using a commercial MATLAB© script optimised to avoid most of the branching in the skeleton. 7. Selection of the free arm ends and measurement of the free arm lengths. 8. Measurement of other parameters of interest like areas, volumes and mean height of the nucleosomes and the octamers (see supplemental materials). These last 5 steps lead to quick and robust measurements. Indeed the use of morphological tools allows parallel calculation simultaneously on all the objects. Moreover, erosion is a good approximation for the inverse operation of the AFM dilatation due to the finite tip radius and leads to a partial removal of the tip effect (66, 67). The longest arm is named L+ and the shortest L-. DNA complexed length is deduced by Lc = Ltot - L- - L+ where Ltot is either 255 bp for short conventional and variant nucleosomes or 356 bp for long conventional nucleosomes. The position of the nucleosome relatively to the DNA template center is calculated as ∆L = (L+ - L-)/2. Notice that the position defined this way corresponds to the location of the most deeply buried base pair, which might differ from dyad axis position (strictly defined for symmetric nucleosomes). Complexed DNA length and nucleosome position distribution construction For the distribution of DNA complexed length, well centered nucleosomes were selected (∆L* - σ∆L/2 ~ 0 bp < ∆L < 12 bp ~ ∆L * + σ∆L/2 for the 255 bp mono-nucleosomes where ∆L* is the most probable nucleosome position and σ∆L is the standard deviation of the ∆L distribution). To construct the histogram a 20 bp-sliding box was used. For each L0 in [0, 300 bp], nucleosomes with a DNA complexed length included in the range [L0 – 10 bp, L0 + 10 bp] were counted. After normalization, a smooth distribution is obtained that represents mathematically the convolution of the real experimental distribution with a rectangular pulse of 20 bp long. To obtain the nucleosome position distribution we have selected nucleosomes with a DNA complexed length Lc in a range of width σLc around L * = 146bp (123 bp ~ L* - σLc < Lc < L * - σLc 169 bp for canonical nucleosomes). Then, the same 20-bp sliding box protocol was used to construct the nucleosome position distribution. The error on the distribution function mean value (standard error) is given by σexp/√N, where σexp is the standard deviation of the experimental distribution, and N the number of analyzed nucleosomes (central limit theorem). 2D distribution Lc/∆∆∆∆L construction To construct the 2D-histogram a 10 bp-sliding box was used. For each coordinates (∆L0, L0) in [0, 75 bp]×[0, 300 bp], nucleosomes with a DNA complexed length included in the range [L0 – 5 bp, L0 + 5 bp] and a position included in the range [∆L0 – 5bp, ∆L0 + 5 bp] were counted. After normalization a smooth distribution is obtained that represents mathematically the convolution of the real experimental 2D-distribution with a 10 bp square rectangular pulse. Reproducibility and experimental errors We have checked that different batches of APTES, nucleosome reconstitutions, ySWI/SNF and mica surfaces lead to similar results for the sliding assays and for the 2D mapping within the experimental uncertainty. Moreover we have checked by image analysis of the same naked DNA on the same surface and within the same experimental conditions (data not shown) that the whole measurement and analysis process have an experimental error of about 10 bp in DNA length measurement. Notice that uncertainty on the mean value of length measurements can be much smaller than this resolution as it is explained in the supplemental material S3. RESULTS AND DISCUSSION Simultaneous measurements of DNA complexed length and nucleosome position. Several biochemical approaches allow accessing either the nucleosome position along a DNA template, or the length of DNA wrapped around the histone octamer, but using AFM, we were able to measure them simultaneously. The results are conveniently plotted as 2D histograms of nucleosome position versus DNA complexed length. For short and long arm mononucleosomes We first investigated the influence of the DNA template length on the nucleosome complexed length distribution for conventional nucleosomes. Indeed, one could expect that the nucleosome positioning efficiency for the 601 DNA template and/or the range of wrapped DNA length could depend on the length of free DNA arms. Using purified conventional recombinant histones, nucleosomes were reconstituted by salt dialysis on 255 bp (short nucleosomes) or 356 bp (long nucleosomes) DNA fragments containing the 601 positioning sequence. Tapping Mode AFM in air was used to visualize the reconstituted particles adsorbed on APTES-mica surfaces and images of 1 µm2 were recorded. A representative image of long mono-nucleosomes (Ltot = 356 bp) is displayed on Figure 1a. Such an image enables to clearly distinguish the nucleosome core particle (red part of the complex : hNCP ~ 2 nm) from the free DNA arms (yellow part of the complex, hDNA ~ 0.7 nm) entering and exiting the complex. Precise measurement of the length of each DNA fragment (respectively L+ and L- for the longer and shorter arm) exiting the nucleosome have been performed. To measure each “arm” of the mono-nucleosome, the octamer part is excluded and the free DNA trajectory is obtained (Fig.1b) using morphological tools avoiding false skeletonization by heuristic algorithm ( cf Material and Methods). From the total DNA length that is un-wrapped around the histone octamer, we get the length of DNA organized by the histone octamer (Lc = Ltot - L+ - L-) as well as the nucleosome position with respect to the center of the sequence (∆L = (L+ - L-)/2). The 2D histogram Lc/ ∆L is plotted on Fig. 1c for 702 conventional short nucleosomes using a 2D sliding box as described in the Material and Methods section. The maximum of the 2D distribution is positioned at L* = 145 bp and ∆L = 15 bp, in qualitative agreement with the DNA template construction. The 2D mapping is an important tool to study nucleosome mobilization (see the SWI/SNF sliding section), since both variables are highly correlated during nucleosome sliding/remodeling. Quantitative information can be however also obtained by projecting such a 2D histogram on each axis. First, we have selected well positioned nucleosomes according to the expected position given by the DNA 601 template construction (0 bp < ∆L < 12 bp for short DNA fragments) and shown their DNA complexed length probability density function (red line, Fig. 1d). This distribution of the DNA length, organized by conventional octamer peaks at L* = 146 ± 2 bp, in quantitative agreement with the crystal structure of the nucleosome (26) and cryoEM measurements (25). The broadness of this distribution (σ = 23bp) might be explained by different nucleosomes wrapping conformations. We will explain later on, how this dispersion relates to DNA-histone interaction energies using a simple model. We have used the same approach to study long nucleosomes (2D histogram not shown). Well positioned long nucleosomes according to the DNA sequence (12 bp < ∆L < 32 bp) have very similar probability distribution (blue line on Fig. 1d) than that obtained for short nucleosomes showing that the free linker DNA does not affect significantly the organization of complexed DNA for such nucleosomes. We now select nucleosomes that have a complexed length in the range L* ± σLc, where σLc is the standard deviation of the Lc distribution, and their position distribution is displayed on Figure 1e. The peak values for each DNA fragment (9 ± 2 bp and 24 ± 2 bp for short and long nucleosomes respectively) is close to the expected value from the DNA template construct (2 bp and 22 bp for short and long DNA fragments respectively). Both distributions have a full width at half maximum that exceeds 20 bp. This width might arise from several features : asymmetric unwrapping of one of the two DNA arms, AFM uncertainty and dispersion in octamer position. However, it is not possible with these measurements to determine what is the contribution of each phenomenon. Next, we can see that the distribution width for longer fragments seems greater. After corrections of artifacts inherent to L+/L- labeling (cf Supplemental Figure 2) these two position distributions are very similar showing that the free linker DNA does not affect either the DNA complexed length nor the positioning of such nucleosomes significantly. We have shown in this section that AFM measurements give comparable estimations with other methods for both the positioning and the DNA wrapping of short 601 mononucleosomes. Furthermore, our experimental approach showed no difference in complexed length probability or nucleosome positioning dynamics for long and short DNA templates. For conventional and H2A.Bbd variant mononucleosomes In order to investigate the influence of the octamer composition on the wrapping of DNA around the histone octamer, a H2A.Bbd histone variant was used instead of conventional H2A, in order to reconstitute mono-nucleosomes on a 255 bp DNA fragment. The H2A.Bbd variant nucleosomes were imaged by AFM (25) and using the same analysis as described above, only the well positioned nucleosomes (∆L < 12 bp) were selected. Their DNA complexed length distribution is plotted on Fig. 2a where it is compared to conventional mononucleosomes reconstituted on the same 601 positioning sequence, 255 bp long, with the same position range selection (∆L < 12 bp). The average length of wrapped DNA is clearly different for the variant H2A.Bbd nucleosomes as the distribution peak value is L*H2A.Bbd = 130 ± 3bp instead of L*H2A = 146 ± 2bp for the conventional nucleosomes. Moreover the standard deviation of the distribution is clearly larger for the H2A.Bbd variant (σ = 41 bp to be compared to σ = 23 bp for the conventional nucleosomes). These differences show that the H2A.Bbd variant nucleosome is a more labile complex with less DNA wrapped around the octamer, in agreement with previous observations by AFM and cryo-EM (25). The difference in DNA complexed length suggests that ∼10 bp at each end of nucleosomal DNA are released from the octamer. Therefore, AFM allows visualizing subtle differences in the nucleosome structure. Finally, the DNA complexed length distribution is asymmetric for canonical nucleosomes. This asymmetry can be quantified by measuring their skewness 3µɶ , defined as: 3 c c 2 22 c c (L L ) ((L L ) ) .We find 3µɶ = -0.57 ± 0.09, the negative sign meaning that nucleosome conformations with sub-complexed DNA, as compared to the mean value 146 bp, are energetically more favorable than with over-complexed DNA. This can be interpreted within the simple model proposed below, based on relevant structural data information (26). Notice that for variant nucleosomes, the complexed length distribution is nearly symmetric ( 3µɶ ≈ 0.01 ± 0.16), and this feature will also be discussed in the modeling section. Simple model of DNA complexed length distribution It has been shown that 14 discrete contacts between DNA and histone octamer are responsible for the stability of the nucleosome (26).The energetic gain at these sites is made through electrostatic interactions and hydrogen bonding. At the length scale of the present analysis, the discreteness of binding sites is not relevant, and it will be replaced by a uniform effective adsorption energy εa< per unit length, in units of kT/bp. The finite number of binding sites, or equivalently the finite DNA length L* complexed through these sites (146 bp for canonical nucleosomes, as determined both by the present experiments and crystal structure), is due to the specific locations of favorable interactions located at the surface of the histone octamer, forming a superhelical trajectory on which DNA is complexed. DNA wrapping around the histone core involves additional bending penalty characterized by the energy per unit length : ε = where Lp is the persistence length of DNA within classical linear elasticity and R the radius of the histone octamer. The stability of the nucleosome requires that the net energy per unit length is negative (energetic gain), and therefore : εb < εa< . The experimental distributions of DNA complexed length show that more DNA can be wrapped around the octamer. For these additional base pairs, the net energy per unit length has to be positive, due mainly to bending cost. However, to allow for the possibility of some residual non specific (mainly electrostatic) attractive interactions beyond the 14 binding sites, the energetic gain of DNA contacting the octamer surface outside of the 14 sites superhelical path has a different value denoted εa> . The difference εa< - εa> is then representative of the specificity of the 14 sites region. Assuming that the energy reference is given by un-complexed straight DNA and octamer, the total energy for nucleosome is given by * * * (sub-complexed nucleosome) (over-complexed nucleosome) ( ) L if L <LE(L ) ( ) L + ( ) (L -L ) if L >L − ⋅ − ⋅ a ab b ε ε ε ε (1) The distribution of DNA complexed length is given by cc -E(L ) / (kT) (L ) ∝P e . It is maximum for the characteristic length L*, which characterizes the region of specific contacts. This length may vary for canonical and variant nucleosomes. The assumptions of energy linearity in wrapped DNA length and of the existence of L*, lead to a double exponential distribution. By construction, one has the following constraints between effective energies εa> < εb < εa< . It should be kept in mind that the effective values εa>, εa< and εb are representative of nucleosomes adsorbed on a charged flat surface. These values might differ for nucleosomes in bulk solution, as discussed below. Extraction of the DNA complexed length parameters It is possible to extract some parameters from each distribution by using the physical model presented below, in order to interpret the experimental distribution of DNA complexed length. We found it more reliable to use global procedure for parameter determination, instead of fitting the multivariate distribution. Since we expect the DNA complexed length distribution to be described by a simple double-exponential model, the probability density function can be written as a skew-Laplace distribution which moments are calculated as : *2 (1 ) 2 2 2 2 c c *2 (1 ) 3 2 3 3 3/ 2 3/ 2 2 3/ 2 L 2 2 , for L>L1 ( ) and then (L L ) 4 (1 ) , for L<L (L L ) 4 50 2 12 48 2 4 (1 )  = = −   = = = − = +  − − + −  = = P L L µ σ ε ε ε ε where L* is the most probable complexed length, ε is the relative asymmetry of the skew- Laplace distribution and σ is the mean decay length. The distribution normalization is taken on full real axis as a first approximation, thus neglecting finite size effects. Given the experimentally determined µ and µ parameters, we extract straightforwardly the parameter L*, ε and σ by numerically solving the equation system (2). Hence, we are able to measure without any fitting the parameters L*, ε and σ by calculating the first three moments µ1, µ2 and µ3 of the DNA complexed length statistical series. In our case we thus have : 2(1 ) 2 and then 1 1 (1 ) 2(1 )  − = − = − + = − = − − = = specific ads a a To see the adequacy of this model with the experimental distribution, the function P(Lc=L) is drawn for the parameters extracted from the experimental data using the same 20 bp-sliding box protocol as for the experimental complexed length distribution (Fig.2) The results are summarized in table 1. The values of energies are expressed in units of kT per binding site, assuming 14 such sites along the 147 base pairs of DNA for canonical nucleosomes. Several comments are to be made on these values. First, the measured characteristic decay lengths corresponding to sub- (L<) and over-complexed (L>) DNA lengths (Table 1, (b) and (d)) are clearly higher than the intrinsic resolution of our AFM measurements (related to the tip size that correspond to ~ 10 bp, as checked by image analysis of the same naked DNA on the same surface and within the same experimental conditions - data not shown) for both conventional and variant nucleosomes, showing therefore the significance of the parameters extracted here. Hence, we are able to quantify the energetic of both sub- and over-complexed DNA length in a mono-nucleosome. For over-complexed DNA length, the energy has been converted artificially into units of kT per binding site for the sake of comparison, although the model assumes that there are no such binding sites beyond the 14 sites found in the crystal structure (26). If one assumes that over-complexed DNA length results solely from bending around the histone core (εa> = 0), the value found for εb leads to a persistence length Lp ~ 3.5 bp, a value definitely too small for double stranded DNA. Even more so, this energy is similar in amplitude to the energy of sub-complexed DNA length but with an opposite sign (Table 1, (c) and (e)). We conclude that it cannot simply be associated to a bending penalty, therefore justifying a posteriori the assumption of residual attractive interaction between DNA extra length and histone octamer. The combination of experimental asymmetry of DNA complexed length distribution and the simple model allows quantifying the specificity of the 14 binding sites in the nucleosomes (Table 1, (f)). In particular this can be interpreted as a rough estimation of non- electrostatic contribution to adhesion energy between DNA and histone octamer. Comparison of model parameters extracted from data. These values have to be compared to other estimates reported in the literature. The net energetic gain per site can be compared to values extracted from experiments done in the group of J. Widom (68-70). The spirit of these experiments was to probe the transient exposure of DNA complexed length in a nucleosome by using different restriction enzymes acting at various well-defined sites along the DNA. The experimental results clearly demonstrate that DNA accessibility is strongly reduced when restriction sites are located far away from entry or exit of nucleosomal DNA, towards the dyad axis. From the experimental data, the authors extract a Boltzmann weight for different site exposures. This distribution should a priori be similar to the DNA complexed length distribution obtained in our work, except that only sub-complexed nucleosomes are probed. However, due to the use of different restriction enzymes with different sizes and mechanisms of action, there is an inherent uncertainty in the assignment of precise DNA complexed length with a free energy of the Boltzmann weight. In other words, only a range of energy per binding site can be extracted from these data. This has to be contrasted with most of previous works using Polach and Widom's data, which quote a single value of 2 kT per binding site (31). The range of net energetic gain we are able to estimate out of these data is between 0,5 to 3 kT per binding site. The value we extracted from our own measurements coincides therefore with the lower bound of this range. This might be due to the difference in the type of experiments used. First, our observations are made on nucleosomes adsorbed on a charged substrate. This might change the energetics of nucleosome opening as compared to its value in solution. A theoretical estimation of this change is currently under progress (Castelnovo et al, work in preparation). Another significant difference between Polach and Widom's experiments and our work is the composition of the buffer, which is known to affect the nucleosome stability. In particular, the buffer used for restriction enzyme assays contains more magnesium ions (about 10 mM MgCl2). The specificity of DNA binding sites on histone octamer, as determined in Table 1 (e) can also be compared to values extracted from X-ray experiments performed in the group of T.J. Richmond (71). Indeed, by counting the hydrogen bonds per binding site found in this structure, one can estimate the specific contribution to the binding energy. These contributions range between 0.8 and 2 kT per binding site (72). Our estimate for conventional nucleosome falls in this range (1.1 kT per binding site). Finally, the comparison between canonical and variant nucleosome shows that both the average complexed length and the energy per binding site are different. The most probable length L* = 127 bp (Table 1 (0)) for the variant claims for either the absence or the strong weakening of at least 2 binding sites. Furthermore, the energy and therefore the stability of the nucleosome for the remaining binding sites is reduced (εH2A.Bbd ~ 2/3 εH2A), in accordance with other experimental observations (16, 25, 73). We have shown in this section that a simple model using a linear energy for the DNA-histone interaction can be used to extract from the AFM data two important energetic parameters : the net energetic gain per site and the specific interaction between the DNA and the histone octamer per site. These values are in good agreement with previous biochemical and X-Ray studies done on conventional nucleosomes and for the first time are measured on a variant nucleosome. Visualization of nucleosome sliding and remodeling by SWI/SNF for conventional and variant nucleosomes. After studying the nucleosomes in their equilibrium state, the same mononucleosomes were visualized in the presence of the SWI/SNF remodeling factor to validate the possibility for this direct imaging approach to acquire new information on the mechanism and dynamics of nucleosome sliding. Centrally positioned conventional and variant mononucleosomes (Ltot = 255 bp) were incubated with SWI/SNF at 29°C in the presence or absence of ATP and then adsorbed on APTES-mica surfaces for AFM visualization. On figure 3, we report AFM images of mononucleosomes incubated without (Fig. 3a) and with (Fig. 3b) ATP for one hour. The sample containing no ATP is the control experiment to account for possible nucleosome thermally driven diffusion when incubated 1 h at 29 °C. The representative chosen set of AFM images of Fig.3 clearly shows that most of the nucleosomes are centered on the DNA template in the negative control (-ATP) whereas they rather exhibit end-position when SWI/SNF and ATP are present. On AFM images, SWI/SNF motor is sometimes visible as a very large proteic complex, and if still attached to nucleosome prevents any image analysis of such objects. Our protocol does not include removing of SWI/SNF before deposition, even if by diluting the nucleosome/motor mix, one could expect detaching of some motors. Therefore, the motor per nucleosome ratio used in the sliding experiments is kept low with respect to biochemical assays (55) (roughly five time less SWI/SNF per nucleosome). Using the same type of image analysis we were able to reconstruct 2D histograms Lc/∆L (using a 2D sliding box as described in the Material and Methods section) at various time steps during nucleosome sliding : 0 (-ATP), 20 min and 1 hour (Fig. 4). We first notice that in the absence of ATP, SWI/SNF has apparently no effect on the Lc/∆L map. The 2D distribution exhibits a single peak corresponding to the canonical nucleosome positioned as expected from the DNA template (α state). As a function of time in the presence of remodeling complex and ATP, new states appear : (β) corresponds to an over-complexed nucleosome having the same mean position ∆L value as (α); this state could result from the capture of extra DNA (a loop of ~ 40 bp) inside the NCP induced by SWI/SNF. (β)-state is spread in the ∆L direction showing that this extra complexed DNA length (~ 40 bp) seems to exist for various positions of the nucleosome (0 < ∆L < 30 bp). (γ) is the slided end-positioned nucleosome (∆L ~ 50 bp) having slightly less DNA wrapped around the histone octamer (Lc ~ 125 bp). The ∆L distance separating (α) and (γ) states is close to the Lc distance between (β) and (α) states, meaning that the slided (γ)-state most likely results from the release of the (β)-state DNA loop (~ 40 bp). The fact that slided nucleosomes are sub- complexed i.e. their dyad has been moved beyond the expected end-position, has already been observed in other biochemical studies (74). Similarly, the anisotropic spreading of the (γ)-peak towards higher ∆L and lower Lc is also consistent with this feature. We cannot exclude that a finite size effect of the DNA template could account for this feature. Finally, (δ) is a wide state with a sub-complexed Lc ~ 75 bp, that could correspond to a tetrasome or hexasome. This state could be due to the loss of one wrapped DNA turn either from the α state or the (γ)-state. Nevertheless, one could notice that the (δ)-state is missing on the ‘+ATP 20 min’ map (Fig. 4b) where only few nucleosomes have been slided (weak γ peak) whereas (δ)-state nucleosomes are clearly visible on Fig. 4c (‘+ATP 1 h’). This tends to show that (δ)- state nucleosomes more likely arise from the loss of one DNA turn of the end-positioned nucleosomes ((γ)-state). We have seen that the 2D-mapping of nucleosome position and DNA complexed length allows characterizing the new states resulting from the ATP-dependent action of SWI/SNF on our 601 nucleosomes : an over-complexed state close to the 601 template center (β), a slided state (γ) and a sub-complexed state (δ). Again, more information can be gained by appropriate projections of these 2D- histograms. Nucleosomes having their DNA complexed length in the range L* ± σLc were selected (L* and σLc are respectively the maximum value and the standard deviation of the corresponding complexed length distribution) and their position distribution is plotted on Fig. 5a. For conventional nucleosomes with SWI/SNF but no ATP, the distributions obtained for nucleosome position (Fig. 5a) and DNA complexed length (Fig. 5b) are very similar to the case without any remodeling complex (Fig. 1b), showing no effect of thermally driven diffusion of mononucleosomes reconstituted on 601 positioning sequence in our conditions. When incubation is increased in the presence of ATP (20 minutes and 1 hour), the position distribution of conventional nucleosomes is clearly changed (Fig. 5a). Indeed, as a function of incubation time, a second peak appears corresponding to the end-positioned nucleosomes (∆L ~ 50 bp, cf (γ)−state in Fig. 4c). After one hour of SWI/SNF action in presence of ATP the second peak height has increased at the expense of the primary peak. This corresponds to the situation were one third of the mono-nucleosomes are positioned at the end of the DNA template. It is interesting to note that during the remodeling factor action we do not see any significant increase in the amount of nucleosomes in an intermediate position (20 bp < ∆L < 40 bp). This provides experimental evidence that this remodeling factor moves centrally positioned nucleosomes directly to the end of our short DNA template. Mainly, two situations can explain the bimodal position distribution of nucleosomes after the action of SWI/SNF. The first hypothesis is that the SWI/SNF complex is a processive molecular motor. As it will not detach from the nucleosome before it reaches the end of the DNA template, the elementary step of the SWI/SNF induced sliding might not be accessible. Indeed, in our experimental conditions, only nucleosomes without SWI/SNF complex attached can be analyzed. The other possibility is that SWI/SNF is weakly processive (SWI/SNF turnover rate is unknown) but with an elementary step of the order of 50 bp, which corresponds to the value measured by us and other approaches (75-77), and happens to be the length of free DNA arms in our case. Therefore, a single step would be enough for the motor to slide a nucleosome to an end-position and release the complex. Nevertheless, another mechanism cannot be excluded by our data, where SWI/SNF action would consist of octamer destabilization followed by thermally driven diffusion towards the end-positioned entropically favored. In this situation, ATP-hydrolysis would only be involved in the nucleosome ‘destabilization’ step. In Fig. 5b, we show projections of the previous 2D-histograms along the DNA complexed length axis without any selection on their position. For conventional nucleosomes in the presence of SWI/SNF but no ATP, the complexed length distribution is similar to case with neither SWI/SNF nor ATP. However, the former distribution is larger due to the contribution of different nucleosome positioning. Then after 20 min, the distribution is broader (roughly twice) and shifted towards higher Lc. This might be attributed to the contributions of the different states (β, γ, δ) identified in the Fig. 4b/c. The increase in Lc mean value is likely due to the statistical weight of the over-complexed (β)-state. The same sliding experiment was performed on H2A.Bbd variant nucleosomes in absence and in presence of ATP and analyzed through the projection of the 2D-histogram Lc/∆L. No significant effect of SWI/SNF complexes in presence of ATP on the position distribution of H2A.Bbd variant nucleosomes is observed (Fig. 5c). This corroborates previous findings using biochemical sliding assay done on 5S and 601 positioning sequence (55). However in AFM measurements, the full position distribution is accessed directly with a resolution better than 10 bp (the size of AFM tip). This variant nucleosome sliding assay shows the reproducibility of our experimental approach as not only the position distribution mean value is constant during one hour in the presence of SWI/SNF and ATP, but also the complete position distribution remains constant (Fig. 5c). Similarly, SWI/SNF in presence of ATP does not seem to influence the DNA complexed length distribution of H2A.Bbd nucleosomes (Fig. 5d). CONCLUSION In summary, we have shown that AFM combined with a systematic computer analysis is a powerful tool to determine the structure of conventional and variant mononucleosomes at equilibrium and after the action of ATP-dependent cellular machineries. With this technique we have quantified simultaneously two important and closely coupled variables : the DNA complexed length and the position of mono-nucleosomes along the 601 DNA template. For each of these two distributions, the most probable value is in perfect agreement with measurements done by other methods that give access to one of these two parameters only. In addition, to explain the experimental complexed length distribution, we have developed a simple model that uses the experimental shape of DNA complexed length distributions to quantify the interaction of DNA with histones. With this model, we extract both the net energetic gain for sub-complexed nucleosomes and the estimation of the non-electrostatic contribution to the adhesion energy between DNA and histone octamer . We further show that H2A.Bbd variant and conventional nucleosomes exhibit clear differences in DNA complexed length and in their ability to be slided by SWI/SNF. Indeed, these variant nucleosomes organize less DNA on average than conventional nucleosomes, and present larger opening and closing fluctuations. Moreover, the whole position distribution as well as complexed length distribution remain unchanged showing H2A.Bbd variant is neither displaced nor remodeled by SWI/SNF complex. Finally, we have plotted Lc/∆L as a 2D map of the nucleosome states. This representation is well suited to highlight the various nucleosome states that appear during the SWI/SNF action. For example, as a function of time, we have evidenced the formation of an over-complexed state followed by the appearance of a slided state. More quantitative information can be obtained by appropriate projections of the 2D-histograms, as for instance the bimodal position distribution induced by SWI/SNF sliding on conventional nucleosomes, suggesting two possible scenarii : a processive action of the molecular motor (no intermediate position visualized) or an elementary stepping length (~ 40 bp) of the size of the free DNA arms (~ 50 bp). The short length of DNA templates and lack of directionality in our position analysis prevent us from discriminating between these two hypotheses, and further experiments on long oriented mononucleosomes are needed to get more insights into the molecular mechanism of SWI/SNF action. The present results as well as preliminary data on longer oriented templates prove nevertheless that this extension will provide useful information on remodeling mechanisms of SWI/SNF. A further perspective of this AFM study will be to test the effect of the flanking DNA sequences on the conformation and dynamics of 601 nucleosomes. Nevertheless, in order to test sequence effect, nucleosomes should be reconstituted on less positioning sequences (5S rDNA for example) or non-positioning sequences, but this will complicate significantly the nucleosome sliding analysis as the initial position distribution of the nucleosome is expected to be broader in this case. ACKNOWLEDGMENTS We thank Dimitar Angelov and Hervé Ménoni for various forms of help with nucleosome reconstitution and sliding assays and Cécile-Marie Doyen for producing H2A.Bbd variant histones. We are grateful to Stefan Dimitrov, Dimitar Angelov, Françoise Argoul, Alain Arneodo, Cédric Vaillant and Phillipe Bouvet for fruitful discussions. We thank Ali Hamiche for providing us with the ySWI/SNF complex. P.St-J. acknowledges CRSNG for financial support. 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TABLE 1 <Lc> (bp) decay length L< (bp) εεεεb - εεεεa< (kT per site) decay length L> (bp) εεεεb - εεεεa> (kT per site) εεεεa< - εεεεa> (kT per site) conventional nucleosome 146 ± 2 22 ± 1.6 -0.479 ± 0.045 17 ± 1.4 0.61 ± 0.064 1.1 ± 0.072 variant nucleosome 127 ± 3 31 ± 1.5 -0.33 ± 0.022 27 ± 1.5 0.39 ± 0.026 0.72 ± 0.021 site exposure model (g) -3 <...< -0.5 crystal structure (h) 147 0.8<…<2 Caption Table1 : Summary of model parameters extracted from experimental data as explained in Materials and methods. All energies are expressed in units of kT per binding site. DNA lengths are expressed in bp. (a) Average complexed length (b) Characteristic length L< of exponential decay towards sub-complexed DNA length. (c) Energy per binding site (1/ L<) for sub-complexed DNA length. (d) Characteristic length L> of exponential decay, towards over-complexed DNA length. (e) Energy per binding site (1/L>) for over-complexed DNA length. (f)Asymmetry of adhesion energy per binding site between sub- and over-complexed DNA length. (g) Range of values extracted from Polach and Widom (27, 69) data using the site exposure model. (h) Range of values extracted from Davey and Richmond (71) data using X-ray crystal structure of the nuclear core particle. Uncertainty values are determined using the central limit theorem and a propagation of uncertainty calculus detailed in supplemental data. N(H2A conventional) = 301 nucleosomes. N(H2A.Bbd variant) = 252 nucleosomes. FIGURE CAPTIONS Figure 1 : AFM visualization of centered mononucleosomes with short and long arms. (a) AFM topography image of mono-nucleosomes reconstituted on 356 bp 601 positioning sequence. Color scale : from 0 to 1.5 nm. X/Y scale bar : 100 nm. (b) Zoom in the AFM topography image of a centered mono-nucleosome and the result of the image analysis. Black line : contour of the mono-nucleosome. Blue point : centroid of the histone octamer. Blue dot circle : excluded area of the histone octamer. Blue line : skeletons of the free DNA arms. Color scale : from 0 to 1.5 nm. X/Y Scale bar : 20 nm. The longest arm is named L+ and the shortest L-. DNA complexed length is deduced by Lc = Ltot - L- - L+ where Ltot is in this case 356 bp. The position of the nucleosome relatively to the center of the sequence is calculated by ∆L = (L+ - L-)/2. (c) 2D histogram Lc/∆L representing the DNA complexed length Lc along with the nucleosome position ∆L for a short DNA fragment of 255 bp (N = 702 nucleosomes). (d) Probability density function of the DNA complexed length Lc for a short DNA fragment (255 bp, purple line) and for a long DNA fragment (356 bp, blue line) obtained by selecting the well positioned nucleosomes (0 < ∆L < 12 bp and 12 < ∆L < 32 bp for the short and long fragments respectively) and projecting the 2D map along the y-axis. (e) Probability density function of the ∆L nucleosome position for a short DNA fragment (255 bp, purple line) and for a long DNA fragment (356 bp, blue line) obtained by selecting nucleosomes having their DNA complexed length Lc in the range 123 bp < Lc < 169 bp for both fragments, and projecting the 2D map along the x-axis. Figure 2 : AFM Visualization of centered H2A.Bbd variant and H2A conventional mononucleosome. (a) Probability density function of the DNA complexed length Lc for a short DNA fragment (255 bp) with conventional H2A (solid thick line) and with variant H2A.Bbd (dotted thick line) nucleosome. Simple model for conventional and variant nucleosomes (respectively solid and dashed thin lines). (b) Description of the model used to measure the DNA-histone adsorption energies per bp (εa< and εa>) and the DNA bending energy per bp (εb) (dotted line). Representation of the model using the 20bp-sliding-box procedure (dotted dashed line). L* corresponds to the most probable DNA complexed length of the distribution. Figure 3 : AFM Visualization of the sliding of centered mononucleosomes by the remodeling complex SWI/SNF. AFM topography image of mononucleosomes reconstituted on 255 bp 601 positioning sequence, incubated at 29°C with SWI/SNF for one hour (a) in the absence and (b) in the presence of ATP. Color scale : from 0 to 1.5 nm. X/Y Scale bar : 150 nm. Zoom in the AFM topography image of a (c) centered mononucleosome and (d) end-positioned mononucleosome the result of the image analysis. Black line : contour of the mono- nucleosome. Blue point : centroid of the histone octamer. Blue line : skeletons of the free DNA arms. Color scale : from 0 to 1.5 nm. X/Y Scale bar : 20 nm. Figure 4 : Evolution of nucleosome Lc/∆∆∆∆L map during nucleosome sliding by SWI/SNF complex for conventional nucleosome. 2D histogram Lc/∆L representing the DNA complexed length Lc along with the nucleosome position ∆L for a conventional nucleosome reconstituted on a short DNA fragment (255 bp) in the presence of remodeling complex SWI/SNF (a) without ATP (1h at 29°C) (b) with ATP (20 min at 29°C) and (c) with ATP (1h at 29°C). (d) Representation of the nucleosome Lc/∆L states for (α), (β), (χ) and (δ) positions as pointed on the 2D maps. N(-ATP, 1h at 29°C) = 692 nucleosomes, N(+ATP, 20 min at 29°C) = 245 nucleosomes, N(+ATP, 1h at 29°C) = 655 nucleosomes. Figure 5 : Evolution of nucleosome position and DNA complexed length distributions during nucleosome sliding by SWI/SNF complex, for conventional and variant nucleosome. Nucleosome position ∆L (a) and DNA complexed length Lc (b) distributions as a function of time (0, 20 min, 1 hour) in the presence of SWI/SNF, for conventional mono-nucleosomes reconstituted on 255 bp long 601 positioning sequence. Nucleosome position ∆L (c) and DNA complexed length Lc (d) distributions as a function of time (0, 1 hour) in the presence of SWI/SNF, for H2A.Bbd variant mono-nucleosomes reconstituted on the same DNA template. The zero time is given by the control in the absence of ATP (solid purple line). For each ∆L position distribution, only nucleosomes having their complexed length in the range Lc * ± σLc are selected. For the sake of figure clarity, error bars are only depicted on 2 distributions of graph (a). Lc = Ltot - L-- L+ ∆∆∆∆L = (L+-L-) / 2 Figure 1 (a)(a) (b)(b) L-=86bp L+=130bp Ltot = 255 bp, 0 < ∆L< 12 bp Ltot = 356 bp, 12 < ∆L< 32 bp 0 0.01 0.02 Probability Ltot = 255 bp, 123 < Lc <169 bp 123 < Lc <169 bp Ltot = 356 bp, Lc (bp) ∆∆ ∆∆ 50 150 250200100 12e-4 probability scale Figure 2 Lcomplexed c( ) L( ) b aP ε εα <+ ⋅ *c cL e if L < L c( ) L( ) b aP ε εα >+ ⋅ *c cL e if L > L -50 0 50 100 150 200 250 300 0.005 0.015 complexed H2A, L =255bp, ∆L<12bp H2A-Bbd, L =255bp, ∆L<12bp Model H2A Model H2A.Bbd + ATP + SWI/SNF - ATP + SWI/SNF Figure 3 Figure 4 ((((a)))) ((((b)))) (α)(α) (α)(α) (α)(α) (β)(β) (γ)(γ) (δ)(δ) LLLLcccc (bp)(bp)(bp)(bp) (γ)(γ) 147bp (601) 12e-4 probability scale 7.5e-4 2.5e-4 7.5e-4 2.5e-4 0 50 100 150 200 250 300 0.002 0.004 0.006 0.008 0.012 -ATP t=1h +ATP t=20min +ATP t=1h H2A nucleosomes 0 25 50 75 100 0.005 0.015 ∆∆∆∆ L (bp) -ATP 1h +ATP 1h H2A.Bbd nucleosomes (d) 0 50 100 150 200 250 300 0.002 0.004 0.006 0.008 0.012 -ATP t=1h +ATP t=1h H2A.Bbd nucleosomes (a) H2A nucleosomes 0 25 50 75 100 0.005 0.015 ∆∆∆∆L (bp) -ATP, t=1h +ATP, t=20min +ATP, t=1h Figure 5

We propose a combined experimental (Atomic Force Microscopy) and theoretical study of the structural and dynamical properties of nucleosomes. In contrast to biochemical approaches, this method allows to determine simultaneously the DNA complexed length distribution and nucleosome position in various contexts. First, we show that differences in the nucleo-proteic structure observed between conventional H2A and H2A.Bbd variant nucleosomes induce quantitative changes in the in the length distribution of DNA complexed with histones. Then, the sliding action of remodeling complex SWI/SNF is characterized through the evolution of the nucleosome position and wrapped DNA length mapping. Using a linear energetic model for the distribution of DNA complexed length, we extract the net wrapping energy of DNA onto the histone octamer, and compare it to previous studies.

Montel_2007_condmat AFM imaging of SWI/SNF action : mapping the nucleosome remodeling and sliding Fabien MONTEL, Emeline FONTAINE, Philippe ST-JEAN, Martin CASTELNOVO and Cendrine FAIVRE-MOSKALENKO Laboratoire Joliot-Curie (CNRS USR 3010) et Laboratoire de Physique (CNRS UMR 5672), Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, 69007 Lyon, France ABSTRACT We propose a combined experimental (Atomic Force Microscopy) and theoretical study of the structural and dynamical properties of nucleosomes. In contrast to biochemical approaches, this method allows to determine simultaneously the DNA complexed length distribution and nucleosome position in various contexts. First, we show that differences in the nucleo-proteic structure observed between conventional H2A and H2A.Bbd variant nucleosomes induce quantitative changes in the in the length distribution of DNA complexed with histones. Then, the sliding action of remodeling complex SWI/SNF is characterized through the evolution of the nucleosome position and wrapped DNA length mapping. Using a linear energetic model for the distribution of DNA complexed length, we extract the net wrapping energy of DNA onto the histone octamer, and compare it to previous studies. Keywords : Atomic Force Microscopy, mono-nucleosome, H2A.Bbd, length distribution of wrapped DNA, nucleosome position distribution, chromatin remodeling factor, histone variant INTRODUCTION DNA is packaged into chromatin in the cell nucleus. The chromatin repeating unit, called the nucleosome, consists of an octamer of the core histones (two each of H2A, H2B, H3 and H4) around which about two superhelical turns of DNA are wrapped (1). The Nucleosome Core Particle (NCP) represents a barrier for the transcription factors binding to their target DNA sequences and interferes with several basic cellular processes (2). Histone modifications, ATP-remodeling machines and the incorporation of histone variants within chromatin are used by the cell to overcome the nucleosomal barrier and modulate DNA accessibility by the control of nucleosome dynamics (3-6). In this work, we use a single molecule technique (Atomic Force Microscopy) to visualize isolated mono-nucleosomes, to quantify the influence of histone octamer composition (H2A-Bbd variant) on the equilibrium nucleosome conformation and to map nucleosome mobility induced by a remodeling complex (SWI/SNF). Chromatin remodeling complexes are used by the cell to overcome the general repression of transcription associated with the DNA organization into chromatin (7-9). In order to destabilize histone-DNA interaction, remodeling factors (like SWI/SNF) consume the energy from ATP hydrolysis to relocate the histone octamer along the DNA sequence (10, 11) and in some cases, the ejection of the octamer from the DNA template is observed (12). The molecular motor SWI/SNF is known to mobilize the histone octamer from a central to an end- position on short DNA templates (13). Nevertheless, the molecular mechanisms involved in the nucleosome remodeling process have not yet been elucidated . Histone variants are nonallelic isoforms of the conventional histones. The function of the different histone variants is far from clear, but the emerging general picture suggests that the incorporation of histone variants (14-19) in the nucleosome has serious impacts on several processes, including transcription and repair, and it may have important epigenetic consequences (20-23). H2A.Bbd (Barr body deficient) is an unusual histone variant whose primary sequence shows only 48% identity compared to its conventional H2A counterpart (24). The current view is that H2A.Bbd is enriched in nucleosomes associated with transcriptionally active regions of the genome (24). In recent studies, the unusual properties of this variant nucleosome were described (16, 25) using a combination of physical methods and molecular biology approaches. Those results were mainly focused on the biological role of the various histone fold domains of H2A.Bbd on the overall structure, stability and dynamics of the nucleosome, whereas we concentrate here on the quantification of the subtle modifications in the nucleosome conformation induced by the presence of this histone variant. Different experimental approaches have been used so far to study the structure and dynamics of the nucleosome, including crystallographic studies by Luger et al. (26) , restriction enzyme accessibility assays (27, 28), and FRET measurements (29, 30). Additionally, physical models (31) and recent computational efforts were developed to describe the nucleosome dynamics and energetics (32-35). Following these numerous contributions, the present study combines experimental (Atomic Force Microscopy) and theoretical tools to bring complementary information regarding the interplay between nucleosome position dynamics and DNA wrapping energetics. Atomic Force Microscopy (36) allows direct visualization of chromatin fibers and isolated nucleosomes (37). Several experimental procedures allow to depose and observe reproducibly, all this without any fixing agent, DNA or chromatin samples (38-44). By scanning the sample with an apex of very high aspect ratio mounted on a flexible lever, the topography of a surface at the nanometric scale can be acquired. Moreover, computer analysis of AFM images enables the extraction of systematic and statistically relevant distributions of structural parameters describing these biological objects (45-47). As nucleosome is a complex and very dynamic structure, it has been observed that, for a given DNA template, the position of the octamer relative to the sequence (13, 48-50) and the length of DNA wrapped around the histone octamer (27-29, 51, 52) both could change drastically in time. This paper is organized as follows. First, we show that mapping the nucleosome position along with the length of DNA complexed with histones within individual nucleosome is a powerful tool to discriminate between conventional and variant nucleosomes. A model is then proposed to explain quantitatively these differences and to calculate the wrapping energy of nucleosomes in each case. Next, we have studied nucleosomes in a more dynamic context by observing the action of chromatin remodeling factor SWI/SNF. To do so, similar mapping of the nucleosome position and DNA complexed length was used to quantify the impact of ATP-activated remodeling and sliding of nucleosomes. The results suggest experimental insights into the processivity of SWI/SNF on mono-nucleosomes. MATERIALS AND METHODS Preparation of DNA fragments The 255 bp and 356 bp DNA fragments, containing the 601 nucleosome positioning sequence(53), were obtained by PCR amplification from plasmid pGem-3Z-601. For the 255 bp template, 147 bp long 601 positioning sequence is flanked by 52 bp on one side and 56 bp on the other side. For the 356 bp template, 147 bp long-601 positioning sequence is flanked by 127 bp on one side and 82 bp on the other side. As both 601 DNA templates are built from the same plasmid, the DNA flanking sequences of the short template are included in the long DNA template. Protein purification, nucleosome reconstitution and remodeling Recombinant Xenopus laevis full-length histone proteins were produced in bacteria and purified as described (54). For the H2A.Bbd protein, the coding sequences for the H2A and for H2A.Bbd were amplified by PCR and introduced in the pET3a vector. Recombinant proteins were purified as previously described (55). Yeast SWI/SNF complex was purified as described previously (56) and its activity was normalized by measuring its effect on the sliding of conventional nucleosomes : 1 unit being defined as the amount of ySWI/SNF required to mobilize 50% of input nucleosomes (~50 ng) at 29°C during 45 minutes. Nucleosome reconstitution was performed by the salt dialysis procedure (57). Nucleosomes reconstituted on a 601 nucleosome positioning sequence (20 ng) were incubated with SWI/SNF as indicated at 29°C and in remodeling buffer (RB) containing 10 mM Tris-HCl, pH = 7.4, 2.5 mM MgCl2, and 1 mM ATP. The reaction was stopped after the time as indicated by diluting about 10 times in TE buffer (Tris- HCl 10 mM, pH = 7.4, EDTA 1 mM) and NaCl 2 mM and deposing the sample onto the functionalized APTES-mica surface. Atomic Force Microscopy and surface preparation For the AFM imaging the conventional and variant nucleosomes were immobilized onto APTES-mica surfaces. The functionalization of freshly cleaved mica disks (muscovite mica, grade V-1, SPI) was obtained by self-assembly of a monolayer of APTES under Argon atmosphere for 2 hours (39). Nucleosomes (DNA concentration ~ 75 ng/µl) were filtered and concentrated using Microcon® centrifugal filters to remove free histones from the solution, and diluted 10 times in TE buffer, just prior to deposition onto APTES-Mica surfaces. A 5 µl droplet of the nucleosome solution is applied on the surface for 1 min, rinsed with 1 mL of milliQ-Ultrapure © water and gently dried by nitrogen flow. The samples were visualized by using a Nanoscope III AFM (Digital Instruments™, Veeco, Santa Barbara, CA). The images were obtained in Tapping Mode in air, with silicon tips (resonant frequency 250-350 kHz) or Diamond Like Carbon Spikes tips (resonant frequency ~150 kHz) at scanning rates of 2 Hz over scan areas of 1 µm wide. This surface functionalization was chosen because it is known to trap 3D conformation of naked DNA molecule on a 2D surface (58, 59). Moreover, under such experimental conditions, rinsing and drying are thought to have little effect on the observed conformation of biomolecule (60). Image analysis We have extracted parameters of interest from the AFM images using a MATLAB© (The Mathworks, Natick, MA) script essentially based on morphological tools such as binary dilatation and erosion (61-64), and height/areas selections . The aim of the first three steps of this algorithm is to select relevant objects : 1. In order to remove the piezoelectric scanner thermal drift, flatten of the image is performed. The use of a height criteria (h>0.5nm where h is the height of the object) allows to avoid the shadow artifact induced by high objects on the image. 2. Building of a binary image using a simple thresholding (h > 0.25 nm where h is the height of the object)) and then selection of the binary objects in the good area range (500 < A < 2000 nm² where A is the area of the object)). 3. Selection of the objects in the good height range using a hysteresis thresholding (65) (hmin1 = 0.25 nm and h 2 = 1.4 nm, where h 1 and h 2 are the height of the two thresholds). These three steps leads to the selection of binary objects whose area is between 500 and 2000 nm² and corresponds in the AFM image to a group of connected pixels whose minimun height is more than 0.25 nm and maximum height is above 1.6 nm. For example a height criterion is used to reject tetrasomes while events with SWI/SNF still complexed with nucleosomes are removed from analysis by a size criterion. The next steps correspond to measurements in itself : 4. Detection of the NCP centroid by shrinking the objects in the binary image. 5. Building of a distance map inside the nucleosome with respect to their NCP centroid using a pseudo-euclidian dilatation based algorithm. 6. Selection of the non-octamer parts of the nucleosomes (d > dc , where d is the constraint distance to the NCP centroid and dc ~ 7.5 nm is the apparent nucleosome radius) and then thinning of the free arm regions using a commercial MATLAB© script optimised to avoid most of the branching in the skeleton. 7. Selection of the free arm ends and measurement of the free arm lengths. 8. Measurement of other parameters of interest like areas, volumes and mean height of the nucleosomes and the octamers (see supplemental materials). These last 5 steps lead to quick and robust measurements. Indeed the use of morphological tools allows parallel calculation simultaneously on all the objects. Moreover, erosion is a good approximation for the inverse operation of the AFM dilatation due to the finite tip radius and leads to a partial removal of the tip effect (66, 67). The longest arm is named L+ and the shortest L-. DNA complexed length is deduced by Lc = Ltot - L- - L+ where Ltot is either 255 bp for short conventional and variant nucleosomes or 356 bp for long conventional nucleosomes. The position of the nucleosome relatively to the DNA template center is calculated as ∆L = (L+ - L-)/2. Notice that the position defined this way corresponds to the location of the most deeply buried base pair, which might differ from dyad axis position (strictly defined for symmetric nucleosomes). Complexed DNA length and nucleosome position distribution construction For the distribution of DNA complexed length, well centered nucleosomes were selected (∆L* - σ∆L/2 ~ 0 bp < ∆L < 12 bp ~ ∆L * + σ∆L/2 for the 255 bp mono-nucleosomes where ∆L* is the most probable nucleosome position and σ∆L is the standard deviation of the ∆L distribution). To construct the histogram a 20 bp-sliding box was used. For each L0 in [0, 300 bp], nucleosomes with a DNA complexed length included in the range [L0 – 10 bp, L0 + 10 bp] were counted. After normalization, a smooth distribution is obtained that represents mathematically the convolution of the real experimental distribution with a rectangular pulse of 20 bp long. To obtain the nucleosome position distribution we have selected nucleosomes with a DNA complexed length Lc in a range of width σLc around L * = 146bp (123 bp ~ L* - σLc < Lc < L * - σLc 169 bp for canonical nucleosomes). Then, the same 20-bp sliding box protocol was used to construct the nucleosome position distribution. The error on the distribution function mean value (standard error) is given by σexp/√N, where σexp is the standard deviation of the experimental distribution, and N the number of analyzed nucleosomes (central limit theorem). 2D distribution Lc/∆∆∆∆L construction To construct the 2D-histogram a 10 bp-sliding box was used. For each coordinates (∆L0, L0) in [0, 75 bp]×[0, 300 bp], nucleosomes with a DNA complexed length included in the range [L0 – 5 bp, L0 + 5 bp] and a position included in the range [∆L0 – 5bp, ∆L0 + 5 bp] were counted. After normalization a smooth distribution is obtained that represents mathematically the convolution of the real experimental 2D-distribution with a 10 bp square rectangular pulse. Reproducibility and experimental errors We have checked that different batches of APTES, nucleosome reconstitutions, ySWI/SNF and mica surfaces lead to similar results for the sliding assays and for the 2D mapping within the experimental uncertainty. Moreover we have checked by image analysis of the same naked DNA on the same surface and within the same experimental conditions (data not shown) that the whole measurement and analysis process have an experimental error of about 10 bp in DNA length measurement. Notice that uncertainty on the mean value of length measurements can be much smaller than this resolution as it is explained in the supplemental material S3. RESULTS AND DISCUSSION Simultaneous measurements of DNA complexed length and nucleosome position. Several biochemical approaches allow accessing either the nucleosome position along a DNA template, or the length of DNA wrapped around the histone octamer, but using AFM, we were able to measure them simultaneously. The results are conveniently plotted as 2D histograms of nucleosome position versus DNA complexed length. For short and long arm mononucleosomes We first investigated the influence of the DNA template length on the nucleosome complexed length distribution for conventional nucleosomes. Indeed, one could expect that the nucleosome positioning efficiency for the 601 DNA template and/or the range of wrapped DNA length could depend on the length of free DNA arms. Using purified conventional recombinant histones, nucleosomes were reconstituted by salt dialysis on 255 bp (short nucleosomes) or 356 bp (long nucleosomes) DNA fragments containing the 601 positioning sequence. Tapping Mode AFM in air was used to visualize the reconstituted particles adsorbed on APTES-mica surfaces and images of 1 µm2 were recorded. A representative image of long mono-nucleosomes (Ltot = 356 bp) is displayed on Figure 1a. Such an image enables to clearly distinguish the nucleosome core particle (red part of the complex : hNCP ~ 2 nm) from the free DNA arms (yellow part of the complex, hDNA ~ 0.7 nm) entering and exiting the complex. Precise measurement of the length of each DNA fragment (respectively L+ and L- for the longer and shorter arm) exiting the nucleosome have been performed. To measure each “arm” of the mono-nucleosome, the octamer part is excluded and the free DNA trajectory is obtained (Fig.1b) using morphological tools avoiding false skeletonization by heuristic algorithm ( cf Material and Methods). From the total DNA length that is un-wrapped around the histone octamer, we get the length of DNA organized by the histone octamer (Lc = Ltot - L+ - L-) as well as the nucleosome position with respect to the center of the sequence (∆L = (L+ - L-)/2). The 2D histogram Lc/ ∆L is plotted on Fig. 1c for 702 conventional short nucleosomes using a 2D sliding box as described in the Material and Methods section. The maximum of the 2D distribution is positioned at L* = 145 bp and ∆L = 15 bp, in qualitative agreement with the DNA template construction. The 2D mapping is an important tool to study nucleosome mobilization (see the SWI/SNF sliding section), since both variables are highly correlated during nucleosome sliding/remodeling. Quantitative information can be however also obtained by projecting such a 2D histogram on each axis. First, we have selected well positioned nucleosomes according to the expected position given by the DNA 601 template construction (0 bp < ∆L < 12 bp for short DNA fragments) and shown their DNA complexed length probability density function (red line, Fig. 1d). This distribution of the DNA length, organized by conventional octamer peaks at L* = 146 ± 2 bp, in quantitative agreement with the crystal structure of the nucleosome (26) and cryoEM measurements (25). The broadness of this distribution (σ = 23bp) might be explained by different nucleosomes wrapping conformations. We will explain later on, how this dispersion relates to DNA-histone interaction energies using a simple model. We have used the same approach to study long nucleosomes (2D histogram not shown). Well positioned long nucleosomes according to the DNA sequence (12 bp < ∆L < 32 bp) have very similar probability distribution (blue line on Fig. 1d) than that obtained for short nucleosomes showing that the free linker DNA does not affect significantly the organization of complexed DNA for such nucleosomes. We now select nucleosomes that have a complexed length in the range L* ± σLc, where σLc is the standard deviation of the Lc distribution, and their position distribution is displayed on Figure 1e. The peak values for each DNA fragment (9 ± 2 bp and 24 ± 2 bp for short and long nucleosomes respectively) is close to the expected value from the DNA template construct (2 bp and 22 bp for short and long DNA fragments respectively). Both distributions have a full width at half maximum that exceeds 20 bp. This width might arise from several features : asymmetric unwrapping of one of the two DNA arms, AFM uncertainty and dispersion in octamer position. However, it is not possible with these measurements to determine what is the contribution of each phenomenon. Next, we can see that the distribution width for longer fragments seems greater. After corrections of artifacts inherent to L+/L- labeling (cf Supplemental Figure 2) these two position distributions are very similar showing that the free linker DNA does not affect either the DNA complexed length nor the positioning of such nucleosomes significantly. We have shown in this section that AFM measurements give comparable estimations with other methods for both the positioning and the DNA wrapping of short 601 mononucleosomes. Furthermore, our experimental approach showed no difference in complexed length probability or nucleosome positioning dynamics for long and short DNA templates. For conventional and H2A.Bbd variant mononucleosomes In order to investigate the influence of the octamer composition on the wrapping of DNA around the histone octamer, a H2A.Bbd histone variant was used instead of conventional H2A, in order to reconstitute mono-nucleosomes on a 255 bp DNA fragment. The H2A.Bbd variant nucleosomes were imaged by AFM (25) and using the same analysis as described above, only the well positioned nucleosomes (∆L < 12 bp) were selected. Their DNA complexed length distribution is plotted on Fig. 2a where it is compared to conventional mononucleosomes reconstituted on the same 601 positioning sequence, 255 bp long, with the same position range selection (∆L < 12 bp). The average length of wrapped DNA is clearly different for the variant H2A.Bbd nucleosomes as the distribution peak value is L*H2A.Bbd = 130 ± 3bp instead of L*H2A = 146 ± 2bp for the conventional nucleosomes. Moreover the standard deviation of the distribution is clearly larger for the H2A.Bbd variant (σ = 41 bp to be compared to σ = 23 bp for the conventional nucleosomes). These differences show that the H2A.Bbd variant nucleosome is a more labile complex with less DNA wrapped around the octamer, in agreement with previous observations by AFM and cryo-EM (25). The difference in DNA complexed length suggests that ∼10 bp at each end of nucleosomal DNA are released from the octamer. Therefore, AFM allows visualizing subtle differences in the nucleosome structure. Finally, the DNA complexed length distribution is asymmetric for canonical nucleosomes. This asymmetry can be quantified by measuring their skewness 3µɶ , defined as: 3 c c 2 22 c c (L L ) ((L L ) ) .We find 3µɶ = -0.57 ± 0.09, the negative sign meaning that nucleosome conformations with sub-complexed DNA, as compared to the mean value 146 bp, are energetically more favorable than with over-complexed DNA. This can be interpreted within the simple model proposed below, based on relevant structural data information (26). Notice that for variant nucleosomes, the complexed length distribution is nearly symmetric ( 3µɶ ≈ 0.01 ± 0.16), and this feature will also be discussed in the modeling section. Simple model of DNA complexed length distribution It has been shown that 14 discrete contacts between DNA and histone octamer are responsible for the stability of the nucleosome (26).The energetic gain at these sites is made through electrostatic interactions and hydrogen bonding. At the length scale of the present analysis, the discreteness of binding sites is not relevant, and it will be replaced by a uniform effective adsorption energy εa< per unit length, in units of kT/bp. The finite number of binding sites, or equivalently the finite DNA length L* complexed through these sites (146 bp for canonical nucleosomes, as determined both by the present experiments and crystal structure), is due to the specific locations of favorable interactions located at the surface of the histone octamer, forming a superhelical trajectory on which DNA is complexed. DNA wrapping around the histone core involves additional bending penalty characterized by the energy per unit length : ε = where Lp is the persistence length of DNA within classical linear elasticity and R the radius of the histone octamer. The stability of the nucleosome requires that the net energy per unit length is negative (energetic gain), and therefore : εb < εa< . The experimental distributions of DNA complexed length show that more DNA can be wrapped around the octamer. For these additional base pairs, the net energy per unit length has to be positive, due mainly to bending cost. However, to allow for the possibility of some residual non specific (mainly electrostatic) attractive interactions beyond the 14 binding sites, the energetic gain of DNA contacting the octamer surface outside of the 14 sites superhelical path has a different value denoted εa> . The difference εa< - εa> is then representative of the specificity of the 14 sites region. Assuming that the energy reference is given by un-complexed straight DNA and octamer, the total energy for nucleosome is given by * * * (sub-complexed nucleosome) (over-complexed nucleosome) ( ) L if L <LE(L ) ( ) L + ( ) (L -L ) if L >L − ⋅ − ⋅ a ab b ε ε ε ε (1) The distribution of DNA complexed length is given by cc -E(L ) / (kT) (L ) ∝P e . It is maximum for the characteristic length L*, which characterizes the region of specific contacts. This length may vary for canonical and variant nucleosomes. The assumptions of energy linearity in wrapped DNA length and of the existence of L*, lead to a double exponential distribution. By construction, one has the following constraints between effective energies εa> < εb < εa< . It should be kept in mind that the effective values εa>, εa< and εb are representative of nucleosomes adsorbed on a charged flat surface. These values might differ for nucleosomes in bulk solution, as discussed below. Extraction of the DNA complexed length parameters It is possible to extract some parameters from each distribution by using the physical model presented below, in order to interpret the experimental distribution of DNA complexed length. We found it more reliable to use global procedure for parameter determination, instead of fitting the multivariate distribution. Since we expect the DNA complexed length distribution to be described by a simple double-exponential model, the probability density function can be written as a skew-Laplace distribution which moments are calculated as : *2 (1 ) 2 2 2 2 c c *2 (1 ) 3 2 3 3 3/ 2 3/ 2 2 3/ 2 L 2 2 , for L>L1 ( ) and then (L L ) 4 (1 ) , for L<L (L L ) 4 50 2 12 48 2 4 (1 )  = = −   = = = − = +  − − + −  = = P L L µ σ ε ε ε ε where L* is the most probable complexed length, ε is the relative asymmetry of the skew- Laplace distribution and σ is the mean decay length. The distribution normalization is taken on full real axis as a first approximation, thus neglecting finite size effects. Given the experimentally determined µ and µ parameters, we extract straightforwardly the parameter L*, ε and σ by numerically solving the equation system (2). Hence, we are able to measure without any fitting the parameters L*, ε and σ by calculating the first three moments µ1, µ2 and µ3 of the DNA complexed length statistical series. In our case we thus have : 2(1 ) 2 and then 1 1 (1 ) 2(1 )  − = − = − + = − = − − = = specific ads a a To see the adequacy of this model with the experimental distribution, the function P(Lc=L) is drawn for the parameters extracted from the experimental data using the same 20 bp-sliding box protocol as for the experimental complexed length distribution (Fig.2) The results are summarized in table 1. The values of energies are expressed in units of kT per binding site, assuming 14 such sites along the 147 base pairs of DNA for canonical nucleosomes. Several comments are to be made on these values. First, the measured characteristic decay lengths corresponding to sub- (L<) and over-complexed (L>) DNA lengths (Table 1, (b) and (d)) are clearly higher than the intrinsic resolution of our AFM measurements (related to the tip size that correspond to ~ 10 bp, as checked by image analysis of the same naked DNA on the same surface and within the same experimental conditions - data not shown) for both conventional and variant nucleosomes, showing therefore the significance of the parameters extracted here. Hence, we are able to quantify the energetic of both sub- and over-complexed DNA length in a mono-nucleosome. For over-complexed DNA length, the energy has been converted artificially into units of kT per binding site for the sake of comparison, although the model assumes that there are no such binding sites beyond the 14 sites found in the crystal structure (26). If one assumes that over-complexed DNA length results solely from bending around the histone core (εa> = 0), the value found for εb leads to a persistence length Lp ~ 3.5 bp, a value definitely too small for double stranded DNA. Even more so, this energy is similar in amplitude to the energy of sub-complexed DNA length but with an opposite sign (Table 1, (c) and (e)). We conclude that it cannot simply be associated to a bending penalty, therefore justifying a posteriori the assumption of residual attractive interaction between DNA extra length and histone octamer. The combination of experimental asymmetry of DNA complexed length distribution and the simple model allows quantifying the specificity of the 14 binding sites in the nucleosomes (Table 1, (f)). In particular this can be interpreted as a rough estimation of non- electrostatic contribution to adhesion energy between DNA and histone octamer. Comparison of model parameters extracted from data. These values have to be compared to other estimates reported in the literature. The net energetic gain per site can be compared to values extracted from experiments done in the group of J. Widom (68-70). The spirit of these experiments was to probe the transient exposure of DNA complexed length in a nucleosome by using different restriction enzymes acting at various well-defined sites along the DNA. The experimental results clearly demonstrate that DNA accessibility is strongly reduced when restriction sites are located far away from entry or exit of nucleosomal DNA, towards the dyad axis. From the experimental data, the authors extract a Boltzmann weight for different site exposures. This distribution should a priori be similar to the DNA complexed length distribution obtained in our work, except that only sub-complexed nucleosomes are probed. However, due to the use of different restriction enzymes with different sizes and mechanisms of action, there is an inherent uncertainty in the assignment of precise DNA complexed length with a free energy of the Boltzmann weight. In other words, only a range of energy per binding site can be extracted from these data. This has to be contrasted with most of previous works using Polach and Widom's data, which quote a single value of 2 kT per binding site (31). The range of net energetic gain we are able to estimate out of these data is between 0,5 to 3 kT per binding site. The value we extracted from our own measurements coincides therefore with the lower bound of this range. This might be due to the difference in the type of experiments used. First, our observations are made on nucleosomes adsorbed on a charged substrate. This might change the energetics of nucleosome opening as compared to its value in solution. A theoretical estimation of this change is currently under progress (Castelnovo et al, work in preparation). Another significant difference between Polach and Widom's experiments and our work is the composition of the buffer, which is known to affect the nucleosome stability. In particular, the buffer used for restriction enzyme assays contains more magnesium ions (about 10 mM MgCl2). The specificity of DNA binding sites on histone octamer, as determined in Table 1 (e) can also be compared to values extracted from X-ray experiments performed in the group of T.J. Richmond (71). Indeed, by counting the hydrogen bonds per binding site found in this structure, one can estimate the specific contribution to the binding energy. These contributions range between 0.8 and 2 kT per binding site (72). Our estimate for conventional nucleosome falls in this range (1.1 kT per binding site). Finally, the comparison between canonical and variant nucleosome shows that both the average complexed length and the energy per binding site are different. The most probable length L* = 127 bp (Table 1 (0)) for the variant claims for either the absence or the strong weakening of at least 2 binding sites. Furthermore, the energy and therefore the stability of the nucleosome for the remaining binding sites is reduced (εH2A.Bbd ~ 2/3 εH2A), in accordance with other experimental observations (16, 25, 73). We have shown in this section that a simple model using a linear energy for the DNA-histone interaction can be used to extract from the AFM data two important energetic parameters : the net energetic gain per site and the specific interaction between the DNA and the histone octamer per site. These values are in good agreement with previous biochemical and X-Ray studies done on conventional nucleosomes and for the first time are measured on a variant nucleosome. Visualization of nucleosome sliding and remodeling by SWI/SNF for conventional and variant nucleosomes. After studying the nucleosomes in their equilibrium state, the same mononucleosomes were visualized in the presence of the SWI/SNF remodeling factor to validate the possibility for this direct imaging approach to acquire new information on the mechanism and dynamics of nucleosome sliding. Centrally positioned conventional and variant mononucleosomes (Ltot = 255 bp) were incubated with SWI/SNF at 29°C in the presence or absence of ATP and then adsorbed on APTES-mica surfaces for AFM visualization. On figure 3, we report AFM images of mononucleosomes incubated without (Fig. 3a) and with (Fig. 3b) ATP for one hour. The sample containing no ATP is the control experiment to account for possible nucleosome thermally driven diffusion when incubated 1 h at 29 °C. The representative chosen set of AFM images of Fig.3 clearly shows that most of the nucleosomes are centered on the DNA template in the negative control (-ATP) whereas they rather exhibit end-position when SWI/SNF and ATP are present. On AFM images, SWI/SNF motor is sometimes visible as a very large proteic complex, and if still attached to nucleosome prevents any image analysis of such objects. Our protocol does not include removing of SWI/SNF before deposition, even if by diluting the nucleosome/motor mix, one could expect detaching of some motors. Therefore, the motor per nucleosome ratio used in the sliding experiments is kept low with respect to biochemical assays (55) (roughly five time less SWI/SNF per nucleosome). Using the same type of image analysis we were able to reconstruct 2D histograms Lc/∆L (using a 2D sliding box as described in the Material and Methods section) at various time steps during nucleosome sliding : 0 (-ATP), 20 min and 1 hour (Fig. 4). We first notice that in the absence of ATP, SWI/SNF has apparently no effect on the Lc/∆L map. The 2D distribution exhibits a single peak corresponding to the canonical nucleosome positioned as expected from the DNA template (α state). As a function of time in the presence of remodeling complex and ATP, new states appear : (β) corresponds to an over-complexed nucleosome having the same mean position ∆L value as (α); this state could result from the capture of extra DNA (a loop of ~ 40 bp) inside the NCP induced by SWI/SNF. (β)-state is spread in the ∆L direction showing that this extra complexed DNA length (~ 40 bp) seems to exist for various positions of the nucleosome (0 < ∆L < 30 bp). (γ) is the slided end-positioned nucleosome (∆L ~ 50 bp) having slightly less DNA wrapped around the histone octamer (Lc ~ 125 bp). The ∆L distance separating (α) and (γ) states is close to the Lc distance between (β) and (α) states, meaning that the slided (γ)-state most likely results from the release of the (β)-state DNA loop (~ 40 bp). The fact that slided nucleosomes are sub- complexed i.e. their dyad has been moved beyond the expected end-position, has already been observed in other biochemical studies (74). Similarly, the anisotropic spreading of the (γ)-peak towards higher ∆L and lower Lc is also consistent with this feature. We cannot exclude that a finite size effect of the DNA template could account for this feature. Finally, (δ) is a wide state with a sub-complexed Lc ~ 75 bp, that could correspond to a tetrasome or hexasome. This state could be due to the loss of one wrapped DNA turn either from the α state or the (γ)-state. Nevertheless, one could notice that the (δ)-state is missing on the ‘+ATP 20 min’ map (Fig. 4b) where only few nucleosomes have been slided (weak γ peak) whereas (δ)-state nucleosomes are clearly visible on Fig. 4c (‘+ATP 1 h’). This tends to show that (δ)- state nucleosomes more likely arise from the loss of one DNA turn of the end-positioned nucleosomes ((γ)-state). We have seen that the 2D-mapping of nucleosome position and DNA complexed length allows characterizing the new states resulting from the ATP-dependent action of SWI/SNF on our 601 nucleosomes : an over-complexed state close to the 601 template center (β), a slided state (γ) and a sub-complexed state (δ). Again, more information can be gained by appropriate projections of these 2D- histograms. Nucleosomes having their DNA complexed length in the range L* ± σLc were selected (L* and σLc are respectively the maximum value and the standard deviation of the corresponding complexed length distribution) and their position distribution is plotted on Fig. 5a. For conventional nucleosomes with SWI/SNF but no ATP, the distributions obtained for nucleosome position (Fig. 5a) and DNA complexed length (Fig. 5b) are very similar to the case without any remodeling complex (Fig. 1b), showing no effect of thermally driven diffusion of mononucleosomes reconstituted on 601 positioning sequence in our conditions. When incubation is increased in the presence of ATP (20 minutes and 1 hour), the position distribution of conventional nucleosomes is clearly changed (Fig. 5a). Indeed, as a function of incubation time, a second peak appears corresponding to the end-positioned nucleosomes (∆L ~ 50 bp, cf (γ)−state in Fig. 4c). After one hour of SWI/SNF action in presence of ATP the second peak height has increased at the expense of the primary peak. This corresponds to the situation were one third of the mono-nucleosomes are positioned at the end of the DNA template. It is interesting to note that during the remodeling factor action we do not see any significant increase in the amount of nucleosomes in an intermediate position (20 bp < ∆L < 40 bp). This provides experimental evidence that this remodeling factor moves centrally positioned nucleosomes directly to the end of our short DNA template. Mainly, two situations can explain the bimodal position distribution of nucleosomes after the action of SWI/SNF. The first hypothesis is that the SWI/SNF complex is a processive molecular motor. As it will not detach from the nucleosome before it reaches the end of the DNA template, the elementary step of the SWI/SNF induced sliding might not be accessible. Indeed, in our experimental conditions, only nucleosomes without SWI/SNF complex attached can be analyzed. The other possibility is that SWI/SNF is weakly processive (SWI/SNF turnover rate is unknown) but with an elementary step of the order of 50 bp, which corresponds to the value measured by us and other approaches (75-77), and happens to be the length of free DNA arms in our case. Therefore, a single step would be enough for the motor to slide a nucleosome to an end-position and release the complex. Nevertheless, another mechanism cannot be excluded by our data, where SWI/SNF action would consist of octamer destabilization followed by thermally driven diffusion towards the end-positioned entropically favored. In this situation, ATP-hydrolysis would only be involved in the nucleosome ‘destabilization’ step. In Fig. 5b, we show projections of the previous 2D-histograms along the DNA complexed length axis without any selection on their position. For conventional nucleosomes in the presence of SWI/SNF but no ATP, the complexed length distribution is similar to case with neither SWI/SNF nor ATP. However, the former distribution is larger due to the contribution of different nucleosome positioning. Then after 20 min, the distribution is broader (roughly twice) and shifted towards higher Lc. This might be attributed to the contributions of the different states (β, γ, δ) identified in the Fig. 4b/c. The increase in Lc mean value is likely due to the statistical weight of the over-complexed (β)-state. The same sliding experiment was performed on H2A.Bbd variant nucleosomes in absence and in presence of ATP and analyzed through the projection of the 2D-histogram Lc/∆L. No significant effect of SWI/SNF complexes in presence of ATP on the position distribution of H2A.Bbd variant nucleosomes is observed (Fig. 5c). This corroborates previous findings using biochemical sliding assay done on 5S and 601 positioning sequence (55). However in AFM measurements, the full position distribution is accessed directly with a resolution better than 10 bp (the size of AFM tip). This variant nucleosome sliding assay shows the reproducibility of our experimental approach as not only the position distribution mean value is constant during one hour in the presence of SWI/SNF and ATP, but also the complete position distribution remains constant (Fig. 5c). Similarly, SWI/SNF in presence of ATP does not seem to influence the DNA complexed length distribution of H2A.Bbd nucleosomes (Fig. 5d). CONCLUSION In summary, we have shown that AFM combined with a systematic computer analysis is a powerful tool to determine the structure of conventional and variant mononucleosomes at equilibrium and after the action of ATP-dependent cellular machineries. With this technique we have quantified simultaneously two important and closely coupled variables : the DNA complexed length and the position of mono-nucleosomes along the 601 DNA template. For each of these two distributions, the most probable value is in perfect agreement with measurements done by other methods that give access to one of these two parameters only. In addition, to explain the experimental complexed length distribution, we have developed a simple model that uses the experimental shape of DNA complexed length distributions to quantify the interaction of DNA with histones. With this model, we extract both the net energetic gain for sub-complexed nucleosomes and the estimation of the non-electrostatic contribution to the adhesion energy between DNA and histone octamer . We further show that H2A.Bbd variant and conventional nucleosomes exhibit clear differences in DNA complexed length and in their ability to be slided by SWI/SNF. Indeed, these variant nucleosomes organize less DNA on average than conventional nucleosomes, and present larger opening and closing fluctuations. Moreover, the whole position distribution as well as complexed length distribution remain unchanged showing H2A.Bbd variant is neither displaced nor remodeled by SWI/SNF complex. Finally, we have plotted Lc/∆L as a 2D map of the nucleosome states. This representation is well suited to highlight the various nucleosome states that appear during the SWI/SNF action. For example, as a function of time, we have evidenced the formation of an over-complexed state followed by the appearance of a slided state. More quantitative information can be obtained by appropriate projections of the 2D-histograms, as for instance the bimodal position distribution induced by SWI/SNF sliding on conventional nucleosomes, suggesting two possible scenarii : a processive action of the molecular motor (no intermediate position visualized) or an elementary stepping length (~ 40 bp) of the size of the free DNA arms (~ 50 bp). The short length of DNA templates and lack of directionality in our position analysis prevent us from discriminating between these two hypotheses, and further experiments on long oriented mononucleosomes are needed to get more insights into the molecular mechanism of SWI/SNF action. The present results as well as preliminary data on longer oriented templates prove nevertheless that this extension will provide useful information on remodeling mechanisms of SWI/SNF. A further perspective of this AFM study will be to test the effect of the flanking DNA sequences on the conformation and dynamics of 601 nucleosomes. Nevertheless, in order to test sequence effect, nucleosomes should be reconstituted on less positioning sequences (5S rDNA for example) or non-positioning sequences, but this will complicate significantly the nucleosome sliding analysis as the initial position distribution of the nucleosome is expected to be broader in this case. ACKNOWLEDGMENTS We thank Dimitar Angelov and Hervé Ménoni for various forms of help with nucleosome reconstitution and sliding assays and Cécile-Marie Doyen for producing H2A.Bbd variant histones. We are grateful to Stefan Dimitrov, Dimitar Angelov, Françoise Argoul, Alain Arneodo, Cédric Vaillant and Phillipe Bouvet for fruitful discussions. We thank Ali Hamiche for providing us with the ySWI/SNF complex. P.St-J. acknowledges CRSNG for financial support. 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TABLE 1 <Lc> (bp) decay length L< (bp) εεεεb - εεεεa< (kT per site) decay length L> (bp) εεεεb - εεεεa> (kT per site) εεεεa< - εεεεa> (kT per site) conventional nucleosome 146 ± 2 22 ± 1.6 -0.479 ± 0.045 17 ± 1.4 0.61 ± 0.064 1.1 ± 0.072 variant nucleosome 127 ± 3 31 ± 1.5 -0.33 ± 0.022 27 ± 1.5 0.39 ± 0.026 0.72 ± 0.021 site exposure model (g) -3 <...< -0.5 crystal structure (h) 147 0.8<…<2 Caption Table1 : Summary of model parameters extracted from experimental data as explained in Materials and methods. All energies are expressed in units of kT per binding site. DNA lengths are expressed in bp. (a) Average complexed length (b) Characteristic length L< of exponential decay towards sub-complexed DNA length. (c) Energy per binding site (1/ L<) for sub-complexed DNA length. (d) Characteristic length L> of exponential decay, towards over-complexed DNA length. (e) Energy per binding site (1/L>) for over-complexed DNA length. (f)Asymmetry of adhesion energy per binding site between sub- and over-complexed DNA length. (g) Range of values extracted from Polach and Widom (27, 69) data using the site exposure model. (h) Range of values extracted from Davey and Richmond (71) data using X-ray crystal structure of the nuclear core particle. Uncertainty values are determined using the central limit theorem and a propagation of uncertainty calculus detailed in supplemental data. N(H2A conventional) = 301 nucleosomes. N(H2A.Bbd variant) = 252 nucleosomes. FIGURE CAPTIONS Figure 1 : AFM visualization of centered mononucleosomes with short and long arms. (a) AFM topography image of mono-nucleosomes reconstituted on 356 bp 601 positioning sequence. Color scale : from 0 to 1.5 nm. X/Y scale bar : 100 nm. (b) Zoom in the AFM topography image of a centered mono-nucleosome and the result of the image analysis. Black line : contour of the mono-nucleosome. Blue point : centroid of the histone octamer. Blue dot circle : excluded area of the histone octamer. Blue line : skeletons of the free DNA arms. Color scale : from 0 to 1.5 nm. X/Y Scale bar : 20 nm. The longest arm is named L+ and the shortest L-. DNA complexed length is deduced by Lc = Ltot - L- - L+ where Ltot is in this case 356 bp. The position of the nucleosome relatively to the center of the sequence is calculated by ∆L = (L+ - L-)/2. (c) 2D histogram Lc/∆L representing the DNA complexed length Lc along with the nucleosome position ∆L for a short DNA fragment of 255 bp (N = 702 nucleosomes). (d) Probability density function of the DNA complexed length Lc for a short DNA fragment (255 bp, purple line) and for a long DNA fragment (356 bp, blue line) obtained by selecting the well positioned nucleosomes (0 < ∆L < 12 bp and 12 < ∆L < 32 bp for the short and long fragments respectively) and projecting the 2D map along the y-axis. (e) Probability density function of the ∆L nucleosome position for a short DNA fragment (255 bp, purple line) and for a long DNA fragment (356 bp, blue line) obtained by selecting nucleosomes having their DNA complexed length Lc in the range 123 bp < Lc < 169 bp for both fragments, and projecting the 2D map along the x-axis. Figure 2 : AFM Visualization of centered H2A.Bbd variant and H2A conventional mononucleosome. (a) Probability density function of the DNA complexed length Lc for a short DNA fragment (255 bp) with conventional H2A (solid thick line) and with variant H2A.Bbd (dotted thick line) nucleosome. Simple model for conventional and variant nucleosomes (respectively solid and dashed thin lines). (b) Description of the model used to measure the DNA-histone adsorption energies per bp (εa< and εa>) and the DNA bending energy per bp (εb) (dotted line). Representation of the model using the 20bp-sliding-box procedure (dotted dashed line). L* corresponds to the most probable DNA complexed length of the distribution. Figure 3 : AFM Visualization of the sliding of centered mononucleosomes by the remodeling complex SWI/SNF. AFM topography image of mononucleosomes reconstituted on 255 bp 601 positioning sequence, incubated at 29°C with SWI/SNF for one hour (a) in the absence and (b) in the presence of ATP. Color scale : from 0 to 1.5 nm. X/Y Scale bar : 150 nm. Zoom in the AFM topography image of a (c) centered mononucleosome and (d) end-positioned mononucleosome the result of the image analysis. Black line : contour of the mono- nucleosome. Blue point : centroid of the histone octamer. Blue line : skeletons of the free DNA arms. Color scale : from 0 to 1.5 nm. X/Y Scale bar : 20 nm. Figure 4 : Evolution of nucleosome Lc/∆∆∆∆L map during nucleosome sliding by SWI/SNF complex for conventional nucleosome. 2D histogram Lc/∆L representing the DNA complexed length Lc along with the nucleosome position ∆L for a conventional nucleosome reconstituted on a short DNA fragment (255 bp) in the presence of remodeling complex SWI/SNF (a) without ATP (1h at 29°C) (b) with ATP (20 min at 29°C) and (c) with ATP (1h at 29°C). (d) Representation of the nucleosome Lc/∆L states for (α), (β), (χ) and (δ) positions as pointed on the 2D maps. N(-ATP, 1h at 29°C) = 692 nucleosomes, N(+ATP, 20 min at 29°C) = 245 nucleosomes, N(+ATP, 1h at 29°C) = 655 nucleosomes. Figure 5 : Evolution of nucleosome position and DNA complexed length distributions during nucleosome sliding by SWI/SNF complex, for conventional and variant nucleosome. Nucleosome position ∆L (a) and DNA complexed length Lc (b) distributions as a function of time (0, 20 min, 1 hour) in the presence of SWI/SNF, for conventional mono-nucleosomes reconstituted on 255 bp long 601 positioning sequence. Nucleosome position ∆L (c) and DNA complexed length Lc (d) distributions as a function of time (0, 1 hour) in the presence of SWI/SNF, for H2A.Bbd variant mono-nucleosomes reconstituted on the same DNA template. The zero time is given by the control in the absence of ATP (solid purple line). For each ∆L position distribution, only nucleosomes having their complexed length in the range Lc * ± σLc are selected. For the sake of figure clarity, error bars are only depicted on 2 distributions of graph (a). Lc = Ltot - L-- L+ ∆∆∆∆L = (L+-L-) / 2 Figure 1 (a)(a) (b)(b) L-=86bp L+=130bp Ltot = 255 bp, 0 < ∆L< 12 bp Ltot = 356 bp, 12 < ∆L< 32 bp 0 0.01 0.02 Probability Ltot = 255 bp, 123 < Lc <169 bp 123 < Lc <169 bp Ltot = 356 bp, Lc (bp) ∆∆ ∆∆ 50 150 250200100 12e-4 probability scale Figure 2 Lcomplexed c( ) L( ) b aP ε εα <+ ⋅ *c cL e if L < L c( ) L( ) b aP ε εα >+ ⋅ *c cL e if L > L -50 0 50 100 150 200 250 300 0.005 0.015 complexed H2A, L =255bp, ∆L<12bp H2A-Bbd, L =255bp, ∆L<12bp Model H2A Model H2A.Bbd + ATP + SWI/SNF - ATP + SWI/SNF Figure 3 Figure 4 ((((a)))) ((((b)))) (α)(α) (α)(α) (α)(α) (β)(β) (γ)(γ) (δ)(δ) LLLLcccc (bp)(bp)(bp)(bp) (γ)(γ) 147bp (601) 12e-4 probability scale 7.5e-4 2.5e-4 7.5e-4 2.5e-4 0 50 100 150 200 250 300 0.002 0.004 0.006 0.008 0.012 -ATP t=1h +ATP t=20min +ATP t=1h H2A nucleosomes 0 25 50 75 100 0.005 0.015 ∆∆∆∆ L (bp) -ATP 1h +ATP 1h H2A.Bbd nucleosomes (d) 0 50 100 150 200 250 300 0.002 0.004 0.006 0.008 0.012 -ATP t=1h +ATP t=1h H2A.Bbd nucleosomes (a) H2A nucleosomes 0 25 50 75 100 0.005 0.015 ∆∆∆∆L (bp) -ATP, t=1h +ATP, t=20min +ATP, t=1h Figure 5

WHEN THE ORBIT ALGEBRA OF GROUP IS AN INTEGRAL DOMAIN? PROOF OF A CONJECTURE OF P.J. CAMERON MAURICE POUZET Abstract. P.J.Cameron introduced the orbit algebra of a permutation group and conjectured that this algebra is an integral domain if and only if the group has no finite orbit. We prove that this conjecture holds and in fact that the age algebra of a relational structure R is an integral domain if and only if R is age-inexhaustible. We deduce these results from a combinatorial lemma asserting that if a product of two non-zero elements of a set algebra is zero then there is a finite common tranversal of their supports. The proof is built on Ramsey theorem and the integrity of a shuffle algebra. Introduction In 1981, P.J.Cameron [4] (see also [9] p.86) associated a graded algebra A[G] to a permutation group G acting on an infinite set E. He formulated two conjectures on the integrity of this algebra. The purpose of this paper is to present a solution to the first of these conjectures. Consequences on the enumeration of finite substructures of a given structure are mentionned. Some problems are stated. 0.1. The conjectures. Here is the content of these conjectures, freely adapted from Cameron’s web page (see Problem 2 [10]). The graded algebra A[G] is the direct sum A[G]n where A[G]n is the set of all G-invariant functions f from the set [E] n of n-element subsets of E into the field C of complex numbers. Multiplication is defined by the rule that if f ∈ A[G]m, g ∈ A[G]n and Q is an (m+ n)-element subset of E then (1) (fg)(Q) := P∈[Q]m f(P )g(Q \ P ) As shown by Cameron, the constant function e in A[G]1 (with value 1 on every one element set) is not a zero-divisor (see Theorem 0.8 below). The group G is entire if A[G] is an integral domain, and strongly entire if A[G]/eA[G] is an integral domain. Date: November 6, 2018. 1991 Mathematics Subject Classification. 03 C13, 03 C52, 05 A16, 05 C30, 20 B27. Key words and phrases. Relational structures, ages, counting functions, oligomorphic groups, age algebra, Ramsey theorem, integral domain. Research done under the auspices of Intas programme 03-51-4110 ”Universal algebra and lattice theory” . http://arxiv.org/abs/0704.1548v1 2 MAURICE POUZET Conjectures 0.1. G is (strongly) entire if and only if it has no finite orbit on E. The condition that G has no finite orbit on E is necessary. We prove that it suffices for G to be entire. As it turns out, our proof extends to the algebra of an age, also invented by Cameron [10]. 0.2. The algebra of an age. A relational structure is a realization of a language whose non-logical symbols are predicates. This is a pair R := (E, (ρi)i∈I) made of a set E and a family of mi-ary relations ρi on E. The set E is the domain or base of R; the family µ := (mi)i∈I is the signature of R. The substructure induced by R on a subset A of E, simply called the restriction of R to A, is the relational structure R↾A := (A, (A mi ∩ ρi)i∈I). Notions of isomorphism, as well as isomorphic type, are defined in natural way (see Subsection 1.1). A map f : [E]m → C, where m is a non negative integer, is R-invariant if f(P ) = f(P ′) whenever the restrictions R|P and R|P ′ are isomorphic. The R-invariant maps can be multiplied. Indeed, it is not difficult to show that if f : [E]m → C and g : [E]n → C are R-invariant, the product defined by Equation (1) is R-invariant. Equipped with this multiplication, the C-vector space spanned by the R-invariant maps becomes a graded algebra, the age algebra of R, that we denote by C.A(R). The name, coined by Cameron, comes from the notion of age defined by Fräıssé [13]. Indeed, the age of R is the collection A(R) of substructures of R induced on the finite subsets of R, isomorphic substructures being identified. And it can be shown that two relational structures with the same age yields the same algebra (up to an isomorphism of graded algebras). The algebra associated to a group is a special case of age algebra. Indeed, to a permutation group G acting on E we may associate a relational structure R with base E such that the G-invariant maps coincide with the R-invariant maps. Our criterium for the integrity of the age algebra is based on the notion of kernel: The kernel of a relational structure R is the subset K(R) of x ∈ E such that A(R|E\{x}) 6= A(R). The emptyness of the kernel R is a necessary condition for the integrity of the age algebra. Indeed, if K(R) 6= ∅, pick x ∈ K(R) and F ∈ [E]<ω such that R↾F ∈ A(R) \ A(R|E\{x}). Let P ∈ [E] <ω. Set f(P ) := 1 if R↾P is isomorphic to R↾F , otherwise set f(P ) := 0. Then f 2 := ff = 0. Theorem 0.2. Let R be a relational structure with possibly infinitely many non iso- morphic types of n-element substructures. The age algebra C.A(R) is an integral domain if and only if the kernel of R is empty. The application to the conjecture of Cameron is immediate. LetG be a permutation group acting on E and let R be a relational structure encoding G. Then, the kernel of R is the union of the finite G-orbits of the one-element sets. Thus, if G has no finite orbit, the kernel of R is empty. Hence from Theorem 0.2, A[G] is an integral domain, as conjectured by Cameron. We deduce Theorem 0.2 from a combinatorial property of a set algebra over a field (Theorem 0.3 below). This property does not depends upon the field, provided that its characteristic is zero. The proof we give in Section 1.1 is an extension of our A CONJECTURE OF P.J. CAMERON 3 1970 proof that the profile of an infinite relational structure does not decrease (see Theorem 0.5 below). The key tool we used then was Ramsey’s theorem presented in terms of a property of almost-chainable relations. Here, these relations are replaced by F −L-invariant relational structures, structures which appeared, under other names, in several of our papers (see [24], [25], [27]). The final step is reminiscent of the proof of the integrity of a shuffle algebra. We introduced the notion of kernel in[24] and studied it in several papers [25] [26], [27] and [29]. As it is easy to see (cf [25][29]), the kernel of a relational structure R is empty if and only if for every finite subset F of E there is a disjoint subset F ′ such that the restrictions R|F and R|F ′ are isomorphic. Hence, relational structures with empty kernel are those for which their age has the disjoint embedding property, meaning that two arbitrary members of the age can be embedded into a third in such a way that their domain are disjoint. In Fräıssé’s terminology, ages with the disjoint embedding property are said inexhaustible and relational structures whose age is inexhaustible are said age-inexhaustible; we say that relational structures with finite kernel are almost age-inexhaustible. 1 0.3. A transversality property of the set algebra. Let K be a field with charac- teristic zero. Let E be a set and let [E]<ω be the set of finite subsets of E (including the empty set ∅). Let K[E] be the set of maps f : [E]<ω → K. Endowed with the usual addition and scalar multiplication of maps, this set is a vector space over K. Let f, g ∈ K[E] and Q ∈ [E]<ω. Set: (2) fg(Q) = P∈[Q]<ω f(P )g(Q \ P ) With this operation added, the above set becomes a K-algebra. This algebra is commutative and it has a unit, denoted by 1. This is the map taking the value 1 on the empty set and the value 0 everywhere else. The set algebra is the subalgebra made of maps f such that f(P ) = 0 for every P ∈ [E]<ω with |P | large enough. This algebra is graded, the homogeneous component of degree n being made of maps which take the value 0 on every subset of size different from n (see Cameron [6]). If f and g belong to two homogeneous components, their product is given by Equation (1), thus an age algebra, or a group algebra, A, as previously defined, is a subalgebra of this set algebra. The set algebra is far from to be an integral domain. But, with the notion of degree, the integrity of A will reduce to the fact that if m and n are two non negative integers and f : [E]m → K, f : [E]n → K are two non-zero maps belonging to A, their product fg is non zero. Let H be a family of subsets of E, a subset T of E is a transversal of H if F ∩T 6= ∅ for every F ∈ H; the transversality of H, denoted τ(H), is the minimum of the cardinalities (possibly infinite) of transversals of H. We make the convention that τ(H) = 0 if H is empty. Let f : [E]m → K, denote supp(f) := {P ∈ [V ]m : f(P ) 6= 0}. 1In order to agree with Fräıssé’s terminology, we disagree with the terminology of our papers, in which inexhaustibility, resp. almost inexhaustibility, is used for relational structures with empty, resp. finite, kernel, rather than for their ages. 4 MAURICE POUZET Here is our combinatorial result: Theorem 0.3. Let m,n be two non negative integers. There is an integer t such that for every set E with at least m + n elements, every field K with characteristic zero, every pair of maps f : [E]m → K, g : [E]n → K such that fg is zero, but f and g are not, then τ(supp(f) ∪ supp(g)) ≤ t. With this result, the proof of Theorem 0.2 is immediate. Indeed, let R be a rela- tional structure with empty kernel. If K.A(R), the age algebra of R over K, is not an integral domain there are two non-zero maps f : [E]m → K, f : [E]n → K belonging to K.A(R), whose product fg is zero. Since K is an integral domain, none of the in- tegers m and n can be zero. Since f is R-invariant, m is positive and the kernel K(R) of R is empty, it turns out that τ(supp(f)) is infinite. Hence τ(supp(f)∪ supp(g)) is infinite, contradicting the conclusion of Theorem 0.3. An other immediate consequence of Theorem 0.3 is the fact, due to Cameron, that on an infinite set E, e is not a zero-divisor (see Theorem 0.8 below). 0.3.1. Existence and values of τ . The fact the size of a transversal can be bounded independently of f and g, and the value of the least upper bound, seem to be of independent interest. So, let τ(m,n) be the least t for which the conclusion of Theorem 0.3 holds. Trivially, we have τ(m,n) = τ(n,m). We have τ(0, n) = τ(m, 0) = 0. Indeed, if m = 0, f is defined on the empty set only, an thus fg(Q) = f(∅)g(Q). Since K has no non zero divisors, fg is non zero provided that f and g are non zero. The fact that there is no pair f, g such that fg is zero, but f and g are not, yields τ(supp(f) ∪ supp(g)) = 0. We have τ(1, n) = 2n (Theorem 2.4). This is a non-trivial fact which essentially amounts to a weighted version of the Gottlieb-Kantor Theorem on incidence matrices ([15], [19], see subsection 0.4 and Theorem 2.3). These are the only exact values we know. We prove that τ(m,n) exists, by supposing that τ(m − 1, n) exists. Our existence proof relies in an essential way on Ramsey theorem. It yields astronomical upper bounds. For example, it yields τ(2, 2) ≤ 2(R2k(4) + 2) , where k = 5 30 and R2k(4) is the Ramsey number equal to the least integer p such that for every colouring of the pairs of {1, . . . , p} into k colors there are four integers whose all pairs have the same colour. The only lower bound we have is τ(2, 2) ≥ 7 and more generally τ(m,n) ≥ (m+ 1)(n + 1)− 2. We cannot preclude a extremely simple upper bound for τ(m,n), eg quadratic in n+m. 0.4. Age algebra and profile of a relational structure. The group agebra was invented by Cameron in order to study the behavior of the function θG which counts for each integer n the number θG(n) of orbits of n-subsets of a set E on which acts a permutation groupG, a function that we call the orbital profile ofG. Groups for which the orbital profile takes only finite values are quite important. Called oligomorphic groups by Cameron, they are an objet of study by itself (see Cameron’s book[5]). We present first some properties of the profile, a counting function somewhat more general. Next, we present the link with the age algebra, then we gives an illustration of Theorem 0.2. We conclude with some problems. A CONJECTURE OF P.J. CAMERON 5 0.4.1. Profile of a relational structure. The profile of a relational structure R with base E is the function ϕR which counts for every integer n the number (possibly infinite) ϕR(n) of substructures of R induced on the n-element subsets, isomorphic substructures being identified. Clearly, if R encodes a permutation groups G, ϕR(n) is the number θG(n) of orbits of n-element subsets of E. If the signature µ is finite (in the sense that I is finite), there are only finitely many relational structures with signature µ on an n-element domain, hence ϕR(n) is necessarily an integer for each integer n. In order to capture examples coming from algebra and group theory, one cannot preclude I to be infinite. But then, ϕR(n) could be an infinite cardinal. As far as one is concerned by the behavior of ϕR, this case can be excluded: Fact 0.4. [28] Let n < |E|. Then (3) ϕR(n) ≤ (n + 1)ϕR(n+ 1) In particular: (4) If ϕR(n) is infinite then ϕR(n + 1) is infinite too and ϕR(n) ≤ ϕR(n+ 1). Inequality (3) can be substantially improved: Theorem 0.5. If R is a relational structure on an infinite set then ϕR is non- decreasing. This result was conjectured with R.Fräıssé [14]. We proved it in 1971; the proof - for a single relation- appeared in 1971 in R.Fräıssé’s book [12], Exercise 8 p. 113; the general case was detailed in [26]. The proof relies on Ramsey theorem [32]. More is true: Theorem 0.6. If R is a relational structure on a set E having at least 2n + m elements then ϕR(n) ≤ ϕR(n+m). Meaning that if |E| := ℓ then ϕR increases up to ; and, for n ≥ ℓ the value in n is at least the value of the symmetric of n w.r.t. ℓ The result is a straightforward consequence of the following property of incidence matrices. Let m,n, ℓ be three non-negative integers and E be an ℓ-element set. Let Mn,n+m be the matrix whose rows are indexed by the n-element subsets P of E and columns by the n +m-element subsets Q of E, the coefficient aP,Q being equal to 1 if P ⊆ Q and equal to 0 otherwise. Theorem 0.7. If 2n+m ≤ l then Mn,n+m has full row rank (over the field of rational numbers). Theorem 0.7 is in W.Kantor 1972 [19], with similar results for affine and vector subspaces of a vector space. Over the last 30 years, it as been applied and redis- covered many times; recently, it was pointed out that it appeared in a 1966 paper of D.H.Gottlieb [15]. Nowadays, this is one of the fundamental tools in algebraic combinatorics. A proof, with a clever argument leading to further developments, was given by Fräıssé in the 1986’s edition of his book, Theory of relations, see [13]. 6 MAURICE POUZET We proved Theorem 0.6 in 1976 [23]. The same conclusion was obtained first for orbits of finite permutation groups by Livingstone and Wagner, 1965 [20], and extended to arbitrary permutation groups by Cameron, 1976 [3]. His proof uses the dual version of Theorem 0.7. Later on, he discovered a nice translation in terms of his age algebra, that we present now. For that, observe that ϕR only depends upon the age of R and, moreover, if ϕR take only integer values, then K.A(R) identifies with the set of (finite) linear combinations of members of A(R). In this case, as pointed out by Cameron, ϕR(n) is the dimension of the homogeneous component of degree n of K.A(R). Let e ∈ K[E] be the map which is 1 on the one-element subsets of E and 0 elsewhere. Let U be the subalgebra generated by e. We can think of e as the sum of isomorphic types of the one-element restrictions of R. Members of U are then of the form λme m + · · ·+ λ1e + λ01 where 1 is the isomorphic type of the empty relational structure and λm, . . . , λ0 are in K. Hence U is graded, with Un, the homogeneous component of degree n, equals to K.en. Here is the Cameron’s result: Theorem 0.8. If R is infinite then, for every u ∈ K.A(R), eu = 0 if and only if u = 0 This innocent looking result implies that ϕR is non decreasing. Indeed, the image of a basis of K.A(R)n by multiplication by e m is an independent subset of K.A(R)n+m. 0.4.2. Growth rate of the profile. Infinite relational structures with a constant profile, equal to 1, were called monomorphic and characterized by R. Fräıssé who proved that they were chainable. Later on, those with bounded profile, called finimorphic, were characterized as almost chainable [14]. Groups with orbital profile equal to 1 were described by P.Cameron in 1976 [3]. From his characterization, Cameron obtained that an orbital profile is ultimately constant, or grows as fast as a linear function with slope 1 The age algebra can be also used to study the growth of the profile. If A is a graded algebra, the Hilbert function hA of A is the function which associates to each integer n the dimension of the homogeneous component of degree n. So, provided that it takes only finite values, the profile ϕR is the Hilbert function of the age algebra C.A(R). In [10], Cameron made the following important observation about the behavior of the Hilbert fonction. Theorem 0.9. Let A be a graded algebra over an algebraically closed field of charac- teristic zero. If A is an integral domain the values of the Hilbert function hA satisfy the inequality (5) hA(n) + hA(m)− 1 ≤ hA(n +m) for all non-negative integers n and m. This result has an immediate consequence on the growth of the profile: Theorem 0.10. [26] The growth of the profile of a relational structure with empty kernel is at least linear provided that it is unbounded. A CONJECTURE OF P.J. CAMERON 7 In fact, provided that the relational structures satisfy some mild conditions, the existence of jumps in the behavior of the profile extends. Let ϕ : N → N and ψ : N → N. Recall that ϕ = O(ψ) and ψ grows as fast as ϕ if ϕ(n) ≤ aψ(n) for some positive real number a and n large enough. We say that ϕ and ψ have the same growth if ϕ grows as fast as ψ and ψ grows as fast as ϕ. The growth of ϕ is polynomial of degree k if ϕ has the same growth as n →֒ nk; in other words there are positive real numbers a and b such that ank ≤ ϕ ≤ bnk for n large enough. Note that the growth of ϕ is as fast as every polynomial if and only if limn→+∞ = +∞ for every non negative integer k. Theorem 0.11. Let R := (E, (ρi)i∈I) be a relational structure. The growth of ϕR is either polynomial or as fast as every polynomial provided that either the signature µ := (ni)i∈I is bounded or the kernel K(R) of R is finite. Theorem 0.11 is in [24]. An outline of the proof is given in [28]. A part appeared in [26], with a detailed proof showing that the growth of unbounded profiles of relational structures with bounded signature is at least linear. The kernel of any relational structure which encodes an oligomorphic permutation group is finite (indeed, as already mentionned, if R encodes a permutation group G acting on a set E then K(R) is the set union of the finite orbits of the one-element subsets of E. Since the number of these orbits is at most θG(1), K(R) is finite if G is oligomorphic). Hence: Corollary 0.12. The orbital profile of an oligomorphic group is either polynomial or faster than every polynomial. For groups, and graphs, there is a much more precise result than Theorem 0.11. It is due to Macpherson, 1985 [22]. Theorem 0.13. The profile of a graph or a permutation groups grows either as a polynomial or as fast as fε, where fε(n) = e , this for every ε > 0. 0.4.3. Growth rate and finite generation. A central question in the study of the profile, raised first by Cameron in the case of oligomorphic groups, is this: Problem 1. If the profile of a relational structures R with finite kernel has polynomial growth, is ϕR(n) ≃ cn k′ for some positive real c and some non-negative integer k′? Let us associate to a relational structure R whose profile takes only finite values its generating series HϕR := ϕR(n)x Problem 2. If R has a finite kernel and ϕR is bounded above by some polynomial, is the series HϕR a rational fraction of the form P (x) (1− x)(1− x2) · · · (1− xk) with P ∈ Z[x]? 8 MAURICE POUZET Under the hypothesis above we do not know if HϕR is a rational fraction. It is well known that if a generating function is of the form P (x) (1−x)(1−x2)···(1−xk) for n large enough, an is a quasi-polynomial of degree k ′, with k′ ≤ k − 1, that is a polynomial ak′(n)n k′ + · · ·+ a0(n) whose coefficients ak′(n), . . . , a0(n) are periodic functions. Hence, a subproblem is: Problem 3. If R has a finite kernel and ϕR is bounded above by some polynomial, is ϕR(n) a quasi-polynomial for n large enough? Remark 0.14. Since the profile is non-decreasing, if ϕR(n) is a quasi-polynomial for n large enough then ak′(n) is eventually constant. Hence the profile has polynomial growth in the sense that ϕR(n) ∼ cn k′ for some positive real c and k′ ∈ N. Thus, in this case, Problem 1 has a positive solution. A special case was solved positively with N.Thiéry [30]. These problems are linked with the structure of the age algebra. Indeed, if a graded algebra A is finitely generated, then, since A is a quotient of a polynomial ring K[x1, . . . , xd], its Hilbert function is bounded above by a polynomial. And, in fact, as it is well known, its Hilbert series is a fraction of form P (x) (1−x)d , thus of the form given in (6). Moreover, one can choose a numerator with non-negative coefficients whenever the algebra is Cohen-Macaulay. Due to Problem 2, one could be tempted to conjecture that these sufficient conditions are necessary in the case of age agebras. Indeed, from Theorem 0.8 one deduces easily: Theorem 0.15. The profile of R is bounded if and only if K.A(R) is finitely generated as a module over U , the graded algebra generated by e. In particular, if one of these equivalent conditions holds, K.A(R) is finitely generated But this case is exceptional. The conjecture can be disproved with tournaments. Indeed, on one hand, there are tournaments whose profile has arbitrarily large poly- nomial growth rate and, on an other hand, the age algebra of a tournament is finitely generated if and only if the profile of the tournament is bounded (this result was obtained with N.Thiery, a proof is presented in [28]). 0.4.4. Initial segments of an age and ideals of a ring. No concrete description of relational structures with bounded signature, or finite kernel, which have polynomial growth is known. In [24] (see also [28]) we proved that if a relational structure R has this property then its age, A(R), is well-quasi-ordered under embeddability, that is every final segment of A(R) is finitely generated, which amounts to the fact that the collection F (A(R)) of final segments of A(R) is noetherian, w.r.t. the inclusion order. Since the fundamental paper of Higman[17], applications of the notion of well-quasi-ordering have proliferated (eg see the Robertson-Seymour’s theorem for an application to graph theory [11] ). Final segments play for posets the same role than ideals for rings. Noticing that an age algebra is finitely generated if and only if it is noetherian, we are lead to have a closer look at the relationship between the basic objects of the theory of relations and of ring theory, particularly ages and ideals. We mention the following result which will be incorporated into a joint paper with N.Thiéry. A CONJECTURE OF P.J. CAMERON 9 Proposition 0.16. Let A be the age of a relational structure R such that the profile of R takes only finite values and K.A be its age algebra. If A′ is an initial segment of A then: (i) The vector subspace J := K.(A \ A′) spanned by A \ A′ is an ideal of K.A. Moreover, the quotient of K.A by J is a ring isomorphic to the ring K.A′. (ii) If this ideal is irreducible then A′ is a subage of A. (iii) This is a prime ideal if and only if A′ is an inexhaustible age. The proof of Item (i) and Item (ii) are immediate. The proof of Item (iii) is essentially based on Theorem 0.2. According to Item (i), F (A) embeds into the collection of ideals of K.A). Conse- quently: Corollary 0.17. If an age algebra is finitely generated then the age is well-quasi- ordered by embeddability. Problem 4. How the finite generation of an age algebra translates in terms of em- beddability between members of the ages? 0.4.5. Links with language theory. In the theory of languages, one of the basic results is that the generating series of a regular language is a rational fraction (see [1]). This result is not far away from our considerations. Indeed, if A is a finite alphabet, with say k elements, and A∗ is the set of words over A, then each word can be viewed as a finite chain coloured by k colors. Hence A∗ can be viewed as the age of the relational structure R made of the chain Q of rational numbers divided into k colors in such a way that, between two distinct rational numbers, all colors appear. Moreover, as pointed out by Cameron [6], the age algebra Q.A(R) is isomorphic to the shuffle algebra over A, an important object in algebraic combinatorics (see [21]). Problem 5. Does the members of the age of a relational structure with polynomial growth can be coded by words forming a regular language? Problem 6. Extend the properties of regular languages to subsets of the collection Ωµ made of isomorphic types of finite relational structures with signature µ. 1. Proof of Theorem 0.3 The proof idea of Theorem 0.3 is very simple and we give it first. We prove the result by induction. We suppose that it holds for pairs (m − 1, n). Now, let f : [E]m → K and g : [E]n → K such that fg is zero, but f and g are not. As already mentionned, m and n are non zero, hence members of supp(f) ∪ supp(g) are non empty. Let sup(f, g) := {(A,B) ∈ supp(f) × supp(g) : A ∩ B = ∅}. We may suppose sup(f, g) 6= ∅, otherwise the conclusion of Theorem 0.3 holds with t := m+ n− 1. For the sake of simplicity, we suppose that K := Q. In this case, we color elements A of [E]m into three colors :-,0, +, according to the value of f(A). We do the same with elements B of [E]n and we color each member (A,B) of supp(f, g) with the colors of its components. With the help of Ramsey’ theorem and a lexicographical ordering, we prove that if the transversality is large enough there is an (m+n)-element subset Q such that all pair (A,B) ∈ supp(f, g)(Q) := supp(f, g)∩ ([Q]m × [Q]n) have 10 MAURICE POUZET the same color. This readily implies that fg(Q) 6= 0, a contradiction. If K 6= Q, we may replace the three colors by five, as the following lemma indicates. Lemma 1.1. Let K be a field with characteristic zero. There is a partition of K∗ := K \ {0} into at most four blocks such that for every integer k and every k-element sequences (α1, . . . , αk) ∈ D k , (β1, . . . , βk) ∈ D ′k, where D, D′ are two blocks of the partition of K∗, then i=1 αiβi ∈ K Proof. This holds trivially if K := C. For an example, divide C∗ into the sets Di := {z ∈ C ∗ : πi ≤ Argz < π(i+1) } (i < 4). If K is arbitrary, use the Com- pactness theorem of first-order logic, under the form of the ”diagram method” of A.Robinson [18]. Namely, to the language of fields, add names for the elements of K, a binary predicate symbol, and axioms, this in such a way that a model, if any, of the resulting theory T will be an extension of K with a partition satisfying the conclusion of the lemma. According to the Compactness theorem of first-order logic, the existence of a model of T , alias the consistency of T , reduces to the consistency of every finite subset A of T . A finite subset A of T leads to a finitely generated subfield of K. Such subfield is isomorphic to a subfield of C (see [18] Example 2, p.99, or [2] Proposition 1, p. 108). This latter subfield equipped with the partition induced by the partition existing on C∗ satisfies the conclusion of the lemma, hence is a model of T , proving that A is consistent. Let T∗ be the set of these four blocks, let T := T∪{0} and let χ be the map from K onto T. 1.1. Invariant relational structures and their age algebra. 1.1.1. Isomorphism, local isomorphism. Let R := (E, (ρi)i∈I) and R ′ := (E ′, (ρ′i)i∈I) be two relational structures having the same signature µ := (mi)i∈I . A map h : E → E ′ is an isomorphism from R onto R′ if (1) h is bijective, (2) (x1, . . . , xmi) ∈ ρi if and only if (h(x1), . . . , h(xmi)) ∈ ρ i for every (x1, . . . , xmi) ∈ Emi , i ∈ I. A partial map of E is a map h from a subset A of E onto a subset A′ of E, these subsets are the domain and codomain of h. A local isomorphism of R if a partial map h which is an isomorphism from R↾A onto R↾A′ (where A and A ′ are the domain an codomain of h). 1.1.2. Invariant relational structures. A chain is a pair L := (C,≤) where ≤ is a linear order on C. Let L be a chain. Let V be a non-empty set, F be a set disjoint from V ×C and let E := F ∪ (V ×C). Let R be a relational structure with base set E. Let r be a non-negative integer, r ≤ |C|. Let X,X ′ ∈ [C]r. Let ℓ be the unique order isomorphism from L↾X onto L↾X′ and let ℓ := 1F ∪ (1V , ℓ) be the partial map such that ℓ(x) = x for x ∈ F and ℓ(x, y) = (x, ℓ(y)) for (x, y) ∈ V ×X . We say that X and X ′ are equivalent if ℓ is an isomorphism of H↾F∪V×X onto H↾F∪V×X′. This defines an equivalence relation on [C] A CONJECTURE OF P.J. CAMERON 11 We say that R is r−F−L-invariant if two arbitrary members of [C]r are equivalent. We say that R is F − L-invariant if it is r − F − L-invariant for every non-negative integer r, r ≤ |C|. It is easy to see that if the signature µ of R is bounded and r :=Max({mi : i ∈ I}), R is F −L-invariant if and only if it is r′ − F − L-invariant for every r′ ≤ r. In fact: Lemma 1.2. If |C| > r :=Max({mi : i ∈ I}), R is F −L-invariant if and only if it is r − F − L-invariant. This is an immediate consequence of the following lemma: Lemma 1.3. If R is r−F −L-invariant and r < |C| then R is r′ −F −L-invariant for all r′ ≤ r. Proof. We only prove that R is (r− 1)− F − L-invariant. This suffices. Let X,X ′ ∈ [C]r−1. Since r < |C|, we may select Z ∈ [C]r such that the last element of Z (w.r.t. the order L) is strictly below some element c ∈ C. Claim 1.4. There are Y, Y ′ ∈ [Z]r−1 which are equivalent to X and X ′ respectively. Proof of Claim 1.4. Extend X and X ′ to two r-element subsets X1 and X 1 of C. Since R is r− F −L-invariant, X1 is equivalent to Z, hence the unique isomorphism from L↾X1 onto L↾Z carries X onto an equivalent subset Y of Z. By the same token, X ′1 is equivalent to a subset Y ′ of Z. Claim 1.5. Y and Y ′ are equivalent. Proof of Claim 1.5. The unique isomorphism from L↾Y ∪{c} onto L↾Y ′∪{c} carries Y onto Y hence T and Y ′ are equivalent. From the two claims above X and X ′ are equivalent. Hence, R is (r− 1)− F −L- invariant. 1.1.3. Coding by words. Let A := P(V ) \ {∅}. Let A∗ := Ap be the set of finite sequences of members of A. A finite sequence u being viewed as a word on the alphabet A, we write it as a juxtaposition of letters and we denote by λ the empty sequence; the length of u, denoted by |u| is the number of its terms. Let p be a non negative integer. If X is a subset of p := {0, . . . , p − 1} and u a word of length p, the restriction of u to X induces a word that we denote by t(u↾X). We suppose that V is finite and we equip A with a linear order. We compare words with the same length with the lexicographical order, denoted by ≤lex. We record without proof the following result. Lemma 1.6. Let p, q be two non negative integers and X be an p-element subset of p + q := {0, . . . , p + q − 1}. The map from Ap × Aq into Ap+q which associates to every pair (u, v) ∈ Ap × Aq the unique word w ∈ Ap+q such that t(w↾X) = u and t(w↾p+q\X) = v is strictly increasing (w.r.t. the lexicographical order). This word w is a shuffle of u and v that we denote uX v. We denote by u ̂ v the largest word of the form uX v. We order A∗ with the radix order defined as follows: if u and v are two distincts words, we set u < v if and only if either |u| < |v| or |u| = |v| et u <lex v. We suppose 12 MAURICE POUZET that F is finite and we order P(F ) in such a way that X < Y implies |X| ≥ |Y |. Finally, we order P(F )×A∗ lexicographically. Let L := (C,≤). Let Q be a finite subset of E := F ∪ (V × C). Let proj(Q) := {i ∈ C : Q ∩ V × {i} 6= ∅}. Let i0, . . . , ip−1 be an enumeration of proj(Q) in an increasing order (w.r.t L) and let w(Q \ F ) be the word u0 . . . up−1 ∈ A ∗ such that Q \ F = u0 ×{i0} ∪ · · · ∪ up−1 × {ip−1}. We set w(Q) := (Q∩ F,w(Q \ F )). If Q is a subset of [E]<ω, we set w(Q) := {w(Q) : Q ∈ Q}. If f : [E]m → K, let lead(f) := −∞ if f = 0 and otherwise let lead(f) be the largest element of w(supp(f)). We show below that this latter parameter behaves as the degree of a polynomial. We start with an easy fact. Lemma 1.7. Let m and n be two non negative integers, A ∈ [E\F ]m and B ∈ [E]n. If |C| ≥ m+n there is A′ ∈ [E\F ]m such that proj(A′)∩proj(B) = ∅ and w(A′) = w(A). Lemma 1.8. Let R be an F − L-invariant structure on E. Let m and n be two non negative integers; let f : [E]m → K, g : [E]n → K be two non zero members of K.A(R). Let A0 ∈ supp(f), and B0 ∈ supp(g) such that w(A0) = lead(f) and w(B0) = lead(g). Suppose that F and V are finite, that |C| ≤ n +m and supp(f) ∩ [E \ F ]m 6= ∅. Then: (7) supp(f, g) 6= ∅. (8) (w(A), w(B)) = (lead(f), lead(g)). for all (A,B) ∈ supp(f, g)(Q0), where w(Q0) = lead(f, g) and lead(f, g) is the largest element of w({A ∪ B : (A,B) ∈ supp(f, g)}). (9) (f(A), g(B)) = (f(A0), g(B0)) for every (A,B) ∈ supp(f, g)(Q0). (10) fg(Q0) = |supp(f, g)(Q0)|f(A0)g(B0). (11) lead(fg) = lead(f, g) = (Q0 ∩ F,w(A0)̂w(B0 \ F )). Proof. (1) Proof of (7). Since supp(f) ∩ [E \ F ]m 6= ∅ and the order on P(F ) decreases with the size, A0 is disjoint from F . Let B ∈ supp(g). According to Lemma 1.7 there is A′ such that proj(A′) ∩ proj(B) = ∅ and w(A′) = w(A0). We have A′ ∩ B = ∅ and, since f ∈ K.A(R) and R is F − L-invariant, f(A′) = f(A0). Thus (A ′, B) ∈ supp(f, g). (2) Proof of (8). Let (A,B) ∈ supp(f, g)(Q0). Since A ∈ supp(f) and B ∈ supp(g), we have trivially: (12) w(A) ≤ lead(f) and w(B) ≤ lead(g). Claim 1.9. B ∩ F = Q0 ∩ F and A ∩ F = ∅. A CONJECTURE OF P.J. CAMERON 13 The pair (A′, B) obtained in the proof of (7) belongs to supp(f, g). Let Q′ := A′ ∪ B. By maximality of w(Q0), we have w(Q ′) ≤ w(Q0). If B ∩ F 6= Q0 ∩ F , then |Q ′ ∩ F | < |Q0 ∩ F |, hence w(Q0) < w(Q ′). A contradiction. The fact that A ∩ F = ∅ follows. Claim 1.10. proj(A) ∩ proj(B) = ∅ and |proj(A)| = |proj(B).| Apply Lemma 1.7. Let A′ such that proj(A′) ∩ proj(B) = ∅ and w(A′) = w(A). Since f ∈ K.A(R) and R is F − L-invariant, f(A′) = f(A) thus (A′, B) ∈ supp(f, g). Set Q′ := A′ ∪ B. We have w(Q′ \ F ) ≤ w(Q0 \ F ) hence |w(Q′)| ≤ |w(Q0)|. Since |w(Q ′ \ F )| = |proj(A′)| + |proj(B)| and |w(Q0\F )| ≤ |proj(A)|+|proj(B)|, we get |w(Q0\F )| = |proj(A)|+|proj(B)|. This proves our claim. Let i0, . . . , ir−1 be an enumeration of proj(Q0 \ F ) in an increasing order. Let X := {j ∈ r : ij ∈ proj(A)}. Since proj(A)∩proj(B) = ∅, we have w(Q0\ F ) = w(A)X w(B \ F ). Since w(A) ≤ w(A0) and |w(A)| = |w(A0)|, Lemma 1.6 yields w(Q0 \ F ) ≤ w(A0)X w(B). As it is easy to see, there is A 0 such that w(A′0) = w(A0) and Q ′ := A′0 ∪ B satisfies w(Q ′ \ F ) = w(A0)X w(B). Since (A′, B) ∈ supp(f, g), we have w(Q′) = w(Q0) by maximality of w(Q0). With Lemma 1.6 again, this yields w(A) = w(A0). Hence w(A) = w(A0). A similar argument yields w(B \F ) = w(B0 \F ) and also w(B \F ) = w(B0 \F ). (3) Proof of (9). Since R is F −L-invariant, from w(A) = lead(f) := w(A0) we get f(A) = f(A0). By the same token, we get g(B) = g(B0). (4) Proof of (10). Since fg(Q0) = (A,B)∈sup(f,g) f(A)g(B) the result follows from (9). (5) Proof of (11). From (10), fg(Q0) 6= 0, the equality lead(fg) = lead(f, g) follows. The remaining equality follows from (8). With this, the proof of the lemma is complete. As far as invariant structures are concerned, we can retain this: Corollary 1.11. Under the hypotheses of Lemma 1.12, fg 6= 0. 1.1.4. An application. Letm,n be two positive integers, E be a set and f : [E]m → K, g : [E]n → K. Let R := (E, (ρ(i,j))(i,j)∈T∗×2)) be the relational structure made of the four m-ary relations ρ(i,0) := {(x1, . . . , xm) : χ ◦ f({x1, . . . , xm}) = i} and the four n- ary relations ρ(i,1) := {(x1, . . . , xn) : χ ◦ g({x1, . . . , xn}) = i}. A map h from a subset A of E onto a subset A′ of E is a local isomorphism of R if χ ◦ f(P ) = χ ◦ f(h[P ]) and χ ◦ f(R) = χ ◦ f(h[R]) for every P ∈ [A]m, every R ∈ [A]n. This fact allows us to consider the pair H := (E, (χ ◦ f, χ ◦ g)) as a relational structure. In the sequel we suppose that E = F ∪ (V × C) with F and V finite; we fix a chain L := (C,≤). Lemma 1.12. Suppose that there are P ∈ supp(f) ∩ [V × C]m and R ∈ supp(g) ∩ [E \ P ]n. If H is F − L-invariant and |C| ≥ m+ n. Then fg 6= 0. Proof. Let lead(f, g) be the largest element of w({A ∪ B : (A,B) ∈ supp(f, g)}) and let Q0 such that w(Q0) = lead(f, g). 14 MAURICE POUZET Claim 1.13. (χ ◦ f(A), χ ◦ g(B)) is constant for (A,B) ∈ sup(f, g)(Q0). Proof of Claim 1.13. Let s : T → K be a section of χ. Let f ′ := s ◦ χ ◦ f and let g′ := s ◦ χ ◦ g. Then f ′, g′ ∈ K.A(H) and supp(f ′, g′) = supp(f, g). According to Equation (9) of Lemma 1.8, (f ′(A), g′(B)) is constant for (A,B) ∈ supp(f ′, g′)(Q0). The result follows. From Lemma 1.1, fg(Q0) := (A,B)∈sup(f,g)(Q0) f(A)g(B) 6= 0. We recall the finite version of the theorem of Ramsey [32], [16]. Theorem 1.14. For every integers r, k, l there is an integer R such that for every partition of the r-element subsets of a R-element set C into k colors there is a l- element subset C ′ of C whose all r-element subsets have the same color. The least integer R for which the conclusion of Theorem 1.14 holds is a Ramsey number that we denote Rrk(l). Let m and n be two non negative integers. Set r := Max({m,n}), s := , k := 5s and, for an integer l, l > r, set ν(l) := Rrk(l). Lemma 1.15. If |F | = n, |V | = m and |C| ≥ ν(l) there is an l-element subset C ′ of C such that H↾F∪V×C′ is F − L↾C′-invariant. The proof is a basic application of Ramsey theorem. We give it for reader conve- nience. See [13] 10.9.4 page 296, or [27] Lemme IV.3.1.1 for a similar result). Proof. The number of equivalence classes on [C]r is at most k := 5s (indeed, this number is bounded by the number of distinct pairs (χ◦f ′, χ◦g′) such that f ′ ∈ K[E g′ ∈ K[E ′]n and |E ′| = n+mr). Thus, according to Theorem 1.14, there is a l-element subset C ′ of C whose all r-element subsets are equivalent. This means that H↾F∪V×C′ is r − F − L↾C′ -invariant. Now, since r < l, Lemma 1.3 asserts that H↾F∪V×C′ is r′ − F − L↾C′-invariant for all r ′ ≤ r. Since the signature of H is bounded by r, H↾F∪V×C′ is F − L↾C′ -invariant from Lemma 1.2. 1.2. The existence of τ(m,n). Let m and n be two non negative integers. Suppose 1 ≤ m ≤ n and that τ(m− 1, n) exists. Let E be a set with at least m+ n elements and f : [E]m → K, g : [E]n → K such that fg is zero, but f and g are not. Lemma 1.16. Let A be a transversal for supp(f). Then there is a transversal B for supp(g) such that: (13) |B \ A| ≤ τ(m− 1, n) Proof. Among the sets P ∈ supp(f) select one, say P0, such that F0 := P0 ∩ A has minimum size, say r0. Case 1. A∪P0 is also a transversal for supp(g). In this case, set B := A∪P0. We have |B \ A| ≤ m− 1, hence inequality (13) follows from inequality (14) below: (14) m− 1 ≤ τ(m− 1, n) This inequality is trivial for m = 1. Let us prove it for m > 1 (a much better inequality is given in Lemma 3.1). Let E ′ be an m + n − 1-element set, let f ′ : [E ′]m−1 → K and g′ : [E ′]n → K, with f ′ non zero on a single m − 1-element set A′, A CONJECTURE OF P.J. CAMERON 15 g′(B′) = 1 if A′ ∩ B′ 6= ∅ and g′(B′) = 0 otherwise. Since m − 1 and n are not 0, f ′ and g′ are not 0. Trivially, f ′g′ = 0 and, as it easy to check, A′ is a transversal for supp(f ′) ∪ supp(g′) having minimum size, hence τ(supp(f ′) ∪ supp(g′)) = m− 1. Case 2. Case 1 does not hold. In this case, pick x0 ∈ F0 and set E ′ := (E \ A) ∪ (F0 \ {x0}). Let f ′ : [E ′]m−1 → K be defined by setting f(P ′) := f(P ′ ∪ {x0}) for all P ′ ∈ [E ′]m−1 and let g′ := g↾[E′]n . Claim 1.17. |E ′| ≥ m+ n− 1 and f ′g′ = 0. Proof of Claim 1.17. The inequality follows from the fact that A ∪ P0 is not a transversal for supp(g). Now, let Q′ ∈ [E ′]m+n−1 and Q := Q′ ∪ {x0}. (15) f ′g′(Q′) = P ′∈[Q′]m−1 f ′(P ′)g′(Q′ \ P ′) = x0∈P∈[Q]m f(P )g(Q \ P ) Since fg = 0 we have: 0 = fg(Q) = P∈[Q]m f(P )g(Q \P ) = x0∈P∈[Q]m f(P )g(Q \P ) + x0 6∈P∈[Q]m f(P )g(Q \P ) If x0 6∈ P , |P ∩ A| < r0, hence f(P ) = 0. This implies that the second term in the last member of (16) is zero, hence the second member of (15) is zero. This proves our claim. From our hypothesis A ∪ P0 is not a tranversal for g. Hence, we have Claim 1.18. f ′ and g′ are not zero. The existence of τ(m − 1, n) insures that there is a transversal H for supp(f ′) ∪ supp(g′) of size at most τ(m − 1, n). The set B := A ∪ H is a transversal for supp(f) ∪ supp(g). Lemma 1.19. Let l be a positive integer. If τ(supp(f) ∪ supp(g)) > n +m(l − 1) + τ(m − 1, n) then for every F ∈ supp(g) there is a subset P ⊆ supp(f) ∩ [E \ F ]m made of at least l pairwise disjoint sets. Proof. Fix F ∈ supp(g). Let P ⊆ supp(f) ∩ [E \ F ]m be a finite subset made of pairwise disjoint sets and let p := |P|. If the conclusion of the lemma does not hold, we have p < l. Select then P with maximum size and set A := F ∪ P. Clearly A is a transversal for supp(f). According to Lemma 1.16 above, τ(supp(f) ∪ supp(g)) ≤ |A|+ τ(m − 1, n) = n +mp + τ(m− 1, n). Thus, according to our hypothese, l ≤ p. A contradiction. Let ϕ(m,n) := n+m(ν(n +m)− 1) + τ(m− 1, n). Lemma 1.20. (17) τ(m,n) ≤ ϕ(m,m) Proof. Suppose τ(supp(f) ∪ supp(g)) > ϕ(m,n). Let F ∈ supp(g). According to Lemma 1.19 there is a subset P ⊆ supp(f) ∩ [E \ F ]m made of at least ν(n + m) pairwise disjoint sets. With no loss of generality, we may suppose that ∪P is a set 16 MAURICE POUZET of the form V × C where |V | = m and |C| = ν(m,n). Let ≤ be a linear order on C and L := (C,≤). According to Lemma 1.15 there is an n +m-element subset C ′ of C such that H↾F∪V×C′ is F − L↾C′-invariant. According to Lemma 1.12 fg 6= 0. A contradiction. With Lemma 1.20, the proof of Theorem 0.3 is complete. Note that ϕ(1, 2) = 1 + R2 (3) + τ(0, n) = R2 (3), whereas τ(1, 2) = 4. Also ϕ(2, 2) = 2R2 (4) + τ(1, 2) = 2(R2 (4) + 2). Our original proof of Theorem 0.3 was a bit simpler. Instead of an m-element set F and several pairwise disjoint n-element sets, we considered several pairwise n+m- element sets. In the particular case of m = n = 2, we got τ(2, 2) ≤ 4R2 (4) + 1. In term of concrete upper-bounds, we are not convinced that the improvement worth the effort. 2. The Gottlieb-Kantor theorem and the case m = 1 Let E be a set. To each x ∈ E associate an indeterminate Xx. Let K[E] be the algebra over the field K of polynomials in these indeterminates. Let f : [E]1 → K. Let Df be the derivation on this algebra which is induced by f , that is Df(Xx) := f({x}) for every x ∈ E. Let ϕ : K[E] → K[E] be the ring homomorphism such that ϕf (1) := 1 and ϕf(Xx) := f({x})Xx. Let e : [E] 1 → K be the constant map equal to 1 and let De be the corresponding derivation. For example Df(XxXyXz) = f({x})XyXz + f({y})XxXz + f({z})XxXy whereas De(XxXyXz) = XyXz + XxXz + XxXy. It is easy to check that: (18) De ◦ ϕf = ϕf ◦Df . Let n be a non negative integer; let K[n][E] be the vector space generated by the monomials made of n distinct variables. From equation (18), we deduce: Corollary 2.1. If f does not take the value zero on [E]1, the surjectivity of the maps from K[n+1][E] into K[n][E] induced by Df and De are equivalent. Suppose that E is finite. In this case, the matrix of the restriction ofDe to K[n+1][E] identifies to Mn,n+1. Thus, according to Theorem 0.7, De is surjective provided that |E| ≥ 2n+ 1. Corollary asserts that in this case, Df is surjective too. This yields: Lemma 2.2. If f does not take the value zero on a subset E ′ of E of size at least 2n + 1 but fg = 0 for some g : [E]n → K then g is zero on the n-element subsets of Proof. Suppose that fg = 0. Let g : K[n][E] → K be the linear form defined by setting g(Πx∈BXx) := g(B) for each B ∈ [E] n. Then g ◦Df is 0 on K[n+1][E]. From Corollary 2, the map from K[n+1][E ′] into K[n][E ′] induced by Df is surjective. Hence g is 0 on K[n][E ′]. Thus g is zero on [E ′]n as claimed. Going a step further, we get a weighted version of Gottlieb-Kantor theorem: Theorem 2.3. Let f : [E]1 → K and g : [E]n → K. If f does not take the value zero on a subset of size at least 2n+ 1 and if fg = 0 then g est identically zero on [E]n. A CONJECTURE OF P.J. CAMERON 17 Proof. Set E ′ := supp(f). According to our hypothesis, E ′ 6= ∅. We prove the lemma by induction on n. If n = 0, pick x ∈ E ′. We have fg({x}) = f({x})g(∅). Since f({x}) 6= 0 and K is an integral domain, g(∅) = 0. Thus g = 0 and the conclusion of the lemma holds for n = 0. Let n ≥ 1. Let B ∈ [E]n. We claim that g(B) = 0. Let F := B \E ′ and r := |F |. If r = 0, that is B ⊆ E ′, we get g(B) = 0 from Lemma 2.2. If r 6= 0, we define gr on [E ′]n−r, setting gr(B ′) := g(B′ ∪ F ) for each B′ ∈ [E ′]n−r. Let Q′ ∈ [E ′]n−r+1 and Q := Q′ ∪ F . We have fgr(Q x∈Q′ f({x})g(Q \ {x}) =∑ x∈Q f({x})g(Q \ {x}) − x∈F f({x})g(Q \ {x}). From our hypothesis on g and the fact that f({x}) = 0 for all x /∈ E ′, both terms on the right hand side of the latter equality are 0, thus fgr(Q ′) = 0. Since |E ′| ≥ 2(n − r) + 1, induction on n applies. Hence gr is 0 on [E ′]n−r. This yields g(B) = 0, proving our claim. Hence the conclusion of the lemma holds for n. Theorem 2.4. τ(1, n) = 2n Proof. Trivially, the formula holds if n = 0. Hence, in the sequel, we suppose n ≥ 1. Claim 2.5. τ(1, n) ≤ 2n Proof of Claim 2.5.Let f and g be non identically zero such that fg = 0. From Theorem 2.3, the support S of f has at most 2n elements. From Lemma 1.16 |supp(f) ∪ (supp(g)| ≤ |S|+ τ(0, n) = |S| ≤ 2n. For the converse inequality, we prove that: Claim 2.6. There is a 2n element set E and a map g : [E]n → K such that eg = 0 and g 6= 0. Proof of Claim 2.6 Let E := {0, 1} × {0, . . . , n− 1}. Set Ei := {0, 1} × {i} for i < n. Let B ∈ [E]n. Set g(B) := 0 if B is not a transversal of the Ei’s, g(B) := −1 if B is a transversal containing an odd number of elements of the form (0, i), g(B) = 1 otherwise. Let Q ∈ [E]n+1. If Q is not a transversal of the Ei’s then g(B) = 0 for every B ∈ [Q]n hence eg(Q) = 0. If Q is a transversal, then there is a unique index i such that Ei ⊆ Q. In this case, the only members of [Q] n on which g is non-zero are Q\{(0, i)} and Q\{(1, i)}; by our choice, they have opposite signs, hence eg(Q) = 0. Since in the example above τ(supp(e)) = 2n, we have τ(1, n) ≥ 2n. With this inequality, the proof of Theorem 2.4 is complete. 3. A lower bound for τ(m,n) Lemma 3.1. τ(m,n) ≥ (m+ 1)(n+ 1)− 2 for all m,n ≥ 1. Proof. For m = 1 this inequality was obtained in Claim 2.6. For the case m > 1, we need the following improvement of Claim 2.6. Claim 3.2. Let n ≥ 1. There is a 2n element set E and a map g : [E]n → K such that eg = 0 and supp(g) = [E]n. Proof of Claim 3.2. Fix a 2n-element set E. From Claim 2.6 and the fact that the symmetric group SE acts transitively on [E] n, we get for each B ∈ [E]n some gB such that egB = 0 and B ∈ supp(gB). Next, we observe that a map g : [E] n → K satisfies 18 MAURICE POUZET eg = 0 if and only if g belongs to the kernel of the linear map T : K[E] → K[E] defined by setting T (g)(Q) := B∈[Q]n g(B) for all g, Q ∈ [E] n+1. To conclude, we apply the claim below with k := Claim 3.3. Let e1 := (1, 0 . . . , 0), . . . , ei := (0, . . . , 1, 0, . . . , 0), . . . , ek := (0, . . . , 1) be the canonical basis of Kk and let H be a subspace of Kk. Then H contains a vector with all its coordinates which are non-zero if and only if for every coordinate i it contains some vector with this coordinate non-zero. Proof of Claim 3.3. This assertion amounts to the fact that a vector space on an infinite field is not the union of finitely many proper subspaces. Let E be the disjoint union of m sets E0, . . . , Ei, . . . Em−1, each of size 2n. For each i, let gi : [Ei] n → K such that B∈[Q]n gi(B) = 0 for each Q ∈ [Ei] n+1 and supp(gi) = [Ei] n (according to Claim 3.2 such a gi exists). Let g : [E] n → K be the ”direct sum” of the gi’s: g(B) := gi(B) if B ∈ [Ei] n, g(B) := 0 otherwise. Let f : [E]m → K defined by setting f(A) := 1 if A ∩ Ei 6= ∅ for all i < m and 0 otherwise. Then, by a similar argument as in Claim 2.6, fg = 0. Next, a transversal of supp(f) must contains some Ei (thus τ(supp(f)) = 2n). And, also, a transversal of supp(g) must be a transversal of each of the supp(gi)’s. Since supp(gi) = [Ei] τ(supp(gi)) = n + 1, hence τ(supp(g)) = (n + 1)m. We get easily that τ(supp(f) ∪ supp(g)) = 2n + (n + 1)(m − 1) = mn + m + n − 1. This completes the proof of Lemma 3.1. Example 3.4. The lemma above gives τ(2, 2) ≥ 7. An example illustrating this inequality is quite simple: let E be made of two squares, let f be the map giving value −1/2 on each side of the squares, value 1 on the diagonals; let g be giving value 1 on each pair meeting the two squares. Then for every x ∈ E, E \ {x} is a minimal transversal of supp(f) ∪ supp(g). References [1] J. Berstel, C. Retenauer. Les séries rationnelles et leurs langages. Études et recherches en Informatique. Masson, Paris, 1984,132 pp. [2] N. Bourbaki. Éléments de mathématiques, Fasc. XI. Algèbre, Chap. V, Actualités scientifiques et industrielles, Hermann, Paris, 1973. [3] P. J. Cameron. Transitivity of permutation groups on unordered sets, Math. Z., 48(1976)127- [4] P.J. Cameron. Orbits of permutation groups on unordered sets. II. J. London Math. Soc., (2) 23 (1981), no. 2, 249–264. [5] P.J. Cameron.Oligomorphic permutation groups. Cambridge University Press, Cambridge, 1990. [6] P.J. Cameron. The algebra of an age. In Model theory of groups and automorphism groups (Blaubeuren, 1995), pages 126–133. Cambridge Univ. Press, Cambridge, 1997. [7] P.J. Cameron. On an algebra related to orbit-counting. J. Group Theory 1 (1998), no. 2, 173– [8] P.J. Cameron. Sequences realized by oligomorphic permutation groups. J. Integer Seq., 3(1):Ar- ticle 00.1.5, 1 HTML document (electronic), 2000. [9] P.J. Cameron. Some counting problems related to permutation groups. Discrete Math., 225(1- 3):77–92, 2000. Formal power series and algebraic combinatorics (Toronto, ON, 1998). [10] P.J. Cameron.Problems on permutation groups, http://www.maths.qmul.ac.uk/∼pjc/pgprob.html. http://www.maths.qmul.ac.uk/~pjc/pgprob.html A CONJECTURE OF P.J. CAMERON 19 [11] R. Diestel. Graph Theory. Springer-Verlag, Heidelberg Graduate Texts in Mathematics, Volume 173, 2005, 431 pages. [12] R. Fräıssé. Cours de logique mathématique. Tome 1: Relation et formule logique. Gauthier- Villars Éditeur, Paris, 1971. [13] R. Fräıssé. Theory of relations. North-Holland Publishing Co., Amsterdam, 2000. [14] R. Fräıssé and Maurice Pouzet. Interprétabilité d’une relation pour une châıne. C. R. Acad. Sci. Paris Sér. A-B, 272:A1624–A1627, 1971. [15] D.H. Gottlieb. A class of incidence matrices, Proc. Amer. Math. Soc., 17(1966)1233-1237. [16] R. Graham, B. Rothschild, J.H. Spencer, Ramsey Theory, John Wiley and Sons, NY (1990). [17] G. Higman. Ordering by divisibility in abstract algebras. Proc. London Math. Soc. (3), 2:326– 336, 1952. [18] W. Hodges, Model Theory, Cambridge University Press, Cambridge, 1993. xiv+772 pp. [19] W.M. Kantor. On incidence matrices of finite projective and affine spaces, Math. Z., 124(1972)315–318. [20] D. Livingstone, A. Wagner, Transitivity of finite permutation groups on unordered sets. Math. Z., 90(1965) 393-403. [21] M. Lothaire. Combinatorics on words. Vol. 17 of Encyclopedia of Mathematics and its Appli- cations. Addison-Wesley, Reading, Mass. Reprinted in the Cambridge Mathematical Library, Cambridge University Press, U.K. 1997. [22] H.D. Macpherson. Growth rates in infinite graphs and permutation groups. Proc. London Math. Soc. (3), 51(2):285–294, 1985. [23] M. Pouzet. Application d’une propriété combinatoire des parties d’un ensemble aux groupes et aux relations. Math. Z., 150(2):117–134, 1976. [24] M. Pouzet. Sur la théorie des relations. Thèse d’état, Université Claude-Bernard, Lyon 1, 1978. [25] M. Pouzet. Relation minimale pour son âge. Z. Math. Logik Grundlag. Math., 25(1979) 315–344. [26] M. Pouzet. Application de la notion de relation presque-enchâınable au dénombrement des restrictions finies d’une relation. Z. Math. Logik Grundlag. Math., 27(4):289–332, 1981. [27] M. Pouzet. Relation impartible. Dissertationnes, 103(1981)1–48. [28] M. Pouzet. The profile of relations. Glob. J. Pure Appl.Math. Volume 2, Number 3 (2007), pp. 237–272 (Proceedings of the 14th Symposium of the Tunisian Mathematical Society held in Hammamet, March 20-23, 2006). ArXiv math.CO/0703211. [29] M. Pouzet and M. Sobrani. Sandwiches of ages. Ann. Pure Appl. Logic, 108 (3)(2001) 295–326. [30] M. Pouzet, N. Thiéry. Some relational structures with polynomial growth and their associ- ated algebras. May 10th, 2005, 19pages, presented at FPSAC for the 75 birthday of A.Garsia. http://arxiv.org/math.CO/0601256. [31] D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra 58 (1979), 432-454. [32] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc., 30 (1930) 264-286. ICJ, Mathématiques, Université Claude-Bernard Lyon1, Domaine de Gerland, Bât. Recherche [B], 50 avenue Tony-Garnier, F69365 Lyon cedex 07, France, e-mail: pouzet@univ- lyon1.fr http://arxiv.org/abs/math/0703211 http://arxiv.org/math.CO/0601256 Introduction 0.1. The conjectures 0.2. The algebra of an age 0.3. A transversality property of the set algebra 0.4. Age algebra and profile of a relational structure 1. Proof of Theorem ?? 1.1. Invariant relational structures and their age algebra 1.2. The existence of (m,n) 2. The Gottlieb-Kantor theorem and the case m=1 3. A lower bound for (m,n) References

P.J.Cameron introduced the orbit algebra of a permutation group and conjectured that this algebra is an integral domain if and only if the group has no finite orbit. We prove that this conjecture holds and in fact that the age algebra of a relational structure $R$ is an integral domain if and only if $R$ is age-inexhaustible. We deduce these results from a combinatorial lemma asserting that if a product of two non-zero elements of a set algebra is zero then there is a finite common tranversal of their supports. The proof is built on Ramsey theorem and the integrity of a shuffle algebra.

Introduction In 1981, P.J.Cameron [4] (see also [9] p.86) associated a graded algebra A[G] to a permutation group G acting on an infinite set E. He formulated two conjectures on the integrity of this algebra. The purpose of this paper is to present a solution to the first of these conjectures. Consequences on the enumeration of finite substructures of a given structure are mentionned. Some problems are stated. 0.1. The conjectures. Here is the content of these conjectures, freely adapted from Cameron’s web page (see Problem 2 [10]). The graded algebra A[G] is the direct sum A[G]n where A[G]n is the set of all G-invariant functions f from the set [E] n of n-element subsets of E into the field C of complex numbers. Multiplication is defined by the rule that if f ∈ A[G]m, g ∈ A[G]n and Q is an (m+ n)-element subset of E then (1) (fg)(Q) := P∈[Q]m f(P )g(Q \ P ) As shown by Cameron, the constant function e in A[G]1 (with value 1 on every one element set) is not a zero-divisor (see Theorem 0.8 below). The group G is entire if A[G] is an integral domain, and strongly entire if A[G]/eA[G] is an integral domain. Date: November 6, 2018. 1991 Mathematics Subject Classification. 03 C13, 03 C52, 05 A16, 05 C30, 20 B27. Key words and phrases. Relational structures, ages, counting functions, oligomorphic groups, age algebra, Ramsey theorem, integral domain. Research done under the auspices of Intas programme 03-51-4110 ”Universal algebra and lattice theory” . http://arxiv.org/abs/0704.1548v1 2 MAURICE POUZET Conjectures 0.1. G is (strongly) entire if and only if it has no finite orbit on E. The condition that G has no finite orbit on E is necessary. We prove that it suffices for G to be entire. As it turns out, our proof extends to the algebra of an age, also invented by Cameron [10]. 0.2. The algebra of an age. A relational structure is a realization of a language whose non-logical symbols are predicates. This is a pair R := (E, (ρi)i∈I) made of a set E and a family of mi-ary relations ρi on E. The set E is the domain or base of R; the family µ := (mi)i∈I is the signature of R. The substructure induced by R on a subset A of E, simply called the restriction of R to A, is the relational structure R↾A := (A, (A mi ∩ ρi)i∈I). Notions of isomorphism, as well as isomorphic type, are defined in natural way (see Subsection 1.1). A map f : [E]m → C, where m is a non negative integer, is R-invariant if f(P ) = f(P ′) whenever the restrictions R|P and R|P ′ are isomorphic. The R-invariant maps can be multiplied. Indeed, it is not difficult to show that if f : [E]m → C and g : [E]n → C are R-invariant, the product defined by Equation (1) is R-invariant. Equipped with this multiplication, the C-vector space spanned by the R-invariant maps becomes a graded algebra, the age algebra of R, that we denote by C.A(R). The name, coined by Cameron, comes from the notion of age defined by Fräıssé [13]. Indeed, the age of R is the collection A(R) of substructures of R induced on the finite subsets of R, isomorphic substructures being identified. And it can be shown that two relational structures with the same age yields the same algebra (up to an isomorphism of graded algebras). The algebra associated to a group is a special case of age algebra. Indeed, to a permutation group G acting on E we may associate a relational structure R with base E such that the G-invariant maps coincide with the R-invariant maps. Our criterium for the integrity of the age algebra is based on the notion of kernel: The kernel of a relational structure R is the subset K(R) of x ∈ E such that A(R|E\{x}) 6= A(R). The emptyness of the kernel R is a necessary condition for the integrity of the age algebra. Indeed, if K(R) 6= ∅, pick x ∈ K(R) and F ∈ [E]<ω such that R↾F ∈ A(R) \ A(R|E\{x}). Let P ∈ [E] <ω. Set f(P ) := 1 if R↾P is isomorphic to R↾F , otherwise set f(P ) := 0. Then f 2 := ff = 0. Theorem 0.2. Let R be a relational structure with possibly infinitely many non iso- morphic types of n-element substructures. The age algebra C.A(R) is an integral domain if and only if the kernel of R is empty. The application to the conjecture of Cameron is immediate. LetG be a permutation group acting on E and let R be a relational structure encoding G. Then, the kernel of R is the union of the finite G-orbits of the one-element sets. Thus, if G has no finite orbit, the kernel of R is empty. Hence from Theorem 0.2, A[G] is an integral domain, as conjectured by Cameron. We deduce Theorem 0.2 from a combinatorial property of a set algebra over a field (Theorem 0.3 below). This property does not depends upon the field, provided that its characteristic is zero. The proof we give in Section 1.1 is an extension of our A CONJECTURE OF P.J. CAMERON 3 1970 proof that the profile of an infinite relational structure does not decrease (see Theorem 0.5 below). The key tool we used then was Ramsey’s theorem presented in terms of a property of almost-chainable relations. Here, these relations are replaced by F −L-invariant relational structures, structures which appeared, under other names, in several of our papers (see [24], [25], [27]). The final step is reminiscent of the proof of the integrity of a shuffle algebra. We introduced the notion of kernel in[24] and studied it in several papers [25] [26], [27] and [29]. As it is easy to see (cf [25][29]), the kernel of a relational structure R is empty if and only if for every finite subset F of E there is a disjoint subset F ′ such that the restrictions R|F and R|F ′ are isomorphic. Hence, relational structures with empty kernel are those for which their age has the disjoint embedding property, meaning that two arbitrary members of the age can be embedded into a third in such a way that their domain are disjoint. In Fräıssé’s terminology, ages with the disjoint embedding property are said inexhaustible and relational structures whose age is inexhaustible are said age-inexhaustible; we say that relational structures with finite kernel are almost age-inexhaustible. 1 0.3. A transversality property of the set algebra. Let K be a field with charac- teristic zero. Let E be a set and let [E]<ω be the set of finite subsets of E (including the empty set ∅). Let K[E] be the set of maps f : [E]<ω → K. Endowed with the usual addition and scalar multiplication of maps, this set is a vector space over K. Let f, g ∈ K[E] and Q ∈ [E]<ω. Set: (2) fg(Q) = P∈[Q]<ω f(P )g(Q \ P ) With this operation added, the above set becomes a K-algebra. This algebra is commutative and it has a unit, denoted by 1. This is the map taking the value 1 on the empty set and the value 0 everywhere else. The set algebra is the subalgebra made of maps f such that f(P ) = 0 for every P ∈ [E]<ω with |P | large enough. This algebra is graded, the homogeneous component of degree n being made of maps which take the value 0 on every subset of size different from n (see Cameron [6]). If f and g belong to two homogeneous components, their product is given by Equation (1), thus an age algebra, or a group algebra, A, as previously defined, is a subalgebra of this set algebra. The set algebra is far from to be an integral domain. But, with the notion of degree, the integrity of A will reduce to the fact that if m and n are two non negative integers and f : [E]m → K, f : [E]n → K are two non-zero maps belonging to A, their product fg is non zero. Let H be a family of subsets of E, a subset T of E is a transversal of H if F ∩T 6= ∅ for every F ∈ H; the transversality of H, denoted τ(H), is the minimum of the cardinalities (possibly infinite) of transversals of H. We make the convention that τ(H) = 0 if H is empty. Let f : [E]m → K, denote supp(f) := {P ∈ [V ]m : f(P ) 6= 0}. 1In order to agree with Fräıssé’s terminology, we disagree with the terminology of our papers, in which inexhaustibility, resp. almost inexhaustibility, is used for relational structures with empty, resp. finite, kernel, rather than for their ages. 4 MAURICE POUZET Here is our combinatorial result: Theorem 0.3. Let m,n be two non negative integers. There is an integer t such that for every set E with at least m + n elements, every field K with characteristic zero, every pair of maps f : [E]m → K, g : [E]n → K such that fg is zero, but f and g are not, then τ(supp(f) ∪ supp(g)) ≤ t. With this result, the proof of Theorem 0.2 is immediate. Indeed, let R be a rela- tional structure with empty kernel. If K.A(R), the age algebra of R over K, is not an integral domain there are two non-zero maps f : [E]m → K, f : [E]n → K belonging to K.A(R), whose product fg is zero. Since K is an integral domain, none of the in- tegers m and n can be zero. Since f is R-invariant, m is positive and the kernel K(R) of R is empty, it turns out that τ(supp(f)) is infinite. Hence τ(supp(f)∪ supp(g)) is infinite, contradicting the conclusion of Theorem 0.3. An other immediate consequence of Theorem 0.3 is the fact, due to Cameron, that on an infinite set E, e is not a zero-divisor (see Theorem 0.8 below). 0.3.1. Existence and values of τ . The fact the size of a transversal can be bounded independently of f and g, and the value of the least upper bound, seem to be of independent interest. So, let τ(m,n) be the least t for which the conclusion of Theorem 0.3 holds. Trivially, we have τ(m,n) = τ(n,m). We have τ(0, n) = τ(m, 0) = 0. Indeed, if m = 0, f is defined on the empty set only, an thus fg(Q) = f(∅)g(Q). Since K has no non zero divisors, fg is non zero provided that f and g are non zero. The fact that there is no pair f, g such that fg is zero, but f and g are not, yields τ(supp(f) ∪ supp(g)) = 0. We have τ(1, n) = 2n (Theorem 2.4). This is a non-trivial fact which essentially amounts to a weighted version of the Gottlieb-Kantor Theorem on incidence matrices ([15], [19], see subsection 0.4 and Theorem 2.3). These are the only exact values we know. We prove that τ(m,n) exists, by supposing that τ(m − 1, n) exists. Our existence proof relies in an essential way on Ramsey theorem. It yields astronomical upper bounds. For example, it yields τ(2, 2) ≤ 2(R2k(4) + 2) , where k = 5 30 and R2k(4) is the Ramsey number equal to the least integer p such that for every colouring of the pairs of {1, . . . , p} into k colors there are four integers whose all pairs have the same colour. The only lower bound we have is τ(2, 2) ≥ 7 and more generally τ(m,n) ≥ (m+ 1)(n + 1)− 2. We cannot preclude a extremely simple upper bound for τ(m,n), eg quadratic in n+m. 0.4. Age algebra and profile of a relational structure. The group agebra was invented by Cameron in order to study the behavior of the function θG which counts for each integer n the number θG(n) of orbits of n-subsets of a set E on which acts a permutation groupG, a function that we call the orbital profile ofG. Groups for which the orbital profile takes only finite values are quite important. Called oligomorphic groups by Cameron, they are an objet of study by itself (see Cameron’s book[5]). We present first some properties of the profile, a counting function somewhat more general. Next, we present the link with the age algebra, then we gives an illustration of Theorem 0.2. We conclude with some problems. A CONJECTURE OF P.J. CAMERON 5 0.4.1. Profile of a relational structure. The profile of a relational structure R with base E is the function ϕR which counts for every integer n the number (possibly infinite) ϕR(n) of substructures of R induced on the n-element subsets, isomorphic substructures being identified. Clearly, if R encodes a permutation groups G, ϕR(n) is the number θG(n) of orbits of n-element subsets of E. If the signature µ is finite (in the sense that I is finite), there are only finitely many relational structures with signature µ on an n-element domain, hence ϕR(n) is necessarily an integer for each integer n. In order to capture examples coming from algebra and group theory, one cannot preclude I to be infinite. But then, ϕR(n) could be an infinite cardinal. As far as one is concerned by the behavior of ϕR, this case can be excluded: Fact 0.4. [28] Let n < |E|. Then (3) ϕR(n) ≤ (n + 1)ϕR(n+ 1) In particular: (4) If ϕR(n) is infinite then ϕR(n + 1) is infinite too and ϕR(n) ≤ ϕR(n+ 1). Inequality (3) can be substantially improved: Theorem 0.5. If R is a relational structure on an infinite set then ϕR is non- decreasing. This result was conjectured with R.Fräıssé [14]. We proved it in 1971; the proof - for a single relation- appeared in 1971 in R.Fräıssé’s book [12], Exercise 8 p. 113; the general case was detailed in [26]. The proof relies on Ramsey theorem [32]. More is true: Theorem 0.6. If R is a relational structure on a set E having at least 2n + m elements then ϕR(n) ≤ ϕR(n+m). Meaning that if |E| := ℓ then ϕR increases up to ; and, for n ≥ ℓ the value in n is at least the value of the symmetric of n w.r.t. ℓ The result is a straightforward consequence of the following property of incidence matrices. Let m,n, ℓ be three non-negative integers and E be an ℓ-element set. Let Mn,n+m be the matrix whose rows are indexed by the n-element subsets P of E and columns by the n +m-element subsets Q of E, the coefficient aP,Q being equal to 1 if P ⊆ Q and equal to 0 otherwise. Theorem 0.7. If 2n+m ≤ l then Mn,n+m has full row rank (over the field of rational numbers). Theorem 0.7 is in W.Kantor 1972 [19], with similar results for affine and vector subspaces of a vector space. Over the last 30 years, it as been applied and redis- covered many times; recently, it was pointed out that it appeared in a 1966 paper of D.H.Gottlieb [15]. Nowadays, this is one of the fundamental tools in algebraic combinatorics. A proof, with a clever argument leading to further developments, was given by Fräıssé in the 1986’s edition of his book, Theory of relations, see [13]. 6 MAURICE POUZET We proved Theorem 0.6 in 1976 [23]. The same conclusion was obtained first for orbits of finite permutation groups by Livingstone and Wagner, 1965 [20], and extended to arbitrary permutation groups by Cameron, 1976 [3]. His proof uses the dual version of Theorem 0.7. Later on, he discovered a nice translation in terms of his age algebra, that we present now. For that, observe that ϕR only depends upon the age of R and, moreover, if ϕR take only integer values, then K.A(R) identifies with the set of (finite) linear combinations of members of A(R). In this case, as pointed out by Cameron, ϕR(n) is the dimension of the homogeneous component of degree n of K.A(R). Let e ∈ K[E] be the map which is 1 on the one-element subsets of E and 0 elsewhere. Let U be the subalgebra generated by e. We can think of e as the sum of isomorphic types of the one-element restrictions of R. Members of U are then of the form λme m + · · ·+ λ1e + λ01 where 1 is the isomorphic type of the empty relational structure and λm, . . . , λ0 are in K. Hence U is graded, with Un, the homogeneous component of degree n, equals to K.en. Here is the Cameron’s result: Theorem 0.8. If R is infinite then, for every u ∈ K.A(R), eu = 0 if and only if u = 0 This innocent looking result implies that ϕR is non decreasing. Indeed, the image of a basis of K.A(R)n by multiplication by e m is an independent subset of K.A(R)n+m. 0.4.2. Growth rate of the profile. Infinite relational structures with a constant profile, equal to 1, were called monomorphic and characterized by R. Fräıssé who proved that they were chainable. Later on, those with bounded profile, called finimorphic, were characterized as almost chainable [14]. Groups with orbital profile equal to 1 were described by P.Cameron in 1976 [3]. From his characterization, Cameron obtained that an orbital profile is ultimately constant, or grows as fast as a linear function with slope 1 The age algebra can be also used to study the growth of the profile. If A is a graded algebra, the Hilbert function hA of A is the function which associates to each integer n the dimension of the homogeneous component of degree n. So, provided that it takes only finite values, the profile ϕR is the Hilbert function of the age algebra C.A(R). In [10], Cameron made the following important observation about the behavior of the Hilbert fonction. Theorem 0.9. Let A be a graded algebra over an algebraically closed field of charac- teristic zero. If A is an integral domain the values of the Hilbert function hA satisfy the inequality (5) hA(n) + hA(m)− 1 ≤ hA(n +m) for all non-negative integers n and m. This result has an immediate consequence on the growth of the profile: Theorem 0.10. [26] The growth of the profile of a relational structure with empty kernel is at least linear provided that it is unbounded. A CONJECTURE OF P.J. CAMERON 7 In fact, provided that the relational structures satisfy some mild conditions, the existence of jumps in the behavior of the profile extends. Let ϕ : N → N and ψ : N → N. Recall that ϕ = O(ψ) and ψ grows as fast as ϕ if ϕ(n) ≤ aψ(n) for some positive real number a and n large enough. We say that ϕ and ψ have the same growth if ϕ grows as fast as ψ and ψ grows as fast as ϕ. The growth of ϕ is polynomial of degree k if ϕ has the same growth as n →֒ nk; in other words there are positive real numbers a and b such that ank ≤ ϕ ≤ bnk for n large enough. Note that the growth of ϕ is as fast as every polynomial if and only if limn→+∞ = +∞ for every non negative integer k. Theorem 0.11. Let R := (E, (ρi)i∈I) be a relational structure. The growth of ϕR is either polynomial or as fast as every polynomial provided that either the signature µ := (ni)i∈I is bounded or the kernel K(R) of R is finite. Theorem 0.11 is in [24]. An outline of the proof is given in [28]. A part appeared in [26], with a detailed proof showing that the growth of unbounded profiles of relational structures with bounded signature is at least linear. The kernel of any relational structure which encodes an oligomorphic permutation group is finite (indeed, as already mentionned, if R encodes a permutation group G acting on a set E then K(R) is the set union of the finite orbits of the one-element subsets of E. Since the number of these orbits is at most θG(1), K(R) is finite if G is oligomorphic). Hence: Corollary 0.12. The orbital profile of an oligomorphic group is either polynomial or faster than every polynomial. For groups, and graphs, there is a much more precise result than Theorem 0.11. It is due to Macpherson, 1985 [22]. Theorem 0.13. The profile of a graph or a permutation groups grows either as a polynomial or as fast as fε, where fε(n) = e , this for every ε > 0. 0.4.3. Growth rate and finite generation. A central question in the study of the profile, raised first by Cameron in the case of oligomorphic groups, is this: Problem 1. If the profile of a relational structures R with finite kernel has polynomial growth, is ϕR(n) ≃ cn k′ for some positive real c and some non-negative integer k′? Let us associate to a relational structure R whose profile takes only finite values its generating series HϕR := ϕR(n)x Problem 2. If R has a finite kernel and ϕR is bounded above by some polynomial, is the series HϕR a rational fraction of the form P (x) (1− x)(1− x2) · · · (1− xk) with P ∈ Z[x]? 8 MAURICE POUZET Under the hypothesis above we do not know if HϕR is a rational fraction. It is well known that if a generating function is of the form P (x) (1−x)(1−x2)···(1−xk) for n large enough, an is a quasi-polynomial of degree k ′, with k′ ≤ k − 1, that is a polynomial ak′(n)n k′ + · · ·+ a0(n) whose coefficients ak′(n), . . . , a0(n) are periodic functions. Hence, a subproblem is: Problem 3. If R has a finite kernel and ϕR is bounded above by some polynomial, is ϕR(n) a quasi-polynomial for n large enough? Remark 0.14. Since the profile is non-decreasing, if ϕR(n) is a quasi-polynomial for n large enough then ak′(n) is eventually constant. Hence the profile has polynomial growth in the sense that ϕR(n) ∼ cn k′ for some positive real c and k′ ∈ N. Thus, in this case, Problem 1 has a positive solution. A special case was solved positively with N.Thiéry [30]. These problems are linked with the structure of the age algebra. Indeed, if a graded algebra A is finitely generated, then, since A is a quotient of a polynomial ring K[x1, . . . , xd], its Hilbert function is bounded above by a polynomial. And, in fact, as it is well known, its Hilbert series is a fraction of form P (x) (1−x)d , thus of the form given in (6). Moreover, one can choose a numerator with non-negative coefficients whenever the algebra is Cohen-Macaulay. Due to Problem 2, one could be tempted to conjecture that these sufficient conditions are necessary in the case of age agebras. Indeed, from Theorem 0.8 one deduces easily: Theorem 0.15. The profile of R is bounded if and only if K.A(R) is finitely generated as a module over U , the graded algebra generated by e. In particular, if one of these equivalent conditions holds, K.A(R) is finitely generated But this case is exceptional. The conjecture can be disproved with tournaments. Indeed, on one hand, there are tournaments whose profile has arbitrarily large poly- nomial growth rate and, on an other hand, the age algebra of a tournament is finitely generated if and only if the profile of the tournament is bounded (this result was obtained with N.Thiery, a proof is presented in [28]). 0.4.4. Initial segments of an age and ideals of a ring. No concrete description of relational structures with bounded signature, or finite kernel, which have polynomial growth is known. In [24] (see also [28]) we proved that if a relational structure R has this property then its age, A(R), is well-quasi-ordered under embeddability, that is every final segment of A(R) is finitely generated, which amounts to the fact that the collection F (A(R)) of final segments of A(R) is noetherian, w.r.t. the inclusion order. Since the fundamental paper of Higman[17], applications of the notion of well-quasi-ordering have proliferated (eg see the Robertson-Seymour’s theorem for an application to graph theory [11] ). Final segments play for posets the same role than ideals for rings. Noticing that an age algebra is finitely generated if and only if it is noetherian, we are lead to have a closer look at the relationship between the basic objects of the theory of relations and of ring theory, particularly ages and ideals. We mention the following result which will be incorporated into a joint paper with N.Thiéry. A CONJECTURE OF P.J. CAMERON 9 Proposition 0.16. Let A be the age of a relational structure R such that the profile of R takes only finite values and K.A be its age algebra. If A′ is an initial segment of A then: (i) The vector subspace J := K.(A \ A′) spanned by A \ A′ is an ideal of K.A. Moreover, the quotient of K.A by J is a ring isomorphic to the ring K.A′. (ii) If this ideal is irreducible then A′ is a subage of A. (iii) This is a prime ideal if and only if A′ is an inexhaustible age. The proof of Item (i) and Item (ii) are immediate. The proof of Item (iii) is essentially based on Theorem 0.2. According to Item (i), F (A) embeds into the collection of ideals of K.A). Conse- quently: Corollary 0.17. If an age algebra is finitely generated then the age is well-quasi- ordered by embeddability. Problem 4. How the finite generation of an age algebra translates in terms of em- beddability between members of the ages? 0.4.5. Links with language theory. In the theory of languages, one of the basic results is that the generating series of a regular language is a rational fraction (see [1]). This result is not far away from our considerations. Indeed, if A is a finite alphabet, with say k elements, and A∗ is the set of words over A, then each word can be viewed as a finite chain coloured by k colors. Hence A∗ can be viewed as the age of the relational structure R made of the chain Q of rational numbers divided into k colors in such a way that, between two distinct rational numbers, all colors appear. Moreover, as pointed out by Cameron [6], the age algebra Q.A(R) is isomorphic to the shuffle algebra over A, an important object in algebraic combinatorics (see [21]). Problem 5. Does the members of the age of a relational structure with polynomial growth can be coded by words forming a regular language? Problem 6. Extend the properties of regular languages to subsets of the collection Ωµ made of isomorphic types of finite relational structures with signature µ. 1. Proof of Theorem 0.3 The proof idea of Theorem 0.3 is very simple and we give it first. We prove the result by induction. We suppose that it holds for pairs (m − 1, n). Now, let f : [E]m → K and g : [E]n → K such that fg is zero, but f and g are not. As already mentionned, m and n are non zero, hence members of supp(f) ∪ supp(g) are non empty. Let sup(f, g) := {(A,B) ∈ supp(f) × supp(g) : A ∩ B = ∅}. We may suppose sup(f, g) 6= ∅, otherwise the conclusion of Theorem 0.3 holds with t := m+ n− 1. For the sake of simplicity, we suppose that K := Q. In this case, we color elements A of [E]m into three colors :-,0, +, according to the value of f(A). We do the same with elements B of [E]n and we color each member (A,B) of supp(f, g) with the colors of its components. With the help of Ramsey’ theorem and a lexicographical ordering, we prove that if the transversality is large enough there is an (m+n)-element subset Q such that all pair (A,B) ∈ supp(f, g)(Q) := supp(f, g)∩ ([Q]m × [Q]n) have 10 MAURICE POUZET the same color. This readily implies that fg(Q) 6= 0, a contradiction. If K 6= Q, we may replace the three colors by five, as the following lemma indicates. Lemma 1.1. Let K be a field with characteristic zero. There is a partition of K∗ := K \ {0} into at most four blocks such that for every integer k and every k-element sequences (α1, . . . , αk) ∈ D k , (β1, . . . , βk) ∈ D ′k, where D, D′ are two blocks of the partition of K∗, then i=1 αiβi ∈ K Proof. This holds trivially if K := C. For an example, divide C∗ into the sets Di := {z ∈ C ∗ : πi ≤ Argz < π(i+1) } (i < 4). If K is arbitrary, use the Com- pactness theorem of first-order logic, under the form of the ”diagram method” of A.Robinson [18]. Namely, to the language of fields, add names for the elements of K, a binary predicate symbol, and axioms, this in such a way that a model, if any, of the resulting theory T will be an extension of K with a partition satisfying the conclusion of the lemma. According to the Compactness theorem of first-order logic, the existence of a model of T , alias the consistency of T , reduces to the consistency of every finite subset A of T . A finite subset A of T leads to a finitely generated subfield of K. Such subfield is isomorphic to a subfield of C (see [18] Example 2, p.99, or [2] Proposition 1, p. 108). This latter subfield equipped with the partition induced by the partition existing on C∗ satisfies the conclusion of the lemma, hence is a model of T , proving that A is consistent. Let T∗ be the set of these four blocks, let T := T∪{0} and let χ be the map from K onto T. 1.1. Invariant relational structures and their age algebra. 1.1.1. Isomorphism, local isomorphism. Let R := (E, (ρi)i∈I) and R ′ := (E ′, (ρ′i)i∈I) be two relational structures having the same signature µ := (mi)i∈I . A map h : E → E ′ is an isomorphism from R onto R′ if (1) h is bijective, (2) (x1, . . . , xmi) ∈ ρi if and only if (h(x1), . . . , h(xmi)) ∈ ρ i for every (x1, . . . , xmi) ∈ Emi , i ∈ I. A partial map of E is a map h from a subset A of E onto a subset A′ of E, these subsets are the domain and codomain of h. A local isomorphism of R if a partial map h which is an isomorphism from R↾A onto R↾A′ (where A and A ′ are the domain an codomain of h). 1.1.2. Invariant relational structures. A chain is a pair L := (C,≤) where ≤ is a linear order on C. Let L be a chain. Let V be a non-empty set, F be a set disjoint from V ×C and let E := F ∪ (V ×C). Let R be a relational structure with base set E. Let r be a non-negative integer, r ≤ |C|. Let X,X ′ ∈ [C]r. Let ℓ be the unique order isomorphism from L↾X onto L↾X′ and let ℓ := 1F ∪ (1V , ℓ) be the partial map such that ℓ(x) = x for x ∈ F and ℓ(x, y) = (x, ℓ(y)) for (x, y) ∈ V ×X . We say that X and X ′ are equivalent if ℓ is an isomorphism of H↾F∪V×X onto H↾F∪V×X′. This defines an equivalence relation on [C] A CONJECTURE OF P.J. CAMERON 11 We say that R is r−F−L-invariant if two arbitrary members of [C]r are equivalent. We say that R is F − L-invariant if it is r − F − L-invariant for every non-negative integer r, r ≤ |C|. It is easy to see that if the signature µ of R is bounded and r :=Max({mi : i ∈ I}), R is F −L-invariant if and only if it is r′ − F − L-invariant for every r′ ≤ r. In fact: Lemma 1.2. If |C| > r :=Max({mi : i ∈ I}), R is F −L-invariant if and only if it is r − F − L-invariant. This is an immediate consequence of the following lemma: Lemma 1.3. If R is r−F −L-invariant and r < |C| then R is r′ −F −L-invariant for all r′ ≤ r. Proof. We only prove that R is (r− 1)− F − L-invariant. This suffices. Let X,X ′ ∈ [C]r−1. Since r < |C|, we may select Z ∈ [C]r such that the last element of Z (w.r.t. the order L) is strictly below some element c ∈ C. Claim 1.4. There are Y, Y ′ ∈ [Z]r−1 which are equivalent to X and X ′ respectively. Proof of Claim 1.4. Extend X and X ′ to two r-element subsets X1 and X 1 of C. Since R is r− F −L-invariant, X1 is equivalent to Z, hence the unique isomorphism from L↾X1 onto L↾Z carries X onto an equivalent subset Y of Z. By the same token, X ′1 is equivalent to a subset Y ′ of Z. Claim 1.5. Y and Y ′ are equivalent. Proof of Claim 1.5. The unique isomorphism from L↾Y ∪{c} onto L↾Y ′∪{c} carries Y onto Y hence T and Y ′ are equivalent. From the two claims above X and X ′ are equivalent. Hence, R is (r− 1)− F −L- invariant. 1.1.3. Coding by words. Let A := P(V ) \ {∅}. Let A∗ := Ap be the set of finite sequences of members of A. A finite sequence u being viewed as a word on the alphabet A, we write it as a juxtaposition of letters and we denote by λ the empty sequence; the length of u, denoted by |u| is the number of its terms. Let p be a non negative integer. If X is a subset of p := {0, . . . , p − 1} and u a word of length p, the restriction of u to X induces a word that we denote by t(u↾X). We suppose that V is finite and we equip A with a linear order. We compare words with the same length with the lexicographical order, denoted by ≤lex. We record without proof the following result. Lemma 1.6. Let p, q be two non negative integers and X be an p-element subset of p + q := {0, . . . , p + q − 1}. The map from Ap × Aq into Ap+q which associates to every pair (u, v) ∈ Ap × Aq the unique word w ∈ Ap+q such that t(w↾X) = u and t(w↾p+q\X) = v is strictly increasing (w.r.t. the lexicographical order). This word w is a shuffle of u and v that we denote uX v. We denote by u ̂ v the largest word of the form uX v. We order A∗ with the radix order defined as follows: if u and v are two distincts words, we set u < v if and only if either |u| < |v| or |u| = |v| et u <lex v. We suppose 12 MAURICE POUZET that F is finite and we order P(F ) in such a way that X < Y implies |X| ≥ |Y |. Finally, we order P(F )×A∗ lexicographically. Let L := (C,≤). Let Q be a finite subset of E := F ∪ (V × C). Let proj(Q) := {i ∈ C : Q ∩ V × {i} 6= ∅}. Let i0, . . . , ip−1 be an enumeration of proj(Q) in an increasing order (w.r.t L) and let w(Q \ F ) be the word u0 . . . up−1 ∈ A ∗ such that Q \ F = u0 ×{i0} ∪ · · · ∪ up−1 × {ip−1}. We set w(Q) := (Q∩ F,w(Q \ F )). If Q is a subset of [E]<ω, we set w(Q) := {w(Q) : Q ∈ Q}. If f : [E]m → K, let lead(f) := −∞ if f = 0 and otherwise let lead(f) be the largest element of w(supp(f)). We show below that this latter parameter behaves as the degree of a polynomial. We start with an easy fact. Lemma 1.7. Let m and n be two non negative integers, A ∈ [E\F ]m and B ∈ [E]n. If |C| ≥ m+n there is A′ ∈ [E\F ]m such that proj(A′)∩proj(B) = ∅ and w(A′) = w(A). Lemma 1.8. Let R be an F − L-invariant structure on E. Let m and n be two non negative integers; let f : [E]m → K, g : [E]n → K be two non zero members of K.A(R). Let A0 ∈ supp(f), and B0 ∈ supp(g) such that w(A0) = lead(f) and w(B0) = lead(g). Suppose that F and V are finite, that |C| ≤ n +m and supp(f) ∩ [E \ F ]m 6= ∅. Then: (7) supp(f, g) 6= ∅. (8) (w(A), w(B)) = (lead(f), lead(g)). for all (A,B) ∈ supp(f, g)(Q0), where w(Q0) = lead(f, g) and lead(f, g) is the largest element of w({A ∪ B : (A,B) ∈ supp(f, g)}). (9) (f(A), g(B)) = (f(A0), g(B0)) for every (A,B) ∈ supp(f, g)(Q0). (10) fg(Q0) = |supp(f, g)(Q0)|f(A0)g(B0). (11) lead(fg) = lead(f, g) = (Q0 ∩ F,w(A0)̂w(B0 \ F )). Proof. (1) Proof of (7). Since supp(f) ∩ [E \ F ]m 6= ∅ and the order on P(F ) decreases with the size, A0 is disjoint from F . Let B ∈ supp(g). According to Lemma 1.7 there is A′ such that proj(A′) ∩ proj(B) = ∅ and w(A′) = w(A0). We have A′ ∩ B = ∅ and, since f ∈ K.A(R) and R is F − L-invariant, f(A′) = f(A0). Thus (A ′, B) ∈ supp(f, g). (2) Proof of (8). Let (A,B) ∈ supp(f, g)(Q0). Since A ∈ supp(f) and B ∈ supp(g), we have trivially: (12) w(A) ≤ lead(f) and w(B) ≤ lead(g). Claim 1.9. B ∩ F = Q0 ∩ F and A ∩ F = ∅. A CONJECTURE OF P.J. CAMERON 13 The pair (A′, B) obtained in the proof of (7) belongs to supp(f, g). Let Q′ := A′ ∪ B. By maximality of w(Q0), we have w(Q ′) ≤ w(Q0). If B ∩ F 6= Q0 ∩ F , then |Q ′ ∩ F | < |Q0 ∩ F |, hence w(Q0) < w(Q ′). A contradiction. The fact that A ∩ F = ∅ follows. Claim 1.10. proj(A) ∩ proj(B) = ∅ and |proj(A)| = |proj(B).| Apply Lemma 1.7. Let A′ such that proj(A′) ∩ proj(B) = ∅ and w(A′) = w(A). Since f ∈ K.A(R) and R is F − L-invariant, f(A′) = f(A) thus (A′, B) ∈ supp(f, g). Set Q′ := A′ ∪ B. We have w(Q′ \ F ) ≤ w(Q0 \ F ) hence |w(Q′)| ≤ |w(Q0)|. Since |w(Q ′ \ F )| = |proj(A′)| + |proj(B)| and |w(Q0\F )| ≤ |proj(A)|+|proj(B)|, we get |w(Q0\F )| = |proj(A)|+|proj(B)|. This proves our claim. Let i0, . . . , ir−1 be an enumeration of proj(Q0 \ F ) in an increasing order. Let X := {j ∈ r : ij ∈ proj(A)}. Since proj(A)∩proj(B) = ∅, we have w(Q0\ F ) = w(A)X w(B \ F ). Since w(A) ≤ w(A0) and |w(A)| = |w(A0)|, Lemma 1.6 yields w(Q0 \ F ) ≤ w(A0)X w(B). As it is easy to see, there is A 0 such that w(A′0) = w(A0) and Q ′ := A′0 ∪ B satisfies w(Q ′ \ F ) = w(A0)X w(B). Since (A′, B) ∈ supp(f, g), we have w(Q′) = w(Q0) by maximality of w(Q0). With Lemma 1.6 again, this yields w(A) = w(A0). Hence w(A) = w(A0). A similar argument yields w(B \F ) = w(B0 \F ) and also w(B \F ) = w(B0 \F ). (3) Proof of (9). Since R is F −L-invariant, from w(A) = lead(f) := w(A0) we get f(A) = f(A0). By the same token, we get g(B) = g(B0). (4) Proof of (10). Since fg(Q0) = (A,B)∈sup(f,g) f(A)g(B) the result follows from (9). (5) Proof of (11). From (10), fg(Q0) 6= 0, the equality lead(fg) = lead(f, g) follows. The remaining equality follows from (8). With this, the proof of the lemma is complete. As far as invariant structures are concerned, we can retain this: Corollary 1.11. Under the hypotheses of Lemma 1.12, fg 6= 0. 1.1.4. An application. Letm,n be two positive integers, E be a set and f : [E]m → K, g : [E]n → K. Let R := (E, (ρ(i,j))(i,j)∈T∗×2)) be the relational structure made of the four m-ary relations ρ(i,0) := {(x1, . . . , xm) : χ ◦ f({x1, . . . , xm}) = i} and the four n- ary relations ρ(i,1) := {(x1, . . . , xn) : χ ◦ g({x1, . . . , xn}) = i}. A map h from a subset A of E onto a subset A′ of E is a local isomorphism of R if χ ◦ f(P ) = χ ◦ f(h[P ]) and χ ◦ f(R) = χ ◦ f(h[R]) for every P ∈ [A]m, every R ∈ [A]n. This fact allows us to consider the pair H := (E, (χ ◦ f, χ ◦ g)) as a relational structure. In the sequel we suppose that E = F ∪ (V × C) with F and V finite; we fix a chain L := (C,≤). Lemma 1.12. Suppose that there are P ∈ supp(f) ∩ [V × C]m and R ∈ supp(g) ∩ [E \ P ]n. If H is F − L-invariant and |C| ≥ m+ n. Then fg 6= 0. Proof. Let lead(f, g) be the largest element of w({A ∪ B : (A,B) ∈ supp(f, g)}) and let Q0 such that w(Q0) = lead(f, g). 14 MAURICE POUZET Claim 1.13. (χ ◦ f(A), χ ◦ g(B)) is constant for (A,B) ∈ sup(f, g)(Q0). Proof of Claim 1.13. Let s : T → K be a section of χ. Let f ′ := s ◦ χ ◦ f and let g′ := s ◦ χ ◦ g. Then f ′, g′ ∈ K.A(H) and supp(f ′, g′) = supp(f, g). According to Equation (9) of Lemma 1.8, (f ′(A), g′(B)) is constant for (A,B) ∈ supp(f ′, g′)(Q0). The result follows. From Lemma 1.1, fg(Q0) := (A,B)∈sup(f,g)(Q0) f(A)g(B) 6= 0. We recall the finite version of the theorem of Ramsey [32], [16]. Theorem 1.14. For every integers r, k, l there is an integer R such that for every partition of the r-element subsets of a R-element set C into k colors there is a l- element subset C ′ of C whose all r-element subsets have the same color. The least integer R for which the conclusion of Theorem 1.14 holds is a Ramsey number that we denote Rrk(l). Let m and n be two non negative integers. Set r := Max({m,n}), s := , k := 5s and, for an integer l, l > r, set ν(l) := Rrk(l). Lemma 1.15. If |F | = n, |V | = m and |C| ≥ ν(l) there is an l-element subset C ′ of C such that H↾F∪V×C′ is F − L↾C′-invariant. The proof is a basic application of Ramsey theorem. We give it for reader conve- nience. See [13] 10.9.4 page 296, or [27] Lemme IV.3.1.1 for a similar result). Proof. The number of equivalence classes on [C]r is at most k := 5s (indeed, this number is bounded by the number of distinct pairs (χ◦f ′, χ◦g′) such that f ′ ∈ K[E g′ ∈ K[E ′]n and |E ′| = n+mr). Thus, according to Theorem 1.14, there is a l-element subset C ′ of C whose all r-element subsets are equivalent. This means that H↾F∪V×C′ is r − F − L↾C′ -invariant. Now, since r < l, Lemma 1.3 asserts that H↾F∪V×C′ is r′ − F − L↾C′-invariant for all r ′ ≤ r. Since the signature of H is bounded by r, H↾F∪V×C′ is F − L↾C′ -invariant from Lemma 1.2. 1.2. The existence of τ(m,n). Let m and n be two non negative integers. Suppose 1 ≤ m ≤ n and that τ(m− 1, n) exists. Let E be a set with at least m+ n elements and f : [E]m → K, g : [E]n → K such that fg is zero, but f and g are not. Lemma 1.16. Let A be a transversal for supp(f). Then there is a transversal B for supp(g) such that: (13) |B \ A| ≤ τ(m− 1, n) Proof. Among the sets P ∈ supp(f) select one, say P0, such that F0 := P0 ∩ A has minimum size, say r0. Case 1. A∪P0 is also a transversal for supp(g). In this case, set B := A∪P0. We have |B \ A| ≤ m− 1, hence inequality (13) follows from inequality (14) below: (14) m− 1 ≤ τ(m− 1, n) This inequality is trivial for m = 1. Let us prove it for m > 1 (a much better inequality is given in Lemma 3.1). Let E ′ be an m + n − 1-element set, let f ′ : [E ′]m−1 → K and g′ : [E ′]n → K, with f ′ non zero on a single m − 1-element set A′, A CONJECTURE OF P.J. CAMERON 15 g′(B′) = 1 if A′ ∩ B′ 6= ∅ and g′(B′) = 0 otherwise. Since m − 1 and n are not 0, f ′ and g′ are not 0. Trivially, f ′g′ = 0 and, as it easy to check, A′ is a transversal for supp(f ′) ∪ supp(g′) having minimum size, hence τ(supp(f ′) ∪ supp(g′)) = m− 1. Case 2. Case 1 does not hold. In this case, pick x0 ∈ F0 and set E ′ := (E \ A) ∪ (F0 \ {x0}). Let f ′ : [E ′]m−1 → K be defined by setting f(P ′) := f(P ′ ∪ {x0}) for all P ′ ∈ [E ′]m−1 and let g′ := g↾[E′]n . Claim 1.17. |E ′| ≥ m+ n− 1 and f ′g′ = 0. Proof of Claim 1.17. The inequality follows from the fact that A ∪ P0 is not a transversal for supp(g). Now, let Q′ ∈ [E ′]m+n−1 and Q := Q′ ∪ {x0}. (15) f ′g′(Q′) = P ′∈[Q′]m−1 f ′(P ′)g′(Q′ \ P ′) = x0∈P∈[Q]m f(P )g(Q \ P ) Since fg = 0 we have: 0 = fg(Q) = P∈[Q]m f(P )g(Q \P ) = x0∈P∈[Q]m f(P )g(Q \P ) + x0 6∈P∈[Q]m f(P )g(Q \P ) If x0 6∈ P , |P ∩ A| < r0, hence f(P ) = 0. This implies that the second term in the last member of (16) is zero, hence the second member of (15) is zero. This proves our claim. From our hypothesis A ∪ P0 is not a tranversal for g. Hence, we have Claim 1.18. f ′ and g′ are not zero. The existence of τ(m − 1, n) insures that there is a transversal H for supp(f ′) ∪ supp(g′) of size at most τ(m − 1, n). The set B := A ∪ H is a transversal for supp(f) ∪ supp(g). Lemma 1.19. Let l be a positive integer. If τ(supp(f) ∪ supp(g)) > n +m(l − 1) + τ(m − 1, n) then for every F ∈ supp(g) there is a subset P ⊆ supp(f) ∩ [E \ F ]m made of at least l pairwise disjoint sets. Proof. Fix F ∈ supp(g). Let P ⊆ supp(f) ∩ [E \ F ]m be a finite subset made of pairwise disjoint sets and let p := |P|. If the conclusion of the lemma does not hold, we have p < l. Select then P with maximum size and set A := F ∪ P. Clearly A is a transversal for supp(f). According to Lemma 1.16 above, τ(supp(f) ∪ supp(g)) ≤ |A|+ τ(m − 1, n) = n +mp + τ(m− 1, n). Thus, according to our hypothese, l ≤ p. A contradiction. Let ϕ(m,n) := n+m(ν(n +m)− 1) + τ(m− 1, n). Lemma 1.20. (17) τ(m,n) ≤ ϕ(m,m) Proof. Suppose τ(supp(f) ∪ supp(g)) > ϕ(m,n). Let F ∈ supp(g). According to Lemma 1.19 there is a subset P ⊆ supp(f) ∩ [E \ F ]m made of at least ν(n + m) pairwise disjoint sets. With no loss of generality, we may suppose that ∪P is a set 16 MAURICE POUZET of the form V × C where |V | = m and |C| = ν(m,n). Let ≤ be a linear order on C and L := (C,≤). According to Lemma 1.15 there is an n +m-element subset C ′ of C such that H↾F∪V×C′ is F − L↾C′-invariant. According to Lemma 1.12 fg 6= 0. A contradiction. With Lemma 1.20, the proof of Theorem 0.3 is complete. Note that ϕ(1, 2) = 1 + R2 (3) + τ(0, n) = R2 (3), whereas τ(1, 2) = 4. Also ϕ(2, 2) = 2R2 (4) + τ(1, 2) = 2(R2 (4) + 2). Our original proof of Theorem 0.3 was a bit simpler. Instead of an m-element set F and several pairwise disjoint n-element sets, we considered several pairwise n+m- element sets. In the particular case of m = n = 2, we got τ(2, 2) ≤ 4R2 (4) + 1. In term of concrete upper-bounds, we are not convinced that the improvement worth the effort. 2. The Gottlieb-Kantor theorem and the case m = 1 Let E be a set. To each x ∈ E associate an indeterminate Xx. Let K[E] be the algebra over the field K of polynomials in these indeterminates. Let f : [E]1 → K. Let Df be the derivation on this algebra which is induced by f , that is Df(Xx) := f({x}) for every x ∈ E. Let ϕ : K[E] → K[E] be the ring homomorphism such that ϕf (1) := 1 and ϕf(Xx) := f({x})Xx. Let e : [E] 1 → K be the constant map equal to 1 and let De be the corresponding derivation. For example Df(XxXyXz) = f({x})XyXz + f({y})XxXz + f({z})XxXy whereas De(XxXyXz) = XyXz + XxXz + XxXy. It is easy to check that: (18) De ◦ ϕf = ϕf ◦Df . Let n be a non negative integer; let K[n][E] be the vector space generated by the monomials made of n distinct variables. From equation (18), we deduce: Corollary 2.1. If f does not take the value zero on [E]1, the surjectivity of the maps from K[n+1][E] into K[n][E] induced by Df and De are equivalent. Suppose that E is finite. In this case, the matrix of the restriction ofDe to K[n+1][E] identifies to Mn,n+1. Thus, according to Theorem 0.7, De is surjective provided that |E| ≥ 2n+ 1. Corollary asserts that in this case, Df is surjective too. This yields: Lemma 2.2. If f does not take the value zero on a subset E ′ of E of size at least 2n + 1 but fg = 0 for some g : [E]n → K then g is zero on the n-element subsets of Proof. Suppose that fg = 0. Let g : K[n][E] → K be the linear form defined by setting g(Πx∈BXx) := g(B) for each B ∈ [E] n. Then g ◦Df is 0 on K[n+1][E]. From Corollary 2, the map from K[n+1][E ′] into K[n][E ′] induced by Df is surjective. Hence g is 0 on K[n][E ′]. Thus g is zero on [E ′]n as claimed. Going a step further, we get a weighted version of Gottlieb-Kantor theorem: Theorem 2.3. Let f : [E]1 → K and g : [E]n → K. If f does not take the value zero on a subset of size at least 2n+ 1 and if fg = 0 then g est identically zero on [E]n. A CONJECTURE OF P.J. CAMERON 17 Proof. Set E ′ := supp(f). According to our hypothesis, E ′ 6= ∅. We prove the lemma by induction on n. If n = 0, pick x ∈ E ′. We have fg({x}) = f({x})g(∅). Since f({x}) 6= 0 and K is an integral domain, g(∅) = 0. Thus g = 0 and the conclusion of the lemma holds for n = 0. Let n ≥ 1. Let B ∈ [E]n. We claim that g(B) = 0. Let F := B \E ′ and r := |F |. If r = 0, that is B ⊆ E ′, we get g(B) = 0 from Lemma 2.2. If r 6= 0, we define gr on [E ′]n−r, setting gr(B ′) := g(B′ ∪ F ) for each B′ ∈ [E ′]n−r. Let Q′ ∈ [E ′]n−r+1 and Q := Q′ ∪ F . We have fgr(Q x∈Q′ f({x})g(Q \ {x}) =∑ x∈Q f({x})g(Q \ {x}) − x∈F f({x})g(Q \ {x}). From our hypothesis on g and the fact that f({x}) = 0 for all x /∈ E ′, both terms on the right hand side of the latter equality are 0, thus fgr(Q ′) = 0. Since |E ′| ≥ 2(n − r) + 1, induction on n applies. Hence gr is 0 on [E ′]n−r. This yields g(B) = 0, proving our claim. Hence the conclusion of the lemma holds for n. Theorem 2.4. τ(1, n) = 2n Proof. Trivially, the formula holds if n = 0. Hence, in the sequel, we suppose n ≥ 1. Claim 2.5. τ(1, n) ≤ 2n Proof of Claim 2.5.Let f and g be non identically zero such that fg = 0. From Theorem 2.3, the support S of f has at most 2n elements. From Lemma 1.16 |supp(f) ∪ (supp(g)| ≤ |S|+ τ(0, n) = |S| ≤ 2n. For the converse inequality, we prove that: Claim 2.6. There is a 2n element set E and a map g : [E]n → K such that eg = 0 and g 6= 0. Proof of Claim 2.6 Let E := {0, 1} × {0, . . . , n− 1}. Set Ei := {0, 1} × {i} for i < n. Let B ∈ [E]n. Set g(B) := 0 if B is not a transversal of the Ei’s, g(B) := −1 if B is a transversal containing an odd number of elements of the form (0, i), g(B) = 1 otherwise. Let Q ∈ [E]n+1. If Q is not a transversal of the Ei’s then g(B) = 0 for every B ∈ [Q]n hence eg(Q) = 0. If Q is a transversal, then there is a unique index i such that Ei ⊆ Q. In this case, the only members of [Q] n on which g is non-zero are Q\{(0, i)} and Q\{(1, i)}; by our choice, they have opposite signs, hence eg(Q) = 0. Since in the example above τ(supp(e)) = 2n, we have τ(1, n) ≥ 2n. With this inequality, the proof of Theorem 2.4 is complete. 3. A lower bound for τ(m,n) Lemma 3.1. τ(m,n) ≥ (m+ 1)(n+ 1)− 2 for all m,n ≥ 1. Proof. For m = 1 this inequality was obtained in Claim 2.6. For the case m > 1, we need the following improvement of Claim 2.6. Claim 3.2. Let n ≥ 1. There is a 2n element set E and a map g : [E]n → K such that eg = 0 and supp(g) = [E]n. Proof of Claim 3.2. Fix a 2n-element set E. From Claim 2.6 and the fact that the symmetric group SE acts transitively on [E] n, we get for each B ∈ [E]n some gB such that egB = 0 and B ∈ supp(gB). Next, we observe that a map g : [E] n → K satisfies 18 MAURICE POUZET eg = 0 if and only if g belongs to the kernel of the linear map T : K[E] → K[E] defined by setting T (g)(Q) := B∈[Q]n g(B) for all g, Q ∈ [E] n+1. To conclude, we apply the claim below with k := Claim 3.3. Let e1 := (1, 0 . . . , 0), . . . , ei := (0, . . . , 1, 0, . . . , 0), . . . , ek := (0, . . . , 1) be the canonical basis of Kk and let H be a subspace of Kk. Then H contains a vector with all its coordinates which are non-zero if and only if for every coordinate i it contains some vector with this coordinate non-zero. Proof of Claim 3.3. This assertion amounts to the fact that a vector space on an infinite field is not the union of finitely many proper subspaces. Let E be the disjoint union of m sets E0, . . . , Ei, . . . Em−1, each of size 2n. For each i, let gi : [Ei] n → K such that B∈[Q]n gi(B) = 0 for each Q ∈ [Ei] n+1 and supp(gi) = [Ei] n (according to Claim 3.2 such a gi exists). Let g : [E] n → K be the ”direct sum” of the gi’s: g(B) := gi(B) if B ∈ [Ei] n, g(B) := 0 otherwise. Let f : [E]m → K defined by setting f(A) := 1 if A ∩ Ei 6= ∅ for all i < m and 0 otherwise. Then, by a similar argument as in Claim 2.6, fg = 0. Next, a transversal of supp(f) must contains some Ei (thus τ(supp(f)) = 2n). And, also, a transversal of supp(g) must be a transversal of each of the supp(gi)’s. Since supp(gi) = [Ei] τ(supp(gi)) = n + 1, hence τ(supp(g)) = (n + 1)m. We get easily that τ(supp(f) ∪ supp(g)) = 2n + (n + 1)(m − 1) = mn + m + n − 1. This completes the proof of Lemma 3.1. Example 3.4. 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Volume 2, Number 3 (2007), pp. 237–272 (Proceedings of the 14th Symposium of the Tunisian Mathematical Society held in Hammamet, March 20-23, 2006). ArXiv math.CO/0703211. [29] M. Pouzet and M. Sobrani. Sandwiches of ages. Ann. Pure Appl. Logic, 108 (3)(2001) 295–326. [30] M. Pouzet, N. Thiéry. Some relational structures with polynomial growth and their associ- ated algebras. May 10th, 2005, 19pages, presented at FPSAC for the 75 birthday of A.Garsia. http://arxiv.org/math.CO/0601256. [31] D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra 58 (1979), 432-454. [32] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc., 30 (1930) 264-286. ICJ, Mathématiques, Université Claude-Bernard Lyon1, Domaine de Gerland, Bât. Recherche [B], 50 avenue Tony-Garnier, F69365 Lyon cedex 07, France, e-mail: pouzet@univ- lyon1.fr http://arxiv.org/abs/math/0703211 http://arxiv.org/math.CO/0601256 Introduction 0.1. The conjectures 0.2. The algebra of an age 0.3. A transversality property of the set algebra 0.4. Age algebra and profile of a relational structure 1. Proof of Theorem ?? 1.1. Invariant relational structures and their age algebra 1.2. The existence of (m,n) 2. The Gottlieb-Kantor theorem and the case m=1 3. A lower bound for (m,n) References

SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS ON C∗-ALGEBRAS JA A JEONG† AND GI HYUN PARK‡ Abstract. If a finite group action α on a unital C∗-algebra M is saturated, the canonical conditional expectation E : M → Mα onto the fixed point algebra is known to be of index finite type with Index(E) = |G| in the sense of Watatani. More generally if a finite dimensional Hopf ∗-algebra A acts on M and the action is saturated, the same is true with Index(E) = dim(A). In this paper we prove that the converse is true. Especially in case M is a commutative C∗-algebra C(X) and α is a finite group action, we give an equivalent condition in order that the expectation E : C(X) → C(X)α is of index finite type, from which we obtain that α is saturated if and only if G acts freely on X. Actions by compact groups are also considered to show that the gauge action γ on a graph C∗-algebra C∗(E) associated with a locally finite directed graph E is saturated. 1. Introduction It is known [17] that if α is an action by a compact group G on a C∗-algebra M , the fixed point algebra Mα is isomorphic to a hereditary subalgebra e(M ×αG)e of the crossed product M ×αG for a projection e in the multiplier algebra of M ×αG. If e(M ×α G)e is full in M ×α G (that is, e(M ×α G)e generates M ×α G as a closed two-sided ideal), the action α is said to be saturated (the notion of saturated action was introduced by Rieffel [14, Chap.7]). Every action α with a simple crossed product M ×α G is obviously saturated. On the other hand, an action of a finite dimensional Hopf ∗-algebra A on a unital C∗-algebra M is considered in [18] and it is shown that if the action is saturated, the canonical conditional expectation E : M → MA onto the fixed point algebra MA is of index finite type in the sense of Watatani [19] and Index(E) = (dimA)1. The main purpose of the present paper is to prove that the converse is also true. We see from our result that for an action α by a finite group G, α is saturated if and only if the canonical expectation E : M → Mα is of index-finite type with index Index(E) = |G|. Besides, we consider actions by compact groups to study the saturation property of a gauge action γ on a C∗-algebra C∗(E) associated with a locally finite directed graph E with no sinks or sources. This paper is organized as follows. In section 2, we review the C∗-basic construction from [19] and finite dimensional Hopf ∗-algebras from [18] setting up our notations. Then we prove in section 3 that if A is a finite dimensional Hopf ∗-algebra acting on a unital C∗-algebra M such that E : M → MA is of index finite type with Index(E) = (dimA)1, then the action is saturated (Theorem 3.3). In section 4, we deal with the crossed product M ×α G by a finite group in detail and give other equivalent conditions in order that α be saturated. From the Research supported by KRF-ABRL-R14(2003-2008)† and Hanshin University Research Grant‡. http://arxiv.org/abs/0704.1549v1 2 JA A JEONG AND GI HYUN PARK conditions one easily see that an action with the Rokhlin property [7] is always saturated. Also we shall show that if M has the cancellation, an action with the tracial Rokhlin property [12] on M is saturated. Note that even for an action α by the finite group Z2, the expectation E : M → Mα may not be of index finite type in general [19, Example 2.8.4]. For a commutative C∗-algebra C(X) and a finite group action α, we give a necessary and sufficient condition that E : C(X) → C(X)α is of index finite type (Theorem 4.10) and provide a formula for Index(E). Then as a corollary we obtain that α is saturated if and only if G acts freely on X . In section 5, we consider a compact group action α and investigate the ideal Jα of M ×α G generated by the hereditary subalgebra e(M ×α G)e. Then we apply the result on Jα to the gauge action on a graph C∗-algebra in section 6. As a generalization of the Cuntz-Krieger algebras [5], the class of graph C∗-algebras C∗(E) associated with directed graphs E has been studied in various directions by considerably many authors (for example see the bibliography in the book [15] by Raeburn). In [9], Kumjian and Pask show among others that if γ is the gauge action on C∗(E), then C∗(E)γ is stably isomorphic to the crossed product C∗(E) ×γ T, which was done by hiring the notions of skew product of graphs and groupoid C∗- algebras. In Theorem 6.3 we shall directly show that the gauge action is actually saturated (this implies that C∗(E)γ and C∗(E)×γ T are stably isomorphic). 2. Preliminaries Watatani’s index theory for C∗-algebras. In [19], Watatani developed the index theory for C∗-algebras, and here we briefly review the basic construction C∗(B, eA). Let B be a C ∗-algebra and A its C∗-subalgebra containing the unit of B. Let E : B → A be a faithful conditional expectation. If there exist finitely many elements {vi}ni=1 in B satisfying the following E(bvi)v viE(v i b), for every b ∈ B, E is said to be of index-finite type and {(vi, v∗i )}ni=1 is called a quasi-basis for E. The positive element i is then the index of E, Index(E), which is known to be an element in the center of B and does not depend on the choice of quasi-bases for E ([19, Proposition 1.2.8]). Let B be the completion of the pre-Hilbert module B0 = {η(b) | b ∈ B} over A with an A-valued inner product 〈η(x), η(y)〉 = E(x∗y), η(x), η(y) ∈ B0. Let LA(B) be the C∗-algebra of all (right) A-module homomorphisms on B with adjoints. For T ∈ LA(B), the norm ‖T ‖ = sup{‖Tx‖ : ‖x‖ = 1} is always bounded. Each b ∈ B is regarded as an operator Lb in LA(B) defined by Lb(η(x)) = η(bx) for η(x) ∈ B0. By eA : B → B we denote the projection in LA(B) such that eA(η(x)) = η(E(x)), η(x) ∈ B0. Then the C∗-basic construction C∗(B, eA) is the C∗-subalgebra of LA(B) in which the linear span of elements LbeALb′ (b, b′ ∈ B) is dense. Finite dimensional Hopf ∗-algebras. As in [18], a finite dimensional Hopf ∗- algebra is a finite matrix pseudogroup of [20]. We review from [18] the definition and some basic properties of a finite dimensional Hopf ∗-algebra which we need in the following section. SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 3 Definition 2.1. ([18, Proposition 2.1]) A finite dimensional C∗-algebra is called a finite dimensional Hopf ∗-algebra if there exist three linear maps, ∆ : A → A⊗A, ǫ : A → C, and S : A → A which satisfy the following properties (i) ∆(comultiplication) and ǫ(counit) are ∗-homomorphisms, and S(antipode) is a ∗-preserving antimultiplicative involution, (ii) ∆(1) = 1⊗ 1, ǫ(1) = 1, S(1) = 1, (iii) (∆⊗ id)∆ = (id⊗∆)∆, (iv) (ǫ ⊗ id)∆ = ∆(ǫ⊗ id), (v) m(S⊗id)(∆(a)) = ǫ(a)1 = m(id⊗S)(∆(a)) for a ∈ A, wherem : A⊗A → A is the multiplication. Proposition 2.2. ([18], [20]) Let A be a finite dimensional Hopf ∗-algebra. Then the following properties hold. (i) For a ∈ A, with the notation ∆(a) = aLi ⊗ aRi , we have ǫ(aLi )a i = a = ǫ(aRi )a aLi S(a i ) = ǫ(a)1 = S(aLi )a aRi S(a i ) = ǫ(a)1 = S(aRi )a (ii) There is a unique normalized trace (called the Haar trace) τ on A such that τ(aLi )a i = τ(a)1 = τ(aRi )a i , a ∈ A. (iii) There exists a minimal central projection e ∈ A (called the distinguished projection) such that ae = ǫ(a)e, a ∈ A. We have ǫ(a) = 1, S(e) = e, and τ(e) = (dimA)−1. 3. Actions by finite dimensional Hopf ∗-algebras Throughout this section A will be a finite dimensional Hopf ∗-algebra. An action of A on a unital C∗-algebra M is a bilinear map · : A × M → M such that for a, b ∈ A, x, y ∈ M , 1 · x = x, a · 1 = ǫ(a)1, ab · x = a · (b · x), a · xy = (aLi · x)(aRi · y), (a · x)∗ = S(a∗) · x∗. 4 JA A JEONG AND GI HYUN PARK Then the crossed product M ⋊A is the algebraic tensor product M ⊗A as a vector space with the following multiplication and ∗-operation: (x⊗ a)(y ⊗ b) := x(aLi · y)⊗ aRi b, (x⊗ a)∗ := (aLi ) ∗ · x∗ ⊗ (aRi )∗. Identifying a ∈ A with 1⊗ a and x ∈ M with x⊗ 1, we see [18] that M ⋊A = span{xa | x ∈ M, a ∈ A}. For the definition of saturated action of A on M , refer to section 4 of [18]. Proposition 3.1. ([18]) Let MA = {x ∈ M | a · x = ǫ(a)x, for all a ∈ A} be the fixed point algebra for the action of A on a unital C∗-algebra M . (i) The action is saturated if and only if M ⋊A = span{xey | x, y ∈ M}, where e ∈ A is the distinguished projection. (ii) The map E : M → MA, E(x) = e · x, is a faithful conditional expectation onto the fixed point algebra such that E((a · x)y) = E(x(S(a) · y)), a ∈ A, x, y ∈ M. (iii) The linear map F : M ⋊A → M , F (xa) = τ(a)x, is a faithful conditional expectation onto M . Recall that M0 := M is an MA-valued inner product module by 〈η(x), η(y)〉MA = E(x∗y) (here we use the convention in [19] for the inner product as in section 2). Since every norm bounded MA-module map on M0 extends uniquely to the Hilbert MA- moduleM, we may identify the ∗-algebra End(M0) (in [18]) of norm bounded right MA-module endomorphisms ofM0 having an adjoint with the C∗-algebra LMA(M) explained in section 2. Remark 3.2. ([19, Proposition 1.3.3]) If E : M → MA is of index-finite type, then C∗(M, eMA) = span{LxeMALy | x, y ∈ M} = LMA(M). In fact, we see from the proof of [19, Proposition 2.1.5] that C∗(M, eMA) contains the unit of LMA(M). Thus the ideal span{LxeMALy | x, y ∈ M} which is dense in C∗(M, eMA) must contain the unit of LMA(M). Theorem 3.3. Let A be a finite dimensional Hopf ∗-algebra acting on a unital C∗-algebra M . Then the following are equivalent: (i) The action is saturated. (ii) E : M → MA is of index finite type with Index(E) = (dimA)1. SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 5 Proof. (i)=⇒ (ii) is shown in [18, Proposition 4.5]. (ii)=⇒ (i). By Remark 3.2, C∗(M, eMA) = span{LxeMALy | x, y ∈ M}. Consider a map ϕ : C∗(M, eMA) → M ⋊A given by LxieMALyi) = xieyi. To see that ϕ is well defined, let LxieMALyi = 0. Then for each z ∈ M , LxieMALyi)(η(z)) = η(xiE(yiz)) = η( xi(e · (yiz))) = 0, hence by the injectivity of η ([19, 2.1]), xi(e·(yiz)) = 0 inM . Since (a·x)e = axe for a ∈ A, a ∈ M (see (7) of [18]), we thus have xi(e · (yiz))e = (xieyi)ze = 0 in M ⋊A for every z ∈ M , which then implies that xieyi)(zez ′) = 0, z, z′ ∈ M. Particulary, ( xieyi)( xieyi) ∗ = 0, so that xieyi = 0 (in M ⋊ A). Thus ϕ is well defined. It is tedious to show that ϕ is a ∗-homomorphism such that the range ϕ(C∗(M, eMA)) = MeM is an ideal of M ⋊A; if x, y, and z ∈ M and a ∈ A, (za)(xey) = (z(a · x))ey ∈ MeM. Hence it suffices to show that ϕ(1) = 1. If {(ui, u∗i )}ni=1 is a quasi-basis for E, then LuieMALu∗i (η(z)) = η(uiE(u i z)) = η(z), z ∈ M, which means that i LuieMALu∗i = 1 ∈ LMA(M). Therefore by Proposition 2.2(iii) and Proposition 3.1(iii) F (ϕ(1)) = F ( i ) = τ(e)uiu i = 1. Since ϕ is a ∗-homomorphism, ϕ(1) is a projection in M⋊A such that F (1−ϕ(1)) = 0. But F is faithful, and ϕ(1) = 1 follows. � 4. Actions by finite groups Throughout this section G will denote a finite group. As is well known the group C∗-algebra C∗(G) generated by the unitaries {λg | g ∈ G} is a finite dimensional Hopf ∗-algebra with ∆(λg) = λg ⊗ λg, ǫ(λg) = 1, S(λg) = λg−1 for λg ∈ C∗(G). The Haar trace τ is given by τ(λg) = διg, where ι is the identity of G, and the distinguished projection is e = 1 g λg. Let α be an action of G on a unital C∗-algebra M . Then it is easy to see that λg · x := αg(x) for g ∈ G, x ∈ M, 6 JA A JEONG AND GI HYUN PARK defines an action of C∗(G) on M . Furthermore M ⋊ C∗(G) is nothing but the usual crossed product M ×α G = span{xλg | x ∈ M, g ∈ G}, and the expectations E : M → Mα(= MC∗(G)), F : M ×α G → M of Proposition 3.1 are given by E(x) = αg(x) and F ( xgλg) = xι (x, xg ∈ M, g ∈ G). (1) Note that for each xhλh ∈ M ×α G and g ∈ G, ‖xg‖ = ‖F (( xhλh)λg−1)‖ ≤ ‖( xhλh)λg−1‖ = ‖ xhλh‖. (2) If Jα denotes the closed ideal ofM×αG generated by the distinguished projection e, then Proposition 3.1(i) says that α is saturated if and only if Jα = M ×α G. We will see in Proposition 5.4 that Jα = span{ xαg(y)λg | x, y ∈ M} = span{ xαg(x ∗)λg | x ∈ M}. (3) The ∗-homomorphism ϕ : C∗(M, eMα) → M ×α G we discussed in the proof of Theorem 3.3 can be rewritten as follows. ϕ(LxeMαLy) = xαg(y)λg, x, y ∈ M (4) because ϕ(LxeMαLy) = xey and e = λg. If {(ui, u∗i )} is a quasi-basis for E, we see from LuieMαLu∗i = 1 and (4) that ϕ(1) = uiαg(u i ))λg (5) is a projection in M ×α G. Recall that ϕ(1) = 1 holds if α is saturated. Theorem 4.1. Let M be a unital C∗-algebra and α be an action of a finite group G on M . Then the following are equivalent: (i) α is saturated, that is, Jα = M ×α G. (ii) E : M → Mα is of index finite type with Index(E) = |G|. (iii) E : M → Mα is of index finite type with Index(E) = |G| and uiαg(u i ) = 0, g 6= ι (6) for a quasi-basis {(ui, u∗i )} for E. (iv) There exist {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} for some m ≥ 1 such that (a) αg(b ) = b , for j = 1, . . . ,m and g, h ∈ G. bjg(b )∗ = δgh. (v) For every ε > 0, there exist {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} for some m ≥ 1 such that ‖αg(bjh)− b ‖ < ε, (b) ‖ bjg(b )∗ − δgh‖ < ε. Proof. (i) ⇐⇒ (ii) follows from Theorem 3.3. (i) =⇒ (iii). If {(ui, u∗i )} is a quasi-basis for E, we have from (5) that uiαg(u i ) = 0 for g 6= ι since ϕ(1) = 1. (iii) =⇒ (ii). Obvious. SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 7 (i) =⇒ (iv). Suppose Jα = M ×α G. By (3) there exist m ∈ N and bj ∈ M , 1 ≤ j ≤ m, such that bjαg(b λg = 1. j = 1 and bjαg(b j ) = 0 for g 6= ι. (7) Set bjg := αg(bj). Then ) = αg(αh(bj)) = αgh(bj) = b bjg(b αg(bj)αh(b bjαg−1h(b = δgh by (7). (iv) =⇒ (v). Obvious. (v) =⇒ (i). Let ε > 0 and let {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} satisfy (a) and (b) of (v). Note that (b) implies ‖bjg‖ < 1 + ε for g ∈ G, 1 ≤ j ≤ m. Indeed from bjg(b ∥ ≤ ‖ bjg(b ∗−1‖ < ε, we have ‖bjg‖2 ≤ ‖ bjg(b ∗‖ < 1+ε. αg((b )∗)λg)− |G| ‖ αg((b )∗)λg − |G| ‖ )∗ − |G| ‖+ ‖ g 6=ι αg((b )∗)λg)‖ )∗ − 1 ‖+ g 6=ι αg((b < ε|G|+ g 6=ι αg((b )∗)− (bj g 6=ι |G|+ |G|2 max ‖bjg‖+ |G|2 |G|+ |G|2(1 + ε) + |G|2 Since αg((b )∗)λg) ∈ Jα and ε can be chosen to be arbitrarily small, we conclude that Jα = M ×α G. � Example 4.2. Let w = be a unitary with wn = 1 and define an automorphism α on M2(C)) by α(a) = waw ∗, a ∈ M2(C). We will show that α is saturated if and only if z2 = −z1. For this, recall from (3) that α is saturated if and only if there exist xj ∈ M2(C), 1 ≤ i ≤ m, satisfying k(x∗j )λk = 1M2(C). (8) 8 JA A JEONG AND GI HYUN PARK Hence, particularly for k = 0, 1, we have xjα(x j ) = With xj = aj bj cj dj and zi = e iθi , i = 1, 2, this means |aj |2 + |bj |2 aj c̄j + bj d̄j cj āj + dj b̄j |cj |2 + |dj |2 |aj |2 + ei(θ2−θ1)|bj |2 ei(θ1−θ2)aj c̄j + bj d̄j |aj |2 + ei(θ2−θ1)dj b̄j ei(θ1−θ2)aj c̄j + |dj |2 . (9) Therefore, by comparing (1,1) entries of each matrices, it follows that if α is satu- rated, then there exist positive real numbers a (= |aj |2) > 0, b (= |bj |2) > 0 such that a+ b = 1 and a+ ei(θ2−θ1)b = 0. (10) Note that b 6= 0 since b = 0 implies a = 0 from |aj |2 + ei(θ2−θ1)dj b̄j = 0 in (9). There are three possible cases for θ1, θ2 as follows. (i) If θ2 − θ1 ≡ 0(mod 2π), that is, α is trivial, then (10) is not possible. (ii) If θ2 − θ1 ≡ π(mod 2π), then and x2 = satisfy (8) with m = 2. Thus α is saturated. (iii) If θ2− θ1 6= 0, π(mod 2π), then (10) is not possible for any a, b > 0. Hence α is not saturated. Remark 4.3. Let α, β ∈ Aut(M) satisfy αn = βn = idM for some n ≥ 1. If there is a unitary u ∈ M such that β = Ad(u) ◦ α, then α and β are said to be exterior equivalent, and if this is the case the crossed products are isomorphic, M×αG ∼= M×βG, [14, p.45]. Example 4.2 says that the property of being saturated may not be preserved under exterior equivalence. Also the case (iii) of Example 4.2 above with w = diag(λ, λ̄), λ = e 3 (hence θ1− θ2 = 2π3 − 6= π(mod 2π)), shows that Index(E) < |G| is possible even when E is of index-finite type. In fact, if and u2 = , then {(ui, u∗i )}2i=1 forms a quasi-basis for E, but Index(E) = 2 < |Z3|. Remark 4.4. Recall that the Rokhlin property and the tracial Rokhlin property (weaker than the Rokhlin property) are defined as follows and considered intensively in [7] and [12], respectively: (a) ([7]) α is said to have the Rokhlin property if for every finite set F ⊂ M , every ε > 0, there are mutually orthogonal projections {eg | g ∈ G} in M such that (i) ‖αg(eh)− egh‖ < ε for g, h ∈ G. (ii) ‖egx− xeg‖ < ε for g ∈ G and all x ∈ F . (iii) g∈G eg = 1. SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 9 (b) ([12]) α is said to have the tracial Rokhlin property if for every finite set F ⊂ M , every ε > 0, every n ∈ N, and every nonzero positive element x ∈ M , there are mutually orthogonal projections {eg | g ∈ G} in M such that: (i) ‖αg(eh)− egh‖ < ε for g, h ∈ G. (ii) ‖egx− xeg‖ < ε for g ∈ G and all x ∈ F . (iii) With e = g∈G eg, the projection 1 − e is Murray-von Neumann equivalent to a projection in the hereditary subalgebra of M generated by x. The following proposition is actually observed in [12, Lemma 1.13], and we put a proof for reader’s convenience. Proposition 4.5. Let M be a unital C∗-algebra and α be an action of a discrete group G on M . Suppose that for every ε > 0 and every finite subset F ⊂ M , there exist a family of projections {eg}g∈G such that (1) ‖αg(eh)− egh‖ < ε. (2) ‖egx− xeg‖ < ε for each x ∈ F . Then α is an outer action. Proof. Suppose there is a unitary u ∈ M such that αg(x) = uxu∗ for every x ∈ M (g 6= ι). Put F = {u} and 0 < ε < 1/4. Then there exist mutually orthogonal projections {eg}g∈G such that ‖αg(eh)− egh‖ < ε < 1/4. Thus ‖egu− ueg‖ < ε < 1/4. Then ‖αg(eι)− ueιu∗‖ = 0. But ‖αg(eι)− ueιu∗‖ = ‖αg(eι)− eg + eg − eι + eι − ueιu∗‖ ≥ ‖eg − eι‖ − ‖αg(eι)− eg‖ − ‖eι − ueι1u∗‖ ≥ 1− 1 which is a contradiction. � Remark 4.6. If M ×αG is simple, α is obviously saturated, and this is the case if G is a finite group, M is α-simple, and T̃(αg) 6= {1} for all g 6= ι ([8, Theorem 3.1]). In particular, α is saturated if M is simple and α is outer. But for a nonsimple M , this may not hold. In fact, if α is an outer action of Zn on M and u is a unitary in M with un = 1 such that the action Ad(u) on M is not saturated (as in Example 4.2), then the action α ⊕ Ad(u) on M ⊕M is outer but not saturated. Now we show that if α satisfies the Rokhlin property (or satisfies the tracial Rokhlin property and M has cancellation) then α is saturated. For this we first review the cancellation property of C∗-algebras. For projections p, q in a C∗-algebra, we write p ⊥ q if pq = 0, and p ∼ q if they are Murrey-von Neumann equivalent. Definition 4.7. A unital C∗-algebra M has the cancellation if, whenever p, q, r are projections in Mn(M) for some n, with p ⊥ r, q ⊥ r, and (p + r) ∼ (q + r), then p ∼ q. 10 JA A JEONG AND GI HYUN PARK Remark 4.8. (1) If M has the cancellation and p, q are projections in M such that (1− p) ∼ (1− q), then p ∼ q ([4, V.2.4.14]). (2) It is well known that every C∗-algebra with stable rank one has the cancel- lation ([4, V.3.1.24]). Proposition 4.9. Let α be an action of a finite group G on a unital C∗-algebra M . Then α is saturated if one of the following holds. (i) α has the Rokhlin property. (ii) α has the tracial Rokhlin property and M has the cancellation. Proof. (i) For an ε > 0, there exist mutually orthogonal projections {eg}g such that g eg = 1 and ‖αg(eh)− egh‖ < ε. Then, with m = 1, the elements b1g := eg satisfy (v) of Theorem 4.1. (ii) Now suppose α has the tracial Rokhlin property and M has the cancellation. We shall show that Jα contains the unit of M ×α G. Let 0 < ε < 1. For each g ∈ G, choose mutually orthogonal projections {eg }h∈G such that ‖αk(egh)− e ‖ < ε 2|G|2 , and put eg := h∈G e . If eg = 1, for some g, then b1h := e (h ∈ G) will satisfy (v) of Theorem 4.1 as in (i). If eg 6= 1 for every g ∈ G, then by the tracial Rokhlin property of α there exist mutually orthogonal projections {fg }h∈G in M such that ‖αk(fgh)− f ‖ < ε 2|G|2 ) ∼ (eg)′ < eg for a subprojection (eg)′ of eg. Put fg := . Then since M has cancellation, it follows that fg ∼ 1−(eg)′ > (1−eg). Let vg ∈ M be a partial isometry satisfying v∗gvg = f g, vgv g = 1− (eg)′, and set xg := (ekhαg(e h) + (1 − ek)vkfkhαg(fkhαg−1 (v∗k)) , g ∈ G. Now we show that the element x := xgλg ∈ Jα satisfies ‖x − 1‖ < ε. In fact, for g 6= ι, ‖xg‖ ≤ ‖(ekhαg(ekh) + (1− ek)vkfkhαg(fkhαg−1(v∗k))‖ ≤ 1|G| ‖(ekhαg(ekh)‖ + ‖fkhαg(fkh )‖ ≤ 1|G| ‖ekh(αg(ekh)− ekgh)‖ + ‖ekhekgh‖+ ‖fkh (αg(fkh )− fkgh)‖+ ‖fkhfkgh‖ ≤ 1|G| 2|G|2 + 2|G|2 ) SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 11 |G| ( (1− ek)vkfkv∗k) (1 − ek)(1 − (ek)′) For the rest of this section we consider a finite group action on a commutative C∗-algebra C(X). If G acts on a compact Hausdorff space X , it induces an action, say α, on C(X) by αg(f)(x) = f(g −1x), f ∈ C(X). For each x ∈ X , let Gx = {g ∈ G : gx = x} be the isotropy group of x and for a subgroup H of G (H < G), put XH = {x ∈ X : Gx = H}. It is readily seen that XH and XH′ are disjoint if H 6= H ′, and X is partitioned as Theorem 4.10. Let X be a compact Hausdorff space and G a finite group acting on X. If α is the induced action of G on C(X), the following are equivalent: (i) E : C(X) → C(X)α is of index finite type. (ii) XH is closed for each H < G. Moreover, if this is case the index of E is Index(E) = χXH , where χXH is the characteristic function on XH . Proof. (i) =⇒ (ii). If E is of index-finite type and {(ui, u∗i )}ki=1 is a quasi-basis for E, then uiE(u i f) = f, that is, ui(x) u∗i (g −1x)f(g−1x) = f(x), (11) for f ∈ C(X) and x ∈ X . For each x ∈ X , choose a continuous function fx ∈ C(X) satisfying fx|Gx\{x} ≡ 0 and fx(x) = 1. Then (11) with fx in place of f gives ui(x) u∗i (g −1x)fx(g = fx(x), (12) and so we have ui(x)u i (x) = 1. (13) 12 JA A JEONG AND GI HYUN PARK To show that each XH is closed, let {xn ∈ XH : n = 1, 2, . . .} be a sequence of elements in XH with limit x ∈ XH′ . Then (11) gives fx(xn) = ui(xn) u∗i (g −1xn)fx(g −1xn) ui(xn) u∗i (g −1xn)fx(g −1xn) + g 6∈H u∗i (g −1xn)fx(g −1xn) ui(xn) |H |u∗i (xn)fx(xn) + g 6∈H u∗i (g −1xn)fx(g −1xn) Taking the limit as n → ∞, we have fx(x) = ui(x) |H |u∗i (x)fx(x) + g 6∈H u∗i (g −1x)fx(g ui(x) |H |u∗i (x)fx(x) + |H ′ \H |u∗i (x)fx(x) |H |+ |H ′ \H | ui(x)u i (x)fx(x). Therefore, comparing with (13), we obtain |H ′| = |H |+ |H ′ \H | since Gx = H ′ and fx(x) = 1. Hence H ⊂ H ′. On the other hand, since Gxn = H , again by (13), ui(xn)u i (xn) = 1 with the limit ui(x)u i (x) = 1 as n → ∞. But also ui(x)u i (x) = 1 by (13), and thus |H | = |H ′|. Consequently we have H = H ′ because H ⊂ H ′. This shows that XH is closed. (ii) =⇒ (i). Assume that XH is closed for every subgroup H of G. Then XH is open since there are only finitely many such subsets. Let UH = {UH,iH : iH = 1, 2, . . . , nH} be an open covering of XH such that x ∈ UH,iH =⇒ g−1x 6∈ UH,iH or g−1x 6∈ XH whenever g−1x 6= x. Let {vH,iH} be a partition of unity subordinate to UH . We understand that the domain of vH,iH is X by assigning 0 to x 6∈ XH . Let uH,iH = vH,iH . We claim that |H | uH,iH , |H | u : H < G, iH = 1, 2, . . . , nH SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 13 is a quasi-basis for E. For f ∈ C(X) and x ∈ X , let F < G and 1 ≤ j ≤ nF be such that x ∈ XF and x ∈ UF,j. Then |H | uH,iHE |H | u |H |uH,iH (x) |H |u (g−1x)f(g−1x) |F |uF,iF (x) u∗F,iF (g −1x)f(g−1x) uF,iF (x) u∗F,iF (g −1x)f(g−1x) uF,iF (x)|F |u∗F,iF (x)f(x) vF,iF (x)f(x) = f(x), as claimed. Recall that an action G on X is free if gx 6= x for g ∈ G, g 6= 1, and x ∈ X Corollary 4.11. Let X be a compact Hausdorff space and G a finite group acting on X. If α is the induced action on C(X), the following are equivalent. (i) G acts freely on X. (ii) E : C(X) → C(X)α is of index-finite type with Index(E) = |G|. (iii) α is saturated. Proof. (i) =⇒ (ii) is proved in [19, Proposition 2.8.1]. To show (ii) =⇒ (i), let E be of index-finite type with Index(E) = |G|. Then from Theorem 4.10, we have |G| = Index(E) = |H |χXH , which implies that H = {ι} is the only subgroup of G such that XH 6= ∅. Hence X = X{ι}, that is, G acts freely on X . (ii) ⇐⇒ (iii) comes from Theorem 4.1. � 5. Saturated actions by compact groups Notation 5.1. Let M be a C∗-algebra and α be an action of a compact group G on M . For x, y ∈ M , define continuous functions fx,y, fx,1, f1,y ∈ C(G,M) from G to 14 JA A JEONG AND GI HYUN PARK M as follows: fx,y(t) = xαt(y), fx,1(t) = x, f1,y(t) = αt(y) for t ∈ G. Then it is easily checked that fx,y = fx,1 ∗ f1,y and f∗x,y = fy∗,x∗ . Recall that C(G,M) is a dense ∗-subalgebra of M ×α G with the multiplication and involution defined by f ∗ g(t) = f(s)αs(g(s −1t))ds, f∗(t) = αt(f(t −1)∗), where dg is the normalized Haar measure ([13, 7.7], [6, 8.3.1]). Hence if G is a finite group, fx,y can be written as fx,y = xαg(y)λg . If M̃ denotes the smallest unitization of M (so M̃ = M if M is unital), the function e : G → M̃, e(s) = 1, for every s ∈ G is a projection of the multiplier algebra of M ×α G ([17]). Proposition 5.2. ([17]) Let α be an action of a compact group G on a C∗-algebra M . Then identifying x ∈ Mα and the constant function in C(G,M) with the value x everywhere we see that x 7→ fx,1 : Mα → e(M ×α G)e is an isomorphism of Mα onto the hereditary subalgebra e(M×αG)e of the crossed product M ×α G. The notion of saturated action is introduced by Rieffel for a compact group action on a C∗-algebra, and we adopt the following equivalent condition as the definition. Definition 5.3. (Rieffel, see [14, 7.1.9 Lemma]) Let M be a C∗-algebra and α be an action of compact group G on M . α is said to be saturated if the linear span of {fa,b | a, b ∈ M} is dense in M ×α G (see Notation 5.1). We denote Jα = span{fa,b | a, b ∈ M}. Proposition 5.4. Let α be an action of a compact group G on a C∗-algebra M . Then Jα is the ideal of M×αG generated by the hereditary subalgebra e(M ×αG)e. Moreover Jα = span{fa,a∗ ∈ C(G,M) | a ∈ M}. Proof. We first show that Jα is an ideal ofM×αG. Let x ∈ C(G,M) and a, b ∈ M . Then x ∗ fa,b ∈ Jα. Indeed, (x ∗ fa,b)(t) = x(s)αs(fa,b(s −1t))ds x(s)αs(a)αt(b)ds x(s)αs(a)ds αt(b), SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 15 hence x∗fa,b = fc,b ∈ Jα, where c = x(s)αs(a)ds ∈ M . Also f∗a,b = fb∗,a∗ implies that Jα = Jα∗ is an ideal of M ×α G. Let J := (M ×α G)e(M ×α G) be the closed ideal generated by e(M ×α G)e. Now we show that Jα ⊂ J . From (fa,b ∗ e)(t) = fa,b(s)αs(e(s −1t)) ds = aαs(b) ds = a αs(b)ds, we have fa,b ∗ e = faE(b),1, where E(b) = αs(b)ds ∈ Mα. Hence for a, b, c, and d in M , we have fa,b ∗ e ∗ fc,d = (fa,b ∗ e) ∗ (fd∗,c∗ ∗ e)∗ = faE(b),1 ∗ (fd∗E(c∗),1)∗ = faE(b),1 ∗ f1,E(c∗)d = faE(b),E(c∗)d, which means that fax,yd ∈ J for any a, d ∈ M and x, y ∈ Mα. Since Mα contains an approximate identity for M , it follows that fa,b ∈ J for a, b ∈ A. For the converse inclusion J ⊂ Jα, note that if x ∈ C(G,M), then (x ∗ e)(t) = x(s)ds for t ∈ G. With notations x′ = x(s)ds and x′′ := αs(x(s −1))ds (∈ M), we see that (x ∗ e ∗ y)(t) = (x ∗ e)(s)αs(y(s−1t))ds αs(y(s −1t))ds = x′αt( αs(y(s −1))ds) = x′αt(y = fx′,y′′(t) belongs to Jα for x, y ∈ C(G,M). Finally the following polarization identity proves the last assertion. aαt(b) = ik(b + ika∗)∗αt(b+ i ka∗). 6. The gauge action γ on a graph C∗-algebra By a (directed) graph E we mean a quadruple E = (E0, E1, r, s) consisting of the vertex set E0, the edge set E1, and the range, source maps r, s : E1 → E0. If each vertex of E emits only finitely many edges E is called row finite and a row finite graph E is locally finite if each vertex receives only finitely many edges. By En we denote the set of all finite paths α = e1 · · · en (r(ei) = s(ei+1), 1 ≤ i ≤ n−1) of length n (|α| = n). Each vertex is regarded as a finite path of length 0. Then E∗ = ∪n≥0En is the set of all finite paths and the maps r and s naturally extend to E∗. A vertex v is called a sink if s−1(v) = ∅ and a source if r−1(v) = ∅. If E is a row finite graph, we call a family {se, pv | e ∈ E1, v ∈ E0} of operators a Cuntz-Krieger(CK) E-family if {se}e are partial isometries and {pv}v are mutually 16 JA A JEONG AND GI HYUN PARK orthogonal projections such that s∗ese = pr(e) and pv = s(e)=v e if s −1(v) 6= ∅. It is now well known that there exists a C∗-algebra C∗(E) generated by a universal CK E-family {se, pv | e ∈ E1, v ∈ E0}, in this case we simply write C∗(E) = C∗(se, pv). For the definition and basic properties of graph C ∗-algebras, see, for example, [1, 2, 10, 11, 15] among others. If α = α1α2 · · ·α|α| (αi ∈ E1) is a finite path, by sα we denote the partial isometry sα1sα2 · · · sα|α| (sv = s∗v = pv, for v ∈ We will consider only locally finite graphs and it is helpful to note the following properties of graph C∗-algebras. Remark 6.1. (i) Let C∗(E) = C∗(se, pv) be the graph C ∗-algebra associated with a row finite graph E, and let α, β ∈ E∗ be finite paths in E. Then s∗αsβ = s∗µ, if α = βµ sν , if β = αν 0, otherwise. Therefore C∗(E) = span{sαs∗β | α, β ∈ E∗}. (ii) Note that sαs β = 0 for α, β ∈ E∗ with r(α) 6= r(β). (iii) If α, β, µ, and ν in En are the paths of same length, β)(sµs ν) = δβ,µsαs Thus for each n ∈ N and a vertex v in a locally finite graph E, we see that span{sαs∗β | α, β ∈ En and r(α) = r(β) = v} is a ∗-algebra which is isomorphic to the full matrix algebraMm = (Mm(C)), where m = ∣{α ∈ En | r(α) = v} Recall that the gauge action γ of T on C∗(E) = C∗(se, pv) is given by γz(se) = zse, γz(pv) = pv, z ∈ T. γ is well defined by the universal property of the CK E-family {se, pv}. Since γz(sαs β)dz = z|α|−|β|(sαs β)dz = 0, |α| 6= |β|, one sees that C∗(E)γ = span{sαs∗β | α, β ∈ E∗, |α| = |β|}. If Z denotes the following graph: • • • • •______ // ______ // ______ //______ // · · · ,· · ·Z : −2 −1 0 1 2 then C∗(Z) is isomorphic to the C∗-algebra K of compact operators on an infinite dimensional separable Hilbert space, hence C∗(Z) is itself a simple AF algebra. But C∗(Z)γ coincides with the commutative subalgebra span{sαs∗α | α ∈ Z∗} which is far from being simple, and thus we know that the simplicity of C∗(E) does not imply that of C∗(E)γ in general. SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 17 In [9], the Cartesian product of two graphs E and F is defined to be the graph E× F = (E0 ×F 0, E1 ×F 1, r, s), where r(e, f) = (r(e), r(f)) and s(e, f) = (s(e), s(f)). Since the graph Z × E has no loops for any row-finite graph E, we know that C∗(E)γ is an AF algebra ([10]) by the following proposition. Proposition 6.2. ([9]) Let E be a row finite graph with no sources. Then the following hold: (a) C∗(E)γ is stably isomorphic to C∗(E)×γ T. (b) C∗(E)×γ T ∼= C∗(Z× E). Now we show that a gauge action is saturated. For this, note that the linear span of the continuous functions of the form t 7→ f(t)x, f ∈ C(G), x ∈ A is dense in C(G,A) [13, 7.6.1]. Hence by Remark 6.1(i), one sees that C∗(E)×γ T = span{znsαs∗β | α, β ∈ E∗ n ∈ Z}. (15) Theorem 6.3. Let E be a locally finite graph with no sinks and no sources. Then the gauge action γ on C∗(E) is saturated. Proof. We show that Jγ = C∗(E)×γ T. By (15) it suffices to see that znsαs β ∈ Jγ for all α, β ∈ E∗, n ≥ 0 (because z−nsαs β = (z ∗ for n ≥ 0). Now fix α, β ∈ E∗ and n ≥ 0. Put l = n− (|α| − |β|). There are two cases. (i) l ≥ 0: One can choose a path µ such that |µ| = l and r(µ) = s(α). Then znsαs β = z l+|α|−|β|s∗µsµsαs β = s µγz(sµαs β) = fs∗µ,sµαs∗β (z), where the function fs∗µ,sµαs∗β belongs to Jγ . (ii) l < 0: Choose a path ν with |ν| = |β|+n and r(ν) = r(α). With a = sαs∗ν , b = sνs β , we have fa,b ∈ Jγ and znsαs β = sαs νγz(sνs β) = fa,b(z). Acknowledgements. The first author would like to thank Hiroyuki Osaka and Ta- motsu Teruya for valuable discussions. References [1] T. Bates, J. H. Hong, I. Raeburn and W. Szymanski, The ideal structure of the C∗-algebras of infinite graphs, Illinois J. Math., 46(2002), 1159–1176. [2] T. Bates, D. Pask, I. Raeburn, and W. Szymanski, The C∗-algebras of row-finite graphs, New York J. Math. 6(2000), 307–324. [3] T. Bates and D. Pask, Flow equivalence of graph algebras, Ergod. Th. & Dynam. Sys. 24(2004), 367–382. [4] B. Blakadar, Operator Algebras, Theory of C∗-algebras and von Neumann algebras, Encyclo- pedia of mathematical sciences 122, Springer, 2006. [5] J. Cuntz and W. Krieger, A class of C∗-algebras and topological Markov chains, Invent. Math. 56(1980), 251–268. [6] P. A. Fillmore, A user’s guide to operator algebras, Canadian Math. Soc. Series of Monographs and Adv. Texts, John Wiley & Sons, Inc. 1996. 18 JA A JEONG AND GI HYUN PARK [7] M. Izumi, Finite group actions on C∗-algebras with the Rohlin property, I, Duke Math. J. 122(2004), no. 2, 233–280. [8] A. Kishimoto, Outer automorphisms and related crossed products of simple C∗-algebras, Comm. Math. Phys. 81(1981), no. 1, 429–435. [9] A. Kumjian and D. Pask, C∗-algebras of directed graphs and group actions, Ergodic Theory Dynam. Systems, 19(1999), 1503–1519. [10] A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184(1998), 161–174. [11] A. Kumjian, D. Pask, I. Raeburn and J. Renault Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144(1997), 505–541. [12] H. Osaka and N. C. Phillips, Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property, Ergod. Th. & Dyn. Sys., to appear. [13] G. K. Pedersen, C∗-algebras and their automorphism groups, London Math. Soc. Monographs No.14, Academic Press 1979. [14] N. C. Phillips, Equivariant K-theory and freness of group actions on C∗-algebras, LNM 1274, Springer, 1987. [15] I. Raeburn, Graph algebras, CBMS. 103, 2005, AMS. [16] J. Renault, A groupoid approach to C∗-algebras, LNM 794, Springer, Berlin, 1980. [17] J. Rosenberg, Appendix to O. Bratteli’s paper on ”crossed products of UHF algebras, Duke Math. J. 46(1979), 25–26. [18] W. Szymański and C. Peligrad, Saturated actions of finite dimensional Hopf ∗-algebras on C∗-algebras, Math. Scand. 75(1994), 217–239. [19] Y. Watatani, Index for C∗-subalgebras, Memoirs of the Amer. Math. Soc. 424, AMS, 1990. [20] S. L. Woronowicz, Compact matrix psedogroups, Comm. Math. Phys. 111(1987), 613–665. Keywords: Finite dimensional Hopf ∗-algebra; saturated action; conditional ex- pectation of index-finite type. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 151–747, Korea E-mail address: jajeong@snu.ac.kr Department of Mathematics, Hanshin University, Osan, 447–791, Korea E-mail address: ghpark@hanshin.ac.kr 1. Introduction 2. Preliminaries 3. Actions by finite dimensional Hopf *-algebras 4. Actions by finite groups 5. Saturated actions by compact groups 6. The gauge action on a graph C*-algebra References

If a finite group action $\alpha$ on a unital $C^*$-algebra $M$ is saturated, the canonical conditional expectation $E:M\to M^\alpha$ onto the fixed point algebra is known to be of index finite type with $Index(E)=|G|$ in the sense of Watatani. More generally if a finite dimensional Hopf $*$-algebra $A$ acts on $M$ and the action is saturated, the same is true with $Index (E)=\dim(A)$. In this paper we prove that the converse is true. Especially in case $M$ is a commutative $C^*$-algebra $C(X)$ and $\alpha$ is a finite group action, we give an equivalent condition in order that the expectation $E:C(X)\to C(X)^\alpha$ is of index finite type, from which we obtain that $\alpha$ is saturated if and only if $G$ acts freely on $X$. Actions by compact groups are also considered to show that the gauge action $\gamma$ on a graph $C^*$-algebra $C^*(E)$ associated with a locally finite directed graph $E$ is saturated.

Introduction It is known [17] that if α is an action by a compact group G on a C∗-algebra M , the fixed point algebra Mα is isomorphic to a hereditary subalgebra e(M ×αG)e of the crossed product M ×αG for a projection e in the multiplier algebra of M ×αG. If e(M ×α G)e is full in M ×α G (that is, e(M ×α G)e generates M ×α G as a closed two-sided ideal), the action α is said to be saturated (the notion of saturated action was introduced by Rieffel [14, Chap.7]). Every action α with a simple crossed product M ×α G is obviously saturated. On the other hand, an action of a finite dimensional Hopf ∗-algebra A on a unital C∗-algebra M is considered in [18] and it is shown that if the action is saturated, the canonical conditional expectation E : M → MA onto the fixed point algebra MA is of index finite type in the sense of Watatani [19] and Index(E) = (dimA)1. The main purpose of the present paper is to prove that the converse is also true. We see from our result that for an action α by a finite group G, α is saturated if and only if the canonical expectation E : M → Mα is of index-finite type with index Index(E) = |G|. Besides, we consider actions by compact groups to study the saturation property of a gauge action γ on a C∗-algebra C∗(E) associated with a locally finite directed graph E with no sinks or sources. This paper is organized as follows. In section 2, we review the C∗-basic construction from [19] and finite dimensional Hopf ∗-algebras from [18] setting up our notations. Then we prove in section 3 that if A is a finite dimensional Hopf ∗-algebra acting on a unital C∗-algebra M such that E : M → MA is of index finite type with Index(E) = (dimA)1, then the action is saturated (Theorem 3.3). In section 4, we deal with the crossed product M ×α G by a finite group in detail and give other equivalent conditions in order that α be saturated. From the Research supported by KRF-ABRL-R14(2003-2008)† and Hanshin University Research Grant‡. http://arxiv.org/abs/0704.1549v1 2 JA A JEONG AND GI HYUN PARK conditions one easily see that an action with the Rokhlin property [7] is always saturated. Also we shall show that if M has the cancellation, an action with the tracial Rokhlin property [12] on M is saturated. Note that even for an action α by the finite group Z2, the expectation E : M → Mα may not be of index finite type in general [19, Example 2.8.4]. For a commutative C∗-algebra C(X) and a finite group action α, we give a necessary and sufficient condition that E : C(X) → C(X)α is of index finite type (Theorem 4.10) and provide a formula for Index(E). Then as a corollary we obtain that α is saturated if and only if G acts freely on X . In section 5, we consider a compact group action α and investigate the ideal Jα of M ×α G generated by the hereditary subalgebra e(M ×α G)e. Then we apply the result on Jα to the gauge action on a graph C∗-algebra in section 6. As a generalization of the Cuntz-Krieger algebras [5], the class of graph C∗-algebras C∗(E) associated with directed graphs E has been studied in various directions by considerably many authors (for example see the bibliography in the book [15] by Raeburn). In [9], Kumjian and Pask show among others that if γ is the gauge action on C∗(E), then C∗(E)γ is stably isomorphic to the crossed product C∗(E) ×γ T, which was done by hiring the notions of skew product of graphs and groupoid C∗- algebras. In Theorem 6.3 we shall directly show that the gauge action is actually saturated (this implies that C∗(E)γ and C∗(E)×γ T are stably isomorphic). 2. Preliminaries Watatani’s index theory for C∗-algebras. In [19], Watatani developed the index theory for C∗-algebras, and here we briefly review the basic construction C∗(B, eA). Let B be a C ∗-algebra and A its C∗-subalgebra containing the unit of B. Let E : B → A be a faithful conditional expectation. If there exist finitely many elements {vi}ni=1 in B satisfying the following E(bvi)v viE(v i b), for every b ∈ B, E is said to be of index-finite type and {(vi, v∗i )}ni=1 is called a quasi-basis for E. The positive element i is then the index of E, Index(E), which is known to be an element in the center of B and does not depend on the choice of quasi-bases for E ([19, Proposition 1.2.8]). Let B be the completion of the pre-Hilbert module B0 = {η(b) | b ∈ B} over A with an A-valued inner product 〈η(x), η(y)〉 = E(x∗y), η(x), η(y) ∈ B0. Let LA(B) be the C∗-algebra of all (right) A-module homomorphisms on B with adjoints. For T ∈ LA(B), the norm ‖T ‖ = sup{‖Tx‖ : ‖x‖ = 1} is always bounded. Each b ∈ B is regarded as an operator Lb in LA(B) defined by Lb(η(x)) = η(bx) for η(x) ∈ B0. By eA : B → B we denote the projection in LA(B) such that eA(η(x)) = η(E(x)), η(x) ∈ B0. Then the C∗-basic construction C∗(B, eA) is the C∗-subalgebra of LA(B) in which the linear span of elements LbeALb′ (b, b′ ∈ B) is dense. Finite dimensional Hopf ∗-algebras. As in [18], a finite dimensional Hopf ∗- algebra is a finite matrix pseudogroup of [20]. We review from [18] the definition and some basic properties of a finite dimensional Hopf ∗-algebra which we need in the following section. SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 3 Definition 2.1. ([18, Proposition 2.1]) A finite dimensional C∗-algebra is called a finite dimensional Hopf ∗-algebra if there exist three linear maps, ∆ : A → A⊗A, ǫ : A → C, and S : A → A which satisfy the following properties (i) ∆(comultiplication) and ǫ(counit) are ∗-homomorphisms, and S(antipode) is a ∗-preserving antimultiplicative involution, (ii) ∆(1) = 1⊗ 1, ǫ(1) = 1, S(1) = 1, (iii) (∆⊗ id)∆ = (id⊗∆)∆, (iv) (ǫ ⊗ id)∆ = ∆(ǫ⊗ id), (v) m(S⊗id)(∆(a)) = ǫ(a)1 = m(id⊗S)(∆(a)) for a ∈ A, wherem : A⊗A → A is the multiplication. Proposition 2.2. ([18], [20]) Let A be a finite dimensional Hopf ∗-algebra. Then the following properties hold. (i) For a ∈ A, with the notation ∆(a) = aLi ⊗ aRi , we have ǫ(aLi )a i = a = ǫ(aRi )a aLi S(a i ) = ǫ(a)1 = S(aLi )a aRi S(a i ) = ǫ(a)1 = S(aRi )a (ii) There is a unique normalized trace (called the Haar trace) τ on A such that τ(aLi )a i = τ(a)1 = τ(aRi )a i , a ∈ A. (iii) There exists a minimal central projection e ∈ A (called the distinguished projection) such that ae = ǫ(a)e, a ∈ A. We have ǫ(a) = 1, S(e) = e, and τ(e) = (dimA)−1. 3. Actions by finite dimensional Hopf ∗-algebras Throughout this section A will be a finite dimensional Hopf ∗-algebra. An action of A on a unital C∗-algebra M is a bilinear map · : A × M → M such that for a, b ∈ A, x, y ∈ M , 1 · x = x, a · 1 = ǫ(a)1, ab · x = a · (b · x), a · xy = (aLi · x)(aRi · y), (a · x)∗ = S(a∗) · x∗. 4 JA A JEONG AND GI HYUN PARK Then the crossed product M ⋊A is the algebraic tensor product M ⊗A as a vector space with the following multiplication and ∗-operation: (x⊗ a)(y ⊗ b) := x(aLi · y)⊗ aRi b, (x⊗ a)∗ := (aLi ) ∗ · x∗ ⊗ (aRi )∗. Identifying a ∈ A with 1⊗ a and x ∈ M with x⊗ 1, we see [18] that M ⋊A = span{xa | x ∈ M, a ∈ A}. For the definition of saturated action of A on M , refer to section 4 of [18]. Proposition 3.1. ([18]) Let MA = {x ∈ M | a · x = ǫ(a)x, for all a ∈ A} be the fixed point algebra for the action of A on a unital C∗-algebra M . (i) The action is saturated if and only if M ⋊A = span{xey | x, y ∈ M}, where e ∈ A is the distinguished projection. (ii) The map E : M → MA, E(x) = e · x, is a faithful conditional expectation onto the fixed point algebra such that E((a · x)y) = E(x(S(a) · y)), a ∈ A, x, y ∈ M. (iii) The linear map F : M ⋊A → M , F (xa) = τ(a)x, is a faithful conditional expectation onto M . Recall that M0 := M is an MA-valued inner product module by 〈η(x), η(y)〉MA = E(x∗y) (here we use the convention in [19] for the inner product as in section 2). Since every norm bounded MA-module map on M0 extends uniquely to the Hilbert MA- moduleM, we may identify the ∗-algebra End(M0) (in [18]) of norm bounded right MA-module endomorphisms ofM0 having an adjoint with the C∗-algebra LMA(M) explained in section 2. Remark 3.2. ([19, Proposition 1.3.3]) If E : M → MA is of index-finite type, then C∗(M, eMA) = span{LxeMALy | x, y ∈ M} = LMA(M). In fact, we see from the proof of [19, Proposition 2.1.5] that C∗(M, eMA) contains the unit of LMA(M). Thus the ideal span{LxeMALy | x, y ∈ M} which is dense in C∗(M, eMA) must contain the unit of LMA(M). Theorem 3.3. Let A be a finite dimensional Hopf ∗-algebra acting on a unital C∗-algebra M . Then the following are equivalent: (i) The action is saturated. (ii) E : M → MA is of index finite type with Index(E) = (dimA)1. SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 5 Proof. (i)=⇒ (ii) is shown in [18, Proposition 4.5]. (ii)=⇒ (i). By Remark 3.2, C∗(M, eMA) = span{LxeMALy | x, y ∈ M}. Consider a map ϕ : C∗(M, eMA) → M ⋊A given by LxieMALyi) = xieyi. To see that ϕ is well defined, let LxieMALyi = 0. Then for each z ∈ M , LxieMALyi)(η(z)) = η(xiE(yiz)) = η( xi(e · (yiz))) = 0, hence by the injectivity of η ([19, 2.1]), xi(e·(yiz)) = 0 inM . Since (a·x)e = axe for a ∈ A, a ∈ M (see (7) of [18]), we thus have xi(e · (yiz))e = (xieyi)ze = 0 in M ⋊A for every z ∈ M , which then implies that xieyi)(zez ′) = 0, z, z′ ∈ M. Particulary, ( xieyi)( xieyi) ∗ = 0, so that xieyi = 0 (in M ⋊ A). Thus ϕ is well defined. It is tedious to show that ϕ is a ∗-homomorphism such that the range ϕ(C∗(M, eMA)) = MeM is an ideal of M ⋊A; if x, y, and z ∈ M and a ∈ A, (za)(xey) = (z(a · x))ey ∈ MeM. Hence it suffices to show that ϕ(1) = 1. If {(ui, u∗i )}ni=1 is a quasi-basis for E, then LuieMALu∗i (η(z)) = η(uiE(u i z)) = η(z), z ∈ M, which means that i LuieMALu∗i = 1 ∈ LMA(M). Therefore by Proposition 2.2(iii) and Proposition 3.1(iii) F (ϕ(1)) = F ( i ) = τ(e)uiu i = 1. Since ϕ is a ∗-homomorphism, ϕ(1) is a projection in M⋊A such that F (1−ϕ(1)) = 0. But F is faithful, and ϕ(1) = 1 follows. � 4. Actions by finite groups Throughout this section G will denote a finite group. As is well known the group C∗-algebra C∗(G) generated by the unitaries {λg | g ∈ G} is a finite dimensional Hopf ∗-algebra with ∆(λg) = λg ⊗ λg, ǫ(λg) = 1, S(λg) = λg−1 for λg ∈ C∗(G). The Haar trace τ is given by τ(λg) = διg, where ι is the identity of G, and the distinguished projection is e = 1 g λg. Let α be an action of G on a unital C∗-algebra M . Then it is easy to see that λg · x := αg(x) for g ∈ G, x ∈ M, 6 JA A JEONG AND GI HYUN PARK defines an action of C∗(G) on M . Furthermore M ⋊ C∗(G) is nothing but the usual crossed product M ×α G = span{xλg | x ∈ M, g ∈ G}, and the expectations E : M → Mα(= MC∗(G)), F : M ×α G → M of Proposition 3.1 are given by E(x) = αg(x) and F ( xgλg) = xι (x, xg ∈ M, g ∈ G). (1) Note that for each xhλh ∈ M ×α G and g ∈ G, ‖xg‖ = ‖F (( xhλh)λg−1)‖ ≤ ‖( xhλh)λg−1‖ = ‖ xhλh‖. (2) If Jα denotes the closed ideal ofM×αG generated by the distinguished projection e, then Proposition 3.1(i) says that α is saturated if and only if Jα = M ×α G. We will see in Proposition 5.4 that Jα = span{ xαg(y)λg | x, y ∈ M} = span{ xαg(x ∗)λg | x ∈ M}. (3) The ∗-homomorphism ϕ : C∗(M, eMα) → M ×α G we discussed in the proof of Theorem 3.3 can be rewritten as follows. ϕ(LxeMαLy) = xαg(y)λg, x, y ∈ M (4) because ϕ(LxeMαLy) = xey and e = λg. If {(ui, u∗i )} is a quasi-basis for E, we see from LuieMαLu∗i = 1 and (4) that ϕ(1) = uiαg(u i ))λg (5) is a projection in M ×α G. Recall that ϕ(1) = 1 holds if α is saturated. Theorem 4.1. Let M be a unital C∗-algebra and α be an action of a finite group G on M . Then the following are equivalent: (i) α is saturated, that is, Jα = M ×α G. (ii) E : M → Mα is of index finite type with Index(E) = |G|. (iii) E : M → Mα is of index finite type with Index(E) = |G| and uiαg(u i ) = 0, g 6= ι (6) for a quasi-basis {(ui, u∗i )} for E. (iv) There exist {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} for some m ≥ 1 such that (a) αg(b ) = b , for j = 1, . . . ,m and g, h ∈ G. bjg(b )∗ = δgh. (v) For every ε > 0, there exist {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} for some m ≥ 1 such that ‖αg(bjh)− b ‖ < ε, (b) ‖ bjg(b )∗ − δgh‖ < ε. Proof. (i) ⇐⇒ (ii) follows from Theorem 3.3. (i) =⇒ (iii). If {(ui, u∗i )} is a quasi-basis for E, we have from (5) that uiαg(u i ) = 0 for g 6= ι since ϕ(1) = 1. (iii) =⇒ (ii). Obvious. SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 7 (i) =⇒ (iv). Suppose Jα = M ×α G. By (3) there exist m ∈ N and bj ∈ M , 1 ≤ j ≤ m, such that bjαg(b λg = 1. j = 1 and bjαg(b j ) = 0 for g 6= ι. (7) Set bjg := αg(bj). Then ) = αg(αh(bj)) = αgh(bj) = b bjg(b αg(bj)αh(b bjαg−1h(b = δgh by (7). (iv) =⇒ (v). Obvious. (v) =⇒ (i). Let ε > 0 and let {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} satisfy (a) and (b) of (v). Note that (b) implies ‖bjg‖ < 1 + ε for g ∈ G, 1 ≤ j ≤ m. Indeed from bjg(b ∥ ≤ ‖ bjg(b ∗−1‖ < ε, we have ‖bjg‖2 ≤ ‖ bjg(b ∗‖ < 1+ε. αg((b )∗)λg)− |G| ‖ αg((b )∗)λg − |G| ‖ )∗ − |G| ‖+ ‖ g 6=ι αg((b )∗)λg)‖ )∗ − 1 ‖+ g 6=ι αg((b < ε|G|+ g 6=ι αg((b )∗)− (bj g 6=ι |G|+ |G|2 max ‖bjg‖+ |G|2 |G|+ |G|2(1 + ε) + |G|2 Since αg((b )∗)λg) ∈ Jα and ε can be chosen to be arbitrarily small, we conclude that Jα = M ×α G. � Example 4.2. Let w = be a unitary with wn = 1 and define an automorphism α on M2(C)) by α(a) = waw ∗, a ∈ M2(C). We will show that α is saturated if and only if z2 = −z1. For this, recall from (3) that α is saturated if and only if there exist xj ∈ M2(C), 1 ≤ i ≤ m, satisfying k(x∗j )λk = 1M2(C). (8) 8 JA A JEONG AND GI HYUN PARK Hence, particularly for k = 0, 1, we have xjα(x j ) = With xj = aj bj cj dj and zi = e iθi , i = 1, 2, this means |aj |2 + |bj |2 aj c̄j + bj d̄j cj āj + dj b̄j |cj |2 + |dj |2 |aj |2 + ei(θ2−θ1)|bj |2 ei(θ1−θ2)aj c̄j + bj d̄j |aj |2 + ei(θ2−θ1)dj b̄j ei(θ1−θ2)aj c̄j + |dj |2 . (9) Therefore, by comparing (1,1) entries of each matrices, it follows that if α is satu- rated, then there exist positive real numbers a (= |aj |2) > 0, b (= |bj |2) > 0 such that a+ b = 1 and a+ ei(θ2−θ1)b = 0. (10) Note that b 6= 0 since b = 0 implies a = 0 from |aj |2 + ei(θ2−θ1)dj b̄j = 0 in (9). There are three possible cases for θ1, θ2 as follows. (i) If θ2 − θ1 ≡ 0(mod 2π), that is, α is trivial, then (10) is not possible. (ii) If θ2 − θ1 ≡ π(mod 2π), then and x2 = satisfy (8) with m = 2. Thus α is saturated. (iii) If θ2− θ1 6= 0, π(mod 2π), then (10) is not possible for any a, b > 0. Hence α is not saturated. Remark 4.3. Let α, β ∈ Aut(M) satisfy αn = βn = idM for some n ≥ 1. If there is a unitary u ∈ M such that β = Ad(u) ◦ α, then α and β are said to be exterior equivalent, and if this is the case the crossed products are isomorphic, M×αG ∼= M×βG, [14, p.45]. Example 4.2 says that the property of being saturated may not be preserved under exterior equivalence. Also the case (iii) of Example 4.2 above with w = diag(λ, λ̄), λ = e 3 (hence θ1− θ2 = 2π3 − 6= π(mod 2π)), shows that Index(E) < |G| is possible even when E is of index-finite type. In fact, if and u2 = , then {(ui, u∗i )}2i=1 forms a quasi-basis for E, but Index(E) = 2 < |Z3|. Remark 4.4. Recall that the Rokhlin property and the tracial Rokhlin property (weaker than the Rokhlin property) are defined as follows and considered intensively in [7] and [12], respectively: (a) ([7]) α is said to have the Rokhlin property if for every finite set F ⊂ M , every ε > 0, there are mutually orthogonal projections {eg | g ∈ G} in M such that (i) ‖αg(eh)− egh‖ < ε for g, h ∈ G. (ii) ‖egx− xeg‖ < ε for g ∈ G and all x ∈ F . (iii) g∈G eg = 1. SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 9 (b) ([12]) α is said to have the tracial Rokhlin property if for every finite set F ⊂ M , every ε > 0, every n ∈ N, and every nonzero positive element x ∈ M , there are mutually orthogonal projections {eg | g ∈ G} in M such that: (i) ‖αg(eh)− egh‖ < ε for g, h ∈ G. (ii) ‖egx− xeg‖ < ε for g ∈ G and all x ∈ F . (iii) With e = g∈G eg, the projection 1 − e is Murray-von Neumann equivalent to a projection in the hereditary subalgebra of M generated by x. The following proposition is actually observed in [12, Lemma 1.13], and we put a proof for reader’s convenience. Proposition 4.5. Let M be a unital C∗-algebra and α be an action of a discrete group G on M . Suppose that for every ε > 0 and every finite subset F ⊂ M , there exist a family of projections {eg}g∈G such that (1) ‖αg(eh)− egh‖ < ε. (2) ‖egx− xeg‖ < ε for each x ∈ F . Then α is an outer action. Proof. Suppose there is a unitary u ∈ M such that αg(x) = uxu∗ for every x ∈ M (g 6= ι). Put F = {u} and 0 < ε < 1/4. Then there exist mutually orthogonal projections {eg}g∈G such that ‖αg(eh)− egh‖ < ε < 1/4. Thus ‖egu− ueg‖ < ε < 1/4. Then ‖αg(eι)− ueιu∗‖ = 0. But ‖αg(eι)− ueιu∗‖ = ‖αg(eι)− eg + eg − eι + eι − ueιu∗‖ ≥ ‖eg − eι‖ − ‖αg(eι)− eg‖ − ‖eι − ueι1u∗‖ ≥ 1− 1 which is a contradiction. � Remark 4.6. If M ×αG is simple, α is obviously saturated, and this is the case if G is a finite group, M is α-simple, and T̃(αg) 6= {1} for all g 6= ι ([8, Theorem 3.1]). In particular, α is saturated if M is simple and α is outer. But for a nonsimple M , this may not hold. In fact, if α is an outer action of Zn on M and u is a unitary in M with un = 1 such that the action Ad(u) on M is not saturated (as in Example 4.2), then the action α ⊕ Ad(u) on M ⊕M is outer but not saturated. Now we show that if α satisfies the Rokhlin property (or satisfies the tracial Rokhlin property and M has cancellation) then α is saturated. For this we first review the cancellation property of C∗-algebras. For projections p, q in a C∗-algebra, we write p ⊥ q if pq = 0, and p ∼ q if they are Murrey-von Neumann equivalent. Definition 4.7. A unital C∗-algebra M has the cancellation if, whenever p, q, r are projections in Mn(M) for some n, with p ⊥ r, q ⊥ r, and (p + r) ∼ (q + r), then p ∼ q. 10 JA A JEONG AND GI HYUN PARK Remark 4.8. (1) If M has the cancellation and p, q are projections in M such that (1− p) ∼ (1− q), then p ∼ q ([4, V.2.4.14]). (2) It is well known that every C∗-algebra with stable rank one has the cancel- lation ([4, V.3.1.24]). Proposition 4.9. Let α be an action of a finite group G on a unital C∗-algebra M . Then α is saturated if one of the following holds. (i) α has the Rokhlin property. (ii) α has the tracial Rokhlin property and M has the cancellation. Proof. (i) For an ε > 0, there exist mutually orthogonal projections {eg}g such that g eg = 1 and ‖αg(eh)− egh‖ < ε. Then, with m = 1, the elements b1g := eg satisfy (v) of Theorem 4.1. (ii) Now suppose α has the tracial Rokhlin property and M has the cancellation. We shall show that Jα contains the unit of M ×α G. Let 0 < ε < 1. For each g ∈ G, choose mutually orthogonal projections {eg }h∈G such that ‖αk(egh)− e ‖ < ε 2|G|2 , and put eg := h∈G e . If eg = 1, for some g, then b1h := e (h ∈ G) will satisfy (v) of Theorem 4.1 as in (i). If eg 6= 1 for every g ∈ G, then by the tracial Rokhlin property of α there exist mutually orthogonal projections {fg }h∈G in M such that ‖αk(fgh)− f ‖ < ε 2|G|2 ) ∼ (eg)′ < eg for a subprojection (eg)′ of eg. Put fg := . Then since M has cancellation, it follows that fg ∼ 1−(eg)′ > (1−eg). Let vg ∈ M be a partial isometry satisfying v∗gvg = f g, vgv g = 1− (eg)′, and set xg := (ekhαg(e h) + (1 − ek)vkfkhαg(fkhαg−1 (v∗k)) , g ∈ G. Now we show that the element x := xgλg ∈ Jα satisfies ‖x − 1‖ < ε. In fact, for g 6= ι, ‖xg‖ ≤ ‖(ekhαg(ekh) + (1− ek)vkfkhαg(fkhαg−1(v∗k))‖ ≤ 1|G| ‖(ekhαg(ekh)‖ + ‖fkhαg(fkh )‖ ≤ 1|G| ‖ekh(αg(ekh)− ekgh)‖ + ‖ekhekgh‖+ ‖fkh (αg(fkh )− fkgh)‖+ ‖fkhfkgh‖ ≤ 1|G| 2|G|2 + 2|G|2 ) SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 11 |G| ( (1− ek)vkfkv∗k) (1 − ek)(1 − (ek)′) For the rest of this section we consider a finite group action on a commutative C∗-algebra C(X). If G acts on a compact Hausdorff space X , it induces an action, say α, on C(X) by αg(f)(x) = f(g −1x), f ∈ C(X). For each x ∈ X , let Gx = {g ∈ G : gx = x} be the isotropy group of x and for a subgroup H of G (H < G), put XH = {x ∈ X : Gx = H}. It is readily seen that XH and XH′ are disjoint if H 6= H ′, and X is partitioned as Theorem 4.10. Let X be a compact Hausdorff space and G a finite group acting on X. If α is the induced action of G on C(X), the following are equivalent: (i) E : C(X) → C(X)α is of index finite type. (ii) XH is closed for each H < G. Moreover, if this is case the index of E is Index(E) = χXH , where χXH is the characteristic function on XH . Proof. (i) =⇒ (ii). If E is of index-finite type and {(ui, u∗i )}ki=1 is a quasi-basis for E, then uiE(u i f) = f, that is, ui(x) u∗i (g −1x)f(g−1x) = f(x), (11) for f ∈ C(X) and x ∈ X . For each x ∈ X , choose a continuous function fx ∈ C(X) satisfying fx|Gx\{x} ≡ 0 and fx(x) = 1. Then (11) with fx in place of f gives ui(x) u∗i (g −1x)fx(g = fx(x), (12) and so we have ui(x)u i (x) = 1. (13) 12 JA A JEONG AND GI HYUN PARK To show that each XH is closed, let {xn ∈ XH : n = 1, 2, . . .} be a sequence of elements in XH with limit x ∈ XH′ . Then (11) gives fx(xn) = ui(xn) u∗i (g −1xn)fx(g −1xn) ui(xn) u∗i (g −1xn)fx(g −1xn) + g 6∈H u∗i (g −1xn)fx(g −1xn) ui(xn) |H |u∗i (xn)fx(xn) + g 6∈H u∗i (g −1xn)fx(g −1xn) Taking the limit as n → ∞, we have fx(x) = ui(x) |H |u∗i (x)fx(x) + g 6∈H u∗i (g −1x)fx(g ui(x) |H |u∗i (x)fx(x) + |H ′ \H |u∗i (x)fx(x) |H |+ |H ′ \H | ui(x)u i (x)fx(x). Therefore, comparing with (13), we obtain |H ′| = |H |+ |H ′ \H | since Gx = H ′ and fx(x) = 1. Hence H ⊂ H ′. On the other hand, since Gxn = H , again by (13), ui(xn)u i (xn) = 1 with the limit ui(x)u i (x) = 1 as n → ∞. But also ui(x)u i (x) = 1 by (13), and thus |H | = |H ′|. Consequently we have H = H ′ because H ⊂ H ′. This shows that XH is closed. (ii) =⇒ (i). Assume that XH is closed for every subgroup H of G. Then XH is open since there are only finitely many such subsets. Let UH = {UH,iH : iH = 1, 2, . . . , nH} be an open covering of XH such that x ∈ UH,iH =⇒ g−1x 6∈ UH,iH or g−1x 6∈ XH whenever g−1x 6= x. Let {vH,iH} be a partition of unity subordinate to UH . We understand that the domain of vH,iH is X by assigning 0 to x 6∈ XH . Let uH,iH = vH,iH . We claim that |H | uH,iH , |H | u : H < G, iH = 1, 2, . . . , nH SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 13 is a quasi-basis for E. For f ∈ C(X) and x ∈ X , let F < G and 1 ≤ j ≤ nF be such that x ∈ XF and x ∈ UF,j. Then |H | uH,iHE |H | u |H |uH,iH (x) |H |u (g−1x)f(g−1x) |F |uF,iF (x) u∗F,iF (g −1x)f(g−1x) uF,iF (x) u∗F,iF (g −1x)f(g−1x) uF,iF (x)|F |u∗F,iF (x)f(x) vF,iF (x)f(x) = f(x), as claimed. Recall that an action G on X is free if gx 6= x for g ∈ G, g 6= 1, and x ∈ X Corollary 4.11. Let X be a compact Hausdorff space and G a finite group acting on X. If α is the induced action on C(X), the following are equivalent. (i) G acts freely on X. (ii) E : C(X) → C(X)α is of index-finite type with Index(E) = |G|. (iii) α is saturated. Proof. (i) =⇒ (ii) is proved in [19, Proposition 2.8.1]. To show (ii) =⇒ (i), let E be of index-finite type with Index(E) = |G|. Then from Theorem 4.10, we have |G| = Index(E) = |H |χXH , which implies that H = {ι} is the only subgroup of G such that XH 6= ∅. Hence X = X{ι}, that is, G acts freely on X . (ii) ⇐⇒ (iii) comes from Theorem 4.1. � 5. Saturated actions by compact groups Notation 5.1. Let M be a C∗-algebra and α be an action of a compact group G on M . For x, y ∈ M , define continuous functions fx,y, fx,1, f1,y ∈ C(G,M) from G to 14 JA A JEONG AND GI HYUN PARK M as follows: fx,y(t) = xαt(y), fx,1(t) = x, f1,y(t) = αt(y) for t ∈ G. Then it is easily checked that fx,y = fx,1 ∗ f1,y and f∗x,y = fy∗,x∗ . Recall that C(G,M) is a dense ∗-subalgebra of M ×α G with the multiplication and involution defined by f ∗ g(t) = f(s)αs(g(s −1t))ds, f∗(t) = αt(f(t −1)∗), where dg is the normalized Haar measure ([13, 7.7], [6, 8.3.1]). Hence if G is a finite group, fx,y can be written as fx,y = xαg(y)λg . If M̃ denotes the smallest unitization of M (so M̃ = M if M is unital), the function e : G → M̃, e(s) = 1, for every s ∈ G is a projection of the multiplier algebra of M ×α G ([17]). Proposition 5.2. ([17]) Let α be an action of a compact group G on a C∗-algebra M . Then identifying x ∈ Mα and the constant function in C(G,M) with the value x everywhere we see that x 7→ fx,1 : Mα → e(M ×α G)e is an isomorphism of Mα onto the hereditary subalgebra e(M×αG)e of the crossed product M ×α G. The notion of saturated action is introduced by Rieffel for a compact group action on a C∗-algebra, and we adopt the following equivalent condition as the definition. Definition 5.3. (Rieffel, see [14, 7.1.9 Lemma]) Let M be a C∗-algebra and α be an action of compact group G on M . α is said to be saturated if the linear span of {fa,b | a, b ∈ M} is dense in M ×α G (see Notation 5.1). We denote Jα = span{fa,b | a, b ∈ M}. Proposition 5.4. Let α be an action of a compact group G on a C∗-algebra M . Then Jα is the ideal of M×αG generated by the hereditary subalgebra e(M ×αG)e. Moreover Jα = span{fa,a∗ ∈ C(G,M) | a ∈ M}. Proof. We first show that Jα is an ideal ofM×αG. Let x ∈ C(G,M) and a, b ∈ M . Then x ∗ fa,b ∈ Jα. Indeed, (x ∗ fa,b)(t) = x(s)αs(fa,b(s −1t))ds x(s)αs(a)αt(b)ds x(s)αs(a)ds αt(b), SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 15 hence x∗fa,b = fc,b ∈ Jα, where c = x(s)αs(a)ds ∈ M . Also f∗a,b = fb∗,a∗ implies that Jα = Jα∗ is an ideal of M ×α G. Let J := (M ×α G)e(M ×α G) be the closed ideal generated by e(M ×α G)e. Now we show that Jα ⊂ J . From (fa,b ∗ e)(t) = fa,b(s)αs(e(s −1t)) ds = aαs(b) ds = a αs(b)ds, we have fa,b ∗ e = faE(b),1, where E(b) = αs(b)ds ∈ Mα. Hence for a, b, c, and d in M , we have fa,b ∗ e ∗ fc,d = (fa,b ∗ e) ∗ (fd∗,c∗ ∗ e)∗ = faE(b),1 ∗ (fd∗E(c∗),1)∗ = faE(b),1 ∗ f1,E(c∗)d = faE(b),E(c∗)d, which means that fax,yd ∈ J for any a, d ∈ M and x, y ∈ Mα. Since Mα contains an approximate identity for M , it follows that fa,b ∈ J for a, b ∈ A. For the converse inclusion J ⊂ Jα, note that if x ∈ C(G,M), then (x ∗ e)(t) = x(s)ds for t ∈ G. With notations x′ = x(s)ds and x′′ := αs(x(s −1))ds (∈ M), we see that (x ∗ e ∗ y)(t) = (x ∗ e)(s)αs(y(s−1t))ds αs(y(s −1t))ds = x′αt( αs(y(s −1))ds) = x′αt(y = fx′,y′′(t) belongs to Jα for x, y ∈ C(G,M). Finally the following polarization identity proves the last assertion. aαt(b) = ik(b + ika∗)∗αt(b+ i ka∗). 6. The gauge action γ on a graph C∗-algebra By a (directed) graph E we mean a quadruple E = (E0, E1, r, s) consisting of the vertex set E0, the edge set E1, and the range, source maps r, s : E1 → E0. If each vertex of E emits only finitely many edges E is called row finite and a row finite graph E is locally finite if each vertex receives only finitely many edges. By En we denote the set of all finite paths α = e1 · · · en (r(ei) = s(ei+1), 1 ≤ i ≤ n−1) of length n (|α| = n). Each vertex is regarded as a finite path of length 0. Then E∗ = ∪n≥0En is the set of all finite paths and the maps r and s naturally extend to E∗. A vertex v is called a sink if s−1(v) = ∅ and a source if r−1(v) = ∅. If E is a row finite graph, we call a family {se, pv | e ∈ E1, v ∈ E0} of operators a Cuntz-Krieger(CK) E-family if {se}e are partial isometries and {pv}v are mutually 16 JA A JEONG AND GI HYUN PARK orthogonal projections such that s∗ese = pr(e) and pv = s(e)=v e if s −1(v) 6= ∅. It is now well known that there exists a C∗-algebra C∗(E) generated by a universal CK E-family {se, pv | e ∈ E1, v ∈ E0}, in this case we simply write C∗(E) = C∗(se, pv). For the definition and basic properties of graph C ∗-algebras, see, for example, [1, 2, 10, 11, 15] among others. If α = α1α2 · · ·α|α| (αi ∈ E1) is a finite path, by sα we denote the partial isometry sα1sα2 · · · sα|α| (sv = s∗v = pv, for v ∈ We will consider only locally finite graphs and it is helpful to note the following properties of graph C∗-algebras. Remark 6.1. (i) Let C∗(E) = C∗(se, pv) be the graph C ∗-algebra associated with a row finite graph E, and let α, β ∈ E∗ be finite paths in E. Then s∗αsβ = s∗µ, if α = βµ sν , if β = αν 0, otherwise. Therefore C∗(E) = span{sαs∗β | α, β ∈ E∗}. (ii) Note that sαs β = 0 for α, β ∈ E∗ with r(α) 6= r(β). (iii) If α, β, µ, and ν in En are the paths of same length, β)(sµs ν) = δβ,µsαs Thus for each n ∈ N and a vertex v in a locally finite graph E, we see that span{sαs∗β | α, β ∈ En and r(α) = r(β) = v} is a ∗-algebra which is isomorphic to the full matrix algebraMm = (Mm(C)), where m = ∣{α ∈ En | r(α) = v} Recall that the gauge action γ of T on C∗(E) = C∗(se, pv) is given by γz(se) = zse, γz(pv) = pv, z ∈ T. γ is well defined by the universal property of the CK E-family {se, pv}. Since γz(sαs β)dz = z|α|−|β|(sαs β)dz = 0, |α| 6= |β|, one sees that C∗(E)γ = span{sαs∗β | α, β ∈ E∗, |α| = |β|}. If Z denotes the following graph: • • • • •______ // ______ // ______ //______ // · · · ,· · ·Z : −2 −1 0 1 2 then C∗(Z) is isomorphic to the C∗-algebra K of compact operators on an infinite dimensional separable Hilbert space, hence C∗(Z) is itself a simple AF algebra. But C∗(Z)γ coincides with the commutative subalgebra span{sαs∗α | α ∈ Z∗} which is far from being simple, and thus we know that the simplicity of C∗(E) does not imply that of C∗(E)γ in general. SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 17 In [9], the Cartesian product of two graphs E and F is defined to be the graph E× F = (E0 ×F 0, E1 ×F 1, r, s), where r(e, f) = (r(e), r(f)) and s(e, f) = (s(e), s(f)). Since the graph Z × E has no loops for any row-finite graph E, we know that C∗(E)γ is an AF algebra ([10]) by the following proposition. Proposition 6.2. ([9]) Let E be a row finite graph with no sources. Then the following hold: (a) C∗(E)γ is stably isomorphic to C∗(E)×γ T. (b) C∗(E)×γ T ∼= C∗(Z× E). Now we show that a gauge action is saturated. For this, note that the linear span of the continuous functions of the form t 7→ f(t)x, f ∈ C(G), x ∈ A is dense in C(G,A) [13, 7.6.1]. Hence by Remark 6.1(i), one sees that C∗(E)×γ T = span{znsαs∗β | α, β ∈ E∗ n ∈ Z}. (15) Theorem 6.3. Let E be a locally finite graph with no sinks and no sources. Then the gauge action γ on C∗(E) is saturated. Proof. We show that Jγ = C∗(E)×γ T. By (15) it suffices to see that znsαs β ∈ Jγ for all α, β ∈ E∗, n ≥ 0 (because z−nsαs β = (z ∗ for n ≥ 0). Now fix α, β ∈ E∗ and n ≥ 0. Put l = n− (|α| − |β|). There are two cases. (i) l ≥ 0: One can choose a path µ such that |µ| = l and r(µ) = s(α). Then znsαs β = z l+|α|−|β|s∗µsµsαs β = s µγz(sµαs β) = fs∗µ,sµαs∗β (z), where the function fs∗µ,sµαs∗β belongs to Jγ . (ii) l < 0: Choose a path ν with |ν| = |β|+n and r(ν) = r(α). With a = sαs∗ν , b = sνs β , we have fa,b ∈ Jγ and znsαs β = sαs νγz(sνs β) = fa,b(z). Acknowledgements. The first author would like to thank Hiroyuki Osaka and Ta- motsu Teruya for valuable discussions. References [1] T. Bates, J. H. Hong, I. Raeburn and W. Szymanski, The ideal structure of the C∗-algebras of infinite graphs, Illinois J. Math., 46(2002), 1159–1176. [2] T. Bates, D. Pask, I. Raeburn, and W. Szymanski, The C∗-algebras of row-finite graphs, New York J. Math. 6(2000), 307–324. [3] T. Bates and D. Pask, Flow equivalence of graph algebras, Ergod. Th. & Dynam. Sys. 24(2004), 367–382. [4] B. Blakadar, Operator Algebras, Theory of C∗-algebras and von Neumann algebras, Encyclo- pedia of mathematical sciences 122, Springer, 2006. [5] J. Cuntz and W. Krieger, A class of C∗-algebras and topological Markov chains, Invent. Math. 56(1980), 251–268. [6] P. A. Fillmore, A user’s guide to operator algebras, Canadian Math. Soc. Series of Monographs and Adv. Texts, John Wiley & Sons, Inc. 1996. 18 JA A JEONG AND GI HYUN PARK [7] M. Izumi, Finite group actions on C∗-algebras with the Rohlin property, I, Duke Math. J. 122(2004), no. 2, 233–280. [8] A. Kishimoto, Outer automorphisms and related crossed products of simple C∗-algebras, Comm. Math. Phys. 81(1981), no. 1, 429–435. [9] A. Kumjian and D. Pask, C∗-algebras of directed graphs and group actions, Ergodic Theory Dynam. Systems, 19(1999), 1503–1519. [10] A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184(1998), 161–174. [11] A. Kumjian, D. Pask, I. Raeburn and J. Renault Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144(1997), 505–541. [12] H. Osaka and N. C. Phillips, Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property, Ergod. Th. & Dyn. Sys., to appear. [13] G. K. Pedersen, C∗-algebras and their automorphism groups, London Math. Soc. Monographs No.14, Academic Press 1979. [14] N. C. Phillips, Equivariant K-theory and freness of group actions on C∗-algebras, LNM 1274, Springer, 1987. [15] I. Raeburn, Graph algebras, CBMS. 103, 2005, AMS. [16] J. Renault, A groupoid approach to C∗-algebras, LNM 794, Springer, Berlin, 1980. [17] J. Rosenberg, Appendix to O. Bratteli’s paper on ”crossed products of UHF algebras, Duke Math. J. 46(1979), 25–26. [18] W. Szymański and C. Peligrad, Saturated actions of finite dimensional Hopf ∗-algebras on C∗-algebras, Math. Scand. 75(1994), 217–239. [19] Y. Watatani, Index for C∗-subalgebras, Memoirs of the Amer. Math. Soc. 424, AMS, 1990. [20] S. L. Woronowicz, Compact matrix psedogroups, Comm. Math. Phys. 111(1987), 613–665. Keywords: Finite dimensional Hopf ∗-algebra; saturated action; conditional ex- pectation of index-finite type. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 151–747, Korea E-mail address: jajeong@snu.ac.kr Department of Mathematics, Hanshin University, Osan, 447–791, Korea E-mail address: ghpark@hanshin.ac.kr 1. Introduction 2. Preliminaries 3. Actions by finite dimensional Hopf *-algebras 4. Actions by finite groups 5. Saturated actions by compact groups 6. The gauge action on a graph C*-algebra References

The electronic structures, the equilibrium geometries and finite temperature properties of Na (n=39-55) Shahab Zorriasatein1,2, Mal-Soon Lee1, and D. G. Kanhere1 Department of Physics, and Center for Modeling and Simulation, University of Pune, Ganeshkhind, Pune–411 007, India Department of Physics, Islamic Azad University, Tehran south branch, Tehran, Iran Density-functional theory has been applied to investigate systematics of sodium clusters Nan in the size range of n= 39-55. A clear evolutionary trend in the growth of their ground-state geometries emerges. The clusters at the beginning of the series (n=39-43) are symmetric and have partial icosahedral (two-shell) structure. The growth then goes through a series of disordered clusters (n=44-52) where the icosahedral core is lost. However, for n ≥53 a three shell icosahedral structure emerges. This change in the nature of the geometry is abrupt. In addition, density-functional molecular dynamics has been used to calculate the specific heat curves for the representative sizes n= 43, 45, 48 and 52. These results along with already available thermodynamic calculations for n= 40, 50, and 55 enable us to carry out a detailed comparison of the heat capacity curves with their respective geometries for the entire series. Our results clearly bring out strong correlation between the evolution of the geometries and the nature of the shape of the heat capacities. The results also firmly establish the size-sensitive nature of the heat capacities in sodium clusters. PACS numbers: 31.15.Ew, 31.15.Qg, 36.40.Ei, 36.40.Qv I. INTRODUCTION Physics and chemistry of clusters are very active ar- eas of research especially because of the emergence of nano science and nano technology.1 Although major ef- forts have been spent into ground-state investigations, finite temperature properties are turning out to be very interesting. Such investigations are challenging, both ex- perimentally as well as theoretically. One of the first de- tailed measurements providing much impetus for theoret- ical work was on free sodium clusters by Haberland and co-workers.2 These measurements reported the melting temperatures (Tm) of sodium clusters in the size range between 55 and 350 and remained unexplained for almost about a decade. The main puzzle was related to the irregular behav- ior of the melting temperature and the absence of any correlation between the peaks and the magic numbers ei- ther geometric or electronic. A good deal of simulation works has been carried out to explain the sodium data, most of the early work being with classical inter atomic potentials.3,4 It turned out that none of these could ob- tain qualitative and quantitative agreement with the ex- perimental data. Thus, it needed an ab initio density- functional method to achieve this. Indeed, much insight and excellent quantitative agreement has been obtained by density-functional molecular dynamics (DFMD) sim- ulations.5,6,7,8,9,10 Recently a very different aspect of finite temperature behavior has been brought out by the experimental and later by theoretical work on gallium clusters.11,12 The experimental reports of Breaux and co-workers11 showed that in the size range of N=30 to 55, free clusters of gal- lium melt much above their bulk melting temperature. Interestingly, their experiment also showed that the na- ture of heat capacity is size sensitive. In fact, addition of even one atom changes the shape of the heat capacity dra- matically, e.g for Ga30 and Ga31. A similar experimental observation has been reported for aluminum clusters in the size range n=49-62.13 Such a size sensitive behavior has also been observed in DFMD simulation of Au19 and Au20. 14 A detailed analysis of the ground-state geome- tries of these clusters brought out the role of order and disorder in their geometries on the shape of the melting curve. A disordered system is shown to display a con- tinuous melting transition leading to a very broad heat capacity curve. However, the effect is subtle and descrip- tion of order and disorder needs careful qualifications. In spite of substantial experimental works on sodium clusters over a period of 10 years and or so, there is no firm and systematic evidence of size sensitivity. This is mainly due to the fact that the reported experimental data2 is for the size range of n=55-350 at discrete sizes. It is necessary to investigate the effect of addition of few atoms in a continuous manner in appropriate size range. However, it is not clear whether larger clusters having sizes of n >100 will also show this effect. An extensive ab initio study on the structural proper- ties of small Na clusters up to N=20 has been reported by Röthlisberger and Andreoni.15 The study reveals that pentagonal motifs dominate the structures above N=7. As expected most of the atoms in these clusters lie on the surface and a discernible core develops after about N=15-16. The shapes after this sizes show signature of icosahedral structures. The finite temperature behavior of sodium clusters in the size range of n=8 to 55 has been reported.7 The study reveals that it is not easy to discern any melt- ing peaks below n=25. However, the simulation data available at rather coarse sizes above n=40 already shows size-sensitive feature. In addition to this feature, our re- cent investigation8 on Na57 and Na58 does bring out the http://arxiv.org/abs/0704.1550v1 role of geometry and electronic structure on the melt- ing. Nevertheless, in order to draw definitive conclusions it is necessary to investigate the effect of addition of a few atoms in a continuous manner in appropriate size range. Therefore, we have chosen the size range of n=39- 55 and have carried out detailed density-functional inves- tigations. The purpose of the present work is two fold. First, to obtain reliable equilibrium geometries for all the clusters in the size range of n=39-55 and to discern evo- lutionary trends. We note that Na40 has a symmetric partially icosahedron core and Na55 is a complete icosa- hedron. Thus it is of considerable interest to examine the growth pattern from n=39 to n=55. The second purpose is to seek correlation between the nature of the ground-state and the evolutionary trends observed in the nature of their specific heats. Towards this end we have carried out extensive finite temperature simulations on representative clusters of size n= 43, 45, 48, and 52. Together with the already published results, this gives us access to specific heats for Na40, Na43, Na45, Na48, Na50, Na52 and Na55, a reasonable representation across the series under investigation. Finally, we note that all the DFMD simulations reported so far have yielded ex- cellent agreement with the experimental data6,8,9 These reports demonstrate the reliability of density-functional molecular dynamics in describing the finite temperature properties. The plan of the paper is as follows. In the next sec- tion (Sec. II) we note the computational details. Sec. III presents equilibrium geometries and their shape system- atic. Sec. IV presents the finite temperatures behavior of Na43, Na45, Na48, Na52 and finally we discuss the corre- lation between the ground states and nature of the spe- cific heats for all the available thermodynamics data. We conclude our discussion in Sec. V. II. COMPUTATIONAL DETAILS We have carried out Born-Oppenheimer molecular dy- namics (BOMD) simulations16 using Vanderbilt’s ultra- soft pseudopotentials17 within the local-density approxi- mation (LDA), as implemented in the VASP package.18 We have optimized about 300 geometries for each of the sodium clusters in the size range between n=39 and n=55. The initial configuration for the optimization of each cluster were obtained by carrying out a constant temperature dynamics simulation of 60 ps each at vari- ous temperatures between 300 to 400 K. For many of the geometries we have also employed basin hopping19 and genetic20,21 algorithms using Gupta potential4 for gen- erating initial guesses. Then we optimized these struc- tures by using the ab initio density-functional method.22 For computing the heat capacities, the iso-kinetic BOMD calculations were carried out at 14 different temperatures for each cluster of Na43, Na45, Na48, Na52 in the range between 100 K and 460 K, each with the time dura- tion of 180 ps or more. Thus, it results in the total simulation time of 2.5 ns per system. In order to get converged heat capacity curve especially in the region of co-existence, more temperatures were required with longer simulation times. We have discarded at least first 30 ps for each temperature for thermalization. To ana- lyze the thermodynamic properties, we first calculate the ionic specific heat by using the Multiple Histogram (MH) technique.23,24 We extract the classical ionic density of states (Ω(E)) of the system, or equivalently the classi- cal ionic entropy, S(E) = kB lnΩ(E), following the MH technique. With S(E) in hand, one can evaluate ther- modynamic averages in a variety of ensembles. We focus in this work on the ionic specific heat. In the canon- ical ensemble, the specific heat is defined as usual by C(T ) = ∂U(T )/∂T , where U(T ) = E p(E, T ) dE is the average total energy. The probability of observing an energy E at a temperature T is given by the Gibbs distribution p(E, T ) = Ω(E) exp(−E/kBT )/Z(T ), with Z(T ) the normalizing canonical partition function. We normalize the calculated canonical specific heat by the zero-temperature classical limit of the rotational plus vi- brational specific heat, i.e., C0 = (3N − 9/2)kB. We have calculated a number of thermodynamic indi- cators such as root-mean-square bond length fluctuations (δrms), mean square displacements (MSD) and radial dis- tribution function (g(r)). The δrms is defined as δrms = N(N − 1) (〈r2ij〉t − 〈rij〉 〈rij〉t , (1) where N is the number of atoms in the system, rij is the distance between atoms i and j, and 〈. . .〉t denotes a time average over the entire trajectory. MSDs for in- dividual atoms is another traditional parameter used for determining phase transition and is defined as, 〈r2I(t)〉 = [RI(t0m + t)−RI(t0m)] where RI is the position of the I th atom and we aver- age over M different time origins t0m spanning over the entire trajectory. The interval between the consecutive t0m for the average was taken to be about 1.5 ps. The MSDs of a cluster indicate the displacement of atoms in the cluster as a function of time. The g(r) is defined as the average number of atoms within the region r and r + dr. We have also calculated the shape deformation param- eter (εdef ), to analyze the shape of the ground state for all the clusters. The εdef is defined as, εdef = Qy +Qz , (3) where Qx ≥ Qy ≥ Qz are the eigenvalues, in descend- ing order, of the quadrupole tensor 39 40 41 42 43 FIG. 1: The ground-state geometries of Nan (n=39-43). 44 45 46 47 48 49 50 51 52 FIG. 2: The ground-state geometries of Nan (n=44-52). Qij = RIiRIj . (4) Here i and j run from 1 to 3, I runs over the number of ions, and RIi is the i th coordinate of ion I relative to the center of mass (COM) of the cluster. A spherical system (Qx = Qy = Qz) has εdef=1 and larger values of εdef indicates deviation of the shape of the cluster from sphericity. 53 54 55 FIG. 3: The ground-state geometries of Nan (n=53-55). 5551474339 Size (No. of Atoms) FIG. 4: The shape deformation parameter for Nan (n=39-55) as a function of cluster size. III. GEOMETRIES The lowest energy geometries of sodium clusters (Nan n=39-55) are shown in Fig. 1 (n=39-43), Fig. 2 (n=44- 52), and Fig. 3 (n=53-55). We have also plotted the shape deformation parameter εdef and the eigenvalues of quadrupole tensor for the ground state geometries of these clusters in Fig. 4 and Fig. 5, respectively. It is convenient to divide these clusters into three groups. The clusters in the first group, shown in Fig. 1 are nearly spherical. The ground-state geometry for Na39 555351494745434139 Cluster Size FIG. 5: The eigenvalues of the quadrupole tensor for Nan (n=39-55) as a function of cluster size. 40200 40200 6040200 ° A Atom number FIG. 6: The distance from the center of mass for each of the atoms ordered in the increasing fashion for Nan (n=39-55). The formation of the shells is evident from the sharp steps. is highly symmetric. This structure has three identical units, each based on the icosahedral motive and these three units are arranged as shown in the Fig. 1. It is interesting to note that an addition of an extra atom changes the structure dramatically. It can be seen that the geometry of Na40 is based on icosahedral structure with missing 12 corner atoms and (111) facet as reported by Rytkönen et al .5 An extra atom added to this struc- ture is accommodated near the surface and deforms the structure slightly. In addition to the deformation the dis- tance between the two shells is reduced by 0.3 Å as com- pared to that in Na40. A single atom added to Na41 is not accommodated in the structure, instead it caps the sur- face. However, the low-lying geometries for Na42 have a spherical shape without any cap (figure not shown). The lowest-energy structure of Na43 shows two caps symmet- rically placed on the opposite side of Na41, accompanied by distortion of the icosahedral core. The second group in Fig. 2 consisting of the clusters with n=44-52 shows sub- stantial distortion of the icosahedral core and even loss of this core structure. These clusters essentially repre- sent the transition region from the two shell icosahedron, Na40 to three shell complete icosahedron, Na55. It can be seen that by adding one atom to Na43 the two shell core is destroyed. The growth from n=44 to n=52 shows successive stages of capping followed by rearrangement in the inner core. There is a dramatic change in the struc- ture as soon as we add one more atom to Na52. All the atoms rearrange to form an icosahedral structure as seen in Na53. Thus, the clusters in the last group namely, Na53 and Na54 differ from a perfect icosahedron of Na55 by an absence of two and one atom(s), respectively (Fig. The nature of the changes in the shape of the clus- ters during the growth can be seen in Fig. 4 and Fig. 5. The shape deformation parameter (εdef ) increases to a value about 2 up to Na52 with slightly higher values for Na45 and Na49 (Fig. 4). However, this value drops suddenly for Na53. It is interesting to examine the three eigenvalues of quadrupole tensor (Qx, Qy, Qz) shown in Fig. 5. It can be seen that two of the eigenvalues are nearly same up to Na52 while the third one continuously grows and indicates that the growth dominantly takes place along one of the directions. A prolate configura- tion has Qz ≫ Qx ≈ Qy. Thus, the majority of the clusters in the second group are prolate. The formation, the destruction and reformation of the shell structure is clearly seen in Fig. 6. In this figure we have plotted the 420340260180100 Temperature (K) 420340260180100 Temperature (K) FIG. 7: (a) The normalized heat capacity and (b) the δrms for Na43. The peak in heat capacity curve is at 270 K. ∆E=0.033 eV ∆E=0.067 eV (a) (b) FIG. 8: Two low lying isomers of Na43. ∆E represents the energy difference with respect to the ground-state. distance of each atom from the center of mass arranged in the increasing fashion. Clearly small rearrangement of atoms yields a change in the structure from Na39 to Na40. The two shell structures are observed till Na43. The for- mation of the shell structure is reflected in the formation of the sharp steps in the graph. As the size increases from Na43 to Na44 the shell structure is destroyed and seen again at Na53. Thus, three shells begin to emerge at Na53. IV. THERMODYNAMICS We have calculated the ionic heat capacity and indi- cators like mean square displacements (MSD) and root- mean-square bond length fluctuations (δrms) for four of the representative clusters in the investigated series which are Na43, Na45, Na48, and Na52. FIG. 9: The radial distribution function calculated for Na43 at five different temperatures. We note that the thermodynamics of Na40 5,7, Na50 and Na55 6,9 has already been reported. Thus it is pos- sible to examine and analyze the systematic variations in the melting characteristic and correlate them with the equilibrium geometries across the entire range of sizes from 40 to 55. The heat capacity and δrms for Na43 are shown in Figs. 7(a) and 7(b), respectively. We also show typical low energy geometries (isomers) for Na43 in Fig. 8. The first isomer shown in Fig. 8(a) has two atoms clos- est to each other capping the surface and second isomer shows a distorted shape and no caps. The heat capacity shows a weak peak around 160 K while the main peak occurs at 270 K. An examination of the motion of ionic trajectory seen as a movie indicates that isomerization (Fig. 8(a)) is responsible for the weak peak. It is interesting to observe the changes of radial dis- tribution function (RDF) as a function of temperature which is shown in Fig. 9. At low temperatures the shell structure is clearly evident. The pattern seen at 180 K and 210 K are mainly due to the fluctuation of the cluster between the ground-state and low-lying states. At 300 K and above the RDF shows the typical melting behavior of a cluster. The δrms in Fig. 7(b) shows the effect of iso- merization around 160 K. It can be seen that the melting 420340260180100 Temperature (K) 420340260180100 Temperature (K) FIG. 10: (a) The normalized heat capacity and (b) the δrms for Na45. The heat capacity curve shows a two stages melting process. 150120906030 Time (ps) 150120906030 Time (ps) FIG. 11: The MSDs of individual atoms calculated for Na45 (a) at 210 K and (b) at 225 K over the last 150 ps. region is of the order of 60 K. Let us recall that Na45, Na48 and Na52 belong to the second class of disordered clusters. Among these Na45 shows some signature of partial shells. The heat capacity and δrms for Na45 are shown in Figs. 10(a) and 10(b), respectively. The heat capacity of Na45 shows a first 420340260180100 Temperature (K) 420340260180100 Temperature (K) 420340260180100 Temperature (K) 420340260180100 Temperature (K) FIG. 12: (a) The normalized heat capacity and (b) the δrms for Na48, (c) The normalized heat capacity and (d) the δrms for Na52. peak around 230 K and a second peak around 300 K. We also show the MSDs of individual atoms at 210 and 225 K in Figs. 11(a) and 11(b), respectively. We have observed the ionic trajectories as a movie in the temperature at 225 K. It turns out that about one third of atoms in the cluster “melt” at this temperature. This is brought out by the contrasting behavior of the MSDs as shown in Figs. 11(a) and 11(b). Interestingly, all these 15 atoms are on the surface, indicating that surface melting takes place first. The peak around 225 K in the heat capacity 400200 Temperature (K) FIG. 13: The normalized heat capacity as a function of tem- perature for Nan, n=40, 43, 45, 48, 50, 52, and 55. is due to partial melting of these surface atoms. We show the δrms as a function of the temperature in Fig. 10(b). There is a sharp increases in δrms around 210 K and a slow rise after 240 K consistent with the behavior of the heat capacity. Thus in Na45 melting takes place in two stages over the range of 120 K. The heat capacity and δrms for Na48 and Na52 are shown in Fig. 12. The heat capacity of Na48 (Fig. 12(a)) is very broad indicating almost continuous phase change starting around 150 k. Thus it is difficult to identify a definite melting temperature. A similar behavior is also seen in that of Na52 except for the small peak seen 180 K due to isomerization (Fig. 12(c)). It is interesting to note that δrms for both clusters also show a gradual rise from about 150 K to 350 K indicating a continuous melting transition as shown in Figs. 12(b) and 12(d). The systematic evolution of heat capacities can be bet- ter appreciated by examination of all the available data (calculated from density-functional simulation). In Fig. 13 we show the specific heats for all the available clusters between Na40 and Na55. The most symmetric cluster Na55 shows the sharpest peak in the heat capacity. The heat capacity of Na40 and Na43 (partial icosahedral struc- tures) have well recognizable peaks which are broader than that of Na55. The disordered phase of the growth is clearly reflected in the very broad heat capacities seen around n=50. V. SUMMARY AND CONCLUSION The ab initio density-functional method has been ap- plied to investigate systematic evolutionary trends in the ground state geometries of the sodium clusters in the size range of n=39-55. The DFMD finite temperature simula- tions have been carried out for representative clusters. A detailed comparison between the heat capacities and the geometries firmly establishes a direct influence of the ge- ometries on the shapes of the heat capacity curves. The heat capacities show size sensitivity. The growth pattern shows a transition from ordered −→ disordered −→ or- dered sequence. The corresponding heat capacities show a transition from peaked to a very broad to peaked se- quence. It is seen that addition of a few atoms changes the shape of heat capacity very significantly. We believe that the size sensitive feature seen our simulation is uni- versal. It may be noted that such a feature has been observed experimentally in the case of gallium and alu- minum and in the case of DFMD simulations for gold. We await the experimental measurements of the heat ca- pacities on the sodium clusters in these range showing the size sensitivity. VI. ACKNOWLEDGMENT We acknowledge partial assistance from the Indo- French center for Promotion of Advance Research (In- dia)/ Center Franco-Indian pour la promotion de la Recherche Avancée (France) (IFC-PAR, project No; 3104-2). We would like to thank Kavita Joshi and Sailaja Krishnamurty for a number of useful discussions. 1 P. G. Reinhard, E. Suraud, Introduction to Cluster Dy- namics (Wiley-VCH, Berlin, 2003) 2 M. Schmidt, R. Kusche, B. von. Issendorff, and H. Haber- land, Nature (London) 393, 238 (1998); M. Schmidt and H. Haberland, C. R. Phys. 3, 327 (2002); H. Haberland, T. Hippler, J. Donges, O. Kostko, M. Schmidt and B. von Issendroff, Phys. Rev. Lett. 94, 035701 (2005) 3 F. Calvo and F. Spiegelmann, J. Chem. Phys. 112, 2888 (2000) 4 Y. Li, E. Blaisten-Barojas, and D. A. Papaconstantopou- los, Phys. Rev. B 57, 15519 (1998). 5 A. Rytkönen, H. Häkkinen, and M. Manninen, Phys. Rev. Lett. 80, 3940 (1998). 6 S. Chacko, D. G. Kanhere, and S. A. Blundell, Phys. Rev. B 71, 155407 (2005). 7 M.-S. Lee, S. Chacko, and D. G. Kanhere, J. Chem. Phys. 123, 164310 (2005). 8 M.-S. Lee and D. G. Kanhere, Phys. Rev. B 75, 125427 (2007). 9 A. Aguado and J. M. López, Phys. Rev. Lett. 94, 233401 (2005). 10 A. Aguado and J. M. López, Phys. Rev. B 74, 115403 (2006). 11 G. A. Breaux, D. A. Hillman, C. M. Neal, R. C. Benirschke, and M. F. Jarrold, J. Am. Chem. Soc. 126, 8628 (2004). 12 K. Joshi, S. Krishnamurty, and D. G. Kanhere, Phys. Rev. Lett. 96, 135703 (2006). 13 G. A. Breaux, C. M. Neal, B. Cao, and M. F. Jarrold, Phys. Rev. Lett. 94, 173401 (2005). 14 S. Krishnamurty, G. Shafai, D. G. Kanhere, and M. J. Ford, cond-mat/0612287 (to be published). 15 U. Röthlisberger and W. Andreoni, J. Chem. Phys. 94, 8129 (1991) 16 M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, Rev. Mod. Phys. 64, 1054 (1992). 17 D. Vanderbilt, Phys. Rev. B 41, 7892 (1990). 18 Vienna ab initio simulation package, Technische Univer- sität Wien (1999); G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). 19 Z. Li and H. A. Scheraga, Proc. Natl. Acad. Sci. 84, 6611 (1987); D. J. Wales and J. P. K. Doye, J. Phys. Chem. A 101, 5111 (1997). 20 M. Iwamatsu, J. Chem. Phys. 112, 10976 (2000) 21 J. A. Niesse and H. R. Mayne, J. Chem. Phys. 105, 4700 (1996) 22 R. G. Parr andW. Yang, The Density Functional Theory of Atoms and Molecules (Oxford university Press, New York, 1989). 23 A. M. Ferrenberg, R. H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988). 24 P. Labastie and R. L. Whetten, Phys. Rev. Lett. 65, 1567 (1990). 25 A. Vichare, D. G. Kanhere, and S. A. Blundell, Phys. Rev. B 64, 045408 (2001). http://arxiv.org/abs/cond-mat/0612287

Density-functional theory has been applied to investigate systematics of sodium clusters Na_n in the size range of n= 39-55. A clear evolutionary trend in the growth of their ground-state geometries emerges. The clusters at the beginning of the series (n=39-43) are symmetric and have partial icosahedral (two-shell) structure. The growth then goes through a series of disordered clusters (n=44-52) where the icosahedral core is lost. However, for n>52 a three shell icosahedral structure emerges. This change in the nature of the geometry is abrupt. In addition, density-functional molecular dynamics has been used to calculate the specific heat curves for the representative sizes n= 43, 45, 48 and 52. These results along with already available thermodynamic calculations for n= 40, 50, and 55 enable us to carry out a detailed comparison of the heat capacity curves with their respective geometries for the entire series. Our results clearly bring out strong correlation between the evolution of the geometries and the nature of the shape of the heat capacities. The results also firmly establish the size-sensitive nature of the heat capacities in sodium clusters.

Introduction to Cluster Dy- namics (Wiley-VCH, Berlin, 2003) 2 M. Schmidt, R. Kusche, B. von. Issendorff, and H. Haber- land, Nature (London) 393, 238 (1998); M. Schmidt and H. Haberland, C. R. Phys. 3, 327 (2002); H. Haberland, T. Hippler, J. Donges, O. Kostko, M. Schmidt and B. von Issendroff, Phys. Rev. Lett. 94, 035701 (2005) 3 F. Calvo and F. Spiegelmann, J. Chem. Phys. 112, 2888 (2000) 4 Y. Li, E. Blaisten-Barojas, and D. A. Papaconstantopou- los, Phys. Rev. B 57, 15519 (1998). 5 A. Rytkönen, H. Häkkinen, and M. Manninen, Phys. Rev. Lett. 80, 3940 (1998). 6 S. Chacko, D. G. Kanhere, and S. A. Blundell, Phys. Rev. B 71, 155407 (2005). 7 M.-S. Lee, S. Chacko, and D. G. Kanhere, J. Chem. Phys. 123, 164310 (2005). 8 M.-S. Lee and D. G. Kanhere, Phys. Rev. B 75, 125427 (2007). 9 A. Aguado and J. M. López, Phys. Rev. Lett. 94, 233401 (2005). 10 A. Aguado and J. M. López, Phys. Rev. B 74, 115403 (2006). 11 G. A. Breaux, D. A. Hillman, C. M. Neal, R. C. Benirschke, and M. F. Jarrold, J. Am. Chem. Soc. 126, 8628 (2004). 12 K. Joshi, S. Krishnamurty, and D. G. Kanhere, Phys. Rev. Lett. 96, 135703 (2006). 13 G. A. Breaux, C. M. Neal, B. Cao, and M. F. Jarrold, Phys. Rev. Lett. 94, 173401 (2005). 14 S. Krishnamurty, G. Shafai, D. G. Kanhere, and M. J. Ford, cond-mat/0612287 (to be published). 15 U. Röthlisberger and W. Andreoni, J. Chem. Phys. 94, 8129 (1991) 16 M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, Rev. Mod. Phys. 64, 1054 (1992). 17 D. Vanderbilt, Phys. Rev. B 41, 7892 (1990). 18 Vienna ab initio simulation package, Technische Univer- sität Wien (1999); G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). 19 Z. Li and H. A. Scheraga, Proc. Natl. Acad. Sci. 84, 6611 (1987); D. J. Wales and J. P. K. Doye, J. Phys. Chem. A 101, 5111 (1997). 20 M. Iwamatsu, J. Chem. Phys. 112, 10976 (2000) 21 J. A. Niesse and H. R. Mayne, J. Chem. Phys. 105, 4700 (1996) 22 R. G. Parr andW. Yang, The Density Functional Theory of Atoms and Molecules (Oxford university Press, New York, 1989). 23 A. M. Ferrenberg, R. H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988). 24 P. Labastie and R. L. Whetten, Phys. Rev. Lett. 65, 1567 (1990). 25 A. Vichare, D. G. Kanhere, and S. A. Blundell, Phys. Rev. B 64, 045408 (2001). http://arxiv.org/abs/cond-mat/0612287

arXiv:0704.1551v1 [quant-ph] 12 Apr 2007 Quantum Zeno Effect in the Decoherent Histories Petros Wallden∗ Abstract The quantum Zeno effect arises due to frequent observation. That implies the existence of some experimenter and its interaction with the system. In this contribution, we examine what happens for a closed system if one considers a quantum Zeno type of question, namely what is the probability of a system, remaining always in a particular subspace. This has implications to the arrival time problem that is also discussed. We employ the decoherent histories approach to quantum theory, as this is the better developed formulation of closed system quantum mechanics, and in particular, dealing with questions that involve time in a non-trivial way. We get a very restrictive decoherence condition, that implies that even if we do introduce an environment, there will be very few cases that we can assign probabilities to these histories, but in those cases, the quantum Zeno effect is still present. 1 Motivation A remarkable property of quantum mechanics, is the so called quantum Zeno effect [1]. This effect, is that frequent observation slow down the evolution of the state, with the limit of continuous observation leading to “freezing” of the state1. This has been experimentally verified. The intuitive explanation, is that the interaction of the observer with the system leads to this apparent paradox. It would therefore be interesting to see whether this effect persists if we consider a closed system. We would try to see what is the probability of a closed system remaining in a particular subspace of its Hilbert space with no external observer. This directly relates to the arrival time problem as well (e.g. [2, 3]). Having said that, we should emphasize that in closed systems, we cannot in general assign probabilities to histories, unless they decohere and it is this property that resolves the apparent paradox that arises. 2 This paper This contribution is largely based on Ref.[3]. In Section 3 we revise the quantum Zeno effect and the decoherent histories, and introduce a new formula for the restricted propagator that will be of use further. In Section 4.1 we see what probabilities we would get if we had decoherence, that highlights the persistence of the quantum Zeno effect. In Section 4.2 we get the decoherence condition that in Section 5 is stressed how restrictive is by considering the arrival time problem. We conclude in Section 6. ∗Raman Research Institute, Theoretical Physics Group; Sadashivanagar, Bangalore - 560 080, India; on leave from: Imperial College, Theoretical Physics Group; Blackett Laboratory, London SW7 2BZ, UK; email: petros.wallden@gmail.com 1To be more precise, restriction to a subspace. http://arxiv.org/abs/0704.1551v1 3 Introductory material 3.1 Quantum Zeno effect In standard Copenhagen quantum mechanics, the measurement is represented by pro- jecting the state to a subspace defined by the eigenstates that correspond to the range of eigenvalues of the measured physical quantity. The latter is represented by a self- adjoint operator. The state, otherwise evolves unitarily: Û(t) = exp(−iĤt), where Ĥ is the Hamiltonian. It is then a mathematical fact, that frequent measurement, of the same quantity (subspace) leads to slow down of the evolution, i.e. decreases the probability that the state evolves outside the subspace in question. This resembles the ancient Greek, Zeno paradox (Zήνων), and thus the name. The continuum measurement limit, leads to zero probability of leaving the observed subspace. The state continues to evolve (unitarily), but restricted in the subspace of observation [4]. This implies that if we project to a one-dimensional subspace, the state stops evolving. In most literature, the question is of a particle decaying or not, so the last comment applies. In particular, the above phenomenon is still present for infinite dimensional Hilbert spaces, but provided that the restricted Hamiltonian (Hr = PHP ) is self-adjoint, as we will see later. 3.2 Decoherent histories Decoherent histories approach to quantum theory is an alternative formulation de- signed to deal with closed systems and it was developed by Griffiths [5], Omnès [6], and Gell-Mann and Hartle [7]. There is no external observer, no a-priori environment- system split. The main mathematical aim of this approach, is to see when is it mean- ingful to assign probabilities to a history of a closed quantum system and of course to determine this probability. Here we will revise the standard non-relativistic quantum mechanics in decoherent histories formulation. To each history (α) corresponds a particular class operator Cα, Cα = Pαne −iH(tn−tn−1)Pαn−1 · · · e −iH(t2−t1)Pα1 (1) Where Pα1 etc are projection operators corresponding to some observable, H is the Hamiltonian, and tn is the total time interval we consider. This class operator corre- sponds to the history, the system is at the subspace spanned by Pα1 at time t1 at Pα2 at time t2 and so on. The probability for this history, provided we had some external observer making the measurement at each time tk would be p(α) = D(α, α) = Tr(CαρC α) (2) where ρ is the initial state. In the case of a closed system, Eq.(2) fails in general to be probability due to interference2. There are, however, certain cases where we can assign probabilities. This happens if for a complete set of histories, they pairwise obey D(α, β) = Tr(CαρC ) = 0 ∀ α 6= β (3) In that case, the complete set of histories is called decoherent set of histories and we can assign to each history of this set the probability of Eq.(2). In order to achieve a set 2The additivity of disjoint regions of the sample space is not satisfied by Eq.(2) of histories that satisfy Eq.(3) in general we need to consider coarse grained histories, or/and very specific initial state ρ3. To sum up, in decoherent histories we need to first construct a class operators that corresponds to the histories of interest4, and then confirm that these histories satisfy Eq.(3). Only then we can give an answer. 3.3 The restricted propagator A mathematical object that will be needed for computing the suitable class operators, is the restricted propagator. This is the propagator restricted to some particular region ∆ (of the configuration space) that corresponds to a subspace of the total Hilbert space denoted by H∆. The most common (but not the most general) is the path integral definition: gr(x, t | x0, t0) = Dx exp(iS[x(t)]) = 〈x|gr(t, t0)|x0〉 (4) The integration is done over paths that remain in the region ∆ during the time interval [t, t0]. The S[x(t)] is as usual the action. The operator form of the above is given by [8, 9]: gr(t, t0) = lim Pe−iH(tn−tn−1)P · · ·Pe−iH(t1−t0)P (5) With tn = t, δt → 0 and n → ∞ simultaneously keeping δt × n = (t − t0). H is the Hamiltonian operator. P is a projection operator on the restricted region ∆. We therefore have gr(x, t | x0, t0) = 〈x|gr(t, t0)|x0〉 (6) Note here that the expression Eq.(5) is the defining one for cases that the restricted region is not a region of the configuration space, but some other subspace of the total Hilbert space H. The differential equation obeyed by the restricted propagator is: −H)gr(t, t0) = [P,H ]gr(t, t0) (7) Which is almost the Schrödinger equation, differing by the commutator of the projec- tion to the restricted region with the Hamiltonian. The most useful form, for our discussion was derived in Ref. [3] gr(t, t0) = P exp (−i(t − t0)PHP )P (8) Note that PHP is the Hamiltonian projected in the subspace H∆. To prove Eq.(8) we multiply Eq. (7) with P we will then get − PHP )gr(t, t0) = 0 (9) using the fact that P [H,P ]P = 0 and that the propagator has a projection P at the final time. This is Schrödinger equation with Hamiltonian PHP . It is evident that 3Note that the interaction of a system with an environment that brings decoherence, in the histories vocabulary, is just a particular type of coarse graining where we ignore the environments degrees of freedom. 4Note that the same classical question can be turned to quantum with several, possibly inequivalent ways. Due to this property, the construction of the suitable class operator is important for questions such as for example, the arrival time or reparametrization invariant questions. this leads to the full propagator in H∆ provided that the operator PHP is self-adjoint in this subspace [4]5. 4 Quantum Zeno histories In this section we will examine the question what is the probability for a system to remain in a particular subspace, during a time interval ∆t = t − t0. We will see the probabilities and decoherence conditions for the general case, and then see what this implies for the arrival time problem, which is just a particular example. 4.1 The class operator and probabilities There are several ways of turning the above classical proposition to a quantum me- chanical one. The most straight forward is the following. We consider a system being in one subspace by projecting to that, and the history of always remaining in that subspace corresponds to the limit of projecting to the region evolving unitarily but for infinitesimal time and then projecting again, i.e. taking the δt between the proposi- tions going to zero. The class operator for remaining always in that subspace follows from Eq.(1) by taking each Pαk being the same (P ) and taking the limit of (tk − tk−1) going to zero for each k. We then have Cα(t, t0) = gr(t, t0) (10) and the class operator for not remaining at this subspace during all the interval is naturally Cβ(t, t0) = g(t, t0)− gr(t, t0) (11) with g(t, t0) = exp(−iH(t− t0)) the full propagator. Let us, for the moment, assume that the initial state |ψ〉 is such, that we do have decoherence. We will return later to see when this is the case. The (candidate) probability is p(α) = 〈ψ|g†r(t, t0)gr(t, t0)|ψ〉 (12) Following Eq.(8) it is clear6 that g†r(t, t0)gr(t, t0) = P (13) which then implies p(α) = 〈ψ|P |ψ〉 (14) For an initial state that is in the subspace defined by P , the probability to remain in this subspace is one. This is the usual account of the quantum Zeno effect. As it is stressed in other literature, to have the quantum Zeno is crucial that the restricted Hamiltonian Hr = PHP to be self-adjoint operator in the subspace. Note, that this only states that the system remain in the subspace, but it does not “freeze” completely and in particular follows unitary evolution in the subspace with Hamiltonian, the restricted 5A detailed proof from Eq.(5) can be found in [3]. 6Provided PHP is self-adjoint in the subspace. This is true for finite dimensional Hilbert spaces and has been shown to be true for regions of the configuration space in a Hamiltonian with at most quadratic momenta [4]. one Hr. The form of Eq.(8) of the restricted propagator makes the latter comment more transparent. 4.2 Decoherence condition All this is well understood for open systems with external observers. To assign the candidate probability of Eq.(12) as a proper probability of a closed system, we need the system to obey the decoherence condition, i.e. D(α, β) = 〈ψ|C βCα|ψ〉 = 0 (15) and this implies that 〈ψ|g†r(t, t0)g(t, t0)|ψ〉 = 〈ψ|P |ψ〉 (16) which is a very restrictive condition and only very few states satisfy this, as we will see in the arrival time example. The condition, essentially states that the overlap of the time evolved state (g(t, t0)|ψ〉) with the state evolved in the subspace (gr(t, t0)|ψ〉) should be the same at the times t0 and t. Given that the restricted Hamiltonian leads, in general, to different evolution, the condition refers only to very special initial states with symmetries, or for particular time intervals ∆t. 5 Arrival time problem The arrival time problem is the following: What is the probability that the system crosses a particular region ∆ of the configuration space, at any time during the time interval ∆t = (t − t0). One can attempt to answer this, by considering what is the probability that the system remains always in the complementary region ∆̄. So if Q is the total configuration space, we have ∆ ∪ ∆̄ = Q and ∆ ∩ ∆̄ = ∅. Taking this approach to the arrival time problem, the relation with the quantum Zeno histories is apparent, since it is just the special case, where the subspace of projection is a region of the configuration space (∆̄) and the Hamiltonian is quadratic in momenta, i.e. |x〉〈x|dx Ĥ = p̂2/2m+ V (x̂) (17) This particular case is infinite dimensional, but as shown in Ref. [4] the restricted Hamiltonian is indeed self-adjoint and the arguments of the previous section apply. Before proceeding further, we should point out that one could construct different class operators that would also correspond to the (classical) arrival time question. For example, one could consider having POVM’s7 instead of projections at each moment of time, or could have a finite (but frequent) number of projections (not taking the limit where δt→ 0). These and other approaches are not discussed here. Let us see now, what the quantum Zeno effect implies about the arrival time. It states that a system initially localized outside ∆ will always remain outside ∆ (if it decoheres) and therefore we can only get zero crossing probabilities. This is definitely surprising, since for a wave packet that is initially localized in ∆̄ and its classical trajectory crosses region ∆, we would expect to get crossing probability one. The resolution comes due to the decoherence condition as will be argued later. 7Positive Operator Valued Measure Returning to the decoherence condition Eq.(16) we see that there is the overlap of the time evolved state with the restricted time evolved state. In the arrival time case, the restricted Hamiltonian corresponds to the Hamiltonian in the restricted region (∆̄) but with infinite potential walls on the boundary (i.e. perfectly reflecting). We then get decoherence in the following four cases. (a) The initial state |ψ〉 is in an energy eigenstate, and it also vanishes on the bound- ary of the region. (b) The restricted propagator can be expressed by the method of images8 and the initial state shares the same symmetry. (c) The full unitary evolution in the time interval ∆t remains in the region ∆̄. (d) Recurrence: Due to the period of the Hamiltonian and the restricted Hamiltonian their overlap happens to be the same after some time t as it was in time t0. This depends sensitively on the time interval and it is thus of less physical significance. It is now apparent that most initial states do not satisfy any of those conditions. In particular, the wavepacket that classically would cross the region ∆, will not satisfy any of these conditions, and we would not be able to assign the candidate probability as a proper one, and thus we avoid the paradox. The introduction of an interacting environment to our system, (that usually produces decoherence by coarse-graining the environment) does not change the probabilities and contrary to the intuitive feeling, it does not provide decoherence for the particular type of question we consider. This still leave us with no answer for any of the cases that the system would classically cross the region. The latter implies, that the straight forward coarse grainings we used, were not general enough to answer fully the arrival time question 9. As a final note, we should point out that the quantum Zeno effect in the decoherent histories, has implications for the decoherent histories approach to the problem of time (e.g. Refs. [9, 3]). 6 Conclusions We examined the quantum Zeno type of histories of a closed system, using the deco- herent histories approach. We show that the quantum Zeno effect is still present, but only for the very few cases that we have decoherence. The situation does not change with the introduction of interacting environment. We see that while in the open sys- tem quantum Zeno, the delay of the evolution arises as interaction with the observer, in the closed system we have the decoherence condition “replacing” the observer and resolving the apparent paradox. Acknowledgments: The author is very grateful to Jonathan J. Halliwell for many useful discussions and suggestions, and would like to thank the organizers for giving the opportunity to give this talk and hosting this very interesting and nice conference. 8Note that the restricted propagator can be expressed using the method of images, if and only if there exist a set of energy eigenstates, vanishing on the boundary, that when projected on the region ∆̄ forms a dense subset of the subspace H , i.e. span H . This is equivalent with requiring that the restricted energy spectrum (i.e. spectrum of the restricted Hamiltonian H ) is a subset of the (unrestricted) energy spectrum, which is not in general the case. 9For more details, examples and discussion see Ref. [3]. References [1] B. Misra and E.C.G. Sudarshan, J. Math. Phys. 18, 756 (1977). [2] J.J.Halliwell and E.Zafiris, Phys.Rev. D57, 3351 (1998). [3] P. Wallden, gr-qc/0607072. [4] P. Facchi, S. Pascazio, A. Scardicchio, and L. S. Schulman, Phys. Rev. A 65, 012108 (2002). [5] R.B. Griffiths, J. Stat. Phys., 36 219, (1984). [6] R. Omnès. J. Stat. Phys. 53, 893 (1988). [7] M. Gell-Mann and J. Hartle, in Complexity, Entropy and the Physics of Infor- mation, SFI Studies in the Science of Complexity, Vol. VIII, edited by W. Zurek, (Addison-Wesley, Reading, 1990). [8] J. J. Halliwell, quant-ph/9506021. [9] J. J. Halliwell and P. Wallden, Phys. Rev. D 73 024011 (2006), gr-qc/0509013.

The quantum Zeno effect arises due to frequent observation. That implies the existence of some experimenter and its interaction with the system. In this contribution, we examine what happens for a closed system if one considers a quantum Zeno type of question, namely: "what is the probability of a system, remaining always in a particular subspace". This has implications to the arrival time problem that is also discussed. We employ the decoherent histories approach to quantum theory, as this is the better developed formulation of closed system quantum mechanics, and in particular, dealing with questions that involve time in a non-trivial way. We get a very restrictive decoherence condition, that implies that even if we do introduce an environment, there will be very few cases that we can assign probabilities to these histories, but in those cases, the quantum Zeno effect is still present.

arXiv:0704.1551v1 [quant-ph] 12 Apr 2007 Quantum Zeno Effect in the Decoherent Histories Petros Wallden∗ Abstract The quantum Zeno effect arises due to frequent observation. That implies the existence of some experimenter and its interaction with the system. In this contribution, we examine what happens for a closed system if one considers a quantum Zeno type of question, namely what is the probability of a system, remaining always in a particular subspace. This has implications to the arrival time problem that is also discussed. We employ the decoherent histories approach to quantum theory, as this is the better developed formulation of closed system quantum mechanics, and in particular, dealing with questions that involve time in a non-trivial way. We get a very restrictive decoherence condition, that implies that even if we do introduce an environment, there will be very few cases that we can assign probabilities to these histories, but in those cases, the quantum Zeno effect is still present. 1 Motivation A remarkable property of quantum mechanics, is the so called quantum Zeno effect [1]. This effect, is that frequent observation slow down the evolution of the state, with the limit of continuous observation leading to “freezing” of the state1. This has been experimentally verified. The intuitive explanation, is that the interaction of the observer with the system leads to this apparent paradox. It would therefore be interesting to see whether this effect persists if we consider a closed system. We would try to see what is the probability of a closed system remaining in a particular subspace of its Hilbert space with no external observer. This directly relates to the arrival time problem as well (e.g. [2, 3]). Having said that, we should emphasize that in closed systems, we cannot in general assign probabilities to histories, unless they decohere and it is this property that resolves the apparent paradox that arises. 2 This paper This contribution is largely based on Ref.[3]. In Section 3 we revise the quantum Zeno effect and the decoherent histories, and introduce a new formula for the restricted propagator that will be of use further. In Section 4.1 we see what probabilities we would get if we had decoherence, that highlights the persistence of the quantum Zeno effect. In Section 4.2 we get the decoherence condition that in Section 5 is stressed how restrictive is by considering the arrival time problem. We conclude in Section 6. ∗Raman Research Institute, Theoretical Physics Group; Sadashivanagar, Bangalore - 560 080, India; on leave from: Imperial College, Theoretical Physics Group; Blackett Laboratory, London SW7 2BZ, UK; email: petros.wallden@gmail.com 1To be more precise, restriction to a subspace. http://arxiv.org/abs/0704.1551v1 3 Introductory material 3.1 Quantum Zeno effect In standard Copenhagen quantum mechanics, the measurement is represented by pro- jecting the state to a subspace defined by the eigenstates that correspond to the range of eigenvalues of the measured physical quantity. The latter is represented by a self- adjoint operator. The state, otherwise evolves unitarily: Û(t) = exp(−iĤt), where Ĥ is the Hamiltonian. It is then a mathematical fact, that frequent measurement, of the same quantity (subspace) leads to slow down of the evolution, i.e. decreases the probability that the state evolves outside the subspace in question. This resembles the ancient Greek, Zeno paradox (Zήνων), and thus the name. The continuum measurement limit, leads to zero probability of leaving the observed subspace. The state continues to evolve (unitarily), but restricted in the subspace of observation [4]. This implies that if we project to a one-dimensional subspace, the state stops evolving. In most literature, the question is of a particle decaying or not, so the last comment applies. In particular, the above phenomenon is still present for infinite dimensional Hilbert spaces, but provided that the restricted Hamiltonian (Hr = PHP ) is self-adjoint, as we will see later. 3.2 Decoherent histories Decoherent histories approach to quantum theory is an alternative formulation de- signed to deal with closed systems and it was developed by Griffiths [5], Omnès [6], and Gell-Mann and Hartle [7]. There is no external observer, no a-priori environment- system split. The main mathematical aim of this approach, is to see when is it mean- ingful to assign probabilities to a history of a closed quantum system and of course to determine this probability. Here we will revise the standard non-relativistic quantum mechanics in decoherent histories formulation. To each history (α) corresponds a particular class operator Cα, Cα = Pαne −iH(tn−tn−1)Pαn−1 · · · e −iH(t2−t1)Pα1 (1) Where Pα1 etc are projection operators corresponding to some observable, H is the Hamiltonian, and tn is the total time interval we consider. This class operator corre- sponds to the history, the system is at the subspace spanned by Pα1 at time t1 at Pα2 at time t2 and so on. The probability for this history, provided we had some external observer making the measurement at each time tk would be p(α) = D(α, α) = Tr(CαρC α) (2) where ρ is the initial state. In the case of a closed system, Eq.(2) fails in general to be probability due to interference2. There are, however, certain cases where we can assign probabilities. This happens if for a complete set of histories, they pairwise obey D(α, β) = Tr(CαρC ) = 0 ∀ α 6= β (3) In that case, the complete set of histories is called decoherent set of histories and we can assign to each history of this set the probability of Eq.(2). In order to achieve a set 2The additivity of disjoint regions of the sample space is not satisfied by Eq.(2) of histories that satisfy Eq.(3) in general we need to consider coarse grained histories, or/and very specific initial state ρ3. To sum up, in decoherent histories we need to first construct a class operators that corresponds to the histories of interest4, and then confirm that these histories satisfy Eq.(3). Only then we can give an answer. 3.3 The restricted propagator A mathematical object that will be needed for computing the suitable class operators, is the restricted propagator. This is the propagator restricted to some particular region ∆ (of the configuration space) that corresponds to a subspace of the total Hilbert space denoted by H∆. The most common (but not the most general) is the path integral definition: gr(x, t | x0, t0) = Dx exp(iS[x(t)]) = 〈x|gr(t, t0)|x0〉 (4) The integration is done over paths that remain in the region ∆ during the time interval [t, t0]. The S[x(t)] is as usual the action. The operator form of the above is given by [8, 9]: gr(t, t0) = lim Pe−iH(tn−tn−1)P · · ·Pe−iH(t1−t0)P (5) With tn = t, δt → 0 and n → ∞ simultaneously keeping δt × n = (t − t0). H is the Hamiltonian operator. P is a projection operator on the restricted region ∆. We therefore have gr(x, t | x0, t0) = 〈x|gr(t, t0)|x0〉 (6) Note here that the expression Eq.(5) is the defining one for cases that the restricted region is not a region of the configuration space, but some other subspace of the total Hilbert space H. The differential equation obeyed by the restricted propagator is: −H)gr(t, t0) = [P,H ]gr(t, t0) (7) Which is almost the Schrödinger equation, differing by the commutator of the projec- tion to the restricted region with the Hamiltonian. The most useful form, for our discussion was derived in Ref. [3] gr(t, t0) = P exp (−i(t − t0)PHP )P (8) Note that PHP is the Hamiltonian projected in the subspace H∆. To prove Eq.(8) we multiply Eq. (7) with P we will then get − PHP )gr(t, t0) = 0 (9) using the fact that P [H,P ]P = 0 and that the propagator has a projection P at the final time. This is Schrödinger equation with Hamiltonian PHP . It is evident that 3Note that the interaction of a system with an environment that brings decoherence, in the histories vocabulary, is just a particular type of coarse graining where we ignore the environments degrees of freedom. 4Note that the same classical question can be turned to quantum with several, possibly inequivalent ways. Due to this property, the construction of the suitable class operator is important for questions such as for example, the arrival time or reparametrization invariant questions. this leads to the full propagator in H∆ provided that the operator PHP is self-adjoint in this subspace [4]5. 4 Quantum Zeno histories In this section we will examine the question what is the probability for a system to remain in a particular subspace, during a time interval ∆t = t − t0. We will see the probabilities and decoherence conditions for the general case, and then see what this implies for the arrival time problem, which is just a particular example. 4.1 The class operator and probabilities There are several ways of turning the above classical proposition to a quantum me- chanical one. The most straight forward is the following. We consider a system being in one subspace by projecting to that, and the history of always remaining in that subspace corresponds to the limit of projecting to the region evolving unitarily but for infinitesimal time and then projecting again, i.e. taking the δt between the proposi- tions going to zero. The class operator for remaining always in that subspace follows from Eq.(1) by taking each Pαk being the same (P ) and taking the limit of (tk − tk−1) going to zero for each k. We then have Cα(t, t0) = gr(t, t0) (10) and the class operator for not remaining at this subspace during all the interval is naturally Cβ(t, t0) = g(t, t0)− gr(t, t0) (11) with g(t, t0) = exp(−iH(t− t0)) the full propagator. Let us, for the moment, assume that the initial state |ψ〉 is such, that we do have decoherence. We will return later to see when this is the case. The (candidate) probability is p(α) = 〈ψ|g†r(t, t0)gr(t, t0)|ψ〉 (12) Following Eq.(8) it is clear6 that g†r(t, t0)gr(t, t0) = P (13) which then implies p(α) = 〈ψ|P |ψ〉 (14) For an initial state that is in the subspace defined by P , the probability to remain in this subspace is one. This is the usual account of the quantum Zeno effect. As it is stressed in other literature, to have the quantum Zeno is crucial that the restricted Hamiltonian Hr = PHP to be self-adjoint operator in the subspace. Note, that this only states that the system remain in the subspace, but it does not “freeze” completely and in particular follows unitary evolution in the subspace with Hamiltonian, the restricted 5A detailed proof from Eq.(5) can be found in [3]. 6Provided PHP is self-adjoint in the subspace. This is true for finite dimensional Hilbert spaces and has been shown to be true for regions of the configuration space in a Hamiltonian with at most quadratic momenta [4]. one Hr. The form of Eq.(8) of the restricted propagator makes the latter comment more transparent. 4.2 Decoherence condition All this is well understood for open systems with external observers. To assign the candidate probability of Eq.(12) as a proper probability of a closed system, we need the system to obey the decoherence condition, i.e. D(α, β) = 〈ψ|C βCα|ψ〉 = 0 (15) and this implies that 〈ψ|g†r(t, t0)g(t, t0)|ψ〉 = 〈ψ|P |ψ〉 (16) which is a very restrictive condition and only very few states satisfy this, as we will see in the arrival time example. The condition, essentially states that the overlap of the time evolved state (g(t, t0)|ψ〉) with the state evolved in the subspace (gr(t, t0)|ψ〉) should be the same at the times t0 and t. Given that the restricted Hamiltonian leads, in general, to different evolution, the condition refers only to very special initial states with symmetries, or for particular time intervals ∆t. 5 Arrival time problem The arrival time problem is the following: What is the probability that the system crosses a particular region ∆ of the configuration space, at any time during the time interval ∆t = (t − t0). One can attempt to answer this, by considering what is the probability that the system remains always in the complementary region ∆̄. So if Q is the total configuration space, we have ∆ ∪ ∆̄ = Q and ∆ ∩ ∆̄ = ∅. Taking this approach to the arrival time problem, the relation with the quantum Zeno histories is apparent, since it is just the special case, where the subspace of projection is a region of the configuration space (∆̄) and the Hamiltonian is quadratic in momenta, i.e. |x〉〈x|dx Ĥ = p̂2/2m+ V (x̂) (17) This particular case is infinite dimensional, but as shown in Ref. [4] the restricted Hamiltonian is indeed self-adjoint and the arguments of the previous section apply. Before proceeding further, we should point out that one could construct different class operators that would also correspond to the (classical) arrival time question. For example, one could consider having POVM’s7 instead of projections at each moment of time, or could have a finite (but frequent) number of projections (not taking the limit where δt→ 0). These and other approaches are not discussed here. Let us see now, what the quantum Zeno effect implies about the arrival time. It states that a system initially localized outside ∆ will always remain outside ∆ (if it decoheres) and therefore we can only get zero crossing probabilities. This is definitely surprising, since for a wave packet that is initially localized in ∆̄ and its classical trajectory crosses region ∆, we would expect to get crossing probability one. The resolution comes due to the decoherence condition as will be argued later. 7Positive Operator Valued Measure Returning to the decoherence condition Eq.(16) we see that there is the overlap of the time evolved state with the restricted time evolved state. In the arrival time case, the restricted Hamiltonian corresponds to the Hamiltonian in the restricted region (∆̄) but with infinite potential walls on the boundary (i.e. perfectly reflecting). We then get decoherence in the following four cases. (a) The initial state |ψ〉 is in an energy eigenstate, and it also vanishes on the bound- ary of the region. (b) The restricted propagator can be expressed by the method of images8 and the initial state shares the same symmetry. (c) The full unitary evolution in the time interval ∆t remains in the region ∆̄. (d) Recurrence: Due to the period of the Hamiltonian and the restricted Hamiltonian their overlap happens to be the same after some time t as it was in time t0. This depends sensitively on the time interval and it is thus of less physical significance. It is now apparent that most initial states do not satisfy any of those conditions. In particular, the wavepacket that classically would cross the region ∆, will not satisfy any of these conditions, and we would not be able to assign the candidate probability as a proper one, and thus we avoid the paradox. The introduction of an interacting environment to our system, (that usually produces decoherence by coarse-graining the environment) does not change the probabilities and contrary to the intuitive feeling, it does not provide decoherence for the particular type of question we consider. This still leave us with no answer for any of the cases that the system would classically cross the region. The latter implies, that the straight forward coarse grainings we used, were not general enough to answer fully the arrival time question 9. As a final note, we should point out that the quantum Zeno effect in the decoherent histories, has implications for the decoherent histories approach to the problem of time (e.g. Refs. [9, 3]). 6 Conclusions We examined the quantum Zeno type of histories of a closed system, using the deco- herent histories approach. We show that the quantum Zeno effect is still present, but only for the very few cases that we have decoherence. The situation does not change with the introduction of interacting environment. We see that while in the open sys- tem quantum Zeno, the delay of the evolution arises as interaction with the observer, in the closed system we have the decoherence condition “replacing” the observer and resolving the apparent paradox. Acknowledgments: The author is very grateful to Jonathan J. Halliwell for many useful discussions and suggestions, and would like to thank the organizers for giving the opportunity to give this talk and hosting this very interesting and nice conference. 8Note that the restricted propagator can be expressed using the method of images, if and only if there exist a set of energy eigenstates, vanishing on the boundary, that when projected on the region ∆̄ forms a dense subset of the subspace H , i.e. span H . This is equivalent with requiring that the restricted energy spectrum (i.e. spectrum of the restricted Hamiltonian H ) is a subset of the (unrestricted) energy spectrum, which is not in general the case. 9For more details, examples and discussion see Ref. [3]. References [1] B. Misra and E.C.G. Sudarshan, J. Math. Phys. 18, 756 (1977). [2] J.J.Halliwell and E.Zafiris, Phys.Rev. D57, 3351 (1998). [3] P. Wallden, gr-qc/0607072. [4] P. Facchi, S. Pascazio, A. Scardicchio, and L. S. Schulman, Phys. Rev. A 65, 012108 (2002). [5] R.B. Griffiths, J. Stat. Phys., 36 219, (1984). [6] R. Omnès. J. Stat. Phys. 53, 893 (1988). [7] M. Gell-Mann and J. Hartle, in Complexity, Entropy and the Physics of Infor- mation, SFI Studies in the Science of Complexity, Vol. VIII, edited by W. Zurek, (Addison-Wesley, Reading, 1990). [8] J. J. Halliwell, quant-ph/9506021. [9] J. J. Halliwell and P. Wallden, Phys. Rev. D 73 024011 (2006), gr-qc/0509013.

Green Function theory vs. Quantum Monte Carlo Calculation for thin magnetic films S. Henning,∗ F. Körmann, J. Kienert, and W. Nolting Lehrstuhl Festkörpertheorie, Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany S. Schwieger Technische Universität Ilmenau, Theoretische Physik I, Postfach 10 05 65, 98684 Ilmenau, Germany (Dated: October 31, 2018) In this work we compare numerically exact Quantum Monte Carlo (QMC) calculations and Green function theory (GFT) calculations of thin ferromagnetic films including second order anisotropies. Thereby we concentrate on easy plane systems, i.e. systems for which the anisotropy favors a magnetization parallel to the film plane. We discuss these systems in perpendicular external field, i.e. B parallel to the film normal. GFT results are in good agreement with QMC for high enough fields and temperatures. Below a critical field or a critical temperature no collinear stable magnetization exists in GFT. On the other hand QMC gives finite magnetization even below those critical values. This indicates that there occurs a transition from non-collinear to collinear configurations with increasing field or temperature. For slightly tilted external fields a rotation of magnetization from out-of-plane to in-plane orientation is found with decreasing temperature. PACS numbers: 75.10.Jm, 75.40.Mg, 75.70.Ak, 75.30.Gw I. INTRODUCTION The fast development of technological applications based on magnetic systems in the last years, e.g. magnetic data storage devices, causes a high interest in thin magnetic films. One precondition for the tech- nological development is the investigation of magnetic anisotropies and spin reorientation transitions connected therewith. Those reorientation transitions can occur from out-of-plane to in-plane or vice versa for increasing film thickness d1, temperature T 2,3,4,5,6,7,8, or external field B0. Quantum Monte Carlo (QMC) calculations give the possibility to compare numerically exact results with analytical approximations. In Ref. 9 the authors inves- tigated a ferromagnetic monolayer including positive second order anisotropy (easy axis perpendicular to the film plane). They discuss the temperature dependence of the magnetization 〈Sz〉(T ) as well as field induced reorientation transitions from out-of-plane to in-plane and compare the QMC results with Green function theory (GFT). They found good agreement in the case of applied external field in the easy direction (here z-axis). However, their GFT fails for external field applied in arbitrary direction, especially in the hard direction (within the film plane). As shown in Ref. 10 for getting closer to the QMC results for magnetic field induced reorientation from out-of-plane to in-plane a more careful treatment of the local anisotropy terms is needed. In Refs. 10,11,12,13 a decoupling scheme was presented which yields excellent agreement with QMC results for out-of-plane systems. The availability of theories such as GFT and their check against state-of-the-art numerical algorithms is highly desirable because of the size limitations of systems where QMC can be performed. On the other hand the extension of GFT from a monolayer (where it can be compared to QMC as in the the present work) to multilayer systems is a straightforward task without further approximations11. Up to now, to our knowledge, there is no comparison between QMC and approximative theories for easy-plane systems and it is not obvious that the theory presented in Refs. 10,11,12,13 can reproduce the QMC results for in-plane systems as accurately as for the out-of-plane case. In contrast to the easy-axis case where a certain direction is preferred by the single ion anisotropy in easy-plane systems the full xy-plane is favored and no particular direction is distinguished within the plane. A magnetic field applied perpendicular to the plane does not destroy the xy-symmetry. For systems exhibiting this kind of symmetry it was shown in a classical treatment that for external fields smaller than a critical field 0 ≤ B < Bcrit (B || z) stable vortices, i.e. a non-collinear arrangement of spins, can exist15,16,17,18,19. These vortices can undergo a Berezinskii-Kosterlitz-Thouless (BKT) transition14. Depending on the strength of the anisotropy K2 there might be vortices with or without a finite z-component of magnetization15. In the small anisotropy case (which is considered in this work, |K2| < 0.1J) there is a finite out-of-plane component and for zero field the two possible directions of magnetization (±z) are en- ergetically degenerate. For increasing magnetic field in z-direction the vortices antiparallel to the field become more and more unstable (heavy vortices). However the so called light vortices (parallel to the field) are stable up to a critical field Bz = Bcrit and contribute a finite z-component to the net-magnetization of the considered system19. The vortices in connection with a finite z-component of the net-magnetization emerge because of two reasons: first the competition between the anisotropy (favoring a orientation of the magnetization within the xy-plane) http://arxiv.org/abs/0704.1552v1 and the external field (favoring a perpendicular magne- tization), and second: the xy-symmetry of the system, which does not allow for a rotated homogeneous phase. In this paper we investigate both aspects, i.e. the field vs. anisotropy competition as well as the symmetry properties in detail for a quantum mechanical system. We will compare the results of QMC and GFT calcula- tions. As explained in more detail below, the QMC al- gorithm used here allows only for an external field applied in z-direction. Thus the xy-symmetry can not be broken and no comparison between xy-symmetric and asymmetric systems is possible. We will use GFT to clarify the influence of this symmetry breaking on the homogeneous phase. On the other hand, the GFT used here is by ansatz limited to the homogeneous phase. Therefore it can not describe a non-collinear (e.g, vortex-) magnetic phase, which is expected for B || z and small field strengths. The breakdown of magnetization in GFT as well as an exposed maximum in the magnetization in QMC at certain critical values of the external field or temperature gives, however, a clear fingerprint of non-collinear configurations, at least if there is no meta-stable homogeneous phase. Below these critical values there will be a finite z-component in QMC and a vanishing magnetization in GFT. For parameters, where both theories are applicable, QMC serves as a test for the approximations needed in In this work we find indications for non-collinear spin configurations below a critical field or temperature for B || z by comparing results of QMC and GFT as explained in the last clause. Above the critical field we obtain good agreement between QMC and GFT results. Breaking the xy-symmetry by adding a small x-component to the external field yields a stable collinear solution in GFT. The z-component of the magnetization in this case is in good agreement with the QMC results calculated with untilted field. Thus we can conclude that except for the restriction to collinear magnetic states GFT describes the competition between external field and anisotropy quite well. The paper is organized as follows: First we ex- plain the basics of the GFT and the QMC calculations. Then we apply both approaches to easy-plane systems in external magnetic fields and report the results of our calculations. II. THEORY A. Green Function Theory In the following we present our theoretical approach us- ing Green function theory. The focus of this work lies on the translational invariant system of a two-dimensional monolayer. Therefore the following Hamiltonian is used: H = − JijSiSj −B Si −K2 (Sz,i) 2. (1) The first term describes the Heisenberg coupling Jij be- tween spins Si and Sj located at sites i and j. The second term contains an external magnetic field B in arbitrary direction (the Landé factor gJ and the Bohr magneton µB are absorbed in B). The third term represents second order lattice anisotropy due to spin-orbit coupling. Sz,i is the z-component of Si (the z-axis of the coordinate system is oriented perpendicular to the film-plane). The lattice anisotropy favours in-plane (K2 < 0) or out-of- plane (K2 > 0) orientation. Our Hamiltonian is similar to that used in Refs. 10,11,13,22,23 for the investigation of the magnetic anisotropy and the field induced reori- entation transition. To simplify calculations we consider nearest neighbor coupling only Jij = J (i), (j) n.n. 0 otherwise. The main idea of the special treatment presented in Refs. 10,11,12,13 is that, before any decoupling is applied, the coordinate system Σ is rotated to a new system Σ′ where the new z′-axis is parallel to the magnetization im- plying a collinear alignment of all spins within the layer. Then a combination of Random Phase approximation (RPA)24 for the nonlocal terms in Eq. (1) (Heisenberg exchange interaction term) and Anderson-Callen approx- imation (AC)25 for the local lattice anisotropy term is applied in the rotated system. After application of the approximation one gets an effective anisotropy Keff (T ) = 2K2 S(S + 1)− 〈S2z′〉 〈Sz′〉 (3) where 〈Sz′〉 is the norm of the magnetization and S is the spin quantum number, that we have chosen to be S = 1 in all our calculations. As shown in comparison with an exact treatment of the local anisotropy term in Ref. 26 this approximation still holds up to anisotropy strengths K2 ∼ 1/2J . Therefore we restrict ourselves in the following to small anisotropies (K2 ≤ 0.1J) as found in most real materials 33. For a magnetic field applied in the xz-plane (B = (Bx, 0, Bz)) our theory gives a condition for the polar angle θ of the magnetization: sin θBz − cos θBx +Keff sin θ cos θ = 0 (4) The uniform magnon energies (q = 0) which dominate the physical behavior of the magnetic system can easily be extracted from the theory12,13: E2q=0 = cos θBz + sin θBx +Keff(cos 2 θ − sin2 θ) cos θBz + sin θBx +Keff cos This result coincides with the spin-wave result13 if one replaces 〈Sz′〉 by the spin quantum number S and Keff by the bare anisotropy constant K2 in Eq. (5). For an easy-plane system (Keff < 0) with external field B in z-direction the polar angle θ of the magnetization34 is given by: cos θ = −B/Keff(T ) for B < |Keff (T )| 1 otherwise By inserting Eq.(6) into Eq.(5) one immediately gets: Keff<0 q=0 (B) = 0 B < |Keff (T )| B +Keff (T ) otherwise. For gapless magnon energies Eq=0 = 0 the magnon oc- cupation number φ diverges (φ → ∞) in film systems with ferromagnetic coupling J > 0 and the magnetiza- tion becomes zero 〈Sz′〉 = 0 in the collinear phase. This can be seen by following an argument of Bloch20 already given in 1930. Since the spin wave dispersion is E ≈ q2 in the vicinity of q = 0 the spin-wave density of states N(E) is independent of E for a two-dimensional system for E close to zero. The excitation of spin-waves at finite temperature leads to a variation of the magnetization of the order: ∆m(T ) ∼ N(E)dE exp(E/kBT )− 1) ∼ kBT exp(x)− 1 . (8) Since the integral in Eq. (8) diverges for T 6= 0 and exited spin-waves lead to a reduction of the magnetization one can conclude that the magnetization should be zero at finite temperature. However for an infinitesimally small contribution of the external field parallel to the plane, i.e. Bx 6= 0, a finite gap in the excitation spectrum at q = 0 opens. This can be seen in Fig.1 where the uniform magnon modes Eq=0(B) are shown for different orientations θB, where θB is the polar angle of the ex- ternal field. The integral (8) is now finite and a stable finite magnetization in the collinear phase having a well defined orientation in the xz-plane is possible. Let us now come back to the case where the applied field is aligned in z-direction. It can be seen from Eq. (7) that for external field B (B || z) larger than a critical field B > Bcrit given by: Bcrit = |Keff (T,B)| (9) a stable collinear solution exists. Since Keff (T ) is a de- creasing function of temperature T a transition from non- collinear to collinear phase with increasing temperature is possible. In Fig. 2 we show the normalized critical field (9) Bcrit/K2 as a function of temperature T . For a constant magnetic field B (B || z) at a temperature T1 with B < Keff (T1, B) no stable collinear phase ex- ist. Then by increasing the temperature up to T2 the ef- fective anisotropy Keff is sufficiently reduced such that 0.01 0.02 0.03 0.04 B / J =0.2° FIG. 1: The energies of the uniform magnon mode Eq=0(B) for different polar angles θB of the external field. Eq=0 is zero below B/J ≈ 0.03 for θB = 0 ◦. The prefactors gJµB and kB are absorbed in B and T respectively. The latter are given in units of the nearest neighbor Heisenberg coupling J . Parameters: S = 1, K/J = −0.03 and T/J = 10−4. FIG. 2: The normalized critical field Bcrit/K2 as a function of temperature. Parameters: S = 1. B > Keff (T2, B), and the collinear phase becomes sta- ble. Before we come to the results let us briefly sketch the main aspects of the QMC. B. QMC In the last section we gave a short description of the theory used to treat a system described by a Hamilto- nian of form (1). This theory applies to the thermody- namic limit (films of infinte size) but contains certain approximations. Additionally the GFT is restricted to ordered phases with a collinear alignment of all spins. Therefore it would be very useful to have exact results at hand to crosscheck the predictions of GFT. A Quantum Monte Carlo method, particularly well suited for spin systems, is the stochastic series expansion (SSE) with directed loop update. We will sketch this method here only briefly as detailed descriptions can be already found elsewhere28,29,30. Our starting point is the series expansion of the parti- tion function Z = Tre−βH = 〈α|(−H)n|α〉 (10) whereH denotes the Hamiltonian, {|α〉} are basis vectors of a proper Hilbert space and β is the inverse tempera- ture. The Hamiltonian is then rewritten in terms of bond Hamiltonians: H = −J Hb (11) where Hb can be further decomposed into a diagonal and an off-diagonal part: HD,b = C + S i(b)S j(b) + bb[S i(b) + S j(b)] (12) +k2b[(S i(b)) 2 + (Szj(b)) HO,b = ] (13) Here we have renormalized the anisotropy constant k2b and the magnetic field bb in such a way that (11) coincides with (1). i(b) and j(b) denotes the lattice sites connected by the bond b and the additional constant C in HD,b will be chosen such that all matrix elements of this term be- come positive, a condition necessary to interpret them as probabilities. Note that for a finite system at finite tem- perature the power series of the partition function can be truncated at a finite cutoff length Λ without intro- ducing any systematic error in practical computations29. Therefore reinserting (11) into (10) and rewriting the re- sult yields: βn(Λ− n)! 〈α|SCΛ |α〉. (14) Here SCΛ denotes a product of operators (operator string) consisting of n non-unity operators and (Λ − n) unity operators H0 = Id which were inserted to get op- erator strings of equal length Λ. In fact it is impossible to evaluate all operator strings in (14). The SSE-QMC replaces such an evaluation there- fore by importance sampling over the strings according to their relative weight. Hence an efficient scheme for gen- erating new operator strings is needed. In the directed loop version of the SSE this is done by dividing the up- date into two parts. In a first step a diagonal update is performed by traversing the operator string and replac- ing some unity operators by diagonal bond operators and vice versa (the probabilities for both substitutions have to fulfill the detailed balance criterion). Then the loop update follows in which new non-diagonal bond opera- tors can appear in the operator string. For details of the update procedure we refer the interested reader to the 0 0.5 1 1.5 2 GFT (RPA+AC) QMC (N=16) QMC (N=32) QMC (N=64) FIG. 3: Magnetization vs. temperature for an out-of-plane easy-axis system (K2 > 0). Straight line: GFT (RPA+AC) result; symbols: QMC results for different system sizes N2. Parameters: S = 1, B/J = 0.01 (B || z) and K2/J = 0.01. according literature28,29,30. A full implementation of the SSE with directed loop update which we have used for all QMC calculations in this work can be found in the ALPS project30,31. Since the SSE-QMC used by us is implemented in z- representation (spin quantization axis along z-axis) in- plane correlation functions e.g. the in-plane magnetiza- tion are not accessible. Further B || z is the only possible field direction in the used QMC implementation because a traverse field (in-plane field component) would lead to non-closing loops (see Ref. 9). III. RESULTS As mentioned in Sec. II A the results for the in-plane systems are very sensitive to the effective anisotropy Keff (T ). This sensitivity of the anisotropy is less pro- nounced for out-of-plane systems (K2 > 0) since the ap- plied field B (B || z) and the intrinsic easy axis are par- allel. In order to test our decoupling scheme (RPA+AC) we first compare GFT and QMC for an out-of-plane system.35 In Fig. 3 the magnetization 〈Sz〉 as a function of tem- perature T is shown. The straight line belongs to the GFT whereas the symbols show the result of the QMC for different system sizes. Let us first comment on finite size effects in the QMC results. It can be seen in Fig. 3 that the QMC results converge for increasing system size N2 (for N ×N square lattice). Indeed forN ≥ 32 the QMC results are unbiased by finite size effects and resulting magnetization curves are almost equal for increasing N ≥ 32. Note that we have omitted error bars in the figures showing QMC results because the relative errors are of the order 10−4. We now compare the GFT with the QMC results (N = 64). For low temperatures (T/J ≤ 0.5) we ob- tain excellent quantitative agreement. This is plausible 0 0.05 0.1 0.15 0.2 QMC (16) QMC (32) QMC (64) QMC (128) FIG. 4: z-component of magnetization as a function of ex- ternal magnetic field for fixed temperature T/J = 0.4. In contrary to the GFT the magnetization obtained by QMC remains finite for all fields. The QMC results are converged for N ≥ 64. Parameters: S = 1, K2/J = −0.06 and θB = 0 0 0.05 0.1 0.15 0.2 GFT (θ = 0°) GFT (θ = 0.5°) QMC (128) FIG. 5: z-component of magnetization vs. external field for T/J = 0.4 with slightly tilted field (θB = 0.5 ◦)in the GFT result (solid line). The dotted line shows GFT result for (θB = 0◦). Other parameters as in Fig. 4. because in this region the GFT result coincides with the result of the spin-wave theory which is known to be re- liable (exact for T = 0) for low temperatures. For the intermediate region T/J = 0.5..1 the RPA slightly un- derestimates the magnetization which was also found in Ref. 9. The opposite is the case in the region near the extrapolated Curie temperature TC 36, where the magne- tization is overestimated. The reason is the presence of longitudinal fluctuations, which play an important role in this region and it is well known that the RPA fails to treat them properly. We consider now the case of in-plane systems (K2 < 0) and applied field in the hard direction (B || z). As al- ready mentioned there is no ’collinear’ magnetization in the GFT for Bz < |Keff (T )|. In Fig. 4 the z-component of the magnetization is shown as a function of the ex- ternal field B for a constant temperature T/J = 0.4. 0 0.05 0.1 0.15 T/J = 0.4 T/J = 0.6 T/J = 0.9 T/J = 1.2 T/J = 1.4 FIG. 6: z-component of magnetization vs. external field for different temperatures T/J and fixed system size N2 (N = 128). Solid lines: GFT (θB = 0.5 ◦), dashed lines: GFT (θB = 0◦) other parameters as in Fig. 4. As in Fig. 3 we see that the QMC results for N ≥ 64 are almost converged and the finite size of the calculated system in QMC should not influence the results anymore. The dotted line marks a critical field Bcrit. For magnetic fields larger than the critical one B > Bcrit we obtain good agreement between QMC and GFT results. Below the critical field B < Bcrit GFT does not yield a stable homogeneous magnetization. However the QMC results show that there is a finite z-component of the magneti- zation in the considered system for 0 ≤ B ≤ Bcrit. In order to compare QMC with GFT results we have tilted the magnetic field by θB = 0.5 ◦ which corresponds to Bx < 10 −2Bz in the GFT. As explained before any symmetry breaking field Bx 6= 0 leads to a stable homo- geneous magnetization with well-defined orientation in the xz-plane. However such a small contribution of the external field within the plane should hardly influence the z-component of the magnetization. This is confirmed by Fig. 5 where we show QMC results (N = 128, θB = 0 as well as the corresponding GFT results with θB = 0 and θB = 0.5 ◦. As expected for |B| > Bcrit the two so- lutions in the GFT are nearly the same and agree well with QMC. Below the critical field only the solution with the slightly tilted field yields a stable homogeneous mag- netization and its z-component compares well with the QMC result in the untilted case. The above results can be interpreted within a semi- classical picture of non-collinear vortex configurations which are stable below a critical field Bcrit in z-direction and contribute a finite z-component to the magnetiza- tion in case of an applied field.19 Despite the lack of di- rect, quantitative access to such states (or correspond- ing physical in-plane observables) within the QMC al- gorithm they are included in principle and one can ob- serve their consequences, namely a finite z-component of the magnetization below the critical GFT field. On the other hand GFT can only describe homogeneous collinear 0 0.5 1 1.5 2 GFT (θ =0.5°) GFT (AC) GFT (MF) QMC (16) QMC (64) QMC (128) QMC (256) FIG. 7: The z-component of magnetization as function of temperature for a fixed external field. Below a critical tem- perature Tcrit there is a breakdown of magnetization in GFT where is no in QMC. Parameters: B/J = 0.03, S = 1, K2/J = −0.06. configurations of spins therefore showing a breakdown of magnetization. However by applying a small field in x- direction the xy-symmetry is broken and the spins ro- tate in the field direction (the vortices vanish) and the collinear phase is retrieved. Our results corroborate this interpretation based on the classical picture. Let us em- phasize that both, GFT for slightly tilted field and QMC for B || z, describe the competition between the external field (which favors magnetization parallel to z) and the anisotropy favoring in-plane magnetization. Comparing the z-components of the magnetization for both cases, one can conclude that the ratio of the competing forces are comparable for QMC and GFT. This indicates that this competition is correctly taken into account in GFT. In Fig. 6 the same field dependence of the z-component of the magnetization is shown for different temperatures. We have plotted the result for the tilted field in case of GFT, the point of breakdown in the untilted case is in- dicated by the dotted line. It can be seen that for higher temperatures no breakdown of collinear magnetization occurs, meaning that the condition for the critical field (B ≤ |Keff (T,B)|) is never fulfilled in this case. The dis- crepancies at intermediate temperatures (T = 0.9..1.2) are due to the RPA decoupling in the GFT as was dis- cussed already. In Figs. 7, 8 and 9 the z-component of the magnetiza- tion is plotted as a function of temperature obtained by GFT (straight line RPA+AC) as well as QMC (symbols) for different system sizes and a constant applied magnetic field. Let us first discuss the qualitative behavior of the mag- netization as a function of temperature which is found in all three figures. For high T (T ≫ Tcrit) the magneti- zation is reduced by thermal fluctuations (where the tail of the curve above T/J ≈ 1.5 is due to the applied ex- ternal field). In the vicinity of Tcrit, T − Tcrit → 0 competition between two effects sets in and has a pro- 0 0.5 1 1.5 2 GFT (θ GFT (θ =0.5°) GFT (two layers) QMC (128) FIG. 8: Same situation as in Fig. 7 for K2/J = −0.04 (other parameters as in Fig.7). The result for a two layer film treated by GFT is plotted also (dashed-dotted line). nounced influence on the magnetization. On the one side the effective anisotropy acts against the external field (Beff = B − |Keff (T )|, (B || z)). The effective anisotropy Keff (T ) is reduced with increasing temper- ature T and thus the effective field Beff increases with T . This effect tends to enhance the magnetization with T . On the other side thermal fluctuations suppress the magnetization with increasing T . The flattening of the magnetization curve near Tcrit is a result of this com- petition. For low temperatures T < Tcrit the effective anisotropy in the GFT cannot be overcome by the exter- nal field (B < |Keff (T )|, (B ||z)). Therefore the collinear magnetization in our approximation vanishes due to the mentioned gapless excitations, in contrast to QMC which yields again a finite magnetization because non-collinear states are taken into account as discussed above. The reduction of the z-component of magnetization in QMC below Tcrit can be pictured classically as the spins being in a non-collinear phase with an angle θ with respect to the z-axis. Since in general anisotropy effects (which fa- vor in-plane magnetization) increase when temperature is lowered the z-component of the magnetization decreases. Now we discuss the three figures in detail. In Fig. 7 we have plotted QMC results for different system size showing again that these are well converged for N ≥ 64. Thus we conclude that the striking difference between GFT and QMC is not a mere finite size effect. The breakdown of magnetization in GFT occurs at a critical temperature Tcrit/J = 0.5 whereas no such breakdown exists in QMC. However the exposed maximum of the magnetization in QMC lies near the breakdown point. The differences between QMC and GFT in the tempera- ture range T/J ≈ 0.3 . . . 1.3 are due to the decoupling of the exchange and anisotropy term in GFT as also seen in Fig. 3. It is worth mentioning that the value of the z-component of the magnetization is nearly the same at the breakdown point in GFT and the maximum in the QMC. Thus we have the result that although GFT can- 0 0.5 1 1.5 2 GFT (<S >; θ =0.5°) GFT (<S >; θ =0.5°) QMC (θ FIG. 9: Same situation as in Fig. 7 for K2/J = −0.01, B/J = 0.005 and slightly tilted field (θB = 0.5 ◦) for the GFT results. not describe the non-collinear phase by ansatz its break- down coincides rather well with the onset of this phase, which we attribute to the maximum of the QMC curve. Fig.8 shows the same situation for a different anisotropy constant K2 = −0.04. The critical temperature is lower than in Fig.7 since the ratio Bz/K2 becomes larger. The tilted field case is also shown for the GFT results. Again the qualitative agreement of the z-component of magne- tization with QMC is good. To confirm this point we have plotted the temperature dependence for an other set of parameters in Fig. 9. There is as good qualitative agreement of the two approaches. Additionally one gets a finite component in x-direction in GFT which is also plotted in the figure. The two effects of the external field vs. anisotropy competition are nicely to be seen: a non- collinear state for B || z (z-component only in QMC but not in GFT) and rotation of magnetization for slightly tilted external field (seen only in GFT). The ratio of the competing forces agree well again in both treatments. In Fig. 7 we have plotted the results of a different decoupling scheme of the anisotropy terms (namely a mean field decoupling, dashed line in Fig. 7). Although the overall characteristic resembles the RPA+AC result (breakdown of magnetization) the mean field results dif- fer extremely from the QMC for a large range of tempera- ture and underestimates the magnetization. This demon- strates the reliability of the Anderson-Callen treatment of the local anisotropy terms presented in Refs. 10,11,12,13. The extension of the GFT method to multi-layer films is straightforward.11 We have also included results for a two-layer film in Fig. 8 for the same parameters as in the monolayer case. One finds that for a double layer magnetism is stabilized, which can be attributed to the increased coordination number and thus higher exchange energy. Just like for a monolayer, one observes a break- down of collinear magnetization at some critical temper- ature. This is due to the fact that the same reasoning regarding the vanishing excitation gap also applies for multilayer (slab) systems32. The effective anisotropy per layer is essentially the same as for a single layer, thus the critical 〈Sz〉-value (magnetization at critical field Bcrit) is practically the same. The critical temperature is higher than that of a monolayer due to the increased magnetic stiffness of the double layer. IV. SUMMARY AND CONCLUSIONS Using GFT and QMC calculations we studied easy- plane systems as well as easy-axis systems with an exter- nal field applied perpendicularly to the film. The GFT treatment of the Hamiltonian Eq. (1) consists of a RPA- decoupling for the nonlocal terms and an AC-decoupling for the local terms performed in a rotated frame, where the new z′-axis is parallel to the magnetization. For the QMC calculations we have used the stochastic series ex- pansion (SSE) with directed loop updates, which is well suited for spin-systems. We have calculated the magnetization as a function of the external field as well as temperature. We found a critical field and critical temperature respectively below which is no magnetization in GFT whereas there is one in QMC. By tilting the field slightly in GFT so that it has a small component in x-direction we get a stable magne- tization even below the critical field or temperature. The z-component of the magnetization in this case coincides well with the z-component obtained by QMC for the un- tilted field confirming that GFT and QMC agree well in the description of the external field vs. anisotropy com- petition. However, this comparison can be only some- what indirect, since QMC has access to the non-collinear (B || z) state only, while GFT is limited to collinear fer- romagnetic states (rotated homogeneous magnetization) found for slightly tilted external fields. For parameters that are accessible by both QMC and GFT (B || z; B > Bcrit(T )) QMC and GFT are in good agreement. Thus one can conclude that the GFT is ap- plicable to the homogeneous phases of systems described by Eq. (1) and can be used also for system configurations not accessible by QMC due to too large system size as e.g. multilayer systems. It would be an interesting task for a succeeding work to extend the GFT in order to get a deeper insight into the non-collinear configurations also. APPENDIX A: MAGNETIZATION ANGLE Here we will discuss the second mathematical solution which occurs besides Eq. 6. For an external field in the z-direction the angle dependent part of the free en- ergy including second order anisotropy can be expanded as1,27: F = −MzBz cos θ − K̃2 cos whereMz is the z-component of the magnetization and K̃2 is the first nonvanishing coefficient in an expansion of the free energy for a system with second order anisotropy. For the equilibrium angle one gets: ∂F (θ) = MzBz sin θ + 2K̃2 cos θ sin θ = 0. (A1) Therefore one gets two solutions for in-plane systems (K̃2 < 0). For sin θ 6= 0 one gets immediately the so- lution of Eq. 6 if 2K̃2/Mz ≡ Keff holds. This is the stable solution. The trivial (second) solution sin θ = 0 is unstable for Bz < |Keff | because ∂2F (θ) |sin θ=0 = < 0 for Bz < |Keff | > 0 otherwise holds. For a detailed discussion of stability conditions in film systems we refer to Refs. 1,27. ∗ Electronic address: henning@physik.hu-berlin.de 1 M. Farle, B. Mirwald-Schulz, A. N. Anisimov, W. Platow, and K. Baberschke, Phys. Rev. B 55, 3708 (1997). 2 A. Hucht and K. D. Usadel, Phys. Rev. B 55, 12309 (1997). 3 P. J. Jensen and K. H. Bennemann, Solid State Comm. 105, 577 (1998), and references therein. 4 R. P. Erickson and D. L. Mills, Phys. Rev. B 44, 11825 (1991). 5 D. K. Morr, P. J. Jensen, and K. H. Bennemann, Surf. Sci. 307-309, 1109 (1994). 6 P. Politi, A. Rettori, M. G. Pini, and D. Pescia, J. Magn. Magn. Mater. 140-144, 647 (1995); A. Abanov, V. Kalatsky, V. L. Pokrovsky and W. M. Saslow, Phys. Rev. B 51, 1023 (1995). 7 A. Hucht, A. Moschel, and K. D. Usadel, J. Magn. Magn. Mater. 148, 32 (1995); S. T. Chui, Phys. Rev. B 50, 12559 (1994). 8 T. Herrmann, M. Potthoff, and W. Nolting, Phys. Rev. B 58, 831 (1998). 9 P. Henelius, P. Fröbrich, P. J. Kuntz, C. Timm, and P. J. Jensen, Phys. Rev. B 66, 094407 (2002). 10 S. Schwieger, J. Kienert, and W. Nolting, Phys. Rev. B 71, 024428 (2005). 11 S. Schwieger, J. Kienert, and W. Nolting, Phys. Rev. B 71, 174441 (2005). 12 F. Körmann, S. Schwieger, J. Kienert, and W. Nolting, Eur. Phys. J. B 53, 463 (2006). 13 M. G. Pini, P. Politi and R. L. Stamps, Phys. Rev. B 72, 014454 (2005). 14 J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973). 15 G. M. Wysin, Phys. Lett. A 240, 95 (1998). 16 E. Yu. Vedmedenko, A. Ghazali, and J. -C. S. Lévy, Phys. Rev. B 59, 3329 (1999). 17 K. W. Lee and C. E. Lee, Phys. Rev. B 70, 144420 (2004). 18 M. Rapini, R. A. Dias, and B. V. Costa, Phys. Rev. B 75, 014425 (2007). 19 B. A. Ivanov and G. M. Wysin, Phys. Rev. B 65, 134434 (2002). 20 F. Bloch, Z. Phys. 61, 206 (1930). 21 P. Bruno, Phys. Rev. B 43, 6015 (1998). 22 P. J. Jensen and K. H. Bennemann, in Magnetism and electronic correlations in local-moment systems, edited by M. Donath, P. A. Dowben and W. Nolting, p.113 (World Scientific, 1998). 23 P. Fröbrich, P. J. Jensen, and P. J. Kuntz, Eur. Phys. J. B 13, 477 (2000). 24 N. N. Bogolyubov and S. V. Tyablikov, Soviet. Phys.- Doklady 4, 589 (1959). 25 F. B. Anderson and H. Callen, Phys. Rev. 136, A1068 (1964). 26 P. Fröbrich and P. J. Kuntz, http://arxiv.org/pdf/cond-mat/0607675. 27 J. Lindner, Ph.D. thesis, Freie Universität Berlin (2002). 28 A. W. Sandvik, Phys. Rev. B 59, R14 157 (1999). 29 O. F. Syljůasen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002). 30 F. Alet, S. Wessel, and M. Troyer, Phys. Rev. E 71, 036706 (2005). 31 ALPS collaboration, J. Phys. Soc. Jpn. Suppl. 74, 30 (2005). Source codes can be obtained from http://alps.comp-phys.org/ 32 A.Gelfert and W.Nolting, Phys. Stat. Sol. B 217, 805 (2000). 33 Besides some rare earth materials where the anisotropy can be of the order of J . 34 For B < |Keff | there is another mathematical solution (sin θ = 0) which however is unstable (see appendix A). 35 Note that a similar result has already been published in Ref. 9. 36 Strictly speaking there is no phase transition because of the applied magnetic field as can be seen from the large tail of the magnetization curve. However one can extract a TC from the curves by extrapolating to the zero field case and additionally to an infinte system size in the QMC calculations. mailto:henning@physik.hu-berlin.de http://arxiv.org/pdf/cond-mat/0607675 http://alps.comp-phys.org/

In this work we compare numerically exact Quantum Monte Carlo (QMC) calculations and Green function theory (GFT) calculations of thin ferromagnetic films including second order anisotropies. Thereby we concentrate on easy plane systems, i.e. systems for which the anisotropy favors a magnetization parallel to the film plane. We discuss these systems in perpendicular external field, i.e. B parallel to the film normal. GFT results are in good agreement with QMC for high enough fields and temperatures. Below a critical field or a critical temperature no collinear stable magnetization exists in GFT. On the other hand QMC gives finite magnetization even below those critical values. This indicates that there occurs a transition from non-collinear to collinear configurations with increasing field or temperature. For slightly tilted external fields a rotation of magnetization from out-of-plane to in-plane orientation is found with decreasing temperature.

Green Function theory vs. Quantum Monte Carlo Calculation for thin magnetic films S. Henning,∗ F. Körmann, J. Kienert, and W. Nolting Lehrstuhl Festkörpertheorie, Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany S. Schwieger Technische Universität Ilmenau, Theoretische Physik I, Postfach 10 05 65, 98684 Ilmenau, Germany (Dated: October 31, 2018) In this work we compare numerically exact Quantum Monte Carlo (QMC) calculations and Green function theory (GFT) calculations of thin ferromagnetic films including second order anisotropies. Thereby we concentrate on easy plane systems, i.e. systems for which the anisotropy favors a magnetization parallel to the film plane. We discuss these systems in perpendicular external field, i.e. B parallel to the film normal. GFT results are in good agreement with QMC for high enough fields and temperatures. Below a critical field or a critical temperature no collinear stable magnetization exists in GFT. On the other hand QMC gives finite magnetization even below those critical values. This indicates that there occurs a transition from non-collinear to collinear configurations with increasing field or temperature. For slightly tilted external fields a rotation of magnetization from out-of-plane to in-plane orientation is found with decreasing temperature. PACS numbers: 75.10.Jm, 75.40.Mg, 75.70.Ak, 75.30.Gw I. INTRODUCTION The fast development of technological applications based on magnetic systems in the last years, e.g. magnetic data storage devices, causes a high interest in thin magnetic films. One precondition for the tech- nological development is the investigation of magnetic anisotropies and spin reorientation transitions connected therewith. Those reorientation transitions can occur from out-of-plane to in-plane or vice versa for increasing film thickness d1, temperature T 2,3,4,5,6,7,8, or external field B0. Quantum Monte Carlo (QMC) calculations give the possibility to compare numerically exact results with analytical approximations. In Ref. 9 the authors inves- tigated a ferromagnetic monolayer including positive second order anisotropy (easy axis perpendicular to the film plane). They discuss the temperature dependence of the magnetization 〈Sz〉(T ) as well as field induced reorientation transitions from out-of-plane to in-plane and compare the QMC results with Green function theory (GFT). They found good agreement in the case of applied external field in the easy direction (here z-axis). However, their GFT fails for external field applied in arbitrary direction, especially in the hard direction (within the film plane). As shown in Ref. 10 for getting closer to the QMC results for magnetic field induced reorientation from out-of-plane to in-plane a more careful treatment of the local anisotropy terms is needed. In Refs. 10,11,12,13 a decoupling scheme was presented which yields excellent agreement with QMC results for out-of-plane systems. The availability of theories such as GFT and their check against state-of-the-art numerical algorithms is highly desirable because of the size limitations of systems where QMC can be performed. On the other hand the extension of GFT from a monolayer (where it can be compared to QMC as in the the present work) to multilayer systems is a straightforward task without further approximations11. Up to now, to our knowledge, there is no comparison between QMC and approximative theories for easy-plane systems and it is not obvious that the theory presented in Refs. 10,11,12,13 can reproduce the QMC results for in-plane systems as accurately as for the out-of-plane case. In contrast to the easy-axis case where a certain direction is preferred by the single ion anisotropy in easy-plane systems the full xy-plane is favored and no particular direction is distinguished within the plane. A magnetic field applied perpendicular to the plane does not destroy the xy-symmetry. For systems exhibiting this kind of symmetry it was shown in a classical treatment that for external fields smaller than a critical field 0 ≤ B < Bcrit (B || z) stable vortices, i.e. a non-collinear arrangement of spins, can exist15,16,17,18,19. These vortices can undergo a Berezinskii-Kosterlitz-Thouless (BKT) transition14. Depending on the strength of the anisotropy K2 there might be vortices with or without a finite z-component of magnetization15. In the small anisotropy case (which is considered in this work, |K2| < 0.1J) there is a finite out-of-plane component and for zero field the two possible directions of magnetization (±z) are en- ergetically degenerate. For increasing magnetic field in z-direction the vortices antiparallel to the field become more and more unstable (heavy vortices). However the so called light vortices (parallel to the field) are stable up to a critical field Bz = Bcrit and contribute a finite z-component to the net-magnetization of the considered system19. The vortices in connection with a finite z-component of the net-magnetization emerge because of two reasons: first the competition between the anisotropy (favoring a orientation of the magnetization within the xy-plane) http://arxiv.org/abs/0704.1552v1 and the external field (favoring a perpendicular magne- tization), and second: the xy-symmetry of the system, which does not allow for a rotated homogeneous phase. In this paper we investigate both aspects, i.e. the field vs. anisotropy competition as well as the symmetry properties in detail for a quantum mechanical system. We will compare the results of QMC and GFT calcula- tions. As explained in more detail below, the QMC al- gorithm used here allows only for an external field applied in z-direction. Thus the xy-symmetry can not be broken and no comparison between xy-symmetric and asymmetric systems is possible. We will use GFT to clarify the influence of this symmetry breaking on the homogeneous phase. On the other hand, the GFT used here is by ansatz limited to the homogeneous phase. Therefore it can not describe a non-collinear (e.g, vortex-) magnetic phase, which is expected for B || z and small field strengths. The breakdown of magnetization in GFT as well as an exposed maximum in the magnetization in QMC at certain critical values of the external field or temperature gives, however, a clear fingerprint of non-collinear configurations, at least if there is no meta-stable homogeneous phase. Below these critical values there will be a finite z-component in QMC and a vanishing magnetization in GFT. For parameters, where both theories are applicable, QMC serves as a test for the approximations needed in In this work we find indications for non-collinear spin configurations below a critical field or temperature for B || z by comparing results of QMC and GFT as explained in the last clause. Above the critical field we obtain good agreement between QMC and GFT results. Breaking the xy-symmetry by adding a small x-component to the external field yields a stable collinear solution in GFT. The z-component of the magnetization in this case is in good agreement with the QMC results calculated with untilted field. Thus we can conclude that except for the restriction to collinear magnetic states GFT describes the competition between external field and anisotropy quite well. The paper is organized as follows: First we ex- plain the basics of the GFT and the QMC calculations. Then we apply both approaches to easy-plane systems in external magnetic fields and report the results of our calculations. II. THEORY A. Green Function Theory In the following we present our theoretical approach us- ing Green function theory. The focus of this work lies on the translational invariant system of a two-dimensional monolayer. Therefore the following Hamiltonian is used: H = − JijSiSj −B Si −K2 (Sz,i) 2. (1) The first term describes the Heisenberg coupling Jij be- tween spins Si and Sj located at sites i and j. The second term contains an external magnetic field B in arbitrary direction (the Landé factor gJ and the Bohr magneton µB are absorbed in B). The third term represents second order lattice anisotropy due to spin-orbit coupling. Sz,i is the z-component of Si (the z-axis of the coordinate system is oriented perpendicular to the film-plane). The lattice anisotropy favours in-plane (K2 < 0) or out-of- plane (K2 > 0) orientation. Our Hamiltonian is similar to that used in Refs. 10,11,13,22,23 for the investigation of the magnetic anisotropy and the field induced reori- entation transition. To simplify calculations we consider nearest neighbor coupling only Jij = J (i), (j) n.n. 0 otherwise. The main idea of the special treatment presented in Refs. 10,11,12,13 is that, before any decoupling is applied, the coordinate system Σ is rotated to a new system Σ′ where the new z′-axis is parallel to the magnetization im- plying a collinear alignment of all spins within the layer. Then a combination of Random Phase approximation (RPA)24 for the nonlocal terms in Eq. (1) (Heisenberg exchange interaction term) and Anderson-Callen approx- imation (AC)25 for the local lattice anisotropy term is applied in the rotated system. After application of the approximation one gets an effective anisotropy Keff (T ) = 2K2 S(S + 1)− 〈S2z′〉 〈Sz′〉 (3) where 〈Sz′〉 is the norm of the magnetization and S is the spin quantum number, that we have chosen to be S = 1 in all our calculations. As shown in comparison with an exact treatment of the local anisotropy term in Ref. 26 this approximation still holds up to anisotropy strengths K2 ∼ 1/2J . Therefore we restrict ourselves in the following to small anisotropies (K2 ≤ 0.1J) as found in most real materials 33. For a magnetic field applied in the xz-plane (B = (Bx, 0, Bz)) our theory gives a condition for the polar angle θ of the magnetization: sin θBz − cos θBx +Keff sin θ cos θ = 0 (4) The uniform magnon energies (q = 0) which dominate the physical behavior of the magnetic system can easily be extracted from the theory12,13: E2q=0 = cos θBz + sin θBx +Keff(cos 2 θ − sin2 θ) cos θBz + sin θBx +Keff cos This result coincides with the spin-wave result13 if one replaces 〈Sz′〉 by the spin quantum number S and Keff by the bare anisotropy constant K2 in Eq. (5). For an easy-plane system (Keff < 0) with external field B in z-direction the polar angle θ of the magnetization34 is given by: cos θ = −B/Keff(T ) for B < |Keff (T )| 1 otherwise By inserting Eq.(6) into Eq.(5) one immediately gets: Keff<0 q=0 (B) = 0 B < |Keff (T )| B +Keff (T ) otherwise. For gapless magnon energies Eq=0 = 0 the magnon oc- cupation number φ diverges (φ → ∞) in film systems with ferromagnetic coupling J > 0 and the magnetiza- tion becomes zero 〈Sz′〉 = 0 in the collinear phase. This can be seen by following an argument of Bloch20 already given in 1930. Since the spin wave dispersion is E ≈ q2 in the vicinity of q = 0 the spin-wave density of states N(E) is independent of E for a two-dimensional system for E close to zero. The excitation of spin-waves at finite temperature leads to a variation of the magnetization of the order: ∆m(T ) ∼ N(E)dE exp(E/kBT )− 1) ∼ kBT exp(x)− 1 . (8) Since the integral in Eq. (8) diverges for T 6= 0 and exited spin-waves lead to a reduction of the magnetization one can conclude that the magnetization should be zero at finite temperature. However for an infinitesimally small contribution of the external field parallel to the plane, i.e. Bx 6= 0, a finite gap in the excitation spectrum at q = 0 opens. This can be seen in Fig.1 where the uniform magnon modes Eq=0(B) are shown for different orientations θB, where θB is the polar angle of the ex- ternal field. The integral (8) is now finite and a stable finite magnetization in the collinear phase having a well defined orientation in the xz-plane is possible. Let us now come back to the case where the applied field is aligned in z-direction. It can be seen from Eq. (7) that for external field B (B || z) larger than a critical field B > Bcrit given by: Bcrit = |Keff (T,B)| (9) a stable collinear solution exists. Since Keff (T ) is a de- creasing function of temperature T a transition from non- collinear to collinear phase with increasing temperature is possible. In Fig. 2 we show the normalized critical field (9) Bcrit/K2 as a function of temperature T . For a constant magnetic field B (B || z) at a temperature T1 with B < Keff (T1, B) no stable collinear phase ex- ist. Then by increasing the temperature up to T2 the ef- fective anisotropy Keff is sufficiently reduced such that 0.01 0.02 0.03 0.04 B / J =0.2° FIG. 1: The energies of the uniform magnon mode Eq=0(B) for different polar angles θB of the external field. Eq=0 is zero below B/J ≈ 0.03 for θB = 0 ◦. The prefactors gJµB and kB are absorbed in B and T respectively. The latter are given in units of the nearest neighbor Heisenberg coupling J . Parameters: S = 1, K/J = −0.03 and T/J = 10−4. FIG. 2: The normalized critical field Bcrit/K2 as a function of temperature. Parameters: S = 1. B > Keff (T2, B), and the collinear phase becomes sta- ble. Before we come to the results let us briefly sketch the main aspects of the QMC. B. QMC In the last section we gave a short description of the theory used to treat a system described by a Hamilto- nian of form (1). This theory applies to the thermody- namic limit (films of infinte size) but contains certain approximations. Additionally the GFT is restricted to ordered phases with a collinear alignment of all spins. Therefore it would be very useful to have exact results at hand to crosscheck the predictions of GFT. A Quantum Monte Carlo method, particularly well suited for spin systems, is the stochastic series expansion (SSE) with directed loop update. We will sketch this method here only briefly as detailed descriptions can be already found elsewhere28,29,30. Our starting point is the series expansion of the parti- tion function Z = Tre−βH = 〈α|(−H)n|α〉 (10) whereH denotes the Hamiltonian, {|α〉} are basis vectors of a proper Hilbert space and β is the inverse tempera- ture. The Hamiltonian is then rewritten in terms of bond Hamiltonians: H = −J Hb (11) where Hb can be further decomposed into a diagonal and an off-diagonal part: HD,b = C + S i(b)S j(b) + bb[S i(b) + S j(b)] (12) +k2b[(S i(b)) 2 + (Szj(b)) HO,b = ] (13) Here we have renormalized the anisotropy constant k2b and the magnetic field bb in such a way that (11) coincides with (1). i(b) and j(b) denotes the lattice sites connected by the bond b and the additional constant C in HD,b will be chosen such that all matrix elements of this term be- come positive, a condition necessary to interpret them as probabilities. Note that for a finite system at finite tem- perature the power series of the partition function can be truncated at a finite cutoff length Λ without intro- ducing any systematic error in practical computations29. Therefore reinserting (11) into (10) and rewriting the re- sult yields: βn(Λ− n)! 〈α|SCΛ |α〉. (14) Here SCΛ denotes a product of operators (operator string) consisting of n non-unity operators and (Λ − n) unity operators H0 = Id which were inserted to get op- erator strings of equal length Λ. In fact it is impossible to evaluate all operator strings in (14). The SSE-QMC replaces such an evaluation there- fore by importance sampling over the strings according to their relative weight. Hence an efficient scheme for gen- erating new operator strings is needed. In the directed loop version of the SSE this is done by dividing the up- date into two parts. In a first step a diagonal update is performed by traversing the operator string and replac- ing some unity operators by diagonal bond operators and vice versa (the probabilities for both substitutions have to fulfill the detailed balance criterion). Then the loop update follows in which new non-diagonal bond opera- tors can appear in the operator string. For details of the update procedure we refer the interested reader to the 0 0.5 1 1.5 2 GFT (RPA+AC) QMC (N=16) QMC (N=32) QMC (N=64) FIG. 3: Magnetization vs. temperature for an out-of-plane easy-axis system (K2 > 0). Straight line: GFT (RPA+AC) result; symbols: QMC results for different system sizes N2. Parameters: S = 1, B/J = 0.01 (B || z) and K2/J = 0.01. according literature28,29,30. A full implementation of the SSE with directed loop update which we have used for all QMC calculations in this work can be found in the ALPS project30,31. Since the SSE-QMC used by us is implemented in z- representation (spin quantization axis along z-axis) in- plane correlation functions e.g. the in-plane magnetiza- tion are not accessible. Further B || z is the only possible field direction in the used QMC implementation because a traverse field (in-plane field component) would lead to non-closing loops (see Ref. 9). III. RESULTS As mentioned in Sec. II A the results for the in-plane systems are very sensitive to the effective anisotropy Keff (T ). This sensitivity of the anisotropy is less pro- nounced for out-of-plane systems (K2 > 0) since the ap- plied field B (B || z) and the intrinsic easy axis are par- allel. In order to test our decoupling scheme (RPA+AC) we first compare GFT and QMC for an out-of-plane system.35 In Fig. 3 the magnetization 〈Sz〉 as a function of tem- perature T is shown. The straight line belongs to the GFT whereas the symbols show the result of the QMC for different system sizes. Let us first comment on finite size effects in the QMC results. It can be seen in Fig. 3 that the QMC results converge for increasing system size N2 (for N ×N square lattice). Indeed forN ≥ 32 the QMC results are unbiased by finite size effects and resulting magnetization curves are almost equal for increasing N ≥ 32. Note that we have omitted error bars in the figures showing QMC results because the relative errors are of the order 10−4. We now compare the GFT with the QMC results (N = 64). For low temperatures (T/J ≤ 0.5) we ob- tain excellent quantitative agreement. This is plausible 0 0.05 0.1 0.15 0.2 QMC (16) QMC (32) QMC (64) QMC (128) FIG. 4: z-component of magnetization as a function of ex- ternal magnetic field for fixed temperature T/J = 0.4. In contrary to the GFT the magnetization obtained by QMC remains finite for all fields. The QMC results are converged for N ≥ 64. Parameters: S = 1, K2/J = −0.06 and θB = 0 0 0.05 0.1 0.15 0.2 GFT (θ = 0°) GFT (θ = 0.5°) QMC (128) FIG. 5: z-component of magnetization vs. external field for T/J = 0.4 with slightly tilted field (θB = 0.5 ◦)in the GFT result (solid line). The dotted line shows GFT result for (θB = 0◦). Other parameters as in Fig. 4. because in this region the GFT result coincides with the result of the spin-wave theory which is known to be re- liable (exact for T = 0) for low temperatures. For the intermediate region T/J = 0.5..1 the RPA slightly un- derestimates the magnetization which was also found in Ref. 9. The opposite is the case in the region near the extrapolated Curie temperature TC 36, where the magne- tization is overestimated. The reason is the presence of longitudinal fluctuations, which play an important role in this region and it is well known that the RPA fails to treat them properly. We consider now the case of in-plane systems (K2 < 0) and applied field in the hard direction (B || z). As al- ready mentioned there is no ’collinear’ magnetization in the GFT for Bz < |Keff (T )|. In Fig. 4 the z-component of the magnetization is shown as a function of the ex- ternal field B for a constant temperature T/J = 0.4. 0 0.05 0.1 0.15 T/J = 0.4 T/J = 0.6 T/J = 0.9 T/J = 1.2 T/J = 1.4 FIG. 6: z-component of magnetization vs. external field for different temperatures T/J and fixed system size N2 (N = 128). Solid lines: GFT (θB = 0.5 ◦), dashed lines: GFT (θB = 0◦) other parameters as in Fig. 4. As in Fig. 3 we see that the QMC results for N ≥ 64 are almost converged and the finite size of the calculated system in QMC should not influence the results anymore. The dotted line marks a critical field Bcrit. For magnetic fields larger than the critical one B > Bcrit we obtain good agreement between QMC and GFT results. Below the critical field B < Bcrit GFT does not yield a stable homogeneous magnetization. However the QMC results show that there is a finite z-component of the magneti- zation in the considered system for 0 ≤ B ≤ Bcrit. In order to compare QMC with GFT results we have tilted the magnetic field by θB = 0.5 ◦ which corresponds to Bx < 10 −2Bz in the GFT. As explained before any symmetry breaking field Bx 6= 0 leads to a stable homo- geneous magnetization with well-defined orientation in the xz-plane. However such a small contribution of the external field within the plane should hardly influence the z-component of the magnetization. This is confirmed by Fig. 5 where we show QMC results (N = 128, θB = 0 as well as the corresponding GFT results with θB = 0 and θB = 0.5 ◦. As expected for |B| > Bcrit the two so- lutions in the GFT are nearly the same and agree well with QMC. Below the critical field only the solution with the slightly tilted field yields a stable homogeneous mag- netization and its z-component compares well with the QMC result in the untilted case. The above results can be interpreted within a semi- classical picture of non-collinear vortex configurations which are stable below a critical field Bcrit in z-direction and contribute a finite z-component to the magnetiza- tion in case of an applied field.19 Despite the lack of di- rect, quantitative access to such states (or correspond- ing physical in-plane observables) within the QMC al- gorithm they are included in principle and one can ob- serve their consequences, namely a finite z-component of the magnetization below the critical GFT field. On the other hand GFT can only describe homogeneous collinear 0 0.5 1 1.5 2 GFT (θ =0.5°) GFT (AC) GFT (MF) QMC (16) QMC (64) QMC (128) QMC (256) FIG. 7: The z-component of magnetization as function of temperature for a fixed external field. Below a critical tem- perature Tcrit there is a breakdown of magnetization in GFT where is no in QMC. Parameters: B/J = 0.03, S = 1, K2/J = −0.06. configurations of spins therefore showing a breakdown of magnetization. However by applying a small field in x- direction the xy-symmetry is broken and the spins ro- tate in the field direction (the vortices vanish) and the collinear phase is retrieved. Our results corroborate this interpretation based on the classical picture. Let us em- phasize that both, GFT for slightly tilted field and QMC for B || z, describe the competition between the external field (which favors magnetization parallel to z) and the anisotropy favoring in-plane magnetization. Comparing the z-components of the magnetization for both cases, one can conclude that the ratio of the competing forces are comparable for QMC and GFT. This indicates that this competition is correctly taken into account in GFT. In Fig. 6 the same field dependence of the z-component of the magnetization is shown for different temperatures. We have plotted the result for the tilted field in case of GFT, the point of breakdown in the untilted case is in- dicated by the dotted line. It can be seen that for higher temperatures no breakdown of collinear magnetization occurs, meaning that the condition for the critical field (B ≤ |Keff (T,B)|) is never fulfilled in this case. The dis- crepancies at intermediate temperatures (T = 0.9..1.2) are due to the RPA decoupling in the GFT as was dis- cussed already. In Figs. 7, 8 and 9 the z-component of the magnetiza- tion is plotted as a function of temperature obtained by GFT (straight line RPA+AC) as well as QMC (symbols) for different system sizes and a constant applied magnetic field. Let us first discuss the qualitative behavior of the mag- netization as a function of temperature which is found in all three figures. For high T (T ≫ Tcrit) the magneti- zation is reduced by thermal fluctuations (where the tail of the curve above T/J ≈ 1.5 is due to the applied ex- ternal field). In the vicinity of Tcrit, T − Tcrit → 0 competition between two effects sets in and has a pro- 0 0.5 1 1.5 2 GFT (θ GFT (θ =0.5°) GFT (two layers) QMC (128) FIG. 8: Same situation as in Fig. 7 for K2/J = −0.04 (other parameters as in Fig.7). The result for a two layer film treated by GFT is plotted also (dashed-dotted line). nounced influence on the magnetization. On the one side the effective anisotropy acts against the external field (Beff = B − |Keff (T )|, (B || z)). The effective anisotropy Keff (T ) is reduced with increasing temper- ature T and thus the effective field Beff increases with T . This effect tends to enhance the magnetization with T . On the other side thermal fluctuations suppress the magnetization with increasing T . The flattening of the magnetization curve near Tcrit is a result of this com- petition. For low temperatures T < Tcrit the effective anisotropy in the GFT cannot be overcome by the exter- nal field (B < |Keff (T )|, (B ||z)). Therefore the collinear magnetization in our approximation vanishes due to the mentioned gapless excitations, in contrast to QMC which yields again a finite magnetization because non-collinear states are taken into account as discussed above. The reduction of the z-component of magnetization in QMC below Tcrit can be pictured classically as the spins being in a non-collinear phase with an angle θ with respect to the z-axis. Since in general anisotropy effects (which fa- vor in-plane magnetization) increase when temperature is lowered the z-component of the magnetization decreases. Now we discuss the three figures in detail. In Fig. 7 we have plotted QMC results for different system size showing again that these are well converged for N ≥ 64. Thus we conclude that the striking difference between GFT and QMC is not a mere finite size effect. The breakdown of magnetization in GFT occurs at a critical temperature Tcrit/J = 0.5 whereas no such breakdown exists in QMC. However the exposed maximum of the magnetization in QMC lies near the breakdown point. The differences between QMC and GFT in the tempera- ture range T/J ≈ 0.3 . . . 1.3 are due to the decoupling of the exchange and anisotropy term in GFT as also seen in Fig. 3. It is worth mentioning that the value of the z-component of the magnetization is nearly the same at the breakdown point in GFT and the maximum in the QMC. Thus we have the result that although GFT can- 0 0.5 1 1.5 2 GFT (<S >; θ =0.5°) GFT (<S >; θ =0.5°) QMC (θ FIG. 9: Same situation as in Fig. 7 for K2/J = −0.01, B/J = 0.005 and slightly tilted field (θB = 0.5 ◦) for the GFT results. not describe the non-collinear phase by ansatz its break- down coincides rather well with the onset of this phase, which we attribute to the maximum of the QMC curve. Fig.8 shows the same situation for a different anisotropy constant K2 = −0.04. The critical temperature is lower than in Fig.7 since the ratio Bz/K2 becomes larger. The tilted field case is also shown for the GFT results. Again the qualitative agreement of the z-component of magne- tization with QMC is good. To confirm this point we have plotted the temperature dependence for an other set of parameters in Fig. 9. There is as good qualitative agreement of the two approaches. Additionally one gets a finite component in x-direction in GFT which is also plotted in the figure. The two effects of the external field vs. anisotropy competition are nicely to be seen: a non- collinear state for B || z (z-component only in QMC but not in GFT) and rotation of magnetization for slightly tilted external field (seen only in GFT). The ratio of the competing forces agree well again in both treatments. In Fig. 7 we have plotted the results of a different decoupling scheme of the anisotropy terms (namely a mean field decoupling, dashed line in Fig. 7). Although the overall characteristic resembles the RPA+AC result (breakdown of magnetization) the mean field results dif- fer extremely from the QMC for a large range of tempera- ture and underestimates the magnetization. This demon- strates the reliability of the Anderson-Callen treatment of the local anisotropy terms presented in Refs. 10,11,12,13. The extension of the GFT method to multi-layer films is straightforward.11 We have also included results for a two-layer film in Fig. 8 for the same parameters as in the monolayer case. One finds that for a double layer magnetism is stabilized, which can be attributed to the increased coordination number and thus higher exchange energy. Just like for a monolayer, one observes a break- down of collinear magnetization at some critical temper- ature. This is due to the fact that the same reasoning regarding the vanishing excitation gap also applies for multilayer (slab) systems32. The effective anisotropy per layer is essentially the same as for a single layer, thus the critical 〈Sz〉-value (magnetization at critical field Bcrit) is practically the same. The critical temperature is higher than that of a monolayer due to the increased magnetic stiffness of the double layer. IV. SUMMARY AND CONCLUSIONS Using GFT and QMC calculations we studied easy- plane systems as well as easy-axis systems with an exter- nal field applied perpendicularly to the film. The GFT treatment of the Hamiltonian Eq. (1) consists of a RPA- decoupling for the nonlocal terms and an AC-decoupling for the local terms performed in a rotated frame, where the new z′-axis is parallel to the magnetization. For the QMC calculations we have used the stochastic series ex- pansion (SSE) with directed loop updates, which is well suited for spin-systems. We have calculated the magnetization as a function of the external field as well as temperature. We found a critical field and critical temperature respectively below which is no magnetization in GFT whereas there is one in QMC. By tilting the field slightly in GFT so that it has a small component in x-direction we get a stable magne- tization even below the critical field or temperature. The z-component of the magnetization in this case coincides well with the z-component obtained by QMC for the un- tilted field confirming that GFT and QMC agree well in the description of the external field vs. anisotropy com- petition. However, this comparison can be only some- what indirect, since QMC has access to the non-collinear (B || z) state only, while GFT is limited to collinear fer- romagnetic states (rotated homogeneous magnetization) found for slightly tilted external fields. For parameters that are accessible by both QMC and GFT (B || z; B > Bcrit(T )) QMC and GFT are in good agreement. Thus one can conclude that the GFT is ap- plicable to the homogeneous phases of systems described by Eq. (1) and can be used also for system configurations not accessible by QMC due to too large system size as e.g. multilayer systems. It would be an interesting task for a succeeding work to extend the GFT in order to get a deeper insight into the non-collinear configurations also. APPENDIX A: MAGNETIZATION ANGLE Here we will discuss the second mathematical solution which occurs besides Eq. 6. For an external field in the z-direction the angle dependent part of the free en- ergy including second order anisotropy can be expanded as1,27: F = −MzBz cos θ − K̃2 cos whereMz is the z-component of the magnetization and K̃2 is the first nonvanishing coefficient in an expansion of the free energy for a system with second order anisotropy. For the equilibrium angle one gets: ∂F (θ) = MzBz sin θ + 2K̃2 cos θ sin θ = 0. (A1) Therefore one gets two solutions for in-plane systems (K̃2 < 0). For sin θ 6= 0 one gets immediately the so- lution of Eq. 6 if 2K̃2/Mz ≡ Keff holds. This is the stable solution. The trivial (second) solution sin θ = 0 is unstable for Bz < |Keff | because ∂2F (θ) |sin θ=0 = < 0 for Bz < |Keff | > 0 otherwise holds. For a detailed discussion of stability conditions in film systems we refer to Refs. 1,27. ∗ Electronic address: henning@physik.hu-berlin.de 1 M. Farle, B. Mirwald-Schulz, A. N. Anisimov, W. Platow, and K. Baberschke, Phys. Rev. B 55, 3708 (1997). 2 A. Hucht and K. D. Usadel, Phys. Rev. B 55, 12309 (1997). 3 P. J. Jensen and K. H. Bennemann, Solid State Comm. 105, 577 (1998), and references therein. 4 R. P. Erickson and D. L. Mills, Phys. Rev. B 44, 11825 (1991). 5 D. K. Morr, P. J. Jensen, and K. H. Bennemann, Surf. Sci. 307-309, 1109 (1994). 6 P. Politi, A. Rettori, M. G. Pini, and D. Pescia, J. Magn. Magn. Mater. 140-144, 647 (1995); A. Abanov, V. Kalatsky, V. L. Pokrovsky and W. M. Saslow, Phys. Rev. B 51, 1023 (1995). 7 A. Hucht, A. 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B 65, 134434 (2002). 20 F. Bloch, Z. Phys. 61, 206 (1930). 21 P. Bruno, Phys. Rev. B 43, 6015 (1998). 22 P. J. Jensen and K. H. Bennemann, in Magnetism and electronic correlations in local-moment systems, edited by M. Donath, P. A. Dowben and W. Nolting, p.113 (World Scientific, 1998). 23 P. Fröbrich, P. J. Jensen, and P. J. Kuntz, Eur. Phys. J. B 13, 477 (2000). 24 N. N. Bogolyubov and S. V. Tyablikov, Soviet. Phys.- Doklady 4, 589 (1959). 25 F. B. Anderson and H. Callen, Phys. Rev. 136, A1068 (1964). 26 P. Fröbrich and P. J. Kuntz, http://arxiv.org/pdf/cond-mat/0607675. 27 J. Lindner, Ph.D. thesis, Freie Universität Berlin (2002). 28 A. W. Sandvik, Phys. Rev. B 59, R14 157 (1999). 29 O. F. Syljůasen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002). 30 F. Alet, S. Wessel, and M. Troyer, Phys. Rev. E 71, 036706 (2005). 31 ALPS collaboration, J. Phys. Soc. Jpn. Suppl. 74, 30 (2005). Source codes can be obtained from http://alps.comp-phys.org/ 32 A.Gelfert and W.Nolting, Phys. Stat. Sol. B 217, 805 (2000). 33 Besides some rare earth materials where the anisotropy can be of the order of J . 34 For B < |Keff | there is another mathematical solution (sin θ = 0) which however is unstable (see appendix A). 35 Note that a similar result has already been published in Ref. 9. 36 Strictly speaking there is no phase transition because of the applied magnetic field as can be seen from the large tail of the magnetization curve. However one can extract a TC from the curves by extrapolating to the zero field case and additionally to an infinte system size in the QMC calculations. mailto:henning@physik.hu-berlin.de http://arxiv.org/pdf/cond-mat/0607675 http://alps.comp-phys.org/

7 Matrix Ordered Operator Algebras. Ekaterina Juschenko, Stanislav Popovych Department of Mathematics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden jushenko@math.chalmers.se stanislp@math.chalmers.se Abstract We study the question when for a given ∗-algebra A a sequence of cones Cn ∈ Mn(A) can be realized as cones of positive operators in a faithful ∗-representation of A on a Hilbert space. A characterization of operator algebras which are completely boundedly isomorphic to ∗-algebras is presented. KEYWORDS: ∗-algebra, operator algebra, C∗-algebra, completely bounded homomorphism, Kadison’s problem. 1 Introduction Effros and Choi gave in [2] an abstract characterization of the self-adjoint sub- spaces S in C∗-algebras with hierarchy of cones of positive elements inMn(S). In Section 2 of the present paper we are concerned with the same question for ∗-subalgebras of C∗-algebras. More precisely, let A be an associative ∗-algebra with unit. In Theorem 2 we present a characterization of the col- lections of cones Cn ⊆Mn(A) for which there exist faithful ∗-representation π of A on a Hilbert space H such that Cn coincides with the cone of positive operators contained in π(n)(Mn(A)). Here π (n)((xi,j)) = (π(xi,j)) for every matrix (xi,j) ∈Mn(A). Note that we do not assume that A has any faithful 0 2000 Mathematics Subject Classification: 46L05, 46L07 (Primary) 47L55, 47L07, 47L30 (Secondary) http://arxiv.org/abs/0704.1553v2 ∗-representation. This follows from the requirements imposed on the cones. In terms close to Effros and Choi we give an abstract characterizations of matrix ordered (not necessary closed) operator ∗-algebras up to complete order ∗-isomorphism. Based on this characterization we study the question when an operator algebra is similar to a C∗-algebra. Let B be a unital (closed) operator algebra in B(H). In [8] C. Le Merdy presented necessary and sufficient conditions for B to be self-adjoint. These conditions involve all completely isometric representations of B on Hilbert spaces. Our characterization is different in the following respect. If S is a bounded invertible operator in B(H) and A is a C∗-algebra in B(H) then the operator algebra S−1AS is not necessarily self-adjoint but only isomorphic to a C∗-algebra via completely bounded isomorphism with completely bounded inverse. By Haagerup’s theorem every completely bounded isomorphism π from a C∗-algebra A to an operator algebra B has the form π(a) = S−1ρ(a)S, a ∈ A, for some ∗-isomorphism ρ : A → B(H) and invertible S ∈ B(H). Thus the question whether an operator algebra B is completely boundedly isomorphic to a C∗-algebra via isomorphism which has completely bounded inverse, is equivalent to the question if there is bounded invertible operator S such that SBS−1 is a C∗-algebra. We will present a criterion for an operator algebra B to be completely boundedly isomorphic to a C∗-algebra in terms of the existence of a collection of cones Cn ∈ Mn(B) satisfying certain axioms (see def. 3). The axioms are derived from the properties of the cones of positive elements of a C∗-algebra preserved under completely bounded isomorphisms. The main results are contained in section 2. We define a ∗-admissible sequence of cones in an operator algebra and present a criterion in Theorem 4 for an operator algebra to be completely boundedly isomorphic to a C∗- algebra. In the last section we consider the operator algebras and collections of cones associated with Kadison similarity problem. 2 Operator realizations of matrix-ordered ∗- algebras. The aim of this section is to give necessary and sufficient conditions on a sequences of cones Cn ⊆ Mn(A)sa for a unital ∗-algebra A such that Cn coincides with the coneMn(A)∩Mn(B(H)) + for some realization of A as a ∗- subalgebra of B(H), where Mn(B(H)) + denotes the set of positive operators acting on Hn = H ⊕ . . .⊕H . In [11] it was proved that a ∗-algebra A with unit e is a ∗-subalgebra of B(H) if and only if there is an algebraically admissible cone on A such that e is an Archimedean order unit. Applying this result to some inductive limit of M2n(A) we obtain the desired characterization in Theorem 2. First we give necessary definitions and fix notations. Let Asa denote the set of self-adjoint elements in A. A subset C ⊂ Asa containing unit e of A is algebraically admissible cone (see [12]) provided that (i) C is a cone in Asa, i.e. λx+ βy ∈ C for all x, y ∈ C and λ ≥ 0, β ≥ 0, λ, β ∈ R; (ii) C ∩ (−C) = {0}; (iii) xCx∗ ⊆ C for every x ∈ A; We call e ∈ Asa an order unit if for every x ∈ Asa there exists r > 0 such that re+ x ∈ C. An order unit e is Archimedean if re+ x ∈ C for all r > 0 implies that x ∈ C In what follows we will need the following. Theorem 1. Let A be a ∗-algebra with unit e and C ⊆ Asa be a cone containing e. If xCx∗ ⊆ C for every x ∈ A and e is an Archimedean order unit then there is a unital ∗-representation π : A → B(H) such that π(C) = π(Asa) ∩ B(H) +. Moreover 1. ‖π(x)‖ = inf{r > 0 : r2 ± x∗x ∈ C}. 2. ker π = {x : x∗x ∈ C ∩ (−C)}. 3. If C ∩ (−C) = {0} then ker π = {0}, ‖π(a)‖ = inf{r > 0 : r ± a ∈ C} for all a = a∗ ∈ A and π(C) = π(A) ∩B(H)+ Proof. Following the same lines as in [11] one obtains that the function ‖ · ‖ : Asa → R+ defined as ‖a‖ = inf{r > 0 : re± a ∈ C} is a seminorm on R-space Asa and |x| = ‖x∗x‖ for x ∈ A defines a pre- C∗-norm on A. If N denote the null-space of | · | then the completion B = A/N with respect to this norm is a C∗-algebra and canonical epimorphism π : A → A/N extends to a unital ∗-homomorphism π : A → B. We can assume without loss of generality that B is a concrete C∗-algebra in B(H) for some Hilbert space H . Thus π : A → B(H) can be regarded as a unital ∗-representation. Clearly, ‖π(x)‖ = |x| for all x ∈ A. This implies 1. To show 2 take x ∈ ker π then ‖π(x)‖ = 0 and re ± x∗x ∈ C for all r > 0. Since e is an Archimedean unit we have x∗x ∈ C ∩ (−C). Conversely if x∗x ∈ C ∩ (−C) then re± x∗x ∈ C, for all r > 0, hence ‖π(x)‖ = 0 and 2 holds. Let us prove that π(C) = π(Asa) ∩ B(H) +. Let x ∈ Asa and π(x) ≥ 0. Then there exists a constant λ > 0 such that ‖λIH − π(x)‖ ≤ λ, hence |λe − x| ≤ λ. Since ‖a‖ ≤ |a| for all self-adjoint a ∈ A, see Lemma 3.3 of [11], we have ‖λe−x‖ ≤ λ. Thus given ε > 0 we have (λ+ε)e±(λe−x) ∈ C. Hence εe+ x ∈ C. Since e is Archimedean x ∈ C. Conversely, let x ∈ C. To show that π(x) ≥ 0 it is sufficient to find λ > 0 such that ‖λIH − π(x)‖ ≤ λ. Since ‖λIH − π(x)‖ = |λe − x| we will prove that |λe − x| ≤ λ for some λ > 0. From the definition of norm | · | we have the following equivalences: |λe− x| ≤ λ ⇔ (λ+ ε)2e− (λe− x)2 ∈ C for all ε > 0 (1) ⇔ ε1e + x(2λe− x) ≥ 0, for all ε1 > 0. (2) By condition (iii) in the definition of algebraically admissible cone we have that xyx ∈ C and yxy ∈ C for every x, y ∈ C. If xy = yx then xy(x + y) ∈ C. Since e is an order unit we can choose r > 0 such that re − x ∈ C. Put y = re − x to obtain rx(re − x) ∈ C. Hence (2) is satisfied with λ = r . Thus ‖λe − π(x)‖ ≤ λ and π(x) ≥ 0, which proves π(C) = π(Asa) ∩ B(H) In particular, for a = a∗ we have ‖π(a)‖ = inf{r > 0 : rIH ± π(a) ∈ π(C)}. (3) We now in a position to prove 3. Suppose that C∩ (−C) = 0. Then ker π is a ∗-ideal and ker π 6= 0 implies that there exists a self-adjoint 0 6= a ∈ ker π, i.e. |a| = 0. Inequality ‖a‖ ≤ |a| implies re± a ∈ C for all r > 0. Since e is Archimedean, ±a ∈ C, i.e. a ∈ C ∩ (−C) and, consequently, a = 0. Since ker π = 0 the inclusion rIH±π(a) ∈ π(C) is equivalent to re±a ∈ C, and by (3), ‖π(a)‖ = inf{r > 0 : re± a ∈ C}. Moreover if π(a) = π(a)∗ then a = a∗. Thus we have π(C) = π(A) ∩ B(H)+. We say that a ∗-algebra A with unit e is a matrix ordered if the following conditions hold: (a) for each n ≥ 1 we are given a cone Cn in Mn(A)sa and e ∈ C1, (b) Cn ∩ (−Cn) = {0} for all n, (c) for all n and m and all A ∈Mn×m(A), we have that A ∗CnA ⊆ Cm, We call e ∈ Asa a matrix order unit provided that for every n ∈ N and every x ∈ Mn(A)sa there exists r > 0 such that ren + x ∈ Cn, where en = e ⊗ In. A matrix order unit is called Archimedean matrix order unit provided that for all n ∈ N inclusion ren + x ∈ Cn for all r > 0 implies that x ∈ Cn. Let π : A → B(H) be a ∗-representation. Define π(n) : Mn(A) → Mn(B(H)) by π (n)((aij)) = (π(aij)). Theorem 2. If A is a matrix-ordered ∗-algebra with a unit e which is Archimedean matrix order unit then there exists a Hilbert spaceH and a faith- ful unital ∗-representation τ : A → B(H), such that τ (n)(Cn) = Mn(τ(A)) for all n. Conversely, every unital ∗-subalgebra D of B(H) is matrix-ordered by cones Mn(D) + = Mn(D) ∩ B(H) + and the unit of this algebra is an Archimedean order unit. Proof. Consider an inductive system of ∗-algebras and unital injective ∗- homomorphisms: φn :M2n(A) →M2n+1(A), φn(a) = for all n ≥ 0, a ∈M2n(A). Let B = lim M2n(A) be the inductive limit of this system. By (c) in the definition of the matrix ordered algebra we have φn(C2n) ⊆ C2n+1. We will identify M2n(A) with a subalgebra of B via canonical inclusions. Let C =⋃ C2n ⊆ Bsa and let e∞ be the unit of B. Let us prove that C is an algebraically admissible cone. Clearly, C satisfies conditions (i) and (ii) of definition of algebraically admissible cone. To prove (iii) suppose that x ∈ B and a ∈ C, then for sufficiently large n we have a ∈ C2n and x ∈ M2n(A). Therefore, by (c), x ∗ax ∈ C. Thus (iii) is proved. Since e is an Archimedean matrix order unit we obviously have that e∞ is also an Archimedean order unit. Thus ∗-algebra B satisfies assumptions of Theorem 1 and there is a faithful ∗-representation π : B → B(H) such that π(C) = π(B) ∩ B(H)+. Let ξn : M2n(A) → B be canonical injections (n ≥ 0). Then τ = π ◦ ξ0 : A → B(H) is an injective ∗-homomorphism. We claim that τ (2 n) is unitary equivalent to π◦ξn. By replacing π with π where α is an infinite cardinal, we can assume that πα is unitary equivalent to π. Since π◦ξn :M2n(A) → B(H) is a ∗-homomorphism there exist unique Hilbert space Kn, ∗-homomorphism ρn : A → B(Kn) and unitary operator Un : Kn ⊗ C 2n → H such that π ◦ ξn = Un(ρn ⊗ idM2n )U For a ∈ A, we have π ◦ ξ0(a) = π ◦ ξn(a⊗ E2n) = Un(ρn(a)⊗ E2n)U where E2n is the identity matrix in M2n(C). Thus τ(a) = U0ρ0(a)U Un(ρn(a) ⊗ E2n)U n. Let ∼ stands for the unitary equivalence of represen- tations. Since π ◦ ξn ∼ ρn ⊗ idM2n and π α ∼ π we have that ραn ⊗ idM2n ∼ πα ◦ ξn ∼ ρn ⊗ idM2n . Hence ρ n ∼ ρn. Thus ρn ⊗ E2n ∼ ρ n ∼ ρn. Consequently ρ0 ∼ ρn and π ◦ ξn ∼ ρ0 ⊗ idM2n ∼ τ ⊗ idM2n . Therefore n) = τ ⊗ idM2n is unitary equivalent to π ◦ ξn. What is left to show is that τ (n)(Cn) = Mn(τ(A)) +. Note that π ◦ ξn(M2n(A))∩B(H) + = π(C2n). Indeed, the inclusion π ◦ ξ(C2n) ⊆M2n(A)∩ B(H)+ is obvious. To show the converse take x ∈M2n(A) such that π(x) ≥ 0. Then x ∈ C∩M2n(A). Using (c) one can easily show that C∩M2n(A) = C2n . Hence π ◦ ξn(M2n(A)) ∩ B(H) + = π(C2n). Since τ (2n) is unitary equivalent to π ◦ ξn we have that τ (2n)(C2n) =M2n(τ(A)) ∩ B(H 2n)+. Let now show that τ (n)(Cn) =Mn(τ(A)) +. For X ∈Mn(A) denote X 0n×(2n−n) 0(2n−n)×n 0(2n−n)×(2n−n) ∈M2n(A). Then, clearly, τ (n)(X) ≥ 0 if and only if τ (2 n)(X̃) ≥ 0. Thus τ (n)(X) ≥ 0 is equivalent to X̃ ∈ C2n which in turn is equivalent to X ∈ Cn by (c). 3 Operator Algebras completely boundedly isomorphic to C∗-algebras. The algebra Mn(B(H)) of n× n matrices with entries in B(H) has a norm ‖ · ‖n via the identification ofMn(B(H)) with B(H n), where Hn is the direct sum of n copies of a Hilbert space H . If A is a subalgebra of B(H) then Mn(A) inherits a norm ‖·‖n via natural inclusion intoMn(B(H)). The norms ‖ · ‖n are called matrix norms on the operator algebra A. In the sequel all operator algebras will be assumed to be norm closed. Operator algebras A and B are called completely boundedly isomorphic if there is a completely bounded isomorphism τ : A → B with completely bounded inverse. The aim of this section is to give necessary and sufficient conditions for an operator algebra to be completely boundedly isomorphic to a C∗-algebra. To do this we introduce a concept of ∗-admissible cones which reflect the properties of the cones of positive elements of a C∗-algebra preserved under completely bounded isomorphism. Definition 3. Let B be an operator algebra with unit e. A sequence Cn ⊆ Mn(B) of closed (in the norm ‖ · ‖n) cones will be called ∗-admissible if it satisfies the following conditions: 1. e ∈ C1; 2. (i) Mn(B) = (Cn − Cn) + i(Cn − Cn), for all n ∈ N, (ii) Cn ∩ (−Cn) = {0}, for all n ∈ N, (iii) (Cn − Cn) ∩ i(Cn − Cn) = {0}, for all n ∈ N; 3. (i) for all c1, c2 ∈ Cn and c ∈ Cn, we have that (c1−c2)c(c1−c2) ∈ Cn, (ii) for all n, m and B ∈Mn×m(C) we have that B ∗CnB ⊆ Cm; 4. there is r > 0 such that for every positive integer n and c ∈ Cn − Cn we have r‖c‖en + c ∈ Cn, 5. there exists a constant K > 0 such that for all n ∈ N and a, b ∈ Cn−Cn we have ‖a‖n ≤ K · ‖a+ ib‖n. Theorem 4. If an operator algebra B has a ∗-admissible sequence of cones then there is a completely bounded isomorphism τ from B onto a C∗-algebra A. If, in addition, one of the following conditions holds (1) there exists r > 0 such that for every n ≥ 1 and c, d ∈ Cn we have ‖c+ d‖ ≥ r‖c‖. (2) ‖(x− iy)(x+ iy)‖ ≥ α‖x− iy‖‖x+ iy‖ for all x, y ∈ Cn − Cn then the inverse τ−1 : A → B is also completely bounded. Conversely, if such isomorphism τ exists then B possesses a ∗-admissible sequence of cones and conditions (1) and (2) are satisfied. The proof will be divided into 4 lemmas. Let {Cn}n≥1 be a ∗-admissible sequence of cones of B. Let B2n =M2n(B), φn : B2n → B2n+1 be unital homomorphisms given by φn(x) = x ∈ B2n . Denote by B∞ = lim−→ B2n the inductive limit of the system (B2n , φn). As all inclusions φn are unital B∞ has a unit, denoted by e∞. Since B∞ can be considered as a subalgebra of a C∗-algebra of the corresponding induc- tive limit of M2n(B(H)) we can define the closure of B∞ in this C ∗-algebra denoted by B∞. Now we will define an involution on B∞. Let ξn : M2n(B) → B∞ be the canonical morphisms. By (3ii), φn(C2n) ⊆ C2n+1 . Hence C = ξn(C2n) is a well defined cone in B∞. Denote by C its completion. By (2i) and (2iii), for every x ∈ B2n , we have x = x1+ ix2 with unique x1, x2 ∈ C2n −C2n . By (3ii) we have ∈ C2n+1 − C2n+1, i = 1, 2. Thus for every x ∈ B∞ we have unique decomposition x = x1+ ix2, x1 ∈ C−C, x2 ∈ C−C. Hence the mapping x 7→ x♯ = x1− ix2 is a well defined involution on B∞. In particular, we have an involution on B which depends only on the cone C1. Lemma 5. Involution on B∞ is defined by the involution on B, i.e. for all A = (aij)i,j ∈M2n(B) A♯ = (a ji)i,j. Proof. Assignment A◦ = (a ji)i,j, clearly, defines an involution on M2n(B). We need to prove that A♯ = A◦. Let A = (aij)i,j ∈ M2n(B) be self-adjoint A ◦ = A. Then A = aii ⊗ Eii + (aij ⊗ Eij + a ij ⊗ Eji) and a ii = aii, for all i. By (3ii) we have aii ⊗ Eii ∈ C2n − C2n . Since aij = a ij + ia ij for some a ij , a ij ∈ C2n − C2n we have aij ⊗Eij + a ij ⊗ Eji = (a ij + ia ij)⊗ Eij + (a ij − ia ij)⊗ Eji = (a′ij ⊗ Eij + a ij ⊗ Eji) + (ia ij ⊗ Eij − ia ij ⊗ Eji) = (Eii + Eji)(a ij ⊗ Eii + a ij ⊗ Ejj)(Eii + Eij) − (a′ij ⊗ Eii + a ij ⊗ Ejj) + (Eii − iEji)(a ij ⊗ Eii + a ij ⊗ Ejj)(Eii + iEij) − (a′′ij ⊗ Eii + a ij ⊗ Ejj) ∈ C2n − C2n . Thus A ∈ C2n − C2n and A ♯ = A. Since for every x ∈ M2n(B) there exist unique x1 = x 1 and x2 = x 2 in M2n(B), such that x = x1 + ix2, and unique x′1 = x 1 and x 2 = x 2 , such that x = x 1 + ix 2, we have that x1 = x 1 = x x2 = x 2 = x 2 and involutions ♯ and ◦ coincide. Lemma 6. Involution x → x♯ is continuous on B∞ and extends to the in- volution on B∞. With respect to this involution C ⊆ (B∞)sa and x ♯Cx ⊆ C for every x ∈ B∞. Proof. Consider a convergent net {xi} ⊆ B∞ with the limit x ∈ B∞. Decom- pose xi = x i with x i ∈ C−C. By (5), the nets {x i} and {x i } are also convergent. Thus x = a+ ib, where a = lim x′i ∈ C − C, b = lim x i ∈ C − C and lim x i = a− ib. Therefore the involution defined on B∞ can be extended by continuity to B∞ by setting x ♯ = a− ib. Under this involution C ⊆ (B∞)sa = {x ∈ B∞ : x = x Let us show that x♯cx ∈ C for every x ∈ B∞ and c ∈ C. Take firstly c ∈ C2n and x ∈ B2n . Then x = x1 + ix2 for some x1, x2 ∈ C2n − C2n and (x1 + ix2) ♯c(x1 + ix2) = (x1 − ix2)c(x1 + ix2) )( −x1 −ix2 ix2 x1 −x1 −ix2 ix2 x1 By (3i), Lemma 5 and (3ii) x♯cx ∈ C2n . Let now c ∈ C and x ∈ B∞. Suppose that ci → c and xi → x, where ci ∈ C, xi ∈ B∞. We can assume that ci, xi ∈ B2ni . Then x icixi ∈ C2ni for all i and since it is convergent we have x♯cx ∈ C. Lemma 7. The unit of B∞ is an Archimedean order unit and (B∞)sa = C − C. Proof. Firstly let us show that e∞ is an order unit. Clearly, (B∞)sa = C − C. For every a ∈ C − C, there is a net ai ∈ C2ni − C2ni convergent to a. Since ‖ai‖ <∞ there exists r1 > 0 such that r1eni − ai ∈ C2ni , i.e. r1e∞− ai ∈ C. Passing to the limit we get r1e∞ − a ∈ C. Replacing a by −a we can find r2 > 0 such that r2e∞ + a ∈ C. If r = max(r1, r2) then re∞ ± a ∈ C. This proves that e∞ is an order unit and that for all a ∈ C − C we have a = re∞ − c for some c ∈ C. Thus C − C ∈ C − C. The converse inclusion, clearly, holds. Thus C − C = C − C. If x ∈ (B∞)sa such that for every r > 0 we have r + x ∈ C then x ∈ C since C is closed. Hence e∞ is an Archimedean order unit. Lemma 8. B∞ ∩ C = C. Proof. Denote by D = lim M2n(B(H)) the C ∗-algebra inductive limit corre- sponding to the inductive system φn and denote φn,m = φm−1 ◦ . . . ◦ φn : M2n(B(H)) → M2m(B(H)). For n < m we identify M2m−n(M2n(B(H))) with M2m(B(H)) by omitting superfluous parentheses in a block matrix B = [Bij ]ij with Bij ∈M2n(B(H)). Denote by Pn,m the operator diag(I, 0, . . . , 0) ∈M2m−n(M2n(B(H))) and set Vn,m = ∑2m−n k=1 Ek,k−1. Here I is the identity matrix in M2n(B(H)) and Ek,k−1 is 2 n×2n block matrix with identity operator at (k, k−1)-entry and all other entries being zero. Define an operator ψn,m([Bij ]) = diag(B11, . . . , B11). It is easy to see that ψn,m([Bij ]) = 2m−n−1∑ (V kn,mPn,m)B(V n,mPn,m) Hence by (3ii) ψn,m(C2m) ⊆ φ(C2n) ⊆ C2m . (4) Clearly, ψn,m is a linear contraction and ψn,m+k ◦ φm,m+k = φm,m+k ◦ ψn,m Hence there is a well defined contraction ψn = lim ψn,m : D → D such that ψn|M2n(B(H)) = idM2n (B(H)), whereM2n(B(H)) is considered as a subalgebra in D. Clearly, ψn(B∞) ⊆ B∞ and ψn|B2n = id. Consider C and C2n as subalgebras in B∞, by (4) we have ψn : C → C2n. To prove that B∞ ∩C = C take c ∈ B∞ ∩C. Then there is a net cj in C such that ‖cj − c‖ → 0. Since c ∈ B∞, c ∈ B2n for some n, and consequently ψn(c) = c. Thus ‖ψn(cj)− c‖ = ‖ψn(cj − c)‖ ≤ ‖cj − c‖. Hence ψn(cj) → c. But ψn(cj) ∈ C2n and the latter is closed. Thus c ∈ C. The converse inclusion is obvious. Remark 9. Note that for every x ∈ D ψn(x) = x. (5) Indeed, for every ε > 0 there is x ∈ M2n(B(H)) such that ‖x − xn‖ < ε. Since ψn is a contraction and ψn(xn) = xn we have ‖ψn(x)− x‖ ≤ ‖ψn(x)− xn‖+ ‖xn − x‖ = ‖ψn(x− xn)‖+ ‖xn − x‖ ≤ 2ε. Since xn ∈ M2n(B(H)) also belong to M2m(B(H)) for all m ≥ n, we have that ‖ψm(x)− x‖ ≤ 2ε. Thus lim ψn(x) = x. Proof of Theorem 4. By Lemma 6 and 7 the cone C and the unit e∞ satisfies all assumptions of Theorem 1. Thus there is a homomorphism τ : B∞ → B(H̃) such that τ(a ♯) = τ(a)∗ for all a ∈ B∞. Since the image of τ is a ∗-subalgebra of B(H̃) we have that τ is bounded by [3, (23.11), p. 81]. The arguments at the end of the proof of Theorem 2 show that the restriction of τ to B2n is unitary equivalent to the 2 n-amplification of τ |B. Thus τ |B is completely bounded. Let us prove that ker(τ) = {0}. By Theorem 2.3 it is sufficient to show that C ∩ (−C) = 0. If c, d ∈ C such that c + d = 0 then c = d = 0. Indeed, for every n ≥ 1, ψn(c) + ψn(d) = 0. By Lemma 8, we have ψn(C) ⊆ C ∩ B2n = C2n . Therefore ψn(c), ψn(d) ∈ C2n . Hence ψn(c) = −ψn(d) ∈ C2n ∩ (−C2n) and, consequently, ψn(c) = ψn(d) = 0. Since ‖ψn(c)−c‖ → 0 and ‖ψn(d)−d‖ → 0 by Remark 9, we have that c = d = 0. If x ∈ C ∩ (−C) then x+ (−x) = 0, x,−x ∈ C and x = 0. Thus τ is injective. We will show that the image of τ is closed if one of the conditions (1) or (2) of the statement holds. Assume firstly that operator algebra B satisfies the first condition. Since τ(B∞) = τ(C)−τ(C)+ i(τ(C)−τ(C)) and τ(C) is exactly the set of positive operators in the image of τ , it is suffices to prove that τ(C) is closed. By Theorem 1.3, for self-adjoint (under involution ♯) x ∈ B∞ we have ‖τ(x)‖ B( eH) = inf{r > 0 : re∞ ± x ∈ C}. If τ(cα) ∈ τ(C) is a Cauchy net in B(H̃) then for every ε > 0 there is γ such that ε ± (cα − cβ) ∈ C when α ≥ γ and β ≥ γ. Since C ∩ B∞ = C, ε ± (cα − cβ) ∈ C. Denote cαβ = ε + (cα − cβ) and dαβ = ε − (cα − cβ). The set of pairs (α, β) is directed if (α, β) ≥ (α1, β1) iff α ≥ α1 and β ≥ β1. Since cαβ + dαβ = 2ε this net converges to zero in the norm of B∞. Thus by assumption 4 in the definition of ∗-admissible sequence of cones, ‖cαβ‖B∞ → 0. This implies that cα is a Cauchy net in B∞. Let c = lim cα. Clearly, c ∈ C. Since τ is continuous ‖τ(cα) − τ(c)‖B∞ → 0. Hence the closure τ(C) is contained in τ(C). By continuity of τ we have τ(C) ⊆ τ(C). Hence τ(C) = τ(C), τ(C) is closed. Let now B satisfy condition (2) of the Theorem. Then for every x ∈ B∞ we have ‖x♯x‖ ≥ α‖x‖‖x♯‖. By [3, theorem 34.3] B∞ admits an equivalent C∗-norm |·|. Since τ is a faithful ∗-representation of the C∗-algebra (B∞, |·|) it is isometric. Therefore τ(B∞) is closed. Let us show that (τ |B) −1 : τ(B) → B is completely bounded. The image A = τ(B∞) is a C ∗-algebra inB(H̃) isomorphic to B∞. By Johnson’s theorem (see [6]), two Banach algebra norms on a semi-simple algebra are equivalent, hence, τ−1 : A → B∞ is bounded homomorphism, say ‖τ −1‖ = R. Let us show that ‖(τ |B) −1‖cb = R. Since τ |B2n = Un(τ |B ⊗ idM2n )U for some unitary Un : K ⊗ C 2n → H̃ we have for any B = [bij ] ∈M2n(B) bij ⊗ Eij‖ ≤ R‖τ( bij ⊗ Eij)‖ = R‖Un( τ(bij)⊗ Eij)U τ(bij)⊗ Eij‖. This is equivalent to τ−1(bij)⊗ eij‖ ≤ R‖ bij ⊗Eij‖, hence ‖(τ−1)2 (B)‖ ≤ R‖B‖. This proves that ‖(τ |B) −1‖cb = R. The converse statement evidently holds with ∗-admissible sequence of cones given by (τ (n))−1(Mn(A) Conditions (1) and (2) were used to prove that the image of isomorphism τ is closed. The natural question one can ask is wether there exists an operator algebra B and isomorphism ρ : B → B(H) with non-closed self-adjoint image. The following example gives the affirmative answer. Example 10. Consider the algebra B = C1([0, 1]) as an operator algebra in C∗-algebra M2(C([0, 1])) via inclusion f(·) 7→ ⊕q∈Q f(q) f ′(q) 0 f(q) The induced norm ‖f‖ = sup (2|f(q)|2 + |f ′(q)|2 + |f ′(q)| 4|f(q)|2 + |f ′(q)|2) satisfies the inequality ‖f‖ ≥ 1√ max{‖f‖∞, ‖f ′‖∞} ≥ ‖f‖1 where ‖f‖1 = ‖f‖∞+‖f ′‖∞ is the standard Banach norm on C 1([0, 1]). Thus B is a closed operator algebra with isometric involution f ♯(x) = f(x), (x ∈ [0, 1]). The identity map C1([0, 1]) → C([0, 1]), f 7→ f is a ∗-isomorphism of B into C∗-algebra with non-closed self-adjoint image. 4 Operator Algebra associated with Kadison’s similarity problem. In 1955 R. Kadison raised the following problem. Is any bounded homomor- phism π of a C∗-algebra A into B(H) similar to a ∗-representation? The similarity above means that there exists invertible operator S ∈ B(H) such that x→ S−1π(x)S is a ∗-representation of A. The following criterion due to Haagerup (see [4]) is widely used in refor- mulations of Kadison’s problem: non-degenerate homomorphism π is similar to a ∗-representation iff π is completely bounded. Moreover the similarity S can be chosen in such a way that ‖S−1‖‖S‖ = ‖π‖cb. The affirmative answer to the Kadison’s problem is obtained in many important cases. In particular, for nuclear A, π is automatically completely bounded with ‖π‖cb ≤ ‖π‖ 2 (see [1]). About recent state of the problem we refer the reader to [9, 5]. We can associate an operator algebra π(B) to every bounded injective homomorphism π of a C∗-algebra A. The fact that π(B) is closed can be seen by restricting π to a nuclear C∗-algebra C∗(x∗x). This restriction is similar to ∗-homomorphism for every x ∈ A which gives the estimate ‖x‖ ≤ ‖π‖3‖π(x)‖ (for details see [10, p. 4]). Denote Cn = π (n)(Mn(A) Let J be an involution in B(H), i.e. self-adjoint operator such that J2 = I. Clearly, J is also a unitary operator. A representation π : A → B(H) of a ∗-algebra A is called J-symmetric if π(a∗) = Jπ(a)∗J . Such representations are natural analogs of ∗-representations for Krein space with indefinite metric [x, y] = 〈Jx, y〉. We will need the following observation due to V. Shulman [13] (see also [7, lemma 9.3, p.131]). If π is an arbitrary representation of A in B(H) then the representation ρ : A → B(H⊕H), a 7→ π(a)⊕π(a∗)∗ is J-symmetric with J(x⊕ y) = y⊕x and representation π is a restriction ρ|K⊕{0}. Moreover, if ρ is similar to ∗-representation then so is π. Clearly the converse is also true, thus π and ρ are simultaneously similar to ∗-representations or not. In sequel for an operator algebra D ∈ B(H) we denote by lim M2n(D) the closure of the algebraic direct limit of ofM2n(D) in the C ∗-algebra direct limit of inductive system M2n(B(H)) with standard inclusions x→ Theorem 11. Let π : A → B(H) be a bounded unital J-symmmetric in- jective homomorphism of a C∗-algebra A and let B = π(A). Then π−1 is a completely bounded homomorphism. Its extension π̃−1 to the homomor- phism between the inductive limits B∞ = lim−→ M2n(B) and A∞ = lim−→ M2n(A) is injective. Proof. Let us show that {Cn}n≥1 is a ∗-admissible sequence of cones. It is routine to verify that conditions (1)-(3) in the definition of ∗-admissible cones are satisfied for {Cn}. To see that condition (4) also holds take B ∈ Cn − Cn and denote r = ‖B‖. Let D ∈ Mn(A)sa be such that B = π (n)(D). Since π(n) : Mn(A) → Mn(B) is algebraic isomorphism it preserves spectra σMn(A)(x) = σMn(B)(π (n)(x)). Since the spectral radius spr(B) ≤ r we have spr(D) ≤ r. Hence ren +D ∈ Mn(A) + because D is self-adjoint. Applying π(n) we get ren +B ∈ Cn which proves condition (4). Since π is J-symmetric ‖π(n)(a)‖ = ‖(J ⊗En)π (n)(a)∗(J ⊗En)‖ = ‖π (n)(a∗)‖ for every a ∈Mn(A), and ‖π(n)(h1)‖ ≤ 1/2(‖π (n)(h1) + iπ (n)(h2)‖+ ‖π (n)(h1)− iπ (n)(h2)‖) = ‖π(n)(h1) + iπ (n)(h2)‖ for all h1, h2 ∈ Cn − Cn. Thus condition (5) is satisfied and {Cn} is ∗- admissible. By Theorem 4, there is an injective bounded homomorphism τ : B∞ → B(H̃) such that its restriction to B is completely bounded, τ(b τ(b)∗ and τn(Cn) = τn(Mn(B)) Denote ρ = τ ◦ π : A → B(H̃). Since ρ is a positive homomorphism, it is a ∗-representation. Moreover, ker ρ = {0} because both π and τ are injective. Therefore ρ−1 is ∗-isomorphism. Since τ : B → B(H̃) extends to an injective homomorphism of inductive limit B∞ and ρ −1 is completely isometric, we have that π−1 = ρ−1 ◦ τ extends to injective homomorphism of B∞. It is also clear that π−1 is completely bounded as a superposition of two completely bounded maps. Remark 12. The first statement of Theorem 11 can be deduced also from [10, Theorem 2.6]. Remark 13. Note that condition (1) and (2) in Theorem 4 for cones Cn from the proof of Theorem 11 is obviously equivalent to π being completely bounded. Acknowledgments. The authors wish to express their thanks to Victor Shulman for helpful comments and providing the reference [13]. The work was written when the second author was visiting Chalmers University of Technology in Göteborg, Sweden. The second author was sup- ported by the Swedish Institute. References [1] J. Bunce, The similarity problem for representations of C∗-algebras, Proc. Amer. Math. Soc. 81 (1981), p. 409-414. [2] M.D. Choi, E.G. Effros, Injectivity and operator spaces. J. Functional Analysis 24 (1977), no. 2, 156–209. [3] R.S. Doran, V.A. Belfi, Characterizations of C∗-algebras. The Gelfand- Năımark theorems. Monographs and Textbooks in Pure and Applied Mathematics, 101. Marcel Dekker, Inc., New York, 1986. xi+426 pp. [4] U. Haagerup, Solution of the similarity problem for cyclic representa- tions of C∗-algebras, Annals of Math. 118 (1983), p. 215-240 [5] D. Hadwin, V. Paulsen, Two reformulations of Kadison’s similarity problem, J. Oper. Theory, Vol. 55, No. 1, (2006), 3-16. [6] B. Johnson The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537-539 [7] E. Kissin, V. Shulman, Representations on Krein spaces and deriva- tions of C∗-algebras, Pitman Monographs and Surveys in Pure and Ap- plied Mathematics 89, 1997 [8] C. Le Merdy, Self adjointness criteria for operator algebras, Arch. Math. 74 (2000), p. 212- 220. [9] G. Pisier, Similarity Problems and Completely Bounded Maps, Springer-Verlag Lecture Notes in Math 1618, 1996 [10] D. Pitts, Norming algebras and automatic complete boundedness of isomorphism of operator algebras, arXiv: math.OA/0609604, 2006 http://arxiv.org/abs/math/0609604 [11] S. Popovych, On O∗-representability and C∗-representability of ∗- algebras, Chalmers & Göteborg University math. preprint 2006:35. [12] R. Powers, Selfadjoint algebras of unbounded operators II, Trans. Amer. Math. Soc. 187 (1974), 261–293. [13] V.S. Shulman, On representations of C∗-algebras on indefinite metric spaces, Mat. Zametki, 22(1977), 583-592 = Math Notes 22(1977) Introduction Operator realizations of matrix-ordered *-algebras. Operator Algebras completely boundedly isomorphic to C*-algebras. Operator Algebra associated with Kadison's similarity problem.

We study the question when for a given *-algebra $\mathcal{A}$ a sequence of cones $C_n\in M_n(\mathcal{A})$ can be realized as cones of positive operators in a faithful *-representation of $\mathcal{A}$ on a Hilbert space. A characterization of operator algebras which are completely boundedly isomorphic to $C\sp*$-algebras is presented.

Introduction Effros and Choi gave in [2] an abstract characterization of the self-adjoint sub- spaces S in C∗-algebras with hierarchy of cones of positive elements inMn(S). In Section 2 of the present paper we are concerned with the same question for ∗-subalgebras of C∗-algebras. More precisely, let A be an associative ∗-algebra with unit. In Theorem 2 we present a characterization of the col- lections of cones Cn ⊆Mn(A) for which there exist faithful ∗-representation π of A on a Hilbert space H such that Cn coincides with the cone of positive operators contained in π(n)(Mn(A)). Here π (n)((xi,j)) = (π(xi,j)) for every matrix (xi,j) ∈Mn(A). Note that we do not assume that A has any faithful 0 2000 Mathematics Subject Classification: 46L05, 46L07 (Primary) 47L55, 47L07, 47L30 (Secondary) http://arxiv.org/abs/0704.1553v2 ∗-representation. This follows from the requirements imposed on the cones. In terms close to Effros and Choi we give an abstract characterizations of matrix ordered (not necessary closed) operator ∗-algebras up to complete order ∗-isomorphism. Based on this characterization we study the question when an operator algebra is similar to a C∗-algebra. Let B be a unital (closed) operator algebra in B(H). In [8] C. Le Merdy presented necessary and sufficient conditions for B to be self-adjoint. These conditions involve all completely isometric representations of B on Hilbert spaces. Our characterization is different in the following respect. If S is a bounded invertible operator in B(H) and A is a C∗-algebra in B(H) then the operator algebra S−1AS is not necessarily self-adjoint but only isomorphic to a C∗-algebra via completely bounded isomorphism with completely bounded inverse. By Haagerup’s theorem every completely bounded isomorphism π from a C∗-algebra A to an operator algebra B has the form π(a) = S−1ρ(a)S, a ∈ A, for some ∗-isomorphism ρ : A → B(H) and invertible S ∈ B(H). Thus the question whether an operator algebra B is completely boundedly isomorphic to a C∗-algebra via isomorphism which has completely bounded inverse, is equivalent to the question if there is bounded invertible operator S such that SBS−1 is a C∗-algebra. We will present a criterion for an operator algebra B to be completely boundedly isomorphic to a C∗-algebra in terms of the existence of a collection of cones Cn ∈ Mn(B) satisfying certain axioms (see def. 3). The axioms are derived from the properties of the cones of positive elements of a C∗-algebra preserved under completely bounded isomorphisms. The main results are contained in section 2. We define a ∗-admissible sequence of cones in an operator algebra and present a criterion in Theorem 4 for an operator algebra to be completely boundedly isomorphic to a C∗- algebra. In the last section we consider the operator algebras and collections of cones associated with Kadison similarity problem. 2 Operator realizations of matrix-ordered ∗- algebras. The aim of this section is to give necessary and sufficient conditions on a sequences of cones Cn ⊆ Mn(A)sa for a unital ∗-algebra A such that Cn coincides with the coneMn(A)∩Mn(B(H)) + for some realization of A as a ∗- subalgebra of B(H), where Mn(B(H)) + denotes the set of positive operators acting on Hn = H ⊕ . . .⊕H . In [11] it was proved that a ∗-algebra A with unit e is a ∗-subalgebra of B(H) if and only if there is an algebraically admissible cone on A such that e is an Archimedean order unit. Applying this result to some inductive limit of M2n(A) we obtain the desired characterization in Theorem 2. First we give necessary definitions and fix notations. Let Asa denote the set of self-adjoint elements in A. A subset C ⊂ Asa containing unit e of A is algebraically admissible cone (see [12]) provided that (i) C is a cone in Asa, i.e. λx+ βy ∈ C for all x, y ∈ C and λ ≥ 0, β ≥ 0, λ, β ∈ R; (ii) C ∩ (−C) = {0}; (iii) xCx∗ ⊆ C for every x ∈ A; We call e ∈ Asa an order unit if for every x ∈ Asa there exists r > 0 such that re+ x ∈ C. An order unit e is Archimedean if re+ x ∈ C for all r > 0 implies that x ∈ C In what follows we will need the following. Theorem 1. Let A be a ∗-algebra with unit e and C ⊆ Asa be a cone containing e. If xCx∗ ⊆ C for every x ∈ A and e is an Archimedean order unit then there is a unital ∗-representation π : A → B(H) such that π(C) = π(Asa) ∩ B(H) +. Moreover 1. ‖π(x)‖ = inf{r > 0 : r2 ± x∗x ∈ C}. 2. ker π = {x : x∗x ∈ C ∩ (−C)}. 3. If C ∩ (−C) = {0} then ker π = {0}, ‖π(a)‖ = inf{r > 0 : r ± a ∈ C} for all a = a∗ ∈ A and π(C) = π(A) ∩B(H)+ Proof. Following the same lines as in [11] one obtains that the function ‖ · ‖ : Asa → R+ defined as ‖a‖ = inf{r > 0 : re± a ∈ C} is a seminorm on R-space Asa and |x| = ‖x∗x‖ for x ∈ A defines a pre- C∗-norm on A. If N denote the null-space of | · | then the completion B = A/N with respect to this norm is a C∗-algebra and canonical epimorphism π : A → A/N extends to a unital ∗-homomorphism π : A → B. We can assume without loss of generality that B is a concrete C∗-algebra in B(H) for some Hilbert space H . Thus π : A → B(H) can be regarded as a unital ∗-representation. Clearly, ‖π(x)‖ = |x| for all x ∈ A. This implies 1. To show 2 take x ∈ ker π then ‖π(x)‖ = 0 and re ± x∗x ∈ C for all r > 0. Since e is an Archimedean unit we have x∗x ∈ C ∩ (−C). Conversely if x∗x ∈ C ∩ (−C) then re± x∗x ∈ C, for all r > 0, hence ‖π(x)‖ = 0 and 2 holds. Let us prove that π(C) = π(Asa) ∩ B(H) +. Let x ∈ Asa and π(x) ≥ 0. Then there exists a constant λ > 0 such that ‖λIH − π(x)‖ ≤ λ, hence |λe − x| ≤ λ. Since ‖a‖ ≤ |a| for all self-adjoint a ∈ A, see Lemma 3.3 of [11], we have ‖λe−x‖ ≤ λ. Thus given ε > 0 we have (λ+ε)e±(λe−x) ∈ C. Hence εe+ x ∈ C. Since e is Archimedean x ∈ C. Conversely, let x ∈ C. To show that π(x) ≥ 0 it is sufficient to find λ > 0 such that ‖λIH − π(x)‖ ≤ λ. Since ‖λIH − π(x)‖ = |λe − x| we will prove that |λe − x| ≤ λ for some λ > 0. From the definition of norm | · | we have the following equivalences: |λe− x| ≤ λ ⇔ (λ+ ε)2e− (λe− x)2 ∈ C for all ε > 0 (1) ⇔ ε1e + x(2λe− x) ≥ 0, for all ε1 > 0. (2) By condition (iii) in the definition of algebraically admissible cone we have that xyx ∈ C and yxy ∈ C for every x, y ∈ C. If xy = yx then xy(x + y) ∈ C. Since e is an order unit we can choose r > 0 such that re − x ∈ C. Put y = re − x to obtain rx(re − x) ∈ C. Hence (2) is satisfied with λ = r . Thus ‖λe − π(x)‖ ≤ λ and π(x) ≥ 0, which proves π(C) = π(Asa) ∩ B(H) In particular, for a = a∗ we have ‖π(a)‖ = inf{r > 0 : rIH ± π(a) ∈ π(C)}. (3) We now in a position to prove 3. Suppose that C∩ (−C) = 0. Then ker π is a ∗-ideal and ker π 6= 0 implies that there exists a self-adjoint 0 6= a ∈ ker π, i.e. |a| = 0. Inequality ‖a‖ ≤ |a| implies re± a ∈ C for all r > 0. Since e is Archimedean, ±a ∈ C, i.e. a ∈ C ∩ (−C) and, consequently, a = 0. Since ker π = 0 the inclusion rIH±π(a) ∈ π(C) is equivalent to re±a ∈ C, and by (3), ‖π(a)‖ = inf{r > 0 : re± a ∈ C}. Moreover if π(a) = π(a)∗ then a = a∗. Thus we have π(C) = π(A) ∩ B(H)+. We say that a ∗-algebra A with unit e is a matrix ordered if the following conditions hold: (a) for each n ≥ 1 we are given a cone Cn in Mn(A)sa and e ∈ C1, (b) Cn ∩ (−Cn) = {0} for all n, (c) for all n and m and all A ∈Mn×m(A), we have that A ∗CnA ⊆ Cm, We call e ∈ Asa a matrix order unit provided that for every n ∈ N and every x ∈ Mn(A)sa there exists r > 0 such that ren + x ∈ Cn, where en = e ⊗ In. A matrix order unit is called Archimedean matrix order unit provided that for all n ∈ N inclusion ren + x ∈ Cn for all r > 0 implies that x ∈ Cn. Let π : A → B(H) be a ∗-representation. Define π(n) : Mn(A) → Mn(B(H)) by π (n)((aij)) = (π(aij)). Theorem 2. If A is a matrix-ordered ∗-algebra with a unit e which is Archimedean matrix order unit then there exists a Hilbert spaceH and a faith- ful unital ∗-representation τ : A → B(H), such that τ (n)(Cn) = Mn(τ(A)) for all n. Conversely, every unital ∗-subalgebra D of B(H) is matrix-ordered by cones Mn(D) + = Mn(D) ∩ B(H) + and the unit of this algebra is an Archimedean order unit. Proof. Consider an inductive system of ∗-algebras and unital injective ∗- homomorphisms: φn :M2n(A) →M2n+1(A), φn(a) = for all n ≥ 0, a ∈M2n(A). Let B = lim M2n(A) be the inductive limit of this system. By (c) in the definition of the matrix ordered algebra we have φn(C2n) ⊆ C2n+1. We will identify M2n(A) with a subalgebra of B via canonical inclusions. Let C =⋃ C2n ⊆ Bsa and let e∞ be the unit of B. Let us prove that C is an algebraically admissible cone. Clearly, C satisfies conditions (i) and (ii) of definition of algebraically admissible cone. To prove (iii) suppose that x ∈ B and a ∈ C, then for sufficiently large n we have a ∈ C2n and x ∈ M2n(A). Therefore, by (c), x ∗ax ∈ C. Thus (iii) is proved. Since e is an Archimedean matrix order unit we obviously have that e∞ is also an Archimedean order unit. Thus ∗-algebra B satisfies assumptions of Theorem 1 and there is a faithful ∗-representation π : B → B(H) such that π(C) = π(B) ∩ B(H)+. Let ξn : M2n(A) → B be canonical injections (n ≥ 0). Then τ = π ◦ ξ0 : A → B(H) is an injective ∗-homomorphism. We claim that τ (2 n) is unitary equivalent to π◦ξn. By replacing π with π where α is an infinite cardinal, we can assume that πα is unitary equivalent to π. Since π◦ξn :M2n(A) → B(H) is a ∗-homomorphism there exist unique Hilbert space Kn, ∗-homomorphism ρn : A → B(Kn) and unitary operator Un : Kn ⊗ C 2n → H such that π ◦ ξn = Un(ρn ⊗ idM2n )U For a ∈ A, we have π ◦ ξ0(a) = π ◦ ξn(a⊗ E2n) = Un(ρn(a)⊗ E2n)U where E2n is the identity matrix in M2n(C). Thus τ(a) = U0ρ0(a)U Un(ρn(a) ⊗ E2n)U n. Let ∼ stands for the unitary equivalence of represen- tations. Since π ◦ ξn ∼ ρn ⊗ idM2n and π α ∼ π we have that ραn ⊗ idM2n ∼ πα ◦ ξn ∼ ρn ⊗ idM2n . Hence ρ n ∼ ρn. Thus ρn ⊗ E2n ∼ ρ n ∼ ρn. Consequently ρ0 ∼ ρn and π ◦ ξn ∼ ρ0 ⊗ idM2n ∼ τ ⊗ idM2n . Therefore n) = τ ⊗ idM2n is unitary equivalent to π ◦ ξn. What is left to show is that τ (n)(Cn) = Mn(τ(A)) +. Note that π ◦ ξn(M2n(A))∩B(H) + = π(C2n). Indeed, the inclusion π ◦ ξ(C2n) ⊆M2n(A)∩ B(H)+ is obvious. To show the converse take x ∈M2n(A) such that π(x) ≥ 0. Then x ∈ C∩M2n(A). Using (c) one can easily show that C∩M2n(A) = C2n . Hence π ◦ ξn(M2n(A)) ∩ B(H) + = π(C2n). Since τ (2n) is unitary equivalent to π ◦ ξn we have that τ (2n)(C2n) =M2n(τ(A)) ∩ B(H 2n)+. Let now show that τ (n)(Cn) =Mn(τ(A)) +. For X ∈Mn(A) denote X 0n×(2n−n) 0(2n−n)×n 0(2n−n)×(2n−n) ∈M2n(A). Then, clearly, τ (n)(X) ≥ 0 if and only if τ (2 n)(X̃) ≥ 0. Thus τ (n)(X) ≥ 0 is equivalent to X̃ ∈ C2n which in turn is equivalent to X ∈ Cn by (c). 3 Operator Algebras completely boundedly isomorphic to C∗-algebras. The algebra Mn(B(H)) of n× n matrices with entries in B(H) has a norm ‖ · ‖n via the identification ofMn(B(H)) with B(H n), where Hn is the direct sum of n copies of a Hilbert space H . If A is a subalgebra of B(H) then Mn(A) inherits a norm ‖·‖n via natural inclusion intoMn(B(H)). The norms ‖ · ‖n are called matrix norms on the operator algebra A. In the sequel all operator algebras will be assumed to be norm closed. Operator algebras A and B are called completely boundedly isomorphic if there is a completely bounded isomorphism τ : A → B with completely bounded inverse. The aim of this section is to give necessary and sufficient conditions for an operator algebra to be completely boundedly isomorphic to a C∗-algebra. To do this we introduce a concept of ∗-admissible cones which reflect the properties of the cones of positive elements of a C∗-algebra preserved under completely bounded isomorphism. Definition 3. Let B be an operator algebra with unit e. A sequence Cn ⊆ Mn(B) of closed (in the norm ‖ · ‖n) cones will be called ∗-admissible if it satisfies the following conditions: 1. e ∈ C1; 2. (i) Mn(B) = (Cn − Cn) + i(Cn − Cn), for all n ∈ N, (ii) Cn ∩ (−Cn) = {0}, for all n ∈ N, (iii) (Cn − Cn) ∩ i(Cn − Cn) = {0}, for all n ∈ N; 3. (i) for all c1, c2 ∈ Cn and c ∈ Cn, we have that (c1−c2)c(c1−c2) ∈ Cn, (ii) for all n, m and B ∈Mn×m(C) we have that B ∗CnB ⊆ Cm; 4. there is r > 0 such that for every positive integer n and c ∈ Cn − Cn we have r‖c‖en + c ∈ Cn, 5. there exists a constant K > 0 such that for all n ∈ N and a, b ∈ Cn−Cn we have ‖a‖n ≤ K · ‖a+ ib‖n. Theorem 4. If an operator algebra B has a ∗-admissible sequence of cones then there is a completely bounded isomorphism τ from B onto a C∗-algebra A. If, in addition, one of the following conditions holds (1) there exists r > 0 such that for every n ≥ 1 and c, d ∈ Cn we have ‖c+ d‖ ≥ r‖c‖. (2) ‖(x− iy)(x+ iy)‖ ≥ α‖x− iy‖‖x+ iy‖ for all x, y ∈ Cn − Cn then the inverse τ−1 : A → B is also completely bounded. Conversely, if such isomorphism τ exists then B possesses a ∗-admissible sequence of cones and conditions (1) and (2) are satisfied. The proof will be divided into 4 lemmas. Let {Cn}n≥1 be a ∗-admissible sequence of cones of B. Let B2n =M2n(B), φn : B2n → B2n+1 be unital homomorphisms given by φn(x) = x ∈ B2n . Denote by B∞ = lim−→ B2n the inductive limit of the system (B2n , φn). As all inclusions φn are unital B∞ has a unit, denoted by e∞. Since B∞ can be considered as a subalgebra of a C∗-algebra of the corresponding induc- tive limit of M2n(B(H)) we can define the closure of B∞ in this C ∗-algebra denoted by B∞. Now we will define an involution on B∞. Let ξn : M2n(B) → B∞ be the canonical morphisms. By (3ii), φn(C2n) ⊆ C2n+1 . Hence C = ξn(C2n) is a well defined cone in B∞. Denote by C its completion. By (2i) and (2iii), for every x ∈ B2n , we have x = x1+ ix2 with unique x1, x2 ∈ C2n −C2n . By (3ii) we have ∈ C2n+1 − C2n+1, i = 1, 2. Thus for every x ∈ B∞ we have unique decomposition x = x1+ ix2, x1 ∈ C−C, x2 ∈ C−C. Hence the mapping x 7→ x♯ = x1− ix2 is a well defined involution on B∞. In particular, we have an involution on B which depends only on the cone C1. Lemma 5. Involution on B∞ is defined by the involution on B, i.e. for all A = (aij)i,j ∈M2n(B) A♯ = (a ji)i,j. Proof. Assignment A◦ = (a ji)i,j, clearly, defines an involution on M2n(B). We need to prove that A♯ = A◦. Let A = (aij)i,j ∈ M2n(B) be self-adjoint A ◦ = A. Then A = aii ⊗ Eii + (aij ⊗ Eij + a ij ⊗ Eji) and a ii = aii, for all i. By (3ii) we have aii ⊗ Eii ∈ C2n − C2n . Since aij = a ij + ia ij for some a ij , a ij ∈ C2n − C2n we have aij ⊗Eij + a ij ⊗ Eji = (a ij + ia ij)⊗ Eij + (a ij − ia ij)⊗ Eji = (a′ij ⊗ Eij + a ij ⊗ Eji) + (ia ij ⊗ Eij − ia ij ⊗ Eji) = (Eii + Eji)(a ij ⊗ Eii + a ij ⊗ Ejj)(Eii + Eij) − (a′ij ⊗ Eii + a ij ⊗ Ejj) + (Eii − iEji)(a ij ⊗ Eii + a ij ⊗ Ejj)(Eii + iEij) − (a′′ij ⊗ Eii + a ij ⊗ Ejj) ∈ C2n − C2n . Thus A ∈ C2n − C2n and A ♯ = A. Since for every x ∈ M2n(B) there exist unique x1 = x 1 and x2 = x 2 in M2n(B), such that x = x1 + ix2, and unique x′1 = x 1 and x 2 = x 2 , such that x = x 1 + ix 2, we have that x1 = x 1 = x x2 = x 2 = x 2 and involutions ♯ and ◦ coincide. Lemma 6. Involution x → x♯ is continuous on B∞ and extends to the in- volution on B∞. With respect to this involution C ⊆ (B∞)sa and x ♯Cx ⊆ C for every x ∈ B∞. Proof. Consider a convergent net {xi} ⊆ B∞ with the limit x ∈ B∞. Decom- pose xi = x i with x i ∈ C−C. By (5), the nets {x i} and {x i } are also convergent. Thus x = a+ ib, where a = lim x′i ∈ C − C, b = lim x i ∈ C − C and lim x i = a− ib. Therefore the involution defined on B∞ can be extended by continuity to B∞ by setting x ♯ = a− ib. Under this involution C ⊆ (B∞)sa = {x ∈ B∞ : x = x Let us show that x♯cx ∈ C for every x ∈ B∞ and c ∈ C. Take firstly c ∈ C2n and x ∈ B2n . Then x = x1 + ix2 for some x1, x2 ∈ C2n − C2n and (x1 + ix2) ♯c(x1 + ix2) = (x1 − ix2)c(x1 + ix2) )( −x1 −ix2 ix2 x1 −x1 −ix2 ix2 x1 By (3i), Lemma 5 and (3ii) x♯cx ∈ C2n . Let now c ∈ C and x ∈ B∞. Suppose that ci → c and xi → x, where ci ∈ C, xi ∈ B∞. We can assume that ci, xi ∈ B2ni . Then x icixi ∈ C2ni for all i and since it is convergent we have x♯cx ∈ C. Lemma 7. The unit of B∞ is an Archimedean order unit and (B∞)sa = C − C. Proof. Firstly let us show that e∞ is an order unit. Clearly, (B∞)sa = C − C. For every a ∈ C − C, there is a net ai ∈ C2ni − C2ni convergent to a. Since ‖ai‖ <∞ there exists r1 > 0 such that r1eni − ai ∈ C2ni , i.e. r1e∞− ai ∈ C. Passing to the limit we get r1e∞ − a ∈ C. Replacing a by −a we can find r2 > 0 such that r2e∞ + a ∈ C. If r = max(r1, r2) then re∞ ± a ∈ C. This proves that e∞ is an order unit and that for all a ∈ C − C we have a = re∞ − c for some c ∈ C. Thus C − C ∈ C − C. The converse inclusion, clearly, holds. Thus C − C = C − C. If x ∈ (B∞)sa such that for every r > 0 we have r + x ∈ C then x ∈ C since C is closed. Hence e∞ is an Archimedean order unit. Lemma 8. B∞ ∩ C = C. Proof. Denote by D = lim M2n(B(H)) the C ∗-algebra inductive limit corre- sponding to the inductive system φn and denote φn,m = φm−1 ◦ . . . ◦ φn : M2n(B(H)) → M2m(B(H)). For n < m we identify M2m−n(M2n(B(H))) with M2m(B(H)) by omitting superfluous parentheses in a block matrix B = [Bij ]ij with Bij ∈M2n(B(H)). Denote by Pn,m the operator diag(I, 0, . . . , 0) ∈M2m−n(M2n(B(H))) and set Vn,m = ∑2m−n k=1 Ek,k−1. Here I is the identity matrix in M2n(B(H)) and Ek,k−1 is 2 n×2n block matrix with identity operator at (k, k−1)-entry and all other entries being zero. Define an operator ψn,m([Bij ]) = diag(B11, . . . , B11). It is easy to see that ψn,m([Bij ]) = 2m−n−1∑ (V kn,mPn,m)B(V n,mPn,m) Hence by (3ii) ψn,m(C2m) ⊆ φ(C2n) ⊆ C2m . (4) Clearly, ψn,m is a linear contraction and ψn,m+k ◦ φm,m+k = φm,m+k ◦ ψn,m Hence there is a well defined contraction ψn = lim ψn,m : D → D such that ψn|M2n(B(H)) = idM2n (B(H)), whereM2n(B(H)) is considered as a subalgebra in D. Clearly, ψn(B∞) ⊆ B∞ and ψn|B2n = id. Consider C and C2n as subalgebras in B∞, by (4) we have ψn : C → C2n. To prove that B∞ ∩C = C take c ∈ B∞ ∩C. Then there is a net cj in C such that ‖cj − c‖ → 0. Since c ∈ B∞, c ∈ B2n for some n, and consequently ψn(c) = c. Thus ‖ψn(cj)− c‖ = ‖ψn(cj − c)‖ ≤ ‖cj − c‖. Hence ψn(cj) → c. But ψn(cj) ∈ C2n and the latter is closed. Thus c ∈ C. The converse inclusion is obvious. Remark 9. Note that for every x ∈ D ψn(x) = x. (5) Indeed, for every ε > 0 there is x ∈ M2n(B(H)) such that ‖x − xn‖ < ε. Since ψn is a contraction and ψn(xn) = xn we have ‖ψn(x)− x‖ ≤ ‖ψn(x)− xn‖+ ‖xn − x‖ = ‖ψn(x− xn)‖+ ‖xn − x‖ ≤ 2ε. Since xn ∈ M2n(B(H)) also belong to M2m(B(H)) for all m ≥ n, we have that ‖ψm(x)− x‖ ≤ 2ε. Thus lim ψn(x) = x. Proof of Theorem 4. By Lemma 6 and 7 the cone C and the unit e∞ satisfies all assumptions of Theorem 1. Thus there is a homomorphism τ : B∞ → B(H̃) such that τ(a ♯) = τ(a)∗ for all a ∈ B∞. Since the image of τ is a ∗-subalgebra of B(H̃) we have that τ is bounded by [3, (23.11), p. 81]. The arguments at the end of the proof of Theorem 2 show that the restriction of τ to B2n is unitary equivalent to the 2 n-amplification of τ |B. Thus τ |B is completely bounded. Let us prove that ker(τ) = {0}. By Theorem 2.3 it is sufficient to show that C ∩ (−C) = 0. If c, d ∈ C such that c + d = 0 then c = d = 0. Indeed, for every n ≥ 1, ψn(c) + ψn(d) = 0. By Lemma 8, we have ψn(C) ⊆ C ∩ B2n = C2n . Therefore ψn(c), ψn(d) ∈ C2n . Hence ψn(c) = −ψn(d) ∈ C2n ∩ (−C2n) and, consequently, ψn(c) = ψn(d) = 0. Since ‖ψn(c)−c‖ → 0 and ‖ψn(d)−d‖ → 0 by Remark 9, we have that c = d = 0. If x ∈ C ∩ (−C) then x+ (−x) = 0, x,−x ∈ C and x = 0. Thus τ is injective. We will show that the image of τ is closed if one of the conditions (1) or (2) of the statement holds. Assume firstly that operator algebra B satisfies the first condition. Since τ(B∞) = τ(C)−τ(C)+ i(τ(C)−τ(C)) and τ(C) is exactly the set of positive operators in the image of τ , it is suffices to prove that τ(C) is closed. By Theorem 1.3, for self-adjoint (under involution ♯) x ∈ B∞ we have ‖τ(x)‖ B( eH) = inf{r > 0 : re∞ ± x ∈ C}. If τ(cα) ∈ τ(C) is a Cauchy net in B(H̃) then for every ε > 0 there is γ such that ε ± (cα − cβ) ∈ C when α ≥ γ and β ≥ γ. Since C ∩ B∞ = C, ε ± (cα − cβ) ∈ C. Denote cαβ = ε + (cα − cβ) and dαβ = ε − (cα − cβ). The set of pairs (α, β) is directed if (α, β) ≥ (α1, β1) iff α ≥ α1 and β ≥ β1. Since cαβ + dαβ = 2ε this net converges to zero in the norm of B∞. Thus by assumption 4 in the definition of ∗-admissible sequence of cones, ‖cαβ‖B∞ → 0. This implies that cα is a Cauchy net in B∞. Let c = lim cα. Clearly, c ∈ C. Since τ is continuous ‖τ(cα) − τ(c)‖B∞ → 0. Hence the closure τ(C) is contained in τ(C). By continuity of τ we have τ(C) ⊆ τ(C). Hence τ(C) = τ(C), τ(C) is closed. Let now B satisfy condition (2) of the Theorem. Then for every x ∈ B∞ we have ‖x♯x‖ ≥ α‖x‖‖x♯‖. By [3, theorem 34.3] B∞ admits an equivalent C∗-norm |·|. Since τ is a faithful ∗-representation of the C∗-algebra (B∞, |·|) it is isometric. Therefore τ(B∞) is closed. Let us show that (τ |B) −1 : τ(B) → B is completely bounded. The image A = τ(B∞) is a C ∗-algebra inB(H̃) isomorphic to B∞. By Johnson’s theorem (see [6]), two Banach algebra norms on a semi-simple algebra are equivalent, hence, τ−1 : A → B∞ is bounded homomorphism, say ‖τ −1‖ = R. Let us show that ‖(τ |B) −1‖cb = R. Since τ |B2n = Un(τ |B ⊗ idM2n )U for some unitary Un : K ⊗ C 2n → H̃ we have for any B = [bij ] ∈M2n(B) bij ⊗ Eij‖ ≤ R‖τ( bij ⊗ Eij)‖ = R‖Un( τ(bij)⊗ Eij)U τ(bij)⊗ Eij‖. This is equivalent to τ−1(bij)⊗ eij‖ ≤ R‖ bij ⊗Eij‖, hence ‖(τ−1)2 (B)‖ ≤ R‖B‖. This proves that ‖(τ |B) −1‖cb = R. The converse statement evidently holds with ∗-admissible sequence of cones given by (τ (n))−1(Mn(A) Conditions (1) and (2) were used to prove that the image of isomorphism τ is closed. The natural question one can ask is wether there exists an operator algebra B and isomorphism ρ : B → B(H) with non-closed self-adjoint image. The following example gives the affirmative answer. Example 10. Consider the algebra B = C1([0, 1]) as an operator algebra in C∗-algebra M2(C([0, 1])) via inclusion f(·) 7→ ⊕q∈Q f(q) f ′(q) 0 f(q) The induced norm ‖f‖ = sup (2|f(q)|2 + |f ′(q)|2 + |f ′(q)| 4|f(q)|2 + |f ′(q)|2) satisfies the inequality ‖f‖ ≥ 1√ max{‖f‖∞, ‖f ′‖∞} ≥ ‖f‖1 where ‖f‖1 = ‖f‖∞+‖f ′‖∞ is the standard Banach norm on C 1([0, 1]). Thus B is a closed operator algebra with isometric involution f ♯(x) = f(x), (x ∈ [0, 1]). The identity map C1([0, 1]) → C([0, 1]), f 7→ f is a ∗-isomorphism of B into C∗-algebra with non-closed self-adjoint image. 4 Operator Algebra associated with Kadison’s similarity problem. In 1955 R. Kadison raised the following problem. Is any bounded homomor- phism π of a C∗-algebra A into B(H) similar to a ∗-representation? The similarity above means that there exists invertible operator S ∈ B(H) such that x→ S−1π(x)S is a ∗-representation of A. The following criterion due to Haagerup (see [4]) is widely used in refor- mulations of Kadison’s problem: non-degenerate homomorphism π is similar to a ∗-representation iff π is completely bounded. Moreover the similarity S can be chosen in such a way that ‖S−1‖‖S‖ = ‖π‖cb. The affirmative answer to the Kadison’s problem is obtained in many important cases. In particular, for nuclear A, π is automatically completely bounded with ‖π‖cb ≤ ‖π‖ 2 (see [1]). About recent state of the problem we refer the reader to [9, 5]. We can associate an operator algebra π(B) to every bounded injective homomorphism π of a C∗-algebra A. The fact that π(B) is closed can be seen by restricting π to a nuclear C∗-algebra C∗(x∗x). This restriction is similar to ∗-homomorphism for every x ∈ A which gives the estimate ‖x‖ ≤ ‖π‖3‖π(x)‖ (for details see [10, p. 4]). Denote Cn = π (n)(Mn(A) Let J be an involution in B(H), i.e. self-adjoint operator such that J2 = I. Clearly, J is also a unitary operator. A representation π : A → B(H) of a ∗-algebra A is called J-symmetric if π(a∗) = Jπ(a)∗J . Such representations are natural analogs of ∗-representations for Krein space with indefinite metric [x, y] = 〈Jx, y〉. We will need the following observation due to V. Shulman [13] (see also [7, lemma 9.3, p.131]). If π is an arbitrary representation of A in B(H) then the representation ρ : A → B(H⊕H), a 7→ π(a)⊕π(a∗)∗ is J-symmetric with J(x⊕ y) = y⊕x and representation π is a restriction ρ|K⊕{0}. Moreover, if ρ is similar to ∗-representation then so is π. Clearly the converse is also true, thus π and ρ are simultaneously similar to ∗-representations or not. In sequel for an operator algebra D ∈ B(H) we denote by lim M2n(D) the closure of the algebraic direct limit of ofM2n(D) in the C ∗-algebra direct limit of inductive system M2n(B(H)) with standard inclusions x→ Theorem 11. Let π : A → B(H) be a bounded unital J-symmmetric in- jective homomorphism of a C∗-algebra A and let B = π(A). Then π−1 is a completely bounded homomorphism. Its extension π̃−1 to the homomor- phism between the inductive limits B∞ = lim−→ M2n(B) and A∞ = lim−→ M2n(A) is injective. Proof. Let us show that {Cn}n≥1 is a ∗-admissible sequence of cones. It is routine to verify that conditions (1)-(3) in the definition of ∗-admissible cones are satisfied for {Cn}. To see that condition (4) also holds take B ∈ Cn − Cn and denote r = ‖B‖. Let D ∈ Mn(A)sa be such that B = π (n)(D). Since π(n) : Mn(A) → Mn(B) is algebraic isomorphism it preserves spectra σMn(A)(x) = σMn(B)(π (n)(x)). Since the spectral radius spr(B) ≤ r we have spr(D) ≤ r. Hence ren +D ∈ Mn(A) + because D is self-adjoint. Applying π(n) we get ren +B ∈ Cn which proves condition (4). Since π is J-symmetric ‖π(n)(a)‖ = ‖(J ⊗En)π (n)(a)∗(J ⊗En)‖ = ‖π (n)(a∗)‖ for every a ∈Mn(A), and ‖π(n)(h1)‖ ≤ 1/2(‖π (n)(h1) + iπ (n)(h2)‖+ ‖π (n)(h1)− iπ (n)(h2)‖) = ‖π(n)(h1) + iπ (n)(h2)‖ for all h1, h2 ∈ Cn − Cn. Thus condition (5) is satisfied and {Cn} is ∗- admissible. By Theorem 4, there is an injective bounded homomorphism τ : B∞ → B(H̃) such that its restriction to B is completely bounded, τ(b τ(b)∗ and τn(Cn) = τn(Mn(B)) Denote ρ = τ ◦ π : A → B(H̃). Since ρ is a positive homomorphism, it is a ∗-representation. Moreover, ker ρ = {0} because both π and τ are injective. Therefore ρ−1 is ∗-isomorphism. Since τ : B → B(H̃) extends to an injective homomorphism of inductive limit B∞ and ρ −1 is completely isometric, we have that π−1 = ρ−1 ◦ τ extends to injective homomorphism of B∞. It is also clear that π−1 is completely bounded as a superposition of two completely bounded maps. Remark 12. The first statement of Theorem 11 can be deduced also from [10, Theorem 2.6]. Remark 13. Note that condition (1) and (2) in Theorem 4 for cones Cn from the proof of Theorem 11 is obviously equivalent to π being completely bounded. Acknowledgments. The authors wish to express their thanks to Victor Shulman for helpful comments and providing the reference [13]. The work was written when the second author was visiting Chalmers University of Technology in Göteborg, Sweden. The second author was sup- ported by the Swedish Institute. References [1] J. Bunce, The similarity problem for representations of C∗-algebras, Proc. Amer. Math. Soc. 81 (1981), p. 409-414. [2] M.D. Choi, E.G. Effros, Injectivity and operator spaces. J. Functional Analysis 24 (1977), no. 2, 156–209. [3] R.S. Doran, V.A. Belfi, Characterizations of C∗-algebras. The Gelfand- Năımark theorems. Monographs and Textbooks in Pure and Applied Mathematics, 101. Marcel Dekker, Inc., New York, 1986. xi+426 pp. [4] U. Haagerup, Solution of the similarity problem for cyclic representa- tions of C∗-algebras, Annals of Math. 118 (1983), p. 215-240 [5] D. Hadwin, V. Paulsen, Two reformulations of Kadison’s similarity problem, J. Oper. Theory, Vol. 55, No. 1, (2006), 3-16. [6] B. Johnson The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537-539 [7] E. Kissin, V. Shulman, Representations on Krein spaces and deriva- tions of C∗-algebras, Pitman Monographs and Surveys in Pure and Ap- plied Mathematics 89, 1997 [8] C. Le Merdy, Self adjointness criteria for operator algebras, Arch. Math. 74 (2000), p. 212- 220. [9] G. Pisier, Similarity Problems and Completely Bounded Maps, Springer-Verlag Lecture Notes in Math 1618, 1996 [10] D. Pitts, Norming algebras and automatic complete boundedness of isomorphism of operator algebras, arXiv: math.OA/0609604, 2006 http://arxiv.org/abs/math/0609604 [11] S. Popovych, On O∗-representability and C∗-representability of ∗- algebras, Chalmers & Göteborg University math. preprint 2006:35. [12] R. Powers, Selfadjoint algebras of unbounded operators II, Trans. Amer. Math. Soc. 187 (1974), 261–293. [13] V.S. Shulman, On representations of C∗-algebras on indefinite metric spaces, Mat. Zametki, 22(1977), 583-592 = Math Notes 22(1977) Introduction Operator realizations of matrix-ordered *-algebras. Operator Algebras completely boundedly isomorphic to C*-algebras. Operator Algebra associated with Kadison's similarity problem.

Properly infinite C(X)-algebras and K1-injectivity Etienne Blanchard, Randi Rohde and Mikael Rørdam Abstract We investigate if a unital C(X)-algebra is properly infinite when all its fibres are prop- erly infinite. We show that this question can be rephrased in several different ways, including the question if every unital properly infinite C∗-algebra is K1-injective. We provide partial answers to these questions, and we show that the general ques- tion on proper infiniteness of C(X)-algebras can be reduced to establishing proper infiniteness of a specific C([0, 1])-algebra with properly infinite fibres. 1 Introduction The problem that we mainly are concerned with in this paper is if any unital C(X)-algebra with properly infinite fibres is itself properly infinite (see Section 2 for a brief introduction to C(X)-algebras). An analogous study was carried out in the recent paper [8] where it was decided when C(X)-algebras, whose fibres are either stable or absorb tensorially a given strongly self-absorbing C∗-algebra, itself has the same property. This was answered in the affirmative in [8] under the crucial assumption that the dimension of the space X is finite, and counterexamples were given in the infinite dimensional case. Along similar lines, Dadarlat, [5], recently proved that C(X)-algebras, whose fibres are Cuntz algebras, are trivial under some K-theoretical conditions provided that the space X is finite dimensional. The property of being properly infinite turns out to behave very differently than the property of being stable or of absorbing a strongly self-absorbing C∗-algebra. It is relative easy to see (Lemma 2.10) that if a fibre Ax of a C(X)-algebra A is properly infinite, then AF is properly infinite for some closed neighborhood F of x. The (possible) obstruction to proper infiniteness of the C(X)-algebra is hence not local. Such an obstruction is also not related to the possible complicated structure of the space X , as we can show that a counterexample, if it exists, can be taken to be a (specific) C([0, 1])-algebra (Example 4.1 and Theorem 5.5). The problem appears to be related with some rather subtle internal structure properties of properly infinite C∗-algebras. Cuntz studied purely infinite—and in the process also properly infinite—C∗-algebras, [4], where he among many other things (he was primarily interested in calculating the http://arxiv.org/abs/0704.1554v1 K-theory of his algebras On) showed that any unital properly infinite C ∗-algebra A is K1- surjective, i.e., the mapping U(A) → K1(A) is onto; and that any purely infinite simple C∗-algebra A is K1-injective, i.e., the mapping U(A)/U 0(A) → K1(A) is injective (and hence an isomorphism). He did not address the question if any properly infinite C∗-al- gebra is K1-injective. That question has not been raised formally to our knowledge—we do so here—but it does appear implicitly, eg. in [10] and in [14], where K1-injectivity of properly infinite C∗-algebras has to be assumed. Proper infiniteness of C∗-algebras has relevance for existence (or rather non-existence) of traces and quasitraces. Indeed, a unital C∗-algebra admits a 2-quasitrace if and only if no matrix algebra over the C∗-algebra is properly infinite, and a unital exact C∗-algebra admits tracial state again if and only if no matrix algebra over the C∗-algebra is properly infinite. In this paper we show that every properly infinite C∗-algebra is K1-injective if and only if every C(X)-algebra with properly infinite fibres itself is properly infinite. We also show that a matrix algebra over any such C(X)-algebra is properly infinite. Examples of unital C∗-algebras A, where Mn(A) is properly infinite for some natural number n ≥ 2 but where Mn−1(A) is not properly infinite, are known, see [12] and [11], but still quite exotic. We relate the question if a given properly infinite C∗-algebra isK1-injective to questions regarding homotopy of projections (Proposition 5.1). In particular we show that our main questions are equivalent to the following question: is any non-trivial projection in the first copy of O∞ in the full unital universal free product O∞∗O∞ homotopic to any (non-trivial) projection in the second copy of O∞? The specific C([0, 1])-algebra, mentioned above, is perhaps not surprisingly a sub-algebra of C([0, 1],O∞ ∗ O∞). Using ideas implicit in Rieffel’s paper, [9], we construct in Section 4 a C(T)-algebra B for each C∗-algebra A and for each unitary u ∈ A for which diag(u, 1) is homotopic to 1M2(A); and B is non-trivial if u is not homotopic to 1A. In this way we relate our question about proper infiniteness of C(X)-algebras to a question about K1-injectivity. The last mentioned author thanks Bruce Blackadar for many inspiring conversations on topics related to this paper. 2 C(X)-algebras with properly infinite fibres A powerful tool in the classification of C∗-algebras is the study of their projections. A projection in a C∗-algebra is said to be infinite if it is equivalent to a proper subprojection of itself, and it is said to be properly infinite if it is equivalent to two mutually orthogonal subprojections of itself. A projection which is not infinite is said to be finite. A unital C∗-algebra is said to be finite, infinite, or properly infinite if its unit is finite, infinite, or properly infinite, respectively. If A is a C∗-algebra for which Mn(A) is finite for all positive integers n, then A is stably finite. In this section we will study stability properties of proper infiniteness under (upper- semi-)continuous deformations using the Cuntz-Toeplitz algebra which is defined as follows. For all integers n ≥ 2 the Cuntz-Toeplitz algebra Tn is the universal C ∗-algebra generated by n isometries s1, . . . , sn satisfying the relation 1 + · · ·+ sns n ≤ 1. Remark 2.1 A unital C∗-algebra A is properly infinite if and only if Tn embeds unitally into A for some n ≥ 2, in which case Tn embeds unitally into A for all n ≥ 2. In order to study deformations of such algebras, let us recall a few notions from the theory of C(X)-algebras. Let X be a compact Hausdorff space and C(X) be the C∗-algebra of continuous func- tions on X with values in the complex field C. Definition 2.2 A C(X)-algebra is a C∗-algebra A endowed with a unital ∗-homomorphism from C(X) to the center of the multiplier C∗-algebra M(A) of A. If A is as above and Y ⊆ X is a closed subset, then we put IY = C0(X \ Y )A, which is a closed two-sided ideal in A. We set AY = A/IY and denote the quotient map by πY . For an element a ∈ A we put aY = πY (a), and if Y consists of a single point x, we will write Ax, Ix, πx and ax in the place of A{x}, I{x}, π{x} and a{x}, respectively. We say that Ax is the fibre of A at x. The function x 7→ ‖ax‖ = inf{‖ [1− f + f(x)]a‖ : f ∈ C(X)} is upper semi-continuous for all a ∈ A (as one can see using the right-hand side identity above). A C(X)-algebra A is said to be continuous (or to be a continuous C∗-bundle over X) if the function x 7→ ‖ax‖ is actually continuous for all element a in A. For any unital C∗-algebra A we let U(A) denote the group of unitary elements in A, U0(A) denotes its connected component containing the unit of A, and Un(A) and U are equal to U(Mn(A)) and U 0(Mn(A)), respectively. An element in a C∗-algebra A is said to be full if it is not contained in any proper closed two-sided ideal in A. It is well-known (see for example [13, Exercise 4.9]) that if p is a properly infinite, full projection in a C∗-algebra A, then e - p, i.e., e is equivalent to a subprojection of p, for every projection e ∈ A. We state below more formally three more or less well-known results that will be used frequently throughout this paper, the first of which is due to Cuntz, [4]. Proposition 2.3 (Cuntz) Let A be a C∗-algebra which contains at least one properly infinite, full projection. (i) Let p and q be properly infinite, full projections in A. Then [p] = [q] in K0(A) if and only if p ∼ q. (ii) For each element g ∈ K0(A) there is a properly infinite, full projection p ∈ A such that g = [p]. The second statement is a variation of the Whitehead lemma. Lemma 2.4 Let A be a unital C∗-algebra. (i) Let v be a partial isometry in A such that 1 − vv∗ and 1 − v∗v are properly infinite and full projections. Then there is a unitary element u in A such that [u] = 0 in K1(A) and v = uv ∗v, i.e., u extends v. (ii) Let u be a unitary element A such that [u] = 0 in K1(A). Suppose there exists a projection p ∈ A such that ‖up− pu‖ < 1 and p and 1 − p are properly infinite and full. Then u belongs to U0(A). Proof: (i). It follows from Proposition 2.3 (i) that 1 − v∗v ∼ 1 − vv∗, so there is a partial isometry w such that 1 − v∗v = w∗w and 1 − vv∗ = ww∗. Now, z = v + w is a unitary element in A with zv∗v = v. The projection 1 − v∗v is properly infinite and full, so 1 - 1 − v∗v, which implies that there is an isometry s in A with ss∗ ≤ 1 − v∗v. As −[z] = [z∗] = [sz∗s∗ + (1 − ss∗)] in K1(A) (see eg. [13, Exercise 8.9 (i)]), we see that u = z(sz∗s∗ + (1− ss∗)) is as desired. (ii). Put x = pup + (1 − p)u(1 − p) and note that ‖u − x‖ < 1. It follows that x is invertible in A and that u ∼h x in GL(A). Let x = v|x| be the polar decomposition of x, where |x| = (x∗x)1/2 and v = x|x|−1 is unitary. Then u ∼h v in U(A) (see eg. [13, Proposition 2.1.8]), and pv = vp. We proceed to show that v belongs to U0(A) (which will entail that u belongs to U0(A)). Write v = v1v2, where v1 = pvp+ (1− p), v2 = p+ (1− p)v(1− p). As 1−p - p we can find a symmetry t in A such that t(1−p)t ≤ p. As t belongs to U0(A) (being a symmetry), we conclude that v2 ∼h tv2t, and one checks that tv2t is of the form w + (1− p) for some unitary w in pAp. It follows that v is homotopic to a unitary of the form v0 + (1− p), where v0 is a unitary in pAp. We can now apply eg. [13, Exercise 8.11] to conclude that v ∼h 1 in U(A). � We remind the reader that if p, q are projections in a unital C∗-algebra A, then p and q are homotopic, in symbols p ∼h q, (meaning that they can be connected by a continuous path of projections in A) if and only if q = upu∗ for some u ∈ U0(A), eg. cf. [13, Proposition 2.2.6]. Proposition 2.5 Let A be a unital C∗-algebra. Let p and q be two properly infinite, full projections in A such that p ∼ q. Suppose that there exists a properly infinite, full projection r ∈ A such that p ⊥ r and q ⊥ r. Then p ∼h q. Proof: Take a partial isometry v0 ∈ A such that v 0v0 = p and v0v 0 = q. Take a subpro- jection r0 of r such that r0 and r− r0 both are properly infinite and full. Put v = v0 + r0. Then vpv∗ = q and vr0 = r0 = r0v. Note that 1 − v ∗v and 1 − vv∗ are properly infi- nite and full (because they dominate the properly infinite, full projection r − r0). Use Lemma 2.4 (i) to extend v to a unitary u ∈ A with [u] = 0 in K1(A). Now, upu ∗ = q and ur0 = vr0 = r0 = r0v = r0u. Hence u ∈ U 0(A) by Lemma 2.4 (ii), and so p ∼h q as desired. Definition 2.6 A unital C∗-algebra A is said to be K1-injective if the natural mapping U(A)/U0(A) → K1(A) is injective. In other words, if A is K1-injective, and if u is a unitary element in A, then u ∼h 1 in U(A) if (and only if) [u] = 0 in K1(A). One could argue thatK1-injectivity should entail that the natural mappings Un(A)/U n(A) → K1(A) be injective for every natural number n. However there seem to be an agreement for defining K1-injectivity as above. As we shall see later, in Proposition 5.2, if A is properly infinite, then the two definitions agree. Proposition 2.7 Let A be a unital C∗-algebra that is the pull-back of two unital, properly infinite C∗-algebras A1 and A2 along the ∗-epimorphisms π1 : A1 → B and π2 : A2 → B: }} ϕ2 π1 A π2~~}} Then M2(A) is properly infinite. Moreover, if B is K1-injective, then A itself is properly infinite. Proof: Take unital embeddings σi : T3 → Ai for i = 1, 2, where T3 is the Cuntz-Toeplitz algebra (defined earlier), and put (π1 ◦ σ1)(tj)(π2 ◦ σ2)(t where t1, t2, t3 are the canonical generators of T3. Note that v is a partial isometry with (π1 ◦ σ1)(tj) = v(π2 ◦ σ2)(tj) for j = 1, 2. As (π1 ◦ σ1)(t3t 3) ≤ 1− vv ∗ and (π2 ◦ σ2)(t3t 1−v∗v, Lemma 2.4 (i) yields a unitary u ∈ B with [u] = 0 in K1(B) and with (π1◦σ1)(tj) = u(π2 ◦ σ2)(tj) for j = 1, 2. If B is K1-injective, then u belongs to U 0(B), whence u lifts to a unitary v ∈ A2. Define σ̃2 : T2 → A2 by σ̃2(tj) = vσ2(tj) for j = 1, 2 (observing that t1, t2 generate T2). Then π1 ◦ σ1 = π2 ◦ σ̃2, which by the universal property of the pull-back implies that σ1 and σ̃2 lift to a (necessarily unital) embedding σ : T2 → A, thus forcing A to be properly infinite. In the general case (where B is not necessarily K1-injective) u may not lift to a unitary element in A2, but diag(u, u) does lift to a unitary element v in M2(A2) by Lemma 2.4 (ii) (applied with p = diag(1, 0)). Define unital embeddings σ̃i : T2 → M2(Ai), i = 1, 2, by σ̃1(tj) = σ1(tj) 0 0 σ1(tj) , σ̃2(tj) = v σ2(tj) 0 0 σ2(tj) for j = 1, 2. As (π1 ⊗ idM2) ◦ σ̃1 = (π2 ⊗ idM2) ◦ σ̃2, the unital embeddings σ̃1 and σ̃2 lift to a (necessarily unital) embedding of T2 into M2(A), thus completing the proof. � Question 2.8 Is the pull-back of any two properly infinite unital C∗-algebras again prop- erly infinite? As mentioned in the introduction, one cannot in general conclude that A is properly infinite if one knows that Mn(A) is properly infinite for some n ≥ 2. One obvious way of obtaining an answer to Question 2.8, in the light of the last state- ment in Proposition 2.7, is to answer the question below in the affirmative: Question 2.9 Is every properly infinite unital C∗-algebra K1-injective? We shall see later, in Section 5, that the two questions above in fact are equivalent. The lemma below, which shall be used several times in this paper, shows that one can lift proper infiniteness from a fibre of a C(X)-algebra to a whole neighborhood of that fibre. Lemma 2.10 Let X be a compact Hausdorff space, let A be a unital C(X)-algebra, let x ∈ X, and suppose that the fibre Ax is properly infinite. Then AF is properly infinite for some closed neighborhood F of x. Proof: Let {Fλ}λ∈Λ be a decreasing net of closed neighborhoods of x ∈ X , fulfilling that⋂ λ∈Λ Fλ = {x}, and set Iλ = C0(X \Fλ)A. Then {Iλ}λ∈Λ is an increasing net of ideals in A, AFλ = A/Iλ, I := λ∈Λ Iλ = C0(X\{x}), and Ax = A/I. By the assumption that Ax is properly infinite there is a unital ∗-homomorphism ψ : T2 → Ax, and since T2 is semi-projective there is a λ0 ∈ Λ and a unital ∗-homomorphism ϕ : T2 → AFλ0 making the diagram // Ax commutative. We can thus take F to be Fλ0 . � Theorem 2.11 Let A be a unital C(X)-algebra where X is a compact Hausdorff space. If all fibres Ax, x ∈ X, are properly infinite, then some matrix algebra over A is properly infinite. Proof: By Lemma 2.10, X can be covered by finitely many closed sets F1, F2, . . . , Fn such that AFj is properly infinite for each j. Put Gj = F1 ∪ F2 ∪ · · · ∪ Fj . For each j = 1, 2, . . . , n− 1 we have a pull-back diagram AGj+1 yyrrr AFj+1 AGj∩Fj+1 We know that M2j−1(AGj ) is properly infinite when j = 1. Proposition 2.7 (applied to the diagram above tensored with M2j−1(C)) tells us that M2j (AGj+1) is properly infinite if M2j−1(AGj) is properly infinite. Hence M2n−1(A) is properly infinite. � Remark 2.12 Uffe Haagerup has suggested another way to prove Theorem 2.11: If no matrix-algebra over A is properly infinite, then there exists a bounded non-zero lower semi-continuous 2-quasi-trace on A, see [7] and [1, page 327], and hence also an extremal 2-quasi-trace. Now, if A is also a C(X)-algebra for some compact Hausdorff space X , this implies that there is a bounded non-zero lower semi-continuous 2-quasitrace on Ax for (at least) one point x ∈ X (see eg. [8, Proposition 3.7]). But then the fibre Ax cannot be properly infinite. Question 2.13 Is any unital C(X)-algebra A properly infinite if all its fibres Ax, x ∈ X , are properly infinite? We shall show in Section 5 that the question above is equivalent to Question 2.8 which again is equivalent to Question 2.9. 3 Lower semi-continuous fields of properly infinite C∗- algebras Let us briefly discuss whether the results from Section 2 can be extended to lower semi- continuous C∗-bundles (A, {σx}) over a compact Hausdorff space X . Recall that any such separable lower semi-continuous C∗-bundle admits a faithful C(X)-linear representation on a Hilbert C(X)-module E such that, for all x ∈ X , the fibre σx(A) is isomorphic to the induced image of A in L(Ex), [2]. Thus, the problem boils down to the following: Given a separable Hilbert C(X)-module E with infinite dimensional fibres Ex, such that the unit p of the C∗-algebra LC(X)(E) of bounded adjointable C(X)-linear operators acting on E has a properly infinite image in L(Ex) for all x ∈ X . Is the projection p itself properly infinite in LC(X)(E)? Dixmier and Douady proved that this is always the case if the space X has finite topological dimension, [6]. But it does not hold anymore in the infinite dimensional case, see [6, §16, Corollaire 1] and [11], not even if X is contractible, [3, Corollary 3.7]. 4 Two examples We describe here two examples of continuous fields; the first is over the interval and the second (which really is a class of examples) is over the circle. Example 4.1 Let (O∞ ∗O∞, (ι1, ι2)) be the universal unital free product of two copies of O∞, and let A be the unital sub-C ∗-algebra of C([0, 1],O∞ ∗ O∞) given by A = {f ∈ C([0, 1],O∞ ∗ O∞) : f(0) ∈ ι1(O∞), f(1) ∈ ι2(O∞)}. Observe that A (in a canonical way) is a C([0, 1])-algebra with fibres ι1(O∞), t = 0, O∞ ∗ O∞, 0 < t < 1, ι2(O∞), t = 1 O∞, t = 0, O∞ ∗ O∞, 0 < t < 1, O∞, t = 1. In particular, all fibres of A are properly infinite. One claim to fame of the example above is that the question below is equivalent to Ques- tion 2.13 above. Hence, to answer Question 2.13 in the affirmative (or in the negative) we need only consider the case where X = [0, 1], and we need only worry about this one particular C([0, 1])-algebra (which of course is bad enough!). Question 4.2 Is the C([0, 1])-algebra A from Example 4.1 above properly infinite? The three equivalent statements in the proposition below will in Section 5 be shown to be equivalent to Question 4.2. Proposition 4.3 The following three statements concerning the C([0, 1])-algebra A and the C∗-algebra (O∞ ∗ O∞, (ι1, ι2)) defined above are equivalent: (i) A contains a non-trivial projection (i.e., a projection other than 0 and 1). (ii) There are non-zero projections p, q ∈ O∞ such that p 6= 1, q 6= 1, and ι1(p) ∼h ι2(q). (iii) Let s be any isometry in O∞. Then ι1(ss ∗) ∼h ι2(ss ∗) in O∞ ∗ O∞. We warn the reader that all three statements above could be false. Proof: (i) ⇒ (ii). Let e be a non-trivial projection in A. Let πt : A → At, t ∈ [0, 1], denote the fibre map. As A ⊆ C([0, 1],O∞ ∗ O∞), the mapping t 7→ πt(e) ∈ O∞ ∗ O∞ is continuous, so in particular, π0(e) ∼h π1(e) in O∞ ∗ O∞. The mappings ι1 and ι2 are injective, so there are projections p, q ∈ O∞ such that π0(e) = ι1(p) and π1(e) = ι2(q). The projections p and q are non-zero because the mapping t 7→ ‖πt(e)‖ is continuous and not constant equal to 0. Similarly, and 1− p and 1− q are non-zero because 1− e is non-zero. (ii) ⇒ (iii). Take non-trivial projections p, q ∈ O∞ such that ι1(p) ∼h ι2(q). Take a unitary v in U0(O∞∗O∞) with ι2(q) = vι1(p)v ∗. Let s ∈ O∞ be an isometry. If s is unitary, then ι1(ss ∗) = 1 = ι2(ss ∗) and there is nothing to prove. Suppose that s is non-unitary. Then ss∗ is homotopic to a subprojection p0 of p and to a subprojection q0 of q (use that p and q are properly infinite and full, then Lemma 2.4 (i), and last the fact that the unitary group of O∞ is connected). Hence ι1(ss ∗) ∼h ι1(p0) ∼h vι1(p0)v ∗ and ι2(ss ∗) ∼h ι2(q0), so we need only show that vι1(p0)v ∗ ∼h ι2(q0). But this follows from Proposition 2.5 with r = 1− ι2(q) = ι2(1− q), as we note that p0 ∼ 1 ∼ q0 in O∞, whence ι2(q0) ∼ ι2(1) = 1 = ι1(1) ∼ ι1(p0) ∼ vι1(p0)v (iii) ⇒ (i). Take a non-unitary isometry s ∈ O∞. Then ι1(ss ∗) ∼h ι2(ss ∗), and so there is a continuous function e : [0, 1] → O∞ ∗O∞ such that e(t) is a projection for all t ∈ [0, 1], e(0) = ι1(ss ∗) and e(1) = ι2(ss ∗). But then e is a non-trivial projection in A. � It follows from Theorem 2.11 that some matrix algebra over A (from Example 4.1) is properly infinite. We can sharpen that statement as follows: Proposition 4.4 M2(A) is properly infinite; and if O∞ ∗O∞ is K1-injective, then A itself is properly infinite. It follows from Theorem 5.5 below that A is properly infinite if and only if O∞ ∗ O∞ is K1-injective. Proof: We have a pull-back diagram A[0, 1 π1/2 %%KK π1/2yysss O∞ ∗ O∞ One can unitally embed O∞ into A[0, 1 ] via ι1, so A[0, 1 ] is properly infinite, and a similar argument shows that A[ 1 ,1] is properly infinite. The two statements now follow from Proposition 2.7. � The example below, which will be the focus of the rest of this section, and in parts also of Section 5, is inspired by arguments from Rieffel’s paper [9]. Example 4.5 Let A be a unital C∗-algebra, and let v be a unitary element in A such that in U2(A). Let t 7→ ut be a continuous path of unitaries in U2(A) such that u0 = 1 and u1 = diag(v, 1). p(t) = ut u∗t ∈M2(A), and note that p(0) = p(1). Identifying, for each C∗-algebra D, C(T, D) with the algebra of all continuous functions f : [0, 1] → D such that f(1) = f(0), we see that p belongs to C(T,M2(A)). Put B = pC(T,M2(A))p, and note that B is a unital (sub-trivial) C(T)-algebra, being a corner of the trivial C(T)- algebra C(T,M2(A)). The fibres of B are Bt = p(t)M2(A)p(t) ∼= A for all t ∈ T. Summing up, for each unital C∗-algebraA, for each unitary v inA for which diag(v, 1) ∼h 1 in U2(A), and for each path t 7→ ut ∈ U2(A) implementing this homotopy we get a C(T)- algebra B with fibres Bt ∼= A. We shall investigate this class of C(T)-algebras below. Lemma 4.6 In the notation of Example 4.5, − p ∼ in C(T,M2(A)). In particular, p is stably equivalent to diag(1, 0). Proof: Put vt = ut , t ∈ [0, 1]. v0 = u0 , v1 = u1 so v belongs to C(T,M2(A)). It is easy to see that v t vt = diag(0, 1) and vtv t = 1 − p(t), and so the lemma is proved. � Proposition 4.7 Let A, v ∈ U(A), and B be as in Example 4.5. Conditions (i) and (ii) below are equivalent for any unital C∗-algebra A, and all three conditions are equivalent if A in addition is assumed to be properly infinite. (i) v ∼h 1 in U(A). (ii) p ∼ diag(1A, 0) in C(T,M2(A)). (iii) The C(T)-algebra B is properly infinite. Proof: (ii) ⇒ (i). Suppose that p ∼ diag(1, 0) in C(T,M2(A)). Then there is a w ∈ C(T,M2(A)) such that and w∗twt = pt for all t ∈ [0, 1] and w1 = w0 (as we identify C(T,M2(A)) with the set of continuous functions f : [0, 1] → M2(A) with f(1) = f(0)). Upon replacing wt with w 0wt we can assume that w1 = w0 = diag(1, 0). Now, with t 7→ ut as in Example 4.5, where t 7→ at is a continuous path of unitaries in A. Because u0 = diag(1, 1) and u1 = diag(v, 1) we see that a0 = 1 and a1 = v, whence v ∼h 1 in U(A). (i) ⇒ (ii). Suppose conversely that v ∼h 1 in U(A). Then we can find a continuous path t 7→ vt ∈ U(A), t ∈ [1 − ε, 1], such that v1−ε = v and v1 = 1 for an ε > 0 (to be determined below). Again with t 7→ ut as in Example 4.5, define ũt = u(1−ε)−1t, 0 ≤ t ≤ 1− ε, diag(vt, 1), 1− ε ≤ t ≤ 1. Then t 7→ ũt is a continuous path of unitaries in U2(A) such that ũ1−ε = u1 = diag(v, 1) and ũ0 = ũ1 = 1. It follows that ũ belongs to C(T,M2(A)). Provided that ε > 0 is chosen small enough we obtain the following inequality: ∥∥∥∥ũt ũ∗t − p(t) ∥∥∥∥ = ∥∥∥∥ũt ũ∗t − ut ∥∥∥∥ < 1 for all t ∈ [0, 1], whence p ∼ ũ diag(1, 0) ũ∗ ∼ diag(1, 0) as desired. (iii) ⇒ (ii). Suppose that B is properly infinite. From Lemma 4.6 we know that [p] = [diag(1A, 0)] in K0(C(T, A)). Because B and A are properly infinite, it follows that p and diag(1A, 0) are properly infinite (and full) projections, and hence they are equivalent by Proposition 2.3 (i). (ii) ⇒ (iii). Since A is properly infinite, diag(1A, 0) and hence p (being equivalent to diag(1A, 0)) are properly infinite (and full) projections, whence B is properly infinite. � We will now use (the ideas behind) Lemma 4.6 and Proposition 4.7 to prove the following general statement about C∗-algebras. Corollary 4.8 Let A be a unital C∗-algebra such that C(T, A) has the cancellation prop- erty. Then A is K1-injective. Proof: It suffices to show that the natural maps Un−1(A)/U n−1(A) → Un(A)/U n(A) are injective for all n ≥ 2. Let v ∈ Un−1(A) be such that diag(v, 1A) ∈ U n(A) and find a continuous path of unitaries t 7→ ut in Un(A) such that u0 = 1Mn(A) = 1Mn−1(A) 0 and u1 = pt = ut 1Mn−1(A) 0 u∗t , t ∈ [0, 1], and note that p0 = p1 so that p defines a projection in C(T,Mn(A)). Repeating the proof of Lemma 4.6 we find that 1Mn(A) − p ∼ diag(0, 1A) in C(T,Mn(A)), whence p ∼ diag(1Mn−1(A), 0) by the cancellation property of C(T, A), where we identify projections in Mn(A) with constant projections in C(T,Mn(A)). The arguments going into the proof of Proposition 4.7 show that v ∼h 1Mn−1(A) in Un−1(A) if (and only if) p ∼ diag(1Mn−1(A), 0). Hence v belongs to U0n−1(A) as desired. � 5 K1-injectivity of properly infinite C ∗-algebras In this section we prove our main result that relate K1-injectivity of arbitrary unital prop- erly infinite C∗-algebras to proper infiniteness of C(X)-algebras and pull-back C∗-algebras. More specifically we shall show that Question 2.9, Question 2.13, Question 2.8, and Ques- tion 4.2 are equivalent. First we reformulate in two different ways the question if a given properly infinite unital C∗-algebra is K1-injective. Proposition 5.1 The following conditions are equivalent for any unital properly infinite C∗-algebra A: (i) A is K1-injective. (ii) Let p, q be projections in A such that p ∼ q and p, q, 1−p, 1−q are properly infinite and full. Then p ∼h q. (iii) Let p and q be properly infinite, full projections in A. There exist properly infinite, full projections p0, q0 ∈ A such that p0 ≤ p, q0 ≤ q, and p0 ∼h q0. Proof: (i) ⇒ (ii). Let p, q be properly infinite, full projections in A with p ∼ q such that 1− p, 1− q are properly infinite and full. Then by Lemma 2.4 (i) there is a unitary v ∈ A such that vpv∗ = q and [v] = 0 in K1(A). By the assumption in (i), v ∈ U 0(A), whence p ∼h q. (ii) ⇒ (i). Let u ∈ U(A) be such that [u] = 0 in K1(A). Take, as we can, a projection p in A such that p and 1 − p are properly infinite and full. Set q = upu∗. Then p ∼h q by (ii), and so there exists a unitary v ∈ U0(A) with p = vqv∗. It follows that pvu = vqv∗vu = v(upu∗)v∗vu = vup. Therefore vu ∈ U0(A) by Lemma 2.4 (ii), which in turn implies that u ∈ U0(A). (ii) ⇒ (iii). Let p, q be properly infinite and full projections in A. There exist mutually orthogonal projections e1, f1 such that e1 ≤ p, f1 ≤ p and e1 ∼ p ∼ f1, and mutually orthogonal projections e2, f2 such that e2 ≤ q, f2 ≤ q and e2 ∼ q ∼ f2. Being equivalent to either p or q, the projections e1, e2, f1 and f2 are properly infinite and full. There are properly infinite, full projections p0 ≤ e1 and q0 ≤ e2 such that [p0] = [q0] = 0 in K0(A) and p0 ∼ q0 (cf. Proposition 2.3). As f1 ≤ 1 − p0 and f2 ≤ 1− q0, we see that 1 − p0 and 1− q0 are properly infinite and full, and so we get p0 ∼h q0 by (ii). (iii) ⇒ (ii). Let p, q be equivalent properly infinite, full projections in A such that 1 − p, 1 − q are properly infinite and full. From (iii) we get properly infinite and full projections p0 ≤ p, q0 ≤ q which satisfy p0 ∼h q0. Thus there is a unitary v ∈ U0(A) such that vp0v ∗ = q0. Upon replacing p by vpv ∗ (as we may do because p ∼h vpv ∗) we can assume that q0 ≤ p and q0 ≤ q. Now, q0 is orthogonal to 1 − p and to 1 − q, and so 1− p ∼h 1− q by Proposition 2.5, whence p ∼h q. � Proposition 5.2 Let A be a unital properly infinite C∗-algebra. The following conditions are equivalent: (i) A is K1-injective, ie., the natural map U(A)/U 0(A) → K1(A) is injective. (ii) The natural map U(A)/U0(A) → U2(A)/U 2 (A) is injective. (iii) The natural maps Un(A)/U n(A) → K1(A) are injective for each natural number n. Proof: (i) ⇒ (ii) holds because the map U(A)/U0(A) → K1(A) factors through the map U(A)/U0(A) → U2(A)/U 2 (A). (ii)⇒ (i). Take u ∈ U(A) and suppose that [u] = 0 inK1(A). Then diag(u, 1A) ∈ U 2 (A) by Lemma 2.4 (ii) (with p = diag(1A, 0)). Hence u ∈ U0(A) by injectivity of the map U(A)/U0(A) → U2(A)/U 2 (A). (i) ⇒ (iii). Let n ≥ 1 be given and consider the natural maps U(A)/U0(A) → Un(A)/U n(A) → K1(A). The first map is onto, as proved by Cuntz in [4], see also [13, Exercise 8.9], and the composition of the two maps is injective by assumption, hence the second map is injective. (iii) ⇒ (i) is trivial. � We give below another application of K1-injectivity for properly infinite C ∗-algebras. First we need a lemma: Lemma 5.3 Let A be a unital, properly infinite C∗-algebra, and let ϕ, ψ : O∞ → A be unital embeddings. Then ψ is homotopic to a unital embedding ψ′ : O∞ → A for which there is a unitary u ∈ A with [u] = 0 in K1(A) and for which ψ ′(sj) = uϕ(sj) for all j (where s1, s2, . . . are the canonical generators of O∞). Proof: For each n set ψ(sj)ϕ(sj) ∗ ∈ A, en = j ∈ O∞. Then vn is a partial isometry in A with vnv n = ψ(en), v nvn = ϕ(en), and ψ(sj) = vnϕ(sj) for j = 1, 2, . . . , n. Since 1− en is full and properly infinite it follows from Lemma 2.4 that each vn extends to a unitary un ∈ A with [un] = 0 in K1(A). In particular, ψ(sj) = unϕ(sj) for j = 1, 2, . . . , n. We proceed to show that n 7→ un extends to a continuous path of unitaries t 7→ ut, for t ∈ [2,∞), such that utϕ(en) = unϕ(en) for t ≥ n + 1. Fix n ≥ 2. To this end it suffices to show that we can find a continuous path t 7→ zt, t ∈ [0, 1], of unitaries in A such that z0 = 1, z1 = u nun+1, and ztϕ(en−1) = ϕ(en−1) (as we then can set ut to be unzt−n for t ∈ [n, n+ 1]). Observe that un+1ϕ(en) = vn+1ϕ(en) = vn = unϕ(en). Set A0 = (1 − ϕ(en−1))A(1 − ϕ(en−1)), and set y = u nun+1(1 − ϕ(en−1)). Then y is a unitary element in A0 and [y] = 0 in K1(A0). Moreover, y commutes with the properly infinite full projection ϕ(en) − ϕ(en−1) ∈ A0. We can therefore use Lemma 2.4 to find a continuous path t 7→ yt of unitaries in A0 such that y0 = 1A0 = 1 − ϕ(en−1) and y1 = y. The continuous path t 7→ zt = yt + ϕ(en−1) is then as desired. For each t ≥ 2 let ψt : O∞ → A be the ∗-homomorphism given by ψt(sj) = utϕ(sj). Then ψt(sj) = ψ(sj) for all t ≥ j + 1, and so it follows that ψt(x) = ψ(x) for all x ∈ O∞. Hence ψ2 is homotopic to ψ, and so we can take ψ ′ to be ψ2. � Proposition 5.4 Any two unital ∗-homomorphisms from O∞ into a unital K1-injective (properly infinite) C∗-algebra are homotopic. Proof: In the light of Lemma 5.3 it suffices to show that if ϕ, ψ : O∞ → A are unital homomorphisms such that, for some unitary u ∈ A with [u] = 0 in K1(A), ψ(sj) = uϕ(sj) for all j, then ψ ∼h ϕ. By assumption, u ∼h 1, so there is a continuous path t 7→ ut of unitaries in A such that u0 = 1 and u1 = u. Letting ϕt : O∞ → A be the ∗-homomorphism given by ϕt(sj) = utϕ(sj) for all j, we get t 7→ ϕt is a continuous path of ∗-homomorphisms connecting ϕ0 = ϕ to ϕ1 = ψ. � Our main theorem below, which in particular implies that Question 2.9, Question 2.13, Question 2.8 and Question 4.2 all are equivalent, also give a special converse to Proposi- tion 5.4: Indeed, with ι1, ι2 : O∞ → O∞ ∗O∞ the two canonical inclusions, if ι1 ∼h ι2, then condition (iv) below holds, whence O∞ ∗ O∞ is K1-injective, which again implies that all unital properly infinite C∗-algebras are K1-injective. Below we retain the convention that O∞ ∗ O∞ is the universal unital free product of two copies of O∞ and that ι1 and ι2 are the two natural inclusions of O∞ into O∞ ∗ O∞. Theorem 5.5 The following statements are equivalent: (i) Every unital, properly infinite C∗-algebra is K1-injective. (ii) For every compact Hausdorff space X, every unital C(X)-algebra A, for which Ax is properly infinite for all x ∈ X, is properly infinite. (iii) Every unital C∗-algebra A, that is the pull-back of two unital, properly infinite C∗- algebras A1 and A2 along ∗-epimorphisms π1 : A1 → B, π2 : A2 → B: }} ϕ2 π1 A π2~~}} is properly infinite. (iv) There exist non-zero projections p, q ∈ O∞ such that p 6= 1, q 6= 1, and ι1(p) ∼h ι2(p) in O∞ ∗ O∞. (v) The specific C([0, 1])-algebra A considered in Example 4.1 (and whose fibres are prop- erly infinite) is properly infinite. (vi) O∞ ∗ O∞ is K1-injective. Note that statement (i) is reformulated in Propositions 5.1, 5.2, and 5.4; and that statement (iv) is reformulated in Proposition 4.3. We warn the reader that all these statements may turn out to be false (in which case, of course, there will be counterexamples to all of them). Proof: (i) ⇒ (iii) follows from Proposition 2.7. (iii) ⇒ (ii). This follows from Lemma 2.10 as in the proof of Theorem 2.11, except that one does not need to pass to matrix algebras. (ii) ⇒ (i). Suppose that A is unital and properly infinite. Take a unitary v ∈ U(A) such that diag(v, 1) ∈ U02 (A). Let B be the C(T)-algebra constructed in Example 4.5 from A, v, and a path of unitaries t 7→ ut connecting 1M2(A) to diag(v, 1). Then Bt ∼= A for all t ∈ T, so all fibres of B are properly infinite. Assuming (ii), we can conclude that B is properly infinite. Proposition 4.7 then yields that v ∈ U0(A). It follows that the natural map U(A)/U0(A) → U2(A)/U 2 (A) is injective, whence A is K1-injective by Proposition 5.2. (ii) ⇒ (v) is trivial (because A is a C([0, 1])-algebra with properly infinite fibres). (v) ⇒ (iv) follows from Proposition 4.3. (iv) ⇒ (i). We show that Condition (iii) of Proposition 4.3 implies Condition (iii) of Proposition 5.1. Let A be a properly infinite C∗-algebra and let p, q be properly infinite, full projections in A. Then there exist (properly infinite, full) projections p0 ≤ p and q0 ≤ q such that p0 ∼ 1 ∼ q0 and such that 1−p0 and 1−q0 are properly infinite and full, cf. Propositions 2.3. Take isometries t1, r1 ∈ A with t1t 1 = p0 and r1r 1 = q0; use the fact that 1 - 1 − p0 and 1 - 1− q0 to find sequences of isometries t2, t3, t4, . . . and r2, r3, r4, . . . in A such that each of the two sequences {tjt j=1 and {rjr j=1 consist of pairwise orthogonal projections. By the universal property of O∞ there are unital ∗-homomorphisms ϕj : O∞ → A, j = 1, 2, such that ϕ1(sj) = tj and ϕ2(sj) = rj, where s1, s2, s3, . . . are the canonical generators of O∞. In particular, ϕ1(s1s 1) = p0 and ϕ2(s1s 1) = q0. By the property of the universal unital free products of C∗-algebras, there is a unique unital ∗-homomorphism ϕ : O∞ ∗ O∞ → A making the diagram O∞ ∗ O∞ ϕ1 %%KK 99ssssssssss ϕ2yysss eeKKKKKKKKKK commutative. It follows that p0 = ϕ(ι1(s1s 1)) and q0 = ϕ(ι2(s1s 1)). By Condition (iii) of Proposition 4.3, ι1(s1s 1) ∼h ι2(s1s 1) in O∞ ∗ O∞, whence p0 ∼h q0 as desired. (i) ⇒ (vi) is trivial. (vi) ⇒ (v) follows from Proposition 4.4. � 6 Concluding remarks We do not know if all unital properly infinite C∗-algebras are K1-injective, but we observe that K1-injectivity is assured in the presence of certain central sequences: Proposition 6.1 Let A be a unital properly infinite C∗-algebras that contains an asymp- totically central sequence {pn} n=1, where pn and 1−pn are properly infinite, full projections for all n. Then A is K1-injective Proof: This follows immediately from Lemma 2.4 (ii). � It remains open if arbitrary C(X)-algebras with properly infinite fibres must be properly infinite. If this fails, then we already have a counterexample of the form B = pC(T, A)p, cf. Example 4.5, for some unital properly infinite C∗-algebra A and for some projection p ∈ C(T, A). (The C∗-algebra B is a C(T)-algebra with fibres Bt ∼= A.) On the other hand, any trivial C(X)-algebra C(X,D) with constant fibre D is clearly properly infinite if its fibre(s) D is unital and properly infinite (because C(X,D) ∼= C(X)⊗ D). We extend this observation in the following easy: Proposition 6.2 Let X be a compact Hausdorff space, let p ∈ C(X,D) be a projection, and consider the sub-trivial C(X)-algebra pC(X,D)p whose fibre at x is equal to p(x)Dp(x). If p is Murray-von Neumann equivalent to a constant projection q, then pC(X,D)p is C(X)-isomorphic to the trivial C(X)-algebra C(X,D0), where D0 = qDq. In this case, pC(X,D)p is properly infinite if and only if D0 is properly infinite. In particular, if X is contractible, then pC(X,D)p is C(X)-isomorphic to a trivial C(X)-algebra for any projection p ∈ C(X,D) and for any C∗-algebra D. Proof: Suppose that p = v∗v and q = vv∗ for some partial isometry v ∈ C(X,D). The map f 7→ vfv∗ defines a C(X)-isomorphism from pC(X,D)p onto qC(X,D)q, and qC(X,D)q = C(X,D0). If X is contractible, then any projection p ∈ C(X,D) is homotopic, and hence equiva- lent, to the constant projection x 7→ p(x0) for any fixed x0 ∈ X . � Remark 6.3 One can elaborate a little more on the construction considered above. Take a unital C∗-algebra D such that for some natural number n ≥ 2, Mn(D) is properly infinite, butMn−1(D) is not properly infinite (see [12] or [11] for such examples). Take any space X , preferably one with highly non-trivial topology, eg. X = Sn, and take, for some k ≥ n, a sufficiently non-trivial n-dimensional projection p in C(X,Mk(D)) such that p(x) is equivalent to the trivial n dimensional projection 1Mn(D) for all x (if X is connected we need only assume that this holds for one x ∈ X). The C(X)-algebra A = pC(X,Mk(D)) p, then has properly infinite fibres Ax = p(x)Dp(x) ∼= Mn(D). Is A always properly infinite? We guess that a possible counterexample to the questions posed in this paper could be of this form (for suitable D, X , and p). Let us end this paper by remarking that the answer to Question 2.13, which asks if any C(X)-algebra with properly infinite fibres is itself properly infinite, does not depend (very much) on X . If it fails, then it fails already for X = [0, 1] (cf. Theorem 5.5), and [0, 1] is a contractible space of low dimension. However, if we make the dimension of X even lower than the dimension of [0, 1], then we do get a positive anwer to our question: Proposition 6.4 Let X be a totally disconnected space, and let A be a C(X)-algebra such that all fibres Ax, x ∈ X, of A are properly infinite. Then A is properly infinite. Proof: Using Lemma 2.10 and the fact that X is totally disconnected we can write X as the disjoint union of clopen sets F1, F2, . . . , Fn such that AFj is properly infinite for all j. A = AF1 ⊕ AF2 ⊕ · · · ⊕AFn , the claim is proved. � References [1] B. Blackadar, D. Handelman, Dimension functions and traces on C∗-algebras, J. Funct. Anal. 45 (1982), 297–340. [2] E. Blanchard, A few remarks on C(X)-algebras, Rev. Roumaine Math. Pures Appl. 45 (2001), 565–576. [3] E. Blanchard, E. Kirchberg, Global Glimm halving for C∗-bundles, J. Op. Th. 52 (2004), 385–420. [4] J. Cuntz, K-theory for certain C∗-algebras, Ann. of Math. 113 (1981), 181–197. [5] M. Dadarlat, Continuous fields of C∗-algebras over finite dimensional spaces, preprint. [6] J. Dixmier, A. Douady, Champs continus d’espaces hilbertiens et de C∗-algèbres, Bull. Soc. Math. France 91 (1963), 227–284. [7] D. Handelman, Homomorphism of C∗-algebras to finite AW ∗ algebras, Michigan Math J. 28 (1981), 229-240. [8] I. Hirshberg, M. Rørdam, W. Winter, C0(X)-algebras, stability and strongly self- absorbing C*-algebras, Preprint July 2006. [9] M. A. Rieffel, The homotopy groups of the unitary groups of non-commutative tori, Journal of Operator Theory, 17-18 (1987), 237-254. [10] M. Rørdam, Classification of inductive limits of Cuntz algebras, J. Reine Angew. Math., 440 (1993), 175–200. [11] M. Rørdam, A simple C∗-algebra with a finite and an infinite projection, Acta Math. 191 (2003), 109–142. [12] M. Rørdam, On sums of finite projections, “Operator algebras and operator theory” (1998), Amer. Math. Soc., Providence, RI, 327–340. [13] M. Rørdam, F. Larsen, N. J. Laustsen, An Introduction to K-theory for C∗-algebras, London Mathematical Society Student Texts 49 (2000) CUP, Cambridge. [14] A. Toms and W. Winter, Strongly self-absorbing C∗-algebras, Transactions AMS (to appear). Projet Algbres d’oprateurs, Institut de Mathmatiques de Jussieu, 175, rue du Chevaleret, F-75013 PARIS, France E-mail address: Etienne.Blanchard@math.jussieu.fr Internet home page: www.math.jussieu.fr/∼blanchar Department of Mathematics, University of Southern Denmark, Odense, Campusvej 55, 5230 Odense M, Denmark E-mail address: rohde@imada.sdu.dk Department of Mathematics, University of Southern Denmark, Odense, Campusvej 55, 5230 Odense M, Denmark E-mail address: mikael@imada.sdu.dk Internet home page: www.imada.sdu.dk/∼mikael/welcome Introduction C(X)-algebras with properly infinite fibres Lower semi-continuous fields of properly infinite C-algebras Two examples K1-injectivity of properly infinite C*-algebras Concluding remarks

We investigate if a unital C(X)-algebra is properly infinite when all its fibres are properly infinite. We show that this question can be rephrased in several different ways, including the question if every unital properly infinite C*-algebra is K_1-injective. We provide partial answers to these questions, and we show that the general question on proper infiniteness of C(X)-algebras can be reduced to establishing proper infiniteness of a specific C([0,1])-algebra with properly infinite fibres.

Introduction The problem that we mainly are concerned with in this paper is if any unital C(X)-algebra with properly infinite fibres is itself properly infinite (see Section 2 for a brief introduction to C(X)-algebras). An analogous study was carried out in the recent paper [8] where it was decided when C(X)-algebras, whose fibres are either stable or absorb tensorially a given strongly self-absorbing C∗-algebra, itself has the same property. This was answered in the affirmative in [8] under the crucial assumption that the dimension of the space X is finite, and counterexamples were given in the infinite dimensional case. Along similar lines, Dadarlat, [5], recently proved that C(X)-algebras, whose fibres are Cuntz algebras, are trivial under some K-theoretical conditions provided that the space X is finite dimensional. The property of being properly infinite turns out to behave very differently than the property of being stable or of absorbing a strongly self-absorbing C∗-algebra. It is relative easy to see (Lemma 2.10) that if a fibre Ax of a C(X)-algebra A is properly infinite, then AF is properly infinite for some closed neighborhood F of x. The (possible) obstruction to proper infiniteness of the C(X)-algebra is hence not local. Such an obstruction is also not related to the possible complicated structure of the space X , as we can show that a counterexample, if it exists, can be taken to be a (specific) C([0, 1])-algebra (Example 4.1 and Theorem 5.5). The problem appears to be related with some rather subtle internal structure properties of properly infinite C∗-algebras. Cuntz studied purely infinite—and in the process also properly infinite—C∗-algebras, [4], where he among many other things (he was primarily interested in calculating the http://arxiv.org/abs/0704.1554v1 K-theory of his algebras On) showed that any unital properly infinite C ∗-algebra A is K1- surjective, i.e., the mapping U(A) → K1(A) is onto; and that any purely infinite simple C∗-algebra A is K1-injective, i.e., the mapping U(A)/U 0(A) → K1(A) is injective (and hence an isomorphism). He did not address the question if any properly infinite C∗-al- gebra is K1-injective. That question has not been raised formally to our knowledge—we do so here—but it does appear implicitly, eg. in [10] and in [14], where K1-injectivity of properly infinite C∗-algebras has to be assumed. Proper infiniteness of C∗-algebras has relevance for existence (or rather non-existence) of traces and quasitraces. Indeed, a unital C∗-algebra admits a 2-quasitrace if and only if no matrix algebra over the C∗-algebra is properly infinite, and a unital exact C∗-algebra admits tracial state again if and only if no matrix algebra over the C∗-algebra is properly infinite. In this paper we show that every properly infinite C∗-algebra is K1-injective if and only if every C(X)-algebra with properly infinite fibres itself is properly infinite. We also show that a matrix algebra over any such C(X)-algebra is properly infinite. Examples of unital C∗-algebras A, where Mn(A) is properly infinite for some natural number n ≥ 2 but where Mn−1(A) is not properly infinite, are known, see [12] and [11], but still quite exotic. We relate the question if a given properly infinite C∗-algebra isK1-injective to questions regarding homotopy of projections (Proposition 5.1). In particular we show that our main questions are equivalent to the following question: is any non-trivial projection in the first copy of O∞ in the full unital universal free product O∞∗O∞ homotopic to any (non-trivial) projection in the second copy of O∞? The specific C([0, 1])-algebra, mentioned above, is perhaps not surprisingly a sub-algebra of C([0, 1],O∞ ∗ O∞). Using ideas implicit in Rieffel’s paper, [9], we construct in Section 4 a C(T)-algebra B for each C∗-algebra A and for each unitary u ∈ A for which diag(u, 1) is homotopic to 1M2(A); and B is non-trivial if u is not homotopic to 1A. In this way we relate our question about proper infiniteness of C(X)-algebras to a question about K1-injectivity. The last mentioned author thanks Bruce Blackadar for many inspiring conversations on topics related to this paper. 2 C(X)-algebras with properly infinite fibres A powerful tool in the classification of C∗-algebras is the study of their projections. A projection in a C∗-algebra is said to be infinite if it is equivalent to a proper subprojection of itself, and it is said to be properly infinite if it is equivalent to two mutually orthogonal subprojections of itself. A projection which is not infinite is said to be finite. A unital C∗-algebra is said to be finite, infinite, or properly infinite if its unit is finite, infinite, or properly infinite, respectively. If A is a C∗-algebra for which Mn(A) is finite for all positive integers n, then A is stably finite. In this section we will study stability properties of proper infiniteness under (upper- semi-)continuous deformations using the Cuntz-Toeplitz algebra which is defined as follows. For all integers n ≥ 2 the Cuntz-Toeplitz algebra Tn is the universal C ∗-algebra generated by n isometries s1, . . . , sn satisfying the relation 1 + · · ·+ sns n ≤ 1. Remark 2.1 A unital C∗-algebra A is properly infinite if and only if Tn embeds unitally into A for some n ≥ 2, in which case Tn embeds unitally into A for all n ≥ 2. In order to study deformations of such algebras, let us recall a few notions from the theory of C(X)-algebras. Let X be a compact Hausdorff space and C(X) be the C∗-algebra of continuous func- tions on X with values in the complex field C. Definition 2.2 A C(X)-algebra is a C∗-algebra A endowed with a unital ∗-homomorphism from C(X) to the center of the multiplier C∗-algebra M(A) of A. If A is as above and Y ⊆ X is a closed subset, then we put IY = C0(X \ Y )A, which is a closed two-sided ideal in A. We set AY = A/IY and denote the quotient map by πY . For an element a ∈ A we put aY = πY (a), and if Y consists of a single point x, we will write Ax, Ix, πx and ax in the place of A{x}, I{x}, π{x} and a{x}, respectively. We say that Ax is the fibre of A at x. The function x 7→ ‖ax‖ = inf{‖ [1− f + f(x)]a‖ : f ∈ C(X)} is upper semi-continuous for all a ∈ A (as one can see using the right-hand side identity above). A C(X)-algebra A is said to be continuous (or to be a continuous C∗-bundle over X) if the function x 7→ ‖ax‖ is actually continuous for all element a in A. For any unital C∗-algebra A we let U(A) denote the group of unitary elements in A, U0(A) denotes its connected component containing the unit of A, and Un(A) and U are equal to U(Mn(A)) and U 0(Mn(A)), respectively. An element in a C∗-algebra A is said to be full if it is not contained in any proper closed two-sided ideal in A. It is well-known (see for example [13, Exercise 4.9]) that if p is a properly infinite, full projection in a C∗-algebra A, then e - p, i.e., e is equivalent to a subprojection of p, for every projection e ∈ A. We state below more formally three more or less well-known results that will be used frequently throughout this paper, the first of which is due to Cuntz, [4]. Proposition 2.3 (Cuntz) Let A be a C∗-algebra which contains at least one properly infinite, full projection. (i) Let p and q be properly infinite, full projections in A. Then [p] = [q] in K0(A) if and only if p ∼ q. (ii) For each element g ∈ K0(A) there is a properly infinite, full projection p ∈ A such that g = [p]. The second statement is a variation of the Whitehead lemma. Lemma 2.4 Let A be a unital C∗-algebra. (i) Let v be a partial isometry in A such that 1 − vv∗ and 1 − v∗v are properly infinite and full projections. Then there is a unitary element u in A such that [u] = 0 in K1(A) and v = uv ∗v, i.e., u extends v. (ii) Let u be a unitary element A such that [u] = 0 in K1(A). Suppose there exists a projection p ∈ A such that ‖up− pu‖ < 1 and p and 1 − p are properly infinite and full. Then u belongs to U0(A). Proof: (i). It follows from Proposition 2.3 (i) that 1 − v∗v ∼ 1 − vv∗, so there is a partial isometry w such that 1 − v∗v = w∗w and 1 − vv∗ = ww∗. Now, z = v + w is a unitary element in A with zv∗v = v. The projection 1 − v∗v is properly infinite and full, so 1 - 1 − v∗v, which implies that there is an isometry s in A with ss∗ ≤ 1 − v∗v. As −[z] = [z∗] = [sz∗s∗ + (1 − ss∗)] in K1(A) (see eg. [13, Exercise 8.9 (i)]), we see that u = z(sz∗s∗ + (1− ss∗)) is as desired. (ii). Put x = pup + (1 − p)u(1 − p) and note that ‖u − x‖ < 1. It follows that x is invertible in A and that u ∼h x in GL(A). Let x = v|x| be the polar decomposition of x, where |x| = (x∗x)1/2 and v = x|x|−1 is unitary. Then u ∼h v in U(A) (see eg. [13, Proposition 2.1.8]), and pv = vp. We proceed to show that v belongs to U0(A) (which will entail that u belongs to U0(A)). Write v = v1v2, where v1 = pvp+ (1− p), v2 = p+ (1− p)v(1− p). As 1−p - p we can find a symmetry t in A such that t(1−p)t ≤ p. As t belongs to U0(A) (being a symmetry), we conclude that v2 ∼h tv2t, and one checks that tv2t is of the form w + (1− p) for some unitary w in pAp. It follows that v is homotopic to a unitary of the form v0 + (1− p), where v0 is a unitary in pAp. We can now apply eg. [13, Exercise 8.11] to conclude that v ∼h 1 in U(A). � We remind the reader that if p, q are projections in a unital C∗-algebra A, then p and q are homotopic, in symbols p ∼h q, (meaning that they can be connected by a continuous path of projections in A) if and only if q = upu∗ for some u ∈ U0(A), eg. cf. [13, Proposition 2.2.6]. Proposition 2.5 Let A be a unital C∗-algebra. Let p and q be two properly infinite, full projections in A such that p ∼ q. Suppose that there exists a properly infinite, full projection r ∈ A such that p ⊥ r and q ⊥ r. Then p ∼h q. Proof: Take a partial isometry v0 ∈ A such that v 0v0 = p and v0v 0 = q. Take a subpro- jection r0 of r such that r0 and r− r0 both are properly infinite and full. Put v = v0 + r0. Then vpv∗ = q and vr0 = r0 = r0v. Note that 1 − v ∗v and 1 − vv∗ are properly infi- nite and full (because they dominate the properly infinite, full projection r − r0). Use Lemma 2.4 (i) to extend v to a unitary u ∈ A with [u] = 0 in K1(A). Now, upu ∗ = q and ur0 = vr0 = r0 = r0v = r0u. Hence u ∈ U 0(A) by Lemma 2.4 (ii), and so p ∼h q as desired. Definition 2.6 A unital C∗-algebra A is said to be K1-injective if the natural mapping U(A)/U0(A) → K1(A) is injective. In other words, if A is K1-injective, and if u is a unitary element in A, then u ∼h 1 in U(A) if (and only if) [u] = 0 in K1(A). One could argue thatK1-injectivity should entail that the natural mappings Un(A)/U n(A) → K1(A) be injective for every natural number n. However there seem to be an agreement for defining K1-injectivity as above. As we shall see later, in Proposition 5.2, if A is properly infinite, then the two definitions agree. Proposition 2.7 Let A be a unital C∗-algebra that is the pull-back of two unital, properly infinite C∗-algebras A1 and A2 along the ∗-epimorphisms π1 : A1 → B and π2 : A2 → B: }} ϕ2 π1 A π2~~}} Then M2(A) is properly infinite. Moreover, if B is K1-injective, then A itself is properly infinite. Proof: Take unital embeddings σi : T3 → Ai for i = 1, 2, where T3 is the Cuntz-Toeplitz algebra (defined earlier), and put (π1 ◦ σ1)(tj)(π2 ◦ σ2)(t where t1, t2, t3 are the canonical generators of T3. Note that v is a partial isometry with (π1 ◦ σ1)(tj) = v(π2 ◦ σ2)(tj) for j = 1, 2. As (π1 ◦ σ1)(t3t 3) ≤ 1− vv ∗ and (π2 ◦ σ2)(t3t 1−v∗v, Lemma 2.4 (i) yields a unitary u ∈ B with [u] = 0 in K1(B) and with (π1◦σ1)(tj) = u(π2 ◦ σ2)(tj) for j = 1, 2. If B is K1-injective, then u belongs to U 0(B), whence u lifts to a unitary v ∈ A2. Define σ̃2 : T2 → A2 by σ̃2(tj) = vσ2(tj) for j = 1, 2 (observing that t1, t2 generate T2). Then π1 ◦ σ1 = π2 ◦ σ̃2, which by the universal property of the pull-back implies that σ1 and σ̃2 lift to a (necessarily unital) embedding σ : T2 → A, thus forcing A to be properly infinite. In the general case (where B is not necessarily K1-injective) u may not lift to a unitary element in A2, but diag(u, u) does lift to a unitary element v in M2(A2) by Lemma 2.4 (ii) (applied with p = diag(1, 0)). Define unital embeddings σ̃i : T2 → M2(Ai), i = 1, 2, by σ̃1(tj) = σ1(tj) 0 0 σ1(tj) , σ̃2(tj) = v σ2(tj) 0 0 σ2(tj) for j = 1, 2. As (π1 ⊗ idM2) ◦ σ̃1 = (π2 ⊗ idM2) ◦ σ̃2, the unital embeddings σ̃1 and σ̃2 lift to a (necessarily unital) embedding of T2 into M2(A), thus completing the proof. � Question 2.8 Is the pull-back of any two properly infinite unital C∗-algebras again prop- erly infinite? As mentioned in the introduction, one cannot in general conclude that A is properly infinite if one knows that Mn(A) is properly infinite for some n ≥ 2. One obvious way of obtaining an answer to Question 2.8, in the light of the last state- ment in Proposition 2.7, is to answer the question below in the affirmative: Question 2.9 Is every properly infinite unital C∗-algebra K1-injective? We shall see later, in Section 5, that the two questions above in fact are equivalent. The lemma below, which shall be used several times in this paper, shows that one can lift proper infiniteness from a fibre of a C(X)-algebra to a whole neighborhood of that fibre. Lemma 2.10 Let X be a compact Hausdorff space, let A be a unital C(X)-algebra, let x ∈ X, and suppose that the fibre Ax is properly infinite. Then AF is properly infinite for some closed neighborhood F of x. Proof: Let {Fλ}λ∈Λ be a decreasing net of closed neighborhoods of x ∈ X , fulfilling that⋂ λ∈Λ Fλ = {x}, and set Iλ = C0(X \Fλ)A. Then {Iλ}λ∈Λ is an increasing net of ideals in A, AFλ = A/Iλ, I := λ∈Λ Iλ = C0(X\{x}), and Ax = A/I. By the assumption that Ax is properly infinite there is a unital ∗-homomorphism ψ : T2 → Ax, and since T2 is semi-projective there is a λ0 ∈ Λ and a unital ∗-homomorphism ϕ : T2 → AFλ0 making the diagram // Ax commutative. We can thus take F to be Fλ0 . � Theorem 2.11 Let A be a unital C(X)-algebra where X is a compact Hausdorff space. If all fibres Ax, x ∈ X, are properly infinite, then some matrix algebra over A is properly infinite. Proof: By Lemma 2.10, X can be covered by finitely many closed sets F1, F2, . . . , Fn such that AFj is properly infinite for each j. Put Gj = F1 ∪ F2 ∪ · · · ∪ Fj . For each j = 1, 2, . . . , n− 1 we have a pull-back diagram AGj+1 yyrrr AFj+1 AGj∩Fj+1 We know that M2j−1(AGj ) is properly infinite when j = 1. Proposition 2.7 (applied to the diagram above tensored with M2j−1(C)) tells us that M2j (AGj+1) is properly infinite if M2j−1(AGj) is properly infinite. Hence M2n−1(A) is properly infinite. � Remark 2.12 Uffe Haagerup has suggested another way to prove Theorem 2.11: If no matrix-algebra over A is properly infinite, then there exists a bounded non-zero lower semi-continuous 2-quasi-trace on A, see [7] and [1, page 327], and hence also an extremal 2-quasi-trace. Now, if A is also a C(X)-algebra for some compact Hausdorff space X , this implies that there is a bounded non-zero lower semi-continuous 2-quasitrace on Ax for (at least) one point x ∈ X (see eg. [8, Proposition 3.7]). But then the fibre Ax cannot be properly infinite. Question 2.13 Is any unital C(X)-algebra A properly infinite if all its fibres Ax, x ∈ X , are properly infinite? We shall show in Section 5 that the question above is equivalent to Question 2.8 which again is equivalent to Question 2.9. 3 Lower semi-continuous fields of properly infinite C∗- algebras Let us briefly discuss whether the results from Section 2 can be extended to lower semi- continuous C∗-bundles (A, {σx}) over a compact Hausdorff space X . Recall that any such separable lower semi-continuous C∗-bundle admits a faithful C(X)-linear representation on a Hilbert C(X)-module E such that, for all x ∈ X , the fibre σx(A) is isomorphic to the induced image of A in L(Ex), [2]. Thus, the problem boils down to the following: Given a separable Hilbert C(X)-module E with infinite dimensional fibres Ex, such that the unit p of the C∗-algebra LC(X)(E) of bounded adjointable C(X)-linear operators acting on E has a properly infinite image in L(Ex) for all x ∈ X . Is the projection p itself properly infinite in LC(X)(E)? Dixmier and Douady proved that this is always the case if the space X has finite topological dimension, [6]. But it does not hold anymore in the infinite dimensional case, see [6, §16, Corollaire 1] and [11], not even if X is contractible, [3, Corollary 3.7]. 4 Two examples We describe here two examples of continuous fields; the first is over the interval and the second (which really is a class of examples) is over the circle. Example 4.1 Let (O∞ ∗O∞, (ι1, ι2)) be the universal unital free product of two copies of O∞, and let A be the unital sub-C ∗-algebra of C([0, 1],O∞ ∗ O∞) given by A = {f ∈ C([0, 1],O∞ ∗ O∞) : f(0) ∈ ι1(O∞), f(1) ∈ ι2(O∞)}. Observe that A (in a canonical way) is a C([0, 1])-algebra with fibres ι1(O∞), t = 0, O∞ ∗ O∞, 0 < t < 1, ι2(O∞), t = 1 O∞, t = 0, O∞ ∗ O∞, 0 < t < 1, O∞, t = 1. In particular, all fibres of A are properly infinite. One claim to fame of the example above is that the question below is equivalent to Ques- tion 2.13 above. Hence, to answer Question 2.13 in the affirmative (or in the negative) we need only consider the case where X = [0, 1], and we need only worry about this one particular C([0, 1])-algebra (which of course is bad enough!). Question 4.2 Is the C([0, 1])-algebra A from Example 4.1 above properly infinite? The three equivalent statements in the proposition below will in Section 5 be shown to be equivalent to Question 4.2. Proposition 4.3 The following three statements concerning the C([0, 1])-algebra A and the C∗-algebra (O∞ ∗ O∞, (ι1, ι2)) defined above are equivalent: (i) A contains a non-trivial projection (i.e., a projection other than 0 and 1). (ii) There are non-zero projections p, q ∈ O∞ such that p 6= 1, q 6= 1, and ι1(p) ∼h ι2(q). (iii) Let s be any isometry in O∞. Then ι1(ss ∗) ∼h ι2(ss ∗) in O∞ ∗ O∞. We warn the reader that all three statements above could be false. Proof: (i) ⇒ (ii). Let e be a non-trivial projection in A. Let πt : A → At, t ∈ [0, 1], denote the fibre map. As A ⊆ C([0, 1],O∞ ∗ O∞), the mapping t 7→ πt(e) ∈ O∞ ∗ O∞ is continuous, so in particular, π0(e) ∼h π1(e) in O∞ ∗ O∞. The mappings ι1 and ι2 are injective, so there are projections p, q ∈ O∞ such that π0(e) = ι1(p) and π1(e) = ι2(q). The projections p and q are non-zero because the mapping t 7→ ‖πt(e)‖ is continuous and not constant equal to 0. Similarly, and 1− p and 1− q are non-zero because 1− e is non-zero. (ii) ⇒ (iii). Take non-trivial projections p, q ∈ O∞ such that ι1(p) ∼h ι2(q). Take a unitary v in U0(O∞∗O∞) with ι2(q) = vι1(p)v ∗. Let s ∈ O∞ be an isometry. If s is unitary, then ι1(ss ∗) = 1 = ι2(ss ∗) and there is nothing to prove. Suppose that s is non-unitary. Then ss∗ is homotopic to a subprojection p0 of p and to a subprojection q0 of q (use that p and q are properly infinite and full, then Lemma 2.4 (i), and last the fact that the unitary group of O∞ is connected). Hence ι1(ss ∗) ∼h ι1(p0) ∼h vι1(p0)v ∗ and ι2(ss ∗) ∼h ι2(q0), so we need only show that vι1(p0)v ∗ ∼h ι2(q0). But this follows from Proposition 2.5 with r = 1− ι2(q) = ι2(1− q), as we note that p0 ∼ 1 ∼ q0 in O∞, whence ι2(q0) ∼ ι2(1) = 1 = ι1(1) ∼ ι1(p0) ∼ vι1(p0)v (iii) ⇒ (i). Take a non-unitary isometry s ∈ O∞. Then ι1(ss ∗) ∼h ι2(ss ∗), and so there is a continuous function e : [0, 1] → O∞ ∗O∞ such that e(t) is a projection for all t ∈ [0, 1], e(0) = ι1(ss ∗) and e(1) = ι2(ss ∗). But then e is a non-trivial projection in A. � It follows from Theorem 2.11 that some matrix algebra over A (from Example 4.1) is properly infinite. We can sharpen that statement as follows: Proposition 4.4 M2(A) is properly infinite; and if O∞ ∗O∞ is K1-injective, then A itself is properly infinite. It follows from Theorem 5.5 below that A is properly infinite if and only if O∞ ∗ O∞ is K1-injective. Proof: We have a pull-back diagram A[0, 1 π1/2 %%KK π1/2yysss O∞ ∗ O∞ One can unitally embed O∞ into A[0, 1 ] via ι1, so A[0, 1 ] is properly infinite, and a similar argument shows that A[ 1 ,1] is properly infinite. The two statements now follow from Proposition 2.7. � The example below, which will be the focus of the rest of this section, and in parts also of Section 5, is inspired by arguments from Rieffel’s paper [9]. Example 4.5 Let A be a unital C∗-algebra, and let v be a unitary element in A such that in U2(A). Let t 7→ ut be a continuous path of unitaries in U2(A) such that u0 = 1 and u1 = diag(v, 1). p(t) = ut u∗t ∈M2(A), and note that p(0) = p(1). Identifying, for each C∗-algebra D, C(T, D) with the algebra of all continuous functions f : [0, 1] → D such that f(1) = f(0), we see that p belongs to C(T,M2(A)). Put B = pC(T,M2(A))p, and note that B is a unital (sub-trivial) C(T)-algebra, being a corner of the trivial C(T)- algebra C(T,M2(A)). The fibres of B are Bt = p(t)M2(A)p(t) ∼= A for all t ∈ T. Summing up, for each unital C∗-algebraA, for each unitary v inA for which diag(v, 1) ∼h 1 in U2(A), and for each path t 7→ ut ∈ U2(A) implementing this homotopy we get a C(T)- algebra B with fibres Bt ∼= A. We shall investigate this class of C(T)-algebras below. Lemma 4.6 In the notation of Example 4.5, − p ∼ in C(T,M2(A)). In particular, p is stably equivalent to diag(1, 0). Proof: Put vt = ut , t ∈ [0, 1]. v0 = u0 , v1 = u1 so v belongs to C(T,M2(A)). It is easy to see that v t vt = diag(0, 1) and vtv t = 1 − p(t), and so the lemma is proved. � Proposition 4.7 Let A, v ∈ U(A), and B be as in Example 4.5. Conditions (i) and (ii) below are equivalent for any unital C∗-algebra A, and all three conditions are equivalent if A in addition is assumed to be properly infinite. (i) v ∼h 1 in U(A). (ii) p ∼ diag(1A, 0) in C(T,M2(A)). (iii) The C(T)-algebra B is properly infinite. Proof: (ii) ⇒ (i). Suppose that p ∼ diag(1, 0) in C(T,M2(A)). Then there is a w ∈ C(T,M2(A)) such that and w∗twt = pt for all t ∈ [0, 1] and w1 = w0 (as we identify C(T,M2(A)) with the set of continuous functions f : [0, 1] → M2(A) with f(1) = f(0)). Upon replacing wt with w 0wt we can assume that w1 = w0 = diag(1, 0). Now, with t 7→ ut as in Example 4.5, where t 7→ at is a continuous path of unitaries in A. Because u0 = diag(1, 1) and u1 = diag(v, 1) we see that a0 = 1 and a1 = v, whence v ∼h 1 in U(A). (i) ⇒ (ii). Suppose conversely that v ∼h 1 in U(A). Then we can find a continuous path t 7→ vt ∈ U(A), t ∈ [1 − ε, 1], such that v1−ε = v and v1 = 1 for an ε > 0 (to be determined below). Again with t 7→ ut as in Example 4.5, define ũt = u(1−ε)−1t, 0 ≤ t ≤ 1− ε, diag(vt, 1), 1− ε ≤ t ≤ 1. Then t 7→ ũt is a continuous path of unitaries in U2(A) such that ũ1−ε = u1 = diag(v, 1) and ũ0 = ũ1 = 1. It follows that ũ belongs to C(T,M2(A)). Provided that ε > 0 is chosen small enough we obtain the following inequality: ∥∥∥∥ũt ũ∗t − p(t) ∥∥∥∥ = ∥∥∥∥ũt ũ∗t − ut ∥∥∥∥ < 1 for all t ∈ [0, 1], whence p ∼ ũ diag(1, 0) ũ∗ ∼ diag(1, 0) as desired. (iii) ⇒ (ii). Suppose that B is properly infinite. From Lemma 4.6 we know that [p] = [diag(1A, 0)] in K0(C(T, A)). Because B and A are properly infinite, it follows that p and diag(1A, 0) are properly infinite (and full) projections, and hence they are equivalent by Proposition 2.3 (i). (ii) ⇒ (iii). Since A is properly infinite, diag(1A, 0) and hence p (being equivalent to diag(1A, 0)) are properly infinite (and full) projections, whence B is properly infinite. � We will now use (the ideas behind) Lemma 4.6 and Proposition 4.7 to prove the following general statement about C∗-algebras. Corollary 4.8 Let A be a unital C∗-algebra such that C(T, A) has the cancellation prop- erty. Then A is K1-injective. Proof: It suffices to show that the natural maps Un−1(A)/U n−1(A) → Un(A)/U n(A) are injective for all n ≥ 2. Let v ∈ Un−1(A) be such that diag(v, 1A) ∈ U n(A) and find a continuous path of unitaries t 7→ ut in Un(A) such that u0 = 1Mn(A) = 1Mn−1(A) 0 and u1 = pt = ut 1Mn−1(A) 0 u∗t , t ∈ [0, 1], and note that p0 = p1 so that p defines a projection in C(T,Mn(A)). Repeating the proof of Lemma 4.6 we find that 1Mn(A) − p ∼ diag(0, 1A) in C(T,Mn(A)), whence p ∼ diag(1Mn−1(A), 0) by the cancellation property of C(T, A), where we identify projections in Mn(A) with constant projections in C(T,Mn(A)). The arguments going into the proof of Proposition 4.7 show that v ∼h 1Mn−1(A) in Un−1(A) if (and only if) p ∼ diag(1Mn−1(A), 0). Hence v belongs to U0n−1(A) as desired. � 5 K1-injectivity of properly infinite C ∗-algebras In this section we prove our main result that relate K1-injectivity of arbitrary unital prop- erly infinite C∗-algebras to proper infiniteness of C(X)-algebras and pull-back C∗-algebras. More specifically we shall show that Question 2.9, Question 2.13, Question 2.8, and Ques- tion 4.2 are equivalent. First we reformulate in two different ways the question if a given properly infinite unital C∗-algebra is K1-injective. Proposition 5.1 The following conditions are equivalent for any unital properly infinite C∗-algebra A: (i) A is K1-injective. (ii) Let p, q be projections in A such that p ∼ q and p, q, 1−p, 1−q are properly infinite and full. Then p ∼h q. (iii) Let p and q be properly infinite, full projections in A. There exist properly infinite, full projections p0, q0 ∈ A such that p0 ≤ p, q0 ≤ q, and p0 ∼h q0. Proof: (i) ⇒ (ii). Let p, q be properly infinite, full projections in A with p ∼ q such that 1− p, 1− q are properly infinite and full. Then by Lemma 2.4 (i) there is a unitary v ∈ A such that vpv∗ = q and [v] = 0 in K1(A). By the assumption in (i), v ∈ U 0(A), whence p ∼h q. (ii) ⇒ (i). Let u ∈ U(A) be such that [u] = 0 in K1(A). Take, as we can, a projection p in A such that p and 1 − p are properly infinite and full. Set q = upu∗. Then p ∼h q by (ii), and so there exists a unitary v ∈ U0(A) with p = vqv∗. It follows that pvu = vqv∗vu = v(upu∗)v∗vu = vup. Therefore vu ∈ U0(A) by Lemma 2.4 (ii), which in turn implies that u ∈ U0(A). (ii) ⇒ (iii). Let p, q be properly infinite and full projections in A. There exist mutually orthogonal projections e1, f1 such that e1 ≤ p, f1 ≤ p and e1 ∼ p ∼ f1, and mutually orthogonal projections e2, f2 such that e2 ≤ q, f2 ≤ q and e2 ∼ q ∼ f2. Being equivalent to either p or q, the projections e1, e2, f1 and f2 are properly infinite and full. There are properly infinite, full projections p0 ≤ e1 and q0 ≤ e2 such that [p0] = [q0] = 0 in K0(A) and p0 ∼ q0 (cf. Proposition 2.3). As f1 ≤ 1 − p0 and f2 ≤ 1− q0, we see that 1 − p0 and 1− q0 are properly infinite and full, and so we get p0 ∼h q0 by (ii). (iii) ⇒ (ii). Let p, q be equivalent properly infinite, full projections in A such that 1 − p, 1 − q are properly infinite and full. From (iii) we get properly infinite and full projections p0 ≤ p, q0 ≤ q which satisfy p0 ∼h q0. Thus there is a unitary v ∈ U0(A) such that vp0v ∗ = q0. Upon replacing p by vpv ∗ (as we may do because p ∼h vpv ∗) we can assume that q0 ≤ p and q0 ≤ q. Now, q0 is orthogonal to 1 − p and to 1 − q, and so 1− p ∼h 1− q by Proposition 2.5, whence p ∼h q. � Proposition 5.2 Let A be a unital properly infinite C∗-algebra. The following conditions are equivalent: (i) A is K1-injective, ie., the natural map U(A)/U 0(A) → K1(A) is injective. (ii) The natural map U(A)/U0(A) → U2(A)/U 2 (A) is injective. (iii) The natural maps Un(A)/U n(A) → K1(A) are injective for each natural number n. Proof: (i) ⇒ (ii) holds because the map U(A)/U0(A) → K1(A) factors through the map U(A)/U0(A) → U2(A)/U 2 (A). (ii)⇒ (i). Take u ∈ U(A) and suppose that [u] = 0 inK1(A). Then diag(u, 1A) ∈ U 2 (A) by Lemma 2.4 (ii) (with p = diag(1A, 0)). Hence u ∈ U0(A) by injectivity of the map U(A)/U0(A) → U2(A)/U 2 (A). (i) ⇒ (iii). Let n ≥ 1 be given and consider the natural maps U(A)/U0(A) → Un(A)/U n(A) → K1(A). The first map is onto, as proved by Cuntz in [4], see also [13, Exercise 8.9], and the composition of the two maps is injective by assumption, hence the second map is injective. (iii) ⇒ (i) is trivial. � We give below another application of K1-injectivity for properly infinite C ∗-algebras. First we need a lemma: Lemma 5.3 Let A be a unital, properly infinite C∗-algebra, and let ϕ, ψ : O∞ → A be unital embeddings. Then ψ is homotopic to a unital embedding ψ′ : O∞ → A for which there is a unitary u ∈ A with [u] = 0 in K1(A) and for which ψ ′(sj) = uϕ(sj) for all j (where s1, s2, . . . are the canonical generators of O∞). Proof: For each n set ψ(sj)ϕ(sj) ∗ ∈ A, en = j ∈ O∞. Then vn is a partial isometry in A with vnv n = ψ(en), v nvn = ϕ(en), and ψ(sj) = vnϕ(sj) for j = 1, 2, . . . , n. Since 1− en is full and properly infinite it follows from Lemma 2.4 that each vn extends to a unitary un ∈ A with [un] = 0 in K1(A). In particular, ψ(sj) = unϕ(sj) for j = 1, 2, . . . , n. We proceed to show that n 7→ un extends to a continuous path of unitaries t 7→ ut, for t ∈ [2,∞), such that utϕ(en) = unϕ(en) for t ≥ n + 1. Fix n ≥ 2. To this end it suffices to show that we can find a continuous path t 7→ zt, t ∈ [0, 1], of unitaries in A such that z0 = 1, z1 = u nun+1, and ztϕ(en−1) = ϕ(en−1) (as we then can set ut to be unzt−n for t ∈ [n, n+ 1]). Observe that un+1ϕ(en) = vn+1ϕ(en) = vn = unϕ(en). Set A0 = (1 − ϕ(en−1))A(1 − ϕ(en−1)), and set y = u nun+1(1 − ϕ(en−1)). Then y is a unitary element in A0 and [y] = 0 in K1(A0). Moreover, y commutes with the properly infinite full projection ϕ(en) − ϕ(en−1) ∈ A0. We can therefore use Lemma 2.4 to find a continuous path t 7→ yt of unitaries in A0 such that y0 = 1A0 = 1 − ϕ(en−1) and y1 = y. The continuous path t 7→ zt = yt + ϕ(en−1) is then as desired. For each t ≥ 2 let ψt : O∞ → A be the ∗-homomorphism given by ψt(sj) = utϕ(sj). Then ψt(sj) = ψ(sj) for all t ≥ j + 1, and so it follows that ψt(x) = ψ(x) for all x ∈ O∞. Hence ψ2 is homotopic to ψ, and so we can take ψ ′ to be ψ2. � Proposition 5.4 Any two unital ∗-homomorphisms from O∞ into a unital K1-injective (properly infinite) C∗-algebra are homotopic. Proof: In the light of Lemma 5.3 it suffices to show that if ϕ, ψ : O∞ → A are unital homomorphisms such that, for some unitary u ∈ A with [u] = 0 in K1(A), ψ(sj) = uϕ(sj) for all j, then ψ ∼h ϕ. By assumption, u ∼h 1, so there is a continuous path t 7→ ut of unitaries in A such that u0 = 1 and u1 = u. Letting ϕt : O∞ → A be the ∗-homomorphism given by ϕt(sj) = utϕ(sj) for all j, we get t 7→ ϕt is a continuous path of ∗-homomorphisms connecting ϕ0 = ϕ to ϕ1 = ψ. � Our main theorem below, which in particular implies that Question 2.9, Question 2.13, Question 2.8 and Question 4.2 all are equivalent, also give a special converse to Proposi- tion 5.4: Indeed, with ι1, ι2 : O∞ → O∞ ∗O∞ the two canonical inclusions, if ι1 ∼h ι2, then condition (iv) below holds, whence O∞ ∗ O∞ is K1-injective, which again implies that all unital properly infinite C∗-algebras are K1-injective. Below we retain the convention that O∞ ∗ O∞ is the universal unital free product of two copies of O∞ and that ι1 and ι2 are the two natural inclusions of O∞ into O∞ ∗ O∞. Theorem 5.5 The following statements are equivalent: (i) Every unital, properly infinite C∗-algebra is K1-injective. (ii) For every compact Hausdorff space X, every unital C(X)-algebra A, for which Ax is properly infinite for all x ∈ X, is properly infinite. (iii) Every unital C∗-algebra A, that is the pull-back of two unital, properly infinite C∗- algebras A1 and A2 along ∗-epimorphisms π1 : A1 → B, π2 : A2 → B: }} ϕ2 π1 A π2~~}} is properly infinite. (iv) There exist non-zero projections p, q ∈ O∞ such that p 6= 1, q 6= 1, and ι1(p) ∼h ι2(p) in O∞ ∗ O∞. (v) The specific C([0, 1])-algebra A considered in Example 4.1 (and whose fibres are prop- erly infinite) is properly infinite. (vi) O∞ ∗ O∞ is K1-injective. Note that statement (i) is reformulated in Propositions 5.1, 5.2, and 5.4; and that statement (iv) is reformulated in Proposition 4.3. We warn the reader that all these statements may turn out to be false (in which case, of course, there will be counterexamples to all of them). Proof: (i) ⇒ (iii) follows from Proposition 2.7. (iii) ⇒ (ii). This follows from Lemma 2.10 as in the proof of Theorem 2.11, except that one does not need to pass to matrix algebras. (ii) ⇒ (i). Suppose that A is unital and properly infinite. Take a unitary v ∈ U(A) such that diag(v, 1) ∈ U02 (A). Let B be the C(T)-algebra constructed in Example 4.5 from A, v, and a path of unitaries t 7→ ut connecting 1M2(A) to diag(v, 1). Then Bt ∼= A for all t ∈ T, so all fibres of B are properly infinite. Assuming (ii), we can conclude that B is properly infinite. Proposition 4.7 then yields that v ∈ U0(A). It follows that the natural map U(A)/U0(A) → U2(A)/U 2 (A) is injective, whence A is K1-injective by Proposition 5.2. (ii) ⇒ (v) is trivial (because A is a C([0, 1])-algebra with properly infinite fibres). (v) ⇒ (iv) follows from Proposition 4.3. (iv) ⇒ (i). We show that Condition (iii) of Proposition 4.3 implies Condition (iii) of Proposition 5.1. Let A be a properly infinite C∗-algebra and let p, q be properly infinite, full projections in A. Then there exist (properly infinite, full) projections p0 ≤ p and q0 ≤ q such that p0 ∼ 1 ∼ q0 and such that 1−p0 and 1−q0 are properly infinite and full, cf. Propositions 2.3. Take isometries t1, r1 ∈ A with t1t 1 = p0 and r1r 1 = q0; use the fact that 1 - 1 − p0 and 1 - 1− q0 to find sequences of isometries t2, t3, t4, . . . and r2, r3, r4, . . . in A such that each of the two sequences {tjt j=1 and {rjr j=1 consist of pairwise orthogonal projections. By the universal property of O∞ there are unital ∗-homomorphisms ϕj : O∞ → A, j = 1, 2, such that ϕ1(sj) = tj and ϕ2(sj) = rj, where s1, s2, s3, . . . are the canonical generators of O∞. In particular, ϕ1(s1s 1) = p0 and ϕ2(s1s 1) = q0. By the property of the universal unital free products of C∗-algebras, there is a unique unital ∗-homomorphism ϕ : O∞ ∗ O∞ → A making the diagram O∞ ∗ O∞ ϕ1 %%KK 99ssssssssss ϕ2yysss eeKKKKKKKKKK commutative. It follows that p0 = ϕ(ι1(s1s 1)) and q0 = ϕ(ι2(s1s 1)). By Condition (iii) of Proposition 4.3, ι1(s1s 1) ∼h ι2(s1s 1) in O∞ ∗ O∞, whence p0 ∼h q0 as desired. (i) ⇒ (vi) is trivial. (vi) ⇒ (v) follows from Proposition 4.4. � 6 Concluding remarks We do not know if all unital properly infinite C∗-algebras are K1-injective, but we observe that K1-injectivity is assured in the presence of certain central sequences: Proposition 6.1 Let A be a unital properly infinite C∗-algebras that contains an asymp- totically central sequence {pn} n=1, where pn and 1−pn are properly infinite, full projections for all n. Then A is K1-injective Proof: This follows immediately from Lemma 2.4 (ii). � It remains open if arbitrary C(X)-algebras with properly infinite fibres must be properly infinite. If this fails, then we already have a counterexample of the form B = pC(T, A)p, cf. Example 4.5, for some unital properly infinite C∗-algebra A and for some projection p ∈ C(T, A). (The C∗-algebra B is a C(T)-algebra with fibres Bt ∼= A.) On the other hand, any trivial C(X)-algebra C(X,D) with constant fibre D is clearly properly infinite if its fibre(s) D is unital and properly infinite (because C(X,D) ∼= C(X)⊗ D). We extend this observation in the following easy: Proposition 6.2 Let X be a compact Hausdorff space, let p ∈ C(X,D) be a projection, and consider the sub-trivial C(X)-algebra pC(X,D)p whose fibre at x is equal to p(x)Dp(x). If p is Murray-von Neumann equivalent to a constant projection q, then pC(X,D)p is C(X)-isomorphic to the trivial C(X)-algebra C(X,D0), where D0 = qDq. In this case, pC(X,D)p is properly infinite if and only if D0 is properly infinite. In particular, if X is contractible, then pC(X,D)p is C(X)-isomorphic to a trivial C(X)-algebra for any projection p ∈ C(X,D) and for any C∗-algebra D. Proof: Suppose that p = v∗v and q = vv∗ for some partial isometry v ∈ C(X,D). The map f 7→ vfv∗ defines a C(X)-isomorphism from pC(X,D)p onto qC(X,D)q, and qC(X,D)q = C(X,D0). If X is contractible, then any projection p ∈ C(X,D) is homotopic, and hence equiva- lent, to the constant projection x 7→ p(x0) for any fixed x0 ∈ X . � Remark 6.3 One can elaborate a little more on the construction considered above. Take a unital C∗-algebra D such that for some natural number n ≥ 2, Mn(D) is properly infinite, butMn−1(D) is not properly infinite (see [12] or [11] for such examples). Take any space X , preferably one with highly non-trivial topology, eg. X = Sn, and take, for some k ≥ n, a sufficiently non-trivial n-dimensional projection p in C(X,Mk(D)) such that p(x) is equivalent to the trivial n dimensional projection 1Mn(D) for all x (if X is connected we need only assume that this holds for one x ∈ X). The C(X)-algebra A = pC(X,Mk(D)) p, then has properly infinite fibres Ax = p(x)Dp(x) ∼= Mn(D). Is A always properly infinite? We guess that a possible counterexample to the questions posed in this paper could be of this form (for suitable D, X , and p). Let us end this paper by remarking that the answer to Question 2.13, which asks if any C(X)-algebra with properly infinite fibres is itself properly infinite, does not depend (very much) on X . If it fails, then it fails already for X = [0, 1] (cf. Theorem 5.5), and [0, 1] is a contractible space of low dimension. However, if we make the dimension of X even lower than the dimension of [0, 1], then we do get a positive anwer to our question: Proposition 6.4 Let X be a totally disconnected space, and let A be a C(X)-algebra such that all fibres Ax, x ∈ X, of A are properly infinite. Then A is properly infinite. Proof: Using Lemma 2.10 and the fact that X is totally disconnected we can write X as the disjoint union of clopen sets F1, F2, . . . , Fn such that AFj is properly infinite for all j. A = AF1 ⊕ AF2 ⊕ · · · ⊕AFn , the claim is proved. � References [1] B. Blackadar, D. Handelman, Dimension functions and traces on C∗-algebras, J. Funct. Anal. 45 (1982), 297–340. [2] E. Blanchard, A few remarks on C(X)-algebras, Rev. Roumaine Math. Pures Appl. 45 (2001), 565–576. [3] E. Blanchard, E. Kirchberg, Global Glimm halving for C∗-bundles, J. Op. Th. 52 (2004), 385–420. [4] J. Cuntz, K-theory for certain C∗-algebras, Ann. of Math. 113 (1981), 181–197. [5] M. Dadarlat, Continuous fields of C∗-algebras over finite dimensional spaces, preprint. [6] J. Dixmier, A. Douady, Champs continus d’espaces hilbertiens et de C∗-algèbres, Bull. Soc. Math. France 91 (1963), 227–284. [7] D. Handelman, Homomorphism of C∗-algebras to finite AW ∗ algebras, Michigan Math J. 28 (1981), 229-240. [8] I. Hirshberg, M. Rørdam, W. Winter, C0(X)-algebras, stability and strongly self- absorbing C*-algebras, Preprint July 2006. [9] M. A. Rieffel, The homotopy groups of the unitary groups of non-commutative tori, Journal of Operator Theory, 17-18 (1987), 237-254. [10] M. Rørdam, Classification of inductive limits of Cuntz algebras, J. Reine Angew. Math., 440 (1993), 175–200. [11] M. Rørdam, A simple C∗-algebra with a finite and an infinite projection, Acta Math. 191 (2003), 109–142. [12] M. Rørdam, On sums of finite projections, “Operator algebras and operator theory” (1998), Amer. Math. Soc., Providence, RI, 327–340. [13] M. Rørdam, F. Larsen, N. J. Laustsen, An Introduction to K-theory for C∗-algebras, London Mathematical Society Student Texts 49 (2000) CUP, Cambridge. [14] A. Toms and W. Winter, Strongly self-absorbing C∗-algebras, Transactions AMS (to appear). Projet Algbres d’oprateurs, Institut de Mathmatiques de Jussieu, 175, rue du Chevaleret, F-75013 PARIS, France E-mail address: Etienne.Blanchard@math.jussieu.fr Internet home page: www.math.jussieu.fr/∼blanchar Department of Mathematics, University of Southern Denmark, Odense, Campusvej 55, 5230 Odense M, Denmark E-mail address: rohde@imada.sdu.dk Department of Mathematics, University of Southern Denmark, Odense, Campusvej 55, 5230 Odense M, Denmark E-mail address: mikael@imada.sdu.dk Internet home page: www.imada.sdu.dk/∼mikael/welcome Introduction C(X)-algebras with properly infinite fibres Lower semi-continuous fields of properly infinite C-algebras Two examples K1-injectivity of properly infinite C*-algebras Concluding remarks

An information-based traffic control in a public conveyance system: reduced clustering and enhanced efficiency Akiyasu Tomoeda and Katsuhiro Nishinari Department of Aeronautics and Astronautics, Faculty of Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan. Debashish Chowdhury Department of Physics, Indian Institute of Technology, Kanpur 208016, India. Andreas Schadschneider Institut für Theoretische Physik, Universität zu Köln D-50937 Köln, Germany (Dated: October 31, 2018) A new public conveyance model applicable to buses and trains is proposed in this paper by using stochastic cellular automaton. We have found the optimal density of vehicles, at which the average velocity becomes maximum, significantly depends on the number of stops and passengers behavior of getting on a vehicle at stops. The efficiency of the hail-and-ride system is also discussed by comparing the different behavior of passengers. Moreover, we have found that a big cluster of vehicles is divided into small clusters, by incorporating information of the number of vehicles between successive stops. I. INTRODUCTION The totally asymmetric simple exclusion process [1, 2, 3] is the simplest model of non-equilibrium systems of in- teracting self-driven particles. Various extensions of this model have been reported in the last few years for captur- ing the essential features of the collective spatio-temporal organizations in wide varieties of systems, including those in vehicular traffic [4, 5, 6, 7, 8]. Traffic of buses and bicy- cles have also been modeled following similar approaches [9, 10]. A simple bus route model [10] exhibits clustering of the buses along the route and the quantitative features of the coarsening of the clusters have strong similarities with coarsening phenomena in many other physical sys- tems. Under normal circumstances, such clustering of buses is undesirable in any real bus route as the effi- ciency of the transport system is adversely affected by clustering. The main aim of this paper is to introduce a traffic control system into the bus route model in such a way that helps in suppressing this tendency of cluster- ing of the buses. This new model exhibits a competition between the two opposing tendencies of clustering and de-clustering which is interesting from the point of view of fundamental physical principles. However, the model may also find application in developing adaptive traffic control systems for public conveyance systems. In some of earlier bus-route models, movement of the buses was monitored on coarse time intervals so that the details of the dynamics of the buses in between two suc- cessive bus stops was not described explicitly. Instead, the movement of the bus from one stop to the next was captured only through probabilities of hopping from one stop to the next; hopping takes place with the lower prob- ability if passengers are waiting at the approaching bus stop [10]. An alternative interpretation of the model is as follows: the passengers could board the bus whenever and wherever they stopped a bus by raising their hand, this is called the hail-and-ride system. Several possible extensions of the bus route model have been reported in the past [11, 12, 13]. For example, in [11], in order to elucidate the connection between the bus route model with parallel updating and the Nagel- Schreckenberg model, two alternative extensions of the latter model with space-/time-dependent hopping rates are proposed. If a bus does not stop at a bus stop, the waiting passengers have to wait further for the next bus; such scenarios were captured in one of the earlier bus route models [12], using modified car-following model. In [13], the bus capacity, as well as the number of pas- sengers getting on and off at each stop, were introduced to make the model more realistic. Interestingly, it has been claimed that the distribution of the time gaps be- tween the arrival of successive buses is described well by the Gaussian Unitary Ensemble of random matrices [14]. In this paper, by extending the model in [10], we sug- gest a new public conveyance model (PCM). Although we refer to each of the public vehicles in this model as a “bus”, the model is equally applicable to train traffic on a given route. In this PCM we can set up arbitrary number of bus stops on the given route. The hail-and- ride system turns out to be a special case of the general PCM. Moreover, in the PCM the duration of the halt of a bus at any arbitrary bus stop depends on the num- ber of waiting passengers. As we shall demonstrate in this paper, the delay in the departure of the buses from crowded bus stops leads to the tendency of the buses to cluster on the route. Furthermore, in the PCM, we also introduce a traffic control system that exploits the in- formation on the number of buses in the “segments” in between successive bus stops; this traffic control system helps in reducing the undesirable tendency of clustering by dispersing the buses more or less uniformly along the http://arxiv.org/abs/0704.1555v2 route. In this study we introduce two different quantitative measures of the efficiency of the bus transport system, and calculate these quantities, both numerically and an- alytically, to determine the conditions under which the system would operate optimally. This paper is organized as follows, in Sec. 2 PCM is in- troduced and we show several simulation results in Sec. 3. The average speed and the number of waiting passengers are studied by mean field analysis in Sec. 4, and conclu- sions are given in Sec. 5. II. A STOCHASTIC CA MODEL FOR PUBLIC CONVEYANCE In this section, we explain the PCM in detail. For the sake of simplicity, we impose periodic boundary condi- tions. Let us imagine that the road is partitioned into L identical cells such that each cell can accommodate at most one bus at a time. Moreover, a total of S (0 ≤ S ≤ L) equispaced cells are identified in the begin- ning as bus stops. Note that, the special case S = L cor- responds to the hail-and-ride system. At any given time step, a passenger arrives with probability f to the sys- tem. Here, we assume that a given passenger is equally likely to arrive at any one of the bus stops with a proba- bility 1/S. Thus, the average number of passengers that arrive at each bus stop per unit time is given by f/S. In contrast to this model, in ref. [15, 16] the passengers were assumed to arrive with probability f at all the bus stops in every time step. Model AModel A Model BModel B BUS BUS BUS BUSBUS BUS )0( =iNQ)0( =iNQ 1H 2H )( qQ > 1 2( )Q H H> > FIG. 1: Schematic illustration of the PCM. In the model A, the hopping probability to the bus stop does not depend on the number of waiting passengers. In contrast, in the model B the hopping probability to the bus stop depends on the number of waiting passengers. Thus if the waiting passengers increase, the hopping probability to the bus stop is decreased. The model A corresponds to those situations where, because of sufficiently large number of broad doors, the time interval during which the doors of the bus remain open after halting at a stop, is independent of the size of waiting crowd of passengers. In contrast, the model B captures those situations where a bus has to halt for a longer period to pick up a larger crowd of waiting pas- sengers. The symbol H is used to denote the hopping probabil- ity of a bus entering into a cell that has been designated as a bus stop. We consider two different forms of H in the two versions of our model which are named as model A and model B. In the model A we assume the form Q no waiting passengers q waiting passengers exist where both Q and q (Q > q) are constants independent of the number of waiting passengers. The form (1) was used in the original formulation of the bus route model by O’Loan et al. [10]. In contrast to most of all the earlier bus route models, we assume in the model B that the maximum number of passengers that can get into one bus at a bus stop is Nmax. Suppose, Ni denotes the number of passengers waiting at the bus stop i (i = 1, · · · , S) at the instant of time when a bus arrives there. In contrast to the form (1) for H in model A, we assume in model B the form min(Ni, Nmax) + 1 where min(Ni, Nmax) is the number of passengers who can get into a bus which arrives at the bus stop i at the instant of time when the number of passengers wait- ing there is Ni. The form (2) is motivated by the com- mon expectation that the time needed for the passengers boarding a bus is proportional to their number. FIG. 1 depicts the hopping probabilities in the two models A and B schematically. The hopping probability of a bus to the cells that are not designated as bus stops is Q; this is already captured by the expressions (1) and (2) since no passenger ever waits at those locations. In principle, the hopping probability H for a real bus would depend also on the number of passengers who get off at the bus stop; in the extreme situations where no passenger waits at a bus stop the hopping probability H would be solely decided by the disembarking passengers. However, in order to keep the model theoretically simple and tractable, we ignore the latter situation and assume that passengers get off only at those stops where waiting passengers get into the bus and that the time taken by the waiting passengers to get into the bus is always adequate for the disembarking passengers to get off the bus. Note that Nmax is the maximum boarding capacity at each bus stop rather than themaximum carrying capacity of each bus. The PCM model reported here can be easily extended to incorporate an additional dynamical variable associated with each bus to account for the instantaneous number of passengers in it. But, for the sake of simplic- ity, such an extension of the model is not reported here. Instead, in the simple version of the PCMmodel reported here, Nmax can be interpreted as the maximum carrying capacity of each bus if we assume that all of the passen- gers on the bus get off whenever it stops. The model is updated according to the following rules. In step 2 − 4, these rules are applied in parallel to all buses and passengers, respectively: 1. Arrival of a passenger A bus stop i (i = 1, · · · , S) is picked up randomly, with probability 1/S, and then the corresponding number of waiting passengers in increased by unity, i.e. Ni → Ni+1, with probability f to account for the arrival of a passenger at the selected bus stop. 2. Bus motion If the cell in front of a bus is not occupied by an- other bus, each bus hops to the next cell with the probability H . Specifically, if passengers do not ex- ist in the next cell in both model A and model B hopping probability equals to Q because Ni equals to 0. Else, if passengers exist in the next cell, the hopping probability equals to q in the model A, whereas in the model B the corresponding hop- ping probability equals to Q/(min(Ni, Nmax) + 1). Note that, when a bus is loaded with passengers to its maximum boarding capacity Nmax, the hopping probability in the model B equals to Q/(Nmax+1), the smallest allowed hopping probability. 3. Boarding a bus When a bus arrives at the i-th (i = 1, · · · , S) bus stop cell, the corresponding number Ni of waiting passengers is updated to max(Ni −Nmax, 0) to ac- count for the passengers boarding the bus. Once the door is closed, no more waiting passenger can get into the bus at the same bus stop although the bus may remain stranded at the same stop for a longer period of time either because of the unavail- ability of the next bus stop or because of the traffic control rule explained next. 4. Bus information update Every bus stop has information Ij (j = 1, · · · , S) which is the number of buses in the segment of the route between the stop j and the next stop j+1 at that instant of time. This information is updated at each time steps. When one bus leaves the j-th bus stop, Ij is increased to Ij + 1. On the other hand, when a bus leaves (j+1)-th bus stop, Ij is reduced to Ij − 1. The desirable value of Ij is I0 = m/S, where m is the total number of buses, for all j so that buses are not clustered in any segment of the route. We implement a traffic control rule based on the information Ij : a bus remains stranded at a stop j as long as Ij exceeds I0. We use the average speed 〈V 〉 of the buses and the number of the waiting passengers 〈N〉 at a bus stop as two quantitative measures of the efficiency of the pub- lic conveyance system under consideration; a higher 〈V 〉 and smaller 〈N〉 correspond to an efficient transportation system. III. COMPUTER SIMULATIONS OF PCM In the simulations we set L = 500, Q = 0.9, q = 0.5 and Nmax = 60. The main parameters of this model, which we varied, are the number of buses (m), the num- ber of bus stops (S) and the probability (f) of arrival of passengers. The number density of buses is defined by ρ = m/L. SPACE SPACE Initial stage late stage FIG. 2: Space-time plots in the model B for the parameter values f = 0.6, S = 5, m = 30. The upper two figures cor- respond to the case where no traffic control system based on the information {I} is operational. The upper left figure cor- responds to the initial stage (from t = 1000 to t = 1500) whereas the upper right plot corresponds to the late stages (from t = 4000 to t = 4500). The lower figures correspond to the case where the information ({I}) based bus-traffic control system is operational (left figure shows data from t = 1000 to t = 1500 while the right figure corresponds to t = 4000 to t = 4500). Clearly, information-based traffic control sys- tem disperses the buses which, in the absence of this control system, would have a tendency to cluster. Typical space-time plots of the model B are given in FIG. 2. If no information-based traffic control system exits, the buses have a tendency to cluster; this phe- nomenon is very simular to that observed in the ant- trail model [15, 16]. However, implementation of the information-based traffic control system restricts the size of such clusters to a maximum of I0 buses in a segment of the route in between two successive bus stops. We study the effects of this control system below by compar- ing the characteristics of two traffic systems one of which includes the information-based control system while the other does not. A. PCM without information-based traffic control In the FIG. 3 - FIG. 8, we plot 〈V 〉 and 〈N〉 against the density of buses for several different values of f . Note that, the FIG. 5 and FIG. 8 corresponds to the hail-and- ride system for models A and B, respectively. 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 FIG. 3: The average speed and the average number of waiting passengers in the model A are plotted against the density for the parameters S = 5 and f = 0.3, 0.6 and 0.9. 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density rs f = 0.3 f = 0.6 f = 0.9 FIG. 4: The plot of 〈V 〉 and 〈N〉 of the model A for S = 50 and f = 0.3, 0.6 and 0.9. 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 FIG. 5: The plot of 〈V 〉 and 〈N〉 of the model A for S = 500(= L) and f = 0.3, 0.6 and 0.9. These figures demonstrate that the average speed 〈V 〉, which is a measure of the efficiency of the bus traffic system, exhibits a maximum at around ρ = 0.2 ∼ 0.3 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 FIG. 6: The plot of 〈V 〉 and 〈N〉 of the model B for S = 5 and f = 0.3, 0.6 and 0.9. 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 FIG. 7: The plot of 〈V 〉 and 〈N〉 of the model B for S = 50 and f = 0.3, 0.6 and 0.9. especially in the model B (comparing FIG. 3 with FIG. 6, it shows the model B (FIG. 6) reflects the bus bunching more clearly than the model A (FIG. 3) especially at large f and small ρ). The average number of waiting passengers 〈N〉, whose inverse is another measure of the efficiency of the bus traffic system, is vanishingly small in the region 0.3 < ρ < 0.7; 〈N〉 increases with decreasing (increasing) ρ in the regime ρ < 0.3 (ρ > 0.7). The average velocity of the model A becomes smaller as S increases in the low density region (see FIG. 3, FIG. 4 and FIG. 5). In contrast, in the model B (FIG. 7 and FIG. 8) we observe that there is no significant differ- ence in the average velocity. Note that the number of waiting passengers is calculated by (total waiting pas- sengers)/(number of bus stops). The total number of waiting passengers in this system is almost the same un- der the case S = 50 and hail-and-ride system S = L in both models. When the parameter S is small (comparing FIG. 3 and FIG. 6), in the model B the waiting passen- gers are larger and the average velocity is smaller than in the model A, since the effect of the delay in getting on a bus is taken into account. In the model B (comparing FIG. 6, FIG. 7 and FIG. 8), the case S = 50 is more effi- cient than S = 5, i.e. the system is likely to become more efficient, as S increases. However, we do not find any sig- nificant variation between S = 50 and S = 500. When S is small, the system becomes more efficient by increasing the number of bus stops. If the number of bus stops in- crease beyond 50, then there is little further variation of the efficiency as S is increased up to the maximum value From FIG. 9, the distribution of 〈N〉 over all the bus stops in the system is shown. We see that the distribution does not show the Zipf’s law, which is sometimes seen in 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 FIG. 8: The plot of 〈V 〉 and 〈N〉 of the model B for S = 500(= L) and f = 0.3, 0.6 and 0.9. 0 10 20 30 40 50 Ranking FIG. 9: The distribution of waiting passengers is plotted against all bus stops for the parameters f = 0.6, B = 50, S = 50. The horizontal line means the ranking, where we arrange the bus stops according to the descending order of natural and social phenomena; frequency of used words [17], population of a city [18], the number of the access to a web site [19], and intervals between successive transit times of the cars of traffic flow [20]. Next, we investigate the optimal density of buses at which the average velocity becomes maximum. The op- timal density depends on Q and is ρ = 0.3 for Q = 0.8 (FIG. 10, see also FIG. 11). In FIG. 10, it is shown that the density corresponding to the maximum veloc- ity shifts to higher values as Q becomes larger. FIG. 11 shows the optimal density of buses in the model B with- out information-based control system. From this figure, we find that the optimal density, for case S = 50, is smaller than that for S = 5. Moreover, for given S, the optimal density decreases with decreasing f . However, for both S = 5 and S = 50, the optimal density corre- sponding to Q = 1.0 is higher for f = 0.6 than that for f = 0.9. What is more effective way of increasing the efficiency of the public conveyance system on a given route by in- creasing the number of buses without increasing the car- rying capacity of each bus, or by increasing the carrying capacity of each bus without recruiting more buses? Or, are these two prescriptions for enhancing efficiency of the public conveyance system equally effective? In order to address these questions, we make a comparative study of two situations on the same route: for example, in the 0 0.2 0.4 0.6 0.8 1 Density Q = 0.8 Q = 1.0 0 0.2 0.4 0.6 0.8 1 Density rs Q = 0.8 Q = 1.0 FIG. 10: The average speed and the average number of wait- ing passengers in the model B are plotted against the density for the parameters f = 0.9, S = 50; the hopping parameters are Q = 0.8 and Q = 1.0. 0.6 0.7 0.8 0.9 1 f = 0.9, S = 5 f = 0.6, S = 5 f = 0.9, S = 50 f = 0.6, S = 50 FIG. 11: The optimal density of buses in the model B is plotted against Q. The parameters are f = 0.9, S = 5 (normal line),f = 0.6, S = 5 (finer broken line), f = 0.9, S = 50 (bold broken line), f = 0.6, S = 50 (longer broken line). first situation the number of buses is 10 and each has a capacity of 60, whereas in the second the number of buses is 5 and each has a capacity of 120. Note that the total carrying capacity of all the buses together is 600 (60×10 and 120× 5 in the two situations), i.e., same in both the situations. But, the number density of the buses in the second situation is just half of that in the first as the length of the bus route is same in both the situations. In FIG. 12, the results for these two cases are plotted; the different scales of density used along the X-axis arises from the differences in the number densities mentioned above. From FIG. 12, we conclude that, at sufficiently low densities, the average velocity is higher for Nmax = 60 compared to those for Nmax = 120. But, in the same regime of the number density of buses, larger number of passengers wait at bus stops when the bus capacity is smaller. Thus, in the region ρ < 0.05, system adminis- trators face a dilemma: if they give priority to the aver- age velocity and decide to choose buses with Nmax = 60, the number of passengers waiting at the bus stops in- creases. On the other hand if they decide to make the passengers happy by reducing their waiting time at the bus stops and, therefore, choose buses with Nmax = 120, the travel time of the passengers after boarding a bus becomes longer. However, at densities ρ > 0.05, the system administra- Density (0.05) (0.1) (0.15) (0.2) (0.25) Density (0.05) (0.1) (0.15) (0.2) (0.25) Density (0.05) (0.1) (0.15) (0.2) (0.25) Density (0.05) (0.1) (0.15) (0.2) (0.25) FIG. 12: Comparison between the case of bus capacity 60 with bus capacity 120. The parameters are Q = 0.9, S = 10, f = 0.6 in the model B without information. The top figure shows the average velocity, the center figure shows waiting passengers and the bottom figure shows the number of con- veyed passengers per unit bus, i.e. this number is calculated by (total number of on-boarding passengers on all buses)/(the number of buses), against the bus density up to 0.5. In each figure, the horizontal axis shows the density; the numbers without parentheses denote the number densities in the case Nmax = 60, whereas the numbers in the parentheses denote the number densities in the case Nmax = 120. tors can satisfy both the criteria, namely, fewer wait- ing passengers and shorter travel times, by one sin- gle choice. In this region of density, the public con- veyance system with Nmax = 60 is more efficient than that with Nmax = 120 because the average velocity is higher and the number of waiting passengers is smaller for Nmax = 60 than for Nmax = 120. Thus, in this regime of bus density, efficiency of the system is enhanced by re- ducing the capacity of individual buses and increasing their number on the same bus route. 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density rs f = 0.3 f = 0.6 f = 0.9 FIG. 13: The plot of 〈V 〉 and 〈N〉 of the model B with infor- mation (S = 5 and f = 0.3, 0.6 and 0.9) 0 0.2 0.4 0.6 0.8 1 Density without info with info 0 0.2 0.4 0.6 0.8 1 Density rs without info with info FIG. 14: The model B with S = 5 and f = 0.9. The left vertical dash line is ρ = 0.28 and the right is ρ = 0.73 in the two figures. B. PCM with information-based traffic control The results for the PCM with information-based traf- fic control system is shown in FIG. 13 and FIG. 14. In the FIG. 13 we plot 〈V 〉 and 〈N〉 against the density of buses for the parameter S = 5. The density correspond- ing to the peak of the average velocity shifts to lower values when the information-based traffic control system is switched on. The data shown in FIG. 14 establish that implemen- tation of the information-based traffic control system does not necessarily always improve the efficiency of the public conveyance system. In fact, in the region 0.3 < ρ < 0.7, the average velocity of the buses is higher if the information-based control system is switched off. Comparing 〈V 〉 and 〈N〉 in FIG. 14, we find that information-based traffic control system can improves the efficiency by reducing the crowd of waiting passengers. But, in the absence of waiting passengers, introduction of the information-based control system adversely affects the efficiency of the public conveyance system by holding up the buses at bus stops when the number of buses in the next segment of the route exceeds I0. 0 10 20 30 40 50 Ranking without information with information FIG. 15: Distribution of headway distance for S = 10, m = 50, f = 0.9 in model B. This figure shows the plot of headway distance against the ranking. Finally, FIG. 15 shows the distribution of headway dis- tance against the ranking, where we arrange the order of magnitude according to the headway distance of buses in descending order. From this figure it is found that the headway distribution is dispersed by the effect of the information. The average headway distance with the information-based traffic control is equal to 8.34, in con- trast to a much shorter value of 0.66 when that control system is switched off. Thus we confirm that the avail- ability of the information Ij and implementation of the traffic control system based on this information, signifi- cantly reduces the undesirable clustering of buses. IV. MEAN FIELD ANALYSIS Let us estimate 〈V 〉 theoretically in the low density limit ρ → 0. Suppose, T is the average time taken by a bus to complete one circuit of the route. In the model A, the number of hops made by a bus with probability q during the time T is S, i.e. the total number of bus stops. Therefore the average period T for a bus in the model A is well approximated by and hence, 〈V 〉 = q(L− S) +QS . (4) In model B, in the low density limit where m buses run practically unhindered and are distributed uniformly in the system without correlations, the average number of passengersN waiting at a bus stop, just before the arrival of the next bus, is . (5) The first factor f/S on the right hand side of the equation (5) is the probability of arrival of passengers per unit time. The second factor on the right hand side of (5) is an estimate of the average time taken by a bus to traverse one segment of the route, i.e. the part of the route between successive bus stops. The last factor in the same equation is the average number of segments of the route in between two successive buses on the same route. Instead of the constant q used in (4) for the evaluation of 〈V 〉 in the model A, we use N + 1 in eq. (4) and eq. (5) for the model B. Then, for the model B, the hopping probability Q is estimated self- consistently solving 〈V 〉 = Q− , (7) (4) and (6) simultaneously. We also obtain, for the model B, the average number of passengers 〈N〉 waiting at a bus stop in the ρ → 0 limit. The average time for moving from one bus stop to the next is ∆t = (L/S − 1)/Q + 1/q̄ and, therefore, we 〈N〉 = (f/S) · (∆t+ 2∆t+ · · ·+ (S − 1)∆t)/S f(S − 1)(q̄(L− S) + SQ) 2S2Qq̄ . (8) As long as the number of waiting passengers does not exceed Nmax, we have observed reasonably good agree- ment between the analytical estimates (4), (8) and the corresponding numerical data obtained from computer simulations. For example, in the model A, we get the es- timates 〈V 〉 = 0.85 and 〈N〉 = 1.71 from the approximate mean field theory for the parameter set S = 50, m = 1, Q = 0.9, q = 0.5, f = 0.3. The corresponding numbers obtained from direct computer simulations of the model A version of PCM are 0.84 and 1.78, respectively. Simi- larly, in the model B under the same conditions, we get 〈V 〉 = 0.60 and 〈N〉 = 2.45 from the mean field theory, while the corresponding numerical values are 0.60 and 2.51, respectively. If we take sufficiently small f ’s, then the mean-field estimates agree almost perfectly with the corresponding simulation data. However, our mean field analysis breaks down when a bus can not pick up all the passengers waiting at a bus stop. V. CONCLUDING DISCUSSIONS In this paper, we have proposed a public conveyance model (PCM) by using stochastic CA. In our PCM, some realistic elements are introduced: e.g., the carrying ca- pacity of a bus, the arbitrary number of bus stops, the halt time of a bus that depends on the number of waiting passengers, and an information-based bus traffic control system which reduces clustering of the buses on the given route. We have obtained quantitative results by using both computer simulations and analytical calculations. In par- ticular, we have introduced two different quantitative measures of the efficiency of the public conveyance sys- tem. We have found that the bus system works efficiently in a region of moderate number density of buses; too many or too few buses drastically reduce the efficiency of the bus-transport system. If the density of the buses is lower than optimal, not only large number of passen- gers are kept waiting at the stops for longer duration, but also the passengers in the buses get a slow ride as buses run slowly because they are slowed down at each stop to pick up the waiting passengers. On the other hand, if the density of the buses is higher than optimal, the mu- tual hindrance created by the buses in the overcrowded route also lowers the efficiency of the transport system. Moreover, we have found that the average velocity in- creases, and the number of waiting passengers decreases, when the information-based bus traffic control system is switched on. However, this enhancement of efficiency of the conveyance system takes place only over a particular range of density; the information-based bus traffic con- trol system does not necessarily improve the efficiency of the system in all possible situations. We have compared two situations where the second situation is obtained from the first one by doubling the carrying capacity of each bus and reducing their number to half the original number on the same route. In the density region ρ > 0.05 the system of Nmax = 60 is more efficient than that with Nmax = 120. However, at small densities (ρ < 0.05), although the average velocity in- creases, the number of waiting passengers also increases, by doubling the carrying capacity from Nmax = 60 to Nmax = 120. Hence, bus-transport system administra- tors would face a dilemma in this region of small density. Finally, in our PCM, the effect of the disembarking passengers on the halt time of the buses has not been captured explicitly. Moreover, this study is restricted to periodic boundary conditions. The clustering of particles occurs not only in a ring-like bus route, but also in shuttle services of buses and trains. Thus it would be interesting to investigate the effects of the information-based traffic control system also on such public transport systems. In a future work, we intend to report the results of our in- vestigations of the model under non-periodic boundary conditions. We hope our model will help in understand- ing the mechanism of congestion in public conveyance system and will provide insight as to the possible ways to reduce undesirable clustering of the vehicles. Acknowledgments: Work of one of the authors (DC) has been supported, in part, by the Council of Scientific and Industrial Research (CSIR), government of India. [1] B. Schmittmann and R.K.P. Zia, in: Phase Transition and Critical Phenomena, Vol. 17, eds. C. Domb and J. L. Lebowitz (Academic Press, 1995). [2] B. Derrida, Phys. Rep. 301, 65 (1998). [3] G. M. Schütz, in Phase Transitions and Critical Phenom- ena, vol. 19 (Acad. Press, 2001). [4] D. Chowdhury, L. Santen and A. Schadschneider, Phys. Rep. 329, 199 (2000). [5] D. Helbing, Rev. Mod. Phys. 73, 1067 (2001). [6] A. Schadschneider, Physica A 313, 153 (2002). [7] T. Nagatani, Rep. Prog. Phys. 65, 1331 (2002). [8] D. Chowdhury, K. Nishinari, L. Santen and A. Schad- schneider, Stochastic Transport in Complex Systems, El- sevier (2008). [9] R. Jiang, B. Jia and Q. S. Wu, J. Phys. A: Math. Gen. 37, 2063 (2004) [10] O.J. O’Loan, M.R. Evans and M.E. Cates, Phys. Rev. E 58, 1404 (1998). [11] D. Chowdhury and R.C. Desai, Eur. Phys. J. B 15, 375 (2000). [12] T. Nagatani, Physica A 287, 302 (2000); Phys. Rev. E 63, 036115 (2001); Physica A 297, 260 (2001). [13] R. Jiang, M-B. Hu, B. Jia, and Q-S. Wu, Eur. Phys. J. B 34, 367 (2003). [14] M. Krbalek and P. Seba, J. Phys. A: Math. Gen. 33, L229 (2000). [15] D. Chowdhury, V. Guttal, K. Nishinari and A. Schad- schneider, J. Phys. A: Math. Gen. 35, L573 (2002). [16] A. Kunwar, A. John, K. Nishinari, A. Schadschneider and D. Chowdhury, J. Phys. Soc. Jpn. 73, 2979 (2004). [17] G. K. 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A new public conveyance model applicable to buses and trains is proposed in this paper by using stochastic cellular automaton. We have found the optimal density of vehicles, at which the average velocity becomes maximum, significantly depends on the number of stops and passengers behavior of getting on a vehicle at stops. The efficiency of the hail-and-ride system is also discussed by comparing the different behavior of passengers. Moreover, we have found that a big cluster of vehicles is divided into small clusters, by incorporating information of the number of vehicles between successive stops.

An information-based traffic control in a public conveyance system: reduced clustering and enhanced efficiency Akiyasu Tomoeda and Katsuhiro Nishinari Department of Aeronautics and Astronautics, Faculty of Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan. Debashish Chowdhury Department of Physics, Indian Institute of Technology, Kanpur 208016, India. Andreas Schadschneider Institut für Theoretische Physik, Universität zu Köln D-50937 Köln, Germany (Dated: October 31, 2018) A new public conveyance model applicable to buses and trains is proposed in this paper by using stochastic cellular automaton. We have found the optimal density of vehicles, at which the average velocity becomes maximum, significantly depends on the number of stops and passengers behavior of getting on a vehicle at stops. The efficiency of the hail-and-ride system is also discussed by comparing the different behavior of passengers. Moreover, we have found that a big cluster of vehicles is divided into small clusters, by incorporating information of the number of vehicles between successive stops. I. INTRODUCTION The totally asymmetric simple exclusion process [1, 2, 3] is the simplest model of non-equilibrium systems of in- teracting self-driven particles. Various extensions of this model have been reported in the last few years for captur- ing the essential features of the collective spatio-temporal organizations in wide varieties of systems, including those in vehicular traffic [4, 5, 6, 7, 8]. Traffic of buses and bicy- cles have also been modeled following similar approaches [9, 10]. A simple bus route model [10] exhibits clustering of the buses along the route and the quantitative features of the coarsening of the clusters have strong similarities with coarsening phenomena in many other physical sys- tems. Under normal circumstances, such clustering of buses is undesirable in any real bus route as the effi- ciency of the transport system is adversely affected by clustering. The main aim of this paper is to introduce a traffic control system into the bus route model in such a way that helps in suppressing this tendency of cluster- ing of the buses. This new model exhibits a competition between the two opposing tendencies of clustering and de-clustering which is interesting from the point of view of fundamental physical principles. However, the model may also find application in developing adaptive traffic control systems for public conveyance systems. In some of earlier bus-route models, movement of the buses was monitored on coarse time intervals so that the details of the dynamics of the buses in between two suc- cessive bus stops was not described explicitly. Instead, the movement of the bus from one stop to the next was captured only through probabilities of hopping from one stop to the next; hopping takes place with the lower prob- ability if passengers are waiting at the approaching bus stop [10]. An alternative interpretation of the model is as follows: the passengers could board the bus whenever and wherever they stopped a bus by raising their hand, this is called the hail-and-ride system. Several possible extensions of the bus route model have been reported in the past [11, 12, 13]. For example, in [11], in order to elucidate the connection between the bus route model with parallel updating and the Nagel- Schreckenberg model, two alternative extensions of the latter model with space-/time-dependent hopping rates are proposed. If a bus does not stop at a bus stop, the waiting passengers have to wait further for the next bus; such scenarios were captured in one of the earlier bus route models [12], using modified car-following model. In [13], the bus capacity, as well as the number of pas- sengers getting on and off at each stop, were introduced to make the model more realistic. Interestingly, it has been claimed that the distribution of the time gaps be- tween the arrival of successive buses is described well by the Gaussian Unitary Ensemble of random matrices [14]. In this paper, by extending the model in [10], we sug- gest a new public conveyance model (PCM). Although we refer to each of the public vehicles in this model as a “bus”, the model is equally applicable to train traffic on a given route. In this PCM we can set up arbitrary number of bus stops on the given route. The hail-and- ride system turns out to be a special case of the general PCM. Moreover, in the PCM the duration of the halt of a bus at any arbitrary bus stop depends on the num- ber of waiting passengers. As we shall demonstrate in this paper, the delay in the departure of the buses from crowded bus stops leads to the tendency of the buses to cluster on the route. Furthermore, in the PCM, we also introduce a traffic control system that exploits the in- formation on the number of buses in the “segments” in between successive bus stops; this traffic control system helps in reducing the undesirable tendency of clustering by dispersing the buses more or less uniformly along the http://arxiv.org/abs/0704.1555v2 route. In this study we introduce two different quantitative measures of the efficiency of the bus transport system, and calculate these quantities, both numerically and an- alytically, to determine the conditions under which the system would operate optimally. This paper is organized as follows, in Sec. 2 PCM is in- troduced and we show several simulation results in Sec. 3. The average speed and the number of waiting passengers are studied by mean field analysis in Sec. 4, and conclu- sions are given in Sec. 5. II. A STOCHASTIC CA MODEL FOR PUBLIC CONVEYANCE In this section, we explain the PCM in detail. For the sake of simplicity, we impose periodic boundary condi- tions. Let us imagine that the road is partitioned into L identical cells such that each cell can accommodate at most one bus at a time. Moreover, a total of S (0 ≤ S ≤ L) equispaced cells are identified in the begin- ning as bus stops. Note that, the special case S = L cor- responds to the hail-and-ride system. At any given time step, a passenger arrives with probability f to the sys- tem. Here, we assume that a given passenger is equally likely to arrive at any one of the bus stops with a proba- bility 1/S. Thus, the average number of passengers that arrive at each bus stop per unit time is given by f/S. In contrast to this model, in ref. [15, 16] the passengers were assumed to arrive with probability f at all the bus stops in every time step. Model AModel A Model BModel B BUS BUS BUS BUSBUS BUS )0( =iNQ)0( =iNQ 1H 2H )( qQ > 1 2( )Q H H> > FIG. 1: Schematic illustration of the PCM. In the model A, the hopping probability to the bus stop does not depend on the number of waiting passengers. In contrast, in the model B the hopping probability to the bus stop depends on the number of waiting passengers. Thus if the waiting passengers increase, the hopping probability to the bus stop is decreased. The model A corresponds to those situations where, because of sufficiently large number of broad doors, the time interval during which the doors of the bus remain open after halting at a stop, is independent of the size of waiting crowd of passengers. In contrast, the model B captures those situations where a bus has to halt for a longer period to pick up a larger crowd of waiting pas- sengers. The symbol H is used to denote the hopping probabil- ity of a bus entering into a cell that has been designated as a bus stop. We consider two different forms of H in the two versions of our model which are named as model A and model B. In the model A we assume the form Q no waiting passengers q waiting passengers exist where both Q and q (Q > q) are constants independent of the number of waiting passengers. The form (1) was used in the original formulation of the bus route model by O’Loan et al. [10]. In contrast to most of all the earlier bus route models, we assume in the model B that the maximum number of passengers that can get into one bus at a bus stop is Nmax. Suppose, Ni denotes the number of passengers waiting at the bus stop i (i = 1, · · · , S) at the instant of time when a bus arrives there. In contrast to the form (1) for H in model A, we assume in model B the form min(Ni, Nmax) + 1 where min(Ni, Nmax) is the number of passengers who can get into a bus which arrives at the bus stop i at the instant of time when the number of passengers wait- ing there is Ni. The form (2) is motivated by the com- mon expectation that the time needed for the passengers boarding a bus is proportional to their number. FIG. 1 depicts the hopping probabilities in the two models A and B schematically. The hopping probability of a bus to the cells that are not designated as bus stops is Q; this is already captured by the expressions (1) and (2) since no passenger ever waits at those locations. In principle, the hopping probability H for a real bus would depend also on the number of passengers who get off at the bus stop; in the extreme situations where no passenger waits at a bus stop the hopping probability H would be solely decided by the disembarking passengers. However, in order to keep the model theoretically simple and tractable, we ignore the latter situation and assume that passengers get off only at those stops where waiting passengers get into the bus and that the time taken by the waiting passengers to get into the bus is always adequate for the disembarking passengers to get off the bus. Note that Nmax is the maximum boarding capacity at each bus stop rather than themaximum carrying capacity of each bus. The PCM model reported here can be easily extended to incorporate an additional dynamical variable associated with each bus to account for the instantaneous number of passengers in it. But, for the sake of simplic- ity, such an extension of the model is not reported here. Instead, in the simple version of the PCMmodel reported here, Nmax can be interpreted as the maximum carrying capacity of each bus if we assume that all of the passen- gers on the bus get off whenever it stops. The model is updated according to the following rules. In step 2 − 4, these rules are applied in parallel to all buses and passengers, respectively: 1. Arrival of a passenger A bus stop i (i = 1, · · · , S) is picked up randomly, with probability 1/S, and then the corresponding number of waiting passengers in increased by unity, i.e. Ni → Ni+1, with probability f to account for the arrival of a passenger at the selected bus stop. 2. Bus motion If the cell in front of a bus is not occupied by an- other bus, each bus hops to the next cell with the probability H . Specifically, if passengers do not ex- ist in the next cell in both model A and model B hopping probability equals to Q because Ni equals to 0. Else, if passengers exist in the next cell, the hopping probability equals to q in the model A, whereas in the model B the corresponding hop- ping probability equals to Q/(min(Ni, Nmax) + 1). Note that, when a bus is loaded with passengers to its maximum boarding capacity Nmax, the hopping probability in the model B equals to Q/(Nmax+1), the smallest allowed hopping probability. 3. Boarding a bus When a bus arrives at the i-th (i = 1, · · · , S) bus stop cell, the corresponding number Ni of waiting passengers is updated to max(Ni −Nmax, 0) to ac- count for the passengers boarding the bus. Once the door is closed, no more waiting passenger can get into the bus at the same bus stop although the bus may remain stranded at the same stop for a longer period of time either because of the unavail- ability of the next bus stop or because of the traffic control rule explained next. 4. Bus information update Every bus stop has information Ij (j = 1, · · · , S) which is the number of buses in the segment of the route between the stop j and the next stop j+1 at that instant of time. This information is updated at each time steps. When one bus leaves the j-th bus stop, Ij is increased to Ij + 1. On the other hand, when a bus leaves (j+1)-th bus stop, Ij is reduced to Ij − 1. The desirable value of Ij is I0 = m/S, where m is the total number of buses, for all j so that buses are not clustered in any segment of the route. We implement a traffic control rule based on the information Ij : a bus remains stranded at a stop j as long as Ij exceeds I0. We use the average speed 〈V 〉 of the buses and the number of the waiting passengers 〈N〉 at a bus stop as two quantitative measures of the efficiency of the pub- lic conveyance system under consideration; a higher 〈V 〉 and smaller 〈N〉 correspond to an efficient transportation system. III. COMPUTER SIMULATIONS OF PCM In the simulations we set L = 500, Q = 0.9, q = 0.5 and Nmax = 60. The main parameters of this model, which we varied, are the number of buses (m), the num- ber of bus stops (S) and the probability (f) of arrival of passengers. The number density of buses is defined by ρ = m/L. SPACE SPACE Initial stage late stage FIG. 2: Space-time plots in the model B for the parameter values f = 0.6, S = 5, m = 30. The upper two figures cor- respond to the case where no traffic control system based on the information {I} is operational. The upper left figure cor- responds to the initial stage (from t = 1000 to t = 1500) whereas the upper right plot corresponds to the late stages (from t = 4000 to t = 4500). The lower figures correspond to the case where the information ({I}) based bus-traffic control system is operational (left figure shows data from t = 1000 to t = 1500 while the right figure corresponds to t = 4000 to t = 4500). Clearly, information-based traffic control sys- tem disperses the buses which, in the absence of this control system, would have a tendency to cluster. Typical space-time plots of the model B are given in FIG. 2. If no information-based traffic control system exits, the buses have a tendency to cluster; this phe- nomenon is very simular to that observed in the ant- trail model [15, 16]. However, implementation of the information-based traffic control system restricts the size of such clusters to a maximum of I0 buses in a segment of the route in between two successive bus stops. We study the effects of this control system below by compar- ing the characteristics of two traffic systems one of which includes the information-based control system while the other does not. A. PCM without information-based traffic control In the FIG. 3 - FIG. 8, we plot 〈V 〉 and 〈N〉 against the density of buses for several different values of f . Note that, the FIG. 5 and FIG. 8 corresponds to the hail-and- ride system for models A and B, respectively. 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 FIG. 3: The average speed and the average number of waiting passengers in the model A are plotted against the density for the parameters S = 5 and f = 0.3, 0.6 and 0.9. 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density rs f = 0.3 f = 0.6 f = 0.9 FIG. 4: The plot of 〈V 〉 and 〈N〉 of the model A for S = 50 and f = 0.3, 0.6 and 0.9. 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 FIG. 5: The plot of 〈V 〉 and 〈N〉 of the model A for S = 500(= L) and f = 0.3, 0.6 and 0.9. These figures demonstrate that the average speed 〈V 〉, which is a measure of the efficiency of the bus traffic system, exhibits a maximum at around ρ = 0.2 ∼ 0.3 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 FIG. 6: The plot of 〈V 〉 and 〈N〉 of the model B for S = 5 and f = 0.3, 0.6 and 0.9. 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 FIG. 7: The plot of 〈V 〉 and 〈N〉 of the model B for S = 50 and f = 0.3, 0.6 and 0.9. especially in the model B (comparing FIG. 3 with FIG. 6, it shows the model B (FIG. 6) reflects the bus bunching more clearly than the model A (FIG. 3) especially at large f and small ρ). The average number of waiting passengers 〈N〉, whose inverse is another measure of the efficiency of the bus traffic system, is vanishingly small in the region 0.3 < ρ < 0.7; 〈N〉 increases with decreasing (increasing) ρ in the regime ρ < 0.3 (ρ > 0.7). The average velocity of the model A becomes smaller as S increases in the low density region (see FIG. 3, FIG. 4 and FIG. 5). In contrast, in the model B (FIG. 7 and FIG. 8) we observe that there is no significant differ- ence in the average velocity. Note that the number of waiting passengers is calculated by (total waiting pas- sengers)/(number of bus stops). The total number of waiting passengers in this system is almost the same un- der the case S = 50 and hail-and-ride system S = L in both models. When the parameter S is small (comparing FIG. 3 and FIG. 6), in the model B the waiting passen- gers are larger and the average velocity is smaller than in the model A, since the effect of the delay in getting on a bus is taken into account. In the model B (comparing FIG. 6, FIG. 7 and FIG. 8), the case S = 50 is more effi- cient than S = 5, i.e. the system is likely to become more efficient, as S increases. However, we do not find any sig- nificant variation between S = 50 and S = 500. When S is small, the system becomes more efficient by increasing the number of bus stops. If the number of bus stops in- crease beyond 50, then there is little further variation of the efficiency as S is increased up to the maximum value From FIG. 9, the distribution of 〈N〉 over all the bus stops in the system is shown. We see that the distribution does not show the Zipf’s law, which is sometimes seen in 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 FIG. 8: The plot of 〈V 〉 and 〈N〉 of the model B for S = 500(= L) and f = 0.3, 0.6 and 0.9. 0 10 20 30 40 50 Ranking FIG. 9: The distribution of waiting passengers is plotted against all bus stops for the parameters f = 0.6, B = 50, S = 50. The horizontal line means the ranking, where we arrange the bus stops according to the descending order of natural and social phenomena; frequency of used words [17], population of a city [18], the number of the access to a web site [19], and intervals between successive transit times of the cars of traffic flow [20]. Next, we investigate the optimal density of buses at which the average velocity becomes maximum. The op- timal density depends on Q and is ρ = 0.3 for Q = 0.8 (FIG. 10, see also FIG. 11). In FIG. 10, it is shown that the density corresponding to the maximum veloc- ity shifts to higher values as Q becomes larger. FIG. 11 shows the optimal density of buses in the model B with- out information-based control system. From this figure, we find that the optimal density, for case S = 50, is smaller than that for S = 5. Moreover, for given S, the optimal density decreases with decreasing f . However, for both S = 5 and S = 50, the optimal density corre- sponding to Q = 1.0 is higher for f = 0.6 than that for f = 0.9. What is more effective way of increasing the efficiency of the public conveyance system on a given route by in- creasing the number of buses without increasing the car- rying capacity of each bus, or by increasing the carrying capacity of each bus without recruiting more buses? Or, are these two prescriptions for enhancing efficiency of the public conveyance system equally effective? In order to address these questions, we make a comparative study of two situations on the same route: for example, in the 0 0.2 0.4 0.6 0.8 1 Density Q = 0.8 Q = 1.0 0 0.2 0.4 0.6 0.8 1 Density rs Q = 0.8 Q = 1.0 FIG. 10: The average speed and the average number of wait- ing passengers in the model B are plotted against the density for the parameters f = 0.9, S = 50; the hopping parameters are Q = 0.8 and Q = 1.0. 0.6 0.7 0.8 0.9 1 f = 0.9, S = 5 f = 0.6, S = 5 f = 0.9, S = 50 f = 0.6, S = 50 FIG. 11: The optimal density of buses in the model B is plotted against Q. The parameters are f = 0.9, S = 5 (normal line),f = 0.6, S = 5 (finer broken line), f = 0.9, S = 50 (bold broken line), f = 0.6, S = 50 (longer broken line). first situation the number of buses is 10 and each has a capacity of 60, whereas in the second the number of buses is 5 and each has a capacity of 120. Note that the total carrying capacity of all the buses together is 600 (60×10 and 120× 5 in the two situations), i.e., same in both the situations. But, the number density of the buses in the second situation is just half of that in the first as the length of the bus route is same in both the situations. In FIG. 12, the results for these two cases are plotted; the different scales of density used along the X-axis arises from the differences in the number densities mentioned above. From FIG. 12, we conclude that, at sufficiently low densities, the average velocity is higher for Nmax = 60 compared to those for Nmax = 120. But, in the same regime of the number density of buses, larger number of passengers wait at bus stops when the bus capacity is smaller. Thus, in the region ρ < 0.05, system adminis- trators face a dilemma: if they give priority to the aver- age velocity and decide to choose buses with Nmax = 60, the number of passengers waiting at the bus stops in- creases. On the other hand if they decide to make the passengers happy by reducing their waiting time at the bus stops and, therefore, choose buses with Nmax = 120, the travel time of the passengers after boarding a bus becomes longer. However, at densities ρ > 0.05, the system administra- Density (0.05) (0.1) (0.15) (0.2) (0.25) Density (0.05) (0.1) (0.15) (0.2) (0.25) Density (0.05) (0.1) (0.15) (0.2) (0.25) Density (0.05) (0.1) (0.15) (0.2) (0.25) FIG. 12: Comparison between the case of bus capacity 60 with bus capacity 120. The parameters are Q = 0.9, S = 10, f = 0.6 in the model B without information. The top figure shows the average velocity, the center figure shows waiting passengers and the bottom figure shows the number of con- veyed passengers per unit bus, i.e. this number is calculated by (total number of on-boarding passengers on all buses)/(the number of buses), against the bus density up to 0.5. In each figure, the horizontal axis shows the density; the numbers without parentheses denote the number densities in the case Nmax = 60, whereas the numbers in the parentheses denote the number densities in the case Nmax = 120. tors can satisfy both the criteria, namely, fewer wait- ing passengers and shorter travel times, by one sin- gle choice. In this region of density, the public con- veyance system with Nmax = 60 is more efficient than that with Nmax = 120 because the average velocity is higher and the number of waiting passengers is smaller for Nmax = 60 than for Nmax = 120. Thus, in this regime of bus density, efficiency of the system is enhanced by re- ducing the capacity of individual buses and increasing their number on the same bus route. 0 0.2 0.4 0.6 0.8 1 Density f = 0.3 f = 0.6 f = 0.9 0 0.2 0.4 0.6 0.8 1 Density rs f = 0.3 f = 0.6 f = 0.9 FIG. 13: The plot of 〈V 〉 and 〈N〉 of the model B with infor- mation (S = 5 and f = 0.3, 0.6 and 0.9) 0 0.2 0.4 0.6 0.8 1 Density without info with info 0 0.2 0.4 0.6 0.8 1 Density rs without info with info FIG. 14: The model B with S = 5 and f = 0.9. The left vertical dash line is ρ = 0.28 and the right is ρ = 0.73 in the two figures. B. PCM with information-based traffic control The results for the PCM with information-based traf- fic control system is shown in FIG. 13 and FIG. 14. In the FIG. 13 we plot 〈V 〉 and 〈N〉 against the density of buses for the parameter S = 5. The density correspond- ing to the peak of the average velocity shifts to lower values when the information-based traffic control system is switched on. The data shown in FIG. 14 establish that implemen- tation of the information-based traffic control system does not necessarily always improve the efficiency of the public conveyance system. In fact, in the region 0.3 < ρ < 0.7, the average velocity of the buses is higher if the information-based control system is switched off. Comparing 〈V 〉 and 〈N〉 in FIG. 14, we find that information-based traffic control system can improves the efficiency by reducing the crowd of waiting passengers. But, in the absence of waiting passengers, introduction of the information-based control system adversely affects the efficiency of the public conveyance system by holding up the buses at bus stops when the number of buses in the next segment of the route exceeds I0. 0 10 20 30 40 50 Ranking without information with information FIG. 15: Distribution of headway distance for S = 10, m = 50, f = 0.9 in model B. This figure shows the plot of headway distance against the ranking. Finally, FIG. 15 shows the distribution of headway dis- tance against the ranking, where we arrange the order of magnitude according to the headway distance of buses in descending order. From this figure it is found that the headway distribution is dispersed by the effect of the information. The average headway distance with the information-based traffic control is equal to 8.34, in con- trast to a much shorter value of 0.66 when that control system is switched off. Thus we confirm that the avail- ability of the information Ij and implementation of the traffic control system based on this information, signifi- cantly reduces the undesirable clustering of buses. IV. MEAN FIELD ANALYSIS Let us estimate 〈V 〉 theoretically in the low density limit ρ → 0. Suppose, T is the average time taken by a bus to complete one circuit of the route. In the model A, the number of hops made by a bus with probability q during the time T is S, i.e. the total number of bus stops. Therefore the average period T for a bus in the model A is well approximated by and hence, 〈V 〉 = q(L− S) +QS . (4) In model B, in the low density limit where m buses run practically unhindered and are distributed uniformly in the system without correlations, the average number of passengersN waiting at a bus stop, just before the arrival of the next bus, is . (5) The first factor f/S on the right hand side of the equation (5) is the probability of arrival of passengers per unit time. The second factor on the right hand side of (5) is an estimate of the average time taken by a bus to traverse one segment of the route, i.e. the part of the route between successive bus stops. The last factor in the same equation is the average number of segments of the route in between two successive buses on the same route. Instead of the constant q used in (4) for the evaluation of 〈V 〉 in the model A, we use N + 1 in eq. (4) and eq. (5) for the model B. Then, for the model B, the hopping probability Q is estimated self- consistently solving 〈V 〉 = Q− , (7) (4) and (6) simultaneously. We also obtain, for the model B, the average number of passengers 〈N〉 waiting at a bus stop in the ρ → 0 limit. The average time for moving from one bus stop to the next is ∆t = (L/S − 1)/Q + 1/q̄ and, therefore, we 〈N〉 = (f/S) · (∆t+ 2∆t+ · · ·+ (S − 1)∆t)/S f(S − 1)(q̄(L− S) + SQ) 2S2Qq̄ . (8) As long as the number of waiting passengers does not exceed Nmax, we have observed reasonably good agree- ment between the analytical estimates (4), (8) and the corresponding numerical data obtained from computer simulations. For example, in the model A, we get the es- timates 〈V 〉 = 0.85 and 〈N〉 = 1.71 from the approximate mean field theory for the parameter set S = 50, m = 1, Q = 0.9, q = 0.5, f = 0.3. The corresponding numbers obtained from direct computer simulations of the model A version of PCM are 0.84 and 1.78, respectively. Simi- larly, in the model B under the same conditions, we get 〈V 〉 = 0.60 and 〈N〉 = 2.45 from the mean field theory, while the corresponding numerical values are 0.60 and 2.51, respectively. If we take sufficiently small f ’s, then the mean-field estimates agree almost perfectly with the corresponding simulation data. However, our mean field analysis breaks down when a bus can not pick up all the passengers waiting at a bus stop. V. CONCLUDING DISCUSSIONS In this paper, we have proposed a public conveyance model (PCM) by using stochastic CA. In our PCM, some realistic elements are introduced: e.g., the carrying ca- pacity of a bus, the arbitrary number of bus stops, the halt time of a bus that depends on the number of waiting passengers, and an information-based bus traffic control system which reduces clustering of the buses on the given route. We have obtained quantitative results by using both computer simulations and analytical calculations. In par- ticular, we have introduced two different quantitative measures of the efficiency of the public conveyance sys- tem. We have found that the bus system works efficiently in a region of moderate number density of buses; too many or too few buses drastically reduce the efficiency of the bus-transport system. If the density of the buses is lower than optimal, not only large number of passen- gers are kept waiting at the stops for longer duration, but also the passengers in the buses get a slow ride as buses run slowly because they are slowed down at each stop to pick up the waiting passengers. On the other hand, if the density of the buses is higher than optimal, the mu- tual hindrance created by the buses in the overcrowded route also lowers the efficiency of the transport system. Moreover, we have found that the average velocity in- creases, and the number of waiting passengers decreases, when the information-based bus traffic control system is switched on. However, this enhancement of efficiency of the conveyance system takes place only over a particular range of density; the information-based bus traffic con- trol system does not necessarily improve the efficiency of the system in all possible situations. We have compared two situations where the second situation is obtained from the first one by doubling the carrying capacity of each bus and reducing their number to half the original number on the same route. In the density region ρ > 0.05 the system of Nmax = 60 is more efficient than that with Nmax = 120. However, at small densities (ρ < 0.05), although the average velocity in- creases, the number of waiting passengers also increases, by doubling the carrying capacity from Nmax = 60 to Nmax = 120. Hence, bus-transport system administra- tors would face a dilemma in this region of small density. Finally, in our PCM, the effect of the disembarking passengers on the halt time of the buses has not been captured explicitly. Moreover, this study is restricted to periodic boundary conditions. The clustering of particles occurs not only in a ring-like bus route, but also in shuttle services of buses and trains. Thus it would be interesting to investigate the effects of the information-based traffic control system also on such public transport systems. In a future work, we intend to report the results of our in- vestigations of the model under non-periodic boundary conditions. We hope our model will help in understand- ing the mechanism of congestion in public conveyance system and will provide insight as to the possible ways to reduce undesirable clustering of the vehicles. Acknowledgments: Work of one of the authors (DC) has been supported, in part, by the Council of Scientific and Industrial Research (CSIR), government of India. [1] B. Schmittmann and R.K.P. Zia, in: Phase Transition and Critical Phenomena, Vol. 17, eds. C. Domb and J. L. Lebowitz (Academic Press, 1995). [2] B. Derrida, Phys. Rep. 301, 65 (1998). [3] G. M. Schütz, in Phase Transitions and Critical Phenom- ena, vol. 19 (Acad. Press, 2001). [4] D. Chowdhury, L. Santen and A. Schadschneider, Phys. Rep. 329, 199 (2000). [5] D. 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A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA NURIT BARNEA AND YUVAL GINOSAR Abstract. The Donald-Flanigan conjecture asserts that for any finite group G and any field k, the group algebra kG can be deformed to a separable algebra. The minimal unsolved instance, namely the quaternion group Q8 over a field k of characteristic 2 was considered as a counterexample. We present here a separable deformation of kQ8. In a sense, the conjecture for any finite group is open again. 1. Introduction In their paper [1], J.D. Donald and F.J. Flanigan conjectured that any group algebra kG of a finite group G over a field k can be deformed to a semisimple algebra even in the modular case, namely where the order of G is not invertible in k. A more customary formulation of the Donald-Flanigan (DF) conjecture is by demanding that the deformed algebra [kG]t should be separable, i.e. it remains semisimple when tensored with the algebraic closure of its base field. If, additionally, the dimensions of the simple components of [kG]t are in one-to-one correspondence with those of the complex group algebra CG, then [kG]t is called a strong solution to the problem. The DF conjecture was solved for groups G which have either a cyclic p-Sylow subgroup over an algebraically closed field [11] or a normal abelian p-Sylow sub- group [5] where p =char(k), and for all but six reflection groups in any characteristic [6, 7, 10]. In [4], it is claimed that the group algebra kQ8, where Q8 = 〈σ, τ |σ 4 = 1, τσ = σ3τ, σ2 = τ2 〉 is the quaternion group of order 8 and k a field of characteristic 2, does not admit a separable deformation. This result allegedly gives a counterexample to the DF conjecture. However, as observed by M. Schaps, the proof apparently contains an error (see §7). The aim of this note is to present a separable deformation of kQ8, where k is any field of characteristic 2, reopening the DF conjecture. 2. Preliminaries Let k[[t]] be the ring of formal power series over k, and let k((t)) be its field of fractions. Recall that the deformed algebra [kG]t has the same underlying k((t))- vector space as k((t))⊗k kG, with multiplication defined on basis elements (2.1) g1 ∗ g2 := g1g2 + Ψi(g1, g2)t i, g1, g2 ∈ G Date: November 13, 2018. http://arxiv.org/abs/0704.1556v1 2 NURIT BARNEA AND YUVAL GINOSAR and extended k((t))-linearly (such that t is central). Here g1g2 is the group multi- plication. The functions Ψi : G×G → kG satisfy certain cohomological conditions induced by the associativity of [kG]t [3, §1 ; §2]. Note that the set of equations (2.1) determines a multiplication on the free k[[t]]-module Λt spanned by the elements {g}g∈G such that kG ≃ Λt/〈tΛt〉 and [kG]t ≃ Λt ⊗k[[t]] k((t)). In a more general context, namely over a domain R which is not necessarily local, the R-module Λt which determines the deformation, is required only to be flat rather than free [2, §1]. In what follows, we shall define the deformed algebra [kG]t by using generators and relations. These will implicitly determine the set of equations (2.1). 3. Sketch of the construction Consider the extension (3.1) [β] : 1 → C4 → Q8 → C2 → 1, where C2 = 〈 τ̄ 〉 acts on C4 = 〈σ 〉 by η : C2 → Aut(C4) η(τ̄ ) : σ 7→ σ3(= σ−1), and the associated 2-cocycle β : C2 × C2 → C4 is given by β(1, 1) = β(1, τ̄) = β(τ̄ , 1) = 1, β(τ̄ , τ̄) = σ2. The group algebra kQ8 (k any field) is isomorphic to the quotient kC4[y; η]/〈 q(y) 〉, where kC4[y; η] is a skew polynomial ring [9, §1.2], whose indeterminate y acts on the ring of coefficients kC4 via the automorphism η(τ̄ ) (extended linearly) and where (3.2) q(y) := y2 − σ2 ∈ kC4[y; η] is central. The above isomorphism is established by identifying τ with the indeter- minate y. Suppose now that Char(k) = 2. The deformed algebra [kQ8]t is constructed as follows. In §4.1 the subgroup algebra kC4 is deformed to a separable algebra [kC4]t which is isomorphic to K⊕k((t))⊕k((t)), where K is a separable field extension of k((t)) of degree 2. The next step (§4.2) is to construct an automorphism ηt of [kC4]t which agrees with the action of C2 on kC4 when specializing t = 0. This action fixes all three primitive idempotents of [kC4]t. By that we obtain the skew polynomial ring [kC4]t[y; ηt]. In §5 we deform q(y) = y2 + σ2 to qt(y), a separable polynomial of degree 2 in the center of [kC4]t[y; ηt]. By factoring out the two-sided ideal generated by qt(y), we establish the defor- mation [kQ8]t := [kC4]t[y; ηt]/〈 qt(y) 〉. In §6 we show that [kQ8]t as above is separable. Moreover, passing to the algebraic closure k((t)) we have [kQ8]t ⊗k((t)) k((t)) ≃ k((t))⊕M2(k((t))). A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 3 This is a strong solution to the DF conjecture since its decomposition to simple components is the same as CQ8 ≃ C⊕M2(C). 4. A Deformation of kC4[y; η] 4.1. We begin by constructing [kC4]t, C4 = 〈σ 〉. Recall that kC4 ≃ k[x]/〈x 4 + 1 〉 by identifying σ with x+ 〈x4+1 〉. We deform the polynomial x4+1 to a separable polynomial pt(x) as follows. Let k[[t]]∗ be the group of invertible elements of k[[t]] and denote by U := {1 + zt|z ∈ k[[t]]∗} its subgroup of 1-units (when k = F2, U is equal to k[[t]] a ∈ k[[t]] \ k[[t]]∗ be a non-zero element, and let b, c, d ∈ U, (c 6= d), such that π(x) := x2 + ax+ b is an irreducible (separable) polynomial in k((t))[x]. Let pt(x) := π(x)(x + c)(x + d) ∈ k((t))[x]. Then the quotient k((t))[x]/〈 pt(x) 〉 is isomorphic to the direct sum K ⊕ k((t)) ⊕ k((t)), where K := k((t))[x]/〈π(x) 〉. The field extension K/k((t)) is separable and of dimension 2. Note that pt=0(x) = x 4+1 and that only lower order terms of the polynomial were deformed. Hence, the quotient k[[t]][x]/〈 pt(x) 〉 is k[[t]]-free and k((t))[x]/〈 pt(x) 〉 indeed defines a deformation [kC4]t of kC4 ≃ k[x]/〈x 4+1 〉. The new multiplication σi∗σj of basis elements (2.1) is determined by identifying σi with x̄i := xi+〈 pt(x) 〉. We shall continue to use the term x̄ in [kC4]t rather than σ. Assume further that there exists w ∈ k[[t]] such that (4.1) (x + w)(x + c)(x+ d) = xπ(x) + a (see example 4.3). Then K ≃ ([kC4]t)e1, where (4.2) e1 = (x̄+ w)(x̄ + c)(x̄ + d) The two other primitive idempotents of [kC4]t are (4.3) e2 = c(x̄ + d)π(x̄) a(c+ d) , e3 = d(x̄ + c)π(x̄) a(c+ d) 4 NURIT BARNEA AND YUVAL GINOSAR 4.2. Let ηt : k((t))[x] → k((t))[x] be an algebra endomorphism determined by its value on the generator x as follows. (4.4) ηt(x) := xπ(x) + x+ a. We compute ηt(π(x)), ηt(x+ c) and ηt(x+ d): ηt(π(x)) = ηt(x) 2 + aηt(x) + b = x 2π(x)2 + x2 + a2 + axπ(x) + ax+ a2 + b = π(x)(x2π(x) + ax+ 1). By (4.1), (4.5) ηt(π(x)) = π(x) + x(x+ w)pt(x) ∈ 〈π(x) 〉. Next, ηt(x+ c) = xπ(x) + x+ a+ c. By (4.1), (4.6) ηt(x+ c) = (x+ c)[(x + w)(x + d) + 1] ∈ 〈x+ c 〉. Similarly, (4.7) ηt(x+ d) = (x+ d)[(x + w)(x + c) + 1] ∈ 〈x+ d 〉. By (4.5), (4.6) and (4.7), we obtain that ηt(pt(x)) ∈ 〈 pt(x) 〉, and hence ηt induces an endomorphism of k((t))[x]/〈 pt(x) 〉 which we continue to denote by ηt. As can easily be verified, the primitive idempotents given in (4.2) and (4.3) are fixed under (4.8) ηt(ei) = ei, i = 1, 2, 3, whereas (4.9) ηt(x̄e1) = ηt(x̄)e1 = (x̄π(x̄) + x̄+ a)e1 = (x̄+ a)e1. Hence, ηt induces an automorphism of K of order 2 while fixing the two copies of k((t)) pointwise. Furthermore, one can easily verify that ηt=0(x̄) = x̄ Consequently, the automorphism ηt of [kC4]t agrees with the automorphism η(τ̄ ) of kC4 when t = 0. The skew polynomial ring [kC4]t[y; ηt] = (k((t))[x]/〈 pt(x) 〉)[y; ηt] is therefore a deformation of kC4[y; η]. Note that by (4.8), the idempotents ei, i = 1, 2, 3 are central in [kC4]t[y; ηt] and hence (4.10) [kC4]t[y; ηt] = [kC4]t[y; ηt]ei. A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 5 4.3. Example. The following is an example for the above construction. t+ t2 + t3 1 + t , b := 1 + t2 + t3, c := 1 + t , d := 1 + t+ t2, w := t. These elements satisfy equation (4.1): (x+ w)(x+ c)(x + d) = (x+ t)(x + 1 + t )(x+ 1 + t+ t2) = x3 + t+ t2 + t3 1 + t x2 + (1 + t2 + t3)x + t+ t2 + t3 1 + t = xπ(x) + a. The polynomial π(x) = x2 + t+ t2 + t3 1 + t x+ 1 + t2 + t3 does not admit roots in k[[t]]/〈 t2 〉, thus it is irreducible over k((t)). 5. A Deformation of q(y) The construction of [kQ8]t will be completed once the product τ̄ ∗ τ̄ is defined. For this purpose the polynomial q(y) (3.2), which determined the ordinary multi- plication τ2, will now be developed in powers of t. For any non-zero element z ∈ k[[t]] \ k[[t]]∗, let (5.1) qt(y) := y 2 + zx̄π(x̄)y + x̄2 + ax̄ ∈ [kC4]t[y; ηt]. Decomposition of (5.1) with respect to the idempotents e1, e2, e3 yields (5.2) qt(y) = (y 2 + b)e1 + [y 2 + zay + c(c+ a)]e2 + [y 2 + zay + d(d+ a)]e3. We now show that qt(y) is in the center of [kC4]t[y; ηt] : First, the leading term y2 is central since the automorphism ηt is of order 2. Next, by (4.8), the free term be1 + c(c + a)e2 + d(d + a)e3 is invariant under the action of ηt and hence central. It is left to check that the term za(e2 + e3)y is central. Indeed, since e2 and e3 are ηt-invariant, then za(e2 + e3)y commutes both with [kC4]t[y; ηt]e2 and [kC4]t[y; ηt]e3. Furthermore, by orthogonality za(e2 + e3)y · [kC4]t[y; ηt]e1 = [kC4]t[y; ηt]e1 · za(e2 + e3)y = 0, and hence za(e2 + e3)y commutes with [kC4]t[y; ηt]. Consequently, 〈 qt(y) 〉 = qt(y)[kC4]t[y; ηt] is a two-sided ideal. Now, as can easily be deduced from (5.1), (5.3) qt=0(y) = y 2 + x̄2 = q(y), where the leading term y2 remains unchanged. Then [kQ8]t := [kC4]t[y; ηt]/〈 qt(y) 〉 is a deformation of kQ8, identifying τ̄ with ȳ := y + 〈 qt(y) 〉. 6 NURIT BARNEA AND YUVAL GINOSAR 6. Separability of [kQ8]t Finally, we need to prove that the deformed algebra [kQ8]t is separable. More- over, we prove that its decomposition to simple components over the algebraic closure of k((t)) resembles that of CQ8. By (4.10), we obtain (6.1) [kQ8]t = [kC4]t[y; ηt]ei/〈 qt(y)ei 〉. We handle the three summands in (6.1) separately: By (5.2), [kC4]t[y; ηt]e1/〈 qt(y)e1 〉 ≃ K[y; ηt]/〈 y 2 + b 〉 ≃ Kf ∗ C2. The rightmost term is the crossed product of the group C2 := 〈 τ̄ 〉 acting faithfully on the field K = [kC4]te1 via ηt (4.9), with a twisting determined by the 2-cocycle f : C2 × C2 → K f(1, 1) = f(1, τ̄) = f(τ̄ , 1) = 1, f(τ̄ , τ̄ ) = b. This is a central simple algebra over the subfield of invariants k((t)) [8, Theorem 4.4.1]. Evidently, this simple algebra is split by k((t)), i.e. (6.2) [kC4]t[y; ηt]e1/〈 qt(y)e1 〉 ⊗k((t)) k((t)) ≃ M2(k((t))). Next, since ηt is trivial on [kC4]te2, the skew polynomial ring [kC4]te2[y; ηt] is actually an ordinary polynomial ring k((t))[y]. Again by (5.2), [kC4]t[y; ηt]e2/〈 qt(y)e2 〉 ≃ k((t))[y]/〈 y 2 + zay + c(c+ a) 〉. Similarly, [kC4]t[y; ηt]e3/〈 qt(y)e3 〉 ≃ k((t))[y]/〈 y 2 + zay + d(d+ a) 〉. The polynomials y2 + zay + c(c+ a) and y2 + zay + d(d + a) are separable (since za is non-zero). Thus, both [kC4]t[y; ηt]e2/〈 qt(y)e2 〉 and [kC4]t[y; ηt]e3/〈 qt(y)e3 〉 are separable k((t))-algebras, and for i = 2, 3 (6.3) [kC4]t[y; ηt]ei/〈 qt(y)ei 〉 ⊗k((t)) k((t)) ≃ k((t)) ⊕ k((t)). Equations (6.1), (6.2) and (6.3) yield [kQ8]t ⊗k((t)) k((t)) ≃ k((t)) ⊕M2(k((t))) as required. 7. Acknowledgement We wish to thank M. Schaps for pointing out to us that there is an error in the attempted proof in [4] that the quaternion group is a counterexample to the DF conjecture. Here is her explanation: The given relations for the group algebra are incorrect. Using the notation in pages 166-7 of [4], if a = 1 + i, b = 1 + j and z = i2 = j2, then ab + ba = ij(1 + z) while a2 = b2 = 1 + z. There is a further error later on when the matrix algebra is deformed to four copies of the field, since a non-commutative algebra can never have a flat deformation to a commutative algebra. A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 7 References [1] J.D. Donald and F.J. Flanigan, A deformation-theoretic version of Maschke’s theorem for modular group algebras: the commutative case, J. Algebra 29 (1974), 98–102. [2] K. Erdmann and M. Schaps, Deformation of tame blocks and related algebras, in: Quantum deformations of algebras and their representations, Israel Math. Conf. Proc., 7, (1993), 25–44. [3] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1964), 59–103. [4] M. Gerstenhaber and A. Giaquinto, Compatible deformations, Contemp. Math. 229, (1998), 159–168. [5] M. Gerstenhaber and M.E. Schaps, The modular version of Maschke’s theorem for normal abelian p-Sylows, J. Pure Appl. Algebra 108 (1996), no. 3, 257–264 [6] M. Gerstenhaber and M.E. Schaps, Hecke algebras, Uqsln, and the Donald-Flanigan conjecture for Sn, Trans. Amer. Math. Soc. 349 (1997), no. 8, 3353–3371. [7] M. Gerstenhaber, A. Giaquinto and M.E. Schaps, The Donald-Flanigan problem for finite reflection groups, Lett. Math. Phys. 56 (2001), no. 1, 41–72. [8] I. N. Herstein, Noncommutative rings, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York 1968. [9] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, John Wiley & Sons, Ltd., Chichester, 1987. [10] M. Peretz and M. Schaps, Hecke algebras and separable deformations of dihedral groups, Far East J. Math. Sci. (FJMS) 1 (1999), no. 1, 17–26. [11] M. Schaps, A modular version of Maschke’s theorem for groups with cyclic p-Sylow subgroups, J. Algebra 163 (1994), no. 3, 623–635. Department of Mathematics, University of Haifa, Haifa 31905, Israel E-mail address: ginosar@math.haifa.ac.il 1. Introduction 2. Preliminaries 3. Sketch of the construction 4. A Deformation of kC4[y;] 4.1. 4.2. 4.3. Example 5. A Deformation of q(y) 6. Separability of [kQ8]t 7. Acknowledgement References

The Donald-Flanigan conjecture asserts that for any finite group and for any field, the corresponding group algebra can be deformed to a separable algebra. The minimal unsolved instance, namely the quaternion group over a field of characteristic 2 was considered as a counterexample. We present here a separable deformation of the quaternion group algebra. In a sense, the conjecture for any finite group is open again.

Introduction In their paper [1], J.D. Donald and F.J. Flanigan conjectured that any group algebra kG of a finite group G over a field k can be deformed to a semisimple algebra even in the modular case, namely where the order of G is not invertible in k. A more customary formulation of the Donald-Flanigan (DF) conjecture is by demanding that the deformed algebra [kG]t should be separable, i.e. it remains semisimple when tensored with the algebraic closure of its base field. If, additionally, the dimensions of the simple components of [kG]t are in one-to-one correspondence with those of the complex group algebra CG, then [kG]t is called a strong solution to the problem. The DF conjecture was solved for groups G which have either a cyclic p-Sylow subgroup over an algebraically closed field [11] or a normal abelian p-Sylow sub- group [5] where p =char(k), and for all but six reflection groups in any characteristic [6, 7, 10]. In [4], it is claimed that the group algebra kQ8, where Q8 = 〈σ, τ |σ 4 = 1, τσ = σ3τ, σ2 = τ2 〉 is the quaternion group of order 8 and k a field of characteristic 2, does not admit a separable deformation. This result allegedly gives a counterexample to the DF conjecture. However, as observed by M. Schaps, the proof apparently contains an error (see §7). The aim of this note is to present a separable deformation of kQ8, where k is any field of characteristic 2, reopening the DF conjecture. 2. Preliminaries Let k[[t]] be the ring of formal power series over k, and let k((t)) be its field of fractions. Recall that the deformed algebra [kG]t has the same underlying k((t))- vector space as k((t))⊗k kG, with multiplication defined on basis elements (2.1) g1 ∗ g2 := g1g2 + Ψi(g1, g2)t i, g1, g2 ∈ G Date: November 13, 2018. http://arxiv.org/abs/0704.1556v1 2 NURIT BARNEA AND YUVAL GINOSAR and extended k((t))-linearly (such that t is central). Here g1g2 is the group multi- plication. The functions Ψi : G×G → kG satisfy certain cohomological conditions induced by the associativity of [kG]t [3, §1 ; §2]. Note that the set of equations (2.1) determines a multiplication on the free k[[t]]-module Λt spanned by the elements {g}g∈G such that kG ≃ Λt/〈tΛt〉 and [kG]t ≃ Λt ⊗k[[t]] k((t)). In a more general context, namely over a domain R which is not necessarily local, the R-module Λt which determines the deformation, is required only to be flat rather than free [2, §1]. In what follows, we shall define the deformed algebra [kG]t by using generators and relations. These will implicitly determine the set of equations (2.1). 3. Sketch of the construction Consider the extension (3.1) [β] : 1 → C4 → Q8 → C2 → 1, where C2 = 〈 τ̄ 〉 acts on C4 = 〈σ 〉 by η : C2 → Aut(C4) η(τ̄ ) : σ 7→ σ3(= σ−1), and the associated 2-cocycle β : C2 × C2 → C4 is given by β(1, 1) = β(1, τ̄) = β(τ̄ , 1) = 1, β(τ̄ , τ̄) = σ2. The group algebra kQ8 (k any field) is isomorphic to the quotient kC4[y; η]/〈 q(y) 〉, where kC4[y; η] is a skew polynomial ring [9, §1.2], whose indeterminate y acts on the ring of coefficients kC4 via the automorphism η(τ̄ ) (extended linearly) and where (3.2) q(y) := y2 − σ2 ∈ kC4[y; η] is central. The above isomorphism is established by identifying τ with the indeter- minate y. Suppose now that Char(k) = 2. The deformed algebra [kQ8]t is constructed as follows. In §4.1 the subgroup algebra kC4 is deformed to a separable algebra [kC4]t which is isomorphic to K⊕k((t))⊕k((t)), where K is a separable field extension of k((t)) of degree 2. The next step (§4.2) is to construct an automorphism ηt of [kC4]t which agrees with the action of C2 on kC4 when specializing t = 0. This action fixes all three primitive idempotents of [kC4]t. By that we obtain the skew polynomial ring [kC4]t[y; ηt]. In §5 we deform q(y) = y2 + σ2 to qt(y), a separable polynomial of degree 2 in the center of [kC4]t[y; ηt]. By factoring out the two-sided ideal generated by qt(y), we establish the defor- mation [kQ8]t := [kC4]t[y; ηt]/〈 qt(y) 〉. In §6 we show that [kQ8]t as above is separable. Moreover, passing to the algebraic closure k((t)) we have [kQ8]t ⊗k((t)) k((t)) ≃ k((t))⊕M2(k((t))). A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 3 This is a strong solution to the DF conjecture since its decomposition to simple components is the same as CQ8 ≃ C⊕M2(C). 4. A Deformation of kC4[y; η] 4.1. We begin by constructing [kC4]t, C4 = 〈σ 〉. Recall that kC4 ≃ k[x]/〈x 4 + 1 〉 by identifying σ with x+ 〈x4+1 〉. We deform the polynomial x4+1 to a separable polynomial pt(x) as follows. Let k[[t]]∗ be the group of invertible elements of k[[t]] and denote by U := {1 + zt|z ∈ k[[t]]∗} its subgroup of 1-units (when k = F2, U is equal to k[[t]] a ∈ k[[t]] \ k[[t]]∗ be a non-zero element, and let b, c, d ∈ U, (c 6= d), such that π(x) := x2 + ax+ b is an irreducible (separable) polynomial in k((t))[x]. Let pt(x) := π(x)(x + c)(x + d) ∈ k((t))[x]. Then the quotient k((t))[x]/〈 pt(x) 〉 is isomorphic to the direct sum K ⊕ k((t)) ⊕ k((t)), where K := k((t))[x]/〈π(x) 〉. The field extension K/k((t)) is separable and of dimension 2. Note that pt=0(x) = x 4+1 and that only lower order terms of the polynomial were deformed. Hence, the quotient k[[t]][x]/〈 pt(x) 〉 is k[[t]]-free and k((t))[x]/〈 pt(x) 〉 indeed defines a deformation [kC4]t of kC4 ≃ k[x]/〈x 4+1 〉. The new multiplication σi∗σj of basis elements (2.1) is determined by identifying σi with x̄i := xi+〈 pt(x) 〉. We shall continue to use the term x̄ in [kC4]t rather than σ. Assume further that there exists w ∈ k[[t]] such that (4.1) (x + w)(x + c)(x+ d) = xπ(x) + a (see example 4.3). Then K ≃ ([kC4]t)e1, where (4.2) e1 = (x̄+ w)(x̄ + c)(x̄ + d) The two other primitive idempotents of [kC4]t are (4.3) e2 = c(x̄ + d)π(x̄) a(c+ d) , e3 = d(x̄ + c)π(x̄) a(c+ d) 4 NURIT BARNEA AND YUVAL GINOSAR 4.2. Let ηt : k((t))[x] → k((t))[x] be an algebra endomorphism determined by its value on the generator x as follows. (4.4) ηt(x) := xπ(x) + x+ a. We compute ηt(π(x)), ηt(x+ c) and ηt(x+ d): ηt(π(x)) = ηt(x) 2 + aηt(x) + b = x 2π(x)2 + x2 + a2 + axπ(x) + ax+ a2 + b = π(x)(x2π(x) + ax+ 1). By (4.1), (4.5) ηt(π(x)) = π(x) + x(x+ w)pt(x) ∈ 〈π(x) 〉. Next, ηt(x+ c) = xπ(x) + x+ a+ c. By (4.1), (4.6) ηt(x+ c) = (x+ c)[(x + w)(x + d) + 1] ∈ 〈x+ c 〉. Similarly, (4.7) ηt(x+ d) = (x+ d)[(x + w)(x + c) + 1] ∈ 〈x+ d 〉. By (4.5), (4.6) and (4.7), we obtain that ηt(pt(x)) ∈ 〈 pt(x) 〉, and hence ηt induces an endomorphism of k((t))[x]/〈 pt(x) 〉 which we continue to denote by ηt. As can easily be verified, the primitive idempotents given in (4.2) and (4.3) are fixed under (4.8) ηt(ei) = ei, i = 1, 2, 3, whereas (4.9) ηt(x̄e1) = ηt(x̄)e1 = (x̄π(x̄) + x̄+ a)e1 = (x̄+ a)e1. Hence, ηt induces an automorphism of K of order 2 while fixing the two copies of k((t)) pointwise. Furthermore, one can easily verify that ηt=0(x̄) = x̄ Consequently, the automorphism ηt of [kC4]t agrees with the automorphism η(τ̄ ) of kC4 when t = 0. The skew polynomial ring [kC4]t[y; ηt] = (k((t))[x]/〈 pt(x) 〉)[y; ηt] is therefore a deformation of kC4[y; η]. Note that by (4.8), the idempotents ei, i = 1, 2, 3 are central in [kC4]t[y; ηt] and hence (4.10) [kC4]t[y; ηt] = [kC4]t[y; ηt]ei. A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 5 4.3. Example. The following is an example for the above construction. t+ t2 + t3 1 + t , b := 1 + t2 + t3, c := 1 + t , d := 1 + t+ t2, w := t. These elements satisfy equation (4.1): (x+ w)(x+ c)(x + d) = (x+ t)(x + 1 + t )(x+ 1 + t+ t2) = x3 + t+ t2 + t3 1 + t x2 + (1 + t2 + t3)x + t+ t2 + t3 1 + t = xπ(x) + a. The polynomial π(x) = x2 + t+ t2 + t3 1 + t x+ 1 + t2 + t3 does not admit roots in k[[t]]/〈 t2 〉, thus it is irreducible over k((t)). 5. A Deformation of q(y) The construction of [kQ8]t will be completed once the product τ̄ ∗ τ̄ is defined. For this purpose the polynomial q(y) (3.2), which determined the ordinary multi- plication τ2, will now be developed in powers of t. For any non-zero element z ∈ k[[t]] \ k[[t]]∗, let (5.1) qt(y) := y 2 + zx̄π(x̄)y + x̄2 + ax̄ ∈ [kC4]t[y; ηt]. Decomposition of (5.1) with respect to the idempotents e1, e2, e3 yields (5.2) qt(y) = (y 2 + b)e1 + [y 2 + zay + c(c+ a)]e2 + [y 2 + zay + d(d+ a)]e3. We now show that qt(y) is in the center of [kC4]t[y; ηt] : First, the leading term y2 is central since the automorphism ηt is of order 2. Next, by (4.8), the free term be1 + c(c + a)e2 + d(d + a)e3 is invariant under the action of ηt and hence central. It is left to check that the term za(e2 + e3)y is central. Indeed, since e2 and e3 are ηt-invariant, then za(e2 + e3)y commutes both with [kC4]t[y; ηt]e2 and [kC4]t[y; ηt]e3. Furthermore, by orthogonality za(e2 + e3)y · [kC4]t[y; ηt]e1 = [kC4]t[y; ηt]e1 · za(e2 + e3)y = 0, and hence za(e2 + e3)y commutes with [kC4]t[y; ηt]. Consequently, 〈 qt(y) 〉 = qt(y)[kC4]t[y; ηt] is a two-sided ideal. Now, as can easily be deduced from (5.1), (5.3) qt=0(y) = y 2 + x̄2 = q(y), where the leading term y2 remains unchanged. Then [kQ8]t := [kC4]t[y; ηt]/〈 qt(y) 〉 is a deformation of kQ8, identifying τ̄ with ȳ := y + 〈 qt(y) 〉. 6 NURIT BARNEA AND YUVAL GINOSAR 6. Separability of [kQ8]t Finally, we need to prove that the deformed algebra [kQ8]t is separable. More- over, we prove that its decomposition to simple components over the algebraic closure of k((t)) resembles that of CQ8. By (4.10), we obtain (6.1) [kQ8]t = [kC4]t[y; ηt]ei/〈 qt(y)ei 〉. We handle the three summands in (6.1) separately: By (5.2), [kC4]t[y; ηt]e1/〈 qt(y)e1 〉 ≃ K[y; ηt]/〈 y 2 + b 〉 ≃ Kf ∗ C2. The rightmost term is the crossed product of the group C2 := 〈 τ̄ 〉 acting faithfully on the field K = [kC4]te1 via ηt (4.9), with a twisting determined by the 2-cocycle f : C2 × C2 → K f(1, 1) = f(1, τ̄) = f(τ̄ , 1) = 1, f(τ̄ , τ̄ ) = b. This is a central simple algebra over the subfield of invariants k((t)) [8, Theorem 4.4.1]. Evidently, this simple algebra is split by k((t)), i.e. (6.2) [kC4]t[y; ηt]e1/〈 qt(y)e1 〉 ⊗k((t)) k((t)) ≃ M2(k((t))). Next, since ηt is trivial on [kC4]te2, the skew polynomial ring [kC4]te2[y; ηt] is actually an ordinary polynomial ring k((t))[y]. Again by (5.2), [kC4]t[y; ηt]e2/〈 qt(y)e2 〉 ≃ k((t))[y]/〈 y 2 + zay + c(c+ a) 〉. Similarly, [kC4]t[y; ηt]e3/〈 qt(y)e3 〉 ≃ k((t))[y]/〈 y 2 + zay + d(d+ a) 〉. The polynomials y2 + zay + c(c+ a) and y2 + zay + d(d + a) are separable (since za is non-zero). Thus, both [kC4]t[y; ηt]e2/〈 qt(y)e2 〉 and [kC4]t[y; ηt]e3/〈 qt(y)e3 〉 are separable k((t))-algebras, and for i = 2, 3 (6.3) [kC4]t[y; ηt]ei/〈 qt(y)ei 〉 ⊗k((t)) k((t)) ≃ k((t)) ⊕ k((t)). Equations (6.1), (6.2) and (6.3) yield [kQ8]t ⊗k((t)) k((t)) ≃ k((t)) ⊕M2(k((t))) as required. 7. Acknowledgement We wish to thank M. Schaps for pointing out to us that there is an error in the attempted proof in [4] that the quaternion group is a counterexample to the DF conjecture. Here is her explanation: The given relations for the group algebra are incorrect. Using the notation in pages 166-7 of [4], if a = 1 + i, b = 1 + j and z = i2 = j2, then ab + ba = ij(1 + z) while a2 = b2 = 1 + z. There is a further error later on when the matrix algebra is deformed to four copies of the field, since a non-commutative algebra can never have a flat deformation to a commutative algebra. A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 7 References [1] J.D. Donald and F.J. Flanigan, A deformation-theoretic version of Maschke’s theorem for modular group algebras: the commutative case, J. Algebra 29 (1974), 98–102. [2] K. Erdmann and M. Schaps, Deformation of tame blocks and related algebras, in: Quantum deformations of algebras and their representations, Israel Math. Conf. Proc., 7, (1993), 25–44. [3] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1964), 59–103. [4] M. Gerstenhaber and A. Giaquinto, Compatible deformations, Contemp. Math. 229, (1998), 159–168. [5] M. Gerstenhaber and M.E. Schaps, The modular version of Maschke’s theorem for normal abelian p-Sylows, J. Pure Appl. Algebra 108 (1996), no. 3, 257–264 [6] M. Gerstenhaber and M.E. Schaps, Hecke algebras, Uqsln, and the Donald-Flanigan conjecture for Sn, Trans. Amer. Math. Soc. 349 (1997), no. 8, 3353–3371. [7] M. Gerstenhaber, A. Giaquinto and M.E. Schaps, The Donald-Flanigan problem for finite reflection groups, Lett. Math. Phys. 56 (2001), no. 1, 41–72. [8] I. N. Herstein, Noncommutative rings, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York 1968. [9] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, John Wiley & Sons, Ltd., Chichester, 1987. [10] M. Peretz and M. Schaps, Hecke algebras and separable deformations of dihedral groups, Far East J. Math. Sci. (FJMS) 1 (1999), no. 1, 17–26. [11] M. Schaps, A modular version of Maschke’s theorem for groups with cyclic p-Sylow subgroups, J. Algebra 163 (1994), no. 3, 623–635. Department of Mathematics, University of Haifa, Haifa 31905, Israel E-mail address: ginosar@math.haifa.ac.il 1. Introduction 2. Preliminaries 3. Sketch of the construction 4. A Deformation of kC4[y;] 4.1. 4.2. 4.3. Example 5. A Deformation of q(y) 6. Separability of [kQ8]t 7. Acknowledgement References

arXiv:0704.1557v1 [math.RA] 12 Apr 2007 On the residue fields of Henselian valued stable fields, II I.D. Chipchakov ∗ Introduction This paper is a continuation of [Ch2]. Let E be a field, Esep a separable closure of E , E∗ the multiplicative group of E , d(E) the class of finite-dimensional central division E -algebras, Br (E) the Brauer group of E , Gal (E) the set of finite Galois extensions of E in Esep , and NG (E) the set of norm groups N(M/E): M ∈ Gal(E) . Let also P be the set of prime numbers, Br (E)p the p -component of Br (E) , for each p ∈ P , P(E) the set of those p ∈ P , for which E is properly included in its maximal p -extension E(p) (in Esep ), and Π(E) the set of all p ′ ∈ P , for which the absolute Galois group GE = G(Esep/E) is of nonzero cohomological p ′ -dimension cd p′(GE) . Recall that E is p -quasilocal, for a given p ∈ P , if p 6∈ P(E) or the relative Brauer group Br (F/E) equals the subgroup pBr(E) = {bp ∈ Br(E): pbp = 0} , for every cyclic extension F of E of degree p . The field E is said to be primarily quasilocal (PQL), if it is p -quasilocal, for all p ∈ P ; E is called quasilocal, if its finite extensions are PQL. We say that E is a strictly PQL-field, if it is PQL and Br (E)p 6= {0} , for every p ∈ P(E) (see [Ch4, Corollary 3.7], for a characterization of the SQL-property). The purpose of this note is to complement the main result of [Ch2] and to shed light on the structure of Br (K) , where K is an absolutely stable field (in the sense of E. Brussel) with a Henselian valuation whose value group is totally indivisible. 1. Subfields of central division algebras over PQL-fields Assume that E is a PQL-field and M ∈ Gal(E) , such that G(M/E) is nilpotent, and let R be an intermediate field of M/E . It follows from Galois theory, [Ch2, Theorem 4.1 (iii)], the Burnside-Wielandt characterization of nilpotent finite groups, and the general properties of central division algebras and their Schur indices (see [KM, Ch. 6, Sect. 2] and [P, Sects. 13.4 and 14.4]) that R embeds as an E -subalgebra in each ∗ Partially supported by Grant MI-1503/2005 of the Bulgarian Foundation for Scien- tific Research. http://arxiv.org/abs/0704.1557v1 algebra D ∈ d(E) of index divisible by [R: E] . In this Section, we demonstrate the optimality of this result in the class of strictly PQL-fields. Theorem 1.1. For each nonnilpotent finite group G , there exists a strictly PQL- field E = E(G) with Br (E) ∼= Q/Z and Gal (E) containing an element M such that G(M/E) ∼= G and M does not embed as an E -subalgebra in any ∆ ∈ d(E) of index equal to [M:E] . Proof. Let P(G) be the set of prime divisors of the order of G . Our argument relies on the existence (cf. [Ch1, Sect. 4]) of an algebraic number field E0 , such that there is M0 ∈ Gal(E0) with G(M0/E0) ∼= G , and E0 possesses a system of (real-valued) valuations {w(p): p ∈ P(G)} satisfying the following conditions: (1.1) (i) w(p) extends the normalized p -adic valuation (in the sense of [CF, Ch. VII]) of the field Q of rational numbers, for each p ∈ P(G) ; (ii) The completion M0,w(p)′ lies in Gal (E0,w(p)) and G(M0,w(p)′/E0,w(p)) is isomor- phic to a Sylow p -subgroup of G(M0/E0) , whenever p ∈ P(G) and w(p)′ extends w(p) on M0 . Let Ap be the maximal abelian p -extension of E0 in M0 , for each p ∈ P(G) . It follows from the nonnilpotency of G , the Burnside-Wielandt theorem and Galois theory that M0/E0 has an intermediate field F 6= E0 such that F ∩ Ap = E , for all p ∈ P(G) (see [Ch1, Sect. 4]). Fix a prime divisor π of [F: E0] as well as Sylow π -subgroups Hπ ∈ SylπG(M0/F) and Gπ ∈ SylπG(M0/E0) with Hπ ⊂ Gπ , and denote by Fπ and Eπ the maximal extensions of E0 in M0 fixed by Hπ and Gπ , respectively. It is clear from (1.1) (ii) and [CF, Ch. II, Theorem 10.2] that Eπ has a valuation t(π) extending w(π) so that M0 ⊗Eπ Eπ,t(π) is a field. This means that t(π) is uniquely (up-to an equivalence) extendable to a valuation µ(π) of M0 , and allows us to identify E0,w(π) with Eπ,t(π) . Using repeatedly the Grunwald-Wang theorem [W] and the normality of maximal subgroups of finite π -groups (cf. [L, Ch. I, Sect. 6]), one also obtains that there is a finite extension K of E0 in E0(π) , such that K⊗E0 E0,w(π) is a field isomorphic to Fπ,ν(π) as an E0 -algebra, where ν(π) is the valuation of Fπ induced by µ(π) . These observations indicate that K ∩M0 = E0 , i.e. the compositum M0K lies in Gal (K) and G((M0K)/K) ∼= G(M0/E0) . At the same time, our argument proves that F is dense in Fπ,ν(p) with respect to the topology induced by ν(p) , K has a unique valuation κ(π) extending w(π) , and (M0K)t(π)′ ∈ Gal(Kκ(π)) with G((M0K)t(π)′/Kκ(π)) ∼= Hπ , for any prolongation t(π)′ of t(π) on M0K . Let now M(K) = {κ(p): p ∈ P} be a system of valuations of K complementing κ(π) so that κ(p) extends w(p) or the normalized p -adic valuation of Q , depending on whether or not p ∈ (P(G) \ {π}) . Applying [Ch3, Theorem 2.2] to K , P and M(K) , and using [Ch3, Lemmas 3.1 and 3.2], one proves the existence of an extension E of K in E0,sep , such that P(E) = P , M0E := M lies in Gal (E) , G(M/E) ∼= G , and E possesses a system {v(p): p ∈ P} of valuations with the following properties: (1.2) For each p ∈ P , v(p) extends κ(p) , Ev(p) is a PQL-field, Ev(p)(p) is E - isomorphic to E(p)⊗E Ev(p) , Br (Ev(p))p 6= {0} and Br (Ev(p′))p = {0} , for every p′ ∈ (P \ {p}) . Moreover, if p ∈ P(G) and v(p)′ is a valuation of M extending v(p) , then Mv(p)′ ∈ Gal(Ev(p)) , G(Mv(p)′/Ev(p)) ∼= G(M0,w(p)′/E0,w(p)): p 6= π , and G(Mv(π)′/Ev(π)) ∼= Hπ . Hence, by [Ch3, Theorem 2.1], E is a nonreal strictly PQL-field. As Ev(π) is PQL, it follows from [Ch2, Theorem 4.1], statements (1.2) and the Brauer-Hasse-Noether and Albert theorem (in the form of [Ch3, Proposition 1.2]) that M splits an algebra ∆ ∈ d(E) of π -primary dimension if and only if ind (∆) divides the order of Hπ . In view of the general theory of simple algebras (see [P, Sects. 13.4 and 14.4]), this proves Theorem 1.1. 2. Divisible and reduced parts of the multiplicative group of an SQL-field A field E is said to be strictly quasilocal (SQL), if its finite extensions are strictly PQL. When this holds and E is almost perfect (in the sense of [Ch2, I, (1.8)]), this Section gives a Galois-theoretic characterization of the maximal divisible subgroup D(E) of E∗ . Theorem 2.1. Let E be an almost perfect SQL-field and N1(E) the intersection of the groups from NG (E) . Then N1(E) = D(E To prove Theorem 2.1 we need the following lemma. Lemma 2.2. Let E and M be fields, M ∈ Gal(E) , and let P(M/E) be the set of prime divisors of [M:E] . Then N(M/E) ⊆ N(M/F) , for every intermediate field F of M/E . Moreover, if Ep is the fixed field of some Sylow p -subgroup of G(M/E) , then N(M/E) = ∩p∈P(M/E)N(M/Ep) . Proof. It is easily verified that if [F: E] = m , {τj : j = 1, . . . ,m} is the set of E - embeddings of F into M , and σu is an automorphism of M extending τu , for each index j , then G(M/E) = ∪mj=1G(M/F)σ j . This yields N E (γ) = N j=1 σ j (γ)) , for any γ ∈ M∗ , which proves that N(M/F) ⊆ N(M/E) . We show that ∩p∈P(M/E) N(M/Ep) := N0(M/E) ⊆ N(M/E) . Take an element α ∈ N0(M/E) and put [Ep: E] = mp , for each p ∈ P(M/E) . It is known that p 6 |mp , for any p ∈ P(M/E) , which implies consecutively that g.c.d.(mp: p ∈ P(M/E)) = 1 , α ∈ E∗ and αmp ∈ N(M/E) , for every p ∈ P(M/E) . Thus it turns out that α ∈ N(M/E) , so Lemma 2.2 is proved. Proof of Theorem 2.1. The inclusion D(E∗) ⊆ N1(E) is obvious, so our objective is to prove the converse. We show that N1(E) ⊆ N1(E)p , for every p ∈ P and n ∈ N . It is clearly sufficient to consider the special case where p ∈ Π(E) . Then there is a finite extension Fp of E in Esep , such that Br (Fp)p 6= {0} and p 6 |[Fp: E] . Note also that each finite extension of Fp in E ∗ is included in a field F̃p ∈ Gal(E) , so it follows from Lemma 2.2 that N1(E) ⊆ N1(Fp) . At the same time, the norm mapping E clearly induces homomorphisms D(F p) → D(E∗) and N1(Fp) → Np(E) (for the latter, see [L, Ch. VIII, Sect. 5]) as well as an automorphism of the maximal p -divisible subgroup of E∗ . These observations allow one to assume additionally that p ∈ P(E) (and Br (E)p 6= {0} ). If p = char(E) , our assertion is implied by [Ch2, Lemma 8.4], so we turn to the case of p 6= char(E) . Let εp be a primitive p -th root of unity in Esep . Then [E(εp): E] divides p− 1 (cf. [L, Ch. VIII, Sect. 3]), whence our considerations reduce further to the case in which εp ∈ E . Now the proof of Theorem 2.1 is completed by applying the following two lemmas. Before stating them, let us recall that the character group C(Y/E) of G(Y/E) is an abelian torsion group, for every Galois extension Y/E (see [K, Ch. 7, Sect. 5]). Therefore, by [F, Theorem 24.5], the divisible part D(Y/E) of C(Y/E) is a direct summand in C(Y/E) , i.e. C(Y/E) is isomorphic to the direct sum D(Y/E)⊕ R(Y/E) , where R(Y/E) ∼= C(Y/E)/D(Y/E) is a maximal reduced subgroup of C(Y/E) . Lemma 2.3. Let E be a p -quasilocal field containing a primitive p -th root of unity ε , for some prime p ∈ P(E) . Assume also that rp(E) is the group of roots of unity in E of p -primary degrees. Then: (i) R(E(p)/E) = {0} if and only if Br (E)p = {0} or rp(E) is infinite; (ii) If Br (E)p 6= {0} , d is the dimension of pBr(E) as a vector space over the field Fp with p elements, and Rp(E) is of order p µ , for some µ ∈ N , then D(E(p)/E) = pµC(E(p)/E) and R(E(p)/E) is presentable as a direct sum of cyclic groups of order pµ , indexed by a set of cardinality d . Proof. Statement (i) is a well-known consequence of Kummer theory, the Merkurjev- Suslin theorem and Galois cohomology (see [MS, (11.5), (16.1)] and [S, Ch. I, 3.4 and 4.2]), so we assume further that d > 0 and rp(E) is of order p µ , for some µ ∈ N . We first show that pµC(E(p)/E) is divisible. Fix a primitive pµ -th root of unity εµ in E and a subset S = {εn: n ∈ N, n ≥ µ+ 1} of E(p) so that εpn+1 = εn , for each index n . Also, let E∞ = E(S) and En = E(εn) , for every integer n > µ . Suppose first that p > 2 or µ ≥ 2 . Then E∞/E is a Zp -extension and En is the unique subextension of E in E∞ of degree p n−µ . Since Zp is a projective profinite group (see [G, Theorem 1]), this enables one to deduce from Galois theory that E(p) possesses a subfield E′ such that E′ ∩ E∞ = E and E∞E′ = E(p) . In addition, it becomes clear that C(E(p)/E) is isomorphic to the direct sum Z(p∞)⊕ C(F/E) , where Z(p∞) is the quasicyclic p -group (identified with the character group of Zp ) and F is the maximal abelian extension of E in E′ . Let now Φ be a cyclic extension of E in F . As E is p -quasilocal, an element βn ∈ E∗n lies in N((ΦEn)/En) , for an arbitrary integer n ≥ µ , if and only if the norm NEnE (βn) ∈ N(Φ/E) . In particular, this proves that εµ ∈ N(Φ/E) if and only if εn ∈ N((ΦEn)/En) . In view of the results of [FSS, Sect. 2], this means that pµC(F/E) = pnC(F/E) , for every index n ≥ µ . In other words, C(F/E) is divisible, so pµC(E(p)/E) has the same property, as claimed. Our objective now is to prove the divisibility of 2C(E(2)/E) , under the hypothesis that p = 2 and µ = 1 . If E∞ = E( −1) = E2 , this is contained in [FSS, Proposi- tion 2], so we assume further that E∞ 6= E2 , i.e. E2 = Eν 6= Eν+1 , for some in- teger ν ≥ 2 . Let R be a cyclic extension of E in E(2) . By Albert’s theorem (see [FSS, Sect. 2]), we have C(R/E) ⊂ 2C(E(2)/E) if and only if −1 ∈ N(R/E) . We show that if C(R/E) ⊂ 2C(E(2)/E) , then εν ∈ N((RE2)/E2) . Since NE2E (εν) equals 1 or −1 , this follows from [Ch2, Lemma 4.2 (iii)] in case R ∩ E2 = E . Suppose further that E2 ⊆ R and fix a generator σ of G(F/E) . Applying [Ch2, Theorem 4.1] one obtains that the cyclic E2 -algebra (R/E2, σ 2, εν) is similar to (R/E, σ, c)⊗E E2 , for some c ∈ E∗ . In view of [P, Sect. 14.7, Proposition b], this ensures that (R/E2, σ 2, εν) ∼= (R/E2, σ2, c) as E2 -algebras. Therefore, [P, Sect. 15.1, Proposition b] yields cε−1ν ∈ N(R/E2) and c2N E (εν) ∈ N(R/E) . Since −1 ∈ N(R/E) , this means that c2 ∈ N(R/E) as well. Observing now that the core- striction homomorphism of Br (E2) into Br (E) maps the similarity class Ac of (R/E2, σ 2, c) into the similarity class of (R/E, σ, c2) (cf. [T, Theorem 2.5]), one con- cludes that Ac lies in the kernel Ker E2/E of this homomorphism. At the same time, by [Ch2, Lemma 4.2 (i)], we have Br (E2)2 ∩KerE2/E = {0} . In particu- lar, Ac = 0 , so it follows from [P, Sect. 15.1, Proposition b] that εν ∈ N(R/E2) , as claimed. The obtained result, combined with [AFSS, Theorem 3], implies that 2C(E(2)/E) = 2νC(E(2)/E) = 4C(E(2)/E) , which proves the divisibility of 2C(E(2)/E) . In order to complete the proof of Lemma 2.3, it remains to be seen that the quotient group C(E(p)/E)/pµC(E(p)/E) is isomorphic to the direct sum of cyclic groups of order pµ , indexed by a set I of cardinality d (see [F, Theorem 24.5]). Since, by Kum- mer’s theory, pµ−1C(E(p)/E) includes the group Xp(E) = {χ ∈ C(E(p)/E): pχ = 0} , it is sufficient to show that (Xp(E) + p µC(E(p)/E))/pµC(E(p)/E) := Xp(E) is iso- morphic to pBr(E) . Applying Albert’s theorem and elementary properties of symbol E -algebras, and taking into account that Xp(E) ∼= Xp(E)/(Xp(E) ∩ pµC(E(p)/E)) , one obtains that Xp(E) ∼= E∗/N(Eµ+1/E) . At the same time, the cyclicity of Eµ+1/E ensures that E∗/N(Eµ+1/E) ∼= Br(Eµ+1/E) (cf. [P, Sect. 15.1, Proposition b]), and by the p -quasilocal property of E , we have Br (Eµ+1/E) = pBr(E) , so our proof is complete. Lemma 2.4. Let E be a p -quasilocal field containing a primitive p -th root of unity ε , and such that Br (E)p 6= {0} , and let Ωp(E) be the set of finite abelian extensions of E in E(p) . Then the intersection Np(E) of the norm groups N(M/E): M ∈ Ωp(E) , coincides with the maximal p -divisible subgroup of E∗ . Proof. It is clearly sufficient to show that Np(E) ⊆ N1(E)p , for each n ∈ N . Denote by rp(E) the group of roots of unity in E of p -primary degrees and fix an algebra D ∈ d(E) of index p (the existence of D is guaranteed by the assumption that Br (E)p 6= {0} and the Merkurjev-Suslin theorem [MS, (16.1)]). As E is p - quasilocal, one obtains from Kummer theory that, for each c ∈ E∗ \ E∗p , there is c′ ∈ E∗ \ E∗p , such that D is E -isomorphic to the symbol E -algebra Aε(c, c′; E) . Hence, c 6∈ N(Ec′/E) , where Ec′ is the extension of E in E(p) obtained by adjunction of a p -th root of c′ (see [P, Sect. 15.1, Proposition b]). The obtained result implies Np(E) ⊆ E∗p . Arguing in a similar manner and applying general properties of cyclic E -algebras (cf. [P, Sect. 15.1, Corollary b]), one deduces that Np(E) ⊆ E∗p , if E contains a primitive pm -th root of unity (and proves the lemma in case rp(E) is infinite). Suppose further that rp(E) is of order p µ , for some µ ∈ N , and take C(E(p)/E) , D(E(p)/E) and R(E(p)/E) as in Lemma 2.3. Also, let c ∈ (E∗pm ∩Np(E)) , for some integer m ≥ µ . We prove Lemma 2.4 by showing that c ∈ (E∗pm+µ ∩Np(E)p ) . Fix a primitive pµ -th root of unity δµ ∈ E , take an element cm ∈ E∗ so that cp m = c , and for any λ ∈ E∗ , denote by Eλ the extension of E in E(p) obtained by adjunction of a pµ -th root of λ . Using again [P, Sect. 15.1, Corollary b], one obtains that cm ∈ N(F/E) whenever F is an intermediate field of a Zp -extension of E . Hence, by Albert’s theorem (see [FSS, Sect. 2]), Kummer theory and the basic properties of symbol E -algebras of dimension p2µ , N(Eδµ)/E) ⊆ N(Ecm)/E) . Since Br (E)p 6= {0} , E admits (one-dimensional) local p -class field theory [Ch4, Theorem 3.1], which means that Ecm ⊆ Eδµ . In view of Kummer theory, the obtained result yields cm = δ m , for some k ∈ N , tm ∈ E∗ . Therefore, we have tp m = (t pm = c . Let now M be an arbitrary finite abelian extension of E in E(p) . Then it follows from Galois theory and Lemma 2.3 that M is a subfield of the compositum of finitely many cyclic extensions Di, i ∈ I , and Rj , j ∈ J , of E in E(p) , such that [Rj : E] = pµ , for every j ∈ J , and Di ⊆ ∆i , where ∆i is a Zp -extension of E in E(p) , for each index i . Hence, by [Ch4, Theorem 3.1], cp m ∈ N(M̃/E) ⊆ N(M/E) , i.e. cp m ∈ Np(E) . Replacing cm by tm and arguing as above, one obtains that t m ∈ Np(E) and c ∈ Np(E)p . Thus the assertion that c ∈ (E∗pm+µ ∩Np(E)p ) is proved. It is now easy to see that Np(E) = Np(E) pn , for any n ∈ N , as required by Lemma 2.4. Remark 2.5. Analyzing the proof of Theorem 2.1, one obtains that its conclu- sion remains valid, if E is a quasilocal perfect field and cd p(GE) 6= 1 , for all p ∈ (P \ char(E)) . One also proves that R(E∗) ∼= Φ(E)⊕ R0(E) , where Φ(E) is a torsion-free group and R0(E) is the direct sum of the groups rp(E) , indexed by those p ∈ Π(E) , for which rp(E) is nontrivial and finite. Moreover, it turns out that if Br (E)p 6= {0} , for every p ∈ (Π(E) \ char(E)) , then D(E∗) equals the intersec- tion of the groups N(M/E) , defined over all M ∈ Gal(E) , such that G(M/E) lies in any fixed class χ of finite groups, which is closed under the formation of sub- groups, homomorphic images and finite direct products, and which contains all finite metabelian groups of orders not divisible by any p ∈ (P \Π(E)) . 3. The reduced part of the Brauer group of an equicharacteristic Henselian valued absolutely stable field with a totally indivisible value group In what follows, our notation agrees with that of Lemma 2.3 and its proof. For each Henselian valued field (K, v) , K̂ and v(K) denote the residue field and the value group of (K, v) , respectively. In this Section we announce the following characterization of the reduced components of the Brauer groups of the fields pointed out in its title: Theorem 3.1. An abelian torsion group T is isomorphic to the reduced part of Br (K) , for some absolutely stable field K = K(T) with a Henselian valuation v such that char (K) = char(K̂) and the value group v(K) is totally indivisible (i.e. v(K) 6= pv(K) , for every p ∈ P ), if and only if the p -component Tp of T is presentable as a direct sum of cyclic groups of one and the same order pnp , for each p ∈ P . The rest of this Section is devoted to the proof of the necessity in Theorem 3.1 (the sufficiency will be proved elsewhere in a more general situation covering the case where char (K) 6= char(K̂) , v is Henselian and discrete, and K̂ is perfect). We begin with a description of some known basic relations between Br (K)p , Br (K̂)p and C(K̂(p)/K̂) (proved for convenience of the reader). Lemma 3.2. Let (K, v) be a Henselian valued field with v(K) 6= pv(K) , for some p ∈ P , and let π be an element of K∗ of value v(π) 6∈ pv(K) . For each χ ∈ C(K̂(p)/K̂) , denote by L̃χ the extension of K̂ in K̂sep corresponding by Ga- lois theory to the kernel Ker (χ) , and by σ̃χ the generator of G(L̃χ/K̂) satis- fying the equality χ(σ̃χ) = (1/[Lχ: K̂]) +Q/Z . Assume also that Lχ is the iner- tial lift of L̃χ (over K̂ ) in Ksep , ηχ is the canonical isomorphism of G(Lχ/K) on G(L̃χ/K̂) , and σχ is the preimage of σ̃χ in G(Lχ/K) with respect to ηχ . Then the mapping Wπ of C(K̂(p)/K̂) into Br (K)p defined by the rule Wπ(χ) = [(Lχ/K, σχ, π)]: χ ∈ C(K̂(p)/K̂) , is an injective group homomorphism. Proof. It is clearly sufficient to show that [(Lχ1+χ2/K, σχ1+χ2 , π)] = [(Lχ1/L, σχ1 , π)] +[(Lχ2/K, σχ2 , π)] , where χ1, χ2 ∈ C(K̂(p)/K̂) , in each of the following special cases: (3.1) (i) χ2 ∈ 〈χ1〉 , i.e. Lχ2 ⊆ Lχ1 ; (ii) 〈χ1〉 ∩ 〈χ2〉 = {0} , i.e. Lχ1 ∩ Lχ2 = K . In case (3.1) (i), this follows from the general properties of cyclic algebras (cf. [P, Sect. 15.1, Corollary b and Proposition b]). Assuming that Lχ1 ∩ Lχ2 = K and [Lχ1 : K] ≥ [Lχ2 : K] , one obtains that the K -algebras (Lχ1/K, σχ2 , π)⊗K (Lχ2/K, σχ2 , π) and (Lχ1+χ2/K, σχ1+χ2 , π)⊗K (Lχ2/K, σχ2 , 1) are isomorphic, which completes our proof. With assumptions being as in the lemma, let v(K) 6= pv(K) , for some prime p 6= char(K̂) , and let Bp be a basis and n(p) the dimension of v(K)/pv(K) as a vector space over Fp . Denote by C(K̂(p)/K̂) n(p) the direct sum of some isomorphic copies of C(K̂(p)/K̂) , indexed by Bp . Fix a linear ordering ≤ on Bp and put Jp = {(cp, dp) ∈ (Bp × Bp): cp < dp} , and in case n(p) ≥ 2 , take a direct sum Rp(K̂) Jp of isomorphic copies of Rp(K̂) , indexed by Jp . Then the Ostrowski- Draxl theorem, the Jacob-Wadsworth decomposition lemmas (see [JW]) and Lemma 3.2 imply the following variant of the Scharlau-Witt theorem [Sch]: (3.2) (i) Br (K)p ∼= Br(K̂)p ⊕ C(K̂(p)/K̂)n(p) unless n(p) ≥ 2 and Rp(K̂) 6= {1} ; (ii) Br (K)p ∼= Br(K̂)p ⊕ C(K̂(p)/K̂)n(p) ⊕ Rp(K̂)Jp , if n(p) ≥ 2 and Rp(K̂) 6= {1} . Suppose now that K is a Henselian valued absolutely stable field such that v(K) is totally indivisible. By [Ch2, Proposition 2.3], K̂ is quasilocal, so the necessity in Theorem 3.1 can be deduced from (3.2), the divisibility of Br (F)q , for any field F of characteristic q > 0 (Witt’s theorem, see [Dr, Sect. 15]), and the following lemma. Before stating it, note that the extension of E in Esep obtained by adjunction of a primitive p -th root of unity, for some p ∈ P , is cyclic of degree dividing p− 1 (cf. [L, Ch. VIII, Sect. 3]). Lemma 3.3. Let E be a quasilocal field not containing a primitive p -th root of unity, for a given prime p 6= char(E) , and let ε be such a root in Esep . Fix a generator ϕ of G(E(ε)/E) , take an integer s so as to satisfy the equality ϕ(ε) = εs , and put Bs = {b ∈ pBr(E(ε)): ϕ(b) = sb} . Then: (i) R(E(p)/E) = {0} if and only if rp(E(ε)) is infinite or the group Bs is trivial; (ii) If Bs is nontrivial and of dimension d as an Fp -vector space, and if rp(E(ε)) is of order pµ , for some µ ∈ N , then D(E(p)/E) = pµC(E(p)/E) and R(E(p)/E) is a direct sum of cyclic groups of order pµ , indexed by a set of cardinality d . Proof. Let Λ = {λ ∈ E(ε)∗: ϕ(λ)λ−s ∈ E(ε)∗p} . It is known (Albert, see [A, Ch. IX, Theorem 15]) that an extension L of E in Esep is cyclic of degree p if and only if L(ε) is generated over E(ε) by a p -th root of an element λ0 ∈ (Λ \ E(ε)∗p) . Hence, the Kummer isomorphism of Xp(E(ε)) on E(ε) ∗/E(ε)∗p induces an isomorphism of Xp(E) on Λ/E(ε) ∗p . Observe also that the similarity class of the symbol E(ε) - algebra Aε(λ1, λ2; E(ε)) lies in Bi whenever λj ∈ Λ , j = 1, 2 . Conversely, by the p -quasilocal property of E(ε) , each element of Bs is presented by a symbol E(ε) - algebra determined by a pair of elements of Λ . The noted results enable one to prove the lemma arguing as in the proof of Lemma 2.3. REFERENCES [A] A.A. ALBERT, Modern Higher Algebra. Univ. of Chicago Press, XIV, Chicago, Ill., 1937. [AFSS] J.K. ARASON, B. FEIN, M. SCHCHER, J. SONN, Cyclic extensions of K( −1) . Trans. Amer. Math. Soc. 313, No 2 (1989), 843-851. [CF] J.W.S. CASSELS, A. FRöHLICH (Eds.), Algebraic Number Theory. Aca- demic Press, London-New York, 1967. [Ch1] I.D. CHIPCHAKOV, On nilpotent Galois groups and the scope of the norm limitation theorem in one-dimensional abstract local class field theory. In: Proc. of ICTAMI 05, Alba Iulia, Romania, 15.9.-18.9, 2005; Acta Univ. Apulensis, No 10 (2005), 149-167. [Ch2] I.D. CHIPCHAKOV, On the residue fields of Henselian valued stable fields. Preprint, v. 3 (to appear at www.math.arXiv.org/math.RA/0412544). [Ch3] I.D. CHIPCHAKOV, Algebraic extensions of global fields admitting one- dimensional local class field theory. Preprint (available at www.arXiv.org/ math.NT/0504021). [Ch4] I.D. CHIPCHAKOV, One-dimensional abstract local class field theory. Preprint, v. 3 (available at www.arXiv.org/math.RA/0506515). [Dr] P.K. DRAXL, Skew Fields. London Math. Soc. Lect. Note Series, 81, Cam- bridge etc., Cambridge Univ. Press, 1983. [FSS] B. FEIN, D. SALTMAN, M. SCHACHER, Heights of cyclic field extensions. Bull. Soc. Math. Belg., Ser. A, 40 (1988), 213-223. [F] L. FUCHS, Infinite Abelian Groups. Academic Press, New York-London, 1970. http://arxiv.org/abs/math/0504021 [G] K.W. GRUENBERG, Projective profinite groups. J. Lond. Math. Soc. 42 (1967), 155-165. [KM] M.I. KARGAPOLOV, Yu.I. MERZLJAKOV, Fundamentals of the Theory of Groups, 2nd Ed. Nauka, Moscow, 1977 (Russian: Engl. Transl. in Graduate Texts in Math. 62, Springer-Verlag, New York-Heidelberg-Berlin, 1979). [K] G. KARPILOVSKY, Topics in Field Theory. North-Holland Math. Stud. 155, Amsterdam, 1989. [JW] B. JACOB, A.R. WADSWORTH, Division algebras over Henselian fields. J. Algebra 128 (1990), 528-579. [L] S. LANG, Algebra, Addison-Wesley, Reading, MA, 1965. [MS] A.S. MERKURJEV, A.A. SUSLIN, K -cohomology of Brauer-Severi varieties and norm residue homomorphisms. Izv. Akad. Nauk SSSR 46 (1982), 1011- 1046 (Russian: Engl. transl. in Math. USSR Izv. 21 (1983), 307-340). [P] R. PIERCE, Associative Algebras. Graduate Texts in Math. 88, Springer- Verlag, New York-Heidelberg-Berlin, 1982. [Sch] W. SCARLAU, Ü ber die Brauer-Gruppe eines Hensel-K ö rpers. Abh. Math. Semin. Univ. Hamb. 33 (1969), 243-249. [S] J.-P. SERRE, Cohomologie Galoisienne. Lect. Notes in Math. 5, Springer- Verlag, Berlin-Heidelberg-New York, 1965. [T] J.-P. TIGNOL, On the corestriction of central simple algebras. Math. Z. 194 (1987), 267-274. [W] S. WANG, On Grunwald’s theorem. Ann. Math. (2), 51 (1950), 471-484. Ivan CHIPCHAKOV Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev Str., bl. 8 1113 SOFIA, Bulgaria

Let $E$ be a primarily quasilocal field, $M/E$ a finite Galois extension and $D$ a central division $E$-algebra of index divisible by $[M\colon E]$. In addition to the main result of Part I, this part of the paper shows that if the Galois group $G(M/E)$ is not nilpotent, then $M$ does not necessarily embed in $D$ as an $E$-subalgebra. When $E$ is quasilocal, we find the structure of the character group of its absolute Galois group; this enables us to prove that if $E$ is strictly quasilocal and almost perfect, then the divisible part of the multiplicative group $E ^{\ast}$ equals the intersection of the norm groups of finite Galois extensions of $E$.

Introduction This paper is a continuation of [Ch2]. Let E be a field, Esep a separable closure of E , E∗ the multiplicative group of E , d(E) the class of finite-dimensional central division E -algebras, Br (E) the Brauer group of E , Gal (E) the set of finite Galois extensions of E in Esep , and NG (E) the set of norm groups N(M/E): M ∈ Gal(E) . Let also P be the set of prime numbers, Br (E)p the p -component of Br (E) , for each p ∈ P , P(E) the set of those p ∈ P , for which E is properly included in its maximal p -extension E(p) (in Esep ), and Π(E) the set of all p ′ ∈ P , for which the absolute Galois group GE = G(Esep/E) is of nonzero cohomological p ′ -dimension cd p′(GE) . Recall that E is p -quasilocal, for a given p ∈ P , if p 6∈ P(E) or the relative Brauer group Br (F/E) equals the subgroup pBr(E) = {bp ∈ Br(E): pbp = 0} , for every cyclic extension F of E of degree p . The field E is said to be primarily quasilocal (PQL), if it is p -quasilocal, for all p ∈ P ; E is called quasilocal, if its finite extensions are PQL. We say that E is a strictly PQL-field, if it is PQL and Br (E)p 6= {0} , for every p ∈ P(E) (see [Ch4, Corollary 3.7], for a characterization of the SQL-property). The purpose of this note is to complement the main result of [Ch2] and to shed light on the structure of Br (K) , where K is an absolutely stable field (in the sense of E. Brussel) with a Henselian valuation whose value group is totally indivisible. 1. Subfields of central division algebras over PQL-fields Assume that E is a PQL-field and M ∈ Gal(E) , such that G(M/E) is nilpotent, and let R be an intermediate field of M/E . It follows from Galois theory, [Ch2, Theorem 4.1 (iii)], the Burnside-Wielandt characterization of nilpotent finite groups, and the general properties of central division algebras and their Schur indices (see [KM, Ch. 6, Sect. 2] and [P, Sects. 13.4 and 14.4]) that R embeds as an E -subalgebra in each ∗ Partially supported by Grant MI-1503/2005 of the Bulgarian Foundation for Scien- tific Research. http://arxiv.org/abs/0704.1557v1 algebra D ∈ d(E) of index divisible by [R: E] . In this Section, we demonstrate the optimality of this result in the class of strictly PQL-fields. Theorem 1.1. For each nonnilpotent finite group G , there exists a strictly PQL- field E = E(G) with Br (E) ∼= Q/Z and Gal (E) containing an element M such that G(M/E) ∼= G and M does not embed as an E -subalgebra in any ∆ ∈ d(E) of index equal to [M:E] . Proof. Let P(G) be the set of prime divisors of the order of G . Our argument relies on the existence (cf. [Ch1, Sect. 4]) of an algebraic number field E0 , such that there is M0 ∈ Gal(E0) with G(M0/E0) ∼= G , and E0 possesses a system of (real-valued) valuations {w(p): p ∈ P(G)} satisfying the following conditions: (1.1) (i) w(p) extends the normalized p -adic valuation (in the sense of [CF, Ch. VII]) of the field Q of rational numbers, for each p ∈ P(G) ; (ii) The completion M0,w(p)′ lies in Gal (E0,w(p)) and G(M0,w(p)′/E0,w(p)) is isomor- phic to a Sylow p -subgroup of G(M0/E0) , whenever p ∈ P(G) and w(p)′ extends w(p) on M0 . Let Ap be the maximal abelian p -extension of E0 in M0 , for each p ∈ P(G) . It follows from the nonnilpotency of G , the Burnside-Wielandt theorem and Galois theory that M0/E0 has an intermediate field F 6= E0 such that F ∩ Ap = E , for all p ∈ P(G) (see [Ch1, Sect. 4]). Fix a prime divisor π of [F: E0] as well as Sylow π -subgroups Hπ ∈ SylπG(M0/F) and Gπ ∈ SylπG(M0/E0) with Hπ ⊂ Gπ , and denote by Fπ and Eπ the maximal extensions of E0 in M0 fixed by Hπ and Gπ , respectively. It is clear from (1.1) (ii) and [CF, Ch. II, Theorem 10.2] that Eπ has a valuation t(π) extending w(π) so that M0 ⊗Eπ Eπ,t(π) is a field. This means that t(π) is uniquely (up-to an equivalence) extendable to a valuation µ(π) of M0 , and allows us to identify E0,w(π) with Eπ,t(π) . Using repeatedly the Grunwald-Wang theorem [W] and the normality of maximal subgroups of finite π -groups (cf. [L, Ch. I, Sect. 6]), one also obtains that there is a finite extension K of E0 in E0(π) , such that K⊗E0 E0,w(π) is a field isomorphic to Fπ,ν(π) as an E0 -algebra, where ν(π) is the valuation of Fπ induced by µ(π) . These observations indicate that K ∩M0 = E0 , i.e. the compositum M0K lies in Gal (K) and G((M0K)/K) ∼= G(M0/E0) . At the same time, our argument proves that F is dense in Fπ,ν(p) with respect to the topology induced by ν(p) , K has a unique valuation κ(π) extending w(π) , and (M0K)t(π)′ ∈ Gal(Kκ(π)) with G((M0K)t(π)′/Kκ(π)) ∼= Hπ , for any prolongation t(π)′ of t(π) on M0K . Let now M(K) = {κ(p): p ∈ P} be a system of valuations of K complementing κ(π) so that κ(p) extends w(p) or the normalized p -adic valuation of Q , depending on whether or not p ∈ (P(G) \ {π}) . Applying [Ch3, Theorem 2.2] to K , P and M(K) , and using [Ch3, Lemmas 3.1 and 3.2], one proves the existence of an extension E of K in E0,sep , such that P(E) = P , M0E := M lies in Gal (E) , G(M/E) ∼= G , and E possesses a system {v(p): p ∈ P} of valuations with the following properties: (1.2) For each p ∈ P , v(p) extends κ(p) , Ev(p) is a PQL-field, Ev(p)(p) is E - isomorphic to E(p)⊗E Ev(p) , Br (Ev(p))p 6= {0} and Br (Ev(p′))p = {0} , for every p′ ∈ (P \ {p}) . Moreover, if p ∈ P(G) and v(p)′ is a valuation of M extending v(p) , then Mv(p)′ ∈ Gal(Ev(p)) , G(Mv(p)′/Ev(p)) ∼= G(M0,w(p)′/E0,w(p)): p 6= π , and G(Mv(π)′/Ev(π)) ∼= Hπ . Hence, by [Ch3, Theorem 2.1], E is a nonreal strictly PQL-field. As Ev(π) is PQL, it follows from [Ch2, Theorem 4.1], statements (1.2) and the Brauer-Hasse-Noether and Albert theorem (in the form of [Ch3, Proposition 1.2]) that M splits an algebra ∆ ∈ d(E) of π -primary dimension if and only if ind (∆) divides the order of Hπ . In view of the general theory of simple algebras (see [P, Sects. 13.4 and 14.4]), this proves Theorem 1.1. 2. Divisible and reduced parts of the multiplicative group of an SQL-field A field E is said to be strictly quasilocal (SQL), if its finite extensions are strictly PQL. When this holds and E is almost perfect (in the sense of [Ch2, I, (1.8)]), this Section gives a Galois-theoretic characterization of the maximal divisible subgroup D(E) of E∗ . Theorem 2.1. Let E be an almost perfect SQL-field and N1(E) the intersection of the groups from NG (E) . Then N1(E) = D(E To prove Theorem 2.1 we need the following lemma. Lemma 2.2. Let E and M be fields, M ∈ Gal(E) , and let P(M/E) be the set of prime divisors of [M:E] . Then N(M/E) ⊆ N(M/F) , for every intermediate field F of M/E . Moreover, if Ep is the fixed field of some Sylow p -subgroup of G(M/E) , then N(M/E) = ∩p∈P(M/E)N(M/Ep) . Proof. It is easily verified that if [F: E] = m , {τj : j = 1, . . . ,m} is the set of E - embeddings of F into M , and σu is an automorphism of M extending τu , for each index j , then G(M/E) = ∪mj=1G(M/F)σ j . This yields N E (γ) = N j=1 σ j (γ)) , for any γ ∈ M∗ , which proves that N(M/F) ⊆ N(M/E) . We show that ∩p∈P(M/E) N(M/Ep) := N0(M/E) ⊆ N(M/E) . Take an element α ∈ N0(M/E) and put [Ep: E] = mp , for each p ∈ P(M/E) . It is known that p 6 |mp , for any p ∈ P(M/E) , which implies consecutively that g.c.d.(mp: p ∈ P(M/E)) = 1 , α ∈ E∗ and αmp ∈ N(M/E) , for every p ∈ P(M/E) . Thus it turns out that α ∈ N(M/E) , so Lemma 2.2 is proved. Proof of Theorem 2.1. The inclusion D(E∗) ⊆ N1(E) is obvious, so our objective is to prove the converse. We show that N1(E) ⊆ N1(E)p , for every p ∈ P and n ∈ N . It is clearly sufficient to consider the special case where p ∈ Π(E) . Then there is a finite extension Fp of E in Esep , such that Br (Fp)p 6= {0} and p 6 |[Fp: E] . Note also that each finite extension of Fp in E ∗ is included in a field F̃p ∈ Gal(E) , so it follows from Lemma 2.2 that N1(E) ⊆ N1(Fp) . At the same time, the norm mapping E clearly induces homomorphisms D(F p) → D(E∗) and N1(Fp) → Np(E) (for the latter, see [L, Ch. VIII, Sect. 5]) as well as an automorphism of the maximal p -divisible subgroup of E∗ . These observations allow one to assume additionally that p ∈ P(E) (and Br (E)p 6= {0} ). If p = char(E) , our assertion is implied by [Ch2, Lemma 8.4], so we turn to the case of p 6= char(E) . Let εp be a primitive p -th root of unity in Esep . Then [E(εp): E] divides p− 1 (cf. [L, Ch. VIII, Sect. 3]), whence our considerations reduce further to the case in which εp ∈ E . Now the proof of Theorem 2.1 is completed by applying the following two lemmas. Before stating them, let us recall that the character group C(Y/E) of G(Y/E) is an abelian torsion group, for every Galois extension Y/E (see [K, Ch. 7, Sect. 5]). Therefore, by [F, Theorem 24.5], the divisible part D(Y/E) of C(Y/E) is a direct summand in C(Y/E) , i.e. C(Y/E) is isomorphic to the direct sum D(Y/E)⊕ R(Y/E) , where R(Y/E) ∼= C(Y/E)/D(Y/E) is a maximal reduced subgroup of C(Y/E) . Lemma 2.3. Let E be a p -quasilocal field containing a primitive p -th root of unity ε , for some prime p ∈ P(E) . Assume also that rp(E) is the group of roots of unity in E of p -primary degrees. Then: (i) R(E(p)/E) = {0} if and only if Br (E)p = {0} or rp(E) is infinite; (ii) If Br (E)p 6= {0} , d is the dimension of pBr(E) as a vector space over the field Fp with p elements, and Rp(E) is of order p µ , for some µ ∈ N , then D(E(p)/E) = pµC(E(p)/E) and R(E(p)/E) is presentable as a direct sum of cyclic groups of order pµ , indexed by a set of cardinality d . Proof. Statement (i) is a well-known consequence of Kummer theory, the Merkurjev- Suslin theorem and Galois cohomology (see [MS, (11.5), (16.1)] and [S, Ch. I, 3.4 and 4.2]), so we assume further that d > 0 and rp(E) is of order p µ , for some µ ∈ N . We first show that pµC(E(p)/E) is divisible. Fix a primitive pµ -th root of unity εµ in E and a subset S = {εn: n ∈ N, n ≥ µ+ 1} of E(p) so that εpn+1 = εn , for each index n . Also, let E∞ = E(S) and En = E(εn) , for every integer n > µ . Suppose first that p > 2 or µ ≥ 2 . Then E∞/E is a Zp -extension and En is the unique subextension of E in E∞ of degree p n−µ . Since Zp is a projective profinite group (see [G, Theorem 1]), this enables one to deduce from Galois theory that E(p) possesses a subfield E′ such that E′ ∩ E∞ = E and E∞E′ = E(p) . In addition, it becomes clear that C(E(p)/E) is isomorphic to the direct sum Z(p∞)⊕ C(F/E) , where Z(p∞) is the quasicyclic p -group (identified with the character group of Zp ) and F is the maximal abelian extension of E in E′ . Let now Φ be a cyclic extension of E in F . As E is p -quasilocal, an element βn ∈ E∗n lies in N((ΦEn)/En) , for an arbitrary integer n ≥ µ , if and only if the norm NEnE (βn) ∈ N(Φ/E) . In particular, this proves that εµ ∈ N(Φ/E) if and only if εn ∈ N((ΦEn)/En) . In view of the results of [FSS, Sect. 2], this means that pµC(F/E) = pnC(F/E) , for every index n ≥ µ . In other words, C(F/E) is divisible, so pµC(E(p)/E) has the same property, as claimed. Our objective now is to prove the divisibility of 2C(E(2)/E) , under the hypothesis that p = 2 and µ = 1 . If E∞ = E( −1) = E2 , this is contained in [FSS, Proposi- tion 2], so we assume further that E∞ 6= E2 , i.e. E2 = Eν 6= Eν+1 , for some in- teger ν ≥ 2 . Let R be a cyclic extension of E in E(2) . By Albert’s theorem (see [FSS, Sect. 2]), we have C(R/E) ⊂ 2C(E(2)/E) if and only if −1 ∈ N(R/E) . We show that if C(R/E) ⊂ 2C(E(2)/E) , then εν ∈ N((RE2)/E2) . Since NE2E (εν) equals 1 or −1 , this follows from [Ch2, Lemma 4.2 (iii)] in case R ∩ E2 = E . Suppose further that E2 ⊆ R and fix a generator σ of G(F/E) . Applying [Ch2, Theorem 4.1] one obtains that the cyclic E2 -algebra (R/E2, σ 2, εν) is similar to (R/E, σ, c)⊗E E2 , for some c ∈ E∗ . In view of [P, Sect. 14.7, Proposition b], this ensures that (R/E2, σ 2, εν) ∼= (R/E2, σ2, c) as E2 -algebras. Therefore, [P, Sect. 15.1, Proposition b] yields cε−1ν ∈ N(R/E2) and c2N E (εν) ∈ N(R/E) . Since −1 ∈ N(R/E) , this means that c2 ∈ N(R/E) as well. Observing now that the core- striction homomorphism of Br (E2) into Br (E) maps the similarity class Ac of (R/E2, σ 2, c) into the similarity class of (R/E, σ, c2) (cf. [T, Theorem 2.5]), one con- cludes that Ac lies in the kernel Ker E2/E of this homomorphism. At the same time, by [Ch2, Lemma 4.2 (i)], we have Br (E2)2 ∩KerE2/E = {0} . In particu- lar, Ac = 0 , so it follows from [P, Sect. 15.1, Proposition b] that εν ∈ N(R/E2) , as claimed. The obtained result, combined with [AFSS, Theorem 3], implies that 2C(E(2)/E) = 2νC(E(2)/E) = 4C(E(2)/E) , which proves the divisibility of 2C(E(2)/E) . In order to complete the proof of Lemma 2.3, it remains to be seen that the quotient group C(E(p)/E)/pµC(E(p)/E) is isomorphic to the direct sum of cyclic groups of order pµ , indexed by a set I of cardinality d (see [F, Theorem 24.5]). Since, by Kum- mer’s theory, pµ−1C(E(p)/E) includes the group Xp(E) = {χ ∈ C(E(p)/E): pχ = 0} , it is sufficient to show that (Xp(E) + p µC(E(p)/E))/pµC(E(p)/E) := Xp(E) is iso- morphic to pBr(E) . Applying Albert’s theorem and elementary properties of symbol E -algebras, and taking into account that Xp(E) ∼= Xp(E)/(Xp(E) ∩ pµC(E(p)/E)) , one obtains that Xp(E) ∼= E∗/N(Eµ+1/E) . At the same time, the cyclicity of Eµ+1/E ensures that E∗/N(Eµ+1/E) ∼= Br(Eµ+1/E) (cf. [P, Sect. 15.1, Proposition b]), and by the p -quasilocal property of E , we have Br (Eµ+1/E) = pBr(E) , so our proof is complete. Lemma 2.4. Let E be a p -quasilocal field containing a primitive p -th root of unity ε , and such that Br (E)p 6= {0} , and let Ωp(E) be the set of finite abelian extensions of E in E(p) . Then the intersection Np(E) of the norm groups N(M/E): M ∈ Ωp(E) , coincides with the maximal p -divisible subgroup of E∗ . Proof. It is clearly sufficient to show that Np(E) ⊆ N1(E)p , for each n ∈ N . Denote by rp(E) the group of roots of unity in E of p -primary degrees and fix an algebra D ∈ d(E) of index p (the existence of D is guaranteed by the assumption that Br (E)p 6= {0} and the Merkurjev-Suslin theorem [MS, (16.1)]). As E is p - quasilocal, one obtains from Kummer theory that, for each c ∈ E∗ \ E∗p , there is c′ ∈ E∗ \ E∗p , such that D is E -isomorphic to the symbol E -algebra Aε(c, c′; E) . Hence, c 6∈ N(Ec′/E) , where Ec′ is the extension of E in E(p) obtained by adjunction of a p -th root of c′ (see [P, Sect. 15.1, Proposition b]). The obtained result implies Np(E) ⊆ E∗p . Arguing in a similar manner and applying general properties of cyclic E -algebras (cf. [P, Sect. 15.1, Corollary b]), one deduces that Np(E) ⊆ E∗p , if E contains a primitive pm -th root of unity (and proves the lemma in case rp(E) is infinite). Suppose further that rp(E) is of order p µ , for some µ ∈ N , and take C(E(p)/E) , D(E(p)/E) and R(E(p)/E) as in Lemma 2.3. Also, let c ∈ (E∗pm ∩Np(E)) , for some integer m ≥ µ . We prove Lemma 2.4 by showing that c ∈ (E∗pm+µ ∩Np(E)p ) . Fix a primitive pµ -th root of unity δµ ∈ E , take an element cm ∈ E∗ so that cp m = c , and for any λ ∈ E∗ , denote by Eλ the extension of E in E(p) obtained by adjunction of a pµ -th root of λ . Using again [P, Sect. 15.1, Corollary b], one obtains that cm ∈ N(F/E) whenever F is an intermediate field of a Zp -extension of E . Hence, by Albert’s theorem (see [FSS, Sect. 2]), Kummer theory and the basic properties of symbol E -algebras of dimension p2µ , N(Eδµ)/E) ⊆ N(Ecm)/E) . Since Br (E)p 6= {0} , E admits (one-dimensional) local p -class field theory [Ch4, Theorem 3.1], which means that Ecm ⊆ Eδµ . In view of Kummer theory, the obtained result yields cm = δ m , for some k ∈ N , tm ∈ E∗ . Therefore, we have tp m = (t pm = c . Let now M be an arbitrary finite abelian extension of E in E(p) . Then it follows from Galois theory and Lemma 2.3 that M is a subfield of the compositum of finitely many cyclic extensions Di, i ∈ I , and Rj , j ∈ J , of E in E(p) , such that [Rj : E] = pµ , for every j ∈ J , and Di ⊆ ∆i , where ∆i is a Zp -extension of E in E(p) , for each index i . Hence, by [Ch4, Theorem 3.1], cp m ∈ N(M̃/E) ⊆ N(M/E) , i.e. cp m ∈ Np(E) . Replacing cm by tm and arguing as above, one obtains that t m ∈ Np(E) and c ∈ Np(E)p . Thus the assertion that c ∈ (E∗pm+µ ∩Np(E)p ) is proved. It is now easy to see that Np(E) = Np(E) pn , for any n ∈ N , as required by Lemma 2.4. Remark 2.5. Analyzing the proof of Theorem 2.1, one obtains that its conclu- sion remains valid, if E is a quasilocal perfect field and cd p(GE) 6= 1 , for all p ∈ (P \ char(E)) . One also proves that R(E∗) ∼= Φ(E)⊕ R0(E) , where Φ(E) is a torsion-free group and R0(E) is the direct sum of the groups rp(E) , indexed by those p ∈ Π(E) , for which rp(E) is nontrivial and finite. Moreover, it turns out that if Br (E)p 6= {0} , for every p ∈ (Π(E) \ char(E)) , then D(E∗) equals the intersec- tion of the groups N(M/E) , defined over all M ∈ Gal(E) , such that G(M/E) lies in any fixed class χ of finite groups, which is closed under the formation of sub- groups, homomorphic images and finite direct products, and which contains all finite metabelian groups of orders not divisible by any p ∈ (P \Π(E)) . 3. The reduced part of the Brauer group of an equicharacteristic Henselian valued absolutely stable field with a totally indivisible value group In what follows, our notation agrees with that of Lemma 2.3 and its proof. For each Henselian valued field (K, v) , K̂ and v(K) denote the residue field and the value group of (K, v) , respectively. In this Section we announce the following characterization of the reduced components of the Brauer groups of the fields pointed out in its title: Theorem 3.1. An abelian torsion group T is isomorphic to the reduced part of Br (K) , for some absolutely stable field K = K(T) with a Henselian valuation v such that char (K) = char(K̂) and the value group v(K) is totally indivisible (i.e. v(K) 6= pv(K) , for every p ∈ P ), if and only if the p -component Tp of T is presentable as a direct sum of cyclic groups of one and the same order pnp , for each p ∈ P . The rest of this Section is devoted to the proof of the necessity in Theorem 3.1 (the sufficiency will be proved elsewhere in a more general situation covering the case where char (K) 6= char(K̂) , v is Henselian and discrete, and K̂ is perfect). We begin with a description of some known basic relations between Br (K)p , Br (K̂)p and C(K̂(p)/K̂) (proved for convenience of the reader). Lemma 3.2. Let (K, v) be a Henselian valued field with v(K) 6= pv(K) , for some p ∈ P , and let π be an element of K∗ of value v(π) 6∈ pv(K) . For each χ ∈ C(K̂(p)/K̂) , denote by L̃χ the extension of K̂ in K̂sep corresponding by Ga- lois theory to the kernel Ker (χ) , and by σ̃χ the generator of G(L̃χ/K̂) satis- fying the equality χ(σ̃χ) = (1/[Lχ: K̂]) +Q/Z . Assume also that Lχ is the iner- tial lift of L̃χ (over K̂ ) in Ksep , ηχ is the canonical isomorphism of G(Lχ/K) on G(L̃χ/K̂) , and σχ is the preimage of σ̃χ in G(Lχ/K) with respect to ηχ . Then the mapping Wπ of C(K̂(p)/K̂) into Br (K)p defined by the rule Wπ(χ) = [(Lχ/K, σχ, π)]: χ ∈ C(K̂(p)/K̂) , is an injective group homomorphism. Proof. It is clearly sufficient to show that [(Lχ1+χ2/K, σχ1+χ2 , π)] = [(Lχ1/L, σχ1 , π)] +[(Lχ2/K, σχ2 , π)] , where χ1, χ2 ∈ C(K̂(p)/K̂) , in each of the following special cases: (3.1) (i) χ2 ∈ 〈χ1〉 , i.e. Lχ2 ⊆ Lχ1 ; (ii) 〈χ1〉 ∩ 〈χ2〉 = {0} , i.e. Lχ1 ∩ Lχ2 = K . In case (3.1) (i), this follows from the general properties of cyclic algebras (cf. [P, Sect. 15.1, Corollary b and Proposition b]). Assuming that Lχ1 ∩ Lχ2 = K and [Lχ1 : K] ≥ [Lχ2 : K] , one obtains that the K -algebras (Lχ1/K, σχ2 , π)⊗K (Lχ2/K, σχ2 , π) and (Lχ1+χ2/K, σχ1+χ2 , π)⊗K (Lχ2/K, σχ2 , 1) are isomorphic, which completes our proof. With assumptions being as in the lemma, let v(K) 6= pv(K) , for some prime p 6= char(K̂) , and let Bp be a basis and n(p) the dimension of v(K)/pv(K) as a vector space over Fp . Denote by C(K̂(p)/K̂) n(p) the direct sum of some isomorphic copies of C(K̂(p)/K̂) , indexed by Bp . Fix a linear ordering ≤ on Bp and put Jp = {(cp, dp) ∈ (Bp × Bp): cp < dp} , and in case n(p) ≥ 2 , take a direct sum Rp(K̂) Jp of isomorphic copies of Rp(K̂) , indexed by Jp . Then the Ostrowski- Draxl theorem, the Jacob-Wadsworth decomposition lemmas (see [JW]) and Lemma 3.2 imply the following variant of the Scharlau-Witt theorem [Sch]: (3.2) (i) Br (K)p ∼= Br(K̂)p ⊕ C(K̂(p)/K̂)n(p) unless n(p) ≥ 2 and Rp(K̂) 6= {1} ; (ii) Br (K)p ∼= Br(K̂)p ⊕ C(K̂(p)/K̂)n(p) ⊕ Rp(K̂)Jp , if n(p) ≥ 2 and Rp(K̂) 6= {1} . Suppose now that K is a Henselian valued absolutely stable field such that v(K) is totally indivisible. By [Ch2, Proposition 2.3], K̂ is quasilocal, so the necessity in Theorem 3.1 can be deduced from (3.2), the divisibility of Br (F)q , for any field F of characteristic q > 0 (Witt’s theorem, see [Dr, Sect. 15]), and the following lemma. Before stating it, note that the extension of E in Esep obtained by adjunction of a primitive p -th root of unity, for some p ∈ P , is cyclic of degree dividing p− 1 (cf. [L, Ch. VIII, Sect. 3]). Lemma 3.3. Let E be a quasilocal field not containing a primitive p -th root of unity, for a given prime p 6= char(E) , and let ε be such a root in Esep . Fix a generator ϕ of G(E(ε)/E) , take an integer s so as to satisfy the equality ϕ(ε) = εs , and put Bs = {b ∈ pBr(E(ε)): ϕ(b) = sb} . Then: (i) R(E(p)/E) = {0} if and only if rp(E(ε)) is infinite or the group Bs is trivial; (ii) If Bs is nontrivial and of dimension d as an Fp -vector space, and if rp(E(ε)) is of order pµ , for some µ ∈ N , then D(E(p)/E) = pµC(E(p)/E) and R(E(p)/E) is a direct sum of cyclic groups of order pµ , indexed by a set of cardinality d . Proof. Let Λ = {λ ∈ E(ε)∗: ϕ(λ)λ−s ∈ E(ε)∗p} . It is known (Albert, see [A, Ch. IX, Theorem 15]) that an extension L of E in Esep is cyclic of degree p if and only if L(ε) is generated over E(ε) by a p -th root of an element λ0 ∈ (Λ \ E(ε)∗p) . Hence, the Kummer isomorphism of Xp(E(ε)) on E(ε) ∗/E(ε)∗p induces an isomorphism of Xp(E) on Λ/E(ε) ∗p . Observe also that the similarity class of the symbol E(ε) - algebra Aε(λ1, λ2; E(ε)) lies in Bi whenever λj ∈ Λ , j = 1, 2 . Conversely, by the p -quasilocal property of E(ε) , each element of Bs is presented by a symbol E(ε) - algebra determined by a pair of elements of Λ . The noted results enable one to prove the lemma arguing as in the proof of Lemma 2.3. REFERENCES [A] A.A. ALBERT, Modern Higher Algebra. Univ. of Chicago Press, XIV, Chicago, Ill., 1937. [AFSS] J.K. ARASON, B. FEIN, M. SCHCHER, J. SONN, Cyclic extensions of K( −1) . Trans. Amer. Math. Soc. 313, No 2 (1989), 843-851. [CF] J.W.S. CASSELS, A. FRöHLICH (Eds.), Algebraic Number Theory. Aca- demic Press, London-New York, 1967. [Ch1] I.D. CHIPCHAKOV, On nilpotent Galois groups and the scope of the norm limitation theorem in one-dimensional abstract local class field theory. In: Proc. of ICTAMI 05, Alba Iulia, Romania, 15.9.-18.9, 2005; Acta Univ. Apulensis, No 10 (2005), 149-167. [Ch2] I.D. 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The Physics of Chromospheric Plasmas ASP Conference Series, Vol. 368, 2007 Petr Heinzel, Ivan Dorotovič and Robert J. Rutten, eds. Chromospheric Cloud-Model Inversion Techniques Kostas Tziotziou National Observatory of Athens, Institute for Space Applications and Remote Sensing, Greece Abstract. Spectral inversion techniques based on the cloud model are ex- tremely useful for the study of properties and dynamics of various chromospheric cloud-like structures. Several inversion techniques are reviewed based on simple (constant source function) and more elaborated cloud models, as well as on grids of synthetic line profiles produced for a wide range of physical parameters by different NLTE codes. Several examples are shown of how such techniques can be used in different chromospheric lines, for the study of structures of the quiet chromosphere, such as mottles/spicules, as well as for active region structures such as fibrils, arch filament systems (AFS), filaments and flares. 1. Introduction Observed intensity line profiles are a function of several parameters describing the three-dimensional solar atmosphere, such as chemical abundance, density, temperature, velocity, magnetic field, microturbulence etc (which one would like to determine), as well as of wavelength, space (solar coordinates) and time. How- ever, due to the large number of parameters that an observed profile depends on, as well as data noise, model atmospheres have to be assumed in order to restrict the number of these unknown parameters. The term “inversion tech- niques” refers to the procedures used for inferring these model parameters from observed profiles. We refer the reader to Mein (2000) for an extended overview of inversion techniques. In this paper, we will review only a class of such inversion techniques known in the solar community as “cloud models”. Cloud models refer to models describing the transfer of radiation through structures located higher up from the solar photosphere, which represents the solar surface, resembling clouds on earth’s sky (see Fig. 1). Such cloud-like structures, when observed from above, would seem to mostly absorb the radi- ation coming from below, an absorption which mostly depends on the optical thickness of the cloud, that is the “transparency” of the cloud to the incident radiation and also on the physical parameters that describe it. The possibility of observed emission from such structures cannot, of course, be excluded when the radiation produced by the cloud-like structure is higher than the absorbed one. The aforementioned processes are described by the radiative transfer equation I(∆λ) = I0(∆λ) e −τ(∆λ) + ∫ τ(∆λ) −t(∆λ) dt , (1) where I(∆λ) is the observed intensity, I0(∆λ) is the reference profile emitted by the background (the incident radiation to the cloud from below), τ(∆λ) is the http://arxiv.org/abs/0704.1558v1 218 Tziotziou Figure 1. Geometry of the cloud model. D is the geometrical thickness of the cloud at height H above the solar surface and V its velocity. From Heinzel et al. (1999). optical thickness and S the source function which is a function of optical depth along the cloud. The first term of the right hand part of the equation represents the absorption of the incident radiation by the cloud, while the second term represents emission by the cloud itself. The simple cloud model method introduced by Beckers (1964) arose from the need to solve fast the radiation transfer equation and deduce the physical parameters that describe the observed structure. Beckers assumed that a) the structure is fully separated from the underlying chromosphere, b) the source function, radial velocity, Doppler width and the absorption coefficient are con- stant along the line-of-sight (hereafter LOS) and c) the background intensity is the same below the structure and the surrounding atmosphere; hence it can be extrapolated from a neighboring to the structure under study region. Under the above assumptions the radiative transfer equation is simplified to I(∆λ) = I0(∆λ) e −τ(∆λ) + S(1− e−τ(∆λ)) (2) and can be rewritten as C(∆λ) = I(∆λ)− I0(∆λ) I0(∆λ) I0(∆λ) (1− e−τ(∆λ)) , (3) where C(∆λ) defines the contrast profile. A Gaussian wavelength dependence is usually assumed for the optical depth as follows τ(∆λ) = τ0 e ∆λ−∆λI , (4) Cloud-model Inversion Techniques 219 where τ0 is the line center optical thickness, ∆λI = λ0v/c is the Doppler shift with λ0 being the line center wavelength, c the speed of light and ∆λD is the Doppler width. The latter depends on temperature T and microturbulent ve- locity ξt through the relationship ∆λD = ξ2t + , (5) where m is the atom rest mass. Other wavelength dependent profiles than the Gaussian one can also be assumed for the optical depth, e.g., a Voigt profile (Tsiropoula et al. 1999). The four adjustable parameters of the model are the source function S, the Doppler width ∆λD, the optical thickness τ0 and the LOS velocity v. All these parameters are assumed to be constant through the structure. There are some crucial assumptions concerning Beckers’ cloud model (hereafter BCM): – the uniform background radiation assumption, which is not always true es- pecially for cloud-like structures that do not reside above quiet Sun regions. Moreover, the background radiation plays an important role in the correct quantitative determination of the physical parameters. – the neglect of incident radiation, the effects of which are of course not directly considered in BCM, but does play an important role in non-Local Thermody- namic Equilibrium (hereafter NLTE) modeling, since it determines the radia- tion field within the structure, that is the excitation and ionization conditions and hence the source function. – the constant source function assumption which is not realistic especially in the optically thick case or not valid in the presence of large velocity gradients. However, the cloud model works quite well for a large number of optically thin structures and can provide useful, reasonable estimates for the physical param- eters that describe them. We refer the reader to Alissandrakis et al. (1990) for a detailed discussion on the validity conditions of BCM for different types of contrast profiles. 2. Cloud Model Variants Since the introduction of the BCM method several improvements have been suggested in the literature. When looking at the radiative transfer equation (Eq. 1), it is obvious that all efforts concentrate on a better description of the source function S which in BCM is considered to be constant. In the following subsections some of these suggested improvements are described. 2.1. Variable source function Mein et al. (1996a) considered a source function that is a function of optical depth and is approximated by a second-order polynomial St = S0 + S1 τ0,max τ0,max 220 Tziotziou Figure 2. Left: Geometry of the cloud model in the case of first-order differ- ential cloud model. From Heinzel et al. (1992). Right: The “3-optical depths” procedure for solving the differential cloud model case which is described in Sect. 3.5 (from Mein & Mein (1988)). with the optical depth at the center of the line τ0 taking values between 0 and the total optical thickness at line center τ0,max, while S0, S1 and S2 are functions of τ0,max. This formulation was further improved by Heinzel et al. (1999), who included also the effect of cloud motion by assuming that S0, S1 and S2 are now not only functions of τ0,max, but also depend on the velocity v of the structure. Tsiropoula et al. (1999) assumed the parabolic formula St = S0 1 + α as an initial condition for the variation of the source function with optical depth, where S0 is the source function at the middle of the structure, τ0 the optical depth at line center, and α a constant expressing the variation of the source function. However, their final results on the dependence of the source on optical depth were in good agreement with the results of Mein et al. (1996a). 2.2. Differential cloud models First and second order differential cloud models (hereafter DCM1 and DCM2) were introduced by Mein & Mein (1988) to account for fast mass flows observed on the disc, where BCM is not valid due to fluctuations of the background and strong velocity gradients along the LOS. DCM1 assumes that the source function S, temperature T and velocity v are constant within a small volume contained between two close lines of sight P and R (see Fig. 2, left panel). If we assume that the variation of the background profile is negligible (I0P ≃ I0R) for such close points then the differential cloud contrast profile can be written as C(P,R, λ) = IP (λ)− IR(λ) IR(λ) IR(λ) (1− e−δτ(λ)) (8) Cloud-model Inversion Techniques 221 δτ(λ) = δτ0 e λ− λ0 − vλ0/c , (9) where ∆λD is the Doppler width. The zero velocity reference wavelength λ0 is obtained by averaging over the whole field of view. DCM1 is a method for suppressing the use of the background radiation. If velocity shears are present between neighboring LOS then DCM2 can be used instead which requires the use of three neighboring LOS. We refer the reader to Mein & Mein (1988) for the precise formulation of DCM2 and to Table 1 of the same paper which summarizes the validity conditions, constrains and results of the two models in comparison to the classical BCM. 2.3. Multi-cloud models The multi-cloud model (Gu et al. 1992, 1996) – hereafter MCM – was intro- duced for the study of asymmetric, non-Gaussian profiles, such as line profiles of post-flare loops, prominences and surges and was based on the BCM and DCMs models. These asymmetric line profiles are assumed to be the result of overlapping of several symmetric Gaussian profiles along the LOS, formed in small radiative elements (clouds) which have a) different or identical physical properties and b) a source function and velocity independent of depth. The profile asymmetry mostly results from the relative Doppler shifts of the different clouds. The total intensity Iλ emitted by m clouds is then given by the relation Iλ = I0,λ e −τλ + Sj(1− e −τλ,j ) exp  , (10) where τλ,0 = 0, I0,λ is the background intensity, τλ = j=1 τλ,j is the total optical depth of the m clouds and τλ,j = τ0,j e λ− λ0 −∆λ0,j ∆λD,j ∆λ0,j = λ0vj/c, Sj, τ0,j, vj , ∆λD,j are respectively the optical depth, Doppler shift, source function, line-center thickness, velocity and Doppler width of the jth cloud. A somewhat similar in philosophy, two-cloud model method was used by Heinzel & Schmieder (1994) in their study of black and white mottles. It was assumed that the LOS intersects two mottles treated as two different clouds c1 and c2 with optical depths τ1 and τ2 respectively. Hence the emerging intensity from the lower mottle I1 is assumed to be the background incident intensity for the second upper mottle. Then, the equations describing the radiation transfer through the two mottles are I2(∆λ) = I1 e −τ2(∆λ) + Ic2(∆λ) I1(∆λ) = I0 e −τ1(∆λ) + Ic1(∆λ) , (12) 222 Tziotziou where I0 is the background chromospheric intensity and Ic1, Ic2 the intensity emitted by the two clouds respectively. The novelty of the method is that for the emitted by the clouds intensity, a grid of 140 NLTE models was used which was computed for prominence-like structures by Gouttebroze et al. (1993). So this method is a combination of MCM with NLTE source function calculations which will be further discussed in Section 2.6. 2.4. The Doppler signal method The Doppler signal method (Georgakilas et al. 1990; Tsiropoula 2000) can be used when filtergrams at two wavelengths −∆λ and +∆λ (blue and red side of the line) are available and a fast determination of mass motions is needed. Then the Doppler signal DS can be defined from the BCM equations as I − 2I0 − e−τ 2− e−τ − e−τ , (13) where ∆I = I(−∆λ) − I(+∆λ), I = I(−∆λ) + I(+∆λ) and τ± = τ(±∆λ). The Doppler signal DS has the same sign as velocity and can be used for a qualitative description of the velocity field. The left hand side of the above equation can be determined by the observations while the right hand clearly does not depend on the source function. Quantitative values for the velocity can be obtained when τ0 < 1; then the Doppler signal equation reduces to τ− − τ+ τ− + τ+ and the velocity v – once DS is calculated from the observations and a value of the Doppler width ∆λD is obtained from the literature or assumed – is given by the equation 1 +DS . (15) 2.5. Avoiding the background profile Liu & Ding (2001) in order to avoid the use of the background profile needed in BCM assumed that it is symmetric, that is I0(∆λ) = I0(−∆λ). Then it can easily be shown that we can obtain the relationship ∆I(∆λ) = I(∆λ)− I(−∆λ) = [I(∆λ)− S][1− eτ(∆λ)−τ(−∆λ)] , (16) which does not require the use of the background for the derivation of the phys- ical parameters. 2.6. NLTE methods As Eq. 1 shows, in the general case, the source function S within a cloud-like structure is not constant, but usually depends on optical depth. In order to cal- culate this dependence, the NLTE radiative transfer problem within the struc- ture has to be solved, taking into account all excitation and ionization conditions within the structure. Several efforts have been undertaken in the past for such Cloud-model Inversion Techniques 223 Figure 3. Geometry of a two-dimensional cloud model slab. The incident radiation comes not only from below, but also from the sides of the structure. From Vial (1982). NLTE calculations, usually for the case of filaments or prominences. Such NLTE calculations started from the one-dimensional regime, where the cloud-like struc- ture is approximated by an infinite one-dimensional slab (see Fig. 1) or a cylin- der. We refer the reader to the works of Heasley et al. (1974), Heasley & Mihalas (1976), Heasley & Milkey (1976), Mozozhenko (1978), Fontenla & Rovira (1985), Heinzel et al. (1987), Gouttebroze et al. (1993), Heinzel (1995), Gouttebroze (2004) for an overview of such one-dimensional NLTE models. The philosophy of two-dimensional NLTE models is similar to the one-dimensional models, but now the cloud-like structure is replaced by a two-dimensional slab or cylinder which is infinite in the third dimension, allowing both vertical, as well as horizontal radiation transport (see Fig. 3). Furthermore, the incident radiation is treated as anisotropic and comes now not only from below, but also from the sides of the structure. We refer the reader to the works of Mihalas et al. (1978), Vial (1982), Paletou et al. (1993), Auer & Paletou (1994), Heinzel & Anzer (2001), Gouttebroze (2005) for an overview of such two-dimensional NLTE models. A general recipe for such NLTE models, which is modified according to the specific needs, i.e. the line profile used and the structure observed, has as follows: – The cloud-like structure is assumed to be a 1-D or 2-D slab or cylinder at a height H above the photosphere. This slab/cylinder can be considered to be either isothermal (e.g., Heinzel 1995) or isothermal and isobaric (e.g., Paletou et al. 1993). – The incident radiation comes in the case of 1-D models only from below and in the case of 2-D models also from the sides and determines the radiation field within the structure, that is all excitation and ionization conditions. – A multi-level atom plus continuum is assumed. The larger the number of atomic levels used, the more computationally demanding the method is. Complete or partial redistribution effects (CRD or PRD) are also assumed 224 Tziotziou depending on the formation properties of the line. Methods with CRD are computationally much faster so sometimes CRD is used but with simulated PRD effects taken into account (e.g., Heinzel 1995). – Some physical parameters are assigned to the slab/cylinder, like temperature T , bulk velocity v, geometrical thickness Z, electronic density Ne or pressure p. Calculations with electronic density are usually faster than calculations with pressure. – The radiative transfer statistical equilibrium equations are numerically solved and the population levels are found and hence the source function as a func- tion of optical depth for a set of selected physical parameters. Once the source function S is obtained as a function of optical depth, Eq. 1 can be solved in order to calculate the emerging observed profile from the structure which is going to be compared to the observed one. 3. Solving the Cloud Model Equations In the following subsections some of the methods used to solve the cloud model equations are reviewed. We remind the reader that whenever the background profile is needed, either the average profile of a quiet Sun region is taken or the average profile of a region close to the structure under study. 3.1. Solving the constant-S case with the “5-point” method Mein et al. (1996a) introduced the “5-point method” for solving the BCM equa- tion with constant S. According to this method five intensities of the observed and the background profile at wavelengths λ1, λ2 (blue wing of the observed profile), λ3, λ4 (red wing of the the observed profile) and the line-center wave- length λ0 are used for solving Eqs. 3 and 4. It is an iterative method that works as follows: – The line-center wavelength λ0 profile and background intensities are used for calculating S, where τ0, ∆λD and v are determined in a previous iteration. At the first step of the iteration some values can be assumed and S can be taken as equal to zero. – Profile and background intensities at wavelengths λ1 and λ3 are used for calculating a new τ0. – Afterwards a new ∆λD is calculated using the other two remaining wave- lengths λ2 and λ4. – Finally a new velocity is calculated from wavelengths λ1, λ2, λ3, and λ4 and then a reconstructed profile obtained using the derived parameters which is compared to the observed one. If any of the departures between the recon- structed and the observed profile is higher than an assumed small threshold value (i.e. 10−4) then the aforementioned procedure is repeated until con- vergence is achieved. If no convergence is gained after a certain number of iterations then it is assumed that no solution exists. We refer the reader to Mein et al. (1996a) for a detailed description of the ana- lytical equations described above. Cloud-model Inversion Techniques 225 3.2. Solving the constant-S case with an iterative least-square fit This method which was used by Alissandrakis et al. (1990) and further described in Tsiropoula et al. (1999) and Tziotziou et al. (2003) fits the observed contrast profile with a curve that results from an iterative least-square procedure for non-linear functions which is repeated until the departures between computed and observed profiles are minimized. The coefficients of the fitted curve are functions of the free parameters of the cloud model. At the beginning of the iteration procedure initial values have to be assumed for the free parameters and especially for the source function S which is usually estimated from some empirical approximate expressions that relate it to the line-center contrast. This method is very accurate and usually converges within a few iterations. The more observed wavelengths used within the profile, the better the determination of the ohysical parameters is. However, as Tziotziou et al. (2003) have reported, the velocity calculation can overshoot producing very high values, if the wings of the profile are not sufficiently covered by observed wavelengths. The suggested way to overcome the problem is to artificially add two extra contrast points near the continuum of the observed profile where the contrast should be in theory equal to zero. This iterative method can also be successfully used not only in the case of a constant source function S, but also for cases with a prescribed expres- sion for the source function, such as the parabolic expression of Eq. 7 used by Tsiropoula et al. (1999). 3.3. Solving the constant-S case with a constrained nonlinear least- square fitting technique The constrained nonlinear least-square fitting technique, used by Chae et al. (2006) for the inversion of a filament with BCM, was introduced by Chae et al. (1998). According to the method a) expectation values pei of the ith free param- eters, b) their uncertainties εi, as well as c) the data to fit are provided (M wave- lengths along the profile) and then a set ofN free parameters p = (p0, p1, ...pN−1) are sought, i.e. p = (S, τ0, λ0,∆λD), that minimize the function H(p) = Cobsj − C j (p) pi − p , (17) where Cobsj and C j are respectively the observed and calculated with the expectation values contrasts and σj the noise in the data. The first term of the sum H represents the data χ2, while the second term the expectation χ2 which regularizes the solution by constraining the probable range of free parameters. For very small values of εi the solution will not be much constrained by the data and will be close to the chosen set of expectation values pei , while for large values of εi it will be mostly constrained by the data and not by the expectation values. We refer the reader to Chae et al. (2006) for a detailed discussion of the effects of constrained fitting. 3.4. Solving the variable-S case Apart from the iterative least-square procedure described above which can be used when the source function varies in a prescribed way, Mein et al. (1996a) 226 Tziotziou have introduced also the “4-point method” for solving the case of a source func- tion that is described by the second order polynomial of Eq. 6. According to the method an intensity I ′(∆λ) can be defined as follows I ′(∆λ) = I(∆λ)− 1− [τ(∆λ) + 1] e−τ(∆λ) τ(∆λ) 2− [τ2(∆λ) + 2τ(∆λ) + 2] e−τ(∆λ) τ2(∆λ) S2 (18) and then the radiative transfer equation reduces to I ′(∆λ) = S0 + (I0 − S0) e −τ(∆λ) . (19) This equation can be solved now using the iteration procedure described in Sect. 3.1, with the modification that I(∆λ) is now replaced by I ′(∆λ) and that the source function calculation in the first step is replaced by the assumed theoretical relation for S given by Eq. 6. 3.5. Solving the DCM cases A method for solving the differential cloud model cases is the “3-optical depths” procedure introduced by Mein & Mein (1988). According to this procedure: – the zero velocity reference is obtained from the average profile over the whole field of view; – a value S is assumed between zero and the line-center intensity (in principle it could even work also for emitting clouds) and a function δτ(λ) is derived from Eq. 8. The latter is characterized by the maximum value δτ0 and δτ1, and the values δτ2 (see right panel of Fig. 2) which correspond to the half widths ∆λ1 and ∆λ2 respectively and are given by the following relations δτ1 = δτ0 e −(∆λ1/∆λD) δτ2 = δτ0 e −(∆λ2/∆λD) ; (20) – the code fits S and ∆λD by the conditions of Eq. 20 coupled with Eq. 8 and the solutions are assumed to be acceptable when the radial velocities v1 and v2, which correspond to widths ∆λ1 and ∆λ2 respectively and are defined as the displacement of the middle of these chords compared with the zero reference position, are not that different. When convergence is achieved the δτ(λ) curve is well represented by a Gaussian and the Doppler width ∆λD is independent of the chord ∆λ. 3.6. Solving the MCM case We refer the reader to the papers by Li & Ding (1992) and Li et al. (1993, 1994) for a detailed description of the methods and mathematical manipulations used for fitting observed profiles with the multi-cloud method, which unfortunately are not easy to concisely describe within a few lines. Cloud-model Inversion Techniques 227 3.7. Using NLTE Methods The most straightforward method for deriving the parameters of an observed structure with NLTE calculations would be the calculation of a grid of models for a wide range of the physical parameters used to describe the structure. How- ever, the calculation of such a grid is computationally demanding, especially in the case when a) a large number of atomic levels is assumed and/or b) partial redistribution effects (PRD) are taken into account and/or c) a two-dimensional geometry is considered. For such cases, either a very small grid of models is con- structed and thus only approximate values for the observed structure are derived or “test and try” methods are used where the user makes a “good guess” for the physical parameter values, proceeds to the respective NLTE calculations, com- pares the derived profile(s) with the observed one(s) and applies the necessary adjustments to the model parameters according to the derived results. However, nowadays the construction of a large grid of models, although time-demanding, becomes more of a common practice with the extended ca- pabilities of modern computers. We refer the reader to Molowny-Horas et al. (2001) and Tziotziou et al. (2001) for two such examples, both considering a one-dimensional isothermal slab for a cloud-like structure, which is the same filament observed and studied in the Hα in Ca II 8542 Å lines respectively. The general methodology used in the case of grids of models is the following: – a grid of synthetic line profiles for a wide range of model parameters is computed using NLTE calculations for the source function, as described in Sect. 2.6; – these synthetic profiles are convolved with the characteristics of the instru- ment used for the observations in order to simulate its effects on the observed profiles; – each observed profile is compared with the whole library of convolved syn- thetic profiles and the best fit is derived, that is the synthetic profile with the smallest departure, and hence the physical parameters that describe it; – an interpolation (linear or parabolic) between neighboring points in the pa- rameter space can also be used, for a more accurate quantitative determina- tion of the physical parameters that best describe the observed profile. Grid models based on NLTE calculations have many advantages since pre- ferred geometries, temperature structures, etc can be used, no iterations are required, errors can be easily defined from the parameter space and inversions are nowadays becoming faster with modern computers. 4. Validity of the Cloud Models The validity of the cloud model used for an inversion obviously strongly de- pends on a) the method used, b) the assumptions that were made for the model atmosphere describing the structure and c) the specific characteristics of the structure under study. Most of the reviewed papers in Sect. 5, concerning ap- plications of different cloud models, have extended discussions on the validity of the cloud model method and the results obtained, as well as the limitations 228 Tziotziou Figure 4. Two left panels: The calculated optical depth τ0,max and velocity v with BCM (constant source function) versus the assumed optical thickness. The dashed curve is the model, the solid curve the inversion. Two right panels: Same plots but with added Gaussian noise. From Mein et al. (1996a). Figure 5. Two left panels: The calculated optical depth τ0,max and velocity v using a cloud model with variable source function (see Eq. 6) depending only on line-center optical thickness versus the assumed optical thickness. Gaussian noise has also been taken into account. Two right panels: Same plots but for an over-estimated chromospheric background profile. FromMein et al. (1996a). of the method for the specific structure. However, below, some studies found in literature about the validity of cloud models are presented. Mein et al. (1996a) presented a rather detailed study about the validity of BCM (constant source function), as well as of cloud models with a variable source function as described in Eq. 6 (depending only on line-center optical thickness) by inverting theoretical profiles produced with a NLTE code and comparing the resulting model parameters from the inversion with the assumed ones. Fig- ure 4 (two left panels) shows the results of the inversion versus the assumed model optical thickness for the BCM inversion (constant source function). The calculated optical thickness is smaller, with the difference increasing with the thickness of the cloud, while the difference in velocity is no more than 20% and only for high values of the thickness. The figure shows that for optically thin structures there is practically no difference in the obtained results. When noise is included (Fig. 4, two right panels) the error increases for increasing thickness but the mean values stay almost the same. Again for optically thin structures the difference in the results is very small. Figure 5 (two left panels) shows the results of the inversion versus the assumed model optical thickness for a cloud model with variable source function Cloud-model Inversion Techniques 229 Figure 6. Comparison of the results obtained with method (a) represented by dots and with method (b) represented by asterisks (see text for de- tails of the methods) with the assumed model values (solid curve). From Heinzel et al. (1999). according to Eq. 6 depending only on line-center optical thickness with an added Gaussian noise; without noise the results are perfectly reproduced. We see that the differences are now almost negligible for a large range of the assumed optical thickness and the parameters are better determined. However, when taking a slightly brighter background (Fig. 5, two right panels) we see that the calculated values of the optical thickness are larger than the assumed ones, while the estimation of velocity is still rather good. This shows the importance of a correct background profile choice in cloud model calculations. Heinzel et al. (1999) has repeated the same exercise (inversion of NLTE synthetic profiles) for a cloud model with a variable source function according to Eq. 6 depending a) only on line-center optical thickness (method a) and b) on line-center optical thickness and velocity (method b). Some of their results are shown in Fig. 6. We see that although with method (a) there are some differences in the calculation of optical thickness, similarly to Mein et al. (1996a), method (b) gives exact solutions. Heinzel et al. (1999) have also applied the two methods in observed profiles of a dark arch filament. Figure 7 shows the comparison of the results obtained with the two methods. We refer also the reader selectively to a) Molowny-Horas et al. (2001) (Fig. 12 of their paper) for a comparison of inversion results for a filament with a NLTE method and a cloud model with a parabolic S, b) Schmieder et al. (2003) (Fig. 16 of their paper) for a comparison of inversion results for a filament with a NLTE 230 Tziotziou Figure 7. Comparison of the results obtained with the two methods (a) and (b) (see text for details) from the inversion of observed profiles of a dark arch filament. Scatter plots are shown for (1) optical thickness, (2) velocity (in km s−1), and (3) Doppler width (in Å). From Heinzel et al. (1999). method and a constant source function cloud model, c) Tsiropoula et al. (1999) (Fig. 5 of their paper) for a comparison of inversion results for mottles for cloud models with a constant and parabolic S, and d) Alissandrakis et al. (1990) (Fig. 8 to 11 of their paper) for a comparison of inversion results for an arch filament system with Beckers’ cloud model, the Doppler signal method and the differential cloud model. 5. Examples of Cloud Model Inversions Cloud models have been so far successfully applied for the derivation of the parameters of several cloud-like solar structures of the quiet Sun, such as mot- tles/spicules, as well of active region structures, such as arch filament systems (AFS), filaments, fibrils, flaring regions, surges etc. Below, some examples of such cloud model inversions are presented. 5.1. Application to filaments Filaments are commonly observed features that appear on the solar disc as dark long structures, lying along longitudinal magnetic field inversion lines. When observed on the limb they are bright and are called prominences. Filaments were some of the first solar structures to be studied with cloud models (see for example Maltby 1976, and references therein). Since then several authors used different cloud models to infer the dynamics and physical parameters of filaments. Mein, Mein & Wiik (1994), for example, studied the dynamical fine structure (threads) of a quiescent filament assuming a number of identical – except for the velocity – threads seen over the chromosphere and using a variant of BCM, while Schmieder et al. (1991) performed a similar study for threads by using the DCM. Morimoto & Kurokawa (2003) developed an interesting method applying BCM to determine the three-dimensional velocity fields of disappearing filaments. Molowny-Horas et al. (2001) and Tziotziou et al. (2001) studied the same filament observed in Hα and Ca II 8542 Å respectively with the Multichannel Subtractive Double Pass (MSDP) spectrograph (Mein 1991, 2002) mounted on the German solar telescope VTT in Tenerife. The filament was studied by using two very large grids of models in Hα and Ca II 8542 Å respectively which were constructed with the NLTE one-dimensional code MALI (Heinzel 1995), as Cloud-model Inversion Techniques 231 Figure 8. Top row: A filament observed in Hα and the two-dimensional parameter distributions derived with a Hα NLTE inversion using a grid of models. From Molowny-Horas et al. (2001). Bottom row: Same filament observed in Ca II 8542 Å and the two-dimensional parameter distributions derived with a Ca II 8542 Å NLTE grid model inversion. From Tziotziou et al. (2001). described in Sect. 2.6 Two-dimensional distributions of the physical parameters were obtained (see Fig. 8) which are not that similar due to the different physical formation properties and formation heights of the two lines. Schmieder et al. (2003) performed a similar NLTE grid inversion of a filament combined with a classical BCM inversion in a multi-wavelength study of filament channels. More recently, Chae et al. (2006) used Hα images obtained with a tunable filter and a BCM inversion to obtain detailed two-dimensional distributions of the physical parameters describing a quiescent filament. 5.2. Application to arch filaments (AFS) Arch filaments systems (AFSs) are low-lying dark loop-like structures formed during the emergence of solar magnetic flux in active regions. Georgakilas et al. (1990) have used the Doppler signal method described in Sect. 2.4 to study mass motions in AFSs observed in Hα, while Alissandrakis et al. (1990) and Tsiropoula et al. (1992) used the standard BCM to obtain the physical param- eters describing arch filament regions observed in the same line (see Fig. 9). An example of the use of the differential cloud model described in Sect. 2.2 for the 232 Tziotziou Figure 9. Contours maps of source function (top right panel) and the ve- locity (bottom right panel) derived with the cloud model for the AFS shown in Hα in the left panel of the figure. From Alissandrakis et al. (1990). study of the dynamics of AFSs can be found in Mein et al. (1996b) who applied the method to Hα observations from a two-telescope coordinated campaign. Fi- nally Mein et al. (2000) present a study of AFSs in Ca II 8542 Å using a fitting done with NLTE synthetic profile calculations – as described in Sect. 2.6 – with the one-dimensional MALI code (Heinzel 1995). 5.3. Application to fibrils Fibrils are small dark structures, belonging to the family of “chromospheric fine structures”, found in active regions surrounding plages or sunspots (penum- bral fibrils). One of the first studies of fibrils was conducted by Bray (1974) who compared observed profiles of fibrils with profiles calculated with BCM. Alissandrakis et al. (1990) used the standard BCM to obtain two-dimensional maps of several physical parameters distributions describing fibrils using Hα ob- servations obtained at Pic du Midi Observatory. Georgakilas et al. (2003) used filtergrams obtained at nine wavelengths along the Hα to study the Evershed flow in sunspots and reconstruct the three-dimensional velocity vector using the Doppler signal method (see Fig. 10), while Tsiropoula (2000) used also the Doppler signal method to determine LOS velocities of dark penumbral fibrils. 5.4. Application to mottles Mottles are small-scale structures (appearing both bright and dark) belonging also to the family of “chromospheric fine structures” and occurring at quiet Sun regions at the boundaries of supergranular cells. Mottles are believed to be the counterparts of limb spicules. They form groups called chains (when they are almost parallel to each other) or rosettes (when they are more or less Cloud-model Inversion Techniques 233 Figure 10. Image of a sunspot observed in Hα (a) and Doppler veloc- ity maps computed with the Doppler signal method from filtergrams in Hα±0.35Å (b), in Hα±0.5Å (c) and in Hα±0.75Å (d). The intensity gray scale bar corresponds to normalized intensities while the Doppler velocity gray scale bars to velocities in km s−1. From Georgakilas et al. (2003). circularly aligned, pointing radially outwards from a central core) depending on their location at the chromospheric network. First cloud studies of mottles started with a controversy about the ability of BCM to explain their contrast profiles. Bray (1973) and Loughhead (1973) who studied bright and dark mottles found that their contrast profiles are in good agreement with BCM. However, Loughhead (1973) used also BCM to deduce that it could not explain the contrast of individual bright and dark mottles observed in Hα near the limb, while Cram (1975) claimed that the parameters inferred from an application of BCM to contrast profiles of chromospheric fine structures are unreliable. Since then cloud models have been established as a reliable method for the study of physical parameters of mottles. Tsiropoula et al. (1999) studied several bright and dark mottles to derive physical parameters assuming a constant as well as a varying source function according to Eq. 7. Tsiropoula & Schmieder (1997) applied Beckers’ cloud model to determine physical parameters in Hα dark mottles of a rosette region, while Tsiropoula et al. (1993, 1994) studied the time evolution and fine structure of a rosette with BCM and first showed an alternating behaviour with time for velocity along mottles (see Fig. 11, left panel). A similar behaviour has also been found by Tziotziou et al. (2003) using BCM for a chain of mottles (see Fig. 11, right panel), while the dynamics of an enhanced network region were also explored in high resolution Hα images by Al et al. (2004). 234 Tziotziou Figure 11. Left panel: Cloud velocity as a function of position and time along the axis of a dark mottle belonging to a rosette. From Tsiropoula et al. (1994). Right panel: Cloud velocity as a function of position and time along the axis of a dark mottle belonging to a chain of mottles. White contours denote downward velocities, black upward velocities, while the thick gray curve is the zero velocity contour. From Tziotziou et al. (2003). 5.5. Application to post-flare loops Post-flare loops are loops generally observed between two-ribbon flares. We refer the reader to Bray & Loughhead (1983) for one of the first post-flare loop studies, who constructed theoretical curves based on the cloud model to fit observed contrast profiles of active region loops. Later Schmieder et al. (1988) and Heinzel et al. (1992) used a differential cloud model to study the structure and dynamics of post-flare loops. Heinzel et al. (1992) also constructed several isobaric and isothermal NLTE models of post-flare loops. Their results were compared by Gu et al. (1997) with two-dimensional maps of Hα post-flare loop cloud parameters obtained using a two-cloud model. Multi-cloud models like the ones described in Sect. 2.3 were used by Liang et al. (2004) to study Hα post- flare loops at the limb (see Fig. 12), by Gu & Ding (2002) for the study of Hα and Ca II 8542 Å post-flare loops and by Dun et al. (2000) for the study of Hβ post-flare loops. Liu & Ding (2001) obtained parameters of Hα post-flare loops using the modified cloud model method presented in Sect. 2.5 that eliminates the use of the background profile while Gu et al. (1992) presented an extensive study using BCM, the differential cloud model and a two-cloud model to study the time evolution of post-flare loops in two-ribbon flares. Finally, we refer the reader to Berlicki et al. (2005) who studied Hα ribbons during the gradual phase of a flare by comparing observed Hα profiles with a grid of synthetic Hα profiles calculated with the NLTE code MALI (Heinzel 1995) which was modified to account for flare conditions. 5.6. Application to surges Surges are large jet-like structures observed in opposite polarity flux emer- gence areas in active regions believed to be supported by magnetic reconnection. Gu et al. (1994) studied a surge on the limb observed in Hα, using a two-cloud model inversion as described in Sect. 2.3 (see Fig 13). The inversion result was Cloud-model Inversion Techniques 235 Figure 12. The distributions of Doppler velocity (in km s−1) derived with a multi-cloud method for Hα limb post-flare loops. Coordinates are in units of arcsec, dashed curves show red-shifted mass motions, while solid curves indicate blue-shifted ones. From Liang et al. (2004). Figure 13. An Hα filtergram of a surge (left panle) and the two-dimensional isocontours of Doppler velocity derived with a two-cloud model. Dashed curves refer to blue-shifted velocities (middle panel), solid curves red-shifted ones (right panel), while the unit of velocity is in km s−1. From Gu et al. (1994). detailed two-dimensional maps of the blue-shifted and red-shifted LOS velocity distributions. 6. Conclusions Several inversion techniques for chromospheric structures based on the cloud model have been reviewed. Cloud models are fast, quite reliable tools for in- ferring the physical parameters describing cloud-like chromospheric structures located above the solar photosphere and being illuminated by a background radi- ation. Cloud model techniques usually provide unique solutions and the results do not differ – in principle – qualitatively, especially for velocity, when using different cloud model techniques. However there can be quantitative differences arising from a) the selection of the background intensity, b) the physical condi- tions and especially the behaviour of the source function within the structure under study, and c) the particular model assumptions. Cloud models are mainly used for absorbing structures, however most of the techniques do work also for 236 Tziotziou line-center contrasts that are slightly higher than zero, indicating an important emission by the structure itself. Several different variants for cloud modeling have been proposed in litera- ture so far that mainly deal with different assumptions or calculations for the source functions and span from the simple BCM that assumes a constant source function to more complicated NLTE calculations of the radiation transfer and hence the source function within the structure. Accordingly, several different techniques – most of them iterative – have been proposed for solving cloud model equations. The latest and more accurate inversion techniques involve the construction of large grids of synthetic profiles, for different geometries and physical conditions, which are used for comparison with observed profiles. Cloud models can be applied with success to several, different in geometry and physical conditions, solar structures both of the quiet Sun, as well as of active regions. The resulting parameter inversions has shed light to several problems involving the physics and dynamics of chromospheric structures. The future of cloud modeling looks even more brighter. New high resolution data from telescopes combined with an always increasing computer power and the continuous development of new, state of the art, NLTE one-dimensional and two-dimensional cloud model codes will provide further detailed insights to the physics and dynamics that govern chromospheric structures. Acknowledgments. KT thanks G. Tsiropoula for constructive comments on the manuscript and acknowledges support by the organizers of the meeting and by Marie Curie European Reintegration Grant MERG-CT-2004-021626. References Al N., Bendlin C., Hirzberger J., Kneer F., Trujillo Bueno J., 2004, A&A 418, 1131 Alissandrakis C. E., Tsiropoula G., Mein P., 1990, A&A 230, 200 Auer L. H., Paletou F., 1994, A&A 284, 675 Beckers J. 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Spectral inversion techniques based on the cloud model are extremely useful for the study of properties and dynamics of various chromospheric cloud-like structures. Several inversion techniques are reviewed based on simple (constant source function) and more elaborated cloud models, as well as on grids of synthetic line profiles produced for a wide range of physical parameters by different NLTE codes. Several examples are shown of how such techniques can be used in different chromospheric lines, for the study of structures of the quiet chromosphere, such as mottles/spicules, as well as for active region structures such as fibrils, arch filament systems (AFS), filaments and flares.

Introduction Observed intensity line profiles are a function of several parameters describing the three-dimensional solar atmosphere, such as chemical abundance, density, temperature, velocity, magnetic field, microturbulence etc (which one would like to determine), as well as of wavelength, space (solar coordinates) and time. How- ever, due to the large number of parameters that an observed profile depends on, as well as data noise, model atmospheres have to be assumed in order to restrict the number of these unknown parameters. The term “inversion tech- niques” refers to the procedures used for inferring these model parameters from observed profiles. We refer the reader to Mein (2000) for an extended overview of inversion techniques. In this paper, we will review only a class of such inversion techniques known in the solar community as “cloud models”. Cloud models refer to models describing the transfer of radiation through structures located higher up from the solar photosphere, which represents the solar surface, resembling clouds on earth’s sky (see Fig. 1). Such cloud-like structures, when observed from above, would seem to mostly absorb the radi- ation coming from below, an absorption which mostly depends on the optical thickness of the cloud, that is the “transparency” of the cloud to the incident radiation and also on the physical parameters that describe it. The possibility of observed emission from such structures cannot, of course, be excluded when the radiation produced by the cloud-like structure is higher than the absorbed one. The aforementioned processes are described by the radiative transfer equation I(∆λ) = I0(∆λ) e −τ(∆λ) + ∫ τ(∆λ) −t(∆λ) dt , (1) where I(∆λ) is the observed intensity, I0(∆λ) is the reference profile emitted by the background (the incident radiation to the cloud from below), τ(∆λ) is the http://arxiv.org/abs/0704.1558v1 218 Tziotziou Figure 1. Geometry of the cloud model. D is the geometrical thickness of the cloud at height H above the solar surface and V its velocity. From Heinzel et al. (1999). optical thickness and S the source function which is a function of optical depth along the cloud. The first term of the right hand part of the equation represents the absorption of the incident radiation by the cloud, while the second term represents emission by the cloud itself. The simple cloud model method introduced by Beckers (1964) arose from the need to solve fast the radiation transfer equation and deduce the physical parameters that describe the observed structure. Beckers assumed that a) the structure is fully separated from the underlying chromosphere, b) the source function, radial velocity, Doppler width and the absorption coefficient are con- stant along the line-of-sight (hereafter LOS) and c) the background intensity is the same below the structure and the surrounding atmosphere; hence it can be extrapolated from a neighboring to the structure under study region. Under the above assumptions the radiative transfer equation is simplified to I(∆λ) = I0(∆λ) e −τ(∆λ) + S(1− e−τ(∆λ)) (2) and can be rewritten as C(∆λ) = I(∆λ)− I0(∆λ) I0(∆λ) I0(∆λ) (1− e−τ(∆λ)) , (3) where C(∆λ) defines the contrast profile. A Gaussian wavelength dependence is usually assumed for the optical depth as follows τ(∆λ) = τ0 e ∆λ−∆λI , (4) Cloud-model Inversion Techniques 219 where τ0 is the line center optical thickness, ∆λI = λ0v/c is the Doppler shift with λ0 being the line center wavelength, c the speed of light and ∆λD is the Doppler width. The latter depends on temperature T and microturbulent ve- locity ξt through the relationship ∆λD = ξ2t + , (5) where m is the atom rest mass. Other wavelength dependent profiles than the Gaussian one can also be assumed for the optical depth, e.g., a Voigt profile (Tsiropoula et al. 1999). The four adjustable parameters of the model are the source function S, the Doppler width ∆λD, the optical thickness τ0 and the LOS velocity v. All these parameters are assumed to be constant through the structure. There are some crucial assumptions concerning Beckers’ cloud model (hereafter BCM): – the uniform background radiation assumption, which is not always true es- pecially for cloud-like structures that do not reside above quiet Sun regions. Moreover, the background radiation plays an important role in the correct quantitative determination of the physical parameters. – the neglect of incident radiation, the effects of which are of course not directly considered in BCM, but does play an important role in non-Local Thermody- namic Equilibrium (hereafter NLTE) modeling, since it determines the radia- tion field within the structure, that is the excitation and ionization conditions and hence the source function. – the constant source function assumption which is not realistic especially in the optically thick case or not valid in the presence of large velocity gradients. However, the cloud model works quite well for a large number of optically thin structures and can provide useful, reasonable estimates for the physical param- eters that describe them. We refer the reader to Alissandrakis et al. (1990) for a detailed discussion on the validity conditions of BCM for different types of contrast profiles. 2. Cloud Model Variants Since the introduction of the BCM method several improvements have been suggested in the literature. When looking at the radiative transfer equation (Eq. 1), it is obvious that all efforts concentrate on a better description of the source function S which in BCM is considered to be constant. In the following subsections some of these suggested improvements are described. 2.1. Variable source function Mein et al. (1996a) considered a source function that is a function of optical depth and is approximated by a second-order polynomial St = S0 + S1 τ0,max τ0,max 220 Tziotziou Figure 2. Left: Geometry of the cloud model in the case of first-order differ- ential cloud model. From Heinzel et al. (1992). Right: The “3-optical depths” procedure for solving the differential cloud model case which is described in Sect. 3.5 (from Mein & Mein (1988)). with the optical depth at the center of the line τ0 taking values between 0 and the total optical thickness at line center τ0,max, while S0, S1 and S2 are functions of τ0,max. This formulation was further improved by Heinzel et al. (1999), who included also the effect of cloud motion by assuming that S0, S1 and S2 are now not only functions of τ0,max, but also depend on the velocity v of the structure. Tsiropoula et al. (1999) assumed the parabolic formula St = S0 1 + α as an initial condition for the variation of the source function with optical depth, where S0 is the source function at the middle of the structure, τ0 the optical depth at line center, and α a constant expressing the variation of the source function. However, their final results on the dependence of the source on optical depth were in good agreement with the results of Mein et al. (1996a). 2.2. Differential cloud models First and second order differential cloud models (hereafter DCM1 and DCM2) were introduced by Mein & Mein (1988) to account for fast mass flows observed on the disc, where BCM is not valid due to fluctuations of the background and strong velocity gradients along the LOS. DCM1 assumes that the source function S, temperature T and velocity v are constant within a small volume contained between two close lines of sight P and R (see Fig. 2, left panel). If we assume that the variation of the background profile is negligible (I0P ≃ I0R) for such close points then the differential cloud contrast profile can be written as C(P,R, λ) = IP (λ)− IR(λ) IR(λ) IR(λ) (1− e−δτ(λ)) (8) Cloud-model Inversion Techniques 221 δτ(λ) = δτ0 e λ− λ0 − vλ0/c , (9) where ∆λD is the Doppler width. The zero velocity reference wavelength λ0 is obtained by averaging over the whole field of view. DCM1 is a method for suppressing the use of the background radiation. If velocity shears are present between neighboring LOS then DCM2 can be used instead which requires the use of three neighboring LOS. We refer the reader to Mein & Mein (1988) for the precise formulation of DCM2 and to Table 1 of the same paper which summarizes the validity conditions, constrains and results of the two models in comparison to the classical BCM. 2.3. Multi-cloud models The multi-cloud model (Gu et al. 1992, 1996) – hereafter MCM – was intro- duced for the study of asymmetric, non-Gaussian profiles, such as line profiles of post-flare loops, prominences and surges and was based on the BCM and DCMs models. These asymmetric line profiles are assumed to be the result of overlapping of several symmetric Gaussian profiles along the LOS, formed in small radiative elements (clouds) which have a) different or identical physical properties and b) a source function and velocity independent of depth. The profile asymmetry mostly results from the relative Doppler shifts of the different clouds. The total intensity Iλ emitted by m clouds is then given by the relation Iλ = I0,λ e −τλ + Sj(1− e −τλ,j ) exp  , (10) where τλ,0 = 0, I0,λ is the background intensity, τλ = j=1 τλ,j is the total optical depth of the m clouds and τλ,j = τ0,j e λ− λ0 −∆λ0,j ∆λD,j ∆λ0,j = λ0vj/c, Sj, τ0,j, vj , ∆λD,j are respectively the optical depth, Doppler shift, source function, line-center thickness, velocity and Doppler width of the jth cloud. A somewhat similar in philosophy, two-cloud model method was used by Heinzel & Schmieder (1994) in their study of black and white mottles. It was assumed that the LOS intersects two mottles treated as two different clouds c1 and c2 with optical depths τ1 and τ2 respectively. Hence the emerging intensity from the lower mottle I1 is assumed to be the background incident intensity for the second upper mottle. Then, the equations describing the radiation transfer through the two mottles are I2(∆λ) = I1 e −τ2(∆λ) + Ic2(∆λ) I1(∆λ) = I0 e −τ1(∆λ) + Ic1(∆λ) , (12) 222 Tziotziou where I0 is the background chromospheric intensity and Ic1, Ic2 the intensity emitted by the two clouds respectively. The novelty of the method is that for the emitted by the clouds intensity, a grid of 140 NLTE models was used which was computed for prominence-like structures by Gouttebroze et al. (1993). So this method is a combination of MCM with NLTE source function calculations which will be further discussed in Section 2.6. 2.4. The Doppler signal method The Doppler signal method (Georgakilas et al. 1990; Tsiropoula 2000) can be used when filtergrams at two wavelengths −∆λ and +∆λ (blue and red side of the line) are available and a fast determination of mass motions is needed. Then the Doppler signal DS can be defined from the BCM equations as I − 2I0 − e−τ 2− e−τ − e−τ , (13) where ∆I = I(−∆λ) − I(+∆λ), I = I(−∆λ) + I(+∆λ) and τ± = τ(±∆λ). The Doppler signal DS has the same sign as velocity and can be used for a qualitative description of the velocity field. The left hand side of the above equation can be determined by the observations while the right hand clearly does not depend on the source function. Quantitative values for the velocity can be obtained when τ0 < 1; then the Doppler signal equation reduces to τ− − τ+ τ− + τ+ and the velocity v – once DS is calculated from the observations and a value of the Doppler width ∆λD is obtained from the literature or assumed – is given by the equation 1 +DS . (15) 2.5. Avoiding the background profile Liu & Ding (2001) in order to avoid the use of the background profile needed in BCM assumed that it is symmetric, that is I0(∆λ) = I0(−∆λ). Then it can easily be shown that we can obtain the relationship ∆I(∆λ) = I(∆λ)− I(−∆λ) = [I(∆λ)− S][1− eτ(∆λ)−τ(−∆λ)] , (16) which does not require the use of the background for the derivation of the phys- ical parameters. 2.6. NLTE methods As Eq. 1 shows, in the general case, the source function S within a cloud-like structure is not constant, but usually depends on optical depth. In order to cal- culate this dependence, the NLTE radiative transfer problem within the struc- ture has to be solved, taking into account all excitation and ionization conditions within the structure. Several efforts have been undertaken in the past for such Cloud-model Inversion Techniques 223 Figure 3. Geometry of a two-dimensional cloud model slab. The incident radiation comes not only from below, but also from the sides of the structure. From Vial (1982). NLTE calculations, usually for the case of filaments or prominences. Such NLTE calculations started from the one-dimensional regime, where the cloud-like struc- ture is approximated by an infinite one-dimensional slab (see Fig. 1) or a cylin- der. We refer the reader to the works of Heasley et al. (1974), Heasley & Mihalas (1976), Heasley & Milkey (1976), Mozozhenko (1978), Fontenla & Rovira (1985), Heinzel et al. (1987), Gouttebroze et al. (1993), Heinzel (1995), Gouttebroze (2004) for an overview of such one-dimensional NLTE models. The philosophy of two-dimensional NLTE models is similar to the one-dimensional models, but now the cloud-like structure is replaced by a two-dimensional slab or cylinder which is infinite in the third dimension, allowing both vertical, as well as horizontal radiation transport (see Fig. 3). Furthermore, the incident radiation is treated as anisotropic and comes now not only from below, but also from the sides of the structure. We refer the reader to the works of Mihalas et al. (1978), Vial (1982), Paletou et al. (1993), Auer & Paletou (1994), Heinzel & Anzer (2001), Gouttebroze (2005) for an overview of such two-dimensional NLTE models. A general recipe for such NLTE models, which is modified according to the specific needs, i.e. the line profile used and the structure observed, has as follows: – The cloud-like structure is assumed to be a 1-D or 2-D slab or cylinder at a height H above the photosphere. This slab/cylinder can be considered to be either isothermal (e.g., Heinzel 1995) or isothermal and isobaric (e.g., Paletou et al. 1993). – The incident radiation comes in the case of 1-D models only from below and in the case of 2-D models also from the sides and determines the radiation field within the structure, that is all excitation and ionization conditions. – A multi-level atom plus continuum is assumed. The larger the number of atomic levels used, the more computationally demanding the method is. Complete or partial redistribution effects (CRD or PRD) are also assumed 224 Tziotziou depending on the formation properties of the line. Methods with CRD are computationally much faster so sometimes CRD is used but with simulated PRD effects taken into account (e.g., Heinzel 1995). – Some physical parameters are assigned to the slab/cylinder, like temperature T , bulk velocity v, geometrical thickness Z, electronic density Ne or pressure p. Calculations with electronic density are usually faster than calculations with pressure. – The radiative transfer statistical equilibrium equations are numerically solved and the population levels are found and hence the source function as a func- tion of optical depth for a set of selected physical parameters. Once the source function S is obtained as a function of optical depth, Eq. 1 can be solved in order to calculate the emerging observed profile from the structure which is going to be compared to the observed one. 3. Solving the Cloud Model Equations In the following subsections some of the methods used to solve the cloud model equations are reviewed. We remind the reader that whenever the background profile is needed, either the average profile of a quiet Sun region is taken or the average profile of a region close to the structure under study. 3.1. Solving the constant-S case with the “5-point” method Mein et al. (1996a) introduced the “5-point method” for solving the BCM equa- tion with constant S. According to this method five intensities of the observed and the background profile at wavelengths λ1, λ2 (blue wing of the observed profile), λ3, λ4 (red wing of the the observed profile) and the line-center wave- length λ0 are used for solving Eqs. 3 and 4. It is an iterative method that works as follows: – The line-center wavelength λ0 profile and background intensities are used for calculating S, where τ0, ∆λD and v are determined in a previous iteration. At the first step of the iteration some values can be assumed and S can be taken as equal to zero. – Profile and background intensities at wavelengths λ1 and λ3 are used for calculating a new τ0. – Afterwards a new ∆λD is calculated using the other two remaining wave- lengths λ2 and λ4. – Finally a new velocity is calculated from wavelengths λ1, λ2, λ3, and λ4 and then a reconstructed profile obtained using the derived parameters which is compared to the observed one. If any of the departures between the recon- structed and the observed profile is higher than an assumed small threshold value (i.e. 10−4) then the aforementioned procedure is repeated until con- vergence is achieved. If no convergence is gained after a certain number of iterations then it is assumed that no solution exists. We refer the reader to Mein et al. (1996a) for a detailed description of the ana- lytical equations described above. Cloud-model Inversion Techniques 225 3.2. Solving the constant-S case with an iterative least-square fit This method which was used by Alissandrakis et al. (1990) and further described in Tsiropoula et al. (1999) and Tziotziou et al. (2003) fits the observed contrast profile with a curve that results from an iterative least-square procedure for non-linear functions which is repeated until the departures between computed and observed profiles are minimized. The coefficients of the fitted curve are functions of the free parameters of the cloud model. At the beginning of the iteration procedure initial values have to be assumed for the free parameters and especially for the source function S which is usually estimated from some empirical approximate expressions that relate it to the line-center contrast. This method is very accurate and usually converges within a few iterations. The more observed wavelengths used within the profile, the better the determination of the ohysical parameters is. However, as Tziotziou et al. (2003) have reported, the velocity calculation can overshoot producing very high values, if the wings of the profile are not sufficiently covered by observed wavelengths. The suggested way to overcome the problem is to artificially add two extra contrast points near the continuum of the observed profile where the contrast should be in theory equal to zero. This iterative method can also be successfully used not only in the case of a constant source function S, but also for cases with a prescribed expres- sion for the source function, such as the parabolic expression of Eq. 7 used by Tsiropoula et al. (1999). 3.3. Solving the constant-S case with a constrained nonlinear least- square fitting technique The constrained nonlinear least-square fitting technique, used by Chae et al. (2006) for the inversion of a filament with BCM, was introduced by Chae et al. (1998). According to the method a) expectation values pei of the ith free param- eters, b) their uncertainties εi, as well as c) the data to fit are provided (M wave- lengths along the profile) and then a set ofN free parameters p = (p0, p1, ...pN−1) are sought, i.e. p = (S, τ0, λ0,∆λD), that minimize the function H(p) = Cobsj − C j (p) pi − p , (17) where Cobsj and C j are respectively the observed and calculated with the expectation values contrasts and σj the noise in the data. The first term of the sum H represents the data χ2, while the second term the expectation χ2 which regularizes the solution by constraining the probable range of free parameters. For very small values of εi the solution will not be much constrained by the data and will be close to the chosen set of expectation values pei , while for large values of εi it will be mostly constrained by the data and not by the expectation values. We refer the reader to Chae et al. (2006) for a detailed discussion of the effects of constrained fitting. 3.4. Solving the variable-S case Apart from the iterative least-square procedure described above which can be used when the source function varies in a prescribed way, Mein et al. (1996a) 226 Tziotziou have introduced also the “4-point method” for solving the case of a source func- tion that is described by the second order polynomial of Eq. 6. According to the method an intensity I ′(∆λ) can be defined as follows I ′(∆λ) = I(∆λ)− 1− [τ(∆λ) + 1] e−τ(∆λ) τ(∆λ) 2− [τ2(∆λ) + 2τ(∆λ) + 2] e−τ(∆λ) τ2(∆λ) S2 (18) and then the radiative transfer equation reduces to I ′(∆λ) = S0 + (I0 − S0) e −τ(∆λ) . (19) This equation can be solved now using the iteration procedure described in Sect. 3.1, with the modification that I(∆λ) is now replaced by I ′(∆λ) and that the source function calculation in the first step is replaced by the assumed theoretical relation for S given by Eq. 6. 3.5. Solving the DCM cases A method for solving the differential cloud model cases is the “3-optical depths” procedure introduced by Mein & Mein (1988). According to this procedure: – the zero velocity reference is obtained from the average profile over the whole field of view; – a value S is assumed between zero and the line-center intensity (in principle it could even work also for emitting clouds) and a function δτ(λ) is derived from Eq. 8. The latter is characterized by the maximum value δτ0 and δτ1, and the values δτ2 (see right panel of Fig. 2) which correspond to the half widths ∆λ1 and ∆λ2 respectively and are given by the following relations δτ1 = δτ0 e −(∆λ1/∆λD) δτ2 = δτ0 e −(∆λ2/∆λD) ; (20) – the code fits S and ∆λD by the conditions of Eq. 20 coupled with Eq. 8 and the solutions are assumed to be acceptable when the radial velocities v1 and v2, which correspond to widths ∆λ1 and ∆λ2 respectively and are defined as the displacement of the middle of these chords compared with the zero reference position, are not that different. When convergence is achieved the δτ(λ) curve is well represented by a Gaussian and the Doppler width ∆λD is independent of the chord ∆λ. 3.6. Solving the MCM case We refer the reader to the papers by Li & Ding (1992) and Li et al. (1993, 1994) for a detailed description of the methods and mathematical manipulations used for fitting observed profiles with the multi-cloud method, which unfortunately are not easy to concisely describe within a few lines. Cloud-model Inversion Techniques 227 3.7. Using NLTE Methods The most straightforward method for deriving the parameters of an observed structure with NLTE calculations would be the calculation of a grid of models for a wide range of the physical parameters used to describe the structure. How- ever, the calculation of such a grid is computationally demanding, especially in the case when a) a large number of atomic levels is assumed and/or b) partial redistribution effects (PRD) are taken into account and/or c) a two-dimensional geometry is considered. For such cases, either a very small grid of models is con- structed and thus only approximate values for the observed structure are derived or “test and try” methods are used where the user makes a “good guess” for the physical parameter values, proceeds to the respective NLTE calculations, com- pares the derived profile(s) with the observed one(s) and applies the necessary adjustments to the model parameters according to the derived results. However, nowadays the construction of a large grid of models, although time-demanding, becomes more of a common practice with the extended ca- pabilities of modern computers. We refer the reader to Molowny-Horas et al. (2001) and Tziotziou et al. (2001) for two such examples, both considering a one-dimensional isothermal slab for a cloud-like structure, which is the same filament observed and studied in the Hα in Ca II 8542 Å lines respectively. The general methodology used in the case of grids of models is the following: – a grid of synthetic line profiles for a wide range of model parameters is computed using NLTE calculations for the source function, as described in Sect. 2.6; – these synthetic profiles are convolved with the characteristics of the instru- ment used for the observations in order to simulate its effects on the observed profiles; – each observed profile is compared with the whole library of convolved syn- thetic profiles and the best fit is derived, that is the synthetic profile with the smallest departure, and hence the physical parameters that describe it; – an interpolation (linear or parabolic) between neighboring points in the pa- rameter space can also be used, for a more accurate quantitative determina- tion of the physical parameters that best describe the observed profile. Grid models based on NLTE calculations have many advantages since pre- ferred geometries, temperature structures, etc can be used, no iterations are required, errors can be easily defined from the parameter space and inversions are nowadays becoming faster with modern computers. 4. Validity of the Cloud Models The validity of the cloud model used for an inversion obviously strongly de- pends on a) the method used, b) the assumptions that were made for the model atmosphere describing the structure and c) the specific characteristics of the structure under study. Most of the reviewed papers in Sect. 5, concerning ap- plications of different cloud models, have extended discussions on the validity of the cloud model method and the results obtained, as well as the limitations 228 Tziotziou Figure 4. Two left panels: The calculated optical depth τ0,max and velocity v with BCM (constant source function) versus the assumed optical thickness. The dashed curve is the model, the solid curve the inversion. Two right panels: Same plots but with added Gaussian noise. From Mein et al. (1996a). Figure 5. Two left panels: The calculated optical depth τ0,max and velocity v using a cloud model with variable source function (see Eq. 6) depending only on line-center optical thickness versus the assumed optical thickness. Gaussian noise has also been taken into account. Two right panels: Same plots but for an over-estimated chromospheric background profile. FromMein et al. (1996a). of the method for the specific structure. However, below, some studies found in literature about the validity of cloud models are presented. Mein et al. (1996a) presented a rather detailed study about the validity of BCM (constant source function), as well as of cloud models with a variable source function as described in Eq. 6 (depending only on line-center optical thickness) by inverting theoretical profiles produced with a NLTE code and comparing the resulting model parameters from the inversion with the assumed ones. Fig- ure 4 (two left panels) shows the results of the inversion versus the assumed model optical thickness for the BCM inversion (constant source function). The calculated optical thickness is smaller, with the difference increasing with the thickness of the cloud, while the difference in velocity is no more than 20% and only for high values of the thickness. The figure shows that for optically thin structures there is practically no difference in the obtained results. When noise is included (Fig. 4, two right panels) the error increases for increasing thickness but the mean values stay almost the same. Again for optically thin structures the difference in the results is very small. Figure 5 (two left panels) shows the results of the inversion versus the assumed model optical thickness for a cloud model with variable source function Cloud-model Inversion Techniques 229 Figure 6. Comparison of the results obtained with method (a) represented by dots and with method (b) represented by asterisks (see text for de- tails of the methods) with the assumed model values (solid curve). From Heinzel et al. (1999). according to Eq. 6 depending only on line-center optical thickness with an added Gaussian noise; without noise the results are perfectly reproduced. We see that the differences are now almost negligible for a large range of the assumed optical thickness and the parameters are better determined. However, when taking a slightly brighter background (Fig. 5, two right panels) we see that the calculated values of the optical thickness are larger than the assumed ones, while the estimation of velocity is still rather good. This shows the importance of a correct background profile choice in cloud model calculations. Heinzel et al. (1999) has repeated the same exercise (inversion of NLTE synthetic profiles) for a cloud model with a variable source function according to Eq. 6 depending a) only on line-center optical thickness (method a) and b) on line-center optical thickness and velocity (method b). Some of their results are shown in Fig. 6. We see that although with method (a) there are some differences in the calculation of optical thickness, similarly to Mein et al. (1996a), method (b) gives exact solutions. Heinzel et al. (1999) have also applied the two methods in observed profiles of a dark arch filament. Figure 7 shows the comparison of the results obtained with the two methods. We refer also the reader selectively to a) Molowny-Horas et al. (2001) (Fig. 12 of their paper) for a comparison of inversion results for a filament with a NLTE method and a cloud model with a parabolic S, b) Schmieder et al. (2003) (Fig. 16 of their paper) for a comparison of inversion results for a filament with a NLTE 230 Tziotziou Figure 7. Comparison of the results obtained with the two methods (a) and (b) (see text for details) from the inversion of observed profiles of a dark arch filament. Scatter plots are shown for (1) optical thickness, (2) velocity (in km s−1), and (3) Doppler width (in Å). From Heinzel et al. (1999). method and a constant source function cloud model, c) Tsiropoula et al. (1999) (Fig. 5 of their paper) for a comparison of inversion results for mottles for cloud models with a constant and parabolic S, and d) Alissandrakis et al. (1990) (Fig. 8 to 11 of their paper) for a comparison of inversion results for an arch filament system with Beckers’ cloud model, the Doppler signal method and the differential cloud model. 5. Examples of Cloud Model Inversions Cloud models have been so far successfully applied for the derivation of the parameters of several cloud-like solar structures of the quiet Sun, such as mot- tles/spicules, as well of active region structures, such as arch filament systems (AFS), filaments, fibrils, flaring regions, surges etc. Below, some examples of such cloud model inversions are presented. 5.1. Application to filaments Filaments are commonly observed features that appear on the solar disc as dark long structures, lying along longitudinal magnetic field inversion lines. When observed on the limb they are bright and are called prominences. Filaments were some of the first solar structures to be studied with cloud models (see for example Maltby 1976, and references therein). Since then several authors used different cloud models to infer the dynamics and physical parameters of filaments. Mein, Mein & Wiik (1994), for example, studied the dynamical fine structure (threads) of a quiescent filament assuming a number of identical – except for the velocity – threads seen over the chromosphere and using a variant of BCM, while Schmieder et al. (1991) performed a similar study for threads by using the DCM. Morimoto & Kurokawa (2003) developed an interesting method applying BCM to determine the three-dimensional velocity fields of disappearing filaments. Molowny-Horas et al. (2001) and Tziotziou et al. (2001) studied the same filament observed in Hα and Ca II 8542 Å respectively with the Multichannel Subtractive Double Pass (MSDP) spectrograph (Mein 1991, 2002) mounted on the German solar telescope VTT in Tenerife. The filament was studied by using two very large grids of models in Hα and Ca II 8542 Å respectively which were constructed with the NLTE one-dimensional code MALI (Heinzel 1995), as Cloud-model Inversion Techniques 231 Figure 8. Top row: A filament observed in Hα and the two-dimensional parameter distributions derived with a Hα NLTE inversion using a grid of models. From Molowny-Horas et al. (2001). Bottom row: Same filament observed in Ca II 8542 Å and the two-dimensional parameter distributions derived with a Ca II 8542 Å NLTE grid model inversion. From Tziotziou et al. (2001). described in Sect. 2.6 Two-dimensional distributions of the physical parameters were obtained (see Fig. 8) which are not that similar due to the different physical formation properties and formation heights of the two lines. Schmieder et al. (2003) performed a similar NLTE grid inversion of a filament combined with a classical BCM inversion in a multi-wavelength study of filament channels. More recently, Chae et al. (2006) used Hα images obtained with a tunable filter and a BCM inversion to obtain detailed two-dimensional distributions of the physical parameters describing a quiescent filament. 5.2. Application to arch filaments (AFS) Arch filaments systems (AFSs) are low-lying dark loop-like structures formed during the emergence of solar magnetic flux in active regions. Georgakilas et al. (1990) have used the Doppler signal method described in Sect. 2.4 to study mass motions in AFSs observed in Hα, while Alissandrakis et al. (1990) and Tsiropoula et al. (1992) used the standard BCM to obtain the physical param- eters describing arch filament regions observed in the same line (see Fig. 9). An example of the use of the differential cloud model described in Sect. 2.2 for the 232 Tziotziou Figure 9. Contours maps of source function (top right panel) and the ve- locity (bottom right panel) derived with the cloud model for the AFS shown in Hα in the left panel of the figure. From Alissandrakis et al. (1990). study of the dynamics of AFSs can be found in Mein et al. (1996b) who applied the method to Hα observations from a two-telescope coordinated campaign. Fi- nally Mein et al. (2000) present a study of AFSs in Ca II 8542 Å using a fitting done with NLTE synthetic profile calculations – as described in Sect. 2.6 – with the one-dimensional MALI code (Heinzel 1995). 5.3. Application to fibrils Fibrils are small dark structures, belonging to the family of “chromospheric fine structures”, found in active regions surrounding plages or sunspots (penum- bral fibrils). One of the first studies of fibrils was conducted by Bray (1974) who compared observed profiles of fibrils with profiles calculated with BCM. Alissandrakis et al. (1990) used the standard BCM to obtain two-dimensional maps of several physical parameters distributions describing fibrils using Hα ob- servations obtained at Pic du Midi Observatory. Georgakilas et al. (2003) used filtergrams obtained at nine wavelengths along the Hα to study the Evershed flow in sunspots and reconstruct the three-dimensional velocity vector using the Doppler signal method (see Fig. 10), while Tsiropoula (2000) used also the Doppler signal method to determine LOS velocities of dark penumbral fibrils. 5.4. Application to mottles Mottles are small-scale structures (appearing both bright and dark) belonging also to the family of “chromospheric fine structures” and occurring at quiet Sun regions at the boundaries of supergranular cells. Mottles are believed to be the counterparts of limb spicules. They form groups called chains (when they are almost parallel to each other) or rosettes (when they are more or less Cloud-model Inversion Techniques 233 Figure 10. Image of a sunspot observed in Hα (a) and Doppler veloc- ity maps computed with the Doppler signal method from filtergrams in Hα±0.35Å (b), in Hα±0.5Å (c) and in Hα±0.75Å (d). The intensity gray scale bar corresponds to normalized intensities while the Doppler velocity gray scale bars to velocities in km s−1. From Georgakilas et al. (2003). circularly aligned, pointing radially outwards from a central core) depending on their location at the chromospheric network. First cloud studies of mottles started with a controversy about the ability of BCM to explain their contrast profiles. Bray (1973) and Loughhead (1973) who studied bright and dark mottles found that their contrast profiles are in good agreement with BCM. However, Loughhead (1973) used also BCM to deduce that it could not explain the contrast of individual bright and dark mottles observed in Hα near the limb, while Cram (1975) claimed that the parameters inferred from an application of BCM to contrast profiles of chromospheric fine structures are unreliable. Since then cloud models have been established as a reliable method for the study of physical parameters of mottles. Tsiropoula et al. (1999) studied several bright and dark mottles to derive physical parameters assuming a constant as well as a varying source function according to Eq. 7. Tsiropoula & Schmieder (1997) applied Beckers’ cloud model to determine physical parameters in Hα dark mottles of a rosette region, while Tsiropoula et al. (1993, 1994) studied the time evolution and fine structure of a rosette with BCM and first showed an alternating behaviour with time for velocity along mottles (see Fig. 11, left panel). A similar behaviour has also been found by Tziotziou et al. (2003) using BCM for a chain of mottles (see Fig. 11, right panel), while the dynamics of an enhanced network region were also explored in high resolution Hα images by Al et al. (2004). 234 Tziotziou Figure 11. Left panel: Cloud velocity as a function of position and time along the axis of a dark mottle belonging to a rosette. From Tsiropoula et al. (1994). Right panel: Cloud velocity as a function of position and time along the axis of a dark mottle belonging to a chain of mottles. White contours denote downward velocities, black upward velocities, while the thick gray curve is the zero velocity contour. From Tziotziou et al. (2003). 5.5. Application to post-flare loops Post-flare loops are loops generally observed between two-ribbon flares. We refer the reader to Bray & Loughhead (1983) for one of the first post-flare loop studies, who constructed theoretical curves based on the cloud model to fit observed contrast profiles of active region loops. Later Schmieder et al. (1988) and Heinzel et al. (1992) used a differential cloud model to study the structure and dynamics of post-flare loops. Heinzel et al. (1992) also constructed several isobaric and isothermal NLTE models of post-flare loops. Their results were compared by Gu et al. (1997) with two-dimensional maps of Hα post-flare loop cloud parameters obtained using a two-cloud model. Multi-cloud models like the ones described in Sect. 2.3 were used by Liang et al. (2004) to study Hα post- flare loops at the limb (see Fig. 12), by Gu & Ding (2002) for the study of Hα and Ca II 8542 Å post-flare loops and by Dun et al. (2000) for the study of Hβ post-flare loops. Liu & Ding (2001) obtained parameters of Hα post-flare loops using the modified cloud model method presented in Sect. 2.5 that eliminates the use of the background profile while Gu et al. (1992) presented an extensive study using BCM, the differential cloud model and a two-cloud model to study the time evolution of post-flare loops in two-ribbon flares. Finally, we refer the reader to Berlicki et al. (2005) who studied Hα ribbons during the gradual phase of a flare by comparing observed Hα profiles with a grid of synthetic Hα profiles calculated with the NLTE code MALI (Heinzel 1995) which was modified to account for flare conditions. 5.6. Application to surges Surges are large jet-like structures observed in opposite polarity flux emer- gence areas in active regions believed to be supported by magnetic reconnection. Gu et al. (1994) studied a surge on the limb observed in Hα, using a two-cloud model inversion as described in Sect. 2.3 (see Fig 13). The inversion result was Cloud-model Inversion Techniques 235 Figure 12. The distributions of Doppler velocity (in km s−1) derived with a multi-cloud method for Hα limb post-flare loops. Coordinates are in units of arcsec, dashed curves show red-shifted mass motions, while solid curves indicate blue-shifted ones. From Liang et al. (2004). Figure 13. An Hα filtergram of a surge (left panle) and the two-dimensional isocontours of Doppler velocity derived with a two-cloud model. Dashed curves refer to blue-shifted velocities (middle panel), solid curves red-shifted ones (right panel), while the unit of velocity is in km s−1. From Gu et al. (1994). detailed two-dimensional maps of the blue-shifted and red-shifted LOS velocity distributions. 6. Conclusions Several inversion techniques for chromospheric structures based on the cloud model have been reviewed. Cloud models are fast, quite reliable tools for in- ferring the physical parameters describing cloud-like chromospheric structures located above the solar photosphere and being illuminated by a background radi- ation. Cloud model techniques usually provide unique solutions and the results do not differ – in principle – qualitatively, especially for velocity, when using different cloud model techniques. However there can be quantitative differences arising from a) the selection of the background intensity, b) the physical condi- tions and especially the behaviour of the source function within the structure under study, and c) the particular model assumptions. Cloud models are mainly used for absorbing structures, however most of the techniques do work also for 236 Tziotziou line-center contrasts that are slightly higher than zero, indicating an important emission by the structure itself. Several different variants for cloud modeling have been proposed in litera- ture so far that mainly deal with different assumptions or calculations for the source functions and span from the simple BCM that assumes a constant source function to more complicated NLTE calculations of the radiation transfer and hence the source function within the structure. Accordingly, several different techniques – most of them iterative – have been proposed for solving cloud model equations. The latest and more accurate inversion techniques involve the construction of large grids of synthetic profiles, for different geometries and physical conditions, which are used for comparison with observed profiles. Cloud models can be applied with success to several, different in geometry and physical conditions, solar structures both of the quiet Sun, as well as of active regions. 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Astronomy & Astrophysics manuscript no. paper˙fin c© ESO 2018 October 30, 2018 Dust covering factor, silicate emission and star formation in luminous QSOs R. Maiolino1, O. Shemmer2, M. Imanishi3, Hagai Netzer4, E. Oliva5, D. Lutz6, and E. Sturm6 1 INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte Porzio Catone, Italy 2 Department of Astronomy and Astrophysics, 525 Davey Laboratory, Pennsylvania State University, University Park, PA 16802 3 National Astronomical Observatory, 2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan 4 School of Physics and Astronomy and the Wise Observatory , Tel-Aviv University, Tel-Aviv 69978, Israel 5 INAF - Telescopio Nazionale Galileo, PO Box 565, 38700 Santa Cruz de La Palma, Tenerife, Spain 6 Max-Planck-Institut für Extraterrestrische Physik, D-85741 Garching, Germany Received ; accepted ABSTRACT We present Spitzer IRS low resolution, mid-IR spectra of a sample of 25 high luminosity QSOs at 2<z<3.5. When combined with archival IRS observations of local, low luminosity type-I active galactic nuclei (AGNs), the sample spans five orders of magnitude in luminosity. We find that the continuum dust thermal emission at λrest = 6.7µm is correlated with the optical luminosity, following the non-linear relation λLλ(6.7µm) ∝ λLλ(5100Å) 0.82 . We also find an anti correlation between λLλ(6.7µm)/λLλ(5100Å) and the [Oiii]λ5007 line luminosity. These effects are interpreted as a decreasing covering factor of the circumnuclear dust as a function of luminosity. Such a result is in agreement with the decreasing fraction of absorbed AGNs as a function of luminosity recently found in various surveys. In particular, while X-ray surveys find a decreasing covering factor of the absorbing gas as a function of luminosity, our data provides an independent and complementary confirmation by finding a decreasing covering factor of dust. We clearly detect the silicate emission feature in the average spectrum, but also in four individual objects. These are the Silicate emission in the most luminous objects obtained so far. When combined with the silicate emission observed in local, low luminosity type-I AGNs, we find that the silicate emission strength is correlated with luminosity. The silicate strength of all type-I AGNs also follows a positive correlation with the black hole mass and with the accretion rate. The Polycyclic Aromatic Hydrocarbon (PAH) emission features, expected from starburst activity, are not detected in the average spectrum of luminous, high-z QSOs. The upper limit inferred from the average spectrum points to a ratio between PAH luminosity and QSO optical luminosity significantly lower than observed in lower luminosity AGNs, implying that the correlation between star formation rate and AGN power saturates at high luminosities. Key words. infrared: galaxies – galaxies: nuclei – galaxies: active – galaxies: Seyfert – galaxies: starburst – quasars: general 1. Introduction The mid-IR (MIR) spectrum of AGNs contains a wealth of in- formation which is crucial to the understanding of their inner region. The observed prominent continuum emission is due to circumnuclear dust heated to a temperature of several hundred degrees by the nuclear, primary optical/UV/X-ray source (pri- marily the central accretion disk); therefore, the MIR contin- uum provides information on the amount and/or covering factor of the circumnuclear dust. The MIR region is also rich of sev- eral emission features which are important tracers of the ISM. Among the dust features, the Polycyclic-Aromatic-Hydrocarbon bands (PAH, whose most prominent feature is at ∼ 7.7µm) are emitted by very small carbon grains excited in the Photo Dissociation Regions, that are tracers of star forming activity (although PAHs may not be reliable SF tracers for compact HII regions or heavily embedded starbursts, Peeters et al., 2004; Förster Schreiber et al., 2004). Additional MIR dust features are the Silicate bands at ∼ 10µm and at ∼ 18µm, often seen in ab- sorption in obscured AGNs and in luminous IR galaxies. Major steps forward in this field were achieved thanks to the Spitzer Space Observatory, and to its infrared spectrome- ter, IRS, which allows a detailed investigation of the MIR spec- tral features in a large number of sources. In particular, IRS Send offprint requests to: R. Maiolino allowed the detection of MIR emission lines in several AGNs (e.g. Armus et al., 2004; Haas et al., 2005; Sturm et al., 2006a; Weedman et al., 2005), the detection of PAHs in local PG QSOs (Schweitzer et al., 2006), the first detection of the Silicate fea- ture in emission (Siebenmorgen et al., 2005; Hao et al., 2005), as well as detailed studies of the silicate strength in various classes of sources (Spoon et al., 2007; Hao et al., 2007; Shi et al., 2006; Imanishi et al., 2007). However, most of the current Spitzer IRS studies have fo- cused on local and modest luminosity AGNs (including low luminosity QSOs), with the exception of a few bright, lensed objects at high redshift (Soifer et al., 2004; Teplitz et al., 2006; Lutz et al., 2007). We have obtained short IRS integrations of a sample of 25 luminous AGNs (hereafter QSOs) at high redshifts with the goal of extending the investigation of the MIR prop- erties to the high luminosity range. The primary goals were to investigate the covering factor of the circumnuclear dust and the dependence of the star formation rate (SFR), as traced by the PAH features, on various quantities such as metallicity, narrow line luminosity, accretion rate and black hole mass. In combi- nation with lower luminosity AGNs obtained by previous IRS studies, our sample spans about 5 orders of magnitude in lumi- nosity. This allows us to look for the dependence of the covering factor on luminosity and black hole mass. We also search for the silicate emission and PAH-related properties although the inte- http://arxiv.org/abs/0704.1559v1 2 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs gration times were too short, in most cases, to unveil the proper- ties of individual sources. In Sect. 2 we discuss the sample selection, the observations and the data reduction. In Sect. 3.1 we describe the spectral analysis and the main observational results, and in Sect. 3.2 we include additional data on local, low luminosity sources from the literature and from the Spitzer archive. The dust covering factor is discussed in Sect. 4.1, the properties of the Silicate emission feature in Sect. 4.2 and the constraints on the star formation in Sect. 4.3. The conclusion are outlined in Sect. 5. Throughout the paper we assume a concordance Λ-cosmology with H0 = 71 km s −1 Mpc−1, Ωm = 0.27 and ΩΛ = 0.73 (Spergel et al., 2003). 2. Sample selection, observations and data reduction High redshift, high luminosity QSOs in our sample were mostly drawn from Shemmer et al. (2004) and from Netzer et al. (2004). The latter papers presented near-IR spectra (optical rest- frame) of a large sample of QSOs at 2<z<3.5, which were used to obtain detailed information on the black hole (BH) mass (by means of the width of the Hβ line), on the accretion rate and on the strength of the narrow emission line [OIII]λ5007. The sample contains also infrared data on two sources from Dietrich et al. (2002) and a few additional QSOs in the same redshift range, for which near-IR spectra where obtained after Shemmer et al. (2004), but unpublished yet. This sample allows us not only to extend the investigation of the MIR properties as a function of luminosity, but also to relate those properties to other physical quantities such as BH mass, accretion rate and luminosity of the narrow line region. In total our sample includes 25 sources which are listed in Table 1. Note that the QSOs in Shemmer et al. (2004) and in Netzer et al. (2004) were extracted from optically or radio selected catalogs, without any pre-selection in terms of mid- or far-IR brightness. Therefore, the sample is not biased in terms of star formation or dust con- tent in the host galaxy. We observed these QSOs with the Long-Low resolution module of the Spitzer Infrared Spectrograph IRS (Houck et al., 2004), covering the wavelength range 22–35µm, in staring mode. Objects were acquired by a blind offset from a nearby, bright 2MASS star, whose location and proper motion were known accurately from the Hipparcos catalog. We adopted the “high accuracy” acquisition procedure, which provides a slit centering good enough to deliver a flux calibration accuracy bet- ter than 5%. The integration time was of 12 minutes on source, with the exception of seven which were observed only 4 minutes each1 We started our reduction from the Basic Calibrated Data (BCD). For each observation, we combined all images with the same position on the slit. Then the sky background was sub- tracted by using pairs of frames where the sources appears at two different positions along the slit. The spectra were cleaned for bad, hot and rogue pixels by using the IRSCLEAN algorithm. The monodimensional spectra were then extracted by means of the SPICE software. 1 More specifically: LBQS0109+0213, [HB89]1318-113, [HB89]1346-036,SBS1425+606,[HB89]2126-158, 2QZJ222006.7- 280324,VV0017. Fig. 1. Average spectrum of all high-z, luminous QSOs in our sample, normalized to the flux at 6.7µm (black solid line). The blue dashed line indicates the power-law fitted to the data at λ < 8µm; the green solid line is the fitted silicate emission and the red, dot-dashed line is the resulting fit to the stacked spec- trum (sum of the power-law and silicate emission). The bottom panel indicates the number of objects contributing to the stacked spectrum at each wavelength. 3. Analysis 3.1. Main observational results All of the objects were clearly detected. In Tab. 2 we list the observed continuum flux densities at the observed wavelength corresponding to λrest = 6.7µm. This wavelength was chosen both because it is directly observed in the spectra of all objects and because it is far from the Silicate feature and in-between PAH features. Thus the determination of L(6.7µm) should be little affected by uncertainties in the subtraction of the star- burst component (see below). For two of the radio-loud objects ([HB89]0123+257 and TON618) the MIR flux lies on the ex- trapolation of the synchrotron radio emission and therefore the former is also probably non-thermal. Since in this paper we are mostly interested in the thermal emission by dust, the latter two objects are not used in the statistical analysis. For the other two radio loud QSOs, the extrapolation of the radio spectrum falls below the observed MIR emission and the latter is little affected by synchrotron contamination. Fig. 1 shows the mean spectrum of all sources in the sample, except for the two which are likely dominated by synchrotron emission. Each spectrum has been normalized to 6.7µm prior to averaging. The bottom panel shows the number of sources contributing to the mean spectrum in different spectral regions. We only consider the rest frame spectral range where at least 5 objects contribute to the mean spectrum. The spectrum at λ < 8µm has been fitted with a simple power-law. While other R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 3 Fig. 2. IRS spectra of four individual high-z luminous QSOs showing evidence for silicate emission. The black solid lines indicate the IRS spectra smoothed with a 5 pixels boxcar. The shaded areas indicate the flux uncertainty. The blue dashed line and the green solid line are the power-law and the silicate emission components of the fits. The black dotted line shows the starburst component, which is formally required by the fit, but statistically not significant. The red dot-dashed lines are the global fits to the observed spectra. workers in this field assumed more complicated continuum (e.g. spline, polynomial) we do not consider it justified given the limited wavelength range of our spectra. The extrapolation of the continuum to 10µm clearly reveals an excess identified with Silicate emission. Fitting and measuring the strength of this fea- ture is not easy given the limited rest-frame spectral coverage of our spectra. Therefore, we resort to the use of templates. In particular, we fit the Silicate feature by using as a template the (continuum-subtracted) silicate feature observed in the aver- age spectrum of local QSOs as obtained by the QUEST project (Schweitzer et al., 2006) and kindly provided by M. Schweitzer. The template Silicate spectrum, with the best fitting scaling fac- tor is shown in green in Fig. 1, while the red dot-dashed line shows the resulting fit including the power-law. We adopt the definition of “silicate strength” given in (Shi et al., 2006) which is the ratio between the maximum of the silicate feature and the interpolated featureless continuum at the same wavelength. In the QSO-QUEST template the maximum of the Silicate feature is at 10.5µm. This wavelength is slightly outside the band cov- ered by our spectra but the uncertainty on the extrapolation is not large (the latter is included in the error estimate of the sil- icate strength). The silicate strength in the mean spectrum is 0.58±0.10 (Tab. 2). We note that the average spectrum does not show evidence for PAH features at 7.7µm and 6.2µm. Such features are ob- served in lower luminosity AGNs. More specifically, a starburst 4 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs template (Sturm et al., 2000) is not required by the fit shown in Fig. 1. In Sect. 4.3 we will infer an upper limit on the PAH lu- minosity and discuss its implication. We clearly detect the blue wing of the silicate feature in four individual spectra, which are shown in Fig. 2. These spectra were fitted with a power-law and a silicate template exactly as the stacked spectrum. The resulting values for the Silicate strength are given in Tab. 2. The presence of silicate emission in all other cases is poorly constrained (or totally unconstrained) either be- cause of low signal-to-noise (S/N) or because of a lack of spec- tral coverage. The one exception is Ton 618 which has a high S/N spectrum and a redshift (z=2.22) appropriate to observe the Silicate 10.5 µm feature. No silicate emission is detected in this case, but note that this is not expected since the MIR radiation of this source is probably dominated by synchrotron emission. Tables 1 and 2 list the more important MIR information on the sources and physical properties deduced from the rest-frame optical spectra and obtained from Shemmer et al. (2004): opti- cal continuum luminosity λLλ(5100Å), [OIII]λ5007 line lumi- nosity, BH mass and Eddington accretion rate L/LEdd. 3.2. A comparison with MIR properties of lower luminosity To compare the MIR properties of our luminous QSOs with those of lower luminosity sources we have included in our study the IRS/MIR spectra of various low redshift, lower luminosity type-I AGNs. We purposely avoid type-II sources because of the additional complication due to absorption along the line of sight. We have used data from Shi et al. (2006) who analyze the in- tensity of the silicate features in several, local AGNs with lumi- nosities ranging from those of nearby Seyfert 1s to intermediate luminosity QSOs. We discarded BAL QSOs (which are known to have intervening gas and dust absorption) as well as dust red- dened type-I nuclei (e.g. 2MASS red QSOs). We also discard those cases (e.g. 3C273) where the optical and MIR continuum is likely dominated by synchrotron radiation. Note that Shi et al. (2006) selected type-I objects with “high brightness” and, there- fore, low-luminosity AGNs tend to be excluded from their sam- Shi et al. (2006) provide a measure of the silicate feature strength (whose definition was adopted also by us). The con- tinuum emission at 6.7µm was measured by us from the archival spectra. We also subtracted from the 6.7µm emission the pos- sible contribution of a starburst component by using the M82 template. We estimate the host galaxy contribution (stellar pho- tospheres) in all sources to be negligible. We include in the sample of local Sy1s also some IRS spectra taken from the sample of Buchanan et al. (2006), whose spectral parameters were determined by us from the archival spectral, in the same manner as for the Shi et al. (2006) spectra. As for the previous sample, we discarded reddened/absorbed sources as well as those affected by synchrotron emission. As discussed in Buchanan et al. (2006), these spectra are affected by significant flux calibration uncertainties, due to the adopted mapping tech- nique. Therefore, the spectra were re-calibrated by using IRAC photometric images. We discarded objects for which IRAC data are not available or not usable (e.g. because saturated). Finally, we also discarded data for which optical spectroscopic data are not available (see below). The mid-IR parameters of the sources in both samples are listed in Tab. 2. Optical data were mostly taken from Marziani et al. (2003) and BH masses and Eddington accretion rates inferred as in Shemmer et al. (2004). The resulting parameters are listed in Table 1. The type-I sources in Shi et al. (2006) and Buchanan et al. (2006) are only used for the investigation of the covering fac- tor and silicate strength, which are the main aims of our work. The Shi et al. (2006) and Buchanan et al. (2006) samples are not suitable for investigating the PAH features because most of these objects are at small distances and the IRS slit misses most of the star formation regions in the host galaxy. For what concerns the the PAH emission, we use the data in Schweitzer et al. (2006) who performed a detailed analysis of the PAH features in their local QSOs sample. The slit losses in those sources are minor. The Schweitzer et al. (2006) sample is also used for the investi- gation of the MIR-to-optical luminosity ratio. The mid-IR data of this sample are not listed in Tab. 2, since such data are already reported in Schweitzer et al. (2006) and in Netzer et al. (2007). 4. Discussion 4.1. Dust covering factor 4.1.1. Covering factor as a function of source luminosity and BH mass The main assumption used here is that the covering factor of the circumnuclear dust is given by the ratio of the thermal infrared emission to the primary AGN radiation. The latter is mostly the “big blue bump” radiation with additional contribution from the optical and X-ray wavelength ranges (Blandford et al., 1990). Determining the integral of the AGN-heated dust emission, and disentangling it from other spectral components is not sim- ple. The FIR emission in type-I AGNs is generally dominated by a starburst component, even in QSOs (Schweitzer et al., 2006). In lower luminosity AGNs, the near-IR emission may be affected by stellar emission in the host galaxy, while in QSOs the near- IR light is often contributed also by the direct primary radia- tion. The MIR range (∼ 4 − 10µm) is where the contrast be- tween AGN-heated dust emission and other components is max- imal. This spectral region contains various spectral features, like PAHs and silicate emission, yet MIR spectra allow us to disen- tangle and remove these components, and determine the hot dust continuum. In particular, by focusing on the continuum emis- sion at 6.7µm, the uncertainty in the removal of PAH emission is minimized, while the contribution from the Silicate emission is totally negligible at this wavelength (note that such a spectral decomposition is unfeasible with photometric data). If the spec- tral shape of the AGN-heated dust does not change from object to object (and in particular it does not change significantly with luminosity), then the 6.7µm emission is a proxy of the global circumnuclear hot dust emission. It is possible to infer a quanti- tative relation between L(6.7µm) and the total AGN-heated hot dust emission through the work of Silva et al. (2004), who use observations of various nearby AGNs to determine their average, nuclear IR SED (divided into absorption classes). From their type I AGNs SED, we find that the integrated nuclear, thermal IR bump is about ∼ 2.7 λLλ(6.7µm). This ratio is also consistent with that found in the QUEST QSO sample, once the contribu- tion by the silicate features is subtracted. Regarding the primary optical-UV radiation, determining its integrated flux would require observations of the entire intrin- sic spectral energy distribution (SED) from the far-UV to the near-IR. This is not available for most sources in our sample. R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 5 Fig. 3. MIR continuum luminosity at 6.7µm versus optical con- tinuum luminosity at 5100Å. Red triangles mark high-z, lumi- nous QSOs and blue diamonds mark low redshift type-I AGNs. The black solid line is a fit to the data, which has a slope with a power index of 0.82. Moreover, there are indications that the SED is luminosity de- pendent (e.g. Scott et al., 2004; Shang et al., 2005) and thus an estimate of the bolometric luminosity of the primary continuum based on the observed luminosity in a certain band is somewhat uncertain. Notwithstanding these limitations, we assume in this work that the optical continuum luminosity can be used as a proxy of the bolometric luminosity of the primary continuum. We use the continuum luminosity at the rest frame wavelength of 5100Å, (λLλ, hereafter L5100) because it is directly measured for all sources in our sample. We can infer the ratio between the bolometric luminosity and L5100 from the mean spectrum of QSOs obtained in recent studies, mostly the results of Scott et al. (2004) and Richards et al. (2006). These studies indicate a bolo- metric correction in the range of 5–9. Here we chose, rather ar- bitrarily, the mean value of 7. The comparison of L(6.7µm) and L5100 is our way of deduc- ing the hot dust covering factor. Fig. 3 shows the λLλ(6.7µm) versus L5100 for our sample. Red triangles are the high-z, lumi- nous QSOs and blue diamonds are local type-I AGNs. Not sur- prisingly, the two quantities show a good correlation. However, the very large luminosity range spanned by our sample allows us to clearly state that the correlation is not linear, but has a slope α = 0.82 ± 0.02 defined by log[λLλ(6.7µm)] = K + α log[λLλ(5100Å)] (1) where K = 8.36±0.80 and luminosities are expressed in erg s−1. This indicating that the MIR, reprocessed emission increases more slowly than the primary luminosity. The same phenomenon is observed in a cleaner way in Fig. 4a, where the ratio between the two continuum luminosi- ties is plotted as a function of L5100. There is a clear anti- correlation between the MIR–to–optical ratio and optical lumi- nosity. Fig. 4b shows the same MIR–to–optical ratio as a func- tion of L([OIII]λ5007) as an alternative tracer of the global AGN luminosity (although the latter is not a linear tracer of the nu- clear luminosity, as discussed in Netzer et al., 2006), which dis- plays the same anti-correlation as for the continuum optical flux. Spearman-rank coefficients and probabilities for these correla- tions are given in Tab. 3. According to the above discussion, an obvious interpretation of the decreasing MIR–to–optical ratio is that the covering factor of the dust surrounding the AGN decreases with luminosity. In particular, if the covering factor is proportional to the MIR–to– optical ratio, then Figs. 4a-b indicate that the dust covering factor decreases by about a factor 10 over the luminosity range probed by us. It is possible to convert the MIR–to–optical luminosity ra- tio into absolute dust covering factor by assuming ratios of broad band to monochromatic continuum luminosities observed in AGNs, as discussed above. In particular, by using the 6.7µm– to–MIR and 5100Å–to–bolometric luminosity ratios reported above, we obtain that the absolute value of the dust covering factor (CF) can be written as: CF(dust) ≈ 0.39 · λLλ(6.7µm) λLλ(5100Å) . (2) In Fig. 4 the axes on the right hand side provide the dust covering factor inferred from the the equation above. A fraction of objects have covering factor formally larger than one, these could be due to uncertainties in the observational data, or nu- clear SED differing from the ones assumed above, or to optical variability, as discussed in Sect. 4.1.2. The dust covering fac- tor ranges from about unity in low luminosity AGNs to about 10% in high luminosity QSOs. As it will be discussed in detail in Sect. 4.1.3, the dust covering factor is expected to be equal to the fraction of type 2 (obscured) AGNs relative to the total AGN population. The finding of a large covering factor in low luminosity AGNs is in agreement with the large fraction of type 2 nuclei observed in local Seyferts (∼ 0.8, Maiolino & Rieke, 1995). A similar result has been obtained, in an independent way, through the finding of a systemic decrease of the the obscured to unobscured AGN ratio as a function of luminosity in various surveys. The comparison with these results will be discussed in more detail in the next section. The physical origin of the decreasing covering factor is still unknown. One possibility is that higher luminosities imply a larger dust sublimation radius: if the obscuring medium is dis- tributed in a disk with constant height, then a larger dust subli- mation radius would automatically give a lower covering factor of dust at higher luminosities (Lawrence, 1991). However, this effect can only explain the decreasing covering factor with lu- minosity for the dusty medium, and not for the gaseous X-ray absorbing medium. Moreover, Simpson (2005) showed that the simple scenario of such a “receding torus” is unable to account for the observed dependence of the type 2 to type 1 AGN ratio as a function of luminosity. Another possibility is that the radiation pressure on dust is stronger, relative to the BH gravitational potential, in luminous AGNs (e.g. Laor & Draine, 1993; Scoville & Norman, 1995), thus sweeping away circumnuclear dust more effectively. In this scenario a more direct relation of the covering factor should be with L/LEdd, rather than with luminosity. Our sample does not show such a relation, as illustrated in Fig. 4c. However, the un- certainties on the accretion rates (horizontal error bar in Fig. 4c) may hamper the identification of such a correlation. Lamastra et al. (2006) proposed that, independently of lumi- nosity, the BH gravitational potential is responsible for flatten- 6 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs Fig. 4. MIR–to–optical continuum luminosity [λLλ(6.7µm)]/[λLλ(5100Å)] versus (a) continuum optical luminosity, (b) [OIII]λ5007 line luminosity, (c) accretion rate L/LEdd and (d) black hole mass. Symbols are the same as in Fig. 3. The magenta square indicates the location of the mean SDSS QSO SED in Richards et al. (2006). The horizontal error bars in panels (c) and (d) indicate con- servative uncertainties on the accretion rates and BH masses. The right hand side axis on each panel shows the circumnuclear dust covering factor inferred from Eq. 2. The dashed lines in panels (a) and (b) shows the fit resulting from the analytical forms in Eqs. 3 and 6, respectively. ing the circumnuclear medium, so that larger BH masses effec- tively produce a lower covering factor. According to this sce- nario, the relation between covering factor and luminosity is only an indirect one, in the sense that more luminous AGN tend to have larger BH masses (if the Eddington accretion rate does not change strongly on average). Fig. 4d shows the MIR–to– optical ratio (and dust covering factor) as a function of BH mass, indicating a clear (anti-)correlation between these two quanti- ties. However, the correlation is not any tighter than the relation with luminosity in Fig. 4a-b, as quantified by the comparison of the Spearman-rank coefficients and probabilities in Tab. 3. The degeneracy between luminosity and BH mass prevents us to dis- criminate which of the two is the physical quantity driving the relation. 4.1.2. Model uncertainties In this section we discuss some possible caveats in our interpre- tation of the MIR–to–optical ratio as an indicator of the hot dust covering factor. Our analysis assumes that the shape of the hot dust IR spec- trum SED is not luminosity dependent. However, an alternative interpretation of the trends observed in Fig. 4 could be that the dust temperature distribution changes with luminosity. Yet, to explain the decreasing 6.7µm to 5100Å flux ratio in terms of dust R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 7 Fig. 5. Mid-IR spectral slope (5–8µm) versus λLλ(6.7µm)/λLλ(5100Å) ratio. Symbols are as in Fig. 3. No clear correlation is observed between these two quan- tities (see also Tab. 3). Note that the exceptional object SDSSJ173352.22+540030.5 is out of scale, and it is discussed in the text. Fig. 6. Optical-to-UV spectral slope (1450–5100Å) versus λLλ(6.7µm)/λLλ(5100Å) ratio. Symbols are as in Fig. 3. The arrow indicates the effect of dust reddening with AV = 2 mag. The data do not show any evidence for dust reddening effects. temperature would require that the circumnuclear dust is cooler at higher luminosities. This, besides being contrary to expecta- tions, is ruled out by the observations which show no clear corre- lation between the mid-IR continuum slope (which is associated with the average dust temperature) and the MIR–to–optical ratio, as illustrated in Fig. 5 and in Tab. 3. The one remarkable excep- tion is SDSSJ173352.22+540030.5,which has the most negative mid-IR continuum slope (αMIR = −2.88, i.e. an inverted spec- trum in λLλ) and the lowest MIR-to-optical ratio of the whole sample (λLλ(6.7µm)/λLλ(5100Å) = 0.22), which is out of scale in Fig. 5. This QSO may be totally devoid of circumnuclear hot dust, and its MIR emission may simply be the continuation of the optical “blue-bump”. Similar high-z QSOs, with an exceptional deficiency of mir-MIR flux, have been reported by Jiang et al. (2006). As explained earlier, there are indications that the UV-optical SED, and hence the bolometric correction based to the observed L5100, are luminosity dependent. If correct, this would mean a smaller bolometric correction for higher L5100 sources which would flatten the relationship found here (i.e. will result in a slower decrease of the covering factor with increasing L5100). However, the expected range (a factor of at most 2 in bolometric correction) is much smaller than the deduced change in covering factor. Variability is an additional potential caveat because of the time delay between the original L5100 “input” and the response of the dusty absorbing “torus”. While we do not have the obser- vations to test this effect (multi-epoch, high-quality optical spec- troscopic data are available only for a few sources in our sam- ple), we note that the location of the 6.7µm emitting gas from the central accretion disk is at least several light years and thus L(MIR) used here reflects the mean L5100 in most sources. We expect that the average luminosity of a large sample will not be affected much by individual source variations. Moreover, in the high luminosity sources of our sample we do not expect much variability (since luminous QSOs are known to show little or no variability). An alternative, possible explanation of the variation of the optical-to-MIR luminosity ratio could be dust extinction affect- ing the observed optical flux. Optical dust absorption increasing towards low luminosities may in principle explain the trends ob- served in Fig. 4. However, we have pre-selected the sample of local QSOs and Sy1s to avoid objects showing any indication of absorption, thus probably shielding us from such spurious ef- fects. Yet, we have further investigated the extinction scenario by analyzing the optical-to-UV continuum shape of our sample. The optical-UV continuum slope does not necessarily trace dust reddening, since intrinsic variations of the continuum shape are known to occur, as discussed above. Variability introduce ad- ditional uncertainties, since optical and UV data are not simul- taneous. However, if the variations of λLλ(6.7µm)/λLλ(5100Å) are mostly due to dust reddening, one would expect the MIR- to-optical ratio to correlate with the optical-UV slope, at least on average. We have compiled UV rest-frame continuum fluxes (at λrest ∼ 1450Å) from spectra in the literature or in the HST archive. By combining such data with the continuum luminosi- ties at λrest = 5100Å we inferred the optical-UV continuum slope αopt−UV defined as 2 Lλ ∝ λ αopt−UV , as listed in Tab. 1. Fig. 6 shows αopt−UV versus λLλ(6.7µm)/λLλ(5100Å). The ar- row indicates the effect of dust reddening with AV = 2 mag (by assuming a SMC extinction curve, as appropriate for type 1 AGNs, Hopkins et al., 2004), which would be required to ac- count for the observed variations of λLλ(6.7µm)/λLλ(5100Å). Fig. 6 does not show evidence for any (positive) correlation be- tween αopt−UV and λLλ(6.7µm)/λLλ(5100Å) (see also Tab 3). In particular, if the variation of MIR-to-optical ratio (spanning more than a factor of ten) was due to dust reddening, we would expect to find ∆αopt−UV > 5 (as indicated by the arrow in Fig. 6), which is clearly not observed. If any, the data show a marginal 2 Our definition of power law index is linked to the αν given in Vanden Berk et al. (2001) by the relation αopt−UV = −(αν + 2). Our dis- tribution of αopt−UV is roughly consistent (within the uncertainties and the scatter) with αν = −0.44 obtained by Vanden Berk et al. (2001) for the SDSS QSO composite spectrum (see also Shemmer et al., 2004). 8 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs anti-correlation between αopt−UV and λLλ(6.7µm)/λLλ(5100Å) (Tab. 3), i.e. opposite to that expected from dust reddening. Finally, possible evolutionary effects on the dust covering factor have not been considered. We have been comparing lo- cal objects with QSOs at z∼2–3 yet assumed that only lumi- nosity or BH mass plays a role. La Franca et al. (2005) and Akylas et al. (2006) find evidence for an increasing fraction of obscured AGNs as a function of redshift, a result which is still debated (see Ueda et al., 2003; Gilli et al., 2007). If the ab- sorbing medium covering factor really increases with redshift, then the actual dependence of the covering factor on luminos- ity would be stronger than shown in Fig. 4. Indeed, according to La Franca et al. (2005) and Akylas et al. (2006), putative low-z counterparts of our high-z QSOs (matching the same luminosi- ties) should be affected by an even lower covering factor. As a consequence, the diagrams in Figs. 4 and 7 should have even steeper trends once the data are corrected for such putative evo- lutionary effects, thus strengthening our conclusions. 4.1.3. Comparison with previous works Wang et al. (2005) used IRAS and ISO mid-IR data of (mostly local) PG quasars to infer their dust covering factor. They find that the covering factor decreases as a function of X-ray lumi- nosity. They probe a narrower luminosity range with respect to our work, nonetheless their results are generally consistent with ours, although with significant scatter. More recently Richards et al. (2006) derived the broad band SED of a large sample of SDSS QSOs by including Spitzer pho- tometric data. Although such data do not have spectroscopic in- formation, they can be used to obtain a rough indication of the dust covering factor in their QSO sample, to be compared with our result. The average optical luminosity of the Richards et al. (2006) sample is 〈log[λLλ(5100Å)]〉 ∼ 45.5 erg s −1. From their mean SED we derive λLλ(6.7µm)/λLλ(5100Å) = 1.17. The cor- responding location on the diagram of Fig. 4a is marked with a magenta square, and it is in agreement with the general trend of our data. In a companion paper, Gallagher et al. (2007) use the same set of data to investigate the MIR-to-optical properties as a func- tion of luminosity. They find a result similar to ours, i.e. the MIR-to-optical ratio decreases with luminosity. However, they interpret such a result as a consequence of dust reddening in the optical, since the effect is stronger in QSOs with redder opti- cal slope. As discussed in the previous section, our data do no support this scenario, at least for our sample. In particular, the analysis of the optical-UV slope indicates that dust reddening does not play a significant role in the variations of the MIR- to-optical luminosity ratio. The discrepancy between our and Gallagher et al. (2007) results may have various explanations. The QSOs in the Gallagher et al. (2007) sample span about two orders of magnitudes in luminosity, while we have seen that to properly quantify the effect a wider luminosity range is re- quired. Moreover, the majority of their sources are clustered around the mean luminosity of 1045.5 erg s−1. In addition, the the lack of spectroscopic information makes it difficult to allow for the presence of other spectral features such as the silicate emission which, as we show later, is luminosity dependent. The lack of spectroscopic information may be an issue specially for samples spanning a wide redshift range (as in Gallagher et al. , 2007), where the photometric bands probe different rest-frame bands. The same concerns applies for the optical luminosities. Our rest-frame continuum optical luminosities are always in- ferred through rest-frame optical spectra, even at high-z (through near-IR spectra). Gallagher et al. (2007) do not probe directly the optical continuum luminosity of high-z sources (at high-z they only have optical and Spitzer data, which probe UV and near-IR rest-frame, respectively). Finally, differences between our and Gallagher et al. (2007) results may be simply due to the different samples. As discussed in the previous section, we avoided dust reddened targets, thus making us little sensitive to extinction effects, while Gallagher et al. (2007) sample may in- clude a larger fraction of reddened objects. A decreasing dust covering factor as a function of lumi- nosity must translate into a decreasing fraction of obscured AGNs as a function of luminosity. The effect has been noted in various X-ray surveys (Ueda et al., 2003; Steffen et al., 2003; La Franca et al., 2005; Akylas et al., 2006; Barger et al., 2005; Tozzi et al., 2006; Simpson, 2005), although the results have been questioned by other authors (e.g. Dwelly & Page, 2006; Treister & Urry, 2005; Wang et al., 2007). The X-ray based studies do not distinguish between dust and gas and thus probe mostly trends of the gaseous absorption. Our result provides an independent confirmation of these trends. Moreover, our find- ings are complementary to those obtained in the X-rays since, instead of the covering factor of gas, we probe the covering fac- tor of dust. In order to compare our findings with the results obtained from X-ray surveys, we have derived the expected fraction of obscured AGNs by fitting the dust covering factor versus lumi- nosity relation with an analytical function. Instead of using the simple power-law illustrated in Fig. 3 (Eq. 1) we fit the depen- dence of the covering factor on luminosity with a broken power- law. The latter analytical function is preferred both because it provides a statistically better fit and because a simple power-law would yield a covering factor larger than unity at low luminosi- ties. As a result we obtain the following best fit for the fraction of obscured AGNs as a function of luminosity: fobsc = 1 +Lopt 0.414 where fobsc is the fraction of obscured AGNs relative to the total Lopt = λLλ(5100Å) [erg s 1045.63 The resulting fit is shown with a dashed line in Fig. 4a. The frac- tion of obscured AGNs as a function of luminosity is also re- ported with a blue, solid line in Fig. 7a. The shaded area reflects the uncertainty in the bolometric correction discussed above. The most recent and most complete investigation on the frac- tion of X-ray obscured AGNs as a function of luminosity has been obtained by Hasinger (2007, in prep., see also Hasinger, 2004) who combined the data from surveys of different areas and limiting fluxes to get a sample of ∼700 objects. We convert from X-ray to optical luminosity by using the non-linear relation obtained by Steffen et al. (2006). The latter use flux densities at 2 keV and 2500Å; we adapt their relation to our reference optical wavelength (5100Å) by assuming the optical-UV spectral slope obtained by Vanden Berk et al. (2001), and to the 2–10 keV inte- grated luminosity (adopted in most X-ray surveys) by assuming a photon index of –1.7, yielding the relation log[L(2 − 10 keV)] = 0.721 · log[λLλ(5100Å)]+ 11.78 (5) (where luminosities are in units of erg s−1). Fig. 7a compares the fraction of obscured AGN obtained by Hasinger (2007) through R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 9 Fig. 7. a) Fraction of obscured AGNs (relative to total) as a function of optical continuum luminosity. The blue line shows the fraction of obscured AGNs inferred from the hot dust covering factor with the analytical form of Eq. 3. The shaded area is the uncertainty resulting from the plausible range of the bolometric correction (see text). Points with error bars are the fraction of X-ray obscured AGNs as a function of X-ray luminosity inferred by Hasinger (2007). The upper scale of the diagram gives the intrinsic hard X-ray luminosity; the (non-linear) correspondence between X-ray and optical luminosity is obtained from Eq. 5. b) Fraction of obscured AGNs as a function of L([OIII]λ5007). The blue line is the fraction of obscured AGNs inferred from the hot dust covering factor with the analytical form of Eq. 6. Points with error bars show the fraction of type 2 AGNs as a function of L([OIII]λ5007) inferred by Simpson (2005). X-ray surveys with our result based on the covering factor of hot dust. Both have the same trends with luminosity, but the fraction of obscured AGN expected from the hot dust cover- ing factor is systematically higher. Such an offset is however ex- pected. Indeed, current high redshift X-ray surveys do not probe the Compton thick population of obscured AGNs since these are heavily absorbed even in the hard X-rays. In local AGNs, Compton thick nuclei are about as numerous as Compton thin ones (Risaliti et al., 1999; Guainazzi et al., 2005; Cappi et al., 2006). The fraction of Compton thick, high luminosity, high redshift AGNs is still debated, but their contribution certainly makes the fraction of X-ray obscured AGNs higher than in- ferred by Hasinger (2007), who can only account for Compton thin sources. The ratio between the dust covering factor curve in Fig. 7a and the X-ray data from Hasinger (2007), indicates that the ratio between the total number of obscured AGNs (includ- ing Compton thick ones) and Compton thin ones is about 2 even at high luminosities, i.e. consistent (within uncertainties) with local, low-luminosity AGNs. An analogous result on the decreasing fraction of obscured AGNs as a function of luminosity was obtained by Simpson (2005) who compared the numbers of type 2 and type 1 AGNs at a given L([OIII]λ5007). To compare with Simpson (2005), we used our sample to derive the following analytical description for the fraction of obscured AGN as a function of L([OIII]λ5007): fobsc = 1 +L[OIII] 0.409 where fobsc is the fraction of obscured AGNs relative to the total L[OIII] = L([OIII]) [erg s−1] 1043.21 The corresponding fit is shown with a dashed line in Fig. 4b, and the fraction of obscured AGNs as a function of L([OIII]) is also shown with a blue line in Fig. 7b. In the latter figure we also compare the fraction of obscured AGNs obtained by Simpson (2005) with our result based on the covering factor of the hot dust. There is a good agreement (within uncertainties) between the fraction of obscured AGNs inferred through the two meth- ods, as expected since both optical surveys and our method probe the covering factor of dust. However, we shall also mention that the results obtained by Simpson (2005) have been questioned by Haas et al. (2005), by arguing that at high luminosities, the de- rived L([OIII]λ5007) may be affected by a large scale absorber. 4.2. Silicate emission In this paper we have presented the most luminous (type 1) QSOs where Silicate emission has been detected so far. When combined with MIR spectra of lower luminosity sources, it is possible to investigate the properties and behavior of this feature over a wide luminosity range. The discovery of silicate emission in the spectrum of (mostly type 1) AGNs obtained by Spitzer was regarded as the so- lution of a long standing puzzle on the properties of the cir- 10 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs Fig. 8. Silicate strength versus (a) L5100, (b) L([OIII]λ5007), (c) normalized accretion rate (L/LEdd) and (d) black hole mass. Symbols are the same as in Fig. 3. The red square indicates the silicate strength measured in the average spectrum. The horizontal, black error bars in panels (c) and (d) indicate conservative uncertainties on the accretion rates and BH masses. cumnuclear dusty medium. Indeed, silicate emission was ex- pected by various models of the dusty torus. However, the ab- sence of clear detections prior to the Spitzer epoch induced various authors to either postulate a very compact and dense torus (e.g. Pier & Krolik, 1993) or different dust compositions (Laor & Draine, 1993; Maiolino et al., 2001a,b). Initial Spitzer detections of silicate emission relaxed the torus model assump- tions (Fritz et al., 2006), but more detailed investigations re- vealed a complex scenario. The detection of silicate emission even in type 2 AGNs (Sturm et al., 2006b; Teplitz et al., 2006; Shi et al., 2006) suggested that part of the silicate emission may originate in the Narrow Line Region (NLR) (Efstathiou, 2006). Further support for a NLR origin of the silicate emission comes from the temperature inferred for the Silicate features, which is much lower (<200 K) than for the circumnuclear dust emitting the featureless MIR continuum (>500 K), as well as from MIR high resolution maps spatially resolving the silicate emission on scales of 100 pc (Schweitzer et al. in prep.). If most of the observed silicate emission originates in the NLR, then the effects of circumnuclear hot dust covering fac- tor should be amplified when looking at the “silicate strength” (which we recall is defined as the ratio of the silicate maximum R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 11 intensity and the featureless hot dust continuum). Indeed, if the covering factor of the circumnuclear dusty torus decreases, it im- plies that the MIR hot dust continuum decreases and the silicate emission increases because a larger volume of the NLR is il- luminated. Both effects go in the same direction of increasing the “silicate strength”. This scenario is made more complex by the tendency of the NLR to disappear at very high luminosities, or to get very dense and not to scale linearly with the nuclear luminosity (Netzer et al., 2004, 2006). Moreover, the schematic division of a silicate feature totally emitted by the NLR and a MIR featureless continuum totally emitted by the inner side of the obscuring torus is probably too simplistic. There must be at least a small contribution to the featureless MIR continuum from dust in the NLR, while some silicate emission is probably also coming from the obscuring torus. However, from a general qualitative point of view we expect a monotonic behavior of the “silicate strength” with the physical quantity responsible for the changes in the hot dust covering factor. Before investigating the various trends of the Silicate strength, we mention that by using the four silicate detections shown in Fig. 2 and listed in Tab. 2, we may in principle intro- duce a bias against weak silicate emitters. Indeed, although we cannot set useful upper limits on the silicate strength in most of the other objects, we have likely missed objects with low sili- cate strength. However, the mean spectrum in Fig. 1 includes all QSOs in our sample, and therefore its silicate strength should be representative of the average Silicate emission in the sample (at least for the objects at z<2.5, which are the ones where the observed band includes the silicate feature, and which are the majority). Figs. 8a-b show the silicate strength of the objects in our combined sample as functions of L5100 and L([OIII]λ5007). The red square indicates the silicate strength in the high-z QSO mean spectrum, while its horizontal bar indicates the range of lumi- nosities spanned by the subsample of objects at z<2.5 (i.e. those contributing to the silicate feature in the mean spectrum). Low redshift AGNs and high redshift QSOs show an apparently clear correlation between silicate strength and luminosity. Although with a significant spread, the Silicate strength is observed to pos- itively correlate also with the accretion rate L/LEdd and with the BH mass, as shown in Figs. 8c-d. Essentially, the correlations observed for the Silicate strength reflects the same correlation observed for the L(6.7µm)/L(5100Å) (with the exception of the accretion rate), in agreement with the idea that also the Silicate strength is a proxy of the covering factor of the circumnuclear hot dust, for the reasons discussed above. Unfortunately, the correlations observed for the Silicate strength do not improve our understanding on the origin of the decreasing covering factor with luminosity, i.e. whether the driv- ing physical quantity is the luminosity itself, the accretion rate or the black hole mass. Formally, the correlation between Silicate strength and optical continuum luminosity is tighter than the oth- ers (Tab. 3), possibly hinting at the luminosity itself as the quan- tity driving the dust covering factor. However, there are a few low luminosity objects, such as a few LINERs, which have large silicate strengths (Sturm et al., 2005) and which clearly deviate from the correlation shown in Fig. 8a, thus questioning the role of luminosity in determining the Silicate strength. In addition, the apparently looser correlations of Silicate strength versus ac- cretion rate and BH mass may simply be due to the additional uncertainties affecting the latter two quantities (horizontal, black error bars in Figs. 8c,d). 4.3. PAHs and star formation The presence and intensity of star formation in QSOs has been a hotly debated issue during the past few years. A major step forward in this debate was achieved by Schweitzer et al. (2006) through the Spitzer IRS detection of PAH features in a sample of nearby QSOs, revealing vigorous star formation in these ob- jects. The analysis also shows that the far-IR emission in these QSOs is dominated by star formation and that the star forming activity correlates with the nuclear AGN power. Here we show in Fig. 9 the latter correlation in terms of PAH(7.7µm) luminosity versus L5100 by using the PAH luminosities from the sample of Schweitzer et al. (2006) and the corresponding optical data from Marziani et al. (2003). Although the large fraction of upper lim- its in the former sample prevents a careful statistical characteri- zation, Fig. 9 shows a general correlation between QSO optical luminosity and starburst activity in the host galaxy as traced by the PAH luminosity. The scale on the right hand side of Fig. 9 translates the 7.7µm PAH luminosity into star formation rate (SFR). This was ob- tained by combining the average L(PAH7.7µm)/L(FIR) ratio ob- tained by Schweitzer et al. (2006) for the starburst dominated QSOs in their sample with the SFR/L(FIR) given in Kennicutt (1998), yielding SFR [M⊙ yr −1] = 3.46 10−42 L(PAH7.7µm) [erg s −1] (8) The average spectrum of high-z, luminous QSOs in Fig. 1 does not show evidence for the presence of PAH features and can only provide an upper limit on the PAH flux relative to the flux at 6.7µm (since all spectra were normalized to the lat- ter wavelength prior to computing the average). However, we can derive an upper limit on the PAH luminosity by assum- ing the average distance of the sources in the sample. The in- ferred upper limit on the PAH luminosity is reported with a red square in Fig. 9a, and it is clearly below the extrapolation of the L(PAH7.7µm) − λLλ(5100Å) relation found for local, low- luminosity QSOs. This is shown more clearly in Fig. 9b which shows the distribution of the ratio L(PAH7.7µm)/λLλ(5100Å) for local QSOs (histogram) and the upper limit inferred from the av- erage spectrum of high-z, luminous QSOs (red solid line). The corresponding upper limit on the SFR is ∼ 700 M⊙ yr Note that certainly there are luminous, high-z QSOs with larger star formation rates (e.g. Bertoldi et al., 2003; Beelen et al., 2006; Lutz et al., 2007). However, since our sam- ple is not pre-selected in terms of MIR or FIR emission, our result is not biased in terms of star formation and dust content, and therefore it is representative of the general high-z, luminous QSO population. Our results indicate that the relation between star forma- tion activity, as traced by the PAH features, and QSO power, as traced by L5100, saturates at high luminosity. This result is not surprising. Indeed, if high-z, luminous QSOs were char- acterized by the same average L(PAH7.7µm)/λLλ(5100Å) ob- served in local QSOs, this would imply huge star formation rates of ∼ 7000 M⊙ yr −1 at λLλ(5100Å) ∼ 10 47 erg s−1. On the contrary, the few high-z QSOs detected at submm-mm wave- lengths have far-IR luminosities corresponding to SFR of about 1000−3000 M⊙ yr −1 (Omont et al., 2003). The majority of high- z QSO (∼70%) are undetected at submm-mm wavelengths. The mean mm-submm fluxes of QSOs in various surveys (includ- ing both detections and non-detections) imply SFRs in the range ∼ 500−1500 M⊙ yr −1 (Omont et al., 2003; Priddey et al., 2003), in fair agreement with our finding, especially if we consider the 12 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs Fig. 9. a) PAH(7.7µm) luminosity as a function of the QSO optical luminosity. Blue diamonds are data from Schweitzer et al. (2006). The red square is the upper limit obtained by the average spectrum of luminous, high-z QSOs. b) Distribution of the PAH(7.7µm) to optical luminosity ratio in the local QSOs sample of Schweitzer et al. (2006). The hatched region indicates upper limits. The red vertical line indicate the upper limit inferred from the average spectrum of luminous QSOs at high-z. uncertainties involved in the two different observational methods to infer the SFR. We also note that a similar, independent result was obtained by Haas et al. (2003), who found that in a large sample of QSOs the ratio between LFIR (powered by star formation) and LB de- creases at high luminosities. The finding of a “saturation” of the relation between star for- mation activity and QSO power may provide an explanation for the evolution of the relation between BH mass and galaxy mass at high redshift. Indeed, Peng et al. (2006) and McLure et al. (2006) found that, for a given BH mass, QSO hosts at z∼2 are characterized by a stellar mass lower than expected from the local BH-galaxy mass relation. In other terms, the BH growth is faster, relative to star formation, in high-z, luminous QSOs. Our result supports this scenario by independently showing that the correlation between star formation and AGN activity breaks down at high luminosities. 5. Conclusions We have presented low resolution, mid-IR Spitzer spectra of a sample of 25 luminous QSOs at high redshifts (2 < z < 3.5). We have combined our data with Spitzer spectra of lower lumi- nosity, type-I AGNs, either published in the literature or in the Spitzer archive. The combined sample spans five orders of mag- nitude in luminosity, and allowed us to investigate the dust prop- erties and star formation rate as a function of luminosity. The spectroscopic information allowed us to disentangle the various spectral components contributing to the MIR band (PAH and sil- icate emission) and to sample the continuum at a specific λrest, in contrast to photometric MIR observations. The main results are: – The mid-IR continuum luminosity at 6.7µm correlates with the optical continuum luminosity but the correlation is not linear. In particular, the ratio λLλ(6.7µm)/λLλ(5100Å) de- creases by about a factor of ten as a function of lumi- nosity over the luminosity range 1042.5 < λLλ(5100Å) < 1047.5 erg s−1. This is interpreted as a reduction of the cov- ering factor of the circumnuclear hot dust as a function of luminosity. This result is in agreement and provides an in- dependent confirmation of the recent findings of a decreas- ing fraction of obscured AGN as a function of luminosity, obtained in X-ray and optical surveys. We stress that while X-ray surveys probe the covering factor of the gas, our result provides an independent confirmation by probing the cover- ing factor of the dust. We have also shown that the dust cov- ering factor, as traced by the λLλ(6.7µm)/λLλ(5100Å) ratio, decreases also as a function of the BH mass. Based on these correlations alone it is not possible to determine whether the physical quantity primarily driving the reduction of the cov- ering factor is the AGN luminosity or the BH mass. – The mean spectrum of the luminous, high-z QSOs in our sample shows a clear silicate emission at λrest ∼ 10µm. Silicate emission is also detected in the individual spectra of four high redshift QSOs. When combined with the spec- tra of local, lower luminosity AGNs we find that the sili- cate strength (defined as the ratio between the maximum of the silicate feature and the extrapolated featureless contin- uum) tend to increase as a function of luminosity. The sili- cate strength correlates positively also with the accretion rate and with the BH mass, albeit with a large scatter. – The mean MIR spectrum of the luminous, high-z QSOs in our sample does not show evidence for PAH emission. Our sample is not pre-selected by the FIR emission and therefore it is not biased in terms star formation. As a consequence, the upper limit on the PAH emission in the total mean spectrum provides a useful, representative upper limit on the SFR in luminous QSOs at high redshifts. We find that the ratio be- tween PAH luminosity and QSO optical luminosity is signifi- cantly lower than observed in local, lower luminosity AGNs, implying that the correlation between star formation rate and R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 13 AGN power probably saturates at high luminosities. 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Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs ‘[HB89]0123+257’ on page 16 ‘HS0211+1858’ on page 16 ‘2QZJ023805.8-274337’ on page 16 ‘SDSSJ024933.42-083454.4’ on page 16 ‘Q0256-0000’ on page 16 ‘Q0302-0019’ on page 16 ‘[HB89]0329-385’ on page 16 ‘SDSSJ100428.43+001825.6’ on page 16 ‘TON618’ on page 16 ‘[HB89]1318-113’ on page 16 ‘[HB89]1346-036’ on page 16 ‘UM629’ on page 16 ‘UM632’ on page 16 ‘BS1425+606’ on page 16 ‘[VCV01]J1649+5303’ on page 16 ‘SDSSJ170102.18+612301.0’ on page 16 ‘SDSSJ173352.22+540030.5’ on page 16 ‘[HB89]2126-158’ on page 16 ‘2QZJ221814.4-300306’ on page 16 ‘2QZJ222006.7-280324’ on page 16 ‘Q2227-3928’ on page 16 ‘[HB89]2254+024’ on page 16 ‘2QZJ234510.3-293155’ on page 16 ‘Mrk335’ on page 16 ‘IIIZw2’ on page 16 ‘PG0050+124’ on page 16 ‘PG0052+251’ on page 16 ‘Fairall9’ on page 16 ‘Mkr79’ on page 16 ‘PG0804+761’ on page 16 ‘Mrk704’ on page 16 ‘PG0953+414’ on page 16 ‘NGC3516’ on page 16 ‘PG1116+215’ on page 16 ‘NGC3783’ on page 16 ‘PG1151+117’ on page 16 ‘NGC4051’ on page 16 ‘PG1211+143’ on page 16 ‘NGC4593’ on page 16 ‘PG1309+355’ on page 16 ‘PG1351+640’ on page 16 ‘IC4329a’ on page 16 ‘NGC5548’ on page 16 ‘Mrk817’ on page 16 ‘Mrk509’ on page 16 ‘Mrk926’ on page 16 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 15 Table 1. Combined sample of high-z luminous QSO, local QSO and Sy1, and physical properties inferred from optical-UV spectra. Name RA(J2000) Dec(J2000) z log(λLλ(5100Å)) logL([OIII]) logM(BH) d L/LdEdd αopt−UV Ref. (erg/s) (erg/s) M⊙ High-z luminous QSOs 2QZJ002830.4-281706 00:12:21.18 −28:36:30.2 2.401 46.59 44.41 9.72 0.35 −1.42 1 LBQS0109+0213 01:12:16.91 +02:29:47.6 2.349 46.81 44.51 10.01 0.30 −1.75 1 [HB89]0123+257a 01:26:42.79 +25:59:01.3 2.369 46.58 44.32 9.10 1.40 −1.65 1 HS0211+1858 02:14:29.70 +19:12:37.0 2.470 46.63 44.38 10.11 0.16 −0.01 3,10 2QZJ023805.8-274337 02:38:05.80 −27:43:37.0 2.471 46.58 <43.71 9.41 0.69 −1.59 1 SDSSJ024933.42-083454.4 02:49:33.41 −08:34:54.4 2.491 46.39 44.12 9.67 0.25 −1.36 1 Q0256-0000 02:59:05.64 +00:11:21.9 3.377 46.99 44.55 10.11 0.19 −0.96 2 Q0302-0019 03:04:49.86 −00:08:13.4 3.286 46.83 45.01 10.11 0.30 −1.66 2 [HB89]0329-385 03:31:06.34 −38:24:04.8 2.435 46.72 44.31 10.11 0.18 −1.79 1 SDSSJ100428.43+001825.6 10:04:28.44 +00:18:25.6 3.040 46.45 44.47 9.34 0.70 −0.70 3,11 TON618a 12:28:24.97 +31:28:37.6 2.226 47.32 <44.12 10.81 0.14 −1.27 1 [HB89]1318-113 13:21:09.38 −11:39:31.6 2.306 46.90 44.32 9.76 0.62 −0.99 1 [HB89]1346-036 13:48:44.08 −03:53:24.9 2.370 46.89 43.73 9.95 0.41 −1.26 1 UM629 14:03:23.39 −00:06:06.9 2.460 46.57 44.41 9.17 1.16 −1.40 1 UM632b 14:04:45.89 −01:30:21.9 2.517 46.55 44.04 9.44 0.61 −1.27 1 SBS1425+606 14:26:56.10 +60:25:50.0 3.202 47.39 45.04 9.83 1.73 −1.45 1 [VCV01]J1649+5303 16:49:14.90 +53:03:16.0 2.260 46.70 44.19 9.99 0.24 −0.86 3,11 SDSSJ170102.18+612301.0 17:01:02.18 +61:23:01.0 2.301 46.35 <43.51 9.73 0.20 −1.48 1 SDSSJ173352.22+540030.5 17:33:52.23 +54:00:30.5 3.428 47.02 44.36 9.58 1.28 −1.51 1 [HB89]2126-158b 21:29:12.17 −15:38:41.0 3.282 47.27 44.66 9.73 1.60 0.72 1 2QZJ221814.4-300306 22:18:14.40 −30:03:06.0 2.389 46.55 43.95 9.28 0.89 −1.27 1 2QZJ222006.7-280324 22:20:06.70 −28:03:23.0 2.414 47.23 44.64 10.21 0.54 −1.28 1 Q2227-3928 22:30:32.95 −39:13:06.8 3.438 46.95 <44.02 10.31 0.19 −1.25 2 [HB89]2254+024 22:57:17.56 +02:43:17.5 2.083 46.46 43.95 9.10 1.08 −1.37 1 2QZJ234510.3-293155 23:45:10.36 −29:31:54.7 2.382 46.33 43.97 9.38 0.42 −1.26 1 High-z QSO aver. (z<2.5)c 46.63 44.07 9.68 0.51 Local QSOs and Sy1s Mrk335 00:06:19.52 +20:12:10.4 0.025 43.62 41.29 7.10 0.28 −2.00 4,14 IIIZw2 00:10:30.80 +10:58:13.0 0.090 44.02 42.25 8.19 0.16 −1.56 4,13 PG0050+124 00:53:34.94 +12:41:36.2 0.058 44.36 41.87 7.09 0.96 −0.91 4,13 PG0052+251 00:54:52.10 +25:25:38.0 0.155 44.46 42.57 8.55 0.21 −2.27 4,13 Fairall9 01:23:45.78 −58:48:20.5 0.046 43.80 41.91 8.27 0.10 −1.99 4,13 Mrk79 07:42:32.79 +49:48:34.7 0.022 43.58 41.37 8.12 0.08 −0.83 9,14 PG0804+761 08:10:58.60 +76:02:42.0 0.100 44.42 42.03 8.08 0.33 −2.64 4,12 Mrk704 09:18:26.00 +16:18:19.2 0.029 43.44 41.18 7.97 0.08 – 4 PG0953+414 09:56:52.40 +41:15:22.0 0.234 44.96 42.69 8.39 0.56 −2.12 4,13 NGC3516 11:06:47.49 +72:34:06.8 0.009 42.81 40.52 7.39 0.06 −1.09 7,8,14 PG1116+215 11:19:08.60 +21:19:18.0 0.176 44.84 42.27 8.27 0.53 −2.58 4,13 NGC3783 11:39:01.72 −37:44:18.9 0.010 43.05 41.10 7.33 0.09 −1.46 4,13 PG1151+117 11:53:49.27 +11:28:30.4 0.176 44.48 42.09 8.31 0.28 −2.50 4,12 NGC4051 12:03:09.61 +44:31:52.8 0.002 41.39 39.64 5.32 0.06 −0.82 5,6,14 PG1211+143 12:14:17.70 +14:03:12.6 0.085 44.58 41.94 7.69 0.68 −1.32 4,13 NGC4593 12:39:39.42 −05:20:39.3 0.009 42.60 40.34 7.40 0.04 −1.26 4,14 PG1309+355 13:12:17.76 +35:15:21.2 0.184 44.50 42.18 8.29 0.30 −2.45 4,12 PG1351+640 13:53:15.80 +63:45:45.4 0.087 44.80 42.52 8.76 0.28 −0.86 4,13 IC4329a 13:49:19.26 −30:18:34.0 0.016 43.13 40.89 7.77 0.06 – 4 NGC5548 14:17:59.53 +25:08:12.4 0.017 43.10 41.15 7.78 0.06 −1.73 4,13 Mrk817 14:36:22.06 +58:47:39.3 0.033 43.96 41.65 8.11 0.16 −0.56 4,14 Mrk509 20:44:09.73 −10:43:24.5 0.034 44.01 42.13 7.87 0.22 −1.69 4,13 Mrk926 23:04:43.47 −08:41:08.6 0.047 43.83 42.29 8.55 0.08 – 4 The following quantities are reported in each column: column 1, object name; columns 2-3, coordinates (J2000); column 4, redshift; column 5, log of the continuum luminosity λLλ (in units of erg/s) at the rest frame wavelength 5100Å; column 6, log of the [OIII]λ5007 emission line luminosity (in units of erg/s); column 7, log of the black hole mass (in units of M⊙); column 8, Eddington ratio Lbol/LEdd; column 9, optical-to-UV (1450Å– 5100Å) continuum slope (Fλ ∝ λ αopt−UV); column 11: reference for the optical and UV data: 1 - Shemmer et al. (2004), Netzer et al. (2004) and therein references for UV data, 2 - Dietrich et al. (2002) and therein references for UV data, 3 - Juarez et al (in prep.), 4 - Marziani et al. (2003), 5 - Suganuma et al. (2006), 6 - Peterson et al. (2000), 7 - Wanders et al. (1993), 8 - Ho & Ulvestad (2001), 9 - Peterson et al. (1998), 10 - Engels et al. (1998), 11 - SDSS DR5 archive, 12- Baskin & Laor (2005), 13 - Evans & Koratkar (2004), 14 - Kaspi et al. (2005). a Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is near or above the observed value, hence the 6.7µm flux is probably dominated by synchrotron emission; these objects will be excluded from statistical analyses. b Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is well below the observed value, hence the 6.7µm flux is likely thermal. c Optical luminosities, black hole mass and Eddington ratio for the stacked spectrum refer to the average values of only the objects at z<2.5, i.e. those who contribute to the Silicate feature observed in the stacked spectrum. d As discussed in Shemmer et al. (2004), the uncertainties on the BH masses and accretion rate are no larger than a factor of two. 16 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs Table 2. Infrared properties of the combined sample of high-z luminous QSO, local QSO and Sy1 Name FMIR αMIR λLλ (6.7µm) λLλ (5100Å) Si strength Ref. (mJy) High-z luminous QSOs 2QZJ002830.4-281706 4.8 -0.59 0.75 1 LBQS0109+0213 8.3 -0.91 0.75 1 [HB89]0123+257a 4.9 -1.69 0.76 1 HS0211+1858 7.3 -1.36 1.09 1 2QZJ023805.8-274337 3.7 -1.04 0.62 0.81±0.16 1 SDSSJ024933.42-083454.4 2.2 -1.26 0.57 1 Q0256-0000 2.7 -1.34 0.30 1 Q0302-0019 3.6 -1.67 0.56 1 [HB89]0329-385 5.5 -1.74 0.65 1 SDSSJ100428.43+001825.6 2.3 -1.51 0.73 1 TON618a 20.1 -1.16 0.51 0.04±0.01 1 [HB89]1318-113 6.6 -1.70 0.47 1 [HB89]1346-036 12.9 -1.68 0.99 1 UM629 3.8 -0.68 0.65 1 UM632b 2.1 -1.77 0.40 1 BS1425+606 23.9 -1.56 0.96 1 [VCV01]J1649+5303 9.4 -1.28 1.03 1 SDSSJ170102.18+612301.0 3.3 -0.66 0.83 1 SDSSJ173352.22+540030.5 2.0 -2.88 0.21 1 [HB89]2126-158b 19.1 -1.18 1.08 1 2QZJ221814.4-300306 4.6 -1.03 0.78 1 2QZJ222006.7-280324 16.0 -1.52 0.58 0.65±0.05 1 Q2227-3928 3.0 -1.70 0.38 1 [HB89]2254+024 3.6 -1.12 0.60 0.63±0.15 1 2QZJ234510.3-293155 4.2 -1.33 1.16 0.92±0.15 1 High-z QSO aver. (z<2.5)c -1.57 0.58±0.10 Local QSOs and Sy1s Mrk335 130. -1.38 1.90 0.25±0.06 3 IIIZw2 52. -1.09 4.02 0.05±0.03 2,3 PG0050+124 245. -0.74 3.55 0.38±0.05 2,3 PG0052+251 28. -1.35 2.40 0.33±0.06 2,3 Fairall9 146. -0.90 4.83 0.21±0.07 2,3 Mkr79 200. -1.03 2.44 0.10±0.06 3 PG0804+761 88. -1.81 3.38 0.60±0.05 2,3 Mrk704 190. -0.95 5.64 0.09±0.06 3 PG0953+414 26. -1.88 1.68 0.40±0.08 2,3 NGC3516 210. -1.05 2.40 0.06±0.05 3 PG1116+215 66. -1.71 3.08 0.22±0.05 2,3 NGC3783 315. -1.08 2.69 -0.01±0.03 2,3 PG1151+117 10. -2.19 1.08 0.36±0.13 2,3 NGC4051 230. -0.43 4.71 0.06±0.05 3 PG1211+143 100. -1.26 1.89 0.55±0.05 2,3 NGC4593 184. -1.14 3.61 0.08±0.05 2,3 PG1309+355 25. -1.24 2.82 0.41±0.07 2,3 PG1351+640 53. -0.89 0.64 1.25±0.05 2,3 IC4329a 487. -0.67 9.00 0.01±0.03 2,3 NGC5548 69. -1.06 1.53 0.27±0.05 2,3 Mrk817 140. -0.65 1.61 0.16±0.06 3 Mrk509 179. -1.29 1.98 0.11±0.04 2,3 Mrk926 55. -1.40 1.77 0.26±0.04 2,3 The following quantities are reported in each column: column 1, object name; column 2, continuum flux density at the observed wavelength corresponding to λrest = 6.7µm (after removing starburst and stellar components, in units of mJy); column 3, power-law index (Fλ ∝ λ α) fitted to the continuum in the 5–8µm range (starburst component–subtracted); column 4, ratio of the continuum emission at 5100Å and at 6.7µm, λLλ(5100Å)/λLλ(6.7µm); column 5, Silicate strength; column 6: reference for the infrared data: 1 - this work (from Spitzer program 20493) ; 2 - Shi et al. (2006); 3 - this work (from Spitzer archival data). a Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is near or above the observed value, hence the 6.7µm flux is probably dominated by synchrotron emission; these objects will be excluded from statistical analyses. b Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is well below the observed value, hence the 6.7µm flux is likely thermal. c The ratio λLλ(5100Å)/λLλ(6.7µm) is not defined for the stacked spectrum, since all spectra were normalized to the 6.7µm flux before stacking. As a consequence, only the Silicate strength (and more generally the continuum shape) has a physical meaning for the stacked spectrum. R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 17 Table 3. Spearman-rank coefficients for the correlations in Figs.4, 8, 5 and 6. log(λLλ(5100Å)) log(L[OIII]) log(L/LEdd) log(MBH) αMIR αopt−UV λLλ (6.7µm) λLλ (5100Å) −0.76 (< 10−6) −0.72 (< 10−6) −0.44 (3 10−4) −0.70 (< 10−6) 0.29 (0.05) −0.20 (0.19) log(Si str.) 0.83 (2 10−5) 0.75 (6 10−6) 0.75 (8 10−6) 0.66 (8 10−4) Numbers in parenthesis give the probability for the correlation coefficient to deviate from zero. Introduction Sample selection, observations and data reduction Analysis Main observational results A comparison with MIR properties of lower luminosity AGNs Discussion Dust covering factor Covering factor as a function of source luminosity and BH mass Model uncertainties Comparison with previous works Silicate emission PAHs and star formation Conclusions

We present Spitzer IRS low resolution, mid-IR spectra of a sample of 25 high luminosity QSOs at 2<z<3.5. When combined with archival IRS observations of local, low luminosity type-I AGNs, the sample spans five orders of magnitude in luminosity. We find that the continuum dust thermal emission at lambda(rest)=6.7um is correlated with the optical luminosity, following the non-linear relation L(6.7um) propto L(5100A)^0.82. We also find an anti correlation between the ratio L(6.7um)/L(5100A) and the [OIII]5007A line luminosity. These effects are interpreted as a decreasing covering factor of the circumnuclear dust as a function of luminosity. Such a result is in agreement with the decreasing fraction of absorbed AGNs as a function of luminosity recently found in various surveys. We clearly detect the silicate emission feature in the average spectrum, but also in four individual objects. These are the Silicate emission in the most luminous objects obtained so far. When combined with the silicate emission observed in local, low luminosity type-I AGNs, we find that the silicate emission strength is correlated with luminosity. The silicate strength of all type-I AGNs also follows a positive correlation with the black hole mass and with the accretion rate. The Polycyclic Aromatic Hydrocarbon (PAH) emission features, expected from starburst activity, are not detected in the average spectrum of luminous, high-z QSOs. The upper limit inferred from the average spectrum points to a ratio between PAH luminosity and QSO optical luminosity significantly lower than observed in lower luminosity AGNs, implying that the correlation between star formation rate and AGN power saturates at high luminosities.

Introduction The mid-IR (MIR) spectrum of AGNs contains a wealth of in- formation which is crucial to the understanding of their inner region. The observed prominent continuum emission is due to circumnuclear dust heated to a temperature of several hundred degrees by the nuclear, primary optical/UV/X-ray source (pri- marily the central accretion disk); therefore, the MIR contin- uum provides information on the amount and/or covering factor of the circumnuclear dust. The MIR region is also rich of sev- eral emission features which are important tracers of the ISM. Among the dust features, the Polycyclic-Aromatic-Hydrocarbon bands (PAH, whose most prominent feature is at ∼ 7.7µm) are emitted by very small carbon grains excited in the Photo Dissociation Regions, that are tracers of star forming activity (although PAHs may not be reliable SF tracers for compact HII regions or heavily embedded starbursts, Peeters et al., 2004; Förster Schreiber et al., 2004). Additional MIR dust features are the Silicate bands at ∼ 10µm and at ∼ 18µm, often seen in ab- sorption in obscured AGNs and in luminous IR galaxies. Major steps forward in this field were achieved thanks to the Spitzer Space Observatory, and to its infrared spectrome- ter, IRS, which allows a detailed investigation of the MIR spec- tral features in a large number of sources. In particular, IRS Send offprint requests to: R. Maiolino allowed the detection of MIR emission lines in several AGNs (e.g. Armus et al., 2004; Haas et al., 2005; Sturm et al., 2006a; Weedman et al., 2005), the detection of PAHs in local PG QSOs (Schweitzer et al., 2006), the first detection of the Silicate fea- ture in emission (Siebenmorgen et al., 2005; Hao et al., 2005), as well as detailed studies of the silicate strength in various classes of sources (Spoon et al., 2007; Hao et al., 2007; Shi et al., 2006; Imanishi et al., 2007). However, most of the current Spitzer IRS studies have fo- cused on local and modest luminosity AGNs (including low luminosity QSOs), with the exception of a few bright, lensed objects at high redshift (Soifer et al., 2004; Teplitz et al., 2006; Lutz et al., 2007). We have obtained short IRS integrations of a sample of 25 luminous AGNs (hereafter QSOs) at high redshifts with the goal of extending the investigation of the MIR prop- erties to the high luminosity range. The primary goals were to investigate the covering factor of the circumnuclear dust and the dependence of the star formation rate (SFR), as traced by the PAH features, on various quantities such as metallicity, narrow line luminosity, accretion rate and black hole mass. In combi- nation with lower luminosity AGNs obtained by previous IRS studies, our sample spans about 5 orders of magnitude in lumi- nosity. This allows us to look for the dependence of the covering factor on luminosity and black hole mass. We also search for the silicate emission and PAH-related properties although the inte- http://arxiv.org/abs/0704.1559v1 2 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs gration times were too short, in most cases, to unveil the proper- ties of individual sources. In Sect. 2 we discuss the sample selection, the observations and the data reduction. In Sect. 3.1 we describe the spectral analysis and the main observational results, and in Sect. 3.2 we include additional data on local, low luminosity sources from the literature and from the Spitzer archive. The dust covering factor is discussed in Sect. 4.1, the properties of the Silicate emission feature in Sect. 4.2 and the constraints on the star formation in Sect. 4.3. The conclusion are outlined in Sect. 5. Throughout the paper we assume a concordance Λ-cosmology with H0 = 71 km s −1 Mpc−1, Ωm = 0.27 and ΩΛ = 0.73 (Spergel et al., 2003). 2. Sample selection, observations and data reduction High redshift, high luminosity QSOs in our sample were mostly drawn from Shemmer et al. (2004) and from Netzer et al. (2004). The latter papers presented near-IR spectra (optical rest- frame) of a large sample of QSOs at 2<z<3.5, which were used to obtain detailed information on the black hole (BH) mass (by means of the width of the Hβ line), on the accretion rate and on the strength of the narrow emission line [OIII]λ5007. The sample contains also infrared data on two sources from Dietrich et al. (2002) and a few additional QSOs in the same redshift range, for which near-IR spectra where obtained after Shemmer et al. (2004), but unpublished yet. This sample allows us not only to extend the investigation of the MIR properties as a function of luminosity, but also to relate those properties to other physical quantities such as BH mass, accretion rate and luminosity of the narrow line region. In total our sample includes 25 sources which are listed in Table 1. Note that the QSOs in Shemmer et al. (2004) and in Netzer et al. (2004) were extracted from optically or radio selected catalogs, without any pre-selection in terms of mid- or far-IR brightness. Therefore, the sample is not biased in terms of star formation or dust con- tent in the host galaxy. We observed these QSOs with the Long-Low resolution module of the Spitzer Infrared Spectrograph IRS (Houck et al., 2004), covering the wavelength range 22–35µm, in staring mode. Objects were acquired by a blind offset from a nearby, bright 2MASS star, whose location and proper motion were known accurately from the Hipparcos catalog. We adopted the “high accuracy” acquisition procedure, which provides a slit centering good enough to deliver a flux calibration accuracy bet- ter than 5%. The integration time was of 12 minutes on source, with the exception of seven which were observed only 4 minutes each1 We started our reduction from the Basic Calibrated Data (BCD). For each observation, we combined all images with the same position on the slit. Then the sky background was sub- tracted by using pairs of frames where the sources appears at two different positions along the slit. The spectra were cleaned for bad, hot and rogue pixels by using the IRSCLEAN algorithm. The monodimensional spectra were then extracted by means of the SPICE software. 1 More specifically: LBQS0109+0213, [HB89]1318-113, [HB89]1346-036,SBS1425+606,[HB89]2126-158, 2QZJ222006.7- 280324,VV0017. Fig. 1. Average spectrum of all high-z, luminous QSOs in our sample, normalized to the flux at 6.7µm (black solid line). The blue dashed line indicates the power-law fitted to the data at λ < 8µm; the green solid line is the fitted silicate emission and the red, dot-dashed line is the resulting fit to the stacked spec- trum (sum of the power-law and silicate emission). The bottom panel indicates the number of objects contributing to the stacked spectrum at each wavelength. 3. Analysis 3.1. Main observational results All of the objects were clearly detected. In Tab. 2 we list the observed continuum flux densities at the observed wavelength corresponding to λrest = 6.7µm. This wavelength was chosen both because it is directly observed in the spectra of all objects and because it is far from the Silicate feature and in-between PAH features. Thus the determination of L(6.7µm) should be little affected by uncertainties in the subtraction of the star- burst component (see below). For two of the radio-loud objects ([HB89]0123+257 and TON618) the MIR flux lies on the ex- trapolation of the synchrotron radio emission and therefore the former is also probably non-thermal. Since in this paper we are mostly interested in the thermal emission by dust, the latter two objects are not used in the statistical analysis. For the other two radio loud QSOs, the extrapolation of the radio spectrum falls below the observed MIR emission and the latter is little affected by synchrotron contamination. Fig. 1 shows the mean spectrum of all sources in the sample, except for the two which are likely dominated by synchrotron emission. Each spectrum has been normalized to 6.7µm prior to averaging. The bottom panel shows the number of sources contributing to the mean spectrum in different spectral regions. We only consider the rest frame spectral range where at least 5 objects contribute to the mean spectrum. The spectrum at λ < 8µm has been fitted with a simple power-law. While other R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 3 Fig. 2. IRS spectra of four individual high-z luminous QSOs showing evidence for silicate emission. The black solid lines indicate the IRS spectra smoothed with a 5 pixels boxcar. The shaded areas indicate the flux uncertainty. The blue dashed line and the green solid line are the power-law and the silicate emission components of the fits. The black dotted line shows the starburst component, which is formally required by the fit, but statistically not significant. The red dot-dashed lines are the global fits to the observed spectra. workers in this field assumed more complicated continuum (e.g. spline, polynomial) we do not consider it justified given the limited wavelength range of our spectra. The extrapolation of the continuum to 10µm clearly reveals an excess identified with Silicate emission. Fitting and measuring the strength of this fea- ture is not easy given the limited rest-frame spectral coverage of our spectra. Therefore, we resort to the use of templates. In particular, we fit the Silicate feature by using as a template the (continuum-subtracted) silicate feature observed in the aver- age spectrum of local QSOs as obtained by the QUEST project (Schweitzer et al., 2006) and kindly provided by M. Schweitzer. The template Silicate spectrum, with the best fitting scaling fac- tor is shown in green in Fig. 1, while the red dot-dashed line shows the resulting fit including the power-law. We adopt the definition of “silicate strength” given in (Shi et al., 2006) which is the ratio between the maximum of the silicate feature and the interpolated featureless continuum at the same wavelength. In the QSO-QUEST template the maximum of the Silicate feature is at 10.5µm. This wavelength is slightly outside the band cov- ered by our spectra but the uncertainty on the extrapolation is not large (the latter is included in the error estimate of the sil- icate strength). The silicate strength in the mean spectrum is 0.58±0.10 (Tab. 2). We note that the average spectrum does not show evidence for PAH features at 7.7µm and 6.2µm. Such features are ob- served in lower luminosity AGNs. More specifically, a starburst 4 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs template (Sturm et al., 2000) is not required by the fit shown in Fig. 1. In Sect. 4.3 we will infer an upper limit on the PAH lu- minosity and discuss its implication. We clearly detect the blue wing of the silicate feature in four individual spectra, which are shown in Fig. 2. These spectra were fitted with a power-law and a silicate template exactly as the stacked spectrum. The resulting values for the Silicate strength are given in Tab. 2. The presence of silicate emission in all other cases is poorly constrained (or totally unconstrained) either be- cause of low signal-to-noise (S/N) or because of a lack of spec- tral coverage. The one exception is Ton 618 which has a high S/N spectrum and a redshift (z=2.22) appropriate to observe the Silicate 10.5 µm feature. No silicate emission is detected in this case, but note that this is not expected since the MIR radiation of this source is probably dominated by synchrotron emission. Tables 1 and 2 list the more important MIR information on the sources and physical properties deduced from the rest-frame optical spectra and obtained from Shemmer et al. (2004): opti- cal continuum luminosity λLλ(5100Å), [OIII]λ5007 line lumi- nosity, BH mass and Eddington accretion rate L/LEdd. 3.2. A comparison with MIR properties of lower luminosity To compare the MIR properties of our luminous QSOs with those of lower luminosity sources we have included in our study the IRS/MIR spectra of various low redshift, lower luminosity type-I AGNs. We purposely avoid type-II sources because of the additional complication due to absorption along the line of sight. We have used data from Shi et al. (2006) who analyze the in- tensity of the silicate features in several, local AGNs with lumi- nosities ranging from those of nearby Seyfert 1s to intermediate luminosity QSOs. We discarded BAL QSOs (which are known to have intervening gas and dust absorption) as well as dust red- dened type-I nuclei (e.g. 2MASS red QSOs). We also discard those cases (e.g. 3C273) where the optical and MIR continuum is likely dominated by synchrotron radiation. Note that Shi et al. (2006) selected type-I objects with “high brightness” and, there- fore, low-luminosity AGNs tend to be excluded from their sam- Shi et al. (2006) provide a measure of the silicate feature strength (whose definition was adopted also by us). The con- tinuum emission at 6.7µm was measured by us from the archival spectra. We also subtracted from the 6.7µm emission the pos- sible contribution of a starburst component by using the M82 template. We estimate the host galaxy contribution (stellar pho- tospheres) in all sources to be negligible. We include in the sample of local Sy1s also some IRS spectra taken from the sample of Buchanan et al. (2006), whose spectral parameters were determined by us from the archival spectral, in the same manner as for the Shi et al. (2006) spectra. As for the previous sample, we discarded reddened/absorbed sources as well as those affected by synchrotron emission. As discussed in Buchanan et al. (2006), these spectra are affected by significant flux calibration uncertainties, due to the adopted mapping tech- nique. Therefore, the spectra were re-calibrated by using IRAC photometric images. We discarded objects for which IRAC data are not available or not usable (e.g. because saturated). Finally, we also discarded data for which optical spectroscopic data are not available (see below). The mid-IR parameters of the sources in both samples are listed in Tab. 2. Optical data were mostly taken from Marziani et al. (2003) and BH masses and Eddington accretion rates inferred as in Shemmer et al. (2004). The resulting parameters are listed in Table 1. The type-I sources in Shi et al. (2006) and Buchanan et al. (2006) are only used for the investigation of the covering fac- tor and silicate strength, which are the main aims of our work. The Shi et al. (2006) and Buchanan et al. (2006) samples are not suitable for investigating the PAH features because most of these objects are at small distances and the IRS slit misses most of the star formation regions in the host galaxy. For what concerns the the PAH emission, we use the data in Schweitzer et al. (2006) who performed a detailed analysis of the PAH features in their local QSOs sample. The slit losses in those sources are minor. The Schweitzer et al. (2006) sample is also used for the investi- gation of the MIR-to-optical luminosity ratio. The mid-IR data of this sample are not listed in Tab. 2, since such data are already reported in Schweitzer et al. (2006) and in Netzer et al. (2007). 4. Discussion 4.1. Dust covering factor 4.1.1. Covering factor as a function of source luminosity and BH mass The main assumption used here is that the covering factor of the circumnuclear dust is given by the ratio of the thermal infrared emission to the primary AGN radiation. The latter is mostly the “big blue bump” radiation with additional contribution from the optical and X-ray wavelength ranges (Blandford et al., 1990). Determining the integral of the AGN-heated dust emission, and disentangling it from other spectral components is not sim- ple. The FIR emission in type-I AGNs is generally dominated by a starburst component, even in QSOs (Schweitzer et al., 2006). In lower luminosity AGNs, the near-IR emission may be affected by stellar emission in the host galaxy, while in QSOs the near- IR light is often contributed also by the direct primary radia- tion. The MIR range (∼ 4 − 10µm) is where the contrast be- tween AGN-heated dust emission and other components is max- imal. This spectral region contains various spectral features, like PAHs and silicate emission, yet MIR spectra allow us to disen- tangle and remove these components, and determine the hot dust continuum. In particular, by focusing on the continuum emis- sion at 6.7µm, the uncertainty in the removal of PAH emission is minimized, while the contribution from the Silicate emission is totally negligible at this wavelength (note that such a spectral decomposition is unfeasible with photometric data). If the spec- tral shape of the AGN-heated dust does not change from object to object (and in particular it does not change significantly with luminosity), then the 6.7µm emission is a proxy of the global circumnuclear hot dust emission. It is possible to infer a quanti- tative relation between L(6.7µm) and the total AGN-heated hot dust emission through the work of Silva et al. (2004), who use observations of various nearby AGNs to determine their average, nuclear IR SED (divided into absorption classes). From their type I AGNs SED, we find that the integrated nuclear, thermal IR bump is about ∼ 2.7 λLλ(6.7µm). This ratio is also consistent with that found in the QUEST QSO sample, once the contribu- tion by the silicate features is subtracted. Regarding the primary optical-UV radiation, determining its integrated flux would require observations of the entire intrin- sic spectral energy distribution (SED) from the far-UV to the near-IR. This is not available for most sources in our sample. R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 5 Fig. 3. MIR continuum luminosity at 6.7µm versus optical con- tinuum luminosity at 5100Å. Red triangles mark high-z, lumi- nous QSOs and blue diamonds mark low redshift type-I AGNs. The black solid line is a fit to the data, which has a slope with a power index of 0.82. Moreover, there are indications that the SED is luminosity de- pendent (e.g. Scott et al., 2004; Shang et al., 2005) and thus an estimate of the bolometric luminosity of the primary continuum based on the observed luminosity in a certain band is somewhat uncertain. Notwithstanding these limitations, we assume in this work that the optical continuum luminosity can be used as a proxy of the bolometric luminosity of the primary continuum. We use the continuum luminosity at the rest frame wavelength of 5100Å, (λLλ, hereafter L5100) because it is directly measured for all sources in our sample. We can infer the ratio between the bolometric luminosity and L5100 from the mean spectrum of QSOs obtained in recent studies, mostly the results of Scott et al. (2004) and Richards et al. (2006). These studies indicate a bolo- metric correction in the range of 5–9. Here we chose, rather ar- bitrarily, the mean value of 7. The comparison of L(6.7µm) and L5100 is our way of deduc- ing the hot dust covering factor. Fig. 3 shows the λLλ(6.7µm) versus L5100 for our sample. Red triangles are the high-z, lumi- nous QSOs and blue diamonds are local type-I AGNs. Not sur- prisingly, the two quantities show a good correlation. However, the very large luminosity range spanned by our sample allows us to clearly state that the correlation is not linear, but has a slope α = 0.82 ± 0.02 defined by log[λLλ(6.7µm)] = K + α log[λLλ(5100Å)] (1) where K = 8.36±0.80 and luminosities are expressed in erg s−1. This indicating that the MIR, reprocessed emission increases more slowly than the primary luminosity. The same phenomenon is observed in a cleaner way in Fig. 4a, where the ratio between the two continuum luminosi- ties is plotted as a function of L5100. There is a clear anti- correlation between the MIR–to–optical ratio and optical lumi- nosity. Fig. 4b shows the same MIR–to–optical ratio as a func- tion of L([OIII]λ5007) as an alternative tracer of the global AGN luminosity (although the latter is not a linear tracer of the nu- clear luminosity, as discussed in Netzer et al., 2006), which dis- plays the same anti-correlation as for the continuum optical flux. Spearman-rank coefficients and probabilities for these correla- tions are given in Tab. 3. According to the above discussion, an obvious interpretation of the decreasing MIR–to–optical ratio is that the covering factor of the dust surrounding the AGN decreases with luminosity. In particular, if the covering factor is proportional to the MIR–to– optical ratio, then Figs. 4a-b indicate that the dust covering factor decreases by about a factor 10 over the luminosity range probed by us. It is possible to convert the MIR–to–optical luminosity ra- tio into absolute dust covering factor by assuming ratios of broad band to monochromatic continuum luminosities observed in AGNs, as discussed above. In particular, by using the 6.7µm– to–MIR and 5100Å–to–bolometric luminosity ratios reported above, we obtain that the absolute value of the dust covering factor (CF) can be written as: CF(dust) ≈ 0.39 · λLλ(6.7µm) λLλ(5100Å) . (2) In Fig. 4 the axes on the right hand side provide the dust covering factor inferred from the the equation above. A fraction of objects have covering factor formally larger than one, these could be due to uncertainties in the observational data, or nu- clear SED differing from the ones assumed above, or to optical variability, as discussed in Sect. 4.1.2. The dust covering fac- tor ranges from about unity in low luminosity AGNs to about 10% in high luminosity QSOs. As it will be discussed in detail in Sect. 4.1.3, the dust covering factor is expected to be equal to the fraction of type 2 (obscured) AGNs relative to the total AGN population. The finding of a large covering factor in low luminosity AGNs is in agreement with the large fraction of type 2 nuclei observed in local Seyferts (∼ 0.8, Maiolino & Rieke, 1995). A similar result has been obtained, in an independent way, through the finding of a systemic decrease of the the obscured to unobscured AGN ratio as a function of luminosity in various surveys. The comparison with these results will be discussed in more detail in the next section. The physical origin of the decreasing covering factor is still unknown. One possibility is that higher luminosities imply a larger dust sublimation radius: if the obscuring medium is dis- tributed in a disk with constant height, then a larger dust subli- mation radius would automatically give a lower covering factor of dust at higher luminosities (Lawrence, 1991). However, this effect can only explain the decreasing covering factor with lu- minosity for the dusty medium, and not for the gaseous X-ray absorbing medium. Moreover, Simpson (2005) showed that the simple scenario of such a “receding torus” is unable to account for the observed dependence of the type 2 to type 1 AGN ratio as a function of luminosity. Another possibility is that the radiation pressure on dust is stronger, relative to the BH gravitational potential, in luminous AGNs (e.g. Laor & Draine, 1993; Scoville & Norman, 1995), thus sweeping away circumnuclear dust more effectively. In this scenario a more direct relation of the covering factor should be with L/LEdd, rather than with luminosity. Our sample does not show such a relation, as illustrated in Fig. 4c. However, the un- certainties on the accretion rates (horizontal error bar in Fig. 4c) may hamper the identification of such a correlation. Lamastra et al. (2006) proposed that, independently of lumi- nosity, the BH gravitational potential is responsible for flatten- 6 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs Fig. 4. MIR–to–optical continuum luminosity [λLλ(6.7µm)]/[λLλ(5100Å)] versus (a) continuum optical luminosity, (b) [OIII]λ5007 line luminosity, (c) accretion rate L/LEdd and (d) black hole mass. Symbols are the same as in Fig. 3. The magenta square indicates the location of the mean SDSS QSO SED in Richards et al. (2006). The horizontal error bars in panels (c) and (d) indicate con- servative uncertainties on the accretion rates and BH masses. The right hand side axis on each panel shows the circumnuclear dust covering factor inferred from Eq. 2. The dashed lines in panels (a) and (b) shows the fit resulting from the analytical forms in Eqs. 3 and 6, respectively. ing the circumnuclear medium, so that larger BH masses effec- tively produce a lower covering factor. According to this sce- nario, the relation between covering factor and luminosity is only an indirect one, in the sense that more luminous AGN tend to have larger BH masses (if the Eddington accretion rate does not change strongly on average). Fig. 4d shows the MIR–to– optical ratio (and dust covering factor) as a function of BH mass, indicating a clear (anti-)correlation between these two quanti- ties. However, the correlation is not any tighter than the relation with luminosity in Fig. 4a-b, as quantified by the comparison of the Spearman-rank coefficients and probabilities in Tab. 3. The degeneracy between luminosity and BH mass prevents us to dis- criminate which of the two is the physical quantity driving the relation. 4.1.2. Model uncertainties In this section we discuss some possible caveats in our interpre- tation of the MIR–to–optical ratio as an indicator of the hot dust covering factor. Our analysis assumes that the shape of the hot dust IR spec- trum SED is not luminosity dependent. However, an alternative interpretation of the trends observed in Fig. 4 could be that the dust temperature distribution changes with luminosity. Yet, to explain the decreasing 6.7µm to 5100Å flux ratio in terms of dust R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 7 Fig. 5. Mid-IR spectral slope (5–8µm) versus λLλ(6.7µm)/λLλ(5100Å) ratio. Symbols are as in Fig. 3. No clear correlation is observed between these two quan- tities (see also Tab. 3). Note that the exceptional object SDSSJ173352.22+540030.5 is out of scale, and it is discussed in the text. Fig. 6. Optical-to-UV spectral slope (1450–5100Å) versus λLλ(6.7µm)/λLλ(5100Å) ratio. Symbols are as in Fig. 3. The arrow indicates the effect of dust reddening with AV = 2 mag. The data do not show any evidence for dust reddening effects. temperature would require that the circumnuclear dust is cooler at higher luminosities. This, besides being contrary to expecta- tions, is ruled out by the observations which show no clear corre- lation between the mid-IR continuum slope (which is associated with the average dust temperature) and the MIR–to–optical ratio, as illustrated in Fig. 5 and in Tab. 3. The one remarkable excep- tion is SDSSJ173352.22+540030.5,which has the most negative mid-IR continuum slope (αMIR = −2.88, i.e. an inverted spec- trum in λLλ) and the lowest MIR-to-optical ratio of the whole sample (λLλ(6.7µm)/λLλ(5100Å) = 0.22), which is out of scale in Fig. 5. This QSO may be totally devoid of circumnuclear hot dust, and its MIR emission may simply be the continuation of the optical “blue-bump”. Similar high-z QSOs, with an exceptional deficiency of mir-MIR flux, have been reported by Jiang et al. (2006). As explained earlier, there are indications that the UV-optical SED, and hence the bolometric correction based to the observed L5100, are luminosity dependent. If correct, this would mean a smaller bolometric correction for higher L5100 sources which would flatten the relationship found here (i.e. will result in a slower decrease of the covering factor with increasing L5100). However, the expected range (a factor of at most 2 in bolometric correction) is much smaller than the deduced change in covering factor. Variability is an additional potential caveat because of the time delay between the original L5100 “input” and the response of the dusty absorbing “torus”. While we do not have the obser- vations to test this effect (multi-epoch, high-quality optical spec- troscopic data are available only for a few sources in our sam- ple), we note that the location of the 6.7µm emitting gas from the central accretion disk is at least several light years and thus L(MIR) used here reflects the mean L5100 in most sources. We expect that the average luminosity of a large sample will not be affected much by individual source variations. Moreover, in the high luminosity sources of our sample we do not expect much variability (since luminous QSOs are known to show little or no variability). An alternative, possible explanation of the variation of the optical-to-MIR luminosity ratio could be dust extinction affect- ing the observed optical flux. Optical dust absorption increasing towards low luminosities may in principle explain the trends ob- served in Fig. 4. However, we have pre-selected the sample of local QSOs and Sy1s to avoid objects showing any indication of absorption, thus probably shielding us from such spurious ef- fects. Yet, we have further investigated the extinction scenario by analyzing the optical-to-UV continuum shape of our sample. The optical-UV continuum slope does not necessarily trace dust reddening, since intrinsic variations of the continuum shape are known to occur, as discussed above. Variability introduce ad- ditional uncertainties, since optical and UV data are not simul- taneous. However, if the variations of λLλ(6.7µm)/λLλ(5100Å) are mostly due to dust reddening, one would expect the MIR- to-optical ratio to correlate with the optical-UV slope, at least on average. We have compiled UV rest-frame continuum fluxes (at λrest ∼ 1450Å) from spectra in the literature or in the HST archive. By combining such data with the continuum luminosi- ties at λrest = 5100Å we inferred the optical-UV continuum slope αopt−UV defined as 2 Lλ ∝ λ αopt−UV , as listed in Tab. 1. Fig. 6 shows αopt−UV versus λLλ(6.7µm)/λLλ(5100Å). The ar- row indicates the effect of dust reddening with AV = 2 mag (by assuming a SMC extinction curve, as appropriate for type 1 AGNs, Hopkins et al., 2004), which would be required to ac- count for the observed variations of λLλ(6.7µm)/λLλ(5100Å). Fig. 6 does not show evidence for any (positive) correlation be- tween αopt−UV and λLλ(6.7µm)/λLλ(5100Å) (see also Tab 3). In particular, if the variation of MIR-to-optical ratio (spanning more than a factor of ten) was due to dust reddening, we would expect to find ∆αopt−UV > 5 (as indicated by the arrow in Fig. 6), which is clearly not observed. If any, the data show a marginal 2 Our definition of power law index is linked to the αν given in Vanden Berk et al. (2001) by the relation αopt−UV = −(αν + 2). Our dis- tribution of αopt−UV is roughly consistent (within the uncertainties and the scatter) with αν = −0.44 obtained by Vanden Berk et al. (2001) for the SDSS QSO composite spectrum (see also Shemmer et al., 2004). 8 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs anti-correlation between αopt−UV and λLλ(6.7µm)/λLλ(5100Å) (Tab. 3), i.e. opposite to that expected from dust reddening. Finally, possible evolutionary effects on the dust covering factor have not been considered. We have been comparing lo- cal objects with QSOs at z∼2–3 yet assumed that only lumi- nosity or BH mass plays a role. La Franca et al. (2005) and Akylas et al. (2006) find evidence for an increasing fraction of obscured AGNs as a function of redshift, a result which is still debated (see Ueda et al., 2003; Gilli et al., 2007). If the ab- sorbing medium covering factor really increases with redshift, then the actual dependence of the covering factor on luminos- ity would be stronger than shown in Fig. 4. Indeed, according to La Franca et al. (2005) and Akylas et al. (2006), putative low-z counterparts of our high-z QSOs (matching the same luminosi- ties) should be affected by an even lower covering factor. As a consequence, the diagrams in Figs. 4 and 7 should have even steeper trends once the data are corrected for such putative evo- lutionary effects, thus strengthening our conclusions. 4.1.3. Comparison with previous works Wang et al. (2005) used IRAS and ISO mid-IR data of (mostly local) PG quasars to infer their dust covering factor. They find that the covering factor decreases as a function of X-ray lumi- nosity. They probe a narrower luminosity range with respect to our work, nonetheless their results are generally consistent with ours, although with significant scatter. More recently Richards et al. (2006) derived the broad band SED of a large sample of SDSS QSOs by including Spitzer pho- tometric data. Although such data do not have spectroscopic in- formation, they can be used to obtain a rough indication of the dust covering factor in their QSO sample, to be compared with our result. The average optical luminosity of the Richards et al. (2006) sample is 〈log[λLλ(5100Å)]〉 ∼ 45.5 erg s −1. From their mean SED we derive λLλ(6.7µm)/λLλ(5100Å) = 1.17. The cor- responding location on the diagram of Fig. 4a is marked with a magenta square, and it is in agreement with the general trend of our data. In a companion paper, Gallagher et al. (2007) use the same set of data to investigate the MIR-to-optical properties as a func- tion of luminosity. They find a result similar to ours, i.e. the MIR-to-optical ratio decreases with luminosity. However, they interpret such a result as a consequence of dust reddening in the optical, since the effect is stronger in QSOs with redder opti- cal slope. As discussed in the previous section, our data do no support this scenario, at least for our sample. In particular, the analysis of the optical-UV slope indicates that dust reddening does not play a significant role in the variations of the MIR- to-optical luminosity ratio. The discrepancy between our and Gallagher et al. (2007) results may have various explanations. The QSOs in the Gallagher et al. (2007) sample span about two orders of magnitudes in luminosity, while we have seen that to properly quantify the effect a wider luminosity range is re- quired. Moreover, the majority of their sources are clustered around the mean luminosity of 1045.5 erg s−1. In addition, the the lack of spectroscopic information makes it difficult to allow for the presence of other spectral features such as the silicate emission which, as we show later, is luminosity dependent. The lack of spectroscopic information may be an issue specially for samples spanning a wide redshift range (as in Gallagher et al. , 2007), where the photometric bands probe different rest-frame bands. The same concerns applies for the optical luminosities. Our rest-frame continuum optical luminosities are always in- ferred through rest-frame optical spectra, even at high-z (through near-IR spectra). Gallagher et al. (2007) do not probe directly the optical continuum luminosity of high-z sources (at high-z they only have optical and Spitzer data, which probe UV and near-IR rest-frame, respectively). Finally, differences between our and Gallagher et al. (2007) results may be simply due to the different samples. As discussed in the previous section, we avoided dust reddened targets, thus making us little sensitive to extinction effects, while Gallagher et al. (2007) sample may in- clude a larger fraction of reddened objects. A decreasing dust covering factor as a function of lumi- nosity must translate into a decreasing fraction of obscured AGNs as a function of luminosity. The effect has been noted in various X-ray surveys (Ueda et al., 2003; Steffen et al., 2003; La Franca et al., 2005; Akylas et al., 2006; Barger et al., 2005; Tozzi et al., 2006; Simpson, 2005), although the results have been questioned by other authors (e.g. Dwelly & Page, 2006; Treister & Urry, 2005; Wang et al., 2007). The X-ray based studies do not distinguish between dust and gas and thus probe mostly trends of the gaseous absorption. Our result provides an independent confirmation of these trends. Moreover, our find- ings are complementary to those obtained in the X-rays since, instead of the covering factor of gas, we probe the covering fac- tor of dust. In order to compare our findings with the results obtained from X-ray surveys, we have derived the expected fraction of obscured AGNs by fitting the dust covering factor versus lumi- nosity relation with an analytical function. Instead of using the simple power-law illustrated in Fig. 3 (Eq. 1) we fit the depen- dence of the covering factor on luminosity with a broken power- law. The latter analytical function is preferred both because it provides a statistically better fit and because a simple power-law would yield a covering factor larger than unity at low luminosi- ties. As a result we obtain the following best fit for the fraction of obscured AGNs as a function of luminosity: fobsc = 1 +Lopt 0.414 where fobsc is the fraction of obscured AGNs relative to the total Lopt = λLλ(5100Å) [erg s 1045.63 The resulting fit is shown with a dashed line in Fig. 4a. The frac- tion of obscured AGNs as a function of luminosity is also re- ported with a blue, solid line in Fig. 7a. The shaded area reflects the uncertainty in the bolometric correction discussed above. The most recent and most complete investigation on the frac- tion of X-ray obscured AGNs as a function of luminosity has been obtained by Hasinger (2007, in prep., see also Hasinger, 2004) who combined the data from surveys of different areas and limiting fluxes to get a sample of ∼700 objects. We convert from X-ray to optical luminosity by using the non-linear relation obtained by Steffen et al. (2006). The latter use flux densities at 2 keV and 2500Å; we adapt their relation to our reference optical wavelength (5100Å) by assuming the optical-UV spectral slope obtained by Vanden Berk et al. (2001), and to the 2–10 keV inte- grated luminosity (adopted in most X-ray surveys) by assuming a photon index of –1.7, yielding the relation log[L(2 − 10 keV)] = 0.721 · log[λLλ(5100Å)]+ 11.78 (5) (where luminosities are in units of erg s−1). Fig. 7a compares the fraction of obscured AGN obtained by Hasinger (2007) through R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 9 Fig. 7. a) Fraction of obscured AGNs (relative to total) as a function of optical continuum luminosity. The blue line shows the fraction of obscured AGNs inferred from the hot dust covering factor with the analytical form of Eq. 3. The shaded area is the uncertainty resulting from the plausible range of the bolometric correction (see text). Points with error bars are the fraction of X-ray obscured AGNs as a function of X-ray luminosity inferred by Hasinger (2007). The upper scale of the diagram gives the intrinsic hard X-ray luminosity; the (non-linear) correspondence between X-ray and optical luminosity is obtained from Eq. 5. b) Fraction of obscured AGNs as a function of L([OIII]λ5007). The blue line is the fraction of obscured AGNs inferred from the hot dust covering factor with the analytical form of Eq. 6. Points with error bars show the fraction of type 2 AGNs as a function of L([OIII]λ5007) inferred by Simpson (2005). X-ray surveys with our result based on the covering factor of hot dust. Both have the same trends with luminosity, but the fraction of obscured AGN expected from the hot dust cover- ing factor is systematically higher. Such an offset is however ex- pected. Indeed, current high redshift X-ray surveys do not probe the Compton thick population of obscured AGNs since these are heavily absorbed even in the hard X-rays. In local AGNs, Compton thick nuclei are about as numerous as Compton thin ones (Risaliti et al., 1999; Guainazzi et al., 2005; Cappi et al., 2006). The fraction of Compton thick, high luminosity, high redshift AGNs is still debated, but their contribution certainly makes the fraction of X-ray obscured AGNs higher than in- ferred by Hasinger (2007), who can only account for Compton thin sources. The ratio between the dust covering factor curve in Fig. 7a and the X-ray data from Hasinger (2007), indicates that the ratio between the total number of obscured AGNs (includ- ing Compton thick ones) and Compton thin ones is about 2 even at high luminosities, i.e. consistent (within uncertainties) with local, low-luminosity AGNs. An analogous result on the decreasing fraction of obscured AGNs as a function of luminosity was obtained by Simpson (2005) who compared the numbers of type 2 and type 1 AGNs at a given L([OIII]λ5007). To compare with Simpson (2005), we used our sample to derive the following analytical description for the fraction of obscured AGN as a function of L([OIII]λ5007): fobsc = 1 +L[OIII] 0.409 where fobsc is the fraction of obscured AGNs relative to the total L[OIII] = L([OIII]) [erg s−1] 1043.21 The corresponding fit is shown with a dashed line in Fig. 4b, and the fraction of obscured AGNs as a function of L([OIII]) is also shown with a blue line in Fig. 7b. In the latter figure we also compare the fraction of obscured AGNs obtained by Simpson (2005) with our result based on the covering factor of the hot dust. There is a good agreement (within uncertainties) between the fraction of obscured AGNs inferred through the two meth- ods, as expected since both optical surveys and our method probe the covering factor of dust. However, we shall also mention that the results obtained by Simpson (2005) have been questioned by Haas et al. (2005), by arguing that at high luminosities, the de- rived L([OIII]λ5007) may be affected by a large scale absorber. 4.2. Silicate emission In this paper we have presented the most luminous (type 1) QSOs where Silicate emission has been detected so far. When combined with MIR spectra of lower luminosity sources, it is possible to investigate the properties and behavior of this feature over a wide luminosity range. The discovery of silicate emission in the spectrum of (mostly type 1) AGNs obtained by Spitzer was regarded as the so- lution of a long standing puzzle on the properties of the cir- 10 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs Fig. 8. Silicate strength versus (a) L5100, (b) L([OIII]λ5007), (c) normalized accretion rate (L/LEdd) and (d) black hole mass. Symbols are the same as in Fig. 3. The red square indicates the silicate strength measured in the average spectrum. The horizontal, black error bars in panels (c) and (d) indicate conservative uncertainties on the accretion rates and BH masses. cumnuclear dusty medium. Indeed, silicate emission was ex- pected by various models of the dusty torus. However, the ab- sence of clear detections prior to the Spitzer epoch induced various authors to either postulate a very compact and dense torus (e.g. Pier & Krolik, 1993) or different dust compositions (Laor & Draine, 1993; Maiolino et al., 2001a,b). Initial Spitzer detections of silicate emission relaxed the torus model assump- tions (Fritz et al., 2006), but more detailed investigations re- vealed a complex scenario. The detection of silicate emission even in type 2 AGNs (Sturm et al., 2006b; Teplitz et al., 2006; Shi et al., 2006) suggested that part of the silicate emission may originate in the Narrow Line Region (NLR) (Efstathiou, 2006). Further support for a NLR origin of the silicate emission comes from the temperature inferred for the Silicate features, which is much lower (<200 K) than for the circumnuclear dust emitting the featureless MIR continuum (>500 K), as well as from MIR high resolution maps spatially resolving the silicate emission on scales of 100 pc (Schweitzer et al. in prep.). If most of the observed silicate emission originates in the NLR, then the effects of circumnuclear hot dust covering fac- tor should be amplified when looking at the “silicate strength” (which we recall is defined as the ratio of the silicate maximum R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 11 intensity and the featureless hot dust continuum). Indeed, if the covering factor of the circumnuclear dusty torus decreases, it im- plies that the MIR hot dust continuum decreases and the silicate emission increases because a larger volume of the NLR is il- luminated. Both effects go in the same direction of increasing the “silicate strength”. This scenario is made more complex by the tendency of the NLR to disappear at very high luminosities, or to get very dense and not to scale linearly with the nuclear luminosity (Netzer et al., 2004, 2006). Moreover, the schematic division of a silicate feature totally emitted by the NLR and a MIR featureless continuum totally emitted by the inner side of the obscuring torus is probably too simplistic. There must be at least a small contribution to the featureless MIR continuum from dust in the NLR, while some silicate emission is probably also coming from the obscuring torus. However, from a general qualitative point of view we expect a monotonic behavior of the “silicate strength” with the physical quantity responsible for the changes in the hot dust covering factor. Before investigating the various trends of the Silicate strength, we mention that by using the four silicate detections shown in Fig. 2 and listed in Tab. 2, we may in principle intro- duce a bias against weak silicate emitters. Indeed, although we cannot set useful upper limits on the silicate strength in most of the other objects, we have likely missed objects with low sili- cate strength. However, the mean spectrum in Fig. 1 includes all QSOs in our sample, and therefore its silicate strength should be representative of the average Silicate emission in the sample (at least for the objects at z<2.5, which are the ones where the observed band includes the silicate feature, and which are the majority). Figs. 8a-b show the silicate strength of the objects in our combined sample as functions of L5100 and L([OIII]λ5007). The red square indicates the silicate strength in the high-z QSO mean spectrum, while its horizontal bar indicates the range of lumi- nosities spanned by the subsample of objects at z<2.5 (i.e. those contributing to the silicate feature in the mean spectrum). Low redshift AGNs and high redshift QSOs show an apparently clear correlation between silicate strength and luminosity. Although with a significant spread, the Silicate strength is observed to pos- itively correlate also with the accretion rate L/LEdd and with the BH mass, as shown in Figs. 8c-d. Essentially, the correlations observed for the Silicate strength reflects the same correlation observed for the L(6.7µm)/L(5100Å) (with the exception of the accretion rate), in agreement with the idea that also the Silicate strength is a proxy of the covering factor of the circumnuclear hot dust, for the reasons discussed above. Unfortunately, the correlations observed for the Silicate strength do not improve our understanding on the origin of the decreasing covering factor with luminosity, i.e. whether the driv- ing physical quantity is the luminosity itself, the accretion rate or the black hole mass. Formally, the correlation between Silicate strength and optical continuum luminosity is tighter than the oth- ers (Tab. 3), possibly hinting at the luminosity itself as the quan- tity driving the dust covering factor. However, there are a few low luminosity objects, such as a few LINERs, which have large silicate strengths (Sturm et al., 2005) and which clearly deviate from the correlation shown in Fig. 8a, thus questioning the role of luminosity in determining the Silicate strength. In addition, the apparently looser correlations of Silicate strength versus ac- cretion rate and BH mass may simply be due to the additional uncertainties affecting the latter two quantities (horizontal, black error bars in Figs. 8c,d). 4.3. PAHs and star formation The presence and intensity of star formation in QSOs has been a hotly debated issue during the past few years. A major step forward in this debate was achieved by Schweitzer et al. (2006) through the Spitzer IRS detection of PAH features in a sample of nearby QSOs, revealing vigorous star formation in these ob- jects. The analysis also shows that the far-IR emission in these QSOs is dominated by star formation and that the star forming activity correlates with the nuclear AGN power. Here we show in Fig. 9 the latter correlation in terms of PAH(7.7µm) luminosity versus L5100 by using the PAH luminosities from the sample of Schweitzer et al. (2006) and the corresponding optical data from Marziani et al. (2003). Although the large fraction of upper lim- its in the former sample prevents a careful statistical characteri- zation, Fig. 9 shows a general correlation between QSO optical luminosity and starburst activity in the host galaxy as traced by the PAH luminosity. The scale on the right hand side of Fig. 9 translates the 7.7µm PAH luminosity into star formation rate (SFR). This was ob- tained by combining the average L(PAH7.7µm)/L(FIR) ratio ob- tained by Schweitzer et al. (2006) for the starburst dominated QSOs in their sample with the SFR/L(FIR) given in Kennicutt (1998), yielding SFR [M⊙ yr −1] = 3.46 10−42 L(PAH7.7µm) [erg s −1] (8) The average spectrum of high-z, luminous QSOs in Fig. 1 does not show evidence for the presence of PAH features and can only provide an upper limit on the PAH flux relative to the flux at 6.7µm (since all spectra were normalized to the lat- ter wavelength prior to computing the average). However, we can derive an upper limit on the PAH luminosity by assum- ing the average distance of the sources in the sample. The in- ferred upper limit on the PAH luminosity is reported with a red square in Fig. 9a, and it is clearly below the extrapolation of the L(PAH7.7µm) − λLλ(5100Å) relation found for local, low- luminosity QSOs. This is shown more clearly in Fig. 9b which shows the distribution of the ratio L(PAH7.7µm)/λLλ(5100Å) for local QSOs (histogram) and the upper limit inferred from the av- erage spectrum of high-z, luminous QSOs (red solid line). The corresponding upper limit on the SFR is ∼ 700 M⊙ yr Note that certainly there are luminous, high-z QSOs with larger star formation rates (e.g. Bertoldi et al., 2003; Beelen et al., 2006; Lutz et al., 2007). However, since our sam- ple is not pre-selected in terms of MIR or FIR emission, our result is not biased in terms of star formation and dust content, and therefore it is representative of the general high-z, luminous QSO population. Our results indicate that the relation between star forma- tion activity, as traced by the PAH features, and QSO power, as traced by L5100, saturates at high luminosity. This result is not surprising. Indeed, if high-z, luminous QSOs were char- acterized by the same average L(PAH7.7µm)/λLλ(5100Å) ob- served in local QSOs, this would imply huge star formation rates of ∼ 7000 M⊙ yr −1 at λLλ(5100Å) ∼ 10 47 erg s−1. On the contrary, the few high-z QSOs detected at submm-mm wave- lengths have far-IR luminosities corresponding to SFR of about 1000−3000 M⊙ yr −1 (Omont et al., 2003). The majority of high- z QSO (∼70%) are undetected at submm-mm wavelengths. The mean mm-submm fluxes of QSOs in various surveys (includ- ing both detections and non-detections) imply SFRs in the range ∼ 500−1500 M⊙ yr −1 (Omont et al., 2003; Priddey et al., 2003), in fair agreement with our finding, especially if we consider the 12 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs Fig. 9. a) PAH(7.7µm) luminosity as a function of the QSO optical luminosity. Blue diamonds are data from Schweitzer et al. (2006). The red square is the upper limit obtained by the average spectrum of luminous, high-z QSOs. b) Distribution of the PAH(7.7µm) to optical luminosity ratio in the local QSOs sample of Schweitzer et al. (2006). The hatched region indicates upper limits. The red vertical line indicate the upper limit inferred from the average spectrum of luminous QSOs at high-z. uncertainties involved in the two different observational methods to infer the SFR. We also note that a similar, independent result was obtained by Haas et al. (2003), who found that in a large sample of QSOs the ratio between LFIR (powered by star formation) and LB de- creases at high luminosities. The finding of a “saturation” of the relation between star for- mation activity and QSO power may provide an explanation for the evolution of the relation between BH mass and galaxy mass at high redshift. Indeed, Peng et al. (2006) and McLure et al. (2006) found that, for a given BH mass, QSO hosts at z∼2 are characterized by a stellar mass lower than expected from the local BH-galaxy mass relation. In other terms, the BH growth is faster, relative to star formation, in high-z, luminous QSOs. Our result supports this scenario by independently showing that the correlation between star formation and AGN activity breaks down at high luminosities. 5. Conclusions We have presented low resolution, mid-IR Spitzer spectra of a sample of 25 luminous QSOs at high redshifts (2 < z < 3.5). We have combined our data with Spitzer spectra of lower lumi- nosity, type-I AGNs, either published in the literature or in the Spitzer archive. The combined sample spans five orders of mag- nitude in luminosity, and allowed us to investigate the dust prop- erties and star formation rate as a function of luminosity. The spectroscopic information allowed us to disentangle the various spectral components contributing to the MIR band (PAH and sil- icate emission) and to sample the continuum at a specific λrest, in contrast to photometric MIR observations. The main results are: – The mid-IR continuum luminosity at 6.7µm correlates with the optical continuum luminosity but the correlation is not linear. In particular, the ratio λLλ(6.7µm)/λLλ(5100Å) de- creases by about a factor of ten as a function of lumi- nosity over the luminosity range 1042.5 < λLλ(5100Å) < 1047.5 erg s−1. This is interpreted as a reduction of the cov- ering factor of the circumnuclear hot dust as a function of luminosity. This result is in agreement and provides an in- dependent confirmation of the recent findings of a decreas- ing fraction of obscured AGN as a function of luminosity, obtained in X-ray and optical surveys. We stress that while X-ray surveys probe the covering factor of the gas, our result provides an independent confirmation by probing the cover- ing factor of the dust. We have also shown that the dust cov- ering factor, as traced by the λLλ(6.7µm)/λLλ(5100Å) ratio, decreases also as a function of the BH mass. Based on these correlations alone it is not possible to determine whether the physical quantity primarily driving the reduction of the cov- ering factor is the AGN luminosity or the BH mass. – The mean spectrum of the luminous, high-z QSOs in our sample shows a clear silicate emission at λrest ∼ 10µm. Silicate emission is also detected in the individual spectra of four high redshift QSOs. When combined with the spec- tra of local, lower luminosity AGNs we find that the sili- cate strength (defined as the ratio between the maximum of the silicate feature and the extrapolated featureless contin- uum) tend to increase as a function of luminosity. The sili- cate strength correlates positively also with the accretion rate and with the BH mass, albeit with a large scatter. – The mean MIR spectrum of the luminous, high-z QSOs in our sample does not show evidence for PAH emission. Our sample is not pre-selected by the FIR emission and therefore it is not biased in terms star formation. As a consequence, the upper limit on the PAH emission in the total mean spectrum provides a useful, representative upper limit on the SFR in luminous QSOs at high redshifts. We find that the ratio be- tween PAH luminosity and QSO optical luminosity is signifi- cantly lower than observed in local, lower luminosity AGNs, implying that the correlation between star formation rate and R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 13 AGN power probably saturates at high luminosities. 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Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs ‘[HB89]0123+257’ on page 16 ‘HS0211+1858’ on page 16 ‘2QZJ023805.8-274337’ on page 16 ‘SDSSJ024933.42-083454.4’ on page 16 ‘Q0256-0000’ on page 16 ‘Q0302-0019’ on page 16 ‘[HB89]0329-385’ on page 16 ‘SDSSJ100428.43+001825.6’ on page 16 ‘TON618’ on page 16 ‘[HB89]1318-113’ on page 16 ‘[HB89]1346-036’ on page 16 ‘UM629’ on page 16 ‘UM632’ on page 16 ‘BS1425+606’ on page 16 ‘[VCV01]J1649+5303’ on page 16 ‘SDSSJ170102.18+612301.0’ on page 16 ‘SDSSJ173352.22+540030.5’ on page 16 ‘[HB89]2126-158’ on page 16 ‘2QZJ221814.4-300306’ on page 16 ‘2QZJ222006.7-280324’ on page 16 ‘Q2227-3928’ on page 16 ‘[HB89]2254+024’ on page 16 ‘2QZJ234510.3-293155’ on page 16 ‘Mrk335’ on page 16 ‘IIIZw2’ on page 16 ‘PG0050+124’ on page 16 ‘PG0052+251’ on page 16 ‘Fairall9’ on page 16 ‘Mkr79’ on page 16 ‘PG0804+761’ on page 16 ‘Mrk704’ on page 16 ‘PG0953+414’ on page 16 ‘NGC3516’ on page 16 ‘PG1116+215’ on page 16 ‘NGC3783’ on page 16 ‘PG1151+117’ on page 16 ‘NGC4051’ on page 16 ‘PG1211+143’ on page 16 ‘NGC4593’ on page 16 ‘PG1309+355’ on page 16 ‘PG1351+640’ on page 16 ‘IC4329a’ on page 16 ‘NGC5548’ on page 16 ‘Mrk817’ on page 16 ‘Mrk509’ on page 16 ‘Mrk926’ on page 16 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 15 Table 1. Combined sample of high-z luminous QSO, local QSO and Sy1, and physical properties inferred from optical-UV spectra. Name RA(J2000) Dec(J2000) z log(λLλ(5100Å)) logL([OIII]) logM(BH) d L/LdEdd αopt−UV Ref. (erg/s) (erg/s) M⊙ High-z luminous QSOs 2QZJ002830.4-281706 00:12:21.18 −28:36:30.2 2.401 46.59 44.41 9.72 0.35 −1.42 1 LBQS0109+0213 01:12:16.91 +02:29:47.6 2.349 46.81 44.51 10.01 0.30 −1.75 1 [HB89]0123+257a 01:26:42.79 +25:59:01.3 2.369 46.58 44.32 9.10 1.40 −1.65 1 HS0211+1858 02:14:29.70 +19:12:37.0 2.470 46.63 44.38 10.11 0.16 −0.01 3,10 2QZJ023805.8-274337 02:38:05.80 −27:43:37.0 2.471 46.58 <43.71 9.41 0.69 −1.59 1 SDSSJ024933.42-083454.4 02:49:33.41 −08:34:54.4 2.491 46.39 44.12 9.67 0.25 −1.36 1 Q0256-0000 02:59:05.64 +00:11:21.9 3.377 46.99 44.55 10.11 0.19 −0.96 2 Q0302-0019 03:04:49.86 −00:08:13.4 3.286 46.83 45.01 10.11 0.30 −1.66 2 [HB89]0329-385 03:31:06.34 −38:24:04.8 2.435 46.72 44.31 10.11 0.18 −1.79 1 SDSSJ100428.43+001825.6 10:04:28.44 +00:18:25.6 3.040 46.45 44.47 9.34 0.70 −0.70 3,11 TON618a 12:28:24.97 +31:28:37.6 2.226 47.32 <44.12 10.81 0.14 −1.27 1 [HB89]1318-113 13:21:09.38 −11:39:31.6 2.306 46.90 44.32 9.76 0.62 −0.99 1 [HB89]1346-036 13:48:44.08 −03:53:24.9 2.370 46.89 43.73 9.95 0.41 −1.26 1 UM629 14:03:23.39 −00:06:06.9 2.460 46.57 44.41 9.17 1.16 −1.40 1 UM632b 14:04:45.89 −01:30:21.9 2.517 46.55 44.04 9.44 0.61 −1.27 1 SBS1425+606 14:26:56.10 +60:25:50.0 3.202 47.39 45.04 9.83 1.73 −1.45 1 [VCV01]J1649+5303 16:49:14.90 +53:03:16.0 2.260 46.70 44.19 9.99 0.24 −0.86 3,11 SDSSJ170102.18+612301.0 17:01:02.18 +61:23:01.0 2.301 46.35 <43.51 9.73 0.20 −1.48 1 SDSSJ173352.22+540030.5 17:33:52.23 +54:00:30.5 3.428 47.02 44.36 9.58 1.28 −1.51 1 [HB89]2126-158b 21:29:12.17 −15:38:41.0 3.282 47.27 44.66 9.73 1.60 0.72 1 2QZJ221814.4-300306 22:18:14.40 −30:03:06.0 2.389 46.55 43.95 9.28 0.89 −1.27 1 2QZJ222006.7-280324 22:20:06.70 −28:03:23.0 2.414 47.23 44.64 10.21 0.54 −1.28 1 Q2227-3928 22:30:32.95 −39:13:06.8 3.438 46.95 <44.02 10.31 0.19 −1.25 2 [HB89]2254+024 22:57:17.56 +02:43:17.5 2.083 46.46 43.95 9.10 1.08 −1.37 1 2QZJ234510.3-293155 23:45:10.36 −29:31:54.7 2.382 46.33 43.97 9.38 0.42 −1.26 1 High-z QSO aver. (z<2.5)c 46.63 44.07 9.68 0.51 Local QSOs and Sy1s Mrk335 00:06:19.52 +20:12:10.4 0.025 43.62 41.29 7.10 0.28 −2.00 4,14 IIIZw2 00:10:30.80 +10:58:13.0 0.090 44.02 42.25 8.19 0.16 −1.56 4,13 PG0050+124 00:53:34.94 +12:41:36.2 0.058 44.36 41.87 7.09 0.96 −0.91 4,13 PG0052+251 00:54:52.10 +25:25:38.0 0.155 44.46 42.57 8.55 0.21 −2.27 4,13 Fairall9 01:23:45.78 −58:48:20.5 0.046 43.80 41.91 8.27 0.10 −1.99 4,13 Mrk79 07:42:32.79 +49:48:34.7 0.022 43.58 41.37 8.12 0.08 −0.83 9,14 PG0804+761 08:10:58.60 +76:02:42.0 0.100 44.42 42.03 8.08 0.33 −2.64 4,12 Mrk704 09:18:26.00 +16:18:19.2 0.029 43.44 41.18 7.97 0.08 – 4 PG0953+414 09:56:52.40 +41:15:22.0 0.234 44.96 42.69 8.39 0.56 −2.12 4,13 NGC3516 11:06:47.49 +72:34:06.8 0.009 42.81 40.52 7.39 0.06 −1.09 7,8,14 PG1116+215 11:19:08.60 +21:19:18.0 0.176 44.84 42.27 8.27 0.53 −2.58 4,13 NGC3783 11:39:01.72 −37:44:18.9 0.010 43.05 41.10 7.33 0.09 −1.46 4,13 PG1151+117 11:53:49.27 +11:28:30.4 0.176 44.48 42.09 8.31 0.28 −2.50 4,12 NGC4051 12:03:09.61 +44:31:52.8 0.002 41.39 39.64 5.32 0.06 −0.82 5,6,14 PG1211+143 12:14:17.70 +14:03:12.6 0.085 44.58 41.94 7.69 0.68 −1.32 4,13 NGC4593 12:39:39.42 −05:20:39.3 0.009 42.60 40.34 7.40 0.04 −1.26 4,14 PG1309+355 13:12:17.76 +35:15:21.2 0.184 44.50 42.18 8.29 0.30 −2.45 4,12 PG1351+640 13:53:15.80 +63:45:45.4 0.087 44.80 42.52 8.76 0.28 −0.86 4,13 IC4329a 13:49:19.26 −30:18:34.0 0.016 43.13 40.89 7.77 0.06 – 4 NGC5548 14:17:59.53 +25:08:12.4 0.017 43.10 41.15 7.78 0.06 −1.73 4,13 Mrk817 14:36:22.06 +58:47:39.3 0.033 43.96 41.65 8.11 0.16 −0.56 4,14 Mrk509 20:44:09.73 −10:43:24.5 0.034 44.01 42.13 7.87 0.22 −1.69 4,13 Mrk926 23:04:43.47 −08:41:08.6 0.047 43.83 42.29 8.55 0.08 – 4 The following quantities are reported in each column: column 1, object name; columns 2-3, coordinates (J2000); column 4, redshift; column 5, log of the continuum luminosity λLλ (in units of erg/s) at the rest frame wavelength 5100Å; column 6, log of the [OIII]λ5007 emission line luminosity (in units of erg/s); column 7, log of the black hole mass (in units of M⊙); column 8, Eddington ratio Lbol/LEdd; column 9, optical-to-UV (1450Å– 5100Å) continuum slope (Fλ ∝ λ αopt−UV); column 11: reference for the optical and UV data: 1 - Shemmer et al. (2004), Netzer et al. (2004) and therein references for UV data, 2 - Dietrich et al. (2002) and therein references for UV data, 3 - Juarez et al (in prep.), 4 - Marziani et al. (2003), 5 - Suganuma et al. (2006), 6 - Peterson et al. (2000), 7 - Wanders et al. (1993), 8 - Ho & Ulvestad (2001), 9 - Peterson et al. (1998), 10 - Engels et al. (1998), 11 - SDSS DR5 archive, 12- Baskin & Laor (2005), 13 - Evans & Koratkar (2004), 14 - Kaspi et al. (2005). a Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is near or above the observed value, hence the 6.7µm flux is probably dominated by synchrotron emission; these objects will be excluded from statistical analyses. b Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is well below the observed value, hence the 6.7µm flux is likely thermal. c Optical luminosities, black hole mass and Eddington ratio for the stacked spectrum refer to the average values of only the objects at z<2.5, i.e. those who contribute to the Silicate feature observed in the stacked spectrum. d As discussed in Shemmer et al. (2004), the uncertainties on the BH masses and accretion rate are no larger than a factor of two. 16 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs Table 2. Infrared properties of the combined sample of high-z luminous QSO, local QSO and Sy1 Name FMIR αMIR λLλ (6.7µm) λLλ (5100Å) Si strength Ref. (mJy) High-z luminous QSOs 2QZJ002830.4-281706 4.8 -0.59 0.75 1 LBQS0109+0213 8.3 -0.91 0.75 1 [HB89]0123+257a 4.9 -1.69 0.76 1 HS0211+1858 7.3 -1.36 1.09 1 2QZJ023805.8-274337 3.7 -1.04 0.62 0.81±0.16 1 SDSSJ024933.42-083454.4 2.2 -1.26 0.57 1 Q0256-0000 2.7 -1.34 0.30 1 Q0302-0019 3.6 -1.67 0.56 1 [HB89]0329-385 5.5 -1.74 0.65 1 SDSSJ100428.43+001825.6 2.3 -1.51 0.73 1 TON618a 20.1 -1.16 0.51 0.04±0.01 1 [HB89]1318-113 6.6 -1.70 0.47 1 [HB89]1346-036 12.9 -1.68 0.99 1 UM629 3.8 -0.68 0.65 1 UM632b 2.1 -1.77 0.40 1 BS1425+606 23.9 -1.56 0.96 1 [VCV01]J1649+5303 9.4 -1.28 1.03 1 SDSSJ170102.18+612301.0 3.3 -0.66 0.83 1 SDSSJ173352.22+540030.5 2.0 -2.88 0.21 1 [HB89]2126-158b 19.1 -1.18 1.08 1 2QZJ221814.4-300306 4.6 -1.03 0.78 1 2QZJ222006.7-280324 16.0 -1.52 0.58 0.65±0.05 1 Q2227-3928 3.0 -1.70 0.38 1 [HB89]2254+024 3.6 -1.12 0.60 0.63±0.15 1 2QZJ234510.3-293155 4.2 -1.33 1.16 0.92±0.15 1 High-z QSO aver. (z<2.5)c -1.57 0.58±0.10 Local QSOs and Sy1s Mrk335 130. -1.38 1.90 0.25±0.06 3 IIIZw2 52. -1.09 4.02 0.05±0.03 2,3 PG0050+124 245. -0.74 3.55 0.38±0.05 2,3 PG0052+251 28. -1.35 2.40 0.33±0.06 2,3 Fairall9 146. -0.90 4.83 0.21±0.07 2,3 Mkr79 200. -1.03 2.44 0.10±0.06 3 PG0804+761 88. -1.81 3.38 0.60±0.05 2,3 Mrk704 190. -0.95 5.64 0.09±0.06 3 PG0953+414 26. -1.88 1.68 0.40±0.08 2,3 NGC3516 210. -1.05 2.40 0.06±0.05 3 PG1116+215 66. -1.71 3.08 0.22±0.05 2,3 NGC3783 315. -1.08 2.69 -0.01±0.03 2,3 PG1151+117 10. -2.19 1.08 0.36±0.13 2,3 NGC4051 230. -0.43 4.71 0.06±0.05 3 PG1211+143 100. -1.26 1.89 0.55±0.05 2,3 NGC4593 184. -1.14 3.61 0.08±0.05 2,3 PG1309+355 25. -1.24 2.82 0.41±0.07 2,3 PG1351+640 53. -0.89 0.64 1.25±0.05 2,3 IC4329a 487. -0.67 9.00 0.01±0.03 2,3 NGC5548 69. -1.06 1.53 0.27±0.05 2,3 Mrk817 140. -0.65 1.61 0.16±0.06 3 Mrk509 179. -1.29 1.98 0.11±0.04 2,3 Mrk926 55. -1.40 1.77 0.26±0.04 2,3 The following quantities are reported in each column: column 1, object name; column 2, continuum flux density at the observed wavelength corresponding to λrest = 6.7µm (after removing starburst and stellar components, in units of mJy); column 3, power-law index (Fλ ∝ λ α) fitted to the continuum in the 5–8µm range (starburst component–subtracted); column 4, ratio of the continuum emission at 5100Å and at 6.7µm, λLλ(5100Å)/λLλ(6.7µm); column 5, Silicate strength; column 6: reference for the infrared data: 1 - this work (from Spitzer program 20493) ; 2 - Shi et al. (2006); 3 - this work (from Spitzer archival data). a Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is near or above the observed value, hence the 6.7µm flux is probably dominated by synchrotron emission; these objects will be excluded from statistical analyses. b Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is well below the observed value, hence the 6.7µm flux is likely thermal. c The ratio λLλ(5100Å)/λLλ(6.7µm) is not defined for the stacked spectrum, since all spectra were normalized to the 6.7µm flux before stacking. As a consequence, only the Silicate strength (and more generally the continuum shape) has a physical meaning for the stacked spectrum. R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 17 Table 3. Spearman-rank coefficients for the correlations in Figs.4, 8, 5 and 6. log(λLλ(5100Å)) log(L[OIII]) log(L/LEdd) log(MBH) αMIR αopt−UV λLλ (6.7µm) λLλ (5100Å) −0.76 (< 10−6) −0.72 (< 10−6) −0.44 (3 10−4) −0.70 (< 10−6) 0.29 (0.05) −0.20 (0.19) log(Si str.) 0.83 (2 10−5) 0.75 (6 10−6) 0.75 (8 10−6) 0.66 (8 10−4) Numbers in parenthesis give the probability for the correlation coefficient to deviate from zero. Introduction Sample selection, observations and data reduction Analysis Main observational results A comparison with MIR properties of lower luminosity AGNs Discussion Dust covering factor Covering factor as a function of source luminosity and BH mass Model uncertainties Comparison with previous works Silicate emission PAHs and star formation Conclusions

Generating entanglement of photon-number states with coherent light via cross-Kerr nonlinearity Zhi-Ming Zhang∗, Jian Yang, and Yafei Yu Laboratory of Photonic Information Technology, School of Information and Photoelectronics, South China Normal University, Guangzhou 510006, China November 25, 2018 Abstract We propose a scheme for generating entangled states of light fields. This scheme only requires the cross-Kerr nonlinear interaction between coherent light-beams, followed by a homodyne de- tection. Therefore, this scheme is within the reach of current technology. We study in detail the generation of the entangled states between two modes, and that among three modes. In addition to the Bell states between two modes and the W states among three modes, we find plentiful new kinds of entangled states. Finally, the scheme can be extend to generate the entangled states among more than three modes. PACS: 03.67.Mn; 42.50.Dv; 42.50.Ct 1 Introduction Entanglement is a characteristic feature of quantum states and has important applications in quantum science and technology, for example, in quantum computation and quantum information [1]. There are a lot of schemes for generating various kinds of entanglement, for example, the entanglement between photons, the entanglement between atoms, the entanglement between trapped ions, and the entanglement between different kinds of particles (for example, between photons and atoms). In addition to the entanglement between two parties, there are also entanglement of multiparties. Among these schemes many use single-photon sources and/or single-photon detectors. Although there are great progresses in the study on these single-photon devices, how to obtain them is still a challenging task. In this paper we propose a simple scheme for generating entangled states of light fields. This scheme only requires the cross-Kerr nonlinear interaction between light fields in coherent states, followed by a homodyne detection. Therefore, this scheme is within the reach of current technology.The basic idea of this scheme is shown in Figure 1. Mode a is a bright beam which is in a coherent state |α〉. Mode b is a weak or bright beam which is also in a coherent state. BS is a 50/50 beam splitter. KM1 and KM2 are Kerr media. HD means homodyne detection [2]. This paper is organized as follows: In section 2 we briefly introduce the cross-Kerr nonlinear inter- action between two field-modes. In section 3 and section 4 we study the generation of entanglement between two modes and that among three modes, respectively. Section 5 is a summary. ∗Corresponding author: zmzhang@scnu.edu.cn http://arxiv.org/abs/0704.1560v1 2 Cross-Kerr nonlinear interaction First, let us briefly review the cross-Kerr nonlinear interaction between a mode A and a mode B. The interaction Hamiltonian has the form [3] HCK = ~Kn̂An̂B, (1) where n̂A and n̂B are the photon-number operator of mode A and mode B, respectively. The coupling coefficient K is proportional to the third-order nonlinear susceptibility χ(3). The time-evolution operator is U (t) = exp = exp {−iKn̂An̂Bt} = exp {−iτ n̂An̂B} = U (τ) , (2) in which τ = Kt = K (l/v), it can be named as the scaled interaction time, or the nonlinear phase shift. Here l is the length of the Kerr medium and v is the velocity of light in the Kerr medium. The cross Kerr nonlinearity has following property U (τ) |n〉B |α〉A = |n〉B ∣αe−inτ , (3) here |n〉 and |α〉 are the photon number state and the coherent state, respectively. 3 Entanglement between two modes Now let us study the generation of the entangled states between two modes. The scheme is shown in Figure 1. Assume that mode a is in a coherent state |α〉 [4]. Mode b is also in a coherent state which is divided by the 50/50 beam splitter BS into two beams b1 and b2, and both b1 and b2 are in coherent state |β〉. We first consider the case of weak coherent state |β〉. In this case we have |β〉 ≈ 1√ 1 + |β|2 (|0〉+ β |1〉) , (4) where |0〉 and |1〉 are the vacuum state and one-photon state, respectively. Let mode a interacts with mode b1 and b2 successively. For simplicity, we assume that both the scaled interaction times are τ , thai is, τ1 = K1t1 = τ2 = K2t2 = τ.The interactions change the state as following way |β〉2 |β〉1 |α〉a → 1 + |β|2 |0〉2 |0〉1 |α〉a + β (|1〉2 |0〉1 + |0〉2 |1〉1) ∣αe−iτ + β2 |1〉2 |1〉1 ∣αe−i2τ where the subscripts 1 and 2 denote modes b1 and b2, respectively. We note that the internal product of coherent states satisfies [4] αe−inτ |αe−i(n+1)τ = e−4|α| 2 sin2(τ/2) ≈ e−|α| 2τ2 , (6) in which we have taken into account the fact that in practice τ is small [3] and therefore sin (τ/2) ≈ τ/2. However, if mode a is bright enough so that |α|2 τ2 ≫ 1, then the two coherent states will be approximately orthogonal. This condition can be easily satisfied in experiments and in following discussions we assume that it is satisfied. In this case, a homodyne detection can distinguish different coherent states [5]. Therefore, when we find that mode a is in the coherent state ∣αe−iτ , then beam b1 and beam b2 will be projected into the entangled state (|1〉2 |0〉1 + |0〉2 |1〉1) , (7) and the probability for getting this entangled state is 2 |β|2 / 1 + |β|2 .This state is one of Bell states [1] and a special case of the NOON states [6]. Now let us consider the general situation in which beam b1 and beam b2 are normal coherent states [4]. In this situation, |β〉 = exp |n〉 . (8) The cross-Kerr interactions transform the state as follows |β〉2 |β〉1 |α〉a = e −|β|2 βm+n√ |m〉2 |n〉1 |α〉a → e−|β| βm+n√ |m〉2 |n〉1 αe−i(m+n)τ . (9) If the homedyne detection finds mode a in the state ∣αe−i(m+n)τ ∣αe−ikτ (k = m+n = 1, 2, ...), then mode b1 and mode b2 will be collapse into the entangled state n! (k − n)! |k − n〉2 |n〉1 (k = 1, 2, ...). (10) Since in this state the sum of photon numbers of the two modes is equal to k, we name this state as the 2-mode k-photon entangled state. The probability for getting this state is exp(−2 |β|2)2 |β|2k . The entanglement property of the states expressed by Eq.(10) can be proved by using following entanglement criteria [7] b+1 b2 > 〈Nb1Nb2〉 , (11) where Nb1(Nb2), b1(b2) and b 2 ) are the photon-number operator, the photon annihilation operator and the photon creation operator of mode b1(b2), respectively. For the states of equation (10), we can find b+1 b2 k2 , and 〈Nb1Nb2〉 = 14k (k − 1) . Therefore the entanglement condition (11) is satisfied, and the states (10) are indeed entangled states. For k = 1, equation (10) reduces to equation (7), and some other examples of the 2-mode k-photon entangled states are listed below. (|2〉2 |0〉1 + |0〉2 |2〉1) + 2 |1〉2 |1〉1 (k = 2) (12) (|3〉2 |0〉1 + |0〉2 |3〉1) + 3 (|2〉2 |1〉1 + |1〉2 |2〉1) (k = 3) (13) Equations (12) and (13) are new kinds of entangled states. Equation (12) can be understood as a superposition of a NOON state (|2〉2 |0〉1 + |0〉2 |2〉1) and a product state |1〉2 |1〉1 ,while equation (13) can be understood as a superposition of a NOON state (|3〉2 |0〉1 + |0〉2 |3〉1) and a NOON − like state (|2〉2 |1〉1 + |1〉2 |2〉1). We also note that in the superposition (13) the probability of getting the state (|2〉2 |1〉1 + |1〉2 |2〉1) is larger than that of getting the state (|3〉2 |0〉1 + |0〉2 |3〉1) .That is, the photons trend to distribute between the two modes symmetrically. The properties and applications of these new kinds of entangled states will be studied in the future. 4 Entanglement among three modes We can extend the scheme above to generate the entanglement among three modes. For this purpose we modify the scheme from Figure 1 to Figure 2, in which BS1 has the reflection/transmission = 1/2 and BS2 has the reflection/transmission = 1/1, so that the three beams b1, b2 and b3 have the same strength, and we assume all of them are in the coherent state |β〉 . We let mode a, in a coherent state |α〉, interacts with modes b1, b2 and b3 successively. And for simplicity, we assume that all of the scaled interaction times are equal, thai is, τ1 = τ2 = τ3 = τ. For the situation in which |β〉 is weak and can be expressed as in equation (4), the interactions transform the states in the following way |β〉3 |β〉2 |β〉1 |α〉a → 1 + |β|2 {|0〉3 |0〉2 |0〉1 |α〉a +β (|1〉3 |0〉2 |0〉1 + |0〉3 |1〉2 |0〉1 + |0〉3 |0〉2 |1〉1) ∣αe−iτ +β2 (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1) ∣αe−i2τ +β3 |1〉3 |1〉2 |1〉1 ∣αe−i3τ }. (14) As discussed above, we assume that different coherent states in above equation are approximately orthogonal, and we can use homodyne detection to distinguish them [5]. If we find that mode a is in state ∣αe−iτ then modes b1, b2 and b3 will be projected to the entangled state (|1〉3 |0〉2 |0〉1 + |0〉3 |1〉2 |0〉1 + |0〉3 |0〉2 |1〉1) , (15) and the probability for obtaining this state is 3 |β|2 / 1 + |β|2 . On the other hand, If we find that mode a is in state ∣αe−i2τ then modes b1, b2 and b3 will be projected to the entangled state (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1) , (16) and the probability for getting this state is 3 |β|4 / 1 + |β|2 . Equations (15) and (16) can be named as 1-photon W state [8] and 2-photon W state, respectively. For the general case in which |β〉 is not very weak we use equation (8). In this case the interactions transform the states as follows: |β〉3 |β〉2 |β〉1 |α〉a = e −3|β|2/2 l,m,n βl+m+n√ l!m!n! |l〉3 |m〉2 |n〉1 |α〉a → e−3|β| l,m,n βl+m+n√ l!m!n! |l〉3 |m〉2 |n〉1 αe−i(l+m+n)τ . (17) If we find that mode a is in the state ∣αe−i(l+m+n)τ ∣αe−ikτ (k = l+m+n = 1, 2, ...), then modes b1, b2 and b3 will be projected to the entangled state (k −m− n)!m!n! |k −m− n〉3 |m〉2 |n〉1 (k = 1, 2, ...). (18) We name this state as the 3-mode k-photon entangled state. The probability for getting this state is exp(−3 |β|2)3 |β|2k .The entanglement property of the states of Eq.(18) can be proved by using following entanglement criteria [7] b+1 b2 > 〈Nb1Nb2〉 and b+2 b3 > 〈Nb2Nb3〉 . (19) For the states (18), we can find b+1 b2 b+2 b3 k2, and 〈Nb1Nb2〉 = 〈Nb2Nb3〉 = 19k (k − 1) .Therefore the entanglement condition (19) is satisfied, and the states (18) are indeed entangled states of three modes. For k = 1, equation (18) reduces to equation (15), and some other examples of the 3-mode k-photon entangled state are as follows: {(|2〉3 |0〉2 |0〉1 + |0〉3 |2〉2 |0〉1 + |0〉3 |0〉2 |2〉1) 2 (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1)} (k = 2) , (20) {(|3〉3 |0〉2 |0〉1 + |0〉3 |3〉2 |0〉1 + |0〉3 |0〉2 |3〉1) 3 (|2〉3 |1〉2 |0〉1 + |2〉3 |0〉2 |1〉1 + |1〉3 |2〉2 |0〉1 + |1〉3 |0〉2 |2〉1 + |0〉3 |2〉2 |1〉1 + |0〉3 |1〉2 |2〉1) 6 |1〉3 |1〉2 |1〉1} (k = 3) . (21) Equation (20) is a superposition of two 2-photon W states. While equation (21) is a superposition of a 3-photon W state (the first line), a product state (the third line), and a state (the second line) which can be expressed as |1〉j |0〉k + |0〉j |1〉k + |1〉i |2〉j |0〉k + |0〉j |2〉k + |0〉i |2〉j |1〉k + |1〉j |2〉k , (22) where the subscripts i = 1,or 2, or 3,and j,k are the other two, respectively. We also note that in the superposition (20) the probability of getting the state (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1) is larger than that of getting the state (|2〉3 |0〉2 |0〉1 + |0〉3 |2〉2 |0〉1 + |0〉3 |0〉2 |2〉1). This shows again that the photons trend to distribute among different modes symmetrically. 5 Summary In summary, we have proposed a scheme for generating entangled states of light fields. This scheme has following advantages: First, the scheme only involves the cross-Kerr nonlinear interaction between coherent light-beams, followed by a homodyne detection. It is not necessary that the cross-Kerr nonlinearity is very large, as long as the coherent light is bright enough. Therefore, this scheme is within the reach of current technology. Second, in addition to the Bell states between two modes and the W states among three modes, plentiful new kinds of entangled states can be generated with this scheme. We also found that in the generated entangled states, the photons have a trend to distribute among different modes symmetrically. Finally, we would like to point out that the scheme can be extend to generate the entangled states among more than three modes. Acknowledgement This work was supported by the National Natural Science Foundation of China under grant nos 60578055 and 10404007. References [1] N.A.Nielsen and I.L.Chuang,Quantum Computation and Quantum Information (Cambridge: Cambridge University Press, 2000 ); D.Bouwmeester, A.Ekert, and A.Zeilinger, The Physics of Quantum Information (Berlin: Springer, 2000). [2] U.Leonhardt,Measuring the Quantum state of light (Cambridge: Cambridge University Press,1997); H.A.Bachor and T.C.Ralph,A Guide to Experiments in Quantum Optics(Weinheim: Wiley-VCH Verlag GmbH & Co.KGaA, 2004). [3] B.C.Sanders and G.J.Milburn, Phys. Rev. A 45, 1919 (1992); C.C.Gerry, Phys. Rev. A 59, 4095 (1999). [4] C.C.Gerry and P.L.Knight, Introductory Quantum Optics (Cambridge: Cambridge University Press, 2005). [5] K.Nemoto and W.J.Munro,Phys. Rev. Lett. 93, 250502 (2004); W.J.Munro,K.Nemoto, R.G.Beauoleil, and T.P.Spiller,Phys. Rev. A 71, 033819 (2005). [6] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996); Z. Y. Ou, Phys. Rev. A 55, 2598 (1997); F.Shafiei, P.Srinivasan, and Z. Y. Ou,Phys. Rev. A 70, 043803 (2004); B.Lin and Z.Y.Ou, Phys. Rev. A 74, 035802 (2006); F.W.Sun, Z.Y.Ou, and G.C.Guo, Phys. Rev. A 73, 023808 (2006). [7] M.Hillery and M.S.Zubairy, Phys. Rev. Lett. 96, 050503 (2006); M.Hillery and M.S.Zubairy, Phys. Rev. A 74, 032333 (2006). [8] W.Dur, G.Vidal, and J.I.Cirac, Phys. Rev. A 62, 062314 (2000); P.Agrawal and A.Pati, Phys. Rev. A 74, 062320 (2006); R.Lohmayer, A.Osterloh, J.Siewert, and A.Uhlmann, Phys. Rev. Lett. 97, 260502 (2006); M.Bourennane, M.Eibl, S.Gaertner, N.Kiesel, C.Kurtsiefer, and H.Weinfurter, Phys. Rev. Lett. 96,100502 (2006); R.Rahimi, A.SaiToh, M.Nakahara, and M.Kitagawa, Phys. Rev. A 75, 032317 (2007). Figure captions Figure 1. Scheme for generating entanglement between two modes. KM:cross-Kerr medium; BS: beam splitter; M: mirror, HD: homodyne detection. Figure 2. Scheme for generating entanglement among three modes. KM:cross-Kerr medium; BS: beam splitter; M: mirror, HD: homodyne detection. Introduction Cross-Kerr nonlinear interaction Entanglement between two modes Entanglement among three modes Summary

We propose a scheme for generating entangled states of light fields. This scheme only requires the cross-Kerr nonlinear interaction between coherent light-beams, followed by a homodyne detection. Therefore, this scheme is within the reach of current technology. We study in detail the generation of the entangled states between two modes, and that among three modes. In addition to the Bell states between two modes and the W states among three modes, we find plentiful new kinds of entangled states. Finally, the scheme can be extend to generate the entangled states among more than three modes.

Introduction Entanglement is a characteristic feature of quantum states and has important applications in quantum science and technology, for example, in quantum computation and quantum information [1]. There are a lot of schemes for generating various kinds of entanglement, for example, the entanglement between photons, the entanglement between atoms, the entanglement between trapped ions, and the entanglement between different kinds of particles (for example, between photons and atoms). In addition to the entanglement between two parties, there are also entanglement of multiparties. Among these schemes many use single-photon sources and/or single-photon detectors. Although there are great progresses in the study on these single-photon devices, how to obtain them is still a challenging task. In this paper we propose a simple scheme for generating entangled states of light fields. This scheme only requires the cross-Kerr nonlinear interaction between light fields in coherent states, followed by a homodyne detection. Therefore, this scheme is within the reach of current technology.The basic idea of this scheme is shown in Figure 1. Mode a is a bright beam which is in a coherent state |α〉. Mode b is a weak or bright beam which is also in a coherent state. BS is a 50/50 beam splitter. KM1 and KM2 are Kerr media. HD means homodyne detection [2]. This paper is organized as follows: In section 2 we briefly introduce the cross-Kerr nonlinear inter- action between two field-modes. In section 3 and section 4 we study the generation of entanglement between two modes and that among three modes, respectively. Section 5 is a summary. ∗Corresponding author: zmzhang@scnu.edu.cn http://arxiv.org/abs/0704.1560v1 2 Cross-Kerr nonlinear interaction First, let us briefly review the cross-Kerr nonlinear interaction between a mode A and a mode B. The interaction Hamiltonian has the form [3] HCK = ~Kn̂An̂B, (1) where n̂A and n̂B are the photon-number operator of mode A and mode B, respectively. The coupling coefficient K is proportional to the third-order nonlinear susceptibility χ(3). The time-evolution operator is U (t) = exp = exp {−iKn̂An̂Bt} = exp {−iτ n̂An̂B} = U (τ) , (2) in which τ = Kt = K (l/v), it can be named as the scaled interaction time, or the nonlinear phase shift. Here l is the length of the Kerr medium and v is the velocity of light in the Kerr medium. The cross Kerr nonlinearity has following property U (τ) |n〉B |α〉A = |n〉B ∣αe−inτ , (3) here |n〉 and |α〉 are the photon number state and the coherent state, respectively. 3 Entanglement between two modes Now let us study the generation of the entangled states between two modes. The scheme is shown in Figure 1. Assume that mode a is in a coherent state |α〉 [4]. Mode b is also in a coherent state which is divided by the 50/50 beam splitter BS into two beams b1 and b2, and both b1 and b2 are in coherent state |β〉. We first consider the case of weak coherent state |β〉. In this case we have |β〉 ≈ 1√ 1 + |β|2 (|0〉+ β |1〉) , (4) where |0〉 and |1〉 are the vacuum state and one-photon state, respectively. Let mode a interacts with mode b1 and b2 successively. For simplicity, we assume that both the scaled interaction times are τ , thai is, τ1 = K1t1 = τ2 = K2t2 = τ.The interactions change the state as following way |β〉2 |β〉1 |α〉a → 1 + |β|2 |0〉2 |0〉1 |α〉a + β (|1〉2 |0〉1 + |0〉2 |1〉1) ∣αe−iτ + β2 |1〉2 |1〉1 ∣αe−i2τ where the subscripts 1 and 2 denote modes b1 and b2, respectively. We note that the internal product of coherent states satisfies [4] αe−inτ |αe−i(n+1)τ = e−4|α| 2 sin2(τ/2) ≈ e−|α| 2τ2 , (6) in which we have taken into account the fact that in practice τ is small [3] and therefore sin (τ/2) ≈ τ/2. However, if mode a is bright enough so that |α|2 τ2 ≫ 1, then the two coherent states will be approximately orthogonal. This condition can be easily satisfied in experiments and in following discussions we assume that it is satisfied. In this case, a homodyne detection can distinguish different coherent states [5]. Therefore, when we find that mode a is in the coherent state ∣αe−iτ , then beam b1 and beam b2 will be projected into the entangled state (|1〉2 |0〉1 + |0〉2 |1〉1) , (7) and the probability for getting this entangled state is 2 |β|2 / 1 + |β|2 .This state is one of Bell states [1] and a special case of the NOON states [6]. Now let us consider the general situation in which beam b1 and beam b2 are normal coherent states [4]. In this situation, |β〉 = exp |n〉 . (8) The cross-Kerr interactions transform the state as follows |β〉2 |β〉1 |α〉a = e −|β|2 βm+n√ |m〉2 |n〉1 |α〉a → e−|β| βm+n√ |m〉2 |n〉1 αe−i(m+n)τ . (9) If the homedyne detection finds mode a in the state ∣αe−i(m+n)τ ∣αe−ikτ (k = m+n = 1, 2, ...), then mode b1 and mode b2 will be collapse into the entangled state n! (k − n)! |k − n〉2 |n〉1 (k = 1, 2, ...). (10) Since in this state the sum of photon numbers of the two modes is equal to k, we name this state as the 2-mode k-photon entangled state. The probability for getting this state is exp(−2 |β|2)2 |β|2k . The entanglement property of the states expressed by Eq.(10) can be proved by using following entanglement criteria [7] b+1 b2 > 〈Nb1Nb2〉 , (11) where Nb1(Nb2), b1(b2) and b 2 ) are the photon-number operator, the photon annihilation operator and the photon creation operator of mode b1(b2), respectively. For the states of equation (10), we can find b+1 b2 k2 , and 〈Nb1Nb2〉 = 14k (k − 1) . Therefore the entanglement condition (11) is satisfied, and the states (10) are indeed entangled states. For k = 1, equation (10) reduces to equation (7), and some other examples of the 2-mode k-photon entangled states are listed below. (|2〉2 |0〉1 + |0〉2 |2〉1) + 2 |1〉2 |1〉1 (k = 2) (12) (|3〉2 |0〉1 + |0〉2 |3〉1) + 3 (|2〉2 |1〉1 + |1〉2 |2〉1) (k = 3) (13) Equations (12) and (13) are new kinds of entangled states. Equation (12) can be understood as a superposition of a NOON state (|2〉2 |0〉1 + |0〉2 |2〉1) and a product state |1〉2 |1〉1 ,while equation (13) can be understood as a superposition of a NOON state (|3〉2 |0〉1 + |0〉2 |3〉1) and a NOON − like state (|2〉2 |1〉1 + |1〉2 |2〉1). We also note that in the superposition (13) the probability of getting the state (|2〉2 |1〉1 + |1〉2 |2〉1) is larger than that of getting the state (|3〉2 |0〉1 + |0〉2 |3〉1) .That is, the photons trend to distribute between the two modes symmetrically. The properties and applications of these new kinds of entangled states will be studied in the future. 4 Entanglement among three modes We can extend the scheme above to generate the entanglement among three modes. For this purpose we modify the scheme from Figure 1 to Figure 2, in which BS1 has the reflection/transmission = 1/2 and BS2 has the reflection/transmission = 1/1, so that the three beams b1, b2 and b3 have the same strength, and we assume all of them are in the coherent state |β〉 . We let mode a, in a coherent state |α〉, interacts with modes b1, b2 and b3 successively. And for simplicity, we assume that all of the scaled interaction times are equal, thai is, τ1 = τ2 = τ3 = τ. For the situation in which |β〉 is weak and can be expressed as in equation (4), the interactions transform the states in the following way |β〉3 |β〉2 |β〉1 |α〉a → 1 + |β|2 {|0〉3 |0〉2 |0〉1 |α〉a +β (|1〉3 |0〉2 |0〉1 + |0〉3 |1〉2 |0〉1 + |0〉3 |0〉2 |1〉1) ∣αe−iτ +β2 (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1) ∣αe−i2τ +β3 |1〉3 |1〉2 |1〉1 ∣αe−i3τ }. (14) As discussed above, we assume that different coherent states in above equation are approximately orthogonal, and we can use homodyne detection to distinguish them [5]. If we find that mode a is in state ∣αe−iτ then modes b1, b2 and b3 will be projected to the entangled state (|1〉3 |0〉2 |0〉1 + |0〉3 |1〉2 |0〉1 + |0〉3 |0〉2 |1〉1) , (15) and the probability for obtaining this state is 3 |β|2 / 1 + |β|2 . On the other hand, If we find that mode a is in state ∣αe−i2τ then modes b1, b2 and b3 will be projected to the entangled state (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1) , (16) and the probability for getting this state is 3 |β|4 / 1 + |β|2 . Equations (15) and (16) can be named as 1-photon W state [8] and 2-photon W state, respectively. For the general case in which |β〉 is not very weak we use equation (8). In this case the interactions transform the states as follows: |β〉3 |β〉2 |β〉1 |α〉a = e −3|β|2/2 l,m,n βl+m+n√ l!m!n! |l〉3 |m〉2 |n〉1 |α〉a → e−3|β| l,m,n βl+m+n√ l!m!n! |l〉3 |m〉2 |n〉1 αe−i(l+m+n)τ . (17) If we find that mode a is in the state ∣αe−i(l+m+n)τ ∣αe−ikτ (k = l+m+n = 1, 2, ...), then modes b1, b2 and b3 will be projected to the entangled state (k −m− n)!m!n! |k −m− n〉3 |m〉2 |n〉1 (k = 1, 2, ...). (18) We name this state as the 3-mode k-photon entangled state. The probability for getting this state is exp(−3 |β|2)3 |β|2k .The entanglement property of the states of Eq.(18) can be proved by using following entanglement criteria [7] b+1 b2 > 〈Nb1Nb2〉 and b+2 b3 > 〈Nb2Nb3〉 . (19) For the states (18), we can find b+1 b2 b+2 b3 k2, and 〈Nb1Nb2〉 = 〈Nb2Nb3〉 = 19k (k − 1) .Therefore the entanglement condition (19) is satisfied, and the states (18) are indeed entangled states of three modes. For k = 1, equation (18) reduces to equation (15), and some other examples of the 3-mode k-photon entangled state are as follows: {(|2〉3 |0〉2 |0〉1 + |0〉3 |2〉2 |0〉1 + |0〉3 |0〉2 |2〉1) 2 (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1)} (k = 2) , (20) {(|3〉3 |0〉2 |0〉1 + |0〉3 |3〉2 |0〉1 + |0〉3 |0〉2 |3〉1) 3 (|2〉3 |1〉2 |0〉1 + |2〉3 |0〉2 |1〉1 + |1〉3 |2〉2 |0〉1 + |1〉3 |0〉2 |2〉1 + |0〉3 |2〉2 |1〉1 + |0〉3 |1〉2 |2〉1) 6 |1〉3 |1〉2 |1〉1} (k = 3) . (21) Equation (20) is a superposition of two 2-photon W states. While equation (21) is a superposition of a 3-photon W state (the first line), a product state (the third line), and a state (the second line) which can be expressed as |1〉j |0〉k + |0〉j |1〉k + |1〉i |2〉j |0〉k + |0〉j |2〉k + |0〉i |2〉j |1〉k + |1〉j |2〉k , (22) where the subscripts i = 1,or 2, or 3,and j,k are the other two, respectively. We also note that in the superposition (20) the probability of getting the state (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1) is larger than that of getting the state (|2〉3 |0〉2 |0〉1 + |0〉3 |2〉2 |0〉1 + |0〉3 |0〉2 |2〉1). This shows again that the photons trend to distribute among different modes symmetrically. 5 Summary In summary, we have proposed a scheme for generating entangled states of light fields. This scheme has following advantages: First, the scheme only involves the cross-Kerr nonlinear interaction between coherent light-beams, followed by a homodyne detection. It is not necessary that the cross-Kerr nonlinearity is very large, as long as the coherent light is bright enough. Therefore, this scheme is within the reach of current technology. Second, in addition to the Bell states between two modes and the W states among three modes, plentiful new kinds of entangled states can be generated with this scheme. We also found that in the generated entangled states, the photons have a trend to distribute among different modes symmetrically. Finally, we would like to point out that the scheme can be extend to generate the entangled states among more than three modes. Acknowledgement This work was supported by the National Natural Science Foundation of China under grant nos 60578055 and 10404007. References [1] N.A.Nielsen and I.L.Chuang,Quantum Computation and Quantum Information (Cambridge: Cambridge University Press, 2000 ); D.Bouwmeester, A.Ekert, and A.Zeilinger, The Physics of Quantum Information (Berlin: Springer, 2000). [2] U.Leonhardt,Measuring the Quantum state of light (Cambridge: Cambridge University Press,1997); H.A.Bachor and T.C.Ralph,A Guide to Experiments in Quantum Optics(Weinheim: Wiley-VCH Verlag GmbH & Co.KGaA, 2004). [3] B.C.Sanders and G.J.Milburn, Phys. Rev. A 45, 1919 (1992); C.C.Gerry, Phys. Rev. A 59, 4095 (1999). [4] C.C.Gerry and P.L.Knight, Introductory Quantum Optics (Cambridge: Cambridge University Press, 2005). [5] K.Nemoto and W.J.Munro,Phys. Rev. Lett. 93, 250502 (2004); W.J.Munro,K.Nemoto, R.G.Beauoleil, and T.P.Spiller,Phys. Rev. A 71, 033819 (2005). [6] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996); Z. Y. Ou, Phys. Rev. A 55, 2598 (1997); F.Shafiei, P.Srinivasan, and Z. Y. Ou,Phys. Rev. A 70, 043803 (2004); B.Lin and Z.Y.Ou, Phys. Rev. A 74, 035802 (2006); F.W.Sun, Z.Y.Ou, and G.C.Guo, Phys. Rev. A 73, 023808 (2006). [7] M.Hillery and M.S.Zubairy, Phys. Rev. Lett. 96, 050503 (2006); M.Hillery and M.S.Zubairy, Phys. Rev. A 74, 032333 (2006). [8] W.Dur, G.Vidal, and J.I.Cirac, Phys. Rev. A 62, 062314 (2000); P.Agrawal and A.Pati, Phys. Rev. A 74, 062320 (2006); R.Lohmayer, A.Osterloh, J.Siewert, and A.Uhlmann, Phys. Rev. Lett. 97, 260502 (2006); M.Bourennane, M.Eibl, S.Gaertner, N.Kiesel, C.Kurtsiefer, and H.Weinfurter, Phys. Rev. Lett. 96,100502 (2006); R.Rahimi, A.SaiToh, M.Nakahara, and M.Kitagawa, Phys. Rev. A 75, 032317 (2007). Figure captions Figure 1. Scheme for generating entanglement between two modes. KM:cross-Kerr medium; BS: beam splitter; M: mirror, HD: homodyne detection. Figure 2. Scheme for generating entanglement among three modes. KM:cross-Kerr medium; BS: beam splitter; M: mirror, HD: homodyne detection. Introduction Cross-Kerr nonlinear interaction Entanglement between two modes Entanglement among three modes Summary

A PARACHUTE FOR THE DEGREE OF A POLYNOMIAL IN ALGEBRAICALLY INDEPENDENT ONES S. VÉNÉREAU Abstract. We give a simpler proof as well as a generalization of the main result of an article of Shestakov and Umirbaev ([3]). This latter article being the first of two that solve a long-standing conjecture about the non-tameness, or ”wildness”, of Nagata’s automorphism. As corollaries we get interesting informations about the leading terms of polynomials forming an automorphism of K[x1, · · · , xn] and reprove the tameness of automorphisms of K[x1, x2]. The following notations are fixed throughout the article: K is a field of cha- racteristic 0 and K[x1, · · · , xn] is the ring of polynomials in the n indetermi- nates x1, · · · , xn with coefficients in K, endowed with the classical degree func- tion: deg. We consider m algebraically independent polynomials in K[x1, · · · , xn]: f1, · · · , fm of respective degrees d1, · · · , dm. There is also, for every polynomial G ∈ K[f1, · · · , fm] a unique one G(X1, · · · , Xm) ∈ K[X1, · · · , Xm], where X1, · · · , Xm are new indeterminates, such that G = G(f1, · · · , fm). By abuse of notation we will write ∂G to denote ∂G (f1, · · · , fm), ∀1 ≤ i ≤ m and degfi G to denote degXi G, the degree of G in Xi. The following definition and properties are only formally new, and come from Definition. We call the parachute of f1, · · · , fm and denote ∇ = ∇(f1, · · · , fm) the integer ∇ = d1 + · · ·+ dm −m− max 1≤i1,··· ,im≤n deg jxi1 ,··· ,xim (f1, · · · , fm) where jxi1 ,··· ,xim (f1, · · · , fm) is the jacobian determinant of f1, · · · , fm with respect to xi1 , · · · , xim that is jxi1 ,··· ,xim (f1, · · · , fm) = det(∂fi/∂xij )i,j. Properties. The parachute of f1, · · · , fm has the following estimate: 0 ≤ ∇ = ∇(f1, · · · , fm) ≤ d1 + · · ·+ dm −m.(1) For any G ∈ K[f1, · · · , fm] and ∀1 ≤ i ≤ m, one has degG ≥ deg ∂G + di −∇ and, inductively, degG ≥ deg ∂ + kdi − k∇, ∀k ≥ 0. Proof. The left minoration 0 ≤ ∇ in (1) is an easy exercise. The right majoration is a direct consequence of the following Fact. The vectors gradf1 = (∂f1/∂x1, · · · , ∂f1/∂xn), · · · , gradfm = (∂fm/∂x1, · · · , ∂fm/∂xn) are linearly independent over K[x1, · · · , xn] therefore the minors of order m of the matrix gradf1 gradfm  are not all 0 and the number max1≤i1,··· ,im≤n deg jxi1 ,··· ,xim (f1, · · · , fm) is non-negative. Proof. As mentioned in [3], it is a well-known fact (see e.g. [1] or [2] for a nice proof) that n rational functions f1, · · · , fn ∈ K(x1, · · · , xn) are algebraically independant http://arxiv.org/abs/0704.1561v2 2 S. VÉNÉREAU if and only if their jacobian determinant is not zero. Our fact is then proved by completing our m algebraically independant polynomials to get n algebraically independant rational functions: the jacobian determinant is not zero and it follows that gradf1, · · · , gradfm must be linearly independant. � It is clearly sufficient to show (2) for i = m. Take any m integers 1 ≤ i1, · · · , im ≤ n. From the definition of jxi1 ,··· ,xim it is clear that deg jxi1 ,··· ,xim (f1, · · · , fm−1, G) ≤ d1 − 1 + · · ·+ dm−1 − 1 + degG− 1 ≤ d1 + · · ·+ dm−1 −m+ degG . On the other hand the chain rule gives jxi1 ,··· ,xim (f1, · · · , fm−1, G) = jxi1 ,··· ,xim (f1, · · · , fm−1, fm) Hence we get deg jxi1 ,··· ,xim (f1, · · · , fm−1, fm) + deg ≤ d1 + · · ·+ dm−1 −m+ degG deg ∂G + dm − (d1 + · · ·+ dm−1 + dm −m) + deg jxi1 ,··· ,xim (f1, · · · , fm) ≤ degG . In particular, when the maximum is realized, +dm−(d1+· · ·+dm−1+dm−m)+ max 1≤i1,··· ,im≤n deg jxi1 ,··· ,xim (f1, · · · , fm) ≤ degG +dm−(d1+· · ·+dm−1+dm−m− max 1≤i1,··· ,im≤ deg jxi1 ,··· ,xim (f1, · · · , fm)) ≤ degG + dm −∇ ≤ degG. In order to state our main theorem one needs to fix some more notations: we denote p̄ the leading term of a polynomial p ∈ K[x1, · · · , xn] and for any subalgebra A ⊂ K[x1, · · · , xn], we denote gr(A) := K[Ā] the subalgebra generated by Ā = {ā|a ∈ A}. We define si, ∀1 ≤ i ≤ m, as the degree of the minimal, if any, polynomial of f̄i over Frac( gr(K[fj ]j 6=i)), the field of fractions of the subalgebra generated by K[fj]j 6=i = K[f1, · · · , fi−1, fi+1, · · · , fm] and as +∞ otherwise. We denote ⌊α⌋ the integral part of a real number α and agree that k/∞ = 0 when 0 ≤ k < ∞. Theorem. Let G be a polynomial in K[f1, · · · , fm]. Then the following minoration holds, ∀1 ≤ i ≤ m, degG ≥ di · degfi G−∇ · ⌊ degfi G Proof. It is of course sufficient to prove it for i = m. First remark that a polynomial m ∈ A[fm], where gi ∈ A := K[f1, · · · , fm−1], has degree strictly smaller than maxi deg gi + i · dm if and only if Ĝ := deg gi+i·dm=max ḡif̄ m = 0 so if sm = +∞, which means such an annihilation cannot occur, then the minoration in the Theorem is clear. Let’s assume now that f̄m does have a minimal polynomial p(f̄m) = 0 with p = p(X) ∈ F [X ] where F is the field of fractions of gr(A) and X a new indeterminate (whence sm := degX p). The following easy lemma constitutes the very improvement with respect to [3]: it simplifies the proof a lot, makes it A PARACHUTE FOR THE DEGREE OF A POLYNOMIAL IN ALGEBRAICALLY INDEPENDENT ONES3 more general and even stronger in the sense that one does not need the estimate (1) anymore. Lemma. Let G = m be in A[fm] and h(X) := deg gi+i·dm=max i ∈ gr(A)[X ] ( hence Ĝ = h(f̄m)). If degG < degfm G ·dm then Ĝ = 0 or, equivalently, h(X) ∈ (p(X)) := p(X) ·F [X ]. Moreover if h′(X) 6= 0, where h′ is the derivative of h, then ∂̂G = h′(f̄m) and more generally, while h(k) 6= 0, one has ∂̂ = h(k)(f̄m). Proof. If degG < degfm G · dm then degG < maxi deg gi + i · dm and Ĝ = 0 as already remarked above. Assume that h′ 6= 0. One has ḡif̄ m = h(f̄m) where I := {i| deg gi + i · dm ≥ deg gj + j · dm ∀j} and iḡif̄ m where I ′ := {i| deg igi + (i− 1) · dm ≥ deg jgj + (j − 1) · dm ∀j}. It remains to notice that I ′ = I ∩ N∗ when this intersection is not empty, which occurs exactly when h′ 6= 0. � Let now k be the maximal number such that h(X) ∈ (p(X)k). Clearly degfm G ≥ deg h ≥ k · deg p = ksm hence k ≤ ⌊ degfm G ⌋. One has h(k) /∈ (p(X)) hence, by the Lemma, ≥ dm · degfm = dm · (degfm G− k) and, by property (2), degG ≥ dm · (degfm G− k) + k · dm − k · ∇ = dm · degfm G− k · ∇. A straightforward computation gives the following Corollary 1. Define, ∀i = 1, · · · ,m, Ni = Ni(f1, · · · , fm) := sidi − ∇. Let G be a polynomial in K[f1, · · · , fm] and, ∀i = 1, · · · ,m, let degfi G = qisi + ri be the euclidean division of degfi G by si. Then the following minoration holds degG ≥ qi ·Ni + ridi. The special case m = 2 corresponds to the main result of [3] (where s1, s2 are easy to compute): Corollary 2. If m = 2, σi := gcd(d1,d2) with (i, j) = (1, 2) and (2, 1) and N := σ1d1 −∇ = σ2d2 −∇ then the following minoration holds, for i = 1, 2, degG ≥ qi ·N + ridi where degfi G = qisi + ri. Proof. Let us prove it for i = 2. By corollary 1 it suffices to prove that s2 ≥ σ2 1: s2 is the degree of the minimal polynomial of f2 over Frac( gr(K[f1])) = Frac(K[f̄1]) = K(f̄1): p(f̄2) = f̄ 2 + ps2−1(f̄1)f̄ 2 + · · ·+ p1(f̄1)f̄2 + p0(f̄1) = 0(3) 1Actually equality holds, as proved in [3] using Zaks Lemma, it is however possible to show it easily and without this result. 4 S. VÉNÉREAU hence ∃0 ≤ i 6= j ≤ s2 such that deg pi(f̄1)f̄ 2 = deg pj(f̄1)f̄ 2 . It follows that i · d2 ≡ jd2 mod d1 whence d1 | (i − j)d2 and i − j ∈ Z gcd(d1,d2) which gives s2 ≥ |i− j| ≥ gcd(d1,d2) = σ2. � Corollary 3. Let G be a polynomial in K[f1, · · · , fm] such that degG = 1. Then, ∀i = 1, · · · ,m, degfi G = 0 or di = 1 or Ni = sidi −∇ ≤ 1. Proof. Otherwise, by corollary 1, degG = 1 ≥ qiNi + ridi ≥ min{Ni, di} ≥ 2, a contradiction. � Corollary 4. Assume m = n and K[f1, · · · , fn] = K[x1, · · · , xn] i.e. f1, · · · , fn define an automorphism (well-known fact). Then ∀i = 1, · · · , n, di = 1 or sidi ≤ d1 + · · · + dn − n + 1. In particular, if dmax ≥ dj , ∀j, and dmax ≥ 2 (i.e. the automorphism is not affine) then smax ≤ n− 1. Proof. One has ∇ = ∇(f1, · · · , fn) = d1+ · · ·+dn−n−deg jx1,··· ,xn(f1, · · · , fn) = d1+ · · ·+dn−n. Moreover ∀j = 1, · · · , n, there exists Gj ∈ K[f1, · · · , fn] such that xj = Gj and, ∀i = 1, · · · , n, degfi Gj ≤ 1 for at least one j = 1, · · · , n otherwise K[f1, · · · , fj−1, fj+1, · · · , fm] = K[x1, · · · , xn] which is impossible. Whence, by corollary 3, di = 1 or sidi ≤ ∇ + 1 = d1 + · · · + dn − n + 1. With dmax one gets smaxdmax ≤ d1 + · · · + dn − n + 1 ≤ ndmax − n + 1 ≤ ndmax − 1 (n ≥ 2) and it follows that smax ≤ n− 1. � Corollary 5 (Tameness Theorem in dimension two). Every automorphism of K[x1, x2] is tame i.e. a product of affine and elementary ones. Recall that an automorphism τ : K[x1, x2] → K[x1, x2] is called elementary when, up to exchanging x1 and x2, τ(x1) = x1 + p(x2) and τ(x2) = x2 for some p(X) ∈ K[X ]. Proof. Let α : K[x1, x2] → K[x1, x2] be an automorphism defined by α(xi) = fi for i = 1, 2. We prove the corollary by induction on d1 + d2 = deg f1 + deg f2. If d1 + d2 = 2 then d1 = d2 = 1 and α is affine. Assume d1 + d2 ≥ 3. Without loss of generality d1 ≤ d2 and d2 ≥ 2 whence, by corollary 4, s2 = 1 and the relation (3) in the proof of corollary 2 becomes: f̄2 = p(f̄1) where p(X) must be of the form p(X) = ps1X s1 ∈ K[X ]. Taking the elementary automorphism τ defined τ(x1) = x1 and τ(x2) = x2 − p(X) one has a new pair f ′1 := ατ(x1) = α(x1) = f1 and f 2 := ατ(x2) = α(x2 − p(X)) = f2− p(f1) with degrees d′1 = d1 and d 2 < d2 hence d 2 < d1+d2. By induction ατ is tame and so is α. � References [1] L. Makar-Limanov, Locally nilpotent derivations, a new ring invariant and applications, preprint. [2] J.T. Yu, On relations between Jacobians and minimal polynomials, Linear Algebra Appl. 221, 1995, 19–29. MR 96c:14014 [3] Shestakov, Ivan P. and Umirbaev, Ualbai U., Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc. 17(1), 2004, 181–196 (electronic). Stéphane Vénéreau Mathematisches Institut Universität Basel Rheinsprung 21, CH-4051 Basel Switzerland stephane.venereau@unibas.ch References

We give a simpler proof as well as a generalization of the main result of an article of Shestakov and Umirbaev. This latter article being the first of two that solve a long-standing conjecture about the non-tameness, or "wildness", of Nagata's automorphism. As corollaries we get interesting informations about the leading terms of polynomials forming an automorphism in any dimension and reprove the tameness of automorphisms in dimension two.

A PARACHUTE FOR THE DEGREE OF A POLYNOMIAL IN ALGEBRAICALLY INDEPENDENT ONES S. VÉNÉREAU Abstract. We give a simpler proof as well as a generalization of the main result of an article of Shestakov and Umirbaev ([3]). This latter article being the first of two that solve a long-standing conjecture about the non-tameness, or ”wildness”, of Nagata’s automorphism. As corollaries we get interesting informations about the leading terms of polynomials forming an automorphism of K[x1, · · · , xn] and reprove the tameness of automorphisms of K[x1, x2]. The following notations are fixed throughout the article: K is a field of cha- racteristic 0 and K[x1, · · · , xn] is the ring of polynomials in the n indetermi- nates x1, · · · , xn with coefficients in K, endowed with the classical degree func- tion: deg. We consider m algebraically independent polynomials in K[x1, · · · , xn]: f1, · · · , fm of respective degrees d1, · · · , dm. There is also, for every polynomial G ∈ K[f1, · · · , fm] a unique one G(X1, · · · , Xm) ∈ K[X1, · · · , Xm], where X1, · · · , Xm are new indeterminates, such that G = G(f1, · · · , fm). By abuse of notation we will write ∂G to denote ∂G (f1, · · · , fm), ∀1 ≤ i ≤ m and degfi G to denote degXi G, the degree of G in Xi. The following definition and properties are only formally new, and come from Definition. We call the parachute of f1, · · · , fm and denote ∇ = ∇(f1, · · · , fm) the integer ∇ = d1 + · · ·+ dm −m− max 1≤i1,··· ,im≤n deg jxi1 ,··· ,xim (f1, · · · , fm) where jxi1 ,··· ,xim (f1, · · · , fm) is the jacobian determinant of f1, · · · , fm with respect to xi1 , · · · , xim that is jxi1 ,··· ,xim (f1, · · · , fm) = det(∂fi/∂xij )i,j. Properties. The parachute of f1, · · · , fm has the following estimate: 0 ≤ ∇ = ∇(f1, · · · , fm) ≤ d1 + · · ·+ dm −m.(1) For any G ∈ K[f1, · · · , fm] and ∀1 ≤ i ≤ m, one has degG ≥ deg ∂G + di −∇ and, inductively, degG ≥ deg ∂ + kdi − k∇, ∀k ≥ 0. Proof. The left minoration 0 ≤ ∇ in (1) is an easy exercise. The right majoration is a direct consequence of the following Fact. The vectors gradf1 = (∂f1/∂x1, · · · , ∂f1/∂xn), · · · , gradfm = (∂fm/∂x1, · · · , ∂fm/∂xn) are linearly independent over K[x1, · · · , xn] therefore the minors of order m of the matrix gradf1 gradfm  are not all 0 and the number max1≤i1,··· ,im≤n deg jxi1 ,··· ,xim (f1, · · · , fm) is non-negative. Proof. As mentioned in [3], it is a well-known fact (see e.g. [1] or [2] for a nice proof) that n rational functions f1, · · · , fn ∈ K(x1, · · · , xn) are algebraically independant http://arxiv.org/abs/0704.1561v2 2 S. VÉNÉREAU if and only if their jacobian determinant is not zero. Our fact is then proved by completing our m algebraically independant polynomials to get n algebraically independant rational functions: the jacobian determinant is not zero and it follows that gradf1, · · · , gradfm must be linearly independant. � It is clearly sufficient to show (2) for i = m. Take any m integers 1 ≤ i1, · · · , im ≤ n. From the definition of jxi1 ,··· ,xim it is clear that deg jxi1 ,··· ,xim (f1, · · · , fm−1, G) ≤ d1 − 1 + · · ·+ dm−1 − 1 + degG− 1 ≤ d1 + · · ·+ dm−1 −m+ degG . On the other hand the chain rule gives jxi1 ,··· ,xim (f1, · · · , fm−1, G) = jxi1 ,··· ,xim (f1, · · · , fm−1, fm) Hence we get deg jxi1 ,··· ,xim (f1, · · · , fm−1, fm) + deg ≤ d1 + · · ·+ dm−1 −m+ degG deg ∂G + dm − (d1 + · · ·+ dm−1 + dm −m) + deg jxi1 ,··· ,xim (f1, · · · , fm) ≤ degG . In particular, when the maximum is realized, +dm−(d1+· · ·+dm−1+dm−m)+ max 1≤i1,··· ,im≤n deg jxi1 ,··· ,xim (f1, · · · , fm) ≤ degG +dm−(d1+· · ·+dm−1+dm−m− max 1≤i1,··· ,im≤ deg jxi1 ,··· ,xim (f1, · · · , fm)) ≤ degG + dm −∇ ≤ degG. In order to state our main theorem one needs to fix some more notations: we denote p̄ the leading term of a polynomial p ∈ K[x1, · · · , xn] and for any subalgebra A ⊂ K[x1, · · · , xn], we denote gr(A) := K[Ā] the subalgebra generated by Ā = {ā|a ∈ A}. We define si, ∀1 ≤ i ≤ m, as the degree of the minimal, if any, polynomial of f̄i over Frac( gr(K[fj ]j 6=i)), the field of fractions of the subalgebra generated by K[fj]j 6=i = K[f1, · · · , fi−1, fi+1, · · · , fm] and as +∞ otherwise. We denote ⌊α⌋ the integral part of a real number α and agree that k/∞ = 0 when 0 ≤ k < ∞. Theorem. Let G be a polynomial in K[f1, · · · , fm]. Then the following minoration holds, ∀1 ≤ i ≤ m, degG ≥ di · degfi G−∇ · ⌊ degfi G Proof. It is of course sufficient to prove it for i = m. First remark that a polynomial m ∈ A[fm], where gi ∈ A := K[f1, · · · , fm−1], has degree strictly smaller than maxi deg gi + i · dm if and only if Ĝ := deg gi+i·dm=max ḡif̄ m = 0 so if sm = +∞, which means such an annihilation cannot occur, then the minoration in the Theorem is clear. Let’s assume now that f̄m does have a minimal polynomial p(f̄m) = 0 with p = p(X) ∈ F [X ] where F is the field of fractions of gr(A) and X a new indeterminate (whence sm := degX p). The following easy lemma constitutes the very improvement with respect to [3]: it simplifies the proof a lot, makes it A PARACHUTE FOR THE DEGREE OF A POLYNOMIAL IN ALGEBRAICALLY INDEPENDENT ONES3 more general and even stronger in the sense that one does not need the estimate (1) anymore. Lemma. Let G = m be in A[fm] and h(X) := deg gi+i·dm=max i ∈ gr(A)[X ] ( hence Ĝ = h(f̄m)). If degG < degfm G ·dm then Ĝ = 0 or, equivalently, h(X) ∈ (p(X)) := p(X) ·F [X ]. Moreover if h′(X) 6= 0, where h′ is the derivative of h, then ∂̂G = h′(f̄m) and more generally, while h(k) 6= 0, one has ∂̂ = h(k)(f̄m). Proof. If degG < degfm G · dm then degG < maxi deg gi + i · dm and Ĝ = 0 as already remarked above. Assume that h′ 6= 0. One has ḡif̄ m = h(f̄m) where I := {i| deg gi + i · dm ≥ deg gj + j · dm ∀j} and iḡif̄ m where I ′ := {i| deg igi + (i− 1) · dm ≥ deg jgj + (j − 1) · dm ∀j}. It remains to notice that I ′ = I ∩ N∗ when this intersection is not empty, which occurs exactly when h′ 6= 0. � Let now k be the maximal number such that h(X) ∈ (p(X)k). Clearly degfm G ≥ deg h ≥ k · deg p = ksm hence k ≤ ⌊ degfm G ⌋. One has h(k) /∈ (p(X)) hence, by the Lemma, ≥ dm · degfm = dm · (degfm G− k) and, by property (2), degG ≥ dm · (degfm G− k) + k · dm − k · ∇ = dm · degfm G− k · ∇. A straightforward computation gives the following Corollary 1. Define, ∀i = 1, · · · ,m, Ni = Ni(f1, · · · , fm) := sidi − ∇. Let G be a polynomial in K[f1, · · · , fm] and, ∀i = 1, · · · ,m, let degfi G = qisi + ri be the euclidean division of degfi G by si. Then the following minoration holds degG ≥ qi ·Ni + ridi. The special case m = 2 corresponds to the main result of [3] (where s1, s2 are easy to compute): Corollary 2. If m = 2, σi := gcd(d1,d2) with (i, j) = (1, 2) and (2, 1) and N := σ1d1 −∇ = σ2d2 −∇ then the following minoration holds, for i = 1, 2, degG ≥ qi ·N + ridi where degfi G = qisi + ri. Proof. Let us prove it for i = 2. By corollary 1 it suffices to prove that s2 ≥ σ2 1: s2 is the degree of the minimal polynomial of f2 over Frac( gr(K[f1])) = Frac(K[f̄1]) = K(f̄1): p(f̄2) = f̄ 2 + ps2−1(f̄1)f̄ 2 + · · ·+ p1(f̄1)f̄2 + p0(f̄1) = 0(3) 1Actually equality holds, as proved in [3] using Zaks Lemma, it is however possible to show it easily and without this result. 4 S. VÉNÉREAU hence ∃0 ≤ i 6= j ≤ s2 such that deg pi(f̄1)f̄ 2 = deg pj(f̄1)f̄ 2 . It follows that i · d2 ≡ jd2 mod d1 whence d1 | (i − j)d2 and i − j ∈ Z gcd(d1,d2) which gives s2 ≥ |i− j| ≥ gcd(d1,d2) = σ2. � Corollary 3. Let G be a polynomial in K[f1, · · · , fm] such that degG = 1. Then, ∀i = 1, · · · ,m, degfi G = 0 or di = 1 or Ni = sidi −∇ ≤ 1. Proof. Otherwise, by corollary 1, degG = 1 ≥ qiNi + ridi ≥ min{Ni, di} ≥ 2, a contradiction. � Corollary 4. Assume m = n and K[f1, · · · , fn] = K[x1, · · · , xn] i.e. f1, · · · , fn define an automorphism (well-known fact). Then ∀i = 1, · · · , n, di = 1 or sidi ≤ d1 + · · · + dn − n + 1. In particular, if dmax ≥ dj , ∀j, and dmax ≥ 2 (i.e. the automorphism is not affine) then smax ≤ n− 1. Proof. One has ∇ = ∇(f1, · · · , fn) = d1+ · · ·+dn−n−deg jx1,··· ,xn(f1, · · · , fn) = d1+ · · ·+dn−n. Moreover ∀j = 1, · · · , n, there exists Gj ∈ K[f1, · · · , fn] such that xj = Gj and, ∀i = 1, · · · , n, degfi Gj ≤ 1 for at least one j = 1, · · · , n otherwise K[f1, · · · , fj−1, fj+1, · · · , fm] = K[x1, · · · , xn] which is impossible. Whence, by corollary 3, di = 1 or sidi ≤ ∇ + 1 = d1 + · · · + dn − n + 1. With dmax one gets smaxdmax ≤ d1 + · · · + dn − n + 1 ≤ ndmax − n + 1 ≤ ndmax − 1 (n ≥ 2) and it follows that smax ≤ n− 1. � Corollary 5 (Tameness Theorem in dimension two). Every automorphism of K[x1, x2] is tame i.e. a product of affine and elementary ones. Recall that an automorphism τ : K[x1, x2] → K[x1, x2] is called elementary when, up to exchanging x1 and x2, τ(x1) = x1 + p(x2) and τ(x2) = x2 for some p(X) ∈ K[X ]. Proof. Let α : K[x1, x2] → K[x1, x2] be an automorphism defined by α(xi) = fi for i = 1, 2. We prove the corollary by induction on d1 + d2 = deg f1 + deg f2. If d1 + d2 = 2 then d1 = d2 = 1 and α is affine. Assume d1 + d2 ≥ 3. Without loss of generality d1 ≤ d2 and d2 ≥ 2 whence, by corollary 4, s2 = 1 and the relation (3) in the proof of corollary 2 becomes: f̄2 = p(f̄1) where p(X) must be of the form p(X) = ps1X s1 ∈ K[X ]. Taking the elementary automorphism τ defined τ(x1) = x1 and τ(x2) = x2 − p(X) one has a new pair f ′1 := ατ(x1) = α(x1) = f1 and f 2 := ατ(x2) = α(x2 − p(X)) = f2− p(f1) with degrees d′1 = d1 and d 2 < d2 hence d 2 < d1+d2. By induction ατ is tame and so is α. � References [1] L. Makar-Limanov, Locally nilpotent derivations, a new ring invariant and applications, preprint. [2] J.T. Yu, On relations between Jacobians and minimal polynomials, Linear Algebra Appl. 221, 1995, 19–29. MR 96c:14014 [3] Shestakov, Ivan P. and Umirbaev, Ualbai U., Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc. 17(1), 2004, 181–196 (electronic). Stéphane Vénéreau Mathematisches Institut Universität Basel Rheinsprung 21, CH-4051 Basel Switzerland stephane.venereau@unibas.ch References

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 8 November 2021 (MN LATEX style file v2.2) Galaxy evolution in the infra-red: comparison of a hierarchical galaxy formation model with SPITZER data C. G. Lacey ⋆,1 C. M. Baugh,1 C.S. Frenk,1 L. Silva,2 G.L. Granato,3 and A. Bressan,3 1Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, UK 2INAF, Osservatorio Astronomico di Trieste, Via Tiepolo 11, I-34131 Trieste, Italy 3INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 2, I-35122 Padova, Italy. 8 November 2021 ABSTRACT We present predictions for the evolution of the galaxy luminosity function, number counts and redshift distributions in the IR based on the ΛCDM cosmological model. We use the combined GALFORM semi-analytical galaxy formation model and GRASIL spectrophotometric code to compute galaxy SEDs including the reprocessing of radiation by dust. The model, which is the same as that in Baugh et al. (2005), assumes two different IMFs: a normal solar neigh- bourhood IMF for quiescent star formation in disks, and a very top-heavy IMF in starbursts triggered by galaxy mergers. We have shown previously that the top-heavy IMF seems to be necessary to explain the number counts of faint sub-mm galaxies. We compare the model with observational data from the Spitzer Space Telescope, with the model parameters fixed at values chosen before Spitzer data became available. We find that the model matches the observed evolution in the IR remarkably well over the whole range of wavelengths probed by Spitzer. In particular, the Spitzer data show that there is strong evolution in the mid-IR galaxy luminosity function over the redshift range z ∼ 0 − 2, and this is reproduced by our model without requiring any adjustment of parameters. On the other hand, a model with a normal IMF in starbursts predicts far too little evolution in the mid-IR luminosity function, and is therefore excluded. Key words: galaxies: evolution – galaxies: formation – galaxies: high-redshift – infrared: galaxies – ISM: dust, extinction 1 INTRODUCTION In recent years, the evolution of galaxies at mid- and far-infrared wavelengths has been opened up for direct observational study by infrared telescopes in space. Already in the 1980s, the IRAS satel- lite surveyed the local universe in the IR, showing that much of present-day star formation is optically obscured, revealing a pop- ulation of luminous and ultra-luminous infrared galaxies (LIRGs with total IR luminosities LIR ∼ 10 − 1012L⊙ and ULIRGs with LIR & 10 12L⊙), and providing the first hints of strong evo- lution in the number density of ULIRGs at recent cosmic epochs (e.g. Wright et al. 1984; Soifer et al. 1987a; Sanders & Mirabel 1996). The next major advance came with the discovery by COBE of the cosmic far-IR background which has an energy density comparable to that in the optical/near-IR background (Puget et al. 1996; Hauser et al. 1998). This implies that, over the history of the universe, as much energy has been emitted by dust in galaxies as reaches us directly in starlight, after dust extinction is taken into account. This discovery made apparent the need to understand the IR as much as the optical emission from galaxies in order to have ⋆ E-mail: Cedric.Lacey@durham.ac.uk (CGL) a complete picture of galaxy evolution. In particular, it is essen- tial to understand IR emission from dust in order to understand the cosmic history of star formation, since most of the radiation from young stars must have been absorbed by dust over the history of the universe, in order to account for the far-IR background (e.g. Hauser et al. 1998). Following these early discoveries, the ISO satellite enabled the first deep surveys of galaxies in the mid- and far-IR. The deep- est of these surveys were in the mid-IR at 15µm, and probed the evolution of LIRGs and ULIRGs out to z ∼ 1, showing strong evolution in these populations, and directly resolving most of the cosmic infrared background at that wavelength (Elbaz et al. 1999, 2002; Gruppioni et al. 2002). Deep ISO surveys in the far-IR at 170µm (Dole et al. 2001; Patris et al. 2003) probed lower red- shifts, z ∼ 0.5. Around the same time, sub-mm observations using the SCUBA instrument on the JCMT revealed a huge population of high-z ULIRGs (Smail, Ivison & Blain 1997; Hughes et al. 1998) which were subsequently found to have a redshift distribution peak- ing at z ∼ 2 (Chapman et al. 2005), confirming the dramatic evo- lution in number density for this population seen at shorter wave- lengths and lower redshifts. The sub-mm galaxies have been stud- c© 0000 RAS http://arxiv.org/abs/0704.1562v2 2 Lacey et al. ied in more detail in subsequent SCUBA surveys (e.g. SHADES, Mortier et al. 2005). Now observations using the Spitzer satellite (Werner et al. 2004), with its hugely increased sensitivity and mapping speed are revolutionizing our knowledge of galaxy evolution at IR wave- lengths from 3.6 to 160 µm. Spitzer surveys have allowed direct determinations of the evolution of the galaxy luminosity func- tion out to z ∼ 1 in the rest-frame near-IR and to z ∼ 2 in the mid-IR (Le Floc’h et al. 2005; Perez-Gonzalez et al. 2005; Babbedge et al. 2006; Franceschini et al. 2006). Individual galax- ies have been detected by Spitzer out to z ∼ 6 (Eyles et al. 2005). In the near future, the Herschel satellite (Pilbratt 2003) should make it possible to measure the far-IR luminosity function out to z ∼ 2, and thus directly measure the total IR luminosities of galax- ies over most of the history of the universe. Accompanying these observational advances, various types of theoretical models have been developed to interpret or explain the observational data on galaxy evolution in the IR. We can distinguish three main classes of model: (a) Purely phenomenological models: In these models, the galaxy luminosity function and its evolution are described by a purely empirical expression, and this is combined with observationally-based templates for the IR spectral energy dis- tribution (SED). The free parameters in the expression for the luminosity function are then chosen to obtain the best match to some set of observational data, such as number counts and redshift distributions in different IR bands. These pa- rameters are purely descriptive and provide little insight into the physical processes which control galaxy evolution. Exam- ples of these models are Pearson & Rowan-Robinson (1996); Xu et al. (1998); Blain et al. (1999); Franceschini et al. (2001); Chary & Elbaz (2001); Rowan-Robinson (2001); Lagache et al. (2003); Gruppioni et al. (2005). (b) Hierarchical galaxy formation models with phenomeno- logical SEDs: In these models, the evolution of the luminosity func- tions of the stellar and total dust emission are calculated from a detailed model of galaxy formation based on the cold dark mat- ter (CDM) model of structure formation, including physical mod- elling of processes such as gas cooling and galaxy mergers. The stellar luminosity of a model galaxy is computed from its star formation history, and the stellar luminosity absorbed by dust, which equals the total IR luminosity emitted by dust, is calculated from this based on some treatment of dust extinction. However, the SED shapes required to calculate the distribution of the dust emission over wavelength from the total IR dust emission are ei- ther observationally-based templates (e.g. Guiderdoni et al. 1998; Devriendt & Guiderdoni 2000) or are purely phenomenological, e.g. a modified Planck function with an empirically chosen dust temperature (e.g. Kaviani et al. 2003). In this approach, the shape of the IR SED assumed for a model galaxy may be incompatible with its other predicted properties, such as its dust mass and radius. (c) Hierarchical galaxy formation models with theoretical SEDs: These models are similar to those of type (b), in that the evolution of the galaxy population is calculated from a detailed physical model of galaxy formation based on CDM, but instead of using phenomenological SEDs for the dust emission, the com- plete SED of each model galaxy, from the far-UV to the radio, is calculated by combining a theoretical stellar population synthesis model for the stellar emission with a theoretical radiative transfer and dust heating model to predict both the extinction of starlight by dust and the IR/sub-mm SED of the dust emission. The ad- vantages of this type of model are that it is completely ab initio, with the maximum possible theoretical self-consistency, and all of the model parameters relate directly to physical processes. For ex- ample, the typical dust temperature and the shape of the SED of dust emission depend on the stellar luminosity and the dust mass, and evolution in all of these quantities is computed self-consistently in this type of model. Following this modelling approach thus al- lows more rigorous testing of the predictions of physical models for galaxy formation against observational data at IR wavelengths, as well as shrinking the parameter space of the predictions. Ex- amples of such models are Granato et al. (2000) and Baugh et al. (2005). (An alternative modelling approach also based on theoret- ical IR SEDs but with a simplified treatment of the assembly of galaxies and halos has been presented by Granato et al. (2004) and Silva et al. (2005).) In this paper, we follow the third approach, with physical mod- elling both of galaxy formation and of the galaxy SEDs, includ- ing the effects of dust. This paper is the third in a series, where we combine the GALFORM semi-analytical model of galaxy forma- tion (Cole et al. 2000) with the GRASILmodel for stellar and dust emission from galaxies (Silva et al. 1998). The GALFORM model computes the evolution of galaxies in the framework of the ΛCDM model for structure formation, based on physical treatments of the assembly of dark matter halos by merging, gas cooling in halos, star formation and supernova feedback, galaxy mergers, and chem- ical enrichment. The GRASILmodel computes the SED of a model galaxy from the far-UV to the radio, based on theoretical models of stellar evolution and stellar atmospheres, radiative transfer through a two-phase dust medium to calculate both the dust extinction and dust heating, and a distribution of dust temperatures in each galaxy calculated from a detailed dust grain model. In the first paper in the series (Granato et al. 2000), we modelled the IR properties of galaxies in the local universe. While this model was very success- ful in explaining observations of the local universe, we found sub- sequently that it failed when confronted with observations of star- forming galaxies at high redshifts, predicting far too few sub-mm galaxies (SMGs) at z ∼ 2 and Lyman-break galaxies (LBGs) at z ∼ 3. Therefore, in the second paper (Baugh et al. 2005), we proposed a new version of the model which assumes a top-heavy IMF in starbursts (with slope x = 0, compared to Salpeter slope x = 1.35), but a normal solar neighbourhood IMF for quiescent star formation. In this new model, the star formation parameters were also changed to force more star formation to happen in bursts. This revised model agreed well with both the number counts and redshift distributions of SMGs detected at 850µm, and with the rest-frame far-UV luminosity function of LBGs at z ∼ 3, while still maintaining consistency with galaxy properties in the local uni- verse such as the optical, near-IR and far-IR luminosity functions, and gas fractions, metallicities, morphologies and sizes. This same model of Baugh et al. (2005) was found by Le Delliou et al. (2005a, 2006) to provide a good match to the ob- served evolution of the population of Lyα-emitting galaxies over the redshift range z ∼ 3−6. Support for the controversial assump- tion of a top-heavy IMF in bursts came from the studies of chem- ical enrichment in this model by Nagashima et al. (2005a,b), who found that the metallicities of both the intergalactic gas in galaxy clusters and the stars in elliptical galaxies were predicted to be sig- nificantly lower than observed values if a normal IMF was assumed for all star formation, but agreed much better if a top-heavy IMF in bursts was assumed, as in Baugh et al. . In this third paper in the series, we extend the Baugh et al. (2005) model to make predic- tions for galaxy evolution in the IR, and compare these predictions with observational data from Spitzer. We emphasize that all of the c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 3 model parameters for the predictions presented in this paper were fixed by Baugh et al. prior to the publication of any results from Spitzer, and we have not tried to obtain a better fit to any of the Spitzer data by adjusting these parameters1. Our goals in this paper are to test our model of galaxy evo- lution with a top-heavy IMF in starbursts against observations of dust-obscured star-forming galaxies over the redshift range z ∼ 0−2, and also to test our predictions for the evolution of the stellar populations of galaxies against observational data in the rest-frame near- and mid-IR. The plan of the paper is as follows: In Section 2, we give an overview of the GALFORM and GRASIL models, fo- cusing on how the predictions we present later on are calculated. In Section 3, we compare the galaxy number counts predicted by our model with observational data in all 7 Spitzer bands, from 3.6 to 160 µm. In Section 4, we investigate galaxy evolution in the IR in more detail, by comparing model predictions directly with galaxy luminosity functions constructed from Spitzer data. In Sec- tion 5, we present the predictions of our model for the evolution of the galaxy stellar mass function and star formation rate distribu- tion, and investigate the insight our model offers on how well stel- lar masses and star formation rates can be estimated from Spitzer data. We present our conclusions in Section 6. In the Appendix, we present model predictions for galaxy redshift distributions in the different Spitzer bands, to assist in interpreting data from different surveys. 2 MODEL In this paper use the GALFORM semi-analytical model to predict the physical properties of the galaxy population at different red- shifts, and combine it with the GRASIL spectrophotometric model to predict the detailed SEDs of model galaxies. Both GALFORM and GRASIL have been described in detail in various previous papers, but since the descriptions of the different model compo- nents, as well as of our particular choice of parameters, are spread among different papers, we give an overview of both of these here. GALFORM is described in §2.1, and GRASIL in §2.2. Particularly important features of our model are the triggering of starbursts by mergers (discussed in §2.1.4) and the assumption of a top-heavy IMF in starbursts (discussed in §2.1.7). We further discuss the choice of model parameters in §2.3. Readers who are already fa- miliar with the Baugh et al. (2005) model can skip straight to the results, starting in §3. 2.1 GALFORM galaxy formation model We compute the formation and evolution of galaxies within the framework of the ΛCDM model of structure formation using the semi-analytical galaxy formation model GALFORM. The general methodology and approximations behind the GALFORM model are set out in detail in Cole et al. (2000) (see also the review by Baugh (2006)). In summary, the GALFORM model follows the main pro- cesses which shape the formation and evolution of galaxies. These include: (i) the collapse and merging of dark matter halos; (ii) the 1 A closely related model of galaxy formation obtained by applying GALFORM principles to the Millennium simulation of Springel et al. (2005) has recently been published by Bower et al. (2006). This model differs from the current one primarily in that it includes feedback from AGN activ- ity, but does not have a top-heavy IMF in bursts. We plan to investigate the IR predictions of this alternative model in a subsequent paper. shock-heating and radiative cooling of gas inside dark halos, lead- ing to the formation of galaxy disks; (iii) quiescent star formation in galaxy disks; (iv) feedback both from supernova explosions and from photo-ionization of the IGM; (v) chemical enrichment of the stars and gas; (vi) galaxy mergers driven by dynamical friction within common dark matter halos, leading to the formation of stel- lar spheroids, and also triggering bursts of star formation. The end product of the calculations is a prediction of the numbers and prop- erties of galaxies that reside within dark matter haloes of different masses. The model predicts the stellar and cold gas masses of the galaxies, along with their star formation and merger histories, their sizes and metallicities. The prescriptions and parameters for the different processes which we use in this paper are identical to those adopted by Baugh et al. (2005), but differ in several important respects from Cole et al. (2000). All of these parameters were chosen by com- parison with pre-Spitzer observational data. The background cos- mology is a spatially flat CDM universe with a cosmological con- stant, with “concordance” parameters Ωm = 0.3, ΩΛ = 0.7, Ωb = 0.04, and h ≡ H0/(100km s −1Mpc−1) = 0.7. The am- plitude of the initial spectrum of density fluctuations is set by the r.m.s. linear fluctuation in a sphere of radius 8h−1Mpc, σ8 = 0.93. For completeness, we now summarize the prescriptions and param- eters used, but give details mainly where they differ from those in Cole et al. (2000), or where they are particularly relevant to pre- dicting IR emission from dust. 2.1.1 Halo assembly histories As in Cole et al. (2000), we describe the assembly histories of dark matter halos through halo merger trees which are calculated using a Monte Carlo method based on the extended Press-Schechter ap- proach (e.g. Lacey & Cole 1993). The process of galaxy forma- tion is then calculated separately for each halo merger tree, follow- ing the baryonic physics in all of the separate branches of the tree. As has been shown by Helly et al. (2003), the statistical properties of galaxies calculated in semi-analytical models using these Monte Carlo merger trees are very similar to those computed using merger trees extracted directly from N-body simulations. 2.1.2 Gas cooling in halos The cooling of gas in halos is calculated using the same simple spherical model as in Cole et al. (2000). The diffuse gas in halos (consisting of all of the gas which has not previously condensed into galaxies) is assumed to be shock-heated to the halo virial tem- perature when the halo is assembled, and then to cool radiatively by atomic processes. The cooling time depends on radius through the gas density profile, which is assumed to have a core radius which grows as gas is removed from the diffuse phase by condensing into galaxies. The gas at some radius r in the halo then cools and col- lapses to the halo centre on a timescale which is the larger of the cooling time tcool and the free-fall time tff at that radius. Thus, for tcool(r) > tff(r), we have hot accretion, and for tcool(r) < tff(r), we have cold accretion 2. In our model, gas only accretes onto the central galaxy in a halo, not onto any satellite galaxies which share 2 Note that contrary to claims by Birnboim & Dekel (2003), the process of “cold accretion”, if not the name, has always been part of semi-analytical models (see Croton et al. (2006) for a detailed discussion) c© 0000 RAS, MNRAS 000, 000–000 4 Lacey et al. that halo. We denote all of the diffuse gas in halos as “hot”, and all of the gas which has condensed into galaxies as “cold”. 2.1.3 Star formation timescale in disks The global rate of star formation ψ in galaxy disks is assumed to be related to the cold gas mass, Mgas, by ψ = Mgas/τ∗,disk, where the star formation timescale is taken to be τ∗,disk = τ∗0 Vc/200 km s , (1) where Vc is the circular velocity of the galaxy disk (at its half- mass radius) and τ∗0 is a constant. We adopt values τ∗0 = 8Gyr and α∗ = −3, chosen to reproduce the observed relation between gas mass and B-band luminosity for present-day disk galaxies. As discussed in Baugh et al. (2005), this assumption means that the disk star formation timescale is independent of redshift (at a given Vc), resulting in disks at high redshift that are much more gas-rich than at low redshift, and have more gas available for star formation in bursts triggered by galaxy mergers at high redshift. 2.1.4 Galaxy mergers and triggering of starbursts In the model, all galaxies originate as central galaxies in some halo, but can then become satellite galaxies if their host halo merges into another halo. Mergers can then occur between satellite and central galaxies within the same halo, after dynamical friction has caused the satellite galaxy to sink to the centre of the halo. Galaxy mergers can produce changes in galaxy morphology and trigger bursts. We classify galaxy mergers according to the ratio of masses (including stars and gas) M2/M1 6 1 of the secondary to primary galaxy involved. We define mergers to be major or minor according to whether M2/M1 > fellip or M2/M1 < fellip (Kauffmann et al. 1993). In major mergers, any stellar disks in either the primary or secondary are assumed to be disrupted, and the stars rearranged into a spheroid. In minor mergers, the stellar disk in the primary galaxy is assumed to remain intact, while all of the stars in the secondary are assumed to be added to the spheroid of the primary. We adopt a threshold fellip = 0.3 for major mergers, consistent with the results of numerical simulations (e.g. Barnes 1998), which reproduces the observed present-day fraction of spheroidal galaxies. We assume that major mergers always trigger a starburst if any gas is present. We also assume that minor mergers can trigger bursts, if they sat- isfy both M2/M1 > fburst and the gas fraction in the disk of the primary galaxy exceeds fgas,crit. Following Baugh et al. (2005), we adopt fburst = 0.05 and fgas,crit = 0.75. The parameters for bursts in minor mergers were motivated by trying to explain the number of sub-mm galaxies. An important consequence of assum- ing eqn.(1) for the star formation timescale in disks, combined with the triggering of starbursts in minor mergers, is that the global star formation rate at high redshifts is dominated by bursts, while that at low redshifts it is dominated by quiescent disks (see Baugh et al. for a detailed discussion of these points). In either kind of starburst, we assume that the burst consumes all of the cold gas in the two galaxies involved in the merger, and that the stars produced are added to the spheroid of the merger rem- nant. During the burst, we assume that star formation proceeds ac- cording to the relation ψ =Mgas/τ∗,burst. For the burst timescale, we assume τ∗,burst = max [fdynτdyn,sph; τ∗,burst,min] , (2) where τdyn,sph is the dynamical time in the newly-formed spheroid. We adopt fdyn = 50 and τ∗,burst,min = 0.2Gyr (these parame- ters were chosen by Baugh et al. (2005) to allow a simultaneous match to the sub-mm number counts and to the local 60µm lu- minosity function). The star formation rate in a burst thus decays exponentially with time after the galaxy merger. It is assumed to be truncated after 3 e-folding times (where the e-folding time takes ac- count of stellar recycling and feedback - see Granato et al. (2000) for details), with the remaining gas being ejected into the galaxy halo at that time. 2.1.5 Feedback from photo-ionization After the intergalactic medium (IGM) has been reionized at redshift zreion, the formation of low-mass galaxies is inhibited, both by the effect of the IGM pressure inhibiting collapse of gas into halos, and by the reduction of gas cooling in halos due to the photo-ionizing background. We model this in a simple way, by assuming that for z < zreion, cooling of gas is completely suppressed in halos with circular velocities Vc < Vcrit. We adopt Vcrit = 60 kms −1, based on the detailed modelling by Benson et al. (2002). We assume in this paper that reionization occurs at zreion = 6, for consistency with Baugh et al. (2005), but increasing this to zreion ∼ 10 in line with the WMAP 3-year estimate of the polarization of the mi- crowave background (Spergel et al. 2006) has no significant effect on the model results presented in this paper. 2.1.6 Feedback from supernovae Photo-ionization feedback only affects very low mass galaxies. More important for most galaxies is feedback from supernova ex- plosions. We assume that energy input from supernovae causes gas to be ejected from galaxies at a rate Ṁej = β(Vc)ψ = [βreh(Vc) + βsw(Vc)] ψ (3) The supernova feedback is assumed to operate for both quiescent star formation in disks and for starbursts triggered by galaxy merg- ers, with the only difference being that we take Vc to be the circular velocity at the half mass radius of the disk in the former case, and at the half-mass radius of the spheroid in the latter case. For sim- plicity, we keep the same feedback parameters for starbursts as for quiescent star formation. The supernova feedback has two components: the reheating term βrehψ describes gas which is reheated and ejected into the galaxy halo, from where it is allowed to cool again after the halo mass has doubled through hierachical mass build-up. For this, we use the parametrization of Cole et al. (2000): βreh = (Vc/Vhot) −αhot , (4) where we adopt parameter values Vhot = 300 kms −1 and αhot = 2. The reheating term has the largest effect on low-mass galaxies, for which ejection of gas from galaxies flattens the faint-end slope of the galaxy luminosity function. The second term βswψ in eqn.(3) is the superwind term, which describes ejection of gas out of the halo rather than just the galaxy. Once ejected, this gas is assumed never to re-accrete onto any halo. We model the superwind ejection efficiency as βsw = fsw min 1, (Vc/Vsw) based on Benson et al. (2003). We adopt parameter values fsw = 2 and Vsw = 200 km s −1, as in Baugh et al. (2005). The superwind term mainly affects higher mass galaxies, where the ejection of gas c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 5 from halos causes an increase in the cooling time of gas in halos by reducing the gas densities. This brings the predicted break at the bright end of the local galaxy luminosity function into agree- ment with observations, as discussed in Benson et al. (2003). The various parameters for supernova feedback are thus chosen in or- der to match the observed present-day optical and near-IR galaxy luminosity functions, as well as the galaxy metallicity-luminosity relation. We note that the galaxy formation model in this paper, un- like some other recent semi-analytical models, does not include AGN feedback. Instead, the role of AGN feedback in reducing the amount of gas cooling to form massive galaxies is taken by su- perwinds driven by supernova explosions. The first semi-analytical model to include AGN feedback was that of Granato et al. (2004), who introduced a detailed model of feedback from QSO winds during the formation phase of supermassive black holes (SMBHs), with the aim of explaining the co-evolution of the spheroidal com- ponents of galaxies and their SMBHs. The predictions of the Granato et al. model for number counts and redshift distributions in the IR have been computed by Silva et al. (2005) using the GRASIL spectrophotometric model, and compared to ISO and Spitzer data. However, the Granato et al. (2004) model has the limitations that it does not include the merging of galaxies or of dark halos, and does not treat the formation and evolution of galac- tic disks. More recently, several semi-analytical models have been published which propose that heating of halo gas by relativistic jets from an AGN in an optically inconspicuous or “radio” mode can balance radiative cooling of gas in high-mass halos, thus sup- pressing hot accretion of gas onto galaxies (Bower et al. 2006; Croton et al. 2006; Cattaneo et al. 2006; Monaco et al. 2007). However, these AGN feedback models differ in detail, and all are fairly schematic. None of these models has been shown to repro- duce the observed number counts and redshifts of the faint sub-mm galaxies. The effects of our superwind feedback are qualitatively quite similar to those of the radio-mode AGN feedback. Both superwind and AGN feedback models contain free parameters, which are ad- justed in order to make the model fit the bright end of the ob- served present-day galaxy luminosity function at optical and near- IR wavelengths. However, since the physical mechanisms are dif- ferent, they make different predictions for how the galaxy lumi- nosity function should evolve with redshift. Current models for the radio-mode AGN feedback are very schematic, but they have the advantage over the superwind model that the energetic constraints are greatly relaxed, since accretion onto black holes can convert mass into energy with a much higher efficiency than can supernova explosions. We will investigate the predictions of models with AGN feedback for the IR and sub-mm evolution of galaxies in a future paper. 2.1.7 The Stellar Initial Mass Function and Chemical Evolution Stars in our model are assumed to form with different Initial Mass Functions (IMFs), depending on whether they form in disks or in bursts. Both IMFs are taken to be piecewise power laws, with slopes x defined by dN/d lnm ∝ m−x, with N the number of stars and m the stellar mass (so the Salpeter slope is x = 1.35), and covering a stellar mass range 0.15 < m < 120M⊙. Quiescent star forma- tion in galaxy disks is assumed to have a solar neighbourhood IMF, for which we use the Kennicutt (1983) paramerization, with slope x = 0.4 for m < M⊙ and x = 1.5 for m > M⊙. (The Kennicutt (1983) IMF is similar to other popular parametrizations of the solar neighbourhood IMF, such as that of Kroupa (2001).) Bursts of star formation triggered by galaxy mergers are assumed to form stars with a top-heavy IMF with slope x = 0. As discussed in detail in Baugh et al. (2005), the top-heavy IMF in bursts was found to be required in order to reproduce the observed number counts and redshift distributions of the faint sub-mm galaxies. Furthermore, as shown by Nagashima et al. (2005a,b), the predicted chemical abundances of the X-ray emitting gas in galaxy clusters and of the stars in elliptical galaxies also agree better with observational data in a model with the top-heavy IMF in bursts, rather than a universal solar neighbourhood IMF. A variety of other observational evidence has accumulated which suggests that the IMF in some environments may be top- heavy compared to the solar neighbourhood IMF. Rieke et al. (1993) argued for a top-heavy IMF in the nearby starburst M82, based on modelling its integrated properties, while Parra et al. (2007) found possible evidence for a top-heavy IMF in the ultra- luminous starburst Arp220 from the relative numbers of super- novae of different types observed at radio wavelengths. Evidence has been found for a top-heavy IMF in some star clusters in in- tensely star-forming regions, both in M82 (e.g. McCrady et al. 2003), and in our own Galaxy (e.g. Figer et al. 1999; Stolte et al. 2005; Harayama et al. 2007). Observations of both the old and young stellar populations in the central 1 pc of our Galaxy also favour a top-heavy IMF (Paumard et al. 2006; Maness et al. 2007). Fardal et al. (2006) found that reconciling measurements of the optical and IR extragalactic background with measurements of the cosmic star formation history also seemed to require an average IMF that was somewhat top-heavy. Finally, van Dokkum (2007) found that reconciling the colour and luminosity evolution of early- type galaxies in clusters also favoured a top-heavy IMF. Larson (1998) summarized other evidence for a top-heavy IMF during the earlier phases of galaxy evolution, and argued that this could be a natural consequence of the temperature-dependence of the Jeans mass for gravitational instability in gas clouds. Larson (2005) ex- tended this to argue that a top-heavy IMF might also be expected in starburst regions, where there is strong heating of the dust by the young stars. In our model, the fraction of star formation occuring in the burst mode increases with redshift (see Baugh et al. (2005)), so the average IMF with which stars are being formed shifts from being close to a solar neighbourhood IMF at the present-day to being very top-heavy at high redshift. In this model, 30% of star formation occured in the burst mode when integrated over the past history of the universe, but only 7% of the current stellar mass was formed in bursts, because of the much larger fraction of mass recycled by dying stars for the top-heavy IMF. We note that our predictions for the IR and sub-mm luminosities of starbursts are not sensitive to the precise form of the top-heavy IMF, but simply require a larger fraction of m ∼ 5− 20M⊙ stars relative to a solar neighbourhood In this paper, we calculate chemical evolution using the instan- taneous recycling approximation, which depends on the total frac- tion of mass recycled from dying stars (R), and the total yield of heavy elements (p). Both of these parameters depend on the IMF. We use the results of stellar evolution computations to calculate values of R and p consistent with each IMF (see Nagashima et al. (2005a) for details of the stellar evolution data used). Thus, we use R = 0.41 and p = 0.023 for the quiescent IMF, and R = 0.91 and p = 0.15 for the burst IMF. Our chemical evolution model then predicts the masses and total metallicities of the gas and stars in each galaxy as a function of time. c© 0000 RAS, MNRAS 000, 000–000 6 Lacey et al. 2.1.8 Galaxy sizes and dust masses For calculating the extinction and emission by dust, it is essential to have an accurate calculation of the dust optical depths in the model galaxies, which in turn depends on the mass of dust and the size of the galaxy. The dust mass is calculated from the gas mass and metallicity predicted by the chemical enrichment model, assuming that the dust-to-gas ratio is proportional to metallicity, normalized to match the local ISM value at solar metallicity. The sizes of galax- ies are computed exactly as in Cole et al. (2000): gas which cools in a halo is assumed to conserve its angular momentum as it col- lapses, forming a rotationally-supported galaxy disk; the radius of this disk is then calculated from its angular momentum, includ- ing the gravity of the disk, spheroid (if any) and dark halo. Galaxy spheroids are built up both from pre-existing stars in galaxy merg- ers, and from the stars formed in bursts triggered by these mergers; the radii of spheroids formed in mergers are computed using an energy conservation argument. In calculating the sizes of disks and spheroids, we include the adiabatic contraction of the dark halo due to the gravity of the baryonic components. This model was tested for disks by Cole et al. (2000) and for spheroids by Almeida et al. (2007) (see also Coenda et al. in preparation, and Gonzalez et al. in preparation). During a burst, we assume that the gas and stars involved in the burst have a distribution with the same half-mass radius as the spheroid (i.e. η = 1 in the notation of Granato et al. (2000), who used a value η = 0.1). 2.2 GRASIL model for stellar and dust emission For each galaxy in our model, we compute the spectral energy dis- tribution using the spectrophotometric model GRASIL (Silva et al. 1998; Granato et al. 2000). GRASIL computes the emission from the stellar population, the absorption and emission of radiation by dust, and also radio emission (thermal and synchrotron) powered by massive stars (Bressan et al. 2002). 2.2.1 SED model The main features of the GRASIL model are as follows: (i) The stars are assumed to have an axisymmetric distribution in a disk and a bulge. Given the distribution of stars in age and metal- licity (obtained from the star formation and chemical enrichment history), the SED of the stellar population is calculated using a population synthesis model based on the Padova stellar evolution tracks and Kurucz model atmospheres (Bressan et al. 1998). This is done separately for the disk and bulge. (ii) The cold gas and dust in a galaxy are assumed to be in a 2-phase medium, consisting of dense gas in giant molecular clouds embed- ded in a lower-density diffuse component. In a quiescent galaxy, the dust and gas are assumed to be confined to the disk, while for a galaxy undergoing a burst, the dust and gas are confined to the spheroidal burst component. (iii) Stars are assumed to be born inside molecular clouds, and then to leak out into the diffuse medium on a timescale tesc. As a result, the youngest and most massive stars are concentrated in the dustiest regions, so they experience larger dust extinctions than older, typ- ically lower-mass stars, and dust in the clouds is also much more strongly heated than dust in the diffuse medium. (iv) The extinction of the starlight by dust is computed using a ra- diative transfer code; this is used also to compute the intensity of the stellar radiation field heating the dust at each point in a galaxy. (v) The dust is modelled as a mixture of graphite and silicate grains with a continuous distribution of grain sizes (varying between 8Å and 0.25 µm), and also Polycyclic Aromatic Hydrocarbon (PAH) molecules with a distribution of sizes. The equilibrium temperature in the local interstellar radiation field is calculated for each type and size of grain, at each point in the galaxy, and this information is then used to calculate the emission from each grain. In the case of very small grains and PAH molecules, temperature fluctuations are important, and the probability distribution of the temperature is cal- culated. The detailed spectrum of the PAH emission is obtained us- ing the PAH cross-sections from Li & Draine (2001), as described in Vega et al. (2005). The grain size distribution is chosen to match the mean dust extinction curve and emissivity in the local ISM, and is not varied, except that the PAH abundance in molecular clouds is assumed to be 10−3 of that in the diffuse medium (Vega et al. 2005). (vi) Radio emission from ionized gas in HII regions and from syn- chrotron radiation from relativistic electrons accelerated in super- nova remnant shocks are calculated as described in Bressan et al. (2002). The output from GRASIL is then the complete SED of a galaxy from the far-UV to the radio (wavelengths 100Å . λ . 1m). The SED of the dust emission is computed as a sum over the different types of grains, having different temperatures depending on their size and their position in the galaxy. The dust SED is thus intrinsically multi-temperature. GRASIL has been shown to give an excellent match to the measured SEDs of both quiescent (e.g. M51) and starburst (e.g. M82) galaxies (Silva et al. 1998; Bressan et al. 2002). The assumption of axisymmetry in GRASIL is a limitation when considering starbursts triggered by galaxy mergers. However, observations of local ULIRGs imply that most of the star formation happens in a single burst component after the galaxy merger is sub- stantially complete, so the assumption of axisymmetry for the burst component may not be so bad. 2.2.2 GRASIL parameters The main parameters in the GRASIL dust model are the fraction fmc of the cold gas which is in molecular clouds, the timescale tesc for newly-formed stars to escape from their parent molecular cloud, and the cloud masses Mc and radii rc in the combination Mc/r which determines the dust optical depth of the clouds. We assume fmc = 0.25, Mc = 10 6M⊙ and rc = 16pc as in Granato et al. (2000), and also adopt the same geometrical parameters as in that paper. We make the following two changes in GRASIL parame- ters relative to Granato et al. , as discussed in Baugh et al. (2005): (a) We assume tesc = 1Myr in both disks and bursts (instead of the Granato et al. values tesc = 2 and 10Myr respectively). This value was chosen in order to obtain a better match of the pre- dicted rest-frame far-UV luminosity function of galaxies at z ∼ 3 to that measured for Lyman-break galaxies. (b) The dust emissivity law in bursts at long wavelengths is modified from ǫν ∝ ν −2 to ǫν ∝ ν −1.5 for λ > 100µm. This was done in order to improve slightly the fit of the model to the observed sub-mm number counts. In applying GRASIL to model the SEDs of a sample of nearby galaxies, Silva et al. (1998) found that a similar modification (to ǫν ∝ ν −1.6) seemed to be required in the case of Arp220 (the only ultra-luminous starburst in their sample), in order to reproduce the observed sub-mm data for that galaxy. This modification in fact has little effect on the IR predictions presented in the present paper, but we retain it for consistency with Baugh et al. (2005). c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 7 2.2.3 Interface with GALFORM For calculating the statistical properties of the galaxy population from the combined GALFORM+GRASIL model, we follow the same strategy as described in Granato et al. (2000). We first run the GALFORM code to generate a large catalogue of model galaxies at any redshift, and then run the GRASIL code on subsamples of these. For the quiescent galaxies, we select a subsample which has equal numbers of galaxies in equal logarithmic bins of stellar mass, while for the bursting galaxies, we select a subsample with equal numbers of galaxies in equal logarithmic bins of burst mass. For the burst sample, we compute SEDs at several different representative stages in the burst evolution, while for the quiescent sample, we only compute SEDs at a single epoch. Using this sampling strat- egy, we obtain a good coverage of all the different masses, types and evolutionary stages of galaxies, while minimizing the compu- tational cost of running the GRASIL code. The statistical properties of the galaxy population are then obtained by assigning the model galaxies appropriate weights depending on their predicted number density in a representative cosmological volume. The outputs from the GALFORM galaxy formation model re- quired by GRASIL to calculate the galaxy SEDs are: the combined star formation history and metallicity distribution for the disk and bulge, the radii of both components, and the total mass of dust. The dust mass is calculated from the mass and metallicity of the cold gas in the galaxy, assuming that the dust-to-gas ratio is proportional to the metallicity. Since the gas mass and metallicity both evolve, so does the dust mass, and this evolution is fully taken into account in GRASIL. For simplicity, we assume that the size distribution of the dust grains and PAH molecules does not evolve, apart from the normalization. Once we have calculated the SEDs for the model galaxies, we compute luminosities in different observed bands (e.g. the optical B-band or the Spitzer 24µm band) by convolving the SED with the filter+detector response function for that band. For computing the predicted fluxes from galaxies in a fixed observer-frame band, we redshift the SED before doing the convolution. The GRASIL code is quite CPU-intensive, requiring several minutes of CPU time per galaxy. Consequently, we are limited to running samples of a few thousand galaxies at each redshift. As a result, quantities such as luminosity functions and redshift distri- butions still show some small amount of noise, rather than being completely smooth curves, as can be seen in many of the figures in this paper. 2.3 Choice of parameters in the GALFORM+GRASIL model The combined GALFORM+GRASIL model has a significant num- ber of parameters, but this is inevitable given the very wide range of physical processes which are included. The parameters are con- strained by requiring the model predictions to reproduce a limited set of observational data - once this is done, there is rather little freedom in the choice of parameters. We have described above how the main parameters are fixed, and more details can be found in Cole et al. (2000) and Baugh et al. (2005). For both of these pa- pers, large grids of GALFORM models were run with different pa- rameters, in order to decide which set of parameters gave the best overall fit to the set of calibrating observational data. These papers also show the effects of varying some of the main model parameters around their best-fit values. The parameters in the standard model for which we present results in this paper were chosen to reproduce the following properties for present-day galaxies: the luminosity functions in the B- and K-bands and at 60µm, the relations between gas mass and luminosity and metallicity and luminosity, the size- luminosity relation for galaxy disks, and the fraction of spheroidal galaxies. In addition, the model was required to reproduce the ob- served rest-frame far-UV (1500Å) luminosity function at z = 3, and the observed sub-mm number counts and redshift distribution at 850µm (Baugh et al. 2005). The sub-mm number counts are the main factor driving the need to include a top-heavy IMF in bursts. The parameters for our standard model are exactly the same as in Baugh et al. (2005), which were chosen before Spitzer data be- came available. Since these parameters were not adjusted to match any data obtained with Spitzer, the predictions of our model in the Spitzer bands are genuine predictions. We could obviously have fine-tuned our parameters in order to match better the observational data we considered in this paper, but this would have conflicted with our main goal, which is to present predictions for a wide set of observable properties based on a single physical model in a series of papers. Since our assumption of a top-heavy IMF in bursts is a con- troversial one, we will also show some predictions from a variant model, which is identical to the standard model, except that we assume the same solar neighbourhood (Kennicutt) IMF in bursts and in disks. Comparing the predictions for the standard and vari- ant models then shows directly the effects of changing the IMF in bursts. We note that the variant model matches the present-day optical and near-IR luminosity functions almost as well as the stan- dard model, though it is a poorer fit to the local 60µm luminosity function for the brightest galaxies (see Fig. 9). The variant model underpredicts the 850µm counts by a factor of 10–30. 3 NUMBER COUNTS We begin our comparison of the predictions of our galaxy forma- tion model against Spitzer data with the galaxy number counts. Fig. 1 shows number counts in the four IRAC bands (3.6, 4.5, 5.8 and 8.0 µm), and Fig. 2 does the same for the three MIPS bands (24, 70 and 160 µm). Each panel is split in two: the upper sub- panel plots the counts per logarithmic flux interval, dN/d lnSν , while the lower sub-panel instead plots SνdN/d lnSν . The latter is designed to take out much of the trend with flux, in order to show more clearly the differences between the model and the on- servational data. In each case we plot three curves for our standard model: the solid blue line shows the total number counts includ- ing both extinction and emission by dust, the solid red line shows the contribution to this from galaxies currently forming stars in a burst, and the solid green line shows the contribution from all other galaxies (star-forming or not), which we denote as “quiescent”. In Fig. 1, we also plot a dashed blue line which shows the predicted total counts if we ignore absorption and emission from interstellar dust (emission from dust in the envelopes of AGB stars is still in- cluded in the stellar contribution, however). In the MIPS bands, the predicted counts are negligible in the absence of interstellar dust, so we do not plot them in Fig. 2. In the lower sub-panels, we also show by a dashed magenta line the prediction from a variant model which assumes a normal (Kennicutt) IMF for all star formation, but is otherwise identical to our standard model (which has a top-heavy IMF in bursts). This variant model fits the local B- and K-band and 60 µm luminosity functions about as well as our standard model, but dramatically underpredicts the 850 µm number counts. The ob- served number counts are shown by black symbols with error bars. Overall, the agreement between the predictions of our stan- c© 0000 RAS, MNRAS 000, 000–000 8 Lacey et al. Figure 1. Galaxy differential number counts in the four IRAC bands. The curves show model predictions, while the symbols with error bars show observational data from Fazio et al. (2004) (with different symbols for data from different survey fields). Each panel is split in two: the upper sub-panel plots the counts as dN/d lnSν vs Sν , while the lower sub-panel plots SνdN/d lnSν (in units mJy deg −2) on the same horizontal scale. The upper sub-panels show four different curves for our standard model - solid blue: total counts including dust extinction and emission; dashed blue: total counts excluding interstellar dust; solid red: ongoing bursts (including dust); solid green: quiescent galaxies (including dust). The lower sub-panels compare the total counts including dust for the standard model (solid blue line) with those for a variant model with a normal IMF for all stars (dashed magenta line). The vertical dashed line shows the estimated confusion limit for the model. (a) 3.6 µm. (b) 4.5 µm. (c) 5.8 µm. (d) 8.0 µm. dard model and the observed counts is remarkably good, when one takes account of the fact that no parameters of the model were ad- justed to improve the fit to any data from Spitzer. Consider first the results for the IRAC bands, shown in Fig. 1. Here, the agreement of the model with observations seems best at 3.6 and 8.0 µm, and somewhat poorer at 5.8 µm. The model predicts somewhat too few objects at fainter fluxes in all of the IRAC bands. Comparing the red and green curves, we see that quiescent galaxies rather than bursts dominate the counts at all observed fluxes in all of the IRAC bands, but especially at the shorter wavelengths, consistent with the expec- tation that at 3.6 and 4.5 µm, we are seeing mostly light from old stellar populations. Comparing the solid and dashed blue lines, we see that the effects of dust are small at 3.6 and 4.5 µm, with a small amount of extinction at faint fluxes (and thus higher average red- shifts), but negligible extinction for brighter fluxes (and thus lower redshifts). On the other hand, dust has large effects at 8.0 µm, with dust emission (due to strong PAH features at λ ∼ 6 − 9µm) be- coming very important at bright fluxes (which correspond to low average redshifts - see Fig. A1(b) in the Appendix). The 8.0 µm counts thus are predicted to be dominated by dust emission from quiescently star-forming galaxies, except at the faintest fluxes. The counts at 5.8 µm show behaviour which is intermediate, with mild emission effects at bright fluxes and mild extinction at faint fluxes. Comparing the solid blue and dashed magenta lines, we see that the predicted number counts in the IRAC bands are almost the same whether or not we assume a top-heavy IMF in bursts, consistent with the counts being dominated by quiescent galaxies. Consider next the results for the MIPS bands, shown in Fig. 2. We again see remarkably good agreement of the standard model with the observational data. The agreement is especially good at faint fluxes (corresponding to higher redshifts). In particular, the model matches well the observed 24 µm counts at the “bump” c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 9 around fluxes Sν ∼ 0.1 − 1mJy. Accurate modelling of the PAH emission features is obviously crucial for modelling the 24 µm number counts, since the PAH features dominate the flux in the 24 µm band as they are redshifted into the band at z & 0.5. On the other hand, the standard model overpredicts the number counts at bright fluxes (corresponding to low redshifts) in all three MIPS bands. The evolution at these wavelengths predicted by our ΛCDM-based model thus seems to be not quite as strong as indi- cated by observations. In the MIPS bands, emission from galaxies is completely dominated by dust, which is why no dashed blue lines are shown in Fig. 2. Comparing the red and green curves, we see that quiescent (but star-forming) galaxies tend to dominate the number counts in these bands at brighter fluxes, and bursts at fainter fluxes. This re- flects the increasing dominance of bursts in the mid- and far-IR luminosity function at higher redshifts. Comparing the solid blue and dashed magenta curves, we see that our standard model with a top-heavy IMF in bursts provides a significantly better overall fit to the observed 24 µm counts than the variant model with a normal IMF in bursts (although at the brightest fluxes, the variant model fits better). The faint number counts at 70 µm also favour the top- heavy IMF model, while the number counts at 160 µm cover a smaller flux range, and do not usefully distinguish between the two variants of our model with different burst IMFs. We can use our model to predict the flux levels at which sources should become confused in the different Spitzer bands. We estimate the confusion limit using the source density cri- terion (e.g. Vaisanen et al. 2001; Dole et al. 2003): if the tele- scope has an FWHM beamwidth of θFWHM , we define the ef- fective beam solid angle as ωbeam = (π/(4 ln 2)) θ FWHM = 1.13θ2FWHM , and then define the confusion limited flux Sconf to be such that N(> Sconf ) = 1/(Nbeamωbeam), where N(> S) is the number per solid angle of sources brighter than flux S. We choose Nbeam = 20 for the number of beams per source, which gives similar results to more detailed analy- ses (e.g. Vaisanen et al. 2001; Dole et al. 2004b). We use values of the beamsize θFWHM = (1.66, 1.72, 1.88, 1.98) arcsec for the four IRAC bands (Fazio et al. 2004b) and (5.6, 16.7, 35.2) arcsec for the three MIPS bands (Dole et al. 2003). Our stan- dard model then predicts confusion-limited fluxes of Sconf = (0.62, 0.62, 0.69, 0.70)µJy in the (3.6, 4.5, 5.8, 8.0)µm IRAC bands, and Sconf = (0.072, 2.6, 43)mJy in the (24, 70, 160)µm MIPS bands. These confusion estimates for the MIPS bands are similar to those of Dole et al. (2004b), which were based on ex- trapolating from the observed counts. These values for the confu- sion limits are indicated in Figs. 1 and 2 by vertical dashed lines. Our galaxy evolution model does not compute the contribution of AGN to the IR luminosities of galaxies. On the other hand, the observed number counts to which we compare include both normal galaxies, in which the IR emission is powered by stellar popula- tions, and AGN, in which there is also IR emission from a dust torus, which is expected to be most prominent in the mid-IR. How- ever, multi-wavelength studies using optical, IR and X-ray data in- dicate that even at 24 µm, the fraction of sources dominated at that wavelength by AGN is only 10-20% (e.g. Franceschini et al. 2005), and the contribution of AGN-dominated sources in the other Spitzer bands is likely to be smaller. Therefore we should not make any serious error by comparing our model predictions directly with the total number counts, as we have done here. Figure 2. Galaxy differential number counts in the three MIPS bands. The curves show model predictions while the symbols with error bars show ob- servational data. The meaning of the different model lines is the same as in Fig. 1. (a) 24 µm, with observational data from Papovich et al. (2004). (b) 70 µm, with observational data from Dole et al. (2004a) (filled sym- bols), Frayer et al. (2006a) (crosses), and Frayer et al. (2006b) (open sym- bols). (c) 160 µm (bottom panel), with observational data from Dole et al. (2004a) (filled symbols) and Frayer et al. (2006a) (crosses). c© 0000 RAS, MNRAS 000, 000–000 10 Lacey et al. Figure 3. Predicted evolution of the galaxy luminosity function in our standard model (including dust) at rest-frame wavelengths of (a) 3.6 and (b) 8.0 µm for redshifts z = 0, 0.5, 1, 1.5, 2 and 3, as shown in the key. 4 EVOLUTION OF THE GALAXY LUMINOSITY FUNCTION While galaxy number counts provide interesting constraints on the- oretical models, it is more physically revealing to compare with galaxy luminosity functions, since these isolate behaviour at partic- ular redshifts, luminosities and rest-frame wavelengths. In the fol- lowing subsections, we compare our model predictions with recent estimates of luminosity function (LF) evolution based on Spitzer data. 4.1 Evolution of the galaxy luminosity function at 3-8 µm We consider first the evolution of the luminosity function in the wavelength range covered by the IRAC bands, i.e. 3.6-8.0 µm. Fig. 3 shows what our standard model with a top-heavy IMF in bursts predicts for LF evolution at rest-frame wavelengths of 3.6 and 8.0 µm for redshifts z = 0 − 3 3. We see that at a rest-frame wavelength of 3.6 µm, the model LF hardly evolves at all over the whole redshift range z = 0 − 3. This lack of evolution appears to be somewhat fortuitous. Galaxy luminosities at a rest-frame wave- length of 3.6 µm are dominated by the emission from moderately old stars, but the stellar mass function in the model evolves quite strongly over the range z = 0 − 3 (as we show in §5). The weak evolution in the 3.6 µm LF results from a cancellation between a declining luminosity-to-stellar-mass ratio with increasing time and increasing stellar masses (see Figs. 13(a) and (e)). On the other hand, at a rest-frame wavelength of 8.0 µm, the model LF becomes significantly brighter in going from z = 0 to z = 3. Galaxy lu- minosities at a rest-frame wavelength of 8.0 µm are dominated by emission from dust heated by young stars, so this evolution reflects the increase in star formation activity with increasing redshift (see Fig. 13(b) in §5). In Fig. 4, we compare the model predictions for evolu- tion of the LF at 3.6 µm with observational estimates from 3 In this figure, and in Figs. 4, 5, 8, and 10, the luminosities Lν are calcu- lated through the corresponding Spitzer passbands. Babbedge et al. (2006) and Franceschini et al. (2006)4. The model predictions are given for redshifts z = 0, 0.5 and 1. For the observational data, the mean redshifts for the different redshift bins used do not exactly coincide with the model redshifts, so we plot them with the model output closest in redshift 5. The observa- tional estimates of the 3.6 µm LF rely on the measured redshifts. In the case of Babbedge et al. (2006), these are mostly photomet- ric, using optical and NIR (including 3.6 and 4.5 µm) fluxes, while for the Franceschini et al. sample, about 50% of the redshifts are spectroscopic and the remainder photometric. In both samples, the measured 3.6 µm fluxes were k-corrected to estimate the rest-frame 3.6 µm luminosities. We see from comparing the blue curve with the observational data in Fig. 4 that the 3.6 µm LF predicted by our standard model is in very good agreement with the observations. In particular, the observational data show very little evolution in the 3.6 µm LF over the redshift range z = 0 − 1. The largest difference seen is at z = 1, where the Babbedge et al. data show a tail of objects to very high luminosities, which is not seen in the model predictions. However, this tail is not seen in the Franceschini et al. data at the same redshift, and is also not present in the observational data at the lower redshifts. More spectroscopic redshifts are needed for the Babbedge et al. sample to clarify whether this high-luminosity tail is real. Comparing the red, green and blue lines for the standard model shows that the model luminosity function is dominated by quiescent galaxies at low luminosity, but the contribution of bursts becomes comparable to that of quiescent galaxies at high luminosi- ties. We have not shown model LFs excluding dust extinction in this figure, since they are almost identical to the predictions includ- ing dust. The dashed magenta lines show the predicted LFs for the 4 Babbedge et al. (2006) also compared their measured LFs at 3.6, 8.0 and 24 µm with predictions from a preliminary version of the model described in this paper 5 Specifically, for z = 0, we compare with the z = 0.1 data from Babbedge et al. , for z = 0.5 we compare with the z = 0.5 data from Babbedge et al. and z = 0.3 data from Franceschini et al. , and for z = 1, we compare with the z = 0.75 (open symbols) and z = 1.25 (filled symbols) data from Babbedge et al. and z = 1.15 data from Franceschini et al. c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 11 Figure 4. Predicted evolution of the galaxy luminosity function at rest- frame 3.6 µm compared to observational data. The different panels show redshifts (a) z = 0, (b) z = 0.5 and (c) z = 1. The predictions for our stan- dard model are shown by the blue line, with the red and green lines showing the separate contributions from ongoing bursts and quiescent galaxies. The dashed magenta line shows the prediction for a variant model with a nor- mal IMF for all stars. The error bars on the model lines indicate the Poisson uncertainties due to the finite number of galaxies simulated. The black sym- bols with error bars show observational data from Babbedge et al. (2006) (open circles and triangles, for z = 0, 0.5 and 1) and Franceschini et al. (2006) (filled squares, for z = 0.5 and 1). Figure 5. Predicted evolution of the galaxy luminosity function at rest- frame 8.0 µm compared to observational data. The different panels show redshifts (a) z = 0, (b) z = 1 and (c) z = 2. The coloured lines showing the model predictions have the same meaning as in Fig. 4. The black sym- bols with error bars show observational data from Babbedge et al. (2006) (open circles for z = 0 and 0.7, triangles for z = 1.2), Huang et al. (2007) (filled circles for z = 0) and Caputi et al. (2007) (filled circles for z = 1 and 2). The observed LFs are for normal galaxies and exclude AGN. c© 0000 RAS, MNRAS 000, 000–000 12 Lacey et al. variant model with a normal IMF in bursts. We see that these differ only slightly from our standard model, but are a somewhat poorer fit to the observational data at higher luminosities. In Fig. 5 we show a similar comparison for the LF evolution at a rest-frame wavelength of 8 µm. The model predictions are given for redshifts z = 0, 1 and 2, and are compared with observational estimates by Huang et al. (2007) (for z ∼ 0), Babbedge et al. (2006) (for z ∼ 0 and z ∼ 1) and Caputi et al. (2007) (for z ∼ 1 and z ∼ 2). These papers all classified objects in their samples as either galaxies or AGN, and then computed separate LFs for the two types of objects 6. Our model does not make any predic- tions for AGN, so we compare our model predictions with the ob- served LFs for objects classified as galaxies only. We see that for redshifts around z = 1, the observed LFs from Babbedge et al. and Caputi et al. are in very poor agreement with each other, with the Caputi et al. LF being around 10 times higher in number den- sity at the same luminosity. This differerence presumably results from some combination of: (a) different methods of classifying ob- jects as galaxies or AGN (Babbedge et al. used only optical and IR fluxes to do this, while Caputi et al. also used X-ray data); (b) different photometric redshift estimators; and (c) different meth- ods for k-correcting luminosities to a rest-frame wavelength of 8 µm. There are smaller differences between the Huang et al. and Babbedge et al. LFs at z ∼ 0. Futher observational investigation appears to be necessary to resolve these issues. Our standard model is in reasonable agreement with the Babbedge et al. observed LF at z ∼ 0, and with the Caputi et al. observed LFs at z ∼ 1 and z ∼ 2, but not with the Babbedge et al. observed LF at z ∼ 1. The comparison with Caputi et al. favours our standard model with a top-heavy IMF in starbursts over the variant model with a normal 4.2 Evolution of the galaxy luminosity function at 12-24 µm In this subsection, we consider the evolution of the galaxy lumi- nosity function at mid-IR wavelengths, and compare with data ob- tained using mainly the MIPS 24 µm band. Fig. 6 shows what our standard model with a top-heavy IMF in bursts predicts for the evolution of the galaxy LF at rest-frame wavelengths of 15 and 24 µm for redshifts z = 0 − 3 7. At rest- frame wavelengths of 15 and 24 µm, galaxy luminosities are typi- cally dominated by the continuum emission from warm dust grains heated by young stars (although PAH emission is also significant at some nearby wavelengths). Fig. 6 shows strong evolution in the model LFs over the redshift range z = 0− 3 at both wavelengths, reflecting both the increase in star formation activity with increas- ing redshift (see Fig. 13(b)) and the increasing dominance of the burst mode of star formation, for which the top-heavy IMF fur- ther boosts the mid- and far-IR luminosities compared to a normal IMF. Comparing Fig. 6 with Fig. 3(a), we also see a difference in the shape of the bright end of the LF: at 3.6 µm, where the LF is dominated by emission from stars, the bright end cuts off roughly 6 Note that a variety of criteria have been used for classifying observed IR sources as AGN or normal galaxies, and these do not all give equivalent results. Even if an object is classified as an AGN, it is also not clear that in all cases the AGN luminosity dominates over that of the host galaxy in all Spitzer bands 7 In this figure, and in Figs. 7 and 8, the 24 µm luminosities are calculated through the corresponding MIPS passband, while the 15 µm luminosities are calculated through a top-hat filter with a fractional width of 10% in wavelength. Figure 8. Predicted evolution of the galaxy luminosity function at rest- frame wavelength 24 µm compared to observational data from Shupe et al. (1998) (at z = 0, open symbols) and from Babbedge et al. (2006) (for the same redshifts as in Fig. 4). The meaning of the curves showing the model predictions is the same as in Fig. 4. (a) z = 0, (b) z = 0.5 and (c) z = 1. c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 13 Figure 6. Predicted evolution of the galaxy luminosity function in our standard model at rest-frame wavelengths(a) 15 µm (left) and (b) 24 µm (right) for redshifts z = 0, 0.5, 1, 1.5, 2 and 3, as shown in the key. exponentially, while at 15 and 24 µm, where the LF is dominated by emission from warm dust, the bright end declines more gradu- ally, roughly as a power-law. This difference reflects the difference in shape of the galaxy stellar mass function (GSMF) and the galaxy star formation rate distribution (GSFRD). The GSMF shows an exponential-like cutoff at high masses, while the GSFRD shows a more gradual cutoff at high SFRs because of starbursts triggered by galaxy mergers (see Figs. 13(a) and (b) in §5). This difference was noticed earlier by observers comparing optical and far-IR LFs of galaxies, but its origin was not understood (Lawrence et al. 1986; Soifer et al. 1987b). In Fig. 7, we compare the model LFs at rest-frame wave- lengths 12 and 15 µm with observational estimates. For z = 0, we plot the observational estimates from Soifer & Neugebauer (1991) and Rush et al. (1993), based on IRAS 12 µm data (with AGN removed). For z = 0.5 − 1 and z = 1.5 − 2.5, we plot the data of Le Floc’h et al. (2005) and Perez-Gonzalez et al. (2005) respectively, which were obtained from galaxy samples selected on Spitzer 24 µm flux. Le Floc’h et al. k-corrected their measured 24 µm fluxes to 15 µm rest-frame luminosities, while Perez-Gonzalez et al. k-corrected to 12 µm rest-frame8. Le Floc’h et al. obtained most of their redshifts from photometric redshifts based on optical data, while Perez-Gonzalez et al. used a new photometric redshift technique based on fitting empirical SEDs to all of the available broad-band data from the far-UV to 24 µm, and also removed “extreme” AGN from their observed LF. Note that the redshifts for the observed LFs do not exactly coincide with model redshifts in all cases, but are close. We see from comparing the blue line to the observational dat- apoints in Fig. 7 that our standard model with a top-heavy IMF in bursts fits the observations remarkably well up to z = 2. In partic- ular, the model matches the strong evolution in the mid-IR LF seen 8 The exact passband used for the model LF in each panel depends on which observational data we are comparing with. For z = 0, we use the IRAS 12 µm passband; at z = 0.5 and z = 1 we use a top-hat passband centred at 15 µm; and at z = 1.5, 2 and 2.5, we use a top-hat passband centred at 12 µm (both top-hat passbands having fractional width 10% in wavelength). in the observational data. The model falls below the observational data at z = 2.5, but here both the photometric redshifts and the k-corrections are probably the most uncertain. The standard model also does not provide a perfect fit to the z = 0 data, predicting somewhat too many very bright galaxies and somewhat too few very faint galaxies (though the latter discrepancy might be affected by local galaxy clustering in the IRAS data). Comparing the red, green and blue lines for the standard model in the figure, we see that the bright end of the 12 or 15 µm LF is dominated by bursts at all redshifts. The figure also shows by a dashed magenta line the predictions for the variant model with a normal IMF in bursts. This latter model predicts much less evolution in the bright end of the LF than is observed. This comparison thus strongly favours the model with the top-heavy IMF in bursts. Finally, in Fig. 8, we carry out a similar comparison of the evo- lution of predicted and observed LFs at a rest-frame wavelength of 24 µm over the redshift range z = 0 − 1, in this case com- paring with observational estimates from Shupe et al. (1998) (for z = 0), based on IRAS data, and from Babbedge et al. (2006) (for z = 0 − 1), based on Spitzer data9. The galaxy redshifts for the Babbedge et al. data were obtained in the same way as for the 3.6 µm LFs shown in Fig. 4, and the luminosities were k-corrected from observer-frame 24 µm to rest-frame 24 µm. The LF plotted from Babbedge et al. is that for normal galaxies, with AGN ex- cluded. The conclusions from comparing the model with the 24 µm LFs are similar to those from the comparison with the 12 and 15 µm LFs. The data favour our standard model over the variant with a normal IMF in bursts (except possibly for z = 0.5), as the latter predicts too little evolution at the bright end. At z = 0, the model fits the 24 µm data rather better than for the corresponding com- parison at 12 µm. On the other hand, at z = 0.5 and z = 1, the model LF is a somewhat worse fit to the observational data at 24 µm than at 15 µm. These differences between the 12/15 and 24 µm comparisons might result from the different photometric red- shifts and k-corrections used in the observational samples in the 9 The model luminosities are all computed through the Spitzer 24 µm pass- band. c© 0000 RAS, MNRAS 000, 000–000 14 Lacey et al. Figure 7. Predicted evolution of the galaxy luminosity function at rest-frame wavelength 12 or 15 µm compared to observational data. The different panels show redshifts: (a) z = 0, (b) z = 0.5, (c) z = 1, (d) z = 1.5, (e) z = 2 and (f) z = 2.5. The meaning of the curves showing the model predictions is the same as in Fig. 4. In panel (a), the predictions at 12µm are compared to observational determinations from Soifer & Neugebauer (1991) (open symbols) and Rush et al. (1993) (filled symbols) based on IRAS data. In panels (b) and (c), the predictions at 15µm are compared to observational data from Le Floc’h et al. (2005). In panels (d), (e) and (f), the predictions at 12µm are compared to observational data from Perez-Gonzalez et al. (2005). c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 15 Figure 9. The predicted galaxy luminosity function at 60 µm compared to observational data from IRAS. The meaning of the different lines is the same as in Fig. 4. The black symbols show observational data from Saunders et al. (1990) (crosses), Soifer & Neugebauer (1991) (open cir- cles), and Takeuchi et al. (2003) (filled circles). two cases. Alternatively, they might result from problems in mod- elling the dust SEDs in the complex mid-IR range. 4.3 Evolution of the galaxy luminosity function at 70-160 µm We now briefly consider the evolution of the luminosity function in the far-IR. The far-IR is the wavelength range where most of the luminosity from dust in normal galaxies is emitted. The lo- cal 60 µm luminosity function was very well measured by sur- veys with IRAS, and so is commonly used as a starting point or benchmark for modelling the evolution of the galaxy population in the far-IR. We therefore present in Fig. 9 the model prediction for the 60 µm luminosity function at z = 0, compared with obser- vational data from Saunders et al. (1990), Soifer & Neugebauer (1991) and Takeuchi et al. (2003). As discussed in Baugh et al. (2005), the local 60 µm luminosity function was used as one of the primary constraints in fixing the parameters of our galaxy forma- tion model, and the figure shows that our standard model provides a good match to the data. The variant model with a normal IMF in bursts underpredicts the abundance of the brightest 60 µm galaxies. In Fig. 10, we show the model predictions for the evolution of the luminosity function in the two longer wavelength MIPS bands, at rest-frame wavelengths of 70 and 160 µm, from z = 0 to z = 3. At 70 µm, the luminosity function at high luminosities is predicted to brighten by about a factor 10 going from z = 0 to z = 2. This is about a factor 2 less than the brightening predicted in the mid-IR at 15 µm (compare to Fig. 6), but nearly a factor 2 more evolution than is predicted at 160 µm. These differences between the amount of evolution seen at different IR wavelengths reflect evolution in the shapes of the SEDs of the galaxies responsible for the bulk of the IR emission. No observational estimates of the evolution of the luminosity function at 70 and 160 µm have yet been published, but they are expected to be forthcoming from ongoing surveys with Spitzer. Figure 11. Predicted evolution of the total mid+far-IR (8-1000 µm) galaxy luminosity function for our standard model, for redshifts z = 0, 1, 2, 3, 4 and 6, as shown in the key. 4.4 Evolution of the total mid+far-IR luminosity function The total mid+far IR luminosity of a galaxy, LIR, integrated over the whole wavelength range 8-1000 µm, is a very good approxi- mation to the total luminosity emitted by interstellar dust grains in all galaxies except those with very small dust contents. In galax- ies with significant star formation, LIR is mostly powered by dust heated by young stars, and so provides a quantitative indicator of the amount of dust-obscured star formation which is independent of the shape of the IR SED (though still subject to uncertainties about the IMF). The evolution of the luminosity function in LIR is therefore a very interesting quantity to compare between models and observations. We show in Fig. 11 what our standard model pre- dicts for the evolution of the IR LF over the range z = 0 − 6. We see that the model predicts substantial evolution in this LF, with the high luminosity end brightening by a factor ∼ 10 from z = 0 to z = 2, followed by a “plateau” from z = 2 to z = 4, and a decline from z = 4 to z = 6. In Fig. 12, we compare our model predictions with existing observational estimates of the total IR LF for z = 0 − 2. These observational estimates are only robust for z = 0, where they are based on IRAS measurements covering the wavelength range 12- 100 µm. At all of the higher redshifts plotted, the observational estimates are based on measurements of the mid-IR luminosity de- rived from Spitzer 24 µm fluxes, converted to total IR luminosi- ties by assuming SED shapes for the mid- to far-IR emission. The bolometric correction from the observed mid-IR luminosity to the inferred total IR luminosity is typically a factor ∼ 10, and is sig- nificantly uncertain. Therefore, the most robust way to compare the models with the observations is to compare them at the mid- IR wavelengths where the measurements are actually made, as we have done in §4.1 and §4.2. Nonetheless, if we take the observa- tional determinations at face value, then we see that observed evo- lution of the total IR LF agrees remarkably well with the predic- tions of our standard model with a top-heavy IMF. On the other hand, the variant model with a normal IMF predicts far too few c© 0000 RAS, MNRAS 000, 000–000 16 Lacey et al. Figure 10. Predicted evolution of the galaxy luminosity function in our standard model (including dust) at rest-frame wavelengths (a) 70 µm and (b) 160 µm, for redshifts z = 0, 0.5, 1, 1.5, 2 and 3, as shown in the key. high LIR galaxies at higher z, and is strongly disfavoured by the existing data. 5 INFERRING STELLAR MASSES AND STAR FORMATION RATES FROM Spitzer DATA In this section, we consider what the models imply about how well we can infer the stellar masses and star formation rates (SFRs) in galaxies from measurements of rest-frame IR luminosities. The top two panels of Fig. 13 show the predicted galaxy stellar mass func- tion (GSMF, left panel) and galaxy star formation rate distribution (GSFRD, right panel), for redshifts z = 0− 6. We see that the pre- dicted stellar mass function shows dramatic evolution over this red- shift range, with a monotonic decline in the number of high-mass galaxies with increasing redshift. On the other hand, the SFR distri- bution shows much less dramatic evolution over this redshift range, with a mild increase in the number of high-SFR objects up to z ∼ 3, followed by a decline above that. The lower four panels in Fig. 13 show the relation in the models between stellar masses and SFRs and rest-frame luminosities at different IR wavelengths. (Note that in all cases, luminosities are measured in units of the bolometric solar luminosity.) The middle and bottom left panels respectively show the mean ratio of luminosity in the rest-frame K (2.2µm) or 3.6 µm bands to stellar mass as a function of stellar mass. The middle and bottom right panels respectively show the mean ratio of total mid+far-IR (8− 1000µm) or rest-frame 15 µm luminosity to SFR as a function of SFR. (The mean L/M∗ or L/SFR ratios plotted are computed by dividing the total luminosity by the total mass or SFR, in each bin of mass or SFR.) The near-IR luminosity is often used as a tracer of stellar mass. The left panels of Fig. 13 show that the L/M∗ ratio varies strongly with redshift, reflecting the difference in the ages of the stellar pop- ulations. At higher redshifts it also shows a significant dependence on stellar mass, presumably reflecting a trend of age with mass. However, the variation of mean L/M∗ with redshift is seen to be much smaller at 2.2 µm than at 3.6 µm, implying that the rest- frame K-band light should provide a more robust estimator of stel- lar mass than the light at longer wavelengths. The differences be- tween L/M∗ values at 2.2 µm and 3.6 µm reflect the larger contri- bution from AGB compared to RGB stars at the longer wavelength. AGB stars have higher masses and younger ages than RGB stars, and so are more sensitive to star formation at recent epochs. The scatter in L/M∗ at a given mass is also found in the models to in- crease with redshift. In the K-band, it increases from ∼ 40% at z ∼ 0 to a factor ∼ 3 at z ∼ 6. The large scatter at high redshifts results in part from having two different IMFs. The luminosity in the mid- and far-IR is widely used as a tracer of dust-obscured star formation (although in galaxies with very low star formation rates, the dust heating can be dominated by older stars). The total mid+far-IR (rest-frame 8-1000 µm) lumi- nosity is expected to provide a more robust tracer of star formation than the luminosity at any single IR wavelength, since the shape of the SED of dust emission depends on the dust temperature distri- bution (as well as on the dust grain properties). This is borne out by our model predictions. The middle right panel of Fig. 13 shows that the LIR/SFR ratio depends weakly on both SFR and red- shift. This behaviour results mostly from having different IMFs in the model in quiescent and bursting galaxies, with the fractional contribution of the bursts increasing both with SFR and with red- shift. If we look at quiescent and bursting galaxies separately, we find roughly constant ratios LIR/SFR ≈ 6 × 10 9h−1L⊙/M⊙ and LIR/SFR = 2× 10 10h−1L⊙/M⊙ respectively, for galaxies where LIR is powered mostly by young stars. However, there is also a trend at lower redshift for LIR/SFR to be larger at lower SFR - this reflects the larger fraction of dust heating from older stars in galaxies with lower SFRs, which more than compensates for the lower average dust obscuration in these galaxies. The lower right panel of Fig. 13 shows that the L/SFR ratio in the mid-IR (in this case at 15 µm in the rest-frame) shows more variation with SFR and redshift than the ratio for the total IR luminosity. This re- flects the variation in the mid- to far-IR SED shapes in the model. The scatter in the L/SFR ratio is roughly a factor 2 around the average relation for the total IR luminosity, but is larger for the 15 µm luminosity. The results of this section illustrate why it is not straightfor- ward to compare theoretical predictions for the evolution of the galaxy stellar mass function and star formation rate distribution (or even the stellar mass and star formation rate densities) with obser- c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 17 Figure 12. Predicted evolution of the total mid+far IR (8-1000µm) galaxy LF compared to observational data. The different panels show redshifts (a) z = 0, (b) z = 0.5, (c) z = 1 and (d) z = 2. For z = 0, we compare with observational data from Sanders et al. (2003) (filled symbols) and Takeuchi et al. (2003) (open symbols, converting his 60 µm LF to a total IR LF assuming a constant conversion factor, LIR/νLν(60µm) = 2.5). We compare with data from Le Floc’h et al. (2005) for z = 0.5 and z = 1 (filled and open symbols), and with Caputi et al. (2007) for z = 1 and z = 2 (crosses). vational estimates. In addition to assumptions about galaxy star for- mation histories and metallicities (for stellar mass estimates), and about the SED shapes for dust emission (for SFR estimates from IR and sub-mm data), observational estimates all rest on some as- sumed form for the IMF. If the IMF assumed in the observational analysis is different from the true IMF, the observational estimates for stellar masses and SFRs can be wrong by large factors. If the IMFs differ only below 1M⊙, then one can apply a simple rescal- ing to relate stellar mass and SFR estimates for different IMFs. However, if our current galaxy formation model is correct, stars form with different IMFs in quiescent disks and in merger-driven bursts, and so no observational estimate based on assuming a sin- gle IMF can give the correct GSMFs and GSFRDs, nor the correct stellar mass and SFR densities. A direct comparison of the GSMF and GSFRD evolution predicted by our model with observational estimates is therefore not meaningful. Instead, the comparison be- tween models and observations must be made via directly observ- able (rather than inferred) quantities, such as the K-band luminosi- ties to constrain stellar masses, and the total IR luminosities to con- strain SFRs. 6 CONCLUSIONS We have computed predictions for the evolution of the galaxy pop- ulation at infrared wavelengths using a detailed model of hierar- chical galaxy formation and of the reprocessing of starlight by dust, and compared these predictions with observational data from the Spitzer Space Telescope. We calculated galaxy formation in the framework of the ΛCDM model using the GALFORM semi- analytical model, which includes physical treatments of the hier- archical assembly of dark matter halos, shock-heating and cool- ing of gas, star formation, feedback from supernova explosions and photo-ionization of the IGM, galaxy mergers and chemical en- richment. We computed the IR luminosities and SEDs of galaxies using the GRASIL multi-wavelength spectrophotometric model, which computes the luminosities of the stellar populations in galax- c© 0000 RAS, MNRAS 000, 000–000 18 Lacey et al. Figure 13. Model predictions for properties related to stellar masses (left column) and star formation rates (right column), for redshifts z = 0, 1, 2, 3, 4, and 6: (a) galaxy stellar mass function (GSMF); (b) galaxy star formation rate distribution (GSFRD); (c) mean ratio of rest-frame K-band luminosity to stellar mass, as a function of stellar mass; (d) mean ratio of total mid+far IR luminosity to SFR, as a function of SFR; (e) mean ratio of rest-frame 3.6 µm luminosity to stellar mass, as a function of stellar mass; (f) mean ratio of rest-frame 15 µm luminosity to SFR, as a function of SFR. (The 15 µm luminosity is here calculated through top-hat filter with a fractional wavelength width of 10%.) c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 19 ies, and then the reprocessing of this radiation by dust, including radiative transfer through a two-phase dust medium, and a self- consistent calculation of the distribution of grain temperatures in each galaxy based on a local balance between heating and cool- ing. The GRASIL model includes a treatment of the emission from PAH molecules, which is essential for understanding the mid-IR emission from galaxies. Our galaxy formation model incorporates two different IMFs: quiescent star formation in galaxy disks occurs with a normal so- lar neighbourhood IMF, but star formation in bursts triggered by galaxy mergers happens with a top-heavy x = 0 IMF. In a previ- ous paper (Baugh et al. 2005), we found that the top-heavy IMF in bursts was required in order that the model reproduces the observed number counts of the faint sub-mm galaxies detected at 850 µm, which are typically ultra-luminous starbursts at z ∼ 2, with total IR luminosities LIR ∼ 10 − 1013L⊙. This conclusion was arrived at following a search of a large grid of model parameters, with the imposition of a variety of detailed observational constraints. The parameters in the Baugh et al. (2005) model were chosen before the publication of any results from Spitzer, without reference to any IR data apart from the local 60 µm luminosity function and the 850 µm galaxy counts. We have kept the same parameter values in the present paper, in order to test what the same model predicts at other wavelengths and other redshifts. By doing this, we hope to address the criticism made of many semi-analytical models that they have no predictive power, because their parameters are always adjusted to match the observational data being analysed at that instant. We first compared the predictions from our model with the galaxy number counts measured in all 7 Spitzer bands, from 3.6 to 160 µm. We found broad agreement between the model and the observations. In the 4 IRAC bands (3.6-8.0 µm), where the counts are mostly dominated by emission from older stellar populations, we found that the predicted counts were insensitive to whether we had a top-heavy or normal IMF in bursts. On the other hand, in the MIPS bands (24-160 µm), where the counts are dominated by emission from dust in star-forming galaxies, the predicted counts are more sensitive to the choice of IMF, and the counts are fit better by the model with a top-heavy IMF. We next investigated the evo- lution of the galaxy luminosity function at IR wavelengths, where several groups have now used Spitzer data to try to measure the evolution of the galaxy luminosity function over the redshift range z ∼ 0− 2, at rest-frame wavelengths from 3.6 to 24 µm. Our model predicts that at mid- and far-IR rest-frame wave- lengths, the luminosity function evolution is very sensitive to the choice of IMF in bursts. We found that our standard model with a top-heavy IMF in bursts fits the measured evolution of the mid-IR luminosity function remarkably well (when allowance is made for complexity of predicting dust emission in the mid-IR), without any adjustment of the parameters. On the other hand, a model with a normal IMF in bursts predicts far too little evolution in the mid-IR luminosity function compared to what is observed. We made a sim- ilar comparison with the evolution of the total IR luminosity func- tion, where in the case of the observations, the total IR luminosities at high redshifts have been inferred from the 24 µm fluxes by fit- ting SEDs, and reached the same conclusion. The evolution of the galaxy luminosity function in the mid-IR found by Spitzer thus sup- ports our original conclusion about the need for a top-heavy IMF in bursts, which was based only on the sub-mm counts. This con- clusion will be further tested by ongoing Spitzer surveys at longer wavelengths. To assist this, we have also presented predictions for the evolution of the luminosity function in the Spitzer 70µm and 160µm bands. We have also presented predictions for the evolution of the stellar mass function and star formation rate distribution of galax- ies. We investigated how the L/M∗ and L/SFR ratios varied with galaxy mass, SFR and redshift in different IR wavelength ranges, and considered the implications for observational estimates of stel- lar masses and SFRs from IR observations. Even in the near-IR, the predicted variations inL/M∗ with mass and redshift can be surpris- ingly large. The variations in L/M∗ are much larger at a rest-frame wavelength of 3.6 µm than at 2.2 µm, implying that the 2.2 µm luminosity is a more robust tracer of stellar mass. Finally, we have presented in an Appendix the predictions of our model for the redshift distributions of galaxies selected at dif- ferent IR fluxes in the Spitzer bands. One significant limitation of our model is that it does not in- clude the effects of AGN. Two effects are relevant here. The first is feedback from AGN on galaxy formation. In several recent galaxy formation models, AGN feedback is invoked to prevent the forma- tion of too many massive galaxies at the present day. In the model presented here, we instead posit feedback from supernova-driven galactic superwinds, which perform a similar role to AGN feed- back in suppressing the formation of very massive galaxies. Both the superwind and AGN feedback models include free parameters which are tuned to give a match to the present-day optical galaxy luminosity function. However, the redshift dependence of the feed- back will be different between our superwind model and the various AGN feedback models, so in general they will all predict different evolution of the galaxy population with redshift. We will investi- gate galaxy evolution in the IR in a model with AGN feedback in a future paper. The second effect of AGN which we have not in- cluded is the emission from AGN and their associated dust tori. In order to compensate for this, we have wherever possible compared our model predictions with observations from which the AGN con- tribution has been subtracted out. This was possible for most of our comparisons of luminosity function evolution. This was not possi- ble for the number counts comparisons, but in this case the contri- bution from AGN is thought (based on observations) to be a small fraction of the total over the flux range explored by Spitzer, even in the mid-IR where the dust tori are the most prominent. We therefore believe that emission from AGN does not seriously affect our con- clusions about the IR evolution of star-forming galaxies. We hope to include AGN emission directly into our models in the future. We have thus shown that Spitzer data provide a stringent test of galaxy formation theory, by probing galaxy evolution, constraining star formation rates and the role of dust to z ∼ 2. We find that an ab initio ΛCDM model gives an acceptable fit to the Spitzer data provided that ∼ 10% of the stars in galaxies today formed in bursts of star formation with a top-heavy IMF. Future facilities like Herschel, SPICA, JWST and ALMA will continue to exploit the valuable information on galaxy formation contained in the IR part of the electromagnetic spectrum. ACKNOWLEDGEMENTS We thank T. Babbedge, K. Caputi, A. Franceschini, E. Le Floch, and P. Perez-Gonzalez, for providing us with their observational data in a convenient form. CMB acknowledges the receipt of a Royal Society University Research Fellowship. CSF is the recip- ient of a Royal Society Wolfson Research Merit Award. This work was also supported by the PPARC rolling grant for extragalactic astronomy and cosmology at Durham. c© 0000 RAS, MNRAS 000, 000–000 20 Lacey et al. REFERENCES Almeida, C., Baugh, C.M., Lacey, C.G, 2007, MNRAS, 376, 1711 Babbedge T. S. R., et al. , 2006, MNRAS, 676 Barnes, J., 1998, in Galaxies: Interactions and Induced Star For- mation, Saas-Fee Advanced Course 26. Lecture Notes 1996. Swiss Society for Astrophysics and Astronomy, XIV, Edited by R. C. Kennicutt, Jr. F. 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S., 1984, Nature 309, Xu, C., Hacking, P.B., Fang, F., Shupe, D.L., Lonsdale, C.J., Lu, N.Y., Helou, G., Stacey, G.J., Ashby, M.L.N., 1998, ApJ, 508, APPENDIX A: REDSHIFT DISTRIBUTIONS In this Appendix, we present some predictions from our standard model for the redshift distributions of galaxies selected at different fluxes in the Spitzer bands. This is principally for completeness, to assist in interpreting data from current surveys, and to assist in planning future surveys based on Spitzer data. The set of redshift distributions at all observed fluxes in principle contains equivalent information to that in the luminosity functions at different wave- lengths and redshifts. However, comparing models with observa- tions via luminosity functions is more physically transparent than making the comparison via redshift distributions, which is why we have presented our results on luminosity functions in the main part of the paper, and why we make only a limited direct comparison with observed redshift distributions in this Appendix. In addition, if one only compares the predicted and observed redshift distribu- tions for galaxies above a single flux limit (e.g. the flux limit of a survey), this has less information than comparing the luminosity functions at different redshifts. We first show in Fig. A1 how the median redshift, and the 10- 90 percentile range, are predicted to change with flux for galax- ies selected in one of the four Spitzer bands 3.6, 8.0, 24 or 70 µm. While at most wavelengths the median redshift is predicted to increase smoothly and monotonically with decreasing flux, this is not true at 24 µm, where there is a bump around Sν ∼ 100µJy. The structure seen for the 24 µm band as compared to the other wavelengths results from different PAH emission features moving through the band with increasing redshift. In Fig. A2, we show the predictions from our standard model for the redshift distributions of galaxies in the four IRAC bands. For each band, we show the redshift distribution for galaxies se- lected to be brighter than Sν > 10µJy in that band. The flux limit Sν > 10µJy has been chosen to match that in the observed deep sample selected at 3.6µm by Franceschini et al. (2006). In each panel, the blue curve shows the predicted dN/dz for all galax- ies, normalized to unit area under the curve, and the red and green curves show the separate contributions of bursting and quiescent galaxies to the total. For 3.6µm, the black line shows the ob- served redshift distribution from Franceschini et al. (2006), which has also been normalized to unit area under the curve. We see that the observed redshift distribution peaks at a slightly higher redshift than in the model. However, the luminosity function evolution de- rived from this same sample is in reasonable agreement with the model, as was already shown in Fig. 4. Franceschini et al. (2006) note that the peak seen in their data at z ∼ 0.8 is partly contributed by large-scale structures in the CDFS field. In Fig. A3, we show predicted redshift distributions for galax- ies selected to be at a set of different fluxes in the four IRAC bands. The curves for the different fluxes are all normalized to have unit area as before, but in this figure the galaxies are selected to be at a particular flux, rather than being brighter than a certain flux. As one would expect, the typical redshift increases as the flux decreases. Figs. A4 and A5 show for the three MIPS bands the equiv- alent of Figs. A2 and A3 for the IRAC bands. In Fig. A4, we show the predicted redshift distributions for galaxies brighter than a particular flux, where this flux limit is taken to be 83 µJy at 24 µm, 10 mJy at 70 µm and 100 mJy at 160 µm. The flux limit at 24 µm has been chosen to match that used in the deep obser- vational samples of Le Floc’h et al. (2005), Perez-Gonzalez et al. (2005) and Caputi et al. (2006a), while the flux limits at 70 µm and 160 µm have been chosen to be roughly 3 times brighter than the source confusion limits in these bands. We see in Fig. A5 that c© 0000 RAS, MNRAS 000, 000–000 http://arxiv.org/abs/0710.0875 22 Lacey et al. Figure A1. Model predictions for the median redshift as a function of flux in four Spitzer bands. (a) 3.6 µm, 8.0 µm, (c) 24 µm, (d) 70 µm. In each panel, the median redshift for galaxies at each flux is shown by a solid line, and the 10- and 90-percentiles are shown by dashed lines. the redshift distributions at 24 µm show much more structure than at other wavelengths. This results from different PAH emission fea- tures moving through the 24 µm band with changing redshift. In Fig. A4(a), we compare the predicted redshift dis- tribution at 24 µm with observational determinations from Perez-Gonzalez et al. (2005) (dashed black line) and Caputi et al. (2006a) (solid black line). The observed distributions have been separately normalized to unit area under the curve, as for the model distribution. Both observed distributions are based pri- marily on photometric redshifts, but the photometric redshifts of Caputi et al. (2006a) are likely to be more accurate than those of Perez-Gonzalez et al. (2005), since the former are based on deeper optical and K-band data than the latter. (Perez-Gonzalez et al. found optical counterparts with BAB . 24.7 or RAB . 23.7 for ∼ 70% of their Sν(24µm) > 83µJy sources, but relied on IRAC fluxes in deriving photo-z’s for the remaining ∼ 30% of their sam- ple. On the other hand, Caputi et al. found K-band counterparts with K(V ega) < 21.5 for 95% of their Sν(24µm) > 80µJy sample, and derived photo-z’s for essentially all of these sources using optical and K-band data alone). Both observed distributions are similar, but the Caputi et al. distribution shows more structure. This is a combination of the effects of more accurate photometric redshifts but also a 9 times smaller survey area, which means that fluctuations due to galaxy clustering are larger. Caputi et al. argue that the separate peaks at z ∼ 0.7 and 1.1 result from large-scale structure, but that the bump at z ∼ 1.9 results from PAH emis- sion features entering the observed 24 µm band. We see that the model also predicts peaks in the redshift distribution at z ∼ 0.3, z ∼ 1 and z ∼ 2, which can be explained by different PAH fea- tures moving through the 24 µm band, although the z ∼ 2 peak is more prominent than is seen in the observational data. Overall, the model redshift distribution at this flux limit is too skewed to high redshift compared to the observations, predicting too few galaxies at z ∼ 0.5− 1, and too many in the peak at z ∼ 2. We investigate further this apparent discrepancy in the 24 µm redshift distribution in Fig. A6, where we show the effects of appar- ent magnitude limits in the R and K-bands on the predicted redshift distributions for Sν(24µm) > 83µJy. In this plot, the redshift distributions are plotted as number per solid angle, without nor- malizing to unit area under the curve. The left and right panels re- spectively have the redshift distributions of Perez-Gonzalez et al. and Caputi et al. overplotted. We concentrate on the comparison c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 23 Figure A2. Predicted galaxy redshift distributions in the four IRAC bands, for galaxies brighter than Sν = 10µJy. (a) 3.6 µm, (b) 4.5 µm, (c) 5.8 µm, and (d) 8.0 µm. The model curves (which all include the effects of dust) are as follows - blue: total; red: ongoing bursts; green: quiescent galaxies. The curves are normalized to unit area under the curve for the total counts. The median (z50) and 10- and 90-percentile (z10, z90) redshifts for the total counts in each band are also given in each panel. For 3.6 µm, the model predictions are compared with observational data from Franceschini et al. (2006) (black dashed line), normalized to unit area as for the models. The error bars plotted on the observational data include Poisson errors only. with Caputi et al. , since this has the simpler sample selection and more accurate redshifts. The model prediction for K < 21.5 (which is the magnitude limit used by Caputi et al. ) is shown by the short-dashed blue line, while the prediction with no limit on the K-magnitude is shown by the solid blue line. The model dN/dz with no limit on the K magnitude is most discrepant with the Caputi et al. data at z ∼ 2, where it predicts ∼ 2 times too many galaxies. This is directly related to the fact that the pre- dicted luminosity function at z = 2 at rest-frame wavelength 8 µm (corresponding to observed wavelength 24 µm) and lumi- nosity ∼ 1011L⊙ is also ∼ 2 times too high compared to what Caputi et al. estimate from their data, as shown in Fig. 5(c). When the effect of the K < 21.5 limit is included, the predicted redshift distribution is closer to the observational data, but only 58% of the model galaxies are brighter than this K-band magnitude limit, as against 95% in the observed sample of Caputi et al. . We conclude that the main reason for the discrepancy between the predicted and observed redshift distributions at 24 µm is that the model predicts a rest-frame 8 µm luminosity function at z ∼ 2 which is somewhat too high at luminosities ∼ 1011L⊙, even though it reproduces quite well the general features of the evolution of the mid-IR luminosity function. c© 0000 RAS, MNRAS 000, 000–000 24 Lacey et al. Figure A3. Predicted galaxy redshift distributions in the four IRAC bands, for different fluxes. (a) 3.6 µm, (b) 4.5 µm, (c) 5.8 µm, and (d) 8.0 µm. In this figure, the redshift distributions are for galaxies at a particular flux. Predictions are shown for fluxes Sν = 0.1, 1, 10, 100 and 1000 µJy, as shown in the key. In all cases, the model curves are normalized to unit area, and include the effects of dust. c© 0000 RAS, MNRAS 000, 000–000 Galaxy evolution in the IR 25 Figure A4. Predicted galaxy redshift distributions in the three MIPS bands, for galaxies brighter than a specified flux. (a) 24 µm, Sν > 83µJy, (b) 70 µm, Sν > 10mJy, and (c) 160 µm, Sν > 100mJy. The model curves are as follows - blue: total; red: ongoing bursts; green: quiescent galax- ies. The curves are normalized to unit area under the curve for the total counts. The median (z50) and 10- and 90-percentile (z10, z90) redshifts for the total counts in each band are also given in each panel. For 24 µm, the model predictions are compared with observational data from Caputi et al. (2006a) (solid black line) and Perez-Gonzalez et al. (2005) (dashed black line), normalized to unit area as for the models. The error bars plotted on the observational data include Poisson errors only for Caputi et al. , but also include errors in photometric redshifts for Perez-Gonzalez et al. Figure A5. Predicted galaxy redshift distributions in the three MIPS bands, for different fluxes. (a) 24 µm, (b) 70 µm, and (c) 160 µm. In this figure, the redshift distributions are for galaxies at a particular flux, as shown in the key in each panel. In all cases, the model curves are normalized to unit area, and include the effects of dust. c© 0000 RAS, MNRAS 000, 000–000 26 Lacey et al. Figure A6. Predicted redshift distributions at 24µm, showing the effects of optical or near-IR magnitude limits. Model galaxies are selected with Sν > 83µJy together with the optical/NIR magnitude limits as shown in the key. The fraction of 24 µm sources brighter than each magnitude limit is also given. (a) R-band magnitude limit. The observed redshift distribution from Perez-Gonzalez et al. (2005) is overplotted in black. Note Le Floc’h et al. (2005) used R < 24 and obtained 54% completeness. (b) K-band magnitude limit. The observed redshift distribution from Caputi et al. (2006a) (with K < 21.5) is overplotted. Magnitudes are on the Vega system. c© 0000 RAS, MNRAS 000, 000–000 Introduction Model GALFORM galaxy formation model GRASIL model for stellar and dust emission Choice of parameters in the GALFORM+GRASIL model Number counts Evolution of the galaxy luminosity function Evolution of the galaxy luminosity function at 3-8 m Evolution of the galaxy luminosity function at 12-24 m Evolution of the galaxy luminosity function at 70-160 m Evolution of the total mid+far-IR luminosity function Inferring stellar masses and star formation rates from Spitzer data Conclusions Redshift distributions

We present predictions for the evolution of the galaxy luminosity function, number counts and redshift distributions in the IR based on the Lambda-CDM cosmological model. We use the combined GALFORM semi-analytical galaxy formation model and GRASIL spectrophotometric code to compute galaxy SEDs including the reprocessing of radiation by dust. The model, which is the same as that in Baugh et al (2005), assumes two different IMFs: a normal solar neighbourhood IMF for quiescent star formation in disks, and a very top-heavy IMF in starbursts triggered by galaxy mergers. We have shown previously that the top-heavy IMF seems to be necessary to explain the number counts of faint sub-mm galaxies. We compare the model with observational data from the SPITZER Space Telescope, with the model parameters fixed at values chosen before SPITZER data became available. We find that the model matches the observed evolution in the IR remarkably well over the whole range of wavelengths probed by SPITZER. In particular, the SPITZER data show that there is strong evolution in the mid-IR galaxy luminosity function over the redshift range z ~ 0-2, and this is reproduced by our model without requiring any adjustment of parameters. On the other hand, a model with a normal IMF in starbursts predicts far too little evolution in the mid-IR luminosity function, and is therefore excluded.

Introduction Model GALFORM galaxy formation model GRASIL model for stellar and dust emission Choice of parameters in the GALFORM+GRASIL model Number counts Evolution of the galaxy luminosity function Evolution of the galaxy luminosity function at 3-8 m Evolution of the galaxy luminosity function at 12-24 m Evolution of the galaxy luminosity function at 70-160 m Evolution of the total mid+far-IR luminosity function Inferring stellar masses and star formation rates from Spitzer data Conclusions Redshift distributions

Use of Triangular Elements for Nearly Exact BEM Solutions Supratik Mukhopadhyay, Nayana Majumdar INO Section, Saha Institute of Nuclear Physics 1/AF, Sector 1, Bidhannagar, Kolkata 700064, WB, India supratik.mukhopadhyay@saha.ac.in, nayana.majumdar@saha.ac.in Abstract A library of C functions yielding exact solutions of potential and flux influences due to uniform surface distribution of singularities on flat triangular and rectangular elements has been developed. This library, ISLES, has been used to develop the neBEM solver that is both precise and fast in solving a wide range of problems of scientific and technological interest. Here we present the exact expressions proposed for computing the influence of uniform singularity distributions on triangular elements and illustrate their accuracy. We also present a study concerning the time taken to evaluate these long and complicated expressions vis a vis that spent in carrying out simple quadratures. Finally, we solve a classic benchmark problem in electrostatics, namely, estimation of the capacitance of a unit square plate raised to unit volt. For this problem, we present the estimated values of capacitance and compare them successfully with some of the most accurate results available in the literature. In addition, we present the variation of the charge density close to the corner of the plate for various degrees of discretization. The variations are found to be smooth and converging. This is in clear contrast to the criticism commonly leveled against usual BEM solvers. Keywords: Boundary element method, triangular element, potential, flux, unit square plate, charge density, capacitance. 1 Introduction One of the elegant methods for solving the Laplace / Poisson equations (normally an integral expression of the inverse square law) is to set up the Boundary Integral Equations (BIE) which lead to the moderately popular Boundary Element Method (BEM). In the forward version of the BEM, surfaces of a given geometry are replaced by a distribution of point singularities such as source / dipole of unknown strengths. The strengths of these singularities are obtained through the satisfaction of a given set of boundary conditions that can be Dirichlet, Neumann or of the Robin type. The numerical implementation requires considerable care [1] because it involves evaluation of singular (weak, strong and hyper) integrals. Some of the notable two- dimensional and three-dimensional approaches are [1] and [2, 3, 4, 5, 6] and the references in these papers. Despite a large body of literature, closed form analytic expressions for computing the effects of distributed singularities are rare [7, 8] and complicated to implement. Thus, for solving difficult but realistic problems involving, for example, sharp edges and corners or thin http://arxiv.org/abs/0704.1563v1 elements, introduction of complicated mathematics and special formulations becomes a necessity [9, 10]. These drawbacks are some of the major reasons behind the relative unpopularity of the BEM despite its significant advantages over domain approaches such as the finite-difference and finite-element methods (FDM and FEM) while solving non-dissipative problems [11]. It is well- understood that most of the difficulties in the available BEM solvers stem from the assumption of nodal concentration of singularities which leads to various mathematical difficulties and to the infamous numerical boundary layers [9, 12]. The Inverse Square Law Exact Solutions (ISLES) library, in contrast, is capable of truly modeling the effect of distributed singularities precisely and, thus, its application is not limited by the proximity of other singular surfaces or their curvature or their size and aspect ratio. The library consists of exact solutions for both potential and flux due to uniform distribution of singularity on flat rectangular and triangular elements. While the rectangular element can be of any arbitrary size [13, 14], the triangular element can be a right angled triangle of arbitrary size [15]. Since any real geometry can be represented through elements of the above two types (or by the triangular type alone), this library can help in developing solvers capable of solving three-dimensional potential problems for any geometry. It may be noted here that any non-right-angled triangle can be easily decomposed in to two right-angled triangles. Thus, the right-angled triangles considered here, in fact, can take care of any three-dimensional geometry. Several difficulties were faced in developing the library which arose due to the various terms of the integrals and also from the approximate nature of computation in digital computers. In this paper, we have discussed these difficulties, solutions adopted at present and possible ways of future improvement. The classic benchmark problem of estimating the capacitance of a unit square plate raised to unit volt has been addressed using a solver based on ISLES, namely, the nearly exact BEM (neBEM) solver. Results obtained using neBEM have been compared with other precise results available in the literature. The comparison clearly indicates the excellent precision and efficiency achievable using ISLES and neBEM. In addition, we have also presented the variation of charge density close to the corner of the square plate. Usually, using BEM, it is difficult to obtain physically consistent results close to these geometric singularities. Wild variations in the magni- tude of the charge density has been observed with the change in the degree of discretization, the reason once again being associated with the nodal model of singularities [16]. In contrast, using neBEM, we have obtained very smooth variation close to the corner. Moreover, the magnitudes of the charge density have been found to be consistently converging to physically realistic values. These results clearly indicate that since the foundation expressions of the solver are exact, it is possible to find the potential and flux accurately in the complete physical domain, includ- ing the critical near-field domain using neBEM. In addition, since singularities are no longer assumed to be nodal and we have the exact expressions for potential and flux throughout the physical domain, the boundary conditions no longer need to be satisfied at special points such as the centroid of an element. Although consequences of this considerable advantage is still under study, it is expected that this feature will allow neBEM to yield accurate estimates for problems involving corners and edges that are very important in a large number of scientific and technological studies. It should be noted here that the exact expressions for triangular elements consist of a signifi- cantly larger number of mathematical operations than those for rectangular elements. Thus, for the solver, it is more economical if we use a mixed mesh of rectangular and triangular elements using rectangular elements as much as possible. However, in the present work, we have inten- tionally concentrated on the performance of the triangular elements and results shown here are those obtained using only triangular elements. 2 Exact Solutions The expressions for potential and flux at a point (X,Y,Z) in free space due to uniform source distributed on a rectangular flat surface having corners situated at (x1, z1) and (x2, z2) has been presented, validated and used in [13, 14] and, thus, is not being repeated here. Here, we present the exact expressions necessary to compute the potential and flux due to a right-angled triangular element of arbitrary size, as shown in Fig.1. It may be noted here that the length in the X direction has been normalized, while that in the Z direction has been allowed to be of any arbitrary magnitude, zM . From the figure, it is easy to see that in order to find out the influence due to triangular element, we have imposed another restriction, namely, the necessity that the X and Z axes coincide with the perpendicular sides of the right-angled triangle. Both these restrictions are trivial and can be taken care of by carrying out suitable scaling and appropriate vector transformations. It may be noted here that closed-form expressions for the influence of rectangular and triangular elements having uniform singularity distributions have been previously presented in [7, 8]. However, in these works, the expressions presented are quite complicated and difficult to implement. In [13] and in the present work, the expressions we have presented are lengthy, but completely straight-forward. As a result, the implementation issues of the present expressions, in terms of the development of the ISLES library and the neBEM solver are managed quite easily. Figure 1: Right-angled triangular element with x-length 1 and an arbitrary z-length, zM ; P is the point where the influence (potential and flux) is being computed It is easy to show that the influence (potential) at a point P (X,Y,Z) due to uniform source distributed on a right-angled triangular element as depicted in Fig.1 can be represented as a multiple of φ(X,Y,Z) = ∫ z(x) dx dz (X − x)2 + Y 2 + (Z − z)2 in which we have assumed that x1 = 0, z1 = 0, x2 = 1 and z2 = zM , as shown in the geometry of the triangular element. The closed-form expression for the potential has been obtained using symbolic integration [17] which was subsequently simplified through substantial effort. It is found to be significantly more complicated in comparison to the expression for rectangular elements presented in [13] and can be written as ( (zMY 2 −XG)(LP1 + LM1 − LP2 − LM2) + i |Y | (zMX +G)(LP1 − LM1 − LP2 + LM2) −S1X(tanh−1( R1 + iI1 D11|Z| ) + tanh−1( R1 − iI1 D11|Z| )− tanh−1( R1 + iI2 D21|Z| )− tanh−1( R1 − iI2 D21|Z| +iS1|Y |(tanh−1( R1 + iI1 D11|Z| )− tanh−1(R1 − iI1 D11|Z| )− tanh−1(R1 + iI2 D21|Z| ) + tanh−1( R1 − iI2 D21|Z| 1 + zM 2 log ( 1 + zM 2D12 − E1√ 1 + zM 2D21 − E2 ) + 2Z log D21 −X + 1 D11 −X ) + C (2) where, D11 = (X − x1)2 + Y 2 + (Z − z1)2; D12 = (X − x1)2 + Y 2 + (Z − z2)2 D21 = (X − x2)2 + Y 2 + (Z − z1)2; I1 = (X − x1) |Y | ; I2 = (X − x2) |Y | S1 = sign(z1 − Z); R1 = Y 2 + (Z − z1)2 E1 = (X + zM 2 − zMZ); E2 = (X − 1− zMZ), G = zM (X − 1) + Z; H1 = Y 2 +G(Z − zM ); H2 = Y 2 +GZ LP1 = G− izM |Y | (H1 +GD12) + i|Y |(E1 − izMD12) −X + i|Y | LM1 = G+ izM |Y | (H1 +GD12)− i|Y |(E1 − izMD12) −X − i|Y | LP2 = G− izM |Y | (H2 +GD21) + i|Y |(E2 − izMD21) 1−X + i|Y | LM2 = G+ izM |Y | (H2 +GD21)− i|Y |(E2 − izMD21) 1−X − i|Y | and C denotes a constant of integration. Similarly, the flux components due to the above singularity distribution can also be repre- sented through closed-form expressions as shown below: Fx = − ( (G)(LP1 + LM1 − LP2 − LM2)− i |Y | (zM )(LP1 − LM1 − LP2 + LM2) +S1(tanh R1 + iI1 D11|Z| ) + tanh−1( R1 − iI1 D11|Z| )− tanh−1( R1 + iI2 D21|Z| )− tanh−1( R1 − iI2 D21|Z| 1 + zM 2 log ( 1 + zM 2D12 − E1√ 1 + zM 2D21 − E2 ) ) + C (3) Fy = − ( (2zMY )(LP1 + LM1 − LP2 − LM2) + i |Y | (Sn(Y )G)(LP1 − LM1 − LP2 + LM2) +iS1Sn(Y )(tanh R1 + iI1 D11|Z| )− tanh−1( R1 − iI1 D11|Z| )− tanh−1( R1 + iI2 D21|Z| ) + tanh−1( R1 − iI2 D21|Z| )) ) + C Fz = − 1 + zM log ( 1 + zM 2D21 − E2√ 1 + zM 2D12 − E1 ) + log D11 −X D21 −X + 1 ) + C (5) where Sn(Y ) implies the sign of the Y-coordinate and C indicates constants of integrations. It is to be noted that the constants of different integrations are not the same. These expression are expected to be useful in the mathematical modeling of physical processes governed by the inverse square laws. Being exact and valid throughout the physical domain, they can be used to formulate versatile solvers to solve multi-scale multi-physics problems governed by the Laplace / Poisson equations involving Dirichlet, Neumann or Robin boundary conditions. 3 Development of the ISLES library Due to the tremendous popularity of the C language we have written the codes in the C pro- gramming language. However, it should be quite simple to translate the library to other popular languages such as FORTRAN or C++, since no special feature of the C language has been used to develop the codes. 3.1 Validation of the exact expressions The expressions for the rectangular element have been validated in detail in [13]. Here, we present the results for triangular elements in fair detail. In Fig.2, we have presented a comparison of potentials evaluated for a unit triangular element by using the exact expressions, as well as by using numerical quadrature of high accuracy. The two results are found to compare very well throughout. Please note that contours have been obtained on the plane of the element, and thus, represents a rather critical situation. Similarly, Fig.3 shows a comparison between the results obtained using closed-form expressions for flux and those obtained using numerical quadrature. The flux considered here is in the Y direction and is along a line beginning from (−2,−2,−2) and ending at (2, 2, 2). The comparison shows the commendable accuracy expected from closed form expressions. In Fig.4(a) and 4(b), the surface plots of potential on the element plane (XZ plane) and Y -flux on the XY plane have been presented from which the expected significant increase in potential and sharp change in the flux value on the element is observed. Thus, by using a small fraction of computational resources in comparison to those consumed in numerical quadratures, ISLES can compute the exact value of potential and flux for singularities distributed on triangular elements. Figure 2: Potential contours on a triangular element computed using exact expressions and by numerical quadrature 3.2 Near-field performance In order to emphasize the accuracy of ISLES, we have considered the following severe situations in the near-field region in which it is observed that the quadratures can match the accuracy of ISLES only when a high degree of discretization is used. Please note that in these cases, the value of zM has been considered to be 10. In Fig.5 we have presented the variation of potential along a line on the element surface running parallel to the Z-axis of the triangular element (see Fig.1) and going through the centroid of the element. It is observed that results obtained using even a 100× 100 quadrature is quite unacceptable. In fact, by zooming on to the image, it can -4 -3 -2 -1 0 1 2 3 4 Distance (a.u.) 100X100 Exact Figure 3: Comparison of flux (in the Y direction) as computed by ISLES and numerical quadra- ture along a diagonal line be found that only the maximum discretization yields results that match closely to the exact solution. It may be noted here that the potential is a relatively easier property to compute. The difficulty of achieving accurate flux estimates is illustrated in the two following figures. The variation of flux in the X-direction along the same line as used in Fig.5 has been presented in Fig.6. Similarly, variation of Y -flux along a diagonal line (beginning at (-10,-10,-10) and ending at (10,10,10) and piercing the element at the centroid) has been presented in Fig.7. From these figures we see that the flux values obtained using the quadrature are always inaccurate even if the discretization is as high as 100 × 100. We also observe that the estimates are locally inaccurate despite the use of very high amount of discretization (200× 200 or 500). Specifically, in the latter figure, even the highest discretization can not match the exact values at the peak, while in the former only the highest one can correctly emulate the sharp change in the flux value. It is also heartening to note that the values from the quadrature using higher amount of discretization consistently converge towards the ISLES values. 3.3 Far field performance It is expected that beyond a certain distance, the effect of the singularity distribution can be considered to be the same as that of a centroidally concentrated singularity or a simple quadrature. The optimized amount of discretization to be used for the quadrature can be determined from a study of the speed of execution of each of the functions in the library and has been presented separately in a following sub-section. If we plan to replace the exact expressions by quadratures (in order to reduce the computational expenses, presumably) beyond a certain given distance, the quadrature should necessarily be efficient enough to justify the replacement. While standard but more elaborate algorithms similar to the fast multipole method (FMM) [18] along with the GMRES [19] matrix solver can lead to further of computational efficiency, the (a) Potential surface (b) Flux surface Figure 4: (a) Potential surface due to a triangular source distribution on the element plane, (b) Flux (in the Y direction) surface due to a triangular source distribution on the XY plane at Z=0 simple approach as outlined above can help in reducing a fair amount of computational effort. In the following, we present the results of numerical experiments that help us in determining the far-field performance of the exact expressions and quadratures of various degrees that, in turn, help us in choosing the more efficient approach for a desired level of accuracy. In Fig.8 we have presented potential values obtained using the exact approach, 100 × 100, 10 × 10 and no discretization, i.e., the usual BEM approximation while using the zeroth order piecewise uniform charge density assumption. The potentials are computed along a diagonal line running from (-1000, -1000, -1000) to (1000, 1000, 1000) which pierces a triangular element of zM = 10. It can be seen that results obtained using the usual BEM approach yields inaccurate results as we move closer than distances of 10 units, while the 10 × 10 discretization yields acceptable results up to a distance of 1.0 unit. In order to visualize the errors incurred due to the use of quadratures, we have plotted Fig.9 where the errors incurred (normalized with respect to the exact value) have been plotted. From this figure we can conclude that for the given diagonal line, the error due to the usual BEM approximation falls below 1% if the distance is larger than 20 units while for the simple 10×10 discretization, it is 2 units. It may be mentioned here that along the axes the error turns out to be significantly more [13] and the limits need to be effectively doubled to achieve the accuracy for all cases possible. Thus, for achieving 1% accuracy, the usual BEM is satisfactory only if the distance of the influenced point is five times the longer side of an element. Please note here that the error drops to 1 out of 106 as the distance becomes fifty times the longer side. Besides proving that the exact expressions work equally well in the near-field as well as the far-field, this fact justifies the usual BEM approach for much of the computational domain leading to substantial savings in computational expenses. The accuracy of the exact expressions used in the ISLES library is confirmed from the above comparisons. However, there are several other important issues related to the development of the library that are discussed below briefly. 3.4 Evaluation of the component functions Many of the irrational and transcendental functions have domains and ranges in which they are defined. Moreover, they are often multiply defined in the complex domain; for example, there 0 2 4 6 8 10 Distance along Z 10by10 50by50 100by100 200by200 500by500 IslesLib AppFlag Figure 5: Variation of potential along a centroidal line on the XZ plane parallel to the Z axis for a triangular element: comparison among values obtained using the exact expressions and numerical quadratures are an infinite number of complex values for the logarithm function. In such cases, only one principal value must be returned by the function. In general, such values cannot be chosen so as to make the range continuous and thus, lines in the domain called branch cuts need to be defined, which in turn define the discontinuities in the range. While evaluating expressions such as the ones displayed in eqns.(2 - 5) a number of such problems are expected to occur. However, when the expressions are analyzed at critical locations such as the corners and edges of the element, it is observed that the terms likely to create difficulties while evaluating potentials are either cancelled out or are themselves multiplied by zero. As a result, at these locations of likely geometric and mathematical singularities, the solution behaves nicely. However, the same is not true for the expressions related to the flux components. For these, we have to deal with branch-cut problems in relation to tanh−1 and problems related to the evaluation of log(0). It should be noted here that these singularities associated to the edges and corners of the elements are of the weak type and it is expected that exact evaluation of these terms as well will be possible through further work. However, difficulties of a different nature crop up in these calculations which can be linked directly to the limitation of the computer itself, namely, round-off errors [15]. These errors can lead to severe problems while handling multi-scale problems such as those described in [13]. A completely different approach is necessary to cope up with these difficulties, for example, the use of extended range arithmetic [20], interval arithmetic [21] or the use of specialized libraries such as the CORE library of the Exact Geometric Computation (EGC) initiative [22]. In the present version of ISLES, a simple approach has been implemented which sets a lower limit to various distance values. Below this value, the distance is considered to be zero. Plan of future improvements in this regard has been kept at a high priority. 5 5.5 6 6.5 7 7.5 8 Distance along Z 10by10 50by50 100by100 200by200 500by500 IslesLib AppFlag Figure 6: Variation of flux in the X direction along a line on the XZ plane parallel to the Z axis for a triangular element: comparison among values obtained using the exact expressions and numerical quadratures 3.4.1 Algorithm As discussed above, there are possibilities of facing problems while using the exact expressions which may be due to the functions being evaluated or due to round-off errors leading to erro- neous results. Moreover, despite providing many checks during the computation there is finite possibility of ending up with a wrong value of a property indicated by its being Nan or inf or potential due to unit positive singularity strength turning out to be negative. In order to maintain the robustness of the library, we have tried to keep checks on the intermediate and final values during the course of the computation. When the results are found to be unsatisfactory, unphysical, we have re-estimated the results by using numerical quadrature and kept a track of the cause by raising a unique approximation flag which is specific for a problem. As a result, the steps for the calculation for a property can be written as follows: • Get the required inputs - geometry of the element and the position where the effect needs to be evaluated; Check whether the element size and distances are large enough so that the results do not suffer from round-off errors. • Check whether the location coincides with one of the special ones, such as corners or edges. • Evaluate the necessary expressions in accordance with the foregoing results. If necessary, consider each term in the expressions separately to sort out difficulties related to singular- ities, branch-cuts or round-off errors. Note that if the multiplier is zero, rest of the term does not need evaluation. • If direct evaluation of the expressions fail, raise a unique approximation flag specific to this problem and term and return the value of the property by using numerical quadrature. -1 -0.5 0 0.5 1 1.5 Distance along diagonal 10by10 50by50 100by100 200by200 500by500 IslesLib AppFlag Figure 7: Comparison of flux (in the Y direction) along a diagonal line piercing the triangular element at the centroid: comparison among values obtained using the exact expressions and numerical quadratures • Compute all the terms and find the final value, Check whether the final value is a number and physically meaningful. If not satisfied, recompute the result using numerical quadra- ture and raise the relevant approximation flag. 3.5 Speed of execution The time taken to compute the potential and flux is an important parameter related to the overall computational efficiency of the codes. This is true despite the fact that, in a typical simulation, the time taken to solve the system of algebraic equation is far greater than the time taken to build the influence coefficient matrix and post-processing. Moreover, the amount of time taken to solve the system of equations tend to increase at a greater rate than the time taken to complete the other two. It should be mentioned here that the time taken in each of these steps can vary to a significant amount depending on the algorithm of the solver. In the present case, the system of equations has been solved using lower upper decomposition using the well known Crout’s partial pivoting. Although this method is known to be very rugged and accurate, it is not efficient as far as number of arithmetic operations, and thus, time is concerned. It is also possible to reduce the time taken to pre-process (generation of mesh and creation of influence matrices), solve the system of algebraic equations and that for post-process (computation of potential and flux at the required locations) can be significantly reduced by adopting faster algorithms, including those involving parallelization. In order to optimize the time taken to generate the influence coefficient matrix and that to carry out the post-processing, we carried out a small numerical study to determine the amount of time taken to complete the various functions being used in ISLES, especially those being used to evaluate the exact expressions and those being used to carry out the quadratures. The results 1 10 100 Exact Usual BEM 10 X 10 100 X 100 Figure 8: Potential along a diagonal through the triangular element computed using exact, 100 × 100, 10 × 10 and usual BEM approach of the study (which was carried out using the linux system command gprof ) has been presented in the following Table1. Table 1: Time taken to evaluate exact expression and various quadratures Method Exact Usual BEM 10× 10 100 × 100 500× 500 Time 0.8 µs 25 ns 1 µs 200 µs 5 ms Please note that the numbers presented in this table are representative and are likely to have statistical fluctuations. However, despite the fluctuations, it may be safely concluded that a quadrature having only 10 × 10 discretization is already consuming time that is comparable to that needed exact evaluation. Thus, the exact expressions, despite their complexity, are extremely efficient in the near-field which can be considered at least as large as 0.5 times the larger side of a triangular element (please refer to Fig.9). In making this statement, we have as- sumed that the required accuracy for generating the influence coefficient matrix and subsequent potential and flux calculations is 1%. This may not be acceptable at all under many practical circumstances, in which case the near-field would imply a larger volume. 3.6 Salient features of ISLES Development of usual BEM solvers are dependent on the two following assumptions: • While computing the influences of the singularities, the singularities are modeled by a sum of known basis functions with constant unknown coefficients. For example, in the constant element approach, the singularities are assumed to be concentrated at the centroid of the element, except for special cases such as self influence. This becomes necessary because 1e-07 1e-06 1e-05 1e-04 0.001 0.01 1 10 100 1000 100 X 100 10 X 10 Usual BEM Figure 9: Error along a diagonal through the triangular element computed using 100 × 100, 10× 10 and usual BEM approach closed form expressions for the influences are not, in general, available for surface elements. An approximate and computationally rather expensive way of circumventing this limitation is to use numerical integration over each element or to use linear or higher order basis functions. • The strengths of the singularities are solved depending upon the boundary conditions, which, in turn, are modeled by the shape functions. For example, in the constant element approach, it is assumed that it is sufficient to satisfy the boundary conditions at the centroids of the elements. In this approach, the position of the singularity and the point where the boundary condition is satisfied for a given element usually matches and is called the collocation point. The first (and possibly, the more damaging) approximation for BEM solvers can be relaxed by using ISLES and can be restated as, • The singularities distributed on the boundary elements are assumed to be uniform on a particular element. The strength of the singularity may change from element to element. This improvement turns out to be very significant as demonstrated in the following section and some of our other studies involving microelectromechanical systems (MEMS) and gas detectors for nuclear applications [13, 14]. Some of the advantages of using ISLES are itemized below: • For a given level of discretization, the estimates are more accurate, • Effective efficiency of the solver improves, as a result, • Large variation of length-scales, aspect ratios can be tackled, • Thinness of members or nearness of surfaces does not pose any problem, • Curvature has no detrimental effect on the solution, • The boundary condition can be satisfied anywhere on the elements, i.e., points other than the centroidal points can be easily used, if necessary (for a corner problem, may be), • The same formulation, library and solver is expected to work in majority of physical situations. As a result, the necessity for specialized formulations of BEM can be greatly minimized. 4 Capacitance of a unit square plate - a classic benchmark prob- Using the neBEM solver, we have computed the capacitance of a unit square conducting plate raised to a unit volt. This problem is still considered to be one of the major unsolved problems of electrostatic theory [23, 8, 26, 16] and no analytical solution for this problem has been obtained so far. The capacitance value estimated by the present method has been compared with very accurate results available in the literature (using BEM and other methods). The results obtained using the neBEM solver is found to be among the most accurate ones available till date as shown in Table.2. Please note that we have not invoked symmetry or used extrapolation techniques to arrive at our result presented in the table. Table 2: Comparison of capacitance values Reference Method Capacitance (pF) / 4 πǫ0 [23] Surface Charge 0.3607 [24] Surface Charge 0.362 [25] Surface Charge 0.367 [8] Refined Surface Charge 0.3667892 ± 1.1× 10−6 and Extrapolation [26] Refined Boundary Element 0.3667874 ± 1× 10−7 and Extrapolation [27] Numerical Path Integration 0.36684 [16] Random Walk 0.36 ± 0.01 This work neBEM 0.3660587 Finally, we consider the corner problem related to the electrostatics of the above conducting plate. Problems of this nature are considered to be challenging for any numerical tool and especially so for the BEM approach. The inadequacy of the BEM approach, especially in solving the present problem, has been mentioned even quite recently [16] in which it has been correctly mentioned that since the method can not extend its mathematical model to include the edges and corners in reality, it is unlikely that it will ever succeed in modeling the edge / corner singularities correctly. As a result, with change in discretization, the properties near these geometric singularities are expected to oscillate significantly leading to erroneous results. However, as discussed above, the neBEM does extend its singularities distributed on the surface elements right till an edge or a corner. Moreover, using neBEM, it is also possible to satisfy the boundary conditions (both potential and flux) as close to the edge / corner as is required. In fact, it should be possible to specify the potentials right at the edge / corner. In the following study, we have presented estimates of charge density very close the flat plate corner as obtained using neBEM. Please note that the boundary conditions have been satisfied at the centroids of each element although we plan to carry out detailed studies of changing the position of these points, especially in relation to problems involving edges / corners. In Fig.10, charge densities very close to the corner of the flat plate estimated by neBEM using various amounts of discretization have been presented. It can be seen that each curve follows the same general trend, does not suffer from any oscillation and seems to be converging to a single curve. This is true despite the fact that there has been almost an order of magnitude variation in the element lengths. Finally, in Fig.11, we present a least-square fitted straight line matching the charge density as obtained the highest discretization in this study. It is found that the slope of the straight line is 0.713567, which compares very well with both old and recent estimates of 0.7034 [28, 29]. This is despite the fact that here we have used a relatively coarse discretization near the corner. It should be mentioned that none of the earlier references cited here used the BEM approach. While the former used a singular perturbation technique, the latter used a diffusion based Monte- Carlo method. Thus, it is extremely encouraging to note that using the neBEM approach, we have been able to match the accuracy of these sophisticated techniques. 0 0.02 0.04 0.06 0.08 0.1 Distance from corner 0.0182 0.0091 0.0067 0.0047 0.0047 Figure 10: Corner charge density estimated by neBEM using various sizes of triangular elements 5 Conclusion An efficient and robust library for solving potential problems in a large variety of science and engineering problems has been developed. Exact closed-form expressions used to develop ISLES have been validated throughout the physical domain (including the critical near-field region) -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 log(r) neBEM Fitted line Figure 11: Variation of charge density with increasing distance from the corner of the unit square plate and a least-square fitted straight line: slope of the fitted line is 0.713567 by comparing these results with results obtained using numerical quadrature of high accuracy. Algorithmic aspects of this development have also been touched upon. A classic benchmark problem of electrostatics has been successfully simulated to very high precision. Charge density values at critical geometric locations like corners have been found to be numerically stable and physically acceptable. Several advantages over usual BEM solvers and other specialized BEM solvers have been briefly mentioned. Work is under way to make the code more robust and efficient through the implementation of more efficient algorithms and parallelization. Acknowledgements We would like to thank Professor Bikas Sinha, Director, SINP and Professor Sudeb Bhat- tacharya, Head, INO Section, SINP for their support and encouragement during the course of this work. References [1] Nagarajan, A., Mukherjee, S., 1993, “A mapping method for numerical evaluation of two- dimensional integrals with 1/r singularity”, Computational Mechanics, 12, pp.19-26. [2] Cruse, T.A., 1969, “Numerical solutions in three dimensional elastostatics”, ( Int. J. Solids Struct., 5, pp.1259-1274. [3] Kutt, H.R., 1975, “The numerical evaluation of principal value integrals by finite-part integration”, Numer. Math., 24, pp.205-210. [4] Lachat, J.C., Watson, J.O., 1976, “Effective numerical treatment of boundary integral equations: A formulation for three dimensional elastostatics”, Int. J. Numer. Meth. Eng., 10, pp.991-1005. [5] Srivastava, R., Contractor, D.N., 1992, “Efficient evaluation of integrals in three- dimensional boundary element method using linear shape functions over plane triangular elements”, Appl. Math. Modelling, 16, pp.282-290. [6] Carini, A., Salvadori, A., 2002, “Analytical integration in 3D BEM: preliminaries”, Com- putational Mechanics, 18, pp.177-185. [7] Newman, J.N., 1986, “Distributions of sources and normal dipoles over a quadrilateral panel”, Jour. of Engg. Math., 20, pp.113-126. [8] Goto, E., Shi, Y. and Yoshida, N., 1992, “Extrapolated surface charge method for capacity calculation of polygons and polyhedra”, Jour. of Comput. Phys., 100, pp.105-115. [9] Chyuan, S-W., Liao, Y-S. and Chen, J-T., 2004, “An efficient technique for solving the arbitrarily multilayered electrostatic problems with singularity arising from a degenerate boundary”, Semicond. Sci. Technol., 19, R47-R58. [10] Bao, Z., Mukherjee, S., 2004, “Electrostatic BEM for MEMS with thin conducting plates and shells”, Eng Analysis Boun Elem, 28, pp.1427-1435. [11] Mukhopadhyay, S., Majumdar, N., 2006, “Effect of finite dimensions on the electric field configuration of cylindrical proportional counters”, IEEE Trans Nucl Sci, 53, pp.539-543. [12] Sladek, V., Sladek, J., 1991, “Elimination of the boundary layer effect in BEM computation of stresses”, Comm. Appl. Num. Meth., 7, pp.539-550. [13] Mukhopadhyay, S., Majumdar, N., 2006, “Computation of 3D MEMS electrostatics using a nearly exact BEM solver”, Eng Anal Boundary Elem, 30, pp.687-696. [14] Majumdar, N., Mukhopadhyay, S., 2006, “Simulation of three-dimensional electrostatic field configuration in wire chambers: A novel approach”, Nucl. Instrum. Meth. Phys. Res. A, 566, pp.489-494. [15] Mukhopadhyay, S., Majumdar, N., 2007, “Use of rectangular and triangular elements for nearly exact BEM solutions”, Emerging Mechanical Technology - Macro to Nano, Research Publishing Services, Chennai, India, pp.107-114 (ISBN: 81-904262-8-1). [16] Wintle, H.J., 2004, “The capacitance of the cube and square plate by random walk meth- ods”, J Electrostatics, 62 pp.51-62. [17] Etter, D.M., 1997, Engineering Problem Solving with MatLab, Prentice Hall, International, Inc., New Jersey 07458, USA. [18] Greengard, L., Rokhlin, V., 1987, “A fast algorithm for particle simulation”, Journal of Computational Physics, 73 pp.325-348. [19] Saad, Y., Schultz, M., 1986, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems”, SIAM J. Sci. Statist. Comput., 7 pp.856-869. [20] Smith, J.M., Olver, F.W., Lozier, D.W., 1981, “Extended-Range Arithmetic and Normal- ized Legendre Polynomial”, ACM Transactions on Mathematical Software, 7, pp.93-105. [21] Alefeld, G., Herzberger, J., 1983, Introduction to interval analysis, Academic Press. [22] http://cs.nyu.edu/exact/ [23] Maxwell, J.C., 1878, Electrical Research of the Honorable Henry Cavendish, p.426, Cam- bridge University Press, Cambridge, UK. [24] Reitan, D.K., Higgins, R.J., 1957, “Accurate determination of capacitance of a thin rect- angular plate”, Trans AIEE, Part 1, 75, pp.761-766. [25] Solomon, L., 1964, C.R.Acad.Sci III, 258, pp.64. [26] Read, F.H., 1997, “Improved extrapolation technique in the boundary element method to find the capacitance of the unit square and cube”, J Comput Phys, 133, pp.1-5. [27] Mansfield, M.L., Douglas, J.F., Garboczi, E.J., 2001, “Intrinsic viscosity and the electrical polarizability of arbitrarily shaped objects”, Phys Rev E, 64, 6, pp.061401-16. [28] Morrison, J.A., Lewis, J.A., 1975, “Charge singularity at the corner of a flat plate”, SIAM J. Appl. Math., 31, no. 2, pp.233-250. [29] Hwang, C-O., Won, T., 2005, “Last-passage algorithms for corner charge singularity of conductors”, Jour. Kor. Phys. Soc., 47, pp.S464-S466. http://cs.nyu.edu/exact/ Introduction Exact Solutions Development of the ISLES library Validation of the exact expressions Near-field performance Far field performance Evaluation of the component functions Algorithm Speed of execution Salient features of ISLES Capacitance of a unit square plate - a classic benchmark problem Conclusion

A library of C functions yielding exact solutions of potential and flux influences due to uniform surface distribution of singularities on flat triangular and rectangular elements has been developed. This library, ISLES, has been used to develop the neBEM solver that is both precise and fast in solving a wide range of problems of scientific and technological interest. Here we present the exact expressions proposed for computing the influence of uniform singularity distributions on triangular elements and illustrate their accuracy. We also present a study concerning the time taken to evaluate these long and complicated expressions \textit{vis a vis} that spent in carrying out simple quadratures. Finally, we solve a classic benchmark problem in electrostatics, namely, estimation of the capacitance of a unit square plate raised to unit volt. For this problem, we present the estimated values of capacitance and compare them successfully with some of the most accurate results available in the literature. In addition, we present the variation of the charge density close to the corner of the plate for various degrees of discretization. The variations are found to be smooth and converging. This is in clear contrast to the criticism commonly leveled against usual BEM solvers.

Introduction One of the elegant methods for solving the Laplace / Poisson equations (normally an integral expression of the inverse square law) is to set up the Boundary Integral Equations (BIE) which lead to the moderately popular Boundary Element Method (BEM). In the forward version of the BEM, surfaces of a given geometry are replaced by a distribution of point singularities such as source / dipole of unknown strengths. The strengths of these singularities are obtained through the satisfaction of a given set of boundary conditions that can be Dirichlet, Neumann or of the Robin type. The numerical implementation requires considerable care [1] because it involves evaluation of singular (weak, strong and hyper) integrals. Some of the notable two- dimensional and three-dimensional approaches are [1] and [2, 3, 4, 5, 6] and the references in these papers. Despite a large body of literature, closed form analytic expressions for computing the effects of distributed singularities are rare [7, 8] and complicated to implement. Thus, for solving difficult but realistic problems involving, for example, sharp edges and corners or thin http://arxiv.org/abs/0704.1563v1 elements, introduction of complicated mathematics and special formulations becomes a necessity [9, 10]. These drawbacks are some of the major reasons behind the relative unpopularity of the BEM despite its significant advantages over domain approaches such as the finite-difference and finite-element methods (FDM and FEM) while solving non-dissipative problems [11]. It is well- understood that most of the difficulties in the available BEM solvers stem from the assumption of nodal concentration of singularities which leads to various mathematical difficulties and to the infamous numerical boundary layers [9, 12]. The Inverse Square Law Exact Solutions (ISLES) library, in contrast, is capable of truly modeling the effect of distributed singularities precisely and, thus, its application is not limited by the proximity of other singular surfaces or their curvature or their size and aspect ratio. The library consists of exact solutions for both potential and flux due to uniform distribution of singularity on flat rectangular and triangular elements. While the rectangular element can be of any arbitrary size [13, 14], the triangular element can be a right angled triangle of arbitrary size [15]. Since any real geometry can be represented through elements of the above two types (or by the triangular type alone), this library can help in developing solvers capable of solving three-dimensional potential problems for any geometry. It may be noted here that any non-right-angled triangle can be easily decomposed in to two right-angled triangles. Thus, the right-angled triangles considered here, in fact, can take care of any three-dimensional geometry. Several difficulties were faced in developing the library which arose due to the various terms of the integrals and also from the approximate nature of computation in digital computers. In this paper, we have discussed these difficulties, solutions adopted at present and possible ways of future improvement. The classic benchmark problem of estimating the capacitance of a unit square plate raised to unit volt has been addressed using a solver based on ISLES, namely, the nearly exact BEM (neBEM) solver. Results obtained using neBEM have been compared with other precise results available in the literature. The comparison clearly indicates the excellent precision and efficiency achievable using ISLES and neBEM. In addition, we have also presented the variation of charge density close to the corner of the square plate. Usually, using BEM, it is difficult to obtain physically consistent results close to these geometric singularities. Wild variations in the magni- tude of the charge density has been observed with the change in the degree of discretization, the reason once again being associated with the nodal model of singularities [16]. In contrast, using neBEM, we have obtained very smooth variation close to the corner. Moreover, the magnitudes of the charge density have been found to be consistently converging to physically realistic values. These results clearly indicate that since the foundation expressions of the solver are exact, it is possible to find the potential and flux accurately in the complete physical domain, includ- ing the critical near-field domain using neBEM. In addition, since singularities are no longer assumed to be nodal and we have the exact expressions for potential and flux throughout the physical domain, the boundary conditions no longer need to be satisfied at special points such as the centroid of an element. Although consequences of this considerable advantage is still under study, it is expected that this feature will allow neBEM to yield accurate estimates for problems involving corners and edges that are very important in a large number of scientific and technological studies. It should be noted here that the exact expressions for triangular elements consist of a signifi- cantly larger number of mathematical operations than those for rectangular elements. Thus, for the solver, it is more economical if we use a mixed mesh of rectangular and triangular elements using rectangular elements as much as possible. However, in the present work, we have inten- tionally concentrated on the performance of the triangular elements and results shown here are those obtained using only triangular elements. 2 Exact Solutions The expressions for potential and flux at a point (X,Y,Z) in free space due to uniform source distributed on a rectangular flat surface having corners situated at (x1, z1) and (x2, z2) has been presented, validated and used in [13, 14] and, thus, is not being repeated here. Here, we present the exact expressions necessary to compute the potential and flux due to a right-angled triangular element of arbitrary size, as shown in Fig.1. It may be noted here that the length in the X direction has been normalized, while that in the Z direction has been allowed to be of any arbitrary magnitude, zM . From the figure, it is easy to see that in order to find out the influence due to triangular element, we have imposed another restriction, namely, the necessity that the X and Z axes coincide with the perpendicular sides of the right-angled triangle. Both these restrictions are trivial and can be taken care of by carrying out suitable scaling and appropriate vector transformations. It may be noted here that closed-form expressions for the influence of rectangular and triangular elements having uniform singularity distributions have been previously presented in [7, 8]. However, in these works, the expressions presented are quite complicated and difficult to implement. In [13] and in the present work, the expressions we have presented are lengthy, but completely straight-forward. As a result, the implementation issues of the present expressions, in terms of the development of the ISLES library and the neBEM solver are managed quite easily. Figure 1: Right-angled triangular element with x-length 1 and an arbitrary z-length, zM ; P is the point where the influence (potential and flux) is being computed It is easy to show that the influence (potential) at a point P (X,Y,Z) due to uniform source distributed on a right-angled triangular element as depicted in Fig.1 can be represented as a multiple of φ(X,Y,Z) = ∫ z(x) dx dz (X − x)2 + Y 2 + (Z − z)2 in which we have assumed that x1 = 0, z1 = 0, x2 = 1 and z2 = zM , as shown in the geometry of the triangular element. The closed-form expression for the potential has been obtained using symbolic integration [17] which was subsequently simplified through substantial effort. It is found to be significantly more complicated in comparison to the expression for rectangular elements presented in [13] and can be written as ( (zMY 2 −XG)(LP1 + LM1 − LP2 − LM2) + i |Y | (zMX +G)(LP1 − LM1 − LP2 + LM2) −S1X(tanh−1( R1 + iI1 D11|Z| ) + tanh−1( R1 − iI1 D11|Z| )− tanh−1( R1 + iI2 D21|Z| )− tanh−1( R1 − iI2 D21|Z| +iS1|Y |(tanh−1( R1 + iI1 D11|Z| )− tanh−1(R1 − iI1 D11|Z| )− tanh−1(R1 + iI2 D21|Z| ) + tanh−1( R1 − iI2 D21|Z| 1 + zM 2 log ( 1 + zM 2D12 − E1√ 1 + zM 2D21 − E2 ) + 2Z log D21 −X + 1 D11 −X ) + C (2) where, D11 = (X − x1)2 + Y 2 + (Z − z1)2; D12 = (X − x1)2 + Y 2 + (Z − z2)2 D21 = (X − x2)2 + Y 2 + (Z − z1)2; I1 = (X − x1) |Y | ; I2 = (X − x2) |Y | S1 = sign(z1 − Z); R1 = Y 2 + (Z − z1)2 E1 = (X + zM 2 − zMZ); E2 = (X − 1− zMZ), G = zM (X − 1) + Z; H1 = Y 2 +G(Z − zM ); H2 = Y 2 +GZ LP1 = G− izM |Y | (H1 +GD12) + i|Y |(E1 − izMD12) −X + i|Y | LM1 = G+ izM |Y | (H1 +GD12)− i|Y |(E1 − izMD12) −X − i|Y | LP2 = G− izM |Y | (H2 +GD21) + i|Y |(E2 − izMD21) 1−X + i|Y | LM2 = G+ izM |Y | (H2 +GD21)− i|Y |(E2 − izMD21) 1−X − i|Y | and C denotes a constant of integration. Similarly, the flux components due to the above singularity distribution can also be repre- sented through closed-form expressions as shown below: Fx = − ( (G)(LP1 + LM1 − LP2 − LM2)− i |Y | (zM )(LP1 − LM1 − LP2 + LM2) +S1(tanh R1 + iI1 D11|Z| ) + tanh−1( R1 − iI1 D11|Z| )− tanh−1( R1 + iI2 D21|Z| )− tanh−1( R1 − iI2 D21|Z| 1 + zM 2 log ( 1 + zM 2D12 − E1√ 1 + zM 2D21 − E2 ) ) + C (3) Fy = − ( (2zMY )(LP1 + LM1 − LP2 − LM2) + i |Y | (Sn(Y )G)(LP1 − LM1 − LP2 + LM2) +iS1Sn(Y )(tanh R1 + iI1 D11|Z| )− tanh−1( R1 − iI1 D11|Z| )− tanh−1( R1 + iI2 D21|Z| ) + tanh−1( R1 − iI2 D21|Z| )) ) + C Fz = − 1 + zM log ( 1 + zM 2D21 − E2√ 1 + zM 2D12 − E1 ) + log D11 −X D21 −X + 1 ) + C (5) where Sn(Y ) implies the sign of the Y-coordinate and C indicates constants of integrations. It is to be noted that the constants of different integrations are not the same. These expression are expected to be useful in the mathematical modeling of physical processes governed by the inverse square laws. Being exact and valid throughout the physical domain, they can be used to formulate versatile solvers to solve multi-scale multi-physics problems governed by the Laplace / Poisson equations involving Dirichlet, Neumann or Robin boundary conditions. 3 Development of the ISLES library Due to the tremendous popularity of the C language we have written the codes in the C pro- gramming language. However, it should be quite simple to translate the library to other popular languages such as FORTRAN or C++, since no special feature of the C language has been used to develop the codes. 3.1 Validation of the exact expressions The expressions for the rectangular element have been validated in detail in [13]. Here, we present the results for triangular elements in fair detail. In Fig.2, we have presented a comparison of potentials evaluated for a unit triangular element by using the exact expressions, as well as by using numerical quadrature of high accuracy. The two results are found to compare very well throughout. Please note that contours have been obtained on the plane of the element, and thus, represents a rather critical situation. Similarly, Fig.3 shows a comparison between the results obtained using closed-form expressions for flux and those obtained using numerical quadrature. The flux considered here is in the Y direction and is along a line beginning from (−2,−2,−2) and ending at (2, 2, 2). The comparison shows the commendable accuracy expected from closed form expressions. In Fig.4(a) and 4(b), the surface plots of potential on the element plane (XZ plane) and Y -flux on the XY plane have been presented from which the expected significant increase in potential and sharp change in the flux value on the element is observed. Thus, by using a small fraction of computational resources in comparison to those consumed in numerical quadratures, ISLES can compute the exact value of potential and flux for singularities distributed on triangular elements. Figure 2: Potential contours on a triangular element computed using exact expressions and by numerical quadrature 3.2 Near-field performance In order to emphasize the accuracy of ISLES, we have considered the following severe situations in the near-field region in which it is observed that the quadratures can match the accuracy of ISLES only when a high degree of discretization is used. Please note that in these cases, the value of zM has been considered to be 10. In Fig.5 we have presented the variation of potential along a line on the element surface running parallel to the Z-axis of the triangular element (see Fig.1) and going through the centroid of the element. It is observed that results obtained using even a 100× 100 quadrature is quite unacceptable. In fact, by zooming on to the image, it can -4 -3 -2 -1 0 1 2 3 4 Distance (a.u.) 100X100 Exact Figure 3: Comparison of flux (in the Y direction) as computed by ISLES and numerical quadra- ture along a diagonal line be found that only the maximum discretization yields results that match closely to the exact solution. It may be noted here that the potential is a relatively easier property to compute. The difficulty of achieving accurate flux estimates is illustrated in the two following figures. The variation of flux in the X-direction along the same line as used in Fig.5 has been presented in Fig.6. Similarly, variation of Y -flux along a diagonal line (beginning at (-10,-10,-10) and ending at (10,10,10) and piercing the element at the centroid) has been presented in Fig.7. From these figures we see that the flux values obtained using the quadrature are always inaccurate even if the discretization is as high as 100 × 100. We also observe that the estimates are locally inaccurate despite the use of very high amount of discretization (200× 200 or 500). Specifically, in the latter figure, even the highest discretization can not match the exact values at the peak, while in the former only the highest one can correctly emulate the sharp change in the flux value. It is also heartening to note that the values from the quadrature using higher amount of discretization consistently converge towards the ISLES values. 3.3 Far field performance It is expected that beyond a certain distance, the effect of the singularity distribution can be considered to be the same as that of a centroidally concentrated singularity or a simple quadrature. The optimized amount of discretization to be used for the quadrature can be determined from a study of the speed of execution of each of the functions in the library and has been presented separately in a following sub-section. If we plan to replace the exact expressions by quadratures (in order to reduce the computational expenses, presumably) beyond a certain given distance, the quadrature should necessarily be efficient enough to justify the replacement. While standard but more elaborate algorithms similar to the fast multipole method (FMM) [18] along with the GMRES [19] matrix solver can lead to further of computational efficiency, the (a) Potential surface (b) Flux surface Figure 4: (a) Potential surface due to a triangular source distribution on the element plane, (b) Flux (in the Y direction) surface due to a triangular source distribution on the XY plane at Z=0 simple approach as outlined above can help in reducing a fair amount of computational effort. In the following, we present the results of numerical experiments that help us in determining the far-field performance of the exact expressions and quadratures of various degrees that, in turn, help us in choosing the more efficient approach for a desired level of accuracy. In Fig.8 we have presented potential values obtained using the exact approach, 100 × 100, 10 × 10 and no discretization, i.e., the usual BEM approximation while using the zeroth order piecewise uniform charge density assumption. The potentials are computed along a diagonal line running from (-1000, -1000, -1000) to (1000, 1000, 1000) which pierces a triangular element of zM = 10. It can be seen that results obtained using the usual BEM approach yields inaccurate results as we move closer than distances of 10 units, while the 10 × 10 discretization yields acceptable results up to a distance of 1.0 unit. In order to visualize the errors incurred due to the use of quadratures, we have plotted Fig.9 where the errors incurred (normalized with respect to the exact value) have been plotted. From this figure we can conclude that for the given diagonal line, the error due to the usual BEM approximation falls below 1% if the distance is larger than 20 units while for the simple 10×10 discretization, it is 2 units. It may be mentioned here that along the axes the error turns out to be significantly more [13] and the limits need to be effectively doubled to achieve the accuracy for all cases possible. Thus, for achieving 1% accuracy, the usual BEM is satisfactory only if the distance of the influenced point is five times the longer side of an element. Please note here that the error drops to 1 out of 106 as the distance becomes fifty times the longer side. Besides proving that the exact expressions work equally well in the near-field as well as the far-field, this fact justifies the usual BEM approach for much of the computational domain leading to substantial savings in computational expenses. The accuracy of the exact expressions used in the ISLES library is confirmed from the above comparisons. However, there are several other important issues related to the development of the library that are discussed below briefly. 3.4 Evaluation of the component functions Many of the irrational and transcendental functions have domains and ranges in which they are defined. Moreover, they are often multiply defined in the complex domain; for example, there 0 2 4 6 8 10 Distance along Z 10by10 50by50 100by100 200by200 500by500 IslesLib AppFlag Figure 5: Variation of potential along a centroidal line on the XZ plane parallel to the Z axis for a triangular element: comparison among values obtained using the exact expressions and numerical quadratures are an infinite number of complex values for the logarithm function. In such cases, only one principal value must be returned by the function. In general, such values cannot be chosen so as to make the range continuous and thus, lines in the domain called branch cuts need to be defined, which in turn define the discontinuities in the range. While evaluating expressions such as the ones displayed in eqns.(2 - 5) a number of such problems are expected to occur. However, when the expressions are analyzed at critical locations such as the corners and edges of the element, it is observed that the terms likely to create difficulties while evaluating potentials are either cancelled out or are themselves multiplied by zero. As a result, at these locations of likely geometric and mathematical singularities, the solution behaves nicely. However, the same is not true for the expressions related to the flux components. For these, we have to deal with branch-cut problems in relation to tanh−1 and problems related to the evaluation of log(0). It should be noted here that these singularities associated to the edges and corners of the elements are of the weak type and it is expected that exact evaluation of these terms as well will be possible through further work. However, difficulties of a different nature crop up in these calculations which can be linked directly to the limitation of the computer itself, namely, round-off errors [15]. These errors can lead to severe problems while handling multi-scale problems such as those described in [13]. A completely different approach is necessary to cope up with these difficulties, for example, the use of extended range arithmetic [20], interval arithmetic [21] or the use of specialized libraries such as the CORE library of the Exact Geometric Computation (EGC) initiative [22]. In the present version of ISLES, a simple approach has been implemented which sets a lower limit to various distance values. Below this value, the distance is considered to be zero. Plan of future improvements in this regard has been kept at a high priority. 5 5.5 6 6.5 7 7.5 8 Distance along Z 10by10 50by50 100by100 200by200 500by500 IslesLib AppFlag Figure 6: Variation of flux in the X direction along a line on the XZ plane parallel to the Z axis for a triangular element: comparison among values obtained using the exact expressions and numerical quadratures 3.4.1 Algorithm As discussed above, there are possibilities of facing problems while using the exact expressions which may be due to the functions being evaluated or due to round-off errors leading to erro- neous results. Moreover, despite providing many checks during the computation there is finite possibility of ending up with a wrong value of a property indicated by its being Nan or inf or potential due to unit positive singularity strength turning out to be negative. In order to maintain the robustness of the library, we have tried to keep checks on the intermediate and final values during the course of the computation. When the results are found to be unsatisfactory, unphysical, we have re-estimated the results by using numerical quadrature and kept a track of the cause by raising a unique approximation flag which is specific for a problem. As a result, the steps for the calculation for a property can be written as follows: • Get the required inputs - geometry of the element and the position where the effect needs to be evaluated; Check whether the element size and distances are large enough so that the results do not suffer from round-off errors. • Check whether the location coincides with one of the special ones, such as corners or edges. • Evaluate the necessary expressions in accordance with the foregoing results. If necessary, consider each term in the expressions separately to sort out difficulties related to singular- ities, branch-cuts or round-off errors. Note that if the multiplier is zero, rest of the term does not need evaluation. • If direct evaluation of the expressions fail, raise a unique approximation flag specific to this problem and term and return the value of the property by using numerical quadrature. -1 -0.5 0 0.5 1 1.5 Distance along diagonal 10by10 50by50 100by100 200by200 500by500 IslesLib AppFlag Figure 7: Comparison of flux (in the Y direction) along a diagonal line piercing the triangular element at the centroid: comparison among values obtained using the exact expressions and numerical quadratures • Compute all the terms and find the final value, Check whether the final value is a number and physically meaningful. If not satisfied, recompute the result using numerical quadra- ture and raise the relevant approximation flag. 3.5 Speed of execution The time taken to compute the potential and flux is an important parameter related to the overall computational efficiency of the codes. This is true despite the fact that, in a typical simulation, the time taken to solve the system of algebraic equation is far greater than the time taken to build the influence coefficient matrix and post-processing. Moreover, the amount of time taken to solve the system of equations tend to increase at a greater rate than the time taken to complete the other two. It should be mentioned here that the time taken in each of these steps can vary to a significant amount depending on the algorithm of the solver. In the present case, the system of equations has been solved using lower upper decomposition using the well known Crout’s partial pivoting. Although this method is known to be very rugged and accurate, it is not efficient as far as number of arithmetic operations, and thus, time is concerned. It is also possible to reduce the time taken to pre-process (generation of mesh and creation of influence matrices), solve the system of algebraic equations and that for post-process (computation of potential and flux at the required locations) can be significantly reduced by adopting faster algorithms, including those involving parallelization. In order to optimize the time taken to generate the influence coefficient matrix and that to carry out the post-processing, we carried out a small numerical study to determine the amount of time taken to complete the various functions being used in ISLES, especially those being used to evaluate the exact expressions and those being used to carry out the quadratures. The results 1 10 100 Exact Usual BEM 10 X 10 100 X 100 Figure 8: Potential along a diagonal through the triangular element computed using exact, 100 × 100, 10 × 10 and usual BEM approach of the study (which was carried out using the linux system command gprof ) has been presented in the following Table1. Table 1: Time taken to evaluate exact expression and various quadratures Method Exact Usual BEM 10× 10 100 × 100 500× 500 Time 0.8 µs 25 ns 1 µs 200 µs 5 ms Please note that the numbers presented in this table are representative and are likely to have statistical fluctuations. However, despite the fluctuations, it may be safely concluded that a quadrature having only 10 × 10 discretization is already consuming time that is comparable to that needed exact evaluation. Thus, the exact expressions, despite their complexity, are extremely efficient in the near-field which can be considered at least as large as 0.5 times the larger side of a triangular element (please refer to Fig.9). In making this statement, we have as- sumed that the required accuracy for generating the influence coefficient matrix and subsequent potential and flux calculations is 1%. This may not be acceptable at all under many practical circumstances, in which case the near-field would imply a larger volume. 3.6 Salient features of ISLES Development of usual BEM solvers are dependent on the two following assumptions: • While computing the influences of the singularities, the singularities are modeled by a sum of known basis functions with constant unknown coefficients. For example, in the constant element approach, the singularities are assumed to be concentrated at the centroid of the element, except for special cases such as self influence. This becomes necessary because 1e-07 1e-06 1e-05 1e-04 0.001 0.01 1 10 100 1000 100 X 100 10 X 10 Usual BEM Figure 9: Error along a diagonal through the triangular element computed using 100 × 100, 10× 10 and usual BEM approach closed form expressions for the influences are not, in general, available for surface elements. An approximate and computationally rather expensive way of circumventing this limitation is to use numerical integration over each element or to use linear or higher order basis functions. • The strengths of the singularities are solved depending upon the boundary conditions, which, in turn, are modeled by the shape functions. For example, in the constant element approach, it is assumed that it is sufficient to satisfy the boundary conditions at the centroids of the elements. In this approach, the position of the singularity and the point where the boundary condition is satisfied for a given element usually matches and is called the collocation point. The first (and possibly, the more damaging) approximation for BEM solvers can be relaxed by using ISLES and can be restated as, • The singularities distributed on the boundary elements are assumed to be uniform on a particular element. The strength of the singularity may change from element to element. This improvement turns out to be very significant as demonstrated in the following section and some of our other studies involving microelectromechanical systems (MEMS) and gas detectors for nuclear applications [13, 14]. Some of the advantages of using ISLES are itemized below: • For a given level of discretization, the estimates are more accurate, • Effective efficiency of the solver improves, as a result, • Large variation of length-scales, aspect ratios can be tackled, • Thinness of members or nearness of surfaces does not pose any problem, • Curvature has no detrimental effect on the solution, • The boundary condition can be satisfied anywhere on the elements, i.e., points other than the centroidal points can be easily used, if necessary (for a corner problem, may be), • The same formulation, library and solver is expected to work in majority of physical situations. As a result, the necessity for specialized formulations of BEM can be greatly minimized. 4 Capacitance of a unit square plate - a classic benchmark prob- Using the neBEM solver, we have computed the capacitance of a unit square conducting plate raised to a unit volt. This problem is still considered to be one of the major unsolved problems of electrostatic theory [23, 8, 26, 16] and no analytical solution for this problem has been obtained so far. The capacitance value estimated by the present method has been compared with very accurate results available in the literature (using BEM and other methods). The results obtained using the neBEM solver is found to be among the most accurate ones available till date as shown in Table.2. Please note that we have not invoked symmetry or used extrapolation techniques to arrive at our result presented in the table. Table 2: Comparison of capacitance values Reference Method Capacitance (pF) / 4 πǫ0 [23] Surface Charge 0.3607 [24] Surface Charge 0.362 [25] Surface Charge 0.367 [8] Refined Surface Charge 0.3667892 ± 1.1× 10−6 and Extrapolation [26] Refined Boundary Element 0.3667874 ± 1× 10−7 and Extrapolation [27] Numerical Path Integration 0.36684 [16] Random Walk 0.36 ± 0.01 This work neBEM 0.3660587 Finally, we consider the corner problem related to the electrostatics of the above conducting plate. Problems of this nature are considered to be challenging for any numerical tool and especially so for the BEM approach. The inadequacy of the BEM approach, especially in solving the present problem, has been mentioned even quite recently [16] in which it has been correctly mentioned that since the method can not extend its mathematical model to include the edges and corners in reality, it is unlikely that it will ever succeed in modeling the edge / corner singularities correctly. As a result, with change in discretization, the properties near these geometric singularities are expected to oscillate significantly leading to erroneous results. However, as discussed above, the neBEM does extend its singularities distributed on the surface elements right till an edge or a corner. Moreover, using neBEM, it is also possible to satisfy the boundary conditions (both potential and flux) as close to the edge / corner as is required. In fact, it should be possible to specify the potentials right at the edge / corner. In the following study, we have presented estimates of charge density very close the flat plate corner as obtained using neBEM. Please note that the boundary conditions have been satisfied at the centroids of each element although we plan to carry out detailed studies of changing the position of these points, especially in relation to problems involving edges / corners. In Fig.10, charge densities very close to the corner of the flat plate estimated by neBEM using various amounts of discretization have been presented. It can be seen that each curve follows the same general trend, does not suffer from any oscillation and seems to be converging to a single curve. This is true despite the fact that there has been almost an order of magnitude variation in the element lengths. Finally, in Fig.11, we present a least-square fitted straight line matching the charge density as obtained the highest discretization in this study. It is found that the slope of the straight line is 0.713567, which compares very well with both old and recent estimates of 0.7034 [28, 29]. This is despite the fact that here we have used a relatively coarse discretization near the corner. It should be mentioned that none of the earlier references cited here used the BEM approach. While the former used a singular perturbation technique, the latter used a diffusion based Monte- Carlo method. Thus, it is extremely encouraging to note that using the neBEM approach, we have been able to match the accuracy of these sophisticated techniques. 0 0.02 0.04 0.06 0.08 0.1 Distance from corner 0.0182 0.0091 0.0067 0.0047 0.0047 Figure 10: Corner charge density estimated by neBEM using various sizes of triangular elements 5 Conclusion An efficient and robust library for solving potential problems in a large variety of science and engineering problems has been developed. Exact closed-form expressions used to develop ISLES have been validated throughout the physical domain (including the critical near-field region) -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 log(r) neBEM Fitted line Figure 11: Variation of charge density with increasing distance from the corner of the unit square plate and a least-square fitted straight line: slope of the fitted line is 0.713567 by comparing these results with results obtained using numerical quadrature of high accuracy. Algorithmic aspects of this development have also been touched upon. A classic benchmark problem of electrostatics has been successfully simulated to very high precision. Charge density values at critical geometric locations like corners have been found to be numerically stable and physically acceptable. Several advantages over usual BEM solvers and other specialized BEM solvers have been briefly mentioned. Work is under way to make the code more robust and efficient through the implementation of more efficient algorithms and parallelization. Acknowledgements We would like to thank Professor Bikas Sinha, Director, SINP and Professor Sudeb Bhat- tacharya, Head, INO Section, SINP for their support and encouragement during the course of this work. References [1] Nagarajan, A., Mukherjee, S., 1993, “A mapping method for numerical evaluation of two- dimensional integrals with 1/r singularity”, Computational Mechanics, 12, pp.19-26. [2] Cruse, T.A., 1969, “Numerical solutions in three dimensional elastostatics”, ( Int. J. Solids Struct., 5, pp.1259-1274. [3] Kutt, H.R., 1975, “The numerical evaluation of principal value integrals by finite-part integration”, Numer. Math., 24, pp.205-210. [4] Lachat, J.C., Watson, J.O., 1976, “Effective numerical treatment of boundary integral equations: A formulation for three dimensional elastostatics”, Int. J. Numer. Meth. Eng., 10, pp.991-1005. [5] Srivastava, R., Contractor, D.N., 1992, “Efficient evaluation of integrals in three- dimensional boundary element method using linear shape functions over plane triangular elements”, Appl. Math. Modelling, 16, pp.282-290. [6] Carini, A., Salvadori, A., 2002, “Analytical integration in 3D BEM: preliminaries”, Com- putational Mechanics, 18, pp.177-185. [7] Newman, J.N., 1986, “Distributions of sources and normal dipoles over a quadrilateral panel”, Jour. of Engg. Math., 20, pp.113-126. [8] Goto, E., Shi, Y. and Yoshida, N., 1992, “Extrapolated surface charge method for capacity calculation of polygons and polyhedra”, Jour. of Comput. Phys., 100, pp.105-115. [9] Chyuan, S-W., Liao, Y-S. and Chen, J-T., 2004, “An efficient technique for solving the arbitrarily multilayered electrostatic problems with singularity arising from a degenerate boundary”, Semicond. Sci. Technol., 19, R47-R58. [10] Bao, Z., Mukherjee, S., 2004, “Electrostatic BEM for MEMS with thin conducting plates and shells”, Eng Analysis Boun Elem, 28, pp.1427-1435. [11] Mukhopadhyay, S., Majumdar, N., 2006, “Effect of finite dimensions on the electric field configuration of cylindrical proportional counters”, IEEE Trans Nucl Sci, 53, pp.539-543. [12] Sladek, V., Sladek, J., 1991, “Elimination of the boundary layer effect in BEM computation of stresses”, Comm. Appl. Num. Meth., 7, pp.539-550. [13] Mukhopadhyay, S., Majumdar, N., 2006, “Computation of 3D MEMS electrostatics using a nearly exact BEM solver”, Eng Anal Boundary Elem, 30, pp.687-696. [14] Majumdar, N., Mukhopadhyay, S., 2006, “Simulation of three-dimensional electrostatic field configuration in wire chambers: A novel approach”, Nucl. Instrum. Meth. Phys. Res. A, 566, pp.489-494. [15] Mukhopadhyay, S., Majumdar, N., 2007, “Use of rectangular and triangular elements for nearly exact BEM solutions”, Emerging Mechanical Technology - Macro to Nano, Research Publishing Services, Chennai, India, pp.107-114 (ISBN: 81-904262-8-1). [16] Wintle, H.J., 2004, “The capacitance of the cube and square plate by random walk meth- ods”, J Electrostatics, 62 pp.51-62. [17] Etter, D.M., 1997, Engineering Problem Solving with MatLab, Prentice Hall, International, Inc., New Jersey 07458, USA. [18] Greengard, L., Rokhlin, V., 1987, “A fast algorithm for particle simulation”, Journal of Computational Physics, 73 pp.325-348. [19] Saad, Y., Schultz, M., 1986, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems”, SIAM J. Sci. Statist. Comput., 7 pp.856-869. [20] Smith, J.M., Olver, F.W., Lozier, D.W., 1981, “Extended-Range Arithmetic and Normal- ized Legendre Polynomial”, ACM Transactions on Mathematical Software, 7, pp.93-105. [21] Alefeld, G., Herzberger, J., 1983, Introduction to interval analysis, Academic Press. [22] http://cs.nyu.edu/exact/ [23] Maxwell, J.C., 1878, Electrical Research of the Honorable Henry Cavendish, p.426, Cam- bridge University Press, Cambridge, UK. [24] Reitan, D.K., Higgins, R.J., 1957, “Accurate determination of capacitance of a thin rect- angular plate”, Trans AIEE, Part 1, 75, pp.761-766. [25] Solomon, L., 1964, C.R.Acad.Sci III, 258, pp.64. [26] Read, F.H., 1997, “Improved extrapolation technique in the boundary element method to find the capacitance of the unit square and cube”, J Comput Phys, 133, pp.1-5. [27] Mansfield, M.L., Douglas, J.F., Garboczi, E.J., 2001, “Intrinsic viscosity and the electrical polarizability of arbitrarily shaped objects”, Phys Rev E, 64, 6, pp.061401-16. [28] Morrison, J.A., Lewis, J.A., 1975, “Charge singularity at the corner of a flat plate”, SIAM J. Appl. Math., 31, no. 2, pp.233-250. [29] Hwang, C-O., Won, T., 2005, “Last-passage algorithms for corner charge singularity of conductors”, Jour. Kor. Phys. Soc., 47, pp.S464-S466. http://cs.nyu.edu/exact/ Introduction Exact Solutions Development of the ISLES library Validation of the exact expressions Near-field performance Far field performance Evaluation of the component functions Algorithm Speed of execution Salient features of ISLES Capacitance of a unit square plate - a classic benchmark problem Conclusion

ENTROPY OF EIGENFUNCTIONS NALINI ANANTHARAMAN, HERBERT KOCH, AND STÉPHANE NONNENMACHER Abstra t. We study the high�energy limit for eigenfun tions of the lapla ian, on a ompa t negatively urved manifold. We review the re ent result of Anantharaman� Nonnenma her [4℄ giving a lower bound on the Kolmogorov�Sinai entropy of semi lassi al measures. The bound proved here improves the result of [4℄ in the ase of variable negative urvature. 1. Motivations The theory of quantum haos tries to understand how the haoti behaviour of a lassi- al Hamiltonian system is re�e ted in its quantum ounterpart. For instan e, let M be a ompa t Riemannian C∞ manifold, with negative se tional urvatures. The geodesi �ow has the Anosov property, whi h is onsidered as the ideal haoti behaviour in the theory of dynami al systems. The orresponding quantum dynami s is the unitary �ow gener- ated by the Lapla e-Beltrami operator on L2(M). One expe ts that the haoti properties of the geodesi �ow in�uen e the spe tral theory of the Lapla ian. The Random Matrix onje ture [7℄ asserts that the large eigenvalues should, after proper unfolding, statisti- ally resemble those of a large random matrix, at least for a generi Anosov metri . The Quantum Unique Ergodi ity onje ture [26℄ (see also [6, 30℄) des ribes the orresponding eigenfun tions ψk: it laims that the probability measure |ψk(x)| 2dx should approa h (in the weak topology) the Riemannian volume, when the eigenvalue tends to in�nity. In fa t a stronger property should hold for the Wigner transform Wψ, a fun tion on the otangent bundle T ∗M , (the lassi al phase spa e) whi h simultaneously des ribes the lo alization of the wave fun tion ψ in position and momentum. We will adopt a semi lassi al point of view, that is onsider the eigenstates of eigenvalue unity of the semi lassi al Lapla ian −~2△, thereby repla ing the high-energy limit by the semi lassi al limit ~ → 0. We denote by (ψk)k∈N an orthonormal basis of L 2(M) made of eigenfun tions of the Lapla ian, and by (− 1 )k∈N the orresponding eigenvalues: (1.1) − ~2k△ψk = ψk, with ~k+1 ≤ ~k . We are interested in the high-energy eigenfun tions of −△, in other words the semi lassi al limit ~k → 0. The Wigner distribution asso iated to an eigenfun tion ψk is de�ned by Wk(a) = 〈Op~k(a)ψk, ψk〉L2(M), a ∈ C ∗M) . Here Op is a quantization pro edure, set at the s ale (wavelength) ~k, whi h asso iates to any smooth phase spa e fun tion a (with ni e behaviour at in�nity) a bounded operator on http://arxiv.org/abs/0704.1564v1 2 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER L2(M). See for instan e [13℄ or [14℄ for various quantizations Op . On a manifold, one an use lo al oordinates to de�ne Op in a �nite system of harts, then glue the obje ts de�ned lo ally thanks to a smooth partition of unity [11℄. For standard quantizations Op the Wigner distribution is of the form Wk(x, ξ) dx dξ, where Wk(x, ξ) is a smooth fun tion on T ∗M , alled the Wigner transform of ψ. If a is a fun tion on the manifold M , Op an be taken as the multipli ation by a, and thus we have Wk(a) = a(x)|ψk(x)| 2dx: the Wigner transform is thus a mi rolo al lift of the density |ψk(x)| . Although the de�nition ofWk depends on a ertain number of hoi es, like the hoi e of lo al oordinates, or of the quantization pro edure (Weyl, anti-Wi k, �right� or �left� quantization...), its asymptoti behaviour when ~k −→ 0 does not. A ordingly, we all semi lassi al measures the limit points of the sequen e (Wk)k∈N, in the distribution topology. In the semi lassi al limit, �quantum me hani s onverges to lassi al me hani s�. We will denote |·|x the norm on T xM given by the metri . The geodesi �ow (g t)t∈R is the Hamiltonian �ow on T ∗M generated by the Hamiltonian H(x, ξ) = |ξ|2x . A quantization of this Hamiltonian is given by the res aled Lapla ian −~ , whi h generates the unitary �ow (U t ) = (exp(it~△ )) a ting on L2(M). The semi lassi al orresponden e of the �ows ) and (gt) is expressed through the Egorov Theorem : Theorem 1.1. Let a ∈ C∞c (T ∗M). Then, for any given t in R, (1.2) ‖U−t (a)U t (a ◦ gt)‖L2(M) = O(~) , ~ → 0 . The onstant implied in the remainder grows (often exponentially) with t, whi h rep- resents a notorious problem when one wants to study the large time behaviour of (U t Typi ally, the quantum- lassi al orresponden e will break down for times t of the order of the Ehrenfest time (3.25). Using (1.2) and other standard semi lassi al arguments, one shows the following : Proposition 1.2. Any semi lassi al measure is a probability measure arried on the energy layer E = H−1(1 ) (whi h oin ides with the unit otangent bundle S∗M). This measure is invariant under the geodesi �ow. Let us all M the set of gt-invariant probability measures on E . This set is onvex and ompa t for the weak topology. If the geodesi �ow has the Anosov property � for instan e if M has negative se tional urvature � that set is very large. The geodesi �ow has ountably many periodi orbits, ea h of them arrying an invariant probability measure. There are many other invariant measures, like the equilibrium states obtained by variational prin iples [19℄, among them the Liouville measure µLiouv, and the measure of maximal entropy. Note that, for all these examples of measures, the geodesi �ow a ts ergodi ally, meaning that these examples are extremal points in M. Our aim is to determine, at least partially, the set Msc formed by all possible semi lassi al measures. By its de�nition, Msc is a losed subset of M, in the weak topology. For manifolds su h that the geodesi �ow is ergodi with respe t to the Liouville measure, it has been known for some time that almost all eigenfun tions be ome equidistributed over E , in the semi lassi al limit. This property is dubbed as Quantum Ergodi ity : ENTROPY OF EIGENFUNCTIONS 3 Theorem 1.3. [27, 32, 11℄ Let M be a ompa t Riemannian manifold, assume that the a tion of the geodesi �ow on E = S∗M is ergodi with respe t to the Liouville measure. Let (ψk)k∈N be an orthonormal basis of L 2(M) onsisting of eigenfun tions of the Lapla ian (1.1), and let (Wk) be the asso iated Wigner distributions on T Then, there exists a subset S ⊂ N of density 1, su h that (1.3) Wk −→µLiouv, k → ∞, k ∈ S. The question of existen e of �ex eptional� subsequen es of eigenstates with a di�erent behaviour is still open. On a negatively urved manifold, the geodesi �ow satis�es the ergodi ity assumption, and in fa t mu h stronger properties : mixing, K�property, et . For su h manifolds, it has been postulated in the Quantum Unique Ergodi ity onje ture [26℄ that the full sequen e of eigenstates be omes semi lassi ally equidistributed over E : one an take S = N in the limit (1.3). In other words, this onje ture states that there exists a unique semi lassi al measure, and Msc = {µLiouv}. So far the most pre ise results on this question were obtained for manifolds M with onstant negative urvature and arithmeti properties: see Rudni k�Sarnak [26℄, Wolpert [31℄. In that very parti ular situation, there exists a ountable ommutative family of self�adjoint operators ommuting with the Lapla ian : the He ke operators. One may thus de ide to restri t the attention to ommon bases of eigenfun tions, often alled �arith- meti � eigenstates, or He ke eigenstates. A few years ago, Lindenstrauss [24℄ proved that any sequen e of arithmeti eigenstates be ome asymptoti ally equidistributed. If there is some degenera y in the spe trum of the Lapla ian, note that it ould be possible that the Quantum Unique Ergodi ity onje tured by Rudni k and Sarnak holds for one orthonormal basis but not for another. On su h arithmeti manifolds, it is believed that the spe trum of the Lapla ian has bounded multipli ity: if this is really the ase, then the semi lassi al equidistribution easily extends to any sequen e of eigenstates. Nevertheless, one may be less optimisti when extending the Quantum Unique Ergod- i ity onje ture to more general systems. One of the simplest example of a symple ti Anosov dynami al system is given by linear hyperboli automorphisms of the 2-torus, e.g. Arnold's � at map� . This system an be quantized into a sequen e ofN×N unitary matri es � the propagators, where N ∼ ~−1 [18℄. The eigenstates of these matri es satisfy a Quantum Ergodi ity theorem similar with Theorem 1.3, meaning that almost all eigen- states be ome equidistributed on the torus in the semi lassi al limit [9℄. Besides, one an hoose orthonormal eigenbases of the propagators, su h that the whole sequen e of eigen- states is semi lassi ally equidistributed [22℄. Still, be ause the spe tra of the propagators are highly degenerate, one an also onstru t sequen es of eigenstates with a di�erent limit measure [15℄, for instan e, a semi lassi al measure onsisting in two ergodi omponents: half of it is the Liouville measure, while the other half is a Dira peak on a single (unsta- ble) periodi orbit. It was also shown that this half-lo alization is maximal for this model [16℄ : a semi lassi al measure annot have more than half its mass arried by a ountable union of periodi orbits. The same type of half-lo alized eigenstates were onstru ted by two of the authors for another solvable model, namely the �Walsh quantization� of the 4 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER baker's map on the torus [3℄; for that model, there exist ergodi semi lassi al measures of purely fra tal type (that is, without any Liouville omponent). Another type of semi las- si al measure was re ently obtained by Kelmer for quantized hyperboli automorphisms on higher-dimensional tori [20℄: it onsists in the Lebesgue measure on some invariant o-isotropi subspa e of the torus. For these Anosov models on tori, the onstru tion of ex eptional eigenstates strongly uses nongeneri algebrai properties of the lassi al and quantized systems, and annot be generalized to nonlinear systems. 2. Main result. In order to understand the set Msc, we will attempt to ompute the Kolmogorov�Sinai entropies of semi lassi al measures. We work on a ompa t Riemannian manifold M of arbitrary dimension, and assume that the geodesi �ow has the Anosov property. A tually, our method an without doubt be adapted to more general Anosov Hamiltonian systems. The Kolmogorov�Sinai entropy, also alled metri entropy, of a (gt)-invariant probability measure µ is a nonnegative number hKS(µ) that des ribes, in some sense, the omplexity of a µ-typi al orbit of the �ow. The pre ise de�nition will be given later, but for the moment let us just give a few fa ts. A measure arried on a losed geodesi has vanishing entropy. In onstant urvature, the entropy is maximal for the Liouville measure. More generally, for any Anosov �ow, the energy layer E is foliated into unstable manifolds of the �ow. An upper bound on the entropy of an invariant probability measure is then provided by the Ruelle inequality: (2.1) hKS(µ) ≤ log Ju(ρ)dµ(ρ) In this inequality, Ju(ρ) is the unstable Ja obian of the �ow at the point ρ ∈ E , de�ned as the Ja obian of the map g−1 restri ted to the unstable manifold at the point g1ρ (note that the average of log Ju over any invariant measure is negative). The equality holds in (2.1) if and only if µ is the Liouville measure on E [23℄. If M has dimension d and has onstant se tional urvature −1, the above inequality just reads hKS(µ) ≤ d− 1. Finally, an important property of the metri entropy is that it is an a�ne fun tional on M. A ording to the Birkho� ergodi theorem, for any µ ∈ M and for µ�almost every ρ ∈ E , the weak limit µρ = lim |t|−→∞ δgsρds exists, and is an ergodi probability measure. We an then write µρdµ(ρ), whi h realizes the ergodi de omposition of µ. The a�neness of the KS entropy means hKS(µ) = hKS(µ ρ)dµ(ρ). ENTROPY OF EIGENFUNCTIONS 5 An obvious onsequen e is the fa t that the range of hKS on M is an interval [0, hmax]. In the whole arti le, we onsider a ertain subsequen e of eigenstates (ψkj )j∈N of the Lapla ian, su h that the orresponding sequen e of Wigner distributions (Wkj) onverges to a semi lassi al measure µ. In the following, the subsequen e (ψkj )j∈N will simply be denoted by (ψ~)~→0, using the slightly abusive notation ψ~ = ψ~kj for the eigenstate ψkj . Ea h eigenstate ψ~ thus satis�es (2.2) (−~2 △−1)ψ~ = 0 . In [2℄ the �rst author proved that the entropy of any µ ∈ Msc is stri tly positive. In [4℄, more expli it lower bounds were obtained. The aim of this paper is to improve the lower bounds of [4℄ into the following Theorem 2.1. Let µ be a semi lassi al measure asso iated to the eigenfun tions of the Lapla ian on M . Then its metri entropy satis�es (2.3) hKS(µ) ≥ log Ju(ρ)dµ(ρ) (d− 1) λmax , where d = dimM and λmax = limt→±∞ log supρ∈E |dg ρ| is the maximal expansion rate of the geodesi �ow on E . In parti ular, if M has onstant se tional urvature −1, we have (2.4) hKS(µ) ≥ In dimension d, we always have log Ju(ρ)dµ(ρ) ≤ (d− 1)λmax , so the above bound is an improvement over the one obtained in [4℄, (2.5) hKS(µ) ≥ log Ju(ρ)dµ(ρ) − (d− 1)λmax . In the ase of onstant or little-varying urvature, the bound (2.4) is mu h sharper than the one proved in [2℄. On the other hand, if the urvature varies a lot (still being negative everywhere), the right hand side of (2.3) may a tually be negative, in whi h ase the bound is trivial. We believe this �problem� to be a te hni al short oming of our method, and a tually onje ture the following bound: (2.6) hKS(µ) ≥ log Ju(ρ)dµ(ρ) Extended to the ase of the quantized torus automorphisms or the Walsh-quantized baker's map, this bound is saturated for the half-lo alized semi lassi al measures onstru ted in [15℄, as well as those obtained in [20, 3℄. This bound allows ertain ergodi omponents to be arried by losed geodesi s, as long as other omponents have positive entropy. This may be ompared with the following result obtained by Bourgain and Lindenstrauss in the ase of arithmeti surfa es : 6 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER Theorem 2.2. [8℄ Let M be a ongruen e arithmeti surfa e, and (ψj) an orthonormal basis of eigenfun tions for the Lapla ian and the He ke operators. Let µ be a orresponding semi lassi al measure, with ergodi de omposition µ = µρdµ(ρ). Then, for µ-almost all ergodi omponents we have hKS(µ ρ) ≥ 1 As dis ussed above, the Liouville measure is the only one satisfying hKS(µ) = log Ju(ρ) dµ(ρ) [23℄, so the Quantum Unique Ergodi ity would be proven in one ould repla e 1/2 by 1 on the right hand side of (2.6). However, we believe that (2.6) is the optimal result that an be obtained without using mu h more pre ise information, like for instan e a sharp ontrol on the spe tral degenera ies, or �ne information on the lengths of losed geodesi s. Indeed, in the above mentioned examples of Anosov systems where the Quantum Unique Ergodi ity onje ture is wrong and the bound (2.6) sharp, the quantum spe trum has very high degenera ies, whi h ould be responsible for the possibility to onstru t ex eptional eigenstates. Su h high degenera ies are not expe ted in the ase of the Lapla ian on a neg- atively urved manifold. For the moment, however, there is no lear understanding of the pre ise relation between spe tral degenera ies and failure of Quantum Unique Ergodi ity. A knowledgements. N.A and S.N. were partially supported by the Agen e Nationale de la Re her he, under the grant ANR-05-JCJC-0107-01. They bene�ted from numerous dis ussions with Y. Colin de Verdière and M. Zworski. S.N. is grateful to the Mathemati al Department in Bonn for its hospitality in De ember 2006. 3. Outline of the proof We start by re alling the de�nition and some properties of the metri entropy asso iated with a probability measure on T ∗M , invariant through the geodesi �ow. In �3.2 we extend the notion of entropy to the quantum framework. Our approa h is semi lassi al, so we want the lassi al and quantum entropies to be onne ted in some way when ~ → 0. The weights appearing in our quantum entropy are estimated in Thm. 3.1, whi h was proven and used in [2℄. In �3.2.1 we also ompare our quantum entropy with several �quantum dynami al entropies� previously de�ned in the literature. The proof of Thm. 2.1 a tually starts in �3.3, where we present the algebrai tool allowing us to take advantage of our estimates (3.9) (or their optimized version given in Thm. 3.5), namely an �entropi un ertainty prin iple� spe i� of the quantum framework. From �3.4 on, we apply this �prin iple� to the quantum entropies appearing in our problem, and pro eed to prove Thm. 2.1. Although the method is basi ally the same as in [4℄, several small modi� ations allow to �nally obtain the improved lower bound (2.3), and also simplify some intermediate proofs, as explained in Remark 3.6. 3.1. De�nition of the metri entropy. In this paper we will meet several types of entropies, all of whi h are de�ned using the fun tion η(s) = −s log s, for s ∈ [0, 1]. We start with the Kolmogorov-Sinai entropy of the geodesi �ow with respe t to an invariant probability measure. Let µ be a probability measure on the otangent bundle T ∗M . Let P = (E1, . . . , EK) be a �nite measurable partition of T ∗M : T ∗M = i=1Ei. We will denote the set of indi es ENTROPY OF EIGENFUNCTIONS 7 {1, . . . , K} = [[1, K]]. The Shannon entropy of µ with respe t to the partition P is de�ned hP(µ) = − µ(Ek) logµ(Ek) = µ(Ek) For any integer n ≥ 1, we denote by P∨n the partition formed by the sets (3.1) E = Eα0 ∩ g −1Eα1 . . . ∩ g −n+1Eαn−1 , where α = (α0, . . . , αn−1) an be any sequen e in [[1, K]] (su h a sequen e is said to be of length |α| = n). The partition P∨n is alled the n-th re�nement of the initial partition P = P∨1. The entropy of µ with respe t to P∨n is denoted by (3.2) hn(µ,P) = hP∨n(µ) = α∈[[1,K]]n If µ is (gt)�invariant, it follows from the onvexity of the logarithm that (3.3) ∀n,m ≥ 1, hn+m(µ,P) ≤ hn(µ,P) + hm(µ,P), in other words the sequen e (hn(µ,P))n∈N is subadditive. The entropy of µ with respe t to the a tion of the geodesi �ow and to the partition P is de�ned by (3.4) hKS(µ,P) = lim hn(µ,P) = inf hn(µ,P) Ea h weight µ(E ) measures the µ�probability to visit su essively Eα0 , Eα1 , . . . , Eαn−1 at times 0, 1, . . . , n − 1 through the geodesi �ow. Roughly speaking, the entropy measures the exponential de ay of these probabilities when n gets large. It is easy to see that hKS(µ,P) ≥ β if there exists C su h that µ(Eα) ≤ C e , for all n and all α ∈ [[1, K]] Finally, the Kolmogorov-Sinai entropy of µ with respe t to the a tion of the geodesi �ow is de�ned as (3.5) hKS(µ) = sup hKS(µ,P), the supremum running over all �nite measurable partitions P. The hoi e to onsider the time 1 of the geodesi �ow in the de�nition (3.1) may seem arbitrary, but the entropy has a natural s aling property : the entropy of µ with respe t to the �ow (gat) is |a|�times its entropy with respe t to (gt). Assume µ is arried on the energy layer E . Due to the Anosov property of the geodesi �ow on E , it is known that the supremum (3.5) is rea hed as soon as the diameter of the partition P ∩ E (that is, the maximum diameter of its elements Ek ∩ E) is small enough. Furthermore, let us assume (without loss of generality) that the inje tivity radius of M is larger than 1. Then, we may restri t our attention to partitions P obtained by lifting on E a partition of the manifoldM , that is take M = k=1Mk and then Ek = T ∗Mk. In fa t, if the diameter of Mk in M is of order ε, then the diameter of the partition P ∨2 ∩E in E is also of order ε. This spe ial hoi e of our partition is not ru ial, but it simpli�es ertain aspe ts of the analysis. 8 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER The existen e of the limit in (3.4), and the fa t that it oin ides with the in�mum, follow from a standard subadditivity argument. It has a ru ial onsequen e : if (µi) is a sequen e of (gt)�invariant probability measures on T ∗M , weakly onverging to a probability µ, and if µ does not harge the boundary of the partition P, we have hKS(µ,P) ≥ lim sup hKS(µi,P) . In parti ular, assume that for i large enough, the following estimates hold : (3.6) ∀n ≥ 1, ∀α ∈ [[1, K]] , µi(Eα) ≤ Ci e −βn , with β independent of i. This implies for i large enough hKS(µi,P) ≥ β, and this estimate goes to the limit to yield hKS(µ) ≥ β. 3.2. From lassi al to quantum dynami al entropy. Sin e our semi lassi al measure µ is de�ned as a limit of Wigner distributions W~, a naive idea would be to estimate from below the KS entropy of W~ and then take the limit ~ → 0. This idea annot work dire tly, be ause the Wigner transformsW~ are neither positive, nor are they (g t)�invariant. Therefore, one annot dire tly use the (formal) integrals W~(Eα) = W~(x, ξ) dx dξ to ompute the entropy of the semi lassi al measure. Instead, the method initiated by the �rst author in [2℄ is based on the following remarks. Ea h integral W~(Eα) an also be written as W~(1lEα) = W~ 1lEα , where 1lEα is the hara teristi fun tion on the set E , that is (3.7) 1lEα = (1lEαn−1 ◦ g n−1)× . . .× (1lEα1 ◦ g)× 1lEα0 . Remember we took Ek = T ∗Mk, where the Mk form a partition of M . From the de�nition of the Wigner distribution, this integral orresponds formally to the overlap 〈ψ~,Op~(1lEα )ψ~〉. Yet, the hara teristi fun tions 1lEα have sharp dis ontinuities, so their quantizations annot be in orporated in a ni e pseudodi�erential al ulus. Besides, the set E is not ompa tly supported, and shrinks in the unstable dire tion when n = |α| −→ +∞, so that the operator Op (1lEα ) is very problemati . We also note that an overlap of the form 〈ψ~,Op~(1lEα)ψ~〉 is a hybrid expression: this is a quantum matrix element of an operator de�ned in terms of the lassi al evolution (3.7). From the point of view of quantum me hani s, it is more natural to onsider, instead, the operator obtained as the produ t of Heisenberg-evolved quantized fun tions, namely (3.8) (U−n+1 Pαn−1U ) (U−n+2 Pαn−2U ) · · · (U−1 Pα1U~)Pα0 . Here we used the shorthand notation Pk = 1lMk , k ∈ [[1, K]] (multipli ation operators). To remedy the fa t that the fun tions 1lMk are not smooth, whi h would prevent us from using a semi lassi al al ulus, we apply a onvolution kernel to smooth them, obtain fun tions 1lsmMk ∈ C ∞(M), and onsider Pk = 1lsmMk (we an do this keeping the property k=1 1l In the following, we will use the notation A(t) = U−t for the Heisenberg evolution of the operator A though the S hrödinger �ow U t = exp(−it~△ ). The norm ‖•‖ will denote ENTROPY OF EIGENFUNCTIONS 9 either the Hilbert norm on L2(M), or the orresponding operator norm. The subsequent �purely quantum� norms were estimated in [2, Thm. 1.3.3℄: Theorem 3.1. (The main estimate [2℄) Set as above Pk = 1lsmMk . For every K > 0, there exists ~K > 0 su h that, uniformly for all ~ < ~K, for all n ≤ K| log ~|, for all (α0, . . . , αn−1) ∈ [[1, K]] (3.9) ‖Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0 ψ~‖ ≤ 2(2π~) −d/2 e− n(1 +O(ε))n. The exponent Λ is given by the �smallest expansion rate�: Λ = − sup log Ju(ρ)dν(ρ) = inf λ+i (γ). The in�mum on the right hand side runs over the set of losed orbits on E , and the λ+i denote the positive Lyapunov exponents along the orbit, that is the logarithms of the expanding eigenvalues of the Poin aré map, divided by the period of the orbit. The parameter ε > 0 is an upper bound on the diameters of the supports of the fun tions 1lsmMk in M . From now on we will all the produ t operator (3.10) P = Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0 , α ∈ [[1, K]] To prove the above estimate, one a tually ontrols the operator norm (3.11) ‖P (χ)‖ ≤ 2(2π~)−d/2 e− n(1 +O(ε))n , where χ ∈ C∞c (E ε) is an energy uto� su h that χ = 1 near E , supported inside a neigh- bourhood Eε = H−1([1 − ε, 1 + ε]) of E . In quantum me hani s, the matrix element 〈ψ~, Pαψ~〉 looks like the �probability�, for a parti le in the state ψ~, to visit su essively the phase spa e regions Eα0 , Eα1 , . . . , Eαn−1 at times 0, 1, . . . , n − 1 of the S hrödinger �ow. Theorem 3.1 implies that this �probability� de ays exponentially fast with n, with rate Λ , but this de ay only starts around the time (3.12) n1 d| log ~| whi h is a kind of �Ehrenfest time� (see (3.25) for another de�nition of Ehrenfest time). Yet, be ause the matrix elements 〈ψ~, Pαψ~〉 are not real in general, they an hardly be used to de�ne a �quantum measure�. Another possibility to de�ne the probability for the parti le to visit the sets Eαk at times k, is to take the squares of the norms appearing in (3.9): (3.13) ‖P 2 = ‖Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0ψ~‖ Now we require the smoothed hara teristi fun tions 1lsmMi to satisfy the identity (3.14) 1lsmMk(x) = 1 for any point x ∈M . 10 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER We denote by Psm the smooth partition of M made by the fun tions (1lsmMk) . The orresponding set of multipli ation operators (Pk) = Pq forms a �quantum partition of unity� : (3.15) P 2k = IdL2 . For any n ≥ 1, we re�ne the quantum partition Pq into (Pα)|α|, as in (3.10). The weights (3.13) exa tly add up to unity, so it makes sense to onsider the entropy (3.16) hn(ψ~,Pq) α∈[[1,K]]n 3.2.1. Conne tion with other quantum entropies. This entropy appears to be a parti ular ase of the �general quantum entropies� des ribed by Sªom zy«ski and �y zkowski [28℄, who already had in mind appli ations to quantum haos. In their terminology, a family of bounded operators π = (πk) k=1 on a Hilbert spa e H satisfying (3.17) π∗k πk = IdH provides an �instrument� whi h, to ea h index k ∈ [[1,N ]], asso iates the following map on density matri es: ρ 7→ I(k)ρ = πk ρ π k , a nonnegative operator with tr(I(k)ρ) ≤ 1 . From a unitary propagator U and its adjoint a tion Uρ = UρU−1, they propose to onstru t the re�ned instrument I(α)ρ = I(αn−1) ◦ · · · U ◦ I(α1) ◦ U ◦ I(α0)ρ = U −n+1 π Un−1 , α ∈ [[1,N ]] where we used (3.10) to re�ne the operators πk into πα. We obtain the probability weights (3.18) tr(I(α)ρ) = tr(π ) , α ∈ [[1,N ]] For any U-invariant density ρ, these weights provide an entropy (3.19) hn(ρ, I) = α∈[[1,N ]]n tr(I(α)ρ) One easily he ks that our quantum partition Pq = (Pk) k=1 satis�es (3.17), and that if one takes ρ = |ψ~〉〈ψ~| the weights tr(I(α)ρ) exa tly orrespond to our weights ‖Pαψ‖ Hen e, the entropy (3.19) oin ides with (3.16). Around the same time, Ali ki and Fannes [1℄ used the same quantum partition (3.17) (whi h they alled ��nite operational partitions of unity�) to de�ne a di�erent type of entropy, now alled the �Ali ki-Fannes entropy� (the de�nition extends to general C∗- dynami al systems). For ea h n ≥ 1 they extend the weights (3.18) to �o�-diagonal entries� to form a N n ×N n density matrix ρn: (3.20) [ρn]α′,α = tr(πα′ ρ π ), α,α′ ∈ [[1,N ]] ENTROPY OF EIGENFUNCTIONS 11 The AF entropy of the system (U , ρ) is then de�ned as follows: take the Von Neumann entropy of these density matri es, hAFn (ρ, π) = tr η(ρn), then take lim supn→∞ hAFn (ρ, π) and �nally take the supremum over all possible �nite operational partitions of unity π. We mention that tra es of the form (3.20) also appear in the �quantum histories� ap- proa h to quantum me hani s (see e.g. [17℄, and [28, Appendix D℄ for referen es). 3.2.2. Naive treatment of the entropy hn(ψ~,Pq). For �xed |α| > 0, the Egorov theorem shows that ‖P onverges to the lassi al weight µ (1lsmMα ) when ~ → 0, so for �xed n > 0 the entropy hn(ψ~,Pq) onverges to hn(µ,Psm), de�ned as in (3.2), the hara teristi fun tions 1lMk being repla ed by their smoothed versions (1l )2. On the other hand, from the estimate (3.11) the entropies hn(ψ~,Pq) satisfy, for ~ small enough, (3.21) hn(ψ~,Pq) ≥ n Λ+O(ε) − d| log ~|+O(1) , for any time n ≤ K| log ~|. For large times n ≈ K| log ~|, this provides a lower bound hn(ψ~,Pq) ≥ Λ +O(ε) +O(1/| log ~|) , whi h looks very promising sin e K an be taken arbitrary large: we ould be tempted to take the semi lassi al limit, and dedu e a lower bound hKS(µ) ≥ Λ. Unfortunately, this does not work, be ause in the range {n > n1} where the estimate (3.21) is useful, the Egorov theorem breaks down, the weights (3.13) do not approximate the lassi al weights µ (1lsmMα ) , and there is no relationship between hn(ψ,Pq) and the lassi al entropies hn(µ,Psm). This breakdown of the quantum- lassi al orresponden e around the Ehrenfest time is ubiquitous for haoti dynami s. It has been observed before when studying the onne tion between the Ali ki-Fannes entropy for the quantized torus automorphisms and the KS entropy of the lassi al dynami s [5℄: the quantum entropies hAFn (ψ~,Pq) follow the lassi al hn(µ,Psm) until the Ehrenfest time (and therefore grow linearly with n), after whi h they �saturate�, to produ e a vanishing entropy lim supn→∞ hAFn (ψ~,Pq). To prove the Theorem 2.1, we will still use the estimates (3.11), but in a more subtle way, namely by referring to an entropi un ertainty prin iple. 3.3. Entropi un ertainty prin iple. The theorem below is an adaptation of the en- tropi un ertainty prin iple onje tured by Deuts h and Kraus [12, 21℄ and proved by Massen and U�nk [25℄. These authors were investigating the theory of measurement in quantum me hani s. Roughly speaking, this result states that if a unitary matrix has �small� entries, then any of its eigenve tors must have a �large� Shannon entropy. Let (H, 〈., .〉) be a omplex Hilbert spa e, and denote ‖ψ‖ = 〈ψ, ψ〉 the asso iated norm. Consider a quantum partition of unity (πk) k=1 on H as in (3.17). If ‖ψ‖ = 1, we de�ne the entropy of ψ with respe t to the partition π as in (3.16), namely hπ(ψ) = 12 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER ‖πk ψ‖ . We extend this de�nition by introdu ing the notion of pressure, asso i- ated to a family v = (vk)k=1,...,N of positive real numbers: the pressure is de�ned by pπ,v(ψ) ‖πk ψ‖ ‖πk ψ‖ 2 log v2k. In Theorem 3.2, we a tually need two partitions of unity (πk) k=1 and (τj) j=1, and two families of weights v = (vk) k=1, w = (wj) j=1, and onsider the orresponding pressures pπ,v(ψ), pτ,w(ψ). Besides the appearan e of the weights v, w, we bring another modi� ation to the statement in [25℄ by introdu ing an auxiliary operator O. Theorem 3.2. [4, Thm. 6.5℄ Let O be a bounded operator and U be an isometry on H. De�ne c (v,w) O (U) = supj,k wj vk ‖τj U π kO‖, and V = maxk vk, W = maxj wj. Then, for any ǫ ≥ 0, for any normalized ψ ∈ H satisfying (3.22) ∀k = 1, . . . ,N , ‖(Id−O) πk ψ‖ ≤ ǫ , the pressures pτ,w , pπ,v satisfy + pπ,v ≥ −2 log (v,w) O (U) +N V W ǫ Example 1. The original result of [25℄ orresponds to the ase where H = CN , O = Id, ǫ = 0, N = M, vk = wj = 1, and the operators πk = τk are the orthogonal proje tors on some orthonormal basis (ek) k=1 of H. In this ase, the theorem asserts that hπ(U ψ) + hπ(ψ) ≥ −2 log c(U) where c(U) = supj,k |〈ek,Uej〉| is the supremum of all matrix elements of U in the orthonor- mal basis (ek). As a spe ial ase, one gets hπ(ψ) ≥ − log c(U) if ψ is an eigenfun tion of 3.4. Applying the entropi un ertainty prin iple to the Lapla ian eigenstates. In this se tion we explain how to use Theorem 3.2 in order to obtain nontrivial information on the quantum entropies (3.16) and then hKS(µ). For this we need to de�ne the data to input in the theorem. Ex ept the Hilbert spa e H = L2(M), all other data depend on the semi lassi al parameter ~: the quantum partition π, the operator O, the positive real number ǫ, the weights (vj), (wk) and the unitary operator U . As explained in se tion 3.2, we partition M into M = ⊔Kk=1Mk, onsider open sets Ωk ⊃⊃Mk (whi h we assume to have diameters ≤ ε), and onsider smoothed hara teristi fun tions 1lsmMk supported respe tively inside Ωk, and satisfying the identity (3.14). The asso iated multipli ation operators on H are form a quantum partition (Pk) k=1, whi h we had alled Pq. To alleviate notations, we will drop the subs ript q. From (3.15), and using the unitarity of U~, one realizes that for any n ≥ 1, the families of operators P∨n = (P ∗ )|α|=n and T ∨n = (P )|α|=n (see (3.10)) make up two quantum partitions of unity as in (3.17), of ardinal Kn. ENTROPY OF EIGENFUNCTIONS 13 3.4.1. Sharp energy lo alization. In the estimate (3.11), we introdu ed an energy uto� χ on a �nite energy strip Eε, with χ ≡ 1 near E . This uto� does not appear in the estimate (3.9), be ause, when applied to the eigenstate ψ~, the operator Op~(χ) essentially a ts like the identity. The estimate (3.11) will a tually not su� e to prove Theorem 2.1. We will need to optimize it by repla ing χ in (3.11) with a �sharp� energy uto�. For some �xed (small) δ ∈ (0, 1), we onsider a smooth fun tion χδ ∈ C ∞(R; [0, 1]), with χδ(t) = 1 for |t| ≤ e and χδ(t) = 0 for |t| ≥ 1. Then, we res ale that fun tion to obtain the following family of ~-dependent uto�s near E : (3.23) ∀~ ∈ (0, 1), ∀n ∈ N, ∀ρ ∈ T ∗M, χ(n)(ρ; ~) e−nδ ~−1+δ(H(ρ)− 1/2) The uto� χ(n) is supported in a tubular neighbourhood of E of width 2~1−δ enδ. We will always assume that this width is << ~1/2 in the semi lassi al limit, whi h is the ase if we ensure that n ≤ Cδ| log ~| for some 0 < Cδ < (2δ) −1−1. In spite of their singular behaviour, these uto�s an be quantized into pseudodi�erential operators Op(χ(n)) des ribed in [4℄ (the quantization uses a pseudodi�erential al ulus adapted to the energy layer E , drawn from [29℄). The eigenstate ψ~ is indeed very lo alized near E , sin e it satis�es (3.24) ‖ Op(χ(0))− 1 ψ~‖ = O(~ ∞) ‖ψ~‖ . In the rest of the paper, we also �x a small δ′ > 0, and all �Ehrenfest time� the ~-dependent integer (3.25) nE(~) ⌊(1− δ′)| log ~| Noti e the resemblan e with the time n1 de�ned in (3.12). The signi� an e of this time s ale will be dis ussed in �3.4.5. The following proposition states that the operators (P ∗ )|α|=nE , almost preserve the en- ergy lo alization of ψ~ : Proposition 3.3. For any L > 0, there exists ~L su h that, for any ~ ≤ ~L, the Lapla ian eigenstate satis�es (3.26) ∀α, |α| = nE , ‖ Op(χ(nE))− Id ψ~‖ ≤ ~ L‖ψ~‖ . We re ognize here a ondition of the form (3.22). 3.4.2. Applying Theorem 3.2: Step 1. We now pre ise some of the data we will use in the entropi un ertainty prin iple, Theorem 3.2. As opposed to the hoi e made in [4℄, we will use two di�erent partitions π, τ . • the quantum partitions π and τ are given respe tively by the families of operators π = P∨nE = (P ∗ )|α|=nE , τ = T ∨nE = (P )|α|=nE . Noti e that these partitions only di�er by the ordering of the operators Pαi(i) inside the produ ts. In the semi lassi al limit, these partitions have ardinality N = KnE ≍ ~−K0 for some �xed K0 > 0. • the isometry will be the propagator at the Ehrenfest time, U = UnE 14 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER • the auxiliarly operator is given as O = Op(χ(nE)), and the error ǫ = ~L, where L will be hosen very large (see �3.4.4). • the weights v will be sele ted in �3.4.4. They will be semi lassi ally tempered, meaning that there exists K1 > 0 su h that, for ~ small enough, all vα, wα are ontained in the interval [1, ~−K1]. The entropy and pressures asso iated with a state ψ ∈ H are given by hπ(ψ) = |α|=nE ,(3.27) pπ,v(ψ) = hπ(ψ)− 2 |α|=nE ψ‖2 log v .(3.28) With respe t to the se ond partition, we have hτ (ψ) = |α|=nE ,(3.29) pτ,w(ψ) = hτ (ψ)− 2 |α|=nE ψ‖2 logw .(3.30) We noti e that the entropy hτ (ψ) exa tly orresponds to the formula (3.16), while hπ(ψ) is built from the norms ψ‖2 = ‖Pα0Pα1(1) · · ·Pαn−1(n− 1)ψ‖ If ψ is an eigenfun tion of U~, the above norm an be obtained from (3.13) by ex hanging U~ with U , and repla ing the sequen e α = (α0, . . . , αn−1) by ᾱ = (αn−1, . . . , α0). So the entropies hπ(ψ) and hτ (ψ) are mapped to one another through the time reversal U~ → U With these data, we draw from Theorem 3.2 the following Corollary 3.4. For ~ > 0 small enough onsider the data π, τ , U , O as de�ned above. (3.31) c O (U) = max |α|=|α′|=nE Op(χ(nE))‖ Then for any normalized state φ satisfying (3.26), pτ,w(U φ) + pπ,v(φ) ≥ −2 log O (U) + h L−K0−2K1 From (3.26), we see that the above orollary applies to the eigenstate ψ~ if ~ is small enough. The reason to take the same value nE for the re�ned partitions P , T ∨nE and the propagator U is the following : the produ ts appearing in c O (U) an be rewritten (with U ≡ U~): ′ UnE P = U−nE+1Pα′ U · · ·UPα′0UPαnE−1U · · ·UPα0 = U ENTROPY OF EIGENFUNCTIONS 15 Thus, the estimate (3.11) with n = 2nE already provides an upper bound for the norms appearing in (3.31) � the repla ement of χ by the sharp uto� χ(nE) does not harm the estimate. To prove Theorem 2.1, we a tually need to improve the estimate (3.11), as was done in [4℄, see Theorem 3.5 below. This improvement is done at two levels: we will use the fa t that the uto�s χ(nE) are sharper than χ, and also the fa t that the expansion rate of the geodesi �ow (whi h governs the upper bound in (3.11)) is not uniform, but depends on the sequen e α. Our hoi e for the weights v will then be guided by the α-dependent upper bounds given in Theorem 3.5. To state that theorem, we introdu e some notations. 3.4.3. Coarse-grained unstable Ja obian. We re all that, for any energy λ > 0, the geodesi �ow gt on the energy layer E(λ) = H−1(λ) ⊂ T ∗M is Anosov, so that the tangent spa e TρE(λ) at ea h ρ ∈ T ∗M , H(ρ) > 0 splits into TρE(λ) = E u(ρ)⊕Es(ρ)⊕ RXH(ρ) where Eu (resp. Es) is the unstable (resp. stable) subspa e. The unstable Ja obian Ju(ρ) is de�ned by Ju(ρ) = det |Eu(g1ρ) (the unstable spa es at ρ and g1ρ are equipped with the indu ed Riemannian metri ). This Ja obian an be �dis retized� as follows in the energy strip Eε ⊃ E . For any pair of indi es (α0, α1) ∈ [[1, K]] , we de�ne (3.32) Ju1 (α0, α1) = sup Ju(ρ) : ρ ∈ T ∗Ωα0 ∩ E ε, g1ρ ∈ T ∗Ωα1 if the set on the right hand side is not empty, and Ju1 (α0, α1) = e otherwise, where R > 0 is a �xed large number. For any sequen e of symbols α of length n, we de�ne (3.33) Jun(α) = Ju1 (α0, α1) · · ·J 1 (αn−2, αn−1) . Although Ju and Ju1 (α0, α1) are not ne essarily everywhere smaller than unity, there exists C, λ+, λ− > 0 su h that, for any n > 0, for any α with |α| = n, (3.34) C−1 e−n(d−1) λ+ ≤ Jun(α) ≤ C e −n(d−1) λ− . One an take λ+ = λmax(1+ε), where λmax is the maximal expanding rate in Theorem. 2.1. We now give our entral estimate, easy to draw from [4, Corollary 3.4℄. Theorem 3.5. Fix small positive onstants ε, δ, δ′ and a onstant 0 < Cδ < (2δ) −1 − 1. Take an open over M = k Ωk of diameter ≤ ε and an asso iated quantum partition P = k=1. There exists ~0 su h that, for any ~ ≤ ~0, for any positive integer n ≤ Cδ| log ~|, and any pair of sequen es α, α of length n, (3.35) ‖P ′ Op(χ(n))‖ = ‖P Op(χ(n))‖ ≤ C ~− −δ enδ Jun(α) J The onstant C only depends on the Riemannian manifold (M, g). If we take n = nE, this takes the form (3.36) ‖P Op(χ(nE))‖ ≤ C ~− d−1+cδ JunE(α) J (α′) , 16 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER where c = 2 + 2λ−1max. The idea of proof in Theorem 3.5 is rather simple, although the te hni al implementation is umbersome. We �rst show that for any normalized state ψ, the state Op(χ(n))ψ an be essentially de omposed into a superposition of ~ −d| suppχ(n)| normalized Lagrangian states, supported on Lagrangian manifolds transverse to the stable foliation. In fa t the Lagrangian states we work with are trun ated δ�fun tions, supported on lagrangians of the form ∪tg tS∗zM . The a tion of the operator U ′ = Pα′ U · · ·UPα0 on su h Lagrangian states an be analyzed through WKB methods, and is simple to understand at the lassi al level : ea h appli ation of the propagator U stret hes the Lagrangian along the unstable dire tion (the rate of stret hing being des ribed by the lo al unstable Ja obian), whereas ea h operator Pk �proje ts� on a pie e of Lagrangian of diameter ε. This iteration of stret hing and utting a ounts for the exponential de ay. The αα -independent fa tor on the right of (3.36) results from adding together the ontributions of all the initial Lagrangian states. Noti e that this prefa tor is smaller than in Theorem. 3.1 due to the ondition Cδ < (2δ) −1 − 1. Remark 3.6. In [4℄ we used the same quantum partition P∨nE for π and τ in Theorem. 3.2. As a result, we needed to estimate from above the norms ‖P ∗ ′ UnE PαOp(χ (nE))‖ (see [4, Theorem. 2.6℄). The proof of this estimate was mu h more involved than the one for (3.36), sin e it required to ontrol long pie es of unstable manifolds. By using instead the two partitions P(n), T (n), we not only prove a more pre ise lower bound (2.3) on the KS entropy, but also short- ir uit some �ne dynami al analysis. 3.4.4. Applying Theorem 3.2: Step 2. There remains to hoose the weights (v ) to use in Theorem 3.2. Our hoi e is guided by the following idea: in (3.31), the weights should balan e the variations (with respe t to α,α′) in the norms, su h as to make all terms in (3.31) of the same order. Using the upper bounds (3.36), we end up with the following hoi e for all α of length nE : = JunE(α) −1/2 . From (3.34), there exists K1 > 0 su h that, for ~ small enough, all the weights are ontained in the interval [1, ~−K1], as announ ed in �3.4.2. Using these weights, the estimate (3.36) implies the following bound on the oe� ient (3.31): ∀~ < ~0, c O (U) ≤ C ~ − d−1+cδ We an now apply Corollary 3.4 to the parti ular ase of the eigenstates ψ~. We hoose L su h that L−K0 − 2K1 > − d−1+cδ , so from Corollary 3.4 we draw the following Proposition 3.7. Let (ψ~)~→0 be our sequen e of eigenstates (2.2). In the semi lassi al limit, the pressures of ψ~ satisfy (3.37) pP∨nE ,v(ψ~) + pT ∨nE ,w(ψ~) ≥ − (d − 1 + cδ)λmax (1− δ′) nE +O(1) . ENTROPY OF EIGENFUNCTIONS 17 If M has onstant urvature we have log Jn ≤ −n(d − 1)λmax(1 − O(ε)) for all α of length n, and the above lower bound an be written hP∨nE (ψ~) + hT ∨nE (ψ~) ≥ (d− 1)λmax 1 +O(ε, δ, δ′) As opposed to (3.21), the above inequality provides a nontrivial lower bound for the quan- tum entropies at the time nE , whi h is smaller than the time n1 of (3.12), and will allow to onne t those entropies to the KS entropy of the semi lassi al measure (see below). 3.4.5. Subadditivity until the Ehrenfest time. Even at the relatively small time nE , the onne tion between the quantum entropy h(ψ~,P ∨nE) and the lassi al h(µ,P∨nEsm ) is not ompletely obvious: both are sums of a large number of terms (≍ ~−K0). Before taking the limit ~ → 0, we will prove that a lower bound of the form (3.37) still holds if we repla e nE ≍ | log ~| by some �xed no ∈ N, and P by the orresponding quantum partition P∨no . The link between quantum pressures at times nE and no is provided by the following subadditivity property, whi h is the semi lassi al analogue of the lassi al subadditivity of pressures for invariant measures (see (3.3)). Proposition 3.8 (Subadditivity). Let δ′ > 0. There is a fun tion R(no, ~), and a real number R > 0 independent of δ′, su h that, for any integer no ≥ 1, lim sup |R(no, ~)| ≤ R and with the following properties. For any small enough ~ > 0, any integers no, n ∈ N with no + n ≤ nE(~), for any ψ~ normalized eigenstate satisfying (2.2), the following inequality holds: pP∨(no+n),v(ψ~) ≤ pP∨no ,v(ψ~) + pP∨n,v(ψ~) +R(no, ~) . The same inequality is satis�ed by the pressures pT ∨n,w(ψ~). To prove this proposition, one uses a re�ned version of Egorov's theorem [10℄ to show that the non� ommutative dynami al system formed by (U t ) a ting (through Heisenberg) on observables supported near E is (approximately) ommutative on time intervals of length nE(~). Pre isely, we showed in [4℄ that, provided ε is small enough, for any a, b ∈ C ∀t ∈ [−nE(~), nE(~)], ‖[Op~(a)(t),Op~(b)]‖ = O(~ cδ′), ~ → 0 , and the implied onstant is uniform with respe t to t. Within that time interval, the oper- ators Pαj (j) appearing in the de�nition of the pressures ommute up to small semi lassi al errors. This almost ommutativity explains why the quantum pressures pP∨n,v(ψ~) satisfy the same subadditivity property as the lassi al entropy (3.3), for times smaller than nE . Thanks to this subadditivity, we may �nish the proof of Theorem. 2.1. Fixing no, using for ea h ~ the Eu lidean division nE(~) = q(~)no + r(~) (with r(~) < no), Proposition 3.8 implies that for ~ small enough, pP∨nE ,v(ψ~) pP∨no ,v(ψ~) pP∨r,v(ψ~) R(no, ~) 18 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER The same inequality is satis�ed by the pressures pT ∨n,w(ψ~). Using (3.37) and the fa t that pP∨r,v(ψ~) stays uniformly bounded when ~ → 0, we �nd (3.38) pP∨no ,v(ψ~) + pT ∨no ,w(ψ~) 2(d− 1 + cδ)λmax 2(1− δ′) 2R(no, ~) +Ono(1/nE) . We are now dealing with quantum partitions P∨no , T ∨no , for n0 ∈ N independent of ~. At this level the quantum and lassi al entropies are related through the (�nite time) Egorov theorem, as we had noti ed in �3.2.2. For any α of length no, the weights ‖Pαψ~‖ both onverge to µ (1lsmMα) , where we re all that 1lsmMα = (1l Mαno−1 ◦ gno−1)× . . .× (1lsmMα1 ◦ g)× 1lsmMα0 Thus, both entropies hP∨no (ψ~), hT ∨no (ψ~) semi lassi ally onverge to the lassi al entropy hno(µ,Psm). As a result, the left hand side of (3.38) onverges to (3.39) 2 hno(µ,Psm) |α|=no (1lsmMα ) log Juno(α) . Sin e µ is gt-invariant and Juno has the multipli ative stru ture (3.33), the se ond term in (3.39) an be simpli�ed: |α|=no (1lsmMα ) log Juno(α) = (no − 1) α0,α1 (1lsmM(α0,α1) log Ju1 (α0, α1) . We have thus obtained the lower bound (3.40) hno(µ,Psm) no − 1 α0,α1 (1lsmM(α0,α1) log Ju1 (α0, α1)− (d− 1 + cδ)λmax 2(1− δ′) At this stage we may forget about δ and δ′. The above lower bound does not depend on the derivatives of the fun tions 1lsmMα , so the same bound arries over if we repla e 1l the hara teristi fun tions 1lMα . We an �nally let no tend to +∞, then let the diameter ε tend to 0. 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Zeldit h, Uniform distribution of the eigenfun tions on ompa t hyperboli surfa es, Duke Math. J. 55, 919�941 (1987) CMLS, É ole Polyte hnique, 91128 Palaiseau, Fran e E-mail address : nalini�math.polyte hnique.fr Mathemati al Institute, University of Bonn, Beringstraÿe 1,D-53115 Bonn, Germany E-mail address : ko h�math.uni-bonn.de Servi e de Physique Théorique, CEA/DSM/PhT, Unité de re her he asso iée au CNRS, CEA/Sa lay, 91191 Gif-sur-Yvette, Fran e E-mail address : snonnenma her� ea.fr 1. Motivations 2. Main result. Acknowledgements 3. Outline of the proof 3.1. Definition of the metric entropy 3.2. From classical to quantum dynamical entropy 3.3. Entropic uncertainty principle 3.4. Applying the entropic uncertainty principle to the Laplacian eigenstates References

We study the high--energy limit for eigenfunctions of the laplacian, on a compact negatively curved manifold. We review the recent result of Anantharaman-Nonnenmacher giving a lower bound on the Kolmogorov-Sinai entropy of semiclassical measures, and improve this lower bound in the case of variable negative curvature.

ENTROPY OF EIGENFUNCTIONS NALINI ANANTHARAMAN, HERBERT KOCH, AND STÉPHANE NONNENMACHER Abstra t. We study the high�energy limit for eigenfun tions of the lapla ian, on a ompa t negatively urved manifold. We review the re ent result of Anantharaman� Nonnenma her [4℄ giving a lower bound on the Kolmogorov�Sinai entropy of semi lassi al measures. The bound proved here improves the result of [4℄ in the ase of variable negative urvature. 1. Motivations The theory of quantum haos tries to understand how the haoti behaviour of a lassi- al Hamiltonian system is re�e ted in its quantum ounterpart. For instan e, let M be a ompa t Riemannian C∞ manifold, with negative se tional urvatures. The geodesi �ow has the Anosov property, whi h is onsidered as the ideal haoti behaviour in the theory of dynami al systems. The orresponding quantum dynami s is the unitary �ow gener- ated by the Lapla e-Beltrami operator on L2(M). One expe ts that the haoti properties of the geodesi �ow in�uen e the spe tral theory of the Lapla ian. The Random Matrix onje ture [7℄ asserts that the large eigenvalues should, after proper unfolding, statisti- ally resemble those of a large random matrix, at least for a generi Anosov metri . The Quantum Unique Ergodi ity onje ture [26℄ (see also [6, 30℄) des ribes the orresponding eigenfun tions ψk: it laims that the probability measure |ψk(x)| 2dx should approa h (in the weak topology) the Riemannian volume, when the eigenvalue tends to in�nity. In fa t a stronger property should hold for the Wigner transform Wψ, a fun tion on the otangent bundle T ∗M , (the lassi al phase spa e) whi h simultaneously des ribes the lo alization of the wave fun tion ψ in position and momentum. We will adopt a semi lassi al point of view, that is onsider the eigenstates of eigenvalue unity of the semi lassi al Lapla ian −~2△, thereby repla ing the high-energy limit by the semi lassi al limit ~ → 0. We denote by (ψk)k∈N an orthonormal basis of L 2(M) made of eigenfun tions of the Lapla ian, and by (− 1 )k∈N the orresponding eigenvalues: (1.1) − ~2k△ψk = ψk, with ~k+1 ≤ ~k . We are interested in the high-energy eigenfun tions of −△, in other words the semi lassi al limit ~k → 0. The Wigner distribution asso iated to an eigenfun tion ψk is de�ned by Wk(a) = 〈Op~k(a)ψk, ψk〉L2(M), a ∈ C ∗M) . Here Op is a quantization pro edure, set at the s ale (wavelength) ~k, whi h asso iates to any smooth phase spa e fun tion a (with ni e behaviour at in�nity) a bounded operator on http://arxiv.org/abs/0704.1564v1 2 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER L2(M). See for instan e [13℄ or [14℄ for various quantizations Op . On a manifold, one an use lo al oordinates to de�ne Op in a �nite system of harts, then glue the obje ts de�ned lo ally thanks to a smooth partition of unity [11℄. For standard quantizations Op the Wigner distribution is of the form Wk(x, ξ) dx dξ, where Wk(x, ξ) is a smooth fun tion on T ∗M , alled the Wigner transform of ψ. If a is a fun tion on the manifold M , Op an be taken as the multipli ation by a, and thus we have Wk(a) = a(x)|ψk(x)| 2dx: the Wigner transform is thus a mi rolo al lift of the density |ψk(x)| . Although the de�nition ofWk depends on a ertain number of hoi es, like the hoi e of lo al oordinates, or of the quantization pro edure (Weyl, anti-Wi k, �right� or �left� quantization...), its asymptoti behaviour when ~k −→ 0 does not. A ordingly, we all semi lassi al measures the limit points of the sequen e (Wk)k∈N, in the distribution topology. In the semi lassi al limit, �quantum me hani s onverges to lassi al me hani s�. We will denote |·|x the norm on T xM given by the metri . The geodesi �ow (g t)t∈R is the Hamiltonian �ow on T ∗M generated by the Hamiltonian H(x, ξ) = |ξ|2x . A quantization of this Hamiltonian is given by the res aled Lapla ian −~ , whi h generates the unitary �ow (U t ) = (exp(it~△ )) a ting on L2(M). The semi lassi al orresponden e of the �ows ) and (gt) is expressed through the Egorov Theorem : Theorem 1.1. Let a ∈ C∞c (T ∗M). Then, for any given t in R, (1.2) ‖U−t (a)U t (a ◦ gt)‖L2(M) = O(~) , ~ → 0 . The onstant implied in the remainder grows (often exponentially) with t, whi h rep- resents a notorious problem when one wants to study the large time behaviour of (U t Typi ally, the quantum- lassi al orresponden e will break down for times t of the order of the Ehrenfest time (3.25). Using (1.2) and other standard semi lassi al arguments, one shows the following : Proposition 1.2. Any semi lassi al measure is a probability measure arried on the energy layer E = H−1(1 ) (whi h oin ides with the unit otangent bundle S∗M). This measure is invariant under the geodesi �ow. Let us all M the set of gt-invariant probability measures on E . This set is onvex and ompa t for the weak topology. If the geodesi �ow has the Anosov property � for instan e if M has negative se tional urvature � that set is very large. The geodesi �ow has ountably many periodi orbits, ea h of them arrying an invariant probability measure. There are many other invariant measures, like the equilibrium states obtained by variational prin iples [19℄, among them the Liouville measure µLiouv, and the measure of maximal entropy. Note that, for all these examples of measures, the geodesi �ow a ts ergodi ally, meaning that these examples are extremal points in M. Our aim is to determine, at least partially, the set Msc formed by all possible semi lassi al measures. By its de�nition, Msc is a losed subset of M, in the weak topology. For manifolds su h that the geodesi �ow is ergodi with respe t to the Liouville measure, it has been known for some time that almost all eigenfun tions be ome equidistributed over E , in the semi lassi al limit. This property is dubbed as Quantum Ergodi ity : ENTROPY OF EIGENFUNCTIONS 3 Theorem 1.3. [27, 32, 11℄ Let M be a ompa t Riemannian manifold, assume that the a tion of the geodesi �ow on E = S∗M is ergodi with respe t to the Liouville measure. Let (ψk)k∈N be an orthonormal basis of L 2(M) onsisting of eigenfun tions of the Lapla ian (1.1), and let (Wk) be the asso iated Wigner distributions on T Then, there exists a subset S ⊂ N of density 1, su h that (1.3) Wk −→µLiouv, k → ∞, k ∈ S. The question of existen e of �ex eptional� subsequen es of eigenstates with a di�erent behaviour is still open. On a negatively urved manifold, the geodesi �ow satis�es the ergodi ity assumption, and in fa t mu h stronger properties : mixing, K�property, et . For su h manifolds, it has been postulated in the Quantum Unique Ergodi ity onje ture [26℄ that the full sequen e of eigenstates be omes semi lassi ally equidistributed over E : one an take S = N in the limit (1.3). In other words, this onje ture states that there exists a unique semi lassi al measure, and Msc = {µLiouv}. So far the most pre ise results on this question were obtained for manifolds M with onstant negative urvature and arithmeti properties: see Rudni k�Sarnak [26℄, Wolpert [31℄. In that very parti ular situation, there exists a ountable ommutative family of self�adjoint operators ommuting with the Lapla ian : the He ke operators. One may thus de ide to restri t the attention to ommon bases of eigenfun tions, often alled �arith- meti � eigenstates, or He ke eigenstates. A few years ago, Lindenstrauss [24℄ proved that any sequen e of arithmeti eigenstates be ome asymptoti ally equidistributed. If there is some degenera y in the spe trum of the Lapla ian, note that it ould be possible that the Quantum Unique Ergodi ity onje tured by Rudni k and Sarnak holds for one orthonormal basis but not for another. On su h arithmeti manifolds, it is believed that the spe trum of the Lapla ian has bounded multipli ity: if this is really the ase, then the semi lassi al equidistribution easily extends to any sequen e of eigenstates. Nevertheless, one may be less optimisti when extending the Quantum Unique Ergod- i ity onje ture to more general systems. One of the simplest example of a symple ti Anosov dynami al system is given by linear hyperboli automorphisms of the 2-torus, e.g. Arnold's � at map� . This system an be quantized into a sequen e ofN×N unitary matri es � the propagators, where N ∼ ~−1 [18℄. The eigenstates of these matri es satisfy a Quantum Ergodi ity theorem similar with Theorem 1.3, meaning that almost all eigen- states be ome equidistributed on the torus in the semi lassi al limit [9℄. Besides, one an hoose orthonormal eigenbases of the propagators, su h that the whole sequen e of eigen- states is semi lassi ally equidistributed [22℄. Still, be ause the spe tra of the propagators are highly degenerate, one an also onstru t sequen es of eigenstates with a di�erent limit measure [15℄, for instan e, a semi lassi al measure onsisting in two ergodi omponents: half of it is the Liouville measure, while the other half is a Dira peak on a single (unsta- ble) periodi orbit. It was also shown that this half-lo alization is maximal for this model [16℄ : a semi lassi al measure annot have more than half its mass arried by a ountable union of periodi orbits. The same type of half-lo alized eigenstates were onstru ted by two of the authors for another solvable model, namely the �Walsh quantization� of the 4 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER baker's map on the torus [3℄; for that model, there exist ergodi semi lassi al measures of purely fra tal type (that is, without any Liouville omponent). Another type of semi las- si al measure was re ently obtained by Kelmer for quantized hyperboli automorphisms on higher-dimensional tori [20℄: it onsists in the Lebesgue measure on some invariant o-isotropi subspa e of the torus. For these Anosov models on tori, the onstru tion of ex eptional eigenstates strongly uses nongeneri algebrai properties of the lassi al and quantized systems, and annot be generalized to nonlinear systems. 2. Main result. In order to understand the set Msc, we will attempt to ompute the Kolmogorov�Sinai entropies of semi lassi al measures. We work on a ompa t Riemannian manifold M of arbitrary dimension, and assume that the geodesi �ow has the Anosov property. A tually, our method an without doubt be adapted to more general Anosov Hamiltonian systems. The Kolmogorov�Sinai entropy, also alled metri entropy, of a (gt)-invariant probability measure µ is a nonnegative number hKS(µ) that des ribes, in some sense, the omplexity of a µ-typi al orbit of the �ow. The pre ise de�nition will be given later, but for the moment let us just give a few fa ts. A measure arried on a losed geodesi has vanishing entropy. In onstant urvature, the entropy is maximal for the Liouville measure. More generally, for any Anosov �ow, the energy layer E is foliated into unstable manifolds of the �ow. An upper bound on the entropy of an invariant probability measure is then provided by the Ruelle inequality: (2.1) hKS(µ) ≤ log Ju(ρ)dµ(ρ) In this inequality, Ju(ρ) is the unstable Ja obian of the �ow at the point ρ ∈ E , de�ned as the Ja obian of the map g−1 restri ted to the unstable manifold at the point g1ρ (note that the average of log Ju over any invariant measure is negative). The equality holds in (2.1) if and only if µ is the Liouville measure on E [23℄. If M has dimension d and has onstant se tional urvature −1, the above inequality just reads hKS(µ) ≤ d− 1. Finally, an important property of the metri entropy is that it is an a�ne fun tional on M. A ording to the Birkho� ergodi theorem, for any µ ∈ M and for µ�almost every ρ ∈ E , the weak limit µρ = lim |t|−→∞ δgsρds exists, and is an ergodi probability measure. We an then write µρdµ(ρ), whi h realizes the ergodi de omposition of µ. The a�neness of the KS entropy means hKS(µ) = hKS(µ ρ)dµ(ρ). ENTROPY OF EIGENFUNCTIONS 5 An obvious onsequen e is the fa t that the range of hKS on M is an interval [0, hmax]. In the whole arti le, we onsider a ertain subsequen e of eigenstates (ψkj )j∈N of the Lapla ian, su h that the orresponding sequen e of Wigner distributions (Wkj) onverges to a semi lassi al measure µ. In the following, the subsequen e (ψkj )j∈N will simply be denoted by (ψ~)~→0, using the slightly abusive notation ψ~ = ψ~kj for the eigenstate ψkj . Ea h eigenstate ψ~ thus satis�es (2.2) (−~2 △−1)ψ~ = 0 . In [2℄ the �rst author proved that the entropy of any µ ∈ Msc is stri tly positive. In [4℄, more expli it lower bounds were obtained. The aim of this paper is to improve the lower bounds of [4℄ into the following Theorem 2.1. Let µ be a semi lassi al measure asso iated to the eigenfun tions of the Lapla ian on M . Then its metri entropy satis�es (2.3) hKS(µ) ≥ log Ju(ρ)dµ(ρ) (d− 1) λmax , where d = dimM and λmax = limt→±∞ log supρ∈E |dg ρ| is the maximal expansion rate of the geodesi �ow on E . In parti ular, if M has onstant se tional urvature −1, we have (2.4) hKS(µ) ≥ In dimension d, we always have log Ju(ρ)dµ(ρ) ≤ (d− 1)λmax , so the above bound is an improvement over the one obtained in [4℄, (2.5) hKS(µ) ≥ log Ju(ρ)dµ(ρ) − (d− 1)λmax . In the ase of onstant or little-varying urvature, the bound (2.4) is mu h sharper than the one proved in [2℄. On the other hand, if the urvature varies a lot (still being negative everywhere), the right hand side of (2.3) may a tually be negative, in whi h ase the bound is trivial. We believe this �problem� to be a te hni al short oming of our method, and a tually onje ture the following bound: (2.6) hKS(µ) ≥ log Ju(ρ)dµ(ρ) Extended to the ase of the quantized torus automorphisms or the Walsh-quantized baker's map, this bound is saturated for the half-lo alized semi lassi al measures onstru ted in [15℄, as well as those obtained in [20, 3℄. This bound allows ertain ergodi omponents to be arried by losed geodesi s, as long as other omponents have positive entropy. This may be ompared with the following result obtained by Bourgain and Lindenstrauss in the ase of arithmeti surfa es : 6 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER Theorem 2.2. [8℄ Let M be a ongruen e arithmeti surfa e, and (ψj) an orthonormal basis of eigenfun tions for the Lapla ian and the He ke operators. Let µ be a orresponding semi lassi al measure, with ergodi de omposition µ = µρdµ(ρ). Then, for µ-almost all ergodi omponents we have hKS(µ ρ) ≥ 1 As dis ussed above, the Liouville measure is the only one satisfying hKS(µ) = log Ju(ρ) dµ(ρ) [23℄, so the Quantum Unique Ergodi ity would be proven in one ould repla e 1/2 by 1 on the right hand side of (2.6). However, we believe that (2.6) is the optimal result that an be obtained without using mu h more pre ise information, like for instan e a sharp ontrol on the spe tral degenera ies, or �ne information on the lengths of losed geodesi s. Indeed, in the above mentioned examples of Anosov systems where the Quantum Unique Ergodi ity onje ture is wrong and the bound (2.6) sharp, the quantum spe trum has very high degenera ies, whi h ould be responsible for the possibility to onstru t ex eptional eigenstates. Su h high degenera ies are not expe ted in the ase of the Lapla ian on a neg- atively urved manifold. For the moment, however, there is no lear understanding of the pre ise relation between spe tral degenera ies and failure of Quantum Unique Ergodi ity. A knowledgements. N.A and S.N. were partially supported by the Agen e Nationale de la Re her he, under the grant ANR-05-JCJC-0107-01. They bene�ted from numerous dis ussions with Y. Colin de Verdière and M. Zworski. S.N. is grateful to the Mathemati al Department in Bonn for its hospitality in De ember 2006. 3. Outline of the proof We start by re alling the de�nition and some properties of the metri entropy asso iated with a probability measure on T ∗M , invariant through the geodesi �ow. In �3.2 we extend the notion of entropy to the quantum framework. Our approa h is semi lassi al, so we want the lassi al and quantum entropies to be onne ted in some way when ~ → 0. The weights appearing in our quantum entropy are estimated in Thm. 3.1, whi h was proven and used in [2℄. In �3.2.1 we also ompare our quantum entropy with several �quantum dynami al entropies� previously de�ned in the literature. The proof of Thm. 2.1 a tually starts in �3.3, where we present the algebrai tool allowing us to take advantage of our estimates (3.9) (or their optimized version given in Thm. 3.5), namely an �entropi un ertainty prin iple� spe i� of the quantum framework. From �3.4 on, we apply this �prin iple� to the quantum entropies appearing in our problem, and pro eed to prove Thm. 2.1. Although the method is basi ally the same as in [4℄, several small modi� ations allow to �nally obtain the improved lower bound (2.3), and also simplify some intermediate proofs, as explained in Remark 3.6. 3.1. De�nition of the metri entropy. In this paper we will meet several types of entropies, all of whi h are de�ned using the fun tion η(s) = −s log s, for s ∈ [0, 1]. We start with the Kolmogorov-Sinai entropy of the geodesi �ow with respe t to an invariant probability measure. Let µ be a probability measure on the otangent bundle T ∗M . Let P = (E1, . . . , EK) be a �nite measurable partition of T ∗M : T ∗M = i=1Ei. We will denote the set of indi es ENTROPY OF EIGENFUNCTIONS 7 {1, . . . , K} = [[1, K]]. The Shannon entropy of µ with respe t to the partition P is de�ned hP(µ) = − µ(Ek) logµ(Ek) = µ(Ek) For any integer n ≥ 1, we denote by P∨n the partition formed by the sets (3.1) E = Eα0 ∩ g −1Eα1 . . . ∩ g −n+1Eαn−1 , where α = (α0, . . . , αn−1) an be any sequen e in [[1, K]] (su h a sequen e is said to be of length |α| = n). The partition P∨n is alled the n-th re�nement of the initial partition P = P∨1. The entropy of µ with respe t to P∨n is denoted by (3.2) hn(µ,P) = hP∨n(µ) = α∈[[1,K]]n If µ is (gt)�invariant, it follows from the onvexity of the logarithm that (3.3) ∀n,m ≥ 1, hn+m(µ,P) ≤ hn(µ,P) + hm(µ,P), in other words the sequen e (hn(µ,P))n∈N is subadditive. The entropy of µ with respe t to the a tion of the geodesi �ow and to the partition P is de�ned by (3.4) hKS(µ,P) = lim hn(µ,P) = inf hn(µ,P) Ea h weight µ(E ) measures the µ�probability to visit su essively Eα0 , Eα1 , . . . , Eαn−1 at times 0, 1, . . . , n − 1 through the geodesi �ow. Roughly speaking, the entropy measures the exponential de ay of these probabilities when n gets large. It is easy to see that hKS(µ,P) ≥ β if there exists C su h that µ(Eα) ≤ C e , for all n and all α ∈ [[1, K]] Finally, the Kolmogorov-Sinai entropy of µ with respe t to the a tion of the geodesi �ow is de�ned as (3.5) hKS(µ) = sup hKS(µ,P), the supremum running over all �nite measurable partitions P. The hoi e to onsider the time 1 of the geodesi �ow in the de�nition (3.1) may seem arbitrary, but the entropy has a natural s aling property : the entropy of µ with respe t to the �ow (gat) is |a|�times its entropy with respe t to (gt). Assume µ is arried on the energy layer E . Due to the Anosov property of the geodesi �ow on E , it is known that the supremum (3.5) is rea hed as soon as the diameter of the partition P ∩ E (that is, the maximum diameter of its elements Ek ∩ E) is small enough. Furthermore, let us assume (without loss of generality) that the inje tivity radius of M is larger than 1. Then, we may restri t our attention to partitions P obtained by lifting on E a partition of the manifoldM , that is take M = k=1Mk and then Ek = T ∗Mk. In fa t, if the diameter of Mk in M is of order ε, then the diameter of the partition P ∨2 ∩E in E is also of order ε. This spe ial hoi e of our partition is not ru ial, but it simpli�es ertain aspe ts of the analysis. 8 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER The existen e of the limit in (3.4), and the fa t that it oin ides with the in�mum, follow from a standard subadditivity argument. It has a ru ial onsequen e : if (µi) is a sequen e of (gt)�invariant probability measures on T ∗M , weakly onverging to a probability µ, and if µ does not harge the boundary of the partition P, we have hKS(µ,P) ≥ lim sup hKS(µi,P) . In parti ular, assume that for i large enough, the following estimates hold : (3.6) ∀n ≥ 1, ∀α ∈ [[1, K]] , µi(Eα) ≤ Ci e −βn , with β independent of i. This implies for i large enough hKS(µi,P) ≥ β, and this estimate goes to the limit to yield hKS(µ) ≥ β. 3.2. From lassi al to quantum dynami al entropy. Sin e our semi lassi al measure µ is de�ned as a limit of Wigner distributions W~, a naive idea would be to estimate from below the KS entropy of W~ and then take the limit ~ → 0. This idea annot work dire tly, be ause the Wigner transformsW~ are neither positive, nor are they (g t)�invariant. Therefore, one annot dire tly use the (formal) integrals W~(Eα) = W~(x, ξ) dx dξ to ompute the entropy of the semi lassi al measure. Instead, the method initiated by the �rst author in [2℄ is based on the following remarks. Ea h integral W~(Eα) an also be written as W~(1lEα) = W~ 1lEα , where 1lEα is the hara teristi fun tion on the set E , that is (3.7) 1lEα = (1lEαn−1 ◦ g n−1)× . . .× (1lEα1 ◦ g)× 1lEα0 . Remember we took Ek = T ∗Mk, where the Mk form a partition of M . From the de�nition of the Wigner distribution, this integral orresponds formally to the overlap 〈ψ~,Op~(1lEα )ψ~〉. Yet, the hara teristi fun tions 1lEα have sharp dis ontinuities, so their quantizations annot be in orporated in a ni e pseudodi�erential al ulus. Besides, the set E is not ompa tly supported, and shrinks in the unstable dire tion when n = |α| −→ +∞, so that the operator Op (1lEα ) is very problemati . We also note that an overlap of the form 〈ψ~,Op~(1lEα)ψ~〉 is a hybrid expression: this is a quantum matrix element of an operator de�ned in terms of the lassi al evolution (3.7). From the point of view of quantum me hani s, it is more natural to onsider, instead, the operator obtained as the produ t of Heisenberg-evolved quantized fun tions, namely (3.8) (U−n+1 Pαn−1U ) (U−n+2 Pαn−2U ) · · · (U−1 Pα1U~)Pα0 . Here we used the shorthand notation Pk = 1lMk , k ∈ [[1, K]] (multipli ation operators). To remedy the fa t that the fun tions 1lMk are not smooth, whi h would prevent us from using a semi lassi al al ulus, we apply a onvolution kernel to smooth them, obtain fun tions 1lsmMk ∈ C ∞(M), and onsider Pk = 1lsmMk (we an do this keeping the property k=1 1l In the following, we will use the notation A(t) = U−t for the Heisenberg evolution of the operator A though the S hrödinger �ow U t = exp(−it~△ ). The norm ‖•‖ will denote ENTROPY OF EIGENFUNCTIONS 9 either the Hilbert norm on L2(M), or the orresponding operator norm. The subsequent �purely quantum� norms were estimated in [2, Thm. 1.3.3℄: Theorem 3.1. (The main estimate [2℄) Set as above Pk = 1lsmMk . For every K > 0, there exists ~K > 0 su h that, uniformly for all ~ < ~K, for all n ≤ K| log ~|, for all (α0, . . . , αn−1) ∈ [[1, K]] (3.9) ‖Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0 ψ~‖ ≤ 2(2π~) −d/2 e− n(1 +O(ε))n. The exponent Λ is given by the �smallest expansion rate�: Λ = − sup log Ju(ρ)dν(ρ) = inf λ+i (γ). The in�mum on the right hand side runs over the set of losed orbits on E , and the λ+i denote the positive Lyapunov exponents along the orbit, that is the logarithms of the expanding eigenvalues of the Poin aré map, divided by the period of the orbit. The parameter ε > 0 is an upper bound on the diameters of the supports of the fun tions 1lsmMk in M . From now on we will all the produ t operator (3.10) P = Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0 , α ∈ [[1, K]] To prove the above estimate, one a tually ontrols the operator norm (3.11) ‖P (χ)‖ ≤ 2(2π~)−d/2 e− n(1 +O(ε))n , where χ ∈ C∞c (E ε) is an energy uto� su h that χ = 1 near E , supported inside a neigh- bourhood Eε = H−1([1 − ε, 1 + ε]) of E . In quantum me hani s, the matrix element 〈ψ~, Pαψ~〉 looks like the �probability�, for a parti le in the state ψ~, to visit su essively the phase spa e regions Eα0 , Eα1 , . . . , Eαn−1 at times 0, 1, . . . , n − 1 of the S hrödinger �ow. Theorem 3.1 implies that this �probability� de ays exponentially fast with n, with rate Λ , but this de ay only starts around the time (3.12) n1 d| log ~| whi h is a kind of �Ehrenfest time� (see (3.25) for another de�nition of Ehrenfest time). Yet, be ause the matrix elements 〈ψ~, Pαψ~〉 are not real in general, they an hardly be used to de�ne a �quantum measure�. Another possibility to de�ne the probability for the parti le to visit the sets Eαk at times k, is to take the squares of the norms appearing in (3.9): (3.13) ‖P 2 = ‖Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0ψ~‖ Now we require the smoothed hara teristi fun tions 1lsmMi to satisfy the identity (3.14) 1lsmMk(x) = 1 for any point x ∈M . 10 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER We denote by Psm the smooth partition of M made by the fun tions (1lsmMk) . The orresponding set of multipli ation operators (Pk) = Pq forms a �quantum partition of unity� : (3.15) P 2k = IdL2 . For any n ≥ 1, we re�ne the quantum partition Pq into (Pα)|α|, as in (3.10). The weights (3.13) exa tly add up to unity, so it makes sense to onsider the entropy (3.16) hn(ψ~,Pq) α∈[[1,K]]n 3.2.1. Conne tion with other quantum entropies. This entropy appears to be a parti ular ase of the �general quantum entropies� des ribed by Sªom zy«ski and �y zkowski [28℄, who already had in mind appli ations to quantum haos. In their terminology, a family of bounded operators π = (πk) k=1 on a Hilbert spa e H satisfying (3.17) π∗k πk = IdH provides an �instrument� whi h, to ea h index k ∈ [[1,N ]], asso iates the following map on density matri es: ρ 7→ I(k)ρ = πk ρ π k , a nonnegative operator with tr(I(k)ρ) ≤ 1 . From a unitary propagator U and its adjoint a tion Uρ = UρU−1, they propose to onstru t the re�ned instrument I(α)ρ = I(αn−1) ◦ · · · U ◦ I(α1) ◦ U ◦ I(α0)ρ = U −n+1 π Un−1 , α ∈ [[1,N ]] where we used (3.10) to re�ne the operators πk into πα. We obtain the probability weights (3.18) tr(I(α)ρ) = tr(π ) , α ∈ [[1,N ]] For any U-invariant density ρ, these weights provide an entropy (3.19) hn(ρ, I) = α∈[[1,N ]]n tr(I(α)ρ) One easily he ks that our quantum partition Pq = (Pk) k=1 satis�es (3.17), and that if one takes ρ = |ψ~〉〈ψ~| the weights tr(I(α)ρ) exa tly orrespond to our weights ‖Pαψ‖ Hen e, the entropy (3.19) oin ides with (3.16). Around the same time, Ali ki and Fannes [1℄ used the same quantum partition (3.17) (whi h they alled ��nite operational partitions of unity�) to de�ne a di�erent type of entropy, now alled the �Ali ki-Fannes entropy� (the de�nition extends to general C∗- dynami al systems). For ea h n ≥ 1 they extend the weights (3.18) to �o�-diagonal entries� to form a N n ×N n density matrix ρn: (3.20) [ρn]α′,α = tr(πα′ ρ π ), α,α′ ∈ [[1,N ]] ENTROPY OF EIGENFUNCTIONS 11 The AF entropy of the system (U , ρ) is then de�ned as follows: take the Von Neumann entropy of these density matri es, hAFn (ρ, π) = tr η(ρn), then take lim supn→∞ hAFn (ρ, π) and �nally take the supremum over all possible �nite operational partitions of unity π. We mention that tra es of the form (3.20) also appear in the �quantum histories� ap- proa h to quantum me hani s (see e.g. [17℄, and [28, Appendix D℄ for referen es). 3.2.2. Naive treatment of the entropy hn(ψ~,Pq). For �xed |α| > 0, the Egorov theorem shows that ‖P onverges to the lassi al weight µ (1lsmMα ) when ~ → 0, so for �xed n > 0 the entropy hn(ψ~,Pq) onverges to hn(µ,Psm), de�ned as in (3.2), the hara teristi fun tions 1lMk being repla ed by their smoothed versions (1l )2. On the other hand, from the estimate (3.11) the entropies hn(ψ~,Pq) satisfy, for ~ small enough, (3.21) hn(ψ~,Pq) ≥ n Λ+O(ε) − d| log ~|+O(1) , for any time n ≤ K| log ~|. For large times n ≈ K| log ~|, this provides a lower bound hn(ψ~,Pq) ≥ Λ +O(ε) +O(1/| log ~|) , whi h looks very promising sin e K an be taken arbitrary large: we ould be tempted to take the semi lassi al limit, and dedu e a lower bound hKS(µ) ≥ Λ. Unfortunately, this does not work, be ause in the range {n > n1} where the estimate (3.21) is useful, the Egorov theorem breaks down, the weights (3.13) do not approximate the lassi al weights µ (1lsmMα ) , and there is no relationship between hn(ψ,Pq) and the lassi al entropies hn(µ,Psm). This breakdown of the quantum- lassi al orresponden e around the Ehrenfest time is ubiquitous for haoti dynami s. It has been observed before when studying the onne tion between the Ali ki-Fannes entropy for the quantized torus automorphisms and the KS entropy of the lassi al dynami s [5℄: the quantum entropies hAFn (ψ~,Pq) follow the lassi al hn(µ,Psm) until the Ehrenfest time (and therefore grow linearly with n), after whi h they �saturate�, to produ e a vanishing entropy lim supn→∞ hAFn (ψ~,Pq). To prove the Theorem 2.1, we will still use the estimates (3.11), but in a more subtle way, namely by referring to an entropi un ertainty prin iple. 3.3. Entropi un ertainty prin iple. The theorem below is an adaptation of the en- tropi un ertainty prin iple onje tured by Deuts h and Kraus [12, 21℄ and proved by Massen and U�nk [25℄. These authors were investigating the theory of measurement in quantum me hani s. Roughly speaking, this result states that if a unitary matrix has �small� entries, then any of its eigenve tors must have a �large� Shannon entropy. Let (H, 〈., .〉) be a omplex Hilbert spa e, and denote ‖ψ‖ = 〈ψ, ψ〉 the asso iated norm. Consider a quantum partition of unity (πk) k=1 on H as in (3.17). If ‖ψ‖ = 1, we de�ne the entropy of ψ with respe t to the partition π as in (3.16), namely hπ(ψ) = 12 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER ‖πk ψ‖ . We extend this de�nition by introdu ing the notion of pressure, asso i- ated to a family v = (vk)k=1,...,N of positive real numbers: the pressure is de�ned by pπ,v(ψ) ‖πk ψ‖ ‖πk ψ‖ 2 log v2k. In Theorem 3.2, we a tually need two partitions of unity (πk) k=1 and (τj) j=1, and two families of weights v = (vk) k=1, w = (wj) j=1, and onsider the orresponding pressures pπ,v(ψ), pτ,w(ψ). Besides the appearan e of the weights v, w, we bring another modi� ation to the statement in [25℄ by introdu ing an auxiliary operator O. Theorem 3.2. [4, Thm. 6.5℄ Let O be a bounded operator and U be an isometry on H. De�ne c (v,w) O (U) = supj,k wj vk ‖τj U π kO‖, and V = maxk vk, W = maxj wj. Then, for any ǫ ≥ 0, for any normalized ψ ∈ H satisfying (3.22) ∀k = 1, . . . ,N , ‖(Id−O) πk ψ‖ ≤ ǫ , the pressures pτ,w , pπ,v satisfy + pπ,v ≥ −2 log (v,w) O (U) +N V W ǫ Example 1. The original result of [25℄ orresponds to the ase where H = CN , O = Id, ǫ = 0, N = M, vk = wj = 1, and the operators πk = τk are the orthogonal proje tors on some orthonormal basis (ek) k=1 of H. In this ase, the theorem asserts that hπ(U ψ) + hπ(ψ) ≥ −2 log c(U) where c(U) = supj,k |〈ek,Uej〉| is the supremum of all matrix elements of U in the orthonor- mal basis (ek). As a spe ial ase, one gets hπ(ψ) ≥ − log c(U) if ψ is an eigenfun tion of 3.4. Applying the entropi un ertainty prin iple to the Lapla ian eigenstates. In this se tion we explain how to use Theorem 3.2 in order to obtain nontrivial information on the quantum entropies (3.16) and then hKS(µ). For this we need to de�ne the data to input in the theorem. Ex ept the Hilbert spa e H = L2(M), all other data depend on the semi lassi al parameter ~: the quantum partition π, the operator O, the positive real number ǫ, the weights (vj), (wk) and the unitary operator U . As explained in se tion 3.2, we partition M into M = ⊔Kk=1Mk, onsider open sets Ωk ⊃⊃Mk (whi h we assume to have diameters ≤ ε), and onsider smoothed hara teristi fun tions 1lsmMk supported respe tively inside Ωk, and satisfying the identity (3.14). The asso iated multipli ation operators on H are form a quantum partition (Pk) k=1, whi h we had alled Pq. To alleviate notations, we will drop the subs ript q. From (3.15), and using the unitarity of U~, one realizes that for any n ≥ 1, the families of operators P∨n = (P ∗ )|α|=n and T ∨n = (P )|α|=n (see (3.10)) make up two quantum partitions of unity as in (3.17), of ardinal Kn. ENTROPY OF EIGENFUNCTIONS 13 3.4.1. Sharp energy lo alization. In the estimate (3.11), we introdu ed an energy uto� χ on a �nite energy strip Eε, with χ ≡ 1 near E . This uto� does not appear in the estimate (3.9), be ause, when applied to the eigenstate ψ~, the operator Op~(χ) essentially a ts like the identity. The estimate (3.11) will a tually not su� e to prove Theorem 2.1. We will need to optimize it by repla ing χ in (3.11) with a �sharp� energy uto�. For some �xed (small) δ ∈ (0, 1), we onsider a smooth fun tion χδ ∈ C ∞(R; [0, 1]), with χδ(t) = 1 for |t| ≤ e and χδ(t) = 0 for |t| ≥ 1. Then, we res ale that fun tion to obtain the following family of ~-dependent uto�s near E : (3.23) ∀~ ∈ (0, 1), ∀n ∈ N, ∀ρ ∈ T ∗M, χ(n)(ρ; ~) e−nδ ~−1+δ(H(ρ)− 1/2) The uto� χ(n) is supported in a tubular neighbourhood of E of width 2~1−δ enδ. We will always assume that this width is << ~1/2 in the semi lassi al limit, whi h is the ase if we ensure that n ≤ Cδ| log ~| for some 0 < Cδ < (2δ) −1−1. In spite of their singular behaviour, these uto�s an be quantized into pseudodi�erential operators Op(χ(n)) des ribed in [4℄ (the quantization uses a pseudodi�erential al ulus adapted to the energy layer E , drawn from [29℄). The eigenstate ψ~ is indeed very lo alized near E , sin e it satis�es (3.24) ‖ Op(χ(0))− 1 ψ~‖ = O(~ ∞) ‖ψ~‖ . In the rest of the paper, we also �x a small δ′ > 0, and all �Ehrenfest time� the ~-dependent integer (3.25) nE(~) ⌊(1− δ′)| log ~| Noti e the resemblan e with the time n1 de�ned in (3.12). The signi� an e of this time s ale will be dis ussed in �3.4.5. The following proposition states that the operators (P ∗ )|α|=nE , almost preserve the en- ergy lo alization of ψ~ : Proposition 3.3. For any L > 0, there exists ~L su h that, for any ~ ≤ ~L, the Lapla ian eigenstate satis�es (3.26) ∀α, |α| = nE , ‖ Op(χ(nE))− Id ψ~‖ ≤ ~ L‖ψ~‖ . We re ognize here a ondition of the form (3.22). 3.4.2. Applying Theorem 3.2: Step 1. We now pre ise some of the data we will use in the entropi un ertainty prin iple, Theorem 3.2. As opposed to the hoi e made in [4℄, we will use two di�erent partitions π, τ . • the quantum partitions π and τ are given respe tively by the families of operators π = P∨nE = (P ∗ )|α|=nE , τ = T ∨nE = (P )|α|=nE . Noti e that these partitions only di�er by the ordering of the operators Pαi(i) inside the produ ts. In the semi lassi al limit, these partitions have ardinality N = KnE ≍ ~−K0 for some �xed K0 > 0. • the isometry will be the propagator at the Ehrenfest time, U = UnE 14 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER • the auxiliarly operator is given as O = Op(χ(nE)), and the error ǫ = ~L, where L will be hosen very large (see �3.4.4). • the weights v will be sele ted in �3.4.4. They will be semi lassi ally tempered, meaning that there exists K1 > 0 su h that, for ~ small enough, all vα, wα are ontained in the interval [1, ~−K1]. The entropy and pressures asso iated with a state ψ ∈ H are given by hπ(ψ) = |α|=nE ,(3.27) pπ,v(ψ) = hπ(ψ)− 2 |α|=nE ψ‖2 log v .(3.28) With respe t to the se ond partition, we have hτ (ψ) = |α|=nE ,(3.29) pτ,w(ψ) = hτ (ψ)− 2 |α|=nE ψ‖2 logw .(3.30) We noti e that the entropy hτ (ψ) exa tly orresponds to the formula (3.16), while hπ(ψ) is built from the norms ψ‖2 = ‖Pα0Pα1(1) · · ·Pαn−1(n− 1)ψ‖ If ψ is an eigenfun tion of U~, the above norm an be obtained from (3.13) by ex hanging U~ with U , and repla ing the sequen e α = (α0, . . . , αn−1) by ᾱ = (αn−1, . . . , α0). So the entropies hπ(ψ) and hτ (ψ) are mapped to one another through the time reversal U~ → U With these data, we draw from Theorem 3.2 the following Corollary 3.4. For ~ > 0 small enough onsider the data π, τ , U , O as de�ned above. (3.31) c O (U) = max |α|=|α′|=nE Op(χ(nE))‖ Then for any normalized state φ satisfying (3.26), pτ,w(U φ) + pπ,v(φ) ≥ −2 log O (U) + h L−K0−2K1 From (3.26), we see that the above orollary applies to the eigenstate ψ~ if ~ is small enough. The reason to take the same value nE for the re�ned partitions P , T ∨nE and the propagator U is the following : the produ ts appearing in c O (U) an be rewritten (with U ≡ U~): ′ UnE P = U−nE+1Pα′ U · · ·UPα′0UPαnE−1U · · ·UPα0 = U ENTROPY OF EIGENFUNCTIONS 15 Thus, the estimate (3.11) with n = 2nE already provides an upper bound for the norms appearing in (3.31) � the repla ement of χ by the sharp uto� χ(nE) does not harm the estimate. To prove Theorem 2.1, we a tually need to improve the estimate (3.11), as was done in [4℄, see Theorem 3.5 below. This improvement is done at two levels: we will use the fa t that the uto�s χ(nE) are sharper than χ, and also the fa t that the expansion rate of the geodesi �ow (whi h governs the upper bound in (3.11)) is not uniform, but depends on the sequen e α. Our hoi e for the weights v will then be guided by the α-dependent upper bounds given in Theorem 3.5. To state that theorem, we introdu e some notations. 3.4.3. Coarse-grained unstable Ja obian. We re all that, for any energy λ > 0, the geodesi �ow gt on the energy layer E(λ) = H−1(λ) ⊂ T ∗M is Anosov, so that the tangent spa e TρE(λ) at ea h ρ ∈ T ∗M , H(ρ) > 0 splits into TρE(λ) = E u(ρ)⊕Es(ρ)⊕ RXH(ρ) where Eu (resp. Es) is the unstable (resp. stable) subspa e. The unstable Ja obian Ju(ρ) is de�ned by Ju(ρ) = det |Eu(g1ρ) (the unstable spa es at ρ and g1ρ are equipped with the indu ed Riemannian metri ). This Ja obian an be �dis retized� as follows in the energy strip Eε ⊃ E . For any pair of indi es (α0, α1) ∈ [[1, K]] , we de�ne (3.32) Ju1 (α0, α1) = sup Ju(ρ) : ρ ∈ T ∗Ωα0 ∩ E ε, g1ρ ∈ T ∗Ωα1 if the set on the right hand side is not empty, and Ju1 (α0, α1) = e otherwise, where R > 0 is a �xed large number. For any sequen e of symbols α of length n, we de�ne (3.33) Jun(α) = Ju1 (α0, α1) · · ·J 1 (αn−2, αn−1) . Although Ju and Ju1 (α0, α1) are not ne essarily everywhere smaller than unity, there exists C, λ+, λ− > 0 su h that, for any n > 0, for any α with |α| = n, (3.34) C−1 e−n(d−1) λ+ ≤ Jun(α) ≤ C e −n(d−1) λ− . One an take λ+ = λmax(1+ε), where λmax is the maximal expanding rate in Theorem. 2.1. We now give our entral estimate, easy to draw from [4, Corollary 3.4℄. Theorem 3.5. Fix small positive onstants ε, δ, δ′ and a onstant 0 < Cδ < (2δ) −1 − 1. Take an open over M = k Ωk of diameter ≤ ε and an asso iated quantum partition P = k=1. There exists ~0 su h that, for any ~ ≤ ~0, for any positive integer n ≤ Cδ| log ~|, and any pair of sequen es α, α of length n, (3.35) ‖P ′ Op(χ(n))‖ = ‖P Op(χ(n))‖ ≤ C ~− −δ enδ Jun(α) J The onstant C only depends on the Riemannian manifold (M, g). If we take n = nE, this takes the form (3.36) ‖P Op(χ(nE))‖ ≤ C ~− d−1+cδ JunE(α) J (α′) , 16 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER where c = 2 + 2λ−1max. The idea of proof in Theorem 3.5 is rather simple, although the te hni al implementation is umbersome. We �rst show that for any normalized state ψ, the state Op(χ(n))ψ an be essentially de omposed into a superposition of ~ −d| suppχ(n)| normalized Lagrangian states, supported on Lagrangian manifolds transverse to the stable foliation. In fa t the Lagrangian states we work with are trun ated δ�fun tions, supported on lagrangians of the form ∪tg tS∗zM . The a tion of the operator U ′ = Pα′ U · · ·UPα0 on su h Lagrangian states an be analyzed through WKB methods, and is simple to understand at the lassi al level : ea h appli ation of the propagator U stret hes the Lagrangian along the unstable dire tion (the rate of stret hing being des ribed by the lo al unstable Ja obian), whereas ea h operator Pk �proje ts� on a pie e of Lagrangian of diameter ε. This iteration of stret hing and utting a ounts for the exponential de ay. The αα -independent fa tor on the right of (3.36) results from adding together the ontributions of all the initial Lagrangian states. Noti e that this prefa tor is smaller than in Theorem. 3.1 due to the ondition Cδ < (2δ) −1 − 1. Remark 3.6. In [4℄ we used the same quantum partition P∨nE for π and τ in Theorem. 3.2. As a result, we needed to estimate from above the norms ‖P ∗ ′ UnE PαOp(χ (nE))‖ (see [4, Theorem. 2.6℄). The proof of this estimate was mu h more involved than the one for (3.36), sin e it required to ontrol long pie es of unstable manifolds. By using instead the two partitions P(n), T (n), we not only prove a more pre ise lower bound (2.3) on the KS entropy, but also short- ir uit some �ne dynami al analysis. 3.4.4. Applying Theorem 3.2: Step 2. There remains to hoose the weights (v ) to use in Theorem 3.2. Our hoi e is guided by the following idea: in (3.31), the weights should balan e the variations (with respe t to α,α′) in the norms, su h as to make all terms in (3.31) of the same order. Using the upper bounds (3.36), we end up with the following hoi e for all α of length nE : = JunE(α) −1/2 . From (3.34), there exists K1 > 0 su h that, for ~ small enough, all the weights are ontained in the interval [1, ~−K1], as announ ed in �3.4.2. Using these weights, the estimate (3.36) implies the following bound on the oe� ient (3.31): ∀~ < ~0, c O (U) ≤ C ~ − d−1+cδ We an now apply Corollary 3.4 to the parti ular ase of the eigenstates ψ~. We hoose L su h that L−K0 − 2K1 > − d−1+cδ , so from Corollary 3.4 we draw the following Proposition 3.7. Let (ψ~)~→0 be our sequen e of eigenstates (2.2). In the semi lassi al limit, the pressures of ψ~ satisfy (3.37) pP∨nE ,v(ψ~) + pT ∨nE ,w(ψ~) ≥ − (d − 1 + cδ)λmax (1− δ′) nE +O(1) . ENTROPY OF EIGENFUNCTIONS 17 If M has onstant urvature we have log Jn ≤ −n(d − 1)λmax(1 − O(ε)) for all α of length n, and the above lower bound an be written hP∨nE (ψ~) + hT ∨nE (ψ~) ≥ (d− 1)λmax 1 +O(ε, δ, δ′) As opposed to (3.21), the above inequality provides a nontrivial lower bound for the quan- tum entropies at the time nE , whi h is smaller than the time n1 of (3.12), and will allow to onne t those entropies to the KS entropy of the semi lassi al measure (see below). 3.4.5. Subadditivity until the Ehrenfest time. Even at the relatively small time nE , the onne tion between the quantum entropy h(ψ~,P ∨nE) and the lassi al h(µ,P∨nEsm ) is not ompletely obvious: both are sums of a large number of terms (≍ ~−K0). Before taking the limit ~ → 0, we will prove that a lower bound of the form (3.37) still holds if we repla e nE ≍ | log ~| by some �xed no ∈ N, and P by the orresponding quantum partition P∨no . The link between quantum pressures at times nE and no is provided by the following subadditivity property, whi h is the semi lassi al analogue of the lassi al subadditivity of pressures for invariant measures (see (3.3)). Proposition 3.8 (Subadditivity). Let δ′ > 0. There is a fun tion R(no, ~), and a real number R > 0 independent of δ′, su h that, for any integer no ≥ 1, lim sup |R(no, ~)| ≤ R and with the following properties. For any small enough ~ > 0, any integers no, n ∈ N with no + n ≤ nE(~), for any ψ~ normalized eigenstate satisfying (2.2), the following inequality holds: pP∨(no+n),v(ψ~) ≤ pP∨no ,v(ψ~) + pP∨n,v(ψ~) +R(no, ~) . The same inequality is satis�ed by the pressures pT ∨n,w(ψ~). To prove this proposition, one uses a re�ned version of Egorov's theorem [10℄ to show that the non� ommutative dynami al system formed by (U t ) a ting (through Heisenberg) on observables supported near E is (approximately) ommutative on time intervals of length nE(~). Pre isely, we showed in [4℄ that, provided ε is small enough, for any a, b ∈ C ∀t ∈ [−nE(~), nE(~)], ‖[Op~(a)(t),Op~(b)]‖ = O(~ cδ′), ~ → 0 , and the implied onstant is uniform with respe t to t. Within that time interval, the oper- ators Pαj (j) appearing in the de�nition of the pressures ommute up to small semi lassi al errors. This almost ommutativity explains why the quantum pressures pP∨n,v(ψ~) satisfy the same subadditivity property as the lassi al entropy (3.3), for times smaller than nE . Thanks to this subadditivity, we may �nish the proof of Theorem. 2.1. Fixing no, using for ea h ~ the Eu lidean division nE(~) = q(~)no + r(~) (with r(~) < no), Proposition 3.8 implies that for ~ small enough, pP∨nE ,v(ψ~) pP∨no ,v(ψ~) pP∨r,v(ψ~) R(no, ~) 18 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER The same inequality is satis�ed by the pressures pT ∨n,w(ψ~). Using (3.37) and the fa t that pP∨r,v(ψ~) stays uniformly bounded when ~ → 0, we �nd (3.38) pP∨no ,v(ψ~) + pT ∨no ,w(ψ~) 2(d− 1 + cδ)λmax 2(1− δ′) 2R(no, ~) +Ono(1/nE) . We are now dealing with quantum partitions P∨no , T ∨no , for n0 ∈ N independent of ~. At this level the quantum and lassi al entropies are related through the (�nite time) Egorov theorem, as we had noti ed in �3.2.2. For any α of length no, the weights ‖Pαψ~‖ both onverge to µ (1lsmMα) , where we re all that 1lsmMα = (1l Mαno−1 ◦ gno−1)× . . .× (1lsmMα1 ◦ g)× 1lsmMα0 Thus, both entropies hP∨no (ψ~), hT ∨no (ψ~) semi lassi ally onverge to the lassi al entropy hno(µ,Psm). As a result, the left hand side of (3.38) onverges to (3.39) 2 hno(µ,Psm) |α|=no (1lsmMα ) log Juno(α) . Sin e µ is gt-invariant and Juno has the multipli ative stru ture (3.33), the se ond term in (3.39) an be simpli�ed: |α|=no (1lsmMα ) log Juno(α) = (no − 1) α0,α1 (1lsmM(α0,α1) log Ju1 (α0, α1) . We have thus obtained the lower bound (3.40) hno(µ,Psm) no − 1 α0,α1 (1lsmM(α0,α1) log Ju1 (α0, α1)− (d− 1 + cδ)λmax 2(1− δ′) At this stage we may forget about δ and δ′. The above lower bound does not depend on the derivatives of the fun tions 1lsmMα , so the same bound arries over if we repla e 1l the hara teristi fun tions 1lMα . We an �nally let no tend to +∞, then let the diameter ε tend to 0. The left hand side onverges to hKS(µ) while, from the de�nition (3.32), the sum in the right hand side of (3.40) onverges to the integral log Ju(ρ)dµ(ρ) as ε → 0, whi h proves (2.3). Referen es [1℄ R. Ali ki and M. Fannes, De�ning quantum dynami al entropy, Lett. Math. Phys. 32 75�82 (1994) [2℄ N. Anantharaman, Entropy and the lo alization of eigenfun tions, to appear in Ann. Math. [3℄ N. Anantharaman and S. Nonnenma her, Entropy of semi lassi al measures of the Walsh- quantized baker's map, Ann. H. Poin aré 8, 37�74 (2007) [4℄ N. Anantharaman, S. Nonnenma her, Half�delo alization of eigenfun tions of the lapla ian on an Anosov manifold, http://hal.ar hives-ouvertes.fr/hal-00104963 [5℄ F. Benatti, V. Cappellini, M. De Co k, M. Fannes and D. Van Peteghem, Classi al Limit of Quantum Dynami al Entropies, Rev. Math. Phys. 15, 1�29 (2003) [6℄ M.V. Berry, Regular and irregular semi lassi al wave fun tions, J.Phys. 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Zeldit h, Uniform distribution of the eigenfun tions on ompa t hyperboli surfa es, Duke Math. J. 55, 919�941 (1987) CMLS, É ole Polyte hnique, 91128 Palaiseau, Fran e E-mail address : nalini�math.polyte hnique.fr Mathemati al Institute, University of Bonn, Beringstraÿe 1,D-53115 Bonn, Germany E-mail address : ko h�math.uni-bonn.de Servi e de Physique Théorique, CEA/DSM/PhT, Unité de re her he asso iée au CNRS, CEA/Sa lay, 91191 Gif-sur-Yvette, Fran e E-mail address : snonnenma her� ea.fr 1. Motivations 2. Main result. Acknowledgements 3. Outline of the proof 3.1. Definition of the metric entropy 3.2. From classical to quantum dynamical entropy 3.3. Entropic uncertainty principle 3.4. Applying the entropic uncertainty principle to the Laplacian eigenstates References

arXiv:0704.1565v2 [hep-ph] 29 Aug 2007 Preprint typeset in JHEP style - PAPER VERSION DESY-07-049 arXiv:0704.1565 [hep-ph] Vector meson production from a polarized nucleon M. Diehl Deutsches Elektronen-Synchroton DESY, 22603 Hamburg, Germany Abstract: We provide a framework to analyze the electroproduction process ep → epρ with a polarized target, writing the angular distribution of the ρ decay products in terms of spin density matrix elements that parameterize the hadronic subprocess γ∗p → ρp. Using the helicity basis for both photon and meson, we find a representation in which the expressions for a polarized and unpolarized target are related by simple substitution rules. Keywords: Lepton-Nucleon Scattering, Spin and Polarization Effects. http://arxiv.org/abs/0704.1565v2 Contents 1. Introduction 1 2. Kinematics and target polarization 2 3. Helicity amplitudes and spin density matrix 4 4. The angular distribution 9 5. Natural and unnatural parity 17 6. Positivity constraints 21 7. Mixing between transverse and longitudinal polarization 23 8. A note on non-resonant contributions 27 9. Summary 28 1. Introduction Exclusive vector meson production has long played an important role in studying the strong interaction. The seminal work [1, 2] has renewed interest in this process, showing that in Bjorken kinematics it provides access to generalized parton distributions and thus to a wealth of information on the structure of the proton. While most theoretical and experimental studies so far are for an unpolarized proton, the particular interest of target polarization became clear when it was pointed out that meson production on a transversely polarized target is sensitive to the nucleon helicity-flip distribution E [3, 4]. This distri- bution offers unique views on the orbital angular momentum carried by partons in the proton [5, 6] and on the correlation between polarization and the spatial distribution of partons [7]. Whereas the corresponding polarization asymmetry in deeply virtual Compton scattering is under better theoretical control, vector meson production has the advantage of a greater sensitivity to the distribution of gluons (which in Compton scattering only enters at next-to-leading order in αs). This holds not only in the high-energy regime but even in a wide range of fixed-target kinematics [8, 9, 10], where polarization measurements are feasible at existing or planned experimental facilities. A different motivation to study polarized exclusive ρ production is that this channel plays a rather prominent role in semi-inclusive pion production [11, 12, 9], which has become a privileged tool to study a variety of spin effects, see e.g. [13]. It is important to identify – 1 – kinematical regions where the exclusive channel ep → epρ → epπ+π− dominates semi- inclusive observables, because in these regions great care must be taken when interpreting the data in terms of semi-inclusive factorization. Even with an unpolarized target, the spin structure of the process ep → epρ→ epπ+π− is very rich, because the angular distribution of the final state contains information on the helicities of the exchanged virtual photon and of the ρ meson, as was worked out in the classical analysis of Schilling and Wolf [14]. Yet more detailed information is available with target polarization [15]. Experiments on unpolarized targets have found that s-channel helicity is approximately conserved in the transition from the γ∗ to the ρ, with helicity changing amplitudes occurring at most at the 10% level [16, 17, 18, 19, 20]. This greatly simplifies the spin structure of the process. The aim of the present paper is to provide an analysis framework for exclusive ρ production on a polarized nucleon target, making as explicit as possible the relation between the angular dependence of the cross section and the helicity amplitudes describing the hadronic subprocess γ∗p→ ρp. We will present our results in a form that emphasizes the close similarity in structure between an unpolarized and a polarized target. Using the helicity basis for both virtual photon and meson, we also provide an alternative to the representation of the unpolarized cross section in [14]. The following section gives the definitions of the kinematics and polarization variables for the reaction under study. In Section 3 we define the helicity amplitudes and the spin density matrix elements describing the process and discuss some of their general properties. In Section 4 we express the angular distribution of the polarized cross section in terms of these spin density matrix elements and point out some salient features of this representa- tion. The simplifications arising from distinguishing natural and unnatural parity exchange in the reaction are discussed in Section 5. A number of positivity bounds relating different spin density matrix elements are given in Section 6. In Section 7 we explain the complica- tions arising from the distinction between target polarization relative to the momentum of either the incident lepton or the virtual photon. The role of non-resonant contributions in π+π− production is briefly discussed in Section 8. Our results are summarized in Section 9. 2. Kinematics and target polarization Let us consider the electroproduction process e(l) + p(p) → e(l′) + p(p′) + ρ(q′) (2.1) followed by the decay ρ(q′) → π+(k) + π−(k′), (2.2) where four-momenta are given in parentheses. Throughout this work we use the one-photon exchange approximation. All or results are equally valid for the production of a φ followed by the decay φ→ K+K−. They also hold if the scattered proton is replaced by an inclusive system X with four-momentum p′, as explained at the end of Section 3. To describe the kinematics we use the conventional variables for deep inelastic pro- cesses, Q2 = −q2, xB = Q2/(2p · q) and y = (p · q)/(p · l). We neglect the lepton mass – 2 – lepton plane hadron plane θγ ST Figure 1: Kinematics of ep → epρ in the target rest frame. ST is the transverse component of the target spin vector w.r.t. the virtual photon direction. throughout and denote the longitudinal lepton beam polarization by Pℓ, with Pℓ = +1 corresponding to a purely right-handed and Pℓ = −1 to a purely left-handed beam. Let us now go to the target rest frame and introduce the right-handed coordinate system (x, y, z) of Fig. 1 such that q points in the positive z direction and l has a positive x component. In this system we have l = |l|(sin θγ , 0, cos θγ) and q = |q|(0, 0, 1), where the angle θγ between l and q is defined to be between 0 and π. In accordance with the Trento convention [21] we define the angle φ between the lepton and the hadron plane as the azimuthal angle of q′ in this coordinate system, and φS as the azimuthal angle of the target spin vector S. Following [22] we write S = (ST cosφS , ST sinφS ,−SL) with 0 ≤ ST ≤ 1 and −1 ≤ SL ≤ 1, so that ST and SL describe transverse and longitudinal polarization with respect to the virtual photon momentum, with SL = 1 corresponding to a right-handed proton in the γ∗p c.m. *γ c.m.p boost π π+ c.m.− Figure 2: Kinematics of the hadronic subprocess γ∗p → ρp followed by the decay ρ → π+π−. The coordinate systems (x, y, z) and (x′, y′, z′) differ from those in Fig. 1. – 3 – To describe the target polarization of a given experimental setup, we introduce an- other right-handed coordinate system (x′, y′, z′) in the target rest frame such that l = |l|(0, 0, 1) and q = |q|(− sin θγ , 0, cos θγ) as shown in Fig. 1. In this system we write S = (PT cosψ,PT sinψ,−PL) with 0 ≤ PT ≤ 1 and −1 ≤ PL ≤ 1, following again [22]. PT and PL describe transverse and longitudinal polarization with respect to the lepton beam direction, with PL = 1 corresponding to a right-handed proton in the ep c.m. The two sets of variables describing the target polarization are related by ST cosφS = cos θγPT cosψ − sin θγPL , ST sinφS = PT sinψ , SL = sin θγPT cosψ + cos θγPL , (2.3) which we will use in Sect. 7. In terms of invariants the mixing angle θγ is given by sin θγ = γ 1− y − 1 1 + γ2 , γ = 2xBMN , (2.4) whereMN is the nucleon mass. In Bjorken kinematics γ is small, and so is sin θγ ≈ γ 1− y. We finally specify the variables describing the vector meson decay (2.2). This is conve- niently done in the π+π− c.m., which can be obtained from the γ∗p c.m. by a boost in the direction of the scattered nucleon as shown in Fig. 2. In the π+π− c.m. we introduce the right-handed coordinate system (x, y, z) shown in Fig. 2, where p′ = |p′|(0, 0,−1) and where the target momentum p has a positive x component. In this system we define ϑ and ϕ as the polar and azimuthal angle of the π+ momentum, i.e. k = |k|(sinϑ cosϕ, sinϑ sinϕ, cos ϑ). The relation between our notation here and the one of Schilling and Wolf is1 φhere = −Φ[14] , ϕhere = φ [14] , ϑhere = θ [14] . (2.5) 3. Helicity amplitudes and spin density matrix The strong-interaction dynamics of the electroproduction process (2.1) is fully contained in the helicity amplitudes for the subprocess γ∗p→ ρp. From these we will construct spin density matrix elements which describe the angular distribution of the overall reaction ep → ep π+π− and its dependence on the target polarization. Since we will deal with interference terms we must specify our phase conventions. We do this in the γ∗p c.m. and use the right-handed coordinate system (x′, y′, z′) shown in Fig. 2. In this system we have q = |q|(0, 0,−1) and q′ = |q′|(sinΘ, 0,− cos Θ), with the scattering angle Θ of the vector meson defined to be between 0 and π. Note that the positive z′ axis points along p rather than q, as is often preferred for theoretical calculations. We specify polarization states of the target proton by two-component spinors χ+1/2 = (1, 0) 1We remark that the expression for sinΦ given in eq. (13) of [14] is incorrect since it is always positive. A correct definition is given in [23]. – 4 – for positive and χ−1/2 = (0, 1) for negative helicity. For the polarization vectors of the virtual photon we choose ε+1 = − 0, 1,−i, 0 , ε−1 = 0, 1, i, 0 εα0 = Nε qα − q p · q p , (3.1) and for the polarization vectors of the ρ e+1 = − 0, cos Θ,−i, sinΘ , e−1 = 0, cos Θ, i, sinΘ eα0 = Ne q′α − q p′ · q′ p , (3.2) where the subscripts indicate helicities. Nε and Ne are positive constants ensuring the proper normalization ε20 = 1 and e 0 = −1 of the longitudinal polarization vectors. In the ρ rest frame and the coordinate system (x, y, z) of Fig. 2, our meson polarization vectors have the standard form e+1 = −(0, 1, i, 0)/ 2, e−1 = (0, 1,−i, 0)/ 2 and e0 = (0, 0, 0, 1). Our phase conventions for the proton and the virtual photon are as in [22]. We now introduce amplitudes T νσµλ for the subprocess γ ∗(µ)+ p(λ) → ρ(ν)+ p(σ) with definite helicities µ, ν, λ, σ. Since the above phase conventions are defined with reference only to momentum vectors of this subprocess, the helicity amplitudes only depend on the photon virtuality, the γ∗p scattering energy and the scattering angle Θ, or equivalently on Q2, xB and t = (p− p′)2. With our phase conventions they obey the usual parity relations T−ν−σ−µ−λ = (−1) ν−µ−σ+λ T νσµλ (3.3) for equal Q2, xB and t on both sides. With these helicity amplitudes we define µµ′,λλ′ = (NT + ǫNL) T νσµλ . (3.4) Regarding the upper indices this is the spin density matrix of the vector meson, whereas the lower indices specify the polarizations in the γ∗p state from which the meson is produced.2 The normalization factors λ,ν,σ ∣T νσ+λ , NL = λ,ν,σ ∣T νσ0λ (3.5) are proportional to the differential cross sections dσT /dt and dσL/dt for transverse and longitudinal photon polarization, respectively, and 1− y − 1 1− y + 1 y2 + 1 (3.6) 2Taking the trace in the meson polarization indices we obtain the relation ρννµµ′,λλ′ ∝ dσ µ′µ/dt between the spin density matrix ρ introduced here and the cross sections and interference terms used in [22]. Compared with [22] we take the opposite order of indices in ρ, so that ν and ν′ appear in the standard order for a spin density matrix. – 5 – is the usual ratio of longitudinal and transverse photon flux. In addition to Q2, xB and t, the spin density matrix elements ρνν µµ′,λλ′ depend on ǫ through the normalization factor (NT + ǫNL). If one can perform a Rosenbluth separation by measuring at different ǫ but equal Q2 and xB, it is advantageous to normalize them instead to NT , NL or NTNL as was done in [14]. It is straightforward to implement such a change in the formulae we give in the following. We find it useful to introduce the combinations µµ′ = µµ′,++ + ρ µµ′,−− , lνν µµ′ = µµ′,++ − ρνν µµ′,−− (3.7) for an unpolarized and a longitudinally polarized target, where for the sake legibility we have labeled the target polarization by ± instead of ±1 . The combinations µµ′ = µµ′,+− + ρ µµ′,−+ , nνν µµ′ = µµ′,+− − ρνν µµ′,−+ (3.8) respectively describe transverse target polarization in the hadron plane (“sideways”) and perpendicular to it (“normal”). One readily finds that the matrices u , l and s are hermi- tian, whereas n is antihermitian, µ′µ = µ′µ = µ′µ = µ′µ = − . (3.9) The diagonal elements uννµµ, l µµ and s µµ are therefore purely real, whereas n µµ is purely imaginary. Furthermore, the parity relations (3.3) translate into u−ν−ν −µ−µ′ = (−1)ν−µ−ν′+µ′ uνν′µµ′ , l−ν−ν −µ−µ′ = −(−1)ν−µ−ν′+µ′ lνν′µµ′ , n−ν−ν −µ−µ′ = (−1) ν−µ−ν′+µ′ nνν µµ′ , s −ν−ν′ −µ−µ′ = −(−1) ν−µ−ν′+µ′ sνν µµ′ . (3.10) As a consequence the matrix elements u−+−+ , u −+ , u 0 0 , u −+ (3.11) are purely real, whereas the corresponding elements of l , s and n are purely imaginary. Both experiment and theory indicate that s-channel helicity is approximately conserved in the γ∗ → ρ transition for small invariant momentum transfer t. Correspondingly, one expects that spin density matrix elements involving the product of two helicity conserving amplitudes are greater than interference terms between a helicity conserving and a helicity changing amplitude, and that those are greater than matrix elements involving the product of two helicity changing amplitudes (where we refer to the helicities of the photon and the ρ but not of the nucleon). Exceptions to this rule are however possible, since two large amplitudes can have a small interference term because of their relative phase, and since there can be cancellation of individually large terms in the linear combinations (3.7) and (3.8) associated with different target polarizations. With this caveat in mind one can readily assess the expected size of the spin density matrix elements (3.7) and (3.8) by comparing the upper with the lower indices. – 6 – Let us now investigate the behavior of our matrix elements for Θ → 0, i.e. in the limit of forward scattering γ∗p→ ρp. To this end we perform a partial wave decomposition T νσµλ (Θ) = tνσµλ(J) d λ−µ,σ−ν(Θ) (3.12) where we have suppressed the dependence of T and the partial wave amplitudes t(J) on Q2 and xB . Using the behavior d m,n(Θ) ∼ Θ|m−n| of the rotation functions for Θ → 0 we readily find µµ′ , l µµ′ ∼ Θp , nνν µµ′ , s µµ′ ∼ Θq (3.13) p ≥ pmin = min σ,λ=±1/2 ∣ν − µ− σ + λ ∣ν ′ − µ′ − σ + λ q ≥ qmin = min σ,λ=±1/2 ∣ν − µ− σ + λ ∣ν ′ − µ′ − σ − λ . (3.14) With Θ ∝ (t0 − t)1/2 for small Θ, we can rewrite (3.13) as µµ′ , l µµ′ ∼ (t0 − t)p/2 , nνν µµ′ , s µµ′ ∼ (t0 − t)q/2 , (3.15) where t0 is the value of t for Θ = 0 at given Q 2 and xB. In Tables 1 and 2 we give the corresponding powers for the linear combinations of spin density matrix elements that will appear in our results for the cross section in Section 4. We have ordered the entries according to the hierarchy discussed after (3.11), listing first terms containing the product of two helicity conserving amplitudes, then terms containing the interference between a helicity conserving and a helicity changing amplitude, and finally terms which only involve helicity changing amplitudes (with helicities always referring to the photon and the ρ but not to the nucleon). We emphasize that certain partial wave amplitudes tνσµλ(J) in (3.12) may be zero or negligibly small for dynamical reasons. The actual powers of (t0 − t)1/2 in (3.15) can thus be larger than the minimum values pmin and qmin required by angular momentum conservation. If there is for instance no s-channel helicity transferred between the proton- proton and the photon-meson transitions, then the relevant powers for n and s are given by q = pmin+1, which is equal to qmin+2 for all but the first four entries in Tables 1 and 2. A concrete realization of this scenario is the calculation in [24], where the proton-proton transition is described by the generalized parton distributions H, E and H̃, Ẽ, which do not allow for helicity transfer to the photon-meson transition. In the limit of large Q2 at fixed xB and t, the proof of the factorization theorem in [2] implies that the transition from a longitudinal photon to a longitudinal ρ becomes dominant, with all other transitions suppressed by powers of 1/Q. In this limit only the spin density matrix elements u 0 00 0 and n 0 0 survive and can be expressed as convolutions of hard-scattering kernels with generalized parton distributions and the light-cone distribution amplitude of the ρ. To leading order in 1/Q one has in particular Imn 0 00 0 u 0 00 0 t0 − t 1− ξ2 Im (1− ξ2) |H|2 − ξ2 + t/(4M2N ) |E|2 − 2ξ2 Re , (3.16) – 7 – matrix elements pmin u 0 0++ + ǫu 0 0 0 u 0+0+ − u 0+ − l u++++ + u ++ + 2ǫu 0 0 l ++ + l u−+−+ l u 0 00+ l u 0+++ − u−0++ + 2Re ǫu 0+0 0 l ++ − l−0++ + 2i Im ǫl 0+0 0 1 u 0+−+ l u 0−0+ − u 0+ − l u−+++ + ǫu 0 0 l ++ + ǫl 0 0 2 u++−+ l u++0+ + u 0+ + l u−+0+ l l 0 0++ 2 u 0 0−+ l u+0−+ l u+−0+ l u+−−+ l Table 1: Minimum values of the powers which control the t→ t0 behavior of combinations of spin density matrix elements u and l as in (3.15). Some of the combinations are purely real or purely imaginary because of the symmetry relations (3.9) and (3.10), whereas others are complex valued. where ξ = xB/(2−xB) and the convolution integrals H and E are for instance given in [22]. Experimental results and phenomenological analysis show however that 1/Q2 suppressed effects can be numerically significant for Q2 of several GeV2, see e.g. [25, 24, 9, 10]. This concerns both power corrections within u 0 00 0 or n 0 0 and formally power suppressed spin density matrix elements such as u++++ or u 0+ . The detailed analysis in [2] reveals that beyond leading-power accuracy in 1/Q, factorization of meson production into a hard- scattering subprocess and nonperturbative quantities pertaining either to the target or to the meson may be broken. On the other hand, factorization based approaches which go beyond leading power in 1/Q and in particular also evaluate transition amplitudes for transverse polarization of the γ∗ or ρ have been phenomenologically rather successful, see e.g. [26, 24] Let us finally generalize our considerations to the process e(l) + p(p) → e(l′) +X(p′) + ρ(q′) , (3.17) where the target proton dissociates into a hadronic system X. In analogy to the elastic case one can introduce helicity amplitudes T and combine them into spin density matrix – 8 – matrix elements qmin n 0 0++ + ǫn 0 0 1 n 0+0+ − n 0+ − s n++++ + n ++ + 2ǫn 0 0 s ++ + s n−+−+ s n 0 00+ s n 0+++ − n−0++ + 2i Im ǫn 0+0 0 s ++ − s−0++ + 2i Im ǫs 0+0 0 0 n 0+−+ s n 0−0+ − n 0+ − s n−+++ + ǫn 0 0 s ++ + ǫs 0 0 1 n++−+ s n++0+ + n 0+ + s n−+0+ s s 0 0++ 1 n 0 0−+ s n+0−+ s n+−0+ s n+−−+ s Table 2: As Table 1 but for combinations of spin density matrix elements n and s . elements µµ′,λλ′ = (NT + ǫNL) ν′σ,X . (3.18) The normalization factors NT and NL are defined as in (3.5) but with an additional sum over all hadronic states X of given invariant mass MX , on which ρ µµ′,λλ′ now depends in addition to Q2, xB , t and ǫ. The combinations (3.7) and (3.8) for different target polarization have the same symmetry properties (3.9) and (3.10) as in the elastic case. Their behavior for t→ t0 can be different, since in (3.14) one must now take the minimum over all possible helicities σ = ±1 , . . . of the hadronic system X. One finds however that the powers pmin and qmin for the combinations of spin density matrix elements in Tables 1 and 2 are the same as in the elastic case. The results in the remainder of this work only depend on the properties (3.9) and (3.10) and thus immediately generalize to the case of target dissociation. 4. The angular distribution The calculation of the cross section for ep → ep π+π− proceeds by using standard methods and we shall only sketch the essential steps. More details are for instance given in [14, 27, – 9 – 22]. With our phase conventions the polarization state of the proton target is described by the spin density matrix τλλ′ = 1 + SL ST e −i(φ−φS) i(φ−φS) 1− SL , (4.1) which is to be contracted with the matrix in (3.4). The result is conveniently expressed in terms of the combinations (3.7) and (3.8) as τλλ′ ρ µµ′,λλ′ = u µµ′ + SL l µµ′ + ST cos(φ− φS) sνν µµ′ − ST sin(φ− φS) inνν µµ′ (4.2) and describes the subprocess γ∗p → ρp. The decay ρ → π+π− is taken into account by multiplication with the spherical harmonics, ρµµ′ = τλλ′ ρ µµ′,λλ′ Y1ν(ϕ, ϑ)Y 1ν′(ϕ, ϑ) , (4.3) where Y1+1 = − sinϑ eiϕ , Y10 = cos ϑ , Y1−1 = sinϑ e−iϕ . (4.4) To obtain the cross section for the overall process ep→ epπ+π− one must finally contract the matrix ρµµ′ in (4.3) with the spin density matrix of the virtual photon. 3 The cross section can be written as dψ dφdϕd(cos ϑ) dxB dQ2 dt (2π)2 dxB dQ2 dt WUU + PℓWLU + SLWUL + PℓSLWLL + STWUT + PℓSTWLT (4.5) dxB dQ2 dt 1− xB , (4.6) where dσT /dt and dσL/dt are the usual γ ∗p cross sections for a transverse and longitudinal photon and an unpolarized proton, with Hand’s convention for virtual photon flux. The angular distribution is described by the quantities WXY , where X specifies the beam and Y the target polarization. The normalization of the unpolarized term WUU is dϕ d(cos ϑ)WUU(φ,ϕ, ϑ) = 1 . (4.7) To limit the length of subsequent expressions, we further decompose the coefficients ac- cording to the ρ polarization and write WXY (φ,ϕ, ϑ) cos2ϑ WLLXY (φ) + 2 cos ϑ sinϑ WLTXY (φ,ϕ) + sin 2ϑ W TTXY (φ,ϕ) (4.8) for X,Y = U,L. The production of a longitudinal ρ is described by WLLXY , the production 3Up to a global factor, the result of this contraction can e.g. be obtained from eq. (3.20) of [27], with ρµµ′ in the present work corresponding to σ µ′µ in [27] and φhere = −ϕ[27]. – 10 – of a transverse ρ (including the interference between positive and negative ρ helicity) by W TTXY , and the interference between longitudinal and transverse ρ polarization by W For a transversely polarized target we have in addition a dependence on φS , WXT (φS , φ, ϕ, ϑ) cos2ϑ WLLXT (φS , φ) + 2 cos ϑ sinϑ WLTXT (φS , φ, ϕ) + sin 2ϑ W TTXT (φS , φ, ϕ) (4.9) with X = U,L. In addition to the angles, all coefficients WXY depend on Q 2, xB and t, which we have not displayed for the sake of legibility. For unpolarized target and beam we have WLLUU(φ) = u 0 0++ + ǫu − 2 cos φ ǫ(1 + ǫ) Reu 0 00+ − cos(2φ) ǫu 0 0−+ , WLTUU (φ,ϕ) = cos(φ+ ϕ) ǫ(1 + ǫ) Re u 0+0+ − u − cosϕ Re u 0+++ − u−0++ + 2ǫu 0+0 0 + cos(2φ+ ϕ) ǫRe u 0+−+ − cos(φ− ϕ) ǫ(1 + ǫ) Re u 0−0+ − u + cos(2φ − ϕ) ǫRe u+0−+ , W TTUU (φ,ϕ) = u++++ + u ++ + 2ǫu cos(2φ+ 2ϕ) ǫu−+−+ − cosφ ǫ(1 + ǫ) Re u++0+ + u + cos(φ+ 2ϕ) ǫ(1 + ǫ) Re u−+0+ − cos(2ϕ) Re u−+++ + ǫu − cos(2φ) ǫRe u++−+ + cos(φ− 2ϕ) ǫ(1 + ǫ) Reu+−0+ + cos(2φ− 2ϕ) ǫu+−−+ . (4.10) Here and in the following we order terms according to the hierarchy discussed after (3.11), as already done in Table 1. The terms independent of φ and ϕ in WLLUU and W UU are related by u++++ + u ++ + 2ǫu 0 0 = 1− u 0 0++ + ǫu , (4.11) which ensures the normalization condition (4.7). The terms for beam polarization with an unpolarized target read WLLLU (φ) = −2 sinφ ǫ(1− ǫ) Imu 0 00+ , WLTLU (φ,ϕ) = sin(φ+ ϕ) ǫ(1− ǫ) Im u 0+0+ − u − sinϕ 1− ǫ2 Im u 0+++ − u−0++ − sin(φ− ϕ) ǫ(1− ǫ) Im u 0−0+ − u W TTLU (φ,ϕ) = − sinφ ǫ(1− ǫ) Im u++0+ + u + sin(φ+ 2ϕ) ǫ(1− ǫ) Imu−+0+ − sin(2ϕ) 1− ǫ2 Imu−+++ + sin(φ− 2ϕ) ǫ(1− ǫ) Imu+−0+ . (4.12) – 11 – The results for longitudinal target polarization are very similar, with WLLUL(φ) = −2 sinφ ǫ(1 + ǫ) Im l 0 00+ − sin(2φ) ǫ Im l 0 0−+ , WLTUL (φ,ϕ) = sin(φ+ ϕ) ǫ(1 + ǫ) Im l 0+0+ − l − sinϕ Im l 0+++ − l−0++ + 2ǫl 0+0 0 + sin(2φ+ ϕ) ǫ Im l 0+−+ − sin(φ− ϕ) ǫ(1 + ǫ) Im l 0−0+ − l + sin(2φ− ϕ) ǫ Im l+0−+ , W TTUL (φ,ϕ) = sin(2φ+ 2ϕ) ǫ Im l−+−+ − sinφ ǫ(1 + ǫ) Im l++0+ + l + sin(φ+ 2ϕ) ǫ(1 + ǫ) Im l−+0+ − sin(2ϕ) Im l−+++ + ǫl − sin(2φ) ǫ Im l++−+ + sin(φ− 2ϕ) ǫ(1 + ǫ) Im l+−0+ + sin(2φ− 2ϕ) ǫ Im l+−−+ (4.13) for an unpolarized beam, and WLLLL (φ) = −2 cosφ ǫ(1− ǫ) Re l 0 00+ + 1− ǫ2 l 0 0++ , WLTLL (φ,ϕ) = cos(φ+ ϕ) ǫ(1− ǫ) Re l 0+0+ − l − cosϕ 1− ǫ2 Re l 0+++ − l−0++ − cos(φ− ϕ) ǫ(1− ǫ) Re l 0−0+ − l W TTLL (φ,ϕ) = 1− ǫ2 1 l++++ + l − cosφ ǫ(1− ǫ) Re l++0+ + l + cos(φ+ 2ϕ) ǫ(1− ǫ) Re l−+0+ − cos(2ϕ) 1− ǫ2 Re l−+++ + cos(φ− 2ϕ) ǫ(1− ǫ) Re l+−0+ (4.14) for beam polarization. In (4.10) to (4.14) we have used the symmetry relations (3.9) and (3.10) to write our results with a minimal set of matrix elements uνν µµ′ or l µµ′ . Although they are a little lengthy, their structure is quite simple: 1. The combinations u++ + u−− , u 0+ − u−0 and u 0− − u+0 and their analogs for l always appear together because the corresponding products of spherical harmonics are identical, Y1+1Y 1+1 = Y1−1Y 1−1 and Y10Y 1+1 = −Y1−1Y ∗10. In some cases the corresponding sum can be simplified using symmetry relations like u++0 0 + u 0 0 = 2u++0 0 , but in others one remains with a linear combination of matrix elements that cannot be separated. With the caveats discussed after (3.11) one finds however that these combinations are dominated by a single term. Exceptions are Re u 0+++ −u−0++ + 2ǫu 0+0 0 and Im l 0+++ − l−0++ + 2ǫl 0+0 0 , each of which contains two interference terms between a helicity conserving and a helicity changing amplitude. – 12 – 2. An angular dependence through (kφ + mϕ) is associated with the interference be- tween transverse and longitudinal ρ polarization for |m| = 1, the interference between positive and negative ρ helicity for |m| = 2, and equal ρ polarization in the amplitude and its conjugate for m = 0. In the same way |k| = 1, |k| = 2 and k = 0 are related to the virtual photon polarization. Notice that for m = 0 one can distinguish transverse and longitudinal ρ production by the ϑ dependence in (4.8), whereas for k = 0 the separation of terms for transverse and longitudinal photons requires variation of ǫ. The beam spin asymmetries WLU and WLL contain no terms with |k| = 2, because there is no term with Pℓ cos 2φ or Pℓ sin 2φ in the spin density matrix of the virtual photon. 3. The unpolarized or doubly polarized terms WUU and WLL depend on Reu or Re l and are even under the reflection (φ,ϕ) → (−φ,−ϕ) of the azimuthal angles, whereas the single spin asymmetriesWLU andWUL depend on Imu or Im l and are odd under (φ,ϕ) → (−φ,−ϕ). This is a consequence of parity and time reversal invariance. 4. As we have written our results, the angular distribution for longitudinal target po- larization can be obtained from the one for an unpolarized target by replacing cos(kφ+mϕ) Re u → sin(kφ+mϕ) Im l , sin(kφ+mϕ) Imu → cos(kφ+mϕ) Re l . (4.15) Terms with k = m = 0 in WUU and WLL are independent of φ and ϕ, and have of course no counterparts inWUL orWLU . This corresponds to 16 terms with a different angular dependence in WUU and 14 terms in WUL, and to 10 terms in WLL and 8 terms in WLU . The symmetry properties (3.9) and (3.10), which we used to obtain (4.10) to (4.14), are identical for uνν µµ′ and in µµ′ , as well as for l µµ′ and s µµ′ . According to (4.2) the cross section for a transversely polarized target can therefore be obtained from the one for longitudinal and no target polarization by the replacements Reu → ST sin(φ− φS) Imn , SL Im l → ST cos(φ− φS) Im s , Imu → −ST sin(φ− φS) Ren , SLRe l → ST cos(φ− φS) Re s . (4.16) We thus simply have WLLUT (φS , φ) = sin(φ− φS) n 0 0++ + ǫn − 2 cos φ ǫ(1 + ǫ) Imn 0 00+ − cos(2φ) ǫ Imn 0 0−+ + cos(φ− φS) −2 sinφ ǫ(1 + ǫ) Im s 0 00+ − sin(2φ) ǫ Im s 0 0−+ WLTUT (φS , φ, ϕ) = sin(φ− φS) cos(φ+ ϕ) ǫ(1 + ǫ) Im n 0+0+ − n − cosϕ Im n 0+++ − n−0++ + 2ǫn 0+0 0 + cos(2φ+ ϕ) ǫ Imn 0+−+ − cos(φ− ϕ) ǫ(1 + ǫ) Im n 0−0+ − n + cos(2φ− ϕ) ǫ Imn+0−+ – 13 – + cos(φ− φS) sin(φ+ ϕ) ǫ(1 + ǫ) Im s 0+0+ − s − sinϕ Im s 0+++ − s−0++ + 2ǫs 0+0 0 + sin(2φ + ϕ) ǫ Im s 0+−+ − sin(φ− ϕ) ǫ(1 + ǫ) Im s 0−0+ − s + sin(2φ− ϕ) ǫ Im s+0−+ W TTUT (φS , φ, ϕ) = sin(φ− φS) n++++ + n ++ + 2ǫn cos(2φ+ 2ϕ) ǫ Imn−+−+ − cosφ ǫ(1 + ǫ) Im n++0+ + n + cos(φ+ 2ϕ) ǫ(1 + ǫ) Imn−+0+ − cos(2ϕ) Im n−+++ + ǫn − cos(2φ) ǫ Imn++−+ + cos(φ− 2ϕ) ǫ(1 + ǫ) Imn+−0+ + cos(2φ− 2ϕ) ǫ Im n+−−+ + cos(φ− φS) sin(2φ+ 2ϕ) ǫ Im s−+−+ − sinφ ǫ(1 + ǫ) Im s++0+ + s + sin(φ+ 2ϕ) ǫ(1 + ǫ) Im s−+0+ − sin(2ϕ) Im s−+++ + ǫs − sin(2φ) ǫ Im s++−+ + sin(φ− 2ϕ) ǫ(1 + ǫ) Im s+−0+ + sin(2φ− 2ϕ) ǫ Im s+−−+ (4.17) for an unpolarized beam, and WLLLT (φS , φ) = sin(φ− φS) 2 sinφ ǫ(1− ǫ) Ren 0 00+ + cos(φ− φS) −2 cos φ ǫ(1− ǫ) Re s 0 00+ + 1− ǫ2 s 0 0++ WLTLT (φS , φ, ϕ) = sin(φ− φS) − sin(φ+ ϕ) ǫ(1− ǫ) Re n 0+0+ − n + sinϕ 1− ǫ2 Re n 0+++ − n−0++ + sin(φ− ϕ) ǫ(1− ǫ) Re n 0−0+ − n + cos(φ− φS) cos(φ+ ϕ) ǫ(1− ǫ) Re s 0+0+ − s − cosϕ 1− ǫ2 Re s 0+++ − s−0++ − cos(φ− ϕ) ǫ(1− ǫ) Re s 0−0+ − s W TTLT (φS , φ, ϕ) = sin(φ− φS) ǫ(1− ǫ) Re n++0+ + n − sin(φ+ 2ϕ) ǫ(1− ǫ) Ren−+0+ + sin(2ϕ) 1− ǫ2 Ren−+++ − sin(φ− 2ϕ) ǫ(1− ǫ) Ren+−0+ + cos(φ− φS) 1− ǫ2 1 s++++ + s – 14 – unpolarized beam polarized beam WUU WUL WUT WLU WLL WLT Re u Im l Imn Im s Imu Re l Ren Re s 15 14 16 14 8 10 8 10 Table 3: Number of linear combinations of spin density matrix elements describing the angular distribution of the cross section (4.5). The number of independent combinations for Reu is one less than for Imn because of the relation (4.11). − cosφ ǫ(1− ǫ) Re s++0+ + s + cos(φ+ 2ϕ) ǫ(1− ǫ) Re s−+0+ − cos(2ϕ) 1− ǫ2 Re s−+++ + cos(φ− 2ϕ) ǫ(1− ǫ) Re s+−0+ (4.18) for beam polarization. With obvious adjustments, the general structure discussed in points 1 to 3 above is found again for a transverse target. Note that the terms u 0 0++ + ǫu 0 0 and u++++ + u ++ + 2ǫu 0 0 in the unpolarized coefficients W UU and W UU add up to 1 according to (4.11), whereas their counterparts Im n 0 0++ + ǫn and Im n++++ + n ++ + 2ǫn WLLUT and W UT are independent quantities. To keep the close similarity between the two cases we have not used (4.11) to simplify (4.10). Since there are two independent transverse polarizations relative to the hadron plane (normal and sideways) we have a rather large number of terms with different angular dependence in (4.17) and (4.18). The single spin asymmetry WUT contains 16 terms with Imn and 14 terms with Im s , whereas the double spin asymmetry WLT contains 8 terms with Ren and 10 terms with Re s . Table 3 lists the number of independent linear combinations of spin density matrix elements describing the angular distribution for the different combinations of beam and target spin. For reasons discussed in Section 5 it is useful to consider the spin density matrices n and s separately. It is then natural to work in the basis of angular functions given by the product of sin(φ − φS) or cos(φ − φS) with sin(kφ +mϕ) or cos(kφ +mϕ). With the replacement rules (4.15) and (4.16) we obtain the combinations sin(φ− φS) cos(kφ+mϕ) Imn + cos(φ− φS) sin(kφ+mϕ) Im s , − sin(φ− φS) sin(kφ+mϕ) Ren + cos(φ− φS) cos(kφ+mϕ) Re s (4.19) in WUT and WLT , respectively. We conclude this section by giving the relation between our spin density matrix ele- ments for an unpolarized target and those in the classical work [14] of Schilling and Wolf. We have u 0 0++ + ǫu 0 0 = r u 0+0+ − u Im r610 − Re r510 u++++ + u ++ + 2ǫu 0 0 = 1− r0400 , u−+−+ = r 1−1 − Im r21−1 , – 15 – Re u 0 00+ = −r500/ u 0+++ − u−0++ + 2ǫu 0+0 0 = 2Re r0410 , Re u 0+−+ = Re r 10 − Im r210 , u 0−0+ − u Im r610 +Re r u−+++ + ǫu = r041−1 , Re u++−+ = r u++0+ + u 2 r511 , Re u−+0+ = Im r61−1 − r51−1 u 0 0−+ = r Re u+0−+ = Re r 10 + Im r Re u+−0+ = − Im r61−1 + r u+−−+ = r 1−1 + Im r 1−1 (4.20) u 0+0+ − u Im r710 +Re r Imu 0 00+ = r u 0+++ − u−0++ = −2 Im r310 , u 0−0+ − u Im r710 − Re r810 Imu−+++ = − Im r31−1 , u++0+ + u 2 r811 , Imu−+0+ = Im r71−1 + r Imu+−0+ = − Im r71−1 − r81−1 2 . (4.21) The lower indices in the matrix elements of Schilling and Wolf refer to the ρ helicity and correspond to the upper indices of u in our notation. Their upper indices correspond to a representation of the virtual photon spin density matrix which refers partly to circular and partly to linear polarization, whereas we use the helicity basis for the photon throughout. The consequences of approximate s-channel helicity conservation are more explicit in our notation: the relation Im r610 ≈ −Re r510 for instance corresponds to u 0+0+ − u u 0−0+ − u ∣. Notice also that the simple relation between single-spin asymmetries and imaginary parts of spin density matrix elements discussed in point 3 above holds in the helicity basis but not for linear polarization. We note that our phase convention (3.1) for the helicity states of the virtual photon differs from the one in [14] by a relative minus sign between transverse and longitudinal polarization, and that our normalization factors NT and NL in (3.5) differ from those in [14] by a factor of two. The combinations of helicity amplitudes corresponding to the spin density matrix elements in (4.20) and (4.21) should be compared according to NT + ǫNL T νσµλ = ηµµ′ NT + ǫNL Tνσ,µλ T ν′σ,µ′λ , (4.22) – 16 – where η0± = η±0 = −1 for the interference of transverse and longitudinal photon polar- ization, and ηµµ′ = +1 in all other cases. 5. Natural and unnatural parity The exclusive process γ∗p → ρp is described by eighteen independent helicity amplitudes, and we have already used approximate s-channel helicity conservation to establish a hierar- chy among these amplitudes and the spin density matrix elements constructed from them. A further dynamical criterion to order these quantities is given by natural and unnatural parity exchange, which we shall now discuss. Following [14] we define amplitudes N for natural and U unnatural parity exchange as linear combinations Nνσµλ = T νσµλ + (−1)ν−µ T−νσ−µλ T νσµλ + (−1)λ−σ T ν−σµ−λ Uνσµλ = T νσµλ − (−1)ν−µ T−νσ−µλ T νσµλ − (−1)λ−σ T ν−σµ−λ . (5.1) With respect to the photon and meson helicity, the amplitudes N have the same symmetry behavior as the amplitudes for γ∗t→ ρt on a spin-zero target t, whereas the corresponding relation for the amplitudes U has an additional minus sign, = (−1)ν−µNνσµλ , U−νσ−µλ = −(−1) ν−µ Uνσµλ . (5.2) For the proton helicity we have relations Nν+µ+ = N µ− and N µ− = −Nν−µ+ for natural parity exchange, compared to Uν+µ+ = −Uν−µ− and Uν+µ− = Uν−µ+ for unnatural parity exchange. This symmetry behavior immediately implies that in a dynamical description using generalized parton distributions, amplitudes N go with distributions H and E, whereas amplitudes U go with distributions H̃ and Ẽ. This is explicitly borne out in the calculation of [24]. Since U 0σ0λ = 0 according to (5.2), unnatural parity exchange amplitudes are power suppressed at large Q2 and the leading-twist factorization theorem [2] only applies to the natural parity exchange amplitudes N 0σ0λ . We remark that in the context of low-energy dynamics t-channel exchange of a pion plays a prominent role for unnatural parity exchange ampli- tudes, see e.g. [15]. This has a natural counterpart in the framework of generalized parton distributions, where pion exchange gives an essential contribution to the distribution Ẽ in the isovector channel [28, 3, 29]. 4The correspondence in (4.20) to (4.22) is obtained from comparing our results (4.10) and (4.12) for the angular distribution with the ones in eqs. (92) and (92a) of [14], together with the relation between spin density matrix elements and helicity amplitudes specified in eq. (91) and Appendix A of [14]. We have not found an explicit specification of the phase convention for the virtual photon polarizations used in [14]. – 17 – For the spin density matrix elements one readily finds µµ′ = (NT + ǫNL) Nνσµ+ + Uνσµ+ µµ′ = (NT + ǫNL) Nνσµ+ + Uνσµ+ µµ′ = (NT + ǫNL) Nνσµ+ + Uνσµ+ µµ′ = (NT + ǫNL) Nνσµ+ + Uνσµ+ . (5.3) The matrix elements u and n hence involve a product of two natural parity exchange amplitudes plus a product of two amplitudes for unnatural parity exchange, whereas l and s involve the interference between natural and unnatural parity exchange [15]. To the extent that amplitudes U are smaller than their counterparts N , one can thus expect that matrix elements l and s are small compared with u and n for equal helicity indices. Exceptions to this guideline are possible since products Nνσµ+ or Nνσµ− have a small real or imaginary part due to the relative phase between the two amplitudes. If amplitudes U are smaller than N , one can furthermore neglect the terms µµ′ = (NT + ǫNL) Uνσµ+ µµ′ = (NT + ǫNL) Uνσµ+ (5.4) involving unnatural parity exchange in the matrix elements u and n . Using the relations (−1)ν−µ u−νν′ = uνν µµ′ − 2ũνν µµ′ (5.5) following from (5.2) and (5.3), we have in particular −u 0+−+ = u 0+++ − 2ũ 0+++ , u−+−+ = u++++ − 2ũ++++ , −u−+0+ = u 0+ − 2ũ 0+ , u −+ = u ++ − 2ũ−+++ . (5.6) This allows us to rewrite WLTUU = − cosϕ Re u 0+++ − u−0++ + 2ǫu 0+0 0 − cos(2φ+ ϕ) ǫRe u 0+++ − 2ũ 0+++ + . . . cos(φ+ ϕ) + . . . cos(φ− ϕ) + . . . cos(2φ− ϕ) , W TTUU = u++++ + u ++ + 2ǫu cos(2φ+ 2ϕ) ǫ u++++ − 2ũ++++ − cosφ ǫ(1 + ǫ) Re u++0+ + u − cos(φ+ 2ϕ) ǫ(1 + ǫ) Re u++0+ − 2ũ − cos(2ϕ) Re u−+++ + ǫu − cos(2φ) ǫRe u−+++ − 2ũ−+++ + . . . cos(φ− 2ϕ) + . . . cos(2φ− 2ϕ) , – 18 – W TTLU = − sinφ ǫ(1− ǫ) Im u++0+ + u − sin(φ+ 2ϕ) ǫ(1− ǫ) Im u++0+ − 2ũ + . . . sin(2ϕ) + . . . sin(φ− 2ϕ) , (5.7) where terms indicated by . . . are the same as in the original expressions (4.10) and (4.12) and have not been repeated for brevity. We see that the coefficients of adjacent terms in (5.7) will be approximately equal to the extent that unnatural parity exchange is suppressed and s-channel helicity approximately conserved. This can be tested experimentally by measuring the angular distribution of the final-state particles. The relations (5.6) and their counterparts for other index combinations can also be used to approximately isolate spin density matrix elements of particular interest. Consider as an example the leading-twist matrix element u 0 00 0 , which in the angular distribution appears only in the combination u 0 0++ + ǫu 0 0 , i.e. together with a matrix element that should be suppressed since it does not conserve s-channel helicity. If unnatural parity exchange is strongly suppressed, an even better approximation for u 0 00 0 can be obtained from the linear combination ǫu 0 00 0 + 2ũ u 0 0++ + ǫu + u 0 0−+ , (5.8) whose r.h.s. can be extracted from the angular distribution. Similarly, one can approxi- mately isolate the matrix element Re u 0+0 0 in the combination ǫRe u 0+0 0 +Re ũ 0+++ − ũ−0++ u 0+++ − u−0++ + 2ǫu 0+0 0 +Re u 0+−+ +Reu . (5.9) Conversely, one can extract from the angular distribution the linear combinations ũ++++ + ũ −− + 2ǫũ 0 0 − 2Re ũ u++++ + u ++ + 2ǫu u−+−+ − 12 u u−+++ + ǫu −Re u++−+ , ũ++0+ + ũ u++0+ + u u−+0+ + u+−0+ , (5.10) which only involve unnatural parity exchange. In a dynamical approach based on gen- eralized parton distributions, these combinations are interesting because they isolate the polarized distributions H̃ and Ẽ and in particular involve these distributions for gluons, which are very hard to access in any other process.5 The price to pay for this is that the corresponding amplitudes are power suppressed and cannot be calculated with the theoretical rigor provided by the leading-twist factorization theorem. On the other hand, phenomenological analysis indicates that a quantitative description of meson production at Q2 of a few GeV2 requires the inclusion of power-suppressed effects also for the leading matrix element u 0 00 0 . The discussion of the matrix elements for transverse target polarization normal to the hadron plane proceeds in full analogy to the unpolarized case. With (−1)ν−µ n−νν′ = nνν µµ′ − 2ñνν µµ′ (5.11) 5In contrast to their quark counterparts, H̃g and Ẽg do not appear in pseudoscalar meson production at leading twist and leading order in αs, see e.g. Section 5.1.1 of [30]. – 19 – we have −n 0+−+ = n 0+++ − 2ñ 0+++ , n−+−+ = n++++ − 2ñ++++ , −n−+0+ = n 0+ − 2ñ 0+ , n −+ = n ++ − 2ñ−+++ (5.12) and can write WLTUT = cos(φ− φS) . . . + sin(φ− φS) − cosϕ Im n 0+++ − n−0++ + 2ǫn 0+0 0 − cos(2φ+ ϕ) ǫ Im n 0+++ − 2ñ 0+++ + . . . cos(φ+ ϕ) + . . . cos(φ− ϕ) + . . . cos(2φ − ϕ) W TTUT = cos(φ− φS) . . . + sin(φ− φS) n++++ + n ++ + 2ǫn cos(2φ+ 2ϕ) ǫ Im n++++ − 2ñ++++ − cosφ ǫ(1 + ǫ) Im n++0+ + n − cos(φ+ 2ϕ) ǫ(1 + ǫ) Im n++0+ − 2ñ − cos(2ϕ) Im n−+++ + ǫn − cos(2φ) ǫ Im n−+++ − 2ñ−+++ + . . . cos(φ− 2ϕ) + . . . cos(2φ − 2ϕ) W TTLT = cos(φ− φS) . . . + sin(φ− φS) ǫ(1− ǫ) Re n++0+ + n + sin(φ+ 2ϕ) ǫ(1− ǫ) Re n++0+ − 2ñ + . . . sin(2ϕ) + . . . sin(φ− 2ϕ) , (5.13) where terms denoted by . . . are as in the original expressions (4.17) and (4.18). Again, the coefficients of adjacent terms should be approximately equal to the extent that unnatural parity exchange is suppressed and s-channel helicity approximately conserved. The matrix elements Imn 0 00 0 and Imn 0 0 can be approximately isolated in ǫ Imn 0 00 0 + 2 Im ñ ++ = Im n 0 0++ + ǫn + Imn 0 0−+ (5.14) ǫ Imn 0+0 0 + Im ñ 0+++ − ñ−0++ n 0+++ − n−0++ + 2ǫn 0+0 0 + Imn 0+−+ + Imn . (5.15) In turn, the linear combinations ñ++++ + ñ −− + 2ǫñ 0 0 − 2ñ n++++ + n ++ + 2ǫn Imn−+−+ − 12 Imn n−+++ + ǫn − Imn++−+ , ñ++0+ + ñ n++0+ + n n−+0+ + n+−0+ (5.16) involve only unnatural parity exchange. – 20 – 6. Positivity constraints From the definition (3.4) of the spin-density matrix elements one readily finds ν′µ′λ′ cνµλ ρ µµ′,λλ′ = (NT + ǫNL) cνµλ T ≥ 0 (6.1) for arbitrary complex numbers cνµλ. Hence ρ µµ′,λλ′ is a positive semidefinite matrix, with row indices specified by {νµλ} and column indices by {ν ′µ′λ′}. This implies inequalities among the spin density matrix elements, which extend those given e.g. in [22, 27]. We do not attempt here to study the bounds following from positivity of the full 18 × 18 matrix ρνν µµ′,λλ′ , which is quite unwieldy. Instead, we consider the subset of matrix elements conserving s-channel helicity for the photon-meson transition and derive a number of simple inequalities, which may be useful in practice. Ordering the row and column indices as {+++}, {0 0+}, {−−+}, {++−}, {0 0−}, {−−−}, we have a positive semidefinite matrix C, which can be written in block form as A+ B+ B− A− (6.2) u++++ + η l u 0+0+ + η l u−+−+ − η l−+−+ u 0+0+ + η l 0 0 u 0+ − η l u−+−+ + η l u 0+0+ − η l u++++ − η l++++ (6.3) s++++ + η n s 0+0+ − η n )∗ −s−+−+ + η n−+−+ s 0+0+ + η n 0+ η n 0 0 −s 0+0+ + η n s−+−+ + η n s 0+0+ + η n )∗ −s++++ + η n++++ , (6.4) where η = ±1. Concentrating first on the matrix elements for an unpolarized or longitu- dinally polarized target, we find that the matrix Aη has eigenvalues whose expressions are very lengthy and therefore restrict our attention to 2×2 submatrices. The matrix obtained from the first and third rows and columns of A+ has eigenvalues u++++ ± u−+−+ l++++ Im l−+−+ , (6.5) whose positivity implies a bound l++++ Im l−+−+ u++++ u−+−+ . (6.6) Similarly, the matrix obtained from the first and second and the matrix obtained from the second and third rows and columns of A+ have respective eigenvalues u++++ + l ++ + u u++++ + l ++ − u 0 00 0 ∣u 0+0+ + l u++++ − l++++ + u 0 00 0 u++++ − l++++ − u 0 00 0 ∣u 0+0+ − l , (6.7) – 21 – whose positivity gives bounds Reu 0+0+ +Re l Imu 0+0+ + Im l )2 ≤ u 0 00 0 u++++ + l Reu 0+0+ − Re l Imu 0+0+ − Im l )2 ≤ u 0 00 0 u++++ − l++++ . (6.8) A weaker condition is obtained by taking the sum of these two bounds, Re l 0+0+ Im l 0+0+ )2 ≤ u 0 00 0 u++++ − Re u 0+0+ Imu 0+0+ . (6.9) The bounds (6.6) and (6.9) have right-hand sides involving only matrix elements accessi- ble with an unpolarized target and constrain the matrix elements for longitudinal target polarization on their left-hand sides. As a second example let us derive conditions which involve only matrix elements u and n . To this end we consider the matrix C′ = 1 C+D†CD (6.10) 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 , (6.11) which is half the sum of the positive semidefinite matrices C and D†CD and hence positive semidefinite itself. One readily finds that matrix elements l and s drop out in C′, which reads u++++ u 0+0+ u−+−+ n n 0+0+ n−+−+ u 0+0+ u 0 0 u 0 0 n u−+−+ u 0+0+ u++++ n n 0+0+ n++++ −n++++ n 0+0+ )∗ −n−+−+ u++++ u 0+0+ u−+−+ −n 0+0+ −n 0 00 0 −n 0 0 u −n−+−+ n 0+0+ )∗ −n++++ u−+−+ u 0+0+ u++++ . (6.12) This matrix has three eigenvalues u++++ − u−+−+ + Imn++++ − Imn−+−+ , u++++ + u −+ + Imn ++ + Imn −+ + u 0 0 + Imn u++++ + u −+ + Imn ++ + Imn −+ − u 0 00 0 − Imn 0 00 0 ∣u 0+0+ − in (6.13) and three further eigenvalues obtained by reversing the sign of all matrix elements n . Their positivity results in the bounds Imn++++ − Imn−+−+ u++++ − u−+−+ (6.14) – 22 – Reu 0+0+ + Imn Imu 0+0+ − Ren u 0 00 0 + Imn u++++ + u −+ + Imn ++ + Imn Reu 0+0+ − Imn Imu 0+0+ +Ren u 0 00 0 − Imn 0 00 0 u++++ + u −+ − Imn++++ − Imn−+−+ . (6.15) Omitting the terms with Imu 0+0+ and Ren 0+ , one obtains bounds involving only matrix elements that are accessible with an unpolarized lepton beam. As we have seen in Section 4, s-channel helicity conserving matrix elements can be extracted from the angular distribution under the approximation that s-channel helicity changing transitions are suppressed. The bounds derived in this section may be used to check the consistency of this approximation. 7. Mixing between transverse and longitudinal polarization So far we have discussed target polarization longitudinal or transverse to the virtual photon direction in the target rest frame, which is natural from the point of view of the strong- interaction dynamics. In an experimental setup one has however definite target polarization with respect to the lepton beam direction. The transformation from one polarization basis to the other is readily performed using the relations (2.3). For a target having longitudinal polarization PL with respect to the lepton beam one finds dφ dϕd(cos ϑ) dxB dQ2 dt dxB dQ2 dt WUU + PL cos θγWUL − sin θγWUT (φS = 0) + PℓWLU + PℓPL cos θγ WLL − sin θγ WLT (φS = 0) . (7.1) Note that in this case the azimuthal angle ψ in (4.5) needs to be defined with respect to some fixed spatial direction in the target rest frame, rather than with respect to the (vanishing) transverse component of the target polarization relative to the lepton beam. We have integrated over this angle in (7.1) because the cross section does not depend on it. For a target having transverse polarization PT with respect to the lepton beam one dφS dφ dϕd(cos ϑ) dxB dQ2 dt (2π)2 dxB dQ2 dt cos θγ 1− sin2θγ sin2φS WUU + PT cos θγ WUT + sin θγ cosφS WUL 1− sin2θγ sin2φS + PℓWLU + PℓPT cos θγ WLT + sin θγ cosφSWLL 1− sin2θγ sin2φS . (7.2) The factor cos θγ /(1 − sin2θγ sin2φS) comes from the change of variables from dψ to dφS – 23 – in the cross section. The relation between these two angles is readily obtained by setting PL = 0 in (2.3) and given in [22]. It is a straightforward (if somewhat lengthy) exercise to insert our results (4.13), (4.14) and (4.17), (4.18) into (7.1) and (7.2) and to rewrite the expressions in terms of a suitable basis of functions depending on the azimuthal angles. Here we only give the combinations needed in (7.2) for a transversely polarized target and an unpolarized beam, cos θγ W UT (φS , φ) + sin θγ cosφS W UL(φ) = sin(φ− φS) cos θγ Im n 0 0++ + ǫn − sin θγ ǫ(1 + ǫ) Im l 0 00+ − cos(2φ) cos θγ ǫ Imn −+ − sin θγ ǫ(1 + ǫ) Im l 0 00+ − 2 cosφ cos θγ ǫ(1 + ǫ) Imn 0 00+ + sin θγ ǫ Im l + cos(φ− φS) − sin(2φ) cos θγ ǫ Im s −+ + sin θγ ǫ(1 + ǫ) Im l 0 00+ − 2 sinφ cos θγ ǫ(1 + ǫ) Im s 0 00+ + sin θγ ǫ Im l sin θγ sin(φS + 2φ) ǫ Im l −+ , (7.3) cos θγW UT (φS , φ, ϕ) + sin θγ cosφSW UL (φ,ϕ) = sin(φ− φS) cos(φ+ ϕ) cos θγ ǫ(1 + ǫ) Im n 0+0+ − n sin θγ l 0+++ − l−0++ + 2ǫl 0+0 0 + ǫ Im l 0+−+ − cos(φ− ϕ) cos θγ ǫ(1 + ǫ) Im n 0−0+ − n sin θγ l 0+++ − l−0++ + 2ǫl 0+0 0 − ǫ Im l+0−+ +cos(2φ+ ϕ) cos θγ ǫ Imn −+ − 12 sin θγ ǫ(1 + ǫ) Im l 0+0+ − l +cos(2φ− ϕ) cos θγ ǫ Imn sin θγ ǫ(1 + ǫ) Im l 0−0+ − l − cosϕ cos θγ Im n 0+++ − n−0++ + 2ǫn 0+0 0 sin θγ ǫ(1 + ǫ) l 0+0+ − l l 0−0+ − l + cos(φ− φS) sin(φ+ ϕ) cos θγ ǫ(1 + ǫ) Im s 0+0+ − s sin θγ l 0+++ − l−0++ + 2ǫl 0+0 0 − ǫ Im l 0+−+ – 24 – − sin(φ− ϕ) cos θγ ǫ(1 + ǫ) Im s 0−0+ − s sin θγ l 0+++ − l−0++ + 2ǫl 0+0 0 + ǫ Im l+0−+ +sin(2φ+ ϕ) cos θγ ǫ Im s sin θγ ǫ(1 + ǫ) Im l 0+0+ − l +sin(2φ− ϕ) cos θγ ǫ Im s −+ − 12 sin θγ ǫ(1 + ǫ) Im l 0−0+ − l − sinϕ cos θγ Im s 0+++ − s−0++ + 2ǫs 0+0 0 sin θγ ǫ(1 + ǫ) l 0+0+ − l l 0−0+ − l sin θγ sin(φS + 2φ+ ϕ) ǫ Im l −+ + sin(φS + 2φ− ϕ) ǫ Im l+0−+ , (7.4) cos θγW UT (φS , φ, ϕ) + sin θγ cosφSW UL (φ,ϕ) = sin(φ− φS) cos θγ Im n++++ + n ++ + 2ǫn sin θγ ǫ(1 + ǫ) Im l++0+ + l − cos(2φ) cos θγ ǫ Imn −+ − 12 sin θγ ǫ(1 + ǫ) Im l++0+ + l − cosφ cos θγ ǫ(1 + ǫ) Im n++0+ + n sin θγ ǫ Im l cos(2φ+ 2ϕ) cos θγ ǫ Imn −+ − sin θγ ǫ(1 + ǫ) Im l−+0+ cos(2φ− 2ϕ) cos θγ ǫ Imn −+ − sin θγ ǫ(1 + ǫ) Im l+−0+ − cos(2ϕ) cos θγ Im n−+++ + ǫn sin θγ ǫ(1 + ǫ) Im l−+0+ + Im l + cos(φ+ 2ϕ) cos θγ ǫ(1 + ǫ) Imn−+0+ + sin θγ ǫ Im l−+−+ + 2 Im l−+++ + ǫl + cos(φ− 2ϕ) cos θγ ǫ(1 + ǫ) Imn+−0+ + sin θγ ǫ Im l+−−+ − 2 Im l−+++ + ǫl + cos(φ− φS) − sin(2φ) cos θγ ǫ Im s sin θγ ǫ(1 + ǫ) Im l++0+ + l − sinφ cos θγ ǫ(1 + ǫ) Im s++0+ + s sin θγ ǫ Im l sin(2φ+ 2ϕ) cos θγ ǫ Im s −+ + sin θγ ǫ(1 + ǫ) Im l−+0+ sin(2φ− 2ϕ) cos θγ ǫ Im s −+ + sin θγ ǫ(1 + ǫ) Im l+−0+ − sin(2ϕ) cos θγ Im s−+++ + ǫs sin θγ ǫ(1 + ǫ) Im l−+0+ − Im l – 25 – + sin(φ+ 2ϕ) cos θγ ǫ(1 + ǫ) Im s−+0+ + sin θγ ǫ Im l−+−+ − 2 Im l−+++ + ǫl + sin(φ− 2ϕ) cos θγ ǫ(1 + ǫ) Im s+−0+ + sin θγ ǫ Im l+−−+ + 2 Im l−+++ + ǫl sin θγ sin(φS + 2φ+ 2ϕ) ǫ Im l −+ + sin(φS + 2φ− 2ϕ) ǫ Im l+−−+ sin θγ sin(φS + 2φ) ǫ Im l −+ . (7.5) Compared with (4.17) and (4.18) we have changed the order of terms such that one readily sees which coefficients cos θγ Imn or cos θγ Im s receive an admixture from the same coef- ficients sin θγ Im l . The terms in the last lines of (7.3) and (7.4) and in the last two lines of (7.5) involve only coefficients sin θγ Im l . They come with an angular dependence which is absent for sin θγ = 0, as is readily seen by rewriting sin(φS + 2φ+mϕ) = − sin(φ− φS) cos(3φ+mϕ) + cos(φ− φS) sin(3φ +mϕ) . (7.6) We see in (7.3) to (7.5) that from the angular dependence of the cross section for transverse target polarization one can extract linear combinations of terms cos θγ Imn and sin θγ Im l or of cos θγ Im s and sin θγ Im l . To separate these terms requires an additional measurement with longitudinal target polarization.6 The expressions (7.3) to (7.5) allow us to see for which terms the admixture of sin θγ Im l terms can be expected to be small, so that Imn and Im s may be determined with reasonable accuracy without such an additional measurement. Let us discuss a few examples. 1. The leading-twist matrix element n 0 00 0 appears in the linear combination c0 = cos θγ Im n 0 0++ + ǫn − sin θγ ǫ(1 + ǫ) Im l 0 00+ (7.7) in (7.3) and thus has an admixture from l 0 00+ , which involves one s-channel helicity changing amplitude. According to Section 5 this admixture is additionally suppressed if unnatural parity exchange is small compared with natural parity exchange. One may also add to c0 the angular coefficient c1 = − cos θγ ǫ Imn 0 0−+ + sin θγ ǫ(1 + ǫ) Im l 0 00+ (7.8) from (7.3), thus trading the admixture of sin θγ l 0+ for an admixture of cos θγ n which involves two s-channel helicity changing amplitudes (but lacks the relative factor tan θγ and is not suppressed by unnatural parity exchange). We remark that the linear combination of matrix elements in (5.14) corresponds to c0 − c1/ǫ, where l 0 00+ does not drop out. Whether c0, c0+ c1 or c0− c1/ǫ gives the best approximation to cos θγ ǫ Imn 0 0 will thus depend on the detailed magnitude of the relevant terms. In practice one might for instance use the difference between these terms as a measure for the uncertainty of this approximation. 6A corresponding separation for semi-inclusive pion production ep → eπX has recently been performed in [31]. – 26 – 2. The s-channel helicity conserving matrix elements n 0+0+ in (7.4) and n ++ , n −+ in (7.5) come together with terms involving at least one s-channel helicity changing amplitude. These admixtures should hence be negligible unless the corresponding s-channel helicity conserving matrix element is small itself. For Imn 0+0+ this may for instance happen because of the relative phase between the interfering amplitudes. 3. The matrix element n 0 00+ in (7.3) comes with an admixture from l −+ , which involves two s-channel helicity changing amplitudes and should hence again be suppressed. In addition, one can extract Im l 0 0−+ from the angular dependence itself, given the last term in (7.3). We remark that the unpolarized analog u 0 00+ of n 0+ has a real part which is experimentally seen to be nonzero [17, 19], providing evidence that s-channel helicity is not strictly conserved in electroproduction. (In the notation of Schilling and Wolf one has r500 = − 2Reu 0 00+ .) 4. The only s-channel helicity conserving matrix elements for sideways transverse target polarization in (7.3) to (7.5) are s 0+0+ and s −+ . They come together with terms involving at least one s-channel helicity changing amplitude, so that the situation is similar to the one in point 2. Note however that in the present case there is no additional suppression of the admixture terms due to unnatural parity exchange, since both s and l contain one unnatural parity exchange amplitude. In these examples one thus has the favorable situation that the admixture from longitudinal polarization terms is probably small and in some cases may even be removed or traded for yet smaller terms. This does not always happen: the matrix elements n 0+−+ and s −+ in (7.4) receive for instance an admixture from the s-channel helicity conserving term l 0+0+ , which may not be small itself, so that from the coefficients of sin(φ − φS) cos(2φ + ϕ) or cos(φ−φS) sin(2φ+ϕ) one cannot directly infer on the matrix elements Imn 0+−+ or Im s 0+−+ . To make a more precise statement about their size one needs independent information on Im l 0+0+ , for instance from the positivity bound (6.9). 8. A note on non-resonant contributions So far we have treated the production of two pions in a two-step picture, where a ρ is first produced in ep → epρ and then decays as ρ→ π+π−. For deriving the angular distribution and polarization dependence we have used that the pion pair is in the L = 1 partial wave, as can be seen in (4.3). We did however not use the narrow-width approximation for the ρ or make any assumption about its line shape. In fact, our results for the angular distribution can readily be used at any given invariant mass mππ of the pion pair, with the ep cross sections on the left- and right-hand sides of (4.5) made differential in mππ. The spin-density matrix ρνν µµ′,λλ′ and its linear combinations u , l , s , n then depend on mππ and refer not to γ∗p → ρp but to γ∗p → π+π− p with π+π− in the L = 1 partial wave. No explicit reference to the ρ resonance needs to be made in this case. The situation is more complicated if one considers other partial waves of the pion pair, which can arise from non-resonant production mechanisms. To describe a general π+π− – 27 – state, one should replace ρνν µµ′,λλ′ with the spin-density matrix ρ νν′,LL′ µµ′,λλ′ for a pion pair with angular momentum L in the amplitude and L′ in the conjugate amplitude. One then has to take YLν(ϕ, ϑ)Y L′ν′(ϕ, ϑ) instead of Y1ν(ϕ, ϑ)Y 1ν′(ϕ, ϑ) in (4.3) and will obviously obtain a different angular dependence of the ep cross section. The distribution in ϕ and ϑ for a pion pair with L = 0, 1, 2 has been discussed in [32]. It is quite simple to test for the presence of L = 0 or L = 2 partial waves in data by using discrete symmetry properties, and for mππ around the ρ mass one can expect that partial waves with L = 3 or higher are strongly phase space suppressed. Since even partial waves of the π+π− system have charge conjugation parity C = +1 and odd partial waves have C = −1, the interference of L = 1 with L = 0 or L = 2 gives rise to terms in the angular distribution which are odd under interchange of the π+ and π− momenta, i.e. under the replacement ϑ→ π − ϑ , ϕ→ ϕ+ π . (8.1) Simple examples are an angular dependence like cos ϑ or like an odd polynomial in cos ϑ. Corresponding observables provide a way to study the L = 0 and L = 2 partial waves as a “signal” interfering with the ρ resonance “background” [33, 34]. This has been used in the experimental analysis [35], which did see such interference away from the ρ resonance peak, whereas close to the peak the predominance of the ρ was too strong to observe a significant contribution from any partial wave with L 6= 1. If on the other hand one is interested in a precise study of the L = 1 component, one can eliminate its interference with even partial waves by symmetrizing the angular distribution according to (8.1). One is then left with contributions from L = 0 and L = 2 in both the amplitude and its conjugate, which should be very small around the ρ peak. 9. Summary We have expressed the fully differential cross section for exclusive ρ production on a po- larized nucleon in terms of spin density matrix element for the subprocess γ∗p → ρp. We work in the helicity basis for both γ∗ and ρ and obtain very similar forms for the unpolar- ized and polarized parts of the cross sections, with the substitution rules (4.15) and (4.16). The terms for transverse target polarization normal to the hadron plane closely resemble those for an unpolarized target, and in both cases the number of independent spin density matrix elements is reduced if one neglects unnatural parity exchange compared with nat- ural parity exchange. The spin density matrix elements for transverse target polarization in the hadron plane closely resemble those for a longitudinally polarized target, with both types of matrix elements involving the interference between natural and unnatural parity exchange. We have given simple positivity bounds which involve only matrix elements for an unpolarized target and either those for longitudinal target polarization or for transverse target polarization normal to the hadron plane. Furthermore, we have investigated the admixture of longitudinal target polarization relative to the virtual photon momentum for a target polarized transversely to the lepton beam. This admixture should be small for the spin density matrix elements which conserve s-channel helicity in the transition from – 28 – γ∗ to ρ, but it may be important for s-channel helicity changing matrix elements. Finally, we have briefly discussed how the results obtained in this paper can be used and extended for analyzing the production of pion pairs not associated with the ρ resonance. Acknowledgments It is a pleasure to thank my colleagues from HERMES for their interest in this work and for many discussions, especially A. Borissov, J. Dreschler, D. Hasch and A. Rostomyan. I also gratefully acknowledge helpful discussions with P. Kroll and A. Schäfer. This work is supported by the Helmholtz Association, contract number VH-NG-004. References [1] A. V. Radyushkin, Phys. Lett. B 385 (1996) 333 [hep-ph/9605431]. [2] J. C. Collins, L. Frankfurt and M. Strikman, Phys. Rev. D 56 (1997) 2982 [hep-ph/9611433]. [3] K. Goeke, M. V. Polyakov and M. Vanderhaeghen, Prog. Part. Nucl. 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We provide a framework to analyze the electroproduction process ep -> ep rho with a polarized target, writing the angular distribution of the rho decay products in terms of spin density matrix elements that parameterize the hadronic subprocess gamma* p -> rho p. Using the helicity basis for both photon and meson, we find a representation in which the expressions for a polarized and unpolarized target are related by simple substitution rules.

Introduction 1 2. Kinematics and target polarization 2 3. Helicity amplitudes and spin density matrix 4 4. The angular distribution 9 5. Natural and unnatural parity 17 6. Positivity constraints 21 7. Mixing between transverse and longitudinal polarization 23 8. A note on non-resonant contributions 27 9. Summary 28 1. Introduction Exclusive vector meson production has long played an important role in studying the strong interaction. The seminal work [1, 2] has renewed interest in this process, showing that in Bjorken kinematics it provides access to generalized parton distributions and thus to a wealth of information on the structure of the proton. While most theoretical and experimental studies so far are for an unpolarized proton, the particular interest of target polarization became clear when it was pointed out that meson production on a transversely polarized target is sensitive to the nucleon helicity-flip distribution E [3, 4]. This distri- bution offers unique views on the orbital angular momentum carried by partons in the proton [5, 6] and on the correlation between polarization and the spatial distribution of partons [7]. Whereas the corresponding polarization asymmetry in deeply virtual Compton scattering is under better theoretical control, vector meson production has the advantage of a greater sensitivity to the distribution of gluons (which in Compton scattering only enters at next-to-leading order in αs). This holds not only in the high-energy regime but even in a wide range of fixed-target kinematics [8, 9, 10], where polarization measurements are feasible at existing or planned experimental facilities. A different motivation to study polarized exclusive ρ production is that this channel plays a rather prominent role in semi-inclusive pion production [11, 12, 9], which has become a privileged tool to study a variety of spin effects, see e.g. [13]. It is important to identify – 1 – kinematical regions where the exclusive channel ep → epρ → epπ+π− dominates semi- inclusive observables, because in these regions great care must be taken when interpreting the data in terms of semi-inclusive factorization. Even with an unpolarized target, the spin structure of the process ep → epρ→ epπ+π− is very rich, because the angular distribution of the final state contains information on the helicities of the exchanged virtual photon and of the ρ meson, as was worked out in the classical analysis of Schilling and Wolf [14]. Yet more detailed information is available with target polarization [15]. Experiments on unpolarized targets have found that s-channel helicity is approximately conserved in the transition from the γ∗ to the ρ, with helicity changing amplitudes occurring at most at the 10% level [16, 17, 18, 19, 20]. This greatly simplifies the spin structure of the process. The aim of the present paper is to provide an analysis framework for exclusive ρ production on a polarized nucleon target, making as explicit as possible the relation between the angular dependence of the cross section and the helicity amplitudes describing the hadronic subprocess γ∗p→ ρp. We will present our results in a form that emphasizes the close similarity in structure between an unpolarized and a polarized target. Using the helicity basis for both virtual photon and meson, we also provide an alternative to the representation of the unpolarized cross section in [14]. The following section gives the definitions of the kinematics and polarization variables for the reaction under study. In Section 3 we define the helicity amplitudes and the spin density matrix elements describing the process and discuss some of their general properties. In Section 4 we express the angular distribution of the polarized cross section in terms of these spin density matrix elements and point out some salient features of this representa- tion. The simplifications arising from distinguishing natural and unnatural parity exchange in the reaction are discussed in Section 5. A number of positivity bounds relating different spin density matrix elements are given in Section 6. In Section 7 we explain the complica- tions arising from the distinction between target polarization relative to the momentum of either the incident lepton or the virtual photon. The role of non-resonant contributions in π+π− production is briefly discussed in Section 8. Our results are summarized in Section 9. 2. Kinematics and target polarization Let us consider the electroproduction process e(l) + p(p) → e(l′) + p(p′) + ρ(q′) (2.1) followed by the decay ρ(q′) → π+(k) + π−(k′), (2.2) where four-momenta are given in parentheses. Throughout this work we use the one-photon exchange approximation. All or results are equally valid for the production of a φ followed by the decay φ→ K+K−. They also hold if the scattered proton is replaced by an inclusive system X with four-momentum p′, as explained at the end of Section 3. To describe the kinematics we use the conventional variables for deep inelastic pro- cesses, Q2 = −q2, xB = Q2/(2p · q) and y = (p · q)/(p · l). We neglect the lepton mass – 2 – lepton plane hadron plane θγ ST Figure 1: Kinematics of ep → epρ in the target rest frame. ST is the transverse component of the target spin vector w.r.t. the virtual photon direction. throughout and denote the longitudinal lepton beam polarization by Pℓ, with Pℓ = +1 corresponding to a purely right-handed and Pℓ = −1 to a purely left-handed beam. Let us now go to the target rest frame and introduce the right-handed coordinate system (x, y, z) of Fig. 1 such that q points in the positive z direction and l has a positive x component. In this system we have l = |l|(sin θγ , 0, cos θγ) and q = |q|(0, 0, 1), where the angle θγ between l and q is defined to be between 0 and π. In accordance with the Trento convention [21] we define the angle φ between the lepton and the hadron plane as the azimuthal angle of q′ in this coordinate system, and φS as the azimuthal angle of the target spin vector S. Following [22] we write S = (ST cosφS , ST sinφS ,−SL) with 0 ≤ ST ≤ 1 and −1 ≤ SL ≤ 1, so that ST and SL describe transverse and longitudinal polarization with respect to the virtual photon momentum, with SL = 1 corresponding to a right-handed proton in the γ∗p c.m. *γ c.m.p boost π π+ c.m.− Figure 2: Kinematics of the hadronic subprocess γ∗p → ρp followed by the decay ρ → π+π−. The coordinate systems (x, y, z) and (x′, y′, z′) differ from those in Fig. 1. – 3 – To describe the target polarization of a given experimental setup, we introduce an- other right-handed coordinate system (x′, y′, z′) in the target rest frame such that l = |l|(0, 0, 1) and q = |q|(− sin θγ , 0, cos θγ) as shown in Fig. 1. In this system we write S = (PT cosψ,PT sinψ,−PL) with 0 ≤ PT ≤ 1 and −1 ≤ PL ≤ 1, following again [22]. PT and PL describe transverse and longitudinal polarization with respect to the lepton beam direction, with PL = 1 corresponding to a right-handed proton in the ep c.m. The two sets of variables describing the target polarization are related by ST cosφS = cos θγPT cosψ − sin θγPL , ST sinφS = PT sinψ , SL = sin θγPT cosψ + cos θγPL , (2.3) which we will use in Sect. 7. In terms of invariants the mixing angle θγ is given by sin θγ = γ 1− y − 1 1 + γ2 , γ = 2xBMN , (2.4) whereMN is the nucleon mass. In Bjorken kinematics γ is small, and so is sin θγ ≈ γ 1− y. We finally specify the variables describing the vector meson decay (2.2). This is conve- niently done in the π+π− c.m., which can be obtained from the γ∗p c.m. by a boost in the direction of the scattered nucleon as shown in Fig. 2. In the π+π− c.m. we introduce the right-handed coordinate system (x, y, z) shown in Fig. 2, where p′ = |p′|(0, 0,−1) and where the target momentum p has a positive x component. In this system we define ϑ and ϕ as the polar and azimuthal angle of the π+ momentum, i.e. k = |k|(sinϑ cosϕ, sinϑ sinϕ, cos ϑ). The relation between our notation here and the one of Schilling and Wolf is1 φhere = −Φ[14] , ϕhere = φ [14] , ϑhere = θ [14] . (2.5) 3. Helicity amplitudes and spin density matrix The strong-interaction dynamics of the electroproduction process (2.1) is fully contained in the helicity amplitudes for the subprocess γ∗p→ ρp. From these we will construct spin density matrix elements which describe the angular distribution of the overall reaction ep → ep π+π− and its dependence on the target polarization. Since we will deal with interference terms we must specify our phase conventions. We do this in the γ∗p c.m. and use the right-handed coordinate system (x′, y′, z′) shown in Fig. 2. In this system we have q = |q|(0, 0,−1) and q′ = |q′|(sinΘ, 0,− cos Θ), with the scattering angle Θ of the vector meson defined to be between 0 and π. Note that the positive z′ axis points along p rather than q, as is often preferred for theoretical calculations. We specify polarization states of the target proton by two-component spinors χ+1/2 = (1, 0) 1We remark that the expression for sinΦ given in eq. (13) of [14] is incorrect since it is always positive. A correct definition is given in [23]. – 4 – for positive and χ−1/2 = (0, 1) for negative helicity. For the polarization vectors of the virtual photon we choose ε+1 = − 0, 1,−i, 0 , ε−1 = 0, 1, i, 0 εα0 = Nε qα − q p · q p , (3.1) and for the polarization vectors of the ρ e+1 = − 0, cos Θ,−i, sinΘ , e−1 = 0, cos Θ, i, sinΘ eα0 = Ne q′α − q p′ · q′ p , (3.2) where the subscripts indicate helicities. Nε and Ne are positive constants ensuring the proper normalization ε20 = 1 and e 0 = −1 of the longitudinal polarization vectors. In the ρ rest frame and the coordinate system (x, y, z) of Fig. 2, our meson polarization vectors have the standard form e+1 = −(0, 1, i, 0)/ 2, e−1 = (0, 1,−i, 0)/ 2 and e0 = (0, 0, 0, 1). Our phase conventions for the proton and the virtual photon are as in [22]. We now introduce amplitudes T νσµλ for the subprocess γ ∗(µ)+ p(λ) → ρ(ν)+ p(σ) with definite helicities µ, ν, λ, σ. Since the above phase conventions are defined with reference only to momentum vectors of this subprocess, the helicity amplitudes only depend on the photon virtuality, the γ∗p scattering energy and the scattering angle Θ, or equivalently on Q2, xB and t = (p− p′)2. With our phase conventions they obey the usual parity relations T−ν−σ−µ−λ = (−1) ν−µ−σ+λ T νσµλ (3.3) for equal Q2, xB and t on both sides. With these helicity amplitudes we define µµ′,λλ′ = (NT + ǫNL) T νσµλ . (3.4) Regarding the upper indices this is the spin density matrix of the vector meson, whereas the lower indices specify the polarizations in the γ∗p state from which the meson is produced.2 The normalization factors λ,ν,σ ∣T νσ+λ , NL = λ,ν,σ ∣T νσ0λ (3.5) are proportional to the differential cross sections dσT /dt and dσL/dt for transverse and longitudinal photon polarization, respectively, and 1− y − 1 1− y + 1 y2 + 1 (3.6) 2Taking the trace in the meson polarization indices we obtain the relation ρννµµ′,λλ′ ∝ dσ µ′µ/dt between the spin density matrix ρ introduced here and the cross sections and interference terms used in [22]. Compared with [22] we take the opposite order of indices in ρ, so that ν and ν′ appear in the standard order for a spin density matrix. – 5 – is the usual ratio of longitudinal and transverse photon flux. In addition to Q2, xB and t, the spin density matrix elements ρνν µµ′,λλ′ depend on ǫ through the normalization factor (NT + ǫNL). If one can perform a Rosenbluth separation by measuring at different ǫ but equal Q2 and xB, it is advantageous to normalize them instead to NT , NL or NTNL as was done in [14]. It is straightforward to implement such a change in the formulae we give in the following. We find it useful to introduce the combinations µµ′ = µµ′,++ + ρ µµ′,−− , lνν µµ′ = µµ′,++ − ρνν µµ′,−− (3.7) for an unpolarized and a longitudinally polarized target, where for the sake legibility we have labeled the target polarization by ± instead of ±1 . The combinations µµ′ = µµ′,+− + ρ µµ′,−+ , nνν µµ′ = µµ′,+− − ρνν µµ′,−+ (3.8) respectively describe transverse target polarization in the hadron plane (“sideways”) and perpendicular to it (“normal”). One readily finds that the matrices u , l and s are hermi- tian, whereas n is antihermitian, µ′µ = µ′µ = µ′µ = µ′µ = − . (3.9) The diagonal elements uννµµ, l µµ and s µµ are therefore purely real, whereas n µµ is purely imaginary. Furthermore, the parity relations (3.3) translate into u−ν−ν −µ−µ′ = (−1)ν−µ−ν′+µ′ uνν′µµ′ , l−ν−ν −µ−µ′ = −(−1)ν−µ−ν′+µ′ lνν′µµ′ , n−ν−ν −µ−µ′ = (−1) ν−µ−ν′+µ′ nνν µµ′ , s −ν−ν′ −µ−µ′ = −(−1) ν−µ−ν′+µ′ sνν µµ′ . (3.10) As a consequence the matrix elements u−+−+ , u −+ , u 0 0 , u −+ (3.11) are purely real, whereas the corresponding elements of l , s and n are purely imaginary. Both experiment and theory indicate that s-channel helicity is approximately conserved in the γ∗ → ρ transition for small invariant momentum transfer t. Correspondingly, one expects that spin density matrix elements involving the product of two helicity conserving amplitudes are greater than interference terms between a helicity conserving and a helicity changing amplitude, and that those are greater than matrix elements involving the product of two helicity changing amplitudes (where we refer to the helicities of the photon and the ρ but not of the nucleon). Exceptions to this rule are however possible, since two large amplitudes can have a small interference term because of their relative phase, and since there can be cancellation of individually large terms in the linear combinations (3.7) and (3.8) associated with different target polarizations. With this caveat in mind one can readily assess the expected size of the spin density matrix elements (3.7) and (3.8) by comparing the upper with the lower indices. – 6 – Let us now investigate the behavior of our matrix elements for Θ → 0, i.e. in the limit of forward scattering γ∗p→ ρp. To this end we perform a partial wave decomposition T νσµλ (Θ) = tνσµλ(J) d λ−µ,σ−ν(Θ) (3.12) where we have suppressed the dependence of T and the partial wave amplitudes t(J) on Q2 and xB . Using the behavior d m,n(Θ) ∼ Θ|m−n| of the rotation functions for Θ → 0 we readily find µµ′ , l µµ′ ∼ Θp , nνν µµ′ , s µµ′ ∼ Θq (3.13) p ≥ pmin = min σ,λ=±1/2 ∣ν − µ− σ + λ ∣ν ′ − µ′ − σ + λ q ≥ qmin = min σ,λ=±1/2 ∣ν − µ− σ + λ ∣ν ′ − µ′ − σ − λ . (3.14) With Θ ∝ (t0 − t)1/2 for small Θ, we can rewrite (3.13) as µµ′ , l µµ′ ∼ (t0 − t)p/2 , nνν µµ′ , s µµ′ ∼ (t0 − t)q/2 , (3.15) where t0 is the value of t for Θ = 0 at given Q 2 and xB. In Tables 1 and 2 we give the corresponding powers for the linear combinations of spin density matrix elements that will appear in our results for the cross section in Section 4. We have ordered the entries according to the hierarchy discussed after (3.11), listing first terms containing the product of two helicity conserving amplitudes, then terms containing the interference between a helicity conserving and a helicity changing amplitude, and finally terms which only involve helicity changing amplitudes (with helicities always referring to the photon and the ρ but not to the nucleon). We emphasize that certain partial wave amplitudes tνσµλ(J) in (3.12) may be zero or negligibly small for dynamical reasons. The actual powers of (t0 − t)1/2 in (3.15) can thus be larger than the minimum values pmin and qmin required by angular momentum conservation. If there is for instance no s-channel helicity transferred between the proton- proton and the photon-meson transitions, then the relevant powers for n and s are given by q = pmin+1, which is equal to qmin+2 for all but the first four entries in Tables 1 and 2. A concrete realization of this scenario is the calculation in [24], where the proton-proton transition is described by the generalized parton distributions H, E and H̃, Ẽ, which do not allow for helicity transfer to the photon-meson transition. In the limit of large Q2 at fixed xB and t, the proof of the factorization theorem in [2] implies that the transition from a longitudinal photon to a longitudinal ρ becomes dominant, with all other transitions suppressed by powers of 1/Q. In this limit only the spin density matrix elements u 0 00 0 and n 0 0 survive and can be expressed as convolutions of hard-scattering kernels with generalized parton distributions and the light-cone distribution amplitude of the ρ. To leading order in 1/Q one has in particular Imn 0 00 0 u 0 00 0 t0 − t 1− ξ2 Im (1− ξ2) |H|2 − ξ2 + t/(4M2N ) |E|2 − 2ξ2 Re , (3.16) – 7 – matrix elements pmin u 0 0++ + ǫu 0 0 0 u 0+0+ − u 0+ − l u++++ + u ++ + 2ǫu 0 0 l ++ + l u−+−+ l u 0 00+ l u 0+++ − u−0++ + 2Re ǫu 0+0 0 l ++ − l−0++ + 2i Im ǫl 0+0 0 1 u 0+−+ l u 0−0+ − u 0+ − l u−+++ + ǫu 0 0 l ++ + ǫl 0 0 2 u++−+ l u++0+ + u 0+ + l u−+0+ l l 0 0++ 2 u 0 0−+ l u+0−+ l u+−0+ l u+−−+ l Table 1: Minimum values of the powers which control the t→ t0 behavior of combinations of spin density matrix elements u and l as in (3.15). Some of the combinations are purely real or purely imaginary because of the symmetry relations (3.9) and (3.10), whereas others are complex valued. where ξ = xB/(2−xB) and the convolution integrals H and E are for instance given in [22]. Experimental results and phenomenological analysis show however that 1/Q2 suppressed effects can be numerically significant for Q2 of several GeV2, see e.g. [25, 24, 9, 10]. This concerns both power corrections within u 0 00 0 or n 0 0 and formally power suppressed spin density matrix elements such as u++++ or u 0+ . The detailed analysis in [2] reveals that beyond leading-power accuracy in 1/Q, factorization of meson production into a hard- scattering subprocess and nonperturbative quantities pertaining either to the target or to the meson may be broken. On the other hand, factorization based approaches which go beyond leading power in 1/Q and in particular also evaluate transition amplitudes for transverse polarization of the γ∗ or ρ have been phenomenologically rather successful, see e.g. [26, 24] Let us finally generalize our considerations to the process e(l) + p(p) → e(l′) +X(p′) + ρ(q′) , (3.17) where the target proton dissociates into a hadronic system X. In analogy to the elastic case one can introduce helicity amplitudes T and combine them into spin density matrix – 8 – matrix elements qmin n 0 0++ + ǫn 0 0 1 n 0+0+ − n 0+ − s n++++ + n ++ + 2ǫn 0 0 s ++ + s n−+−+ s n 0 00+ s n 0+++ − n−0++ + 2i Im ǫn 0+0 0 s ++ − s−0++ + 2i Im ǫs 0+0 0 0 n 0+−+ s n 0−0+ − n 0+ − s n−+++ + ǫn 0 0 s ++ + ǫs 0 0 1 n++−+ s n++0+ + n 0+ + s n−+0+ s s 0 0++ 1 n 0 0−+ s n+0−+ s n+−0+ s n+−−+ s Table 2: As Table 1 but for combinations of spin density matrix elements n and s . elements µµ′,λλ′ = (NT + ǫNL) ν′σ,X . (3.18) The normalization factors NT and NL are defined as in (3.5) but with an additional sum over all hadronic states X of given invariant mass MX , on which ρ µµ′,λλ′ now depends in addition to Q2, xB , t and ǫ. The combinations (3.7) and (3.8) for different target polarization have the same symmetry properties (3.9) and (3.10) as in the elastic case. Their behavior for t→ t0 can be different, since in (3.14) one must now take the minimum over all possible helicities σ = ±1 , . . . of the hadronic system X. One finds however that the powers pmin and qmin for the combinations of spin density matrix elements in Tables 1 and 2 are the same as in the elastic case. The results in the remainder of this work only depend on the properties (3.9) and (3.10) and thus immediately generalize to the case of target dissociation. 4. The angular distribution The calculation of the cross section for ep → ep π+π− proceeds by using standard methods and we shall only sketch the essential steps. More details are for instance given in [14, 27, – 9 – 22]. With our phase conventions the polarization state of the proton target is described by the spin density matrix τλλ′ = 1 + SL ST e −i(φ−φS) i(φ−φS) 1− SL , (4.1) which is to be contracted with the matrix in (3.4). The result is conveniently expressed in terms of the combinations (3.7) and (3.8) as τλλ′ ρ µµ′,λλ′ = u µµ′ + SL l µµ′ + ST cos(φ− φS) sνν µµ′ − ST sin(φ− φS) inνν µµ′ (4.2) and describes the subprocess γ∗p → ρp. The decay ρ → π+π− is taken into account by multiplication with the spherical harmonics, ρµµ′ = τλλ′ ρ µµ′,λλ′ Y1ν(ϕ, ϑ)Y 1ν′(ϕ, ϑ) , (4.3) where Y1+1 = − sinϑ eiϕ , Y10 = cos ϑ , Y1−1 = sinϑ e−iϕ . (4.4) To obtain the cross section for the overall process ep→ epπ+π− one must finally contract the matrix ρµµ′ in (4.3) with the spin density matrix of the virtual photon. 3 The cross section can be written as dψ dφdϕd(cos ϑ) dxB dQ2 dt (2π)2 dxB dQ2 dt WUU + PℓWLU + SLWUL + PℓSLWLL + STWUT + PℓSTWLT (4.5) dxB dQ2 dt 1− xB , (4.6) where dσT /dt and dσL/dt are the usual γ ∗p cross sections for a transverse and longitudinal photon and an unpolarized proton, with Hand’s convention for virtual photon flux. The angular distribution is described by the quantities WXY , where X specifies the beam and Y the target polarization. The normalization of the unpolarized term WUU is dϕ d(cos ϑ)WUU(φ,ϕ, ϑ) = 1 . (4.7) To limit the length of subsequent expressions, we further decompose the coefficients ac- cording to the ρ polarization and write WXY (φ,ϕ, ϑ) cos2ϑ WLLXY (φ) + 2 cos ϑ sinϑ WLTXY (φ,ϕ) + sin 2ϑ W TTXY (φ,ϕ) (4.8) for X,Y = U,L. The production of a longitudinal ρ is described by WLLXY , the production 3Up to a global factor, the result of this contraction can e.g. be obtained from eq. (3.20) of [27], with ρµµ′ in the present work corresponding to σ µ′µ in [27] and φhere = −ϕ[27]. – 10 – of a transverse ρ (including the interference between positive and negative ρ helicity) by W TTXY , and the interference between longitudinal and transverse ρ polarization by W For a transversely polarized target we have in addition a dependence on φS , WXT (φS , φ, ϕ, ϑ) cos2ϑ WLLXT (φS , φ) + 2 cos ϑ sinϑ WLTXT (φS , φ, ϕ) + sin 2ϑ W TTXT (φS , φ, ϕ) (4.9) with X = U,L. In addition to the angles, all coefficients WXY depend on Q 2, xB and t, which we have not displayed for the sake of legibility. For unpolarized target and beam we have WLLUU(φ) = u 0 0++ + ǫu − 2 cos φ ǫ(1 + ǫ) Reu 0 00+ − cos(2φ) ǫu 0 0−+ , WLTUU (φ,ϕ) = cos(φ+ ϕ) ǫ(1 + ǫ) Re u 0+0+ − u − cosϕ Re u 0+++ − u−0++ + 2ǫu 0+0 0 + cos(2φ+ ϕ) ǫRe u 0+−+ − cos(φ− ϕ) ǫ(1 + ǫ) Re u 0−0+ − u + cos(2φ − ϕ) ǫRe u+0−+ , W TTUU (φ,ϕ) = u++++ + u ++ + 2ǫu cos(2φ+ 2ϕ) ǫu−+−+ − cosφ ǫ(1 + ǫ) Re u++0+ + u + cos(φ+ 2ϕ) ǫ(1 + ǫ) Re u−+0+ − cos(2ϕ) Re u−+++ + ǫu − cos(2φ) ǫRe u++−+ + cos(φ− 2ϕ) ǫ(1 + ǫ) Reu+−0+ + cos(2φ− 2ϕ) ǫu+−−+ . (4.10) Here and in the following we order terms according to the hierarchy discussed after (3.11), as already done in Table 1. The terms independent of φ and ϕ in WLLUU and W UU are related by u++++ + u ++ + 2ǫu 0 0 = 1− u 0 0++ + ǫu , (4.11) which ensures the normalization condition (4.7). The terms for beam polarization with an unpolarized target read WLLLU (φ) = −2 sinφ ǫ(1− ǫ) Imu 0 00+ , WLTLU (φ,ϕ) = sin(φ+ ϕ) ǫ(1− ǫ) Im u 0+0+ − u − sinϕ 1− ǫ2 Im u 0+++ − u−0++ − sin(φ− ϕ) ǫ(1− ǫ) Im u 0−0+ − u W TTLU (φ,ϕ) = − sinφ ǫ(1− ǫ) Im u++0+ + u + sin(φ+ 2ϕ) ǫ(1− ǫ) Imu−+0+ − sin(2ϕ) 1− ǫ2 Imu−+++ + sin(φ− 2ϕ) ǫ(1− ǫ) Imu+−0+ . (4.12) – 11 – The results for longitudinal target polarization are very similar, with WLLUL(φ) = −2 sinφ ǫ(1 + ǫ) Im l 0 00+ − sin(2φ) ǫ Im l 0 0−+ , WLTUL (φ,ϕ) = sin(φ+ ϕ) ǫ(1 + ǫ) Im l 0+0+ − l − sinϕ Im l 0+++ − l−0++ + 2ǫl 0+0 0 + sin(2φ+ ϕ) ǫ Im l 0+−+ − sin(φ− ϕ) ǫ(1 + ǫ) Im l 0−0+ − l + sin(2φ− ϕ) ǫ Im l+0−+ , W TTUL (φ,ϕ) = sin(2φ+ 2ϕ) ǫ Im l−+−+ − sinφ ǫ(1 + ǫ) Im l++0+ + l + sin(φ+ 2ϕ) ǫ(1 + ǫ) Im l−+0+ − sin(2ϕ) Im l−+++ + ǫl − sin(2φ) ǫ Im l++−+ + sin(φ− 2ϕ) ǫ(1 + ǫ) Im l+−0+ + sin(2φ− 2ϕ) ǫ Im l+−−+ (4.13) for an unpolarized beam, and WLLLL (φ) = −2 cosφ ǫ(1− ǫ) Re l 0 00+ + 1− ǫ2 l 0 0++ , WLTLL (φ,ϕ) = cos(φ+ ϕ) ǫ(1− ǫ) Re l 0+0+ − l − cosϕ 1− ǫ2 Re l 0+++ − l−0++ − cos(φ− ϕ) ǫ(1− ǫ) Re l 0−0+ − l W TTLL (φ,ϕ) = 1− ǫ2 1 l++++ + l − cosφ ǫ(1− ǫ) Re l++0+ + l + cos(φ+ 2ϕ) ǫ(1− ǫ) Re l−+0+ − cos(2ϕ) 1− ǫ2 Re l−+++ + cos(φ− 2ϕ) ǫ(1− ǫ) Re l+−0+ (4.14) for beam polarization. In (4.10) to (4.14) we have used the symmetry relations (3.9) and (3.10) to write our results with a minimal set of matrix elements uνν µµ′ or l µµ′ . Although they are a little lengthy, their structure is quite simple: 1. The combinations u++ + u−− , u 0+ − u−0 and u 0− − u+0 and their analogs for l always appear together because the corresponding products of spherical harmonics are identical, Y1+1Y 1+1 = Y1−1Y 1−1 and Y10Y 1+1 = −Y1−1Y ∗10. In some cases the corresponding sum can be simplified using symmetry relations like u++0 0 + u 0 0 = 2u++0 0 , but in others one remains with a linear combination of matrix elements that cannot be separated. With the caveats discussed after (3.11) one finds however that these combinations are dominated by a single term. Exceptions are Re u 0+++ −u−0++ + 2ǫu 0+0 0 and Im l 0+++ − l−0++ + 2ǫl 0+0 0 , each of which contains two interference terms between a helicity conserving and a helicity changing amplitude. – 12 – 2. An angular dependence through (kφ + mϕ) is associated with the interference be- tween transverse and longitudinal ρ polarization for |m| = 1, the interference between positive and negative ρ helicity for |m| = 2, and equal ρ polarization in the amplitude and its conjugate for m = 0. In the same way |k| = 1, |k| = 2 and k = 0 are related to the virtual photon polarization. Notice that for m = 0 one can distinguish transverse and longitudinal ρ production by the ϑ dependence in (4.8), whereas for k = 0 the separation of terms for transverse and longitudinal photons requires variation of ǫ. The beam spin asymmetries WLU and WLL contain no terms with |k| = 2, because there is no term with Pℓ cos 2φ or Pℓ sin 2φ in the spin density matrix of the virtual photon. 3. The unpolarized or doubly polarized terms WUU and WLL depend on Reu or Re l and are even under the reflection (φ,ϕ) → (−φ,−ϕ) of the azimuthal angles, whereas the single spin asymmetriesWLU andWUL depend on Imu or Im l and are odd under (φ,ϕ) → (−φ,−ϕ). This is a consequence of parity and time reversal invariance. 4. As we have written our results, the angular distribution for longitudinal target po- larization can be obtained from the one for an unpolarized target by replacing cos(kφ+mϕ) Re u → sin(kφ+mϕ) Im l , sin(kφ+mϕ) Imu → cos(kφ+mϕ) Re l . (4.15) Terms with k = m = 0 in WUU and WLL are independent of φ and ϕ, and have of course no counterparts inWUL orWLU . This corresponds to 16 terms with a different angular dependence in WUU and 14 terms in WUL, and to 10 terms in WLL and 8 terms in WLU . The symmetry properties (3.9) and (3.10), which we used to obtain (4.10) to (4.14), are identical for uνν µµ′ and in µµ′ , as well as for l µµ′ and s µµ′ . According to (4.2) the cross section for a transversely polarized target can therefore be obtained from the one for longitudinal and no target polarization by the replacements Reu → ST sin(φ− φS) Imn , SL Im l → ST cos(φ− φS) Im s , Imu → −ST sin(φ− φS) Ren , SLRe l → ST cos(φ− φS) Re s . (4.16) We thus simply have WLLUT (φS , φ) = sin(φ− φS) n 0 0++ + ǫn − 2 cos φ ǫ(1 + ǫ) Imn 0 00+ − cos(2φ) ǫ Imn 0 0−+ + cos(φ− φS) −2 sinφ ǫ(1 + ǫ) Im s 0 00+ − sin(2φ) ǫ Im s 0 0−+ WLTUT (φS , φ, ϕ) = sin(φ− φS) cos(φ+ ϕ) ǫ(1 + ǫ) Im n 0+0+ − n − cosϕ Im n 0+++ − n−0++ + 2ǫn 0+0 0 + cos(2φ+ ϕ) ǫ Imn 0+−+ − cos(φ− ϕ) ǫ(1 + ǫ) Im n 0−0+ − n + cos(2φ− ϕ) ǫ Imn+0−+ – 13 – + cos(φ− φS) sin(φ+ ϕ) ǫ(1 + ǫ) Im s 0+0+ − s − sinϕ Im s 0+++ − s−0++ + 2ǫs 0+0 0 + sin(2φ + ϕ) ǫ Im s 0+−+ − sin(φ− ϕ) ǫ(1 + ǫ) Im s 0−0+ − s + sin(2φ− ϕ) ǫ Im s+0−+ W TTUT (φS , φ, ϕ) = sin(φ− φS) n++++ + n ++ + 2ǫn cos(2φ+ 2ϕ) ǫ Imn−+−+ − cosφ ǫ(1 + ǫ) Im n++0+ + n + cos(φ+ 2ϕ) ǫ(1 + ǫ) Imn−+0+ − cos(2ϕ) Im n−+++ + ǫn − cos(2φ) ǫ Imn++−+ + cos(φ− 2ϕ) ǫ(1 + ǫ) Imn+−0+ + cos(2φ− 2ϕ) ǫ Im n+−−+ + cos(φ− φS) sin(2φ+ 2ϕ) ǫ Im s−+−+ − sinφ ǫ(1 + ǫ) Im s++0+ + s + sin(φ+ 2ϕ) ǫ(1 + ǫ) Im s−+0+ − sin(2ϕ) Im s−+++ + ǫs − sin(2φ) ǫ Im s++−+ + sin(φ− 2ϕ) ǫ(1 + ǫ) Im s+−0+ + sin(2φ− 2ϕ) ǫ Im s+−−+ (4.17) for an unpolarized beam, and WLLLT (φS , φ) = sin(φ− φS) 2 sinφ ǫ(1− ǫ) Ren 0 00+ + cos(φ− φS) −2 cos φ ǫ(1− ǫ) Re s 0 00+ + 1− ǫ2 s 0 0++ WLTLT (φS , φ, ϕ) = sin(φ− φS) − sin(φ+ ϕ) ǫ(1− ǫ) Re n 0+0+ − n + sinϕ 1− ǫ2 Re n 0+++ − n−0++ + sin(φ− ϕ) ǫ(1− ǫ) Re n 0−0+ − n + cos(φ− φS) cos(φ+ ϕ) ǫ(1− ǫ) Re s 0+0+ − s − cosϕ 1− ǫ2 Re s 0+++ − s−0++ − cos(φ− ϕ) ǫ(1− ǫ) Re s 0−0+ − s W TTLT (φS , φ, ϕ) = sin(φ− φS) ǫ(1− ǫ) Re n++0+ + n − sin(φ+ 2ϕ) ǫ(1− ǫ) Ren−+0+ + sin(2ϕ) 1− ǫ2 Ren−+++ − sin(φ− 2ϕ) ǫ(1− ǫ) Ren+−0+ + cos(φ− φS) 1− ǫ2 1 s++++ + s – 14 – unpolarized beam polarized beam WUU WUL WUT WLU WLL WLT Re u Im l Imn Im s Imu Re l Ren Re s 15 14 16 14 8 10 8 10 Table 3: Number of linear combinations of spin density matrix elements describing the angular distribution of the cross section (4.5). The number of independent combinations for Reu is one less than for Imn because of the relation (4.11). − cosφ ǫ(1− ǫ) Re s++0+ + s + cos(φ+ 2ϕ) ǫ(1− ǫ) Re s−+0+ − cos(2ϕ) 1− ǫ2 Re s−+++ + cos(φ− 2ϕ) ǫ(1− ǫ) Re s+−0+ (4.18) for beam polarization. With obvious adjustments, the general structure discussed in points 1 to 3 above is found again for a transverse target. Note that the terms u 0 0++ + ǫu 0 0 and u++++ + u ++ + 2ǫu 0 0 in the unpolarized coefficients W UU and W UU add up to 1 according to (4.11), whereas their counterparts Im n 0 0++ + ǫn and Im n++++ + n ++ + 2ǫn WLLUT and W UT are independent quantities. To keep the close similarity between the two cases we have not used (4.11) to simplify (4.10). Since there are two independent transverse polarizations relative to the hadron plane (normal and sideways) we have a rather large number of terms with different angular dependence in (4.17) and (4.18). The single spin asymmetry WUT contains 16 terms with Imn and 14 terms with Im s , whereas the double spin asymmetry WLT contains 8 terms with Ren and 10 terms with Re s . Table 3 lists the number of independent linear combinations of spin density matrix elements describing the angular distribution for the different combinations of beam and target spin. For reasons discussed in Section 5 it is useful to consider the spin density matrices n and s separately. It is then natural to work in the basis of angular functions given by the product of sin(φ − φS) or cos(φ − φS) with sin(kφ +mϕ) or cos(kφ +mϕ). With the replacement rules (4.15) and (4.16) we obtain the combinations sin(φ− φS) cos(kφ+mϕ) Imn + cos(φ− φS) sin(kφ+mϕ) Im s , − sin(φ− φS) sin(kφ+mϕ) Ren + cos(φ− φS) cos(kφ+mϕ) Re s (4.19) in WUT and WLT , respectively. We conclude this section by giving the relation between our spin density matrix ele- ments for an unpolarized target and those in the classical work [14] of Schilling and Wolf. We have u 0 0++ + ǫu 0 0 = r u 0+0+ − u Im r610 − Re r510 u++++ + u ++ + 2ǫu 0 0 = 1− r0400 , u−+−+ = r 1−1 − Im r21−1 , – 15 – Re u 0 00+ = −r500/ u 0+++ − u−0++ + 2ǫu 0+0 0 = 2Re r0410 , Re u 0+−+ = Re r 10 − Im r210 , u 0−0+ − u Im r610 +Re r u−+++ + ǫu = r041−1 , Re u++−+ = r u++0+ + u 2 r511 , Re u−+0+ = Im r61−1 − r51−1 u 0 0−+ = r Re u+0−+ = Re r 10 + Im r Re u+−0+ = − Im r61−1 + r u+−−+ = r 1−1 + Im r 1−1 (4.20) u 0+0+ − u Im r710 +Re r Imu 0 00+ = r u 0+++ − u−0++ = −2 Im r310 , u 0−0+ − u Im r710 − Re r810 Imu−+++ = − Im r31−1 , u++0+ + u 2 r811 , Imu−+0+ = Im r71−1 + r Imu+−0+ = − Im r71−1 − r81−1 2 . (4.21) The lower indices in the matrix elements of Schilling and Wolf refer to the ρ helicity and correspond to the upper indices of u in our notation. Their upper indices correspond to a representation of the virtual photon spin density matrix which refers partly to circular and partly to linear polarization, whereas we use the helicity basis for the photon throughout. The consequences of approximate s-channel helicity conservation are more explicit in our notation: the relation Im r610 ≈ −Re r510 for instance corresponds to u 0+0+ − u u 0−0+ − u ∣. Notice also that the simple relation between single-spin asymmetries and imaginary parts of spin density matrix elements discussed in point 3 above holds in the helicity basis but not for linear polarization. We note that our phase convention (3.1) for the helicity states of the virtual photon differs from the one in [14] by a relative minus sign between transverse and longitudinal polarization, and that our normalization factors NT and NL in (3.5) differ from those in [14] by a factor of two. The combinations of helicity amplitudes corresponding to the spin density matrix elements in (4.20) and (4.21) should be compared according to NT + ǫNL T νσµλ = ηµµ′ NT + ǫNL Tνσ,µλ T ν′σ,µ′λ , (4.22) – 16 – where η0± = η±0 = −1 for the interference of transverse and longitudinal photon polar- ization, and ηµµ′ = +1 in all other cases. 5. Natural and unnatural parity The exclusive process γ∗p → ρp is described by eighteen independent helicity amplitudes, and we have already used approximate s-channel helicity conservation to establish a hierar- chy among these amplitudes and the spin density matrix elements constructed from them. A further dynamical criterion to order these quantities is given by natural and unnatural parity exchange, which we shall now discuss. Following [14] we define amplitudes N for natural and U unnatural parity exchange as linear combinations Nνσµλ = T νσµλ + (−1)ν−µ T−νσ−µλ T νσµλ + (−1)λ−σ T ν−σµ−λ Uνσµλ = T νσµλ − (−1)ν−µ T−νσ−µλ T νσµλ − (−1)λ−σ T ν−σµ−λ . (5.1) With respect to the photon and meson helicity, the amplitudes N have the same symmetry behavior as the amplitudes for γ∗t→ ρt on a spin-zero target t, whereas the corresponding relation for the amplitudes U has an additional minus sign, = (−1)ν−µNνσµλ , U−νσ−µλ = −(−1) ν−µ Uνσµλ . (5.2) For the proton helicity we have relations Nν+µ+ = N µ− and N µ− = −Nν−µ+ for natural parity exchange, compared to Uν+µ+ = −Uν−µ− and Uν+µ− = Uν−µ+ for unnatural parity exchange. This symmetry behavior immediately implies that in a dynamical description using generalized parton distributions, amplitudes N go with distributions H and E, whereas amplitudes U go with distributions H̃ and Ẽ. This is explicitly borne out in the calculation of [24]. Since U 0σ0λ = 0 according to (5.2), unnatural parity exchange amplitudes are power suppressed at large Q2 and the leading-twist factorization theorem [2] only applies to the natural parity exchange amplitudes N 0σ0λ . We remark that in the context of low-energy dynamics t-channel exchange of a pion plays a prominent role for unnatural parity exchange ampli- tudes, see e.g. [15]. This has a natural counterpart in the framework of generalized parton distributions, where pion exchange gives an essential contribution to the distribution Ẽ in the isovector channel [28, 3, 29]. 4The correspondence in (4.20) to (4.22) is obtained from comparing our results (4.10) and (4.12) for the angular distribution with the ones in eqs. (92) and (92a) of [14], together with the relation between spin density matrix elements and helicity amplitudes specified in eq. (91) and Appendix A of [14]. We have not found an explicit specification of the phase convention for the virtual photon polarizations used in [14]. – 17 – For the spin density matrix elements one readily finds µµ′ = (NT + ǫNL) Nνσµ+ + Uνσµ+ µµ′ = (NT + ǫNL) Nνσµ+ + Uνσµ+ µµ′ = (NT + ǫNL) Nνσµ+ + Uνσµ+ µµ′ = (NT + ǫNL) Nνσµ+ + Uνσµ+ . (5.3) The matrix elements u and n hence involve a product of two natural parity exchange amplitudes plus a product of two amplitudes for unnatural parity exchange, whereas l and s involve the interference between natural and unnatural parity exchange [15]. To the extent that amplitudes U are smaller than their counterparts N , one can thus expect that matrix elements l and s are small compared with u and n for equal helicity indices. Exceptions to this guideline are possible since products Nνσµ+ or Nνσµ− have a small real or imaginary part due to the relative phase between the two amplitudes. If amplitudes U are smaller than N , one can furthermore neglect the terms µµ′ = (NT + ǫNL) Uνσµ+ µµ′ = (NT + ǫNL) Uνσµ+ (5.4) involving unnatural parity exchange in the matrix elements u and n . Using the relations (−1)ν−µ u−νν′ = uνν µµ′ − 2ũνν µµ′ (5.5) following from (5.2) and (5.3), we have in particular −u 0+−+ = u 0+++ − 2ũ 0+++ , u−+−+ = u++++ − 2ũ++++ , −u−+0+ = u 0+ − 2ũ 0+ , u −+ = u ++ − 2ũ−+++ . (5.6) This allows us to rewrite WLTUU = − cosϕ Re u 0+++ − u−0++ + 2ǫu 0+0 0 − cos(2φ+ ϕ) ǫRe u 0+++ − 2ũ 0+++ + . . . cos(φ+ ϕ) + . . . cos(φ− ϕ) + . . . cos(2φ− ϕ) , W TTUU = u++++ + u ++ + 2ǫu cos(2φ+ 2ϕ) ǫ u++++ − 2ũ++++ − cosφ ǫ(1 + ǫ) Re u++0+ + u − cos(φ+ 2ϕ) ǫ(1 + ǫ) Re u++0+ − 2ũ − cos(2ϕ) Re u−+++ + ǫu − cos(2φ) ǫRe u−+++ − 2ũ−+++ + . . . cos(φ− 2ϕ) + . . . cos(2φ− 2ϕ) , – 18 – W TTLU = − sinφ ǫ(1− ǫ) Im u++0+ + u − sin(φ+ 2ϕ) ǫ(1− ǫ) Im u++0+ − 2ũ + . . . sin(2ϕ) + . . . sin(φ− 2ϕ) , (5.7) where terms indicated by . . . are the same as in the original expressions (4.10) and (4.12) and have not been repeated for brevity. We see that the coefficients of adjacent terms in (5.7) will be approximately equal to the extent that unnatural parity exchange is suppressed and s-channel helicity approximately conserved. This can be tested experimentally by measuring the angular distribution of the final-state particles. The relations (5.6) and their counterparts for other index combinations can also be used to approximately isolate spin density matrix elements of particular interest. Consider as an example the leading-twist matrix element u 0 00 0 , which in the angular distribution appears only in the combination u 0 0++ + ǫu 0 0 , i.e. together with a matrix element that should be suppressed since it does not conserve s-channel helicity. If unnatural parity exchange is strongly suppressed, an even better approximation for u 0 00 0 can be obtained from the linear combination ǫu 0 00 0 + 2ũ u 0 0++ + ǫu + u 0 0−+ , (5.8) whose r.h.s. can be extracted from the angular distribution. Similarly, one can approxi- mately isolate the matrix element Re u 0+0 0 in the combination ǫRe u 0+0 0 +Re ũ 0+++ − ũ−0++ u 0+++ − u−0++ + 2ǫu 0+0 0 +Re u 0+−+ +Reu . (5.9) Conversely, one can extract from the angular distribution the linear combinations ũ++++ + ũ −− + 2ǫũ 0 0 − 2Re ũ u++++ + u ++ + 2ǫu u−+−+ − 12 u u−+++ + ǫu −Re u++−+ , ũ++0+ + ũ u++0+ + u u−+0+ + u+−0+ , (5.10) which only involve unnatural parity exchange. In a dynamical approach based on gen- eralized parton distributions, these combinations are interesting because they isolate the polarized distributions H̃ and Ẽ and in particular involve these distributions for gluons, which are very hard to access in any other process.5 The price to pay for this is that the corresponding amplitudes are power suppressed and cannot be calculated with the theoretical rigor provided by the leading-twist factorization theorem. On the other hand, phenomenological analysis indicates that a quantitative description of meson production at Q2 of a few GeV2 requires the inclusion of power-suppressed effects also for the leading matrix element u 0 00 0 . The discussion of the matrix elements for transverse target polarization normal to the hadron plane proceeds in full analogy to the unpolarized case. With (−1)ν−µ n−νν′ = nνν µµ′ − 2ñνν µµ′ (5.11) 5In contrast to their quark counterparts, H̃g and Ẽg do not appear in pseudoscalar meson production at leading twist and leading order in αs, see e.g. Section 5.1.1 of [30]. – 19 – we have −n 0+−+ = n 0+++ − 2ñ 0+++ , n−+−+ = n++++ − 2ñ++++ , −n−+0+ = n 0+ − 2ñ 0+ , n −+ = n ++ − 2ñ−+++ (5.12) and can write WLTUT = cos(φ− φS) . . . + sin(φ− φS) − cosϕ Im n 0+++ − n−0++ + 2ǫn 0+0 0 − cos(2φ+ ϕ) ǫ Im n 0+++ − 2ñ 0+++ + . . . cos(φ+ ϕ) + . . . cos(φ− ϕ) + . . . cos(2φ − ϕ) W TTUT = cos(φ− φS) . . . + sin(φ− φS) n++++ + n ++ + 2ǫn cos(2φ+ 2ϕ) ǫ Im n++++ − 2ñ++++ − cosφ ǫ(1 + ǫ) Im n++0+ + n − cos(φ+ 2ϕ) ǫ(1 + ǫ) Im n++0+ − 2ñ − cos(2ϕ) Im n−+++ + ǫn − cos(2φ) ǫ Im n−+++ − 2ñ−+++ + . . . cos(φ− 2ϕ) + . . . cos(2φ − 2ϕ) W TTLT = cos(φ− φS) . . . + sin(φ− φS) ǫ(1− ǫ) Re n++0+ + n + sin(φ+ 2ϕ) ǫ(1− ǫ) Re n++0+ − 2ñ + . . . sin(2ϕ) + . . . sin(φ− 2ϕ) , (5.13) where terms denoted by . . . are as in the original expressions (4.17) and (4.18). Again, the coefficients of adjacent terms should be approximately equal to the extent that unnatural parity exchange is suppressed and s-channel helicity approximately conserved. The matrix elements Imn 0 00 0 and Imn 0 0 can be approximately isolated in ǫ Imn 0 00 0 + 2 Im ñ ++ = Im n 0 0++ + ǫn + Imn 0 0−+ (5.14) ǫ Imn 0+0 0 + Im ñ 0+++ − ñ−0++ n 0+++ − n−0++ + 2ǫn 0+0 0 + Imn 0+−+ + Imn . (5.15) In turn, the linear combinations ñ++++ + ñ −− + 2ǫñ 0 0 − 2ñ n++++ + n ++ + 2ǫn Imn−+−+ − 12 Imn n−+++ + ǫn − Imn++−+ , ñ++0+ + ñ n++0+ + n n−+0+ + n+−0+ (5.16) involve only unnatural parity exchange. – 20 – 6. Positivity constraints From the definition (3.4) of the spin-density matrix elements one readily finds ν′µ′λ′ cνµλ ρ µµ′,λλ′ = (NT + ǫNL) cνµλ T ≥ 0 (6.1) for arbitrary complex numbers cνµλ. Hence ρ µµ′,λλ′ is a positive semidefinite matrix, with row indices specified by {νµλ} and column indices by {ν ′µ′λ′}. This implies inequalities among the spin density matrix elements, which extend those given e.g. in [22, 27]. We do not attempt here to study the bounds following from positivity of the full 18 × 18 matrix ρνν µµ′,λλ′ , which is quite unwieldy. Instead, we consider the subset of matrix elements conserving s-channel helicity for the photon-meson transition and derive a number of simple inequalities, which may be useful in practice. Ordering the row and column indices as {+++}, {0 0+}, {−−+}, {++−}, {0 0−}, {−−−}, we have a positive semidefinite matrix C, which can be written in block form as A+ B+ B− A− (6.2) u++++ + η l u 0+0+ + η l u−+−+ − η l−+−+ u 0+0+ + η l 0 0 u 0+ − η l u−+−+ + η l u 0+0+ − η l u++++ − η l++++ (6.3) s++++ + η n s 0+0+ − η n )∗ −s−+−+ + η n−+−+ s 0+0+ + η n 0+ η n 0 0 −s 0+0+ + η n s−+−+ + η n s 0+0+ + η n )∗ −s++++ + η n++++ , (6.4) where η = ±1. Concentrating first on the matrix elements for an unpolarized or longitu- dinally polarized target, we find that the matrix Aη has eigenvalues whose expressions are very lengthy and therefore restrict our attention to 2×2 submatrices. The matrix obtained from the first and third rows and columns of A+ has eigenvalues u++++ ± u−+−+ l++++ Im l−+−+ , (6.5) whose positivity implies a bound l++++ Im l−+−+ u++++ u−+−+ . (6.6) Similarly, the matrix obtained from the first and second and the matrix obtained from the second and third rows and columns of A+ have respective eigenvalues u++++ + l ++ + u u++++ + l ++ − u 0 00 0 ∣u 0+0+ + l u++++ − l++++ + u 0 00 0 u++++ − l++++ − u 0 00 0 ∣u 0+0+ − l , (6.7) – 21 – whose positivity gives bounds Reu 0+0+ +Re l Imu 0+0+ + Im l )2 ≤ u 0 00 0 u++++ + l Reu 0+0+ − Re l Imu 0+0+ − Im l )2 ≤ u 0 00 0 u++++ − l++++ . (6.8) A weaker condition is obtained by taking the sum of these two bounds, Re l 0+0+ Im l 0+0+ )2 ≤ u 0 00 0 u++++ − Re u 0+0+ Imu 0+0+ . (6.9) The bounds (6.6) and (6.9) have right-hand sides involving only matrix elements accessi- ble with an unpolarized target and constrain the matrix elements for longitudinal target polarization on their left-hand sides. As a second example let us derive conditions which involve only matrix elements u and n . To this end we consider the matrix C′ = 1 C+D†CD (6.10) 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 , (6.11) which is half the sum of the positive semidefinite matrices C and D†CD and hence positive semidefinite itself. One readily finds that matrix elements l and s drop out in C′, which reads u++++ u 0+0+ u−+−+ n n 0+0+ n−+−+ u 0+0+ u 0 0 u 0 0 n u−+−+ u 0+0+ u++++ n n 0+0+ n++++ −n++++ n 0+0+ )∗ −n−+−+ u++++ u 0+0+ u−+−+ −n 0+0+ −n 0 00 0 −n 0 0 u −n−+−+ n 0+0+ )∗ −n++++ u−+−+ u 0+0+ u++++ . (6.12) This matrix has three eigenvalues u++++ − u−+−+ + Imn++++ − Imn−+−+ , u++++ + u −+ + Imn ++ + Imn −+ + u 0 0 + Imn u++++ + u −+ + Imn ++ + Imn −+ − u 0 00 0 − Imn 0 00 0 ∣u 0+0+ − in (6.13) and three further eigenvalues obtained by reversing the sign of all matrix elements n . Their positivity results in the bounds Imn++++ − Imn−+−+ u++++ − u−+−+ (6.14) – 22 – Reu 0+0+ + Imn Imu 0+0+ − Ren u 0 00 0 + Imn u++++ + u −+ + Imn ++ + Imn Reu 0+0+ − Imn Imu 0+0+ +Ren u 0 00 0 − Imn 0 00 0 u++++ + u −+ − Imn++++ − Imn−+−+ . (6.15) Omitting the terms with Imu 0+0+ and Ren 0+ , one obtains bounds involving only matrix elements that are accessible with an unpolarized lepton beam. As we have seen in Section 4, s-channel helicity conserving matrix elements can be extracted from the angular distribution under the approximation that s-channel helicity changing transitions are suppressed. The bounds derived in this section may be used to check the consistency of this approximation. 7. Mixing between transverse and longitudinal polarization So far we have discussed target polarization longitudinal or transverse to the virtual photon direction in the target rest frame, which is natural from the point of view of the strong- interaction dynamics. In an experimental setup one has however definite target polarization with respect to the lepton beam direction. The transformation from one polarization basis to the other is readily performed using the relations (2.3). For a target having longitudinal polarization PL with respect to the lepton beam one finds dφ dϕd(cos ϑ) dxB dQ2 dt dxB dQ2 dt WUU + PL cos θγWUL − sin θγWUT (φS = 0) + PℓWLU + PℓPL cos θγ WLL − sin θγ WLT (φS = 0) . (7.1) Note that in this case the azimuthal angle ψ in (4.5) needs to be defined with respect to some fixed spatial direction in the target rest frame, rather than with respect to the (vanishing) transverse component of the target polarization relative to the lepton beam. We have integrated over this angle in (7.1) because the cross section does not depend on it. For a target having transverse polarization PT with respect to the lepton beam one dφS dφ dϕd(cos ϑ) dxB dQ2 dt (2π)2 dxB dQ2 dt cos θγ 1− sin2θγ sin2φS WUU + PT cos θγ WUT + sin θγ cosφS WUL 1− sin2θγ sin2φS + PℓWLU + PℓPT cos θγ WLT + sin θγ cosφSWLL 1− sin2θγ sin2φS . (7.2) The factor cos θγ /(1 − sin2θγ sin2φS) comes from the change of variables from dψ to dφS – 23 – in the cross section. The relation between these two angles is readily obtained by setting PL = 0 in (2.3) and given in [22]. It is a straightforward (if somewhat lengthy) exercise to insert our results (4.13), (4.14) and (4.17), (4.18) into (7.1) and (7.2) and to rewrite the expressions in terms of a suitable basis of functions depending on the azimuthal angles. Here we only give the combinations needed in (7.2) for a transversely polarized target and an unpolarized beam, cos θγ W UT (φS , φ) + sin θγ cosφS W UL(φ) = sin(φ− φS) cos θγ Im n 0 0++ + ǫn − sin θγ ǫ(1 + ǫ) Im l 0 00+ − cos(2φ) cos θγ ǫ Imn −+ − sin θγ ǫ(1 + ǫ) Im l 0 00+ − 2 cosφ cos θγ ǫ(1 + ǫ) Imn 0 00+ + sin θγ ǫ Im l + cos(φ− φS) − sin(2φ) cos θγ ǫ Im s −+ + sin θγ ǫ(1 + ǫ) Im l 0 00+ − 2 sinφ cos θγ ǫ(1 + ǫ) Im s 0 00+ + sin θγ ǫ Im l sin θγ sin(φS + 2φ) ǫ Im l −+ , (7.3) cos θγW UT (φS , φ, ϕ) + sin θγ cosφSW UL (φ,ϕ) = sin(φ− φS) cos(φ+ ϕ) cos θγ ǫ(1 + ǫ) Im n 0+0+ − n sin θγ l 0+++ − l−0++ + 2ǫl 0+0 0 + ǫ Im l 0+−+ − cos(φ− ϕ) cos θγ ǫ(1 + ǫ) Im n 0−0+ − n sin θγ l 0+++ − l−0++ + 2ǫl 0+0 0 − ǫ Im l+0−+ +cos(2φ+ ϕ) cos θγ ǫ Imn −+ − 12 sin θγ ǫ(1 + ǫ) Im l 0+0+ − l +cos(2φ− ϕ) cos θγ ǫ Imn sin θγ ǫ(1 + ǫ) Im l 0−0+ − l − cosϕ cos θγ Im n 0+++ − n−0++ + 2ǫn 0+0 0 sin θγ ǫ(1 + ǫ) l 0+0+ − l l 0−0+ − l + cos(φ− φS) sin(φ+ ϕ) cos θγ ǫ(1 + ǫ) Im s 0+0+ − s sin θγ l 0+++ − l−0++ + 2ǫl 0+0 0 − ǫ Im l 0+−+ – 24 – − sin(φ− ϕ) cos θγ ǫ(1 + ǫ) Im s 0−0+ − s sin θγ l 0+++ − l−0++ + 2ǫl 0+0 0 + ǫ Im l+0−+ +sin(2φ+ ϕ) cos θγ ǫ Im s sin θγ ǫ(1 + ǫ) Im l 0+0+ − l +sin(2φ− ϕ) cos θγ ǫ Im s −+ − 12 sin θγ ǫ(1 + ǫ) Im l 0−0+ − l − sinϕ cos θγ Im s 0+++ − s−0++ + 2ǫs 0+0 0 sin θγ ǫ(1 + ǫ) l 0+0+ − l l 0−0+ − l sin θγ sin(φS + 2φ+ ϕ) ǫ Im l −+ + sin(φS + 2φ− ϕ) ǫ Im l+0−+ , (7.4) cos θγW UT (φS , φ, ϕ) + sin θγ cosφSW UL (φ,ϕ) = sin(φ− φS) cos θγ Im n++++ + n ++ + 2ǫn sin θγ ǫ(1 + ǫ) Im l++0+ + l − cos(2φ) cos θγ ǫ Imn −+ − 12 sin θγ ǫ(1 + ǫ) Im l++0+ + l − cosφ cos θγ ǫ(1 + ǫ) Im n++0+ + n sin θγ ǫ Im l cos(2φ+ 2ϕ) cos θγ ǫ Imn −+ − sin θγ ǫ(1 + ǫ) Im l−+0+ cos(2φ− 2ϕ) cos θγ ǫ Imn −+ − sin θγ ǫ(1 + ǫ) Im l+−0+ − cos(2ϕ) cos θγ Im n−+++ + ǫn sin θγ ǫ(1 + ǫ) Im l−+0+ + Im l + cos(φ+ 2ϕ) cos θγ ǫ(1 + ǫ) Imn−+0+ + sin θγ ǫ Im l−+−+ + 2 Im l−+++ + ǫl + cos(φ− 2ϕ) cos θγ ǫ(1 + ǫ) Imn+−0+ + sin θγ ǫ Im l+−−+ − 2 Im l−+++ + ǫl + cos(φ− φS) − sin(2φ) cos θγ ǫ Im s sin θγ ǫ(1 + ǫ) Im l++0+ + l − sinφ cos θγ ǫ(1 + ǫ) Im s++0+ + s sin θγ ǫ Im l sin(2φ+ 2ϕ) cos θγ ǫ Im s −+ + sin θγ ǫ(1 + ǫ) Im l−+0+ sin(2φ− 2ϕ) cos θγ ǫ Im s −+ + sin θγ ǫ(1 + ǫ) Im l+−0+ − sin(2ϕ) cos θγ Im s−+++ + ǫs sin θγ ǫ(1 + ǫ) Im l−+0+ − Im l – 25 – + sin(φ+ 2ϕ) cos θγ ǫ(1 + ǫ) Im s−+0+ + sin θγ ǫ Im l−+−+ − 2 Im l−+++ + ǫl + sin(φ− 2ϕ) cos θγ ǫ(1 + ǫ) Im s+−0+ + sin θγ ǫ Im l+−−+ + 2 Im l−+++ + ǫl sin θγ sin(φS + 2φ+ 2ϕ) ǫ Im l −+ + sin(φS + 2φ− 2ϕ) ǫ Im l+−−+ sin θγ sin(φS + 2φ) ǫ Im l −+ . (7.5) Compared with (4.17) and (4.18) we have changed the order of terms such that one readily sees which coefficients cos θγ Imn or cos θγ Im s receive an admixture from the same coef- ficients sin θγ Im l . The terms in the last lines of (7.3) and (7.4) and in the last two lines of (7.5) involve only coefficients sin θγ Im l . They come with an angular dependence which is absent for sin θγ = 0, as is readily seen by rewriting sin(φS + 2φ+mϕ) = − sin(φ− φS) cos(3φ+mϕ) + cos(φ− φS) sin(3φ +mϕ) . (7.6) We see in (7.3) to (7.5) that from the angular dependence of the cross section for transverse target polarization one can extract linear combinations of terms cos θγ Imn and sin θγ Im l or of cos θγ Im s and sin θγ Im l . To separate these terms requires an additional measurement with longitudinal target polarization.6 The expressions (7.3) to (7.5) allow us to see for which terms the admixture of sin θγ Im l terms can be expected to be small, so that Imn and Im s may be determined with reasonable accuracy without such an additional measurement. Let us discuss a few examples. 1. The leading-twist matrix element n 0 00 0 appears in the linear combination c0 = cos θγ Im n 0 0++ + ǫn − sin θγ ǫ(1 + ǫ) Im l 0 00+ (7.7) in (7.3) and thus has an admixture from l 0 00+ , which involves one s-channel helicity changing amplitude. According to Section 5 this admixture is additionally suppressed if unnatural parity exchange is small compared with natural parity exchange. One may also add to c0 the angular coefficient c1 = − cos θγ ǫ Imn 0 0−+ + sin θγ ǫ(1 + ǫ) Im l 0 00+ (7.8) from (7.3), thus trading the admixture of sin θγ l 0+ for an admixture of cos θγ n which involves two s-channel helicity changing amplitudes (but lacks the relative factor tan θγ and is not suppressed by unnatural parity exchange). We remark that the linear combination of matrix elements in (5.14) corresponds to c0 − c1/ǫ, where l 0 00+ does not drop out. Whether c0, c0+ c1 or c0− c1/ǫ gives the best approximation to cos θγ ǫ Imn 0 0 will thus depend on the detailed magnitude of the relevant terms. In practice one might for instance use the difference between these terms as a measure for the uncertainty of this approximation. 6A corresponding separation for semi-inclusive pion production ep → eπX has recently been performed in [31]. – 26 – 2. The s-channel helicity conserving matrix elements n 0+0+ in (7.4) and n ++ , n −+ in (7.5) come together with terms involving at least one s-channel helicity changing amplitude. These admixtures should hence be negligible unless the corresponding s-channel helicity conserving matrix element is small itself. For Imn 0+0+ this may for instance happen because of the relative phase between the interfering amplitudes. 3. The matrix element n 0 00+ in (7.3) comes with an admixture from l −+ , which involves two s-channel helicity changing amplitudes and should hence again be suppressed. In addition, one can extract Im l 0 0−+ from the angular dependence itself, given the last term in (7.3). We remark that the unpolarized analog u 0 00+ of n 0+ has a real part which is experimentally seen to be nonzero [17, 19], providing evidence that s-channel helicity is not strictly conserved in electroproduction. (In the notation of Schilling and Wolf one has r500 = − 2Reu 0 00+ .) 4. The only s-channel helicity conserving matrix elements for sideways transverse target polarization in (7.3) to (7.5) are s 0+0+ and s −+ . They come together with terms involving at least one s-channel helicity changing amplitude, so that the situation is similar to the one in point 2. Note however that in the present case there is no additional suppression of the admixture terms due to unnatural parity exchange, since both s and l contain one unnatural parity exchange amplitude. In these examples one thus has the favorable situation that the admixture from longitudinal polarization terms is probably small and in some cases may even be removed or traded for yet smaller terms. This does not always happen: the matrix elements n 0+−+ and s −+ in (7.4) receive for instance an admixture from the s-channel helicity conserving term l 0+0+ , which may not be small itself, so that from the coefficients of sin(φ − φS) cos(2φ + ϕ) or cos(φ−φS) sin(2φ+ϕ) one cannot directly infer on the matrix elements Imn 0+−+ or Im s 0+−+ . To make a more precise statement about their size one needs independent information on Im l 0+0+ , for instance from the positivity bound (6.9). 8. A note on non-resonant contributions So far we have treated the production of two pions in a two-step picture, where a ρ is first produced in ep → epρ and then decays as ρ→ π+π−. For deriving the angular distribution and polarization dependence we have used that the pion pair is in the L = 1 partial wave, as can be seen in (4.3). We did however not use the narrow-width approximation for the ρ or make any assumption about its line shape. In fact, our results for the angular distribution can readily be used at any given invariant mass mππ of the pion pair, with the ep cross sections on the left- and right-hand sides of (4.5) made differential in mππ. The spin-density matrix ρνν µµ′,λλ′ and its linear combinations u , l , s , n then depend on mππ and refer not to γ∗p → ρp but to γ∗p → π+π− p with π+π− in the L = 1 partial wave. No explicit reference to the ρ resonance needs to be made in this case. The situation is more complicated if one considers other partial waves of the pion pair, which can arise from non-resonant production mechanisms. To describe a general π+π− – 27 – state, one should replace ρνν µµ′,λλ′ with the spin-density matrix ρ νν′,LL′ µµ′,λλ′ for a pion pair with angular momentum L in the amplitude and L′ in the conjugate amplitude. One then has to take YLν(ϕ, ϑ)Y L′ν′(ϕ, ϑ) instead of Y1ν(ϕ, ϑ)Y 1ν′(ϕ, ϑ) in (4.3) and will obviously obtain a different angular dependence of the ep cross section. The distribution in ϕ and ϑ for a pion pair with L = 0, 1, 2 has been discussed in [32]. It is quite simple to test for the presence of L = 0 or L = 2 partial waves in data by using discrete symmetry properties, and for mππ around the ρ mass one can expect that partial waves with L = 3 or higher are strongly phase space suppressed. Since even partial waves of the π+π− system have charge conjugation parity C = +1 and odd partial waves have C = −1, the interference of L = 1 with L = 0 or L = 2 gives rise to terms in the angular distribution which are odd under interchange of the π+ and π− momenta, i.e. under the replacement ϑ→ π − ϑ , ϕ→ ϕ+ π . (8.1) Simple examples are an angular dependence like cos ϑ or like an odd polynomial in cos ϑ. Corresponding observables provide a way to study the L = 0 and L = 2 partial waves as a “signal” interfering with the ρ resonance “background” [33, 34]. This has been used in the experimental analysis [35], which did see such interference away from the ρ resonance peak, whereas close to the peak the predominance of the ρ was too strong to observe a significant contribution from any partial wave with L 6= 1. If on the other hand one is interested in a precise study of the L = 1 component, one can eliminate its interference with even partial waves by symmetrizing the angular distribution according to (8.1). One is then left with contributions from L = 0 and L = 2 in both the amplitude and its conjugate, which should be very small around the ρ peak. 9. Summary We have expressed the fully differential cross section for exclusive ρ production on a po- larized nucleon in terms of spin density matrix element for the subprocess γ∗p → ρp. We work in the helicity basis for both γ∗ and ρ and obtain very similar forms for the unpolar- ized and polarized parts of the cross sections, with the substitution rules (4.15) and (4.16). The terms for transverse target polarization normal to the hadron plane closely resemble those for an unpolarized target, and in both cases the number of independent spin density matrix elements is reduced if one neglects unnatural parity exchange compared with nat- ural parity exchange. The spin density matrix elements for transverse target polarization in the hadron plane closely resemble those for a longitudinally polarized target, with both types of matrix elements involving the interference between natural and unnatural parity exchange. We have given simple positivity bounds which involve only matrix elements for an unpolarized target and either those for longitudinal target polarization or for transverse target polarization normal to the hadron plane. Furthermore, we have investigated the admixture of longitudinal target polarization relative to the virtual photon momentum for a target polarized transversely to the lepton beam. 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Astronomy & Astrophysics manuscript no. attIarxiv c© ESO 2018 October 31, 2018 The effect of the solar corona on the attenuation of small-amplitude prominence oscillations I. Longitudinal magnetic field R. Soler, R. Oliver, and J. L. Ballester Departament de Fı́sica, Universitat de les Illes Balears, E-07122, Palma de Mallorca, Spain e-mail: [roberto.soler;ramon.oliver;dfsjlb0]@uib.es Received xxxx; accepted xxxx ABSTRACT Context. One of the typical features shown by observations of solar prominence oscillations is that they are damped in time and that the values of the damping times are usually between one and three times the corresponding oscillatory period. However, the mechanism responsible for the attenuation is still not well-known. Aims. Thermal conduction, optically thin or thick radiation and heating are taken into account in the energy equation, and their role on the attenuation of prominence oscillations is evaluated. Methods. The dispersion relation for linear non-adiabatic magnetoacoustic waves is derived considering an equilibrium made of a prominence plasma slab embedded in an unbounded corona. The magnetic field is orientated along the direction parallel to the slab axis and has the same strength in all regions. By solving the dispersion relation for a fixed wavenumber, a complex oscillatory frequency is obtained, and the period and the damping time are computed. Results. The effect of conduction and radiation losses is different for each magnetoacoustic mode and depends on the wavenumber. In the observed range of wavelengths the internal slow mode is attenuated by radiation from the prominence plasma, the fast mode by the combination of prominence radiation and coronal conduction and the external slow mode by coronal conduction. The consideration of the external corona is of paramount importance in the case of the fast and external slow modes, whereas it does not affect the internal slow modes at all. When a thinner slab representing a filament thread is considered the fast mode is less attenuatted whereas both internal and external slow modes are not affected. Conclusions. Non-adiabatic effects are efficient damping mechanisms for magnetoacoustic modes, and the values of the obtained damping times are compatible with those observed. Key words. Sun: oscillations – Sun: magnetic fields – Sun: corona – Sun: prominences 1. Introduction Prominences are dense coronal structures which appear as thin, dark filaments on the solar disc when observed in Hα. On the contrary, they show up as bright objects above the solar limb. The coronal magnetic field is responsible for the support of prominences against gravity, and it also plays a fundamental role in the thermal confinement of the cool prominence plasma em- bedded in the much hotter coronal environment. Nevertheless, the structure, orientation and strength of the magnetic field in prominences and the surrounding corona is still enigmatic and not well-known. High resolution observations reveal that promi- nences are composed by numerous very thin, thread-like struc- tures, called fibrils, piled up to form the body of the prominence (Lin et al. 2003; Lin et al. 2005, Lin et al. 2007) and measures also indicate that magnetic field lines are orientated along these thin threads. The observational evidence of small-amplitude oscillations in quiescent solar prominences goes back to 40 years ago (Harvey 1969). The amplitude of these oscillations typically goes from less than 0.1 km s−1 to 2–3 km s−1, and have been historically classified, according to their periods, in short- (P < 10 min), intermediate- (10 min < P < 40 min) and long-period Send offprint requests to: R. Soler oscillations (P > 40 min), although very short-periods of less than 1 min (Balthasar et al. 1993) and extreme ultra-long-periods of more than 8 hours (Foullon et al. 2004) have been reported. Nevertheless, the value of the period seems not to be related with the nature or the source of the trigger and probably is linked to the prominence eigenmode that is excited. There are also a few determinations of the wavelength and phase speed of standing oscillations and propagating waves in large regions of promi- nences (Molowny-Horas et al. 1997; Terradas et al. 2002) and in single filament threads (Lin et al. 2007). On the other hand, several observations (Molowny-Horas et al. 1999; Terradas et al. 2002) have informed about the evidence of the attenuation of the oscillations in Doppler velocity time series, which is a common feature observed in large areas. By fitting a sinusoidal function multiplied by a factor exp(−t/τD) to the Doppler series, these authors have obtained values of the damping time, τD, which are usually between 1 and 3 times the corresponding oscilla- tory period. The reader is referred to some recent reviews for more information about the observational background (Oliver & Ballester 2002, Wiehr 2004, Engvold 2004, Ballester 2006). From the theoretical point of view, small-amplitude promi- nence oscillations can be interpreted in terms of linear magne- tohydrodynamic (MHD) waves. Although, there is a wide bib- liography of works that investigate the ideal MHD wave modes http://arxiv.org/abs/0704.1566v2 2 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I supported by prominence models (see Oliver & Ballester 2002 for an extensive review of theoretical studies), the investigation of the wave damping has been broached in few papers. By re- moving the ideal assumption and including dissipative terms in the basic MHD equations, several works have studied the atten- uation of prominence oscillations considering radiative losses based on the Newtonian law of cooling with a constant relax- ation time (Terradas et al 2001), or performing a more complete treatment of non-adiabatic effects, assuming optically thin radi- ation, heating and thermal conduction (Carbonell et al. 2004; Terradas et al. 2005). The main conclusion of these previous studies is that only the slow wave is attenuated by thermal effects, radiation being the dominant damping mechanism in the range of typically observed wavelengths in prominences, but the fast wave remains practically undamped. On the other hand, Forteza et al. (2007) proposed ion-neutral collisions as a damping mechanism on the basis that prominences are partially ionised plasmas, but they found that this mechanism is only ef- ficient in attenuating the fast mode in quasi-neutral plasmas, the slow mode being almost unaffected. In the light of these referred studies, it is likely that non- adiabatic effects are the best candidates for the damping of small-amplitude oscillations, at least for slow modes. However, previous results do not asses the influence of the corona. The main aim of the present work is to perform a step forward in the investigation of the effect of non-adiabatic mechanisms (radia- tion losses, thermal conduction and heating) on the time damp- ing of prominence oscillations. We consider a slab model with a longitudinal magnetic field and take into account the external coronal medium. So, we explore for the first time the joint ef- fect of prominence and coronal mechanisms on the attenuation of oscillations. The magnetoacoustic normal modes of this equi- librium have been previously investigated by Edwin & Roberts (1982) and Joarder & Roberts (1992) in the adiabatic case. Later, a revision of these works has been done in Soler et al. (2007), hereafter Paper I, and the normal modes have been studied and reclassified according to their magnetoacoustic properties. This paper is organised as follows. The description of the equilibrium model and the linear non-adiabatic wave equations are given in Sect. 2, whereas the dispersion relation for the mag- netoacoustic modes is derived in Sect. 3. Then, the results are plotted and investigated in Sect. 4. Finally, Sect. 5 contains the conclusions of this work. 2. Equilibrium and basic equations Our equilibrium configuration (Fig. 1) is made of a homoge- neous plasma layer with prominence conditions (density ρp and temperature Tp) embedded in an unbounded corona (density ρc and temperature Tc). The coronal density is computed by fix- ing the coronal temperature and imposing pressure continuity across the interfaces. The magnetic field is B0 = B0êx, with B0 a constant everywhere. Both media are unlimited in the x- and y-directions. The half-width of the prominence slab is zp. The basic magnetohydrodynamic equations for the discus- sion of non-adiabatic processes are: + ρ∇ · v = 0, (1) = −∇p + (∇ × B) × B, (2) + (γ − 1)[ρL(ρ, T ) − ∇ · (κ · ∇T )] = 0, (3) Fig. 1. Sketch of the equilibrium. Table 1. Parameter values of the radiative loss function corresponding to the considered regimes. The three promi- nence regimes represent different plasma optical thicknesses. Prominence (1) regime corresponds to an optically thin plasma whereas Prominence (2) and Prominence (3) regimes represent greater optical thicknesses. All quantities are expressed in MKS units. Regime χ∗ α Reference Prominence (1) 1.76 × 10−13 7.4 Hildner (1974) Prominence (2) 1.76 × 10−53 17.4 Milne et al. (1979) Prominence (3) 7.01 × 10−104 30 Rosner et al. (1978) Corona 1.97 × 1024 −1 Hildner (1974) = ∇ × (v × B), (4) ∇ · B = 0, (5) , (6) where D + v · ∇ is the material derivative for time vari- ations following the motion and all quantities have their usual meaning. Equation (3) is the energy equation, which in the present form takes into account non-adiabatic effects (radiation losses, thermal conduction and heating) and whose terms are ex- plained in detail in Carbonell et al. (2004) and Terradas et al. (2005). Following these works, only thermal conduction par- allel to the magnetic field is assumed and we use the typical value for the parallel conductivity in prominence and coronal applications, κ‖ = 10 −11T 5/2 W m−1 K−1. Radiative losses and heating are evaluated together through the heat-loss function, L(ρ, T ) = χ∗ρTα − hρaT b, where radiation is parametrised with χ∗ and α (see Table 1) and the heating scenario is given by ex- ponents a and b. The heating mechanisms taken into account in this work are (Rosner et al. 1978; Dahlburg & Mariska 1988): – constant heating per unit volume (a = b = 0); – constant heating per unit mass (a = 1, b = 0); – heating by coronal current dissipation (a = b = 1); – heating by Alfvén mode/mode conversion (a = b = 7/6); – heating by Alfvén mode/anomalous conduction damping (a = 1/2, b = −1/2). Following the same process as in Carbonell et al. (2004), we consider small perturbations from the equilibrium state, lin- earise the basic Eqs. (1)–(6) and obtain their Eqs. (9)–(14). Since R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 3 our model is unlimited in the x- and y-directions, we assume all perturbations are in the form f1(z) exp i(ωt+ kxx+ kyy), and con- sidering only motions and propagation in the xz-plane (vy = 0, ky = 0), which excludes Alfvén waves, the linearised equations become iωρ1 + ρ0 ikxvx + = 0, (7) iωρ0vx = −ikx p1, (8) iωρ0vz = − ikxB1z − , (9) p1 − c = −(γ − 1) k2xκ‖T1 + ωρρ1 + ωT T1 , (10) iωB1x = −B0 , (11) iωB1z = B0ikxvz, (12) where c2s = is the adiabatic sound speed squared and L + ρ0Lρ , ωT ≡ T0LT , Lρ, LT being the partial derivatives of the heat-loss function with respect to density and temperature, respectively, , LT ≡ Now, it is possible to eliminate all perturbations in favour of vz to obtain a single differential equation + k2z vz = 0, (13) in which k2z = ω2 − k2xv ω2 − k2xΛ v2A + Λ ω2 − k2xc̃ ) , (14) where v2A = is the Alfvén speed squared. Λ2 and c̃2T are the modified sound and cusp (or tube) speed squared, respectively, (γ − 1) x + ωT − ωρ + iγω (γ − 1) κ‖k2x + ωT , (15) c̃2T ≡ v2A + Λ . (16) Expressions for the perturbations in terms of vz are given in App. A. In all the following formulae, subscripts p or c denote quantities computed using prominence or coronal values, respec- tively. 3. Dispersion relation We impose some restrictions on the solutions of Eq. (13) in or- der to obtain the dispersion relation for the linear non-adiabatic magnetoacoustic waves. We restrict this analysis to body waves which are evanescent in the corona, since we are looking for solutions which are essentially confined to the slab. For such so- lutions, vz(z) is of the form vz(z) = A1 exp z + zp , if z ≤ −zp, A2 cos + A3 sin , if −zp ≤ z ≤ zp, A4 exp z − zp , if z ≥ zp. withℜ(kzp) > 0 andℜ(kzc) > 0. Imposing continuity of vz and the total (gas plus magnetic) pressure perturbation across the interfaces, we find four alge- braic relations between the constants A1, A2, A3 and A4. The non-trivial solution of this system gives us the dispersion rela- Ac − ω kzpzp Ap − ω kzc = 0, (18) where cot/tan terms and ± signs are related with the symmetry of the perturbations. The cot term and the + sign correspond to kink modes (A3 = 0), whereas the tan term and the − sign correspond to sausage modes (A2 = 0). The dispersion relation for the magnetoacoustic waves pre- sented in Eq. (18) is equivalent to the relation investigated in Edwin & Roberts (1982) and Joarder & Roberts (1992), and re- vised Paper I, in the case of adiabatic perturbations, since all non-adiabatic terms are now enclosed in kzp and kzc through Eq. (14). 4. Results Now, we assume Prominence (1) conditions inside the slab (i.e. an optically thin prominence) and a heating mechanism given by a = b = 0. Unless otherwise stated, the following equilibrium parameters are considered in all computations: Tp = 8000 K, ρp = 5 × 10 −11 kg m−3, Tc = 10 6 K, ρc = 2.5 × 10 −13 kg m−3, B0 = 5 G and zp = 3000 km. The solution of the dispersion relation (Eq. [18]) for a fixed real kx gives us a complex fre- quency ω = ωR + iωI. We then compute the oscillatory period, the damping time and the ratio of the damping time to the pe- riod because this is an important quantity from the observational point of view, , τD = Since we are interested in studying the behaviour of the most relevant solutions of the dispersion relation, we only compute the results for the fundamental modes, which are labelled, accord- ing to the classification of Paper I, as internal and external slow modes and fast modes. The band structure described in Paper I is slightly modified when non-adiabatic terms are considered (see Fig. 2). The phase speed of the internal slow modes is now en- closed in the bandℜ(c̃Tp) < ωR/kx < ℜ(Λp). The adiabatic fast modes exist in two separated bands in the phase speed diagram due to the presence of a forbidden region (cTc < ωR/kx < csc), but now the forbidden band is avoided and a continuous fast mode is found with vAp < ωR/kx < vAc. Finally, and like in the adiabatic case, among the external slow modes only the fun- damental kink one exists as a non-leaky solution in a restricted 4 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I wavenumber range and couples with the fundamental fast kink mode. Its phase speed is ωR/kx ≈ ℜ(Λc). Therefore, we see that in the non-adiabatic case Λ plays the role of cs in the adiabatic case. Fig. 2. Phase speed versus the dimensionless wavenumber for the three fundamental oscillatory modes. Solid lines denote kink modes whereas dotted lines correspond to sausage modes. The shaded zones are projections of the forbidden (or leaky) regions on the plane of this diagram. Note that the vertical axis is not drawn to scale. In Fig. 3 P, τD and τD/P are represented for the fundamental modes and for a range of the longitudinal wavenumber between 10−10 m−1 and 103 m−1. The shaded zones correspond to wave- lengths between 5 × 103 km and 105 km, the typically observed values. It turns out that the values of the period are very sim- ilar to those obtained in the adiabatic case (Joarder & Roberts 1992; Paper I). The damping time presents a strong dependence with the wavenumber and its behaviour is very different from one mode to another. This fact suggests that the non-adiabatic mechanisms can affect each mode in a different way (Carbonell et al. 2004). This is studied in detail in Sect 4.1. Observations show that prominence oscillations are typically attenuated in a few periods (Terradas et al. 2002), so a damping time of the order of the period is expected. In our results, the fundamental modes present values of τD/P in the range 1 to 10 in the observed wave- length region, which is in agreement with observations. 4.1. Regions of dominance of the damping mechanisms The importance of the different non-adiabatic terms included in the energy equation (Eq. [3]) depends on the wavenumber. In order to know which is the range of dominance of each mech- anism, we compare the damping time obtained when consider- ing all non-adiabatic terms (displayed in the middle column of Fig. 3) with the results obtained when a specific mechanism is removed from the energy equation. With this analysis, we are able to know where the omitted mechanism has an appreciable effect on the damping. The results of these computations for the fundamental kink modes (Fig. 4) are summarised as follows: – The fundamental internal slow kink mode is not affected by the mechanisms related with the corona. This is a conse- quence of the nature of this mode, which propagates strictly along the prominence without disturbing the corona (see Fig. 4, top row, of Paper I). For this reason, in the adiabatic case it is also independent of the coronal conditions. On the other hand, the prominence-related mechanisms show different ef- fects in two different ranges of kx. For kx . 10 −3 m−1 promi- nence radiation dominates, while for kx & 10 −3 m−1 promi- nence conduction is the dominant mechanism. Beginning from small values of the wavenumber, prominence radiation becomes more efficient as kx grows and the damping time falls following a power law until kx ≈ 10 −5 m−1, where τD saturates in a plateau between kx ≈ 10 −5 and kx ≈ 10 −3 m−1. Then, prominence conduction becomes the dominant mech- anism and the damping time falls again until kx ≈ 10 −1 m−1 where a new plateau begins. This last part of the curve cor- responds to the isothermal or superconductive regime, in which the amplitude of the temperature perturbation drops dramatically (Carbonell et al. 2006). Prominence radiation is responsible for the attenuation of the slow mode in the observed wavelength range. An approximate dispersion re- lation for the internal slow modes is included in App. B. – The fundamental fast kink mode is affected by the four mechanisms. For kx . 3 × 10 −9 m−1 coronal radiation dom- inates but for 3 × 10−9 m−1 . kx . 5 × 10 −7 m−1 the effect of coronal conduction grows and becomes the main damping mechanism. Then, for kx & 5×10 −7 m−1 the corona loses dra- matically its influence and prominence mechanisms become responsible for the attenuation of this mode. First, promi- nence radiation is dominant in the range 5 × 10−7 m−1 . kx . 10 −3 m−1, then prominence conduction governs the wave damping for kx & 10 −3 m−1 and finally the isother- mal regime begins for kx ≈ 10 0 m−1. The minimum of τD occurs into the coronal conduction regime, for the value of kx which corresponds to the coupling with the external slow mode. The transition between the coronal conduction regime and the prominence radiation regime occurs in the observed wavelength range. The reason for the sensitivity of the fast mode damping time on prominence and coronal conditions is that this wave has a considerable amplitude both inside the prominence and in the corona, the later becoming more im- portant for long wavelengths (see the second and third rows of Fig. 4 of Paper I). – The behaviour of the damping time of the fundamental exter- nal slow kink mode is entirely dominated by coronal mech- anisms whereas the prominence mechanisms do not affect it at all. This behaviour is a result of the negligible amplitude of this wave in the prominence (see the fourth and fifth rows of Fig. 4 of Paper I). For kx . 3× 10 −9 m−1 coronal radiation dominates, but for shorter wavelengths coronal conduction becomes more relevant and is responsible for the damping in the observed wavelength range until the frequency cut-off is reached. At the cut-off, τD has a value of the order of the period. Regarding the fundamental sausage modes, the behaviour of the internal slow sausage mode is exactly that of the slow kink mode, so no additional comments are needed. The fundamen- tal fast sausage mode (Fig. 5) presents the same scheme as the fundamental fast kink mode for kx & 10 −8 m−1. The main dif- ference between the fast kink and sausage modes happens in the observed wavelength range, where the effect of coronal conduc- tion on the sausage mode is less relevant. If coronal conduction is omitted, the fundamental fast sausage mode is not able to tra- verse the forbidden region in the dispersion diagram and then shows frequency cut-offs as in the adiabatic case. This means R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 5 Fig. 3. Period (left), damping time (centre) and ratio of the damping time to the period (right) versus the longitudinal wavenumber for the fundamental oscillatory modes. Upper panels: internal slow kink (solid line), fast kink (dotted line) and external slow kink (dashed line). Lower panels: internal slow sausage (solid line) and fast sausage (dotted line). Shaded zones correspond to those wavelengths typically observed. Note the cut-offs of the external slow kink mode and the fast sausage mode. Prominence (1) radiation conditions have been taken for the prominence plasma and the heating scenario is given by a = b = 0. Fig. 4. Damping time versus the longitudinal wavenumber for the three fundamental kink oscillatory modes: internal slow (left), fast (centre) and external slow (right). Different linestyles represent the omitted mechanism: all mechanisms considered (solid line), prominence conduction eliminated (dotted line), prominence radiation eliminated (dashed line), coronal conduction eliminated (dot- dashed line) and coronal radiation eliminated (three dot-dashed line). Prominence (1) radiation conditions have been taken for the prominence plasma and the heating scenario is given by a = b = 0. that coronal conduction causes the fast mode to cross the forbid- den region in the dispersion diagram in the non-adiabatic case. Approximate values of kx for which the transitions between regimes take place can be computed by following a process simi- lar to that in Carbonell et al. (2006). The thermal ratio, d, and the radiation ratio, r, quantify the importance of thermal conduction and radiation, respectively (De Moortel & Hood 2004), (γ − 1)κ‖T0ρ0 γ2 p20τs τcond , (19) (γ − 1)τsρ , (20) where τs is the sound travel time and τcond and τrad are character- istic conductive and radiative time scales. Taking τs = 2π/k the value of k∗ for which the condition d = r is satisfied is k∗ = 2πρ0 χ∗Tα−10 . (21) Now, we use prominence values to compute k∗ for the promi- nence radiation–prominence conduction transition (k∗p), and 6 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I Fig. 6. Damping time versus the longitudinal wavenumber for the fundamental internal slow kink mode (left), the fundamental fast kink mode (centre) and the fundamental external slow kink mode (right). The different linestyles represent different values of the prominence temperature: Tp = 8000 K (solid line), Tp = 5000 K (dotted line) and Tp = 13000 K (dashed line). The heating scenario is given by a = b = 0 and the optical thickness for the prominence plasma is Prominence (1). Fig. 7. Same as Fig. 6 with ρp = 5 × 10−11 kg m−3 (solid line), ρp = 2 × 10−11 kg m−3 (dotted line) and ρp = 10−10 kg m−3 (dashed line). Fig. 8. Same as Fig. 6 with B0 = 5 G (solid line), B0 = 2 G (dotted line) and B0 = 10 G (dashed line). coronal values for the coronal radiation–coronal conduction transition (k∗c). This gives the values k p ≈ 1.7 × 10 −3 m−1, and k∗c ≈ 2.2 × 10 −8 m−1. For the transition of the fast kink mode between the coronal conduction and the prominence radiation regimes, the boundary wavenumber k∗p↔c can be roughly calcu- lated by imposing dc = rp, that gives k∗p↔c = 2πρp cscχ∗pT cspκ‖cTc , (22) and whose numerical value is k∗p↔c ≈ 1.4 × 10 −6 m−1. All these wavenumbers for the transitions between different regimes are independent of the wave type, be it fast or slow, internal or ex- ternal (this agrees with Figs. 4 and 5). On the other hand, the be- ginning of the isothermal regime can be estimated by following Porter et al. (1994). Considering c2sp/v Ap ≪ 1 and the approxi- mations ωR ≈ kxcsp for the slow wave and ωR ≈ kxvAp for the fast wave, the critical wavenumber is kcrit−slow = 2ρpkBcsp κ‖pmp cos θ , (23) for the internal slow mode, and kcrit−fast = 2ρpkBvAp κ‖pmp cos2 θ , (24) R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 7 Fig. 5. Same as Fig. 4 for the fundamental fast sausage mode. for the fast mode, where mp is the proton mass, kB is the Boltzmann constant and θ is the angle between B and k. Taking cos θ = 1 for simplicity, the approximate critical values are kcrit−slow ≈ 1.7 × 10 −1 m−1 and kcrit−fast ≈ 9.1 × 10 −1 m−1. We note that all these approximate values describe correctly the tran- sitions between the diverse regimes shown in Figs. 4 and 5, but their numerical values overestimate by almost an order of mag- nitude the actual critical wavenumbers. 4.2. Exploring the parameter space 4.2.1. Dependence on the equilibrium physical conditions In this section, we compute the solutions for different values of the equilibrium physical conditions. We only present the re- sults for the fundamental kink modes since they are equivalent to those of sausage modes. Figures 6, 7 and 8 display the damping time as function of kx for some selected values of the prominence temperature, the prominence density and the magnetic field, re- spectively. For the internal slow mode, a decrease of the prominence temperature or the prominence density raises the position of the radiative plateau and increases its length. The opposite be- haviour is seen when the density or the temperature are in- creased. However, the value of the magnetic field does not in- fluence the attenuation of this mode, such as expected for a slow wave. Increasing the value of the prominence temperature causes a vertical displacement of τD of the fast mode in those regions in which prominence mechanisms dominate. The value of the prominence density has a smaller effect and its main influence is in changing the coupling point with the external slow mode, which moves to higher kx for greater values of the density. The magnetic field strength has a more complex effect on τD and also modifies the coupling point. Finally, the external slow mode is only slightly affected by a modification of the prominence physical parameters since it is mainly dominated by coronal conditions, and the influence of the magnetic field is very small due to the slow-like magnetoa- coustic character of this solution. 4.2.2. Dependence on the prominence optical thickness The optically thin radiation assumption is a reasonable approxi- mation in a plasma with coronal conditions but prominence plas- mas often are optically thick. In this section we compare the results obtained considering different optical thicknesses for the prominence plasma (see Fig. 9 for the fundamental kink modes). The results corresponding to the slow sausage mode have not been plotted since they are equivalent to those obtained for slow kink mode; those for the fundamental fast sausage mode, how- ever, are displayed in Fig. 10. The variation of the prominence optical thickness modi- fies the prominence conduction–prominence radiation critical wavenumber, k∗p (see analytical approximation of Eq. [21]). For the internal slow mode, an increase in the optical thickness raises the position of the radiative plateau and shifts it to smaller wavenumbers. This fact causes an a priori surprising result in the observed wavelength range, since τD has a smaller value for opti- cally thick radiation, Prominence (3), than for optically thin radi- ation, Prominence (1). Regarding fast modes, the damping time increases when the optical thickness is increased, but only in the region in which prominence radiation dominates. The value of τD inside the observed wavelength range is partially affected and raises an order of magnitude for Prominence (3) conditions in comparison with the results for Prominence (1) conditions. Finally, the damping time of the external slow mode is not af- fected by the prominence optical thickness since it is entirely dominated by the corona, as it has been noticed in Sect. 4.1. Fig. 10. Same as Fig. 9 for the fundamental fast sausage mode. 4.2.3. Dependence on the heating scenario Now, we compute the damping time for the five possible heating scenarios. For simplicity, we only consider the fundamental kink modes (Fig. 11). Carbonell et al. (2004) showed that in a plasma with prominence conditions the different heating scenarios have no significant influence on the damping time. Nevertheless, in coronal conditions wave instabilities can appear depending on the heating mechanism. In our results, we see that the heating scenario affects the value of τD only in the ranges of kx in which radiation is the dominant damping mechanism. The heating sce- nario has a negligible effect when prominence radiation domi- nates, since τD is only slightly modified. On the contrary, wave instabilities appear in those regions in which coronal radiation dominates. Thermal destabilisation occurs when the imaginary part of the frequency becomes negative, so oscillations are not attenuated but amplified in time. Instabilities only occur in the fundamental fast kink and the external slow modes for very small values of kx, outside the observed wavelength range. 8 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I Fig. 9. Same as Fig. 6 with the prominence optical thickness given by Prominence (1) (solid line), Prominence (2) (dotted line) and Prominence (3) (dashed line) conditions. Fig. 11. Same as Fig. 6 with the heating scenario given by a = b = 0 (solid line); a = 1, b = 0 (dotted line); a = b = 1 (dashed line); a = b = 7/6 (dot-dashed line); a = 1/2, b = −1/2 (three dot-dashed line). 4.3. Comparison with the solution for an isolated slab In order to assess the effects arising from the presence of two different media in the equilibrium, a comparison between the previous results and those corresponding to a single medium is suitable. So, we consider a simpler equilibrium made of an iso- lated prominence slab with the magnetic field parallel to its axis. The external medium is not taken into account. Magnetoacoustic non-adiabatic perturbations are governed by Eq. (13), and rigid boundary conditions for vz are imposed at the edges of the promi- nence slab, vz(−zp) = vz(zp) = 0. (25) Then, the solution is of the form vz(z) = C1 cos +C2 sin , (26) and after imposing boundary conditions (Eq. [25]), we deduce the dispersion relation for the magnetoacoustic slow and fast non-adiabatic waves, kzpzp = π, (n = 0, 1, 2, . . .), (27) for the kink modes, and kzpzp = nπ, (n = 1, 2, 3, . . .), (28) for the sausage modes. Inserting expressions (14) and (15) for kzp and Λp respectively, one can rewrite the dispersion relations (27) and (28) as polynomial equations in ω. See App. C for the details. Next, considering only the fundamental kink modes for sim- plicity, we compute the period and the damping time and com- pare with those obtained when the surrounding corona is taken into account (Fig. 12). We see that there is a perfect agree- ment between both results in the case of the internal slow mode, whereas the solutions for the fast mode only coincide for inter- mediate and large wavenumbers, and show an absolutely differ- ent behaviour in the observed wavelength range and for smaller wavenumbers. Additionally, one must bear in mind that the ex- ternal slow mode exists because of the presence of the coronal medium, hence it is not supported by an isolated slab. In Paper I we proved that the internal slow mode is essen- tially confined within the prominence slab and that the effect of the corona on its oscillatory period can be neglected. Now, we see that the corona has no influence on the damping time either. On the other hand, the confinement of the fast mode is poor for small wavenumbers, the isolated slab approximation not being valid. As it has been noted in Section 4.1, the corona has an es- sential effect on the attenuation of the fast mode in the observed wavelength range. 4.4. Application to a prominence fibril Since magnetic field lines are orientated along fibrils, our model can also be applied to study the oscillatory modes supported by a single prominence fibril. In order to perform this investigation, we reduce the slab half-width, zp, to a value according to the typical observed size of filament threads, which is between 0.2 to 0.6 arcsec (Lin et al. 2005). Since these values are close to the resolution limit of present-day telescopes, it is likely that thinner R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 9 Fig. 12. Comparison between the solutions for a prominence plus corona system and for an isolated slab with prominence conditions. The upper panels correspond to the fundamental in- ternal slow kink mode and the lower panels to the fundamental fast kink mode. The solid lines are the solutions for a prominence plus corona equilibrium whereas the dotted lines with diamonds represent the solutions for an isolated slab. Prominence (1) pa- rameters and a = b = 0 have been used in the computations. threads could exist. So, assuming now zp = 30 km, we compute P, τD and τD/P for the fundamental kink modes and compare these results with those obtained for zp = 3000 km. Such as displayed in Fig. 13, both internal and external slow modes are not affected by the width of the prominence slab since they are essentially polarised along the x-direction and so they are not influenced by the equilibrium structure in the z-direction. Nevertheless, the location of the cut-off of the external slow mode and the coupling point with the fast mode are shifted to larger values of kx when the slab width is reduced. On the other hand, the fast mode, which is responsible for transverse motions, is highly influenced by the value of zp. The τD curve for the fast mode is displaced to larger values of kx when smaller zp is considered. This causes that higher values of τD/P are obtained in the observed wavelength range. Hence, these results suggest that local prominence oscillations related with transverse fast modes supported by a single fibril could be less affected by non- adiabatic mechanisms than global fast modes supported by the whole or large regions of the prominence. However, according to the results pointed out by Dı́az et al. (2005) and Dı́az & Roberts (2006), large groups of fibrils tend to oscillate together since the separation between individual fibrils is of the order of their thick- ness. In a very rough approximation one can consider that a thick prominence slab could represent many near threads which oscil- late together and that the larger the slab width, the more threads fit inside it. So, our results show that the slab size (i.e. the number of threads which oscillate together in this rough approximation) has important repercussions on the damping time of collective transverse oscillations, hence the oscillations could be more at- tenuated when the number of oscillating threads is larger. This affirmation should be verified by investigating the damping in multifibril models. 5. Conclusions In this paper, we have studied the time damping of magne- toacoustic waves in a prominence-corona system considering non-adiabatic terms (thermal conduction, radiation losses and heating) in the energy equation. Small amplitude perturbations have been assumed, so the linearised non-adiabatic MHD equa- tions have been considered and the dispersion relation for the slow and fast magnetoacoustic modes has been found assuming evanescent-like perturbations in the coronal medium. Finally, the damping time of the fundamental oscillatory modes has been computed and the relevance of each non-adiabatic mechanism on the attenuation has been assessed. Next, we summarise the main conclusions of this work: 1. Non-adiabatic effects are an efficient mechanism to obtain small ratios of the damping time to the period in the range of typically observed wavelengths of small-amplitude promi- nence oscillations. 2. The mechanism responsible for the attenuation of oscilla- tions is different for each magnetoacoustic mode and de- pends on the wavenumber. 3. The damping of the internal slow mode is dominated by prominence-related mechanisms, prominence radiation be- ing responsible for the attenuation in the observed wave- length range. Such as happens in the adiabatic case (see Paper I) the corona does not affect the slow mode at all, and these results are in perfect agreement with those for an iso- lated prominence slab. 4. The attenuation of the fast mode in the observed wavelength range is governed by a combined effect of prominence radia- tion and coronal conduction. The presence of the corona is of paramount importance to explain the behaviour of the damp- ing time for small wavenumbers within the observed range of wavelengths. Non-adiabatic mechanisms in both the promi- nence and the corona are significant because the fast mode achieves large amplitudes in both regions. 5. Since the external slow mode is principally supported by the corona, its damping time is entirely governed by coro- nal mechanisms, coronal conduction being the dominant one in the observed wavelength range. 6. The consideration of different optical thicknesses for the prominence plasma causes an important variation of the damping time of the internal slow and fast modes in the ob- served wavelength range. Hence a precise knowledge of the radiative processes of prominence plasmas is needed to ob- tain more realistic theoretical results. 7. The heating scenario has a negligible effect on the damp- ing time of all solutions in the observed wavelength range. Depending on the scenario considered, thermal instabilities can appear for small values of the wavenumber, in which coronal radiation dominates. 8. The width of the prominence slab does not affect the results for both internal and external slow modes. However, fast modes are less attenuated in the range of observed wave- lengths when thinner slabs or filaments threads are consid- ered. Taking into account the results in the observed range of wavelengths, one can conclude that radiative effects of the prominence plasma are responsible for the attenuation of the in- ternal slow modes, which can be connected with intermediate- and long-period prominence oscillations, whereas a combined effect of prominence radiation and coronal thermal conduction 10 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I Fig. 13. Period (left), damping time (centre) and ratio of the damping time to the period (right) versus kx for the fundamental kink oscillatory modes: internal slow (top panels), fast (mid panels) and external slow (bottom panels). Solid lines correspond to zp = 3000 km whereas dotted lines correspond to zp = 30 km. Prominence (1) radiation conditions have been taken for the prominence plasma and the heating scenario is given by a = b = 0. governs the damping of fast modes, whose periods are compati- ble with those of short-period oscillations. Acknowledgements. The authors acknowledge the financial support received from the Spanish Ministerio de Ciencia y Tecnologı́a under grant AYA2006- 07637. R. Soler thanks the Conselleria d’Economia, Hisenda i Innovació for a fellowship. Appendix A: Expressions for the perturbations Combining Eqs. (7)–(12), one can obtain the expressions for the perturbed quantities as functions of vz and its derivative −ikxΛ ω2 − k2xΛ , (A.1) ω2 − k2xΛ , (A.2) iωρ0Λ ω2 − k2xΛ , (A.3) ω2 − k2xΛ , (A.4) B1x = , (A.5) B1z = vz. (A.6) Now, we write the expressions for the perturbations to the mag- netic pressure, p1m, and the total pressure, p1T, p1m = B1x = , (A.7) p1T = p1 + p1m = ω2 − k2xv . (A.8) In the limit Λ → cs (i.e. in the absence of conduction, radiation losses and heating), all the expressions reduce to those corre- sponding to the adiabatic case. R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 11 Appendix B: Approximate dispersion relation for the internal slow modes Internal slow modes are almost non-dispersive and for adia- batic perturbations a good approximation for the frequency is ω ≈ cspkx, csp being the prominence sound speed. In the non- adiabatic case, we can consider the equivalence between cs and Λ to propose ω ≈ Λpkx as an approximate dispersion relation. Taking into account Eq. (15) for Λ, the approximate dispersion relation for the internal slow modes is a third order polynomial in ω, − iBω2 − k2xc spω + i Ak2x = 0, (B.1) A = (γ − 1) κ̂‖pk x + ωTp − ωρp , (B.2) B = (γ − 1) κ̂‖pk x + ωTp , (B.3) κ̂‖p = κ‖p In Fig. B.1 a comparison between the exact and approximate solutions is displayed and a perfect agreement is seen. Fig. B.1. Period (left) and damping time (right) versus the lon- gitudinal wavenumber for the fundamental internal slow kink mode. The solid line corresponds to the exact solution and the di- amonds correspond to the approximate solution. Prominence (1) parameters and a = b = 0 have been used in the computations. Appendix C: Dispersion relation for an isolated slab We here deduce a polynomial dispersion relation for the magne- toacoustic normal modes of a slab with a longitudinal magnetic field. Taking Eqs. (27) and (28) as the dispersion relations for the kink and sausage modes, respectively, one can replace kz and Λ with their correspondent expressions (Eqs. [14]–[15]), and the following fifth order polynomial equation is found, − iBω4 − v2A + c v2AB +A + v2Ac iAv2Ac = 0, (C.1) = k2x + (n + 1/2)2 π2 , (n = 0, 1, 2, . . .), for the kink modes, and = k2x + , (n = 1, 2, 3, . . .), for the sausage modes. Quantities A and B are given by Eqs. (B.2) and (B.3), respectively. References Ballester, J. L. 2006, Phil. Trans. R. Soc. A, 364, 405 Balthasar, H., Wiehr, E., Schleicher, H. & Wöhl, H. 1993, A&A, 277, 635 Carbonell, M., Oliver, R. & Ballester, J. L. 2004, A&A, 415, 739 Carbonell, M., Terradas, J., Oliver, R. & Ballester, J. L. 2006, A&A, 460, 573 Dahlburg, R. B. & Mariska, J. T. 1988, Sol. Phys., 117, 51 De Moortel, I. & Hood, A. W. 2004, A&A, 415, 705 Dı́az, A J., Oliver R. & Ballester, J. L. 2005, ApJ, 440, 1167 Dı́az, A. J. & Roberts, B. 2006, Sol. Phys., 236, 111 Edwin, P. M. & Roberts, B. 1982, Sol. Phys., 76, 239 Engvold, O. 2004, Proc. IAU Collq. on Multiwavelength investigations of solar activity (eds. A. V. Stepanov, E. E. Benevolenskaya & A. G. Kosovichev), Forteza, P., Oliver, R., Ballester, J. L. & Khodachenko, M. L. 2007, A&A, 461, Foullon, C., Verwichte, E. & Nakariakov, V. M. 2004, A&A, 427, L5 Harvey, J. 1969, Ph.D. thesis, University of Colorado, USA Hildner, E. 1974, Sol. Phys., 35, 123 Joarder, P. S. & Roberts, B. 1992, A&A, 256, 264 Lin, Y., Engvold, O. & Wiik, J. E. 2003, Sol. Phys., 216, 109 Lin, Y. et al. 2005, Sol. Phys., 226, 239 Lin, Y., Engvold, O., Rouppe van der Voort, L. H. M. & van Noort, M. 2007, Sol. Phys., in press Milne, A. M., Priest, E. R. & Roberts, B. 1979, ApJ, 232, 304 Molowny-Horas, R., Oliver, R., Ballester, J. L. & Baudin, F. 1997, Sol. Phys., 172, 181 Molowny-Horas, R., Heinzel, P., Mein, P. & Mein, N. 1999, A&A, 345, 618 Oliver, R. & Ballester, J. L. 2002, Sol. Phys., 206, 45 Porter, L. J., Klimchuk, J. A. & Sturrock, P. A. 1994, ApJ, 435, 482 Rosner, R., Tucker, W. H. & Vaiana, G. S. 1978, ApJ, 220, 643 Soler, R., Oliver, R. & Ballester, J. L. 2007, Sol. Phys., submitted (Paper I) Terradas, J., Oliver, R. & Ballester, J. L. 2001, A&A, 378, 635 Terradas, J., Molowny-Horas, R., Wiehr, E. et al. 2002, A&A, 393, 637 Terradas, J., Carbonell, M., Oliver, R. & Ballester, J. L. 2005, A&A, 434, 741 Wiehr, E. 2004, Proc. SOHO 13, ESA SP-547, 185 Introduction Equilibrium and basic equations Dispersion relation Results Regions of dominance of the damping mechanisms Exploring the parameter space Dependence on the equilibrium physical conditions Dependence on the prominence optical thickness Dependence on the heating scenario Comparison with the solution for an isolated slab Application to a prominence fibril Conclusions Expressions for the perturbations Approximate dispersion relation for the internal slow modes Dispersion relation for an isolated slab

Context. One of the typical features shown by observations of solar prominence oscillations is that they are damped in time and that the values of the damping times are usually between one and three times the corresponding oscillatory period. However, the mechanism responsible for the attenuation is still not well-known. Aims. Thermal conduction, optically thin or thick radiation and heating are taken into account in the energy equation, and their role on the attenuation of prominence oscillations is evaluated. Methods. The dispersion relation for linear non-adiabatic magnetoacoustic waves is derived considering an equilibrium made of a prominence plasma slab embedded in an unbounded corona. The magnetic field is orientated along the direction parallel to the slab axis and has the same strength in all regions. By solving the dispersion relation for a fixed wavenumber, a complex oscillatory frequency is obtained, and the period and the damping time are computed. Results. The effect of conduction and radiation losses is different for each magnetoacoustic mode and depends on the wavenumber. In the observed range of wavelengths the internal slow mode is attenuated by radiation from the prominence plasma, the fast mode by the combination of prominence radiation and coronal conduction and the external slow mode by coronal conduction. The consideration of the external corona is of paramount importance in the case of the fast and external slow modes, whereas it does not affect the internal slow modes at all. Conclusions. Non-adiabatic effects are efficient damping mechanisms for magnetoacoustic modes, and the values of the obtained damping times are compatible with those observed.

Introduction Prominences are dense coronal structures which appear as thin, dark filaments on the solar disc when observed in Hα. On the contrary, they show up as bright objects above the solar limb. The coronal magnetic field is responsible for the support of prominences against gravity, and it also plays a fundamental role in the thermal confinement of the cool prominence plasma em- bedded in the much hotter coronal environment. Nevertheless, the structure, orientation and strength of the magnetic field in prominences and the surrounding corona is still enigmatic and not well-known. High resolution observations reveal that promi- nences are composed by numerous very thin, thread-like struc- tures, called fibrils, piled up to form the body of the prominence (Lin et al. 2003; Lin et al. 2005, Lin et al. 2007) and measures also indicate that magnetic field lines are orientated along these thin threads. The observational evidence of small-amplitude oscillations in quiescent solar prominences goes back to 40 years ago (Harvey 1969). The amplitude of these oscillations typically goes from less than 0.1 km s−1 to 2–3 km s−1, and have been historically classified, according to their periods, in short- (P < 10 min), intermediate- (10 min < P < 40 min) and long-period Send offprint requests to: R. Soler oscillations (P > 40 min), although very short-periods of less than 1 min (Balthasar et al. 1993) and extreme ultra-long-periods of more than 8 hours (Foullon et al. 2004) have been reported. Nevertheless, the value of the period seems not to be related with the nature or the source of the trigger and probably is linked to the prominence eigenmode that is excited. There are also a few determinations of the wavelength and phase speed of standing oscillations and propagating waves in large regions of promi- nences (Molowny-Horas et al. 1997; Terradas et al. 2002) and in single filament threads (Lin et al. 2007). On the other hand, several observations (Molowny-Horas et al. 1999; Terradas et al. 2002) have informed about the evidence of the attenuation of the oscillations in Doppler velocity time series, which is a common feature observed in large areas. By fitting a sinusoidal function multiplied by a factor exp(−t/τD) to the Doppler series, these authors have obtained values of the damping time, τD, which are usually between 1 and 3 times the corresponding oscilla- tory period. The reader is referred to some recent reviews for more information about the observational background (Oliver & Ballester 2002, Wiehr 2004, Engvold 2004, Ballester 2006). From the theoretical point of view, small-amplitude promi- nence oscillations can be interpreted in terms of linear magne- tohydrodynamic (MHD) waves. Although, there is a wide bib- liography of works that investigate the ideal MHD wave modes http://arxiv.org/abs/0704.1566v2 2 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I supported by prominence models (see Oliver & Ballester 2002 for an extensive review of theoretical studies), the investigation of the wave damping has been broached in few papers. By re- moving the ideal assumption and including dissipative terms in the basic MHD equations, several works have studied the atten- uation of prominence oscillations considering radiative losses based on the Newtonian law of cooling with a constant relax- ation time (Terradas et al 2001), or performing a more complete treatment of non-adiabatic effects, assuming optically thin radi- ation, heating and thermal conduction (Carbonell et al. 2004; Terradas et al. 2005). The main conclusion of these previous studies is that only the slow wave is attenuated by thermal effects, radiation being the dominant damping mechanism in the range of typically observed wavelengths in prominences, but the fast wave remains practically undamped. On the other hand, Forteza et al. (2007) proposed ion-neutral collisions as a damping mechanism on the basis that prominences are partially ionised plasmas, but they found that this mechanism is only ef- ficient in attenuating the fast mode in quasi-neutral plasmas, the slow mode being almost unaffected. In the light of these referred studies, it is likely that non- adiabatic effects are the best candidates for the damping of small-amplitude oscillations, at least for slow modes. However, previous results do not asses the influence of the corona. The main aim of the present work is to perform a step forward in the investigation of the effect of non-adiabatic mechanisms (radia- tion losses, thermal conduction and heating) on the time damp- ing of prominence oscillations. We consider a slab model with a longitudinal magnetic field and take into account the external coronal medium. So, we explore for the first time the joint ef- fect of prominence and coronal mechanisms on the attenuation of oscillations. The magnetoacoustic normal modes of this equi- librium have been previously investigated by Edwin & Roberts (1982) and Joarder & Roberts (1992) in the adiabatic case. Later, a revision of these works has been done in Soler et al. (2007), hereafter Paper I, and the normal modes have been studied and reclassified according to their magnetoacoustic properties. This paper is organised as follows. The description of the equilibrium model and the linear non-adiabatic wave equations are given in Sect. 2, whereas the dispersion relation for the mag- netoacoustic modes is derived in Sect. 3. Then, the results are plotted and investigated in Sect. 4. Finally, Sect. 5 contains the conclusions of this work. 2. Equilibrium and basic equations Our equilibrium configuration (Fig. 1) is made of a homoge- neous plasma layer with prominence conditions (density ρp and temperature Tp) embedded in an unbounded corona (density ρc and temperature Tc). The coronal density is computed by fix- ing the coronal temperature and imposing pressure continuity across the interfaces. The magnetic field is B0 = B0êx, with B0 a constant everywhere. Both media are unlimited in the x- and y-directions. The half-width of the prominence slab is zp. The basic magnetohydrodynamic equations for the discus- sion of non-adiabatic processes are: + ρ∇ · v = 0, (1) = −∇p + (∇ × B) × B, (2) + (γ − 1)[ρL(ρ, T ) − ∇ · (κ · ∇T )] = 0, (3) Fig. 1. Sketch of the equilibrium. Table 1. Parameter values of the radiative loss function corresponding to the considered regimes. The three promi- nence regimes represent different plasma optical thicknesses. Prominence (1) regime corresponds to an optically thin plasma whereas Prominence (2) and Prominence (3) regimes represent greater optical thicknesses. All quantities are expressed in MKS units. Regime χ∗ α Reference Prominence (1) 1.76 × 10−13 7.4 Hildner (1974) Prominence (2) 1.76 × 10−53 17.4 Milne et al. (1979) Prominence (3) 7.01 × 10−104 30 Rosner et al. (1978) Corona 1.97 × 1024 −1 Hildner (1974) = ∇ × (v × B), (4) ∇ · B = 0, (5) , (6) where D + v · ∇ is the material derivative for time vari- ations following the motion and all quantities have their usual meaning. Equation (3) is the energy equation, which in the present form takes into account non-adiabatic effects (radiation losses, thermal conduction and heating) and whose terms are ex- plained in detail in Carbonell et al. (2004) and Terradas et al. (2005). Following these works, only thermal conduction par- allel to the magnetic field is assumed and we use the typical value for the parallel conductivity in prominence and coronal applications, κ‖ = 10 −11T 5/2 W m−1 K−1. Radiative losses and heating are evaluated together through the heat-loss function, L(ρ, T ) = χ∗ρTα − hρaT b, where radiation is parametrised with χ∗ and α (see Table 1) and the heating scenario is given by ex- ponents a and b. The heating mechanisms taken into account in this work are (Rosner et al. 1978; Dahlburg & Mariska 1988): – constant heating per unit volume (a = b = 0); – constant heating per unit mass (a = 1, b = 0); – heating by coronal current dissipation (a = b = 1); – heating by Alfvén mode/mode conversion (a = b = 7/6); – heating by Alfvén mode/anomalous conduction damping (a = 1/2, b = −1/2). Following the same process as in Carbonell et al. (2004), we consider small perturbations from the equilibrium state, lin- earise the basic Eqs. (1)–(6) and obtain their Eqs. (9)–(14). Since R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 3 our model is unlimited in the x- and y-directions, we assume all perturbations are in the form f1(z) exp i(ωt+ kxx+ kyy), and con- sidering only motions and propagation in the xz-plane (vy = 0, ky = 0), which excludes Alfvén waves, the linearised equations become iωρ1 + ρ0 ikxvx + = 0, (7) iωρ0vx = −ikx p1, (8) iωρ0vz = − ikxB1z − , (9) p1 − c = −(γ − 1) k2xκ‖T1 + ωρρ1 + ωT T1 , (10) iωB1x = −B0 , (11) iωB1z = B0ikxvz, (12) where c2s = is the adiabatic sound speed squared and L + ρ0Lρ , ωT ≡ T0LT , Lρ, LT being the partial derivatives of the heat-loss function with respect to density and temperature, respectively, , LT ≡ Now, it is possible to eliminate all perturbations in favour of vz to obtain a single differential equation + k2z vz = 0, (13) in which k2z = ω2 − k2xv ω2 − k2xΛ v2A + Λ ω2 − k2xc̃ ) , (14) where v2A = is the Alfvén speed squared. Λ2 and c̃2T are the modified sound and cusp (or tube) speed squared, respectively, (γ − 1) x + ωT − ωρ + iγω (γ − 1) κ‖k2x + ωT , (15) c̃2T ≡ v2A + Λ . (16) Expressions for the perturbations in terms of vz are given in App. A. In all the following formulae, subscripts p or c denote quantities computed using prominence or coronal values, respec- tively. 3. Dispersion relation We impose some restrictions on the solutions of Eq. (13) in or- der to obtain the dispersion relation for the linear non-adiabatic magnetoacoustic waves. We restrict this analysis to body waves which are evanescent in the corona, since we are looking for solutions which are essentially confined to the slab. For such so- lutions, vz(z) is of the form vz(z) = A1 exp z + zp , if z ≤ −zp, A2 cos + A3 sin , if −zp ≤ z ≤ zp, A4 exp z − zp , if z ≥ zp. withℜ(kzp) > 0 andℜ(kzc) > 0. Imposing continuity of vz and the total (gas plus magnetic) pressure perturbation across the interfaces, we find four alge- braic relations between the constants A1, A2, A3 and A4. The non-trivial solution of this system gives us the dispersion rela- Ac − ω kzpzp Ap − ω kzc = 0, (18) where cot/tan terms and ± signs are related with the symmetry of the perturbations. The cot term and the + sign correspond to kink modes (A3 = 0), whereas the tan term and the − sign correspond to sausage modes (A2 = 0). The dispersion relation for the magnetoacoustic waves pre- sented in Eq. (18) is equivalent to the relation investigated in Edwin & Roberts (1982) and Joarder & Roberts (1992), and re- vised Paper I, in the case of adiabatic perturbations, since all non-adiabatic terms are now enclosed in kzp and kzc through Eq. (14). 4. Results Now, we assume Prominence (1) conditions inside the slab (i.e. an optically thin prominence) and a heating mechanism given by a = b = 0. Unless otherwise stated, the following equilibrium parameters are considered in all computations: Tp = 8000 K, ρp = 5 × 10 −11 kg m−3, Tc = 10 6 K, ρc = 2.5 × 10 −13 kg m−3, B0 = 5 G and zp = 3000 km. The solution of the dispersion relation (Eq. [18]) for a fixed real kx gives us a complex fre- quency ω = ωR + iωI. We then compute the oscillatory period, the damping time and the ratio of the damping time to the pe- riod because this is an important quantity from the observational point of view, , τD = Since we are interested in studying the behaviour of the most relevant solutions of the dispersion relation, we only compute the results for the fundamental modes, which are labelled, accord- ing to the classification of Paper I, as internal and external slow modes and fast modes. The band structure described in Paper I is slightly modified when non-adiabatic terms are considered (see Fig. 2). The phase speed of the internal slow modes is now en- closed in the bandℜ(c̃Tp) < ωR/kx < ℜ(Λp). The adiabatic fast modes exist in two separated bands in the phase speed diagram due to the presence of a forbidden region (cTc < ωR/kx < csc), but now the forbidden band is avoided and a continuous fast mode is found with vAp < ωR/kx < vAc. Finally, and like in the adiabatic case, among the external slow modes only the fun- damental kink one exists as a non-leaky solution in a restricted 4 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I wavenumber range and couples with the fundamental fast kink mode. Its phase speed is ωR/kx ≈ ℜ(Λc). Therefore, we see that in the non-adiabatic case Λ plays the role of cs in the adiabatic case. Fig. 2. Phase speed versus the dimensionless wavenumber for the three fundamental oscillatory modes. Solid lines denote kink modes whereas dotted lines correspond to sausage modes. The shaded zones are projections of the forbidden (or leaky) regions on the plane of this diagram. Note that the vertical axis is not drawn to scale. In Fig. 3 P, τD and τD/P are represented for the fundamental modes and for a range of the longitudinal wavenumber between 10−10 m−1 and 103 m−1. The shaded zones correspond to wave- lengths between 5 × 103 km and 105 km, the typically observed values. It turns out that the values of the period are very sim- ilar to those obtained in the adiabatic case (Joarder & Roberts 1992; Paper I). The damping time presents a strong dependence with the wavenumber and its behaviour is very different from one mode to another. This fact suggests that the non-adiabatic mechanisms can affect each mode in a different way (Carbonell et al. 2004). This is studied in detail in Sect 4.1. Observations show that prominence oscillations are typically attenuated in a few periods (Terradas et al. 2002), so a damping time of the order of the period is expected. In our results, the fundamental modes present values of τD/P in the range 1 to 10 in the observed wave- length region, which is in agreement with observations. 4.1. Regions of dominance of the damping mechanisms The importance of the different non-adiabatic terms included in the energy equation (Eq. [3]) depends on the wavenumber. In order to know which is the range of dominance of each mech- anism, we compare the damping time obtained when consider- ing all non-adiabatic terms (displayed in the middle column of Fig. 3) with the results obtained when a specific mechanism is removed from the energy equation. With this analysis, we are able to know where the omitted mechanism has an appreciable effect on the damping. The results of these computations for the fundamental kink modes (Fig. 4) are summarised as follows: – The fundamental internal slow kink mode is not affected by the mechanisms related with the corona. This is a conse- quence of the nature of this mode, which propagates strictly along the prominence without disturbing the corona (see Fig. 4, top row, of Paper I). For this reason, in the adiabatic case it is also independent of the coronal conditions. On the other hand, the prominence-related mechanisms show different ef- fects in two different ranges of kx. For kx . 10 −3 m−1 promi- nence radiation dominates, while for kx & 10 −3 m−1 promi- nence conduction is the dominant mechanism. Beginning from small values of the wavenumber, prominence radiation becomes more efficient as kx grows and the damping time falls following a power law until kx ≈ 10 −5 m−1, where τD saturates in a plateau between kx ≈ 10 −5 and kx ≈ 10 −3 m−1. Then, prominence conduction becomes the dominant mech- anism and the damping time falls again until kx ≈ 10 −1 m−1 where a new plateau begins. This last part of the curve cor- responds to the isothermal or superconductive regime, in which the amplitude of the temperature perturbation drops dramatically (Carbonell et al. 2006). Prominence radiation is responsible for the attenuation of the slow mode in the observed wavelength range. An approximate dispersion re- lation for the internal slow modes is included in App. B. – The fundamental fast kink mode is affected by the four mechanisms. For kx . 3 × 10 −9 m−1 coronal radiation dom- inates but for 3 × 10−9 m−1 . kx . 5 × 10 −7 m−1 the effect of coronal conduction grows and becomes the main damping mechanism. Then, for kx & 5×10 −7 m−1 the corona loses dra- matically its influence and prominence mechanisms become responsible for the attenuation of this mode. First, promi- nence radiation is dominant in the range 5 × 10−7 m−1 . kx . 10 −3 m−1, then prominence conduction governs the wave damping for kx & 10 −3 m−1 and finally the isother- mal regime begins for kx ≈ 10 0 m−1. The minimum of τD occurs into the coronal conduction regime, for the value of kx which corresponds to the coupling with the external slow mode. The transition between the coronal conduction regime and the prominence radiation regime occurs in the observed wavelength range. The reason for the sensitivity of the fast mode damping time on prominence and coronal conditions is that this wave has a considerable amplitude both inside the prominence and in the corona, the later becoming more im- portant for long wavelengths (see the second and third rows of Fig. 4 of Paper I). – The behaviour of the damping time of the fundamental exter- nal slow kink mode is entirely dominated by coronal mech- anisms whereas the prominence mechanisms do not affect it at all. This behaviour is a result of the negligible amplitude of this wave in the prominence (see the fourth and fifth rows of Fig. 4 of Paper I). For kx . 3× 10 −9 m−1 coronal radiation dominates, but for shorter wavelengths coronal conduction becomes more relevant and is responsible for the damping in the observed wavelength range until the frequency cut-off is reached. At the cut-off, τD has a value of the order of the period. Regarding the fundamental sausage modes, the behaviour of the internal slow sausage mode is exactly that of the slow kink mode, so no additional comments are needed. The fundamen- tal fast sausage mode (Fig. 5) presents the same scheme as the fundamental fast kink mode for kx & 10 −8 m−1. The main dif- ference between the fast kink and sausage modes happens in the observed wavelength range, where the effect of coronal conduc- tion on the sausage mode is less relevant. If coronal conduction is omitted, the fundamental fast sausage mode is not able to tra- verse the forbidden region in the dispersion diagram and then shows frequency cut-offs as in the adiabatic case. This means R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 5 Fig. 3. Period (left), damping time (centre) and ratio of the damping time to the period (right) versus the longitudinal wavenumber for the fundamental oscillatory modes. Upper panels: internal slow kink (solid line), fast kink (dotted line) and external slow kink (dashed line). Lower panels: internal slow sausage (solid line) and fast sausage (dotted line). Shaded zones correspond to those wavelengths typically observed. Note the cut-offs of the external slow kink mode and the fast sausage mode. Prominence (1) radiation conditions have been taken for the prominence plasma and the heating scenario is given by a = b = 0. Fig. 4. Damping time versus the longitudinal wavenumber for the three fundamental kink oscillatory modes: internal slow (left), fast (centre) and external slow (right). Different linestyles represent the omitted mechanism: all mechanisms considered (solid line), prominence conduction eliminated (dotted line), prominence radiation eliminated (dashed line), coronal conduction eliminated (dot- dashed line) and coronal radiation eliminated (three dot-dashed line). Prominence (1) radiation conditions have been taken for the prominence plasma and the heating scenario is given by a = b = 0. that coronal conduction causes the fast mode to cross the forbid- den region in the dispersion diagram in the non-adiabatic case. Approximate values of kx for which the transitions between regimes take place can be computed by following a process simi- lar to that in Carbonell et al. (2006). The thermal ratio, d, and the radiation ratio, r, quantify the importance of thermal conduction and radiation, respectively (De Moortel & Hood 2004), (γ − 1)κ‖T0ρ0 γ2 p20τs τcond , (19) (γ − 1)τsρ , (20) where τs is the sound travel time and τcond and τrad are character- istic conductive and radiative time scales. Taking τs = 2π/k the value of k∗ for which the condition d = r is satisfied is k∗ = 2πρ0 χ∗Tα−10 . (21) Now, we use prominence values to compute k∗ for the promi- nence radiation–prominence conduction transition (k∗p), and 6 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I Fig. 6. Damping time versus the longitudinal wavenumber for the fundamental internal slow kink mode (left), the fundamental fast kink mode (centre) and the fundamental external slow kink mode (right). The different linestyles represent different values of the prominence temperature: Tp = 8000 K (solid line), Tp = 5000 K (dotted line) and Tp = 13000 K (dashed line). The heating scenario is given by a = b = 0 and the optical thickness for the prominence plasma is Prominence (1). Fig. 7. Same as Fig. 6 with ρp = 5 × 10−11 kg m−3 (solid line), ρp = 2 × 10−11 kg m−3 (dotted line) and ρp = 10−10 kg m−3 (dashed line). Fig. 8. Same as Fig. 6 with B0 = 5 G (solid line), B0 = 2 G (dotted line) and B0 = 10 G (dashed line). coronal values for the coronal radiation–coronal conduction transition (k∗c). This gives the values k p ≈ 1.7 × 10 −3 m−1, and k∗c ≈ 2.2 × 10 −8 m−1. For the transition of the fast kink mode between the coronal conduction and the prominence radiation regimes, the boundary wavenumber k∗p↔c can be roughly calcu- lated by imposing dc = rp, that gives k∗p↔c = 2πρp cscχ∗pT cspκ‖cTc , (22) and whose numerical value is k∗p↔c ≈ 1.4 × 10 −6 m−1. All these wavenumbers for the transitions between different regimes are independent of the wave type, be it fast or slow, internal or ex- ternal (this agrees with Figs. 4 and 5). On the other hand, the be- ginning of the isothermal regime can be estimated by following Porter et al. (1994). Considering c2sp/v Ap ≪ 1 and the approxi- mations ωR ≈ kxcsp for the slow wave and ωR ≈ kxvAp for the fast wave, the critical wavenumber is kcrit−slow = 2ρpkBcsp κ‖pmp cos θ , (23) for the internal slow mode, and kcrit−fast = 2ρpkBvAp κ‖pmp cos2 θ , (24) R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 7 Fig. 5. Same as Fig. 4 for the fundamental fast sausage mode. for the fast mode, where mp is the proton mass, kB is the Boltzmann constant and θ is the angle between B and k. Taking cos θ = 1 for simplicity, the approximate critical values are kcrit−slow ≈ 1.7 × 10 −1 m−1 and kcrit−fast ≈ 9.1 × 10 −1 m−1. We note that all these approximate values describe correctly the tran- sitions between the diverse regimes shown in Figs. 4 and 5, but their numerical values overestimate by almost an order of mag- nitude the actual critical wavenumbers. 4.2. Exploring the parameter space 4.2.1. Dependence on the equilibrium physical conditions In this section, we compute the solutions for different values of the equilibrium physical conditions. We only present the re- sults for the fundamental kink modes since they are equivalent to those of sausage modes. Figures 6, 7 and 8 display the damping time as function of kx for some selected values of the prominence temperature, the prominence density and the magnetic field, re- spectively. For the internal slow mode, a decrease of the prominence temperature or the prominence density raises the position of the radiative plateau and increases its length. The opposite be- haviour is seen when the density or the temperature are in- creased. However, the value of the magnetic field does not in- fluence the attenuation of this mode, such as expected for a slow wave. Increasing the value of the prominence temperature causes a vertical displacement of τD of the fast mode in those regions in which prominence mechanisms dominate. The value of the prominence density has a smaller effect and its main influence is in changing the coupling point with the external slow mode, which moves to higher kx for greater values of the density. The magnetic field strength has a more complex effect on τD and also modifies the coupling point. Finally, the external slow mode is only slightly affected by a modification of the prominence physical parameters since it is mainly dominated by coronal conditions, and the influence of the magnetic field is very small due to the slow-like magnetoa- coustic character of this solution. 4.2.2. Dependence on the prominence optical thickness The optically thin radiation assumption is a reasonable approxi- mation in a plasma with coronal conditions but prominence plas- mas often are optically thick. In this section we compare the results obtained considering different optical thicknesses for the prominence plasma (see Fig. 9 for the fundamental kink modes). The results corresponding to the slow sausage mode have not been plotted since they are equivalent to those obtained for slow kink mode; those for the fundamental fast sausage mode, how- ever, are displayed in Fig. 10. The variation of the prominence optical thickness modi- fies the prominence conduction–prominence radiation critical wavenumber, k∗p (see analytical approximation of Eq. [21]). For the internal slow mode, an increase in the optical thickness raises the position of the radiative plateau and shifts it to smaller wavenumbers. This fact causes an a priori surprising result in the observed wavelength range, since τD has a smaller value for opti- cally thick radiation, Prominence (3), than for optically thin radi- ation, Prominence (1). Regarding fast modes, the damping time increases when the optical thickness is increased, but only in the region in which prominence radiation dominates. The value of τD inside the observed wavelength range is partially affected and raises an order of magnitude for Prominence (3) conditions in comparison with the results for Prominence (1) conditions. Finally, the damping time of the external slow mode is not af- fected by the prominence optical thickness since it is entirely dominated by the corona, as it has been noticed in Sect. 4.1. Fig. 10. Same as Fig. 9 for the fundamental fast sausage mode. 4.2.3. Dependence on the heating scenario Now, we compute the damping time for the five possible heating scenarios. For simplicity, we only consider the fundamental kink modes (Fig. 11). Carbonell et al. (2004) showed that in a plasma with prominence conditions the different heating scenarios have no significant influence on the damping time. Nevertheless, in coronal conditions wave instabilities can appear depending on the heating mechanism. In our results, we see that the heating scenario affects the value of τD only in the ranges of kx in which radiation is the dominant damping mechanism. The heating sce- nario has a negligible effect when prominence radiation domi- nates, since τD is only slightly modified. On the contrary, wave instabilities appear in those regions in which coronal radiation dominates. Thermal destabilisation occurs when the imaginary part of the frequency becomes negative, so oscillations are not attenuated but amplified in time. Instabilities only occur in the fundamental fast kink and the external slow modes for very small values of kx, outside the observed wavelength range. 8 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I Fig. 9. Same as Fig. 6 with the prominence optical thickness given by Prominence (1) (solid line), Prominence (2) (dotted line) and Prominence (3) (dashed line) conditions. Fig. 11. Same as Fig. 6 with the heating scenario given by a = b = 0 (solid line); a = 1, b = 0 (dotted line); a = b = 1 (dashed line); a = b = 7/6 (dot-dashed line); a = 1/2, b = −1/2 (three dot-dashed line). 4.3. Comparison with the solution for an isolated slab In order to assess the effects arising from the presence of two different media in the equilibrium, a comparison between the previous results and those corresponding to a single medium is suitable. So, we consider a simpler equilibrium made of an iso- lated prominence slab with the magnetic field parallel to its axis. The external medium is not taken into account. Magnetoacoustic non-adiabatic perturbations are governed by Eq. (13), and rigid boundary conditions for vz are imposed at the edges of the promi- nence slab, vz(−zp) = vz(zp) = 0. (25) Then, the solution is of the form vz(z) = C1 cos +C2 sin , (26) and after imposing boundary conditions (Eq. [25]), we deduce the dispersion relation for the magnetoacoustic slow and fast non-adiabatic waves, kzpzp = π, (n = 0, 1, 2, . . .), (27) for the kink modes, and kzpzp = nπ, (n = 1, 2, 3, . . .), (28) for the sausage modes. Inserting expressions (14) and (15) for kzp and Λp respectively, one can rewrite the dispersion relations (27) and (28) as polynomial equations in ω. See App. C for the details. Next, considering only the fundamental kink modes for sim- plicity, we compute the period and the damping time and com- pare with those obtained when the surrounding corona is taken into account (Fig. 12). We see that there is a perfect agree- ment between both results in the case of the internal slow mode, whereas the solutions for the fast mode only coincide for inter- mediate and large wavenumbers, and show an absolutely differ- ent behaviour in the observed wavelength range and for smaller wavenumbers. Additionally, one must bear in mind that the ex- ternal slow mode exists because of the presence of the coronal medium, hence it is not supported by an isolated slab. In Paper I we proved that the internal slow mode is essen- tially confined within the prominence slab and that the effect of the corona on its oscillatory period can be neglected. Now, we see that the corona has no influence on the damping time either. On the other hand, the confinement of the fast mode is poor for small wavenumbers, the isolated slab approximation not being valid. As it has been noted in Section 4.1, the corona has an es- sential effect on the attenuation of the fast mode in the observed wavelength range. 4.4. Application to a prominence fibril Since magnetic field lines are orientated along fibrils, our model can also be applied to study the oscillatory modes supported by a single prominence fibril. In order to perform this investigation, we reduce the slab half-width, zp, to a value according to the typical observed size of filament threads, which is between 0.2 to 0.6 arcsec (Lin et al. 2005). Since these values are close to the resolution limit of present-day telescopes, it is likely that thinner R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 9 Fig. 12. Comparison between the solutions for a prominence plus corona system and for an isolated slab with prominence conditions. The upper panels correspond to the fundamental in- ternal slow kink mode and the lower panels to the fundamental fast kink mode. The solid lines are the solutions for a prominence plus corona equilibrium whereas the dotted lines with diamonds represent the solutions for an isolated slab. Prominence (1) pa- rameters and a = b = 0 have been used in the computations. threads could exist. So, assuming now zp = 30 km, we compute P, τD and τD/P for the fundamental kink modes and compare these results with those obtained for zp = 3000 km. Such as displayed in Fig. 13, both internal and external slow modes are not affected by the width of the prominence slab since they are essentially polarised along the x-direction and so they are not influenced by the equilibrium structure in the z-direction. Nevertheless, the location of the cut-off of the external slow mode and the coupling point with the fast mode are shifted to larger values of kx when the slab width is reduced. On the other hand, the fast mode, which is responsible for transverse motions, is highly influenced by the value of zp. The τD curve for the fast mode is displaced to larger values of kx when smaller zp is considered. This causes that higher values of τD/P are obtained in the observed wavelength range. Hence, these results suggest that local prominence oscillations related with transverse fast modes supported by a single fibril could be less affected by non- adiabatic mechanisms than global fast modes supported by the whole or large regions of the prominence. However, according to the results pointed out by Dı́az et al. (2005) and Dı́az & Roberts (2006), large groups of fibrils tend to oscillate together since the separation between individual fibrils is of the order of their thick- ness. In a very rough approximation one can consider that a thick prominence slab could represent many near threads which oscil- late together and that the larger the slab width, the more threads fit inside it. So, our results show that the slab size (i.e. the number of threads which oscillate together in this rough approximation) has important repercussions on the damping time of collective transverse oscillations, hence the oscillations could be more at- tenuated when the number of oscillating threads is larger. This affirmation should be verified by investigating the damping in multifibril models. 5. Conclusions In this paper, we have studied the time damping of magne- toacoustic waves in a prominence-corona system considering non-adiabatic terms (thermal conduction, radiation losses and heating) in the energy equation. Small amplitude perturbations have been assumed, so the linearised non-adiabatic MHD equa- tions have been considered and the dispersion relation for the slow and fast magnetoacoustic modes has been found assuming evanescent-like perturbations in the coronal medium. Finally, the damping time of the fundamental oscillatory modes has been computed and the relevance of each non-adiabatic mechanism on the attenuation has been assessed. Next, we summarise the main conclusions of this work: 1. Non-adiabatic effects are an efficient mechanism to obtain small ratios of the damping time to the period in the range of typically observed wavelengths of small-amplitude promi- nence oscillations. 2. The mechanism responsible for the attenuation of oscilla- tions is different for each magnetoacoustic mode and de- pends on the wavenumber. 3. The damping of the internal slow mode is dominated by prominence-related mechanisms, prominence radiation be- ing responsible for the attenuation in the observed wave- length range. Such as happens in the adiabatic case (see Paper I) the corona does not affect the slow mode at all, and these results are in perfect agreement with those for an iso- lated prominence slab. 4. The attenuation of the fast mode in the observed wavelength range is governed by a combined effect of prominence radia- tion and coronal conduction. The presence of the corona is of paramount importance to explain the behaviour of the damp- ing time for small wavenumbers within the observed range of wavelengths. Non-adiabatic mechanisms in both the promi- nence and the corona are significant because the fast mode achieves large amplitudes in both regions. 5. Since the external slow mode is principally supported by the corona, its damping time is entirely governed by coro- nal mechanisms, coronal conduction being the dominant one in the observed wavelength range. 6. The consideration of different optical thicknesses for the prominence plasma causes an important variation of the damping time of the internal slow and fast modes in the ob- served wavelength range. Hence a precise knowledge of the radiative processes of prominence plasmas is needed to ob- tain more realistic theoretical results. 7. The heating scenario has a negligible effect on the damp- ing time of all solutions in the observed wavelength range. Depending on the scenario considered, thermal instabilities can appear for small values of the wavenumber, in which coronal radiation dominates. 8. The width of the prominence slab does not affect the results for both internal and external slow modes. However, fast modes are less attenuated in the range of observed wave- lengths when thinner slabs or filaments threads are consid- ered. Taking into account the results in the observed range of wavelengths, one can conclude that radiative effects of the prominence plasma are responsible for the attenuation of the in- ternal slow modes, which can be connected with intermediate- and long-period prominence oscillations, whereas a combined effect of prominence radiation and coronal thermal conduction 10 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I Fig. 13. Period (left), damping time (centre) and ratio of the damping time to the period (right) versus kx for the fundamental kink oscillatory modes: internal slow (top panels), fast (mid panels) and external slow (bottom panels). Solid lines correspond to zp = 3000 km whereas dotted lines correspond to zp = 30 km. Prominence (1) radiation conditions have been taken for the prominence plasma and the heating scenario is given by a = b = 0. governs the damping of fast modes, whose periods are compati- ble with those of short-period oscillations. Acknowledgements. The authors acknowledge the financial support received from the Spanish Ministerio de Ciencia y Tecnologı́a under grant AYA2006- 07637. R. Soler thanks the Conselleria d’Economia, Hisenda i Innovació for a fellowship. Appendix A: Expressions for the perturbations Combining Eqs. (7)–(12), one can obtain the expressions for the perturbed quantities as functions of vz and its derivative −ikxΛ ω2 − k2xΛ , (A.1) ω2 − k2xΛ , (A.2) iωρ0Λ ω2 − k2xΛ , (A.3) ω2 − k2xΛ , (A.4) B1x = , (A.5) B1z = vz. (A.6) Now, we write the expressions for the perturbations to the mag- netic pressure, p1m, and the total pressure, p1T, p1m = B1x = , (A.7) p1T = p1 + p1m = ω2 − k2xv . (A.8) In the limit Λ → cs (i.e. in the absence of conduction, radiation losses and heating), all the expressions reduce to those corre- sponding to the adiabatic case. R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 11 Appendix B: Approximate dispersion relation for the internal slow modes Internal slow modes are almost non-dispersive and for adia- batic perturbations a good approximation for the frequency is ω ≈ cspkx, csp being the prominence sound speed. In the non- adiabatic case, we can consider the equivalence between cs and Λ to propose ω ≈ Λpkx as an approximate dispersion relation. Taking into account Eq. (15) for Λ, the approximate dispersion relation for the internal slow modes is a third order polynomial in ω, − iBω2 − k2xc spω + i Ak2x = 0, (B.1) A = (γ − 1) κ̂‖pk x + ωTp − ωρp , (B.2) B = (γ − 1) κ̂‖pk x + ωTp , (B.3) κ̂‖p = κ‖p In Fig. B.1 a comparison between the exact and approximate solutions is displayed and a perfect agreement is seen. Fig. B.1. Period (left) and damping time (right) versus the lon- gitudinal wavenumber for the fundamental internal slow kink mode. The solid line corresponds to the exact solution and the di- amonds correspond to the approximate solution. Prominence (1) parameters and a = b = 0 have been used in the computations. Appendix C: Dispersion relation for an isolated slab We here deduce a polynomial dispersion relation for the magne- toacoustic normal modes of a slab with a longitudinal magnetic field. Taking Eqs. (27) and (28) as the dispersion relations for the kink and sausage modes, respectively, one can replace kz and Λ with their correspondent expressions (Eqs. [14]–[15]), and the following fifth order polynomial equation is found, − iBω4 − v2A + c v2AB +A + v2Ac iAv2Ac = 0, (C.1) = k2x + (n + 1/2)2 π2 , (n = 0, 1, 2, . . .), for the kink modes, and = k2x + , (n = 1, 2, 3, . . .), for the sausage modes. Quantities A and B are given by Eqs. (B.2) and (B.3), respectively. References Ballester, J. L. 2006, Phil. Trans. R. Soc. A, 364, 405 Balthasar, H., Wiehr, E., Schleicher, H. & Wöhl, H. 1993, A&A, 277, 635 Carbonell, M., Oliver, R. & Ballester, J. L. 2004, A&A, 415, 739 Carbonell, M., Terradas, J., Oliver, R. & Ballester, J. L. 2006, A&A, 460, 573 Dahlburg, R. B. & Mariska, J. T. 1988, Sol. Phys., 117, 51 De Moortel, I. & Hood, A. W. 2004, A&A, 415, 705 Dı́az, A J., Oliver R. & Ballester, J. L. 2005, ApJ, 440, 1167 Dı́az, A. J. & Roberts, B. 2006, Sol. Phys., 236, 111 Edwin, P. M. & Roberts, B. 1982, Sol. Phys., 76, 239 Engvold, O. 2004, Proc. IAU Collq. on Multiwavelength investigations of solar activity (eds. A. V. Stepanov, E. E. Benevolenskaya & A. G. Kosovichev), Forteza, P., Oliver, R., Ballester, J. L. & Khodachenko, M. L. 2007, A&A, 461, Foullon, C., Verwichte, E. & Nakariakov, V. M. 2004, A&A, 427, L5 Harvey, J. 1969, Ph.D. thesis, University of Colorado, USA Hildner, E. 1974, Sol. Phys., 35, 123 Joarder, P. S. & Roberts, B. 1992, A&A, 256, 264 Lin, Y., Engvold, O. & Wiik, J. E. 2003, Sol. Phys., 216, 109 Lin, Y. et al. 2005, Sol. Phys., 226, 239 Lin, Y., Engvold, O., Rouppe van der Voort, L. H. M. & van Noort, M. 2007, Sol. Phys., in press Milne, A. M., Priest, E. R. & Roberts, B. 1979, ApJ, 232, 304 Molowny-Horas, R., Oliver, R., Ballester, J. L. & Baudin, F. 1997, Sol. Phys., 172, 181 Molowny-Horas, R., Heinzel, P., Mein, P. & Mein, N. 1999, A&A, 345, 618 Oliver, R. & Ballester, J. L. 2002, Sol. Phys., 206, 45 Porter, L. J., Klimchuk, J. A. & Sturrock, P. A. 1994, ApJ, 435, 482 Rosner, R., Tucker, W. H. & Vaiana, G. S. 1978, ApJ, 220, 643 Soler, R., Oliver, R. & Ballester, J. L. 2007, Sol. Phys., submitted (Paper I) Terradas, J., Oliver, R. & Ballester, J. L. 2001, A&A, 378, 635 Terradas, J., Molowny-Horas, R., Wiehr, E. et al. 2002, A&A, 393, 637 Terradas, J., Carbonell, M., Oliver, R. & Ballester, J. L. 2005, A&A, 434, 741 Wiehr, E. 2004, Proc. SOHO 13, ESA SP-547, 185 Introduction Equilibrium and basic equations Dispersion relation Results Regions of dominance of the damping mechanisms Exploring the parameter space Dependence on the equilibrium physical conditions Dependence on the prominence optical thickness Dependence on the heating scenario Comparison with the solution for an isolated slab Application to a prominence fibril Conclusions Expressions for the perturbations Approximate dispersion relation for the internal slow modes Dispersion relation for an isolated slab

arXiv:0704.1567v2 [gr-qc] 15 Apr 2007 Energy and Momentum Distributions of Kantowski and Sachs Space-time Ragab M. Gad1 and A. Fouad Mathematics Department, Faculty of Science, Minia University, 61915 El-Minia, EGYPT. Abstract We use the Einstein, Bergmann-Thomson, Landau-Lifshitz and Pa- papetrou energy-momentum complexes to calculate the energy and mo- mentum distributions of Kantowski and Sachs space-time. We show that the Einstein and Bergmann-Thomson definitions furnish a consis- tent result for the energy distribution, but the definition of Landau- Lifshitz do not agree with them. We show that a signature switch should affect about everything including energy distribution in the case of Einstein and Papapetrou prescriptions but not in Bergmann- Thomson and Landau-Lifshitz prescriptions. 1 Introduction One of the most interesting and intricate problems still unsolved since the outset of general relativity is the energy-momentum localization. Einstein himself proposed the first energy-momentum complex in an attempt to de- fine the local distribution of energy and momentum [1]. After this attempt, a plethora of different energy-momentum complexes were proposed, including formulations by Tolman [2], Papapetrou [3], Møller [4], Landau and Lifshitz [5], Weinberg [6], Bergmann-Thomson [7] and others. This approach was abandoned for a long time due to severe criticism for a number of reasons. Recently, Virbhadra re-opened the subject of energy-momentum com- plexes [8]. He pointed out that though these complexes are non-tensors, they yield reasonable and consistent results for a given space-time. Aguir- regabiria et al [9] found that for any metric of the Kerr-Schild class, sev- eral different definitions of the energy-momentum complex yield precisely the same results.Virbhadra [10] investigated whether or not these energy momentum complexes lead to the same results for the most general non- static spherically symmetric metric and found that they disagree. He noted that the energy-momentum complexes of Landau and Lifshitz, Papapetrou 1Email Address: ragab2gad@hotmail.com http://arxiv.org/abs/0704.1567v2 and Weinberg give the same results as in the Einstein definition if the cal- culations are performed in Kerr-Schild Cartesian coordinates. However, these energy-momentum complexes disagree if computations are done in ”Schwarzschild Cartesian” coordinates2. In a detail study of the question, Xulu [11] has confirmed this suggestion. He obtained the energy distribu- tion for the most general non-static spherically symmetric using Møller’s definition and found different results in general from those obtained us- ing Einstein’s definition. These results agree for the Schwarzschild, Vaidya and Janis-Newmann-Winicour space-times, but disagree for the Reissner- Nordström space-time. Many authors had similarly successfully applied the aforementioned energy-momentum complexes to various black hole configu- rations [12]. It has been remained a controversial problem whether or not energy and momentum are localizable. There are different opinions on this subject. Contradicting the viewpoint of Misner et al. [13] that the energy is local- izable only for spherical systems, Cooperstock and Sarracino [14] argued that if the energy localization is meaningful for spherical systems then it is meaningful for all systems. Bondi [15] expressed that a non-localizable form of energy is inadmissible in relativity and its location can in prin- ciple be found. These contradictory viewpoints bear significantly on the study of gravitational waves. It is an interesting question whether or not gravitational waves have energy and momentum content. In a series of pa- pers, Cooperstock [16] hypothesized that in a curved space-time energy and momentum are confined to the region of non-vanishing energy-momentum tensor T ab and consequently the gravitational waves are not carriers of energy and momentum in vacuum space-times. This hypothesis has neither been proved nor disproved. There are many results supporting this hypothesis (see for example, [17, 18]). In this paper we evaluate the energy and momentum distributions of the Kantowski and Sachs space-time, using Einstein, Bergmann-Thomson, Landau-Lifshitz and Papapetrou energy-momentum complexes. Through this paper we use G = 1 and c = 1 units and follow the conven- tion that Latin indices take value from 0 to 3 and Greek indices take value from 1 to 3. 2Schwarzschild metric in “Schwarzschild Cartesian coordinates” is obtained by trans- forming this metric (in usual Schwarzschild coordinates {t, r, θ, φ}) to {t, x, y, z} using x = r sin θ cosφ, y = r sin θ sinφ, z = r cos θ. 2 Kantowski and Sachs Space-time The standard representation of Kantowski and Sachs space-times are given by [19] dS2 = dt2 −A2(t)dr2 −B2(t)(d2θ + sin2 θd2φ), (2.1) where the functions A(t) and B(t) are to be determined from the field equa- tions. The solutions of the Einstein field equations for the above metric were con- sidered with dust source [19], but they were generalized (in fact earlier) to the general perfect fluid source [21]. From the geometrically point of view, this line element admits a four parameter continuous group of isometries which acts on space-like hypersur- face, and has no three parameter subgroup that would be simply transitive on the orbits (for more detailed description see Kantowski and Sachs [19] and Collins [20]). From the physical point of view, the metric (2.1) automatically defines an energy-momentum tensor of a fluid with anisotropic pressure, and the coor- dinates of (2.1) are comoving. The rotation and acceleration are zero, but if the source is to be a perfect fluid, then the shear is necessarily non-zero. It is well known that if the calculations are performed in quasi-Cartesian coordinates, all the energy-momentum complexes give meaningful results. According to the following transformations x2 + y2 + z2, φ = arctan( the line element (2.1) written in terms of quasi-Cartesian coordinates reads: dS2 = dt2+ (dx2+ dy2+ dz2)− (xdx+ ydy+ zdz) . (2.2) For the above metric the determinant of the metric tensor and the con- travariant components of the tensor are given, respectively, as follows det(g) = −A g00 = 1, g11 = x2 g12 = xy g13 = xz g22 = y2 g23 = yz g33 = z2 (2.3) 3 Einstein’s Energy-momentum Complex The energy-momentum complex as defined by Einstein [1] is given by θki = Hkli,l, (3.4) where the Einstein’s superpotential Hkli is of the form Hkli = −H [− g(gknglm − glngkm)],m. (3.5) and θ0α are the energy and momentum density components, respectively. The Einstein energy-momentum satisfies the local conservation law The energy and momentum in the Einstein’s prescription are given by ∫ ∫ ∫ θ0i dx 1dx2dx3. (3.6) Using the Gauss theorem we obtain H0αi nαds, (3.7) where nα = ( ) are the components of a normal vector over an in- finitesimal surface element ds = r2 sin θdθdφ. The required non zero components of Hkli for the line element (2.1) are given by A2r2 +B2 A2r2 +B2 A2r2 +B2 (ȦB −AḂ)− (ȦB +AḂ) = H02 = 2xyB (ȦB −AḂ), = H03 = 2xzB (ȦB −AḂ), (ȦB −AḂ)− (ȦB +AḂ) = H03 = 2yzB (ȦB −AḂ), (ȦB −AḂ)− (ȦB +AḂ) (3.8) Using the components (3.8) we obtain the components of energy and mo- mentum densities in the form 8πAr4 (A2r2 −B2), (ȦB +AḂ), (ȦB +AḂ), (ȦB +AḂ). (3.9) Using equations (3.8) in equation (3.7), the energy and momentum distri- butions are the following EEin = P0 = (A2r2 +B2), P1 = P2 = P3 = 0. We notice that if the signature of the space-time under study is changed from +2 to -2, we find that the values of energy and momentum densities as well as the energy distribution are changed from positive to negative. 4 The Energy-Momentum Complex of Bergmann- Thomson The Bergmann-Thomson energy-momentum complex [7] is given by [gilBkml ],m, (4.1) where Bkml = gkngmp − gmngkp B00 and B0α are the energy and momentum density components. The energy and momentum are given by P i = ∫ ∫ ∫ Bi0dx1dx2dx3. (4.2) Using the Gauss theorem we have P i = Bi0αnαdS. (4.3) In order to calculate the energy and momentum distributions for the space-time under consideration, using Bergmann-Thomson energy-momentum complex, we require the following non-vanishing components of Bkml A2r2 +B2 A2r2 +B2 A2r2 +B2 (ȦB −AḂ)− (ȦB +AḂ) = B02 (ȦB −AḂ), = B03 = 2xzB (ȦB −AḂ), (ȦB −AḂ)− (ȦB +AḂ) = B03 = 2yzB (ȦB −AḂ), (ȦB −AḂ)− (ȦB +AḂ) (4.4) Using the components (4.4) in (4.1), the components of energy and mo- mentum densities are as follows 00 = 1 8πAr4 (A2r2 −B2), 01 = − x Ȧr2 + B (2AḂ − ȦB 02 = − y Ȧr2 + B (2AḂ − ȦB 03 = − z Ȧr2 + B (2AḂ − ȦB (4.5) Using equations (4.4) in equation (4.3), we obtain the energy and momentum distributions in the following form EBerg = P (A2r2 +B2), P1 = P2 = P3 = 0. The above energy density and energy distribution are agreement with that obtained before, using Einstein’s energy-momentum complex. In the case of Bergmann’s energy-momentum complex, we notice that a signature switch do not affect about everything including energy distribution. Consequently, the Einstein and Bergmann-Thomson prescription do not give the same results when the signature of the space-time under study is -2. 5 Landau-Lifshitz’s Energy-momentum Complex Landau-Lifshitz’s energy-momentum complex [5] is given by Lij = , (5.1) where Sikjl = −g(gijgkl − gilgkj). (5.2) Lij is symmetric in its indices, L00 is the energy density and L0α are the momentum (energy current) density components. Sikjl has the symmetries of the Riemann curvature tensor. The energy and momentum are given by P i = ∫ ∫ ∫ Li0dx1dx2dx3. (5.3) Using the Gauss theorem we have P i = Si0αnαdS. (5.4) The required non-vanishing components of Sikjl are S0101 = B2 (A2r2 −B2)− A S0102 = B2(A2r2 −B2), S0103 = xz B2(A2r2 −B2), S0202 = B2 (A2r2 −B2)− A S0203 = B2(A2r2 −B2), S0303 = B2 (A2r2 −B2)− A (5.5) Using these components in equation (5.1), we obtained the energy and mo- mentum densities are L00 = − B [A2r2 + 3B2], L10 = − xB [A(AḂ + ȦB)r2 + 2B2Ḃ], L20 = − yB [A(AḂ + ȦB)r2 + 2B2Ḃ], L30 = − zB [A(AḂ + ȦB)r2 + 2B2Ḃ]. (5.6) Using equations (5.1) in equation (5.4), we obtain the energy and momentum distributions ELL = (A2r2 +B2). (5.7) P 1LL = P LL = P LL = 0. (5.8) The above results do not agree with the results obtained before, using Ein- stein and Bergmann-Thomson energy-momentum complexes. A signature switch does not affect about everything including energy distribution. 6 Papapetrou’s Energy-momentum Complex The symmetric energy-momentum complex of Papapetrou [3] is given by Ωij = , (6.1) where Υijkl = −g(gijηkl − gikηjl + gklηij − gjlηik), (6.2) and ηik is the Minkowski metric with signature +2. Ω00 and Ωα0 are the energy and momentum density components. The energy and momentum, using the Papapetrou prescription are given by P i = ∫ ∫ ∫ Ωi0dx1dx2dx3. (6.3) Using the Gauss theorem we obtain P i = Υi0αl,l nαdS. (6.4) The non-vanishing components of Υijkl are as follows Υ0011 = 1 (A2r2 −B2)−A2 Υ0012 = xy (A2r2 −B2), Υ0013 = xz (A2r2 −B2), Υ0022 = 1 (A2r2 −B2)−A2 Υ0023 = (A2r2 −B2), Υ0033 = 1 (A2r2 −B2)−A2 (6.5) Using these components in (6.1), we get the following energy and momentum density components Ω00 = A [r2 −B2], Ω10 = − x [r2Ȧ+B(ȦB + 2AḂ)], Ω20 = − y [r2Ȧ+B(ȦB + 2AḂ)], Ω30 = − z [r2Ȧ+B(ȦB + 2AḂ)]. (6.6) Using equations (6.5) in equation (6.4), we obtain the energy and momentum distributions (r2 +B2) (6.7) P 1 = P 2 = P 3 = 0 (6.8) The above results do not agree with the results obtained before, using Ein- stein, Bergmann-Thomson and Landau and Lifshitz energy-momentum com- plexes. A signature switch should affect about everything including energy distribution. Discussion We investigated the energy and momentum (due to matter plus fields in- cluding gravity) distribution of the Kantowski and Sachs space-time using the Einstein, Bergmann-Thomson, Landau-Lifshitz and Papapetrou energy- momentum complexes. We found that the quantities of energy and mo- mentum densities as well as energy distribution are well-defined and well- behaved. For the space-time under consideration, we found that the energy- momentum complexes of Einstein and Bergmann-Thomson give the same results, while Landau-Lifshitz and Papapetrou do not give the same results and not agree with the aforementioned complexes. We have shown that a signature switch affects about every thing (by changing the sign of the values of energy and momentum densities as well as energy distribution) including energy distribution. These changes occur in the case of Einstein and Pa- papetrou prescriptions but not in Bergmann-Thomson and Landau-Lifshitz prescriptions. References [1] A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 778 (1915). [2] R. C. Tolman, ”Relativity, Thermodynamics and Cosmology, (Oxford University Press, Oxford), p. 227 (1934). [3] A. Papapetrou, Proc. R. Ir. Acad. A52, 11 (1948). [4] C. Møller, Ann. Phys. (NY) 4, 347 (1958). [5] L. D. Landau and E. M. Lifshitz, ”The Classical Theory of Fields”, (Addison-Wesley Press, Reading, MA) p. 317 (1951). [6] S. Weinberg, ”Gravitation and Cosmology: Principles and Applications of General Theory of Relativity” ( Wiley, New York) 165 (1972). [7] P. G. Bergmann and R. Thompson, Phys. Rev. 89, 400 (1953). [8] K. S. Virbhadra, Phys. Rev D41, 1086 (1990).; K. S. Virbhadra, Phys. Rev. D42, 1066 (1990).; K. S. Virbhadra, Phys. Rev. D42, 2919 (1990).; K. S. Virbhadra, Pramana-J. Phys. 45, 215 (1995); N. Rosen, K.S. Virbhadra, Gen. Rel. Grav. 25, 429 (1993). [9] J. M. Aguirregabiria, A. Chamorro and K.S. Virbhadra, Gen. Relativ. Gravit. 28, 1393 (1996). [10] K. S. Virbhadra , Phys. Rev. D60, 104041 (1999). [11] S. S. Xulu, Astrophys.Space Sci. 283, 23 (2003). [12] R.M. Gad, Astrophys. Space Sci. 293, 453 (2004); R.M. Gad, Astro- phys. Space Sci. 295, 459 (2005); R.M. Gad, Mod. Phys. Lett. A19, 1847 (2004); R.M. Gad, Gen. Relativ. Gravit., 38, 417 (2006); R. M. 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Yang, FizikaB 14, 3 (2005); I. Radinschi, Rom. J. Phys. 50, 57 (2005); I. Radinski and Th. Grammenos, Int. J. Mod. Phys. A 21, 2853 (2006); I.-C. Yang, Chin. J. Phys. 38, 1040 (2000); O. Aydogdu and M. Salti, Prog. Theor. Phys. 115, 63 (2006); O. Aydogdu, M. Salti and M. Korunur, Acta Phys. Slov. 55, 537 (2005); M. Salti, Acta Phys. Slov. 55, 563 (2005); M. Salti, Nuovo Cim. 120B, 53 (2005); O. Aydogdu and M. Salti, Astrophys. Space Sci. 299, 227 (2005); M. Salti, Mod. Phys. Lett. A 20, 2175 (2005); M. Salti and A. Havare, Int. J. Mod. Phys. A 20, 2169 (2005); A. Havare, M. Korunur and M. Salti, Astrophys. Space Sci. 301, 43 (2006); O. Patashnick, Int. J. Mod. Phys. D 14, 1607 (2005); T. Grammenos, Mod. Phys. Lett. A 20, 1741 (2005). [13] C. W. Misner, K. S. Thorne and J. A. Wheeler, ”Gravitation” (Freeman W. H and Co., NY) p. 603 (1973). [14] F. I. Cooperstock and R. S. Sarracino, J. Phys. A 11 877 (1978). [15] H. Bondi, Proc. R. Soc. London A 427, 249 (1990). [16] F. I. Cooperstock, Found. Phys. 22, 1011 (1992); in Topics in Quantum Gravity and Beyond: Pepers in Honor of L, Witten eds. F. Mansouri and J. J. Scanio (World Scientific, Singapore, (1993)) 201; In Relativis- tic Astrophysics and Cosmology, eds. Buitrago et al. (World Scientific, Singapore, (1997))61; Annals Phys. 282, 115 (2000). [17] S. S. Xulu, Mod. Phys. Lett. A15, 1511; Astrophys. and Space Science 283, 23 (2003). [18] R. M. Gad, Astrophys. space Sci., 295, 451 (2005). [19] R. Kantowski and R. K. Sachs, J. Math. Phys. 7, 443 (1966). [20] C. B. Collins, J. Math. Phys. 18, 2116 (1977). [21] A. S. Kompaneets and Chernov A. S., Sov. Phys. JETP 20, 1303 (1965).

We use the Einstein, Bergmann-Thomson, Landau-Lifshitz and Papapetrou energy-momentum complexes to calculate the energy and momentum distributions of Kantowski and Sachs space-time. We show that the Einstein and Bergmann-Thomson definitions furnish a consistent result for the energy distribution, but the definition of Landau-Lifshitz do not agree with them. We show that a signature switch should affect about everything including energy distribution in the case of Einstein and Papapetrou prescriptions but not in Bergmann-Thomson and Landau-Lifshitz prescriptions.

Introduction One of the most interesting and intricate problems still unsolved since the outset of general relativity is the energy-momentum localization. Einstein himself proposed the first energy-momentum complex in an attempt to de- fine the local distribution of energy and momentum [1]. After this attempt, a plethora of different energy-momentum complexes were proposed, including formulations by Tolman [2], Papapetrou [3], Møller [4], Landau and Lifshitz [5], Weinberg [6], Bergmann-Thomson [7] and others. This approach was abandoned for a long time due to severe criticism for a number of reasons. Recently, Virbhadra re-opened the subject of energy-momentum com- plexes [8]. He pointed out that though these complexes are non-tensors, they yield reasonable and consistent results for a given space-time. Aguir- regabiria et al [9] found that for any metric of the Kerr-Schild class, sev- eral different definitions of the energy-momentum complex yield precisely the same results.Virbhadra [10] investigated whether or not these energy momentum complexes lead to the same results for the most general non- static spherically symmetric metric and found that they disagree. He noted that the energy-momentum complexes of Landau and Lifshitz, Papapetrou 1Email Address: ragab2gad@hotmail.com http://arxiv.org/abs/0704.1567v2 and Weinberg give the same results as in the Einstein definition if the cal- culations are performed in Kerr-Schild Cartesian coordinates. However, these energy-momentum complexes disagree if computations are done in ”Schwarzschild Cartesian” coordinates2. In a detail study of the question, Xulu [11] has confirmed this suggestion. He obtained the energy distribu- tion for the most general non-static spherically symmetric using Møller’s definition and found different results in general from those obtained us- ing Einstein’s definition. These results agree for the Schwarzschild, Vaidya and Janis-Newmann-Winicour space-times, but disagree for the Reissner- Nordström space-time. Many authors had similarly successfully applied the aforementioned energy-momentum complexes to various black hole configu- rations [12]. It has been remained a controversial problem whether or not energy and momentum are localizable. There are different opinions on this subject. Contradicting the viewpoint of Misner et al. [13] that the energy is local- izable only for spherical systems, Cooperstock and Sarracino [14] argued that if the energy localization is meaningful for spherical systems then it is meaningful for all systems. Bondi [15] expressed that a non-localizable form of energy is inadmissible in relativity and its location can in prin- ciple be found. These contradictory viewpoints bear significantly on the study of gravitational waves. It is an interesting question whether or not gravitational waves have energy and momentum content. In a series of pa- pers, Cooperstock [16] hypothesized that in a curved space-time energy and momentum are confined to the region of non-vanishing energy-momentum tensor T ab and consequently the gravitational waves are not carriers of energy and momentum in vacuum space-times. This hypothesis has neither been proved nor disproved. There are many results supporting this hypothesis (see for example, [17, 18]). In this paper we evaluate the energy and momentum distributions of the Kantowski and Sachs space-time, using Einstein, Bergmann-Thomson, Landau-Lifshitz and Papapetrou energy-momentum complexes. Through this paper we use G = 1 and c = 1 units and follow the conven- tion that Latin indices take value from 0 to 3 and Greek indices take value from 1 to 3. 2Schwarzschild metric in “Schwarzschild Cartesian coordinates” is obtained by trans- forming this metric (in usual Schwarzschild coordinates {t, r, θ, φ}) to {t, x, y, z} using x = r sin θ cosφ, y = r sin θ sinφ, z = r cos θ. 2 Kantowski and Sachs Space-time The standard representation of Kantowski and Sachs space-times are given by [19] dS2 = dt2 −A2(t)dr2 −B2(t)(d2θ + sin2 θd2φ), (2.1) where the functions A(t) and B(t) are to be determined from the field equa- tions. The solutions of the Einstein field equations for the above metric were con- sidered with dust source [19], but they were generalized (in fact earlier) to the general perfect fluid source [21]. From the geometrically point of view, this line element admits a four parameter continuous group of isometries which acts on space-like hypersur- face, and has no three parameter subgroup that would be simply transitive on the orbits (for more detailed description see Kantowski and Sachs [19] and Collins [20]). From the physical point of view, the metric (2.1) automatically defines an energy-momentum tensor of a fluid with anisotropic pressure, and the coor- dinates of (2.1) are comoving. The rotation and acceleration are zero, but if the source is to be a perfect fluid, then the shear is necessarily non-zero. It is well known that if the calculations are performed in quasi-Cartesian coordinates, all the energy-momentum complexes give meaningful results. According to the following transformations x2 + y2 + z2, φ = arctan( the line element (2.1) written in terms of quasi-Cartesian coordinates reads: dS2 = dt2+ (dx2+ dy2+ dz2)− (xdx+ ydy+ zdz) . (2.2) For the above metric the determinant of the metric tensor and the con- travariant components of the tensor are given, respectively, as follows det(g) = −A g00 = 1, g11 = x2 g12 = xy g13 = xz g22 = y2 g23 = yz g33 = z2 (2.3) 3 Einstein’s Energy-momentum Complex The energy-momentum complex as defined by Einstein [1] is given by θki = Hkli,l, (3.4) where the Einstein’s superpotential Hkli is of the form Hkli = −H [− g(gknglm − glngkm)],m. (3.5) and θ0α are the energy and momentum density components, respectively. The Einstein energy-momentum satisfies the local conservation law The energy and momentum in the Einstein’s prescription are given by ∫ ∫ ∫ θ0i dx 1dx2dx3. (3.6) Using the Gauss theorem we obtain H0αi nαds, (3.7) where nα = ( ) are the components of a normal vector over an in- finitesimal surface element ds = r2 sin θdθdφ. The required non zero components of Hkli for the line element (2.1) are given by A2r2 +B2 A2r2 +B2 A2r2 +B2 (ȦB −AḂ)− (ȦB +AḂ) = H02 = 2xyB (ȦB −AḂ), = H03 = 2xzB (ȦB −AḂ), (ȦB −AḂ)− (ȦB +AḂ) = H03 = 2yzB (ȦB −AḂ), (ȦB −AḂ)− (ȦB +AḂ) (3.8) Using the components (3.8) we obtain the components of energy and mo- mentum densities in the form 8πAr4 (A2r2 −B2), (ȦB +AḂ), (ȦB +AḂ), (ȦB +AḂ). (3.9) Using equations (3.8) in equation (3.7), the energy and momentum distri- butions are the following EEin = P0 = (A2r2 +B2), P1 = P2 = P3 = 0. We notice that if the signature of the space-time under study is changed from +2 to -2, we find that the values of energy and momentum densities as well as the energy distribution are changed from positive to negative. 4 The Energy-Momentum Complex of Bergmann- Thomson The Bergmann-Thomson energy-momentum complex [7] is given by [gilBkml ],m, (4.1) where Bkml = gkngmp − gmngkp B00 and B0α are the energy and momentum density components. The energy and momentum are given by P i = ∫ ∫ ∫ Bi0dx1dx2dx3. (4.2) Using the Gauss theorem we have P i = Bi0αnαdS. (4.3) In order to calculate the energy and momentum distributions for the space-time under consideration, using Bergmann-Thomson energy-momentum complex, we require the following non-vanishing components of Bkml A2r2 +B2 A2r2 +B2 A2r2 +B2 (ȦB −AḂ)− (ȦB +AḂ) = B02 (ȦB −AḂ), = B03 = 2xzB (ȦB −AḂ), (ȦB −AḂ)− (ȦB +AḂ) = B03 = 2yzB (ȦB −AḂ), (ȦB −AḂ)− (ȦB +AḂ) (4.4) Using the components (4.4) in (4.1), the components of energy and mo- mentum densities are as follows 00 = 1 8πAr4 (A2r2 −B2), 01 = − x Ȧr2 + B (2AḂ − ȦB 02 = − y Ȧr2 + B (2AḂ − ȦB 03 = − z Ȧr2 + B (2AḂ − ȦB (4.5) Using equations (4.4) in equation (4.3), we obtain the energy and momentum distributions in the following form EBerg = P (A2r2 +B2), P1 = P2 = P3 = 0. The above energy density and energy distribution are agreement with that obtained before, using Einstein’s energy-momentum complex. In the case of Bergmann’s energy-momentum complex, we notice that a signature switch do not affect about everything including energy distribution. Consequently, the Einstein and Bergmann-Thomson prescription do not give the same results when the signature of the space-time under study is -2. 5 Landau-Lifshitz’s Energy-momentum Complex Landau-Lifshitz’s energy-momentum complex [5] is given by Lij = , (5.1) where Sikjl = −g(gijgkl − gilgkj). (5.2) Lij is symmetric in its indices, L00 is the energy density and L0α are the momentum (energy current) density components. Sikjl has the symmetries of the Riemann curvature tensor. The energy and momentum are given by P i = ∫ ∫ ∫ Li0dx1dx2dx3. (5.3) Using the Gauss theorem we have P i = Si0αnαdS. (5.4) The required non-vanishing components of Sikjl are S0101 = B2 (A2r2 −B2)− A S0102 = B2(A2r2 −B2), S0103 = xz B2(A2r2 −B2), S0202 = B2 (A2r2 −B2)− A S0203 = B2(A2r2 −B2), S0303 = B2 (A2r2 −B2)− A (5.5) Using these components in equation (5.1), we obtained the energy and mo- mentum densities are L00 = − B [A2r2 + 3B2], L10 = − xB [A(AḂ + ȦB)r2 + 2B2Ḃ], L20 = − yB [A(AḂ + ȦB)r2 + 2B2Ḃ], L30 = − zB [A(AḂ + ȦB)r2 + 2B2Ḃ]. (5.6) Using equations (5.1) in equation (5.4), we obtain the energy and momentum distributions ELL = (A2r2 +B2). (5.7) P 1LL = P LL = P LL = 0. (5.8) The above results do not agree with the results obtained before, using Ein- stein and Bergmann-Thomson energy-momentum complexes. A signature switch does not affect about everything including energy distribution. 6 Papapetrou’s Energy-momentum Complex The symmetric energy-momentum complex of Papapetrou [3] is given by Ωij = , (6.1) where Υijkl = −g(gijηkl − gikηjl + gklηij − gjlηik), (6.2) and ηik is the Minkowski metric with signature +2. Ω00 and Ωα0 are the energy and momentum density components. The energy and momentum, using the Papapetrou prescription are given by P i = ∫ ∫ ∫ Ωi0dx1dx2dx3. (6.3) Using the Gauss theorem we obtain P i = Υi0αl,l nαdS. (6.4) The non-vanishing components of Υijkl are as follows Υ0011 = 1 (A2r2 −B2)−A2 Υ0012 = xy (A2r2 −B2), Υ0013 = xz (A2r2 −B2), Υ0022 = 1 (A2r2 −B2)−A2 Υ0023 = (A2r2 −B2), Υ0033 = 1 (A2r2 −B2)−A2 (6.5) Using these components in (6.1), we get the following energy and momentum density components Ω00 = A [r2 −B2], Ω10 = − x [r2Ȧ+B(ȦB + 2AḂ)], Ω20 = − y [r2Ȧ+B(ȦB + 2AḂ)], Ω30 = − z [r2Ȧ+B(ȦB + 2AḂ)]. (6.6) Using equations (6.5) in equation (6.4), we obtain the energy and momentum distributions (r2 +B2) (6.7) P 1 = P 2 = P 3 = 0 (6.8) The above results do not agree with the results obtained before, using Ein- stein, Bergmann-Thomson and Landau and Lifshitz energy-momentum com- plexes. A signature switch should affect about everything including energy distribution. Discussion We investigated the energy and momentum (due to matter plus fields in- cluding gravity) distribution of the Kantowski and Sachs space-time using the Einstein, Bergmann-Thomson, Landau-Lifshitz and Papapetrou energy- momentum complexes. We found that the quantities of energy and mo- mentum densities as well as energy distribution are well-defined and well- behaved. For the space-time under consideration, we found that the energy- momentum complexes of Einstein and Bergmann-Thomson give the same results, while Landau-Lifshitz and Papapetrou do not give the same results and not agree with the aforementioned complexes. We have shown that a signature switch affects about every thing (by changing the sign of the values of energy and momentum densities as well as energy distribution) including energy distribution. These changes occur in the case of Einstein and Pa- papetrou prescriptions but not in Bergmann-Thomson and Landau-Lifshitz prescriptions. References [1] A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 778 (1915). [2] R. C. Tolman, ”Relativity, Thermodynamics and Cosmology, (Oxford University Press, Oxford), p. 227 (1934). [3] A. Papapetrou, Proc. R. Ir. Acad. A52, 11 (1948). [4] C. Møller, Ann. Phys. (NY) 4, 347 (1958). [5] L. D. Landau and E. M. 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Green functions and nonlinear systems: Short time expansion Marco Frasca∗ Via Erasmo Gattamelata, 3 00176 Roma (Italy) (Dated: February 1, 2008) We show that Green function methods can be straightforwardly applied to nonlinear equations appearing as the leading order of a short time expansion. Higher order corrections can be then computed giving a satisfactory agreement with numerical results. The relevance of these results relies on the possibility of fully exploiting a gradient expansion in both classical and quantum field theory granting the existence of a strong coupling expansion. Having a Green function in this regime in quantum field theory amounts to obtain the corresponding spectrum of the theory. I. INTRODUCTION Nonlinear equations represent a class of very difficult mathematical problems to manage by analytical methods. A lot of fundamental aspects of physics are described by these equations making not easy their understanding due to the lack of useful techniques. In this paper we present an approach that is based on an unexpected result for Green functions. First hints in this direction were obtained in [1, 2] where we showed that for a part of the integration interval, a nonlinear differential equation like φ̈ + φ3 = j, being j a source term, can be solved by its Green function G̈ + G3 = δ(t) as φ(t) ≈ G(t − t′)j(t′)dt′. Although this solution was put forward, the knowledge of this result is hardly useful unless we are not able to understand how to get higher order corrections. The aim of this paper is to give a proper understanding of this solution and to give a technique to get higher order corrections in order to improve it. We will show that it represents a short time solution. Then, a form factor described by polynomial terms in time can correct properly the propagator to improve in some cases this approximation. The reason to give such a solution relies on the possibility to treat strong coupled quantum field theories that at the leading order produce nonlinear equations driven by a source. Strongly coupled theory can be managed by a gradient expansion that is the dual perturbation series to a weak coupling expansion as we proved in [1, 3, 4]. By “dual” we mean that two series can be obtained by simply interchanging the terms of the expansion producing in a case a series with an expansion parameter being the inverse of the expansion parameter of the other series, the asymptotic series so obtained holding in the proper limit where this parameter gives a converging expansion (coupling going to infinity in a case while going to zero in the other). This approach is true for any differential equation set and we applied it also in general relativity [3] obtaining a sound proof of the Belinski-Khalatnikov-Lifshitz conjecture [5, 6, 7] as this is a result of a gradient expansion. A strongly coupled system in quantum mechanics is known to be a classical system as was firstly shown by Simon [8]. We revised this approach in [9] where we have seen that the gradient expansion for the Schrödinger equation, also known as Wigner-Kirkwood expansion, gives rise to a Thomas-Fermi approximation to the leading order for a many-body system [10] and has the same eigenvalue expansion as for a WKB approximation. Wigner-Kirkwood expansion is indeed the gradient expansion of the Schrödinger equation. Gradient expansions in quantum field theories were not widely used before while their proper understanding is not that easy. Our aim in this paper is to fully exploit this perturbation approach and its application in quantum field theory wherever possible. This method may pave the way to manage analytically some problems that now appear difficult to manage also in a wide variety of fields where nonlinear equations are at the foundations. The paper is structured in the following way. In section II we show how to derive a gradient expansion out of a duality principle in perturbation theory with the proper understanding of the expansion parameter. In section III we present the main motivation for this paper, that is the continuum limit of a scalar quantum field theory giving rise to a model nonlinear equation we will use throughout the paper. In section IV we present the method firstly applied to a simple Riccati equation having a known analytical solution and then we generalize our method to the case of the leading order equation of a scalar field theory. In section V we give the numerical results showing how the marcofrasca@mclink.it http://arXiv.org/abs/0704.1568v3 mailto:marcofrasca@mclink.it approximation improves with higher order corrections varying also the forcing term into the equation. In section VI we compare our approach with functional iteration method, a well known method used to analyze non-linear differential equations. This will give a proper understanding of the speed of convergence of our method. Finally, in section VII conclusions are given. II. DUAL EXPANSION FOR NONLINEAR PDES In order to make the paper self-contained we present here some material already given in Ref.[4]. We specialize the presentation to a λφ4 model that is our reference model. The Hamiltonian of the model is given by dD−1x (∇φ)2 + V (φ) being D the spacetime dimensionality and V (φ) = 1 φ2 + λ φ4 and we take the case of a single component for the sake of simplicity. Hamilton equations are ∂tφ = π (2) ∂tπ = ∇2φ − φ − λφ3. We can see at glance that we can chose to do perturbation theory by two different choices. One can take either λφ3 or ∇2φ − φ as a small term. What we want to understand is the link between the two series with respect to the parameter λ. By choosing λφ3 as a small term one gets the small perturbation series ∂tφ0 = π0 (3) ∂tφ1 = π1 ∂tφ2 = π2 ∂tπ0 = ∇2φ0 − φ0 ∂tπ1 = ∇2φ1 − φ1 − φ30 ∂tπ2 = ∇2φ2 − φ2 − 3φ20φ1 where it easily seen that the free theory, �φ0 + φ0 = 0, is the leading order solution. Our aim is to derive a dual perturbation series to this one meaning by this that we want a series with a development parameter going as 1 In order to reach our aim, following the principle of duality in perturbation theory [11] we put λt (4) π2 + . . . φ = φ0 + φ2 + . . . . The following non trivial set of equations is obtained ∂τφ0 = π0 (5) ∂τφ1 = π1 ∂τφ2 = π2 ∂τπ0 = −φ30 ∂τπ1 = ∇2φ0 − φ0 − 3φ20φ1 ∂τπ2 = ∇2φ1 − φ1 − 3φ0φ21 − 3φ20φ2 whose solution proves the existence of a dual perturbation series for the classical λφ4 theory. We easily realize that this set of equations would have been obtained if one takes as a small term ∇2φ − φ giving rise in this case to a gradient expansion, that is a series having derivatives in space as small terms. So, strong coupling expansion and weak coupling expansion are related by the duality principle in perturbation theory [11] producing in the former case a gradient expansion. This result can be easily generalized to any kind of PDE [3, 4]. The point to be noted is that to have an analytical result for a strong coupling expansion we have to solve a nonlinear differential equation that in this case is given by ∂2τφ0 + φ 0 = 0. (6) Things can be more involved when a source term is present as is generally the case in quantum field theory and a meaning should be attached to the leading order equation ∂2τφ0 + φ 0 = j. (7) being j a source term. The aim of this paper is to show that indeed an approach through Green functions is applicable in these cases, that is, as already shown by numerical methods in [1, 2], a first approximated solution is given by dτ ′G(τ − τ ′)j(τ ′) (8) being G(τ) the Green function solving the equation ∂2τG(τ) + G(τ) 3 = δ(τ). (9) We will give in this paper a general approach to compute higher order corrections to this result. We will note that the method can be applicable when a solution is known to an equation like τG(τ) + F (G(τ)) = aδ(τ) (10) being a a constant and F (G(τ)) a generic term. Otherwise we are not able to get analytical results and we have to resort to numerical methods. Anyhow, the situation is favorable for the most common models. III. GRADIENT EXPANSION AND QUANTUM FIELD THEORY Quantum field theory of the model we are considering is given by the generating functional Z[j] = [dφ]e{i (∂tφ) 4+jφ]}e{−i (∇φ)2+ 1 2]} (11) that we have written separating the spatial part from the rest. We did this in order to derive the strong coupling expansion to this case as already done in sec.II for the classical model. By doing the expansion, considering the gradient term 1 (∇φ)2 + 1 φ2 as small, the leading order term to be computed is Z0[j] = [dφ]e{i (∂tφ) 4+jφ]} (12) and in the end we are left with the equation to solve ∂2t φ + λφ 3 = j (13) that is the leading order of our gradient expansion as already seen in sec.II. The applicability of the Green function method implies that also in a strong coupling regime one can obtain information on the spectrum of the theory in this limit. We can exploit this point easily for a our case. Firstly, we use the mass µ0 of the theory to make all adimensional putting x → µ0x, φ2 → µ2−D0 φ2 and introducing the coupling constant g = λ . Then, let us consider the equation ∂2t G + gG 3 = δ(t) (14) that has solution [1] G(t) = θ(t) being θ(t) the Heaviside function and sn a Jacobi elliptical function. Being the equation second order we have that also the time reversed solution holds. It is known [12] that the following series holds for this Jacobi function sn(u, i) = (−1)ne−(n+ 12 )π 1 + e−(2n+1)π (2n + 1) 2K(i) being K(i) = 1+sin2 θ ≈ 1.3111028777 a constant. Then the mass spectrum of the theory in the limit of a very large g is given by En = (2n + 1) 2K(i) 4 µ0 that we can recognize as the one of a harmonic oscillator. So, the main physical motivation to study our approach through Green functions for nonlinear equations is to have a deeper understanding in quantum field theories but the method is rather general and could find applications in a lot of other fields. IV. GREEN FUNCTION METHOD FOR NONLINEAR DIFFERENTIAL EQUATIONS In order to make our approach as clearer as possible, we consider the trivial problem of a Riccati equation ẏ + y2 = 1 (17) with the initial condition y(0) = 0. The solution is given by y(t) = tanh(t). A Green function is easy to compute for this equation being given by G(t) = θ(t) 1 + t . (18) So, let us consider the following small time expansion as a solution of the above Riccati equation y(t) ≈ 1 + t − t′ 1 + t − t′ (t − t′) (19) 1 + t − t′ (t − t′)2 + c 1 + t − t′ (t − t′)3 + . . . being a, b and c constants to be computed. In order to compute these constants we consider the equation we started with and compute all the derivatives till the order we are interested in. Then, we compare these derivatives with the one obtained through equation (19) fixing in this way the values of the constants to make them equal. So, from eq.(19) one gets y(t) ≈ ln(t + 1) + a[− ln(t + 1) + t] + b[ln(t + 1) + t − t] + c[− ln(t + 1) + t + t − t ] + . . . (20) and from this we can compute y(0), ẏ(0), ÿ(0) and so on. From the Riccati equation we have y(0) = 0, ẏ(0) = 1, ÿ(0) = 0 and so on giving finally a = 1, b = −1 and c = −1 for our case yielding y(t) = t − t that are the first two terms of the Taylor series of the tanh(t), the exact solution of the equation. From this exercise we learn that the series (19) is a small time series solution of the original equation and that the convergence may be really slow. It is a rather interesting aspect of this approach that the Green function method has such a way to be applied to nonlinear equations. The case we considered here is a rather trivial one but things are made more interesting for the case of a λφ4 when we go to a numerical comparison. We want to apply the above approach to the case of equation (13). So, let us seek a solution in the form (properly normalized by µ0) φ(t) ≈ (t − t′), i j(t′)dt′ (21) (t − t′), i (t − t′)4j(t′)dt′ (t − t′), i (t − t′)6j(t′)dt′ + . . . 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Numerical 2 terms 1 term FIG. 1: Comparison for a driving source j(t) = sin(2πt). 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Numerical 2 terms 3 terms 1 term FIG. 2: Comparison for a driving source j(t) = exp(−t). and after computing derivatives of this equation and eq.(13) we get easily a = g and b = − g [j(0)]2. This gives the result we aimed for. We have got the proper expansion by Green function method of a solution to a nonlinear differential equation. What we want to see is how good is this approximation when compared to numerical results. This will be shown in the following section. V. NUMERICAL RESULTS In order to verify the quality of our approximation we solve the equation (13) for two different driving sources and take the coupling constant g = 1. Firstly we considered j(t) = sin(2πt) and the results are given in fig.1. In this case we can only have a first order correction as j(0) = 0. The agreement is very satisfactory till the end of the integration interval. The second case we considered j(t) = e−t permits to introduce another correction term but we notice no significant improvement due to the slow convergence of the approximation as can be seen from fig.2. The quality of the approximation depends on j(t) that can make very demanding the need for higher order correc- tions. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Numerical Functional Iteration Green Function FIG. 3: Comparison between our approach and functional iteration method. VI. COMPARISON WITH OTHER METHODS There exist different techniques to manage non-linear differential equations. In order to have a proper comparison we have to limit our analysis to small time methods. In this sense, the most similar approach to ours is the functional iteration method [13]. This method proves to have a rapid convergence to a good approximant of the true solution when the equation is not stiff. This is exactly our case. So, let us consider the equation φ̈(t) + φ3(t) = sin(2πt) (22) where dot means a time derivative. Functional iteration method implies that we solve the above equation iteratively. We assume φ(0) = 0 and φ̇(0) = 0. We take as zero order iterate φ0(t) = φ(0) = 0 and then for the successive iterates we have φ̈ν+1(t) = −φ3ν(t) + sin(2πt) (23) starting with ν = 0. Already at the second iterate we get a very good approximation to the true solution in the range we are considering. Then, we can compare this approximation with our method considering two terms. The results are presented in fig.3. This result shows that functional iteration method has a faster convergence and at least another term should be computed with our approach to reach an identical precision in the required range. This result should also be expected on the ground of efficiency of iterative methods with respect to series solutions. So, in order to decide the proper method to use one should properly analyze the problem at hand. VII. CONCLUSIONS We have shown an approximation method to solve nonlinear differential equations using Green function methods. This method proves to be a small time expansion and the convergence in some cases may turn out really slow. The main point to be emphasized is the unexpected utility of this approach generally assumed to hold only for linear differential equations. This implies that a gradient expansion for nonlinear PDE can also be applied successfully and a quantum field theory obtained. In this latter case one should consider that a gradient expansion is a strong coupling expansion and then, information in this regime of the corresponding quantum field theory is given. This yields another method to approach these problems generally very difficulty to manage with analytical methods. [1] M. Frasca, Phys. Rev. D 73, 027701 (2006); Erratum-ibid., 049902 (2006). [2] M. Frasca, Mod. Phys. Lett. A 22, 1293 (2007). [3] M. Frasca, Int. J. Mod. Phys. D 15, 1373 (2006). [4] M. Frasca, Int. J. Mod. Phys. A 22, 1441 (2007). [5] I. M. Kalathnikov, and E. M. Lifshitz, Phys. Rev. Lett. 24, 76 (1970). [6] V. A. Belinski, I. M. Kalathnikov, and E. M. Lifshitz, Adv. Phys. 19, 525 (1970). [7] V. A. Belinski, I. M. Kalathnikov, and E. M. Lifshitz, Adv. Phys. 31, 639 (1982). [8] B. Simon, Functional Integration and Quantum Physics, (AMS, Providence, 2005). [9] M. Frasca, Proc. R. Soc. A 463, 2195 (2007). [10] P. Ring and P. Schuck, The Nuclear Many-Body Problem, (Springer, Berlin, 1980). [11] M. Frasca, Phys. Rev. A 58, 3439 (1998). [12] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, 2000). [13] D. Zwillinger, Handbook of Differential Equations, (Academic Press, San Diego, 1989). Introduction Dual expansion for nonlinear PDEs Gradient expansion and quantum field theory Green function method for nonlinear differential equations Numerical results Comparison with other methods Conclusions References

We show that Green function methods can be straightforwardly applied to nonlinear equations appearing as the leading order of a short time expansion. Higher order corrections can be then computed giving a satisfactory agreement with numerical results. The relevance of these results relies on the possibility of fully exploiting a gradient expansion in both classical and quantum field theory granting the existence of a strong coupling expansion. Having a Green function in this regime in quantum field theory amounts to obtain the corresponding spectrum of the theory.

Introduction Dual expansion for nonlinear PDEs Gradient expansion and quantum field theory Green function method for nonlinear differential equations Numerical results Comparison with other methods Conclusions References

One-way permutations, computational asymmetry and distortion Jean-Camille Birget ∗ October 30, 2018 Abstract Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of one-way transformations. We introduce a computational asymmetry function that measures the amount of one-wayness of permutations. We also introduce the word-length asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to word-length. Moreover, boolean circuits have a representation in terms of elements of a Thompson group, in such a way that circuit size is polynomially equivalent to word-length. We show that circuits built with gates that are not constrained to have fixed-length inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixed-length inputs and outputs). Finally, we show that the computational asymmetry function is closely related to certain distortion functions: The computational asymmetry function is polynomially equivalent to the distortion of the path length in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of certain Thompson monoids. We also show that the results of Razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions. 1 Introduction The existence of one-way functions, i.e., functions that are “easy to evaluate” but “hard to invert”, is a major open problem. Much of cryptography depends on one-way functions; moreover, indirectly, their existence is connected to the question whether P is different from NP. In this paper we give some connections between these questions and some group-theoretic concepts: (1) We continue the work of [7], [8], and [9], on the relation between combinational circuits, on the one hand, and Thompson groups and monoids on the other hand. We give a representation of any circuit by a word over the Thompson group, such that circuit size is polynomially equivalent to word-length. (2) We establish connections between the existence of one-way permutations and the distortion func- tion in a certain Thompson group. Distortion is an important concept in metric spaces (e.g., Bourgain [10]) and in combinatorial group theory (e.g., Gromov [17], Farb [14]). Overview: Subsections 1.1 - 1.6 of the present Section define and motivate the concepts used: One-way functions and one-way permutations; computational asymmetry; word-length asymmetry; reversible computing; distortion; Thompson groups and monoids. In Section 2 we show that circuits can be represented by elements of Thompson monoids: A boolean circuit is equivalent to a word over a fixed generating set ∗Supported by NSF grant CCR-0310793. Some of the results of this paper were presented at the AMS Section Meeting, Oct. 21-23, 2005, Lincoln, Nebraska (http://www.ams.org/amsmtgs/2117 program.html), and at the conference “Various Faces of Cryptography”, 10 Nov. 2006 at City College of CUNY, New York. http://arxiv.org/abs/0704.1569v1 http://www.ams.org/amsmtgs/2117_program.html of a Thompson monoid, with circuit size being equal (or linearly equivalent) to word-length over the generating set. The Thompson monoids that appear here are monoid generalizations of the Thompson group G2,1, obtained when bijections are generalized to partial functions [9]. Section 3 shows that computational asymmetry and word-length asymmetry (for the Thompson groups and monoids) are linearly related. In Section 4 we give a representation of arbitrary (not necessarily bijective) circuits by elements of the Thompson group G2,1; circuit size is polynomially equivalent to word-length over a certain generating set in the Thompson group. In Section 5 we show that the computational asymmetry function of permutations is polynomially related to a certain distortion in a Thompson group. Section 6 contains miscellaneous results, in particular that the work of Razborov and others on monotone circuit complexity leads to exponential lower bounds on certain distortion functions. 1.1 One-way functions and one-way permutations Intuitively, a one-way function is a function f (mapping words to words, over a finite alphabet), such that f is “easy to evaluate” (i.e., given x0 in the domain, it is “easy” to compute f(x0)), but “hard to invert” (i.e., given y0 in the range, it is “hard” to find any x0 such that f(x0) = y0). The concept was introduced by Diffie and Hellman [13]. There are many ways of defining the words “easy” and “hard”, and accordingly there exist many different rigorous notions of a one-way function, all corresponding to a similar intuition. It remains an open problem whether one-way functions exist, for any “reasonable” definition. Moreover, for certain definitional choices, this problem is a generalization of the famous question whether P 6= NP [16, 34, 11]. We will base our one-way functions on combinational circuits and their size. The size of a circuit will also be called its complexity. Below, {0, 1}n (for any integer n ≥ 0) denotes the set of all bitstrings of length n. A combinational circuit with input-output function f : {0, 1}m → {0, 1}n is an acyclic boolean circuit with m input wires (or “input ports”) and n output wires (or “output ports”). The circuit is made from gates of type not, and, or, fork, as well as wire-crossings or wire-swappings. These gates are very traditional and are defined as follows. and: (x1, x2) ∈ {0, 1}2 7−→ y ∈ {0, 1}, where y = 1 if x1 = x2 = 1, and y = 0 otherwise. or: (x1, x2) ∈ {0, 1}2 7−→ y ∈ {0, 1}, where y = 0 if x1 = x2 = 0, and y = 1 otherwise. not: x ∈ {0, 1} 7−→ y ∈ {0, 1}, where y = 0 if x = 1, y = 1 otherwise. fork: x ∈ {0, 1} 7−→ (x, x) ∈ {0, 1}2. Another gate that is often used is the exclusive-or gate, xor: (x1, x2) ∈ {0, 1}2 7−→ y ∈ {0, 1}, where y = 1 if x1 6= x2, and y = 0 otherwise. The wire-swapping of the ith and jth wire (i < j) is described by the bit transposition (or bit position transposition) τi,j : uxivxjw ∈ {0, 1}ℓ 7−→ uxjvxiw ∈ {0, 1}ℓ, where |u| = i− 1, |v| = j − i− 1, |w| = ℓ− j − 1. The fork and wire-swapping operations, although heavily used, are usually not explicitly called “gates”; but because of their important role we will need to consider them explicitly. Other notations for the gates: and(x1, x2) = x1 ∧ x2, or(x1, x2) = x1 ∨ x2, not(x) = x, xor(x1, x2) = x1 ⊕ x2. A combinational circuit for a function f : {0, 1}m → {0, 1}n is defined by an acyclic directed graph drawn in the plane (with crossing of edges allowed). In the circuit drawing, the m input ports are vertices lined up in a vertical column on the left end of the circuit, and the n output ports are vertices lined up in a vertical column on the right end of the circuit. The input and output ports and the gates of the circuit (including the fork gates, but not the wire transpositions) form the vertices of the circuit graph. We often view the circuit as cut into vertical slices. A slice can be any collection of gates and wires in the circuit such that no gate in a slice is an ancestor of another gate in the same slice, and no wire in a slice is an ancestor of another wire in the same slice (unless these two wires are an input wire and an output wire of a same gate). Two slices do not overlap, and every wire and every gate belongs to some slice. For more details on combinational circuits, see [32, 43, 11]. The size of a combinational circuit is defined to be the number of gates in the circuit, including forks and wire-swappings, as well as the input ports and the output ports. For a function f : {0, 1}m → {0, 1}n, the circuit complexity (denoted C(f)) is the smallest size of any combinational circuit with input-output function f . A cause of confusion about gates in a circuit is that gates of a certain type (e.g., and) are tradi- tionally considered the same, no matter where they occur in the circuit. However, gates applied to different wires in a circuit are different functions; e.g., for the and gate, (x1, x2, x3) 7→ (x1 ∧ x2, x3) is a different function than (x1, x2, x3) 7→ (x1, x2 ∧ x3). 1.2 Computational Asymmetry Computational asymmetry is the core property of one-way functions. Below we will define computa- tional asymmetry in a quantitative way, and in a later Section we will relate it to the group-theoretic notion of distortion. For the existence of one-way functions, it is mainly the relation between the circuit complexity C(f) of f and the circuit complexity C(f−1) of f−1 that matters, not the complexities of f and of f−1 themselves. Indeed, a classical padding argument can be used: If we add C(f) “identity wires” to a circuit for f , then the resulting circuit has linear size as a function of its number of input wires; see Proposition 1.2 below. (An identity wire is a wire that goes directly from an input port to an output port, without being connected to any gate.) In [11] (page 230) Boppana and Lagarias considered logC(f ′)/logC(f) as a measure of one- wayness; here, f ′ denotes an inverse of f , i.e., any function such that f ◦ f ′ ◦ f = f . Massey and Hiltgen [25, 19] introduced the phrases complexity asymmetry and computational asymmetry for injec- tive functions, in reference to the situation where the circuit complexities C(f) and C(f−1) are very different. The concept of computational asymmetry can be generalized to arbitrary (non-injective) functions, with the meaning that for every inverse f ′ of f , C(f) and C(f ′) are very different. In [25] Massey made the following observation. For any large-enough fixed m and for almost all permutations f of {0, 1}m, the circuit complexities C(f) and C(f−1) are very similar: C(f) ≤ C(f−1) ≤ 10 C(f) Massey’s proof is adapted from the Shannon lower bound [35] and the Lupanov upper bound [23] (see also [19], [32]), from which it follows that almost all functions and almost all permutations (and their inverses) have circuit complexity close to the Shannon bounds. Massey’s observation can be extended to the set of all functions f : {0, 1}m → {0, 1}n, i.e., for almost all f and for every inverse f ′ of f , the complexities C(f) and C(f ′) are within constant factors of each other. Hence, computationally asymmetric permutations are rare among the boolean permutations overall (and similarly for functions). This is an interesting fact about computational asymmetry, but by itself it does not imply anything about the existence or non-existence of one-way functions, not even heuristically. Indeed, Massey proved his linear relation C(f) = Θ(C(f ′)) in the situation where C(f) = Θ(2m), and then uses the fact that the condition C(f) = Θ(2m) holds for almost all boolean permutations and for almost all boolean functions. But there also exist functions with C(f) = O(mk), with k a small constant. In particular, one-way functions (if they exist) have small circuits; by definition, one-way functions violate the condition C(f) = Θ(2m). A well-known candidate for a one-way permutation is the following. For a large prime number p and a primitive root r modulo p, consider the map x ∈ {0, 1, . . . , p−2} 7−→ rx−1 ∈ {0, 1, . . . , p−2}. This is a permutation whose inverse, known as the discrete logarithm, is believed to be difficult to compute. Measuring computational asymmetry: Let S{0,1}m denote the set of all permutations of {0, 1}m, i.e., S{0,1}m is the symmetric group. We will measure the computational asymmetry of all permutations of {0, 1}m (for all m > 0) by defining a computational asymmetry function, as follows. A function a : N → N is an upper bound on the computational asymmetry function iff for all all m > 0 and all permutations f of {0, 1}m we have: C(f−1) ≤ a . The computational asymmetry function α of the boolean permutations is the least such function a(.). Hence: Definition 1.1 The computational asymmetry function α of the boolean permutations is defined as follows for all s ∈ N : α(s) = max C(f−1) : C(f) ≤ s, f ∈ S{0,1}m , m > 0 Note that in this definition we look at all combinational circuits, for all permutations in m>0 S{0,1}m ; we don’t need to work with non-uniform or uniform families of circuits. Computational asymmetry is closely related to one-wayness, as the next proposition shows. Proposition 1.2 (1) For infinitely many n we have: There exists a permutation fn of {0, 1}n such that fn is computed by a circuit of size ≤ 3n, but f−1n has no circuit of size < α(n). (2) Suppose that α is exponential, i.e., there is k > 1 such that for all n, α(n) ≥ kn. Then k ≤ 2, and there is a constant c > 1 such that we have: For every integer n ≥ 1 there exists a permutation Fn of {0, 1}n which is computed by a circuit of size ≤ c n, but F−1n has no circuit of size < kn. Proof. (1) By the definition of α, for every m > 0 there exists a permutation F of {0, 1}m such that F is computed by a circuit of some size CF , but F −1 has no circuit of size < α(CF ). Let n = CF , and let us consider the function fn : {0, 1}CF → {0, 1}CF defined by fn : (x,w) 7−→ (F (x), w), for all x ∈ {0, 1}m and w ∈ {0, 1}CF−m. Then fn(x,w) is computed by a circuit of size CF + 2 (CF − m); the term “2 (CF −m)” comes from counting the input-output wires of w. Hence fn has a circuit of size ≤ 3n. On the other hand, (y,w) 7−→ f−1n (y,w) = (F−1(y), w) is not computed by any circuit of size < α(CF ), so f−1n has no circuit of size < α(n). (2) For every n ≥ 1 there exists a permutation F of {0, 1}n such that F is computed by a circuit of some size CF , and F −1 has a circuit of size CF−1 = α(CF ) ≥ kCF ; moreover, F−1 has no circuit of size < α(CF ). Thus, k CF ≤ CF−1 ≤ 2n (1 + co lognn ), for some constant co > 1; the latter inequality comes from the Lupanov upper bound [23] (or see Theorem 2.13.2 in [32]). Hence, k ≤ 2 and n ≤ CF ≤ 1log2 k n + c1 , for some constant c1 > 0. Hence, for all n ≥ 1 there exists a permutation F of {0, 1}n with circuit size CF ∈ [n, 1log2 k · n+ c1 · ], such that CF−1 = α(CF ) ≥ kCF ≥ kn. ✷ We will show later that the computational asymmetry function is closely related to the distortion of certain groups within certain monoids. Remarks: Although in this paper we only use the computational asymmetry function of the boolean permuta- tions, the concept can be generalized. Let Inj({0, 1}m, {0, 1}n) denote the set of all injective functions {0, 1}m → {0, 1}n. The computational asymmetry function αinj of the injective boolean functions is defined by αinj(s) = max C(f−1) : C(f) ≤ s, f ∈ Inj({0, 1}m, {0, 1}n), m > 0, n > 0 More generally, let ({0, 1}n){0,1}m denote the set of all functions {0, 1}m → {0, 1}n. The computa- tional asymmetry of all finite boolean functions is defined by αfunc(s) = max C(f ′) : C(f) ≤ s, ff ′f = f, f, f ′ ∈ ({0, 1}n){0,1}m , n > 0,m > 0 When we compare functions we will be mostly interested in their asymptotic growth pattern. Hence we will often use the big-O notation, and the following definitions. By definition, two functions f1 : N → N and f2 : N → N are linearly equivalent iff there are constants c0, c1, c2 > 0 such that for all n ≥ c0 : f1(n) ≤ c1 f2(c1n) and f2(n) ≤ c2 f1(c2n). Notation: f1 ≃lin f2. Two functions f1 and f2 (from N to N) are called polynomially equivalent iff there are constants c0, c1, c2, d, e > 0 such that for all n ≥ c0 : f1(n) ≤ c1 f2(c1nd)d and f2(n) ≤ c2 f1(c2ne)e. Notation: f1 ≃poly f2. 1.3 Wordlength asymmetry We introduce an algebraic notion that looks very similar to computational asymmetry: Definition 1.3 Let G be a group, let M be a monoid with generating set Γ (finite or infinite), and suppose G ⊆M . The word-length asymmetry function of G within M (over Γ) is λ(n) = max{ |g−1|Γ : |g|Γ ≤ n, g ∈ G}. The word-length asymmetry function λ depends on G, M , Γ, and the embedding of G in M . Consider the right Cayley graph of the monoid M with generating set Γ; its vertex set is M and the edges have the form x γ−→ γx (for x ∈ M , γ ∈ Γ). For x, y ∈ M , the directed distance d(x, y) in the Cayley graph is the shortest length over all paths from x to y in the Cayley graph; if no path from x to y exists, the directed distance is infinite. By “path” we always mean directed path. Lemma 1.4 Under the above conditions on G, M , Γ, we have for every g ∈ G : d(1, g−1) = d(g,1) and d(1, g) = d(g−1,1). Proof. Let η : Γ∗ →M be the map that evaluates generator sequences in M . If v ∈ Γ∗ is the label of a shortest path from 1 to g−1 in the Cayley graph then g ·η(v) = 1 inM , hence η(v) = g−1. Therefore, the path starting at g and labeled by v ends at 1; hence d(g,1) ≤ |v| = d(1, g−1). In a similar way one proves that d(1, g−1) ≤ d(g,1). The equality d(1, g) = d(g−1,1) is also proved in a similar way. Since |g|Γ is the distance d(1, g) in the graph of M , and since |g−1|Γ = d(1, g−1) = d(g,1), the word-length asymmetry also measures the asymmetry of the directed distance, to or from the identity element 1 in the Cayley graph of M , restricted to vertices in the subgroup G. For distances to or from the identity element of M it does not matter whether we consider the left Caley graph or the right Caley graph. 1.4 Computational asymmetry and reversible computing Reversible computing deals with the following questions: If a function f is injective (or bijective) and computable, can f be computed in such a way that each elementary computation step is injective (respectively bijective)? And if such injective (or bijective) computations are possible, what is their complexity, compared to the usual (non-injective) complexity? One of the main results is the following (Bennett’s theorem [4, 5], and earlier work of Lecerf [22]): Let f be an injective function, and assume f and f−1 are computable by deterministic Turing machines with time complexity Tf (.), respectively Tf−1(.). Then f (and also f −1) is computable by a reversible Turing machine (in which every transition is deterministic and injective) with time complexity O(Tf + Tf−1). Note that only injectiveness (not bijectiveness) is used here. Bennett’s theorem has the following important consequence, which relates reversible computing to one-way functions: Injective one-way functions exist iff there exist injective functions that have efficient traditional algorithms but that do not have efficient reversible algorithms. Toffoli representation Remarkably, it is possible to “simulate” any function f : {0, 1}m → {0, 1}n (injective or not, one- way or not) by a bijective circuit; a circuit is called bijective iff the circuit is made from bijective gates. Here, bijective circuits will be built from the wire swapping operations and the following bijective gates: not (negation), c-not (the Controlled Not, also called “Feynman gate”) defined by (x1, x2) ∈ {0, 1}2 7−→ (x1, x1 ⊕ x2) ∈ {0, 1}2, and cc-not (the Doubly Controlled Not, also called “Toffoli gate”) defined by (x1, x2, x3) ∈ {0, 1}3 7−→ (x1, x2, (x1 ∧ x2)⊕ x3) ∈ {0, 1}3. Theorem 1.5 (Toffoli [40]). For every boolean function f : {0, 1}m → {0, 1}n there exists a bijective boolean circuit βf (over the bijective gates not, c-not, cc-not, and wire transpositions), with input- output function βf : x 0 n ∈ {0, 1}m+n 7−→ f(x) x ∈ {0, 1}n+m. In other words, f(x) consists of the projection onto the first n bits of βf (x 0 n); equivalently, f(.) = projn ◦ βf ◦ concat0n(.), where projn projects a string of length n+m to the first n bits, and concat0n concatenates 0n to the right of a string. See Theorems 4.1, 5.3 and 5.4 of [40], and see Fig. 1 below. ✲ f(x) Fig. 1: Toffoli representation of the function f . The Toffoli representation contains two non-bijective actions: The projection at the output, and the forced setting of the value of some of the input wires. Toffoli’s proofs and constructions are based on truth tables, and he does not prove anything about the circuit size of βf (counting the bijective gates), compared to the circuit size of f . The following gives a polynomial bound on the size of the bijective circuit, at the expense of a large number of input- and output-wires. Theorem 1.6 (E. Fredkin, T. Toffoli [15]). For every boolean function f : {0, 1}m → {0, 1}n with circuit size C(f) there exists a bijective boolean circuit Bf (over a bounded collection of bijective gates, e.g., not, c-not, cc-not, and wire transpositions), with input-output function Bf : x 0 n+C(f) ∈ {0, 1}m+n+C(f) 7−→ f(x) z(x) ∈ {0, 1}m+n+C(f) for some z(x) ∈ {0, 1}m+C(f). If g : {0, 1}m → {0, 1}m is a permutation then there exists a bijective boolean circuit Ug (over bijective gates), with input-output function Ug : x 1 m 0m+C ∈ {0, 1}3m+C 7−→ g(x) g(x) x 0C ∈ {0, 1}3m+C where C = max{C(g), C(g−1)}, and g(x) is the bitwise complement of g(x). Later we will introduce another reversible representation of boolean functions by bijective gates; we will need only one 0-wire, but the gates will be taken from the Thompson group G2,1, i.e., we will also use non-length-preserving transformations of bitstrings (Theorems 4.1 and 4.2 below). 1.5 Distortion We will prove later (Theorem 5.10) that computational asymmetry has a lot to do with distortion, a concept introduced into group theory by Gromov [17] and Farb [14]. Distortion is already known to have connections with isoperimetric functions (see [28], [29], [24]). A somewhat different problem about distortion (for finite metric spaces) was tackled by Bourgain [10]. We will use a slightly more general notion of distortion, based on (possibly directed) countably infinite rooted graphs, and their (directed) path metric. A weighted directed graph is a structure (V,E, ω) where V is a set (called the vertex set), E ⊆ V ×V (called the edge set), and ω : E 7−→ R>0 is a function (called the weight function); note that every edge has a strictly positive weight. It is sometimes convenient to define ω(u, v) = ∞ when (u, v) ∈ V ×V −E. A path in (V,E) is a sequence of edges (ui, vi) (1 ≤ i ≤ n) such that ui+1 = vi for all i < n, and such that all elements in {ui : 1 ≤ i ≤ n} ∪ {vn} are distinct; u1 is called the start vertex of this path, and vn is called the end vertex of this path; the sum of weights i=1 ω(ui, vi) over the edges in the path is called the length of the path. Here we do not consider any paths with infinitely many edges; but we allow V and E to be countably infinite. A vertex w2 is said to be reachable from a vertex w1 in (V,E) iff there exists a path with start vertex w1 and end vertex w2. If w2 is reachable from w1 then the minimum length over all paths from w1 to w2 is called the directed distance from w1 to w2, denoted d(w1, w2); since we only consider finite paths here, this minimum exists. If w2 is not reachable from w1 then we define d(w1, w2) to be ∞. Clearly we have w1 = w2 iff d(w1, w2) = 0, and for all u, v, w ∈ V , d(u,w) ≤ d(u, v) + d(v,w). In a directed graph, the function d(., .) need not be symmetric. The function d : V ×V → R≥0 ∪{∞} is called the directed path metric of (V,E, ω). A rooted directed weighted graph is a structure (V,E, ω, r) where (V,E, ω) is a directed weighted graph, r ∈ V , and all vertices in V are reachable from r. A set M with a function d : M × M → R≥0 ∪ {∞}, satisfying the two axioms w1 = w2 iff d(w1, w2) = 0, and d(u,w) ≤ d(u, v) + d(v,w), will be called directed metric space (a.k.a. quasi-metric space). Any subset G embedded in a directed metric space M becomes a directed metric space by using the directed distance of M . We call this the directed distance on G inherited from M . If G ⊆ V for a rooted directed weighted graph (V,E, ω, r), we consider the function ℓ : g ∈ G 7−→ d(r, g) ∈ R≥0, which we call the directed length function on G inherited from (V,E, ω, r). (The value ∞ will not appear here since all of G is reachable from r.) We now define distortion in a very general way. Intuitively, distortion in a set is a quantitative comparison between two (directed) length functions that are defined on the same set. Definition 1.7 Let G be a set, and let ℓ1 and ℓ2 be two functions G → R≥0. The distortion of ℓ1 with respect to ℓ2 is the function δℓ1,ℓ2 : R≥0 → R≥0 defined by δℓ1,ℓ2(n) = max{ℓ1(g) : g ∈ G, ℓ2(g) ≤ n}. We will also use the notation δ[ℓ1, ℓ2](.) for δℓ1,ℓ2(.). When we consider a distortion δℓ1,ℓ2(.) we often assume that ℓ2 ≤ ℓ1 or ℓ2 ≤ O(ℓ1); this insures that the distortion is at least linear, i.e., δℓ1,ℓ2(n) ≥ c n, for some constant c > 0. We will only deal with functions obtained from the lengths of finite paths in countable directed graphs, so in that case the functions ℓi are discrete, and the distortion function exists. The next Lemma generalizes the distortion result of Prop. 4.2 of [14]. Lemma 1.8 Let G be a set and consider three functions ℓ3, ℓ2, ℓ1 : G→ R≥0 such that ℓ1(.) ≥ ℓ2(.) ≥ ℓ3(.). Then the corresponding distortions satisfy: δℓ1,ℓ3(.) ≤ δℓ1,ℓ2 ◦ δℓ2,ℓ3(.). Proof. The inequalities ℓ1(.) ≥ ℓ2(.) ≥ ℓ3(.) guarantee that the three distortions δℓ1,ℓ3 , δℓ1,ℓ2 , and δℓ2,ℓ3 are at least as large as the identity map. By definition, δℓ1,ℓ2 δℓ2,ℓ3(n) = max{ℓ1(x) : x ∈ G, ℓ2(x) ≤ δℓ2,ℓ3(n)} = max ℓ1(x) : x ∈ G, ℓ2(x) ≤ max{ℓ2(z) : z ∈ G, ℓ3(z) ≤ n} = max ℓ1(x) : x ∈ G, (∃z ∈ G) ℓ2(x) ≤ ℓ2(z) and ℓ3(z) ≤ n ≥ max{ℓ1(x) : x ∈ G, ℓ3(x) ≤ n} = δℓ1,ℓ3(n). The last inequality follows from the fact that if ℓ3(x) ≤ n then for some z (e.g., for z = x): ℓ2(x) ≤ ℓ2(z) and ℓ3(z) ≤ n. ✷ Examples of distortion: Distortion and asymmetry are unifying concepts that apply to many fields. 1. Gromov distortion: Let G be a subgroup of a group H, with generating sets ΓG, respectively ΓH , such that ΓG ⊆ ΓH , and such that ΓG = Γ−1G and ΓH = Γ H . This determines a Cayley graph for G and a Cayley graph for H. Now we have two distance functions on G, one obtained from the Cayley graph of G itself (based on ΓG), and the other inherited from the embedding of G in H. See [17], [10], and [14]. The Gromov distortion function is a natural measure of the difficulty of the generalized word problem. A very important case is when both ΓG and ΓH are finite. Here are some results for that case: Theorem of Ol′shanskii and Sapir [29] (making precise and proving the outline on pp. 66-67 in [17]): All Dehn functions of finitely presented groups (and “approximately all” time complexity functions of nondeterministic Turing machines) are Gromov distortion functions of finitely generated subgroups of FG2×FG2; here, FG2 denotes the 2-generated free group. Moreover, in [6] it was proved that FG2×FG2 is embeddable with linear distortion in the Thompson group G2,1. So the theorem of Ol′shanskii and Sapir also holds for the finitely generated subgroups of G2,1. Actually, Gromov [17] and Bourgain [10] defined the distortion to be 1 ·max{|g|ΓG : |g|ΓH ≤ n, g ∈ G}, i.e., they use an extra factor 1 . However, the connections between distortion, the generalized word problem, and complexity (as we just saw, and will further see in the present paper) are more direct without the factor 1 2. Bourgain’s distortion theorem: Given a finite metric space G with n elements, the aim is to find embeddings of G into a finite-dimensional euclidean space. The two distances of G are its given distance and the inherited euclidean distance. In this problem the goal is to have small distortion, as a function of the cardinality of G, while also keeping the dimension of the euclidean space small. Bourgain [10] found a bound O(n log n) for the distortion (or “O(log n)” in Bourgain’s and Gromov’s terminology). This is an important result. See also [21], [2], [3]. 3. Generator distortion: A variant of Gromov’s distortion is obtained when G = H, but ΓG $ ΓH . So here we look at the distorting effect of a change of generators in a given group. When ΓG and ΓH are both finite the generator distortion is linear; however, when ΓG is finite and ΓH is infinite the distortion becomes interesting. E.g., for the Thompson group G2,1 let us take ΓG to be any finite generating set, and for ΓH let us take ΓG ∪ {τi,j : 1 ≤ i < j}; here τi,j is the position transposition defined earlier. Then the generator distortion is exponential (see [7]). Also, the word problem of G2,1 over any finite generating set ΓG is in P, but the word problem of G2,1 over ΓG ∪ {τi,j : 1 ≤ i < j} is coNP-complete (see [7] and [8]). 4. Monoids and directed distance: Gromov’s distortion and the generator distortion can be generalized to monoids. We repeat what we said about Gromov distortion, but G and H are now monoids, and ΓG, respectively ΓH , are monoid generating sets which are used to define monoid Cayley graphs. We will use the left Cayley graphs. We assume ΓG ⊆ ΓH . In each Cayley graph there is a directed distance, defined by the lengths of directed paths. The monoid G now has two directed distance functions, the distance in the Cayley graph ofG itself, and the directed distance thatG inherits from its embedding into the Cayley graph of H. We denote the word-length of g ∈ G over ΓG by |g|G; this is the minimum length of all words over ΓG that represent g; it is also the length of a shortest path from the identity to g in the Cayley graph of G. Similarly, we denote the word-length of h ∈ H over ΓH by |h|H . The definition of the distortion becomes: δ(n) = max{|g|G : g ∈ G, |g|H ≤ n}. 5. Schreier graphs: Let G, H, and F be groups, where F is a subgroup of H. Let ΓH be a generating set of ΓH , and assume ΓH = Γ H . We can define the Schreier left coset graph of H/F over the generating set ΓH , and the distance function dH/F (., .) in this coset graph. By definition, this Schreier graph has vertex set H/F (i.e., the left cosets, of the form h · F with h ∈ H), and it has directed edges of the form h · F γ−→ γg · F , for h ∈ H, γ ∈ ΓH . The graph is symmetric; for every edge as above there is an opposite edge γh ·F γ −→ h ·F . Because of symmetry the Schreier graph has a (symmetric) distance function based on path length, dH/F (., .) : H/F ×H/F → N. Next, assume that G is embedded into H/F by some injective function G →֒ H/F . Such an embedding happens, e.g., if G and F are subgroups of H such that G∩F = {1}. Indeed, in that case each coset in H/F contains at most one element of G (since g1F = g2F implies g 2 g1 ∈ F ∩G = {1}). The group G now inherits a distance function from the path length in the Schreier graph of H/F . Comparing this distance with other distances in G leads to distortion functions. E.g., if the group G is also embedded in a monoid M with monoid generating set ΓM , this leads to the following distortion function: δG(n) = max{dH/F (F, gF ) : g ∈ G, |g|M ≤ n}. It will turn out that for appropriate choices of G,F,H, ΓH , and ΓM , this last distortion is polyno- mially related to the computational asymmetry function α of boolean permutations (Theorem 5.10). 6. Asymmetry functions: We already saw the computational asymmetry function of combinational circuits, and the word-length asymmetry function of a group embedded in a monoid. More generally, in any quasi-metric space (S, d), where d(., .) is a directed distance function, an asymmetry function A : R≥0 → R≥0 can be defined by A(n) = max{d(x2, x1) : x1, x2 ∈ S, d(x1, x2) ≤ n}. This asymmetry function can also be viewed as the distortion of drev with respect to d in S; here drev denotes the reverse directed distance, defined by drev(x1, x2) = d(x2, x1). 7. Other distortions: - Distortion can compare lengths of proofs (or lengths of expressions) in various, more or less pow- erful proof systems (respectively description languages). Distortion can also compare the duration of computations or of rewriting processes in various models of computation. Hence, many (perhaps all) notions of complexity are examples of distortion. Distortion is an algebraic or geometric representation (or cause) of complexity. - Instead of length and distance, other measures (e.g., volumes in higher dimension, energy, action, entropy, etc.) could be used. 1.6 Thompson-Higman groups and monoids The Thompson groups, introduced by Richard J. Thompson [38, 26, 39], are finitely presented infinite groups that act as bijections between certain subsets of {0, 1}∗. So, the elements of the Thompson groups are transformations of bitstrings, and hence they are related to input-output maps of boolean circuits. In this subsection we define the Thompson group G2,1 (also known as “V ”), as well as its generalization (by Graham Higman [18]) to the group Gk,1 that partially acts on A ∗, for any finite alphabet A of size k ≥ 2. We will follow the presentation of [6] (see also [8] and [7]); another reference is [33], which is also based on string transformations but with a different terminology; the classical references [38, 26, 39, 18, 12] do not describe the Thompson groups by transformations of finite strings. Because of our interest in strings and in circuits, we also use generalizations of the Thompson groups to monoids, as introduced in [9]. Some preliminary definitions, all fairly standard, are needed in order to define the Thompson- Higman group Gk,1. First, we pick any alphabet A of cardinality |A| = k. By A∗ we denote the set of all finite words (or “strings”) over A; the empty word ε is also in A∗. We denote the length of w ∈ A∗ by |w| and we let An denote the set of words of length n. We denote the concatenation of two words u, v ∈ A∗ by uv or by u · v; the concatenation of two subsets B,C ⊆ A∗ is defined by BC = {uv : u ∈ B, v ∈ C}. A right ideal of A∗ is a subset R ⊆ A∗ such that RA∗ ⊆ R. A generating set of a right ideal R is, by definition, a set C such that R is equal to the intersection of all right ideals that contain C; equivalently, C generates R (as a right ideal) iff R = CA∗. A right ideal R is called essential iff R has a non-empty intersection with every right ideal of A∗. For u, v ∈ A∗, we call u a prefix of v iff there exists z ∈ A∗ such that uz = v. A prefix code is a subset C ⊆ A∗ such that no element of C is a prefix of another element of C. A prefix code C over A is maximal iff C is not a strict subset of any other prefix code over A. It is easy to prove that a right ideal R has a unique minimal (under inclusion) generating set CR, and that CR is a prefix code; moreover, CR is a maximal prefix code iff R is an essential right ideal. For a partial function f : A∗ → A∗ we denote the domain by Dom(f) and the image (range) by Im(f). A restriction of f is any partial function f1 : A ∗ → A∗ such that Dom(f1) ⊆ Dom(f), and such that f1(x) = f(x) for all x ∈ Dom(f1). An extension of f is any partial function of which f is a restriction. An isomorphism between right ideals R1, R2 of A ∗ is a bijection ϕ : R1 → R2 such that for all r1 ∈ R1 and all z ∈ A∗: ϕ(r1z) = ϕ(r1) · z. The isomorphism ϕ is uniquely determined by a bijection between the prefix codes that minimally generate R1, respectively R2. One can prove [39, 33, 6] that every isomorphism ϕ between essential right ideals has a unique maximal extension (within the category of isomorphisms between essential right ideals of A∗); we denote this unique maximal extension by max(ϕ). Now, finally, we define the Thompson-Higman group Gk,1: It consists of all maximally extended isomorphisms between finitely generated essential right ideals of A∗. The multiplication consists of composition followed by maximum extension: ϕ · ψ = max(ϕ ◦ ψ). Note that Gk,1 acts partially and faithfully on A∗ on the left. Every element ϕ ∈ Gk,1 can be described by a bijection between two finite maximal prefix codes; this bijection can be described concretely by a finite function table. When ϕ is described by a maximally extended isomorphism between essential right ideals, ϕ : R1 → R2, we call the minimum generating set of R1 the domain code of ϕ, and denote it by domC(ϕ); similarly, the minimum generating set of R2 is called the image code of ϕ, denoted by imC(ϕ). Thompson and Higman proved that Gk,1 is finitely presented. Also, when k is even Gk,1 is a simple group, and when k is odd Gk,1 has a simple normal subgroup of index 2. In [6] it was proved that the word problem of Gk,1 over any finite generating set is in P (in fact, more strongly, in the parallel complexity class AC1). In [8, 7] it was proved that the word problem of Gk,1 over Γ∪{τi,j : 1 ≤ i < j} is coNP-complete, where Γ is any finite generating set of Gk,1, and where τi,j is the position transposition introduced in Subsection 1.1. Because of connections with circuits we consider the subgroup lpGk,1 of all length-preserving elements of Gk,1; more precisely, lpGk,1 = {ϕ ∈ Gk,1 : ∀x ∈ Dom(ϕ), |x| = |ϕ(x)|}. See [8] for a study of lpGk,1 and some of its properties. In particular, it was proved that lpGk,1 is a direct limit of finite alternating groups, and that lpG2,1 is generated by the set {N,C, T} ∪ {τi,i+1 : 1 ≤ i}, where N : x1w 7→ x1w, C : x1x2w 7→ x1 (x2 ⊕ x1)w, and T : x1x2x3w 7→ x1x2 (x3 ⊕ (x2 ∧ x1))w (for x1, x2, x3 ∈ {0, 1} and w ∈ {0, 1}∗). Thus (recalling Subsection 1.4), N,C, T are the not, c-not, cc-not gates, applied to the first (left-most) bits of a binary string. It is known that the gates not, c-not, cc-not, together with the wire-swappings, form a complete set of gates for bijective circuits (see [36, 40, 15]); hence, lpG2,1 is closely related to the field of reversible computing. It is natural to generalize the bijections between finite maximal prefix codes to functions between finite prefix codes. Following [9] we will define below the Thompson-Higman monoids Mk,1. First, some preliminary definitions. A right-ideal homomorphism of A∗ is a total function ϕ : R1 → A∗ such that R1 is a right ideal, and such that for all r1 ∈ R1 and all z ∈ A∗: ϕ(r1z) = ϕ(r1) · z. It is easy to prove that Im(ϕ) is then also a right ideal of A∗. From now on we will write a right-ideal homomorphism as a total surjective function ϕ : R1 → R2, where both R1 and R2 are right ideals. The homomorphism ϕ is uniquely determined by a total surjective function f : P1 → S2, with P1, S2 ⊂ A∗ where P1 is the prefix code (not necessarily maximal) that generates R1 as a right ideal, and where S2 is a set (not necessarily a prefix code) that generates R2 as a right ideal; f can be described by a finite function table. For two sets X,Y , we say that X and Y “intersect” iff X ∩ Y 6= ∅. We say that a right ideal R′1 is essential in a right ideal R1 iff R 1 intersects every right ideal that R1 intersects. An essential restriction of a right-ideal homomorphism ϕ : R1 → R2 is a right ideal-homomorphism Φ : R′1 → R′2 such that R′1 is essential in R1, and for all x 1 ∈ R′1: ϕ(x′1) = Φ(x′1). In that case we also say that ϕ is an essential extension of Φ. If Φ is an essential restriction of ϕ then R′2 = Im(Φ) will automatically be essential in R2 = Im(ϕ). Indeed, if I is any no-empty right subideal of R1 then I ∩R′1 6= ∅, hence ∅ 6= Φ(I ∩ R′1) ⊆ Φ(I) ∩ Φ(R′1) = Φ(I) ∩ R′2; moreover, any non-empty right subideal J of R2 is of the form J = Φ(I), where I = Φ−1(J) is a non-empty right subideal of R1; hence, for any non-empty right subideal J of R2, ∅ 6= J ∩R′2. The free monoid A∗ can be pictured by its right Cayley graph, which is easily seen to be the infinite regular k-ary tree with vertex set A∗ and edge set {(v, va) : v ∈ A∗, a ∈ A}. We simply call this the tree of A∗. It is a directed, rooted tree, with all paths directed away from the root ε (the empty word); by “path” we will always mean a directed path. Many of the previously defined concepts can be reformulated more intuitively in the context of the tree of A∗: A word v is a prefix of a word w iff v is an ancestor of w in the tree. A set P is a prefix code iff no two elements of P are on a common path. A set R is a right ideal iff any path that starts in R has all its vertices in R. The prefix code that generates R consists of the elements of R that are maximal (within R) in the prefix order, i.e., maximally close (along paths) to the root ε. A finitely generated right ideal R is essential iff every infinite path eventually reaches R (and then stays in it from there on). Similarly, a finite prefix code P is maximal iff any infinite path starting at the root eventually intersects P . For two finitely generated right ideals R′, R with R′ ⊂ R we have: R′ is essential in R iff any infinite path starting in R eventually reaches R′ (and then stays in it from there on). Assume now that a total order a1 < a2 < . . . < ak has been chosen for the alphabet A; this means that the tree of A∗ is now an oriented rooted tree, i.e., the children of each vertex v have a total order va1 < va2 < . . . < vak. The following can be proved (see [9], Prop. 1.4(1)): Φ is an essential restriction of ϕ iff Φ can be obtained from ϕ by starting from the table of ϕ and applying a finite number of restriction steps of the following form: “replace (x, y) in a table by {(xa1, ya1), . . . , (xak, yak)}”. In the tree of A∗ this means that x and y are replaced by their children xa1, . . . , xak, respectively ya1, . . . , yak, paired according to the order on the children. One can also prove (see [9], Remark after Prop. 1.4): Every right ideal homomorphism ϕ with table P → S has an essential restriction ϕ′ that has a table P ′ → Q′ such that both P ′ and Q′ are prefix codes. An important fact is the following (see [9], Prop. 1.4(2)): Every homomorphism between finitely generated right ideals of A∗ has a unique maximal essential extension; we call it the maximum essential extension of Φ and denote it by max(Φ). Finally here is the definition of the Thompson-Higman monoid: Mk,1 consists of all maximum es- sential extensions of homomorphisms between finitely generated right ideals of A∗. The multiplication is composition followed by maximum essential extension. One can prove the following, which implies associativity: For all right ideal homomorphisms ϕ1, ϕ2 : max(ϕ2 ◦ ϕ1) = max(max(ϕ2) ◦ ϕ1) = max(ϕ2 ◦max(ϕ1)). In [9] the following are proved about the Thompson-Higman monoid Mk,1: • The Thompson-Higman group Gk,1 is the group of invertible elements of the monoid Mk,1. • Mk,1 is finitely generated. • The word problem of Mk,1 over any finite generating set is in P. • The word problem of Mk,1 over a generating set Γ ∪ {τi,j : 1 ≤ i < j}, where Γ is any finite generating set of Mk,1, is coNP-complete. 2 Boolean functions as elements of Thompson monoids The input-output functions of digital circuits map bitstrings of some fixed length to bitstrings of a fixed length (possibly different from the input length). In other words, circuits have input-output maps that are total functions of the form f : {0, 1}m → {0, 1}n for some m,n > 0. The Thompson-Higman monoid Mk,1 has an interesting submonoid that corresponds to fixed-length maps, defined as follows. Definition 2.1 (the submonoid lepMk,1). Let ϕ : PA ∗ → QA∗ be a right-ideal homomorphism, where P,Q ⊂ A∗ are finite prefix codes, and where P is a maximal prefix code. Then ϕ is called length equality preserving iff for all x1, x2 ∈ Dom(ϕ) : |x1| = |x2| implies |ϕ(x1)| = |ϕ(x2)|. The submonoid lepMk,1 of Mk,1 consists of those elements of Mk,1 that can be represented by length-equality preserving right-ideal homomorphisms. It is easy to check that an essential restriction of an element of lepMk,1 is again in lepMk,1, so lepMk,1 is well defined as a subset of Mk,1; moreover, one can easily check that lepMk,1 is closed under composition, so lepMk,1 is indeed a submonoid of Mk,1. For ϕ ∈ Mk,1 we have ϕ ∈ lepMk,1 iff there exist m > 0 and n > 0 such that Am ⊂ Dom(ϕ) and ϕ(Am) ⊆ An. So (by means of an essential restriction, if necessary), ϕ can be represented by a function table Am → Q ⊆ An with a fixed input length and a fixed output length (but the input and output lengths can be different). The motivation for studying the monoid lepMk,1 is the following. Every boolean function f : {0, 1}m → {0, 1}n (for any m,n > 0) determines an element of lepMk,1, and conversely, this element of lepMk,1 determines f when restricted to {0, 1}m. By considering all boolean functions as elements of lepMk,1 we gain the ability to compose arbitrary boolean functions, even if their domain and range “do not match”. Moreover, in lepMk,1 we are able to generate all boolean functions from gates by using ordinary functional composition (instead of graph-based circuit lay-outs). The following remains open: Question: Is lepMk,1 finitely generated? However we can find nice infinite generating sets, in connection with circuits. Proposition 2.2 (Generators of lepMk,1). The monoid lepMk,1 has a generating set of the form Γ ∪ {τi,i+1 : 1 ≤ i}, for some finite subset Γ ⊂ lepMk,1. Proof. We only prove the result for k = 2; a similar reasoning works for all k (using k-ary logic). It is a classical fact that any function f : {0, 1}m → {0, 1}n can be implemented by a combinational circuit that uses copies of and, or, not, fork and wire-crossings. So all we need to do is to express theses gates, at any place in the circuit, by a finite subset of lepM2,1 and by positions transpositions τi,i+1. For each gate g ∈ {and, or} we define an element γg ∈ lepMk,1 by γg : x1x2w ∈ {0, 1}m 7−→ g(x1, x2) w ∈ {0, 1}m−1. Similarly we define γnot, γfork ∈ lepMk,1 by γnot : x1w ∈ {0, 1}m 7−→ x1 w ∈ {0, 1}m, γfork : x1w ∈ {0, 1}m 7−→ x1 x1 w ∈ {0, 1}m+1. For each g ∈ {and, or, not, fork}, γg transforms only the first one or two boolean variables, and leaves the other boolean variables unchanged. We also need to simulate the effect of a gate g on any variable xi or pair of variables xixi+1, i.e., we need to construct the map uxixi+1v ∈ {0, 1}m 7−→ u g(xi, xi+1) v ∈ {0, 1}m−1 (and similarly in case where g is not or fork). For this, we apply wire-transpositions to move xixi+1 to the wire-positions 1 and 2, then we apply γg, then we apply more wire-transpositions in order to move g(x1, x2) back to position i. Thus the effect of any gate anywhere in the circuit can be expressed as a composition of γg and position transpositions in {τi,i+1 : 1 ≤ i}. ✷ Proposition 2.3 (Change of generators of lepMk,1). Let {τi,i+1 : 1 ≤ i} be denoted by τ . If Γ,Γ′ ⊂ lepMk,1 are two finite sets such that Γ∪τ and Γ′∪τ generate lepMk,1, then the word-length over Γ ∪ τ is linearly related to the word-length over Γ′ ∪ τ . In other words, there are constants c′ ≥ c ≥ 1 such that for all m ∈ lepMk,1 : |m|Γ∪τ ≤ c · |m|Γ′∪τ ≤ c′ · |m|Γ∪τ . Proof. Since Γ is finite, the elements of Γ can be expressed by a finite set of words of bounded length (≤ c) over Γ′ ∪ τ . Thus, every word of length n over Γ∪ τ is equivalent to a word of length ≤ c n over Γ′ ∪ τ . This proves the first inequality. A similar reasoning proves the second inequality. ✷ Proposition 2.4 (Circuit size vs. lepM2,1 word-length). Let ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j} be a generating set of lepM2,1 with ΓlepM2,1 finite. Let f : {0, 1}m → Q (⊆ {0, 1}n) be a function defining an element of lepM2,1, and let |f |lepM2,1 the word-length of f over the generating set ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j}. Let |Cf | be the circuit size of f (using any finite universal set of gates and wire-swappings). Then |f |lepM2,1 and |Cf | are linearly related. More precisely, for some constants c1 ≥ co ≥ 1 : |Cf | ≤ co · |f |lepM2,1 ≤ c1 · |Cf |. Proof. For the proof we assume that the set of gates for circuits (not counting the wire-transpositions) is ΓlepM2,1 . If we make a different choice for the universal set of gates for circuits, and a different choice for the finite portion ΓlepM2,1 of the generating set of lepM2,1 then the inequalities remain the same, except for the constants c1, co. The inequality |Cf | ≤ |f |lepM2,1 is obvious, since a word w over ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j} is automatically a circuit of size |w|. For the other inequality, we want to simulate each gate of the circuit Cf by a word over ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j}. The reasoning is the same for every gate, so let us just focus on an or gate. The essential difference between circuit gates and elements of lepM2,1 is that in a circuit, a gate (with 2 input wires, for example) can be applied to any two wires in the circuit; on the other hand, the functions in lepM2,1 are applied to the first few wires. However, the circuit gate or, applied to (i, i+1) can be simulated by an element of ΓlepM2,1 and a few wire transpositions, since we have: ori,i+1(.) = γor ◦ τ2,i+1 ◦ τ1,i(.). The output wire of ori,i+1(.) is wire number i, whereas the output wire of γor ◦ τ2,i+1 ◦ τ1,i(.) is wire number 1. However, instead of permuting all the wires in order to place the output of γor τ2,i+1 τ1,i(.) on wire i, we just leave the output of γor τ2,i+1 τ1,i(.) on wire 1 for now. The simulation of the next gate will then use appropriate transpositions τ2,j · τ1,k for fetch the correct input wires for the next gate. Thus, each gate of Cf is simulated by one function in ΓlepM2,1 and a bounded number of wire-transpositions in {τi,j : 1 ≤ i < j}. At the output end of the circuit, a permutation of the n output wires is needed in order to send the outputs to the correct wires; any permutation of n elements can be realized with < n (≤ |Cf |) transpositions. (The inequality n ≤ |Cf | holds because since we count the output ports in the circuit size.) ✷ Remark. The above Proposition motivates our choice of generating set of the form Γ∪{τi,j : 1 ≤ i < j} (with Γ finite) for lepMk,1; in particular, it motivates the inclusion of all the position transpositions τi,j in the generating set. The Proposition also motivates the definition of word-length in which τi,j has word-length 1 for all j > i ≥ 1. Next we will study the distortion of lepMk,1 in Mk,1. We first need some Lemmas. Lemma 2.5 (Lemma 3.3 in [6]). If P,Q,R ⊆ A∗ are such that PA∗ ∩QA∗ = RA∗ and R is a prefix code, then R ⊆ P ∪Q. Proof. For any r ∈ R there are p ∈ P, q ∈ Q and v,w ∈ A∗ such that r = pv = qw. Hence p is a prefix of q or q is a prefix of p. Let us assume p is a prefix of q = px, for some x ∈ A∗ (the other case is similar) Hence q = px ∈ PA∗ ∩QA∗ = RA∗, and q is a prefix of r = qw. Since R is a prefix code, r = q, hence r ∈ Q. ✷ Lemma 2.6 Let P,Q ⊂ A∗ be finite prefix codes, and let θ : PA∗ → QA∗ be a right-ideal homomor- phism with domain PA∗ and image QA∗. Let S be a prefix code with S ⊂ QA∗. Then θ−1(S) is a prefix code and θ−1(SA∗) = θ−1(S) A∗. Proof. First, θ−1(S) is a prefix code. Indeed, if we had x1 = x2u for some x1, x2 ∈ θ−1(S) with u non-empty, then θ(x1) = θ(x2) u. This would contradict the assumption that S is a prefix code. Second, θ−1(S) ⊂ θ−1(SA∗), hence θ−1(S) A∗ ⊆ θ−1(SA∗), since θ−1(SA∗) is a right ideal. (Recall that the inverse image of a right ideal under a right-ideal homomorphism is a right ideal.) We also want to show that θ−1(SA∗) ⊆ θ−1(S) A∗. Let x ∈ θ−1(SA∗). So, θ(x) = sv for some s ∈ S, v ∈ A∗, and s = qu for some q ∈ Q, u ∈ A∗. Since θ(x) = quv, we have x = puv for some p ∈ P with θ(p) = q. Hence θ(pu) = qu = s. Therefore, x = puv with pu ∈ θ−1(s) ⊆ θ−1(S), hence x ∈ θ−1(S) A∗. ✷ Notation: For a right-ideal homomorphism ϕ : Dom(ϕ) = PA∗ → Im(ϕ) = QA∗, where P,Q ⊂ A∗ are finite prefix codes, we define ℓ(ϕ) = max{|z| : z ∈ P ∪Q}, For any finite prefix code C ⊂ A∗ we define ℓ(C) = max{|z| : z ∈ C}. Lemma 2.7 Let ϕ : Dom(ϕ) = PA∗ → Im(ϕ) = QA∗ be a right-ideal homomorphism, where P and Q are finite prefix codes. Let R ⊂ A∗ be any finite prefix code. Then we have: (1) ℓ(ϕ−1(R)) < ℓ(ϕ) + ℓ(R), (2) ℓ(ϕ(R)) < ℓ(ϕ) + ℓ(R). Proof. (1) Let r ∈ R ∩ Im(ϕ). Then every element of ϕ−1(r) has the form p1w for some p1 ∈ P and w ∈ A∗ such that r = q1w for some q1 ∈ Q (with ϕ(p1) = q1). Hence |p1w| = |p1| + |r| − |q1| = |r|+ |p1| − |q1|. Moreover, |r| ≤ ℓ(R) and |p1| − |q1| < ℓ(ϕ), so |p1w| < ℓ(R) + ℓ(ϕ). (2) If r ∈ R ∩Dom(ϕ) then ϕ(r) has the form q1v for some q1 ∈ Q and v ∈ A∗ such that r = p1w for some p1 ∈ P (with ϕ(p1) = q1). Hence |q1v| = |q1| + |r| − |p1| = |r| + |q1| − |p1|. Moreover, |r| ≤ ℓ(R) and |q1| − |p1| < ℓ(ϕ), so |q1w| < ℓ(R) + ℓ(ϕ). ✷ For any right-ideal homomorphisms ϕi (with i = 1, . . . , N), the composite map ϕN ◦ . . . ◦ ϕ1(.) is a right-ideal homomorphism. We say that right-ideal homomorphisms Φi (with i = 1, . . . , N) are directly composable iff Dom(Φi+1) = Im(Φi), for i = 1, . . . , N − 1. The next Lemma shows that we can replace composition by direct composition. Lemma 2.8 Let ϕi : Dom(ϕi) = PiA ∗ → Im(ϕi) = QiA∗ be a right-ideal homomorphism (for i = 1, . . . , N), where Pi and Qi are finite prefix codes. Then each ϕi has a (not necessarily essential) restriction to a right-ideal homomorphism Φi with the following properties: • ΦN ◦ . . . ◦ Φ1(.) = ϕN ◦ . . . ◦ ϕ1(.); • Dom(Φi+1) = Im(Φi), for i = 1, . . . , N − 1; • ℓ(Φi) ≤ j=1 ℓ(ϕj) for every i = 1, . . . , N . Proof. We use induction on N . For N = 1 there is nothing to prove. So we let N > 1 and we assume that the Lemma holds for ϕi : PiA ∗ → QiA∗ with i = 2, . . . , N , i.e., we assume that each ϕi (for i = 2, . . . , N) has a restriction ϕ i : P ∗ → Q′iA∗ such that ϕ′N ◦ . . . ◦ ϕ′2 = ϕN ◦ . . . ◦ ϕ2, P ′i+1 = Q i (for i = 2, . . . , N − 1), and ℓ(ϕ′i) ≤ j=2 ℓ(ϕj) for every i = 2, . . . , N . From P i+1 = Q (for i = 2, . . . , N − 1) it follows that ℓ(ϕ′N ◦ . . . ◦ ϕ′2) ≤ max{ℓ(ϕ′i) : i = 2, . . . , N} ≤ j=2 ℓ(ϕj). Using the notation ϕ′ [N,2] for ϕ′N ◦ . . . ◦ ϕ′2 we have Dom(ϕ′[N,2]) = P2A ∗ and Im(ϕ′ [N,2] ) = QNA When we compose ϕ1 and ϕ [N,2] we obtain ϕ−11 (Q1A ∗ ∩ P2A∗) Φ1−→ Q1A∗ ∩ P2A∗ [N,2]−→ ϕ′ [N,2] ∗ ∩ P2A∗). In this diagram, Φ1 is the restriction of ϕ1 to the domain ϕ 1 (Q1A ∗∩P2A∗) and image Q1A∗∩P2A∗; and Φ′ [N,2] is the restriction of ϕ′ [N,2] to the domain Q1A ∗ ∩ P2A∗ and image ϕ′[N,2](Q1A ∗ ∩ P2A∗). Hence, Φ′ [N,2] ◦Φ1 = ϕ′[N,2] ◦ϕ1, and Dom(Φ [N,2] ) = Im(Φ1) (= Q1A ∗ ∩P2A∗). So Φ1 and Φ′[N,2] are directly composable. By Lemma 2.5 there is a prefix code S ⊂ A∗ such that SA∗ = Q1A∗ ∩ P2A∗ and S ⊆ Q1 ∪ P2. Hence, ℓ(S) ≤ max{ℓ(Q1), ℓ(P2)} ≤ max{ℓ(ϕ1), ℓ(ϕ′2)} ≤ max{ℓ(ϕ1), j=2 ℓ(ϕj)} ≤ j=1 ℓ(ϕj). It follows also that ϕ−11 (Q1A ∗∩P2A∗) = ϕ−11 (SA∗) = ϕ 1 (S) A ∗ (the latter equality is from Lemma 2.6). Since S ⊆ Q1 ∪ P2 implies ϕ−11 (S) ⊆ ϕ 1 (Q1) ∪ ϕ 1 (P2) = P1 ∪ ϕ 1 (P2), we have ℓ(ϕ 1 (S)) ≤ max{ℓ(P1), ℓ(ϕ−11 (P2))}. Obviously, ℓ(P1) ≤ ℓ(ϕ1). Moreover, by Lemma 2.7, ℓ(ϕ 1 (P2)) ≤ ℓ(ϕ1) + ℓ(P2). Since ℓ(P2) ≤ ℓ(ϕ′2) ≤ j=2 ℓ(ϕj) (the latter “≤” by induction), we have ℓ(ϕ 1 (S)) ≤ ℓ(ϕ1)+ j=2 ℓ(ϕj) = j=1 ℓ(ϕj). Since the domain code of Φ1 is ϕ 1 (S) and its image code is S, we conclude that ℓ(Φ1) ≤ j=1 ℓ(ϕj). Let us now consider any Φ′ [i,2] , for i = 1, . . . , N . By definition, Φ′ [i,2] is the restriction of ϕ′i ◦ . . . ◦ϕ′2 to the domain SA∗. So the domain code of Φ′ [i,2] is S, and we just proved that ℓ(S) ≤ j=1 ℓ(ϕj). The image code of Φ′ [i,2] is ϕ′i ◦ . . . ◦ ϕ′2(S). Since S ⊆ Q1 ∪ P2 we have ϕ′i ◦ . . . ◦ ϕ′2(S) ⊆ ϕ′i ◦ . . . ◦ ϕ′2(Q1) ∪ ϕ′i ◦ . . . ◦ ϕ′2(P2) = ϕ′i ◦ . . . ◦ ϕ′2(Q1) ∪ Q′i. Therefore: ℓ(ϕ′i ◦ . . . ◦ ϕ′2(S)) ≤ max{ℓ(ϕ′i ◦ . . . ◦ ϕ′2(Q1)), ℓ(Q′i)}. We have ℓ(Q′i) ≤ ℓ(ϕ′i) ≤ j=2 ℓ(ϕj) (the last “≤” by induction). By Lemma 2.7, ℓ(ϕ′i ◦ . . . ◦ ϕ′2(Q1)) ≤ ℓ(ϕ′i ◦ . . . ◦ ϕ′2) + ℓ(Q1) ≤ ℓ(ϕ′i ◦ . . . ◦ ϕ′2) + ℓ(ϕ1). And ℓ(ϕ′i ◦ . . . ◦ ϕ′2) ≤ max{ℓ(ϕ′j) : j = 2, . . . , i}, because Dom(ϕ′r+1) = Im(ϕ′r) for all r = 2, . . . , N − 1. And by induction, ℓ(ϕ′j) ≤ j=2 ℓ(ϕj). Hence, ℓ(ϕ i ◦ . . . ◦ ϕ′2(Q1)) ≤ j=1 ℓ(ϕj). Thus, ℓ(Φ′ [i,2] j=1 ℓ(ϕj) for every i = 2, . . . , N . Finally, we factor Φ′ [N,2] as Φ′ [N,2] = ΦN ◦ . . . ◦Φ2, where Φi (for i = 2, . . . , N) is defined to be the restriction of ϕ′i to the domain ϕ i−1 ◦ . . . ◦ ϕ′2(SA∗) (= Φ′[i−1,2](SA ∗)). Since Dom(ϕ′r+1) = Im(ϕ (for all r = 2, . . . , N − 1), the domain of ϕ′i is equal to the image of ϕ′i−1 ◦ . . . ◦ ϕ′2. So, the domain code of Φi is ϕ i−1 ◦ . . . ◦ϕ′2(S), and its image code is ϕ′i ◦ϕ′i−1 ◦ . . . ◦ϕ′2(S). Since we already proved that ℓ(ϕ′i ◦ . . . ◦ ϕ′2(S)) ≤ j=1 ℓ(ϕj) (for all i), it follows that ℓ(Φi) ≤ j=1 ℓ(ϕj). ✷ In the next theorem we show that the distortion of lepMk,1 in Mk,1 is at most quadratic (over the generators considered so far, which include the bit position transpositions). Combined with Proposi- tion 2.4, this means the following: Assume circuits are built with gates that are not constrained to have fixed-length inputs and outputs, but assume the input-output function has fixed-length inputs and outputs. Then the resulting circuits are not much more compact than conventional circuits, built from gates that have fixed-length inputs and outputs (we gain at most a square-root in size). Theorem 2.9 (Distortion of lepMk,1 in Mk,1). The word-length (or Cayley graph) distortion of lepMk,1 in Mk,1 has a quadratic upper bound; in other words, for all x ∈ lepMk,1: |x|lepMk,1 ≤ c · (|x|Mk,1)2 where c ≥ 1 is a constant. Here the generating sets used are ΓMk,1 ∪ {τi,j : 1 ≤ i < j} for Mk,1, and ΓlepMk,1 ∪ {τi,j : 1 ≤ i < j} for lepMk,1, where ΓMk,1 and ΓlepMk,1 are finite. By |x|Mk,1 and |x|lepMk,1 we denote the word-length of x over ΓMk,1 ∪ {τi,j : 1 ≤ i < j}, respectively ΓlepMk,1 ∪ {τi,j : 1 ≤ i < j}. Proof. We only prove the result for k = 2; a similar proof applies for any k. We abbreviate the set {τi,j : 1 ≤ i < j} by τ . The choice of the finite sets ΓMk,1 and ΓlepMk,1 does not matter (it only affects the constant c in the Theorem. By Corollary 3.6 in [9] we can choose ΓMk,1 so that each γ ∈ ΓMk,1 satisfies the following (recall that ℓ(S) denotes the length of the longest words in a set S): domC(γ) ∪ imC(γ) ≤ 2, and ∣|γ(x)| − |x| ∣ ≤ 1 for all x ∈ Dom(γ). Let ϕ ∈ lepMk,1, and let w = αN . . . α1 be a shortest word over the generating set ΓMk,1 ∪ τ of Mk,1, representing ϕ. So N = |ϕ|Mk,1 . We restrict each partial function αi to a partial function α′i such that imC(α′i) = domC(α i+1) for i = 1, . . . , N−1, according to Lemma 2.8. Hence, αN ◦ . . .◦α1(.) = α′N ◦ . . . ◦ α′1(.), and ℓ(α′i) ≤ j=1 ℓ(αj) for every i = 1, . . . , N . Then αN ◦ . . . ◦ α1(.) is a function {0, 1}m {0, 1}∗ → Q {0, 1}∗, representing ϕ, and we will identify αN ◦ . . . ◦ α1(.) with ϕ. It follows that domC(α′1) = domC(ϕ) = {0, 1}m, and imC(α′N ) = imC(ϕ) = Q ⊆ {0, 1}n. More generally, it follows that imC(α′i ◦ . . . ◦ α′1) = imC(α′i), and domC(α′N ◦ . . . ◦ α′i) = domC(α′i). Since ℓ(α′i) ≤ j=1 ℓ(αj), and ℓ(αj) ≤ 2 for all j, we have for every i = 1, . . . , N : ℓ(α′i) ≤ 2N . From here on we will simply denote ℓ(α′i) by ℓi. Now, we will replace each α i ∈ Mk,1 by βi ∈ lepMk,1, such that domC(βi) = {0, 1}ℓi , and imC(βi) ⊆ {0, 1}ℓi+1 ; so βi is length-equality preserving. This will be done by artificially lengthening those words in domC(α′i) that have length < ℓi and those words in imC(α′i) that have length < ℓi+1. Moreover, we make βi defined on all of {0, 1}ℓi . In detail, βi is defined as follows: • If ℓi ≤ ℓi+1 : βi(u z) = v z 0 ℓi+1−ℓi−|v|+|u| for all u ∈ domC(α′i), and z ∈ {0, 1}ℓi−|u|; here v = α′i(u); βi(x) = x 0 ℓi+1−ℓi for all x 6∈ Dom(α′i), |x| = ℓi. • If ℓi > ℓi+1 : βi(u z1 z2) = v z1 for all u ∈ domC(α′i) and all z1, z2 ∈ {0, 1}∗ with |z1| = ℓi+1 − |v|, |z2| = ℓi − ℓi+1 + |v| − |u|; here, v = α′i(u); βi(x1 x2) = x1 for all x1, x2 ∈ {0, 1}∗ such that x1x2 6∈ Dom(α′i), with |x1| = ℓi+1, |x2| = ℓi − ℓi+1. Claim. βN ◦ . . . ◦ β1(.) = ϕ. Proof of the Claim: We observe first that domC(β1) = domC(α 1) (= domC(ϕ) = {0, 1}m). Next, assume by induction that for every x ∈ {0, 1}m : α′i−1 ◦ . . . ◦ α′1(x) = u is a prefix of βi−1 ◦ . . . ◦ β1(x) = u z. Then βi(u z) = v z 0 ℓi+1−ℓi−|v|+|u| (if ℓi ≤ ℓi+1); or βi(u z) = v z1 (if ℓi ≥ ℓi+1, with |z1| = ℓi+1 − |v| and z = z1z2). In either case we find that α′i(α′i−1 ◦ . . . ◦ α′1(x)) = v is a prefix of βi(βi−1 ◦ . . . ◦ β1(x)) = βi(u z). Hence, when i = N we obtain for any x ∈ {0, 1}m: βN ◦ . . . ◦ β1(x) = y s is a prefix of α′N ◦ . . . ◦ α′1(x) = ϕ(x) = y for some y and s with |y s| = ℓN = n. Since y ∈ imC(ϕ) ⊆ {0, 1}n we conclude that s is empty, hence βN ◦ . . . ◦ β1(x) = α′N ◦ . . . ◦ α′1(x). [End, proof of Claim.] At this point we have expressed ϕ as a product of N elements βi ∈ lepMk,1, where N = |ϕ|Mk,1 . We now want to find the word-length of each βi over ΓlepMk,1 ∪ τ , in order to find an upper bound on the total word-length of ϕ over ΓlepMk,1 ∪ τ . As we saw above, ℓi ≤ 2N for every i = 1, . . . , N . We examine each generator in ΓMk,1 ∪ τ . If αi ∈ τ then βi ∈ τ , so in this case |βi|lepMk,1 = 1. Suppose now that αi ∈ ΓMk,1 . By Proposition 2.4 it is sufficient to construct a circuit that computes βi; the circuit can then be immediately translated into a word over ΓlepMk,1 ∪ τ with linear increase in length. Since domC(αi) ⊆ {0, 1}≤2, we can restrict αi so that its domain code becomes a subset of {0, 1}2; next, we extend αi to a map α i that acts as the identity map on {0, 1}2 where αi was undefined. The image code of α′′i is a subset of {0, 1}≤3. In order to compute βi we first introduce a circuit C(α′′i ) that computes α′′i . A difficulty here is that α i does not produce fixed-length outputs in general, whereas C(α′′i ) has to work with fixed-length inputs and outputs; so the output of C(α i ) represents the output of α′′i indirectly, as follows: The circuit C(α′′i ) has two input bits u = u1u2 ∈ {0, 1}2, and 5 output bits: First there are 3 output bits 03−|v| v ∈ {0, 1}3, where v = α′′i (u); second, there are two more output bits, c1c2 ∈ {0, 1}2, defined by c1c2 = bin(3− |v|) (the binary representation of the non-negative integer 3− |v|). Hence, c1c2 = 00 if |v| = 3, c1c2 = 01 if |v| = 2, c1c2 = 10 if |v| = 1; since |v| > 0, the value c1c2 = 11 will not occur. Thus c1c2 0 3−|v| v contains the same information as v, but has the advantage of having a fixed length (always 5). The circuit C(α′′i ) can be built with a small constant number of and, or, not, fork gates, and we will not need to know the details. We now build a circuit for βi. • Circuit for βi if ℓi ≤ ℓi+1: On input u z ∈ {0, 1}ℓi (with u ∈ {0, 1}2), we want to produce the output v z 0ℓi+1−ℓi−|v|+|u|, where v = α′′i (u). We first apply the circuit C(α′′i ), thus obtaining c1c2 0 3−|v| v z. Then we apply two fork operations (always to the last bit in z) to produce c1c2 0 3−|v| v z b b, where b is the last bit of z. Applying a negation to the first b and an and operation, we obtain c1c2 0 3−|v| v z 0. Applying ℓi+1 − ℓi − 1 more fork operations to the last 0 yields c1c2 0 3−|v| v z 0ℓi+1−ℓi−1. Next, we want to move 03−|v| to the right of the output, in order to obtain c1c2 v z 0 3−|v|+ℓi+1−ℓi−1. For this effect we introduce a controlled cycle. Let κ : x1x2x3 ∈ {0, 1}3 7−→ x3x1x2 be the usual cyclic permutations of 3 bit positions. The controlled cycle acts as the identity map when c1c2 = 00 or 11, τ1,2 when c1c2 = 01, and κ when c1c2 = 10. More precisely, κc : c1c2 x1x2x3 ∈ {0, 1}5 7−→ c1c2 x1x2x3 if c1c2 = 00 or 11, c1c2 x2x1x3 if c1c2 = 01, c1c2 x3x1x2 if c1c2 = 10. We apply ℓi copies of κc(c1, c2, ., ., .) (all controlled by the same value of c1c2) to 0 3−|v| v z. The first κc(c1, c2, ., ., .) is applied to the 3 bits 0 3−|v| v, producing 3 bits y1y2y3; the second κc(c1, c2, ., ., .) is applied to y2y3 and the first bit of z, producing 3 bits y 3; the third κc(c1, c2, ., ., .) is applied to 3 and the second bit of z, etc. So, each one of the ℓi copies of κc acts one bit further down than the previous copy of κc. This will yield c1c2 v z 0 3−|v|+ℓi+1−ℓi−1. Finally, to make c1c2 disappear, we apply two fork operations to c1, then a negation and an and, to make a 0 appear. We combine this 0 with c1 and c2 by and gates, thus transforming 0c1c2 into 0. Finally, an or operation between this 0 and the first bit of v makes this 0 disappear. The number of gates used to compute βi is O(ℓi+1 + ℓi), which is ≤ O(N). • Circuit for βi if ℓi > ℓi+1: On input u z ∈ {0, 1}ℓi (with u ∈ {0, 1}2), we want to produce the output v z1, where v = α′′i (u). We first apply the circuit C(α′′i ), which yields the output c1c2 0 3−|v| v z. Now we want to erase the ℓi− ℓi+1+1 last bits of z. For this we apply two fork operations to the last bit of z (let’s call it b), then a negation and an and, to make a 0 appear. We combine this 0 with the last ℓi − ℓi+1 bits of z, using that many and gates, turning all these bits into a single 0; finally, an or operation between this 0 and the bit of the remainder of z makes this 0 disappear. At this point, the output is c1c2 0 3−|v| v Z1, where Z1 is the prefix of length ℓi+1 − 1 of z. Next, we apply O(ℓi+1) position transpositions to Z1 in order move the two last bits of Z1 to the front of Z1. Let b1b2 be the last two bits of Z1; so, Z1 = z0b1b2 (where z0 is the prefix of length ℓi+1−3 of z); at this point, the output of the circuit is c1c2 0 3−|v| v b1b2 z0. We now introduce a fixed small circuit with 7 input bits and 5 output bits, defined by the following input-output map: ωc : c1c2 x1x2x3 b1b2 ∈ {0, 1}7 7−→ c1c2 x1x2x3 if c1c2 = 00 or 11, c1c2 x1x2 b1 if c1c2 = 01, c1c2 x3 b1b2 if c1c2 = 10. When this map is applied to c1c2 0 3−|v| v b1b2 the output is therefore given by ωc : c1c2 0 3−|v| v b1b2 ∈ {0, 1}7 7−→ c1c2 v if |v| = 3, c1c2 v b1 if |v| = 2, c1c2 v b1b2 if |v| = 1. A circuit for ωc can be built with a small fixed number of and, or, not, fork gates, and we will not need to know the details. After applying ωc to c1c2 0 3−|v| v b1b2 z0 the output has length ℓi+1 + 2; the “+2” comes from c1c2. The output is c1c2 v z0, or c1c2 v b1 z0, or c1c2 v b1b2 z0, depending on whether |v| = 3, 2, or 1. We need to move b1b2 or b1 (or nothing) back to the right-most positions of z0. We do this by applying ℓi+1 copies of the controlled cycle κc(c1, c2, ., ., .) (all copies controlled by the same value of c1c2). We proceed in the same way as when we used κc in the previous case, and we obtain the output c1c2 v z0 (if |v| = 3), or c1c2 v z0 b1 (if |v| = 2), or c1c2 v z0 b1b2 (if |v| = 1). Finally, we erase c1c2 in the same way as in the previous case, thus obtaining the final output. The number of gates used to compute βi is O(ℓi+1 + ℓi) ≤ O(N). This completes the constuction of a circuit for βi. Through this circuit, βi : {0, 1}ℓi → {0, 1}ℓi+1 is expressed as a word over the generating set ΓlepMk,1 ∪ τ , of length ≤ O(ℓi+1 + ℓi) ≤ O(N). Since we have described ϕ as a product of N = |ϕ|Mk,1 elements βi ∈ lepMk,1, each of word-length O(N), we conclude that ϕ has word-length ≤ O(N2) over the generating set ΓlepMk,1 ∪ τ of lepMk,1. Question: Does the distortion of lepMk,1 inMk,1 (over the generators of Theorem 2.9) have an upper bound that is less than quadratic? 3 Wordlength asymmetry vs. computational asymmetry Proposition 3.1 The word-length asymmetry function λ of the Thompson group lpG2,1 within the Thompson monoid lepM2,1 is linearly equivalent to the computational asymmetry function α: α ≃lin λ. Here the generating set used for lepM2,1 is ΓlepM2,1 ∪ {τi,j : 0 ≤ i < j}, where ΓlepM2,1 is finite. The gates used for circuits are any finite universal set of gates, together with the wire-swapping operations {τi,j : 0 ≤ i < j}. We can choose ΓlepM2,1 to consist exactly of the gates used in the circuits; then α = λ. Proof. For any g ∈ lpG2,1 we have C(g−1) ≤ c0 · |g−1|lepM2,1 ≤ c0 · λ(|g|lepM2,1) ≤ c0 · λ(c1 · C(g)). The first and last “≤” come from Prop. 2.4 (since lpG2,1 ⊂ lepM2,1), and the middle “≤” comes from the definition of λ; c0 and c1 are positive constants. Hence, α(n) ≤ c0 · λ(c1 n) for all n. In a very similar way we prove that λ(n) ≤ c′0 · α(c′1 n) for some positive constants c′0, c′1. ✷ Proposition 3.2 The word-length asymmetry function λM2,1 of the Thompson group lpG2,1 within the Thompson monoid M2,1 is polynomially equivalent to the word-length asymmetry function λlepM2,1 of lpG2,1 within the Thompson monoid lepM2,1. More precisely we have for all n : λM2,1(n) ≤ c0 · λlepM2,1(c1 n2), λlepM2,1(n) ≤ c′0 · λM2,1(c where c0, c1, c 1 are positive constants. Here the generating set used for lepM2,1 is ΓlepM2,1 ∪ {τi,j : 0 ≤ i < j}, where ΓlepM2,1 is finite. The generating set used for M2,1 is ΓM2,1 ∪ {τi,j : 0 ≤ i < j}, where ΓM2,1 is a finite generating set of M2,1. Proof. For any g ∈ lpG2,1 we have |g−1|M2,1 ≤ c0 · |g−1|lepM2,1 ≤ c0 · λlepM2,1(|g|lepM2,1) ≤ c0 · λlepM2,1(c1 · |g|2M2,1). The first “≤” holds because lpG2,1 ⊂ lepM2,1 ⊂M2,1 and because of the choice of the generating sets. The second “≤” holds by the definition of λlepM2,1 . The third “≤” comes from the quadratic distortion of lepM2,1 in M2,1 (Theorem 2.9). For the same reasons we also have the following: |g−1|lepM2,1 ≤ c′0 · |g−1|2M2,1 ≤ c 0 · (λM2,1(|g|M2,1))2 ≤ c′0 · (λM2,1(c1 · |g|lepM2,1))2 where c′0, c 1 are positive constants. ✷ 4 Reversible representation over the Thompson groups Theorems 4.1 and 4.2 below introduce a representation of elements of the Thompson monoid lepM2,1 by elements of the Thompson group G2,1, in analogy with the Toffoli representation (Theorem 1.5 above), and the Fredkin representation (Theorem 1.6 above). Our representation preserves complexity, up to a polynomial change, and uses only one constant-0 input. Note that although the functions and circuits considered here use fixed-length inputs and outputs, the representations is over the Thompson group G2,1, which includes functions with variable-length inputs and outputs. In the Theorem below, ΓG2,1 is any finite generating set of G2,1. We denote the length of a word w by |w|, and we denote the size of a circuit C by |C|. The gates and, or, not will also be denoted respectively by ∧,∨,¬. We distinguish between a word Wf (over a generating set of G2,1) and the element wf of G2,1 represented by Wf . Theorem 4.1 (Representation of boolean functions by the Thompson group). Let f : {0, 1}m → {0, 1}n be any total function and let Cf be a minimum-size circuit (made of ∧,∨,¬, fork- gates and wire-swappings τi,j) that computes f . Then there exists a word Wf over the generating set ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1 such that: • For all x ∈ {0, 1}m: wf (0x) = 0 f(x) x, where wf is the element of G2,1 represented by Wf . • The length of the word Wf is bounded by |Wf | ≤ O(|Cf |4). • The largest subscript of any transposition τi,i+1 occurring in Wf has an upper bound ≤ |Cf |2 + 2. Proof. Wire-swappings in circuits are represented by the position transpositions τi,i+1 ∈ G2,1. The gates not, or, and and of circuits are represented by the following elements of G2,1: , ϕ∨ = 0x1x2 1x1x2 (x1 ∨ x2)x1x2 (x1 ∨ x2 )x1x2 , ϕ∧ = 0x1x2 1x1x2 (x1 ∧ x2)x1x2 (x1 ∧ x2 )x1x2 where x1, x2 range over {0, 1}. Hence the domain and image codes of ϕ∨ and ϕ∧ are all equal to {0, 1}3. To represent fork we use the following element, in which we recognize σ ∈ F2,1, one of the commonly used generators of the Thompson group F2,1: 0 10 11 00 01 1 00 01 10 11 000 001 01 1 Note that σ agrees with fork only on input 0, but that is all we will need. By its very essense, the forking operation cannot be represented by a length-equality preserving element of G2,1, because G2,1∩ lepM2,1 = lpG2,1 (the group of length-preserving elements of G2,1). A small remark: In [6, 7, 8], what we call “σ” here, was called “σ−1”. We will occasionally use the wire-swapping τi,j (1 ≤ i < j); note that τi,j can be expressed in terms of transpositions of neighboring wires as follows: τi,j(.) = τi,i+1 τi+1,i+2 . . . τj−2,j−1 τj−1,j τj−2,j−1 . . . τi+1,i+2 τi,i+1(.) so the word-length of τi,j over {τℓ,ℓ+1 : 1 ≤ ℓ} is ≤ 2(j − i)− 1. For x = x1 . . . xm ∈ {0, 1}m and f(x) = y = y1 . . . yn ∈ {0, 1}n, we will construct a word Wf over the generators ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1, such that Wf defines the map wf (.) : 0x 7→ 0 f(x) x. The circuit Cf is partitioned into slices cℓ (ℓ = 1, . . . , L). Two gates g1 and g2 are in the same slice iff the length of the longest path from g1 to any input port is the same as the length of the longest path from g2 to any input port. We assume that Cf is strictly layered, i.e., each gate in slice cℓ only has in-wires coming from slice cℓ−1, and out-wires going toward slice cℓ+1, for all ℓ. To make a circuit C strictly layered we need to add at most |C|2 identity gates (see p. 52 in [7]). The input-output map of slice cℓ has the form cℓ(.) : y (ℓ−1) = y (ℓ−1) 1 . . . y (ℓ−1) nℓ−1 ∈ {0, 1}nℓ−1 7−→ y(ℓ) = y 1 . . . y nℓ ∈ {0, 1}nℓ . Then y(0) = x and y(L) = y, where x ∈ {0, 1}m is the input and y ∈ {0, 1}n is the output of Cf . Each slice is a circuit of depth 1. Before studying in more detail how Cf is built from slices, let us see how a slice is built from gates (inductively, one gate at a time). Let C be a depth-1 circuit with k + 1 gates, obtained by adding one gate to a depth-1 circuit K with k gates. Let K(.) : x1 . . . xm 7−→ y1 . . . yn be the input-output map of the circuit K. Assume by induction that K is represented by a word WK over the generating set ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1. The input-output map of WK is, by induction hypothesis, wK(.) : 0x1 . . . xm 7−→ 0 y1 . . . yn x1 . . . xm. The word WC that represents C over G2,1 is obtained as follows from WK ; there are several cases, depending on the gate that is added to K to obtain C. Case 1: An identity-gate (or a not-gate) is added to K to form C, i.e., C(.) : x1 . . . xmxm+1 7−→ y1 . . . ynxm+1 (or, C(.) : x1 . . . xmxm+1 7−→ y1 . . . ynxm+1). Then WC is given by wC : 0x1x2 . . . xmxm+1 σ7−→ 00x1x2 . . . xmxm+1 τ3,m+37−→ 00xm+1 x2 . . . xmx1 ϕ∨7−→ xm+10xm+1 x2 . . . xmx1 τ3,m+37−→ xm+1 0x1x2 . . . xmxm+1 π7−→ 0x1x2 . . . xmxm+1xm+1 wK7−→ 0 y1 . . . yn x1 . . . xmxm+1xm+1 π′7−→ 0 y1 . . . yn xm+1 x1 . . . xmxm+1 , where π(.) = τm+1,m+2 . . . τ2,3 τ1,2(.) shifts xm+1 from position 1 to position m+ 2, while shifting 0x1 . . . xm one position to the left; and π ′(.) = τm+2,m+3 . . . τn+m+1,n+m+2 τn+m+2,n+m+3(.) shifts xm+1 from position n+m+ 3 to position n+ 2, while shifting x1 . . . xm one position to the right. So, WC = π ′ WK π τ3,m+3 ϕ∨ τ3,m+3 σ, noting that functions act on the left. Thus, |WC | = |WK |+m+n+5 if we use all of {τi,j : 1 ≤ i < j} in the generating set; over {τi,i+1 : 1 ≤ i}, τ3,m+3 has length ≤ 2m− 1, hence |WC | ≤ 3m + n + 4. If we denote the maximum index in the transpositions occurring in WC by JC then we have JC = max{JK , n+m+ 3}. In case a not-gate is added (instead of an identity gate), ϕ∨ is replaced by ϕ¬ ϕ∨ in WC , and the result is similar. Case 2: An and-gate (or an or-gate) is added to K to form C, i.e., C(.) : x1 . . . xmxm+1xm+2 7−→ y1 . . . yn (xm+1 ∧ xm+2) (or, C(.) : x1 . . . xmxm+1xm+2 7−→ y1 . . . yn (xm+1 ∨ xm+2)). Then WC is given by wC : 0x1x2 . . . xmxm+1xm+2 σ7−→ 00x1x2 . . . xmxm+1xm+2 τ2,m+37−→ τ3,m+47−→ 0xm+1xm+2 x2 . . . xm0x1 ϕ∧7−→ (xm+1 ∧ xm+2) xm+1xm+2 x2 . . . xm0x1 τ2,m+37−→ τ3,m+47−→ (xm+1 ∧ xm+2) 0x1x2 . . . xmxm+1xm+2 π7−→ 0x1x2 . . . xm (xm+1 ∧ xm+2) xm+1xm+2 wK7−→ 0 y1 . . . yn x1x2 . . . xm (xm+1 ∧ xm+2) xm+1xm+2 π′7−→ 0 y1 . . . yn (xm+1 ∧ xm+2) x1x2 . . . xmxm+1xm+2 , where π = τm+1,m+2 . . . τ2,3 τ1,2 shifts (xm+1∧xm+2) from position 1 to positionm+2, while shifting 0x1x2 . . . xm one position to the left; and π ′ = τm+2,m+3 . . . τm+n+1,m+n+2 shifts (xm+1 ∧ xm+2) from position n+m+ 2 to position m+ 2, while shifting x1 . . . xm one position to the right. So, WC = π ′ WK π τ3,m+4 τ2,m+3 ϕ∧ τ3,m+4 τ2,m+3 σ, hence |WC | = |WK | + n +m + 7 if all of {τi,j : 1 ≤ i < j} is used in the generating set; over {τi,i+1 : 1 ≤ i}, τ3,m+4 and τ2,m+3 have length ≤ 2(m+ 1)− 1, so |WC | ≤ |WK |+ 5m+ n+ 9. Moreover, JC = max{JK , m+ n+ 2}. Case 3: A fork-gate is added to K to form C, i.e., C(.) : x1 . . . xmxm+1 7−→ y1 . . . yn xm+1xm+1. Then WC is given by wC : 0x1x2 . . . xmxm+1 σ27−→ 000x1x2 . . . xmxm+1 τ3,m+47−→ 00xm+1x1x2 . . . xm0 ϕ∨7−→ xm+10xm+1x1x2 . . . xm0 τ1,m+47−→ 00xm+1x1x2 . . . xmxm+1 ϕ∨7−→ xm+10xm+1x1x2 . . . xmxm+1 0x1x2 . . . xmxm+1xm+1xm+1 wK7−→ 0 y1 . . . yn x1x2 . . . xm xm+1xm+1xm+1 π′7−→ 0 y1 . . . yn xm+1xm+1 x1x2 . . . xmxm+1 , where π = τm+3,m+4 . . . τ1,2 τm+3,m+4 . . . τ3,4 shifts the two copies of xm+1 at the left end from positions 1 and 3 to positions m+ 3 and m+ 4, while shifting 0 to position 1 and shifting x1 . . . xm two positions to the left; and π′ = τm+3,m+4 . . . τm+n+2,m+n+3 τm+2,m+3 . . . τm+n+1,m+n+2 shifts xm+1xm+1 from positions m + n + 2 and m + n + 3 to positions m + 2 and m + 3, while shifting x1 . . . xm two positions to the right. So, WC = π ′ WK π ϕ∨ τ1,m+4 ϕ∨ τ3,m+4 σ 2, hence |WC | = |WK | + 2m + n + 10, if all of {τi,j : 1 ≤ i < j} is used in the generating set; over {τi,i+1 : 1 ≤ i}, τ1,m+4 has length ≤ 2(m+3)−1 and τ3,m+4 has length ≤ 2m−1. Hence, |WC | ≤ |WK |+6m+n+14. Moreover, JC = max{JK , m+n+3}. In all cases, |WC | ≤ |WK |+c·(m+n+1) (for some constant c > 1), and JC ≤ max{JK , n+m+3}. Thus, each slice cℓ, with input-output map cℓ(.) : y (ℓ−1) 7−→ y(ℓ), is represented by a word Wcℓ with map wcℓ(.) : 0 y (ℓ−1) 7−→ 0 y(ℓ) y(ℓ−1), such that |Wcℓ| ≤ c · (n2ℓ−1 + n2ℓ) (for some constant c > 1), and Jcℓ ≤ nℓ−1 + nℓ + c. Regarding wire-crossings, we do not include them into other slices; we put the wire-crossings into pure wire-crossing slices. So we consider two kinds of slices: Slices entirely made of wire-crossings and identities, slices without any wire-crossings. Wire-crossings in circuits are identical to the group elements τi,i+1. We now construct the word Wf from the words Wcℓ (ℓ = 1, . . . , L). First observe that since the map wcℓ(.) is a right-ideal isomorphism (being an element of G2,1), we not only have wcℓ(.) : 0 y (ℓ−1) 7−→ 0 y(ℓ)y(ℓ−1) but also wcℓ(.) : 0 y (ℓ−1)y(ℓ−2) . . . y(1)y(0) 7−→ 0 y(ℓ)y(ℓ−1)y(ℓ−2) . . . y(1)y(0). Then, by concatenating all Wcℓ (and by recalling that y = y (L) and x = y(0)) we obtain wcL wcL−1 . . . wc2 wc1(.) : 0x 7−→ 0 y y(L−1) . . . y(2) y(1) x. Let πCf be the position permutation that shifts y right to the positions just right of x: πCf : 0 y y (L−1) . . . y(2) y(1) x 7−→ 0 y(L−1) . . . y(2) y(1) x y. Observe that for (WcL−1 . . . Wc2 Wc1) −1 we have (wcL−1 . . . wc2 wc1) −1(.) : 0 y(L−1) . . . y(2) y(1) x y 7−→ 0x y. Then we have: wcL wcL−1 . . . wc2 wc1 πCf (wcL−1 . . . wc2 wc1) −1(.) : 0x 7−→ 0x y . By using the position permutation πm,n : 0x y 7−→ 0 y x, we now see how to define Wf : Wf = πm,n WcL WcL−1 . . . Wc2 Wc1 πCf (WcL−1 . . . Wc2 Wc1) Then we have: wf (.) : 0x 7−→ 0 y x, where y = f(x). Finally, we need to examine the length of the word Wf in terms of the size of the circuit Cf that computes f : {0, 1}m → {0, 1}n. The position permutation πm,n shifts the n = |y| letters of y to the left over the m = |x| positions of x. So, πm,n can be written as the product of nm transpositions in {τi,i+1 : 1 ≤ i}, with maximum subscript Jπm,n ≤ m+ n+ 1. The position permutation πCf shifts y to the right from positions in the interval [2, n + 1] within the string 0 y y(L−1) . . . y(2) y(1) x to positions in the interval [2 + i=0 ni, 2 + i=0 ni] within the string 0 y(L−1) . . . y(2) y(1) x y. Note that i=0 ni = |Cf | (the size of the circuit Cf ), and nL = |y| = n, n0 = |x| = m. We shift y starting with the right-most letters of y. This takes i=0 ni = n (|Cf |−n) transpositions in {τi,i+1 : 1 ≤ i}, with maximum subscript JπCf = |Cf |+2. We saw already that |Wcℓ | ≤ c (n2ℓ−1 + n2ℓ), and Jcℓ ≤ nℓ−1 + nℓ + c, for some constant c > 1. Note that i=0 n i ≤ ( i=0 ni) 2 = |Cf |2. Hence we have: |Wf | ≤ co |Cf |2, for some constant co > 1. Moreover, the largest subscript in any transposition occurring in Wf is JWf ≤ |Cf |+ 2. Recall that we assumed that our circuit Cf was strictly layered, and that the circuit size has to be squared (at most) in order to make the circuit strictly layered. Thus, if Cf was originally not strictly layered, our bounds become |Wf | ≤ co |Cf |4, and JWf ≤ |Cf |2 + 2. ✷ The next theorem gives a representation of a boolean permutation by an element of the Thompson group G2,1; the main point of the theorem is the polynomial bound on the word-length in terms of circuit size. Theorem 4.2 (Representation of permutations by the Thompson group). Let g : {0, 1}m → {0, 1}m be any permutation and let Cg and Cg−1 be minimum-size circuits that compute g, respectively g−1. Then there exists a word W(g,g−1) over the generating set ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1, representing an element w(g,g−1) ∈ G2,1 such that: • For all x ∈ Dom(g) and all y ∈ Im(g): w(g,g−1)(0x) = 0 g(x), and (w(g,g−1)) −1(0 y) = 0 g−1(y), where (w(g,g−1)) −1 ∈ G2,1 is represented by the free-group inverse (W(g,g−1))−1 of the word W(g,g−1). • w(g,g−1)(.) and (w(g,g−1))−1 stabilize both 0 {0, 1}∗ and 1 {0, 1}∗. • We have a length upper bound |W(g,g−1)| = |(W(g,g−1))−1| ≤ O(|Cg|4 + |Cg−1 |4). • The largest subscript of transpositions τi,i+1 occurring in W(g,g−1) is ≤ max{|Cg|2, |Cg−1 |2} +2. Note that we distinguish between the word W(g,g−1) (over a generating set of G2,1) and the element w(g,g−1) of G2,1 represented by W(g,g−1). Also, note that although g is length-preserving (g ∈ lpG2,1), w(g,g−1) ∈ G2,1 is not length-preserving. Proof. Consider the position permutation π : 0 y x 7−→ 0x y, for all x, y ∈ {0, 1}m; we express π as a composition of ≤ m2 position transpositions of the form τi,i+1. Let Wg be the word constructed in Theorem 4.1 for g, and let Wg−1 be the word constructed for g −1. We define W(g,g−1) by W(g,g−1) = (Wg−1) −1 π Wg. Then for all x ∈ Dom(g) we have: w(g,g−1) : 0x 7−→ 0 y, where y = g(x). More precisely, for all x ∈ domC(g), wg−−→ 0 g(x) x = 0 y x π−→ 0 x y = 0 g−1(y) −−−−−−→ 0 y = 0 g(x). Since domC(g) is a maximal prefix code, w(g,g−1) maps 0 {0, 1}∗ into 0 {0, 1}∗ (where defined). Similarly, for all y ∈ Im(g) = Dom(g−1) we have: (w(g,g−1))−1 : 0 y 7−→ 0x, where x = g−1(y), y = g(x). Since domC(g−1) is a maximal prefix code, (w(g,g−1)) −1 maps 0 {0, 1}∗ into 0 {0, 1}∗ (where defined). Hence, elements of 0 {0, 1}∗ are never images of 1 {0, 1}∗. Thus, 1 {0, 1}∗ is also stabilized by w(g,g−1) and by (w(g,g−1)) The length of the wordW(g,g−1) is bounded as follows: We have |Wg| ≤ co |Cg|4, and |(Wg−1)−1| = |Wg−1 | ≤ co |Cg−1 |4, by Theorem 4.1. Moreover, π can be expressed as the composition of ≤ m2 (< |Cg|2) transpositions in {τi,i+1 : 1 ≤ i}. The bound on the subscripts also follows from Theorem 4.1. ✷ 5 Distortion vs. computational asymmetry We show in this Section that the computational asymmetry function α(.) is polynomially related to a certain distortion of the group lpG2,1. By Theorem 4.2, for every element g ∈ lpG2,1 there is an element w(g,g−1) ∈ G2,1 which agrees with g on 0 {0, 1}∗, and which stabilizes 0 {0, 1}∗ and 1 {0, 1}∗ . The main property of W(g,g−1) is that its length is polynomially bounded by the circuit sizes of g and g−1; that fact will be crucial later. First we want to study how w(g,g−1) is related to g. Recall that we distinguish between the word W(g,g−1) (over a generating set of G2,1) and the element w(g,g−1) of G2,1 represented by W(g,g−1). Theorem 4.2 inspires the following concepts. Definition 5.1 Let G be a subgroup of G2,1. For any prefix codes P1, . . . , Pk ⊂ {0, 1}∗, the joint stabilizer (in G) of the right ideals P1{0, 1}∗, . . . , Pk{0, 1}∗ is defined by StabG(P1, . . . , Pk) = g ∈ G : g(Pi{0, 1}∗) ⊆ Pi{0, 1}∗ for every i = 1, . . . , k The fixator (in G) of P1{0, 1}∗ is defined by FixG(P1) = g ∈ G : g(x) = x for all x ∈ P1{0, 1}∗) The fixator is also called “point-wise stabilizer”. The following is an easy consequence of the definition: FixG(Pi) is a subgroup of G (⊆ G2,1), for i = 1, . . . , k. If the prefix codes P1, . . . , Pk are such that the right ideals P1{0, 1}∗, . . . , Pk{0, 1}∗ are two-by-two disjoint, and such that P1 ∪ . . . ∪ Pk is a maximal prefix code, then StabG(P1, . . . , Pk) is closed under inverse. Hence in this case StabG(P1, . . . , Pk) is a subgroup of G. In particular, we will consider the following groups: • The joint stabilizer of 0 {0, 1}∗ and 1 {0, 1}∗, StabG(0, 1) = g ∈ G : g(0 {0, 1}∗) ⊆ 0 {0, 1}∗ and g(1 {0, 1}∗) ⊆ 1 {0, 1}∗ • The fixator of 0 {0, 1}∗, FixG(0) = {g ∈ G : g(x) = x for all x ∈ 0 {0, 1}∗}. • The fixator of 1 {0, 1}∗, FixG(1) = {g ∈ G : g(x) = x for all x ∈ 1 {0, 1}∗}. Clearly, FixG(0) and FixG(1) are subgroups of StabG(0, 1). Lemma 5.2 (Self-embeddings of G2,1). Let G be a subgroup of G2,1. Then G is isomorphic to FixG(1) and to FixG(0) by the following isomorphisms: Λ0 : g ∈ G 7−→ (g)0 ∈ FixG(1) Λ1 : g ∈ G 7−→ (g)1 ∈ FixG(0) where (g)0 and (g)1 defined as follows for any g ∈ G2,1: (g)0 : 0x ∈ 0 {0, 1}∗ 7−→ 0 g(x) 1x ∈ 1 {0, 1}∗ 7−→ 1x (g)1 : 1x ∈ 1 {0, 1}∗ 7−→ 1 g(x) 0x ∈ 0 {0, 1}∗ 7−→ 0x Proof. It is straightforward to verify that Λ0 and Λ1 are injective homomorphisms. That Λ0 is onto FixG(1) can be seen from the fact that every element of FixG(1) has a table of the form 0x1 . . . 0xn 1 0y1 . . . 0yn 1 where {x1, . . . , xn} and {y1, . . . , yn} are two maximal prefix codes, and x1 . . . xn y1 . . . yn is an arbitrary element of G. ✷ Lemma 5.3 Let G be a subgroup of G2,1. Then the direct product G×G is isomorphic to StabG(0, 1) by the isomorphism Λ : (f, g) ∈ G×G 7−→ 0x 7→ 0 f(x), 1x 7→ 1 g(x) ∈ StabG(0, 1). Proof. It is straightforward to verify that Λ is a homomorphism. That Λ is onto StabG(0, 1) and injective follows from the fact that every element of StabG(0, 1) has a table of the form 0x1 . . . 0xm 1x 1 . . . 1x 0y1 . . . 0ym 1y 1 . . . 1y where {x1, . . . , xm}, {y1, . . . , ym}, {x′1, . . . , x′n}, and {y′1, . . . , y′n}, are maximal prefix codes, and x1 . . . xm y1 . . . ym x′1 . . . x y′1 . . . y are arbitrary elements of G (⊆ G2,1). ✷ Lemmas 5.2 and 5.3 reveal certain self-similarity properties of the Thompson group G2,1. (Self- similarity of groups with total action on an infinite tree is an important subject, see [27]. However, the action of G2,1 is partial, so much of the known theory does not apply directly.) The stabilizer and the fixators above have some interesting properties. Lemma 5.4 . (1) For all f, g ∈ G: (f)0 (g)1 = (g)1 (f)0 (i.e., the commutator of FixG(0) and FixG(1) is the identity). (2) FixG(0) · FixG(1) = StabG(0, 1) and FixG(0) ∩ FixG(1) = 1; (3) StabG(0, 1) is the internal direct product of FixG(0) and FixG(1). (This is equivalent to the combination of (1) and (2).) (4) For all f, g ∈ G: Λ(f, g) = Λ0(f) · Λ1(g), Λ0(f) = Λ(f,1), and Λ1(g) = Λ(1, g). Moreover, FixG(0) = Λ1(G), FixG(1) = Λ0(G), and StabG(0, 1) = Λ(G×G). Proof. The proof is a straightforward verification. ✷ Lemma 5.5 For every position transposition τi,j, with 1 ≤ i < j, we have (τi,j)0 = τ2,i+1 ◦ τ3,j+1◦ (τ1,2)0 ◦ τ3,j+1 ◦ τ2,i+1. Hence, assuming (τ1,2)0 ∈ ΓG2,1 , and abbreviating {τi,j : 0 < i < j} by τ , we have: |(τi,j)0|ΓG2,1∪τ ≤ 5. Proof. Recall that for (τ1,2)0 we have, by definition, (τ1,2)0(1w) = 1w, and (τ1,2)0(0x2x3w) = 0x3x2w, for all w ∈ {0, 1}∗ and x2, x3 ∈ {0, 1}. The proof of the Lemma is a straightforward verification. ✷ Now we arrive at the relation between w(g,g−1) and g. Lemma 5.6 For all g ∈ lpG2,1 the following relation holds between g and w(g,g−1) : w(g,g−1) · (g)−10 , (g) 0 · w(g,g−1) ∈ FixlpG2,1(0). Equivalently, (g)0 · FixlpG2,1(0) = w(g,g−1) · FixlpG2,1(0), and FixlpG2,1(0) · (g)0 = FixlpG2,1(0) · w(g,g−1) . Proof. By Theorem 4.2 we have w(g,g−1)(0x) = 0 g(x) for all x ∈ Dom(g). So, w(g,g−1) and (g)0 act in the same way on 0 {0, 1}∗ . Also, both w(g,g−1) and (g)0 map 0 {0, 1}∗ into 0 {0, 1}∗, and both map 1 {0, 1}∗ into 1 {0, 1}∗. The Lemma follows from this. ✷ We abbreviate {τi,j : 0 < i < j} by τ . The element w(g,g−1) of G2,1, represented by the word W(g,g−1), belongs to StablpG2,1(0, 1) as we saw in Theorem 4.2. However, the word W(g,g−1) itself is a sequence over the generating set ΓG2,1 ∪ τ of G2,1. Therefore, in order to follow the action of W(g,g−1) and of its prefixes we need to take Fix(0) as a subgroup of G2,1. This leads us to the Schreier left coset graph of FixG2,1(0) within G2,1, over the generating set ΓG2,1 ∪ τ . By definition this Schreier graph has vertex set G2,1/FixG2,1(0), i.e., the left cosets, of the form g · FixG2,1(0) with g ∈ G2,1. And it has directed edges of the form g ·FixG2,1(0) γ−→ γg ·FixG2,1(0) for g ∈ G2,1, γ ∈ ΓG2,1 ∪ τ . Lemma 5.6 implies that for all g ∈ lpG2,1, (g)0 · FixG2,1(0) = w(g,g−1) · FixG2,1(0). We assume that ΓG2,1 = Γ , so the Schreier graph is symmetric, and hence it has a distance function based on path length; we denote this distance by dG/F (., .) : G2,1/FixG2,1(0)×G2,1/FixG2,1(0) −→ N. Lemma 5.7 There are injective morphisms g ∈ lpG2,1 →֒ g ∈ G2,1 ≃−→ (g)0 ∈ FixG2,1(1) ≃−→ (g)0 · FixG2,1(0) ∈ StabG2,1(0, 1)/FixG2,1(0), and an inclusion map (g)0 · FixG2,1(0) ∈ StabG2,1(0, 1)/FixG2,1(0) →֒ (g)0 · FixG2,1(0) ∈ G2,1/FixG2,1(0). In particular, g ∈ G2,1 7−→ (g)0 · FixG2,1(0) ∈ G2,1/FixG2,1(0) is an embedding of G2,1, as a set, into the vertex set G2,1/FixG2,1(0) of the Schreier graph. Proof. Recall that the map Λ0 : g ∈ G2,1 7−→ (g)0 ∈ FixG2,1(1) is a bijective morphism (Lemma 5.2). Also, the map u ∈ FixG2,1(1) 7−→ u · FixG2,1(0) ∈ G2,1/FixG2,1(0) is injective; indeed, if u · FixG2,1(0) = v · FixG2,1(0) with u, v ∈ FixG2,1(1) then v−1u ∈ FixG2,1(0) ∩ FixG2,1(1) = {1}. The map g ∈ G2,1 7−→ (g)0 · FixG2,1(0) ∈ StabG2,1(0, 1)/FixG2,1(0) is a surjective group homomorphism since FixG2,1(0) is a normal subgroup of StabG2,1(0, 1). Since FixG2,1(0)∩FixG(1) = {1}, this homomorphism is injective from FixG2,1(1) onto StabG2,1(0, 1)/FixG2,1(0). The combination of these maps provides an isomorphism from G2,1 onto StabG2,1(0, 1)/FixG2,1(0). Hence we also have an embedding of G2,1, as a set, into the vertex set G2,1/FixG2,1(0) of the Schreier graph. ✷ Since by Lemma 5.7 we can consider G2,1 as a subset of the vertex set G2,1/FixG2,1(0) of the Schreier graph, the path-distance dG/F (., .) on G2,1/FixG2,1(0) leads to a distance on G2,1, inherited from dG/F (., .) : Definition 5.8 For all g, g′ ∈ G2,1 the Schreier graph distance inherited by G2,1 is D(g, g′) = dG/F (g)0 · FixG2,1(0), (g′)0 · FixG2,1(0) The comparison of the Schreier graph distance D(., .) on lpG2,1 with the word-length that lpG2,1 inherits from its embedding into lepM2,1 leads to the following distortion of lpG2,1: Definition 5.9 In lpG2,1 we consider the distortion ∆(n) = max{D(1, g) : |g|lepM2,1 ≤ n, g ∈ lpG2,1}. We now state and prove the main theorem relating ∆(.) and α. Recall that α(.) is the computational asymmetry function of boolean permutations, defined in terms of circuit size. Theorem 5.10 (Computational asymmetry vs. distortion). The computational asymmetry function α(.) and the distortion ∆(.) of lpG2,1 are polynomially related. More precisely, for all n ∈ N : )1/2 ≤ c′ ·∆(n) ≤ c n4 + c · α(c n) where c ≥ c′ ≥ 1 are constants. Proof. The Theorem follows immediately from Lemmas 5.11 and 5.12. ✷ Lemma 5.11. There is a constant c ≥ 1 such that for all n ∈ N : ∆(n) ≤ c n4 + c · α(c n) Proof. By Lemma 5.6, (g)0 · FixG2,1(0) = w(g,g−1) · FixG2,1(0), hence FixG2,1(0), (g)0 · FixG2,1(0) FixG2,1(0), w(g,g−1) · FixG2,1(0) Since the wordW(g,g−1) and the Schreier graph use the same generating set, namely ΓG2,1 ∪ τ , we have FixG2,1(0), w(g,g−1) · FixG2,1(0) ≤ |W(g,g−1)|. By Theorem 4.2, |W(g,g−1)| ≤ O(|Cg|4 + |Cg−1 |4). And by the definition of the computational asymmetry function, |Cg−1 | ≤ α(|Cg |). Hence FixG2,1(0), (g)0 · FixG2,1(0) ≤ O(|Cg|4 + |Cg−1 |4) ≤ O |Cg|4 + α(|Cg|)4 By Proposition 2.4, |Cg| = O(|g|lepM2,1). Hence, for some constants c′′, c′ ≥ 1, FixG2,1(0), (g)0 · FixG2,1(0) ≤ c′ · |g|4 lepM2,1 + c′ · α(c′′ · |g|lepM2,1)4. Thus, FixG2,1(0), (g)0 · FixG2,1(0) : |g|lepM2,1 ≤ n, g ∈ lpG2,1 ≤ c′ n4 + c′ α(c′′ n)4. By Definition 5.9 of the distortion function ∆ we have therefore ∆(n) ≤ c′ n4 + c′ α(c′′ n)4. This proves the Lemma. ✷ Lemma 5.12 There is a constant c ≥ 1 such that for all n ∈ N : α(n) ≤ c ·∆(c n)2. Proof. We first prove the following. Claim: For every g ∈ lpG2,1, the inverse permutation g−1 can be computed by a circuit Cg−1 of size |Cg−1 | ≤ c · d FixG2,1(0), (g)0 · FixG2,1(0) , for some constant c ≥ 1. Proof of the Claim: There is a wordW ′ of length |W ′| = d FixG2,1(0), (g)0 ·FixG2,1(0) over ΓG2,1∪ τ that labels a shortest path from FixG2,1(0) to (g)0·FixG2,1(0) in the Schreier graph ofG2,1/FixG2,1(0). Let W = (W ′)−1 (the free-group inverse of W ′), so |W | = |W ′|. Let w be the element of G2,1 represented by W . Then W labels a shortest path from FixG2,1(0) to (g −1)0 · FixG2,1(0) in the Schreier graph of G2,1/FixG2,1(0); this path has length |W | = |W ′| = d FixG2,1(0), (g)0 · FixG2,1(0) FixG2,1(0), (g−1)0 · FixG2,1(0) We have w · FixG2,1(0) = (g−1)0 · FixG2,1(0), thus for all x ∈ {0, 1}∗ : w(0x) = 0 g−1(x). We now take the word VWU over the generating set ΓM2,1 ∪ τ of the monoid M2,1, where we choose the words U and V to be U = (and, not, fork, fork), and V = (or). The functions and, not, fork, or were defined in Subsection 1.1. Then for all x = x1 . . . xn ∈ {0, 1}∗, with x1, . . . , xn ∈ {0, 1}, we have x1 . . . xn fork−→ x1 x1 . . . xn fork−→ not−→ x1 x1 x1 . . . xn and−→ 0x1 . . . xn = 0x W−→ 0 g−1(x) or−→ g−1(x). The last or combines 0 and the first bit of g−1(x), and this makes 0 disappear. Thus overall, VWU(x) = g−1(x). The length is |V WU | = |W |+ 5. Since g−1 ∈ lpG2,1 ⊂ lepM2,1, Theorem 2.9 implies that there exists a word Z over the generators ΓlepM2,1 ∪ τ of lepM2,1 such that (1) |Z| ≤ c1 · |VWU |2, for some constant c1 ≥ 1, and (2) Z represents the same element of lepM2,1 as VWU , namely g Moreover, by Prop. 2.4, the word Z can be transformed into a circuit of size ≤ c2 · |Z| (for some constant c2 ≥ 1). This proves that there is a circuit Cg−1 for g−1 of size |Cg−1 | ≤ c · |W |2 (for some constant c ≥ 1). Since we saw that |W | = dG/F (FixG2,1(0), (g)0 · FixG2,1(0)), the Claim follows. [End, Proof of the Claim.] By definition, D(1, g) = dG/F (FixG2,1(0), (g)0 · FixG2,1(0)). Hence, by the Claim above: |Cg−1 | ≤ c · D(1, g) By Prop. 2.4 the word-length in lepM2,1 and the circuit size are linearly related; hence |g|lepM2,1 ≤ c0 |Cg|, for some constant c0 ≥ 1. Therefore, α(n) = max{|Cg−1 | : |Cg| ≤ n, g ∈ lpG2,1} ≤ max{|Cg−1 | : |g|lepM2,1 ≤ c0 n, g ∈ lpG2,1} ≤ max D(1, g) : |g|lepM2,1 ≤ c0 n, g ∈ lpG2,1 ≤ c · ∆(c0 n) This proves the Lemma. ✷ 6 Other bounds and distortions 6.1 Other distortions in the Thompson groups and monoids The next proposition gives more upper bounds on the computational asymmetry function α. Proposition 6.1. Assume ΓlepG2,1 ⊂ ΓlepM2,1 ⊂ ΓM2,1 . Let δlpG,lepM = δ |.|ΓlpG2,1∪τ , |.|ΓlepM2,1∪τ be the distortion function of lpG2,1 in the Thompson monoid lepM2,1, based on word-length. Similarly, let δlpG,M = δ |.|ΓlpG2,1∪τ , |.|ΓM2,1∪τ be the distortion function of lpG2,1 in the Thompson monoid M2,1. Then for some constant c ≥ 1 and for all n ∈ N, α(n) ≤ c · δlpG,lepM (c n) ≤ c · δlpG,M (c n). Proof. We first prove that δlpG,lepM (n) ≤ δlpG,M (n). Recall that by definition, δlpG,lepM (n) = max{|g|lpG2,1 : g ∈ lpG2,1, |g|lepM2,1 ≤ n}, and similarly for δlpG,M(n). Since ΓlepM2,1 ⊂ ΓM2,1 we have |x|lepM2,1 ≤ |x|M2,1 . Hence, {|g|lpG2,1 : g ∈ lpG2,1, |g|lepM2,1 ≤ n} ⊆ {|g|lpG2,1 : g ∈ lpG2,1, |g|M2,1 ≤ n}. By taking max over each of these two sets it follows that δlpG,lepM (n) ≤ δlpG,M(n). Next we prove that α(n) ≤ c·δlpG,lepM (c n). For any g ∈ lpG2,1 we have C(g−1) ≤ O(|g−1|lepM2,1), by Prop. 3.2. Moreover, |g−1|lepM2,1 ≤ |g−1|lpG2,1 since lpG2,1 is a subgroup of lepM2,1, and since the generating set used for lpG2,1 (including all τi,j) is a subset of the generating set used for lepM2,1. For any group with generating set closed under inverse we have |g−1|G = |g|G. And by the definition of the distortion δlpG,lepM we have |g|lpG2,1 ≤ δlpG,lepM (|g|lepM2,1). And again, by Prop. 3.2, |g|lepM2,1 ≤ O(C(g)). Putting all this together we have C(g−1) ≤ c1 · |g−1|lepM2,1 ≤ c1 · |g−1|lpG2,1 = c1 · |g|lpG2,1 ≤ c1 · δlpG,lepM (|g|lepM2,1) ≤ c1 · δlpG,lepM (c2 C(g)). Thus, c1 ·δlpG,lepM (c2 C(g)) is an upper bound on C(g−1). Since, by definition, α(C(g)) is the smallest upper bound on C(g−1), it follows that α(C(g)) ≤ c1 · δlpG,lepM (c2 C(g)). ✷ Recall that in the definition 5.9 of the distortion ∆ we compared D(., .) with the word-length in lepM2,1. If, instead, we compare D(., .) with the word-length inM2,1 we obtain the following distortion of lpG2,1 : δ(n) = max{D(1, g) : |g|M2,1 ≤ n, g ∈ lpG2,1}. Proposition 6.2 The distortion functions ∆(.) and δ(.) are polynomially related. More precisely, there are constants c′, c1, c2 ≥ 1 such that for all n ∈ N: ∆(n) ≤ c1 δ(n) ≤ c2 ∆(c′ n2). Proof. Let’s assume first that ΓlepM2,1 ⊆ ΓM2,1 , from which it follows that |g|M2,1 ≤ |g|M2,1 . Therefore, {D(1, g) : |g|lepM2,1 ≤ n} ⊆ {D(1, g) : |g|M2,1 ≤ n}. Hence, ∆(n) ≤ δ(n). By Theorem 2.9, |g|lepM2,1 ≤ c · |g|2M2,1 . So, {D(1, g) : |g|M2,1 ≤ n} ⊆ {D(1, g) : |g|lepM2,1 ≤ c n Hence, δ(n) ≤ ∆(c n2). When we do not have ΓlepM2,1 ⊆ ΓM2,1 , the constants in the theorem change, but the statement remains the same. ✷ 6.2 Monotone boolean functions and distortion On {0, 1}∗ we can define the product order, also called “bit-wise order”. It is a partial order (and in fact, a lattice order), denoted by “�”, and defined as follows. First, 0 ≺ 1; next, for any u, v ∈ {0, 1}∗ we have u � v iff |u| = |v| and ui � vi for all i = 1, . . . , |u|, where ui (or vi) denotes the ith bit of u (respectively v). By definition, a partial function f : {0, 1}∗ → {0, 1}∗ is monotone (also called “product-order preserving”) iff for all u, v ∈ Dom(f) : u � v implies f(u) � f(v). The following fact is well known (see e.g., [43] Section 4.5): A function f : {0, 1}m → {0, 1}n is monotone iff f can be computed by a combinational circuit that only uses gates of type and, or, fork, and wire-swappings; i.e., not is absent. A circuit of this restricted type is called a monotone circuit. Razborov [30] proved super-polynomial lower bounds for the size of monotone circuits that solve the clique problem, and in [31] he proved super-polynomial lower bounds for the size of monotone circuits that solve the perfect matching problem for bipartite graphs; the latter problem is in P. Tardos [37], based on work by Alon and Boppana [1], gave an exponential lower bound for the size of monotone circuits that solve a problem in P; see also [42] (Chapter 14 by Boppana and Sipser). Thus, there exist problems that can be solved by polynomial-size circuits but for which monotone circuits must have exponential size. In particular (for some constants b > 1, c > 0), there are infinitely many monotone functions fn : {0, 1}n → {0, 1}n such that fn has a combinational circuit of size ≤ nc, but fn has no monotone circuit of size ≤ bn. Based on an alphabet A = {a1, . . . , ak} with a1 ≺ a2 ≺ . . . ≺ ak we define a partial function f : A∗ → A∗ to be monotone iff f preserves the product order of A∗. The monotone functions enable us to define the following submonoid of the Thompson-Higman monoid lepMk,1 : monMk,1 = {ϕ ∈ lepMk,1 : ϕ can be represented by a monotone function P → Q, where P and Q are prefix codes, with P maximal }. An essential extension or restriction of an element of monMk,1 is again in monMk,1, so this set is well-defined as a subset of lepMk,1. It is easily seen to be closed under composition, so monMk,1 is a submonoid of lepMk,1. We saw that all monotone finite functions have circuits made from gates of type and, or, fork. Hence monM2,1 has the following generating set: {and, or, fork} ∪ {τi,j : j > i ≥ 1}. The results about monotone circuit size imply the following distortion result. Again, “exponential” refers to a function with a lower bound of the form n ∈ N 7−→ exp( c c′ n), for some constants c′ > 0 and c ≥ 1. Proposition 6.3 Consider the monoid monM2,1 over the generating set {and, or, fork} ∪ {τi,j : j > i ≥ 1}, and the monoid lepM2,1 over the generating set ΓlepM2,1 ∪ {τi,j : j > i ≥ 1}, where ΓlepM2,1 is finite. Then monM2,1 has exponential word-length distortion in lepM2,1. Proof. Let Γmon = {and, or, fork}. By Prop. 2.4 we have |f |ΓlepM2,1∪τ = |Cf |, where |Cf | denotes the ordinary circuit size of f . By a similar argument we obtain: |f |Γmon∪τ = |monCf |, where |monCf | denotes the monotone circuit size of f . We saw that as a consequence of the work of Razborov, Alon, Boppana, and Tardos, there exists an infinite set of monotone functions that have polynomial-size circuits but whose monotone circuit-size is exponential. 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Wegener, The complexity of boolean functions, Wiley/Teubner (1987). Jean-Camille Birget Dept. of Computer Science Rutgers University at Camden Camden, NJ 08102, USA birget@camden.rutgers.edu Introduction One-way functions and one-way permutations Computational Asymmetry Wordlength asymmetry Computational asymmetry and reversible computing Distortion Thompson-Higman groups and monoids Boolean functions as elements of Thompson monoids Wordlength asymmetry vs. computational asymmetry Reversible representation over the Thompson groups Distortion vs. computational asymmetry Other bounds and distortions Other distortions in the Thompson groups and monoids Monotone boolean functions and distortion

Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of one-way transformations. We introduce a computational asymmetry function that measures the amount of one-wayness of permutations. We also introduce the word-length asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to word-length. Moreover, boolean circuits have a representation in terms of elements of a Thompson group, in such a way that circuit size is polynomially equivalent to word-length. We show that circuits built with gates that are not constrained to have fixed-length inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixed-length inputs and outputs). Finally, we show that the computational asymmetry function is closely related to certain distortion functions: The computational asymmetry function is polynomially equivalent to the distortion of the path length in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of certain Thompson monoids. We also show that the results of Razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions.

Introduction The existence of one-way functions, i.e., functions that are “easy to evaluate” but “hard to invert”, is a major open problem. Much of cryptography depends on one-way functions; moreover, indirectly, their existence is connected to the question whether P is different from NP. In this paper we give some connections between these questions and some group-theoretic concepts: (1) We continue the work of [7], [8], and [9], on the relation between combinational circuits, on the one hand, and Thompson groups and monoids on the other hand. We give a representation of any circuit by a word over the Thompson group, such that circuit size is polynomially equivalent to word-length. (2) We establish connections between the existence of one-way permutations and the distortion func- tion in a certain Thompson group. Distortion is an important concept in metric spaces (e.g., Bourgain [10]) and in combinatorial group theory (e.g., Gromov [17], Farb [14]). Overview: Subsections 1.1 - 1.6 of the present Section define and motivate the concepts used: One-way functions and one-way permutations; computational asymmetry; word-length asymmetry; reversible computing; distortion; Thompson groups and monoids. In Section 2 we show that circuits can be represented by elements of Thompson monoids: A boolean circuit is equivalent to a word over a fixed generating set ∗Supported by NSF grant CCR-0310793. Some of the results of this paper were presented at the AMS Section Meeting, Oct. 21-23, 2005, Lincoln, Nebraska (http://www.ams.org/amsmtgs/2117 program.html), and at the conference “Various Faces of Cryptography”, 10 Nov. 2006 at City College of CUNY, New York. http://arxiv.org/abs/0704.1569v1 http://www.ams.org/amsmtgs/2117_program.html of a Thompson monoid, with circuit size being equal (or linearly equivalent) to word-length over the generating set. The Thompson monoids that appear here are monoid generalizations of the Thompson group G2,1, obtained when bijections are generalized to partial functions [9]. Section 3 shows that computational asymmetry and word-length asymmetry (for the Thompson groups and monoids) are linearly related. In Section 4 we give a representation of arbitrary (not necessarily bijective) circuits by elements of the Thompson group G2,1; circuit size is polynomially equivalent to word-length over a certain generating set in the Thompson group. In Section 5 we show that the computational asymmetry function of permutations is polynomially related to a certain distortion in a Thompson group. Section 6 contains miscellaneous results, in particular that the work of Razborov and others on monotone circuit complexity leads to exponential lower bounds on certain distortion functions. 1.1 One-way functions and one-way permutations Intuitively, a one-way function is a function f (mapping words to words, over a finite alphabet), such that f is “easy to evaluate” (i.e., given x0 in the domain, it is “easy” to compute f(x0)), but “hard to invert” (i.e., given y0 in the range, it is “hard” to find any x0 such that f(x0) = y0). The concept was introduced by Diffie and Hellman [13]. There are many ways of defining the words “easy” and “hard”, and accordingly there exist many different rigorous notions of a one-way function, all corresponding to a similar intuition. It remains an open problem whether one-way functions exist, for any “reasonable” definition. Moreover, for certain definitional choices, this problem is a generalization of the famous question whether P 6= NP [16, 34, 11]. We will base our one-way functions on combinational circuits and their size. The size of a circuit will also be called its complexity. Below, {0, 1}n (for any integer n ≥ 0) denotes the set of all bitstrings of length n. A combinational circuit with input-output function f : {0, 1}m → {0, 1}n is an acyclic boolean circuit with m input wires (or “input ports”) and n output wires (or “output ports”). The circuit is made from gates of type not, and, or, fork, as well as wire-crossings or wire-swappings. These gates are very traditional and are defined as follows. and: (x1, x2) ∈ {0, 1}2 7−→ y ∈ {0, 1}, where y = 1 if x1 = x2 = 1, and y = 0 otherwise. or: (x1, x2) ∈ {0, 1}2 7−→ y ∈ {0, 1}, where y = 0 if x1 = x2 = 0, and y = 1 otherwise. not: x ∈ {0, 1} 7−→ y ∈ {0, 1}, where y = 0 if x = 1, y = 1 otherwise. fork: x ∈ {0, 1} 7−→ (x, x) ∈ {0, 1}2. Another gate that is often used is the exclusive-or gate, xor: (x1, x2) ∈ {0, 1}2 7−→ y ∈ {0, 1}, where y = 1 if x1 6= x2, and y = 0 otherwise. The wire-swapping of the ith and jth wire (i < j) is described by the bit transposition (or bit position transposition) τi,j : uxivxjw ∈ {0, 1}ℓ 7−→ uxjvxiw ∈ {0, 1}ℓ, where |u| = i− 1, |v| = j − i− 1, |w| = ℓ− j − 1. The fork and wire-swapping operations, although heavily used, are usually not explicitly called “gates”; but because of their important role we will need to consider them explicitly. Other notations for the gates: and(x1, x2) = x1 ∧ x2, or(x1, x2) = x1 ∨ x2, not(x) = x, xor(x1, x2) = x1 ⊕ x2. A combinational circuit for a function f : {0, 1}m → {0, 1}n is defined by an acyclic directed graph drawn in the plane (with crossing of edges allowed). In the circuit drawing, the m input ports are vertices lined up in a vertical column on the left end of the circuit, and the n output ports are vertices lined up in a vertical column on the right end of the circuit. The input and output ports and the gates of the circuit (including the fork gates, but not the wire transpositions) form the vertices of the circuit graph. We often view the circuit as cut into vertical slices. A slice can be any collection of gates and wires in the circuit such that no gate in a slice is an ancestor of another gate in the same slice, and no wire in a slice is an ancestor of another wire in the same slice (unless these two wires are an input wire and an output wire of a same gate). Two slices do not overlap, and every wire and every gate belongs to some slice. For more details on combinational circuits, see [32, 43, 11]. The size of a combinational circuit is defined to be the number of gates in the circuit, including forks and wire-swappings, as well as the input ports and the output ports. For a function f : {0, 1}m → {0, 1}n, the circuit complexity (denoted C(f)) is the smallest size of any combinational circuit with input-output function f . A cause of confusion about gates in a circuit is that gates of a certain type (e.g., and) are tradi- tionally considered the same, no matter where they occur in the circuit. However, gates applied to different wires in a circuit are different functions; e.g., for the and gate, (x1, x2, x3) 7→ (x1 ∧ x2, x3) is a different function than (x1, x2, x3) 7→ (x1, x2 ∧ x3). 1.2 Computational Asymmetry Computational asymmetry is the core property of one-way functions. Below we will define computa- tional asymmetry in a quantitative way, and in a later Section we will relate it to the group-theoretic notion of distortion. For the existence of one-way functions, it is mainly the relation between the circuit complexity C(f) of f and the circuit complexity C(f−1) of f−1 that matters, not the complexities of f and of f−1 themselves. Indeed, a classical padding argument can be used: If we add C(f) “identity wires” to a circuit for f , then the resulting circuit has linear size as a function of its number of input wires; see Proposition 1.2 below. (An identity wire is a wire that goes directly from an input port to an output port, without being connected to any gate.) In [11] (page 230) Boppana and Lagarias considered logC(f ′)/logC(f) as a measure of one- wayness; here, f ′ denotes an inverse of f , i.e., any function such that f ◦ f ′ ◦ f = f . Massey and Hiltgen [25, 19] introduced the phrases complexity asymmetry and computational asymmetry for injec- tive functions, in reference to the situation where the circuit complexities C(f) and C(f−1) are very different. The concept of computational asymmetry can be generalized to arbitrary (non-injective) functions, with the meaning that for every inverse f ′ of f , C(f) and C(f ′) are very different. In [25] Massey made the following observation. For any large-enough fixed m and for almost all permutations f of {0, 1}m, the circuit complexities C(f) and C(f−1) are very similar: C(f) ≤ C(f−1) ≤ 10 C(f) Massey’s proof is adapted from the Shannon lower bound [35] and the Lupanov upper bound [23] (see also [19], [32]), from which it follows that almost all functions and almost all permutations (and their inverses) have circuit complexity close to the Shannon bounds. Massey’s observation can be extended to the set of all functions f : {0, 1}m → {0, 1}n, i.e., for almost all f and for every inverse f ′ of f , the complexities C(f) and C(f ′) are within constant factors of each other. Hence, computationally asymmetric permutations are rare among the boolean permutations overall (and similarly for functions). This is an interesting fact about computational asymmetry, but by itself it does not imply anything about the existence or non-existence of one-way functions, not even heuristically. Indeed, Massey proved his linear relation C(f) = Θ(C(f ′)) in the situation where C(f) = Θ(2m), and then uses the fact that the condition C(f) = Θ(2m) holds for almost all boolean permutations and for almost all boolean functions. But there also exist functions with C(f) = O(mk), with k a small constant. In particular, one-way functions (if they exist) have small circuits; by definition, one-way functions violate the condition C(f) = Θ(2m). A well-known candidate for a one-way permutation is the following. For a large prime number p and a primitive root r modulo p, consider the map x ∈ {0, 1, . . . , p−2} 7−→ rx−1 ∈ {0, 1, . . . , p−2}. This is a permutation whose inverse, known as the discrete logarithm, is believed to be difficult to compute. Measuring computational asymmetry: Let S{0,1}m denote the set of all permutations of {0, 1}m, i.e., S{0,1}m is the symmetric group. We will measure the computational asymmetry of all permutations of {0, 1}m (for all m > 0) by defining a computational asymmetry function, as follows. A function a : N → N is an upper bound on the computational asymmetry function iff for all all m > 0 and all permutations f of {0, 1}m we have: C(f−1) ≤ a . The computational asymmetry function α of the boolean permutations is the least such function a(.). Hence: Definition 1.1 The computational asymmetry function α of the boolean permutations is defined as follows for all s ∈ N : α(s) = max C(f−1) : C(f) ≤ s, f ∈ S{0,1}m , m > 0 Note that in this definition we look at all combinational circuits, for all permutations in m>0 S{0,1}m ; we don’t need to work with non-uniform or uniform families of circuits. Computational asymmetry is closely related to one-wayness, as the next proposition shows. Proposition 1.2 (1) For infinitely many n we have: There exists a permutation fn of {0, 1}n such that fn is computed by a circuit of size ≤ 3n, but f−1n has no circuit of size < α(n). (2) Suppose that α is exponential, i.e., there is k > 1 such that for all n, α(n) ≥ kn. Then k ≤ 2, and there is a constant c > 1 such that we have: For every integer n ≥ 1 there exists a permutation Fn of {0, 1}n which is computed by a circuit of size ≤ c n, but F−1n has no circuit of size < kn. Proof. (1) By the definition of α, for every m > 0 there exists a permutation F of {0, 1}m such that F is computed by a circuit of some size CF , but F −1 has no circuit of size < α(CF ). Let n = CF , and let us consider the function fn : {0, 1}CF → {0, 1}CF defined by fn : (x,w) 7−→ (F (x), w), for all x ∈ {0, 1}m and w ∈ {0, 1}CF−m. Then fn(x,w) is computed by a circuit of size CF + 2 (CF − m); the term “2 (CF −m)” comes from counting the input-output wires of w. Hence fn has a circuit of size ≤ 3n. On the other hand, (y,w) 7−→ f−1n (y,w) = (F−1(y), w) is not computed by any circuit of size < α(CF ), so f−1n has no circuit of size < α(n). (2) For every n ≥ 1 there exists a permutation F of {0, 1}n such that F is computed by a circuit of some size CF , and F −1 has a circuit of size CF−1 = α(CF ) ≥ kCF ; moreover, F−1 has no circuit of size < α(CF ). Thus, k CF ≤ CF−1 ≤ 2n (1 + co lognn ), for some constant co > 1; the latter inequality comes from the Lupanov upper bound [23] (or see Theorem 2.13.2 in [32]). Hence, k ≤ 2 and n ≤ CF ≤ 1log2 k n + c1 , for some constant c1 > 0. Hence, for all n ≥ 1 there exists a permutation F of {0, 1}n with circuit size CF ∈ [n, 1log2 k · n+ c1 · ], such that CF−1 = α(CF ) ≥ kCF ≥ kn. ✷ We will show later that the computational asymmetry function is closely related to the distortion of certain groups within certain monoids. Remarks: Although in this paper we only use the computational asymmetry function of the boolean permuta- tions, the concept can be generalized. Let Inj({0, 1}m, {0, 1}n) denote the set of all injective functions {0, 1}m → {0, 1}n. The computational asymmetry function αinj of the injective boolean functions is defined by αinj(s) = max C(f−1) : C(f) ≤ s, f ∈ Inj({0, 1}m, {0, 1}n), m > 0, n > 0 More generally, let ({0, 1}n){0,1}m denote the set of all functions {0, 1}m → {0, 1}n. The computa- tional asymmetry of all finite boolean functions is defined by αfunc(s) = max C(f ′) : C(f) ≤ s, ff ′f = f, f, f ′ ∈ ({0, 1}n){0,1}m , n > 0,m > 0 When we compare functions we will be mostly interested in their asymptotic growth pattern. Hence we will often use the big-O notation, and the following definitions. By definition, two functions f1 : N → N and f2 : N → N are linearly equivalent iff there are constants c0, c1, c2 > 0 such that for all n ≥ c0 : f1(n) ≤ c1 f2(c1n) and f2(n) ≤ c2 f1(c2n). Notation: f1 ≃lin f2. Two functions f1 and f2 (from N to N) are called polynomially equivalent iff there are constants c0, c1, c2, d, e > 0 such that for all n ≥ c0 : f1(n) ≤ c1 f2(c1nd)d and f2(n) ≤ c2 f1(c2ne)e. Notation: f1 ≃poly f2. 1.3 Wordlength asymmetry We introduce an algebraic notion that looks very similar to computational asymmetry: Definition 1.3 Let G be a group, let M be a monoid with generating set Γ (finite or infinite), and suppose G ⊆M . The word-length asymmetry function of G within M (over Γ) is λ(n) = max{ |g−1|Γ : |g|Γ ≤ n, g ∈ G}. The word-length asymmetry function λ depends on G, M , Γ, and the embedding of G in M . Consider the right Cayley graph of the monoid M with generating set Γ; its vertex set is M and the edges have the form x γ−→ γx (for x ∈ M , γ ∈ Γ). For x, y ∈ M , the directed distance d(x, y) in the Cayley graph is the shortest length over all paths from x to y in the Cayley graph; if no path from x to y exists, the directed distance is infinite. By “path” we always mean directed path. Lemma 1.4 Under the above conditions on G, M , Γ, we have for every g ∈ G : d(1, g−1) = d(g,1) and d(1, g) = d(g−1,1). Proof. Let η : Γ∗ →M be the map that evaluates generator sequences in M . If v ∈ Γ∗ is the label of a shortest path from 1 to g−1 in the Cayley graph then g ·η(v) = 1 inM , hence η(v) = g−1. Therefore, the path starting at g and labeled by v ends at 1; hence d(g,1) ≤ |v| = d(1, g−1). In a similar way one proves that d(1, g−1) ≤ d(g,1). The equality d(1, g) = d(g−1,1) is also proved in a similar way. Since |g|Γ is the distance d(1, g) in the graph of M , and since |g−1|Γ = d(1, g−1) = d(g,1), the word-length asymmetry also measures the asymmetry of the directed distance, to or from the identity element 1 in the Cayley graph of M , restricted to vertices in the subgroup G. For distances to or from the identity element of M it does not matter whether we consider the left Caley graph or the right Caley graph. 1.4 Computational asymmetry and reversible computing Reversible computing deals with the following questions: If a function f is injective (or bijective) and computable, can f be computed in such a way that each elementary computation step is injective (respectively bijective)? And if such injective (or bijective) computations are possible, what is their complexity, compared to the usual (non-injective) complexity? One of the main results is the following (Bennett’s theorem [4, 5], and earlier work of Lecerf [22]): Let f be an injective function, and assume f and f−1 are computable by deterministic Turing machines with time complexity Tf (.), respectively Tf−1(.). Then f (and also f −1) is computable by a reversible Turing machine (in which every transition is deterministic and injective) with time complexity O(Tf + Tf−1). Note that only injectiveness (not bijectiveness) is used here. Bennett’s theorem has the following important consequence, which relates reversible computing to one-way functions: Injective one-way functions exist iff there exist injective functions that have efficient traditional algorithms but that do not have efficient reversible algorithms. Toffoli representation Remarkably, it is possible to “simulate” any function f : {0, 1}m → {0, 1}n (injective or not, one- way or not) by a bijective circuit; a circuit is called bijective iff the circuit is made from bijective gates. Here, bijective circuits will be built from the wire swapping operations and the following bijective gates: not (negation), c-not (the Controlled Not, also called “Feynman gate”) defined by (x1, x2) ∈ {0, 1}2 7−→ (x1, x1 ⊕ x2) ∈ {0, 1}2, and cc-not (the Doubly Controlled Not, also called “Toffoli gate”) defined by (x1, x2, x3) ∈ {0, 1}3 7−→ (x1, x2, (x1 ∧ x2)⊕ x3) ∈ {0, 1}3. Theorem 1.5 (Toffoli [40]). For every boolean function f : {0, 1}m → {0, 1}n there exists a bijective boolean circuit βf (over the bijective gates not, c-not, cc-not, and wire transpositions), with input- output function βf : x 0 n ∈ {0, 1}m+n 7−→ f(x) x ∈ {0, 1}n+m. In other words, f(x) consists of the projection onto the first n bits of βf (x 0 n); equivalently, f(.) = projn ◦ βf ◦ concat0n(.), where projn projects a string of length n+m to the first n bits, and concat0n concatenates 0n to the right of a string. See Theorems 4.1, 5.3 and 5.4 of [40], and see Fig. 1 below. ✲ f(x) Fig. 1: Toffoli representation of the function f . The Toffoli representation contains two non-bijective actions: The projection at the output, and the forced setting of the value of some of the input wires. Toffoli’s proofs and constructions are based on truth tables, and he does not prove anything about the circuit size of βf (counting the bijective gates), compared to the circuit size of f . The following gives a polynomial bound on the size of the bijective circuit, at the expense of a large number of input- and output-wires. Theorem 1.6 (E. Fredkin, T. Toffoli [15]). For every boolean function f : {0, 1}m → {0, 1}n with circuit size C(f) there exists a bijective boolean circuit Bf (over a bounded collection of bijective gates, e.g., not, c-not, cc-not, and wire transpositions), with input-output function Bf : x 0 n+C(f) ∈ {0, 1}m+n+C(f) 7−→ f(x) z(x) ∈ {0, 1}m+n+C(f) for some z(x) ∈ {0, 1}m+C(f). If g : {0, 1}m → {0, 1}m is a permutation then there exists a bijective boolean circuit Ug (over bijective gates), with input-output function Ug : x 1 m 0m+C ∈ {0, 1}3m+C 7−→ g(x) g(x) x 0C ∈ {0, 1}3m+C where C = max{C(g), C(g−1)}, and g(x) is the bitwise complement of g(x). Later we will introduce another reversible representation of boolean functions by bijective gates; we will need only one 0-wire, but the gates will be taken from the Thompson group G2,1, i.e., we will also use non-length-preserving transformations of bitstrings (Theorems 4.1 and 4.2 below). 1.5 Distortion We will prove later (Theorem 5.10) that computational asymmetry has a lot to do with distortion, a concept introduced into group theory by Gromov [17] and Farb [14]. Distortion is already known to have connections with isoperimetric functions (see [28], [29], [24]). A somewhat different problem about distortion (for finite metric spaces) was tackled by Bourgain [10]. We will use a slightly more general notion of distortion, based on (possibly directed) countably infinite rooted graphs, and their (directed) path metric. A weighted directed graph is a structure (V,E, ω) where V is a set (called the vertex set), E ⊆ V ×V (called the edge set), and ω : E 7−→ R>0 is a function (called the weight function); note that every edge has a strictly positive weight. It is sometimes convenient to define ω(u, v) = ∞ when (u, v) ∈ V ×V −E. A path in (V,E) is a sequence of edges (ui, vi) (1 ≤ i ≤ n) such that ui+1 = vi for all i < n, and such that all elements in {ui : 1 ≤ i ≤ n} ∪ {vn} are distinct; u1 is called the start vertex of this path, and vn is called the end vertex of this path; the sum of weights i=1 ω(ui, vi) over the edges in the path is called the length of the path. Here we do not consider any paths with infinitely many edges; but we allow V and E to be countably infinite. A vertex w2 is said to be reachable from a vertex w1 in (V,E) iff there exists a path with start vertex w1 and end vertex w2. If w2 is reachable from w1 then the minimum length over all paths from w1 to w2 is called the directed distance from w1 to w2, denoted d(w1, w2); since we only consider finite paths here, this minimum exists. If w2 is not reachable from w1 then we define d(w1, w2) to be ∞. Clearly we have w1 = w2 iff d(w1, w2) = 0, and for all u, v, w ∈ V , d(u,w) ≤ d(u, v) + d(v,w). In a directed graph, the function d(., .) need not be symmetric. The function d : V ×V → R≥0 ∪{∞} is called the directed path metric of (V,E, ω). A rooted directed weighted graph is a structure (V,E, ω, r) where (V,E, ω) is a directed weighted graph, r ∈ V , and all vertices in V are reachable from r. A set M with a function d : M × M → R≥0 ∪ {∞}, satisfying the two axioms w1 = w2 iff d(w1, w2) = 0, and d(u,w) ≤ d(u, v) + d(v,w), will be called directed metric space (a.k.a. quasi-metric space). Any subset G embedded in a directed metric space M becomes a directed metric space by using the directed distance of M . We call this the directed distance on G inherited from M . If G ⊆ V for a rooted directed weighted graph (V,E, ω, r), we consider the function ℓ : g ∈ G 7−→ d(r, g) ∈ R≥0, which we call the directed length function on G inherited from (V,E, ω, r). (The value ∞ will not appear here since all of G is reachable from r.) We now define distortion in a very general way. Intuitively, distortion in a set is a quantitative comparison between two (directed) length functions that are defined on the same set. Definition 1.7 Let G be a set, and let ℓ1 and ℓ2 be two functions G → R≥0. The distortion of ℓ1 with respect to ℓ2 is the function δℓ1,ℓ2 : R≥0 → R≥0 defined by δℓ1,ℓ2(n) = max{ℓ1(g) : g ∈ G, ℓ2(g) ≤ n}. We will also use the notation δ[ℓ1, ℓ2](.) for δℓ1,ℓ2(.). When we consider a distortion δℓ1,ℓ2(.) we often assume that ℓ2 ≤ ℓ1 or ℓ2 ≤ O(ℓ1); this insures that the distortion is at least linear, i.e., δℓ1,ℓ2(n) ≥ c n, for some constant c > 0. We will only deal with functions obtained from the lengths of finite paths in countable directed graphs, so in that case the functions ℓi are discrete, and the distortion function exists. The next Lemma generalizes the distortion result of Prop. 4.2 of [14]. Lemma 1.8 Let G be a set and consider three functions ℓ3, ℓ2, ℓ1 : G→ R≥0 such that ℓ1(.) ≥ ℓ2(.) ≥ ℓ3(.). Then the corresponding distortions satisfy: δℓ1,ℓ3(.) ≤ δℓ1,ℓ2 ◦ δℓ2,ℓ3(.). Proof. The inequalities ℓ1(.) ≥ ℓ2(.) ≥ ℓ3(.) guarantee that the three distortions δℓ1,ℓ3 , δℓ1,ℓ2 , and δℓ2,ℓ3 are at least as large as the identity map. By definition, δℓ1,ℓ2 δℓ2,ℓ3(n) = max{ℓ1(x) : x ∈ G, ℓ2(x) ≤ δℓ2,ℓ3(n)} = max ℓ1(x) : x ∈ G, ℓ2(x) ≤ max{ℓ2(z) : z ∈ G, ℓ3(z) ≤ n} = max ℓ1(x) : x ∈ G, (∃z ∈ G) ℓ2(x) ≤ ℓ2(z) and ℓ3(z) ≤ n ≥ max{ℓ1(x) : x ∈ G, ℓ3(x) ≤ n} = δℓ1,ℓ3(n). The last inequality follows from the fact that if ℓ3(x) ≤ n then for some z (e.g., for z = x): ℓ2(x) ≤ ℓ2(z) and ℓ3(z) ≤ n. ✷ Examples of distortion: Distortion and asymmetry are unifying concepts that apply to many fields. 1. Gromov distortion: Let G be a subgroup of a group H, with generating sets ΓG, respectively ΓH , such that ΓG ⊆ ΓH , and such that ΓG = Γ−1G and ΓH = Γ H . This determines a Cayley graph for G and a Cayley graph for H. Now we have two distance functions on G, one obtained from the Cayley graph of G itself (based on ΓG), and the other inherited from the embedding of G in H. See [17], [10], and [14]. The Gromov distortion function is a natural measure of the difficulty of the generalized word problem. A very important case is when both ΓG and ΓH are finite. Here are some results for that case: Theorem of Ol′shanskii and Sapir [29] (making precise and proving the outline on pp. 66-67 in [17]): All Dehn functions of finitely presented groups (and “approximately all” time complexity functions of nondeterministic Turing machines) are Gromov distortion functions of finitely generated subgroups of FG2×FG2; here, FG2 denotes the 2-generated free group. Moreover, in [6] it was proved that FG2×FG2 is embeddable with linear distortion in the Thompson group G2,1. So the theorem of Ol′shanskii and Sapir also holds for the finitely generated subgroups of G2,1. Actually, Gromov [17] and Bourgain [10] defined the distortion to be 1 ·max{|g|ΓG : |g|ΓH ≤ n, g ∈ G}, i.e., they use an extra factor 1 . However, the connections between distortion, the generalized word problem, and complexity (as we just saw, and will further see in the present paper) are more direct without the factor 1 2. Bourgain’s distortion theorem: Given a finite metric space G with n elements, the aim is to find embeddings of G into a finite-dimensional euclidean space. The two distances of G are its given distance and the inherited euclidean distance. In this problem the goal is to have small distortion, as a function of the cardinality of G, while also keeping the dimension of the euclidean space small. Bourgain [10] found a bound O(n log n) for the distortion (or “O(log n)” in Bourgain’s and Gromov’s terminology). This is an important result. See also [21], [2], [3]. 3. Generator distortion: A variant of Gromov’s distortion is obtained when G = H, but ΓG $ ΓH . So here we look at the distorting effect of a change of generators in a given group. When ΓG and ΓH are both finite the generator distortion is linear; however, when ΓG is finite and ΓH is infinite the distortion becomes interesting. E.g., for the Thompson group G2,1 let us take ΓG to be any finite generating set, and for ΓH let us take ΓG ∪ {τi,j : 1 ≤ i < j}; here τi,j is the position transposition defined earlier. Then the generator distortion is exponential (see [7]). Also, the word problem of G2,1 over any finite generating set ΓG is in P, but the word problem of G2,1 over ΓG ∪ {τi,j : 1 ≤ i < j} is coNP-complete (see [7] and [8]). 4. Monoids and directed distance: Gromov’s distortion and the generator distortion can be generalized to monoids. We repeat what we said about Gromov distortion, but G and H are now monoids, and ΓG, respectively ΓH , are monoid generating sets which are used to define monoid Cayley graphs. We will use the left Cayley graphs. We assume ΓG ⊆ ΓH . In each Cayley graph there is a directed distance, defined by the lengths of directed paths. The monoid G now has two directed distance functions, the distance in the Cayley graph ofG itself, and the directed distance thatG inherits from its embedding into the Cayley graph of H. We denote the word-length of g ∈ G over ΓG by |g|G; this is the minimum length of all words over ΓG that represent g; it is also the length of a shortest path from the identity to g in the Cayley graph of G. Similarly, we denote the word-length of h ∈ H over ΓH by |h|H . The definition of the distortion becomes: δ(n) = max{|g|G : g ∈ G, |g|H ≤ n}. 5. Schreier graphs: Let G, H, and F be groups, where F is a subgroup of H. Let ΓH be a generating set of ΓH , and assume ΓH = Γ H . We can define the Schreier left coset graph of H/F over the generating set ΓH , and the distance function dH/F (., .) in this coset graph. By definition, this Schreier graph has vertex set H/F (i.e., the left cosets, of the form h · F with h ∈ H), and it has directed edges of the form h · F γ−→ γg · F , for h ∈ H, γ ∈ ΓH . The graph is symmetric; for every edge as above there is an opposite edge γh ·F γ −→ h ·F . Because of symmetry the Schreier graph has a (symmetric) distance function based on path length, dH/F (., .) : H/F ×H/F → N. Next, assume that G is embedded into H/F by some injective function G →֒ H/F . Such an embedding happens, e.g., if G and F are subgroups of H such that G∩F = {1}. Indeed, in that case each coset in H/F contains at most one element of G (since g1F = g2F implies g 2 g1 ∈ F ∩G = {1}). The group G now inherits a distance function from the path length in the Schreier graph of H/F . Comparing this distance with other distances in G leads to distortion functions. E.g., if the group G is also embedded in a monoid M with monoid generating set ΓM , this leads to the following distortion function: δG(n) = max{dH/F (F, gF ) : g ∈ G, |g|M ≤ n}. It will turn out that for appropriate choices of G,F,H, ΓH , and ΓM , this last distortion is polyno- mially related to the computational asymmetry function α of boolean permutations (Theorem 5.10). 6. Asymmetry functions: We already saw the computational asymmetry function of combinational circuits, and the word-length asymmetry function of a group embedded in a monoid. More generally, in any quasi-metric space (S, d), where d(., .) is a directed distance function, an asymmetry function A : R≥0 → R≥0 can be defined by A(n) = max{d(x2, x1) : x1, x2 ∈ S, d(x1, x2) ≤ n}. This asymmetry function can also be viewed as the distortion of drev with respect to d in S; here drev denotes the reverse directed distance, defined by drev(x1, x2) = d(x2, x1). 7. Other distortions: - Distortion can compare lengths of proofs (or lengths of expressions) in various, more or less pow- erful proof systems (respectively description languages). Distortion can also compare the duration of computations or of rewriting processes in various models of computation. Hence, many (perhaps all) notions of complexity are examples of distortion. Distortion is an algebraic or geometric representation (or cause) of complexity. - Instead of length and distance, other measures (e.g., volumes in higher dimension, energy, action, entropy, etc.) could be used. 1.6 Thompson-Higman groups and monoids The Thompson groups, introduced by Richard J. Thompson [38, 26, 39], are finitely presented infinite groups that act as bijections between certain subsets of {0, 1}∗. So, the elements of the Thompson groups are transformations of bitstrings, and hence they are related to input-output maps of boolean circuits. In this subsection we define the Thompson group G2,1 (also known as “V ”), as well as its generalization (by Graham Higman [18]) to the group Gk,1 that partially acts on A ∗, for any finite alphabet A of size k ≥ 2. We will follow the presentation of [6] (see also [8] and [7]); another reference is [33], which is also based on string transformations but with a different terminology; the classical references [38, 26, 39, 18, 12] do not describe the Thompson groups by transformations of finite strings. Because of our interest in strings and in circuits, we also use generalizations of the Thompson groups to monoids, as introduced in [9]. Some preliminary definitions, all fairly standard, are needed in order to define the Thompson- Higman group Gk,1. First, we pick any alphabet A of cardinality |A| = k. By A∗ we denote the set of all finite words (or “strings”) over A; the empty word ε is also in A∗. We denote the length of w ∈ A∗ by |w| and we let An denote the set of words of length n. We denote the concatenation of two words u, v ∈ A∗ by uv or by u · v; the concatenation of two subsets B,C ⊆ A∗ is defined by BC = {uv : u ∈ B, v ∈ C}. A right ideal of A∗ is a subset R ⊆ A∗ such that RA∗ ⊆ R. A generating set of a right ideal R is, by definition, a set C such that R is equal to the intersection of all right ideals that contain C; equivalently, C generates R (as a right ideal) iff R = CA∗. A right ideal R is called essential iff R has a non-empty intersection with every right ideal of A∗. For u, v ∈ A∗, we call u a prefix of v iff there exists z ∈ A∗ such that uz = v. A prefix code is a subset C ⊆ A∗ such that no element of C is a prefix of another element of C. A prefix code C over A is maximal iff C is not a strict subset of any other prefix code over A. It is easy to prove that a right ideal R has a unique minimal (under inclusion) generating set CR, and that CR is a prefix code; moreover, CR is a maximal prefix code iff R is an essential right ideal. For a partial function f : A∗ → A∗ we denote the domain by Dom(f) and the image (range) by Im(f). A restriction of f is any partial function f1 : A ∗ → A∗ such that Dom(f1) ⊆ Dom(f), and such that f1(x) = f(x) for all x ∈ Dom(f1). An extension of f is any partial function of which f is a restriction. An isomorphism between right ideals R1, R2 of A ∗ is a bijection ϕ : R1 → R2 such that for all r1 ∈ R1 and all z ∈ A∗: ϕ(r1z) = ϕ(r1) · z. The isomorphism ϕ is uniquely determined by a bijection between the prefix codes that minimally generate R1, respectively R2. One can prove [39, 33, 6] that every isomorphism ϕ between essential right ideals has a unique maximal extension (within the category of isomorphisms between essential right ideals of A∗); we denote this unique maximal extension by max(ϕ). Now, finally, we define the Thompson-Higman group Gk,1: It consists of all maximally extended isomorphisms between finitely generated essential right ideals of A∗. The multiplication consists of composition followed by maximum extension: ϕ · ψ = max(ϕ ◦ ψ). Note that Gk,1 acts partially and faithfully on A∗ on the left. Every element ϕ ∈ Gk,1 can be described by a bijection between two finite maximal prefix codes; this bijection can be described concretely by a finite function table. When ϕ is described by a maximally extended isomorphism between essential right ideals, ϕ : R1 → R2, we call the minimum generating set of R1 the domain code of ϕ, and denote it by domC(ϕ); similarly, the minimum generating set of R2 is called the image code of ϕ, denoted by imC(ϕ). Thompson and Higman proved that Gk,1 is finitely presented. Also, when k is even Gk,1 is a simple group, and when k is odd Gk,1 has a simple normal subgroup of index 2. In [6] it was proved that the word problem of Gk,1 over any finite generating set is in P (in fact, more strongly, in the parallel complexity class AC1). In [8, 7] it was proved that the word problem of Gk,1 over Γ∪{τi,j : 1 ≤ i < j} is coNP-complete, where Γ is any finite generating set of Gk,1, and where τi,j is the position transposition introduced in Subsection 1.1. Because of connections with circuits we consider the subgroup lpGk,1 of all length-preserving elements of Gk,1; more precisely, lpGk,1 = {ϕ ∈ Gk,1 : ∀x ∈ Dom(ϕ), |x| = |ϕ(x)|}. See [8] for a study of lpGk,1 and some of its properties. In particular, it was proved that lpGk,1 is a direct limit of finite alternating groups, and that lpG2,1 is generated by the set {N,C, T} ∪ {τi,i+1 : 1 ≤ i}, where N : x1w 7→ x1w, C : x1x2w 7→ x1 (x2 ⊕ x1)w, and T : x1x2x3w 7→ x1x2 (x3 ⊕ (x2 ∧ x1))w (for x1, x2, x3 ∈ {0, 1} and w ∈ {0, 1}∗). Thus (recalling Subsection 1.4), N,C, T are the not, c-not, cc-not gates, applied to the first (left-most) bits of a binary string. It is known that the gates not, c-not, cc-not, together with the wire-swappings, form a complete set of gates for bijective circuits (see [36, 40, 15]); hence, lpG2,1 is closely related to the field of reversible computing. It is natural to generalize the bijections between finite maximal prefix codes to functions between finite prefix codes. Following [9] we will define below the Thompson-Higman monoids Mk,1. First, some preliminary definitions. A right-ideal homomorphism of A∗ is a total function ϕ : R1 → A∗ such that R1 is a right ideal, and such that for all r1 ∈ R1 and all z ∈ A∗: ϕ(r1z) = ϕ(r1) · z. It is easy to prove that Im(ϕ) is then also a right ideal of A∗. From now on we will write a right-ideal homomorphism as a total surjective function ϕ : R1 → R2, where both R1 and R2 are right ideals. The homomorphism ϕ is uniquely determined by a total surjective function f : P1 → S2, with P1, S2 ⊂ A∗ where P1 is the prefix code (not necessarily maximal) that generates R1 as a right ideal, and where S2 is a set (not necessarily a prefix code) that generates R2 as a right ideal; f can be described by a finite function table. For two sets X,Y , we say that X and Y “intersect” iff X ∩ Y 6= ∅. We say that a right ideal R′1 is essential in a right ideal R1 iff R 1 intersects every right ideal that R1 intersects. An essential restriction of a right-ideal homomorphism ϕ : R1 → R2 is a right ideal-homomorphism Φ : R′1 → R′2 such that R′1 is essential in R1, and for all x 1 ∈ R′1: ϕ(x′1) = Φ(x′1). In that case we also say that ϕ is an essential extension of Φ. If Φ is an essential restriction of ϕ then R′2 = Im(Φ) will automatically be essential in R2 = Im(ϕ). Indeed, if I is any no-empty right subideal of R1 then I ∩R′1 6= ∅, hence ∅ 6= Φ(I ∩ R′1) ⊆ Φ(I) ∩ Φ(R′1) = Φ(I) ∩ R′2; moreover, any non-empty right subideal J of R2 is of the form J = Φ(I), where I = Φ−1(J) is a non-empty right subideal of R1; hence, for any non-empty right subideal J of R2, ∅ 6= J ∩R′2. The free monoid A∗ can be pictured by its right Cayley graph, which is easily seen to be the infinite regular k-ary tree with vertex set A∗ and edge set {(v, va) : v ∈ A∗, a ∈ A}. We simply call this the tree of A∗. It is a directed, rooted tree, with all paths directed away from the root ε (the empty word); by “path” we will always mean a directed path. Many of the previously defined concepts can be reformulated more intuitively in the context of the tree of A∗: A word v is a prefix of a word w iff v is an ancestor of w in the tree. A set P is a prefix code iff no two elements of P are on a common path. A set R is a right ideal iff any path that starts in R has all its vertices in R. The prefix code that generates R consists of the elements of R that are maximal (within R) in the prefix order, i.e., maximally close (along paths) to the root ε. A finitely generated right ideal R is essential iff every infinite path eventually reaches R (and then stays in it from there on). Similarly, a finite prefix code P is maximal iff any infinite path starting at the root eventually intersects P . For two finitely generated right ideals R′, R with R′ ⊂ R we have: R′ is essential in R iff any infinite path starting in R eventually reaches R′ (and then stays in it from there on). Assume now that a total order a1 < a2 < . . . < ak has been chosen for the alphabet A; this means that the tree of A∗ is now an oriented rooted tree, i.e., the children of each vertex v have a total order va1 < va2 < . . . < vak. The following can be proved (see [9], Prop. 1.4(1)): Φ is an essential restriction of ϕ iff Φ can be obtained from ϕ by starting from the table of ϕ and applying a finite number of restriction steps of the following form: “replace (x, y) in a table by {(xa1, ya1), . . . , (xak, yak)}”. In the tree of A∗ this means that x and y are replaced by their children xa1, . . . , xak, respectively ya1, . . . , yak, paired according to the order on the children. One can also prove (see [9], Remark after Prop. 1.4): Every right ideal homomorphism ϕ with table P → S has an essential restriction ϕ′ that has a table P ′ → Q′ such that both P ′ and Q′ are prefix codes. An important fact is the following (see [9], Prop. 1.4(2)): Every homomorphism between finitely generated right ideals of A∗ has a unique maximal essential extension; we call it the maximum essential extension of Φ and denote it by max(Φ). Finally here is the definition of the Thompson-Higman monoid: Mk,1 consists of all maximum es- sential extensions of homomorphisms between finitely generated right ideals of A∗. The multiplication is composition followed by maximum essential extension. One can prove the following, which implies associativity: For all right ideal homomorphisms ϕ1, ϕ2 : max(ϕ2 ◦ ϕ1) = max(max(ϕ2) ◦ ϕ1) = max(ϕ2 ◦max(ϕ1)). In [9] the following are proved about the Thompson-Higman monoid Mk,1: • The Thompson-Higman group Gk,1 is the group of invertible elements of the monoid Mk,1. • Mk,1 is finitely generated. • The word problem of Mk,1 over any finite generating set is in P. • The word problem of Mk,1 over a generating set Γ ∪ {τi,j : 1 ≤ i < j}, where Γ is any finite generating set of Mk,1, is coNP-complete. 2 Boolean functions as elements of Thompson monoids The input-output functions of digital circuits map bitstrings of some fixed length to bitstrings of a fixed length (possibly different from the input length). In other words, circuits have input-output maps that are total functions of the form f : {0, 1}m → {0, 1}n for some m,n > 0. The Thompson-Higman monoid Mk,1 has an interesting submonoid that corresponds to fixed-length maps, defined as follows. Definition 2.1 (the submonoid lepMk,1). Let ϕ : PA ∗ → QA∗ be a right-ideal homomorphism, where P,Q ⊂ A∗ are finite prefix codes, and where P is a maximal prefix code. Then ϕ is called length equality preserving iff for all x1, x2 ∈ Dom(ϕ) : |x1| = |x2| implies |ϕ(x1)| = |ϕ(x2)|. The submonoid lepMk,1 of Mk,1 consists of those elements of Mk,1 that can be represented by length-equality preserving right-ideal homomorphisms. It is easy to check that an essential restriction of an element of lepMk,1 is again in lepMk,1, so lepMk,1 is well defined as a subset of Mk,1; moreover, one can easily check that lepMk,1 is closed under composition, so lepMk,1 is indeed a submonoid of Mk,1. For ϕ ∈ Mk,1 we have ϕ ∈ lepMk,1 iff there exist m > 0 and n > 0 such that Am ⊂ Dom(ϕ) and ϕ(Am) ⊆ An. So (by means of an essential restriction, if necessary), ϕ can be represented by a function table Am → Q ⊆ An with a fixed input length and a fixed output length (but the input and output lengths can be different). The motivation for studying the monoid lepMk,1 is the following. Every boolean function f : {0, 1}m → {0, 1}n (for any m,n > 0) determines an element of lepMk,1, and conversely, this element of lepMk,1 determines f when restricted to {0, 1}m. By considering all boolean functions as elements of lepMk,1 we gain the ability to compose arbitrary boolean functions, even if their domain and range “do not match”. Moreover, in lepMk,1 we are able to generate all boolean functions from gates by using ordinary functional composition (instead of graph-based circuit lay-outs). The following remains open: Question: Is lepMk,1 finitely generated? However we can find nice infinite generating sets, in connection with circuits. Proposition 2.2 (Generators of lepMk,1). The monoid lepMk,1 has a generating set of the form Γ ∪ {τi,i+1 : 1 ≤ i}, for some finite subset Γ ⊂ lepMk,1. Proof. We only prove the result for k = 2; a similar reasoning works for all k (using k-ary logic). It is a classical fact that any function f : {0, 1}m → {0, 1}n can be implemented by a combinational circuit that uses copies of and, or, not, fork and wire-crossings. So all we need to do is to express theses gates, at any place in the circuit, by a finite subset of lepM2,1 and by positions transpositions τi,i+1. For each gate g ∈ {and, or} we define an element γg ∈ lepMk,1 by γg : x1x2w ∈ {0, 1}m 7−→ g(x1, x2) w ∈ {0, 1}m−1. Similarly we define γnot, γfork ∈ lepMk,1 by γnot : x1w ∈ {0, 1}m 7−→ x1 w ∈ {0, 1}m, γfork : x1w ∈ {0, 1}m 7−→ x1 x1 w ∈ {0, 1}m+1. For each g ∈ {and, or, not, fork}, γg transforms only the first one or two boolean variables, and leaves the other boolean variables unchanged. We also need to simulate the effect of a gate g on any variable xi or pair of variables xixi+1, i.e., we need to construct the map uxixi+1v ∈ {0, 1}m 7−→ u g(xi, xi+1) v ∈ {0, 1}m−1 (and similarly in case where g is not or fork). For this, we apply wire-transpositions to move xixi+1 to the wire-positions 1 and 2, then we apply γg, then we apply more wire-transpositions in order to move g(x1, x2) back to position i. Thus the effect of any gate anywhere in the circuit can be expressed as a composition of γg and position transpositions in {τi,i+1 : 1 ≤ i}. ✷ Proposition 2.3 (Change of generators of lepMk,1). Let {τi,i+1 : 1 ≤ i} be denoted by τ . If Γ,Γ′ ⊂ lepMk,1 are two finite sets such that Γ∪τ and Γ′∪τ generate lepMk,1, then the word-length over Γ ∪ τ is linearly related to the word-length over Γ′ ∪ τ . In other words, there are constants c′ ≥ c ≥ 1 such that for all m ∈ lepMk,1 : |m|Γ∪τ ≤ c · |m|Γ′∪τ ≤ c′ · |m|Γ∪τ . Proof. Since Γ is finite, the elements of Γ can be expressed by a finite set of words of bounded length (≤ c) over Γ′ ∪ τ . Thus, every word of length n over Γ∪ τ is equivalent to a word of length ≤ c n over Γ′ ∪ τ . This proves the first inequality. A similar reasoning proves the second inequality. ✷ Proposition 2.4 (Circuit size vs. lepM2,1 word-length). Let ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j} be a generating set of lepM2,1 with ΓlepM2,1 finite. Let f : {0, 1}m → Q (⊆ {0, 1}n) be a function defining an element of lepM2,1, and let |f |lepM2,1 the word-length of f over the generating set ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j}. Let |Cf | be the circuit size of f (using any finite universal set of gates and wire-swappings). Then |f |lepM2,1 and |Cf | are linearly related. More precisely, for some constants c1 ≥ co ≥ 1 : |Cf | ≤ co · |f |lepM2,1 ≤ c1 · |Cf |. Proof. For the proof we assume that the set of gates for circuits (not counting the wire-transpositions) is ΓlepM2,1 . If we make a different choice for the universal set of gates for circuits, and a different choice for the finite portion ΓlepM2,1 of the generating set of lepM2,1 then the inequalities remain the same, except for the constants c1, co. The inequality |Cf | ≤ |f |lepM2,1 is obvious, since a word w over ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j} is automatically a circuit of size |w|. For the other inequality, we want to simulate each gate of the circuit Cf by a word over ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j}. The reasoning is the same for every gate, so let us just focus on an or gate. The essential difference between circuit gates and elements of lepM2,1 is that in a circuit, a gate (with 2 input wires, for example) can be applied to any two wires in the circuit; on the other hand, the functions in lepM2,1 are applied to the first few wires. However, the circuit gate or, applied to (i, i+1) can be simulated by an element of ΓlepM2,1 and a few wire transpositions, since we have: ori,i+1(.) = γor ◦ τ2,i+1 ◦ τ1,i(.). The output wire of ori,i+1(.) is wire number i, whereas the output wire of γor ◦ τ2,i+1 ◦ τ1,i(.) is wire number 1. However, instead of permuting all the wires in order to place the output of γor τ2,i+1 τ1,i(.) on wire i, we just leave the output of γor τ2,i+1 τ1,i(.) on wire 1 for now. The simulation of the next gate will then use appropriate transpositions τ2,j · τ1,k for fetch the correct input wires for the next gate. Thus, each gate of Cf is simulated by one function in ΓlepM2,1 and a bounded number of wire-transpositions in {τi,j : 1 ≤ i < j}. At the output end of the circuit, a permutation of the n output wires is needed in order to send the outputs to the correct wires; any permutation of n elements can be realized with < n (≤ |Cf |) transpositions. (The inequality n ≤ |Cf | holds because since we count the output ports in the circuit size.) ✷ Remark. The above Proposition motivates our choice of generating set of the form Γ∪{τi,j : 1 ≤ i < j} (with Γ finite) for lepMk,1; in particular, it motivates the inclusion of all the position transpositions τi,j in the generating set. The Proposition also motivates the definition of word-length in which τi,j has word-length 1 for all j > i ≥ 1. Next we will study the distortion of lepMk,1 in Mk,1. We first need some Lemmas. Lemma 2.5 (Lemma 3.3 in [6]). If P,Q,R ⊆ A∗ are such that PA∗ ∩QA∗ = RA∗ and R is a prefix code, then R ⊆ P ∪Q. Proof. For any r ∈ R there are p ∈ P, q ∈ Q and v,w ∈ A∗ such that r = pv = qw. Hence p is a prefix of q or q is a prefix of p. Let us assume p is a prefix of q = px, for some x ∈ A∗ (the other case is similar) Hence q = px ∈ PA∗ ∩QA∗ = RA∗, and q is a prefix of r = qw. Since R is a prefix code, r = q, hence r ∈ Q. ✷ Lemma 2.6 Let P,Q ⊂ A∗ be finite prefix codes, and let θ : PA∗ → QA∗ be a right-ideal homomor- phism with domain PA∗ and image QA∗. Let S be a prefix code with S ⊂ QA∗. Then θ−1(S) is a prefix code and θ−1(SA∗) = θ−1(S) A∗. Proof. First, θ−1(S) is a prefix code. Indeed, if we had x1 = x2u for some x1, x2 ∈ θ−1(S) with u non-empty, then θ(x1) = θ(x2) u. This would contradict the assumption that S is a prefix code. Second, θ−1(S) ⊂ θ−1(SA∗), hence θ−1(S) A∗ ⊆ θ−1(SA∗), since θ−1(SA∗) is a right ideal. (Recall that the inverse image of a right ideal under a right-ideal homomorphism is a right ideal.) We also want to show that θ−1(SA∗) ⊆ θ−1(S) A∗. Let x ∈ θ−1(SA∗). So, θ(x) = sv for some s ∈ S, v ∈ A∗, and s = qu for some q ∈ Q, u ∈ A∗. Since θ(x) = quv, we have x = puv for some p ∈ P with θ(p) = q. Hence θ(pu) = qu = s. Therefore, x = puv with pu ∈ θ−1(s) ⊆ θ−1(S), hence x ∈ θ−1(S) A∗. ✷ Notation: For a right-ideal homomorphism ϕ : Dom(ϕ) = PA∗ → Im(ϕ) = QA∗, where P,Q ⊂ A∗ are finite prefix codes, we define ℓ(ϕ) = max{|z| : z ∈ P ∪Q}, For any finite prefix code C ⊂ A∗ we define ℓ(C) = max{|z| : z ∈ C}. Lemma 2.7 Let ϕ : Dom(ϕ) = PA∗ → Im(ϕ) = QA∗ be a right-ideal homomorphism, where P and Q are finite prefix codes. Let R ⊂ A∗ be any finite prefix code. Then we have: (1) ℓ(ϕ−1(R)) < ℓ(ϕ) + ℓ(R), (2) ℓ(ϕ(R)) < ℓ(ϕ) + ℓ(R). Proof. (1) Let r ∈ R ∩ Im(ϕ). Then every element of ϕ−1(r) has the form p1w for some p1 ∈ P and w ∈ A∗ such that r = q1w for some q1 ∈ Q (with ϕ(p1) = q1). Hence |p1w| = |p1| + |r| − |q1| = |r|+ |p1| − |q1|. Moreover, |r| ≤ ℓ(R) and |p1| − |q1| < ℓ(ϕ), so |p1w| < ℓ(R) + ℓ(ϕ). (2) If r ∈ R ∩Dom(ϕ) then ϕ(r) has the form q1v for some q1 ∈ Q and v ∈ A∗ such that r = p1w for some p1 ∈ P (with ϕ(p1) = q1). Hence |q1v| = |q1| + |r| − |p1| = |r| + |q1| − |p1|. Moreover, |r| ≤ ℓ(R) and |q1| − |p1| < ℓ(ϕ), so |q1w| < ℓ(R) + ℓ(ϕ). ✷ For any right-ideal homomorphisms ϕi (with i = 1, . . . , N), the composite map ϕN ◦ . . . ◦ ϕ1(.) is a right-ideal homomorphism. We say that right-ideal homomorphisms Φi (with i = 1, . . . , N) are directly composable iff Dom(Φi+1) = Im(Φi), for i = 1, . . . , N − 1. The next Lemma shows that we can replace composition by direct composition. Lemma 2.8 Let ϕi : Dom(ϕi) = PiA ∗ → Im(ϕi) = QiA∗ be a right-ideal homomorphism (for i = 1, . . . , N), where Pi and Qi are finite prefix codes. Then each ϕi has a (not necessarily essential) restriction to a right-ideal homomorphism Φi with the following properties: • ΦN ◦ . . . ◦ Φ1(.) = ϕN ◦ . . . ◦ ϕ1(.); • Dom(Φi+1) = Im(Φi), for i = 1, . . . , N − 1; • ℓ(Φi) ≤ j=1 ℓ(ϕj) for every i = 1, . . . , N . Proof. We use induction on N . For N = 1 there is nothing to prove. So we let N > 1 and we assume that the Lemma holds for ϕi : PiA ∗ → QiA∗ with i = 2, . . . , N , i.e., we assume that each ϕi (for i = 2, . . . , N) has a restriction ϕ i : P ∗ → Q′iA∗ such that ϕ′N ◦ . . . ◦ ϕ′2 = ϕN ◦ . . . ◦ ϕ2, P ′i+1 = Q i (for i = 2, . . . , N − 1), and ℓ(ϕ′i) ≤ j=2 ℓ(ϕj) for every i = 2, . . . , N . From P i+1 = Q (for i = 2, . . . , N − 1) it follows that ℓ(ϕ′N ◦ . . . ◦ ϕ′2) ≤ max{ℓ(ϕ′i) : i = 2, . . . , N} ≤ j=2 ℓ(ϕj). Using the notation ϕ′ [N,2] for ϕ′N ◦ . . . ◦ ϕ′2 we have Dom(ϕ′[N,2]) = P2A ∗ and Im(ϕ′ [N,2] ) = QNA When we compose ϕ1 and ϕ [N,2] we obtain ϕ−11 (Q1A ∗ ∩ P2A∗) Φ1−→ Q1A∗ ∩ P2A∗ [N,2]−→ ϕ′ [N,2] ∗ ∩ P2A∗). In this diagram, Φ1 is the restriction of ϕ1 to the domain ϕ 1 (Q1A ∗∩P2A∗) and image Q1A∗∩P2A∗; and Φ′ [N,2] is the restriction of ϕ′ [N,2] to the domain Q1A ∗ ∩ P2A∗ and image ϕ′[N,2](Q1A ∗ ∩ P2A∗). Hence, Φ′ [N,2] ◦Φ1 = ϕ′[N,2] ◦ϕ1, and Dom(Φ [N,2] ) = Im(Φ1) (= Q1A ∗ ∩P2A∗). So Φ1 and Φ′[N,2] are directly composable. By Lemma 2.5 there is a prefix code S ⊂ A∗ such that SA∗ = Q1A∗ ∩ P2A∗ and S ⊆ Q1 ∪ P2. Hence, ℓ(S) ≤ max{ℓ(Q1), ℓ(P2)} ≤ max{ℓ(ϕ1), ℓ(ϕ′2)} ≤ max{ℓ(ϕ1), j=2 ℓ(ϕj)} ≤ j=1 ℓ(ϕj). It follows also that ϕ−11 (Q1A ∗∩P2A∗) = ϕ−11 (SA∗) = ϕ 1 (S) A ∗ (the latter equality is from Lemma 2.6). Since S ⊆ Q1 ∪ P2 implies ϕ−11 (S) ⊆ ϕ 1 (Q1) ∪ ϕ 1 (P2) = P1 ∪ ϕ 1 (P2), we have ℓ(ϕ 1 (S)) ≤ max{ℓ(P1), ℓ(ϕ−11 (P2))}. Obviously, ℓ(P1) ≤ ℓ(ϕ1). Moreover, by Lemma 2.7, ℓ(ϕ 1 (P2)) ≤ ℓ(ϕ1) + ℓ(P2). Since ℓ(P2) ≤ ℓ(ϕ′2) ≤ j=2 ℓ(ϕj) (the latter “≤” by induction), we have ℓ(ϕ 1 (S)) ≤ ℓ(ϕ1)+ j=2 ℓ(ϕj) = j=1 ℓ(ϕj). Since the domain code of Φ1 is ϕ 1 (S) and its image code is S, we conclude that ℓ(Φ1) ≤ j=1 ℓ(ϕj). Let us now consider any Φ′ [i,2] , for i = 1, . . . , N . By definition, Φ′ [i,2] is the restriction of ϕ′i ◦ . . . ◦ϕ′2 to the domain SA∗. So the domain code of Φ′ [i,2] is S, and we just proved that ℓ(S) ≤ j=1 ℓ(ϕj). The image code of Φ′ [i,2] is ϕ′i ◦ . . . ◦ ϕ′2(S). Since S ⊆ Q1 ∪ P2 we have ϕ′i ◦ . . . ◦ ϕ′2(S) ⊆ ϕ′i ◦ . . . ◦ ϕ′2(Q1) ∪ ϕ′i ◦ . . . ◦ ϕ′2(P2) = ϕ′i ◦ . . . ◦ ϕ′2(Q1) ∪ Q′i. Therefore: ℓ(ϕ′i ◦ . . . ◦ ϕ′2(S)) ≤ max{ℓ(ϕ′i ◦ . . . ◦ ϕ′2(Q1)), ℓ(Q′i)}. We have ℓ(Q′i) ≤ ℓ(ϕ′i) ≤ j=2 ℓ(ϕj) (the last “≤” by induction). By Lemma 2.7, ℓ(ϕ′i ◦ . . . ◦ ϕ′2(Q1)) ≤ ℓ(ϕ′i ◦ . . . ◦ ϕ′2) + ℓ(Q1) ≤ ℓ(ϕ′i ◦ . . . ◦ ϕ′2) + ℓ(ϕ1). And ℓ(ϕ′i ◦ . . . ◦ ϕ′2) ≤ max{ℓ(ϕ′j) : j = 2, . . . , i}, because Dom(ϕ′r+1) = Im(ϕ′r) for all r = 2, . . . , N − 1. And by induction, ℓ(ϕ′j) ≤ j=2 ℓ(ϕj). Hence, ℓ(ϕ i ◦ . . . ◦ ϕ′2(Q1)) ≤ j=1 ℓ(ϕj). Thus, ℓ(Φ′ [i,2] j=1 ℓ(ϕj) for every i = 2, . . . , N . Finally, we factor Φ′ [N,2] as Φ′ [N,2] = ΦN ◦ . . . ◦Φ2, where Φi (for i = 2, . . . , N) is defined to be the restriction of ϕ′i to the domain ϕ i−1 ◦ . . . ◦ ϕ′2(SA∗) (= Φ′[i−1,2](SA ∗)). Since Dom(ϕ′r+1) = Im(ϕ (for all r = 2, . . . , N − 1), the domain of ϕ′i is equal to the image of ϕ′i−1 ◦ . . . ◦ ϕ′2. So, the domain code of Φi is ϕ i−1 ◦ . . . ◦ϕ′2(S), and its image code is ϕ′i ◦ϕ′i−1 ◦ . . . ◦ϕ′2(S). Since we already proved that ℓ(ϕ′i ◦ . . . ◦ ϕ′2(S)) ≤ j=1 ℓ(ϕj) (for all i), it follows that ℓ(Φi) ≤ j=1 ℓ(ϕj). ✷ In the next theorem we show that the distortion of lepMk,1 in Mk,1 is at most quadratic (over the generators considered so far, which include the bit position transpositions). Combined with Proposi- tion 2.4, this means the following: Assume circuits are built with gates that are not constrained to have fixed-length inputs and outputs, but assume the input-output function has fixed-length inputs and outputs. Then the resulting circuits are not much more compact than conventional circuits, built from gates that have fixed-length inputs and outputs (we gain at most a square-root in size). Theorem 2.9 (Distortion of lepMk,1 in Mk,1). The word-length (or Cayley graph) distortion of lepMk,1 in Mk,1 has a quadratic upper bound; in other words, for all x ∈ lepMk,1: |x|lepMk,1 ≤ c · (|x|Mk,1)2 where c ≥ 1 is a constant. Here the generating sets used are ΓMk,1 ∪ {τi,j : 1 ≤ i < j} for Mk,1, and ΓlepMk,1 ∪ {τi,j : 1 ≤ i < j} for lepMk,1, where ΓMk,1 and ΓlepMk,1 are finite. By |x|Mk,1 and |x|lepMk,1 we denote the word-length of x over ΓMk,1 ∪ {τi,j : 1 ≤ i < j}, respectively ΓlepMk,1 ∪ {τi,j : 1 ≤ i < j}. Proof. We only prove the result for k = 2; a similar proof applies for any k. We abbreviate the set {τi,j : 1 ≤ i < j} by τ . The choice of the finite sets ΓMk,1 and ΓlepMk,1 does not matter (it only affects the constant c in the Theorem. By Corollary 3.6 in [9] we can choose ΓMk,1 so that each γ ∈ ΓMk,1 satisfies the following (recall that ℓ(S) denotes the length of the longest words in a set S): domC(γ) ∪ imC(γ) ≤ 2, and ∣|γ(x)| − |x| ∣ ≤ 1 for all x ∈ Dom(γ). Let ϕ ∈ lepMk,1, and let w = αN . . . α1 be a shortest word over the generating set ΓMk,1 ∪ τ of Mk,1, representing ϕ. So N = |ϕ|Mk,1 . We restrict each partial function αi to a partial function α′i such that imC(α′i) = domC(α i+1) for i = 1, . . . , N−1, according to Lemma 2.8. Hence, αN ◦ . . .◦α1(.) = α′N ◦ . . . ◦ α′1(.), and ℓ(α′i) ≤ j=1 ℓ(αj) for every i = 1, . . . , N . Then αN ◦ . . . ◦ α1(.) is a function {0, 1}m {0, 1}∗ → Q {0, 1}∗, representing ϕ, and we will identify αN ◦ . . . ◦ α1(.) with ϕ. It follows that domC(α′1) = domC(ϕ) = {0, 1}m, and imC(α′N ) = imC(ϕ) = Q ⊆ {0, 1}n. More generally, it follows that imC(α′i ◦ . . . ◦ α′1) = imC(α′i), and domC(α′N ◦ . . . ◦ α′i) = domC(α′i). Since ℓ(α′i) ≤ j=1 ℓ(αj), and ℓ(αj) ≤ 2 for all j, we have for every i = 1, . . . , N : ℓ(α′i) ≤ 2N . From here on we will simply denote ℓ(α′i) by ℓi. Now, we will replace each α i ∈ Mk,1 by βi ∈ lepMk,1, such that domC(βi) = {0, 1}ℓi , and imC(βi) ⊆ {0, 1}ℓi+1 ; so βi is length-equality preserving. This will be done by artificially lengthening those words in domC(α′i) that have length < ℓi and those words in imC(α′i) that have length < ℓi+1. Moreover, we make βi defined on all of {0, 1}ℓi . In detail, βi is defined as follows: • If ℓi ≤ ℓi+1 : βi(u z) = v z 0 ℓi+1−ℓi−|v|+|u| for all u ∈ domC(α′i), and z ∈ {0, 1}ℓi−|u|; here v = α′i(u); βi(x) = x 0 ℓi+1−ℓi for all x 6∈ Dom(α′i), |x| = ℓi. • If ℓi > ℓi+1 : βi(u z1 z2) = v z1 for all u ∈ domC(α′i) and all z1, z2 ∈ {0, 1}∗ with |z1| = ℓi+1 − |v|, |z2| = ℓi − ℓi+1 + |v| − |u|; here, v = α′i(u); βi(x1 x2) = x1 for all x1, x2 ∈ {0, 1}∗ such that x1x2 6∈ Dom(α′i), with |x1| = ℓi+1, |x2| = ℓi − ℓi+1. Claim. βN ◦ . . . ◦ β1(.) = ϕ. Proof of the Claim: We observe first that domC(β1) = domC(α 1) (= domC(ϕ) = {0, 1}m). Next, assume by induction that for every x ∈ {0, 1}m : α′i−1 ◦ . . . ◦ α′1(x) = u is a prefix of βi−1 ◦ . . . ◦ β1(x) = u z. Then βi(u z) = v z 0 ℓi+1−ℓi−|v|+|u| (if ℓi ≤ ℓi+1); or βi(u z) = v z1 (if ℓi ≥ ℓi+1, with |z1| = ℓi+1 − |v| and z = z1z2). In either case we find that α′i(α′i−1 ◦ . . . ◦ α′1(x)) = v is a prefix of βi(βi−1 ◦ . . . ◦ β1(x)) = βi(u z). Hence, when i = N we obtain for any x ∈ {0, 1}m: βN ◦ . . . ◦ β1(x) = y s is a prefix of α′N ◦ . . . ◦ α′1(x) = ϕ(x) = y for some y and s with |y s| = ℓN = n. Since y ∈ imC(ϕ) ⊆ {0, 1}n we conclude that s is empty, hence βN ◦ . . . ◦ β1(x) = α′N ◦ . . . ◦ α′1(x). [End, proof of Claim.] At this point we have expressed ϕ as a product of N elements βi ∈ lepMk,1, where N = |ϕ|Mk,1 . We now want to find the word-length of each βi over ΓlepMk,1 ∪ τ , in order to find an upper bound on the total word-length of ϕ over ΓlepMk,1 ∪ τ . As we saw above, ℓi ≤ 2N for every i = 1, . . . , N . We examine each generator in ΓMk,1 ∪ τ . If αi ∈ τ then βi ∈ τ , so in this case |βi|lepMk,1 = 1. Suppose now that αi ∈ ΓMk,1 . By Proposition 2.4 it is sufficient to construct a circuit that computes βi; the circuit can then be immediately translated into a word over ΓlepMk,1 ∪ τ with linear increase in length. Since domC(αi) ⊆ {0, 1}≤2, we can restrict αi so that its domain code becomes a subset of {0, 1}2; next, we extend αi to a map α i that acts as the identity map on {0, 1}2 where αi was undefined. The image code of α′′i is a subset of {0, 1}≤3. In order to compute βi we first introduce a circuit C(α′′i ) that computes α′′i . A difficulty here is that α i does not produce fixed-length outputs in general, whereas C(α′′i ) has to work with fixed-length inputs and outputs; so the output of C(α i ) represents the output of α′′i indirectly, as follows: The circuit C(α′′i ) has two input bits u = u1u2 ∈ {0, 1}2, and 5 output bits: First there are 3 output bits 03−|v| v ∈ {0, 1}3, where v = α′′i (u); second, there are two more output bits, c1c2 ∈ {0, 1}2, defined by c1c2 = bin(3− |v|) (the binary representation of the non-negative integer 3− |v|). Hence, c1c2 = 00 if |v| = 3, c1c2 = 01 if |v| = 2, c1c2 = 10 if |v| = 1; since |v| > 0, the value c1c2 = 11 will not occur. Thus c1c2 0 3−|v| v contains the same information as v, but has the advantage of having a fixed length (always 5). The circuit C(α′′i ) can be built with a small constant number of and, or, not, fork gates, and we will not need to know the details. We now build a circuit for βi. • Circuit for βi if ℓi ≤ ℓi+1: On input u z ∈ {0, 1}ℓi (with u ∈ {0, 1}2), we want to produce the output v z 0ℓi+1−ℓi−|v|+|u|, where v = α′′i (u). We first apply the circuit C(α′′i ), thus obtaining c1c2 0 3−|v| v z. Then we apply two fork operations (always to the last bit in z) to produce c1c2 0 3−|v| v z b b, where b is the last bit of z. Applying a negation to the first b and an and operation, we obtain c1c2 0 3−|v| v z 0. Applying ℓi+1 − ℓi − 1 more fork operations to the last 0 yields c1c2 0 3−|v| v z 0ℓi+1−ℓi−1. Next, we want to move 03−|v| to the right of the output, in order to obtain c1c2 v z 0 3−|v|+ℓi+1−ℓi−1. For this effect we introduce a controlled cycle. Let κ : x1x2x3 ∈ {0, 1}3 7−→ x3x1x2 be the usual cyclic permutations of 3 bit positions. The controlled cycle acts as the identity map when c1c2 = 00 or 11, τ1,2 when c1c2 = 01, and κ when c1c2 = 10. More precisely, κc : c1c2 x1x2x3 ∈ {0, 1}5 7−→ c1c2 x1x2x3 if c1c2 = 00 or 11, c1c2 x2x1x3 if c1c2 = 01, c1c2 x3x1x2 if c1c2 = 10. We apply ℓi copies of κc(c1, c2, ., ., .) (all controlled by the same value of c1c2) to 0 3−|v| v z. The first κc(c1, c2, ., ., .) is applied to the 3 bits 0 3−|v| v, producing 3 bits y1y2y3; the second κc(c1, c2, ., ., .) is applied to y2y3 and the first bit of z, producing 3 bits y 3; the third κc(c1, c2, ., ., .) is applied to 3 and the second bit of z, etc. So, each one of the ℓi copies of κc acts one bit further down than the previous copy of κc. This will yield c1c2 v z 0 3−|v|+ℓi+1−ℓi−1. Finally, to make c1c2 disappear, we apply two fork operations to c1, then a negation and an and, to make a 0 appear. We combine this 0 with c1 and c2 by and gates, thus transforming 0c1c2 into 0. Finally, an or operation between this 0 and the first bit of v makes this 0 disappear. The number of gates used to compute βi is O(ℓi+1 + ℓi), which is ≤ O(N). • Circuit for βi if ℓi > ℓi+1: On input u z ∈ {0, 1}ℓi (with u ∈ {0, 1}2), we want to produce the output v z1, where v = α′′i (u). We first apply the circuit C(α′′i ), which yields the output c1c2 0 3−|v| v z. Now we want to erase the ℓi− ℓi+1+1 last bits of z. For this we apply two fork operations to the last bit of z (let’s call it b), then a negation and an and, to make a 0 appear. We combine this 0 with the last ℓi − ℓi+1 bits of z, using that many and gates, turning all these bits into a single 0; finally, an or operation between this 0 and the bit of the remainder of z makes this 0 disappear. At this point, the output is c1c2 0 3−|v| v Z1, where Z1 is the prefix of length ℓi+1 − 1 of z. Next, we apply O(ℓi+1) position transpositions to Z1 in order move the two last bits of Z1 to the front of Z1. Let b1b2 be the last two bits of Z1; so, Z1 = z0b1b2 (where z0 is the prefix of length ℓi+1−3 of z); at this point, the output of the circuit is c1c2 0 3−|v| v b1b2 z0. We now introduce a fixed small circuit with 7 input bits and 5 output bits, defined by the following input-output map: ωc : c1c2 x1x2x3 b1b2 ∈ {0, 1}7 7−→ c1c2 x1x2x3 if c1c2 = 00 or 11, c1c2 x1x2 b1 if c1c2 = 01, c1c2 x3 b1b2 if c1c2 = 10. When this map is applied to c1c2 0 3−|v| v b1b2 the output is therefore given by ωc : c1c2 0 3−|v| v b1b2 ∈ {0, 1}7 7−→ c1c2 v if |v| = 3, c1c2 v b1 if |v| = 2, c1c2 v b1b2 if |v| = 1. A circuit for ωc can be built with a small fixed number of and, or, not, fork gates, and we will not need to know the details. After applying ωc to c1c2 0 3−|v| v b1b2 z0 the output has length ℓi+1 + 2; the “+2” comes from c1c2. The output is c1c2 v z0, or c1c2 v b1 z0, or c1c2 v b1b2 z0, depending on whether |v| = 3, 2, or 1. We need to move b1b2 or b1 (or nothing) back to the right-most positions of z0. We do this by applying ℓi+1 copies of the controlled cycle κc(c1, c2, ., ., .) (all copies controlled by the same value of c1c2). We proceed in the same way as when we used κc in the previous case, and we obtain the output c1c2 v z0 (if |v| = 3), or c1c2 v z0 b1 (if |v| = 2), or c1c2 v z0 b1b2 (if |v| = 1). Finally, we erase c1c2 in the same way as in the previous case, thus obtaining the final output. The number of gates used to compute βi is O(ℓi+1 + ℓi) ≤ O(N). This completes the constuction of a circuit for βi. Through this circuit, βi : {0, 1}ℓi → {0, 1}ℓi+1 is expressed as a word over the generating set ΓlepMk,1 ∪ τ , of length ≤ O(ℓi+1 + ℓi) ≤ O(N). Since we have described ϕ as a product of N = |ϕ|Mk,1 elements βi ∈ lepMk,1, each of word-length O(N), we conclude that ϕ has word-length ≤ O(N2) over the generating set ΓlepMk,1 ∪ τ of lepMk,1. Question: Does the distortion of lepMk,1 inMk,1 (over the generators of Theorem 2.9) have an upper bound that is less than quadratic? 3 Wordlength asymmetry vs. computational asymmetry Proposition 3.1 The word-length asymmetry function λ of the Thompson group lpG2,1 within the Thompson monoid lepM2,1 is linearly equivalent to the computational asymmetry function α: α ≃lin λ. Here the generating set used for lepM2,1 is ΓlepM2,1 ∪ {τi,j : 0 ≤ i < j}, where ΓlepM2,1 is finite. The gates used for circuits are any finite universal set of gates, together with the wire-swapping operations {τi,j : 0 ≤ i < j}. We can choose ΓlepM2,1 to consist exactly of the gates used in the circuits; then α = λ. Proof. For any g ∈ lpG2,1 we have C(g−1) ≤ c0 · |g−1|lepM2,1 ≤ c0 · λ(|g|lepM2,1) ≤ c0 · λ(c1 · C(g)). The first and last “≤” come from Prop. 2.4 (since lpG2,1 ⊂ lepM2,1), and the middle “≤” comes from the definition of λ; c0 and c1 are positive constants. Hence, α(n) ≤ c0 · λ(c1 n) for all n. In a very similar way we prove that λ(n) ≤ c′0 · α(c′1 n) for some positive constants c′0, c′1. ✷ Proposition 3.2 The word-length asymmetry function λM2,1 of the Thompson group lpG2,1 within the Thompson monoid M2,1 is polynomially equivalent to the word-length asymmetry function λlepM2,1 of lpG2,1 within the Thompson monoid lepM2,1. More precisely we have for all n : λM2,1(n) ≤ c0 · λlepM2,1(c1 n2), λlepM2,1(n) ≤ c′0 · λM2,1(c where c0, c1, c 1 are positive constants. Here the generating set used for lepM2,1 is ΓlepM2,1 ∪ {τi,j : 0 ≤ i < j}, where ΓlepM2,1 is finite. The generating set used for M2,1 is ΓM2,1 ∪ {τi,j : 0 ≤ i < j}, where ΓM2,1 is a finite generating set of M2,1. Proof. For any g ∈ lpG2,1 we have |g−1|M2,1 ≤ c0 · |g−1|lepM2,1 ≤ c0 · λlepM2,1(|g|lepM2,1) ≤ c0 · λlepM2,1(c1 · |g|2M2,1). The first “≤” holds because lpG2,1 ⊂ lepM2,1 ⊂M2,1 and because of the choice of the generating sets. The second “≤” holds by the definition of λlepM2,1 . The third “≤” comes from the quadratic distortion of lepM2,1 in M2,1 (Theorem 2.9). For the same reasons we also have the following: |g−1|lepM2,1 ≤ c′0 · |g−1|2M2,1 ≤ c 0 · (λM2,1(|g|M2,1))2 ≤ c′0 · (λM2,1(c1 · |g|lepM2,1))2 where c′0, c 1 are positive constants. ✷ 4 Reversible representation over the Thompson groups Theorems 4.1 and 4.2 below introduce a representation of elements of the Thompson monoid lepM2,1 by elements of the Thompson group G2,1, in analogy with the Toffoli representation (Theorem 1.5 above), and the Fredkin representation (Theorem 1.6 above). Our representation preserves complexity, up to a polynomial change, and uses only one constant-0 input. Note that although the functions and circuits considered here use fixed-length inputs and outputs, the representations is over the Thompson group G2,1, which includes functions with variable-length inputs and outputs. In the Theorem below, ΓG2,1 is any finite generating set of G2,1. We denote the length of a word w by |w|, and we denote the size of a circuit C by |C|. The gates and, or, not will also be denoted respectively by ∧,∨,¬. We distinguish between a word Wf (over a generating set of G2,1) and the element wf of G2,1 represented by Wf . Theorem 4.1 (Representation of boolean functions by the Thompson group). Let f : {0, 1}m → {0, 1}n be any total function and let Cf be a minimum-size circuit (made of ∧,∨,¬, fork- gates and wire-swappings τi,j) that computes f . Then there exists a word Wf over the generating set ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1 such that: • For all x ∈ {0, 1}m: wf (0x) = 0 f(x) x, where wf is the element of G2,1 represented by Wf . • The length of the word Wf is bounded by |Wf | ≤ O(|Cf |4). • The largest subscript of any transposition τi,i+1 occurring in Wf has an upper bound ≤ |Cf |2 + 2. Proof. Wire-swappings in circuits are represented by the position transpositions τi,i+1 ∈ G2,1. The gates not, or, and and of circuits are represented by the following elements of G2,1: , ϕ∨ = 0x1x2 1x1x2 (x1 ∨ x2)x1x2 (x1 ∨ x2 )x1x2 , ϕ∧ = 0x1x2 1x1x2 (x1 ∧ x2)x1x2 (x1 ∧ x2 )x1x2 where x1, x2 range over {0, 1}. Hence the domain and image codes of ϕ∨ and ϕ∧ are all equal to {0, 1}3. To represent fork we use the following element, in which we recognize σ ∈ F2,1, one of the commonly used generators of the Thompson group F2,1: 0 10 11 00 01 1 00 01 10 11 000 001 01 1 Note that σ agrees with fork only on input 0, but that is all we will need. By its very essense, the forking operation cannot be represented by a length-equality preserving element of G2,1, because G2,1∩ lepM2,1 = lpG2,1 (the group of length-preserving elements of G2,1). A small remark: In [6, 7, 8], what we call “σ” here, was called “σ−1”. We will occasionally use the wire-swapping τi,j (1 ≤ i < j); note that τi,j can be expressed in terms of transpositions of neighboring wires as follows: τi,j(.) = τi,i+1 τi+1,i+2 . . . τj−2,j−1 τj−1,j τj−2,j−1 . . . τi+1,i+2 τi,i+1(.) so the word-length of τi,j over {τℓ,ℓ+1 : 1 ≤ ℓ} is ≤ 2(j − i)− 1. For x = x1 . . . xm ∈ {0, 1}m and f(x) = y = y1 . . . yn ∈ {0, 1}n, we will construct a word Wf over the generators ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1, such that Wf defines the map wf (.) : 0x 7→ 0 f(x) x. The circuit Cf is partitioned into slices cℓ (ℓ = 1, . . . , L). Two gates g1 and g2 are in the same slice iff the length of the longest path from g1 to any input port is the same as the length of the longest path from g2 to any input port. We assume that Cf is strictly layered, i.e., each gate in slice cℓ only has in-wires coming from slice cℓ−1, and out-wires going toward slice cℓ+1, for all ℓ. To make a circuit C strictly layered we need to add at most |C|2 identity gates (see p. 52 in [7]). The input-output map of slice cℓ has the form cℓ(.) : y (ℓ−1) = y (ℓ−1) 1 . . . y (ℓ−1) nℓ−1 ∈ {0, 1}nℓ−1 7−→ y(ℓ) = y 1 . . . y nℓ ∈ {0, 1}nℓ . Then y(0) = x and y(L) = y, where x ∈ {0, 1}m is the input and y ∈ {0, 1}n is the output of Cf . Each slice is a circuit of depth 1. Before studying in more detail how Cf is built from slices, let us see how a slice is built from gates (inductively, one gate at a time). Let C be a depth-1 circuit with k + 1 gates, obtained by adding one gate to a depth-1 circuit K with k gates. Let K(.) : x1 . . . xm 7−→ y1 . . . yn be the input-output map of the circuit K. Assume by induction that K is represented by a word WK over the generating set ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1. The input-output map of WK is, by induction hypothesis, wK(.) : 0x1 . . . xm 7−→ 0 y1 . . . yn x1 . . . xm. The word WC that represents C over G2,1 is obtained as follows from WK ; there are several cases, depending on the gate that is added to K to obtain C. Case 1: An identity-gate (or a not-gate) is added to K to form C, i.e., C(.) : x1 . . . xmxm+1 7−→ y1 . . . ynxm+1 (or, C(.) : x1 . . . xmxm+1 7−→ y1 . . . ynxm+1). Then WC is given by wC : 0x1x2 . . . xmxm+1 σ7−→ 00x1x2 . . . xmxm+1 τ3,m+37−→ 00xm+1 x2 . . . xmx1 ϕ∨7−→ xm+10xm+1 x2 . . . xmx1 τ3,m+37−→ xm+1 0x1x2 . . . xmxm+1 π7−→ 0x1x2 . . . xmxm+1xm+1 wK7−→ 0 y1 . . . yn x1 . . . xmxm+1xm+1 π′7−→ 0 y1 . . . yn xm+1 x1 . . . xmxm+1 , where π(.) = τm+1,m+2 . . . τ2,3 τ1,2(.) shifts xm+1 from position 1 to position m+ 2, while shifting 0x1 . . . xm one position to the left; and π ′(.) = τm+2,m+3 . . . τn+m+1,n+m+2 τn+m+2,n+m+3(.) shifts xm+1 from position n+m+ 3 to position n+ 2, while shifting x1 . . . xm one position to the right. So, WC = π ′ WK π τ3,m+3 ϕ∨ τ3,m+3 σ, noting that functions act on the left. Thus, |WC | = |WK |+m+n+5 if we use all of {τi,j : 1 ≤ i < j} in the generating set; over {τi,i+1 : 1 ≤ i}, τ3,m+3 has length ≤ 2m− 1, hence |WC | ≤ 3m + n + 4. If we denote the maximum index in the transpositions occurring in WC by JC then we have JC = max{JK , n+m+ 3}. In case a not-gate is added (instead of an identity gate), ϕ∨ is replaced by ϕ¬ ϕ∨ in WC , and the result is similar. Case 2: An and-gate (or an or-gate) is added to K to form C, i.e., C(.) : x1 . . . xmxm+1xm+2 7−→ y1 . . . yn (xm+1 ∧ xm+2) (or, C(.) : x1 . . . xmxm+1xm+2 7−→ y1 . . . yn (xm+1 ∨ xm+2)). Then WC is given by wC : 0x1x2 . . . xmxm+1xm+2 σ7−→ 00x1x2 . . . xmxm+1xm+2 τ2,m+37−→ τ3,m+47−→ 0xm+1xm+2 x2 . . . xm0x1 ϕ∧7−→ (xm+1 ∧ xm+2) xm+1xm+2 x2 . . . xm0x1 τ2,m+37−→ τ3,m+47−→ (xm+1 ∧ xm+2) 0x1x2 . . . xmxm+1xm+2 π7−→ 0x1x2 . . . xm (xm+1 ∧ xm+2) xm+1xm+2 wK7−→ 0 y1 . . . yn x1x2 . . . xm (xm+1 ∧ xm+2) xm+1xm+2 π′7−→ 0 y1 . . . yn (xm+1 ∧ xm+2) x1x2 . . . xmxm+1xm+2 , where π = τm+1,m+2 . . . τ2,3 τ1,2 shifts (xm+1∧xm+2) from position 1 to positionm+2, while shifting 0x1x2 . . . xm one position to the left; and π ′ = τm+2,m+3 . . . τm+n+1,m+n+2 shifts (xm+1 ∧ xm+2) from position n+m+ 2 to position m+ 2, while shifting x1 . . . xm one position to the right. So, WC = π ′ WK π τ3,m+4 τ2,m+3 ϕ∧ τ3,m+4 τ2,m+3 σ, hence |WC | = |WK | + n +m + 7 if all of {τi,j : 1 ≤ i < j} is used in the generating set; over {τi,i+1 : 1 ≤ i}, τ3,m+4 and τ2,m+3 have length ≤ 2(m+ 1)− 1, so |WC | ≤ |WK |+ 5m+ n+ 9. Moreover, JC = max{JK , m+ n+ 2}. Case 3: A fork-gate is added to K to form C, i.e., C(.) : x1 . . . xmxm+1 7−→ y1 . . . yn xm+1xm+1. Then WC is given by wC : 0x1x2 . . . xmxm+1 σ27−→ 000x1x2 . . . xmxm+1 τ3,m+47−→ 00xm+1x1x2 . . . xm0 ϕ∨7−→ xm+10xm+1x1x2 . . . xm0 τ1,m+47−→ 00xm+1x1x2 . . . xmxm+1 ϕ∨7−→ xm+10xm+1x1x2 . . . xmxm+1 0x1x2 . . . xmxm+1xm+1xm+1 wK7−→ 0 y1 . . . yn x1x2 . . . xm xm+1xm+1xm+1 π′7−→ 0 y1 . . . yn xm+1xm+1 x1x2 . . . xmxm+1 , where π = τm+3,m+4 . . . τ1,2 τm+3,m+4 . . . τ3,4 shifts the two copies of xm+1 at the left end from positions 1 and 3 to positions m+ 3 and m+ 4, while shifting 0 to position 1 and shifting x1 . . . xm two positions to the left; and π′ = τm+3,m+4 . . . τm+n+2,m+n+3 τm+2,m+3 . . . τm+n+1,m+n+2 shifts xm+1xm+1 from positions m + n + 2 and m + n + 3 to positions m + 2 and m + 3, while shifting x1 . . . xm two positions to the right. So, WC = π ′ WK π ϕ∨ τ1,m+4 ϕ∨ τ3,m+4 σ 2, hence |WC | = |WK | + 2m + n + 10, if all of {τi,j : 1 ≤ i < j} is used in the generating set; over {τi,i+1 : 1 ≤ i}, τ1,m+4 has length ≤ 2(m+3)−1 and τ3,m+4 has length ≤ 2m−1. Hence, |WC | ≤ |WK |+6m+n+14. Moreover, JC = max{JK , m+n+3}. In all cases, |WC | ≤ |WK |+c·(m+n+1) (for some constant c > 1), and JC ≤ max{JK , n+m+3}. Thus, each slice cℓ, with input-output map cℓ(.) : y (ℓ−1) 7−→ y(ℓ), is represented by a word Wcℓ with map wcℓ(.) : 0 y (ℓ−1) 7−→ 0 y(ℓ) y(ℓ−1), such that |Wcℓ| ≤ c · (n2ℓ−1 + n2ℓ) (for some constant c > 1), and Jcℓ ≤ nℓ−1 + nℓ + c. Regarding wire-crossings, we do not include them into other slices; we put the wire-crossings into pure wire-crossing slices. So we consider two kinds of slices: Slices entirely made of wire-crossings and identities, slices without any wire-crossings. Wire-crossings in circuits are identical to the group elements τi,i+1. We now construct the word Wf from the words Wcℓ (ℓ = 1, . . . , L). First observe that since the map wcℓ(.) is a right-ideal isomorphism (being an element of G2,1), we not only have wcℓ(.) : 0 y (ℓ−1) 7−→ 0 y(ℓ)y(ℓ−1) but also wcℓ(.) : 0 y (ℓ−1)y(ℓ−2) . . . y(1)y(0) 7−→ 0 y(ℓ)y(ℓ−1)y(ℓ−2) . . . y(1)y(0). Then, by concatenating all Wcℓ (and by recalling that y = y (L) and x = y(0)) we obtain wcL wcL−1 . . . wc2 wc1(.) : 0x 7−→ 0 y y(L−1) . . . y(2) y(1) x. Let πCf be the position permutation that shifts y right to the positions just right of x: πCf : 0 y y (L−1) . . . y(2) y(1) x 7−→ 0 y(L−1) . . . y(2) y(1) x y. Observe that for (WcL−1 . . . Wc2 Wc1) −1 we have (wcL−1 . . . wc2 wc1) −1(.) : 0 y(L−1) . . . y(2) y(1) x y 7−→ 0x y. Then we have: wcL wcL−1 . . . wc2 wc1 πCf (wcL−1 . . . wc2 wc1) −1(.) : 0x 7−→ 0x y . By using the position permutation πm,n : 0x y 7−→ 0 y x, we now see how to define Wf : Wf = πm,n WcL WcL−1 . . . Wc2 Wc1 πCf (WcL−1 . . . Wc2 Wc1) Then we have: wf (.) : 0x 7−→ 0 y x, where y = f(x). Finally, we need to examine the length of the word Wf in terms of the size of the circuit Cf that computes f : {0, 1}m → {0, 1}n. The position permutation πm,n shifts the n = |y| letters of y to the left over the m = |x| positions of x. So, πm,n can be written as the product of nm transpositions in {τi,i+1 : 1 ≤ i}, with maximum subscript Jπm,n ≤ m+ n+ 1. The position permutation πCf shifts y to the right from positions in the interval [2, n + 1] within the string 0 y y(L−1) . . . y(2) y(1) x to positions in the interval [2 + i=0 ni, 2 + i=0 ni] within the string 0 y(L−1) . . . y(2) y(1) x y. Note that i=0 ni = |Cf | (the size of the circuit Cf ), and nL = |y| = n, n0 = |x| = m. We shift y starting with the right-most letters of y. This takes i=0 ni = n (|Cf |−n) transpositions in {τi,i+1 : 1 ≤ i}, with maximum subscript JπCf = |Cf |+2. We saw already that |Wcℓ | ≤ c (n2ℓ−1 + n2ℓ), and Jcℓ ≤ nℓ−1 + nℓ + c, for some constant c > 1. Note that i=0 n i ≤ ( i=0 ni) 2 = |Cf |2. Hence we have: |Wf | ≤ co |Cf |2, for some constant co > 1. Moreover, the largest subscript in any transposition occurring in Wf is JWf ≤ |Cf |+ 2. Recall that we assumed that our circuit Cf was strictly layered, and that the circuit size has to be squared (at most) in order to make the circuit strictly layered. Thus, if Cf was originally not strictly layered, our bounds become |Wf | ≤ co |Cf |4, and JWf ≤ |Cf |2 + 2. ✷ The next theorem gives a representation of a boolean permutation by an element of the Thompson group G2,1; the main point of the theorem is the polynomial bound on the word-length in terms of circuit size. Theorem 4.2 (Representation of permutations by the Thompson group). Let g : {0, 1}m → {0, 1}m be any permutation and let Cg and Cg−1 be minimum-size circuits that compute g, respectively g−1. Then there exists a word W(g,g−1) over the generating set ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1, representing an element w(g,g−1) ∈ G2,1 such that: • For all x ∈ Dom(g) and all y ∈ Im(g): w(g,g−1)(0x) = 0 g(x), and (w(g,g−1)) −1(0 y) = 0 g−1(y), where (w(g,g−1)) −1 ∈ G2,1 is represented by the free-group inverse (W(g,g−1))−1 of the word W(g,g−1). • w(g,g−1)(.) and (w(g,g−1))−1 stabilize both 0 {0, 1}∗ and 1 {0, 1}∗. • We have a length upper bound |W(g,g−1)| = |(W(g,g−1))−1| ≤ O(|Cg|4 + |Cg−1 |4). • The largest subscript of transpositions τi,i+1 occurring in W(g,g−1) is ≤ max{|Cg|2, |Cg−1 |2} +2. Note that we distinguish between the word W(g,g−1) (over a generating set of G2,1) and the element w(g,g−1) of G2,1 represented by W(g,g−1). Also, note that although g is length-preserving (g ∈ lpG2,1), w(g,g−1) ∈ G2,1 is not length-preserving. Proof. Consider the position permutation π : 0 y x 7−→ 0x y, for all x, y ∈ {0, 1}m; we express π as a composition of ≤ m2 position transpositions of the form τi,i+1. Let Wg be the word constructed in Theorem 4.1 for g, and let Wg−1 be the word constructed for g −1. We define W(g,g−1) by W(g,g−1) = (Wg−1) −1 π Wg. Then for all x ∈ Dom(g) we have: w(g,g−1) : 0x 7−→ 0 y, where y = g(x). More precisely, for all x ∈ domC(g), wg−−→ 0 g(x) x = 0 y x π−→ 0 x y = 0 g−1(y) −−−−−−→ 0 y = 0 g(x). Since domC(g) is a maximal prefix code, w(g,g−1) maps 0 {0, 1}∗ into 0 {0, 1}∗ (where defined). Similarly, for all y ∈ Im(g) = Dom(g−1) we have: (w(g,g−1))−1 : 0 y 7−→ 0x, where x = g−1(y), y = g(x). Since domC(g−1) is a maximal prefix code, (w(g,g−1)) −1 maps 0 {0, 1}∗ into 0 {0, 1}∗ (where defined). Hence, elements of 0 {0, 1}∗ are never images of 1 {0, 1}∗. Thus, 1 {0, 1}∗ is also stabilized by w(g,g−1) and by (w(g,g−1)) The length of the wordW(g,g−1) is bounded as follows: We have |Wg| ≤ co |Cg|4, and |(Wg−1)−1| = |Wg−1 | ≤ co |Cg−1 |4, by Theorem 4.1. Moreover, π can be expressed as the composition of ≤ m2 (< |Cg|2) transpositions in {τi,i+1 : 1 ≤ i}. The bound on the subscripts also follows from Theorem 4.1. ✷ 5 Distortion vs. computational asymmetry We show in this Section that the computational asymmetry function α(.) is polynomially related to a certain distortion of the group lpG2,1. By Theorem 4.2, for every element g ∈ lpG2,1 there is an element w(g,g−1) ∈ G2,1 which agrees with g on 0 {0, 1}∗, and which stabilizes 0 {0, 1}∗ and 1 {0, 1}∗ . The main property of W(g,g−1) is that its length is polynomially bounded by the circuit sizes of g and g−1; that fact will be crucial later. First we want to study how w(g,g−1) is related to g. Recall that we distinguish between the word W(g,g−1) (over a generating set of G2,1) and the element w(g,g−1) of G2,1 represented by W(g,g−1). Theorem 4.2 inspires the following concepts. Definition 5.1 Let G be a subgroup of G2,1. For any prefix codes P1, . . . , Pk ⊂ {0, 1}∗, the joint stabilizer (in G) of the right ideals P1{0, 1}∗, . . . , Pk{0, 1}∗ is defined by StabG(P1, . . . , Pk) = g ∈ G : g(Pi{0, 1}∗) ⊆ Pi{0, 1}∗ for every i = 1, . . . , k The fixator (in G) of P1{0, 1}∗ is defined by FixG(P1) = g ∈ G : g(x) = x for all x ∈ P1{0, 1}∗) The fixator is also called “point-wise stabilizer”. The following is an easy consequence of the definition: FixG(Pi) is a subgroup of G (⊆ G2,1), for i = 1, . . . , k. If the prefix codes P1, . . . , Pk are such that the right ideals P1{0, 1}∗, . . . , Pk{0, 1}∗ are two-by-two disjoint, and such that P1 ∪ . . . ∪ Pk is a maximal prefix code, then StabG(P1, . . . , Pk) is closed under inverse. Hence in this case StabG(P1, . . . , Pk) is a subgroup of G. In particular, we will consider the following groups: • The joint stabilizer of 0 {0, 1}∗ and 1 {0, 1}∗, StabG(0, 1) = g ∈ G : g(0 {0, 1}∗) ⊆ 0 {0, 1}∗ and g(1 {0, 1}∗) ⊆ 1 {0, 1}∗ • The fixator of 0 {0, 1}∗, FixG(0) = {g ∈ G : g(x) = x for all x ∈ 0 {0, 1}∗}. • The fixator of 1 {0, 1}∗, FixG(1) = {g ∈ G : g(x) = x for all x ∈ 1 {0, 1}∗}. Clearly, FixG(0) and FixG(1) are subgroups of StabG(0, 1). Lemma 5.2 (Self-embeddings of G2,1). Let G be a subgroup of G2,1. Then G is isomorphic to FixG(1) and to FixG(0) by the following isomorphisms: Λ0 : g ∈ G 7−→ (g)0 ∈ FixG(1) Λ1 : g ∈ G 7−→ (g)1 ∈ FixG(0) where (g)0 and (g)1 defined as follows for any g ∈ G2,1: (g)0 : 0x ∈ 0 {0, 1}∗ 7−→ 0 g(x) 1x ∈ 1 {0, 1}∗ 7−→ 1x (g)1 : 1x ∈ 1 {0, 1}∗ 7−→ 1 g(x) 0x ∈ 0 {0, 1}∗ 7−→ 0x Proof. It is straightforward to verify that Λ0 and Λ1 are injective homomorphisms. That Λ0 is onto FixG(1) can be seen from the fact that every element of FixG(1) has a table of the form 0x1 . . . 0xn 1 0y1 . . . 0yn 1 where {x1, . . . , xn} and {y1, . . . , yn} are two maximal prefix codes, and x1 . . . xn y1 . . . yn is an arbitrary element of G. ✷ Lemma 5.3 Let G be a subgroup of G2,1. Then the direct product G×G is isomorphic to StabG(0, 1) by the isomorphism Λ : (f, g) ∈ G×G 7−→ 0x 7→ 0 f(x), 1x 7→ 1 g(x) ∈ StabG(0, 1). Proof. It is straightforward to verify that Λ is a homomorphism. That Λ is onto StabG(0, 1) and injective follows from the fact that every element of StabG(0, 1) has a table of the form 0x1 . . . 0xm 1x 1 . . . 1x 0y1 . . . 0ym 1y 1 . . . 1y where {x1, . . . , xm}, {y1, . . . , ym}, {x′1, . . . , x′n}, and {y′1, . . . , y′n}, are maximal prefix codes, and x1 . . . xm y1 . . . ym x′1 . . . x y′1 . . . y are arbitrary elements of G (⊆ G2,1). ✷ Lemmas 5.2 and 5.3 reveal certain self-similarity properties of the Thompson group G2,1. (Self- similarity of groups with total action on an infinite tree is an important subject, see [27]. However, the action of G2,1 is partial, so much of the known theory does not apply directly.) The stabilizer and the fixators above have some interesting properties. Lemma 5.4 . (1) For all f, g ∈ G: (f)0 (g)1 = (g)1 (f)0 (i.e., the commutator of FixG(0) and FixG(1) is the identity). (2) FixG(0) · FixG(1) = StabG(0, 1) and FixG(0) ∩ FixG(1) = 1; (3) StabG(0, 1) is the internal direct product of FixG(0) and FixG(1). (This is equivalent to the combination of (1) and (2).) (4) For all f, g ∈ G: Λ(f, g) = Λ0(f) · Λ1(g), Λ0(f) = Λ(f,1), and Λ1(g) = Λ(1, g). Moreover, FixG(0) = Λ1(G), FixG(1) = Λ0(G), and StabG(0, 1) = Λ(G×G). Proof. The proof is a straightforward verification. ✷ Lemma 5.5 For every position transposition τi,j, with 1 ≤ i < j, we have (τi,j)0 = τ2,i+1 ◦ τ3,j+1◦ (τ1,2)0 ◦ τ3,j+1 ◦ τ2,i+1. Hence, assuming (τ1,2)0 ∈ ΓG2,1 , and abbreviating {τi,j : 0 < i < j} by τ , we have: |(τi,j)0|ΓG2,1∪τ ≤ 5. Proof. Recall that for (τ1,2)0 we have, by definition, (τ1,2)0(1w) = 1w, and (τ1,2)0(0x2x3w) = 0x3x2w, for all w ∈ {0, 1}∗ and x2, x3 ∈ {0, 1}. The proof of the Lemma is a straightforward verification. ✷ Now we arrive at the relation between w(g,g−1) and g. Lemma 5.6 For all g ∈ lpG2,1 the following relation holds between g and w(g,g−1) : w(g,g−1) · (g)−10 , (g) 0 · w(g,g−1) ∈ FixlpG2,1(0). Equivalently, (g)0 · FixlpG2,1(0) = w(g,g−1) · FixlpG2,1(0), and FixlpG2,1(0) · (g)0 = FixlpG2,1(0) · w(g,g−1) . Proof. By Theorem 4.2 we have w(g,g−1)(0x) = 0 g(x) for all x ∈ Dom(g). So, w(g,g−1) and (g)0 act in the same way on 0 {0, 1}∗ . Also, both w(g,g−1) and (g)0 map 0 {0, 1}∗ into 0 {0, 1}∗, and both map 1 {0, 1}∗ into 1 {0, 1}∗. The Lemma follows from this. ✷ We abbreviate {τi,j : 0 < i < j} by τ . The element w(g,g−1) of G2,1, represented by the word W(g,g−1), belongs to StablpG2,1(0, 1) as we saw in Theorem 4.2. However, the word W(g,g−1) itself is a sequence over the generating set ΓG2,1 ∪ τ of G2,1. Therefore, in order to follow the action of W(g,g−1) and of its prefixes we need to take Fix(0) as a subgroup of G2,1. This leads us to the Schreier left coset graph of FixG2,1(0) within G2,1, over the generating set ΓG2,1 ∪ τ . By definition this Schreier graph has vertex set G2,1/FixG2,1(0), i.e., the left cosets, of the form g · FixG2,1(0) with g ∈ G2,1. And it has directed edges of the form g ·FixG2,1(0) γ−→ γg ·FixG2,1(0) for g ∈ G2,1, γ ∈ ΓG2,1 ∪ τ . Lemma 5.6 implies that for all g ∈ lpG2,1, (g)0 · FixG2,1(0) = w(g,g−1) · FixG2,1(0). We assume that ΓG2,1 = Γ , so the Schreier graph is symmetric, and hence it has a distance function based on path length; we denote this distance by dG/F (., .) : G2,1/FixG2,1(0)×G2,1/FixG2,1(0) −→ N. Lemma 5.7 There are injective morphisms g ∈ lpG2,1 →֒ g ∈ G2,1 ≃−→ (g)0 ∈ FixG2,1(1) ≃−→ (g)0 · FixG2,1(0) ∈ StabG2,1(0, 1)/FixG2,1(0), and an inclusion map (g)0 · FixG2,1(0) ∈ StabG2,1(0, 1)/FixG2,1(0) →֒ (g)0 · FixG2,1(0) ∈ G2,1/FixG2,1(0). In particular, g ∈ G2,1 7−→ (g)0 · FixG2,1(0) ∈ G2,1/FixG2,1(0) is an embedding of G2,1, as a set, into the vertex set G2,1/FixG2,1(0) of the Schreier graph. Proof. Recall that the map Λ0 : g ∈ G2,1 7−→ (g)0 ∈ FixG2,1(1) is a bijective morphism (Lemma 5.2). Also, the map u ∈ FixG2,1(1) 7−→ u · FixG2,1(0) ∈ G2,1/FixG2,1(0) is injective; indeed, if u · FixG2,1(0) = v · FixG2,1(0) with u, v ∈ FixG2,1(1) then v−1u ∈ FixG2,1(0) ∩ FixG2,1(1) = {1}. The map g ∈ G2,1 7−→ (g)0 · FixG2,1(0) ∈ StabG2,1(0, 1)/FixG2,1(0) is a surjective group homomorphism since FixG2,1(0) is a normal subgroup of StabG2,1(0, 1). Since FixG2,1(0)∩FixG(1) = {1}, this homomorphism is injective from FixG2,1(1) onto StabG2,1(0, 1)/FixG2,1(0). The combination of these maps provides an isomorphism from G2,1 onto StabG2,1(0, 1)/FixG2,1(0). Hence we also have an embedding of G2,1, as a set, into the vertex set G2,1/FixG2,1(0) of the Schreier graph. ✷ Since by Lemma 5.7 we can consider G2,1 as a subset of the vertex set G2,1/FixG2,1(0) of the Schreier graph, the path-distance dG/F (., .) on G2,1/FixG2,1(0) leads to a distance on G2,1, inherited from dG/F (., .) : Definition 5.8 For all g, g′ ∈ G2,1 the Schreier graph distance inherited by G2,1 is D(g, g′) = dG/F (g)0 · FixG2,1(0), (g′)0 · FixG2,1(0) The comparison of the Schreier graph distance D(., .) on lpG2,1 with the word-length that lpG2,1 inherits from its embedding into lepM2,1 leads to the following distortion of lpG2,1: Definition 5.9 In lpG2,1 we consider the distortion ∆(n) = max{D(1, g) : |g|lepM2,1 ≤ n, g ∈ lpG2,1}. We now state and prove the main theorem relating ∆(.) and α. Recall that α(.) is the computational asymmetry function of boolean permutations, defined in terms of circuit size. Theorem 5.10 (Computational asymmetry vs. distortion). The computational asymmetry function α(.) and the distortion ∆(.) of lpG2,1 are polynomially related. More precisely, for all n ∈ N : )1/2 ≤ c′ ·∆(n) ≤ c n4 + c · α(c n) where c ≥ c′ ≥ 1 are constants. Proof. The Theorem follows immediately from Lemmas 5.11 and 5.12. ✷ Lemma 5.11. There is a constant c ≥ 1 such that for all n ∈ N : ∆(n) ≤ c n4 + c · α(c n) Proof. By Lemma 5.6, (g)0 · FixG2,1(0) = w(g,g−1) · FixG2,1(0), hence FixG2,1(0), (g)0 · FixG2,1(0) FixG2,1(0), w(g,g−1) · FixG2,1(0) Since the wordW(g,g−1) and the Schreier graph use the same generating set, namely ΓG2,1 ∪ τ , we have FixG2,1(0), w(g,g−1) · FixG2,1(0) ≤ |W(g,g−1)|. By Theorem 4.2, |W(g,g−1)| ≤ O(|Cg|4 + |Cg−1 |4). And by the definition of the computational asymmetry function, |Cg−1 | ≤ α(|Cg |). Hence FixG2,1(0), (g)0 · FixG2,1(0) ≤ O(|Cg|4 + |Cg−1 |4) ≤ O |Cg|4 + α(|Cg|)4 By Proposition 2.4, |Cg| = O(|g|lepM2,1). Hence, for some constants c′′, c′ ≥ 1, FixG2,1(0), (g)0 · FixG2,1(0) ≤ c′ · |g|4 lepM2,1 + c′ · α(c′′ · |g|lepM2,1)4. Thus, FixG2,1(0), (g)0 · FixG2,1(0) : |g|lepM2,1 ≤ n, g ∈ lpG2,1 ≤ c′ n4 + c′ α(c′′ n)4. By Definition 5.9 of the distortion function ∆ we have therefore ∆(n) ≤ c′ n4 + c′ α(c′′ n)4. This proves the Lemma. ✷ Lemma 5.12 There is a constant c ≥ 1 such that for all n ∈ N : α(n) ≤ c ·∆(c n)2. Proof. We first prove the following. Claim: For every g ∈ lpG2,1, the inverse permutation g−1 can be computed by a circuit Cg−1 of size |Cg−1 | ≤ c · d FixG2,1(0), (g)0 · FixG2,1(0) , for some constant c ≥ 1. Proof of the Claim: There is a wordW ′ of length |W ′| = d FixG2,1(0), (g)0 ·FixG2,1(0) over ΓG2,1∪ τ that labels a shortest path from FixG2,1(0) to (g)0·FixG2,1(0) in the Schreier graph ofG2,1/FixG2,1(0). Let W = (W ′)−1 (the free-group inverse of W ′), so |W | = |W ′|. Let w be the element of G2,1 represented by W . Then W labels a shortest path from FixG2,1(0) to (g −1)0 · FixG2,1(0) in the Schreier graph of G2,1/FixG2,1(0); this path has length |W | = |W ′| = d FixG2,1(0), (g)0 · FixG2,1(0) FixG2,1(0), (g−1)0 · FixG2,1(0) We have w · FixG2,1(0) = (g−1)0 · FixG2,1(0), thus for all x ∈ {0, 1}∗ : w(0x) = 0 g−1(x). We now take the word VWU over the generating set ΓM2,1 ∪ τ of the monoid M2,1, where we choose the words U and V to be U = (and, not, fork, fork), and V = (or). The functions and, not, fork, or were defined in Subsection 1.1. Then for all x = x1 . . . xn ∈ {0, 1}∗, with x1, . . . , xn ∈ {0, 1}, we have x1 . . . xn fork−→ x1 x1 . . . xn fork−→ not−→ x1 x1 x1 . . . xn and−→ 0x1 . . . xn = 0x W−→ 0 g−1(x) or−→ g−1(x). The last or combines 0 and the first bit of g−1(x), and this makes 0 disappear. Thus overall, VWU(x) = g−1(x). The length is |V WU | = |W |+ 5. Since g−1 ∈ lpG2,1 ⊂ lepM2,1, Theorem 2.9 implies that there exists a word Z over the generators ΓlepM2,1 ∪ τ of lepM2,1 such that (1) |Z| ≤ c1 · |VWU |2, for some constant c1 ≥ 1, and (2) Z represents the same element of lepM2,1 as VWU , namely g Moreover, by Prop. 2.4, the word Z can be transformed into a circuit of size ≤ c2 · |Z| (for some constant c2 ≥ 1). This proves that there is a circuit Cg−1 for g−1 of size |Cg−1 | ≤ c · |W |2 (for some constant c ≥ 1). Since we saw that |W | = dG/F (FixG2,1(0), (g)0 · FixG2,1(0)), the Claim follows. [End, Proof of the Claim.] By definition, D(1, g) = dG/F (FixG2,1(0), (g)0 · FixG2,1(0)). Hence, by the Claim above: |Cg−1 | ≤ c · D(1, g) By Prop. 2.4 the word-length in lepM2,1 and the circuit size are linearly related; hence |g|lepM2,1 ≤ c0 |Cg|, for some constant c0 ≥ 1. Therefore, α(n) = max{|Cg−1 | : |Cg| ≤ n, g ∈ lpG2,1} ≤ max{|Cg−1 | : |g|lepM2,1 ≤ c0 n, g ∈ lpG2,1} ≤ max D(1, g) : |g|lepM2,1 ≤ c0 n, g ∈ lpG2,1 ≤ c · ∆(c0 n) This proves the Lemma. ✷ 6 Other bounds and distortions 6.1 Other distortions in the Thompson groups and monoids The next proposition gives more upper bounds on the computational asymmetry function α. Proposition 6.1. Assume ΓlepG2,1 ⊂ ΓlepM2,1 ⊂ ΓM2,1 . Let δlpG,lepM = δ |.|ΓlpG2,1∪τ , |.|ΓlepM2,1∪τ be the distortion function of lpG2,1 in the Thompson monoid lepM2,1, based on word-length. Similarly, let δlpG,M = δ |.|ΓlpG2,1∪τ , |.|ΓM2,1∪τ be the distortion function of lpG2,1 in the Thompson monoid M2,1. Then for some constant c ≥ 1 and for all n ∈ N, α(n) ≤ c · δlpG,lepM (c n) ≤ c · δlpG,M (c n). Proof. We first prove that δlpG,lepM (n) ≤ δlpG,M (n). Recall that by definition, δlpG,lepM (n) = max{|g|lpG2,1 : g ∈ lpG2,1, |g|lepM2,1 ≤ n}, and similarly for δlpG,M(n). Since ΓlepM2,1 ⊂ ΓM2,1 we have |x|lepM2,1 ≤ |x|M2,1 . Hence, {|g|lpG2,1 : g ∈ lpG2,1, |g|lepM2,1 ≤ n} ⊆ {|g|lpG2,1 : g ∈ lpG2,1, |g|M2,1 ≤ n}. By taking max over each of these two sets it follows that δlpG,lepM (n) ≤ δlpG,M(n). Next we prove that α(n) ≤ c·δlpG,lepM (c n). For any g ∈ lpG2,1 we have C(g−1) ≤ O(|g−1|lepM2,1), by Prop. 3.2. Moreover, |g−1|lepM2,1 ≤ |g−1|lpG2,1 since lpG2,1 is a subgroup of lepM2,1, and since the generating set used for lpG2,1 (including all τi,j) is a subset of the generating set used for lepM2,1. For any group with generating set closed under inverse we have |g−1|G = |g|G. And by the definition of the distortion δlpG,lepM we have |g|lpG2,1 ≤ δlpG,lepM (|g|lepM2,1). And again, by Prop. 3.2, |g|lepM2,1 ≤ O(C(g)). Putting all this together we have C(g−1) ≤ c1 · |g−1|lepM2,1 ≤ c1 · |g−1|lpG2,1 = c1 · |g|lpG2,1 ≤ c1 · δlpG,lepM (|g|lepM2,1) ≤ c1 · δlpG,lepM (c2 C(g)). Thus, c1 ·δlpG,lepM (c2 C(g)) is an upper bound on C(g−1). Since, by definition, α(C(g)) is the smallest upper bound on C(g−1), it follows that α(C(g)) ≤ c1 · δlpG,lepM (c2 C(g)). ✷ Recall that in the definition 5.9 of the distortion ∆ we compared D(., .) with the word-length in lepM2,1. If, instead, we compare D(., .) with the word-length inM2,1 we obtain the following distortion of lpG2,1 : δ(n) = max{D(1, g) : |g|M2,1 ≤ n, g ∈ lpG2,1}. Proposition 6.2 The distortion functions ∆(.) and δ(.) are polynomially related. More precisely, there are constants c′, c1, c2 ≥ 1 such that for all n ∈ N: ∆(n) ≤ c1 δ(n) ≤ c2 ∆(c′ n2). Proof. Let’s assume first that ΓlepM2,1 ⊆ ΓM2,1 , from which it follows that |g|M2,1 ≤ |g|M2,1 . Therefore, {D(1, g) : |g|lepM2,1 ≤ n} ⊆ {D(1, g) : |g|M2,1 ≤ n}. Hence, ∆(n) ≤ δ(n). By Theorem 2.9, |g|lepM2,1 ≤ c · |g|2M2,1 . So, {D(1, g) : |g|M2,1 ≤ n} ⊆ {D(1, g) : |g|lepM2,1 ≤ c n Hence, δ(n) ≤ ∆(c n2). When we do not have ΓlepM2,1 ⊆ ΓM2,1 , the constants in the theorem change, but the statement remains the same. ✷ 6.2 Monotone boolean functions and distortion On {0, 1}∗ we can define the product order, also called “bit-wise order”. It is a partial order (and in fact, a lattice order), denoted by “�”, and defined as follows. First, 0 ≺ 1; next, for any u, v ∈ {0, 1}∗ we have u � v iff |u| = |v| and ui � vi for all i = 1, . . . , |u|, where ui (or vi) denotes the ith bit of u (respectively v). By definition, a partial function f : {0, 1}∗ → {0, 1}∗ is monotone (also called “product-order preserving”) iff for all u, v ∈ Dom(f) : u � v implies f(u) � f(v). The following fact is well known (see e.g., [43] Section 4.5): A function f : {0, 1}m → {0, 1}n is monotone iff f can be computed by a combinational circuit that only uses gates of type and, or, fork, and wire-swappings; i.e., not is absent. A circuit of this restricted type is called a monotone circuit. Razborov [30] proved super-polynomial lower bounds for the size of monotone circuits that solve the clique problem, and in [31] he proved super-polynomial lower bounds for the size of monotone circuits that solve the perfect matching problem for bipartite graphs; the latter problem is in P. Tardos [37], based on work by Alon and Boppana [1], gave an exponential lower bound for the size of monotone circuits that solve a problem in P; see also [42] (Chapter 14 by Boppana and Sipser). Thus, there exist problems that can be solved by polynomial-size circuits but for which monotone circuits must have exponential size. In particular (for some constants b > 1, c > 0), there are infinitely many monotone functions fn : {0, 1}n → {0, 1}n such that fn has a combinational circuit of size ≤ nc, but fn has no monotone circuit of size ≤ bn. Based on an alphabet A = {a1, . . . , ak} with a1 ≺ a2 ≺ . . . ≺ ak we define a partial function f : A∗ → A∗ to be monotone iff f preserves the product order of A∗. The monotone functions enable us to define the following submonoid of the Thompson-Higman monoid lepMk,1 : monMk,1 = {ϕ ∈ lepMk,1 : ϕ can be represented by a monotone function P → Q, where P and Q are prefix codes, with P maximal }. An essential extension or restriction of an element of monMk,1 is again in monMk,1, so this set is well-defined as a subset of lepMk,1. It is easily seen to be closed under composition, so monMk,1 is a submonoid of lepMk,1. We saw that all monotone finite functions have circuits made from gates of type and, or, fork. Hence monM2,1 has the following generating set: {and, or, fork} ∪ {τi,j : j > i ≥ 1}. The results about monotone circuit size imply the following distortion result. Again, “exponential” refers to a function with a lower bound of the form n ∈ N 7−→ exp( c c′ n), for some constants c′ > 0 and c ≥ 1. Proposition 6.3 Consider the monoid monM2,1 over the generating set {and, or, fork} ∪ {τi,j : j > i ≥ 1}, and the monoid lepM2,1 over the generating set ΓlepM2,1 ∪ {τi,j : j > i ≥ 1}, where ΓlepM2,1 is finite. Then monM2,1 has exponential word-length distortion in lepM2,1. Proof. Let Γmon = {and, or, fork}. By Prop. 2.4 we have |f |ΓlepM2,1∪τ = |Cf |, where |Cf | denotes the ordinary circuit size of f . By a similar argument we obtain: |f |Γmon∪τ = |monCf |, where |monCf | denotes the monotone circuit size of f . We saw that as a consequence of the work of Razborov, Alon, Boppana, and Tardos, there exists an infinite set of monotone functions that have polynomial-size circuits but whose monotone circuit-size is exponential. 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Razborov, “Lower bounds for the monotone complexity of some boolean functions”, Doklady Akademii Nauk SSSR 281(4) (1985) 798-801. (English transl.: Soviet Mathematical Doklady 31 (1985) 354-357.) http://www.iacr.org/publications/dl/ [31] A.A. Razborov, “Lower bounds of monotone complexity of the logical permanent function”, Matematich- eskie Zametki 37(6) (1985) 887-900. (English transl.: Mathematical Notes of the Academy of Sciences of the USSR 37 (1985) 485-493.) [32] J.E. Savage, Models of Computation, Addison-Wesley (1998). [33] E.A. Scott, “A construction which can be used to produce finitely presented infinite simple groups”, J. of Algebra 90 (1984) 294-322. [34] A. Selman, “A survey of one-way functions in complexity theory”, Mathematical Systems Theory 25 (1992) 203-221. [35] C.E. Shannon, “The synthesis of two-terminal switching circuits”, Bell System Technical J. 28 (1949) 59-98. [36] V. Shende, A. Prasad, I. Markov, J. Hayes, “Synthesis of reversible logic circuits”, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 22(6) (2003) 710-722. [37] E. Tardos, “The gap between monotone and non-monotone circuit complexity is exponential”, Combina- torica 7(4) (1987) 141-142. [38] Richard J. Thompson, Manuscript (1960s). [39] R.J. Thompson, “Embeddings into finitely generated simple groups which preserve the word problem”, in Word Problems II, (S. Adian, W. Boone, G. Higman, editors), North-Holland (1980) pp. 401-441. [40] T. Toffoli, “Reversible computing”, MIT Laboratory for Computer Science, Technical Memo MIT/LCS/TM-151 (1980). [41] T. Toffoli, “Reversible computing”, Automata, Languages and Programming (7th Colloquium), Lecture Notes in Computer Science 85 (July 1980) 623-644. (Abridged version of [40].) [42] J. van Leeuwen (editor), Handbook of Theoretical Computer Science, volume A, MIT Press and Elsevier (1990). [43] I. Wegener, The complexity of boolean functions, Wiley/Teubner (1987). Jean-Camille Birget Dept. of Computer Science Rutgers University at Camden Camden, NJ 08102, USA birget@camden.rutgers.edu Introduction One-way functions and one-way permutations Computational Asymmetry Wordlength asymmetry Computational asymmetry and reversible computing Distortion Thompson-Higman groups and monoids Boolean functions as elements of Thompson monoids Wordlength asymmetry vs. computational asymmetry Reversible representation over the Thompson groups Distortion vs. computational asymmetry Other bounds and distortions Other distortions in the Thompson groups and monoids Monotone boolean functions and distortion

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 29 October 2018 (MN LATEX style file v2.2) A method for the direct determination of the surface gravities of transiting extrasolar planets John Southworth⋆, Peter J Wheatley and Giles Sams Department of Physics, University of Warwick, Coventry, CV4 7AL, UK 29 October 2018 ABSTRACT We show that the surface gravity of a transiting extrasolar planet can be calculated from only the spectroscopic orbit of its parent star and the analysis of its transit light curve. This does not require additional constraints, such as are often inferred from theoretical stellar models or model atmospheres. The planet’s surface gravity can therefore be measured precisely and from only directly observable quantities. We outline the method and apply it to the case of the first known transiting extrasolar planet, HD 209458b. We find a surface gravity of gp = 9.28± 0.15m s −2, which is an order of magnitude more precise than the best available measurements of its mass, radius and density. This confirms that the planet has a much lower surface gravity that that predicted by published theoretical models of gas giant planets. We apply our method to all fourteen known transiting extrasolar planets and find a significant correlation between surface gravity and orbital period, which is related to the known correlation between mass and period. This correlation may be the underlying effect as surface gravity is a fundamental parameter in the evaporation of planetary atmo- spheres. Key words: stars: planetary systems — stars: individual: HD 209458 — stars: bina- ries: eclipsing — stars: binaries: spectroscopic — methods: data analysis 1 INTRODUCTION Since the discovery that the star HD209458 is eclipsed by a planet in a short-period orbit (Henry et al. 2000; Charbonneau et al. 2000) it has become possible to derive the basic astrophysical properties of extrasolar planets and compare these quantities with theoretical predictions (e.g. Baraffe et al. 2003). However, the absolute masses, radii and density of the transiting planet cannot be calculated directly from the transit light curve and the velocity variation of the parent star, so extra information is required in order to obtain them. Additional constraints can be found from spectral analysis of the parent star or by imposing a theo- retical stellar mass–radius relation (Cody & Sasselov 2002), but this causes a dependence on theoretical stellar models or model atmospheres. The uncertainties in these constraints dominate the overall errors in mass, radius and density (e.g. Konacki et al. 2004), limiting the accuracy with which prop- erties of the star and planet can be measured. In this work we show that the surface gravity of the planet can be measured directly using only the transit light curve and the radial velocity amplitude of the parent star. No additional information is required and so accurate and ⋆ E-mail: j.k.taylor@warwick.ac.uk precise surface gravity values can be obtained. As theoretical studies often supply predicted values for the surface gravities of planetary objects (e.g., Baraffe et al. 2003) we propose that this quantity is very well suited for comparing observa- tion with theory. In addition, the surface gravity is an im- portant parameter in constructing theoretical models of the atmospheres of planets (Marley et al. 1999; Hubbard et al. 2001). After deriving an equation for surface gravity in terms of directly observed parameters, we illustrate this concept by studying HD209458. We then apply it to the other known transiting extradsolar planets using results available in the literature. The planet HD209458 b is known to be over- sized for its mass and to be strongly irradiated by the star it orbits. An excellent transit light curve was obtained for HD209458 by Brown et al. (2001), who used the HST/STIS spectrograph to obtain high-precision photometry cover- ing several different transit events. Precise radial velocity studies of HD209458 are also available (Henry et al. 2000; Mazeh et al. 2000; Naef et al. 2004). c© 0000 RAS http://arxiv.org/abs/0704.1570v1 2 Southworth, Wheatley & Sams Table 1. Results of the modelling of the HST/STIS light curve of HD209458. The upper part of the table gives the optimised parameters and the lower part gives quantities calculated from these parameters. The midpoint of the transit, TMin I, is expressed in HJD − 2 400 000. Limb darkening Linear Quadratic Square-root Adopted parameters r⋆+rp 0.12889± 0.00042 0.12771± 0.00049 0.12799± 0.00050 k 0.12260± 0.00011 0.12097± 0.00025 0.12051± 0.00037 i (deg.) 86.472± 0.040 86.665± 0.054 86.689± 0.060 86.677± 0.060 T0 51659.936716± 0.000021 51659.936712± 0.000021 51659.936711± 0.000021 u⋆ 0.494± 0.004 0.297± 0.027 −0.312± 0.127 v⋆ 0.338± 0.047 1.356± 0.218 r⋆ 0.11481± 0.00036 0.11393± 0.00042 0.11418± 0.00042 0.11405± 0.00042 rp 0.01408± 0.00006 0.01378± 0.00007 0.01376± 0.00008 0.01377± 0.00008 1.146 1.056 1.054 2 SURFACE GRAVITY MEASUREMENT The fractional radii of the star and the planet in the system are defined as where a is the orbital semi-major axis, and R⋆ and Rp are the absolute radii of the star and planet, respectively. r⋆ and rp can be directly determined from a transit light curve. The mass function of a spectroscopic binary is given by (e.g. Hilditch 2001): f(Mp) = (1− e2) p sin (M⋆ +Mp)2 whereK⋆ is the velocity amplitude of the star, e is the orbital eccentricity, P is the orbital period, i is the orbital inclina- tion, and M⋆ and Mp are the masses of the star and planet respectively. Including Kepler’s Third Law and solving for the sum of the masses of the two components gives: (M⋆ +Mp) 2πGM 3p sin (1− e2) 2K 3⋆ P (2π)4a6 G2P 4 By substituting Rp = arp into the definition of surface grav- ity and replacing a using Eq. 3 we find that the surface grav- ity of the planet, gp is given by: (1− e2) r 2p sin i Eq. 4 shows that we are able to calculate the surface gravity of a transiting extrasolar planet from the quantities P , K⋆, e, i and rp. This can be understood intuitively be- cause both the radial velocity motion of the star and the planet’s surface gravity are manifestations of the accelera- tion due to the gravity of the planet. A discussion in the context of eclipsing binaries is given by Southworth et al. (2004b). To measure gp using Eq. 4, the orbital period, P , can be obtained from either radial velocities or light curves of the system, and is typically determined very precisely compared to the other measurable quantities. The radial velocities also give e and K⋆, whilst the quantities i and rp can be obtained directly from the transit light curve. Note that it is also pos- sible to constrain the orbital eccentricity from observations of the secondary eclipse of a system. 3 APPLICATION TO HD209458B In order to measure the surface gravity for HD209458 b we need to know rp and i. These quantities are standard param- eters in the analysis of transit light curves. We have chosen to obtain them by modelling the high-precision HST/STIS light curve presented by Brown et al. (2001). We followed Brown et al. by rejecting data from the first HST orbit of each observed transit. To model the photometric data we used the jktebop code1 (Southworth et al. 2004a), which is a modified version of the ebop program (Popper & Etzel 1981; Etzel 1981). Giménez (2006) has shown that ebop is very well suited to the analysis of the light curves of transiting extrasolar planets. Importantly for this application, jktebop has been extended to treat limb darkening (LD) using several non- linear LD laws (Southworth et al. 2007). It also includes Monte Carlo and bootstrapping simulation algorithms for error analysis (Southworth et al. 2004a,b). ebop and jkte- bop model the two components of an eclipsing system using biaxial ellipsoids (Nelson & Davis 1972; Etzel 1975), so al- low for the deformation of the bodies from a spherical shape. When modelling the data we adopted the precise orbital period of 3.52474859 days given by Knutson et al. (2007). We fitted for the sum of the fractional radii, r⋆+rp, the ratio of the radii, k = , the orbital inclination, and the midpoint of a transit, T0. We also fitted for the LD coefficients of the star, rather than fixing them at values calculated using model atmospheres, to avoid introducing a dependence on theoretical predictions. The linear and non- linear LD coefficients are denoted by u⋆ and v⋆, respectively. We assumed that the planet contributes no light at the optical wavelengths considered here (see Wittenmyer et al. 2005) and that the orbit is circular (see Laughlin et al. 2005; Deming et al. 2005; Winn et al. 2005). Given suggestions that the choice of LD law can influence the derived inclina- tion (Winn et al. 2005), we obtained solutions for the linear, quadratic and square-root laws (Southworth et al. 2007). A mass ratio of 0.00056 was used (Knutson et al. 2007) but large changes in this parameter have a negligible effect on the solution. We have calculated robust 1 σ error estimates us- jktebop is written in fortran77 and the source code is avail- able at http://www.astro.keele.ac.uk/∼jkt/ c© 0000 RAS, MNRAS 000, 000–000 http://www.astro.keele.ac.uk/~jkt/ Direct determination of surface gravities of exoplanets 3 Figure 1. Best fit to the HST transit light curve of HD209458 using the quadratic LD law. The residuals of the fit are shown offset downwards by 0.02 in flux for clarity. ing Monte Carlo simulations (see Press et al. 1992, p.684; Southworth et al. 2004a), which we have previously found to provide very reliable results (Southworth et al. 2005a,b). This procedure assumes that systematic errors are negligi- ble. We find no reason to suspect that significant systematic errors remain in the HST light curve after the processing of this data described by Brown et al. (2001) (see Fig. 1). The best-fitting parameters of the transit light curve are given in Table 1. The best-fitting model using the quadratic LD law is shown in Fig. 1 along with the residuals of that fit. The solution using linear LD can be rejected as its reduced 2 is substantially larger than for the other two solutions. The quadratic and square-root LD laws give very similar solutions with reduced χ2 values close to one. For our final results we adopt the mean for each parameter along with uncertainties from the Monte Carlo simulations (Table 1). These are in good agreement with the light curve solutions obtained by Giménez (2006) and Mandel & Agol (2002), both of which used the approximation that the planet is spherical. With the orbital period given by Knutson et al. (2007), the stellar velocity ampitude K⋆ = 85.1 ± 1.0m s −2 from Naef et al. (2004), and the results of our light curve analysis (Table 1) we find the surface gravity of HD209458 b to be gp = 9.28±0.15m s −2. In this case the total uncertainty in gp is due to almost equal contributions from the uncertainties in K⋆ and rp. 4 APPLICATION TO ALL KNOWN TRANSITING EXTRASOLAR PLANETS We have calculated the surface gravity values for each of the known transiting extrasolar planets (apart from HD209458), using data taken from the literature (Table 2). In several cases (marked with asterisks in Table 2) it was not possible Table 2. Surface gravity values for the known transiting extra- solar planets. These have been calculated using Eq. 4 with input parameters taken from the literature. System Surface gravity Literature references m s−2 rp and i K⋆ HD 189733 21.5± 3.5 1 2 HD 209458 9.28± 0.15 3 4 OGLE-TR-10 4.5± 2.1 5 6 OGLE-TR-56 17.9± 1.9 5 2 OGLE-TR-111 13.3± 4.2 7 8 TrES-1 16.1± 1.0 9 10 WASP-1 10.6± 1.7 11 12 ∗ HAT-P-1 7.1± 1.1 13 13 ∗ XO-1 13.3± 2.5 14 14 ∗ HD 149026 16.4± 2.5 15 15 ∗ OGLE-TR-113 28.3± 4.4 16 17 ∗ OGLE-TR-132 18.0± 6.0 18 17 ∗ TrES-2 20.7± 2.6 19 19 ∗ WASP-2 20.1± 2.7 20 12 ∗ The surface gravity values for these objects have larger error estimates than are needed, because their fractional radii are not available in the literature. In these cases we have had to calculate them from Rp and a, which are less certain than rp because of the need to adopt an additional constraints to calculate them (see text). References: (1) Winn et al. (2007c); (2) Bouchy et al. (2005); (3) This work; (4) Naef et al. (2004); (5) Pont et al. (2007); (6) Konacki et al. (2005); (7) Winn et al. (2007a); (8) Pont et al. (2004); (9) Winn et al. (2007b); (10) Alonso et al. (2004); (11) Shporer et al. (2007); (12) Cameron et al. (2007); (13) Bakos et al. (2007); (14) McCullough et al. (2006); (15) Sato et al. (2005); (16) Gillon et al. (2006); (17) Bouchy et al. (2004); (18) Gillon et al. (2007); (19) O’Donovan et al. (2006); (20) Charbonneau et al. (2007). to obtain rp directly from the results available in the litera- ture. In these cases it had to be calculated from Rp and a, resulting in an increased uncertainty. This is because rp is a parameter obtainable directly from a transit light curve, whereas additional constraints (for example using theoreti- cal stellar models) are needed to calculate a and Rp. The orbital periods and surface gravities of all fourteen transiting extrasolar planets are plotted in Fig. 2, and show that these quantities are correlated. The linear Pearson cor- relation coefficient of these data is r = −0.70, indicating that the correlation is significant at better than the 0.5% level. This correlation is certainly related to that found by Mazeh et al. (2005) between the orbital periods and masses of the six transiting extrasolar planets then known. How- ever, it may be that surface gravity, rather than mass or radius, is the main parameter correlated with orbital period for these objects. Theoretical calculations have shown that surface gravity is a fundamental parameter in the evapora- tion rates of the atmospheres of irradiated gas giant planets (Lammer et al. 2003). 5 SUMMARY AND DISCUSSION We have shown that the surface gravity of transiting extraso- lar planets can be measured from analysis of the light curve c© 0000 RAS, MNRAS 000, 000–000 4 Southworth, Wheatley & Sams Figure 2. Comparison between the surface gravities and orbital periods of the known transiting exoplanets. Filled and open circles denote the systems in the upper and lower halves of Table 2, re- spectively. The errorbars for HD209458 (P = 3.52 d) are smaller than the plotted symbol. and a spectroscopic orbit of the parent star. We have anal- ysed the HST/STIS light curve of HD209458 (Brown et al. 2001) with the jktebop code. By combining the results of the light curve analysis with published spectroscopy (Naef et al. 2004) we find that the planet has a surface grav- ity of gp = 9.28 ± 0.15m s −2. We stress that this measure- ment does not depend on theoretical stellar evolutionary models or model atmospheres. In Fig. 3 we have plotted theoretical isochrones for ages of 0.5 to 10 Gyr from Baraffe et al. (2003) against the mass and surface gravity of HD209458 b, adopting a mass of Mp = 0.64±0.06 MJup from Knutson et al. (2007). The dis- crepancy between the observed and predicted surface gravity can clearly be seen. Note that these models do not include the effects of irradiation from the star. The density of a transiting extrasolar planet is of- ten used to compare observation with theory, but is typi- cally measured with a much lower precision than its sur- face gravity, given the same dataset. For example, the den- sity of HD209458 b derived by Knutson et al. (2007) is 345±50 kgm−3. Using the mass and radius given by Knutson et al. leads to gp = 9.1±0.9m s −2, where the uncertainty has been calculated by simple error propagation. These quanti- ties are both much less precise and require more elaborate calculations than using Eq. 4 to find the surface gravity: gp = 9.28± 0.15m s We have applied Eq. 4 to each of the known transiting extrasolar planets (Table 2). The resulting surface gravities show a clear correlation with orbital period (Fig. 2) which is connected with the known correlation between orbital period and mass for these objects. We propose that surface gravity may be the underlying parameter of the correlation due to its influence on the evaporation rates of the atmospheres of short-period giant planets. Figure 3. Plot of mass versus surface gravity for HD209458 b compared to the theoretical model predictions of Baraffe et al. (2003) for ages 0.5, 1.0, 5.0 and 10.0 Gyr (from lower to higher log g). As gp can be very precisely measured, and can be di- rectly compared with theoretical models and used to con- struct model atmospheres of the planet, we propose that it is an important parameter in our understanding of short- period extrasolar giant planets. In the near future, the high- precision light curves obtained by the CoRoT and Kepler satellites will allow accurate surface gravity values to be ob- tained for the terretrial-mass transiting extrasolar planets which these satellites should find. 6 ACKNOWLEDGEMENTS We are grateful to David Charbonneau for making the HST/STIS light curve of HD209458 available on his website (http://cfa-www.harvard.edu/∼dcharbon/frames.html), to Pierre Maxted for discussions, and to the referee whose report contributed significantly to improvements in this work. JS acknowledges financial support from PPARC in the form of a postdoctoral research assistant position. The fol- lowing internet-based resources were used in research for this paper: the NASA Astrophysics Data System; the SIM- BAD database operated at CDS, Strasbourg, France; and the arχiv scientific paper preprint service operated by Cor- nell University. 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A., et al., 2005, ApJ, 632, 1157 c© 0000 RAS, MNRAS 000, 000–000 http://arxiv.org/abs/astro-ph/0702192 http://arxiv.org/abs/astro-ph/0610556 Introduction Surface gravity measurement Application to HD209458b Application to all known transiting extrasolar planets Summary and discussion Acknowledgements

We show that the surface gravity of a transiting extrasolar planet can be calculated from only the spectroscopic orbit of its parent star and the analysis of its transit light curve. This does not require additional constraints, such as are often inferred from theoretical stellar models or model atmospheres. The planet's surface gravity can therefore be measured precisely and from only directly observable quantities. We outline the method and apply it to the case of the first known transiting extrasolar planet, HD 209458b. We find a surface gravity of g_p = 9.28 +/- 0.15 m/s, which is an order of magnitude more precise than the best available measurements of its mass, radius and density. This confirms that the planet has a much lower surface gravity that that predicted by published theoretical models of gas giant planets. We apply our method to all fourteen known transiting extrasolar planets and find a significant correlation between surface gravity and orbital period, which is related to the known correlation between mass and period. This correlation may be the underlying effect as surface gravity is a fundamental parameter in the evaporation of planetary atmospheres.

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On restrictions of balanced 2-interval graphs On restri tions of balan ed 2-interval graphs Philippe Gambette LIAFA, UMR CNRS 7089, Université Paris 7, Fran e Département Informatique, ENS Ca han, Fran e gambette�liafa.jussieu.fr Stéphane Vialette LRI, UMR CNRS 8623, Université Paris-Sud 11, Fran e vialette�lri.fr Abstra t The lass of 2-interval graphs has been introdu ed for modelling s heduling and allo ation problems, and more re ently for spe i� bioinformati problems. Some of those appli ations imply restri tions on the 2-interval graphs, and justify the introdu tion of a hierar hy of sub lasses of 2-interval graphs that generalize line graphs: balan ed 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instan es that show that all the in lusions are stri t. We extend the NP- ompleteness proof of re ognizing 2-interval graphs to the re ognition of balan ed 2-interval graphs. Finally we give hints on the omplexity of unit 2-interval graphs re ognition, by studying relationships with other graph lasses: proper ir ular-ar , quasi-line graphs, K1,5-free graphs, . . . Keywords: 2-interval graphs, graph lasses, line graphs, quasi-line graphs, law-free graphs, ir ular interval graphs, proper ir ular-ar graphs, bioinformati s, s heduling. 1 2-interval graphs and restri tions The interval number of a graph, and the lasses of k-interval graphs have been introdu ed as a generalization of the lass of interval graphs by M Guigan [M G77℄ in the ontext of s heduling and allo ation problems. Re ently, bioinformati s problems have renewed interest in the lass of 2-interval graphs (ea h vertex is asso iated to a pair of disjoint intervals and edges denote interse tion between two su h pairs). Indeed, a pair of intervals an model two asso iated tasks in s heduling [BYHN 06℄, but also two similar segments of DNA in the ontext of DNA omparison [JMT92℄, or two omplementary segments of RNA for RNA se ondary stru ture predi tion and omparison [Via04℄. (a) (b) ( ) Figure 1: Heli es in a RNA se ondary stru ture (a) an be modeled as a set of balan ed 2- intervals among all 2-intervals orresponding to omplementary and inverted pairs of letter sequen es (b), or as an independent subset in the balan ed asso iated 2-interval graph ( ). http://arxiv.org/abs/0704.1571v2 RNA (ribonu lei a id) are polymers of nu leotides linked in a hain through phosphodiester bonds. Unlike DNA, RNAs are usually single stranded, but many RNAmole ules have se ondary stru ture in whi h intramole ular loops are formed by omplementary base pairing. RNA se - ondary stru ture is generally divided into heli es ( ontiguous base pairs), and various kinds of loops (unpaired nu leotides surrounded by heli es). The stru tural stability and fun tion of non- oding RNA (n RNA) genes are largely determined by the formation of stable se ondary stru tures through omplementary bases, and hen e n RNA genes a ross di�erent spe ies are most similar in the pattern of nu leotide omplementarity rather than in the genomi sequen e. This motivates the use of 2-intervals for modelling RNA se ondary stru tures: ea h helix of the stru ture is modeled by a 2-interval. Moreover, the fa t that these 2-intervals are usually required to be disjoint in the stru ture naturally suggests the use of 2-interval graphs. Fur- thermore, aiming at better modelling RNA se ondary stru tures, it was suggested in [CHLV05℄ to fo us on balan ed 2-interval sets (ea h 2-interval is omposed of two equal length intervals) and their asso iated interse tion graphs referred as balan ed 2-interval graphs. Indeed, heli es in RNA se ondary stru tures are most of the time omposed of equal length ontiguous base pairs parts. To the best of our knowledge, nothing is known on the lass of balan ed 2-interval graphs. Sharper restri tions have also been introdu ed in s heduling, where it is possible to on- sider tasks whi h all have the same duration, that is 2-interval whose intervals have the same length [BYHN 06, Kar05℄. This motivates the study of the lasses of unit 2-interval graphs, and (x, x)-interval graphs. In this paper, we onsider these sub lasses of interval graphs, and in parti ular we address the problem of re ognizing them. A graph G = (V,E) of order n is a 2-interval graph if it is the interse tion graph of a set of n unions of two disjoint intervals on the real line, that is ea h vertex orresponds to a union of two disjoint intervals Ik = Ik ∪ Ikr , k ∈ J1, nK (l for � left� and r for �right�), and there is an edge between Ij and Ik i� Ij ∩ Ik 6= ∅. Note that for the sake of simpli ity we use the same letter to denote a vertex and its orresponding 2-interval. A set of 2-intervals orresponding to a graph G is alled a realization of G. The set of all intervals, k=1{I , Ikr }, is alled the ground set of G (or the ground set of {I1, . . . , In}). The lass of 2-interval graphs is a generalization of interval graphs, and also ontains all ir ular-ar graphs (interse tion graphs of ar s of a ir le), outerplanar graphs (have a planar embedding with all verti es around one of the fa es [KW99℄), ubi graphs (maximum degree 3 [GW80℄), and line graphs (interse tion graphs of edges of a graph). Unfortunately, most lassi al graph ombinatorial problems turn out to be NP- omplete for 2-interval graphs: re ognition [WS84℄, maximum independent set [BNR96, Via01℄, ol- oration [Via01℄, . . . Surprisingly enough, the omplexity of the maximum lique problem for 2-interval graphs is still open (although it has been re ently proven to be NP- omplete for 3-interval graphs [BHLR07℄). For pra ti al appli ation, restri ted 2-interval graphs are needed. A 2-interval graph is said to be balan ed if it has a 2-interval realization in whi h ea h 2-interval is omposed of two intervals of the same length [CHLV05℄, unit if it has a 2-interval realization in whi h all intervals of the ground set have length 1 [BFV04℄, and is alled a (x, x)-interval graph if it has a 2-interval realization in whi h all intervals of the ground set are open, have integer endpoints, and length x [BYHN+06, Kar05℄. In the following se tions, we will study those restri tions of 2-interval graphs, and their position in the hierar hy of graph lasses illustrated in Figure 2. Note that all (x, x)-interval graphs are unit 2-interval graphs, and that all unit 2-interval graphs are balan ed 2-interval graphs. We an also noti e that (1, 1)-interval graphs are exa tly line graphs: ea h interval of length 1 of the ground set an be onsidered as the vertex of a root graph and ea h 2-interval as an edge in the root graph. This implies for example that the Figure 2: Graph lasses related to 2-interval graphs and its restri tions. A lass pointing towards another stri tly ontains it, and the dashed lines mean that there is no in lusion relationship be- tween the two. Dark lasses orrespond to lasses not yet present in the ISGCI Database [BLS oloration problem is also NP- omplete for (2, 2)-interval graphs and wider lasses of graphs. It is also known that the omplexity of the maximum independent set problem is NP- omplete on (2, 2)-interval graphs [BNR96℄. Re ognition of (1, 2)-union graphs, a related lass (restri tion of multitra k interval graphs), was also re ently proven NP- omplete [HK06℄. 2 Useful gadgets for 2-interval graphs and restri tions For proving hardness of re ognizing 2-interval graphs, West and Shmoys onsidered in [WS84℄ the omplete bipartite graph K5,3 as a useful 2-interval gadget. Indeed, all realizations of this graph are ontiguous, that is, for any realization, the union of all intervals in its ground set is an interval. Thus, by putting edges between some verti es of a K5,3 and another vertex v, we an for e one interval of the 2-interval v (or just one extremity of this interval) to be blo ked inside the realization of K5,3. It is not di� ult to see that K5,3 has a balan ed 2-interval realization, for example the one in Figure 3. (a) (b) ( ) Figure 3: The omplete bipartite graph K5,3 (a,b) has a balan ed 2-interval realization ( ): verti es of S5 are asso iated to balan ed 2-intervals of length 7, and verti es of S3 are asso iated to balan ed 2-intervals of length 11. Any realization of this graph is ontiguous, i.e., the union of all 2-intervals is an interval. However, K5,3 is not a unit 2-interval graph. Indeed, ea h 2-interval I = Il∪Ir orresponding to a degree 5 vertex interse t 5 disjoint 2-intervals, and hen e one of Il or Ir interse t at least 3 intervals, whi h is impossible for unit intervals. Therefore, we introdu e the new gadget K4,4−e whi h is a (2, 2)-interval graph with only ontiguous realizations. (a) (b) ( ) Figure 4: The graph K4,4 − e (a), a ni er representation (b), and a 2-interval realization with open intervals of length 2 ( ). Property 1. Any 2-interval realization of K4,4 − e is ontiguous. Proof. Write G = (V,E) the graph K4,4− e. To study all possible realizations of G, let us study all possible realizations of G[V − I8]. As 2-intervals I1, I2, I3 and I4 are disjoints, their ground set I = {[li, ri], 1 ≤ i ≤ 8, ri < li+1} is a set of eight disjoint intervals. The ground set Imobile of I , I6 and I7 is a set of six disjoint intervals. Let xi be the number of intervals of Imobile interse ting i ≤ 8 intervals of . We have dire tly: x0 + x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 = |Imobile| = 6. (1) As there are 12 edges in G[V \{v8}] whi h is bipartite, we also have: x1 + 2x2 + 3x3 + 4x4 + 5x5 + 6x6 + 7x7 + 8x8 ≥ 12. (2) Finally, to build a realization of G from a realization of G[V \{v8}] , one must pla e I so as to interse t three disjoint intervals of I . Thus one of the intervals of I8 interse ts at least two intervals ]lk, rk[ and ]ll, rl[ (k < l) of I�xed. So there is �a hole between those two intervals�, for example [rk, lk+1], whi h is in luded in one of the intervals of I . So we noti e that I8 has to �ll one of the seven holes of I . Thus, the intervals of I mobile an not �ll more than six holes, and the observation that an interval interse ting i onse utive intervals (for i ≥ 1) �lls i− 1 holes, we get: x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 7x8 ≤ 6. (3) Equations 1, 2 and 3 are ne essary for any valid realization of G[V \{v8}] whi h gives a valid realization of G. Let's suppose by ontradi tion that the union of all intervals of the ground set of G is not an interval. Then there is a hole, that is an interval in luded in the overing interval of {I1, . . . , I8}, whi h interse t no Ii. We pro eed like for equation 3, with the onstraint that another hole annot be �lled by the intervals of I mobile , so we get instead: x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 7x8 ≤ 5. (4) By adding 1 and 4, and subtra ting 2, we get x0 ≤ −1 : impossible! So we have proved that the union of all intervals of the ground set of any realization of G is indeed an interval. 3 Balan ed 2-interval graphs We show in this se tion that the lass of balan ed 2-interval graphs is stri tly in luded in the lass of 2-interval graphs, and stri tly ontains ir ular-ar graphs. Moreover, we prove that re ognizing balan ed 2-interval graphs is as hard as re ognizing (general) 2-interval graphs. Property 2. The lass of balan ed 2-interval graphs is stri tly in luded in the lass of 2-interval graphs. Proof. We build a 2-interval graph that has no balan ed 2-interval realization. Let's onsider a hain of gadgets K5,3 (introdu ed in previous se tion) to whi h we add three verti es I , I2, and I3 as illustrated in Figure 5. Figure 5: An example of unbalan ed 2-interval graph (a) : any realization groups intervals of the seven K5,3 in a blo k, and the hain of seven blo ks reates six �holes� between them, whi h make it impossible to balan e the lengths of the three 2-intervals I1, I2, and I3. In any realization, the presen e of holes showed by rosses in the Figure gives the following inequalities for any realization: l(Il 2) < l(Il 1), l(Il 3) < l(Ir 2), and l(Ir 1) < l(Ir 3) (or if the realization of the hain of K5,3 appears in the symmetri al order: l(Il 1) < l(Il 3), l(Ir 3) < l(Il and l(Ir 2) < l(Ir 1)). If this realization was balan ed, then we would have l(Il 1) = l(Ir 3) = l(Il 3) < l(Ir 2) = l(Il 2) (or for the symmetri al ase: l(Ir 1) = l(Il 1) < l(Il 3) = l(Ir 2) = l(Ir 2)) : impossible! So this graph has no balan ed 2-interval realization although it has a 2-interval generalization. Theorem 1. Re ognizing balan ed 2-interval graphs is an NP- omplete problem. Proof. We just adapt the proof of West and Shmoys [WS84, GW95℄. The problem of determining if there is a Hamiltonian y le in a 3-regular triangle-free graph is proven NP- omplete, by redu tion from the more general problem without the no triangle restri tion. So we redu e the problem of Hamiltonian y le in a 3-regular triangle-free graph to balan ed 2-interval re ognition. Let G = (V,E) be a 3-regular triangle-free graph. We build a graph G′ whi h has a 2- interval realization (a spe ial one, very spe i� , alled H-representation and whi h we prove to be balan ed) i� G has a Hamiltonian y le. The onstru tion of G′, illustrated in Figure 6(a) is almost identi al to the one by West and Shmoys, so we just prove that G′ has a balan ed realization, shown in Figure 6 (b), by omputing lengths for ea h interval to ensure it. All K5,3 have a balan ed realization as shown Figure 6: There is a balan ed 2-interval of G′ (whi h has been dilated in the drawing to remain readable) i� there is an H-representation (that is a realization where the left intervals of all 2-intervals are grouped together in a ontiguous blo k) for its indu ed subgraph G i� there is a Hamiltonian y le in G. in se tion 1 of total length 79, in parti ular H3. We an thus a�e t length 83 to the intervals of v0. The intervals of the other vi an have length 3, and their M(vi) length 79, so through the omputation illustrated in Figure 6, intervals of z an have length 80 + 82 + 2(n − 1) + 3, that is 163 + 2n. We dilate H1 until a hole between two onse utive intervals of its S3 an ontain an interval of z, that is until the hole has length 165 + 2n : so after this dilating, H1 has length 79(165 + 2n). Finally if G has a Hamiltonian y le, then we have found a balan ed 2-interval realization of G of total length 13, 273 + 241n. It is known that ir ular-ar graphs are 2-interval graphs, they are also balan ed 2-interval. Property 3. The lass of ir ular-ar graphs is stri tly in luded in the lass of balan ed 2- interval graphs. Proof. The transformation is simple: if we have a ir ular-ar representation of a graph G = (V,E), then we hoose some point P of the ir le. We partition V in V1∪V2, where P interse ts all the ar s orresponding to verti es of V1 and none of the ar s of the verti es of V2. Then we ut the ir le at point P to map it to a line segment: every ar of V2 be omes an interval, and every ar of V1 be omes a 2-interval. To obtain a balan ed realization we just ut in half the intervals of V2 to obtain two intervals of equal length for ea h. And for ea h 2-interval [g(Il), d(Il)] ∪ [g(Ir), d(Ir)] of V1, as both intervals are lo ated on one of the extremities of the realization, we an in rease the length of the shortest so that it rea hes the length of the longest without hanging interse tions with the other intervals. The in lusion is stri t be ause K2,3 is a balan ed 2-interval graph (as a subgraph of K5,3 for example) but is not a ir ular-ar graph (we an �nd two C4 in K2,3, and only one an be realized with a ir ular-ar representation). 4 Unit 2-interval and (x,x)-interval graphs Property 4. Let x ∈ N, x ≥ 2. The lass of (x, x)-interval graphs is stri tly in luded in the lass of (x+ 1, x+ 1)-interval graphs. Proof. We �rst prove that an interval graph with a representation where all intervals have length k (and integer open bounds) has a representation where all intervals have length k + 1. We use the following algorithm. Let S be initialized as the set of all intervals of length k, and let T be initially the empty set. As long as S is not empty, let I = [a, b] be the left-most interval of S, remove from S ea h interval [α, β] su h that α < b (in luding I), add [α, β + 1] to T , and translate by +1 all the remaining intervals in S. When S is empty, the interse tion graph of T , where all intervals have length k + 1 is the same as the interse tion graph for the original S. We also build for ea h x ≥ 2 a (x + 1, x + 1)-interval graph whi h is not a (x, x)-interval graph. We onsider the bipartite graph K2x and a perfe t mat hing {(vi, v i), i ∈ J1, xK}. We all K ′x the graph obtained from K2x with the following transformations, illustrated in Figure 7(a): remove edges (vi, v i) of the perfe t mat hing, add four graphs K4,4−e alled X1, X2, X3, X4 (for ea h Xi, we all v and vir the verti es of degree 3), link v r and v , link all vi to v r and v , link all v′i to v and v3r , and �nally add a vertex a (resp. b) linked to all vi, v i, and to two adja ent verti es of X1 (resp. X4) of degree 4. We illustrate in Figure 7(b) that K x has a realization with intervals of length x+ 1. We an prove by indu tion on x that K ′x has no realization with intervals of length x: it is rather te hni al, so we just give the idea. Any realization of K ′x for es the blo k X2 to share an extremity with the blo k X3, so ea h 2-interval v i has one interval interse ting the other extremity of X2, and the other interse ting the other extremity of X3. Then onstraints on the position of verti es vi for e their intervals to appear as two �stairways� as shown in Figure 7(b). So v1r must ontain x extremities of intervals whi h have to be di�erent, so it must have length x+ 1. Figure 7: The graph K ′4 (a) is (5,5)-interval but not (4,4)-interval. The omplexity of re ognizing unit 2-interval graphs and (x, x)-interval graphs remains open, however the following shows a relationship between those omplexities. Lemma 1. {unit 2-interval graphs} = {(x, x)-interval graphs}. Proof. The ⊃ part is trivial. To prove ⊂, let G = (V,E) be a unit 2-interval graph. Then it has a realization with |V | = n 2-intervals, that is 2n intervals of the ground set. So we onsider the interval graph of the ground set, whi h is a unit interval graph. There is a linear time algorithm based on breadth-�rst sear h to ompute a realization of su h a graph where interval endpoints are rational, with denominator 2n [CKN+95℄. So by dilating by a fa tor 2n su h a realization, we obtain a realization of G where intervals of the ground set have length 2n. Theorem 2. If re ognizing (x, x)-interval graphs is polynomial for any integer x then re ognizing unit 2-interval graphs is polynomial. 5 Investigating the omplexity of unit 2-interval graphs In this se tion we show that all proper ir ular-ar graphs ( ir ular-ar graphs su h that no ar is in luded in another in the representation) are unit 2-interval graphs, and we study a lass of graphs whi h generalizes quasi-line graphs and ontains unit 2-interval graphs. Re all that, a ording to Property 3, ir ular-ar graphs are balan ed 2-interval graphs. However, ir ular-ar graphs are not ne essarily unit 2-interval graphs. Property 5. The lass of proper ir ular-ar graphs is stri tly in luded in the lass of unit 2-interval graphs. Proof. As in the proof of Property 3, we hoose a point P on the ir le of the representation of a proper ir ular-ar graph G, and maps the ut ir le into a line segment. We extend the outer extremities of intervals that have been ut so that no interval ontains another. Thus we obtain a set of 2-intervals for ar s ontaining P , and a set I of intervals for ar s not ontaining P . For ea h interval of I, we add a new interval disjoint of any other to get a 2-interval. If we onsider the interse tion graph of the ground set of su h a representation, it is a proper interval graph. So it is also a unit interval graph [Rob69℄, whi h provides a unit 2-interval representation of G. To omplete the proof, we noti e that the domino (two y les C4 having an edge in ommon) is a unit 2-interval graph but not a ir ular-ar graph. Quasi-line graphs are those graphs whose verti es are bisimpli ial, i.e., the losed neighbor- hood of ea h vertex is the union of two liques. This graph lass has been introdu ed as a gener- alization of line graphs and a useful sub lass of law-free graphs [Ben81, FFR97, CS05, KR07℄. Following the example of quasi-line graphs that generalize line graphs, we introdu e here a new lass of graphs for generalizing unit 2-interval graphs. Let k ∈ N∗. A graph G = (V,E) is all-k-simpli ial if the neighborhood of ea h vertex v ∈ V an be partitioned into at most k liques. The lass of quasi-line graphs is thus exa tly the lass of all-2-simpli ial graphs. Noti e that this de�nition is equivalent to the following: in the omplement graph of G, for ea h vertex u, the verti es that are not in the neighborhood of u are k- olorable. Property 6. The lass of unit 2-interval graphs is stri tly in luded in the lass of all-4-simpli ial graphs. Proof. The in lusion is trivial. What is left is to show that the in lusion is stri t. Consider the following graph whi h is all-4-simpli ial but not unit 2-interval: start with the y le C4, all its verti es vi, i ∈ J1, 4K, add four K4,4 − e gadgets alled Xi, and for ea h i we onne t the vertex vi to two onne ted verti es of degree 4 in Xi. This graph is ertainly all-4-simpli ial. But if we try to build a 2-interval realization of this graph, then ea h of the 2-intervals vk has an interval trapped into the blo k Xk. So ea h 2-interval vk has only one interval to realize the interse tions with the other vi: this is impossible as we have to realize a C4 whi h has no interval representation. Property 7. The lass of law-free graphs is not in luded in the lass of all-4-simpli ial graphs. Proof. The Kneser Graph KG(7, 2) is triangle-free, but not 4- olorable [Lov78℄. We onsider the graph obtained by adding an isolated vertex v and then taking the omplement graph, i.e., KG(7, 2) ⊎ {v}. It is law-free as KG(7, 2) is triangle-free. And if it was all-4-simpli ial, then the neighborhood of v in KG(7, 2) ⊎ {v}, that is KG(7, 2), would be a union of at most four liques, so KG(7, 2) would be 4- olorable: impossible so this graph is law-free but not all-4-simpli ial. Property 8. The lass of all-k-simpli ial graphs is stri tly in luded in the lass of K1,k+1-free graphs. Proof. If a graph G ontains K1,k+1, then it has a vertex with k + 1 independent neighbors, and hen e G is not all-k-simpli ial. The wheel W2k+1 is a simple example of a graph whi h is K1,k+1-free but in whi h the enter an not have its neighborhood (a C2k+1) partitioned into k liques or less. Unfortunately, all-k-simpli ial graphs do not have a ni e stru ture whi h ould help unit 2-interval graph re ognition. Theorem 3. Re ognizing all-k-simpli ial graphs is NP- omplete for k ≥ 3. Proof. We redu e from the Graph k- olorability problem, whi h is known to be NP- omplete for k ≥ 3 [Kar72℄. Let G = (V,E) be a graph, and let G′ be the omplement graph of G to whi h we add a universal vertex v. We laim that G is k- olorable i� G′ is all-k-simpli ial. If G is k- olorable, then the non-neighborhood of any vertex in G is k- olorable, so the neighborhood of any vertex in G is a union of at most k liques. And the neighborhood of v is also a union of at most k liques, so G′ is all-k-simpli ial. Conversely, if G′ is all-k-simpli ial, then in parti ular the neighborhood of v is a union of at most k liques. Let's partition it into k vertex-disjoint liques X1, . . . ,Xk. Then, oloring G su h that two verti es have the same olor i� they are in the same Xi leads to a valid k- oloring of G. 6 Con lusion Motivated by pra ti al appli ations in s heduling and omputational biology, we fo used in this paper on balan ed 2-interval graphs and unit 2-intervals graphs. Also, we introdu ed two natural new lasses: (x, x)-interval graphs and all-k-simpli ial graphs. We mention here some dire tions for future works. First, the omplexity of re ognizing unit 2-interval graphs and (x, x)-interval graphs remains open. Se ond, the relationships between quasi-line graphs and sub lasses of balan ed 2-intervals graphs still have to be investigated. Last, sin e most problems remains NP-hard for balan ed 2-interval graphs, there is thus a natural interest in investigating the omplexity and approximation of lassi al optimization problems on unit 2-interval graphs and (x, x)-interval graphs. A knowledgments We are grateful to Vin ent Limouzy in parti ular for bringing to our attention the lass of quasi-line graphs, and Mi hael Rao and Mi hel Habib for useful dis ussions. Referen es [Ben81℄ A. Ben Rebea. Étude des stables dans les graphes quasi-adjoints. PhD thesis, Université de Grenoble, 1981. [BFV04℄ G. Blin, G. Fertin, and S. Vialette. New results for the 2-interval pattern problem. In Pro eedings of the 15 Symposium on Combinatorial Pattern Mat hing (CPM'04), 2004. [BHLR07℄ A. Butman, D. Hermelin, M. Lewenstein, and D. Rawitz. Optimization problems in multiple-interval graphs. In Pro eedings of the 18 Annual Symposium On Dis rete Al- gorithms (SODA'07), pages 268�277, 2007. ℄ A. Brandstädt, V. B. Le, T. Szym zak, F. Siegemund, H.N. de Ridder, S. Knorr, M. Rze- hak, M. Mowitz, and N. Ryabova. ISGCI: Information System on Graph Class In lusions. http://wwwteo.informatik.uni-rosto k.de/isg i/ lasses. gi. [BNR96℄ V. Bafna, B. O. Narayanan, and R. Ravi. Nonoverlapping lo al alignments (weighted independent sets of axis-parallel re tangles). Dis rete Applied Math., 71(1):41�54, 1996. [BYHN 06℄ R. Bar-Yehuda, M. M. Halldórson, J. Naor, H. Sha hnai, and I. Shapira. S heduling split intervals. SIAM Journal on Computing, 36(1):1�15, 2006. [CHLV05℄ M. Cro hemore, D. Hermelin, G. M. Landau, and S. Vialette. Approximating the 2-interval pattern problem. In Pro eedings of the 13 Annual European Symposium on Algorithms (ESA'05), volume 3669 of LNCS, pages 426�437, 2005. 95℄ D. G. Corneil, H. Kim, S. Natarajan, S. Olariu, and A. P. Sprague. Simple linear time re ognition of unit interval graphs. Information Pro essing Letters, 55:99�104, 1995. [CS05℄ M. Chudnovsky and P. Seymour. The stru ture of law-free graphs. In Surveys in Com- binatori s, volume 327 of London. Math. So . Le ture Notes, pages 153�172. Cambridge University Press, 2005. [FFR97℄ R. Faudree, E. Flandrin, and Z. Ryjá£ek. Claw-free graphs - a survey. Dis rete Mathemati s, 164:87�147, 1997. [GW80℄ J. R. Griggs and D. B. West. Extremal values of the interval number of a graph. SIAM Journal on Algebrai and Dis rete Methods, 1:1�7, 1980. [GW95℄ A. Gyárfás and D. B. West. Multitra k interval graphs. Congress Numerantium, 109:109� 116, 1995. [HK06℄ M. M. Halldórsson and R. K. Karlsson. Strip graphs: Re ognition and s heduling. 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Presentation at NSF-CBMS Conferen e at Colby College. [Rob69℄ F. S. Roberts. Indi�eren e graphs. In Proof Te hniques in Graph Theory, Pro eedings of the Se ond Ann Arbor Graph Theory Conferen e, pages 139�146, 1969. [Via01℄ S. Vialette. Aspe ts algorithmiques de la prédi tion des stru tures se ondaires d'ARN. PhD thesis, Université Paris 7, 2001. [Via04℄ S. Vialette. On the omputational omplexity of 2-interval pattern mat hing. Theoreti al Computer S ien e, 312(2-3):223�249, 2004. [WS84℄ D. B. West and D. B. Shmoys. Re ognizing graphs with �xed interval number is NP- omplete. Dis rete Applied Math., 8:295�305, 1984. http://wwwteo.informatik.uni-rostock.de/isgci/classes.cgi 7 Appendix We give the detailed proofs of Theorem 1 and Property 4. Proof of Theorem 1. Let G = (V,E) be a 3-regular triangle-free graph. We build a graph G′ whi h has a 2-interval realization (a spe ial one, very spe i� , and whi h we prove to be balan ed) i� G has a Hamiltonian y le. First we will detail how we build G′ starting from the graph G, and adding some verti es, in parti ular K5,3 gadgets. The idea is that the edges of G will partition into a Hamiltonian y le and a perfe t mat hing i� all 2-intervals of the realization of G′ an have their left inter- val realizing the Hamiltonian y le, and their right interval realizing the perfe t mat hing. A realization with su h a pla ement of the intervals is alled an �H-representation� of G. We pro eed as illustrated in Figure 6. We hoose some vertex of G that we all v0 (whi h will be the �origin� of the Hamiltonian y le), and the other are alled v1, . . . , vn. For ea h vertex vi of G we link it to a vertex of the S5 of a K5,3 alled M(vi) (whi h will blo k one of the four extremities of the 2-interval vi). We link all verti es to a new vertex z, whi h is linked to no M(v) ex ept M(v0) (thus the interval of ea h vi interse ting M(vi), for i 6= 0, won't interse t z). We add three K5,3, H1, H2 and H3 : two verti es of the S5 of H1 are linked to z, a third one is linked to one vertex of the S5 of H2, one vertex of the S5 of H3 is linked to z, and all verti es of H3 to v0. To explain this onstru tion in detail, we study the realization of G′, if we suppose it is a (balan ed) 2-interval graph, and we prove that it leads us to �nd a Hamiltonian y le in G. As the realization of H1 and H2 are two ontiguous blo ks of intervals then one of their extremities must interse t. As z is linked to two disjoint verti es of H1, both intervals of z are used to realize those interse tions. But one interval of z that we all zr, also has to interse t one vertex of H3 whi h is not linked to H1, so zr interse ts the se ond extremity of the blo k H1 (the �rst extremity being o upied by the extremity of H2). And as zr interse ts only one interval of H3, it must be the extremity of H3. The other interval of z is ontained in the blo k H1, thus an't interse t M(v0) neither all the verti es vi, so all those 2-intervals interse t zr. And as none of them interse t H3 ex ept v0, all of them ex ept v0 have an interval ontained in zr, that we all vi,g. The other interval of ea h vi is linked to a K5,3 so it has one extremity o upied by K5,3, and the other one is free. Conversely, if G has a Hamiltonian y le, then it is possible to �nd a H-representation, su h that all the onstraints indu ed by the edges of G′ are respe ted, as illustrated with the realization in Figure 6. We have already proved that this realization an be balan ed. Proof of Property 4. In the following, as we only onsidering the interval of vi or vir lo ated at one extremity of the blo k Xi, and not the one inside, we will use v and vir to denote those extremity intervals. For ea h vertex vi, we all vi,l its left interval and vi,r its right interval. We do the same for v′i, and all l(I) the left extremity of any interval I. We prove by indu tion that the graph K ′x is (x + 1, x + 1)-interval but not (x, x)-interval, and that for any unit 2-interval realization, there exists an order σ ∈ Sx su h that : • either l(vσ(x),l) < . . . < l(vσ(1),l) < l(v σ(x),l ) < . . . < l(v′ σ(1),l ) and l(v′ σ(x),r ) < . . . < σ(1),r ) < l(vσ(x),r) < . . . < l(vσ(1),r), • or the symmetri ase: l(vσ(1),l) < . . . < l(vσ(x),l) < l(v σ(1),l ) < . . . < l(v′ σ(x),l ) and σ(1),r ) < . . . < l(v′ σ(x),r ) < l(vσ(1),r) < . . . < l(vσ(x),r). Those two equalities orrespond in fa t to the �two stairways stru ture� whi h appears in Fig- ure 7. Base ase : we study all possible unit 2-interval realizations of K ′2 to prove that one of the expe ted inequalities is always true. We also prove that K ′2 has no (2,2)-interval realization. First re all that realizations of Xi subgraphs an only be blo ks of ontiguous intervals. The edge between v2r and v for es the two blo ks of X2 and X3 to be ontiguous, with intervals and v3r at their extremities. Ea h 2-interval v i must interse t both v and v3r , so one of its intervals interse ts v2 and the other interse ts v3r . Thus, one same interval of v i an not interse t both a and b whi h are disjoint, so a interse ts one interval of v′i (say the one interse ting v , the other ase being treated symmetri ally) and b interse ts the other one (so, the one interse ting v3r ). Ea h vi has to interse t both a and b, so it has to interse t a with its �rst interval and b with the se ond. But 2-interval vi must also interse t v r and v whi h are both disjoint and disjoint to a and b. So one interval of ea h vi must interse t v r and the other one must interse t So we have shown that any unit 2-interval realization of K ′2 has the following aspe t (or the symmetri ) : the extremity of the blo k X1 interse ting all vi whi h interse t a (or b) whi h interse ts all v′i, whi h interse t the extremity X2 (or X3) whi h interse ts the extremity of X3 (or X2), whi h interse ts all v i, whi h interse t b (or a), whi h interse ts all vi, whi h interse t the extremity of X4. Now we suppose, by ontradi tion, that there exists a (2,2)-interval realization of K ′2. v an interval of length 2, but one of its two parts of length one has to interse t an element of X1. The other has to interse t both v1 and v2. As neither v1 nor v2 an interse t other intervals of X1, then the �rst interval of v1 and v2 is the same interval. By pro eeding the same way on X4 and v4 , we obtain that the se ond interval of v1 and v2 is the same interval, so v1 and v2 should orrespond to the same 2-interval: it ontradi ts with the fa t that verti es v1 and v2 have a di�erent neighborhood. So K ′2 has no (2,2)-interval realization. To obtain the expe ted inequalities, we have to analyze the possible positions of all vi and v′i. We only treat the �rst two inequalities as the se ond ase is symmetri . Suppose that l(v2,l) < l(v1,l). As v1 and v 1 are non adja ent, then interval v1,l is stri tly on the left of v′1,l, so v2,l is stri tly on the left of v 1,l. Thus those two intervals do not interse t. But v2 and v 1 are adja ent, so v2 and v 1 must have interse ting right intervals. But then we have l(v′2,r) < l(v 1,r) < l(v2,r) < l(v1,r), and the right intervals of v 2 and v1 an not interse t. We dedu e their left intervals interse t, so l(v2,l) < l(v1,l) < l(v 2,l) < l(v 1,l). If we suppose that l(v1,l) < l(v2,l), we get as well that l(v 1,r) < l(v 2,r) < l(v1,r) < l(v2,r) and l(v1,l) < l(v2,l) < l(v 1,l) < l(v 2,l). So for any unit 2-interval realization of K 2 there exists an order σ = 12 or σ = 21 su h that: • either l(vσ(2),l) < l(vσ(1),l) < l(v σ(2),l ) < l(v′ σ(1),l ) and l(v′ σ(2),r ) < l(v′ σ(1),r ) < l(vσ(2),r) < l(vσ(1),r), • or the symmetri inequalities. Re ursion: suppose that for some x, K ′x−1 is not (x−1, x−1)-interval but is (x, x)-interval, and that any (x, x)-interval realization veri�es one of the expe ted inequalities. Graph K ′x−1 is an indu e subgraph of K x = (V,E) : K x−1 = K x[V \ {vx, v x}]. So by the indu tion hypothesis, there exists an order σ ∈ Sx−1 su h that for any unit 2-interval realization of K ′x : • either l(vσ(x−1),l) < . . . < l(vσ(1),l) < l(v σ(x−1),l ) < . . . < l(v′ σ(1),l ) and l(v′ σ(x−1),r ) < . . . < σ(1),r ) < l(vσ(x−1),r) < . . . < l(vσ(1),r), • or the symmetri ase: l(vσ(1),l) < . . . < l(vσ(x−1),l) < l(v σ(1),l ) < . . . < l(v′ σ(x−1),l ) and σ(1),r ) < . . . < l(v′ σ(x−1),r ) < l(vσ(1),r) < . . . < l(vσ(x−1),r). The position of vx and v x remains to be determined. We treat only the ase where the �rst two inequalities are true, as the se ond ase is symmetri . As vx and v r are adja ent, and v σ(x−1) and v1r are not, then l(v r ) < l(vx,l) < l(v σ(x−1),l ). So we de�ne j the following way: vσ(j),l is the leftmost interval su h that l(vx,l) ≤ l(vσ(j),l). if there is none, we say j = 0. Then we all σ′ ∈ Sx the permutation de�ned by: σ′(i) = σ(i) if i < j, σ′(j + 1) = x, σ′(i) = σ(i− 1) if i > j. Then we dire tly get inequalities: • l(v1r ) < l(vσ′(x),l) < . . . < l(vσ′(j+1),l) ≤ l(vx,l) < l(vσ′(j−1),l) < . . . < l(vσ′(1),l) < σ′(x),l ) < . . . < l(v′ σ′(j+1),l ) < l(v′ σ′(j−1),l ) < . . . < l(v′ σ′(1),l • l(v′ σ′(x),r ) < . . . < l(v′ σ′(j+1),r ) < l(v′ σ′(j−1),r ) < . . . < l(v′ σ′(1),r ) < l(vσ′(x),r) < . . . < l(vσ′(j+1),r) < l(vσ′(j−1),r) < . . . < l(vσ′(1),r) We obtain the expe ted inequalities by reasoning the same way as in the end of the base ase. So in parti ular we have l(vσ(x),l) < . . . < l(vσ(1),l) and v r must interse t all those vi for i ∈ J1, xK, but also an interval of X1 whi h interse ts none of the vi. So it must have length x+ 1, thus K ′x is not a (x, x)-interval graph Con lusion: As the base ase and the re ursion has been proved, expe ted properties of the graph K ′x are true for any x ≥ 2. 2-interval graphs and restrictions Useful gadgets for 2-interval graphs and restrictions Balanced 2-interval graphs Unit 2-interval and (x,x)-interval graphs Investigating the complexity of unit 2-interval graphs Conclusion Appendix

The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatic problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval graphs that generalize line graphs: balanced 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instances that show that all the inclusions are strict. We extend the NP-completeness proof of recognizing 2-interval graphs to the recognition of balanced 2-interval graphs. Finally we give hints on the complexity of unit 2-interval graphs recognition, by studying relationships with other graph classes: proper circular-arc, quasi-line graphs, K_{1,5}-free graphs, ...

On restrictions of balanced 2-interval graphs On restri tions of balan ed 2-interval graphs Philippe Gambette LIAFA, UMR CNRS 7089, Université Paris 7, Fran e Département Informatique, ENS Ca han, Fran e gambette�liafa.jussieu.fr Stéphane Vialette LRI, UMR CNRS 8623, Université Paris-Sud 11, Fran e vialette�lri.fr Abstra t The lass of 2-interval graphs has been introdu ed for modelling s heduling and allo ation problems, and more re ently for spe i� bioinformati problems. Some of those appli ations imply restri tions on the 2-interval graphs, and justify the introdu tion of a hierar hy of sub lasses of 2-interval graphs that generalize line graphs: balan ed 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instan es that show that all the in lusions are stri t. We extend the NP- ompleteness proof of re ognizing 2-interval graphs to the re ognition of balan ed 2-interval graphs. Finally we give hints on the omplexity of unit 2-interval graphs re ognition, by studying relationships with other graph lasses: proper ir ular-ar , quasi-line graphs, K1,5-free graphs, . . . Keywords: 2-interval graphs, graph lasses, line graphs, quasi-line graphs, law-free graphs, ir ular interval graphs, proper ir ular-ar graphs, bioinformati s, s heduling. 1 2-interval graphs and restri tions The interval number of a graph, and the lasses of k-interval graphs have been introdu ed as a generalization of the lass of interval graphs by M Guigan [M G77℄ in the ontext of s heduling and allo ation problems. Re ently, bioinformati s problems have renewed interest in the lass of 2-interval graphs (ea h vertex is asso iated to a pair of disjoint intervals and edges denote interse tion between two su h pairs). Indeed, a pair of intervals an model two asso iated tasks in s heduling [BYHN 06℄, but also two similar segments of DNA in the ontext of DNA omparison [JMT92℄, or two omplementary segments of RNA for RNA se ondary stru ture predi tion and omparison [Via04℄. (a) (b) ( ) Figure 1: Heli es in a RNA se ondary stru ture (a) an be modeled as a set of balan ed 2- intervals among all 2-intervals orresponding to omplementary and inverted pairs of letter sequen es (b), or as an independent subset in the balan ed asso iated 2-interval graph ( ). http://arxiv.org/abs/0704.1571v2 RNA (ribonu lei a id) are polymers of nu leotides linked in a hain through phosphodiester bonds. Unlike DNA, RNAs are usually single stranded, but many RNAmole ules have se ondary stru ture in whi h intramole ular loops are formed by omplementary base pairing. RNA se - ondary stru ture is generally divided into heli es ( ontiguous base pairs), and various kinds of loops (unpaired nu leotides surrounded by heli es). The stru tural stability and fun tion of non- oding RNA (n RNA) genes are largely determined by the formation of stable se ondary stru tures through omplementary bases, and hen e n RNA genes a ross di�erent spe ies are most similar in the pattern of nu leotide omplementarity rather than in the genomi sequen e. This motivates the use of 2-intervals for modelling RNA se ondary stru tures: ea h helix of the stru ture is modeled by a 2-interval. Moreover, the fa t that these 2-intervals are usually required to be disjoint in the stru ture naturally suggests the use of 2-interval graphs. Fur- thermore, aiming at better modelling RNA se ondary stru tures, it was suggested in [CHLV05℄ to fo us on balan ed 2-interval sets (ea h 2-interval is omposed of two equal length intervals) and their asso iated interse tion graphs referred as balan ed 2-interval graphs. Indeed, heli es in RNA se ondary stru tures are most of the time omposed of equal length ontiguous base pairs parts. To the best of our knowledge, nothing is known on the lass of balan ed 2-interval graphs. Sharper restri tions have also been introdu ed in s heduling, where it is possible to on- sider tasks whi h all have the same duration, that is 2-interval whose intervals have the same length [BYHN 06, Kar05℄. This motivates the study of the lasses of unit 2-interval graphs, and (x, x)-interval graphs. In this paper, we onsider these sub lasses of interval graphs, and in parti ular we address the problem of re ognizing them. A graph G = (V,E) of order n is a 2-interval graph if it is the interse tion graph of a set of n unions of two disjoint intervals on the real line, that is ea h vertex orresponds to a union of two disjoint intervals Ik = Ik ∪ Ikr , k ∈ J1, nK (l for � left� and r for �right�), and there is an edge between Ij and Ik i� Ij ∩ Ik 6= ∅. Note that for the sake of simpli ity we use the same letter to denote a vertex and its orresponding 2-interval. A set of 2-intervals orresponding to a graph G is alled a realization of G. The set of all intervals, k=1{I , Ikr }, is alled the ground set of G (or the ground set of {I1, . . . , In}). The lass of 2-interval graphs is a generalization of interval graphs, and also ontains all ir ular-ar graphs (interse tion graphs of ar s of a ir le), outerplanar graphs (have a planar embedding with all verti es around one of the fa es [KW99℄), ubi graphs (maximum degree 3 [GW80℄), and line graphs (interse tion graphs of edges of a graph). Unfortunately, most lassi al graph ombinatorial problems turn out to be NP- omplete for 2-interval graphs: re ognition [WS84℄, maximum independent set [BNR96, Via01℄, ol- oration [Via01℄, . . . Surprisingly enough, the omplexity of the maximum lique problem for 2-interval graphs is still open (although it has been re ently proven to be NP- omplete for 3-interval graphs [BHLR07℄). For pra ti al appli ation, restri ted 2-interval graphs are needed. A 2-interval graph is said to be balan ed if it has a 2-interval realization in whi h ea h 2-interval is omposed of two intervals of the same length [CHLV05℄, unit if it has a 2-interval realization in whi h all intervals of the ground set have length 1 [BFV04℄, and is alled a (x, x)-interval graph if it has a 2-interval realization in whi h all intervals of the ground set are open, have integer endpoints, and length x [BYHN+06, Kar05℄. In the following se tions, we will study those restri tions of 2-interval graphs, and their position in the hierar hy of graph lasses illustrated in Figure 2. Note that all (x, x)-interval graphs are unit 2-interval graphs, and that all unit 2-interval graphs are balan ed 2-interval graphs. We an also noti e that (1, 1)-interval graphs are exa tly line graphs: ea h interval of length 1 of the ground set an be onsidered as the vertex of a root graph and ea h 2-interval as an edge in the root graph. This implies for example that the Figure 2: Graph lasses related to 2-interval graphs and its restri tions. A lass pointing towards another stri tly ontains it, and the dashed lines mean that there is no in lusion relationship be- tween the two. Dark lasses orrespond to lasses not yet present in the ISGCI Database [BLS oloration problem is also NP- omplete for (2, 2)-interval graphs and wider lasses of graphs. It is also known that the omplexity of the maximum independent set problem is NP- omplete on (2, 2)-interval graphs [BNR96℄. Re ognition of (1, 2)-union graphs, a related lass (restri tion of multitra k interval graphs), was also re ently proven NP- omplete [HK06℄. 2 Useful gadgets for 2-interval graphs and restri tions For proving hardness of re ognizing 2-interval graphs, West and Shmoys onsidered in [WS84℄ the omplete bipartite graph K5,3 as a useful 2-interval gadget. Indeed, all realizations of this graph are ontiguous, that is, for any realization, the union of all intervals in its ground set is an interval. Thus, by putting edges between some verti es of a K5,3 and another vertex v, we an for e one interval of the 2-interval v (or just one extremity of this interval) to be blo ked inside the realization of K5,3. It is not di� ult to see that K5,3 has a balan ed 2-interval realization, for example the one in Figure 3. (a) (b) ( ) Figure 3: The omplete bipartite graph K5,3 (a,b) has a balan ed 2-interval realization ( ): verti es of S5 are asso iated to balan ed 2-intervals of length 7, and verti es of S3 are asso iated to balan ed 2-intervals of length 11. Any realization of this graph is ontiguous, i.e., the union of all 2-intervals is an interval. However, K5,3 is not a unit 2-interval graph. Indeed, ea h 2-interval I = Il∪Ir orresponding to a degree 5 vertex interse t 5 disjoint 2-intervals, and hen e one of Il or Ir interse t at least 3 intervals, whi h is impossible for unit intervals. Therefore, we introdu e the new gadget K4,4−e whi h is a (2, 2)-interval graph with only ontiguous realizations. (a) (b) ( ) Figure 4: The graph K4,4 − e (a), a ni er representation (b), and a 2-interval realization with open intervals of length 2 ( ). Property 1. Any 2-interval realization of K4,4 − e is ontiguous. Proof. Write G = (V,E) the graph K4,4− e. To study all possible realizations of G, let us study all possible realizations of G[V − I8]. As 2-intervals I1, I2, I3 and I4 are disjoints, their ground set I = {[li, ri], 1 ≤ i ≤ 8, ri < li+1} is a set of eight disjoint intervals. The ground set Imobile of I , I6 and I7 is a set of six disjoint intervals. Let xi be the number of intervals of Imobile interse ting i ≤ 8 intervals of . We have dire tly: x0 + x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 = |Imobile| = 6. (1) As there are 12 edges in G[V \{v8}] whi h is bipartite, we also have: x1 + 2x2 + 3x3 + 4x4 + 5x5 + 6x6 + 7x7 + 8x8 ≥ 12. (2) Finally, to build a realization of G from a realization of G[V \{v8}] , one must pla e I so as to interse t three disjoint intervals of I . Thus one of the intervals of I8 interse ts at least two intervals ]lk, rk[ and ]ll, rl[ (k < l) of I�xed. So there is �a hole between those two intervals�, for example [rk, lk+1], whi h is in luded in one of the intervals of I . So we noti e that I8 has to �ll one of the seven holes of I . Thus, the intervals of I mobile an not �ll more than six holes, and the observation that an interval interse ting i onse utive intervals (for i ≥ 1) �lls i− 1 holes, we get: x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 7x8 ≤ 6. (3) Equations 1, 2 and 3 are ne essary for any valid realization of G[V \{v8}] whi h gives a valid realization of G. Let's suppose by ontradi tion that the union of all intervals of the ground set of G is not an interval. Then there is a hole, that is an interval in luded in the overing interval of {I1, . . . , I8}, whi h interse t no Ii. We pro eed like for equation 3, with the onstraint that another hole annot be �lled by the intervals of I mobile , so we get instead: x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 7x8 ≤ 5. (4) By adding 1 and 4, and subtra ting 2, we get x0 ≤ −1 : impossible! So we have proved that the union of all intervals of the ground set of any realization of G is indeed an interval. 3 Balan ed 2-interval graphs We show in this se tion that the lass of balan ed 2-interval graphs is stri tly in luded in the lass of 2-interval graphs, and stri tly ontains ir ular-ar graphs. Moreover, we prove that re ognizing balan ed 2-interval graphs is as hard as re ognizing (general) 2-interval graphs. Property 2. The lass of balan ed 2-interval graphs is stri tly in luded in the lass of 2-interval graphs. Proof. We build a 2-interval graph that has no balan ed 2-interval realization. Let's onsider a hain of gadgets K5,3 (introdu ed in previous se tion) to whi h we add three verti es I , I2, and I3 as illustrated in Figure 5. Figure 5: An example of unbalan ed 2-interval graph (a) : any realization groups intervals of the seven K5,3 in a blo k, and the hain of seven blo ks reates six �holes� between them, whi h make it impossible to balan e the lengths of the three 2-intervals I1, I2, and I3. In any realization, the presen e of holes showed by rosses in the Figure gives the following inequalities for any realization: l(Il 2) < l(Il 1), l(Il 3) < l(Ir 2), and l(Ir 1) < l(Ir 3) (or if the realization of the hain of K5,3 appears in the symmetri al order: l(Il 1) < l(Il 3), l(Ir 3) < l(Il and l(Ir 2) < l(Ir 1)). If this realization was balan ed, then we would have l(Il 1) = l(Ir 3) = l(Il 3) < l(Ir 2) = l(Il 2) (or for the symmetri al ase: l(Ir 1) = l(Il 1) < l(Il 3) = l(Ir 2) = l(Ir 2)) : impossible! So this graph has no balan ed 2-interval realization although it has a 2-interval generalization. Theorem 1. Re ognizing balan ed 2-interval graphs is an NP- omplete problem. Proof. We just adapt the proof of West and Shmoys [WS84, GW95℄. The problem of determining if there is a Hamiltonian y le in a 3-regular triangle-free graph is proven NP- omplete, by redu tion from the more general problem without the no triangle restri tion. So we redu e the problem of Hamiltonian y le in a 3-regular triangle-free graph to balan ed 2-interval re ognition. Let G = (V,E) be a 3-regular triangle-free graph. We build a graph G′ whi h has a 2- interval realization (a spe ial one, very spe i� , alled H-representation and whi h we prove to be balan ed) i� G has a Hamiltonian y le. The onstru tion of G′, illustrated in Figure 6(a) is almost identi al to the one by West and Shmoys, so we just prove that G′ has a balan ed realization, shown in Figure 6 (b), by omputing lengths for ea h interval to ensure it. All K5,3 have a balan ed realization as shown Figure 6: There is a balan ed 2-interval of G′ (whi h has been dilated in the drawing to remain readable) i� there is an H-representation (that is a realization where the left intervals of all 2-intervals are grouped together in a ontiguous blo k) for its indu ed subgraph G i� there is a Hamiltonian y le in G. in se tion 1 of total length 79, in parti ular H3. We an thus a�e t length 83 to the intervals of v0. The intervals of the other vi an have length 3, and their M(vi) length 79, so through the omputation illustrated in Figure 6, intervals of z an have length 80 + 82 + 2(n − 1) + 3, that is 163 + 2n. We dilate H1 until a hole between two onse utive intervals of its S3 an ontain an interval of z, that is until the hole has length 165 + 2n : so after this dilating, H1 has length 79(165 + 2n). Finally if G has a Hamiltonian y le, then we have found a balan ed 2-interval realization of G of total length 13, 273 + 241n. It is known that ir ular-ar graphs are 2-interval graphs, they are also balan ed 2-interval. Property 3. The lass of ir ular-ar graphs is stri tly in luded in the lass of balan ed 2- interval graphs. Proof. The transformation is simple: if we have a ir ular-ar representation of a graph G = (V,E), then we hoose some point P of the ir le. We partition V in V1∪V2, where P interse ts all the ar s orresponding to verti es of V1 and none of the ar s of the verti es of V2. Then we ut the ir le at point P to map it to a line segment: every ar of V2 be omes an interval, and every ar of V1 be omes a 2-interval. To obtain a balan ed realization we just ut in half the intervals of V2 to obtain two intervals of equal length for ea h. And for ea h 2-interval [g(Il), d(Il)] ∪ [g(Ir), d(Ir)] of V1, as both intervals are lo ated on one of the extremities of the realization, we an in rease the length of the shortest so that it rea hes the length of the longest without hanging interse tions with the other intervals. The in lusion is stri t be ause K2,3 is a balan ed 2-interval graph (as a subgraph of K5,3 for example) but is not a ir ular-ar graph (we an �nd two C4 in K2,3, and only one an be realized with a ir ular-ar representation). 4 Unit 2-interval and (x,x)-interval graphs Property 4. Let x ∈ N, x ≥ 2. The lass of (x, x)-interval graphs is stri tly in luded in the lass of (x+ 1, x+ 1)-interval graphs. Proof. We �rst prove that an interval graph with a representation where all intervals have length k (and integer open bounds) has a representation where all intervals have length k + 1. We use the following algorithm. Let S be initialized as the set of all intervals of length k, and let T be initially the empty set. As long as S is not empty, let I = [a, b] be the left-most interval of S, remove from S ea h interval [α, β] su h that α < b (in luding I), add [α, β + 1] to T , and translate by +1 all the remaining intervals in S. When S is empty, the interse tion graph of T , where all intervals have length k + 1 is the same as the interse tion graph for the original S. We also build for ea h x ≥ 2 a (x + 1, x + 1)-interval graph whi h is not a (x, x)-interval graph. We onsider the bipartite graph K2x and a perfe t mat hing {(vi, v i), i ∈ J1, xK}. We all K ′x the graph obtained from K2x with the following transformations, illustrated in Figure 7(a): remove edges (vi, v i) of the perfe t mat hing, add four graphs K4,4−e alled X1, X2, X3, X4 (for ea h Xi, we all v and vir the verti es of degree 3), link v r and v , link all vi to v r and v , link all v′i to v and v3r , and �nally add a vertex a (resp. b) linked to all vi, v i, and to two adja ent verti es of X1 (resp. X4) of degree 4. We illustrate in Figure 7(b) that K x has a realization with intervals of length x+ 1. We an prove by indu tion on x that K ′x has no realization with intervals of length x: it is rather te hni al, so we just give the idea. Any realization of K ′x for es the blo k X2 to share an extremity with the blo k X3, so ea h 2-interval v i has one interval interse ting the other extremity of X2, and the other interse ting the other extremity of X3. Then onstraints on the position of verti es vi for e their intervals to appear as two �stairways� as shown in Figure 7(b). So v1r must ontain x extremities of intervals whi h have to be di�erent, so it must have length x+ 1. Figure 7: The graph K ′4 (a) is (5,5)-interval but not (4,4)-interval. The omplexity of re ognizing unit 2-interval graphs and (x, x)-interval graphs remains open, however the following shows a relationship between those omplexities. Lemma 1. {unit 2-interval graphs} = {(x, x)-interval graphs}. Proof. The ⊃ part is trivial. To prove ⊂, let G = (V,E) be a unit 2-interval graph. Then it has a realization with |V | = n 2-intervals, that is 2n intervals of the ground set. So we onsider the interval graph of the ground set, whi h is a unit interval graph. There is a linear time algorithm based on breadth-�rst sear h to ompute a realization of su h a graph where interval endpoints are rational, with denominator 2n [CKN+95℄. So by dilating by a fa tor 2n su h a realization, we obtain a realization of G where intervals of the ground set have length 2n. Theorem 2. If re ognizing (x, x)-interval graphs is polynomial for any integer x then re ognizing unit 2-interval graphs is polynomial. 5 Investigating the omplexity of unit 2-interval graphs In this se tion we show that all proper ir ular-ar graphs ( ir ular-ar graphs su h that no ar is in luded in another in the representation) are unit 2-interval graphs, and we study a lass of graphs whi h generalizes quasi-line graphs and ontains unit 2-interval graphs. Re all that, a ording to Property 3, ir ular-ar graphs are balan ed 2-interval graphs. However, ir ular-ar graphs are not ne essarily unit 2-interval graphs. Property 5. The lass of proper ir ular-ar graphs is stri tly in luded in the lass of unit 2-interval graphs. Proof. As in the proof of Property 3, we hoose a point P on the ir le of the representation of a proper ir ular-ar graph G, and maps the ut ir le into a line segment. We extend the outer extremities of intervals that have been ut so that no interval ontains another. Thus we obtain a set of 2-intervals for ar s ontaining P , and a set I of intervals for ar s not ontaining P . For ea h interval of I, we add a new interval disjoint of any other to get a 2-interval. If we onsider the interse tion graph of the ground set of su h a representation, it is a proper interval graph. So it is also a unit interval graph [Rob69℄, whi h provides a unit 2-interval representation of G. To omplete the proof, we noti e that the domino (two y les C4 having an edge in ommon) is a unit 2-interval graph but not a ir ular-ar graph. Quasi-line graphs are those graphs whose verti es are bisimpli ial, i.e., the losed neighbor- hood of ea h vertex is the union of two liques. This graph lass has been introdu ed as a gener- alization of line graphs and a useful sub lass of law-free graphs [Ben81, FFR97, CS05, KR07℄. Following the example of quasi-line graphs that generalize line graphs, we introdu e here a new lass of graphs for generalizing unit 2-interval graphs. Let k ∈ N∗. A graph G = (V,E) is all-k-simpli ial if the neighborhood of ea h vertex v ∈ V an be partitioned into at most k liques. The lass of quasi-line graphs is thus exa tly the lass of all-2-simpli ial graphs. Noti e that this de�nition is equivalent to the following: in the omplement graph of G, for ea h vertex u, the verti es that are not in the neighborhood of u are k- olorable. Property 6. The lass of unit 2-interval graphs is stri tly in luded in the lass of all-4-simpli ial graphs. Proof. The in lusion is trivial. What is left is to show that the in lusion is stri t. Consider the following graph whi h is all-4-simpli ial but not unit 2-interval: start with the y le C4, all its verti es vi, i ∈ J1, 4K, add four K4,4 − e gadgets alled Xi, and for ea h i we onne t the vertex vi to two onne ted verti es of degree 4 in Xi. This graph is ertainly all-4-simpli ial. But if we try to build a 2-interval realization of this graph, then ea h of the 2-intervals vk has an interval trapped into the blo k Xk. So ea h 2-interval vk has only one interval to realize the interse tions with the other vi: this is impossible as we have to realize a C4 whi h has no interval representation. Property 7. The lass of law-free graphs is not in luded in the lass of all-4-simpli ial graphs. Proof. The Kneser Graph KG(7, 2) is triangle-free, but not 4- olorable [Lov78℄. We onsider the graph obtained by adding an isolated vertex v and then taking the omplement graph, i.e., KG(7, 2) ⊎ {v}. It is law-free as KG(7, 2) is triangle-free. And if it was all-4-simpli ial, then the neighborhood of v in KG(7, 2) ⊎ {v}, that is KG(7, 2), would be a union of at most four liques, so KG(7, 2) would be 4- olorable: impossible so this graph is law-free but not all-4-simpli ial. Property 8. The lass of all-k-simpli ial graphs is stri tly in luded in the lass of K1,k+1-free graphs. Proof. If a graph G ontains K1,k+1, then it has a vertex with k + 1 independent neighbors, and hen e G is not all-k-simpli ial. The wheel W2k+1 is a simple example of a graph whi h is K1,k+1-free but in whi h the enter an not have its neighborhood (a C2k+1) partitioned into k liques or less. Unfortunately, all-k-simpli ial graphs do not have a ni e stru ture whi h ould help unit 2-interval graph re ognition. Theorem 3. Re ognizing all-k-simpli ial graphs is NP- omplete for k ≥ 3. Proof. We redu e from the Graph k- olorability problem, whi h is known to be NP- omplete for k ≥ 3 [Kar72℄. Let G = (V,E) be a graph, and let G′ be the omplement graph of G to whi h we add a universal vertex v. We laim that G is k- olorable i� G′ is all-k-simpli ial. If G is k- olorable, then the non-neighborhood of any vertex in G is k- olorable, so the neighborhood of any vertex in G is a union of at most k liques. And the neighborhood of v is also a union of at most k liques, so G′ is all-k-simpli ial. Conversely, if G′ is all-k-simpli ial, then in parti ular the neighborhood of v is a union of at most k liques. Let's partition it into k vertex-disjoint liques X1, . . . ,Xk. Then, oloring G su h that two verti es have the same olor i� they are in the same Xi leads to a valid k- oloring of G. 6 Con lusion Motivated by pra ti al appli ations in s heduling and omputational biology, we fo used in this paper on balan ed 2-interval graphs and unit 2-intervals graphs. Also, we introdu ed two natural new lasses: (x, x)-interval graphs and all-k-simpli ial graphs. We mention here some dire tions for future works. First, the omplexity of re ognizing unit 2-interval graphs and (x, x)-interval graphs remains open. Se ond, the relationships between quasi-line graphs and sub lasses of balan ed 2-intervals graphs still have to be investigated. Last, sin e most problems remains NP-hard for balan ed 2-interval graphs, there is thus a natural interest in investigating the omplexity and approximation of lassi al optimization problems on unit 2-interval graphs and (x, x)-interval graphs. A knowledgments We are grateful to Vin ent Limouzy in parti ular for bringing to our attention the lass of quasi-line graphs, and Mi hael Rao and Mi hel Habib for useful dis ussions. Referen es [Ben81℄ A. Ben Rebea. Étude des stables dans les graphes quasi-adjoints. PhD thesis, Université de Grenoble, 1981. [BFV04℄ G. Blin, G. Fertin, and S. Vialette. New results for the 2-interval pattern problem. 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First we will detail how we build G′ starting from the graph G, and adding some verti es, in parti ular K5,3 gadgets. The idea is that the edges of G will partition into a Hamiltonian y le and a perfe t mat hing i� all 2-intervals of the realization of G′ an have their left inter- val realizing the Hamiltonian y le, and their right interval realizing the perfe t mat hing. A realization with su h a pla ement of the intervals is alled an �H-representation� of G. We pro eed as illustrated in Figure 6. We hoose some vertex of G that we all v0 (whi h will be the �origin� of the Hamiltonian y le), and the other are alled v1, . . . , vn. For ea h vertex vi of G we link it to a vertex of the S5 of a K5,3 alled M(vi) (whi h will blo k one of the four extremities of the 2-interval vi). We link all verti es to a new vertex z, whi h is linked to no M(v) ex ept M(v0) (thus the interval of ea h vi interse ting M(vi), for i 6= 0, won't interse t z). We add three K5,3, H1, H2 and H3 : two verti es of the S5 of H1 are linked to z, a third one is linked to one vertex of the S5 of H2, one vertex of the S5 of H3 is linked to z, and all verti es of H3 to v0. To explain this onstru tion in detail, we study the realization of G′, if we suppose it is a (balan ed) 2-interval graph, and we prove that it leads us to �nd a Hamiltonian y le in G. As the realization of H1 and H2 are two ontiguous blo ks of intervals then one of their extremities must interse t. As z is linked to two disjoint verti es of H1, both intervals of z are used to realize those interse tions. But one interval of z that we all zr, also has to interse t one vertex of H3 whi h is not linked to H1, so zr interse ts the se ond extremity of the blo k H1 (the �rst extremity being o upied by the extremity of H2). And as zr interse ts only one interval of H3, it must be the extremity of H3. The other interval of z is ontained in the blo k H1, thus an't interse t M(v0) neither all the verti es vi, so all those 2-intervals interse t zr. And as none of them interse t H3 ex ept v0, all of them ex ept v0 have an interval ontained in zr, that we all vi,g. The other interval of ea h vi is linked to a K5,3 so it has one extremity o upied by K5,3, and the other one is free. Conversely, if G has a Hamiltonian y le, then it is possible to �nd a H-representation, su h that all the onstraints indu ed by the edges of G′ are respe ted, as illustrated with the realization in Figure 6. We have already proved that this realization an be balan ed. Proof of Property 4. In the following, as we only onsidering the interval of vi or vir lo ated at one extremity of the blo k Xi, and not the one inside, we will use v and vir to denote those extremity intervals. For ea h vertex vi, we all vi,l its left interval and vi,r its right interval. We do the same for v′i, and all l(I) the left extremity of any interval I. We prove by indu tion that the graph K ′x is (x + 1, x + 1)-interval but not (x, x)-interval, and that for any unit 2-interval realization, there exists an order σ ∈ Sx su h that : • either l(vσ(x),l) < . . . < l(vσ(1),l) < l(v σ(x),l ) < . . . < l(v′ σ(1),l ) and l(v′ σ(x),r ) < . . . < σ(1),r ) < l(vσ(x),r) < . . . < l(vσ(1),r), • or the symmetri ase: l(vσ(1),l) < . . . < l(vσ(x),l) < l(v σ(1),l ) < . . . < l(v′ σ(x),l ) and σ(1),r ) < . . . < l(v′ σ(x),r ) < l(vσ(1),r) < . . . < l(vσ(x),r). Those two equalities orrespond in fa t to the �two stairways stru ture� whi h appears in Fig- ure 7. Base ase : we study all possible unit 2-interval realizations of K ′2 to prove that one of the expe ted inequalities is always true. We also prove that K ′2 has no (2,2)-interval realization. First re all that realizations of Xi subgraphs an only be blo ks of ontiguous intervals. The edge between v2r and v for es the two blo ks of X2 and X3 to be ontiguous, with intervals and v3r at their extremities. Ea h 2-interval v i must interse t both v and v3r , so one of its intervals interse ts v2 and the other interse ts v3r . Thus, one same interval of v i an not interse t both a and b whi h are disjoint, so a interse ts one interval of v′i (say the one interse ting v , the other ase being treated symmetri ally) and b interse ts the other one (so, the one interse ting v3r ). Ea h vi has to interse t both a and b, so it has to interse t a with its �rst interval and b with the se ond. But 2-interval vi must also interse t v r and v whi h are both disjoint and disjoint to a and b. So one interval of ea h vi must interse t v r and the other one must interse t So we have shown that any unit 2-interval realization of K ′2 has the following aspe t (or the symmetri ) : the extremity of the blo k X1 interse ting all vi whi h interse t a (or b) whi h interse ts all v′i, whi h interse t the extremity X2 (or X3) whi h interse ts the extremity of X3 (or X2), whi h interse ts all v i, whi h interse t b (or a), whi h interse ts all vi, whi h interse t the extremity of X4. Now we suppose, by ontradi tion, that there exists a (2,2)-interval realization of K ′2. v an interval of length 2, but one of its two parts of length one has to interse t an element of X1. The other has to interse t both v1 and v2. As neither v1 nor v2 an interse t other intervals of X1, then the �rst interval of v1 and v2 is the same interval. By pro eeding the same way on X4 and v4 , we obtain that the se ond interval of v1 and v2 is the same interval, so v1 and v2 should orrespond to the same 2-interval: it ontradi ts with the fa t that verti es v1 and v2 have a di�erent neighborhood. So K ′2 has no (2,2)-interval realization. To obtain the expe ted inequalities, we have to analyze the possible positions of all vi and v′i. We only treat the �rst two inequalities as the se ond ase is symmetri . Suppose that l(v2,l) < l(v1,l). As v1 and v 1 are non adja ent, then interval v1,l is stri tly on the left of v′1,l, so v2,l is stri tly on the left of v 1,l. Thus those two intervals do not interse t. But v2 and v 1 are adja ent, so v2 and v 1 must have interse ting right intervals. But then we have l(v′2,r) < l(v 1,r) < l(v2,r) < l(v1,r), and the right intervals of v 2 and v1 an not interse t. We dedu e their left intervals interse t, so l(v2,l) < l(v1,l) < l(v 2,l) < l(v 1,l). If we suppose that l(v1,l) < l(v2,l), we get as well that l(v 1,r) < l(v 2,r) < l(v1,r) < l(v2,r) and l(v1,l) < l(v2,l) < l(v 1,l) < l(v 2,l). So for any unit 2-interval realization of K 2 there exists an order σ = 12 or σ = 21 su h that: • either l(vσ(2),l) < l(vσ(1),l) < l(v σ(2),l ) < l(v′ σ(1),l ) and l(v′ σ(2),r ) < l(v′ σ(1),r ) < l(vσ(2),r) < l(vσ(1),r), • or the symmetri inequalities. Re ursion: suppose that for some x, K ′x−1 is not (x−1, x−1)-interval but is (x, x)-interval, and that any (x, x)-interval realization veri�es one of the expe ted inequalities. Graph K ′x−1 is an indu e subgraph of K x = (V,E) : K x−1 = K x[V \ {vx, v x}]. So by the indu tion hypothesis, there exists an order σ ∈ Sx−1 su h that for any unit 2-interval realization of K ′x : • either l(vσ(x−1),l) < . . . < l(vσ(1),l) < l(v σ(x−1),l ) < . . . < l(v′ σ(1),l ) and l(v′ σ(x−1),r ) < . . . < σ(1),r ) < l(vσ(x−1),r) < . . . < l(vσ(1),r), • or the symmetri ase: l(vσ(1),l) < . . . < l(vσ(x−1),l) < l(v σ(1),l ) < . . . < l(v′ σ(x−1),l ) and σ(1),r ) < . . . < l(v′ σ(x−1),r ) < l(vσ(1),r) < . . . < l(vσ(x−1),r). The position of vx and v x remains to be determined. We treat only the ase where the �rst two inequalities are true, as the se ond ase is symmetri . As vx and v r are adja ent, and v σ(x−1) and v1r are not, then l(v r ) < l(vx,l) < l(v σ(x−1),l ). So we de�ne j the following way: vσ(j),l is the leftmost interval su h that l(vx,l) ≤ l(vσ(j),l). if there is none, we say j = 0. Then we all σ′ ∈ Sx the permutation de�ned by: σ′(i) = σ(i) if i < j, σ′(j + 1) = x, σ′(i) = σ(i− 1) if i > j. Then we dire tly get inequalities: • l(v1r ) < l(vσ′(x),l) < . . . < l(vσ′(j+1),l) ≤ l(vx,l) < l(vσ′(j−1),l) < . . . < l(vσ′(1),l) < σ′(x),l ) < . . . < l(v′ σ′(j+1),l ) < l(v′ σ′(j−1),l ) < . . . < l(v′ σ′(1),l • l(v′ σ′(x),r ) < . . . < l(v′ σ′(j+1),r ) < l(v′ σ′(j−1),r ) < . . . < l(v′ σ′(1),r ) < l(vσ′(x),r) < . . . < l(vσ′(j+1),r) < l(vσ′(j−1),r) < . . . < l(vσ′(1),r) We obtain the expe ted inequalities by reasoning the same way as in the end of the base ase. So in parti ular we have l(vσ(x),l) < . . . < l(vσ(1),l) and v r must interse t all those vi for i ∈ J1, xK, but also an interval of X1 whi h interse ts none of the vi. So it must have length x+ 1, thus K ′x is not a (x, x)-interval graph Con lusion: As the base ase and the re ursion has been proved, expe ted properties of the graph K ′x are true for any x ≥ 2. 2-interval graphs and restrictions Useful gadgets for 2-interval graphs and restrictions Balanced 2-interval graphs Unit 2-interval and (x,x)-interval graphs Investigating the complexity of unit 2-interval graphs Conclusion Appendix

Exchange parameters from approximate self-interaction correction scheme A. Akande and S. Sanvito School of Physics and CRANN, Trinity College, Dublin 2, Ireland (Dated: November 28, 2018) The approximate atomic self-interaction corrections (ASIC) method to density functional theory is put to the test by calculating the exchange interaction for a number of prototypical materials, critical to local exchange and correlation functionals. ASIC total energy calculations are mapped onto an Heisenberg pair-wise interaction and the exchange constants J are compared to those obtained with other methods. In general the ASIC scheme drastically improves the bandstructure, which for almost all the cases investigated resemble closely available photo-emission data. In contrast the results for the exchange parameters are less satisfactory. Although ASIC performs reasonably well for systems where the magnetism originates from half-filled bands, it suffers from similar problems than those of LDA for other situations. In particular the exchange constants are still overestimated. This reflects a subtle interplay between exchange and correlation energy, not captured by the ASIC. PACS numbers: I. INTRODUCTION Theoretical studies based on density functional the- ory (DFT) [1, 2] have given remarkable insights into the electronic and magnetic properties of both molecules and solids [3]. In particular, a number of these studies attempt to quantitatively describe the magnetic inter- action in a broad range of systems including transition metals [4], hypothetical atomic chains [5, 6], ionic solids [7, 8, 9], transition metal oxides [10, 11] and transition metal polynuclear complexes [12, 13, 14]. DFT uses an effective single-particle picture where spin symmetry is generally broken. For this reason exchange parameters J are conventionally extracted by using a mapping proce- dure, where total energy calculations are fitted to a classi- cal Heisenberg Hamiltonian [4, 15]. This is then used for evaluating the Curie or Néel temperatures, the magnetic susceptibility and for interpreting neutron diffraction ex- periments [16]. Notably, the accuracy and reliability of the numerical values of the J ’s depend on the functional used for the ex- change and correlation (XC) energy, being this the only approximated part of the DFT total energy [17]. Cal- culations based on well-known local functionals, namely the local density approximation (LDA) and the gener- alised gradient approximation (GGA), are successful with itinerant magnetism in transition metals [4], but largely over-estimates the Heisenberg exchange parameters in many other situations [7, 8, 9, 11, 14]. Additional cor- rections based on the kinetic energy density (metaGGA) [18] marginally improves the agreement with experiments [9], although an extensive investigation over several solid state systems has not been carried out so far. These failures are usually connected to the local char- acter of the LDA, which is only weakly modified by con- structing XC potentials including the gradient, or higher derivative of the charge density. A direct consequence is that the charge density is artificially delocalized in space, leading to an erroneous alignment of the magnetic bands. These are also artificially broadened. A typical example is that of NiO, which LDA predicts as Mott-Hubbard in- stead of charge-transfer insulator. Clearly a qualitative failure in describing the ground state results in an erro- neous prediction of the exchange parameters. One of the reasons behind the inability of LDA and GGA of describing localized charge densities is attributed to the presence of the self-interaction error (SIE) [19]. This originates from the spurious Coulomb interaction of an electron with itself, which is inherent to local func- tionals. Hartree-Fock (HF) methods, in the unrestricted or spin polarised form, are SIE free and produce sys- tematic improved J parameters. However, these meth- ods lack of correlation and usually overcorrect. A typ- ical example is the antiferromagnetic insulator KNiF3 for which HF predicts a nearest neighbour J of around 2 meV [7, 20, 21, 22, 23] against an experimental value of 8.6 meV [24]. Direct SIE subtraction, convention- ally called self-interaction corrected (SIC) LDA, also im- proves the results and seems to be less affected by over- correction [5, 25]. Similarly hybrid-functionals, which mix portions of HF exchange with the local density ap- proximation of DFT, perform better than local function- als and in several situations return values for J in close agreement with experiments [7, 8]. It is important to note that both methods based non- local exchange or SIC, are computationally demanding and thus their application to the solid state remains rather limited. It is then crucial to develop practical com- putational schemes able to provide a good estimate of the exchange parameters for those systems critical to LDA, which at the same time are not numerically intensive. Based on the idea that most of the SIE originates from highly localized states, with a charge distribution resem- bling those of free atoms, Vogel et al. [26] proposed a sim- ple SIC scheme where the corrections are approximated by a simple on-site term. This method was then gener- alized to fractional occupation by Filippetti and Spaldin [27] and then implemented in a localized atomic orbital code for large scaling by Pemmaraju et al. [28]. Despite its simplicity the method has been successfully applied to a number of interesting physical systems including , tran- sition metal monoxides [27, 29], silver halides [30], no- ble metal oxides [31], ferroelectric materials [27, 32, 33], high-k materials [34], diluted magnetic semiconductors [35, 36] and also to quantum transport [37, 38]. The method is strictly speaking not variational, in the sense that a functional generating the ASIC potential via variational principle is not available. However, since typ- ically the LDA energy is a good approximation of the exact DFT energy, although the LDA potential is rather different from the exact KS potential, a “practical” def- inition of total energy can be provided. In this work we evaluate the ability of this approximated energy in de- scribing exchange parameters for a variety of magnetic systems. II. THE ATOMIC SIC METHOD The seminal work of Perdew and Zunger [19] pioneered the modern theory of SIC. The main idea is that of sub- tracting directly the spurious SI for each Kohn-Sham (KS) orbital ψn. The SIC-LDA [39] XC energy thus writes ESICxc [ρ ↑, ρ↓] = ELDAxc [ρ ↑, ρ↓]− occupied∑ δSICn , (1) where ELDAxc [ρ ↑, ρ↓] is the LDA-XC energy and δSICn is the sum of the self-Hartree and self-XC energy associated to the charge density ρσn = |ψσn|2 of the fully occupied KS orbital ψσn δSICn = U [ρ n] + E xc [ρ n, 0] . (2) Here U is the Hartree energy and σ is the spin index. The search for the energy minimum is not trivial, since ESICxc is not invariant under unitary rotations of the occu- pied KS orbitals. As a consequence the KS method be- comes either non-orthogonal or size-inconsistent. These problems however can be avoided [40, 41, 42] by intro- ducing a second set of orbitals φσn related to the canonical KS orbitals by a unitary transformation M ψσn = Mσnmφ m . (3) The functional can then be minimized by varying both the orbitals ψ and the unitary transformationM, leading to a system of equations n = (H 0 + ∆v n = � σ,SIC n , (4) ψσn = Mnmφσm , (5) ∆vSICn = MnmvSICm vSICm P̂ m , (6) where Hσ0 is the LDA Hamiltonian, P̂ n(r) = φσm(r)〈φσm|ψσn〉 and vSICn = u([ρn]; r) + vσ,LDAxc ([ρ↑n, 0]; r), with u and vσ,LDAxc the Hatree and LDA-XC potential respectively. In equation (4) we have used the fact that at the en- ergy minimum the matrix of SIC KS-eigenvalues �σ,SICnm is diagonalized by the KS orbitals ψn. Importantly such minimization scheme can be readily applied to extended systems, without loosing the Bloch representation of the KS orbitals [43, 44]. The ASIC method consists in taking two drastic ap- proximations in equation (4). First we assume that the orbitals φm, that minimize the SIC functional are atomic- like orbitals Φσm (ASIC orbitals) thus∑ vSICm (r)P̂ m → α ṽσSICm (r)P̂ m , (7) where ṽσSICm (r) and P̂ m are the SIC potential and the projector associated to the atomic orbital Φσm. Secondly we replace the non-local projector P̂Φm with its expecta- tion value in such a way that the final ASIC potential reads vσASIC(r) = α ṽσSICm (r)p m , (8) where pσm is the orbital occupation (essentially the spin- resolved Mülliken orbital population) of Φm. Note that in the final expression for the potential a factor α appears. This is an empirical scaling term that accounts for the fact that the ASIC orbital Φ in general do not coincide with those that minimize the SIC func- tional (1). By construction α = 1 in the single particle limit, while it vanishes for the homogeneous electron gas. Although in general 0 < α < 1, extensive testing [28] demonstrates that a value around 1 describes well ionic solids and molecules, while a value around 1/2 is enough for mid- to wide-gap insulators. In the following we will label with ASIC1/2 and ASIC1 calculations obtained re- spectively with α = 1/2 and α = 1. Finally we make a few comments over the total energy. As pointed out in the introduction the present theory is not variational since the KS potential cannot be related to a functional by a variational principle. However, since typical LDA energies are more accurate than their corre- sponding KS potentials, we use the expression of equation (1) as suitable energy. In this case the orbital densities entering the SIC are those given by the ASIC orbital Φ. Moreover, in presenting the data, we will distinguish re- sults obtained by using the SIC energy (1) from those obtained simply from the LDA functional evaluated at the ASIC density, i.e. without including the δn correc- tions (2). III. RESULTS All our results have been obtained with an implemen- tation of the ASIC method [28] based on the DFT code Siesta [45]. Siesta is an advanced DFT code using pseu- dopotentials and an efficient numerical atomic orbital ba- sis set. In order to compare the exchange parameters ob- tained with different XC functionals we consider the LDA parameterization of Ceperly and Alder [46], the GGA functional obtained by combining Becke exchange [47] with Lee-Yang-Parr correlation [48] (BLYP), the nonem- pirical Purdew, Burke and Ernzerhof (PBE) GGA [49], and the ASIC scheme as implemented in reference [28]. Calculations are performed for different systems crit- ical to LDA and GGA, ranging from molecules to ex- tended solids. These include hypothetical H-He atomic chains, the ionic solid KNiF3 and the transition metal monoxides MnO and NiO. DFT total energy calculations are mapped onto an effective pairwise Heisenberg Hamil- tonian HH = − Jnm~Sn · ~Sm , (9) where the sums runs over all the possible pairs of spins. In doing this we wish to stress that the mapping is a con- venient way of comparing total energies of different mag- netic configurations calculated with different function- als. In this spirit the controversy around using the spin- projected (Heisenberg mapping) or the non-projected scheme is immaterial [5, 50, 51]. A. H-He chain As an example of molecular systems, we consider H- He monoatomic chains at a inter-atomic separation of 1.625 Å (see figure 1). This is an important benchmark for DFT since the wave-function is expected to be rather localized and therefore to be badly described by local XC functionals. In addition the system is simple enough to be accessible by accurate quantum chemistry calcula- tions. As basis set we use two radial functions (double-ζ) for the s and p angular momenta of both H and He, while the density of the real-space grid converges the self-consistent calculation at 300 Ry. Here we consider all possible Heisenberg parameters. Thus the triangular molecule (Fig.1a) has only one nearest neighbour param- eter Ja12, the 5-atom chain (Fig.1b) has both first J 12 and second neighbour Jb13 parameters, and the 7-atom chain (Fig.1c) has three parameters describing respectively the nearest neighbour interaction with peripheral atoms Jc12, the nearest neighbour interaction between the two middle atoms Jc23 and the second neighbour interaction J Following reference [5], accurate calculations based on second-order perturbation theory (CASPT2) [12] are used as comparison. The quality of each particular func- tionals is measured as the relative mean deviation of the nearest neighbour exchange parameters only (Ja12, J FIG. 1: (Color on-line) H-He-H chains at an inter-atomic dis- tance of 1.625Å. Method Ja12 J 13 δ (%) CASPT2 -24 -74 -0.7 -74 -79 -0.7 0 SIC-B3LYP -31 -83 -0.2 -83 -88 -0.3 16 LDA -68 -232 -6 -234 -260 -6 210 PBE -60 -190 -1.8 -190 -194 -1.6 152 BLYP -62 -186 -2 -186 -200 -1 147 ASIC1 -36 -112 -1 -110 -122 -0.6 51 ASIC1/2 -44 -152 -1 -152 -168 -1.4 101 ASIC∗1 -40 -128 -0.6 -128 -142 -1.0 73 ASIC∗1/2 -50 -170 -1.4 -170 -190 -1.8 127 TABLE I: Calculated J values (in meV) for the three different H–He chains shown in Fig.1. The CASPT2 values are from reference [12], while the SIC-B3LYP are from reference [5]. The last two rows correspond to J values obtained from the LDA energy calculated at the ASIC density. Jc12, J 23), since those are the largest ones |Ji − JCASPT2i | |JCASPT2i | . (10) Our calculated J values and their relative δ are pre- sented in table I, where we also include results for a fully self-consistent SIC calculation over the B3LYP functional (SIC-B3LYP) [5]. It comes without big surprise that the LDA systematically overestimates all the exchange pa- rameters with errors up to a factor 6 for the smaller J (Jb13 and J 13) and an average error δ for the largest J of about 200%. Standard GGA corrections considerably improve the description although the J ’s are still system- atically larger than those obtained with CASPT2. Note that the results seem rather independent of the particular GGA parameterization, with PBE and BLYB producing similar exchange constants. This is in good agreement with previous calculations [5]. SIC in general dramatically improves the LDA and GGA description and our results for ASIC1 are reason- ably close to those obtained with the full self-consistent procedure (SIC-B3LYP). This is an interesting result, considering that our ASIC starts from a local exchange functional, while B3LYP already contains non-local con- tributions. We also evaluate the J parameters by using the LDA energy evaluated at the ASIC density (last two rows in table I). In general this procedure gives J ’s larger than those obtained by using the energy of equation (1), meaning that the δSICn contributions reduce the J values. It is then clear that the ASIC scheme systematically improves the J values as compared to local functionals. The agreement however is not as good as the one ob- tained by using a fully self-consistent SIC scheme, mean- ing that for this molecular system the ASIC orbitals are probably still not localized enough. This can alter the actual contribution of δSICn to the total energy and there- fore the exchange parameters. B. Ionic antiferromagnets: KNiF3 Motivated by the substantial improvement of ASIC over LDA, we then investigate its performances for real solid-state systems, starting from KNiF3. This is a pro- totypical Heisenberg antiferromagnet with strong ionic character, a material for which our ASIC approxima- tion is expected to work rather well [28]. It is also a well studied material, both experimentally [24, 52] and theoretically [7, 9, 21, 22], allowing us extensive com- parisons. The KNiF3 has cubic perovskite-like structure with the nickel atoms at the edges of the cube, flourine atoms at the sides and potassium atoms at the center (see Fig.2). At low temperature, KNiF3 is a type II antiferro- magnetic insulator consisting of ferromagnetic (111) Ni planes aligned antiparallel to each other. For our calcu- lations we use a double-ζ polarized basis for the s and p orbitals of K, Ni and F, a double-ζ for the 3d of K and Ni, and a single-ζ for the 3d of F. Finally, we use 5×5×5 k- points in the full Brillouin zone and the real-space mesh cutoff is 550 Ry. Note that the configuration used to generate the pseudopotential is that of Ni2+, 4s13d7. We first consider the band-structure as obtained with LDA and ASIC. For comparison we also include results obtained with LDA+U [53, 54] as implemented in Siesta [55]. In this case we correct only the Ni d shell and we fix the Hubbard-U and Hund’s exchange-J parameters by fitting the experimental lattice constant (a0 = 4.014 Å). The calculated values are U=8 eV and J=1 eV. The bands obtained with the three methods and the corre- sponding orbital projected density of states (DOS) are presented in figures 3 and 4 respectively. All the three functionals describe KNiF3 as an insula- tor with bandgaps respectively of 1.68 eV (LDA), 3.19 eV (ASIC1), and 5.0 eV (LDA+U). An experimental value for the gap is not available and therefore a comparison cannot be made. In the case of LDA and ASIC the gap is formed between Ni states, with conductance band bot- FIG. 2: (Color on-line) Cubic perovskite structure of KNiF3. Color code: blue=Ni, red=F, Green=K. FIG. 3: Band structure for type II antiferromagnetic KNiF3 obtained with a) LDA, b) ASIC1 and c) LDA+U (U=8 eV and J=1 eV). The valence band top is aligned at E=EF=0 eV (horizontal line). tom well described by eg orbitals. These are progressively moved upwards in energy by the SIC, but still occupy the gap. Such feature is modified by LDA+U which pushes the unoccupied eg states above the conductance band minimum, which is now dominated by K 4s orbitals. In more detail the valence band is characterized by a low-lying K 3p band and by a mixed Ni-3d/F 2p. While the K 3p band is extremely localized and does not present substantial additional orbital components the amount of mixing and the broadening of the Ni-3d/F 2p varies with the functionals used. In particular both LDA and ASIC predict that the Ni 3d component occupies the high en- FIG. 4: (Color on-line) DOS for type II antiferromagnetic KNiF3 obtained with a) LDA, b) ASIC1 and c) LDA+U (U=8 eV and J=1 eV). The valence band top is aligned at E=0 eV (vertical line). The experimental UPS spectrum from reference [56] is also presented (thick green line). The relative binding energy is shifted in order to match the K 3p peak. ergy part of the band, while the F 2p the lower. For both the total bandwidth is rather similar and it is about 9- 10 eV. In contrast LDA+U offers a picture where the Ni-F hybridization spread across the whole bandwidth, which is now reduced to less than 7 eV. Experimentally, ultraviolet photoemission spec- troscopy (UPS) study of the whole KMF3 (M: Mn, Fe, Co, Ni, Cu, Zn) series [56] gives us a spectrum dominated by two main peaks: a low K 3p peak and broad band mainly attributed to F 2p. These two spectroscopical features are separated by a binding energy of about 10 eV. In addition the 10 eV wide F 2p band has some fine structure related to various Ni d multiplets. An analysis based on averaging the multiplet structure [56] locates the occupied Ni d states at a bounding energy about 3 eV smaller than that of the F 2p band. In figure 4 we superimpose the experimental UPS spectrum to our calculated DOS, with the convention of aligning in each case the sharp K 3p peak. It is then clear that ASIC provides in general a better agreement with the UPS data. In particular both the Ni-3d/F 2p bandwidth and the position of the Fermi en- ergy (EF) with respect to the K 3p peak are correctly predicted. This is an improvement over LDA, which de- scribes well the Ni-3d/F 2p band, but positions the K 3p states too close to EF. For this reason, when we align the Method a0 Jth P d Jex P LDA 3.951 46.12 (53.1) 1.829 40.4 1.834 PBE 4.052 33.98 (37.0) 1.813 36.48 1.808 BLYP 4.091 31.10 (37.6) 1.821 36.72 1.812 ASIC1/2 3.960 40.83 1.876 36.14 1.878 ASIC1 3.949 36.22 1.907 30.45 1.914 ASIC∗1/2 3.969 43.44 1.876 38.57 1.878 ASIC∗1 3.949 39.80 1.907 33.56 1.914 LDA+U 4.007 12.55 10.47 1.940 TABLE II: Calculated J parameters (in meV) and the Mülliken magnetic moment for Ni 3d (Pd) in KNiF3. The ex- perimental values for J and a0 are 8.2±0.6 meV and 4.014Å respectively while the values in brackets are those from refer- ence [9]. In the table we report values evaluated at the theo- retical (Jth and P d ) and experimental (Jex and P d ) lattice constant. ASIC∗1/2 and ASIC 1 are obtained from the LDA energies evaluated at the ASIC density. UPS spectrum at the K 3p position, this extends over EF. Finally in the case of LDA+U , there is a substantial mis- alignment between the UPS data and our DOS. LDA+U in fact erroneously pushes part of the Ni d mainfold below the F 2p DOS, which now forms a rather narrow band. We now turn our attention to total energy related quantities. In table II we present the theoretical equi- librium lattice constant a0 and the Heisenberg exchange parameter J for all the functionals used. Experimentally we have J=8.2± 0.6 meV [24]. The values of a0 and J are calculated for the type II antiferromagnetic ground state, by constructing a supercell along the (111) direction. Im- portantly values of J obtained by considering a supercell along the (100) direction, i.e. by imposing antiferromag- netic alignment between ferromagnetic (100) planes (type I antiferromagnet), yield essentially the same result, con- firming the fact that the interaction is effectively only extending to nearest neighbors. Furthermore we report results obtained both at the theoretical equilibrium lat- tice constant (Jth) and at the experimental one (Jex). Also in this case local XC functionals largely overesti- mate J , with errors for Jex going from a factor 8 (LDA) to a factor 4.5 (GGA-type). ASIC improves these val- ues, although only marginally, and our best agreement is found for ASIC1, while ASIC1/2 is substantially iden- tical to GGA. Interestingly the ASIC1 performance is rather similar, if not better, to that of meta-GGA func- tionals [9]. The situation is however worsened when we consider J parameters obtained at the theoretical lattice constant. The ASIC-calculated a0 are essentially identi- cal to those from LDA and about 2% shorter than those from GGA. Since J depends rather severely on the lattice parameter we find that at the theoretical lattice constant GGA-functionals perform actually better than our ASIC. Finally, also in this case the J ’s obtained by simply us- ing the LDA energies are larger than those calculated by including the SIC corrections (see equation 1). In general the improvement of the J parameter is cor- related to an higher degree of electron localization, in particular of the Ni d shell. In table II the magnetic mo- ment of the Ni d shell Pd, obtained from the Mülliken population, is reported. This increases systematically when going from LDA to GGA to ASIC approaching the atomic value expected from Ni2+. Our best result is obtained with LDA+U , which re- turns an exchange of 10.47 meV for the same U and J that fit the experimental lattice constant. This is some- how superior performance of LDA+U with respect to ASIC should not be surprising and it is partially related to an increased localization. The Ni ions d shell in oc- tahedral coordination splits into t2g and eg states, which further split according to Hund’s rule. The t2g states are all filled, while for the eg only the majority are. By look- ing at the LDA DOS one can recognize the occupied t↑2g orbitals (we indicate majority and minority spins respec- tively with ↑ and ↓) at -3 eV, the e↑g at -2 eV and the t at about 0 eV, while the empty e↓g are at between 1 and 3 eV above the valence band maximum. The local Hund’s split can be estimated from the e↑g- e↓g separation. The ASIC scheme corrects only occupied states [57], and therefore it enhances the local exchange by only a downshift of the valence band. From the DOS of figure 4 it is clear that this is only a small contri- bution. In contrast the LDA+U scheme also corrects empty states, effectively pushing upwards in energy the e↓g band. The net result is that of a much higher degree of localization of the d shell with a consequent reduction of the Ni-Ni exchange. This is similar to the situation de- scribed by the Hartree-Fock method, which however re- turns exchange parameters considerably smaller than the experimental value [20, 21, 22, 23]. Interestingly hybrid functionals [7] have the right mixture of non-local ex- change and electron correlation and produce J ’s in close agreement with the experiments. We further investigate the magnetic interaction by evaluating J as a function of the lattice constant. Ex- perimentally this can be achieved by replacing K with Rb and Tl, and indeed de Jongh and Block [58] early suggested a d−α power law with α = 12± 2. Our calcu- lated J as a function of the lattice constant d for LDA, GGA, ASIC1 and LDA+U (U=8 eV and J=1 eV) are presented in figure ??. For all the four functionals in- vestigated J varies as a power law, although the calcu- lated exponents are rather different: 8.6 for LDA, 9.1 for GGA, 11.3 for ASIC1 and 14.4 for LDA+U . This further confirms the strong underestimation of the ex- change constants from local functionals. Clearly the rel- ative difference between the J obtained with different functionals becomes less pronounced for small d, where the hybridization increases and local functionals perform better. Note that only ASIC1 is compatible with the ex- perimental exponent of 12 ± 2, being the one evaluated from LDA+U too large. Importantly we do not expect to extrapolate the LDA+U value at any distance, since FIG. 5: J as a function of the lattice constant for LDA, GGA, ASIC1 and LDA+U (U=8 eV and J=1 eV). The symbols are our calculate value while the solid lines represent the best power-law fit. the screening of the parameters U and J changes with the lattice constant. In conclusion for the critical case of KNiF3 the ASIC method appears to improve the LDA results. This is es- sentially due to the better degree of localization achieved by the ASIC as compared with standard local function- als. However, while the improvement over the bandstruc- ture is substantial, it is only marginal for energy-related quantities. The main contribution to the total energy in the ASIC scheme comes from the LDA functional, which is now evaluated at the ASIC density. This is not suf- ficient for improving the exchange parameters, which in contrast need at least a portion of non-local exchange. C. Transition metal monoxides Another important test for the ASIC method is that of transition metal monoxides. These have been extensively studied both experimentally and theoretically and they are the prototypical materials for which the LDA appears completely inadequate. In this work we consider MnO and NiO, which have respectively half-filled and partially- filled 3d shells. They both crystallize in the rock-salt structure and in the ground state they are both type- II antiferromagnetic insulators. The Néel’s temperatures are 116 K and 525 K respectively for MnO and NiO. In all our calculations we consider double-ζ polarised basis for the s and p shell of Ni, Mn and O, double-ζ for the Ni and Mn 3d orbitals, and single-ζ for the empty 3d of O. We sample 6×6×6 k-points in the full Brillouin zone of both the cubic and rhombohedral cell describing respectively type I and type II antiferromagnetism. Finally the real- space mesh cutoff is 500 Ry. The calculated band structures obtained from LDA, ASIC1/2 and ASIC1 are shown in figures 6 and 7 for MnO and NiO respectively. These have been already discussed extensively in the context of the ASIC method [27, 28] and here we report only the main features. For both the materials LDA already predicts an insulating behavior, although the calculated gaps are rather small and the nature of the gaps is not what experimentally found. In both cases the valence band top has an almost pure d component, which suggests these materials to be small gap Mott-Hubbard insulators. The ASIC downshifts the occupied d bands which now hybridize with the O-p man- ifold. The result is a systematic increase of the band-gap which is more pronounced as the parameter α goes from 1/2 to 1. Importantly, as noted already before [28], the experimental band-gap is obtained for α ∼ 1/2. FIG. 6: Calculated band structure for the type II anti- ferromagnetic MnO obtained from a) LDA, b) ASIC1/2 and c) ASIC1. The valence band top is aligned at 0 eV (horizontal line). We then moved to calculating the exchange parame- ters. In this case we extend the Heisenberg model to second nearest neighbors, by introducing the first (J1) and second (J2) neighbor exchange parameters. These are evaluated from total energy calculations for a ferro- magnetic and both type II and type I antiferromagnetic alignments. Our calculated results, together with a few selected data available from the literature are presented in table III. Let us first focus our attention to MnO. In this case both the J ’s are rather small and positive (antiferro- magnetic coupling is favorite), in agreement with the Goodenough-Kanamori rules [59] and the rather low Néel temperature. Direct experimental measurements of the exchange parameters are not available and the com- monly accepted values are those obtained by fitting the magnetic susceptibility with semi-empirical methods [60]. Importantly this fit gives almost identical first and second nearest neighbour exchange constants. In contrast all the exchange functionals we have investigated offer a picture where J2 is always approximately twice as large as J1. FIG. 7: Calculated band structure for the type II anti- ferromagnetic NiO obtained from a) LDA, b) ASIC1/2 and c) ASIC1. The valence band top is aligned at 0 eV (horizon- tal line). Method MnO NiO J1 J2 Pd J1 J2 Pd LDA 1.0 2.5 4.49 (4.38) 13.0 -94.4 1.41 (1.41) PBE 1.5 2.5 4.55 (4.57) 7.0 -86.8 1.50 (1.59) ASIC1/2 1.15 2.44 4.72 (4.77) 6.5 -67.3 1.72 (1.77) ASIC1 0.65 1.81 4.84 (4.86) 3.8 -41.8 1.83 (1.84) ASIC∗1/2 1.27 2.65 4.72 (4.77) 7.1 -74.6 1.72 (1.77) ASIC∗1 0.69 2.03 4.84 (4.86) 4.4 -47.9 1.83 (1.84) SE1a 0.86 0.95 HFb 0.22 0.36 B3LYPc 0.81 1.71 PBE0b 0.96 1.14 B3LYPd 2.4 -26.7 HFd 0.8 -4.6 SIC-LDAe 2.3 -12 Expt.f 1.4 -19.8 Expt.g 1.4 -17.0 TABLE III: Calculated J1 and J2 in meV for MnO and NiO. Pd is the magnetic moment of the d shell calculated from the type II antiferromagnetic phase. Values in bracket are for Pd evaluated from the ferromagnetic ground state. ASIC∗1/2 and ASIC∗1 are obtained from the LDA energies evaluated at the ASIC density. a) Ref. [60], b) Ref. [61], c) Ref. [62], d) Ref. [11], e) Ref. [25], f) Ref. [64], g) Ref. [65] This gives us a reasonably accurate value of J1 for LDA and GGA, but J2 is overestimated by approximately a factor 2, in agreement with previous calculations [10]. ASIC systematically improves the LDA/GGA descrip- tion, by reducing both J1 and J2. This is related to the enhanced localization of the Mn d electrons achieved by the ASIC, as it can be seen by comparing the Mn d mag- netic moments (Pd) calculated for different functionals (see table III). Thus ASIC1, which provides the largest magnetic moment, gives also J ’s in closer agreement with the experimental values, while ASIC1/2 is not very dif- ferent from LDA. Importantly for half-filling, as in MnO, the ASIC scheme for occupied states is fundamentally analogous to the LDA+U method, with the advantage that the U parameter does not need to be evaluated. Finally, at variance with KNiF3, it does not seem that a portion of exact exchange is strictly necessary in this case. Hartree- Fock [61] results in a dramatic underestimation of the J parameters, while B3LYP [62] is essentially very similar to ASIC1. Curiously the best results available in the lit- erature [61] are obtained with the PBE0 functional [63], which combines 25% of exact-exchange with GGA. The situation for NiO is rather different. The ex- perimentally available data [64, 65] show antiferromag- netic nearest neighbour and ferromagnetic second near- est neighbour exchange parameters. The magnitude is also rather different with |J2| > 10 |J1|. Standard lo- cal functionals (LDA and GGA) fail badly and overes- timate both the J ’s by more than a factor 5. ASIC in general reduces the exchange constants and drasti- cally improves the agreement between theory and exper- iments. In particular ASIC1 gives exchange parameters only about twice as large as those measured experimen- tally. A better understanding can be obtained by looking at the orbital-resolved DOS for the Ni d and the O p orbitals (figure 8) as calculated from LDA and ASIC. There are two main differences between the LDA and the ASIC results. First there is an increase of the fundamental band-gap from 0.54 eV for LDA to 3.86 eV for ASIC1/2 to 6.5 eV for ASIC1. Secondly there is change in the relative energy positioning of the Ni d and O p contributions to the valence band. In LDA the top of the valence band is Ni d in nature, with the O p dominated part of the DOS lying between 4 eV and 8 eV from the valence band top. ASIC corrects this feature and for ASIC1/2 the O p and Ni d states are well mixed across the whole bandwidth. A further increase of the ASIC corrections (α = 1) leads to a further downshift of the Ni d band, whose contribution becomes largely suppressed close to the valence band-top. Thus, increasing the portion of ASIC pushes NiO further into the charge transfer regime. Interestingly, although ASIC1/2 gives the best band- structure, the J ’s obtained with ASIC1 are in better agreement with the experiments. This is somehow sim- ilar to what observed when hybrid functionals are put to the test. Moreira et al. demonstrated [11] that J ’s in close agreement with experiments can be obtained by using 35% Hartree-Fock exchange in LDA. Moreover, in close analogy to the ASIC behaviour, as the fraction of exact exchange increases from LDA to purely Hartree- Fock, the exchange constants decrease while the band- gap gets larger. However, while the best J ’s are obtained with 35% exchange, a gap close to the experimental one FIG. 8: Calculated orbital resolved DOS for type II anti- ferromagnetic NiO obtained with a) LDA, b) ASIC1/2 and c) ASIC1. The valence band top is aligned at 0 eV. is obtained with B3LYP, which in turns overestimates the J ’s. This remarks the subtile interplay between exchange and correlations in describing the magnetic interaction of this complex material. Finally, it is worth remarking that a fully self-consistent SIC [25] seems to overcorrect the J ’s, while still presenting the erroneous separation between the Ni d and O p states. IV. CONCLUSIONS In conclusions the approximated expression for the ASIC total energy is put to the test of calculating ex- change parameters for a variety of materials, where local and gradient-corrected XC functionals fail rather badly. This has produced mixed results. On the one hand, the general bandstructure and in particular the valence band, is considerably improved and resembles closely data from photo-emission. On the other hand, the exchange con- stants are close to experiments only for the case when the magnetism originates from half-filled shells. 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Jones, Phys. Rev. 139, A1313 (1965). [61] C. Franchini, V. Bayer, R. Podloucky, J. Paier and G. Kresse, Phys. Reb. B 72, 045132 (2005). [62] X. Fenf, Phys. Rev. B 69, 155107 (2004). [63] M. Ernzerhof and G.E. Scuseria, J. chem. Phys. 110, 5029 (1999). [64] M. T. Hutching and E. J. Samuelsen, Phys. Rev. B 6, 3447 (1972). [65] R. Shanker and R. A. Singh, Phys. Rev. B 7, 5000 (1973). [66] S. Hüfner and T. Riesterer, Phys. Rev. B 33, 7267 (1986). Introduction The atomic SIC method Results H-He chain Ionic antiferromagnets: KNiF3 Transition metal monoxides Conclusions Acknowledgements References

The approximate atomic self-interaction corrections (ASIC) method to density functional theory is put to the test by calculating the exchange interaction for a number of prototypical materials, critical to local exchange and correlation functionals. ASIC total energy calculations are mapped onto an Heisenberg pair-wise interaction and the exchange constants J are compared to those obtained with other methods. In general the ASIC scheme drastically improves the bandstructure, which for almost all the cases investigated resemble closely available photo-emission data. In contrast the results for the exchange parameters are less satisfactory. Although ASIC performs reasonably well for systems where the magnetism originates from half-filled bands, it suffers from similar problems than those of LDA for other situations. In particular the exchange constants are still overestimated. This reflects a subtle interplay between exchange and correlation energy, not captured by the ASIC.

Introduction The atomic SIC method Results H-He chain Ionic antiferromagnets: KNiF3 Transition metal monoxides Conclusions Acknowledgements References

Astronomy & Astrophysics manuscript no. p0603 c© ESO 2018 November 29, 2018 Generalized ǫ-law The role of unphysical source terms in resonance line polarization transfer and its importance as an additional test of NLTE radiative transfer codes. J. Štěpán1,2 and V. Bommier2 1 Astronomical Institute, Academy of Sciences of the Czech Republic, 251 65 Ondřejov, Czech Republic e-mail: stepan@asu.cas.cz 2 LERMA, Observatoire de Paris – Meudon, CNRS UMR 8112, 5, Place Jules Janssen, 92195 Meudon Cedex, France e-mail: [jiri.stepan;V.Bommier]@obspm.fr Received 4 October 2006 / Accepted 26 March 2007 ABSTRACT Context. A derivation of a generalized ǫ-law for nonthermal collisional rates of excitation by charged perturbers is presented. Aims. Aim of this paper is to find a more general analytical expression for a surface value of the source function which can be used as an addtional tool for verification of the non-LTE radiative transfer codes. Methods. Under the impact approximation hypothesis, static, one-dimensional, plane-parallel atmosphere, constant magnetic field of arbitrary strength and direction, two-level atom model with unpolarized lower level and stimulated emission neglected, we introduce the unphysical terms into the equations of statistical equilibrium and solve the appropriate non-LTE integral equations. Results. We derive a new analytical condition for the surface values of the source function components expressed in the basis of irreducible spherical tensors. Key words. line: formation – polarization – radiative transfer 1. Introduction In the series of papers of Landi Degl’Innocenti et al. (1991a,b), Landi Degl’Innocenti & Bommier (1994) (from now on refer- enced as Paper I), the general formalism of resonance line po- larization scattering for a two-level atom has been developed. The non-LTE problem of the 2nd kind for an arbitrary magnetic field, three-dimensional geometry of the medium and arbitrary irradiation by external sources has been discussed. The effect of inelastic collisions with charged perturbers has been considered for the particular case of a relative Maxwellian velocity distribu- tion. Paper I analysed the analytical properties of the solutions in the particular case of a one-dimensional, semi-infinite, static at- mosphere with a constant magnetic field of arbitrary strength and direction and assuming zero external irradiation of the at- mosphere. They derived a generalization of the well known law (e.g. Avrett & Hummer 1965; Mihalas 1970; Hubený 1987) for the case of polarized radiation and extended the previous re- sults of Ivanov (1990) who studied scattering in a non-magnetic regime. In most cases of practical interest the polarization degree is rather small. The purpose of this paper is to find a new analytical solution of the non-LTE problem in unphysical conditions in or- der to better verify the accuracy of the polarized radiation trans- fer codes. This is done by introduction of an unphysical source term in the polarization into the equations of statistical equilib- rium. Such a generalization can be useful in testing the accuracy of the radiative transfer codes whose purpose is to deal with the Send offprint requests to: J. Štěpán non-thermal collisional processes (for instance in the impact po- larization studies of solar flares). Following the approach of the papers quoted above, we adopt the formalism of density matrix in the representation of irre- ducible tensorial operators (e.g. Fano 1957). We consider the lower level with total angular momentum j to be unpolarized. This level is completely described by the overall population which is set to 1 for normalization reasons. The upper level with angular momentum j′ is described by the multipole components of the statistical tensor ρKQ. Coherences between different levels j and j′ are neglected but coherences between Zeeman sublevels of level j′ are in general taken into account. The calculation is performed in the Wien limit of line frequency whose assump- tion makes it possible to neglect stimulated emission effect, and to preserve the linearity of the non-LTE problem. 2. Equations of statistical equilibrium The suitable coordinate system Σ0 for atomic state description is the one with the z-axis directed along the magnetic field (see Figure 1). Radiative rate contributions to the evolution of statistical op- erator ρKQ are given by (Landi Degl’Innocenti 1985)   = −iA j′ jΓQρKQ − A j′ jρ w(K)j′ j (−1) 2 j′ + 1 B j j′J −Q. (1) http://arxiv.org/abs/0704.1573v1 2 J. Štěpán and V. Bommier: Generalized ǫ-law In this equation A j′ j (B j j′) is the Einstein coefficient of spon- taneous emission (absorption) from level j′ ( j) to level j ( j′). Γ = 2πg j′νL/A j′ j with g j′ being the Landé factor of the level j′ and νL is the Larmor frequency. The transition-dependent nu- merical factor w(K)j′ j has been defined by Landi Degl’Innocenti (1984) as have the irreducible components of the mean radiation tensor J Q. Besides the radiative rates, collisional rates have to be considered in the statistical equilibrium, because the source of radiation in a semi-infinite atmosphere is the collisional exci- tation followed by radiative de-excitation. Thus, the source term of the radiative transfer equation originates in the inelastic col- lision effect. As the purpose of the present paper is to consider unphysical source terms in the non-zero ranks (K,Q) of the ir- reducible tensorial operator basis T KQ , we will introduce an un- physical (K,Q)-dependence to the inelastic collisional rates of the statistical equilibrium equation below. The purpose here is not to thus describe anisotropic collisions, which would require a proper formalism that is out of the scope of the present paper (see, for instance, Landi Degl’Innocenti & Landolfi (2004) for a two-level atom, and Derouich (2006), for polarization trans- fer rates in a multi-level atom due to isotropic collisions). We introduce as usual the depolarizing rate due to isotropic elastic collisions. Thus, the contribution of collisional rates reads   = (C j j′ ) Q − (C j′ j) Q − D Q. (2) The terms (C j j′ ) Q and (C j′ j) Q on the right-hand side of equation (2) are the multipole components of collisional rates of excita- tion and relaxation respectively. D(K) is the depolarization rate due to elastic collisions.1 The radiative and collisional rates can be added under the impact approximation hypothesis (Bommier & Sahal-Bréchot 1991) dρKQ/dt = [dρ Q/dt]RAD + [dρ Q/dt]COLL. Using the equa- tions (1), (2), and the condition for static atmosphere, dρKQ/dt = 0, we obtain the equations of statistical equilibrium [iA j′ jΓQ + A j′ j + (C j′ j) Q + D (K)]ρKQ w(K)j′ j (−1) 2 j′ + 1 B j j′J −Q + (C j j′) Q. (3) By applying the relation between Einstein coefficients for spon- taneous emission and absorption, B j j′ = 2 j′ + 1 2 j + 1 2hν30 A j′ j, (4) and dividing the formula (3) by A j′ j, we obtain the equation (1 + ǫKQ + δ + iΓQ)ρKQ = (−1)Qw(K)j′ j J 2 j′ + 1 2 j + 1 2hν30 (C j j′ ) A j′ j′ . (5) One can introduce the dimensionless parameter of the depolar- ization rate A j′ j , (6) 1 This process cannot change a total population of the level. Therefore it is always D(0) = 0. We take formally into account only the depolarization rate D(K) to use a formalism coherent with the previ- ous papers. A general treatment of physically more relevant transfer of multipole components of the upper level is out of scope of this paper. Fig. 1. The reference frame Σ1 has its Z-axis oriented vertically with respect to the atmosphere, while the z-axis of the reference frame Σ0 is parallel to the direction of magnetic field B. The axes X and x lie in the same plane defined by Z-axis and B; the axes Y and y are defined to complement the right-handed orthogonal coordinate systems. and the irreducible tensor which plays the role of generalized photon destruction probability (CRj′ j) A j′ j′ . (7) If the relation (CRj′ j) Q , 0 is satisfied we may define the quantity B(KQ) = 2hν30 2 j + 1 2 j′ + 1 (C j j′) (CRj′ j) . (8) It is easy to show (see below) that in the particular case of a Maxwellian velocity distribution of colliders the relation B(00) = BP is satisfied, where BP is the Planck function in the Wien limit at given temperature. Using the definition of irreducible compo- nents of the two-level source function (cf. Paper I) S KQ = 2hν30 2 j + 1 2 j′ + 1 Q, (9) we obtain the statistical equilibrium equations in the compact (1 + ǫKQ + δ + iΓQ)S KQ = w(K)j′ j (−1) −Q + ǫ . (10) J. Štěpán and V. Bommier: Generalized ǫ-law 3 3. Solution of the Wiener-Hopf equations From now on we reduce our analysis to the case of semi-infinite, plane-parallel geometry with constant magnetic field along the atmosphere. The velocity distribution and volume density of col- liders is also constant along the atmosphere but it is in general non-thermal. The only position coordinate is the common line optical depth τ. Following the procedure of Paper I a formal solution of ra- diative transfer equation is substituted into the definition of ten- sor J Q; after that we obtain a set of integral Wiener–Hopf equa- tions of the 2nd kind, (1 + ǫKQ + δ + iΓQ)S KQ(τ) K̃KQ,K′Q′ (τ, τ ′)S K Q′ (τ ′)dτ′ + ǫKQ B , (11) which describe coupling of the tensors ρKQ(τ) at differ- ent optical depths via radiation. Several important prop- erties of kernels K̃KQ,K′Q′ (τ, τ ′) have been discussed by Landi Degl’Innocenti et al. (1990) and in Paper I Using their in- dexing notation one can rewrite the equation (11) in the short- handed form aiS i(τ) = Ki j(|τ − τ′|)S j(τ′)dτ′ + bi, (12) ai = 1 + ǫ Q + δ + iΓQ, (13) bi = ǫ . (14) The index i in these expressions runs between the limits 1 and N, where N is the number of KQ-multipoles. In the following we briefly repeat the derivation performed by Frisch & Frisch (1975) emphasizing the differences due to presence of bi terms. Calculation of the derivative of (12) with respect to τ, split- ting the integral on the right-hand side into two parts, multiplica- tion of the equation by S i(τ), summation over index i, and finally integration with respect to τ leads to the set of equations S i(τ) dS i(τ) S j(0) Ki j(τ)S i(τ)dτ dτS i(τ) dτ′Ki j(|τ′ − τ|) dS j(τ . (15) The left-hand side of (15) is easily evaluated as S i(∞)2 − S i(0)2 The first term on the right-hand side of (15) is evaluated using the kernels symmetry Ki j(t) = K ji(t) and the equation (12), so that we obtain S i(0) [aiS i(0) − bi] , (17) while the second term equals S i(∞)2 − S i(0)2 bi [S i(∞) − S i(0)] . (18) We put these results into (15) to get aiS i(0) biS i(∞). (19) Calculation of the limit τ → ∞ of both sides of the equation (12) leads to the set of linear algebraic equations for the compo- nents of source function tensor in the infinite depth: a jδi j − Ki j(t)dt S j(∞) = bi. (20) We can solve these equations and write S(∞) = L−1b, (21) where S is the formal vector of S i components, b is the formal vector of bi components, and the elements of matrix L are de- fined by relation {L}i j = a jδi j − Ki j(t)dt. (22) Establishing a new matrix ℓ = L−1 and substituting (21) into (19) leads to the generalized form of the ǫ-law aiS i(0) bib jℓi j. (23) 4. Particular solutions Setting the special conditions for magnetic field and collisional rates, one recovers the less general but more common and ex- plicit formulations of the ǫ-law than the one given by (23). In the following sections we will verify this result in the limit- ing conditions assumed in recent papers and we will analyse the simple examples of non-thermal collisional excitation. 4.1. Maxwellian velocity distribution of colliders In the case of Maxwellian velocity distribution of colliders, re- laxation rates of all multipole components ρKQ are the same: (CRj′ j) Q = C j′ j, (24) where CRj′ j is the usual relaxation rate for collisional deexcitation from j′ to j. For excitation rates one has (C j j′) C j j′√ 2 j′ + 1 δK0δQ0, (25) where the factor (2 j′ + 1)−1/2 has been introduced to make a connection with the usual collisional rate C j j′ of standard unpo- larized theory. In this isotropic case, there is no collisional exci- tation of higher ranks of density matrix. From the assumption of thermodynamic equilibrium one has C j j′ CRj′ j 2 j′ + 1 2 j + 1 e−hν0/kBT , (26) where kB stands for the Boltzmann constant and T for a tem- perature of the atmosphere. From (24) and (7) it is evident that ǫKQ = ǫ for all possible K and Q, where ǫ is the common photon destruction probability. Further B(KQ) = BPδK0δQ0. (27) 4 J. Štěpán and V. Bommier: Generalized ǫ-law Substituting the rates (24) and (25) into the formula (22) and employing the general identity −∞ Ki1(t)dt = δi1 (see Paper I) together with bi = δi1, we recover form (23) the formula (16) of the previously cited paper: (1 + ǫ + δ(K) + iΓQ)[S KQ(0)] = ǫB2P. (28) Assuming that there is zero magnetic field, i.e. Γ = 0, the source function tensor reduces due to symmetry reasons to the two non-vanishing components S 00 and S 0 in the reference frame Σ1. This reference frame is suitable for descriptions of the atomic system under these conditions, so that we may identify Σ0 ≡ Σ1, with X and Y axes oriented arbitrary in the plane parallel to atmo- spheric surface. Further, assuming that there is no depolarization of the upper level (δ(K) = 0), we realize from (28): S 00(0) S 20(0) 1 + ǫ ǫ′BP, (29) which is the same result derived in different notation by Ivanov (1990). For simplicity the common alternative to the photon de- struction probability has been introduced: ǫ′ = ǫ/(1 + ǫ). If depolarization of the upper level is high enough to destroy atomic level polarization (δ(K) → ∞ for K > 0), or the upper level is unpolarizable, the common ǫ-law for scalar radiation is recovered, S 00(0) = 1 + ǫ ǫ′BP. (30) 4.2. Anisotropic alignment (de)excitation The relation ǫKQ = ǫ is not in general satisfied for all the mul- tipoles because the relaxation of the ρKQ state depends on the velocity distribution of colliders. In the following text we will neglect the effects of magnetic field. Let us assume an example of a relative velocity distribution of particles that is axially symmetric with the axis of symmetry parallel to the vertical of the atmosphere (so that it is as in the former case Σ0 ≡ Σ1) and that the collisional interaction can be fully described by only the first two even multipole components of this distribution. Thanks to these assumptions the only non- vanishing excitation collisional rates are (C j j′ ) 0 and (C j j′ ) 0, the relaxation rates (CRj′ j) 0 and (C j′ j) 0 and for the same reasons the only non-zero source function components are S 00 and S An explicit evaluation of the integrals of kernels∫ ∞ −∞ K̃KQ,K′Q′ (τ, τ ′)dτ′ under these conditions shows that the only non-zero ones are given by (A5) and (A12) of Landi Degl’Innocenti et al. (1991b). In our notation they read K̃00,00(τ, τ ′)dτ′ = 1, (31) K̃20,20(τ, τ ′)dτ′ = W2, (32) with W2 = (w j′ j) 2. Substituting these results into (23) we see that (1 + ǫ00 )(S + (1 + ǫ20 )(S (00))2 + (ǫ20 B (20))2 1 + ǫ20 − 10 W2 . (33) To check the validity of polarized radiative transfer codes, it is advantageous if one can verify that the transfer of higher ranks of the radiation tensor is accurate enough. In the realistic scattering polarization models the polarization degree does not exceed a few percent so that |S 00(0)| ≫ |S Q(0)|. By setting ar- bitrary (even unphysical) collisional rates it is possible to verify transfer codes in conditions with |S 00| ≪ |S To privilege transfer in higher ranks of the radiation tensor one can artificially suppress the excitation rate (C j j′) 0. In the extremal case one can set (C j j′) 0 → 0. The easiest way to do this is the formal interchange of the role of excitation rates of population and alignment, i.e. (C j j′ ) 0 ↔ (C j j′) 0 of the original Maxwellian velocity distribution: (C j j′) 0 = 0, (C j j′ ) C j j′√ 2 j′ + 1 (no collisional excitation to upper level population) and the re- laxation rates set to the Maxwellian ones (CRj′ j) 0 = (C j′ j) 0 = C j′ j. (35) In this case we have B(00) = 0, B(20) = BP, (36) and again 0 = ǫ 0 = ǫ. (37) Substituting this into (33) we find out the ǫ-law in the form [S 00(0)] 2 + [S 20(0)] 1 − 710 W2(1 − ǫ BP. (38) The particular collisional rates (34) are in fact arbitrary and have been chosen to obtain a formula similar to the one of the Maxwellian distribution case. This relation is useful to test polarized radiative transfer codes, because in this unphysical case S 20(0) is the largest term, unlike the physical case where the largest term is S 00(0) and S is only a few percent of it. By applying Eq. (38) the test is much more sensitive to the polarization, and the polarization is better tested. We have thus successfully tested a multilevel non-LTE radiative transfer code that we are developing, but this code and its results are the subjects of a forthcoming paper. 5. Conclusions We have derived a more general formulation of the so-called√ ǫ-law of radiation transfer. This analytical condition couples the value of source function tensor of a two-level atom with other physical properties of the atmosphere. The simplest re- sult obtained in conditions of a non-magnetic, isothermal, plane- parallel, semi-infinite atmosphere with thermal velocity distri- bution of particles and unpolarized atomic levels (e.g. Mihalas 1970) has been generalized by Ivanov (1990) to account for scat- tering of polarized radiation and polarized upper atomic level. Further generalizations done in Paper I, which account for a magnetic field of arbitrary strength and direction, has been ex- tended in the present paper to account for non-thermal colli- sional interactions. It was done by introducing the tensor of the photon destruction probability ǫKQ and by defining the function B(KQ). The resulting formula (23) reduces to the cases mentioned above if the physical conditions become more symmetric. On J. Štěpán and V. Bommier: Generalized ǫ-law 5 the other hand, situations with a high degree of perturbers ve- locity distribution anisotropy and especially ones with unphysi- cal collisional rates result in a wide range of models which can be calculated both numerically and analytically. Thus they offer new possibilities for verification of the non-LTE radiation trans- fer codes. References Avrett, E. H. & Hummer, D. G. 1965, MNRAS, 130, 295 Bommier, V. & Sahal-Bréchot, S. 1991, Ann. Phys. Fr., 16, 555 Derouich, M. 2006, A&A, 449, 1 Fano, U. 1957, Rev. Mod. Phys., 29, 74 Frisch, U. & Frisch, H. 1975, MNRAS, 173, 167 Hubený, I. 1987, A&A, 185, 332 Ivanov, V. V. 1990, Soviet Astronomy, 34, 621 Landi Degl’Innocenti, E. 1984, Sol. Phys., 91, 1 Landi Degl’Innocenti, E. 1985, Sol. Phys., 102, 1 Landi Degl’Innocenti, E. & Bommier, V. 1994, A&A, 284, 865 Landi Degl’Innocenti, E., Bommier, V., & Sahal-Bréchot, S. 1990, A&A, 235, Landi Degl’Innocenti, E., Bommier, V., & Sahal-Bréchot, S. 1991a, A&A, 244, Landi Degl’Innocenti, E., Bommier, V., & Sahal-Bréchot, S. 1991b, A&A, 244, Landi Degl’Innocenti, E. & Landolfi, M. 2004, Polarization in Spectral Lines (Kluwer acad. Publ.) Mihalas, D. 1970, Stellar Atmospheres (W. H. Freeman and Company) Introduction Equations of statistical equilibrium Solution of the Wiener-Hopf equations Particular solutions Maxwellian velocity distribution of colliders Anisotropic alignment (de)excitation Conclusions

Context. A derivation of a generalized sqrt(epsilon)-law for nonthermal collisional rates of excitation by charged perturbers is presented. Aims. Aim of this paper is to find a more general analytical expression for a surface value of the source function which can be used as an addtional tool for verification of the non-LTE radiative transfer codes. Methods. Under the impact approximation hypothesis, static, one-dimensional, plane-parallel atmosphere, constant magnetic field of arbitrary strength and direction, two-level atom model with unpolarized lower level and stimulated emission neglected, we introduce the unphysical terms into the equations of statistical equilibrium and solve the appropriate non-LTE integral equations. Results. We derive a new analytical condition for the surface values of the source function components expressed in the basis of irreducible spherical tensors.

Introduction In the series of papers of Landi Degl’Innocenti et al. (1991a,b), Landi Degl’Innocenti & Bommier (1994) (from now on refer- enced as Paper I), the general formalism of resonance line po- larization scattering for a two-level atom has been developed. The non-LTE problem of the 2nd kind for an arbitrary magnetic field, three-dimensional geometry of the medium and arbitrary irradiation by external sources has been discussed. The effect of inelastic collisions with charged perturbers has been considered for the particular case of a relative Maxwellian velocity distribu- tion. Paper I analysed the analytical properties of the solutions in the particular case of a one-dimensional, semi-infinite, static at- mosphere with a constant magnetic field of arbitrary strength and direction and assuming zero external irradiation of the at- mosphere. They derived a generalization of the well known law (e.g. Avrett & Hummer 1965; Mihalas 1970; Hubený 1987) for the case of polarized radiation and extended the previous re- sults of Ivanov (1990) who studied scattering in a non-magnetic regime. In most cases of practical interest the polarization degree is rather small. The purpose of this paper is to find a new analytical solution of the non-LTE problem in unphysical conditions in or- der to better verify the accuracy of the polarized radiation trans- fer codes. This is done by introduction of an unphysical source term in the polarization into the equations of statistical equilib- rium. Such a generalization can be useful in testing the accuracy of the radiative transfer codes whose purpose is to deal with the Send offprint requests to: J. Štěpán non-thermal collisional processes (for instance in the impact po- larization studies of solar flares). Following the approach of the papers quoted above, we adopt the formalism of density matrix in the representation of irre- ducible tensorial operators (e.g. Fano 1957). We consider the lower level with total angular momentum j to be unpolarized. This level is completely described by the overall population which is set to 1 for normalization reasons. The upper level with angular momentum j′ is described by the multipole components of the statistical tensor ρKQ. Coherences between different levels j and j′ are neglected but coherences between Zeeman sublevels of level j′ are in general taken into account. The calculation is performed in the Wien limit of line frequency whose assump- tion makes it possible to neglect stimulated emission effect, and to preserve the linearity of the non-LTE problem. 2. Equations of statistical equilibrium The suitable coordinate system Σ0 for atomic state description is the one with the z-axis directed along the magnetic field (see Figure 1). Radiative rate contributions to the evolution of statistical op- erator ρKQ are given by (Landi Degl’Innocenti 1985)   = −iA j′ jΓQρKQ − A j′ jρ w(K)j′ j (−1) 2 j′ + 1 B j j′J −Q. (1) http://arxiv.org/abs/0704.1573v1 2 J. Štěpán and V. Bommier: Generalized ǫ-law In this equation A j′ j (B j j′) is the Einstein coefficient of spon- taneous emission (absorption) from level j′ ( j) to level j ( j′). Γ = 2πg j′νL/A j′ j with g j′ being the Landé factor of the level j′ and νL is the Larmor frequency. The transition-dependent nu- merical factor w(K)j′ j has been defined by Landi Degl’Innocenti (1984) as have the irreducible components of the mean radiation tensor J Q. Besides the radiative rates, collisional rates have to be considered in the statistical equilibrium, because the source of radiation in a semi-infinite atmosphere is the collisional exci- tation followed by radiative de-excitation. Thus, the source term of the radiative transfer equation originates in the inelastic col- lision effect. As the purpose of the present paper is to consider unphysical source terms in the non-zero ranks (K,Q) of the ir- reducible tensorial operator basis T KQ , we will introduce an un- physical (K,Q)-dependence to the inelastic collisional rates of the statistical equilibrium equation below. The purpose here is not to thus describe anisotropic collisions, which would require a proper formalism that is out of the scope of the present paper (see, for instance, Landi Degl’Innocenti & Landolfi (2004) for a two-level atom, and Derouich (2006), for polarization trans- fer rates in a multi-level atom due to isotropic collisions). We introduce as usual the depolarizing rate due to isotropic elastic collisions. Thus, the contribution of collisional rates reads   = (C j j′ ) Q − (C j′ j) Q − D Q. (2) The terms (C j j′ ) Q and (C j′ j) Q on the right-hand side of equation (2) are the multipole components of collisional rates of excita- tion and relaxation respectively. D(K) is the depolarization rate due to elastic collisions.1 The radiative and collisional rates can be added under the impact approximation hypothesis (Bommier & Sahal-Bréchot 1991) dρKQ/dt = [dρ Q/dt]RAD + [dρ Q/dt]COLL. Using the equa- tions (1), (2), and the condition for static atmosphere, dρKQ/dt = 0, we obtain the equations of statistical equilibrium [iA j′ jΓQ + A j′ j + (C j′ j) Q + D (K)]ρKQ w(K)j′ j (−1) 2 j′ + 1 B j j′J −Q + (C j j′) Q. (3) By applying the relation between Einstein coefficients for spon- taneous emission and absorption, B j j′ = 2 j′ + 1 2 j + 1 2hν30 A j′ j, (4) and dividing the formula (3) by A j′ j, we obtain the equation (1 + ǫKQ + δ + iΓQ)ρKQ = (−1)Qw(K)j′ j J 2 j′ + 1 2 j + 1 2hν30 (C j j′ ) A j′ j′ . (5) One can introduce the dimensionless parameter of the depolar- ization rate A j′ j , (6) 1 This process cannot change a total population of the level. Therefore it is always D(0) = 0. We take formally into account only the depolarization rate D(K) to use a formalism coherent with the previ- ous papers. A general treatment of physically more relevant transfer of multipole components of the upper level is out of scope of this paper. Fig. 1. The reference frame Σ1 has its Z-axis oriented vertically with respect to the atmosphere, while the z-axis of the reference frame Σ0 is parallel to the direction of magnetic field B. The axes X and x lie in the same plane defined by Z-axis and B; the axes Y and y are defined to complement the right-handed orthogonal coordinate systems. and the irreducible tensor which plays the role of generalized photon destruction probability (CRj′ j) A j′ j′ . (7) If the relation (CRj′ j) Q , 0 is satisfied we may define the quantity B(KQ) = 2hν30 2 j + 1 2 j′ + 1 (C j j′) (CRj′ j) . (8) It is easy to show (see below) that in the particular case of a Maxwellian velocity distribution of colliders the relation B(00) = BP is satisfied, where BP is the Planck function in the Wien limit at given temperature. Using the definition of irreducible compo- nents of the two-level source function (cf. Paper I) S KQ = 2hν30 2 j + 1 2 j′ + 1 Q, (9) we obtain the statistical equilibrium equations in the compact (1 + ǫKQ + δ + iΓQ)S KQ = w(K)j′ j (−1) −Q + ǫ . (10) J. Štěpán and V. Bommier: Generalized ǫ-law 3 3. Solution of the Wiener-Hopf equations From now on we reduce our analysis to the case of semi-infinite, plane-parallel geometry with constant magnetic field along the atmosphere. The velocity distribution and volume density of col- liders is also constant along the atmosphere but it is in general non-thermal. The only position coordinate is the common line optical depth τ. Following the procedure of Paper I a formal solution of ra- diative transfer equation is substituted into the definition of ten- sor J Q; after that we obtain a set of integral Wiener–Hopf equa- tions of the 2nd kind, (1 + ǫKQ + δ + iΓQ)S KQ(τ) K̃KQ,K′Q′ (τ, τ ′)S K Q′ (τ ′)dτ′ + ǫKQ B , (11) which describe coupling of the tensors ρKQ(τ) at differ- ent optical depths via radiation. Several important prop- erties of kernels K̃KQ,K′Q′ (τ, τ ′) have been discussed by Landi Degl’Innocenti et al. (1990) and in Paper I Using their in- dexing notation one can rewrite the equation (11) in the short- handed form aiS i(τ) = Ki j(|τ − τ′|)S j(τ′)dτ′ + bi, (12) ai = 1 + ǫ Q + δ + iΓQ, (13) bi = ǫ . (14) The index i in these expressions runs between the limits 1 and N, where N is the number of KQ-multipoles. In the following we briefly repeat the derivation performed by Frisch & Frisch (1975) emphasizing the differences due to presence of bi terms. Calculation of the derivative of (12) with respect to τ, split- ting the integral on the right-hand side into two parts, multiplica- tion of the equation by S i(τ), summation over index i, and finally integration with respect to τ leads to the set of equations S i(τ) dS i(τ) S j(0) Ki j(τ)S i(τ)dτ dτS i(τ) dτ′Ki j(|τ′ − τ|) dS j(τ . (15) The left-hand side of (15) is easily evaluated as S i(∞)2 − S i(0)2 The first term on the right-hand side of (15) is evaluated using the kernels symmetry Ki j(t) = K ji(t) and the equation (12), so that we obtain S i(0) [aiS i(0) − bi] , (17) while the second term equals S i(∞)2 − S i(0)2 bi [S i(∞) − S i(0)] . (18) We put these results into (15) to get aiS i(0) biS i(∞). (19) Calculation of the limit τ → ∞ of both sides of the equation (12) leads to the set of linear algebraic equations for the compo- nents of source function tensor in the infinite depth: a jδi j − Ki j(t)dt S j(∞) = bi. (20) We can solve these equations and write S(∞) = L−1b, (21) where S is the formal vector of S i components, b is the formal vector of bi components, and the elements of matrix L are de- fined by relation {L}i j = a jδi j − Ki j(t)dt. (22) Establishing a new matrix ℓ = L−1 and substituting (21) into (19) leads to the generalized form of the ǫ-law aiS i(0) bib jℓi j. (23) 4. Particular solutions Setting the special conditions for magnetic field and collisional rates, one recovers the less general but more common and ex- plicit formulations of the ǫ-law than the one given by (23). In the following sections we will verify this result in the limit- ing conditions assumed in recent papers and we will analyse the simple examples of non-thermal collisional excitation. 4.1. Maxwellian velocity distribution of colliders In the case of Maxwellian velocity distribution of colliders, re- laxation rates of all multipole components ρKQ are the same: (CRj′ j) Q = C j′ j, (24) where CRj′ j is the usual relaxation rate for collisional deexcitation from j′ to j. For excitation rates one has (C j j′) C j j′√ 2 j′ + 1 δK0δQ0, (25) where the factor (2 j′ + 1)−1/2 has been introduced to make a connection with the usual collisional rate C j j′ of standard unpo- larized theory. In this isotropic case, there is no collisional exci- tation of higher ranks of density matrix. From the assumption of thermodynamic equilibrium one has C j j′ CRj′ j 2 j′ + 1 2 j + 1 e−hν0/kBT , (26) where kB stands for the Boltzmann constant and T for a tem- perature of the atmosphere. From (24) and (7) it is evident that ǫKQ = ǫ for all possible K and Q, where ǫ is the common photon destruction probability. Further B(KQ) = BPδK0δQ0. (27) 4 J. Štěpán and V. Bommier: Generalized ǫ-law Substituting the rates (24) and (25) into the formula (22) and employing the general identity −∞ Ki1(t)dt = δi1 (see Paper I) together with bi = δi1, we recover form (23) the formula (16) of the previously cited paper: (1 + ǫ + δ(K) + iΓQ)[S KQ(0)] = ǫB2P. (28) Assuming that there is zero magnetic field, i.e. Γ = 0, the source function tensor reduces due to symmetry reasons to the two non-vanishing components S 00 and S 0 in the reference frame Σ1. This reference frame is suitable for descriptions of the atomic system under these conditions, so that we may identify Σ0 ≡ Σ1, with X and Y axes oriented arbitrary in the plane parallel to atmo- spheric surface. Further, assuming that there is no depolarization of the upper level (δ(K) = 0), we realize from (28): S 00(0) S 20(0) 1 + ǫ ǫ′BP, (29) which is the same result derived in different notation by Ivanov (1990). For simplicity the common alternative to the photon de- struction probability has been introduced: ǫ′ = ǫ/(1 + ǫ). If depolarization of the upper level is high enough to destroy atomic level polarization (δ(K) → ∞ for K > 0), or the upper level is unpolarizable, the common ǫ-law for scalar radiation is recovered, S 00(0) = 1 + ǫ ǫ′BP. (30) 4.2. Anisotropic alignment (de)excitation The relation ǫKQ = ǫ is not in general satisfied for all the mul- tipoles because the relaxation of the ρKQ state depends on the velocity distribution of colliders. In the following text we will neglect the effects of magnetic field. Let us assume an example of a relative velocity distribution of particles that is axially symmetric with the axis of symmetry parallel to the vertical of the atmosphere (so that it is as in the former case Σ0 ≡ Σ1) and that the collisional interaction can be fully described by only the first two even multipole components of this distribution. Thanks to these assumptions the only non- vanishing excitation collisional rates are (C j j′ ) 0 and (C j j′ ) 0, the relaxation rates (CRj′ j) 0 and (C j′ j) 0 and for the same reasons the only non-zero source function components are S 00 and S An explicit evaluation of the integrals of kernels∫ ∞ −∞ K̃KQ,K′Q′ (τ, τ ′)dτ′ under these conditions shows that the only non-zero ones are given by (A5) and (A12) of Landi Degl’Innocenti et al. (1991b). In our notation they read K̃00,00(τ, τ ′)dτ′ = 1, (31) K̃20,20(τ, τ ′)dτ′ = W2, (32) with W2 = (w j′ j) 2. Substituting these results into (23) we see that (1 + ǫ00 )(S + (1 + ǫ20 )(S (00))2 + (ǫ20 B (20))2 1 + ǫ20 − 10 W2 . (33) To check the validity of polarized radiative transfer codes, it is advantageous if one can verify that the transfer of higher ranks of the radiation tensor is accurate enough. In the realistic scattering polarization models the polarization degree does not exceed a few percent so that |S 00(0)| ≫ |S Q(0)|. By setting ar- bitrary (even unphysical) collisional rates it is possible to verify transfer codes in conditions with |S 00| ≪ |S To privilege transfer in higher ranks of the radiation tensor one can artificially suppress the excitation rate (C j j′) 0. In the extremal case one can set (C j j′) 0 → 0. The easiest way to do this is the formal interchange of the role of excitation rates of population and alignment, i.e. (C j j′ ) 0 ↔ (C j j′) 0 of the original Maxwellian velocity distribution: (C j j′) 0 = 0, (C j j′ ) C j j′√ 2 j′ + 1 (no collisional excitation to upper level population) and the re- laxation rates set to the Maxwellian ones (CRj′ j) 0 = (C j′ j) 0 = C j′ j. (35) In this case we have B(00) = 0, B(20) = BP, (36) and again 0 = ǫ 0 = ǫ. (37) Substituting this into (33) we find out the ǫ-law in the form [S 00(0)] 2 + [S 20(0)] 1 − 710 W2(1 − ǫ BP. (38) The particular collisional rates (34) are in fact arbitrary and have been chosen to obtain a formula similar to the one of the Maxwellian distribution case. This relation is useful to test polarized radiative transfer codes, because in this unphysical case S 20(0) is the largest term, unlike the physical case where the largest term is S 00(0) and S is only a few percent of it. By applying Eq. (38) the test is much more sensitive to the polarization, and the polarization is better tested. We have thus successfully tested a multilevel non-LTE radiative transfer code that we are developing, but this code and its results are the subjects of a forthcoming paper. 5. Conclusions We have derived a more general formulation of the so-called√ ǫ-law of radiation transfer. This analytical condition couples the value of source function tensor of a two-level atom with other physical properties of the atmosphere. The simplest re- sult obtained in conditions of a non-magnetic, isothermal, plane- parallel, semi-infinite atmosphere with thermal velocity distri- bution of particles and unpolarized atomic levels (e.g. Mihalas 1970) has been generalized by Ivanov (1990) to account for scat- tering of polarized radiation and polarized upper atomic level. Further generalizations done in Paper I, which account for a magnetic field of arbitrary strength and direction, has been ex- tended in the present paper to account for non-thermal colli- sional interactions. It was done by introducing the tensor of the photon destruction probability ǫKQ and by defining the function B(KQ). The resulting formula (23) reduces to the cases mentioned above if the physical conditions become more symmetric. On J. Štěpán and V. Bommier: Generalized ǫ-law 5 the other hand, situations with a high degree of perturbers ve- locity distribution anisotropy and especially ones with unphysi- cal collisional rates result in a wide range of models which can be calculated both numerically and analytically. Thus they offer new possibilities for verification of the non-LTE radiation trans- fer codes. References Avrett, E. H. & Hummer, D. G. 1965, MNRAS, 130, 295 Bommier, V. & Sahal-Bréchot, S. 1991, Ann. Phys. Fr., 16, 555 Derouich, M. 2006, A&A, 449, 1 Fano, U. 1957, Rev. Mod. Phys., 29, 74 Frisch, U. & Frisch, H. 1975, MNRAS, 173, 167 Hubený, I. 1987, A&A, 185, 332 Ivanov, V. V. 1990, Soviet Astronomy, 34, 621 Landi Degl’Innocenti, E. 1984, Sol. Phys., 91, 1 Landi Degl’Innocenti, E. 1985, Sol. Phys., 102, 1 Landi Degl’Innocenti, E. & Bommier, V. 1994, A&A, 284, 865 Landi Degl’Innocenti, E., Bommier, V., & Sahal-Bréchot, S. 1990, A&A, 235, Landi Degl’Innocenti, E., Bommier, V., & Sahal-Bréchot, S. 1991a, A&A, 244, Landi Degl’Innocenti, E., Bommier, V., & Sahal-Bréchot, S. 1991b, A&A, 244, Landi Degl’Innocenti, E. & Landolfi, M. 2004, Polarization in Spectral Lines (Kluwer acad. Publ.) Mihalas, D. 1970, Stellar Atmospheres (W. H. Freeman and Company) Introduction Equations of statistical equilibrium Solution of the Wiener-Hopf equations Particular solutions Maxwellian velocity distribution of colliders Anisotropic alignment (de)excitation Conclusions

Sharp dark-mode resonances in planar metamaterials with broken structural symmetry V. A. Fedotov,1, ∗ M. Rose,1 S. L. Prosvirnin,2 N. Papasimakis,1 and N. I. Zheludev1, † Optoelectronics Research Centre, University of Southampton, SO17 1BJ, UK Institute of Radio Astronomy, National Academy of Sciences of Ukraine, Kharkov, 61002, Ukraine (Dated: October 29, 2018) We report that resonant response with a very high quality factor can be achieved in a planar metamaterial by introducing symmetry breaking in the shape of its structural elements, which enables excitation of dark modes, i.e. modes that are weakly coupled to free space. PACS numbers: 78.67.-n, 42.70.-a, 42.25.Bs Metamaterials research has attracted tremendous amount of attention in the recent years. The interest is mainly driven by the opportunity of achieving new elec- tromagnetic properties, some with no analog in naturally available materials. Extraordinary transmission [1], ar- tificial magnetism and negative refraction [2], invisible metal [3], magnetic mirror [4], asymmetric transmission [5] and cloaking [6] are just few examples of the new phenomena emerged from the development of artificially structured matter. The exotic and often dramatic physics predicted for metamaterials is underpinned by the resonant nature of their response and therefore achieving resonances with high quality factors is essential in order to make metama- terials’ performance efficient. However, resonance quality factors (that is the resonant frequency over width of the resonance) demonstrated by conventional metamaterials are often limited to rather small values. This comes from the fact that resonating structural elements of metama- terials are strongly coupled to free-space and therefore suffer significant losses due to radiation. Furthermore, conventional metamaterials are often composed of sub- wavelength particles that are simply unable to provide large-volume confinement of electromagnetic field neces- sary to support high-Q resonances. As recent theoretical analysis showed, high-Q resonances involving dark (or closed) modes are nevertheless possible in metamaterials if certain small asymmetries are introduced in the shape of their structural elements [7]. In this Letter we report the observation of exception- ally narrow resonant responses in transmission and reflec- tion of planar metamaterial achieved through introduc- ing asymmetry into its structural elements. The appear- ance of narrow resonances is attributed to the excitation of otherwise forbidden anti-symmetric modes, that are weakly coupled to free-space (”dark modes”). Metamaterials that were used in our experiments con- sisted of identical sub-wavelength metallic ”inclusions” structured in the form of asymmetrically split rings (ASR), which were arranged in a periodic array and placed on a thin dielectric substrate (see Fig. 1). ASR- patterns were etched from 35 µm copper cladding cov- ering IS620 PCB substrate of 1.5 mm thickness. Each copper split ring had the radius of 6 mm and width of 0.8 mm and occupied a square translation cell of 15×15mm (see Fig. 1). Such periodic structure does not diffract normal incident electromagnetic radiation for fre- quencies lower than 20 GHz. The overall size of the sam- ples used were approximately 220× 220 mm. Transmis- sion and reflection of a single sheet of this meta-material were measured in an anechoic chamber under normal in- cidence conditions using broadband horn antennas. FIG. 1: (Color online) Fragments of planar metamaterials with asymmetrically split copper rings. The dashed boxes indicate elementary translation cells of the structures. We studied structures with two different types of asym- metry designated as type A and B in Fig. 1. The rings of type A had two equal splits dividing them into pairs of arcs of different length corresponding to 140 and 160 deg (see Fig. 1A). The rings of type B were split along their diameter into two equal parts but had splits of different length corresponding to 10 and 30 deg (see Fig. 1B). http://arxiv.org/abs/0704.1577v1 Transmission and reflection properties of structures of both types depended strongly on the polarization state of incident electromagnetic waves. The most dramatic spectral selectivity was observed for electrical field being perpendicular to the mirror line of the asymmetrically split rings, which corresponded to x-polarization in the case of structure A and y-polarization for structure B (as defined in Fig. 1). For the orthogonal polarizations the ASR-structures did not show any spectral features originating from asymmetrical structuring. FIG. 2: (Color online) (a) Normal incidence reflection and transmission spectra of A-type metamaterial (presented in Fig. 1A) for x-polarization: solid line - experiment, filled cir- cles - theory (method of moments), empty circles - theory for reference structure with symmetrically split rings. (b) x- Component of the instantaneous current distribution in the asymmetrically split rings corresponding to resonant features I, II and III as marked in section (a). Arrows indicate in- stantaneous directions of the current flow, while their length corresponds to the current strength. The results of reflection and transmission measure- ments of metamaterial A obtained for x-polarization are presented in Fig. 2a. The reflection spectrum reveals an ultra-sharp resonance near 6 GHz (marked as II), where reflectivity losses exceed -10 dB. It is accompanied by two much weaker resonances (marked as I and III) cor- responding to reflection peaks at about 5.5 and 7.0 GHz respectively. The sharp spectral response in reflection is matched by a very narrow transmission peak reaching -3 dB and having the width of only 0.27 GHz as mea- sured at 3 dB below the maximum. The quality factor Q of such response is 20, which is larger than that of the most metamaterials based on lossy PCB substrates by at least one order of magnitude. On both sides of the peak the transmission decreases resonantly to about -35 dB at frequencies corresponding to reflection maxima. Fig. 3a presents transmission and reflection spectra of B-type metamaterial measured for y-polarization. A very narrow resonant transmission dip can be seen near 5.5 GHz, where transmission drops to about -5 dB. The corresponding reflection spectrum shows an usually sharp roll-off (I-II) between -4 and -14 dB spanning only 0.13 GHz at around the same frequency. At the fre- quency of about 11.5 GHz the ASR-structure exhibits its fundamental reflection resonance (marked as III) where the wavelength of excitation becomes equal to the length of the arcs. To understand the resonst nature of the response, the ASR-structures were modelled using the method of mo- ments. It is a well established numerical method, which involves solving the integral equation for the surface cur- rent induced in the metal pattern by the field of the in- cident wave. This is followed by calculations of scattered fields as a superposition of partial spatial waves. The metal pattern is treated as a perfect conductor, while the substrate is assumed to be a lossy dielectric. For both transmission and reflection the theoretical calcula- tions show a very good agreement with the experimental results assuming ǫ = 4.07+ i ·0.05 (see Fig. 2 and 3, filled circles). For comparison we also modelled metamaterial composed of split rings with no structural asymmetry, i.e. equally split along their diameter. Our calculations indicate that for both polarizations the response of such structure is free form sharp high-Q resonant feature (see Fig. 2 and 3, open circles). The origin of the unusually strong and narrow spec- tral responses of the ASR-structures can be traced to so-called ”dark modes”, i.e. electromagnetic modes that are weakly coupled to free-space. It is this property of the dark modes that allows in principal to achieve high qual- ity resonances in very thin structures [7]. These modes are usually forbidden but can be excited in a planar meta- material if, for example, its particles have certain struc- tural asymmetry. Our calculations showed that in the case of structure A an anti-symmetric current mode can dominate the usual symmetric one: at the high-Q transmission resonance, as shown in Fig. 2b (II), two parts of the ring are excited in anti-phase while currents have almost the same ampli- tude. The scattered electromagnetic fields produced by such current configuration are very weak, which dramat- ically reduces coupling to free-space and therefore radia- FIG. 3: (Color online) (a) Normal incidence reflection and transmission spectra of B-type metamaterial (presented in Fig. 1B) for y-polarization: solid line - experiment, filled cir- cles - theory (method of moments), empty circles - theory for reference structure with symmetrically split rings. (b) y- Component of the instantaneous current distribution in the asymmetrically split rings corresponding to resonant features I, II and III as marked in section (a). Arrows indicate in- stantaneous directions of the current flow, while their length corresponds to the current strength. tion losses. As a consequence, the strength of the induced currents can reach very high values and therefore ensures high quality factor of the response. At the reflection res- onances, in contrast to the ”dark mode” regime, currents in both sections of the asymmetrically split ring oscillate in phase but excitation of one of the sections dominates the other (see Fig. 2b (I and III)). Importantly, the ampli- tudes of the currents in this case are significantly smaller than in ”dark mode” resonance, which yields lower Q- factors for this type of the response. If the structural asymmetry is removed the anti-symmetric current mode becomes forbidden while two reflection resonances de- generate to a single low-Q resonance state where both parts of the ring are excited equally. Thus introduction of asymmetry in the split-ring structure effectively allows to create a very narrow pass-band inside its transmission stop-band. This effect is somewhat analogous to appear- ance of an allowed state in the bandgap of photonic crys- tals due to structural defects. Interpretation of the results obtained for structure B appears to be slightly more elaborate. From the symme- try of the split rings it follows that for y-polarized excita- tion at any frequency current distribution in the opposite sections of the ring should have equal y-components os- cillating in phase and equal x-components oscillating in anti-phase. The net current in the ring has therefore al- ways zero x-component, while its y-component can not be fully compensated due to the structural asymmetry. At low frequencies the net y-component is small but it increases significantly as the frequency of excitation ap- proaches 5.5 GHz. At this frequency the wavelength becomes equal to circumference of the split ring and, as shown in Fig. 3b (I), the right side of the ring dominates its left side oscillating in anti-phase. The later results in a resonant increase of the metamaterial reflection (see Fig. 3a). Immediately above this resonance contributions of both sides of the ASR-particle are still in anti-phase but become nearly identical (see Fig. 3b (II)) making the y-component of the induced net current almost zero and therefore dramatically reducing radiation losses (reflec- tion). Further increase of the excitation frequency leads to rise of the reflection until the fundamental resonance of the ASR-structure is reached where the correspond- ing wavelength is equal to the length of the arcs. In this case both sides of the ASR-particle oscillate in phase and equally contribute to electromagnetic field scattering (see Fig. 3b (III)). The quality factor of the dark-mode resonances will in- crease on reducing the degree of asymmetry of metama- terial particles and in case of low dissipative losses can be made exceptionally high. In the microwave region met- als are almost perfect conductors and the main source of dissipative losses is the substrate material (dielectrics). Therefore significantly higher resonance quality factors can be achieved for a free-standing thin metal film, which is patterned complimentary to ASR-structure, i.e. pe- riodically perforated with ASR-openings. In the visi- ble and IR spectral ranges, however, losses in metals dominate and therefore nano-scaled versions of the orig- inal metal-dielectric ASR-structures would perform bet- ter. According to our estimates Q-factor of such ASR- nanostructures in the near-IR can be as high as 6. In summary, we experimentally and theoretically showed that a new type of planar metamaterials com- posed of asymmetrically split rings exhibit unusually strong high-Q resonances and provide for extremely nar- row transmission and reflection pass- and stop-bands. The metamaterials’ response has a quality factor of about 20, which is one order of magnitude larger than the typ- ical value for many conventional metamaterials. This is achieved via weak coupling between ”dark modes” in the resonant inclusions of the ASR-metamaterial and free- space, while weak symmetry breaking enables excitation of so-called ”dark modes”. Achieving the ”dark mode” resonances will be especially important for metamate- rials in the optical part of the spectrum, where losses are significant and unavoidable. In a certain way such symmetry-breaking resonances in meta-materials resem- bles the recently identified spectral lines of plasmon ab- sorbtion of shell nanoparticle appearing due to asymme- try [8]. The authors would like to acknowledge the financial support of the EPSRC (UK) and Metamorphose NoE. ∗ Electronic address: n.i.zheludev@soton.ac.uk † URL: www.nanophotonics.org.uk [1] W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003). [2] D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004). [3] V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, Phys. Rev. E 72, 056613 (2005). [4] A. S. Schwanecke, V. A. Fedotov, V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, J. Opt. A 9, L01 (2007). [5] V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, Phys. Rev. Lett. 97, 167401 (2006). [6] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006). [7] S. Prosvirnin and S. Zouhdi, in Advances in Electromag- netics of Complex Media and Metamaterials, edited by S. Zouhdi and et al. (Kluwer Academic Publishers, Printed in the Netherlands, 2003), pp. 281–290. [8] H. Wang, Y. Wu, B. Lassiter, C. L. Nehl, J. H. Hafner, P. Nordlander, and N. J. Halas, in Proceedings of the Na- tional Academy of Science of the United States of America (2006), vol. 103, p. 10856. mailto:n.i.zheludev@soton.ac.uk www.nanophotonics.org.uk

We report that resonant response with a very high quality factor can be achieved in a planar metamaterial by introducing symmetry breaking in the shape of its structural elements, which enables excitation of dark modes, i.e. modes that are weakly coupled to free space.

Sharp dark-mode resonances in planar metamaterials with broken structural symmetry V. A. Fedotov,1, ∗ M. Rose,1 S. L. Prosvirnin,2 N. Papasimakis,1 and N. I. Zheludev1, † Optoelectronics Research Centre, University of Southampton, SO17 1BJ, UK Institute of Radio Astronomy, National Academy of Sciences of Ukraine, Kharkov, 61002, Ukraine (Dated: October 29, 2018) We report that resonant response with a very high quality factor can be achieved in a planar metamaterial by introducing symmetry breaking in the shape of its structural elements, which enables excitation of dark modes, i.e. modes that are weakly coupled to free space. PACS numbers: 78.67.-n, 42.70.-a, 42.25.Bs Metamaterials research has attracted tremendous amount of attention in the recent years. The interest is mainly driven by the opportunity of achieving new elec- tromagnetic properties, some with no analog in naturally available materials. Extraordinary transmission [1], ar- tificial magnetism and negative refraction [2], invisible metal [3], magnetic mirror [4], asymmetric transmission [5] and cloaking [6] are just few examples of the new phenomena emerged from the development of artificially structured matter. The exotic and often dramatic physics predicted for metamaterials is underpinned by the resonant nature of their response and therefore achieving resonances with high quality factors is essential in order to make metama- terials’ performance efficient. However, resonance quality factors (that is the resonant frequency over width of the resonance) demonstrated by conventional metamaterials are often limited to rather small values. This comes from the fact that resonating structural elements of metama- terials are strongly coupled to free-space and therefore suffer significant losses due to radiation. Furthermore, conventional metamaterials are often composed of sub- wavelength particles that are simply unable to provide large-volume confinement of electromagnetic field neces- sary to support high-Q resonances. As recent theoretical analysis showed, high-Q resonances involving dark (or closed) modes are nevertheless possible in metamaterials if certain small asymmetries are introduced in the shape of their structural elements [7]. In this Letter we report the observation of exception- ally narrow resonant responses in transmission and reflec- tion of planar metamaterial achieved through introduc- ing asymmetry into its structural elements. The appear- ance of narrow resonances is attributed to the excitation of otherwise forbidden anti-symmetric modes, that are weakly coupled to free-space (”dark modes”). Metamaterials that were used in our experiments con- sisted of identical sub-wavelength metallic ”inclusions” structured in the form of asymmetrically split rings (ASR), which were arranged in a periodic array and placed on a thin dielectric substrate (see Fig. 1). ASR- patterns were etched from 35 µm copper cladding cov- ering IS620 PCB substrate of 1.5 mm thickness. Each copper split ring had the radius of 6 mm and width of 0.8 mm and occupied a square translation cell of 15×15mm (see Fig. 1). Such periodic structure does not diffract normal incident electromagnetic radiation for fre- quencies lower than 20 GHz. The overall size of the sam- ples used were approximately 220× 220 mm. Transmis- sion and reflection of a single sheet of this meta-material were measured in an anechoic chamber under normal in- cidence conditions using broadband horn antennas. FIG. 1: (Color online) Fragments of planar metamaterials with asymmetrically split copper rings. The dashed boxes indicate elementary translation cells of the structures. We studied structures with two different types of asym- metry designated as type A and B in Fig. 1. The rings of type A had two equal splits dividing them into pairs of arcs of different length corresponding to 140 and 160 deg (see Fig. 1A). The rings of type B were split along their diameter into two equal parts but had splits of different length corresponding to 10 and 30 deg (see Fig. 1B). http://arxiv.org/abs/0704.1577v1 Transmission and reflection properties of structures of both types depended strongly on the polarization state of incident electromagnetic waves. The most dramatic spectral selectivity was observed for electrical field being perpendicular to the mirror line of the asymmetrically split rings, which corresponded to x-polarization in the case of structure A and y-polarization for structure B (as defined in Fig. 1). For the orthogonal polarizations the ASR-structures did not show any spectral features originating from asymmetrical structuring. FIG. 2: (Color online) (a) Normal incidence reflection and transmission spectra of A-type metamaterial (presented in Fig. 1A) for x-polarization: solid line - experiment, filled cir- cles - theory (method of moments), empty circles - theory for reference structure with symmetrically split rings. (b) x- Component of the instantaneous current distribution in the asymmetrically split rings corresponding to resonant features I, II and III as marked in section (a). Arrows indicate in- stantaneous directions of the current flow, while their length corresponds to the current strength. The results of reflection and transmission measure- ments of metamaterial A obtained for x-polarization are presented in Fig. 2a. The reflection spectrum reveals an ultra-sharp resonance near 6 GHz (marked as II), where reflectivity losses exceed -10 dB. It is accompanied by two much weaker resonances (marked as I and III) cor- responding to reflection peaks at about 5.5 and 7.0 GHz respectively. The sharp spectral response in reflection is matched by a very narrow transmission peak reaching -3 dB and having the width of only 0.27 GHz as mea- sured at 3 dB below the maximum. The quality factor Q of such response is 20, which is larger than that of the most metamaterials based on lossy PCB substrates by at least one order of magnitude. On both sides of the peak the transmission decreases resonantly to about -35 dB at frequencies corresponding to reflection maxima. Fig. 3a presents transmission and reflection spectra of B-type metamaterial measured for y-polarization. A very narrow resonant transmission dip can be seen near 5.5 GHz, where transmission drops to about -5 dB. The corresponding reflection spectrum shows an usually sharp roll-off (I-II) between -4 and -14 dB spanning only 0.13 GHz at around the same frequency. At the fre- quency of about 11.5 GHz the ASR-structure exhibits its fundamental reflection resonance (marked as III) where the wavelength of excitation becomes equal to the length of the arcs. To understand the resonst nature of the response, the ASR-structures were modelled using the method of mo- ments. It is a well established numerical method, which involves solving the integral equation for the surface cur- rent induced in the metal pattern by the field of the in- cident wave. This is followed by calculations of scattered fields as a superposition of partial spatial waves. The metal pattern is treated as a perfect conductor, while the substrate is assumed to be a lossy dielectric. For both transmission and reflection the theoretical calcula- tions show a very good agreement with the experimental results assuming ǫ = 4.07+ i ·0.05 (see Fig. 2 and 3, filled circles). For comparison we also modelled metamaterial composed of split rings with no structural asymmetry, i.e. equally split along their diameter. Our calculations indicate that for both polarizations the response of such structure is free form sharp high-Q resonant feature (see Fig. 2 and 3, open circles). The origin of the unusually strong and narrow spec- tral responses of the ASR-structures can be traced to so-called ”dark modes”, i.e. electromagnetic modes that are weakly coupled to free-space. It is this property of the dark modes that allows in principal to achieve high qual- ity resonances in very thin structures [7]. These modes are usually forbidden but can be excited in a planar meta- material if, for example, its particles have certain struc- tural asymmetry. Our calculations showed that in the case of structure A an anti-symmetric current mode can dominate the usual symmetric one: at the high-Q transmission resonance, as shown in Fig. 2b (II), two parts of the ring are excited in anti-phase while currents have almost the same ampli- tude. The scattered electromagnetic fields produced by such current configuration are very weak, which dramat- ically reduces coupling to free-space and therefore radia- FIG. 3: (Color online) (a) Normal incidence reflection and transmission spectra of B-type metamaterial (presented in Fig. 1B) for y-polarization: solid line - experiment, filled cir- cles - theory (method of moments), empty circles - theory for reference structure with symmetrically split rings. (b) y- Component of the instantaneous current distribution in the asymmetrically split rings corresponding to resonant features I, II and III as marked in section (a). Arrows indicate in- stantaneous directions of the current flow, while their length corresponds to the current strength. tion losses. As a consequence, the strength of the induced currents can reach very high values and therefore ensures high quality factor of the response. At the reflection res- onances, in contrast to the ”dark mode” regime, currents in both sections of the asymmetrically split ring oscillate in phase but excitation of one of the sections dominates the other (see Fig. 2b (I and III)). Importantly, the ampli- tudes of the currents in this case are significantly smaller than in ”dark mode” resonance, which yields lower Q- factors for this type of the response. If the structural asymmetry is removed the anti-symmetric current mode becomes forbidden while two reflection resonances de- generate to a single low-Q resonance state where both parts of the ring are excited equally. Thus introduction of asymmetry in the split-ring structure effectively allows to create a very narrow pass-band inside its transmission stop-band. This effect is somewhat analogous to appear- ance of an allowed state in the bandgap of photonic crys- tals due to structural defects. Interpretation of the results obtained for structure B appears to be slightly more elaborate. From the symme- try of the split rings it follows that for y-polarized excita- tion at any frequency current distribution in the opposite sections of the ring should have equal y-components os- cillating in phase and equal x-components oscillating in anti-phase. The net current in the ring has therefore al- ways zero x-component, while its y-component can not be fully compensated due to the structural asymmetry. At low frequencies the net y-component is small but it increases significantly as the frequency of excitation ap- proaches 5.5 GHz. At this frequency the wavelength becomes equal to circumference of the split ring and, as shown in Fig. 3b (I), the right side of the ring dominates its left side oscillating in anti-phase. The later results in a resonant increase of the metamaterial reflection (see Fig. 3a). Immediately above this resonance contributions of both sides of the ASR-particle are still in anti-phase but become nearly identical (see Fig. 3b (II)) making the y-component of the induced net current almost zero and therefore dramatically reducing radiation losses (reflec- tion). Further increase of the excitation frequency leads to rise of the reflection until the fundamental resonance of the ASR-structure is reached where the correspond- ing wavelength is equal to the length of the arcs. In this case both sides of the ASR-particle oscillate in phase and equally contribute to electromagnetic field scattering (see Fig. 3b (III)). The quality factor of the dark-mode resonances will in- crease on reducing the degree of asymmetry of metama- terial particles and in case of low dissipative losses can be made exceptionally high. In the microwave region met- als are almost perfect conductors and the main source of dissipative losses is the substrate material (dielectrics). Therefore significantly higher resonance quality factors can be achieved for a free-standing thin metal film, which is patterned complimentary to ASR-structure, i.e. pe- riodically perforated with ASR-openings. In the visi- ble and IR spectral ranges, however, losses in metals dominate and therefore nano-scaled versions of the orig- inal metal-dielectric ASR-structures would perform bet- ter. According to our estimates Q-factor of such ASR- nanostructures in the near-IR can be as high as 6. In summary, we experimentally and theoretically showed that a new type of planar metamaterials com- posed of asymmetrically split rings exhibit unusually strong high-Q resonances and provide for extremely nar- row transmission and reflection pass- and stop-bands. The metamaterials’ response has a quality factor of about 20, which is one order of magnitude larger than the typ- ical value for many conventional metamaterials. This is achieved via weak coupling between ”dark modes” in the resonant inclusions of the ASR-metamaterial and free- space, while weak symmetry breaking enables excitation of so-called ”dark modes”. Achieving the ”dark mode” resonances will be especially important for metamate- rials in the optical part of the spectrum, where losses are significant and unavoidable. In a certain way such symmetry-breaking resonances in meta-materials resem- bles the recently identified spectral lines of plasmon ab- sorbtion of shell nanoparticle appearing due to asymme- try [8]. The authors would like to acknowledge the financial support of the EPSRC (UK) and Metamorphose NoE. ∗ Electronic address: n.i.zheludev@soton.ac.uk † URL: www.nanophotonics.org.uk [1] W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003). [2] D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004). [3] V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, Phys. Rev. E 72, 056613 (2005). [4] A. S. Schwanecke, V. A. Fedotov, V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, J. Opt. A 9, L01 (2007). [5] V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, Phys. Rev. Lett. 97, 167401 (2006). [6] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006). [7] S. Prosvirnin and S. Zouhdi, in Advances in Electromag- netics of Complex Media and Metamaterials, edited by S. Zouhdi and et al. (Kluwer Academic Publishers, Printed in the Netherlands, 2003), pp. 281–290. [8] H. Wang, Y. Wu, B. Lassiter, C. L. Nehl, J. H. Hafner, P. Nordlander, and N. J. Halas, in Proceedings of the Na- tional Academy of Science of the United States of America (2006), vol. 103, p. 10856. mailto:n.i.zheludev@soton.ac.uk www.nanophotonics.org.uk

Mon. Not. R. Astron. Soc. 000, 1–11 (2007) Printed 3 December 2018 (MN LATEX style file v2.2) Proper motion L and T dwarf candidate members of the Pleiades S. L. Casewell1⋆, P. D. Dobbie1,2, S. T. Hodgkin3, E. Moraux4, R. F. Jameson1, N. C. Hambly5, J. Irwin3 and N. Lodieu6,1 1Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK 2Anglo-Australian Observatory, PO Box 296, Epping NSW 1710 Australia 3CASU, Institute of Astronomy,University of Cambridge, Maddingley Road, Cambridge, CB3 0HA, UK 4Laboratoire d’Astrophysique, Observatoire de Grenoble, Université Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France 5Scottish Universities Physics Alliance (SUPA), Institute for Astronomy, School of Physics, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ 6 Instituto de Astrofı́sica de Canarias, Vı́a Láctea s/n, E-38205 La Laguna, Tenerife, Spain January 2007 ABSTRACT We present the results of a deep optical-near-infrared multi-epoch survey covering 2.5 square degrees of the Pleiades open star cluster to search for new very-low-mass brown dwarf mem- bers. A significant (∼ 5 year) epoch difference exists between the optical (CFH12k I-, Z- band) and near infrared (UKIRT WFCAM J-band) observations. We construct I,I-Z and Z,Z-J colour magnitude diagrams to select candidate cluster members. Proper motions are computed for all candidate members and compared to the background field objects to further refine the sample. We recover all known cluster members within the area of our survey. In addition, we have discovered 9 new candidate brown dwarf cluster members. The 7 faintest candidates have red Z-J colours and show blue near-infrared colours. These are consistent with being L and T-type Pleiads. Theoretical models predict their masses to be around 11MJup. Key words: stars: low-mass, brown dwarfs, open clusters and associa- tions:individual:Pleiades 1 INTRODUCTION The initial mass spectrum (IMS), the number of objects manufac- tured per unit mass interval, is an outcome of the star formation process which can be constrained via observation. Consequently, empirical determinations of the form of the IMS can be used to crit- ically examine our theoretical understanding of the complexities of star formation. In recent years there has been a particular empha- sis on building a solid comprehension of the mechanisms by which very-low-mass stars, brown dwarfs and free-floating planetary mass objects form (e.g. Boss 2001; Bate 2004; Goodwin, Whitworth & Ward-Thompson 2004; Whitworth & Goodwin 2005). Neverthe- less, one key question which remains unanswered is what is the lowest possible mass of object that can be manufactured by the star formation process? From a theoretical stance, traditional models predict that if substellar objects form like stars, via the fragmenta- tion and collapse of molecular clouds, then there is a strict lower mass limit to their manufacture of 0.007-0.010 M⊙ (Padoan & Nordlund, 2002). This is set by the rate at which the gas can radiate away the heat released by the compression (e.g. Low & Lynden- Bell, 1976). However, in more elaborate theories, magnetic fields could cause rebounds in collapsing cloud cores which might lead ⋆ E-mail: slc25@star.le.ac.uk to the decompressional cooling of the primordial gas, a lowering of the Jeans mass and hence the production of gravitationally bound fragments with masses of only ∼0.001 M⊙ (Boss, 2001). In con- trast, if feedback from putative winds and outflows driven by the onset of deuterium burning play a role, the smallest objects which form via the star formation process may be restricted to masses equal to or greater than the deuterium burning limit (∼0.013 M⊙; Adams & Fatuzzo, 1996). Recent work on very young clusters (τ<10 Myrs) and star formation regions e.g. σ-Orionis, the Trapezium, IC348 and Up- per Sco (Béjar et al., 2001; Muench et al., 2002; Muench et al., 2003; Lodieu et al., 2007a) suggests that the initial mass function continues slowly rising down to masses of the order M∼0.01 M⊙, at least in these environments. Indeed, it has been claimed that an object with a mass as low as 2-3 MJup has been unearthed in σ- Ori (Zapatero-Osorio et al, 2002). However, the cluster member- ship of σ-Ori 70 is disputed by Burgasser et al. (2004). Further- more, mass estimates for such young substellar objects derived by comparing their observed properties to the predictions of theoreti- cal evolutionary tracks remain somewhat controversial. Baraffe et al. (2002) have shown that to robustly model the effective temper- ature and luminosity of a low mass object with an age less than ∼1 Myr, evolutionary calculations need to be coupled to detailed simulations of the collapse and accretion phase of star formation. c© 2007 RAS http://arxiv.org/abs/0704.1578v2 2 S. L. Casewell et al. As the current generation of evolutionary models start from arbi- trary initial conditions, theoretical predictions for ages less than a few Myrs must be treated with a fair degree of caution. Indeed, the few available dynamical mass measurements of pre-main sequence objects indicate that models tend to underestimate mass by a few tens of percent in the range 0.3<∼M<∼1.0 M⊙ (see Hillenbrand & White, 2004 for review). A recent dynamical mass measurement of the 50-125 Myrs old object AB Dor C (spectral type ∼M8), the first for a pre-main sequence object with M<0.3 M⊙, suggests that the discrepancy between model predictions and reality might be even larger at lower masses, with the former underestimating mass by a factor 2-3 at M∼0.1 M⊙ (Close et al., 2005). However, this conclu- sion is dependent on the assumed age of AB Dor, which is currently a matter of great contention (Luhman, Stauffer & Mamajek, 2005; Janson et al., 2006). On the positive side, Zapatero-Osorio et al., (2004) have determined the masses of the brown dwarf binary com- ponents of GJ 569 Bab and their luminosities and effective temper- atures are in agreement with theoretical predictions, for an age of 300 Myr. More recently, Stassun, Mathieu & Valenti (2006) discuss an eclipsing brown dwarf binary in the Orion nebula star forming region and find the large radii predicted by theory for a very young dwarf. Surprisingly, they find that the secondary is hotter than the more massive primary. Clearly further work is still needed to sup- port the predictions of theoretical models. It is clearly important to search for the lowest mass objects, not only in the young clusters, but also in more mature clusters, such as the Pleiades. The results of previous surveys of the Pleiades indicate that the present day cluster mass function, across the stel- lar/substellar boundary and down to M∼0.02 M⊙ (based on the evolutionary models of the Lyon Group), can be represented by a slowly rising power law model, dN/dM∝M−α . For example, from their Canada-France-Hawaii Telescope (CFHT) survey conducted at R and I and covering 2.5 sq. degrees, Bouvier et al. (1998) identi- fied 17 candidate brown dwarfs (IC>17.8) and derived a power law index of α=0.6. From their 1.1 sq degrees Isaac Newton Telescope (INT) survey conducted at I and Z, with follow-up work undertaken at K, Dobbie et al. (2002) unearthed 16 candidate substellar mem- bers and found a power law of index α=0.8 to be compatible with their data. Jameson et al. (2002) showed that a powerlaw of index α=0.41±0.08 was consistent with the observed mass function over the range 0.3>∼M>∼0.035 M⊙. This study used a sample of 49 prob- able brown dwarf members assembled from the four most extensive CCD surveys of the cluster available at the time, the International Time Project survey (Zapatero Osorio et al., 1998), the CFHT sur- vey (Bouvier et al., 1998; Moraux, Bouvier & Stauffer, 2001), the Burrell Schmidt survey (Pinfield et al., 2000) and the INT survey (Dobbie et al., 2002). The CFHT survey was subsequently extended to an area of 6.4 sq. degrees (at I and Z) and unearthed a total of 40 candidate brown dwarfs. Moraux et al. (2003) applied statistical arguments to account for non-members in their sample and derived a power law index of α=0.6. Most recently, Bihain et al. (2006) have used deep R, I, J and K band photometry and proper motion measurements to unearth 6 robust L type Pleiades members in an area of 1.8 sq. degrees with masses in the range 0.04-0.02 M⊙ and derived a power law index of α=0.5±0.2. Here we report the results of a new optical/infrared survey of 2.5 sq. degrees of the Pleiades, the aim of which is to extend empir- ical constraints on the cluster mass function down to the planetary mass regime (M∼0.01 M⊙). In the next section we describe the ob- servations acquired/used as part for this study, their reduction, their calibration and their photometric completeness. In subsequent sec- tions we describe how we have identified candidate brown dwarf members on the basis of colours and proper motions. We use our new results to constrain the form of the cluster mass function and conclude by briefly discussing our findings in the context of star formation models. 2 OBSERVATIONS, DATA REDUCTION AND SURVEY COMPLETENESS 2.1 The J band imaging and its reduction Approximately 3.0 square degrees of the Pleiades cluster was ob- served in the J band using the Wide Field Camera (WFCAM) on the United Kingdom Infrared Telescope (UKIRT) between the dates of 29/09/2005 and 08/01/2006. WFCAM is a near infrared imager consisting of 4 Rockwell Hawaii-II (HgCdTe 2048x2048) arrays with 0.4” pixels, arranged such that 4 separate pointings (pawprints) can be tiled together to cover a 0.75 sq. degree region of sky (see http://www.ukidss.org/technical/technical.html for dia- gram). A total of four tiles were observed in a mixture of photo- metric and non-photometric conditions but in seeing of typically ≈ 1.0 arcsecond or better. To ensure that the images were properly sampled we employed the 2×2 microstep mode. The locations on the sky of our tiles (shown in Figure 1) were chosen to provide maximum overlap with the optical fields surveyed in 2000 by the Canada-France-Hawaii telescope and CFH12k camera but also to avoid bright stars and areas of significant interstellar extinction. The images were reduced at the Cambridge Astronomical Sur- vey Unit (CASU) using procedures which have been custom writ- ten for the treatment of WFCAM data. In brief, each frame was de- biased, dark corrected and then flat fielded. The individual dithered images were stacked before having an object detection routine run on them. The detection procedure employs a core radius of 5 pixels, and identifies objects as islands of more than 4 interconnected pix- els with flux >1.5σ above the background level. The frames were astrometrically calibrated using point sources in the Two micron All Sky Survey (2MASS) catalogue. These solutions, in general, had a scatter of less than 0.1 arcseconds. The photometric calibra- tion employed by the CASU pipeline also relies on 2MASS data (there are typically hundreds of 2MASS calibrators per detector) and is found to be accurate to ≈2% in good conditions (see Warren et al., 2007, Hodgkin et al., 2007 for details). In measuring our photometry we used an aperture of 2”, which is approximately twice the core radius of point sources. This 2” di- ameter of the aperture is also twice the seeing FWHM. The reduc- tion pipeline also attempts to classify each source depending on its morphology (e.g. galaxy, star, noise). However, at the limit of the data this classification becomes less reliable. Therefore, in our sub- sequent analysis we chose to define as stellar all objects which lie within 3 sigma of the stellar locus, where sigma is defined accord- ing to Irwin et al. (in prep). 2.2 The far-red optical imaging and a new reduction As part of this work we have used a subset (2.54 square degrees) of the far-red optical data obtained in the course of the IZ survey of the Pleiades conducted in 2000 by Moraux et al. (2003). The rele- vant CFH12k data were extracted from the Canadian Astrophysical Data Center archive and were reprocessed at Cambridge University using the CASU optical imaging pipeline (Irwin & Lewis, 2001). In brief, these data were bias subtracted and corrected for non- linearity prior to flat fielding. Fringe maps, which were constructed c© 2007 RAS, MNRAS 000, 1–11 http://www.ukidss.org/technical/technical.html Proper motion L and T dwarf candidate members of the Pleiades 3 Figure 1. The regions imaged at I, Z and J with the CFHT and UKIRT. The CFH12k pointings (light rectangular outlines) are labelled alphabetically as in Moraux et al. (2003), while the WFCAM tiles (bold square outlines) are labelled numerically, ranging from 1 to 4. Note that the observations avoid the region of high reddening to the south of the Merope and the bright stars in vicinity of the cluster centre. for each photometric band from images obtained during the observ- ing run, were used to remove the effects of interference between night sky lines in the CCD substrate. Subsequently, sources at a level of significant of 3σ or greater were morphologically classified and aperture photometry obtained for each. A World Coordinate System (WCS) was determined for each frame by cross-correlating these sources with the Automated Plate Measuring (APM) machine catalogue (Irwin, 1985). The approximately 100 common objects per CCD chip lead to an internal accuracy of typically better than 0.3 ”. The photometry was calibrated onto a CFH12k I and Z natu- ral system using stars with near zero colour (B-V-R-I≈0) in Landolt standard field SA98 (Landolt, 1992) which was observed the same nights as the science data. The systematic errors in the photome- try were calculated by comparing the photometry of overlapping fields as in Moraux et al. (2003). The photometry was found to be accurate to ≈3%. 2.3 The completeness of datasets To estimate the completeness of our IR images, we injected fake stars with magnitudes in the range J=12-22 into each of the 16 chips of every WFCAM frame and re-ran the object detection soft- ware with the same parameters that were used to detect the real sources. To avoid significantly increasing the density of all sources in the data we inserted only 200 fake stars per chip in a given run. To provide meaningful statistics we repeated this whole procedure ten times. Subsequently, we calculated percentage completeness at a given magnitude by taking the ratio of the number of fake stars recovered to the number of fake stars injected into a given magni- tude bin (and multiplying by 100). We note that a 100% recovery rate was never achieved at any magnitude since a small proportion of the fake stars always fell sufficiently close to other sources to be overlooked by the object detection algorithm. This method was also applied to determine the completeness of the I and Z band CFH12k data. However, the magnitude range of the fake stars was adjusted to be consistent with the different saturation and faint end magnitude limits of these data. The results of this procedure for all 3 photometric bands are shown in Table 1. A glance at this table indicates that the IR data are in general 90% complete to J≈19.7, although Field 3 is slightly less deep, due to moonlight and poor seeing. In this case the proximity of the moon led to higher background counts. The I data are typically 90% and 50% complete to I=22.5 and 23.5 respectively. The cor- responding completeness limits for the Z band data are Z=21.5 and 22.5 respectively. 3 ANALYSIS OF THE DATA 3.1 Photometric selection of candidate cluster members An initial photometrically culled sample of candidate brown dwarfs has been obtained from the I,I-Z colour-magnitude diagram (Fig- c© 2007 RAS, MNRAS 000, 1–11 4 S. L. Casewell et al. 0 1 2 3 4 5 Figure 2. The I,I-Z CMD for the whole of field 1. The solid line is the NEXTGEN model, and the dotted line the DUSTY model. The small points are all objects that were classed as stellar in both I and Z data. The crosses are all objects that met the following selection criteria: classed as stellar in both I and Z data, for 16.5 < I < 22.5, they must lie no more than 0.25 magnitudes to the left of the DUSTY isochrone, for I>22.5, they must lie to the right of the line, I-Z= (I-19.0)/3.5. The filled squares are the previously identified cluster candidate members from Bihain et al. (2006), Moraux et al. (2003) and Bouvier et al. (1998), plotted to highlight the cluster sequence. 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 3. The Z,Z-J CMD for the whole of the survey. The solid line is the NEXTGEN model, and the dotted line the DUSTY model. The crosses are all the objects selected from the I,I-Z (crosses on Figure 2.). The filled diamonds are all objects that met our selection criteria from the I,I-Z,and Z, Z-J CMDs. These were selected for proper motion analysis, and were found to be non members. The filled squares are our candidate cluster members (objects that remained after proper motion analysis). The squares are our ZJ only candidates for all four fields that remained after proper motion analysis. The previously identified probable members from Bihain et al. (2006), Moraux et al. (2003) and Bouvier et al. (1998) that remained after our proper motion analysis are identified by open circles around the plotted symbols. c© 2007 RAS, MNRAS 000, 1–11 Proper motion L and T dwarf candidate members of the Pleiades 5 Table 1. 50 and 90% completeness figures for the optical and infrared fields. The positioning of these fields is shown in Figure 1. Note that while WFCAM field 1 corresponds to CFHT fields B, C, R and Q, the individual pawprints, do not correspond on a one to one basis - i.e. field1 00 does not correspond to field Field name I Z WFCAM tile name WFCAM pawprint name J 50% 90% 50% 90% 50% 90% B 23.2 22.5 22.3 21.5 field1 00 20.9 19.9 C 23.7 22.6 22.6 21.6 field1 01 20.9 20.1 R 24.0 23.0 22.9 21.6 field1 10 20.9 19.8 Q 23.7 22.5 22.7 21.6 field1 11 20.9 19.7 K 23.6 22.5 23.0 21.9 field2 00 20.9 19.7 L 24.0 22.7 23.0 21.8 field2 01 20.9 19.9 D 23.7 22.4 23.0 21.7 field2 10 21.0 19.9 field2 11 20.9 19.7 U 23.5 22.5 22.9 21.7 field3 00 19.5 18.8 V 23.8 22.5 22.7 21.7 field3 01 19.0 17.7 T 23.6 22.5 22.6 21.5 field3 10 19.6 18.6 field3 11 18.9 17.7 I 23.9 22.3 23.1 22.0 field4 00 20.8 19.7 G 23.7 22.7 23.4 22.3 field4 01 20.8 19.7 field4 10 20.8 19.7 field4 11 20.8 19.7 1 2 3 4 5 6 7 Figure 4. The J,I-J CMD for the whole of the survey. The solid line is the NEXTGEN model, and the dotted line the DUSY model. The crosses are all the objects selected from the I,I-Z (crosses on Figure 2.). The filled diamonds are all objects that met our selection criteria from the I,I-Z,and Z, Z-J CMDs. These were selected for proper motion analysis, and were found to be non members. The filled squares are our candidate cluster members (objects that remained after proper motion analysis). The previously identified members from Bihain et al. (2006), Moraux et al. (2003) and Bouvier et al. (1998) that remained after our proper motion analysis are identified by open circles around the plotted symbols. ure 2) where the 120 Myr NEXTGEN (Baraffe et al., 1998) and DUSTY (Chabrier et al., 2000) model isochrones (modified to take into account the Pleiades distance of 134 pc e.g. Percival, Salaris & Groenewegen, 2005) served as a guide to the location of the Pleiades sequence. With the uncertainties in both the photometry and the age of the cluster in mind, we selected all objects classed as stellar in both the I and Z data, which in the magnitude range 16.5 < I < 22.5 lay no more than 0.25 magnitudes to the left of the DUSTY isochrone. All the candidate Pleiads found by Moraux et al. (2003) and Bihain et al. (2006) lay within ±0.25 magnitudes of the DUSTY model. Thus our selection criterion is 0.25 magni- tudes to the left of the DUSTY model. Below I=22.5, the DUSTY model is not red enough to account for known field stars, and so is inappropriate in this effective temperature regime. We have cal- culated an approximate field star sequence from Tinney, Burgasser & Kirkpatrick (2003) and Hawley et al (2002) and lowered it by 2 magnitudes. This results in the line I-Z= (I-19.0)/3.5. This se- lection is conservative, and is particularly aimed at removing the c© 2007 RAS, MNRAS 000, 1–11 6 S. L. Casewell et al. bulk of the red tail of the background stars. Subsequently, the initial list of candidates was cross-correlated with our J band photometric catalogue (using a matching radius of 2 arcseconds) and a refined photometrically culled sample obtained using the Z,Z-J colour- magnitude diagram (Figure 3). These objects are also shown on the J, I-J colour-magnitude diagram (Figure 4). As before, the 120 Myr model isochrones were used as a guide to the location of the cluster sequence. With the photometric uncertainties in mind, all candi- dates with Z620 were retained. All candidates with 20<Z<21 and Z-J>1.6 were also retained. Finally, all candidates with Z>21 and Z-J>1.9 were retained. These constraints are conservative and are based on the field L and T dwarfs sequence (Z-J>3, Chiu et al., 2006) since the DUSTY models are known to be inappropriate in this effective temperature regime. Since our survey is limited by the depth of the I band data, all candidates with Z>20 and no I band counterpart were also kept. 3.2 Refining the sample using astrometric measurements To weed out non-members we have measured the proper motion of each candidate brown dwarf, using the Z and J band data where the epoch difference was 5 years. In this process, only objects lying within 2 arcminutes of each candidate were chosen as potential as- trometric reference stars. This compromise provided a sufficiently large number of sources but at the same time minimised the effects of non-linear distortions in the images. Furthermore, objects with large ellipticity (>0.2), classed by the photometric pipeline as non- stellar in the Z band data and with Z<16 or Z>20 were rejected. This ensured that, in the main, the astrometric reference sources were not of very low S/N in the J band or saturated in the optical data. These criteria generally provided at least 20 suitable stars per candidate brown dwarf. Six coefficient transforms between the epoch 1 Z band im- ages and the epoch 2 J band images were calculated using routines drawn from the STARLINK SLALIB package. The iterative fitting rejects objects having residuals greater than 3σ, where σ is robustly calculated as the median of absolute deviation of the reference star residuals, scaled by the appropriate factor (1.48) to yield an equiv- alent RMS. Once the routine had converged the relative proper mo- tions in pixels were calculated by dividing the fitting residuals of each candidate by the epoch difference. For our data the epoch difference is approximately 5 years. Subsequently, the astrometric motion in milliarcseconds per year in RA and DEC was derived by folding these values through the World Coordinate System trans- form matrix of the relevant WFCAM image. To estimate the errors on our proper motions measurements, we have injected fake stars into both the Z and J band data, in a sim- ilar fashion to that described in section 2.3. However, here we have determined the difference between the inserted position and the photometric pipeline estimate of the centroid of each star. Assum- ing that the differences between these two locations are normally distributed, we have divided the fake stars into 3 magnitude bins in each photometric band (Z621, 23>Z>21, 24>Z>23, 21>J>17) and fit 2d Gaussians to estimate the 1-sigma centroiding uncertainty as a function of source brightness. We find that in the Z band data, for objects with magnitudes Z621, the centroiding uncertainty is equivalent to 3 mas yr−1 in each axis, while for objects with 23>Z>21 this number increases to 8 mas yr−1. For our faintest Z band objects, 24>Z>23, the cen- troiding uncertainty is equivalent to 12 mas yr−1 in each axis. In the J band data, for objects with magnitudes 21>J>17, the cen- troiding uncertainty is equivalent to 5 mas yr−1 in each axis. Thus for our brightest candidates (Z<21, J<19), the quadratic sum of the Z and J band centroiding errors is less than or comparable to the RMS of the residuals of the linear transform fit, which is typ- ically 5-10 mas yr−1 in each axis. We adopt this latter quantity as the proper motion uncertainty in both the RA and DEC directions for these objects. It is worth noting at this point that both the stars and brown dwarfs of the Pleiades appear to be in a state of dynam- ical relation (e.g. Pinfield et al. 1998, Jameson et al. 2002), where the velocity dispersion of the members is proportional to 1/M0.5, where M is mass. Based on an extrapolation of the data in Figure 4 of Pinfield et al. (1998), we would expect our lowest mass brown dwarf members (0.01-0.02M⊙) to have velocity dispersion of ∼ 7 mas yr−1. This velocity dispersion should be added quadratically to the above uncertainties. Our final adopted proper motion selec- tion, effectively a radius of 14 mas yr−1 , is described below, and the velocity dispersion is small compared to this. We fitted the proper motions of all of our photometric candi- dates (excluding the ZJ only candidates) with a 2D Gaussian, which centred around 1.1, -7 mas yr−1. This Gaussian had a σ of 14.0. We were not able to fit two Gaussians, one to the background stars and one to the Pleiades dwarfs, as described in Moraux et al.(2003), since only ≈ 30 objects have the correct proper motion for cluster membership. Consequently, we only selected objects to be proper motion members if they had proper motions that fell within 1σ of the proper motion of the cluster at +20.0, -40.0 mas yr−1 (Jones 1981; Hambly, Jameson & Hawkins, 1991; Moraux et al., 2001). We required the selection criteria to be 1σ, as extending this to 2σ, would seriously overlap with the field stars centred on 0,0. We did however extend the selection criteria to 1.5σ, which yielded 14 ad- ditional objects, however all were rejected due to their bright, but blue (I≈17.0, I-Z<1.0) positions on the I,I-Z CMD, which led us to believe that they were field objects. We also attempted to tighten our selection criteria to a circle with radius 10 mas yr−1. This se- lection meant that we lost as possible members objects PLZJ 78, 9, 77, 23 (see Table 4). PLIZ 79, 9 and 77 have all been identified and confirmed as proper motion members by Bihain et al. (2006), Moraux et al. (2001), and Bouvier et al. (1998). Unfortunately, as we cannot fit two Gaussians to our data, we cannot calculate a prob- ability of membership for these objects by the standard method as defined by Sanders (1971). The proper motion measurements may be found in Table 4, as well as the I, Z, J, H and K magnitudes for these candidate members to the cluster. We have attempted to use control data to determine the level of contamination within our data, however, the numbers involved are very small, so any calculated probability will be rather uncer- tain. We used as controls, two circles of radius 14.0 mas yr−1, at the same distance from 0,0 proper motion as the Pleiades. We then separated the data into one magnitude bins, and calculated the prob- ability for each magnitude bin, using equation 1. Pmembership = Ncluster −Ncontrol Ncluster Where Pmembership is the probability of membership for that mag- nitude bin, Ncluster is the number of stars and contaminants within the cluster circle in that magnitude bin. Ncontrol is the number of dwarfs in the control circle of proper motion space, see Figure 4. Ncluster - Ncontrol is the number of Pleiads. It can be seen that the probability depends on where the control circle is located. Thus as well as using control circles, we use an annulus and scale down the count to an area equal to that of a control circle. Note that Figure 5 is for all of the magnitude bins together. Figure 6 is the same as Figure 5, but for the ZJ selected objects only. The statistics are c© 2007 RAS, MNRAS 000, 1–11 Proper motion L and T dwarf candidate members of the Pleiades 7 -100 -50 0 50 100 µαcos(δ) mas/yr Figure 5. Proper motion vector diagram of the photometrically selected candidate members. The filled triangles are candidate and known cluster members. The filled diamonds and filled circles are the two separate control clusters used. The annulus used for the radial method is also plotted. -100 -50 0 50 100 µαcos(δ) mas/yr Figure 6. Proper motion vector diagram of the photometrically selected candidate members. The filled triangles are candidate cluster members se- lected from the Z,Z-J CMD only. The filled diamonds and filled circles are the two separate control clusters used. The annulus used for the radial method is also plotted. much poorer for the individual magnitude bins and the probabili- ties are correspondingly more uncertain. It can be seen in Figure 4 that there is not a symmetrical distribution of proper motions. In fact the distribution in the Vector point diagram, is a classical ”velocity ellipsoid” displaced from zero by reflex motion from the Sun’s peculiar velocity, and happens to be in the direction of the Pleiades proper motion vector. We have therefore probably under- estimated the contamination, as the annulus method of calculating probabilities assumes that the vector point diagram has a circularly symmetric distribution of objects. These probabilities are shown in Table 2, and probabilities derived in the same way but for the ZJ only candidates can be found in Table 3. An alternative approach to estimating the contamination is the use the field L and T dwarf luminosity functions. Chabrier (2005) gives the T dwarf luminosity function as being 10−3 dwarfs/pc3/unit J mag interval. Our 7 L and T dwarf candidates cover a total of 0.7 mag in the J band. Note PLZJ 323 and 23 may be late L dwarfs but we include them in this analysis. The volume of space we use is 836 pc2, based on 2.5 square degrees and a distance to the Pleiades of 134±30 pc (Percival et al., 2005). This distance range corresponds to a distance modulus range of ±0.5 magni- tudes, which is generous, given that the sequence shown in figure 8 is clearly narrower than ±0.5 magnitudes. Thus the expected num- ber of contaminating field dwarfs is 0.6. In addition to this, field T dwarfs are unlikely to have the same proper motion as the Pleiades, thus reducing the 0.6 further. For the field L dwarfs with MJ≈13.0 (i.e. J≈18.5 at the distance of the Pleiades) the luminosity func- tion is 3×10−4 dwarfs/pc3/unit J mag interval (Chabrier, 2005). A similar calculation then gives 0.25 contaminating L dwarfs which should be further reduced by considering proper motions. It is thus clear that the field luminosity function indicates that contamination by field L and T dwarfs should be negligible. 4 RESULTS Most of these objects, except two bright objects and the faintest seven have been documented before in surveys - Moraux et al (2003) and Bihain et al (2006). We recovered all of these objects within our overlapping area, and none were rejected by our IZ pho- tometric selection. The objects we recovered were BRB 4, 8, 17, 13, 19, 21, 22, 27 and 28 and PLIZ 2, 3, 5, 6, 13, 14, 19, 20, 26, 28, 31, 34, 35 and 36. PLIZ 18, 27 and 39 were found to have no J counter- part in our catalogues. Of these objects, BRB 19 and PLIZ 14 and 26 met by our selection criteria on the Z, Z-J CMD, however they were too blue in their Z-J colour for their place on the sequence. Out of the remaining objects we find that we agree with the proper motion measurements as calculated by Bihain et al.(2006) for PLIZ 28, which we believe is a member of the cluster. We agree with Bi- hain et al.(2006) over their candidates BRB 13 and BRB 19 that they are not proper motion members to the cluster, however we dis- agree with their proper motion measurement for BRB 19. We also find that PLIZ 5 is a non member to the cluster - ie its proper mo- tion measurement is not within 14 mas yr−1 of the cluster proper motion value. We find that PLIZ 14 and 26 are not proper motion members to the cluster, as well as not having met our selection cri- teria. PLIZ 26 was found to have a proper motion measurement of 35.73±9.00, -25.83±6.96, which did not fall within 14 mas y−1 of the cluster, and also missed the selection made with the wider circle (21 mas yr−1) as well. We find that PLIZ 19, 20, 34 and 36 are not proper motion members to the cluster. However this means we disagree with Moraux et al. (2003), over their object PLIZ 20. They find a proper motion of 25.6±7.3, -44.7±7.4 mas yr−1 for it. Our proper motion measurement is 0.88± 15.86, -0.92±8.42 mas yr−1. It is possible that this object has been adversely affected by its position on the edge of one of the WFCAM chips, thus reducing the number of reference stars used to calculate its proper motion. An alternative method of measuring the proper motion using all the objects on the same chip produced a measurement of 19.14±11.06, -28.989±11.94 mas yr−1. This value does meet our selection cri- teria, and has been previously accepted as a member. We suggest PLIZ 20 is likely to be a member because of this. We find that PLIZ 2, 3, 6, 31 and 35 are all proper motion members to the cluster. In addition to this, we find 2 brighter new candidate members to the cluster. These objects are bright enough to have appeared in previous surveys, and in the UKIDSS Galactic cluster survey (GCS). We also have 2 fainter new members to the c© 2007 RAS, MNRAS 000, 1–11 Table 4. Name,coordinates, Z, I, J, H and K magnitudes for our members to the cluster. The errors quoted are internal (from photon counting). The systematic calibration errors are 2% in the J, H and K wavebands (Warren et al., 2007), and 3% in the I and Z wavebands. The J, H and K magnitudes are on the MKO system. Previously discovered members also also have their other known names listed from Moraux et al. (2003), Bihain et al. (2006) and Bouvier et al. (1998). The H and K band magnitudes are taken from the UKIDSS Galactic Cluster Survey with the exceptions of PLZJ 23, 93, 721 and 235 which have their H band magnitudes listed from our H survey. The K band magnitude for PLZJ 93 is from our UFTI photometry, and PLZJ 23 is from LIRIS service time. The final 5 objects in the table are our candidates selected from the ZJ data only. Name Alternate RA dec µαcosδ µδ I Z J H K name J2000.0 mas yr−1 PLZJ 29 BRB4 03 44 23.23 +25 38 45.11 23.40±8.24 -48.51±6.34 17.005 ± 0.001 16.163 ± 0.001 14.732 ± 0.001 14.132±0.004 13.744±0.004 PLZJ 56 03 44 53.51 +25 36 19.46 19.68±7.34 -35.63±5.29 17.012 ± 0.001 16.351 ± 0.001 15.250 ± 0.001 14.650±0.005 14.342±0.006 PLZJ 45 BRB8, CFHT-PL-7 03 52 58.2 +24 17 31.57 19.72±4.95 -42.37±7.44 17.101 ± 0.001 16.417 ± 0.001 15.247 ± 0.001 14.614±0.005 14.251±0.006 PLZJ 50 03 43 55.98 +25 36 25.45 13.48±8.24 -35.65±5.38 17.239 ± 0.001 16.496 ± 0.001 15.268 ± 0.001 14.693±0.006 14.319±0.006 PLZJ 60 CFHT-PL-10 03 44 32.32 +25 25 18.06 16.93±7.76 -43.15±5.72 17.592 ± 0.001 16.810 ± 0.001 15.460 ± 0.001 14.884±0.007 14.465±0.006 PLZJ 78 PLIZ2 03 55 23.07 +24 49 05.18 19.72±10.06 -29.74±10.45 17.719 ± 0.001 16.948 ± 0.001 15.574 ± 0.001 14.963±0.007 14.552±0.007 PLZJ 46 PLIZ3, BRB11 03 52 67.20 +24 16 01.00 19.55±5.15 -42.58±7.57 17.742 ± 0.001 16.945 ± 0.001 15.583 ± 0.001 14.966±0.007 14.503±0.008 PLZJ 9 PLIZ6,BRB9 03 53 55.09 +23 23 36.38 24.13±13.83 -50.10±22.71 17.752 ± 0.001 16.804 ± 0.001 15.222 ± 0.001 14.548±0.005 14.054±0.005 PLZJ 11 PLIZ20 03 54 05.33 +23 33 59.71 9.14±11.06 -28.98±11.94 19.571±0.004 18.563±0.004 16.691±0.005 15.980±0.016 15.436±0.016 PLZJ 77 PLIZ28,BRB18 03 54 10.04 +23 17 52.28 12.01±14.59 -51.60±15.84 20.760 ± 0.010 19.728 ± 0.010 17.647 ± 0.010 16.789±0.031 16.131±0.030 PLZJ 21 PLIZ31 03 51 47.65 +24 39 59.18 17.84±9.41 -44.92±8.03 20.944±0.014 19.762±0.013 17.575±0.012 16.774±0.026 16.089±0.028 PLZJ 10 PLIZ35,BRB15 03 52 31.19 +24 46 29.61 15.84±8.88 -49.34±6.24 21.293±0.018 20.292±0.016 18.181±0.022 17.118±0.041 16.506±0.0416 PLZJ 4 BRB21 03 54 10.25 +23 41 40.67 29.74±13.17 -38.46±8.88 21.322 ± 0.010 20.215 ± 0.013 18.171 ± 0.010 17.141±0.045 16.377±0.039 PLZJ 61 BRB22 03 44 31.27 +25 35 14.97 25.82±7.89 -40.21±8.47 22.043 ± 0.030 20.782 ± 0.026 18.298 ± 0.020 17.393±0.059 16.657±0.04 PLZJ 32 BRB27 03 44 27.27 +25 44 41.99 25.03±11.52 -38.65±23.46 22.235 ± 0.040 20.962 ± 0.029 18.871 ± 0.030 17.793±0.094 16.950±0.070 PLZJ 37 BRB28 03 52 54.92 +24 37 18.85 18.13±11.53 -48.68±11.38 22.452 ± 0.05 21.216 ± 0.041 18.839 ± 0.030 17.742±0.071 16.921±0.058 PLZJ 23 03 51 53.38 +24 38 12.11 20.75±10.51 -50.05±9.96 23.541 ± 0.140 22.187 ± 0.112 19.960 ± 0.100 19.362±0.100 18.510±0.030 PLZJ 93 03 55 13.00 +24 36 15.8 13.11±14.36 -33.77±12.97 24.488 ± 0.370 22.592 ± 0.164 19.968 ± 0.080 19.955±0.100 19.420 ±0.100 PLZJ 323 03 43 55.27 +25 43 26.2 29.87±12.05 -39.37±11.70 - 21.597±0.054 19.613±0.076 - - PLZJ 721 03 55 07.14 +24 57 22.34 19.18±22.23 -40.70±12.38 - 22.195±0.092 20.248±0.116 20.417±0.123 - PLZJ 235 03 52 32.57 +24 44 36.64 20.92±12.16 -45.84±11.75 - 22.339±0.115 20.039±0.112 20.245±0.127 - PLZJ 112 03 53 19.37 +24 53 31.85 8.56±14.08 -34.59±19.99 - 22.532±0.116 20.281±0.143 - - PLZJ 100 03 47 19.19 +25 50 53.3 20.23±14.27 -37.28±23.82 - 23.563±0.373 20.254±0.114 - - 000,1 Proper motion L and T dwarf candidate members of the Pleiades 9 Table 2. Probability of membership, magnitude range for our methods of calculating probabilities of membership using the annulus as well as the two control areas. Probability Probability Probability Magnitude range annulus µαcosδ=-20 mas yr−1 µδ=-40 mas yr −1 µαcosδ=+40 mas yr−1 µδ=-20 mas yr 0.67 0.25 0.0 16 - 17 0.82 0.66 0.0 17 - 18 0.88 1.00 0.0 18 - 19 0.84 1.00 0.0 19 - 20 1.00 1.00 1.00 20 - 21 0.88 0.50 1.00 21 - 22 0.61 1.00 0.00 22 - 23 Table 3. Probability of membership, magnitude range for our methods of calculating probabilities of membership using the annulus as well as the two control areas for our candidates selected from the ZJ data only. Probability Probability Probability Magnitude range annulus µαcosδ=-20 mas yr−1 µδ=-40 mas yr −1 µαcosδ=+40 mas yr−1 µδ=-20 mas yr 0.61 1.00 1.00 21 - 22 0.35 0.67 0.33 22 - 23 -0.16 -2.00 0.00 23 - 24 cluster, and 5 objects selected using the ZJ photometry only. All of the objects identified as cluster members in this work are presented in Table 4. Two WFCAM tiles, 1 and 4, (see Figure 1) also had deep H band photometry. These tiles were observed at the same time as the J band imaging, and were observed under the same conditions, but with the exception that microstepping was not used. These data were reduced using the same pipeline as the J band data, but the photometry and object detection used a core radius of 2.5 pixels in this case. Fortunately these tiles also covered our faintest, previously undiscovered Pleiades candidates, PLZJ 23 and PLZJ 93, as well as two of the candidates selected from the ZJ data only, PLZJ 721 and 235. The UKIDSS Galactic Cluster survey (GCS) has also covered the entire area at J, H and K. The UKIDSS data are reduced using the same pipeline as the WFCAM data (see Dye et al, 2006 for details of the pipeline). We also have used UKIRT service time to measure photom- etry for PLZJ 93 in the K band. This observation was taken on 09/09/2006 in seeing of better than 1.1” using the UKIRT Fast Track Imager (UFTI), with a five point dither pattern. The data were reduced using the ORAC-DR pipeline, and the photometry was calibrated using UKIRT Faint Standard 115. The K band photometry for PLZJ 23 was obtained on the night of 05/03/2007 using the long slit intermediate resolution spectro- graph (LIRIS) on the William Hershel Telescope in service time, using a nine point dither pattern in seeing of ≈ 0.9”. The data were reduced using IRAF and astrometrically and photometrically cali- brated using 2MASS. The colour transforms presented in Carpen- ter, (2001) were used to calculate the K band magnitude from the KS magnitude. Thus we have I, Z, J, H and K band photometry for the major- ity of our Pleiades candidates. However H or K band photometry is still needed for PLZJ 323, 721, 235, 112 and 100, (see Table 4). Figures 7 and 8 show the K, J-K and H, J-H, colour magnitude diagrams, together with the NEXTGEN (Baraffe et al, 1998) and DUSTY (Chabrier et al, 2000) models for the Pleiades age of 120 Myrs (Stauffer et al 1998). The candidate members listed in Table 4 are also plotted in Figures 3 and 4 for clarity. In both of these diagrams the M dwarf tail, the redward L dwarf sequence and the L to T blueward transition sequence are clear. The L-T transition sequence of course only has two objects plotted on it on Figure 7 as we have no K band photometry for the ZJ candidates. As ex- pected the K, J-K diagram gives the best differentiation between the sequences. The redward L sequence in this diagram agrees with that found by Lodieu et al,(2007b) derived from a much greater area of the Pleiades by the UKIDSS GCS. The GCS is not sensi- tive enough to see the L-T blueward transition sequence however. The K, J-K diagram also shows the separation between single and binary dwarfs quite clearly. Note that the DUSTY theoretical track is too flat compared to our empirical sequence, see figures 7 and 8. PLZJ 23, 93, 721 and 235 have J-H colours of 0.60, 0.00, - 0.17 and -0.21 respectively. Comparing these colours with the spec- tral type colour relations of field dwarfs described in Leggett et al. (2002), yields estimated spectral types of T1.5, T4.5, T6 and T6 respectively. PLZJ 93 has J-K=0.60 which gives a spectral type of T3 (Leggett et al., 2002), which is consistent with the spectral type derived from the J-H colour (T4.5), within the errors. We also can calculate a H-K colour for this dwarf of 0.6, however the H-K colour is not a good choice for spectral typing, for instance, H- K=0.6 covers a range of spectral types from L1 to T3 (Chiu et al., 2006). The Z-J colour is also not a good choice of colour for mea- suring spectral types until the later T dwarfs (>T2)(Hawley et al., 2002). PLZJ 23 has J-K=1.45, which gives a spectral type of be- tween L8 and T1. The H-K colour for this dwarf is 0.85. We may thus assume that PLZJ 23 has a spectral type between L8 and T1.5, and likewise that PLZJ 93 has a spectral type of between T3 and T5 to take into account the photometric errors. It should be noted that the Z band quoted in Hawley et al., (2002) is for the Sloan filter system, and so for this reason we have not chosen to use it to spectral type our objects. We believe that the J-H colour gives the best estimate available to us of spectral types. Two of the three candidate members without H band photometry PLZJ 112 and 100 have fainter J magnitudes than PLZJ 23 and 93, and so it is likely that they are also T dwarfs. PLZJ 323 is brighter and is therefore c© 2007 RAS, MNRAS 000, 1–11 10 S. L. Casewell et al. Table 4. 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure 7. The K,J-K CMD for our candidate cluster members. The solid line is the NEXTGEN model of Baraffe et al (1998), and the dotted line is the DUSTY model of Chabrier et al. (2000). The filled squares are the candidates identified by Moraux et al. (2003), the filled triangles are the candidates identified by Bihain et al. (2006), the object enclosed by the open circle is CFHT-PL-10 identified by Bouvier et al. (1998). The objects marked by small points are our new candidate members. One of our T dwarf candidates, PLZJ 93, is found to the bottom of the plot, with a J-K of ≈ 0.6. PLZJ 23 is also present with a J-K of 1.45. probably a late L dwarf. Indeed our faintest candidate at Z, PLZJ 100, may be a very late T dwarf, however this assumption is made using its Z-J colour, which is very red. Using J magnitudes and the COND models of Baraffe et al. (2003) for 120 Myrs (the DUSTY models are no longer appropriate for calculating masses for objects this faint in the Pleiades), we calculate masses of ≈ 11 MJup for PLZJ 23, 93, 323, 721, 235, 112 and 100. More photometry in the H and K bands is clearly needed to improve and extend these esti- mates of the spectral types. 5 MASS SPECTRUM To calculate the mass spectrum, we first divided the sample into single dwarfs or single dwarfs with possible low mass companions and dwarfs that are close to 0.75 magnitudes above the single star sequence in the K, J-K colour magnitude diagram. The latter we assume to be equal mass binaries and count them as dwarfs with masses the same as a dwarf on the single dwarf sequence below them. From Figures 3, 4, 7 and 8 it can be seen that there are 2 such binaries all with J-K ≈ 1. Dwarfs with J-K <1.2 are as- signed masses using their H magnitudes and the NEXTGEN mod- els (Baraffe et al. 1998). For 1.2<J-K<2.0 we use the DUSTY models (Chabrier et al., 2000) and the J-H colour to assign a mass. Finally the T dwarf masses were calculated from their J magni- tudes and the COND models (Baraffe et al., 2003). The masses were binned into three mass intervals, covering the low, medium and high mass ranges and the numbers per bin are weighted by the probabilities of membership calculated using the annulus, and the bin width has been taken into account. The candidate members with negative probabilities are obviously omitted from the mass spectrum. The resultant mass spectrum is shown in Figure 9. The -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Figure 8. The H,J-H CMD for our candidate cluster members. The filled squares are the candidates identified by Moraux et al. (2003), the filled trian- gles are the candidates identified by Bihain et al. (2006), the object enclosed by the open circle is CFHT-PL-10 identified by Bouvier et al. (1998). The objects marked by small points are our new candidate members.The filled diamonds are the two candidates with H magnitudes selected from the ZJ data only. The solid line is the NEXTGEN model of Baraffe et al (1998), and the dotted line is the DUSTY model of Chabrier et al. (2000). errors are poissonian. Clearly the statistics are very poor, due to the small number of objects being dealt with. Using linear regres- sion we have fitted our data to the relationship dN/dM∝M−α , and calculate α=0.35±0.31. This is lower but still in agreement with values presented in the literature (within 1σ), however the error on this value is large, and the statistics are poor due to the small num- bers involved. If we take into account the fact that the last mass bin is only 50% complete (using Tables 1 and 4), then the lowest mass bin can be increased by 50% to compensate. If we then fit these data, we derive a value for α of 0.62±0.14, which is in agree- ment with the literature. Alternatively, we can discount this final low mass bin as being incomplete and simply omit it from the fit. In this case we calculate a value for α of 0.86. We have only dis- played the mass spectrum for the cluster in the area and magnitude surveyed. This is to avoid trying to take into account biases caused by some areas being more studied than others, and also because we are only adding a maximum of 9 objects to the mass spectrum, 7 of which have low probabilities of membership and small masses, and so are not likely to affect previous results a large amount. The mass spectrum appears to be rising towards the lowest masses, but this is not statistically significant due to the large error bars. 6 CONCLUSIONS We have confirmed a number of L dwarf candidates in the Pleiades. However the main result in this paper is the discovery of seven L and T dwarf Pleiads of masses ≈ 11 MJup, below the 13 MJup deuterium burning limit that is often used, somewhat artificially as the upper bound for planetary masses. Further H and K band pho- tometry, currently lacking for some of these candidates, will im- prove confidence in their membership of the cluster. Planetary mass brown dwarfs have, of course, been claimed in the Orion nebula c© 2007 RAS, MNRAS 000, 1–11 Proper motion L and T dwarf candidate members of the Pleiades 11 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 log mass (Solar masses) Figure 9. The mass spectrum for our Pleiades candidate members. The mass bin is in units of M⊙. The solid line is the fit to the data, (α=0.35±0.31). (Lucas & Roche 2000) and in the σ-Ori cluster (Zapatero-Osorio et al., 2002). These clusters both have very young ages and may also have a spread of ages (Béjar et al., 2001), making mass determi- nations somewhat uncertain. Lodieu et al. (2006, 2007a) have also found planetary mass brown dwarfs in the Upper Scorpius Associ- ation which has an age of 5 Myrs (Preibisch & Zinnecker, 2002). At very young ages the theoretical models may have significant er- rors when used to assign masses (Baraffe et al., 2002). Our result is the first detection of planetary mass objects in a mature cluster whose age is well established. It strengthens the case that the star formation process can produce very low mass objects. 7 ACKNOWLEDGEMENTS SLC, NL,and PDD acknowledge funding from PPARC. We also acknowledge the Canadian Astronomy Data Centre, which is oper- ated by the Dominion Astrophysical Observatory for the National Research Council of Canada’s Herzberg Institute of Astrophysics. This work has been based on observations obtained at the Canada- France-Hawaii Telescope (CFHT) which is operated by the Na- tional Research Council of Canada, the Institut National des Sci- ences de l’Univers of the Centre National de la Recherche Scien- tifique of France, and the University of Hawaii. 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Lodieu6,1 1Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK 2Anglo-Australian Observatory, PO Box 296, Epping NSW 1710 Australia 3CASU, Institute of Astronomy,University of Cambridge, Maddingley Road, Cambridge, CB3 0HA, UK 4Laboratoire d’Astrophysique, Observatoire de Grenoble, Université Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France 5Scottish Universities Physics Alliance (SUPA), Institute for Astronomy, School of Physics, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ 6 Instituto de Astrofı́sica de Canarias, Vı́a Láctea s/n, E-38205 La Laguna, Tenerife, Spain 3 December 2018 The paper “Proper motion L and T dwarf candidate members of the Pleiades” was published in the Monthly Notices of the Royal Astronomical Society, 2007, 378, 1131. It has come to our attention that there were errors in Table 4 as regards the Right Ascension of the candidate coordinates. Table 4 should read as follows. This has no impact on the scientific results presented in the paper. c© 2009 RAS http://arxiv.org/abs/0704.1578v2 Table 4Name,coordinates, Z, I, J, H and K magnitudes for our members to the cluster. The errors quoted are internal (from photon counting). The systematic calibration errors are 2% in the J, H and K wavebands (Warren et al., 2007), and 3% in the I and Z wavebands. The J, H and K magnitudes are on the MKO system. Previously discovered members also also have their other known names listed from Moraux et al. (2003), Bihain et al. (2006) and Bouvier et al. (1998). The H and K band magnitudes are taken from the UKIDSS Galactic Cluster Survey with the exceptions of PLZJ 23, 93, 721 and 235 which have their H band magnitudes listed from our H survey. The K band magnitude for PLZJ 93 is from our UFTI photometry, and PLZJ 23 is from LIRIS service time. The final 5 objects in the table are our candidates selected from the ZJ data only. Name Alternate RA dec µαcosδ µδ I Z J H K name J2000.0 mas yr−1 PLZJ 29 BRB4 03 44 23.23 +25 38 45.11 23.40±8.24 -48.51±6.34 17.005 ± 0.001 16.163 ± 0.001 14.732 ± 0.001 14.132±0.004 13.744±0.004 PLZJ 56 03 44 53.51 +25 36 19.46 19.68±7.34 -35.63±5.29 17.012 ± 0.001 16.351 ± 0.001 15.250 ± 0.001 14.650±0.005 14.342±0.006 PLZJ 45 BRB8, CFHT-PL-7 03 52 05.82 +24 17 31.57 19.72±4.95 -42.37±7.44 17.101 ± 0.001 16.417 ± 0.001 15.247 ± 0.001 14.614±0.005 14.251±0.006 PLZJ 50 03 43 55.98 +25 36 25.45 13.48±8.24 -35.65±5.38 17.239 ± 0.001 16.496 ± 0.001 15.268 ± 0.001 14.693±0.006 14.319±0.006 PLZJ 60 CFHT-PL-10 03 44 32.32 +25 25 18.06 16.93±7.76 -43.15±5.72 17.592 ± 0.001 16.810 ± 0.001 15.460 ± 0.001 14.884±0.007 14.465±0.006 PLZJ 78 PLIZ2 03 55 23.07 +24 49 05.18 19.72±10.06 -29.74±10.45 17.719 ± 0.001 16.948 ± 0.001 15.574 ± 0.001 14.963±0.007 14.552±0.007 PLZJ 46 PLIZ3, BRB11 03 52 06.71 +24 16 00.99 19.55±5.15 -42.58±7.57 17.742 ± 0.001 16.945 ± 0.001 15.583 ± 0.001 14.966±0.007 14.503±0.008 PLZJ 9 PLIZ6,BRB9 03 53 55.09 +23 23 36.38 24.13±13.83 -50.10±22.71 17.752 ± 0.001 16.804 ± 0.001 15.222 ± 0.001 14.548±0.005 14.054±0.005 PLZJ 11 PLIZ20 03 54 05.33 +23 33 59.71 9.14±11.06 -28.98±11.94 19.571±0.004 18.563±0.004 16.691±0.005 15.980±0.016 15.436±0.016 PLZJ 77 PLIZ28,BRB18 03 54 14.04 +23 17 52.28 12.01±14.59 -51.60±15.84 20.760 ± 0.010 19.728 ± 0.010 17.647 ± 0.010 16.789±0.031 16.131±0.030 PLZJ 21 PLIZ31 03 51 47.65 +24 39 59.18 17.84±9.41 -44.92±8.03 20.944±0.014 19.762±0.013 17.575±0.012 16.774±0.026 16.089±0.028 PLZJ 10 PLIZ35,BRB15 03 52 39.13 +24 46 29.61 15.84±8.88 -49.34±6.24 21.293±0.018 20.292±0.016 18.181±0.022 17.118±0.041 16.506±0.0416 PLZJ 4 BRB21 03 54 10.25 +23 41 40.67 29.74±13.17 -38.46±8.88 21.322 ± 0.010 20.215 ± 0.013 18.171 ± 0.010 17.141±0.045 16.377±0.039 PLZJ 61 BRB22 03 44 31.27 +25 35 14.97 25.82±7.89 -40.21±8.47 22.043 ± 0.030 20.782 ± 0.026 18.298 ± 0.020 17.393±0.059 16.657±0.04 PLZJ 32 BRB27 03 44 27.27 +25 44 41.99 25.03±11.52 -38.65±23.46 22.235 ± 0.040 20.962 ± 0.029 18.871 ± 0.030 17.793±0.094 16.950±0.070 PLZJ 37 BRB28 03 52 54.92 +24 37 18.85 18.13±11.53 -48.68±11.38 22.452 ± 0.05 21.216 ± 0.041 18.839 ± 0.030 17.742±0.071 16.921±0.058 PLZJ 23 03 51 53.38 +24 38 12.11 20.75±10.51 -50.05±9.96 23.541 ± 0.140 22.187 ± 0.112 19.960 ± 0.100 19.362±0.100 18.510±0.030 PLZJ 93 03 55 13.00 +24 36 15.8 13.11±14.36 -33.77±12.97 24.488 ± 0.370 22.592 ± 0.164 19.968 ± 0.080 19.955±0.100 19.420 ±0.100 PLZJ 323 03 43 55.27 +25 43 26.2 29.87±12.05 -39.37±11.70 - 21.597±0.054 19.613±0.076 - - PLZJ 721 03 55 07.14 +24 57 22.34 19.18±22.23 -40.70±12.38 - 22.195±0.092 20.248±0.116 20.417±0.123 - PLZJ 235 03 52 32.57 +24 44 36.64 20.92±12.16 -45.84±11.75 - 22.339±0.115 20.039±0.112 20.245±0.127 - PLZJ 112 03 53 19.37 +24 53 31.85 8.56±14.08 -34.59±19.99 - 22.532±0.116 20.281±0.143 - - PLZJ 100 03 47 19.19 +25 20 53.3 20.23±14.27 -37.28±23.82 - 23.563±0.373 20.254±0.114 - - 000–000 000,1 Introduction Observations, Data Reduction and Survey Completeness The J band imaging and its reduction The far-red optical imaging and a new reduction The completeness of datasets Analysis of the data Photometric selection of candidate cluster members Refining the sample using astrometric measurements Results Mass spectrum Conclusions Acknowledgements

We present the results of a deep optical-near-infrared multi-epoch survey covering 2.5 square degrees of the Pleiades open star cluster to search for new very-low-mass brown dwarf members. A significant (~ 5 year) epoch difference exists between the optical (CFH12k I-, Z-band) and near infrared (UKIRT WFCAM J-band) observations. We construct I,I-Z and Z,Z-J colour magnitude diagrams to select candidate cluster members. Proper motions are computed for all candidate members and compared to the background field objects to further refine the sample. We recover all known cluster members within the area of our survey. In addition, we have discovered 9 new candidate brown dwarf cluster members. The 7 faintest candidates have red Z-J colours and show blue near-infrared colours. These are consistent with being L and T-type Pleiads. Theoretical models predict their masses to be around 11 Jupiter masses. There is 1 errata for this paper

Introduction Observations, Data Reduction and Survey Completeness The J band imaging and its reduction The far-red optical imaging and a new reduction The completeness of datasets Analysis of the data Photometric selection of candidate cluster members Refining the sample using astrometric measurements Results Mass spectrum Conclusions Acknowledgements

Astronomy & Astrophysics manuscript no. aguerri˙rv c© ESO 2019 August 20, 2019 A Study of Catalogued Nearby Galaxy Clusters in the SDSS-DR4 I. Cluster Global Properties J. A. L. Aguerri, R. Sánchez-Janssen & C. Muñoz-Tuñón Instituto de Astrofı́sica de Canarias C/ Vı́a Láctea s/n, 38200 La Laguna, Spain. e-mail: jalfonso@iac.es, ruben@iac.es, cmt@iac.es Received ; accepted ABSTRACT Context. Large surveys as the Sloan Digital Sky Survey have made available large amounts of spectroscopic and photometric data of galaxies, providing important information for the study of galaxy evolution in dense environments. Aims. We have selected a sample of 88 nearby (z < 0.1) galaxy clusters from the SDSS-DR4 with redshift information for the cluster members. In particular, we focus our results on the galaxy morphological distribution, the velocity dispersion profiles, and the fraction of blue galaxies in clusters. Methods. Cluster membership was determined using the available velocity information. We have derived global properties for each cluster, such as their mean recessional velocity, velocity dispersion, and virial radii. Cluster galaxies have been grouped in two families according to their u − r colours. Results. The total sample consists of 10865 galaxies. As expected, the highest fraction of galaxies (62%) turned to be early-type (red) ones, being located at smaller distances from the cluster centre and showing lower velocity dispersions than late-type (blue) ones. The brightest cluster galaxies are located in the innermost regions and show the smallest velocity dispersions. Early-type galaxies also show constant velocity dispersion profiles inside the virial radius and a mild decline in the outermost regions. In contrast, late-type galaxies show always decreasing velocity dispersions profiles. No correlation has been found between the fraction of blue galaxies and cluster global properties,such as cluster velocity dispersion and galaxy concentration. In contrast, we found correlation between the X-ray luminosity and the fraction of blue galaxies. Conclusions. These results indicate that early- and late-type galaxies may have had different evolution. Thus, blue galaxies are located in more anisotropic and radial orbits than early-type ones. Their star formation seems to be independent of the cluster global properties in low mass clusters, but not for the most massive ones. We consider that it is unlikely that the whole blue population consists of recent arrivals to the cluster. These observational results suggest that the global environment could be important for driving the evolution of galaxies in the most massive cluster (σ > 800 km s−1). However, the local environment could play a key role in galaxy evolution for low mass clusters. Key words. galaxies: clusters: general 1. Introduction The large amount of spectroscopic and photometric data ob- tained during the last years by surveys such as the Sloan Digital Sky Survey (SDSS; York et al. 2000) or the 2dF Galaxy Redshift Survey (2dFGRS; Colless et al. 2001) have opened a new horizon for the study of galaxy evolution, and in partic- ular in the study of galaxy clusters. It is well known that the environment plays an important role in the evolution of galax- ies, and it is one of the keys that a good galaxy evolution the- ory should address. There are several physical mechanisms, not present in the field, which can dramatically transform galaxies in high density environments. Galaxies in clusters can evolve due to, e.g., dynamical friction, which can slow down the more massive galaxies, circularise their orbits and enhance their merger rate (den Hartog & Katgert 1996; Mamon 1992). Send offprint requests to: J. A. L. Aguerri Interactions with other galaxies and with the cluster grav- itational potential can disrupt the outermost regions of the galaxies and produce galaxy morphological transformations from late- to early-types (Moore et al. 1996), or even change massive galaxies into dwarf ones (Mastropietro et al. 2005). Swept of cold gas produced by ram pressure stripping (Gunn & Gott 1972 ; Quilis et al. 2000) or swept of the hot gas reservoirs (Bekki et al. 2002) can alter the star formation rate (SFR) of galaxies in clusters. But it is still a matter of debate which of these mechanisms is the main responsible of the galaxy evolution in galaxy clusters (see Goto 2005). Nevertheless, it is clear that all of these mechanisms transform galaxies from late- to early-types, and can produce the different segregations observed in galaxy clusters. One of the first segregations discovered in galaxy clus- ters was the morphological one. The first evidences of such segregation date from Curtis (1918) and Hubble & Humason http://arxiv.org/abs/0704.1579v1 2 Aguerri et al.: Global Properties of Nearby Galaxy Clusters (1931), and was quantified by Oemler (1974) and Melnick & Sargent (1977). In a thorough work, Dressler (1980) analysed a sample of 55 nearby galaxy clusters, contain- ing over 6000 galaxies, and observed that elliptical and S0 galaxies represent the largest fraction of galaxies lo- cated in the innermost and denser regions of galaxy clus- ters. In contrast, the outskirts of the clusters were domi- nated by spiral galaxies. In more distant clusters the frac- tion of E galaxies is as large or larger than in low-redshift clusters, but the S0 fraction is smaller (Dressler et al. 1997; Fasano et al. 2000). This has been interpreted as an evolution with redshift, being late-type galaxies transformed into early- type ones. Segregations in velocity space have also been ob- served in galaxy clusters. Early observations found that E and S0 galaxies showed smaller velocity dispersions than spi- rals and irregulars (Tammann 1972; Melnick & Sargent 1977; Moss & Dickens 1977). This has also been confirmed by other authors during the last two decades (Sodre et al. 1989; Biviano et al. 1992; Andreon et al. 1996; Stein 1997). The data from the ENACS survey (Katgert et al. 1998) produced a large sample of galaxies with spectroscopic redshifts and shed more light to this problem. Thus, Adami et al. (1998) studied a sam- ple of 2000 galaxies, confirming early findings that the ve- locity dispersion of galaxies increases along the Hubble se- quence: E/S0 galaxies show smaller velocity dispersions than early- and late-type spirals. This segregation was also observed in the velocity dispersion profiles (VDPs): late-type galaxies have decreasing VDPs, while E, S0 and early spirals show al- most flat VDPs (Adami et al. 1998). The different kinematics shown by the different types of galaxies was analysed in more detail by Biviano & Katgert (2004) who found that the ve- locity segregation of the different Hubble types is due to dif- ferences in orbits. Thus, early-type spirals have isotropic or- bits, while late-type ones are located in more anisotropic or- bits. The observed morphological and velocity segregation in clusters have been usually used to conclude that late-type spi- ral galaxies in clusters are recent arrivals to the cluster potential (Stein 1997; Adami et al. 1998). Star formation in galaxies is also affected by the envi- ronment. Butcher & Oemler (1984) found that the fraction of blue galaxies, fb, in clusters is smaller than in the field and evolves with redshift: more distant clusters show larger values of fb. This trend was interpreted as an evolutionary effect of the SFR in galaxy clusters. But the significant increase of new data has made it clear that the Butcher-Oemler effect is not only an evolutionary trend. A large scatter in the values of fb has been observed in narrow redshift ranges (Smail et al. 1998; Margoniner & de Carvalho 2000; Goto et al. 2003), which suggests that the variation of fb is influenced by environmental effects. In the past, many authors have tried to find correlations of fb with cluster properties, such as X-ray luminosity (Andreon & Ettori 1999; Smail et al. 1998; Fairley et al. 2002), luminosity limit and clustercentric distance (Ellingson et al. 2001; Goto et al. 2003; De Propris et al. 2004), richness (Margoniner et al. 2001; De Propris et al. 2004), cluster con- centration (Butcher & Oemler 1984; De Propris et al. 2004), presence of substructure (Metevier et al. 2000) or cluster velocity dispersion (De Propris et al. 2004). Some of these works found correlations between fb and the cluster envi- ronment while others did not, being such connection still a matter of debate. However, these works were usually done using small and heterogeneous cluster samples (but see e.g., De Propris et al. 2004). Environmental effects have also been invoked in order to explain the differences between the photometrical compo- nents of cluster and field spiral galaxies. Thus, it has been observed that the scale-lengths of the disks of spiral galax- ies in the Coma cluster are smaller than those of similar galaxies in the field (Gutiérrez et al. 2004; Aguerri et al. 2004). Interactions between galaxies or with the cluster potential can disrupt the disks of spiral galaxies in clusters. They can be strong enough for transforming bright late-type spi- ral galaxies in dwarfs (Aguerri et al. 2005a). The disrupted material would be part of the intracluster light already de- tected in some nearby galaxy clusters (Arnaboldi et al. 2002; Arnaboldi et al. 2004 ; Aguerri et al. 2005b) and galaxy groups (Castro-Rodrı́guez et al. 2003; Aguerri et al. 2006). The observational results summarised before illustrate the important role played by environment in galaxy evolution. They also indicate that late-type and early-type galaxies in clusters are two different families of objects with differ- ent properties, which points to different origins or evolution. Nevertheless, the main mechanisms responsible of this differ- ent evolution still remain unknown. In the present paper, we study one of the largest and more homogeneous galaxy cluster sample available in the literature. We have obtained the cluster membership, mean velocity, velocity dispersion, virial radius and positions for a sample of 88 clusters located at z < 0.1. We have investigated the main properties of a large sample of early (red) and late (blue) types of galaxies, such as their lo- cation within the cluster, their mean velocity dispersion, their VDPs, the LX −σ relation, and the fraction of blue galaxies for each cluster. This work provides important information about the properties of galaxies in nearby clusters, which will be useful in order to put constraints on cosmological models of cluster formation. This is the first paper of a series in which we will analyse the properties of the dwarf galaxy population (Sánchez-Janssen et al. in preparation), substructure in galaxy clusters (Aguerri et al. in preparation), and composite luminos- ity function of galaxy clusters (Sánchez-Janssen et al., in prepa- ration). The paper is organised as follows. Section 2 shows the dis- cussion about the galaxy cluster sample. The cluster member- ship and cluster global parameters are presented in Section 3. The results obtained about the morphological segregation, ve- locity dispersion profiles, LX − σ relation, and the fraction of blue galaxies are given in Sections 4, 5, 6 and 7, respectively. The discussion and conclusions are presented in Sections 8 and 9, respectively. Throughout this work we have used the cos- mological parameters: Ho = 75 km s −1 Mpc−1, Ωm = 0.3 and ΩΛ = 0.7. Aguerri et al.: Global Properties of Nearby Galaxy Clusters 3 2. Galaxy cluster Sample We have used photometric and spectroscopic data of ob- jects classified as galaxies from the SDSS-DR4, an imag- ing and spectroscopic survey of a large area in the sky (York et al. 2000). The imaging survey was carried out through five broad-band filters, ugriz, spanning the range from 3000 to 10000 Å, reaching a limiting r-band mag- nitude ≈ 22.2 with 95% completeness, and covering an area of 6670 deg2 (Adelman-McCarthy et al. 2006). A se- ries of pipelines process the imaging data and perfom the astrometric calibration (Pier et al. 2003), the photometric re- duction (Lupton et al. 2002) and the photometric calibration (Hogg et al. 2001). Objects brighter than mr = 17.77 were se- lected as possible targets for the spectroscopic survey, covering an area of 4783 deg2 of the sky for the DR4. The spectroscopic data were obtained with optical fibers with a diameter of 3 the focal plane, resulting in an spectral covering in the wave- length range 3800–9200 Å with a resolution of λ/∆λ ≈ 2000. Our sample consists of all clusters with known redshift at z < 0.1 from the catalogues of Abell et al. (1989), Zwicky et al. (1961), Böhringer et al.(2000) and Voges et al. (1999) that have been mapped by the SDSS-DR4. We downloaded only those galaxies located within a radius of 4.5 Mpc around the centres of the galaxy clusters. Only those clusters with more than 30 galaxies with spectroscopic data in the searching radius were considered, resulting in a sample formed by 240 clusters following the previous criteria. The SDSS-DR4 spectroscopic galaxy target selection was done by an automatic algorithm (see Strauss et al. 2002). The main galaxy sample consists of galaxies with r-band Petrosian magnitudes brighter than 17.77 and r-band Petrosian half-light surface brightness brighter than 24.5 mag arcsec−2. The completeness of this sample is high, exceeding 99% (see Strauss et al. 2002). However, some of the selected spectroscopic targets were not observed at the end. This incompleteness has several causes, including the fact that two spectroscopic fibers cannot be placed closer than 55 given plate, possible gaps between the plates, fibers that fall out of their holes, and so on. According to these reasons, we expect that the incompleteness of the spectroscopic data will be more important for bright galaxies in high density environments such as galaxy clusters. Figure 1 shows the mean completeness1 of the SDSS-DR4 spectroscopic data as a function of the r-band magnitude for the selected galaxies, where a fast increment to- wards faint magnitudes can be observed. In order to avoid pos- sible effects on the results due to this effect, we have completed the spectroscopic SDSS-DR4 observations with the data avail- able at the Nasa Extragalactic Database (NED). Figure 1 also shows the mean completeness as a function of r-band magni- tude after the spectroscopic data from NED were included in the sample. Notice that the new mean completeness is almost constant (≈ 85%) for all magnitudes brighter than mr = 17.77. We have made a second selection of the clusters by considering only those from our original list with completeness larger than 70% for galaxies brighter than 17.77 in the r-band. 1 We have defined the spectroscopic completeness per magnitude bin as the ratio of the number of galaxies with spectroscopic data to the number of galaxies with photometric information. Fig. 1. Mean completeness of the cluster sample as a function of the r-band magnitude. Diamonds represent the spectroscopic data from SDSS-DR4 and black circles after the completion with data from NED. 3. Cluster Membership Clusters properties such as the mean cluster velocity, the ve- locity dispersion, the cluster centre or the virial radius can be significantly affected by projection effects. Several methods have been developed during decades in order to obtain reliable galaxy cluster membership and avoid the presence of interlop- ers. They can be classified in two families. First, those algo- rithms that use only the information in the velocity space, e.g. 3σ-clipping techniques (Yahil & Vidal 1977), gapping proce- dures (Beers et al. 1990; Zabludoff et al. 1990, hereafter ZHG algorithm) or the KMM algorithm (Ashman et al. 1994). The other family corresponds to those algorithms which use infor- mation of both position and velocity, such as the methods de- signed by Fadda et al. (1996), den Hartog & Katgert (1996), or Rines et al. (2003). The cluster membership in our sample was obtained using a combination of two algorithms. A first rough cluster mem- bership determination was obtained using the ZHG algorithm, which in a second step was then refined using the KMM al- gorithm. The ZHG algorithm is a typical gapping procedure which determines the cluster membership by the exclusion of those galaxies located at more than a certain velocity distance (∆v) from the nearest galaxy in the velocity space. Then, the mean velocity (vm) and velocity dispersion (σ) of the remain- ing galaxies are calculated. After sorting objects with velocities greater than vm, any galaxy separated in velocity more than σ from the previous one is classified as non member. The same is done for those galaxies with velocities less than vm. The pro- cess is repeated several times and finally the mean cluster ve- locity (vc) and the cluster velocity dispersion (σc) are obtained. Zabludoff et al. (1990) pointed out that this method lacks statis- tical rigour and tends to give overestimated values of σc. One of the disadvantages of this method is that the results obtained strongly depend on the chosen value of ∆v. Large values of ∆v imply that a large fraction of interlopers are identified as clus- ter members. On the contrary, small values of ∆v result in the 4 Aguerri et al.: Global Properties of Nearby Galaxy Clusters lost of cluster galaxies. We have investigated the variation of σc for different values of ∆v, obtaining that ∆v=500 km s is an appropriate value for our clusters. This method has also the advantage that has an easy implementation and does not re- quire too much computational time. Recently, it has been used in works involving a large number of clusters, such as those from the 2dFGRS (De Propris et al. 2003). The ZHG algorithm splits the velocity histograms in different galaxy groups, be- ing one of them located at the catalogued redshift of the clus- ter. That group was taken and analysed in more detail with the KMM algorithm. In the few cases where there was no galaxy group located at the catalogued redshift we identified the most significant groups having z < 0.1 as the cluster itself. The KMM algorithm (Ashman et al. 1994) estimates the statistical significance of bi-modality in a dataset. We have run it to the group of galaxies given by the ZHG algorithm which contains the catalogued redshift of the cluster. The KMM algo- rithm gives us the compatibility of the velocity distribution of such group of galaxies with a single or multiple Gaussian dis- tribution. We considered three different cases which are sum- marised in Fig. 2: – Single cluster: the velocity distribution of the galaxies is compatible with a single Gaussian, e.g. Abell 757. – Cluster with substructure: the velocity distribution is com- patible with multiple groups. We identified the cluster as the group with the largest number of galaxies plus those groups which mean velocities lie within 3σ from the mean velocity of the largest one2, e.g. Abell 1003. – Cluster with contamination: the velocity distribution is compatible with the presence of several groups, but the mean velocities of the smaller groups deviate more than 3σ from the most populated one, which we identify as the cluster itself, e.g. Abell 168 . We have explored the differences in the values of vc and σc if we consider as interlopers those groups of galaxies located at a velocity distance larger than 1σ or 3σ from the mean velocity of the main galaxy group. We obtained that the differences in vc and σc in 90% of the clusters are less than 20%. The remaining 10% of the clusters are those with significant structure in the velocity distribution, being most of them more than one cluster along the line of sight. Thus, we have adopted 3σ as the default except for those clusters with significant differences between 1σ and 3σ, for which we have measured the mean velocity and velocity dispersion of the cluster adopting the criteria of 1σ. Through all of this process, the determination of vc and σc was done using the biweight robust estimator of Beers et al. (1990). 3.1. Cluster global parameters Once the cluster membership was determined, we obtained the global parameters of each cluster, i.e., mean velocity (vc), ve- locity dispersion (σc), cluster centre, and the radius r200. All of these parameters were computed using only the cluster mem- bers. 2 In this case σ is the velocity dispersion of the largest group of galaxies. Fig. 2. Velocity histograms of three representative clusters of the sample. The vertical full lines represent the mean velocity of the different groups of galaxies in which KMM algorithm has divided the velocity histogram. The dotted vertical lines represent vc ± 3σc. The determination of the cluster centre is important in or- der to accurately compute the other parameters of the clusters. The centre of the cluster is determined by the potential well, which can be traced by the position of the peak of the X-ray lu- minosity of the cluster. That peak was considered as the centre of those clusters from our sample with X-ray measurements in the literature. Unfortunately, not all the clusters from the sam- ple have X-ray data. In that case, the centre of these clusters was determined by the peak of the galaxy surface density3. For those clusters with X-ray data we have compared the centres given by the peaks of X-ray luminosity and galaxy surface den- sity, obtaining a mean difference of 150 kpc. Analytic models (Gott 1972) and simulations (Cole & Lacey 1996) indicate that the virialized mass of clusters is generally contained inside the surface where the mean inner density is 200ρc, where ρc is the critical density of the Universe. The radius of that surface is called r200. We have computed the r200 for our clusters using the same approximation as Carlberg et al. (1997): r200 = 10 H(zc) , (1) where H(zc) is the Hubble constant at the cluster redshift 3 The galaxy surface density was computed using the algorithm de- signed by Pisani (1996). Aguerri et al.: Global Properties of Nearby Galaxy Clusters 5 The previous global parameters of the clusters (vc, σc, r200 and centre) were obtained as described above but in a recur- rent way. In a first step, they where determined using all cluster member galaxies around 4.5 Mpc from the centre of the clus- ter. After this step we recalculated the parameters using only those galaxies located inside r200. The method was repeated several times until the difference in the parameters obtained in two consecutive steps was less than 5%. Three or four iterations were usually enough for reaching the convergence. In order to obtain reliable parameters of the clusters, those with less than 15 galaxies within r200 were removed from our list. This re- sults in a final sample formed by 110 nearby galaxy clusters. Table 1 shows the sample of galaxy clusters and their global parameters. The columns of Table 1 represent: (1) galaxy clus- ter name, (2, 3) cluster centres (α (J2000), δ (J2000)), (4) mean radial velocity, (5) cluster velocity dispersion, (6) r200 radius, (7) number of galaxies within r200, and (8) spectroscopic com- pleteness. For 6 clusters (Abell 1003, Abell 1032, Abell 1459, Abell 2023, Abell 2241 and ZwCl1316.4-0044) large differences in the mean recessional velocity have been found between the val- ues given in Table 2 and those from NED. These are the clusters with no significant galaxy group at the catalogued redshift (see Section 3). In order to consider the possible influence of neighbouring clusters on the global properties of the sample we searched in the surroundings of each cluster for the presence of compan- ions. Following Biviano & Girardi (2003), we have considered that two clusters, i and j, are in interaction when: |vi − vj| < 3(σi + σj) Ri,j < 2(r200,i + r200,j), (2) where Ri, j is the projected distance between the centres of the clusters and vi, j, σi, j, r200,i, j their respective mean velocities, velocity dispersions and r200. We found 16 couples of clusters in interaction according to the previous criteria. The remaining sample (88 clusters) followed the isolation criteria, and will be used in the analysis presented in the following sections. Figure 3 shows the sky distribution of the cluster members and the galaxy velocities as a function of clustercentric distance for a sample of 8 clusters. Red points represent the galaxies taken as cluster members while black points are interlopers. Notice the large number of interlopers in some of the galaxy clusters, such as Abell 1291, Abell 1383, Abell 2244. Some of them, Abell 1291 and Abell 1383, were not included in the final isolated sample due to the presence of companions. 3.2. Corrections to line-of-sight velocities Line-of-sight velocities of galaxies in clusters were corrected by two effects: cosmological redshift and global velocity field. We should take into account that we will compare the veloc- ity dispersion of clusters located at different redshifts. Thus, for each galaxy we have 1 + zobs = (1 + zc)(1 + zgal) (Danese et al. 1980), being zobs the apparent redshift of the galaxies, zc the cosmological redshift of the cluster, and zgal the redshift of the galaxy respect to the cluster centre. This cor- rection can affect up to 10% for the most distant clusters in our sample. Galaxy clusters are frequently part of larger cosmological structures such as filaments, superclusters or multiple systems, which can affect the velocity field resulting in a modified clus- ter velocity dispersion. The interaction between galaxy clus- ters can also produce distorted velocity fields. We have inves- tigated the importance of these effects in the velocity field of our clusters by making a least-square fit to the radial velocities of cluster galaxies with respect to their position in the plane of the sky (see den Hartog & Katgert 1996; Girardi et al. 1996). For each fit we computed the coefficient of multiple determi- nation, R2. In order to test the significance of the fitted veloc- ity gradients, we run 1000 Monte Carlo simulations for each cluster for which the correlation between position and veloc- ity was removed. This was achieved by shuffling the veloc- ities of the galaxies with respect to their positions. We de- fined the significance of velocity gradients as the fraction of Monte Carlo simulations with R2 smaller than the observed one. This correction of the velocity field was applied to those cluster in which the significance of velocity gradients is larger than 99% ( 30% of the total sample). However, this correction has small effects both in the shape of the velocity dispersion profiles and on the total velocity dispersion (the mean abso- lute correction was about 40 ± 15 km s−1). This is in agree- ment with similar corrections applied in other cluster samples (den Hartog & Katgert 1996; Girardi et al. 1996). 3.3. Comparison with other methods Some of the clusters presented in our sample have been previ- ously studied by other authors. However, we have avoided com- paring our results with those from the literature given the differ- ent datasets used. In order to compare our cluster membership method with others proposed in the literature, we have com- puted σc of our clusters with two more methods: a 3σ-clipping and the method proposed by Fadda et al. (1996). The median absolute difference between our σc and those computed by the 3σ-clipping method is only 17 km s−1. Only 10% of the clus- ters show important diferences (∆σc > 200 km s −1) in the com- putation of the velocity dispersion of the cluster with the two methods. They correspond to those clusters affected by large amount of structure along the line of sight. The 3σ-clipping method gives for these clusters considerably larger values of σc than ours. Differences were larger when we compared with Fadda’s method. In this case the mean absolute difference in σc between the two methods was 84 km s −1 and 80% of the clusters show differences smaller than 200 km s−1. Recently, Popesso et al. (2006) have obtained the values of σc for a sample of Abell clusters using SDSS-DR4 data, for which cluster membership was obtained using the selection al- gorithm of Katgert et al. (2004). The median absolute differ- ence between our and their σc is 45 km s −1 for the 28 clusters in common. Only for 4 clusters (Abell 1750, Abell 1773, Abell 2244 and Abell 2255) the absolute differences in σc is larger than 200 km s−1. We have also compared our results with those given in the cluster catalogue presented by Miller et al. (2005). We found 16 clusters in common, being 74 km s−1 the median absolute dif- 6 Aguerri et al.: Global Properties of Nearby Galaxy Clusters Fig. 3. Galaxy surface density (images) and radial velocity versus distance to the cluster center for the galaxy cluster member (red points) of a subsample of 8 clusters. The overplotted circle have a radius equal to r200 for each cluster. The black points represent interloper galaxies. ference between our and their σc. In this case, 3 clusters show an absolute differences in σc larger than 200 km s We can conclude that in most of the cases our cluster mem- bership method reported values of σc similar to those given by other methods. Only for 10-20% of the clusters the absolute differences in σc between our method and the others is larger than 200 km s−1. For these clusters the structure along the line of sight is the responsible of the difference, being our σc values smaller than the others. 3.4. Lx-σ relation We can learn about the nature of cluster assembly by studying the relations between cluster observables. One of the most uni- versals is the well known relation between the cluster X-ray lu- minosity and the velocity dispersion of its galaxies (LX ∝ σbc). Cluster formation models predict that if the only energy source in the cluster comes from the gravitational collapse, then b ≈ 4. This relation has been studied in the literature by many authors using different cluster samples, finding values of b between 2.9 and 5.3 (Edge & Stewart 1991; Quintana & Melnick 1982; Mulchaey & Zabludoff 1998; Mahdavi & Geller 2001; Girardi & Mezzetti 2001; Borgani et al. 1999; Xue & Wu 2000; Ortiz-Gil et al. 2004; Hilton et al. 2005). The study of the LX − σc relation in our cluster sample will be also useful as another check for the values of σc we have derived. We have X-ray data for 48 galaxy clusters from our sample. The X-ray data have been obtained from Ebeling et al. (1998), Böhringer et al.(2000), Ebeling et al.(2000) and Ledlow et al.(2003), and the X-ray luminosities are measured in the ROSAT band (0.1-2.4 keV). Figure 4 shows the LX−σ relation for this subset with avail- able X-ray data in the literature. The Spearman coefficient of the relation is 0.56 and the significance from zero correlation is greater than 3σ. This indicates the existence of a correlation between LX and σc for the clusters of our sample. We used the bivariate correlated errors and intrinsic scatter (BCES) bisector method of Akritas & Bershady (1996) to obtain the coefficient and power-law slope estimates of the relation. This fitting tech- nique takes into account errors in both variables and intrinsic scatter. The LX − σc relation for our clusters is given by: LX(0.1 − 2.4 keV) = 1033.7±1.2σ3.9±0.4 (3) Aguerri et al.: Global Properties of Nearby Galaxy Clusters 7 Fig. 4. LX−σ relation for the 48 galaxy clusters with X-ray data in the ROSAT band (0.1-2.4 keV) from our sample. The full line represents the best fit using the BCES bisector algorithm (see text for more details). Fig. 5. Absolute r-band magnitude as a function of redshift for the galaxies of our cluster sample. This result is in very good agreement with another mea- surement of this relation using the same ROSAT band (0.1-2.4 keV) for the X-ray data and the same fitting algorithm (see Hilton et al. 2005). 3.5. Redshift distribution and sample completeness The 88 isolated galaxy clusters are located in a redshift range between 0.02 and 0.1, with an average redshift of 0.071. Figure 5 shows the absolute r-band magnitude (Mr) as a function of the redshift for the galaxies in our cluster sample4. It is clear that the completeness magnitude is a function of redshift. This figure shows that the full sample is complete for galaxies brighter than Mr = −20.0. The lack of completeness for fainter galaxies will be taken into account in the subsequent analysis. 4 See section 3 for the explanation of the computation of the abso- lute magnitudes of the galaxies. 4. Morphological Segregation Light concentration or colours have been used extensively in the literature in order to classify galaxies. Shimasaku et al. (2001) and Strateva et al. (2001) using SDSS data, found that the ratio of Petrosian 50 percent light radius to Petrosian 90 percent light radius, Cin, measured in the r-band image was a useful index for quantifying galaxy morphology. Strateva et al. (2001) also found that the colour u − r = -2.22 efficiently sep- arates early- and late-type galaxies at z < 0.4. We have used colours for classifying galaxies, because properties such as ve- locity dispersion in galaxy clusters are better correlated with galaxy colours than galaxy morphology (Goto 2005). The mag- nitude of the galaxies were corrected by two effects: Galactic absorption and k−correction. The Galactic absorption in the different filters was obtained from the dust maps of Schlegel et al. (1998). We applied the k−correction using the kcor- rect.v4 1 4 code by Blanton et al. (2003) in order to obtain the rest-frame magnitudes of the galaxies for the different band- passes. Once these two corrections were done, we classified the galaxies in red (u − r ≥ 2.22) and blue ones (u − r < 2.22). The galaxy data was downloaded from the SDSS database according to a metric criteria: we downloaded the information of all galaxies located within a radius of 4.5 Mpc at each galaxy cluster redshift. This means that we are mapping different phys- ical regions for each cluster. In order to avoid this problem we have studied the ratio rmax/r200 for each cluster, being rmax the maximum distance of a galaxy from the cluster centre for each galaxy cluster. We have obtained that all clusters of our sample reach rmaxr200 = 2, and 50% of them reach Our sample of galaxies consists of 6880 galaxies located within a radius 2 × r200, being 62% of them red galaxies and 38% blue ones. If we consider all galaxies within 5 × r200 then the sample has 10865 galaxies, being 55% and 45% red and blue galaxies, respectively. The red and blue galaxies were also grouped in three categories according to their r-band magni- tude: Mr < M r − 1, M r − 1 < Mr < M r + 1, and Mr > M r + 1 The first group contains the brightest members of the clusters, the third group contains the so-called dwarf population and the second one is formed by normal bright galaxies. Table 2 shows the median location, r-band absolute magnitude, velocity dis- persion and local density6 of the different galaxy groups. In general, red galaxies are brighter than blue ones, and are also located closer to the cluster centre at higher local density re- gions. The two families of galaxies present different kinemat- ics, in the sense that red galaxies show a smaller velocity dis- persion than blue ones. This different kinematic between red and blue galaxies has also been seen in other studies, and have been interpreted as red and blue galaxies having different kind of orbits, being the orbits of blue galaxies more anisotropic than the red ones (Adami et al. 1998; Biviano & Katgert 2004). Other authors interpret this difference in velocity dispersion as an evidence that ram pressure is not playing an important role in galaxy evolution in clusters. In contrast, tidal interactions 5 M∗r − 5log(h) = −20.04, Blanton et al. (2005) 6 The local surface density (Σ) was computed with the 10 nearest neighbours to each galaxy belonging to the cluster. 8 Aguerri et al.: Global Properties of Nearby Galaxy Clusters should be the dominant mechanism (Goto 2005). All of these properties are independent of the sampled area. It is also interesting that red dwarf galaxies are located at similar environments as the brightest red ones: close to the cluster centre in high local density regions (see also Hogg et al. 2004). But the red dwarf population shows a larger velocity dispersion than the brightest red galaxies. Biviano & Katgert (2004) found that the brightest cluster members were not in equilibrium with the cluster potential. They are especial galaxies that could have formed close to the cluster centre or have fallen to this region due to dynamical friction. In contrast, dynamical friction is not so efficient in the dwarf population, so that the main presence of these galaxies in the central regions of the clusters should be due to their origin. The discussion about the properties and origin of the dwarf population will be given in another paper (Sánchez-Janssen et al., in preparation). 5. Velocity Dispersion Profiles The adopted cluster velocity dispersion was calculated with the galaxies located within the r200 radius of each cluster. But, how does σ depend on the clustercentric distance in our sample?. This can be answered by studying the integrated velocity dis- persion profiles (VDPs) of the clusters. These profiles also pro- vide information about the dynamical properties of the galax- ies. Thus, a system with galaxies predominantly in radial or- bits produces an outwards declining VDP, while the opposite behaviour suggests instead that the galactic orbits are largely circular. In contrast, constant VDPs are characteristic of an isotropic distribution of velocities (Solanes et al. 2001). Figure 6 shows the VDPs for some of the clusters in our sample. They show the velocity dispersion of the cluster at a given radius evaluated using all the galaxies within that radius, without any restriction in their luminosities. The errors showed in Fig. 6 were computed using the approximation given by Danese et al. (1980). In order to classify the VDPs of our clusters, we computed the velocity dispersion (σi, i = 1, 2, 3, 4, 5) of the galaxies in the clusters located within five different radius: 0.4×r200, 0.6×r200, 2×r200, 3×r200 and 4×r200, respectively. We compared these values with σc, given in Table 1. The resulting mean ratios σi/σc were: 1.02± 0.04, 1.01±0.01, 0.97±0.01, 0.94±0.02 and 0.94±0.02, for i = 1, 2, 3, 4, 5, respectively. These values indi- cate that within r200 the VDPs of the total galaxy cluster popu- lation are consistent with being flat. The mean variation of the VDPs inside r200 is only 2%. The values of σi/σc, i = 3, 4, 5 show that, outside r200 the VDPs slowly decrease. The mean variation of the VDPs outside r200 is −6%. No differences in the ratios σi/σc have been found when we have divided the galaxy sample between bright (Mr < M r +1) and dwarf (Mr > M r +1) galaxies. This flat behaviour of the VDPs inside r200 suggests that galaxies in these areas have an isotropic distribution of ve- locities. In contrast, the decline with radius of VDPs outside r200 points to radial orbits (Solanes et al. 2001). Figure 6 also shows the VDPs of early- (red) and late-type (blue) galaxies. In most profiles the velocity dispersion of blue galaxies is larger than the corresponding one for early-type ones. We have also analysed the shape of VDPs of blue and Fig. 7. Histograms of the ratios σi/σc, i = 1, 2, 3, 4, 5 for the galaxies in the clusters. The black full line represent all galax- ies, the blue and red lines correspond to late- and early-type ones. See text for more details. red galaxies as we did for the total sample. For red galaxies, we obtained that σi/σc,r are 1.04± 0.03, 1.03±0.03, 0.97±0.02, 0.96±0.02 and 0.96±0.03, for i = 1, 2, 3, 4, 5, respectively. The values of σi/σc,b for the blue galaxies are: 1.15± 0.07, 1.04±0.03, 0.95±0.04 and 0.93±0.04 and 0.92±0.04, respec- tively. In those computations, σc,r and σc,b represent the ve- locity dispersion of the red and blue galaxies within a radius equal to r200, respectively. Figure 7 show the distribution of σi/σc, i = 1, 2, 3, 4, 5 for the blue, red and the total galaxy sam- The VDPs have been studied in the literature by sev- eral authors. Most of them conclude that for large radii (r > 1 Mpc) the VDPs are flat (Girardi & Mezzetti 2001; Rines & Diaferio 2006; Fadda et al. 1996; Muriel et al. 2002). This is consistent with the mild decrease that we have found in our clusters. The VDPs for red galaxies in our sample are almost flat outside r200. This is not the case of the VDPs of blue galaxies which clearly decrease with radius outside r200. In the inner regions (r < r200) the VDPs of the total sample and those corresponding to the red galaxies are flat. In con- trast, the VDPs of blue galaxies decrease with radius. Different authors show that VDPs can decrease or increase with radius. den Hartog & Katgert (1996) made a thorough study and found that the variations of the VDPs in the innermost regions of clus- ters (r < 0.5 Mpc) are real and not due to noise or bad centre election. We have re-computedσ1/σc andσ2/σc only for those clusters with X-ray centres, and our results did not significantly change. Thus, we can conclude that in our galaxy cluster sam- ple only blue galaxies show increasing VDPs towards the cen- tre of the cluster, while red galaxies show flat VDPs. Aguerri et al.: Global Properties of Nearby Galaxy Clusters 9 Table 1. Main properties of the different types of galaxies Galaxies within r/r200 < 5 < r/r200 > < Mr > < σ > < log(Σ) > Ngal u − r < 2.22 1.85±0.02 -19.65±0.01 1.04±0.01 0.46±0.08 4937 u − r < 2.22 & Mr < M∗r − 1 1.79±0.11 -21.54±0.02 1.08±0.09 0.40±0.25 94 u − r < 2.22 & M∗r − 1 < Mr < M∗r + 1 1.90±0.02 -20.00±0.01 1.03±0.01 0.38±0.08 3126 u − r < 2.22 & Mr > M∗r + 1 1.74±0.03 -18.74±0.02 1.05±0.02 0.64±0.14 1717 u − r ≥ 2.22 1.03±0.01 -20.16±0.01 0.90±0.01 0.80±0.05 5928 u − r ≥ 2.22 & Mr < M∗r − 1 0.95±0.05 -21.62±0.02 0.78±0.03 0.89±0.14 537 u − r ≥ 2.22 & M∗r − 1 < Mr < M r + 1 1.10±0.02 -20.20±0.01 0.91±0.01 0.75±0.06 4592 u − r ≥ 2.22 & Mr > M∗r + 1 0.85±0.04 -18.91±0.02 0.90±0.03 1.06±0.12 729 Galaxies within r/r200 < 2 < r/r200 > < Mr > < σ > < log(Σ) > Ngal u − r < 2.22 0.97±0.01 -19.61±0.02 1.08±0.02 0.80±0.07 2636 u − r < 2.22 & Mr < M∗r − 1 1.15±0.07 -21.56±0.03 1.18±0.13 0.62±0.24 54 u − r < 2.22 & M∗r − 1 < Mr < M∗r + 1 1.04±0.01 -19.96±0.01 1.08±0.02 0.70±0.07 1648 u − r < 2.22 & Mr > M∗r + 1 0.85±0.01 -18.70±0.02 1.07±0.03 1.02±0.09 934 u − r ≥ 2.22 0.67±0.01 -20.14±0.01 0.91±0.01 0.98±0.04 4244 u − r ≥ 2.22 & Mr < M∗r − 1 0.57±0.03 -21.65±0.02 0.80±0.03 1.02±0.13 397 u − r ≥ 2.22 & M∗r − 1 < Mr < M r + 1 0.69±0.01 -20.19±0.01 0.92±0.01 0.94±0.05 3239 u − r ≥ 2.22 & Mr > M∗r + 1 0.61±0.02 -18.92±0.02 0.89±0.03 1.19±0.10 608 The previous findings can also be seen in Fig 8. We show the VDPs of the different galaxy classes for the normalised cluster, which was obtained by normalising the scales and ve- locities of each galaxy of the sample. Thus, the radial distance of each galaxy to the cluster centre was scaled by r200 of the corresponding cluster, and the relative velocity of each galaxy cluster was normalised by the velocity dispersion of the clus- ter. Figure 8 shows the VDPs which correspond to the total, bright (Mr < M r + 1) and dwarf (Mr > M r + 1) galaxy sam- ples. We have also distinguished between red and blue objects. The VDPs of the total galaxy sample indicate that blue galax- ies have always larger velocity dispersion than red ones. They also show always decreasing VDPs, while red ones have almost constant and slowly decrease VDPs inside and outside r200, re- spectively. These features can also be seen in the VDPs of the bright galaxy sample. In contrast, red and blue dwarfs show decreasing VDPs inside r200. The shape of the VDPs can provide information about the dynamical state of the galaxies. Thus, clusters with galaxies predominantly in radial orbits produce an outwards declining VDP. This is the case of the blue galaxies of our sample, which is in agreement with previous findings (Biviano & Katgert 2004; Adami et al. 1998). We have also obtained that the red dwarf galaxies inside r200 has an outwards declining VDP. This would imply that this kind of galaxies may also be located in radial orbits. In contrast, constant VDPs im- ply an isotropic distribution of velocities (Solanes et al. 2001). This is the case of the red bright galaxy population inside r200. 6. Fraction of blue galaxies Butcher & Oemler (1984) observed that the fraction of blue galaxies ( fb) in clusters evolves with redshift, in the sense that galaxy clusters located at medium redshift have a larger fb than nearby ones. This has been usually interpreted as an evolution- ary trend in clusters. But it is a matter of debate which is the role played by the environment in the change of the fraction of blue galaxies. We have computed fb in our sample of galaxy clusters, studying the variation with z and the possible influence of the environment. 6.1. Adopted aperture and limiting magnitude The original analysis of Butcher & Oemler (1984) defined blue galaxies as those within a radius containing 30 per cent of the cluster population, being brighter than Mv = −20 and bluer by 0.2 mag in B − V than the colour-magnitude relation defined by the cluster early-type galaxies. It has been noticed by sev- eral authors that the fraction of blue galaxies strongly depends on the magnitude limit and the clustercentric distance used (Ellingson et al. 2001; Goto et al. 2003; De Propris et al. 2004; Andreon et al. 2006). They observed that fb grows when the magnitude limit is fainter and the aperture is larger, reflect- ing the existence of a large fraction of blue faint galaxies in the outer regions of the clusters. De Propris et al. (2004) con- sidered appropriate to measure fb in apertures based on clus- ter physical properties. They used r200 as the aperture radius where they measured fb for their clusters. We have adopted also this radius in order to determine fb in our galaxy clus- ters. As it was previously commented, fb depends also on the adopted limiting magnitude of the galaxies in clusters. It should be noticed that as we move to higher redshifts we systematically lose faint galaxies (see Fig 2). Our clusters spread in a redshift range 0.02 < z < 0.1, and only galax- ies brighter than Mr = −20.0 (≈ M∗r + 0.5) can be observed at all redshifts. For this reason, we have adopted this abso- lute magnitude as the limiting magnitude for the computa- tion of fb. This ensures us to work with a complete galaxy sample at all redshifts. Other authors adopted fainter limiting magnitudes, e.g. M∗ + 1.5 (De Propris et al. 2004) or M∗ + 3 (Margoniner & de Carvalho 2000). If there is a large number of blue galaxies at faint magnitudes, we expect that our val- ues of fb will be smaller than those reported by the previous authors. 10 Aguerri et al.: Global Properties of Nearby Galaxy Clusters Fig. 6. Velocity dispersion profiles of some clusters of our sample. The black symbols represent the velocity dispersion profile taking into account all types of galaxies. Blue and red symbols represent the velocity dispersion profiles corresponding to blue and red galaxies, respectively. 6.2. Colour-magnitude diagrams We determined the g − r versus r colour-magnitude diagrams for all the clusters in our sample. The colour-magnitude rela- tion was measured by a robust fitting routine by minimising the absolute deviation in g−r colour, using only early-type galaxies located within an aperture of radius equal to r200. The galaxy types were determined according to the u−r colour and the light galaxy concentration parameter, Cin. These two criteria allow us to identify the most reliable sample of E/S0 galaxies (see Shimasaku et al. 2001; Strateva et al. 2001). Thus, we consid- ered early-type galaxies those with u − r ≥ 2.22 and Cin < 0.4. Figure 9 shows the colour-magnitude diagrams of four repre- sentative galaxy clusters. The colour-magnitude relation fitted in each case is also overploted. Figure 9 (left column) also shows the histograms of the colour distribution, marginalised over the fitted colour-magnitude relation. The average of the slopes of the colour-magnitude relations of the early-type galaxies of the clusters is -0.014±0.008. This slope is within the errors in agreement with the slope obtained by Gallazzi et al. (2006) for a large sample of galaxies using SDSS data. It is also in agreement with the average B−R slope obtained by De Propris et al. (2004) for a sample of galaxy clusters from 2dFGRS. 6.3. Calculation the blue fraction of galaxies As we explained before, the blue fraction of galaxies was com- puted using only those galaxies brighter than Mr = −20 and located within an aperture of radius r200. In the present study we only used spectroscopically confirmed galaxy cluster mem- bers. This should not bias our results, especially due to our high completeness. Figure 10 presents fb as a function of redshift. The errors of fb were computed according to the prescription given by De Propris et al. (2004). We observe no evolution of fb with redshift, which means that our sample is ideal to study the effects of the environment on fb. We have considered three cluster properties (concentration, velocity dispersion and X-ray luminosity) of each cluster in or- der to analyse the dependence of fb on the environment. The concentration parameter was computed following the prescrip- tion of De Propris et al. (2004), i.e. C = log(r60/r20), where r60 and r20 are the radii containing 60 and 20 per cent of the cluster galaxies, respectively. The velocity dispersion of the clusters Aguerri et al.: Global Properties of Nearby Galaxy Clusters 11 Fig. 8. Velocity dispersion profiles of the galaxies of the normalised cluster. The total galaxy population is showed in the top panel. Bright galaxies (Mr < M r + 1) are in the middle panel, and the bottom panel shows the VDPs corresponding to the dwarf galaxy sample (Mr > M r + 1). The VDP of the total, blue and red galaxy samples are represented by black, blue and red colours, respectively (see text for more details). Fig. 10. The fraction of blue galaxies ( fb) as a function of red- shift of the galaxy clusters. was taken from Table 1. The X-ray luminosities were obtained from the literature (Ebeling et al. 1998, Böhringer et al.2000, Ebeling et al. 2000 and Ledlow et al. 2003), being measured in the ROSAT band (0.1-2.4 keV). We only found X-ray data for 48 clusters of the sample. Figure 11 shows the dependence of the fraction of blue galaxies on concentration, cluster velocity dispersion and X- ray luminosity. The non-parametric Spearman test returns that fb has a low correlation with concentration and velocity dis- persion. The fraction of blue galaxies correlates best with the velocity dispersion, but the significance of the correlation is 2.6σ. In contrast, the Spearman test shows correlation between fb and X-ray luminosity, being the significance of this correla- tion just 3σ. Notice that the points are distributed in the LX − fb plane following a triangular shape. Clusters with large X-ray luminosity (LX(0.1 − 2.4keV) > 1045ergs−1) show small frac- tions of blue galaxies (less than 10%). Nevertheless, those clus- ters with small X-ray lumisosity show small and large fraction of blue galaxies. This correlation could indicate that there is a threshold over which cluster environment can affect the galaxy colours, and play a role in the galaxy evolution. This means that, according with our LX −σ relation, the evolution of galax- ies could be driven by the cluster environment for those clusters 12 Aguerri et al.: Global Properties of Nearby Galaxy Clusters Fig. 9. Colour-magnitude relation (left) and histograms (right) of marginalised colour distribution for four representative clusters at different redshifts of our cluster sample. The full line in left panels represent the fitted colour-magnitude relation. The vertical point lines in right panel represent the blue/red separation in the Butcher-Oemler effect. The red points are the galaxies with u − r ≥ 2.22 and Cin < 0.4 (see text for more details). with velocity dispersion larger than σ ≈ 800 km s−1. Recently, (Popesso et al. 2006) found a similar correlation between LX and fb for a larger cluster sample. The shape of the our LX − fb correlation is similar to the correlation between cluster veloc- ity dispersion and the fraction of [OII] emitters for clusters at low redshift reported by Poggianti et al. (2006). They found that clusters with σ > 550 km s−1 have a constant low fraction (less than 30%) of [OII] emiters. In contrast, those clusters with smaller σ show large and small fractions of [OII] emiters. We have recomputed fb taking into account those galax- ies within an aperture of radius equal to r200 and brighter than Mr = −19.5. We restricted the analysis only to those clus- ters with z < 0.05, because our sample is complete down to Mr = −19.5 in this redshift range. In this case the number of cluster decreases to 13. We have again studied the correlations of fb with galaxy concentration, velocity dispersion and X-ray luminosity, obtaining similar correlations as with the full sam- 7. Discussion From the study presented in this paper, most of the galaxies (62%) located in the central regions of galaxy clusters (r/r200 < 2) are early-type galaxies (see section 4). In contrast, the field population is dominated by late-type galaxies. In the literature it is also well established that the colour of galaxies in clus- ters and field is different, an indication of the low star forma- tion activity found in cluster galaxies (e.g. Balogh et al. 1998; Lewis et al. 2002; Gómez et al. 2003). These differences in morphology and stellar content between field and cluster galax- ies suggest different evolutionary processes. The facts that late- type galaxies show larger velocity dispersions and are located at larger distances from the cluster centre than early-type ones have been interpreted as late-type galaxies being recent ar- rivals to the cluster potential, forming a non-relaxed group of galaxies moving in more radial orbits than early-type ones (e.g. Stein 1997; Adami et al. 1998). As late-type galaxies fall into the cluster potential and encounter denser environments, they evolve to early-type ones. The results presented in the present work are in agreement with previous findings. However, as pointed out by Goto (2005), this would imply that a large frac- tion of galaxies (≈ 40% according to our sample) should be recent arrivals to the cluster, a possibility that seems unlikely. Goto (2005) concluded that the different observational proper- ties between red and blue galaxies may indicate which is the main mechanism driving the evolution of galaxies in clusters. Gas stripping, mergers and interactions with other galaxies and with the cluster potential are the main mechanisms which are able to transform galaxies in clusters, making late-type galax- ies lose their gas content, stop their star formation, circularise their orbits and transform their morphology from disk-like ob- jects to spheroids. All of these mechanisms affect galaxies in Aguerri et al.: Global Properties of Nearby Galaxy Clusters 13 Fig. 11. The fraction of blue galaxies ( fb) as a function of galaxy distribution (top), cluster velocity dispersion (middle), and X-ray luminosity (bottom) of the galaxy clusters. clusters but, can we infer from the observational results which is the dominant one?. It should be noted that the different mechanisms of galaxy evolution have very different time-scales. While gas stripping has a very short time-scale (≈ 50 Myr, Quilis et al. 2000), the galaxy infall process can take ≈ 1 Gyr. The different mecha- nisms also have different underlying physics. Thus, ram pres- sure stripping is proportional to the density of the intracluster medium (ICM) and to the square of the velocity of the galaxy. In contrast, dynamical interactions are more efficient when the relative velocity of galaxies is smaller (Mamon 1992). This means that gas stripping is stronger in the cluster centres and for galaxies with high velocities, while dynamical interactions should be more efficient for galaxies with smaller velocity dis- persions. Numerical simulations have shown that most of the galaxies inside the virial radius have already been through the cluster core more than once (Mamon et al. 2004). If gas strip- ping were the main mechanism driving galaxy evolution in clusters, according to the short time scale of this process, only few blue (late-type) galaxies should be observed in the central regions of clusters. Moreover, gas stripping is also stronger in galaxies with larger velocity dispersion which means that late- type galaxies should be more affected by this mechanism than early-type ones. Based on these considerations, Goto (2005) concluded that gas stripping is not the main responsible mech- anism driving the evolution of galaxies in clusters. Instead, galaxies in clusters evolve mainly by dynamical interactions. We can add to Goto’s discussion that if gas stripping were the main galactic evolution mechanism in clusters, then the frac- tion of blue galaxies should depend on the cluster mass as the temperature and density of the gas increases with the cluster mass. According to our results, this is true for those clusters with large X-ray luminosities. In contrast, the cluster environ- ment is not so important in driving the evolution of galax- ies in low mass clusters. Thus, gas stripping may not be the main responsible mechanism transforming late-type to early- type galaxies in low mass clusters, but could be important in the most massive ones. This does not mean that gas stripping is absent in the evolution of galaxies in clusters; some clear ex- amples of gas stripping have been observed in galaxies in Virgo (Kenney et al. 2004). Dynamical interactions include both interactions with the cluster potential and with other galaxies. These effects can trig- ger temporary star formation in cluster galaxies (Fujita 1998), which can be analysed by studying their colour distribution. These interactions can also disrupt stars from galaxies, form- ing at the beginning long tidal tails that subsequently will be diluted and will form the diffuse light observed in some nearby clusters like Virgo (see Aguerri et al. 2005b, and ref- erences therein). These effects will be more important in those galaxies with smaller relative velocities. Fujita (1998) con- clude that if the tidal effects enhance the SFR in the galaxies, then the bluest galaxies should be located close to the clus- ter centre (within ≈ 300 kpc), whereas they should be in the outer parts of the cluster if the SFR is induced by galaxy- galaxy encounters. We have investigated the fraction of blue 14 Aguerri et al.: Global Properties of Nearby Galaxy Clusters galaxies in our clusters located within 300 kpc from the cen- tre of the cluster. The sample has been divided in bright and dwarf galaxies (Mr < M r + 1 and Mr > M r + 1, respec- tively). We have obtained that 40% of the blue bright galax- ies and 30% of the blue dwarf ones are located at smaller dis- tance than 300 kpc from the cluster center. This means that tidal interactions with the cluster potential are not the respon- sible mechanism for the formation of most of the blue galax- ies in our clusters. The lack of blue galaxies in the central re- gions of clusters has been observed also in nearby clusters like Coma (Aguerri et al. 2004) as well as in other distant clusters (Rakos et al. 1997; Abraham et al. 1996; Balogh et al. 1997). These evidences indicate that the evolution of galaxies in clusters could be driven by the cluster environment in the most massive ones, but galaxies in low mass clusters could mainly evolve due to the local environment. 8. Conclusions In the present paper we have analysed the main properties of the galaxies of one of the largest (10865 galaxies) and homo- geneous sample presented in the literature. The galaxies have been grouped in two families according to their u − r colour. Those galaxies with u − r ≥ 2.22 formed the red (early-type) family, and those with u − r < 2.22 the blue (late-type) one. We have derived the position, velocity dispersion, and VDPs of both families of galaxies, obtaining: – Within 2×r200, 62% and 38% of the galaxies turned to be red and blue, respectively. – The median positions and velocity dispersions are smaller for red galaxies than for blue ones. – Bright (Mr < M∗r −1) and dwarf (Mr > M r +1) red galaxies are located at smaller distances than the blue ones, sharing the same cluster environment. – The brightest cluster members (Mr < −21.0) show smaller velocity dispersions than the remaining. – The VDPs of the total galaxy cluster population are con- stant with radius in the central regions of the clusters (r < r200), while slowly decrease in the outermost regions (r ≥ r200). The red galaxy population have also flat VDPs in the central regions (r < r200). In contrast, the VDPs of blue galaxies grow towards the cluster centre. In the outer re- gions (r > r200), the VDPs of red galaxies decline smoothly with radius, while for blue ones the decrement is faster. This indicates that the galaxies in the outermost regions of the clusters are dominated by the blue population, and have more radial and anisotropic orbits than galaxies in the inner regions dominated by the red population. – The fraction of blue galaxies in our cluster sample does not correlate with cluster global properties, such as the concen- tration of the galaxy distribution and cluster velocity disper- sion. However, we found a correlation between the X-ray luminosity and the fraction of blue galaxies. Those clusters with LX(0.1 − 2.4keV) > 1045 erg s−1 have a low fraction of blue galaxies (less than 10%). In contrast, clusters with low of X-ray luminosity show large and small fractions of blue galaxies. This could indicate that the star formation in cluster galaxies may be regulated by global cluster proper- ties for clusters with LX(0.1 − 2.4keV) > 1045 erg s−1, i.e. those clusters with σc > 800 km s All these results are in agreement with previous findings from other cluster samples, indicating that red and blue galax- ies have different evolution in galaxy clusters. We have dis- cussed these results according to the different galaxy transfor- mation mechanisms presented in galaxy clusters, concluding the local environment plays a key role in galaxy evolution in low mass clusters, while the evolution of galaxies in massive clusters could be driven by the global cluster environment. Acknowledgements. We wish to thank to the anonymous referee for useful coments which have improved this manuscript. We also acknowledge financial support by the Spanish Ministerio de Ciencia y Tecnologı́a grants AYA2004-08260. We would like also to thank T. Beers for providing us with copy of his code ROSTAT, and K. M. Ashman and S. Zepf for making their KMM code available to us. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, Cambridge University, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck- Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington. 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Cluster characteristics Name α (J2000) δ (J2000) vc σc r200 Ngal C (degrees) (degrees) (km s−1) (km s−1) (Mpc) Abell0085 10.4571 -9.30694 16633+40−29 979 −39 2.10 273 0.93 Abell0117 14.0100 -10.0022 16568+31−42 531 −27 1.19 60 0.88 Abell0152 17.5229 13.9804 17888+67−34 538 −38 1.12 27 0.85 Abell0168 18.7429 0.365833 13534+23−14 578 −28 1.19 106 0.88 Abell0257 27.3396 14.0372 21060+47−21 381 −44 0.81 26 0.94 Abell0602 118.341 29.3717 18202+34−51 834 −61 1.87 78 0.74 Abell0628 122.543 35.2958 25139+24−89 666 −38 1.47 43 0.83 Abell0671 127.121 30.4169 14599+19−33 610 −33 1.42 72 0.89 Abell0690 129.815 28.9033 23689+44−23 395 −24 0.85 22 0.95 Abell0695 130.309 32.4174 20251+46−37 456 −32 1.04 16 0.86 Abell0699 131.236 27.7508 25375+35−49 438 −37 0.91 19 0.73 Abell0724 134.600 38.5137 28134+25−55 433 −32 1.00 29 0.94 Abell0727 134.976 39.4389 28571+54−14 423 −29 0.96 33 0.96 Abell0757 138.277 47.7036 15402+27−36 409 −30 0.84 30 0.85 Abell0779 139.962 33.7714 6921+13−33 336 −21 0.79 57 0.75 Abell0819 143.076 9.68861 22872+39−50 536 −37 1.19 31 0.94 Abell0883 147.822 5.48799 21750+109−30 523 −58 1.17 18 0.91 Abell0971 154.997 40.9925 27809+54−43 816 −61 1.88 40 0.87 Abell0999 155.842 12.8466 9618+10−54 271 −17 0.60 25 0.90 Abell1003 156.235 47.8442 18762+44−84 617 −34 1.37 29 0.94 Abell1016 156.762 10.9780 9629+34−9 259 −17 0.60 25 0.91 Abell1024 157.096 3.76341 22067+40−20 532 −34 1.26 35 0.90 Abell1032 157.547 4.03417 20008+26−24 355 −32 0.77 25 0.89 Abell1035 158.092 40.1817 20270+36−34 575 −45 1.34 49 0.97 Abell1066 159.911 5.17444 20708+4−81 826 −44 1.71 95 0.92 Abell1142 165.229 10.5477 10601+30−25 557 −38 1.33 59 0.88 Abell1149 165.769 7.57833 21479+2−64 352 −31 0.85 26 0.94 Abell1169 166.967 43.9506 17532+24−35 433 −32 0.91 35 0.94 Abell1173 167.328 41.5624 22789+23−69 611 −41 1.33 35 0.95 Abell1189 167.775 1.09899 28860+109−60 807 −88 1.76 41 0.94 Abell1190 167.902 40.8417 22610+17−39 706 −30 1.50 77 0.92 Abell1205 168.328 2.53867 22852+17−53 890 −53 2.04 83 0.84 Abell1215 170.100 4.34280 14747+25−10 214 −39 0.48 17 0.89 Abell1238 170.711 1.09389 22140+9−54 544 −41 1.30 60 0.91 Abell1270 172.366 54.0428 20728+49−36 569 −39 1.31 43 0.99 Abell1291 173.092 55.9783 17144+43−60 720 −36 1.58 45 0.97 Abell1318 173.883 55.0767 17185+54−22 360 −25 0.81 22 0.96 Abell1346 175.304 5.74613 29523+41−25 790 −62 1.69 66 0.85 Abell1377 176.883 55.7597 15378+35−33 671 −37 1.47 69 0.96 Abell1383 176.973 54.7089 17855+38−35 456 −23 1.02 35 0.95 Abell1385 177.017 11.5864 25337+49−44 609 −45 1.34 22 0.86 Abell1390 177.378 12.3034 25101+48−29 483 −41 1.06 27 0.86 Abell1424 179.361 5.12000 22736+33−39 617 −43 1.46 63 0.95 Abell1436 180.095 56.2314 19432+39−77 712 −44 1.51 66 0.96 Abell1452 180.802 51.6642 18609+56−58 533 −30 1.30 18 0.91 Abell1459 181.108 1.88281 6010+37−21 527 −48 1.18 65 0.95 Abell1507 183.766 59.8947 18009+9−70 379 −36 0.85 23 0.91 Abell1516 184.729 5.24731 23019+24−84 720 −43 1.49 60 0.90 Abell1552 187.392 11.7733 26495+64−42 442 −32 0.93 20 0.88 Abell1564 188.720 1.78056 23763+30−64 641 −62 1.51 46 0.91 Abell1616 191.817 55.0006 24882+65−93 565 −45 1.32 29 0.87 Abell1620 192.510 -1.53764 25400+64−38 829 −43 1.76 58 0.88 Abell1630 192.942 4.59694 19458+36−33 444 −37 0.98 30 0.90 18 Aguerri et al.: Global Properties of Nearby Galaxy Clusters Table 2. continued. Name α (J2000) δ (J2000) vc σc r200 Ngal C (degrees) (degrees) (km s−1) (km s−1) (Mpc) Abell1650 194.672 -1.76417 25138+86−18 790 −47 1.61 63 0.85 Abell1663 195.717 -2.51782 24953+60−20 729 −40 1.54 72 0.88 Abell1692 198.060 -0.976000 25395+51−47 607 −43 1.32 40 0.89 Abell1728 200.876 11.2960 26977+95−22 824 −62 1.88 50 0.70 Abell1750 202.709 -1.86389 26259+19−14 518 −33 1.15 35 0.95 Abell1767 204.024 59.2042 21174+39−32 885 −40 2.05 109 0.94 Abell1773 205.533 2.24805 23544+31−48 481 −31 1.09 32 0.87 Abell1780 206.149 2.86750 23285+35−22 624 −53 1.45 53 0.90 Abell1783 205.848 55.6261 20550+48−11 383 −32 0.94 33 0.94 Abell1809 208.245 5.16139 23815+39−31 737 −47 1.68 89 0.82 Abell1885 213.431 43.6634 26793+33−50 541 −56 1.23 22 0.92 Abell1999 223.522 54.2682 29841+11−57 463 −54 1.07 24 0.92 Abell2018 225.266 47.2831 26246+93−5 635 −37 1.45 39 0.88 Abell2023 227.496 2.98910 27743+48−18 516 −73 1.12 23 0.86 Abell2026 227.106 -0.267500 27188+62−37 747 −49 1.50 43 0.91 Abell2030 227.844 -0.0857717 27399+38−27 495 −45 1.10 38 0.91 Abell2061 230.317 30.6122 23646+24−19 622 −32 1.43 98 0.83 Abell2067 230.780 30.8703 23039+33−34 917 −46 2.19 118 0.85 Abell2092 233.348 31.1475 20000+42−17 458 −30 0.93 41 0.86 Abell2110 234.953 30.7173 29250+94−28 622 −49 1.25 21 0.83 Abell2122 236.259 36.1161 19793+28−42 826 −47 1.80 91 0.92 Abell2124 236.263 36.1172 19783+41−32 826 −47 1.78 90 0.92 Abell2145 240.094 33.2306 26583+80−23 632 −56 1.40 24 0.87 Abell2149 240.350 53.9061 19564+63−44 459 −27 1.01 20 0.85 Abell2169 243.422 49.1261 17286+30−50 521 −33 1.24 40 0.82 Abell2175 245.132 29.8953 28876+61−64 878 −57 1.76 58 0.87 Abell2199 247.154 39.5244 9118+15−30 747 −19 1.77 269 0.92 Abell2241 254.928 32.6161 29403+70−68 806 −62 1.61 37 0.90 Abell2244 255.663 34.0411 28927+50−57 428 −49 0.99 23 0.94 Abell2245 255.640 33.5056 25686+40−49 535 −39 1.28 39 0.95 Abell2255 258.222 64.0653 24052+19−39 883 −35 1.86 184 0.91 Abell2428 334.065 -9.34139 25207+15−68 433 −37 0.95 26 0.87 Abell2670 358.556 -10.4133 22755+29−20 642 −29 1.46 137 0.94 MACSJ0810.3+4216 122.600 42.2733 19193+44−25 505 −35 1.23 32 0.84 MACSJ1440.0+3707 220.011 37.0839 29402+88−46 587 −49 1.31 18 0.91 NSCJ152902+524945 232.309 52.8433 22063+101−7 652 −43 1.4 45 0.89 NSCJ161123+365846 242.854 36.9700 20221+39−25 485 −39 1.17 30 0.86 RBS1385 215.969 40.2619 24544+40−78 419 −36 0.84 16 0.91 RXCJ0137.2-0912 24.3137 -9.20277 12169+27−30 453 −30 0.93 49 0.92 RXCJ0828.6+3025 127.162 30.4280 14630+50−25 628 −33 1.45 76 0.89 RXCJ0953.6+0142 148.393 1.70550 29450+24−63 584 −59 1.30 22 0.96 RXCJ1115.5+5426 168.887 54.4350 20965+35−66 639 −38 1.35 50 0.94 RXCJ1121.7+0249 170.428 2.81840 14807+25−14 567 −41 1.40 73 0.85 RXCJ1351.7+4622 207.940 46.3668 18937+32−40 531 −27 1.16 40 0.94 RXCJ1424.8+0240 216.159 2.75677 16337+44−57 539 −36 1.19 22 0.91 RXJ1017.7-0002 154.452 -0.0595327 19169+44−28 413 −39 0.90 16 0.91 RXJ1022.1+3830 155.583 38.5308 16300+37−38 591 −33 1.35 51 0.82 RXJ1053.7+5450 163.449 54.8500 21623+23−49 665 −45 1.52 46 0.86 WBL238 146.732 54.4183 13995+51−17 602 −30 1.30 44 0.87 WBL518 220.179 3.45305 8141+19−28 454 −20 1.03 103 0.85 ZwCl0027.0-0036 7.31721 -0.183598 17994+18−44 465 −36 0.99 36 0.85 ZwCl0743.5+3110 116.655 31.0136 17419+49−89 694 −45 1.40 29 0.82 ZwCl1207.5+0542 182.578 5.38500 23137+43−36 580 −40 1.20 40 0.91 Aguerri et al.: Global Properties of Nearby Galaxy Clusters 19 Table 2. continued. Name α (J2000) δ (J2000) vc σc r200 Ngal C (degrees) (degrees) (km s−1) (km s−1) (Mpc) ZwCl1215.1+0400 184.422 3.66040 23229+22−40 955 −39 2.17 130 0.90 ZwCl1316.4-0044 199.816 -0.907816 24972+56−31 557 −20 1.16 38 0.87 ZwCl1730.4+5829 261.856 58.4749 8379+26−36 491 −22 1.02 33 0.86 Introduction Galaxy cluster Sample Cluster Membership Cluster global parameters Corrections to line-of-sight velocities Comparison with other methods Lx- relation Redshift distribution and sample completeness Morphological Segregation Velocity Dispersion Profiles Fraction of blue galaxies Adopted aperture and limiting magnitude Colour-magnitude diagrams Calculation the blue fraction of galaxies Discussion Conclusions

We have selected a sample of 88 nearby (z<0.1) galaxy clusters from the SDSS-DR4 with redshift information for the cluster members. We have derived global properties for each cluster, such as their mean recessional velocity, velocity dispersion, and virial radii. Cluster galaxies have been grouped in two families according to their $u-r$ colours. The total sample consists of 10865 galaxies. As expected, the highest fraction of galaxies (62%) turned to be early-type (red) ones, being located at smaller distances from the cluster centre and showing lower velocity dispersions than late-type (blue) ones. The brightest cluster galaxies are located in the innermost regions and show the smallest velocity dispersions. Early-type galaxies also show constant velocity dispersion profiles inside the virial radius and a mild decline in the outermost regions. In contrast, late-type galaxies show always decreasing velocity dispersions profiles. No correlation has been found between the fraction of blue galaxies and cluster global properties, such as cluster velocity dispersion and galaxy concentration. In contrast, we found correlation between the X-ray luminosity and the fraction of blue galaxies. These results indicate that early- and late-type galaxies may have had different evolution. Thus, blue galaxies are located in more anisotropic and radial orbits than early-type ones. Their star formation seems to be independent of the cluster global properties in low mass clusters, but not for the most massive ones. These observational results suggest that the global environment could be important for driving the evolution of galaxies in the most massive cluster ($\sigma > 800$ km s$^{-1}$). However, the local environment could play a key role in galaxy evolution for low mass clusters.

Introduction The large amount of spectroscopic and photometric data ob- tained during the last years by surveys such as the Sloan Digital Sky Survey (SDSS; York et al. 2000) or the 2dF Galaxy Redshift Survey (2dFGRS; Colless et al. 2001) have opened a new horizon for the study of galaxy evolution, and in partic- ular in the study of galaxy clusters. It is well known that the environment plays an important role in the evolution of galax- ies, and it is one of the keys that a good galaxy evolution the- ory should address. There are several physical mechanisms, not present in the field, which can dramatically transform galaxies in high density environments. Galaxies in clusters can evolve due to, e.g., dynamical friction, which can slow down the more massive galaxies, circularise their orbits and enhance their merger rate (den Hartog & Katgert 1996; Mamon 1992). Send offprint requests to: J. A. L. Aguerri Interactions with other galaxies and with the cluster grav- itational potential can disrupt the outermost regions of the galaxies and produce galaxy morphological transformations from late- to early-types (Moore et al. 1996), or even change massive galaxies into dwarf ones (Mastropietro et al. 2005). Swept of cold gas produced by ram pressure stripping (Gunn & Gott 1972 ; Quilis et al. 2000) or swept of the hot gas reservoirs (Bekki et al. 2002) can alter the star formation rate (SFR) of galaxies in clusters. But it is still a matter of debate which of these mechanisms is the main responsible of the galaxy evolution in galaxy clusters (see Goto 2005). Nevertheless, it is clear that all of these mechanisms transform galaxies from late- to early-types, and can produce the different segregations observed in galaxy clusters. One of the first segregations discovered in galaxy clus- ters was the morphological one. The first evidences of such segregation date from Curtis (1918) and Hubble & Humason http://arxiv.org/abs/0704.1579v1 2 Aguerri et al.: Global Properties of Nearby Galaxy Clusters (1931), and was quantified by Oemler (1974) and Melnick & Sargent (1977). In a thorough work, Dressler (1980) analysed a sample of 55 nearby galaxy clusters, contain- ing over 6000 galaxies, and observed that elliptical and S0 galaxies represent the largest fraction of galaxies lo- cated in the innermost and denser regions of galaxy clus- ters. In contrast, the outskirts of the clusters were domi- nated by spiral galaxies. In more distant clusters the frac- tion of E galaxies is as large or larger than in low-redshift clusters, but the S0 fraction is smaller (Dressler et al. 1997; Fasano et al. 2000). This has been interpreted as an evolution with redshift, being late-type galaxies transformed into early- type ones. Segregations in velocity space have also been ob- served in galaxy clusters. Early observations found that E and S0 galaxies showed smaller velocity dispersions than spi- rals and irregulars (Tammann 1972; Melnick & Sargent 1977; Moss & Dickens 1977). This has also been confirmed by other authors during the last two decades (Sodre et al. 1989; Biviano et al. 1992; Andreon et al. 1996; Stein 1997). The data from the ENACS survey (Katgert et al. 1998) produced a large sample of galaxies with spectroscopic redshifts and shed more light to this problem. Thus, Adami et al. (1998) studied a sam- ple of 2000 galaxies, confirming early findings that the ve- locity dispersion of galaxies increases along the Hubble se- quence: E/S0 galaxies show smaller velocity dispersions than early- and late-type spirals. This segregation was also observed in the velocity dispersion profiles (VDPs): late-type galaxies have decreasing VDPs, while E, S0 and early spirals show al- most flat VDPs (Adami et al. 1998). The different kinematics shown by the different types of galaxies was analysed in more detail by Biviano & Katgert (2004) who found that the ve- locity segregation of the different Hubble types is due to dif- ferences in orbits. Thus, early-type spirals have isotropic or- bits, while late-type ones are located in more anisotropic or- bits. The observed morphological and velocity segregation in clusters have been usually used to conclude that late-type spi- ral galaxies in clusters are recent arrivals to the cluster potential (Stein 1997; Adami et al. 1998). Star formation in galaxies is also affected by the envi- ronment. Butcher & Oemler (1984) found that the fraction of blue galaxies, fb, in clusters is smaller than in the field and evolves with redshift: more distant clusters show larger values of fb. This trend was interpreted as an evolutionary effect of the SFR in galaxy clusters. But the significant increase of new data has made it clear that the Butcher-Oemler effect is not only an evolutionary trend. A large scatter in the values of fb has been observed in narrow redshift ranges (Smail et al. 1998; Margoniner & de Carvalho 2000; Goto et al. 2003), which suggests that the variation of fb is influenced by environmental effects. In the past, many authors have tried to find correlations of fb with cluster properties, such as X-ray luminosity (Andreon & Ettori 1999; Smail et al. 1998; Fairley et al. 2002), luminosity limit and clustercentric distance (Ellingson et al. 2001; Goto et al. 2003; De Propris et al. 2004), richness (Margoniner et al. 2001; De Propris et al. 2004), cluster con- centration (Butcher & Oemler 1984; De Propris et al. 2004), presence of substructure (Metevier et al. 2000) or cluster velocity dispersion (De Propris et al. 2004). Some of these works found correlations between fb and the cluster envi- ronment while others did not, being such connection still a matter of debate. However, these works were usually done using small and heterogeneous cluster samples (but see e.g., De Propris et al. 2004). Environmental effects have also been invoked in order to explain the differences between the photometrical compo- nents of cluster and field spiral galaxies. Thus, it has been observed that the scale-lengths of the disks of spiral galax- ies in the Coma cluster are smaller than those of similar galaxies in the field (Gutiérrez et al. 2004; Aguerri et al. 2004). Interactions between galaxies or with the cluster potential can disrupt the disks of spiral galaxies in clusters. They can be strong enough for transforming bright late-type spi- ral galaxies in dwarfs (Aguerri et al. 2005a). The disrupted material would be part of the intracluster light already de- tected in some nearby galaxy clusters (Arnaboldi et al. 2002; Arnaboldi et al. 2004 ; Aguerri et al. 2005b) and galaxy groups (Castro-Rodrı́guez et al. 2003; Aguerri et al. 2006). The observational results summarised before illustrate the important role played by environment in galaxy evolution. They also indicate that late-type and early-type galaxies in clusters are two different families of objects with differ- ent properties, which points to different origins or evolution. Nevertheless, the main mechanisms responsible of this differ- ent evolution still remain unknown. In the present paper, we study one of the largest and more homogeneous galaxy cluster sample available in the literature. We have obtained the cluster membership, mean velocity, velocity dispersion, virial radius and positions for a sample of 88 clusters located at z < 0.1. We have investigated the main properties of a large sample of early (red) and late (blue) types of galaxies, such as their lo- cation within the cluster, their mean velocity dispersion, their VDPs, the LX −σ relation, and the fraction of blue galaxies for each cluster. This work provides important information about the properties of galaxies in nearby clusters, which will be useful in order to put constraints on cosmological models of cluster formation. This is the first paper of a series in which we will analyse the properties of the dwarf galaxy population (Sánchez-Janssen et al. in preparation), substructure in galaxy clusters (Aguerri et al. in preparation), and composite luminos- ity function of galaxy clusters (Sánchez-Janssen et al., in prepa- ration). The paper is organised as follows. Section 2 shows the dis- cussion about the galaxy cluster sample. The cluster member- ship and cluster global parameters are presented in Section 3. The results obtained about the morphological segregation, ve- locity dispersion profiles, LX − σ relation, and the fraction of blue galaxies are given in Sections 4, 5, 6 and 7, respectively. The discussion and conclusions are presented in Sections 8 and 9, respectively. Throughout this work we have used the cos- mological parameters: Ho = 75 km s −1 Mpc−1, Ωm = 0.3 and ΩΛ = 0.7. Aguerri et al.: Global Properties of Nearby Galaxy Clusters 3 2. Galaxy cluster Sample We have used photometric and spectroscopic data of ob- jects classified as galaxies from the SDSS-DR4, an imag- ing and spectroscopic survey of a large area in the sky (York et al. 2000). The imaging survey was carried out through five broad-band filters, ugriz, spanning the range from 3000 to 10000 Å, reaching a limiting r-band mag- nitude ≈ 22.2 with 95% completeness, and covering an area of 6670 deg2 (Adelman-McCarthy et al. 2006). A se- ries of pipelines process the imaging data and perfom the astrometric calibration (Pier et al. 2003), the photometric re- duction (Lupton et al. 2002) and the photometric calibration (Hogg et al. 2001). Objects brighter than mr = 17.77 were se- lected as possible targets for the spectroscopic survey, covering an area of 4783 deg2 of the sky for the DR4. The spectroscopic data were obtained with optical fibers with a diameter of 3 the focal plane, resulting in an spectral covering in the wave- length range 3800–9200 Å with a resolution of λ/∆λ ≈ 2000. Our sample consists of all clusters with known redshift at z < 0.1 from the catalogues of Abell et al. (1989), Zwicky et al. (1961), Böhringer et al.(2000) and Voges et al. (1999) that have been mapped by the SDSS-DR4. We downloaded only those galaxies located within a radius of 4.5 Mpc around the centres of the galaxy clusters. Only those clusters with more than 30 galaxies with spectroscopic data in the searching radius were considered, resulting in a sample formed by 240 clusters following the previous criteria. The SDSS-DR4 spectroscopic galaxy target selection was done by an automatic algorithm (see Strauss et al. 2002). The main galaxy sample consists of galaxies with r-band Petrosian magnitudes brighter than 17.77 and r-band Petrosian half-light surface brightness brighter than 24.5 mag arcsec−2. The completeness of this sample is high, exceeding 99% (see Strauss et al. 2002). However, some of the selected spectroscopic targets were not observed at the end. This incompleteness has several causes, including the fact that two spectroscopic fibers cannot be placed closer than 55 given plate, possible gaps between the plates, fibers that fall out of their holes, and so on. According to these reasons, we expect that the incompleteness of the spectroscopic data will be more important for bright galaxies in high density environments such as galaxy clusters. Figure 1 shows the mean completeness1 of the SDSS-DR4 spectroscopic data as a function of the r-band magnitude for the selected galaxies, where a fast increment to- wards faint magnitudes can be observed. In order to avoid pos- sible effects on the results due to this effect, we have completed the spectroscopic SDSS-DR4 observations with the data avail- able at the Nasa Extragalactic Database (NED). Figure 1 also shows the mean completeness as a function of r-band magni- tude after the spectroscopic data from NED were included in the sample. Notice that the new mean completeness is almost constant (≈ 85%) for all magnitudes brighter than mr = 17.77. We have made a second selection of the clusters by considering only those from our original list with completeness larger than 70% for galaxies brighter than 17.77 in the r-band. 1 We have defined the spectroscopic completeness per magnitude bin as the ratio of the number of galaxies with spectroscopic data to the number of galaxies with photometric information. Fig. 1. Mean completeness of the cluster sample as a function of the r-band magnitude. Diamonds represent the spectroscopic data from SDSS-DR4 and black circles after the completion with data from NED. 3. Cluster Membership Clusters properties such as the mean cluster velocity, the ve- locity dispersion, the cluster centre or the virial radius can be significantly affected by projection effects. Several methods have been developed during decades in order to obtain reliable galaxy cluster membership and avoid the presence of interlop- ers. They can be classified in two families. First, those algo- rithms that use only the information in the velocity space, e.g. 3σ-clipping techniques (Yahil & Vidal 1977), gapping proce- dures (Beers et al. 1990; Zabludoff et al. 1990, hereafter ZHG algorithm) or the KMM algorithm (Ashman et al. 1994). The other family corresponds to those algorithms which use infor- mation of both position and velocity, such as the methods de- signed by Fadda et al. (1996), den Hartog & Katgert (1996), or Rines et al. (2003). The cluster membership in our sample was obtained using a combination of two algorithms. A first rough cluster mem- bership determination was obtained using the ZHG algorithm, which in a second step was then refined using the KMM al- gorithm. The ZHG algorithm is a typical gapping procedure which determines the cluster membership by the exclusion of those galaxies located at more than a certain velocity distance (∆v) from the nearest galaxy in the velocity space. Then, the mean velocity (vm) and velocity dispersion (σ) of the remain- ing galaxies are calculated. After sorting objects with velocities greater than vm, any galaxy separated in velocity more than σ from the previous one is classified as non member. The same is done for those galaxies with velocities less than vm. The pro- cess is repeated several times and finally the mean cluster ve- locity (vc) and the cluster velocity dispersion (σc) are obtained. Zabludoff et al. (1990) pointed out that this method lacks statis- tical rigour and tends to give overestimated values of σc. One of the disadvantages of this method is that the results obtained strongly depend on the chosen value of ∆v. Large values of ∆v imply that a large fraction of interlopers are identified as clus- ter members. On the contrary, small values of ∆v result in the 4 Aguerri et al.: Global Properties of Nearby Galaxy Clusters lost of cluster galaxies. We have investigated the variation of σc for different values of ∆v, obtaining that ∆v=500 km s is an appropriate value for our clusters. This method has also the advantage that has an easy implementation and does not re- quire too much computational time. Recently, it has been used in works involving a large number of clusters, such as those from the 2dFGRS (De Propris et al. 2003). The ZHG algorithm splits the velocity histograms in different galaxy groups, be- ing one of them located at the catalogued redshift of the clus- ter. That group was taken and analysed in more detail with the KMM algorithm. In the few cases where there was no galaxy group located at the catalogued redshift we identified the most significant groups having z < 0.1 as the cluster itself. The KMM algorithm (Ashman et al. 1994) estimates the statistical significance of bi-modality in a dataset. We have run it to the group of galaxies given by the ZHG algorithm which contains the catalogued redshift of the cluster. The KMM algo- rithm gives us the compatibility of the velocity distribution of such group of galaxies with a single or multiple Gaussian dis- tribution. We considered three different cases which are sum- marised in Fig. 2: – Single cluster: the velocity distribution of the galaxies is compatible with a single Gaussian, e.g. Abell 757. – Cluster with substructure: the velocity distribution is com- patible with multiple groups. We identified the cluster as the group with the largest number of galaxies plus those groups which mean velocities lie within 3σ from the mean velocity of the largest one2, e.g. Abell 1003. – Cluster with contamination: the velocity distribution is compatible with the presence of several groups, but the mean velocities of the smaller groups deviate more than 3σ from the most populated one, which we identify as the cluster itself, e.g. Abell 168 . We have explored the differences in the values of vc and σc if we consider as interlopers those groups of galaxies located at a velocity distance larger than 1σ or 3σ from the mean velocity of the main galaxy group. We obtained that the differences in vc and σc in 90% of the clusters are less than 20%. The remaining 10% of the clusters are those with significant structure in the velocity distribution, being most of them more than one cluster along the line of sight. Thus, we have adopted 3σ as the default except for those clusters with significant differences between 1σ and 3σ, for which we have measured the mean velocity and velocity dispersion of the cluster adopting the criteria of 1σ. Through all of this process, the determination of vc and σc was done using the biweight robust estimator of Beers et al. (1990). 3.1. Cluster global parameters Once the cluster membership was determined, we obtained the global parameters of each cluster, i.e., mean velocity (vc), ve- locity dispersion (σc), cluster centre, and the radius r200. All of these parameters were computed using only the cluster mem- bers. 2 In this case σ is the velocity dispersion of the largest group of galaxies. Fig. 2. Velocity histograms of three representative clusters of the sample. The vertical full lines represent the mean velocity of the different groups of galaxies in which KMM algorithm has divided the velocity histogram. The dotted vertical lines represent vc ± 3σc. The determination of the cluster centre is important in or- der to accurately compute the other parameters of the clusters. The centre of the cluster is determined by the potential well, which can be traced by the position of the peak of the X-ray lu- minosity of the cluster. That peak was considered as the centre of those clusters from our sample with X-ray measurements in the literature. Unfortunately, not all the clusters from the sam- ple have X-ray data. In that case, the centre of these clusters was determined by the peak of the galaxy surface density3. For those clusters with X-ray data we have compared the centres given by the peaks of X-ray luminosity and galaxy surface den- sity, obtaining a mean difference of 150 kpc. Analytic models (Gott 1972) and simulations (Cole & Lacey 1996) indicate that the virialized mass of clusters is generally contained inside the surface where the mean inner density is 200ρc, where ρc is the critical density of the Universe. The radius of that surface is called r200. We have computed the r200 for our clusters using the same approximation as Carlberg et al. (1997): r200 = 10 H(zc) , (1) where H(zc) is the Hubble constant at the cluster redshift 3 The galaxy surface density was computed using the algorithm de- signed by Pisani (1996). Aguerri et al.: Global Properties of Nearby Galaxy Clusters 5 The previous global parameters of the clusters (vc, σc, r200 and centre) were obtained as described above but in a recur- rent way. In a first step, they where determined using all cluster member galaxies around 4.5 Mpc from the centre of the clus- ter. After this step we recalculated the parameters using only those galaxies located inside r200. The method was repeated several times until the difference in the parameters obtained in two consecutive steps was less than 5%. Three or four iterations were usually enough for reaching the convergence. In order to obtain reliable parameters of the clusters, those with less than 15 galaxies within r200 were removed from our list. This re- sults in a final sample formed by 110 nearby galaxy clusters. Table 1 shows the sample of galaxy clusters and their global parameters. The columns of Table 1 represent: (1) galaxy clus- ter name, (2, 3) cluster centres (α (J2000), δ (J2000)), (4) mean radial velocity, (5) cluster velocity dispersion, (6) r200 radius, (7) number of galaxies within r200, and (8) spectroscopic com- pleteness. For 6 clusters (Abell 1003, Abell 1032, Abell 1459, Abell 2023, Abell 2241 and ZwCl1316.4-0044) large differences in the mean recessional velocity have been found between the val- ues given in Table 2 and those from NED. These are the clusters with no significant galaxy group at the catalogued redshift (see Section 3). In order to consider the possible influence of neighbouring clusters on the global properties of the sample we searched in the surroundings of each cluster for the presence of compan- ions. Following Biviano & Girardi (2003), we have considered that two clusters, i and j, are in interaction when: |vi − vj| < 3(σi + σj) Ri,j < 2(r200,i + r200,j), (2) where Ri, j is the projected distance between the centres of the clusters and vi, j, σi, j, r200,i, j their respective mean velocities, velocity dispersions and r200. We found 16 couples of clusters in interaction according to the previous criteria. The remaining sample (88 clusters) followed the isolation criteria, and will be used in the analysis presented in the following sections. Figure 3 shows the sky distribution of the cluster members and the galaxy velocities as a function of clustercentric distance for a sample of 8 clusters. Red points represent the galaxies taken as cluster members while black points are interlopers. Notice the large number of interlopers in some of the galaxy clusters, such as Abell 1291, Abell 1383, Abell 2244. Some of them, Abell 1291 and Abell 1383, were not included in the final isolated sample due to the presence of companions. 3.2. Corrections to line-of-sight velocities Line-of-sight velocities of galaxies in clusters were corrected by two effects: cosmological redshift and global velocity field. We should take into account that we will compare the veloc- ity dispersion of clusters located at different redshifts. Thus, for each galaxy we have 1 + zobs = (1 + zc)(1 + zgal) (Danese et al. 1980), being zobs the apparent redshift of the galaxies, zc the cosmological redshift of the cluster, and zgal the redshift of the galaxy respect to the cluster centre. This cor- rection can affect up to 10% for the most distant clusters in our sample. Galaxy clusters are frequently part of larger cosmological structures such as filaments, superclusters or multiple systems, which can affect the velocity field resulting in a modified clus- ter velocity dispersion. The interaction between galaxy clus- ters can also produce distorted velocity fields. We have inves- tigated the importance of these effects in the velocity field of our clusters by making a least-square fit to the radial velocities of cluster galaxies with respect to their position in the plane of the sky (see den Hartog & Katgert 1996; Girardi et al. 1996). For each fit we computed the coefficient of multiple determi- nation, R2. In order to test the significance of the fitted veloc- ity gradients, we run 1000 Monte Carlo simulations for each cluster for which the correlation between position and veloc- ity was removed. This was achieved by shuffling the veloc- ities of the galaxies with respect to their positions. We de- fined the significance of velocity gradients as the fraction of Monte Carlo simulations with R2 smaller than the observed one. This correction of the velocity field was applied to those cluster in which the significance of velocity gradients is larger than 99% ( 30% of the total sample). However, this correction has small effects both in the shape of the velocity dispersion profiles and on the total velocity dispersion (the mean abso- lute correction was about 40 ± 15 km s−1). This is in agree- ment with similar corrections applied in other cluster samples (den Hartog & Katgert 1996; Girardi et al. 1996). 3.3. Comparison with other methods Some of the clusters presented in our sample have been previ- ously studied by other authors. However, we have avoided com- paring our results with those from the literature given the differ- ent datasets used. In order to compare our cluster membership method with others proposed in the literature, we have com- puted σc of our clusters with two more methods: a 3σ-clipping and the method proposed by Fadda et al. (1996). The median absolute difference between our σc and those computed by the 3σ-clipping method is only 17 km s−1. Only 10% of the clus- ters show important diferences (∆σc > 200 km s −1) in the com- putation of the velocity dispersion of the cluster with the two methods. They correspond to those clusters affected by large amount of structure along the line of sight. The 3σ-clipping method gives for these clusters considerably larger values of σc than ours. Differences were larger when we compared with Fadda’s method. In this case the mean absolute difference in σc between the two methods was 84 km s −1 and 80% of the clusters show differences smaller than 200 km s−1. Recently, Popesso et al. (2006) have obtained the values of σc for a sample of Abell clusters using SDSS-DR4 data, for which cluster membership was obtained using the selection al- gorithm of Katgert et al. (2004). The median absolute differ- ence between our and their σc is 45 km s −1 for the 28 clusters in common. Only for 4 clusters (Abell 1750, Abell 1773, Abell 2244 and Abell 2255) the absolute differences in σc is larger than 200 km s−1. We have also compared our results with those given in the cluster catalogue presented by Miller et al. (2005). We found 16 clusters in common, being 74 km s−1 the median absolute dif- 6 Aguerri et al.: Global Properties of Nearby Galaxy Clusters Fig. 3. Galaxy surface density (images) and radial velocity versus distance to the cluster center for the galaxy cluster member (red points) of a subsample of 8 clusters. The overplotted circle have a radius equal to r200 for each cluster. The black points represent interloper galaxies. ference between our and their σc. In this case, 3 clusters show an absolute differences in σc larger than 200 km s We can conclude that in most of the cases our cluster mem- bership method reported values of σc similar to those given by other methods. Only for 10-20% of the clusters the absolute differences in σc between our method and the others is larger than 200 km s−1. For these clusters the structure along the line of sight is the responsible of the difference, being our σc values smaller than the others. 3.4. Lx-σ relation We can learn about the nature of cluster assembly by studying the relations between cluster observables. One of the most uni- versals is the well known relation between the cluster X-ray lu- minosity and the velocity dispersion of its galaxies (LX ∝ σbc). Cluster formation models predict that if the only energy source in the cluster comes from the gravitational collapse, then b ≈ 4. This relation has been studied in the literature by many authors using different cluster samples, finding values of b between 2.9 and 5.3 (Edge & Stewart 1991; Quintana & Melnick 1982; Mulchaey & Zabludoff 1998; Mahdavi & Geller 2001; Girardi & Mezzetti 2001; Borgani et al. 1999; Xue & Wu 2000; Ortiz-Gil et al. 2004; Hilton et al. 2005). The study of the LX − σc relation in our cluster sample will be also useful as another check for the values of σc we have derived. We have X-ray data for 48 galaxy clusters from our sample. The X-ray data have been obtained from Ebeling et al. (1998), Böhringer et al.(2000), Ebeling et al.(2000) and Ledlow et al.(2003), and the X-ray luminosities are measured in the ROSAT band (0.1-2.4 keV). Figure 4 shows the LX−σ relation for this subset with avail- able X-ray data in the literature. The Spearman coefficient of the relation is 0.56 and the significance from zero correlation is greater than 3σ. This indicates the existence of a correlation between LX and σc for the clusters of our sample. We used the bivariate correlated errors and intrinsic scatter (BCES) bisector method of Akritas & Bershady (1996) to obtain the coefficient and power-law slope estimates of the relation. This fitting tech- nique takes into account errors in both variables and intrinsic scatter. The LX − σc relation for our clusters is given by: LX(0.1 − 2.4 keV) = 1033.7±1.2σ3.9±0.4 (3) Aguerri et al.: Global Properties of Nearby Galaxy Clusters 7 Fig. 4. LX−σ relation for the 48 galaxy clusters with X-ray data in the ROSAT band (0.1-2.4 keV) from our sample. The full line represents the best fit using the BCES bisector algorithm (see text for more details). Fig. 5. Absolute r-band magnitude as a function of redshift for the galaxies of our cluster sample. This result is in very good agreement with another mea- surement of this relation using the same ROSAT band (0.1-2.4 keV) for the X-ray data and the same fitting algorithm (see Hilton et al. 2005). 3.5. Redshift distribution and sample completeness The 88 isolated galaxy clusters are located in a redshift range between 0.02 and 0.1, with an average redshift of 0.071. Figure 5 shows the absolute r-band magnitude (Mr) as a function of the redshift for the galaxies in our cluster sample4. It is clear that the completeness magnitude is a function of redshift. This figure shows that the full sample is complete for galaxies brighter than Mr = −20.0. The lack of completeness for fainter galaxies will be taken into account in the subsequent analysis. 4 See section 3 for the explanation of the computation of the abso- lute magnitudes of the galaxies. 4. Morphological Segregation Light concentration or colours have been used extensively in the literature in order to classify galaxies. Shimasaku et al. (2001) and Strateva et al. (2001) using SDSS data, found that the ratio of Petrosian 50 percent light radius to Petrosian 90 percent light radius, Cin, measured in the r-band image was a useful index for quantifying galaxy morphology. Strateva et al. (2001) also found that the colour u − r = -2.22 efficiently sep- arates early- and late-type galaxies at z < 0.4. We have used colours for classifying galaxies, because properties such as ve- locity dispersion in galaxy clusters are better correlated with galaxy colours than galaxy morphology (Goto 2005). The mag- nitude of the galaxies were corrected by two effects: Galactic absorption and k−correction. The Galactic absorption in the different filters was obtained from the dust maps of Schlegel et al. (1998). We applied the k−correction using the kcor- rect.v4 1 4 code by Blanton et al. (2003) in order to obtain the rest-frame magnitudes of the galaxies for the different band- passes. Once these two corrections were done, we classified the galaxies in red (u − r ≥ 2.22) and blue ones (u − r < 2.22). The galaxy data was downloaded from the SDSS database according to a metric criteria: we downloaded the information of all galaxies located within a radius of 4.5 Mpc at each galaxy cluster redshift. This means that we are mapping different phys- ical regions for each cluster. In order to avoid this problem we have studied the ratio rmax/r200 for each cluster, being rmax the maximum distance of a galaxy from the cluster centre for each galaxy cluster. We have obtained that all clusters of our sample reach rmaxr200 = 2, and 50% of them reach Our sample of galaxies consists of 6880 galaxies located within a radius 2 × r200, being 62% of them red galaxies and 38% blue ones. If we consider all galaxies within 5 × r200 then the sample has 10865 galaxies, being 55% and 45% red and blue galaxies, respectively. The red and blue galaxies were also grouped in three categories according to their r-band magni- tude: Mr < M r − 1, M r − 1 < Mr < M r + 1, and Mr > M r + 1 The first group contains the brightest members of the clusters, the third group contains the so-called dwarf population and the second one is formed by normal bright galaxies. Table 2 shows the median location, r-band absolute magnitude, velocity dis- persion and local density6 of the different galaxy groups. In general, red galaxies are brighter than blue ones, and are also located closer to the cluster centre at higher local density re- gions. The two families of galaxies present different kinemat- ics, in the sense that red galaxies show a smaller velocity dis- persion than blue ones. This different kinematic between red and blue galaxies has also been seen in other studies, and have been interpreted as red and blue galaxies having different kind of orbits, being the orbits of blue galaxies more anisotropic than the red ones (Adami et al. 1998; Biviano & Katgert 2004). Other authors interpret this difference in velocity dispersion as an evidence that ram pressure is not playing an important role in galaxy evolution in clusters. In contrast, tidal interactions 5 M∗r − 5log(h) = −20.04, Blanton et al. (2005) 6 The local surface density (Σ) was computed with the 10 nearest neighbours to each galaxy belonging to the cluster. 8 Aguerri et al.: Global Properties of Nearby Galaxy Clusters should be the dominant mechanism (Goto 2005). All of these properties are independent of the sampled area. It is also interesting that red dwarf galaxies are located at similar environments as the brightest red ones: close to the cluster centre in high local density regions (see also Hogg et al. 2004). But the red dwarf population shows a larger velocity dispersion than the brightest red galaxies. Biviano & Katgert (2004) found that the brightest cluster members were not in equilibrium with the cluster potential. They are especial galaxies that could have formed close to the cluster centre or have fallen to this region due to dynamical friction. In contrast, dynamical friction is not so efficient in the dwarf population, so that the main presence of these galaxies in the central regions of the clusters should be due to their origin. The discussion about the properties and origin of the dwarf population will be given in another paper (Sánchez-Janssen et al., in preparation). 5. Velocity Dispersion Profiles The adopted cluster velocity dispersion was calculated with the galaxies located within the r200 radius of each cluster. But, how does σ depend on the clustercentric distance in our sample?. This can be answered by studying the integrated velocity dis- persion profiles (VDPs) of the clusters. These profiles also pro- vide information about the dynamical properties of the galax- ies. Thus, a system with galaxies predominantly in radial or- bits produces an outwards declining VDP, while the opposite behaviour suggests instead that the galactic orbits are largely circular. In contrast, constant VDPs are characteristic of an isotropic distribution of velocities (Solanes et al. 2001). Figure 6 shows the VDPs for some of the clusters in our sample. They show the velocity dispersion of the cluster at a given radius evaluated using all the galaxies within that radius, without any restriction in their luminosities. The errors showed in Fig. 6 were computed using the approximation given by Danese et al. (1980). In order to classify the VDPs of our clusters, we computed the velocity dispersion (σi, i = 1, 2, 3, 4, 5) of the galaxies in the clusters located within five different radius: 0.4×r200, 0.6×r200, 2×r200, 3×r200 and 4×r200, respectively. We compared these values with σc, given in Table 1. The resulting mean ratios σi/σc were: 1.02± 0.04, 1.01±0.01, 0.97±0.01, 0.94±0.02 and 0.94±0.02, for i = 1, 2, 3, 4, 5, respectively. These values indi- cate that within r200 the VDPs of the total galaxy cluster popu- lation are consistent with being flat. The mean variation of the VDPs inside r200 is only 2%. The values of σi/σc, i = 3, 4, 5 show that, outside r200 the VDPs slowly decrease. The mean variation of the VDPs outside r200 is −6%. No differences in the ratios σi/σc have been found when we have divided the galaxy sample between bright (Mr < M r +1) and dwarf (Mr > M r +1) galaxies. This flat behaviour of the VDPs inside r200 suggests that galaxies in these areas have an isotropic distribution of ve- locities. In contrast, the decline with radius of VDPs outside r200 points to radial orbits (Solanes et al. 2001). Figure 6 also shows the VDPs of early- (red) and late-type (blue) galaxies. In most profiles the velocity dispersion of blue galaxies is larger than the corresponding one for early-type ones. We have also analysed the shape of VDPs of blue and Fig. 7. Histograms of the ratios σi/σc, i = 1, 2, 3, 4, 5 for the galaxies in the clusters. The black full line represent all galax- ies, the blue and red lines correspond to late- and early-type ones. See text for more details. red galaxies as we did for the total sample. For red galaxies, we obtained that σi/σc,r are 1.04± 0.03, 1.03±0.03, 0.97±0.02, 0.96±0.02 and 0.96±0.03, for i = 1, 2, 3, 4, 5, respectively. The values of σi/σc,b for the blue galaxies are: 1.15± 0.07, 1.04±0.03, 0.95±0.04 and 0.93±0.04 and 0.92±0.04, respec- tively. In those computations, σc,r and σc,b represent the ve- locity dispersion of the red and blue galaxies within a radius equal to r200, respectively. Figure 7 show the distribution of σi/σc, i = 1, 2, 3, 4, 5 for the blue, red and the total galaxy sam- The VDPs have been studied in the literature by sev- eral authors. Most of them conclude that for large radii (r > 1 Mpc) the VDPs are flat (Girardi & Mezzetti 2001; Rines & Diaferio 2006; Fadda et al. 1996; Muriel et al. 2002). This is consistent with the mild decrease that we have found in our clusters. The VDPs for red galaxies in our sample are almost flat outside r200. This is not the case of the VDPs of blue galaxies which clearly decrease with radius outside r200. In the inner regions (r < r200) the VDPs of the total sample and those corresponding to the red galaxies are flat. In con- trast, the VDPs of blue galaxies decrease with radius. Different authors show that VDPs can decrease or increase with radius. den Hartog & Katgert (1996) made a thorough study and found that the variations of the VDPs in the innermost regions of clus- ters (r < 0.5 Mpc) are real and not due to noise or bad centre election. We have re-computedσ1/σc andσ2/σc only for those clusters with X-ray centres, and our results did not significantly change. Thus, we can conclude that in our galaxy cluster sam- ple only blue galaxies show increasing VDPs towards the cen- tre of the cluster, while red galaxies show flat VDPs. Aguerri et al.: Global Properties of Nearby Galaxy Clusters 9 Table 1. Main properties of the different types of galaxies Galaxies within r/r200 < 5 < r/r200 > < Mr > < σ > < log(Σ) > Ngal u − r < 2.22 1.85±0.02 -19.65±0.01 1.04±0.01 0.46±0.08 4937 u − r < 2.22 & Mr < M∗r − 1 1.79±0.11 -21.54±0.02 1.08±0.09 0.40±0.25 94 u − r < 2.22 & M∗r − 1 < Mr < M∗r + 1 1.90±0.02 -20.00±0.01 1.03±0.01 0.38±0.08 3126 u − r < 2.22 & Mr > M∗r + 1 1.74±0.03 -18.74±0.02 1.05±0.02 0.64±0.14 1717 u − r ≥ 2.22 1.03±0.01 -20.16±0.01 0.90±0.01 0.80±0.05 5928 u − r ≥ 2.22 & Mr < M∗r − 1 0.95±0.05 -21.62±0.02 0.78±0.03 0.89±0.14 537 u − r ≥ 2.22 & M∗r − 1 < Mr < M r + 1 1.10±0.02 -20.20±0.01 0.91±0.01 0.75±0.06 4592 u − r ≥ 2.22 & Mr > M∗r + 1 0.85±0.04 -18.91±0.02 0.90±0.03 1.06±0.12 729 Galaxies within r/r200 < 2 < r/r200 > < Mr > < σ > < log(Σ) > Ngal u − r < 2.22 0.97±0.01 -19.61±0.02 1.08±0.02 0.80±0.07 2636 u − r < 2.22 & Mr < M∗r − 1 1.15±0.07 -21.56±0.03 1.18±0.13 0.62±0.24 54 u − r < 2.22 & M∗r − 1 < Mr < M∗r + 1 1.04±0.01 -19.96±0.01 1.08±0.02 0.70±0.07 1648 u − r < 2.22 & Mr > M∗r + 1 0.85±0.01 -18.70±0.02 1.07±0.03 1.02±0.09 934 u − r ≥ 2.22 0.67±0.01 -20.14±0.01 0.91±0.01 0.98±0.04 4244 u − r ≥ 2.22 & Mr < M∗r − 1 0.57±0.03 -21.65±0.02 0.80±0.03 1.02±0.13 397 u − r ≥ 2.22 & M∗r − 1 < Mr < M r + 1 0.69±0.01 -20.19±0.01 0.92±0.01 0.94±0.05 3239 u − r ≥ 2.22 & Mr > M∗r + 1 0.61±0.02 -18.92±0.02 0.89±0.03 1.19±0.10 608 The previous findings can also be seen in Fig 8. We show the VDPs of the different galaxy classes for the normalised cluster, which was obtained by normalising the scales and ve- locities of each galaxy of the sample. Thus, the radial distance of each galaxy to the cluster centre was scaled by r200 of the corresponding cluster, and the relative velocity of each galaxy cluster was normalised by the velocity dispersion of the clus- ter. Figure 8 shows the VDPs which correspond to the total, bright (Mr < M r + 1) and dwarf (Mr > M r + 1) galaxy sam- ples. We have also distinguished between red and blue objects. The VDPs of the total galaxy sample indicate that blue galax- ies have always larger velocity dispersion than red ones. They also show always decreasing VDPs, while red ones have almost constant and slowly decrease VDPs inside and outside r200, re- spectively. These features can also be seen in the VDPs of the bright galaxy sample. In contrast, red and blue dwarfs show decreasing VDPs inside r200. The shape of the VDPs can provide information about the dynamical state of the galaxies. Thus, clusters with galaxies predominantly in radial orbits produce an outwards declining VDP. This is the case of the blue galaxies of our sample, which is in agreement with previous findings (Biviano & Katgert 2004; Adami et al. 1998). We have also obtained that the red dwarf galaxies inside r200 has an outwards declining VDP. This would imply that this kind of galaxies may also be located in radial orbits. In contrast, constant VDPs im- ply an isotropic distribution of velocities (Solanes et al. 2001). This is the case of the red bright galaxy population inside r200. 6. Fraction of blue galaxies Butcher & Oemler (1984) observed that the fraction of blue galaxies ( fb) in clusters evolves with redshift, in the sense that galaxy clusters located at medium redshift have a larger fb than nearby ones. This has been usually interpreted as an evolution- ary trend in clusters. But it is a matter of debate which is the role played by the environment in the change of the fraction of blue galaxies. We have computed fb in our sample of galaxy clusters, studying the variation with z and the possible influence of the environment. 6.1. Adopted aperture and limiting magnitude The original analysis of Butcher & Oemler (1984) defined blue galaxies as those within a radius containing 30 per cent of the cluster population, being brighter than Mv = −20 and bluer by 0.2 mag in B − V than the colour-magnitude relation defined by the cluster early-type galaxies. It has been noticed by sev- eral authors that the fraction of blue galaxies strongly depends on the magnitude limit and the clustercentric distance used (Ellingson et al. 2001; Goto et al. 2003; De Propris et al. 2004; Andreon et al. 2006). They observed that fb grows when the magnitude limit is fainter and the aperture is larger, reflect- ing the existence of a large fraction of blue faint galaxies in the outer regions of the clusters. De Propris et al. (2004) con- sidered appropriate to measure fb in apertures based on clus- ter physical properties. They used r200 as the aperture radius where they measured fb for their clusters. We have adopted also this radius in order to determine fb in our galaxy clus- ters. As it was previously commented, fb depends also on the adopted limiting magnitude of the galaxies in clusters. It should be noticed that as we move to higher redshifts we systematically lose faint galaxies (see Fig 2). Our clusters spread in a redshift range 0.02 < z < 0.1, and only galax- ies brighter than Mr = −20.0 (≈ M∗r + 0.5) can be observed at all redshifts. For this reason, we have adopted this abso- lute magnitude as the limiting magnitude for the computa- tion of fb. This ensures us to work with a complete galaxy sample at all redshifts. Other authors adopted fainter limiting magnitudes, e.g. M∗ + 1.5 (De Propris et al. 2004) or M∗ + 3 (Margoniner & de Carvalho 2000). If there is a large number of blue galaxies at faint magnitudes, we expect that our val- ues of fb will be smaller than those reported by the previous authors. 10 Aguerri et al.: Global Properties of Nearby Galaxy Clusters Fig. 6. Velocity dispersion profiles of some clusters of our sample. The black symbols represent the velocity dispersion profile taking into account all types of galaxies. Blue and red symbols represent the velocity dispersion profiles corresponding to blue and red galaxies, respectively. 6.2. Colour-magnitude diagrams We determined the g − r versus r colour-magnitude diagrams for all the clusters in our sample. The colour-magnitude rela- tion was measured by a robust fitting routine by minimising the absolute deviation in g−r colour, using only early-type galaxies located within an aperture of radius equal to r200. The galaxy types were determined according to the u−r colour and the light galaxy concentration parameter, Cin. These two criteria allow us to identify the most reliable sample of E/S0 galaxies (see Shimasaku et al. 2001; Strateva et al. 2001). Thus, we consid- ered early-type galaxies those with u − r ≥ 2.22 and Cin < 0.4. Figure 9 shows the colour-magnitude diagrams of four repre- sentative galaxy clusters. The colour-magnitude relation fitted in each case is also overploted. Figure 9 (left column) also shows the histograms of the colour distribution, marginalised over the fitted colour-magnitude relation. The average of the slopes of the colour-magnitude relations of the early-type galaxies of the clusters is -0.014±0.008. This slope is within the errors in agreement with the slope obtained by Gallazzi et al. (2006) for a large sample of galaxies using SDSS data. It is also in agreement with the average B−R slope obtained by De Propris et al. (2004) for a sample of galaxy clusters from 2dFGRS. 6.3. Calculation the blue fraction of galaxies As we explained before, the blue fraction of galaxies was com- puted using only those galaxies brighter than Mr = −20 and located within an aperture of radius r200. In the present study we only used spectroscopically confirmed galaxy cluster mem- bers. This should not bias our results, especially due to our high completeness. Figure 10 presents fb as a function of redshift. The errors of fb were computed according to the prescription given by De Propris et al. (2004). We observe no evolution of fb with redshift, which means that our sample is ideal to study the effects of the environment on fb. We have considered three cluster properties (concentration, velocity dispersion and X-ray luminosity) of each cluster in or- der to analyse the dependence of fb on the environment. The concentration parameter was computed following the prescrip- tion of De Propris et al. (2004), i.e. C = log(r60/r20), where r60 and r20 are the radii containing 60 and 20 per cent of the cluster galaxies, respectively. The velocity dispersion of the clusters Aguerri et al.: Global Properties of Nearby Galaxy Clusters 11 Fig. 8. Velocity dispersion profiles of the galaxies of the normalised cluster. The total galaxy population is showed in the top panel. Bright galaxies (Mr < M r + 1) are in the middle panel, and the bottom panel shows the VDPs corresponding to the dwarf galaxy sample (Mr > M r + 1). The VDP of the total, blue and red galaxy samples are represented by black, blue and red colours, respectively (see text for more details). Fig. 10. The fraction of blue galaxies ( fb) as a function of red- shift of the galaxy clusters. was taken from Table 1. The X-ray luminosities were obtained from the literature (Ebeling et al. 1998, Böhringer et al.2000, Ebeling et al. 2000 and Ledlow et al. 2003), being measured in the ROSAT band (0.1-2.4 keV). We only found X-ray data for 48 clusters of the sample. Figure 11 shows the dependence of the fraction of blue galaxies on concentration, cluster velocity dispersion and X- ray luminosity. The non-parametric Spearman test returns that fb has a low correlation with concentration and velocity dis- persion. The fraction of blue galaxies correlates best with the velocity dispersion, but the significance of the correlation is 2.6σ. In contrast, the Spearman test shows correlation between fb and X-ray luminosity, being the significance of this correla- tion just 3σ. Notice that the points are distributed in the LX − fb plane following a triangular shape. Clusters with large X-ray luminosity (LX(0.1 − 2.4keV) > 1045ergs−1) show small frac- tions of blue galaxies (less than 10%). Nevertheless, those clus- ters with small X-ray lumisosity show small and large fraction of blue galaxies. This correlation could indicate that there is a threshold over which cluster environment can affect the galaxy colours, and play a role in the galaxy evolution. This means that, according with our LX −σ relation, the evolution of galax- ies could be driven by the cluster environment for those clusters 12 Aguerri et al.: Global Properties of Nearby Galaxy Clusters Fig. 9. Colour-magnitude relation (left) and histograms (right) of marginalised colour distribution for four representative clusters at different redshifts of our cluster sample. The full line in left panels represent the fitted colour-magnitude relation. The vertical point lines in right panel represent the blue/red separation in the Butcher-Oemler effect. The red points are the galaxies with u − r ≥ 2.22 and Cin < 0.4 (see text for more details). with velocity dispersion larger than σ ≈ 800 km s−1. Recently, (Popesso et al. 2006) found a similar correlation between LX and fb for a larger cluster sample. The shape of the our LX − fb correlation is similar to the correlation between cluster veloc- ity dispersion and the fraction of [OII] emitters for clusters at low redshift reported by Poggianti et al. (2006). They found that clusters with σ > 550 km s−1 have a constant low fraction (less than 30%) of [OII] emiters. In contrast, those clusters with smaller σ show large and small fractions of [OII] emiters. We have recomputed fb taking into account those galax- ies within an aperture of radius equal to r200 and brighter than Mr = −19.5. We restricted the analysis only to those clus- ters with z < 0.05, because our sample is complete down to Mr = −19.5 in this redshift range. In this case the number of cluster decreases to 13. We have again studied the correlations of fb with galaxy concentration, velocity dispersion and X-ray luminosity, obtaining similar correlations as with the full sam- 7. Discussion From the study presented in this paper, most of the galaxies (62%) located in the central regions of galaxy clusters (r/r200 < 2) are early-type galaxies (see section 4). In contrast, the field population is dominated by late-type galaxies. In the literature it is also well established that the colour of galaxies in clus- ters and field is different, an indication of the low star forma- tion activity found in cluster galaxies (e.g. Balogh et al. 1998; Lewis et al. 2002; Gómez et al. 2003). These differences in morphology and stellar content between field and cluster galax- ies suggest different evolutionary processes. The facts that late- type galaxies show larger velocity dispersions and are located at larger distances from the cluster centre than early-type ones have been interpreted as late-type galaxies being recent ar- rivals to the cluster potential, forming a non-relaxed group of galaxies moving in more radial orbits than early-type ones (e.g. Stein 1997; Adami et al. 1998). As late-type galaxies fall into the cluster potential and encounter denser environments, they evolve to early-type ones. The results presented in the present work are in agreement with previous findings. However, as pointed out by Goto (2005), this would imply that a large frac- tion of galaxies (≈ 40% according to our sample) should be recent arrivals to the cluster, a possibility that seems unlikely. Goto (2005) concluded that the different observational proper- ties between red and blue galaxies may indicate which is the main mechanism driving the evolution of galaxies in clusters. Gas stripping, mergers and interactions with other galaxies and with the cluster potential are the main mechanisms which are able to transform galaxies in clusters, making late-type galax- ies lose their gas content, stop their star formation, circularise their orbits and transform their morphology from disk-like ob- jects to spheroids. All of these mechanisms affect galaxies in Aguerri et al.: Global Properties of Nearby Galaxy Clusters 13 Fig. 11. The fraction of blue galaxies ( fb) as a function of galaxy distribution (top), cluster velocity dispersion (middle), and X-ray luminosity (bottom) of the galaxy clusters. clusters but, can we infer from the observational results which is the dominant one?. It should be noted that the different mechanisms of galaxy evolution have very different time-scales. While gas stripping has a very short time-scale (≈ 50 Myr, Quilis et al. 2000), the galaxy infall process can take ≈ 1 Gyr. The different mecha- nisms also have different underlying physics. Thus, ram pres- sure stripping is proportional to the density of the intracluster medium (ICM) and to the square of the velocity of the galaxy. In contrast, dynamical interactions are more efficient when the relative velocity of galaxies is smaller (Mamon 1992). This means that gas stripping is stronger in the cluster centres and for galaxies with high velocities, while dynamical interactions should be more efficient for galaxies with smaller velocity dis- persions. Numerical simulations have shown that most of the galaxies inside the virial radius have already been through the cluster core more than once (Mamon et al. 2004). If gas strip- ping were the main mechanism driving galaxy evolution in clusters, according to the short time scale of this process, only few blue (late-type) galaxies should be observed in the central regions of clusters. Moreover, gas stripping is also stronger in galaxies with larger velocity dispersion which means that late- type galaxies should be more affected by this mechanism than early-type ones. Based on these considerations, Goto (2005) concluded that gas stripping is not the main responsible mech- anism driving the evolution of galaxies in clusters. Instead, galaxies in clusters evolve mainly by dynamical interactions. We can add to Goto’s discussion that if gas stripping were the main galactic evolution mechanism in clusters, then the frac- tion of blue galaxies should depend on the cluster mass as the temperature and density of the gas increases with the cluster mass. According to our results, this is true for those clusters with large X-ray luminosities. In contrast, the cluster environ- ment is not so important in driving the evolution of galax- ies in low mass clusters. Thus, gas stripping may not be the main responsible mechanism transforming late-type to early- type galaxies in low mass clusters, but could be important in the most massive ones. This does not mean that gas stripping is absent in the evolution of galaxies in clusters; some clear ex- amples of gas stripping have been observed in galaxies in Virgo (Kenney et al. 2004). Dynamical interactions include both interactions with the cluster potential and with other galaxies. These effects can trig- ger temporary star formation in cluster galaxies (Fujita 1998), which can be analysed by studying their colour distribution. These interactions can also disrupt stars from galaxies, form- ing at the beginning long tidal tails that subsequently will be diluted and will form the diffuse light observed in some nearby clusters like Virgo (see Aguerri et al. 2005b, and ref- erences therein). These effects will be more important in those galaxies with smaller relative velocities. Fujita (1998) con- clude that if the tidal effects enhance the SFR in the galaxies, then the bluest galaxies should be located close to the clus- ter centre (within ≈ 300 kpc), whereas they should be in the outer parts of the cluster if the SFR is induced by galaxy- galaxy encounters. We have investigated the fraction of blue 14 Aguerri et al.: Global Properties of Nearby Galaxy Clusters galaxies in our clusters located within 300 kpc from the cen- tre of the cluster. The sample has been divided in bright and dwarf galaxies (Mr < M r + 1 and Mr > M r + 1, respec- tively). We have obtained that 40% of the blue bright galax- ies and 30% of the blue dwarf ones are located at smaller dis- tance than 300 kpc from the cluster center. This means that tidal interactions with the cluster potential are not the respon- sible mechanism for the formation of most of the blue galax- ies in our clusters. The lack of blue galaxies in the central re- gions of clusters has been observed also in nearby clusters like Coma (Aguerri et al. 2004) as well as in other distant clusters (Rakos et al. 1997; Abraham et al. 1996; Balogh et al. 1997). These evidences indicate that the evolution of galaxies in clusters could be driven by the cluster environment in the most massive ones, but galaxies in low mass clusters could mainly evolve due to the local environment. 8. Conclusions In the present paper we have analysed the main properties of the galaxies of one of the largest (10865 galaxies) and homo- geneous sample presented in the literature. The galaxies have been grouped in two families according to their u − r colour. Those galaxies with u − r ≥ 2.22 formed the red (early-type) family, and those with u − r < 2.22 the blue (late-type) one. We have derived the position, velocity dispersion, and VDPs of both families of galaxies, obtaining: – Within 2×r200, 62% and 38% of the galaxies turned to be red and blue, respectively. – The median positions and velocity dispersions are smaller for red galaxies than for blue ones. – Bright (Mr < M∗r −1) and dwarf (Mr > M r +1) red galaxies are located at smaller distances than the blue ones, sharing the same cluster environment. – The brightest cluster members (Mr < −21.0) show smaller velocity dispersions than the remaining. – The VDPs of the total galaxy cluster population are con- stant with radius in the central regions of the clusters (r < r200), while slowly decrease in the outermost regions (r ≥ r200). The red galaxy population have also flat VDPs in the central regions (r < r200). In contrast, the VDPs of blue galaxies grow towards the cluster centre. In the outer re- gions (r > r200), the VDPs of red galaxies decline smoothly with radius, while for blue ones the decrement is faster. This indicates that the galaxies in the outermost regions of the clusters are dominated by the blue population, and have more radial and anisotropic orbits than galaxies in the inner regions dominated by the red population. – The fraction of blue galaxies in our cluster sample does not correlate with cluster global properties, such as the concen- tration of the galaxy distribution and cluster velocity disper- sion. However, we found a correlation between the X-ray luminosity and the fraction of blue galaxies. Those clusters with LX(0.1 − 2.4keV) > 1045 erg s−1 have a low fraction of blue galaxies (less than 10%). In contrast, clusters with low of X-ray luminosity show large and small fractions of blue galaxies. This could indicate that the star formation in cluster galaxies may be regulated by global cluster proper- ties for clusters with LX(0.1 − 2.4keV) > 1045 erg s−1, i.e. those clusters with σc > 800 km s All these results are in agreement with previous findings from other cluster samples, indicating that red and blue galax- ies have different evolution in galaxy clusters. We have dis- cussed these results according to the different galaxy transfor- mation mechanisms presented in galaxy clusters, concluding the local environment plays a key role in galaxy evolution in low mass clusters, while the evolution of galaxies in massive clusters could be driven by the global cluster environment. Acknowledgements. We wish to thank to the anonymous referee for useful coments which have improved this manuscript. We also acknowledge financial support by the Spanish Ministerio de Ciencia y Tecnologı́a grants AYA2004-08260. We would like also to thank T. Beers for providing us with copy of his code ROSTAT, and K. M. Ashman and S. Zepf for making their KMM code available to us. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, Cambridge University, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck- Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington. 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Cluster characteristics Name α (J2000) δ (J2000) vc σc r200 Ngal C (degrees) (degrees) (km s−1) (km s−1) (Mpc) Abell0085 10.4571 -9.30694 16633+40−29 979 −39 2.10 273 0.93 Abell0117 14.0100 -10.0022 16568+31−42 531 −27 1.19 60 0.88 Abell0152 17.5229 13.9804 17888+67−34 538 −38 1.12 27 0.85 Abell0168 18.7429 0.365833 13534+23−14 578 −28 1.19 106 0.88 Abell0257 27.3396 14.0372 21060+47−21 381 −44 0.81 26 0.94 Abell0602 118.341 29.3717 18202+34−51 834 −61 1.87 78 0.74 Abell0628 122.543 35.2958 25139+24−89 666 −38 1.47 43 0.83 Abell0671 127.121 30.4169 14599+19−33 610 −33 1.42 72 0.89 Abell0690 129.815 28.9033 23689+44−23 395 −24 0.85 22 0.95 Abell0695 130.309 32.4174 20251+46−37 456 −32 1.04 16 0.86 Abell0699 131.236 27.7508 25375+35−49 438 −37 0.91 19 0.73 Abell0724 134.600 38.5137 28134+25−55 433 −32 1.00 29 0.94 Abell0727 134.976 39.4389 28571+54−14 423 −29 0.96 33 0.96 Abell0757 138.277 47.7036 15402+27−36 409 −30 0.84 30 0.85 Abell0779 139.962 33.7714 6921+13−33 336 −21 0.79 57 0.75 Abell0819 143.076 9.68861 22872+39−50 536 −37 1.19 31 0.94 Abell0883 147.822 5.48799 21750+109−30 523 −58 1.17 18 0.91 Abell0971 154.997 40.9925 27809+54−43 816 −61 1.88 40 0.87 Abell0999 155.842 12.8466 9618+10−54 271 −17 0.60 25 0.90 Abell1003 156.235 47.8442 18762+44−84 617 −34 1.37 29 0.94 Abell1016 156.762 10.9780 9629+34−9 259 −17 0.60 25 0.91 Abell1024 157.096 3.76341 22067+40−20 532 −34 1.26 35 0.90 Abell1032 157.547 4.03417 20008+26−24 355 −32 0.77 25 0.89 Abell1035 158.092 40.1817 20270+36−34 575 −45 1.34 49 0.97 Abell1066 159.911 5.17444 20708+4−81 826 −44 1.71 95 0.92 Abell1142 165.229 10.5477 10601+30−25 557 −38 1.33 59 0.88 Abell1149 165.769 7.57833 21479+2−64 352 −31 0.85 26 0.94 Abell1169 166.967 43.9506 17532+24−35 433 −32 0.91 35 0.94 Abell1173 167.328 41.5624 22789+23−69 611 −41 1.33 35 0.95 Abell1189 167.775 1.09899 28860+109−60 807 −88 1.76 41 0.94 Abell1190 167.902 40.8417 22610+17−39 706 −30 1.50 77 0.92 Abell1205 168.328 2.53867 22852+17−53 890 −53 2.04 83 0.84 Abell1215 170.100 4.34280 14747+25−10 214 −39 0.48 17 0.89 Abell1238 170.711 1.09389 22140+9−54 544 −41 1.30 60 0.91 Abell1270 172.366 54.0428 20728+49−36 569 −39 1.31 43 0.99 Abell1291 173.092 55.9783 17144+43−60 720 −36 1.58 45 0.97 Abell1318 173.883 55.0767 17185+54−22 360 −25 0.81 22 0.96 Abell1346 175.304 5.74613 29523+41−25 790 −62 1.69 66 0.85 Abell1377 176.883 55.7597 15378+35−33 671 −37 1.47 69 0.96 Abell1383 176.973 54.7089 17855+38−35 456 −23 1.02 35 0.95 Abell1385 177.017 11.5864 25337+49−44 609 −45 1.34 22 0.86 Abell1390 177.378 12.3034 25101+48−29 483 −41 1.06 27 0.86 Abell1424 179.361 5.12000 22736+33−39 617 −43 1.46 63 0.95 Abell1436 180.095 56.2314 19432+39−77 712 −44 1.51 66 0.96 Abell1452 180.802 51.6642 18609+56−58 533 −30 1.30 18 0.91 Abell1459 181.108 1.88281 6010+37−21 527 −48 1.18 65 0.95 Abell1507 183.766 59.8947 18009+9−70 379 −36 0.85 23 0.91 Abell1516 184.729 5.24731 23019+24−84 720 −43 1.49 60 0.90 Abell1552 187.392 11.7733 26495+64−42 442 −32 0.93 20 0.88 Abell1564 188.720 1.78056 23763+30−64 641 −62 1.51 46 0.91 Abell1616 191.817 55.0006 24882+65−93 565 −45 1.32 29 0.87 Abell1620 192.510 -1.53764 25400+64−38 829 −43 1.76 58 0.88 Abell1630 192.942 4.59694 19458+36−33 444 −37 0.98 30 0.90 18 Aguerri et al.: Global Properties of Nearby Galaxy Clusters Table 2. continued. Name α (J2000) δ (J2000) vc σc r200 Ngal C (degrees) (degrees) (km s−1) (km s−1) (Mpc) Abell1650 194.672 -1.76417 25138+86−18 790 −47 1.61 63 0.85 Abell1663 195.717 -2.51782 24953+60−20 729 −40 1.54 72 0.88 Abell1692 198.060 -0.976000 25395+51−47 607 −43 1.32 40 0.89 Abell1728 200.876 11.2960 26977+95−22 824 −62 1.88 50 0.70 Abell1750 202.709 -1.86389 26259+19−14 518 −33 1.15 35 0.95 Abell1767 204.024 59.2042 21174+39−32 885 −40 2.05 109 0.94 Abell1773 205.533 2.24805 23544+31−48 481 −31 1.09 32 0.87 Abell1780 206.149 2.86750 23285+35−22 624 −53 1.45 53 0.90 Abell1783 205.848 55.6261 20550+48−11 383 −32 0.94 33 0.94 Abell1809 208.245 5.16139 23815+39−31 737 −47 1.68 89 0.82 Abell1885 213.431 43.6634 26793+33−50 541 −56 1.23 22 0.92 Abell1999 223.522 54.2682 29841+11−57 463 −54 1.07 24 0.92 Abell2018 225.266 47.2831 26246+93−5 635 −37 1.45 39 0.88 Abell2023 227.496 2.98910 27743+48−18 516 −73 1.12 23 0.86 Abell2026 227.106 -0.267500 27188+62−37 747 −49 1.50 43 0.91 Abell2030 227.844 -0.0857717 27399+38−27 495 −45 1.10 38 0.91 Abell2061 230.317 30.6122 23646+24−19 622 −32 1.43 98 0.83 Abell2067 230.780 30.8703 23039+33−34 917 −46 2.19 118 0.85 Abell2092 233.348 31.1475 20000+42−17 458 −30 0.93 41 0.86 Abell2110 234.953 30.7173 29250+94−28 622 −49 1.25 21 0.83 Abell2122 236.259 36.1161 19793+28−42 826 −47 1.80 91 0.92 Abell2124 236.263 36.1172 19783+41−32 826 −47 1.78 90 0.92 Abell2145 240.094 33.2306 26583+80−23 632 −56 1.40 24 0.87 Abell2149 240.350 53.9061 19564+63−44 459 −27 1.01 20 0.85 Abell2169 243.422 49.1261 17286+30−50 521 −33 1.24 40 0.82 Abell2175 245.132 29.8953 28876+61−64 878 −57 1.76 58 0.87 Abell2199 247.154 39.5244 9118+15−30 747 −19 1.77 269 0.92 Abell2241 254.928 32.6161 29403+70−68 806 −62 1.61 37 0.90 Abell2244 255.663 34.0411 28927+50−57 428 −49 0.99 23 0.94 Abell2245 255.640 33.5056 25686+40−49 535 −39 1.28 39 0.95 Abell2255 258.222 64.0653 24052+19−39 883 −35 1.86 184 0.91 Abell2428 334.065 -9.34139 25207+15−68 433 −37 0.95 26 0.87 Abell2670 358.556 -10.4133 22755+29−20 642 −29 1.46 137 0.94 MACSJ0810.3+4216 122.600 42.2733 19193+44−25 505 −35 1.23 32 0.84 MACSJ1440.0+3707 220.011 37.0839 29402+88−46 587 −49 1.31 18 0.91 NSCJ152902+524945 232.309 52.8433 22063+101−7 652 −43 1.4 45 0.89 NSCJ161123+365846 242.854 36.9700 20221+39−25 485 −39 1.17 30 0.86 RBS1385 215.969 40.2619 24544+40−78 419 −36 0.84 16 0.91 RXCJ0137.2-0912 24.3137 -9.20277 12169+27−30 453 −30 0.93 49 0.92 RXCJ0828.6+3025 127.162 30.4280 14630+50−25 628 −33 1.45 76 0.89 RXCJ0953.6+0142 148.393 1.70550 29450+24−63 584 −59 1.30 22 0.96 RXCJ1115.5+5426 168.887 54.4350 20965+35−66 639 −38 1.35 50 0.94 RXCJ1121.7+0249 170.428 2.81840 14807+25−14 567 −41 1.40 73 0.85 RXCJ1351.7+4622 207.940 46.3668 18937+32−40 531 −27 1.16 40 0.94 RXCJ1424.8+0240 216.159 2.75677 16337+44−57 539 −36 1.19 22 0.91 RXJ1017.7-0002 154.452 -0.0595327 19169+44−28 413 −39 0.90 16 0.91 RXJ1022.1+3830 155.583 38.5308 16300+37−38 591 −33 1.35 51 0.82 RXJ1053.7+5450 163.449 54.8500 21623+23−49 665 −45 1.52 46 0.86 WBL238 146.732 54.4183 13995+51−17 602 −30 1.30 44 0.87 WBL518 220.179 3.45305 8141+19−28 454 −20 1.03 103 0.85 ZwCl0027.0-0036 7.31721 -0.183598 17994+18−44 465 −36 0.99 36 0.85 ZwCl0743.5+3110 116.655 31.0136 17419+49−89 694 −45 1.40 29 0.82 ZwCl1207.5+0542 182.578 5.38500 23137+43−36 580 −40 1.20 40 0.91 Aguerri et al.: Global Properties of Nearby Galaxy Clusters 19 Table 2. continued. Name α (J2000) δ (J2000) vc σc r200 Ngal C (degrees) (degrees) (km s−1) (km s−1) (Mpc) ZwCl1215.1+0400 184.422 3.66040 23229+22−40 955 −39 2.17 130 0.90 ZwCl1316.4-0044 199.816 -0.907816 24972+56−31 557 −20 1.16 38 0.87 ZwCl1730.4+5829 261.856 58.4749 8379+26−36 491 −22 1.02 33 0.86 Introduction Galaxy cluster Sample Cluster Membership Cluster global parameters Corrections to line-of-sight velocities Comparison with other methods Lx- relation Redshift distribution and sample completeness Morphological Segregation Velocity Dispersion Profiles Fraction of blue galaxies Adopted aperture and limiting magnitude Colour-magnitude diagrams Calculation the blue fraction of galaxies Discussion Conclusions

Optical implementation and entanglement distribution in Gaussian valence bond states Gerardo Adesso Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom; Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, 00185 Roma, Italy; Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy Marie Ericsson Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom PACS numbers: 42.50.Dv, 03.67.Mn, 03.67.Hk, 03.65.Ud. Abstract. We study Gaussian valence bond states of continuous variable systems, obtained as the outputs of projection operations from an ancillary space of M infinitely entangled bonds connecting neighboring sites, applied at each of N sites of an harmonic chain. The entanglement distribution in Gaussian valence bond states can be controlled by varying the input amount of entanglement engineered in a (2M + 1)-mode Gaussian state known as the building block, which is isomorphic to the projector applied at a given site. We show how this mechanism can be interpreted in terms of multiple entanglement swapping from the chain of ancillary bonds, through the building blocks. We provide optical schemes to produce bisymmetric three-mode Gaussian building blocks (which correspond to a single bond, M = 1), and study the entanglement structure in the output Gaussian valence bond states. The usefulness of such states for quantum communication protocols with continuous variables, like telecloning and teleportation networks, is finally discussed. http://arxiv.org/abs/0704.1580v1 Optical implementation and entanglement distribution in Gaussian valence bond states 2 1. Introduction Quantum information aims at the treatment and transport of information in ways forbidden by classical physics. For this goal, continuous variables (CV) of atoms and light have emerged as a powerful tool [1]. In this context, entanglement is an essential resource. Recently, the valence bond formalism, originally developed for spin systems [2], has been generalized to the CV scenario [3, 4] for the special class of Gaussian states, which play a central role in theoretical and practical CV quantum information and communication [5]. In this work we analyze feasible implementations of Gaussian valence bond states (GVBS) for quantum communication between many users in a CV setting, as enabled by their peculiar structure of distributed entanglement [4]. After recalling the necessary notation (Sec. 2) and the construction of Gaussian valence bond states [3] (Sec. 3), we discuss the characterization of entanglement and its distribution in such states as regulated by the entanglement properties of simpler states involved in the valence bond construction [4] (Sec. 4). We then focus on the realization of GVBS by means of quantum optics, provide a scheme for their state engineering (Sec. 5), and discuss the applications of such resources in the context of CV telecloning [6, 7] on multimode harmonic rings (Sec. 6). 2. Continuous variable systems and Gaussian states A CV system [1, 5] is described by a Hilbert space H = i=1 Hi resulting from the tensor product of infinite dimensional Fock spaces Hi’s. Let ai and a†i be the annihilation and creation operators acting on Hi (ladder operators), and q̂i = (ai + a†i ) and p̂i = (ai − a i )/i be the related quadrature phase operators. Let R̂ = (x̂1, p̂1, . . . , q̂N , p̂N ) denote the vector of the operators q̂i and p̂i. The canonical commutation relations for the R̂i can be expressed in terms of the symplectic form Ω as [R̂i, R̂j ] = 2iΩij , with Ω ≡ ω⊕N , ω ≡ The state of a CV system can be equivalently described by quasi-probability distributions defined on the 2N -dimensional space associated to the quadratic form Ω, known as quantum phase space. In the phase space picture, the tensor productH = i Hi of the Hilbert spaces Hi’s of the N modes results in the direct sum Λ = i Λi of the phase spaces Λi’s. States with Gaussian quasi-probability distributions are referred to as Gaussian states. Such states are at the heart of information processing in CV systems [1, 5] and are the subject of our analysis. By definition, a Gaussian state is completely characterized by the first and second statistical moments of the field operators, which will be denoted, respectively, by the vector of first moments R̄ ≡ 〈R̂1〉, 〈R̂2〉, . . . , 〈R̂2N−1〉, 〈R̂2N 〉 and the covariance matrix (CM) γ of elements γij ≡ 〈R̂iR̂j + R̂jR̂i〉 − 〈R̂i〉〈R̂j〉 . (1) Coherent states, resulting from the application of displacement operatorsDY = e iY TΩR̂ (Y ∈ R2n) to the vacuum state, are Gaussian states with CM γ = 1 and first statistical moments R̄ = Y . First moments can be arbitrarily adjusted by local unitary operations (displacements), which cannot affect any property related to entropy or entanglement. They Optical implementation and entanglement distribution in Gaussian valence bond states 3 can thus be assumed zero without any loss of generality. A N -mode Gaussian state will be completely characterized by its real, symmetric, 2N × 2N CM γ. The canonical commutation relations and the positivity of the density matrix ρ of a Gaussian state imply the bona fide condition γ + iΩ ≥ 0 , (2) as a necessary and sufficient constraint the matrix γ has to fulfill to be a CM corresponding to a physical state [8, 9]. Note that the previous condition is necessary for the CM of any (generally non Gaussian) state, as it generalizes to many modes the Robertson-Schrödinger uncertainty relation [10]. A major role in the theoretical and experimental manipulation of Gaussian states is played by unitary operations which preserve the Gaussian character of the states on which they act. Such operations are all those generated by terms of the first and second order in the field operators. As a consequence of the Stone-Von Neumann theorem, any such operation at the Hilbert space level corresponds, in phase space, to a symplectic transformation, i.e. to a linear transformation S which preserves the symplectic form Ω, so that Ω = STΩS, i.e. it preserves the commutators between the different operators. Symplectic transformations on a 2N -dimensional phase space form the (real) symplectic group, denoted by Sp(2N,R). Such transformations act linearly on first moments and “by congruence” on the CM (i.e. so that γ 7→ SγST ). One has DetS = 1, ∀S ∈ Sp(2N,R). A crucial symplectic operation is the one achieving the normal mode decomposition. Due to Williamson theorem [11], any N -mode Gaussian state can be symplectically diagonalized in phase space, so that its CM is brought in the form ν, such that SγST = ν, with ν = diag {ν1, ν1, . . . νN , νN}. The set {νi} of the positive-defined eigenvalues of |iΩγ| constitutes the symplectic spectrum of γ and its elements, the so-called symplectic eigenvalues, must fulfill the conditions νi ≥ 1, following from the uncertainty principle Eq. (2) and ensuring positivity of the density matrix ρ corresponding to γ. Ideal beam splitters, phase shifters and squeezers are described by symplectic transformations. In particular, a phase-free two-mode squeezing transformation, which corresponds to squeezing the first mode (say i) in one quadrature (say momentum, p̂i) and the second mode (say j) in the orthogonal quadrature (say position, q̂j) with the same degree of squeezing r, can be represented in phase space by the symplectic transformation Sij(r) = diag{exp r, exp−r, exp−r, exp r} . (3) These trasformations occur for instance in parametric down conversions [12]. Another important example of symplectic operation is the ideal (phase-free) beam splitter, which acts on a pair of modes i and j as [13] B̂ij(θ) : âi 7→ âi cos θ + âj sin θ âj 7→ âi sin θ − âj cos θ and corresponds to a rotation in phase space of the form Bij(θ) = cos(θ) 0 sin(θ) 0 0 cos(θ) 0 sin(θ) sin(θ) 0 − cos(θ) 0 0 sin(θ) 0 − cos(θ) . (4) The transmittivity τ of the beam splitter is given by τ = cos2(θ) so that a 50:50 beam splitter (τ = 1/2) amounts to a phase-space rotation of π/4. Optical implementation and entanglement distribution in Gaussian valence bond states 4 The combined application of a two-mode squeezing and a 50:50 beam splitter realizes the entangling twin-beam transformation [14] Tij(r) = Bij(π/4) · Sij(r) , (5) which, if applied to two uncorrelated vacuum modes i and j (whose initial CM is the identity matrix), results in the production of a pure two-mode squeezed Gaussian state with CM σi,j(r) = Tij(r)T ij (r) given by σi,j(r) = cosh(2r) 0 sinh(2r) 0 0 cosh(2r) 0 − sinh(2r) sinh(2r) 0 cosh(2r) 0 0 − sinh(2r) 0 cosh(2r) . (6) The CV entanglement in the state σi,j(r) increases unboundedly as a function of r, and in the limit r → ∞ Eq. (6) approaches the (unnormalizable) Einstein-Podolski-Rosen (EPR) state [15], simultaneous eigenstate of relative position and total momentum of the two modes i and j. Concerning entanglement in general, the “positivity of partial transposition” (PPT) criterion states that a Gaussian CM γ is separable (with respect to a 1×N bipartition) if and only if the partially transposed CM γ̃ satisfies the uncertainty principle Eq. (2) [9, 16]. In phase space, partial transposition amounts to a mirror reflection of one quadrature associated to the single-mode partition. If {ν̃i} is the symplectic spectrum of the partially transposed CM γ̃, then a (N + 1)-mode Gaussian state with CM γ is separable if and only if ν̃i ≥ 1 ∀ i. A proper measure of CV entanglement is the logarithmic negativityEN [17], which is readily computed in terms of the symplectic spectrum ν̃i of γ̃ as EN = − i: ν̃i<1 log ν̃i . (7) Such an entanglement monotone [18] quantifies the extent to which the PPT condition ν̃i ≥ 1 is violated. For 1 × N Gaussian states, only the smallest symplectic eigenvalue ν̃− of the partially transposed CM can be smaller than one [10], thus simplifying the expression of EN : then the PPT criterion simply yields that γ is entangled as soon as ν̃− < 1, and infinite entanglement (accompanied by infinite energy in the state) is reached for ν̃− → 0+. For 1× 1 Gaussian states γi,j symmetric under mode permutations, the entanglement of formation EF is computable as well via the formula [19] EF (γi,j) = max{0, f(ν̃ − )} , (8) f(x) = (1 + x)2 (1 + x)2 − (1− x) (1− x)2 Being a monotonically decreasing function of the smallest symplectic eigenvalue ν̃i,j− of the partial transpose γ̃i,j of γi,j , the entanglement of formation is completely equivalent to the logarithmic negativity in this case. For a two-mode state, ν̃i,j can be computed from the symplectic invariants of the state [20], and experimentally estimated with measures of global and local purities [21] (the purity µ = Tr ρ2 of a Gaussian state ρ with CM γ is equal to µ = (Det γ)−1/2). 3. Gaussian valence bond states Let us review the basic definitions and notations for GVBS, as adopted in Ref. [4]. The so-called matrix product Gaussian states introduced in Ref. [3] are N -mode states obtained Optical implementation and entanglement distribution in Gaussian valence bond states 5Optical implementation and entanglement distribution in Gaussian valence bond states i i+1 Figure 1. Gaussian valence bond states. Γin is the state of N EPR bonds and γ is the three- mode building block. After the EPR measurements (depicted as curly brackets), the chain of modes γ collapses into a Gaussian valence bond state with global state Γout. See also Ref. [4]. by taking a fixed number, M , of infinitely entangled ancillary bonds (EPR pairs) shared by adjacent sites, and applying an arbitrary 2M → 1 Gaussian operation on each site i = 1, . . . , N . Such a construction, more properly definable as a “valence bond” picture for Gaussian states, can be better understood by resorting to the Jamiolkowski isomorphism between quantum operations and quantum states [22]. In this framework, one starts with a chain ofN Gaussian states of 2M +1 modes (the building blocks). The global Gaussian state of the chain is described by a CM Γ = i=1 γ [i]. As the interest in GVBS lies mainly in their connections with ground states of Hamiltonians invariant under translation [3], we can focus on pure (Detγ [i] = 1), translationally invariant (γ[i] ≡ γ ∀i) GVBS. Moreover, in this work we consider single-bonded GVBS, i.e. withM = 1. This is also physically motivated in view of experimental implementations of GVBS, as more than one EPR bond would result in a building block with five or more correlated modes, which appears technologically demanding. Under the considered prescriptions, the building block γ is a pure Gaussian state of three modes. As we aim to construct a translationally invariant state, it is convenient to consider a γ whose first two modes, which will be combined with two identical halves of consecutive EPR bonds (see Fig. 3), have the same reduced CM. This yields a pure, three-mode Gaussian building block with the property of being bisymmetric [23], that is with a CM invariant under permutation of the first two modes. This choice of the building block is further justified by the fact that, among all pure three-mode Gaussian states, bisymmetric states maximize the genuine tripartite entanglement [24]: no entanglement is thus wasted in the projection process. The 6 × 6 CM γ of the building block can be written as follows in terms of 2 × 2 submatrices, γs εss εsx εTss γs εsx εTsx ε sx γx . (9) The 4×4 CM of the first two modes (each of them having reduced CM γs) will be denoted by γss, and will be regarded as the input port of the building block. On the other hand, the CM γx of mode 3 will play the role of the output port. The intermodal correlations are encoded in the off-diagonal ε matrices. Without loss of generality, we can assume γ to be, up to local unitary operations, in the standard form [24] with γs = diag{s, s} , γx = diag{x, x} , (10) Optical implementation and entanglement distribution in Gaussian valence bond states 6 εss = diag{t+, t−} , εsx = diag{u+, u−} ; x2 − 1± 16s4 − 8(x2 + 1)s2 + (x2 − 1)2 x2 − 1 (x − 2s)2 − 1± (x+ 2s)2 − 1 The valence bond construction works as follows (see Fig. 3). The global CM Γ = i=1 γ acts as the projector from the state Γ in of the N ancillary EPR pairs, to the final N -mode GVBS Γout. This is realized by collapsing the state Γin, transposed in phase space, with the ‘input port’ Γss = i γss of Γ, so that the ‘output port’ Γx = i γx turns into the desired Γout. Here collapsing means that, at each site, the two two-mode states, each constituted by one mode (1 or 2) of γss and one half of the EPR bond between site i and its neighbor (i − 1 or i + 1, respectively), undergo an “EPR measurement” i.e. are projected onto the infinitely entangled EPR state [22, 3]. An EPR pair between modes i and j can be described, see Eq. (6), as a two-mode squeezed state σi,j(r) in the limit of infinite squeezing (r → ∞). The input state is then Γin = limr→∞ i σi,i+1(r), where we have set periodic boundary conditions so that N + 1 = 1 in labeling the sites. The projection corresponds mathematically to taking a Schur complement (see Refs. [4, 3, 22] for details), yielding an output pure GVBS of N modes on a ring with a CM out = Γx − ΓTsx(Γss + θΓ θ)−1Γsx , (11) where Γsx = i γsx, and θ = i diag{1, −1, 1, −1} represents transposition in phase space (q̂i → q̂i, p̂i → −p̂i). Within the building block picture, the valence bond construction can be in toto understood as a multiple CV entanglement swapping [25], as shown in Fig. 3: the GVBS is created as the entanglement in the bonds is swapped to the chain of output modes via teleportation [26] through the input port of the building blocks. It is thus clear that at a given initialization of the output port (i.e. at fixed x), changing the properties of the input port (i.e. varying s), which corresponds to implementing different Gaussian projections from the ancillary space to the physical one, will affect the structure and entanglement properties of the target GVBS. This link is explored in the following section. 4. Entanglement distribution In Ref. [4] the quantum correlations of GVBS of the form Eq. (11) have been studied, and related to the entanglement properties of the building block γ. Let us first recall the characterization of entanglement in the latter. As a consequence of the uncertainty principle Eq. (2), the CM Eq. (9) of the building block describes a physical state if [24] x ≥ 1 , s ≥ smin ≡ . (12) Let us keep the output parameter x fixed. Straightforward applications of the PPT separability conditions, and consequent calculations of the logarithmic negativity Eq. (7), reveal that the entanglement in the CM γss of the first two modes (input port) is monotonically increasing as a function of s, ranging from the case s = smin when γss is separable to the limit s → ∞ when the block γss is infinitely entangled. Accordingly, the entanglement between each of the first two modes γs of γ and the third one γx decreases with s. One can also show that the genuine tripartite entanglement in the building block increases with the difference s − smin [24]. The entanglement properties of the building block are summarized in Fig. 4. Optical implementation and entanglement distribution in Gaussian valence bond states 7 Figure 2. How a Gaussian valence bond state is created via continuous-variable entanglement swapping. At each step, Alice attempts to teleport her mode 0 (half of an EPR bond, depicted in yellow) to Bob, exploiting as an entangled resource two of the three modes of the building block (denoted at each step by 1 and 2). The curly bracket denotes homodyne detection, which together with classical communication and conditional displacement at Bob’s side achieves teleportation. The state will be approximately recovered in mode 2, owned by Bob. Since mode 0, at each step, is entangled with the respective half of an EPR bond, the process swaps entanglement from the ancillary chain of the EPR bonds to the modes in the building block. The picture has to be followed column-wise. For ease of clarity, we depict the process as constituted by two sequences: in the first sequence [frames (1) to (4)] modes 1 and 2 are the two input modes of the building block (depicted in blue); in the second sequence [frames (5) to (8)] modes 1 and 2 are respectively an input and an output mode of the building block. As a result of the multiple entanglement swapping [frame (9)] the chain of the output modes (depicted in red), initially in a product state, is transformed into a translationally invariant Gaussian valence bond state, possessing in general multipartite entanglement among all the modes (depicted in magenta). The main question addressed in Ref. [4] is how the initial entanglement in the building block γ redistributes in the Gaussian MPS Γout. The answer is that the more entanglement one prepares in the input port γss, the longer the range of pairwise quantum correlations in the output GVBS is, as pictorially shown in Fig. 4. In more detail, let us consider first a building block γ with s = smin = (x + 1)/2. In this case, a separability analysis shows that, for an arbitrary numberN of modes in the GVBS chain, the target state Γout exhibits bipartite entanglement only between nearest neighbor modes, for any value of x > 1 (for x = 1 we trivially obtain a product state). In fact, each reduced two-mode block γouti,j is separable for |i− j| > 1. With increasing s in the choice of the building block, one finds that in the target GVBS the correlations start to extend smoothly to distant modes. A series of thresholds sk can be Optical implementation and entanglement distribution in Gaussian valence bond states 8 Figure 3. Entanglement properties of the three-mode building block γ, Eq. (9), of the Gaussian valence bond construction, as functions of the standard form covariances x and d ≡ s − smin. (a) Bipartite entanglement, as quantified by the logarithmic negativity, between the first two input-port modes 1 and 2; (b) Bipartite entanglement, as quantified by the logarithmic negativity, between each of the first two modes and the output-port mode 3; (c) Genuine tripartite entanglement, as quantified by the residual Gaussian contangle [27, 24], among all the three modes. found such that for s > sk, two given modes i and j with |i − j| ≤ k are entangled. While trivially s1(x) = smin for anyN (notice that nearest neighbors are entangled also for s = s1), the entanglement boundaries for k > 1 are in general different functions of x, depending on the number of modes. We observe however a certain regularity in the process: sk(x,N) always increases with the integer k. Very remarkably, this means that the maximum range of bipartite entanglement between two modes, or equivalently the maximum distribution of multipartite entanglement, in a GVBS on a translationally invariant ring, is monotonically related to the amount of entanglement in the reduced two-mode input port of the building block [4]. Moreover, no complete transfer of entanglement to more distant modes occurs: closer sites remain still entangled even when correlations between farther pairs arise. The most interesting feature is perhaps obtained when infinite entanglement is fed in the input port (s → ∞): in this limit, the output GVBS turns out to be a fully symmetric, permutation-invariant, N -mode Gaussian state. This means that each individual mode is equally entangled with any other, no matter how distant they are [4]. These states, being thus Optical implementation and entanglement distribution in Gaussian valence bond states 9 Figure 4. Pictorial representation of the entanglement between a probe (green) mode and its neighbor (magenta) modes on an harmonic ring with an underlying valence bond structure. As soon as the parameter s (encoding entanglement in the input port of the valence bond building block) is increased, pairwise entanglement between the probe mode and its farther and farther neighbors gradually appears in the corresponding output Gaussian valence bond states. By translational invariance, each mode exhibits the same entanglement structure with its respective neighbors. In the limit s → ∞, every single mode becomes equally entangled with every other single mode on the ring, independently of their relative distance: the Gaussian valence bond state is in this case fully symmetric. built by a symmetric distribution of infinite pairwise entanglement among multiple modes, achieve maximum genuine multiparty entanglement among all Gaussian states (at a given energy) while keeping the strongest possible bipartite one in any pair, a property known as monogamous but promiscuous entanglement sharing [27]. Keeping Fig. 3 in mind, we can conclude that having the two input modes initially entangled in the building blocks, increases the efficiency of the entanglement-swapping mechanism, inducing correlations between distant modes on the GVBS chain, which enable to store and distribute joint information. In the asymptotic limit of an infinitely entangled input port of the building block, the entanglement range in the target GVBS states is engineered to be maximum, and communication between any two modes, independently of their distance, is enabled nonclassically. In the next sections, we investigate the possibility of producing GVBS with linear optics, and discuss with a specific example the usefulness of such resource states for multiparty CV quantum communication protocols such as telecloning [6] and teleportation networks [13]. 5. Optical implementation of Gaussian valence bond states The power of describing the production of GVBS in terms of physical states, the building blocks, rather than in terms of arbitrary non-unitary Gaussian maps, lies not only in the immediacy of the analytical treatment. From a practical point of view, the recipe of Fig. 3 can be directly implemented to produce GVBS experimentally in the domain of quantum optics. We first note that the EPR measurements are realized by the standard toolbox of a beamsplitter plus homodyne detection [22], as demonstrated in several CV teleportation experiments [28]. The next ingredient to produce aN -mode GVBS is constituted byN copies of the three- mode building block γ. We provide here an easy scheme (see also Refs. [6, 29]) to realize bisymmetric three-mode Gaussian states of the form Eq. (9). As shown in Fig. 5(a), one can start from three vacuum modes and first apply a twin-beam operation to modes 1 and 3, characterized by a squeezing r13, then apply another twin-beam operation to modes 1 and 2, Optical implementation and entanglement distribution in Gaussian valence bond states 10Optical implementation and entanglement distribution in Gaussian valence bond states 50:50 γγγγ0000 γγγγ0000 γγγγ0000 Figure 5. Optical production of bisymmetric three-mode Gaussian states, used as buildingFigure 5. Optical production of bisymmetric three-mode Gaussian states, used as building blocks for the valence bond construction. (a) Three initial vacuum modes are entangled through two sequential twin-beam boxes, the first (parametrized by a squeezing degree r13) acting on modes 1 and 3, and the second (parametrized by a squeezing degree r12) acting on the transformed mode 1 and mode 2. The output is a pure three-mode Gaussian state whose covariance matrix is equivalent, up to local unitary operations, to the standard form given in Eq. (9). (b) Detail of the entangling twin-beam transformation. One input mode is squeezed in a quadrature, say momentum, of a degree r (this transformation is denoted by stretching arrows→| |←); the other input mode is squeezed in the orthogonal quadrature, say position, of the same amount (this anti-squeezing transformation is denoted by the corresponding rotated symbol). Then the two squeezed modes are combined at a 50:50 beam-splitter. If the input modes are both in the vacuum state, the output is a pure two-mode squeezed Gaussian state, with entanglement proportional to the degree of squeezing r. parametrized by r12. The symplectic operation describing the twin-beam transformation (two- mode squeezing plus balanced beam splitter) is given by Eq. (5) and pictorially represented in Fig. 5(b). The output of this optical network is a pure, bisymmetric, three-mode Gaussian state with a CM γB = T12(r12)T13(r13)T 13(r13)T 12(r12) of the form Eq. (9), with γs = diag e−2r12 e4r12 cosh (2r13) + 1 e−2r12 cosh (2r13) + e γx = diag {cosh (2r13) , cosh (2r13)} , εss = diag e−2r12 e4r12 cosh (2r13)− 1 e−2r12 cosh (2r13) − e4r12 εsx = diag 2er12 cosh (r13) sinh (r13) , − 2e−r12 cosh (r13) sinh (r13) By means of local symplectic operations (unitary on the Hilbert space), like additional single- mode squeezings, the CM γB can be brought in the standard form of Eq. (10), from which Optical implementation and entanglement distribution in Gaussian valence bond states 11 one has r13 = arccos x+ 1√ , r12 = arccos −x3 + 2x2 + 4s2x− x For a given r13 (i.e. at fixed x), the quantity r12 is a monotonic function of the standard-form covariance s, so this squeezing parameter which enters in the production of the building block (see Fig. 5) directly regulates the entanglement distribution in the target GVBS, as discussed in Sec. 4. The only unfeasible part of the scheme seems constituted by the ancillary EPR pairs. But are infinitely entangled bonds truly necessary? In Ref. [4] the possibility is considered of using a Γin given by the direct sum of two-mode squeezed states of Eq. (6), but with finite r. Repeating the previous analysis to investigate the entanglement properties of the resulting GVBS with finitely entangled bonds, it is found that, at fixed (x, s), the entanglement in the various partitions is degraded as r decreases, as somehow expected. Crucially, this does not affect the connection between input entanglement and output correlation length. Numerical investigations show that, while the thresholds sk for the onset of entanglement between distant pairs are quantitatively modified – a bigger s is required at a given x to compensate the less entangled bonds – the overall structure stays untouched. This ensures that the possibility of engineering the entanglement structure in GVBS via the properties of the building block is robust against imperfect resources, definitely meaning that the presented scheme is feasible. Alternatively, one could from the beginning observe that the triples consisting of two projective measurements and one EPR pair can be replaced by a single projection onto the EPR state, applied at each site i between the input mode 2 of the building block and the consecutive input mode 1 of the building block of site i+1 [3]. The output of all the homodyne measurements will conditionally realize the target GVBS. 6. Telecloning with Gaussian valence bond resources The protocol of CV quantum telecloning [6] amongN parties is defined as a process in which one of them (Alice) owns an unknown coherent state, and wants to distribute her state to all the other N − 1 remote parties. The telecloning is achieved by a succession of standard two-party CV teleportations [26] between the sender Alice and each of the N − 1 remote receivers, exploiting each time the corresponding reduced two-mode state shared as resource by the selected pair of parties. The 1 → 2 CV telecloning of unknown coherent states has been recently demonstrated experimentally [7]. The no-cloning theorem [30] yields that the N − 1 remote clones can resemble the original input state only to a certain extent. The fidelity, which quantifies the success of a teleportation experiment, is defined as F ≡ 〈ψin|ρout|ψin〉, where “in” and “out” denote the input and the output state. F reaches unity only for a perfect state transfer, ρout = |ψin〉〈ψin|. Without using entanglement, by purely classical communication, an average fidelity of Fcl = 1/2 is the best that can be achieved if the alphabet of input states includes all coherent states with even weight [31]. The sufficient fidelity criterion states that, if teleportation is performed with F > Fcl, then the two parties exploited an entangled state [31]. The converse is generally false, i.e. some entangled resources may yield lower-than-classical fidelities. In Ref. [32] it has been shown, however, that if the fidelity is optimized over all possible local unitary operations performed on the shared Gaussian resource (which preserve entanglement by definition), then it becomes equivalent, both qualitatively and quantitatively, to the entanglement in the resource. Optical implementation and entanglement distribution in Gaussian valence bond states 12 Let us also recall that the fidelity of CV two-user teleportation [26] of arbitrary single- mode Gaussian states with CM γin (equal to the identity for coherent states) exploiting two- mode Gaussian resources with CM γab = γa εab εTab γb , can be computed [33] as F = 2√ , Σ ≡ 2γin + ξγaξ + γb + ξεab + εTabξ , (14) with ξ = diag{−1 , 1}. We can now consider the general setting of 1 → N − 1 telecloning, where N parties share a N -mode GVBS as an entangled resource, and one of them plays the role of Alice (the sender) distributing imperfect copies of unknown coherent states to all the N − 1 receivers. For any N , the fidelity can be easily computed from the reduced two-mode CMs via Eq. (14) and will depend, for translationally invariant states, on the relative distance between the two considered modes. In this work we focus on a practical example of a GVBS on a translationally invariant harmonic ring, with N = 6 modes. As shown in the previous section, these states can be produced with the current optical technology. They are completely characterized, up to local unitary operations, by a 12 × 12 CM analytically obtained from Eq. (11) by considering the building block in standard form Eq. (9), whose elements are algebraic functions of s and x here omitted for brevity (as no particular insight is gained from their explicit expressions). First of all we can construct the reduced CMs γouti,i+k of two modes with distance k, and evaluate for each k the respective symplectic eigenvalue ν̃i,i+k− of the corresponding partial transpose. The entanglement condition s > sk will correspond to the inequality ν̃ i,i+k − < 1. With this conditions one finds that s2(x) is the only acceptable solution to the equation: 72s 12(x2+1)s6+(−34x4+28x2−34)s4+(x6−5x4−5x2+1)s2+(x2−1)2(x4−6x2+1) = 0, while for the next-next-nearest neighbors threshold one has simply s3(x) = x. This enables us to classify the entanglement distribution and, more specifically, to observe the interaction scale in the GVBS Γout: as discussed in Sec. 4 and explicitly shown in Ref. [4], by increasing initial entanglement in γss one can gradually switch on pairwise quantum correlations between more and more distant sites. Accordingly, it is now interesting to test whether this entanglement is useful to achieve nonclassical telecloning towards distant receivers. In this specific instance, Alice will send two identical (approximate) clones to her nearest neighbors, two other identical clones (with in principle different fidelity than the previous case) to her next-nearest neighbors, and one final clone to the most distant site. The fidelities for the three transmissions can be computed from Eq. (14) and are plotted in Fig. 6(a). For s = smin, obviously, only the two nearest neighbor clones can be teleported with nonclassical fidelity, as the reduced states of more distant pairs are separable. With increasing s also the state transfer to more distant sites is enabled with nonclassical efficiency, but not in the whole region of the space of parameters s and x in which the corresponding two-mode resources are entangled. As mentioned before, one can optimize the telecloning fidelity considering resources prepared in a different way but whose CM can be brought by local unitary operations (single- mode symplectic transformations) in the standard form of Eq. (11). For GVBS resources, this local-unitary freedom can be transferred to the preparation of the building block. A more general γ locally equivalent to the standard form given in Eq. (10), can be realized by complementing the presented state engineering scheme for the three-mode building block as in Eq. (13) [see Fig. 5(a)], with additional single-mode rotations and squeezing transformations aimed at increasing the output fidelity in the target GVBS states, while keeping both the entanglement in the building block and consequently the entanglement in the final GVBS unchanged by definition. Optical implementation and entanglement distribution in Gaussian valence bond states 13 Figure 6. 1 → 5 quantum telecloning of unknown coherent states exploiting a six-mode translationally invariant Gaussian valence bond state as a shared resource. Alice owns mode i. Fidelities F for distributing clones to modes j such as k = |i − j| are plotted for k = 1 [(a),(d)]; k = 2 [(b),(e)]; and k = 3 [(c),(f)], as functions of the local invariants s and x of the building block. In the first row [(a)–(c)] the fidelities are achieved exploiting the non-optimized Gaussian valence bond resource in standard form. In the second row [(d)– (f)] fidelities optimized over local unitary operations on the resource are displayed, which are equivalent to the entanglement in the corresponding reduced two-mode states (see, as a comparison, Fig. 3 in Ref. [4]). Only nonclassical values of the fidelities (F > 0.5) are shown. The optimal telecloning fidelity, obtained in this way exploiting the results of Ref. [32], is plotted in Fig. 6(b) for the three teleportations between modes i and j with k = |i − j| = 1, 2, 3. In this case, one immediately recovers a non-classical fidelity as soon as the separability condition s ≤ sk is violated in the corresponding resources. Moreover, the optimal telecloning fidelity at a given k is itself a quantitative measure of the entanglement in the reduced two-mode resource, being equal to [32] Foptk = 1/(1 + ν̃ i,i+k − ) , (15) where ν̃i,i+k− is the smallest symplectic eigenvalue of the partially transposed CM in the corresponding bipartition. The optimal fidelity is thus completely equivalent to the entanglement of formation Eq. (8) and to the logarithmic negativity Eq. (7). In the limit s → ∞, as discussed in Sec. 4, the GVBS become fully permutation- invariant for anyN . Consequently, the (optimized and non-optimized) telecloning fidelity for distributing coherent states is equal for any pair of sender-receiver parties. These resources are thus useful for 1 → N−1 symmetric telecloning. However, due to the monogamy constraints on distribution of CV entanglement [27], this two-party fidelity will decrease with increasing N , vanishing in the limit N → ∞ where the resources become completely separable. In this Optical implementation and entanglement distribution in Gaussian valence bond states 14 respect, it is worth pointing out that the fully symmetric GVBS resources are more useful for teleportation networks [13, 34], where N − 2 parties first perform local measurements (momentum detections) on their single-mode portion of the entangled resource to concentrate as much entanglement as possible onto the two-mode state of Alice and Bob, who can accomplish non-classical teleportation (after the outcomes of the N − 2 measurements are classically communicated to Bob). In this case, the optimal fidelity of N -user teleportation network is an estimator of multipartite entanglement in the shared N -mode resource [32], which is indeed a GVBS obtained from an infinitely entangled building block. 7. Conclusion The valence bond picture is a valuable framework to study the structure of correlations in quantum states of harmonic lattices. In fact, the motivation for such a formalism is quite different from the finite-dimensional case, where valence bond/matrix product states are useful to efficiently approximate ground states of N -body systems – generally described by a number of parameters exponential in N – with polynomial resources [2]. In continuous variable systems, the key feature of GVBS lies in the understanding of their entanglement distribution as governed by the properties of simpler structures [4]. This has also experimental implications giving a robust recipe to engineer correlations in many-body Gaussian states from feasible operations on the building blocks. We have provided a simple scheme to produce bisymmetric three-mode building blocks with linear optics, and discussed the subsequent implementation of the valence bond construction. We have also investigated the usefulness of such GVBS as resources for nonclassical communication, like telecloning of unknown coherent states to distant receivers on a harmonic ring. It would be interesting to employ the valence bond picture to describe quantum computation with continuous-variable cluster states [35], and to devise efficient protocols for its optical implementation. Acknowledgments This work is supported by MIUR (Italy) and by the European Union through the Integrated Project RESQ (IST-2001-37559), QAP (IST-3-015848), SCALA (CT-015714), and SECOQC. References [1] S. L. Braunstein and P. van Loock, Rev. Mod. 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Adesso, A. Serafini, and F. Illuminati, New J. Phys. 9, 60 (2007). [30] W. K. Wootters and W. H. Zurek, Nature 299, 802 (1982). [31] S. L. Braunstein, C. A. Fuchs and H. J. Kimble, J. Mod. Opt. 47, 267 (2000); K. Hammerer, M. M. Wolf, E. S. Polzik, and J. I. Cirac, Phys. Rev. Lett. 94, 150503 (2005). [32] G. Adesso and F. Illuminati, Phys. Rev. Lett. 95, 150503 (2005). [33] J. Fiurás̆ek, Phys. Rev. A 66, 012304 (2002). [34] H. Yonezawa, T. Aoki, and A. Furusawa, Nature 431, 430 (2004). [35] N. C. Menicucci et al., Phys. Rev. Lett. 97, 110501 (2005); P. Van Loock, J. Opt. Soc. Am. B 24, 340 (2007). http://arxiv.org/abs/quant-ph/0703277 Introduction Continuous variable systems and Gaussian states Gaussian valence bond states Entanglement distribution Optical implementation of Gaussian valence bond states Telecloning with Gaussian valence bond resources Conclusion

We study Gaussian valence bond states of continuous variable systems, obtained as the outputs of projection operations from an ancillary space of M infinitely entangled bonds connecting neighboring sites, applied at each of $N$ sites of an harmonic chain. The entanglement distribution in Gaussian valence bond states can be controlled by varying the input amount of entanglement engineered in a (2M+1)-mode Gaussian state known as the building block, which is isomorphic to the projector applied at a given site. We show how this mechanism can be interpreted in terms of multiple entanglement swapping from the chain of ancillary bonds, through the building blocks. We provide optical schemes to produce bisymmetric three-mode Gaussian building blocks (which correspond to a single bond, M=1), and study the entanglement structure in the output Gaussian valence bond states. The usefulness of such states for quantum communication protocols with continuous variables, like telecloning and teleportation networks, is finally discussed.

Introduction Quantum information aims at the treatment and transport of information in ways forbidden by classical physics. For this goal, continuous variables (CV) of atoms and light have emerged as a powerful tool [1]. In this context, entanglement is an essential resource. Recently, the valence bond formalism, originally developed for spin systems [2], has been generalized to the CV scenario [3, 4] for the special class of Gaussian states, which play a central role in theoretical and practical CV quantum information and communication [5]. In this work we analyze feasible implementations of Gaussian valence bond states (GVBS) for quantum communication between many users in a CV setting, as enabled by their peculiar structure of distributed entanglement [4]. After recalling the necessary notation (Sec. 2) and the construction of Gaussian valence bond states [3] (Sec. 3), we discuss the characterization of entanglement and its distribution in such states as regulated by the entanglement properties of simpler states involved in the valence bond construction [4] (Sec. 4). We then focus on the realization of GVBS by means of quantum optics, provide a scheme for their state engineering (Sec. 5), and discuss the applications of such resources in the context of CV telecloning [6, 7] on multimode harmonic rings (Sec. 6). 2. Continuous variable systems and Gaussian states A CV system [1, 5] is described by a Hilbert space H = i=1 Hi resulting from the tensor product of infinite dimensional Fock spaces Hi’s. Let ai and a†i be the annihilation and creation operators acting on Hi (ladder operators), and q̂i = (ai + a†i ) and p̂i = (ai − a i )/i be the related quadrature phase operators. Let R̂ = (x̂1, p̂1, . . . , q̂N , p̂N ) denote the vector of the operators q̂i and p̂i. The canonical commutation relations for the R̂i can be expressed in terms of the symplectic form Ω as [R̂i, R̂j ] = 2iΩij , with Ω ≡ ω⊕N , ω ≡ The state of a CV system can be equivalently described by quasi-probability distributions defined on the 2N -dimensional space associated to the quadratic form Ω, known as quantum phase space. In the phase space picture, the tensor productH = i Hi of the Hilbert spaces Hi’s of the N modes results in the direct sum Λ = i Λi of the phase spaces Λi’s. States with Gaussian quasi-probability distributions are referred to as Gaussian states. Such states are at the heart of information processing in CV systems [1, 5] and are the subject of our analysis. By definition, a Gaussian state is completely characterized by the first and second statistical moments of the field operators, which will be denoted, respectively, by the vector of first moments R̄ ≡ 〈R̂1〉, 〈R̂2〉, . . . , 〈R̂2N−1〉, 〈R̂2N 〉 and the covariance matrix (CM) γ of elements γij ≡ 〈R̂iR̂j + R̂jR̂i〉 − 〈R̂i〉〈R̂j〉 . (1) Coherent states, resulting from the application of displacement operatorsDY = e iY TΩR̂ (Y ∈ R2n) to the vacuum state, are Gaussian states with CM γ = 1 and first statistical moments R̄ = Y . First moments can be arbitrarily adjusted by local unitary operations (displacements), which cannot affect any property related to entropy or entanglement. They Optical implementation and entanglement distribution in Gaussian valence bond states 3 can thus be assumed zero without any loss of generality. A N -mode Gaussian state will be completely characterized by its real, symmetric, 2N × 2N CM γ. The canonical commutation relations and the positivity of the density matrix ρ of a Gaussian state imply the bona fide condition γ + iΩ ≥ 0 , (2) as a necessary and sufficient constraint the matrix γ has to fulfill to be a CM corresponding to a physical state [8, 9]. Note that the previous condition is necessary for the CM of any (generally non Gaussian) state, as it generalizes to many modes the Robertson-Schrödinger uncertainty relation [10]. A major role in the theoretical and experimental manipulation of Gaussian states is played by unitary operations which preserve the Gaussian character of the states on which they act. Such operations are all those generated by terms of the first and second order in the field operators. As a consequence of the Stone-Von Neumann theorem, any such operation at the Hilbert space level corresponds, in phase space, to a symplectic transformation, i.e. to a linear transformation S which preserves the symplectic form Ω, so that Ω = STΩS, i.e. it preserves the commutators between the different operators. Symplectic transformations on a 2N -dimensional phase space form the (real) symplectic group, denoted by Sp(2N,R). Such transformations act linearly on first moments and “by congruence” on the CM (i.e. so that γ 7→ SγST ). One has DetS = 1, ∀S ∈ Sp(2N,R). A crucial symplectic operation is the one achieving the normal mode decomposition. Due to Williamson theorem [11], any N -mode Gaussian state can be symplectically diagonalized in phase space, so that its CM is brought in the form ν, such that SγST = ν, with ν = diag {ν1, ν1, . . . νN , νN}. The set {νi} of the positive-defined eigenvalues of |iΩγ| constitutes the symplectic spectrum of γ and its elements, the so-called symplectic eigenvalues, must fulfill the conditions νi ≥ 1, following from the uncertainty principle Eq. (2) and ensuring positivity of the density matrix ρ corresponding to γ. Ideal beam splitters, phase shifters and squeezers are described by symplectic transformations. In particular, a phase-free two-mode squeezing transformation, which corresponds to squeezing the first mode (say i) in one quadrature (say momentum, p̂i) and the second mode (say j) in the orthogonal quadrature (say position, q̂j) with the same degree of squeezing r, can be represented in phase space by the symplectic transformation Sij(r) = diag{exp r, exp−r, exp−r, exp r} . (3) These trasformations occur for instance in parametric down conversions [12]. Another important example of symplectic operation is the ideal (phase-free) beam splitter, which acts on a pair of modes i and j as [13] B̂ij(θ) : âi 7→ âi cos θ + âj sin θ âj 7→ âi sin θ − âj cos θ and corresponds to a rotation in phase space of the form Bij(θ) = cos(θ) 0 sin(θ) 0 0 cos(θ) 0 sin(θ) sin(θ) 0 − cos(θ) 0 0 sin(θ) 0 − cos(θ) . (4) The transmittivity τ of the beam splitter is given by τ = cos2(θ) so that a 50:50 beam splitter (τ = 1/2) amounts to a phase-space rotation of π/4. Optical implementation and entanglement distribution in Gaussian valence bond states 4 The combined application of a two-mode squeezing and a 50:50 beam splitter realizes the entangling twin-beam transformation [14] Tij(r) = Bij(π/4) · Sij(r) , (5) which, if applied to two uncorrelated vacuum modes i and j (whose initial CM is the identity matrix), results in the production of a pure two-mode squeezed Gaussian state with CM σi,j(r) = Tij(r)T ij (r) given by σi,j(r) = cosh(2r) 0 sinh(2r) 0 0 cosh(2r) 0 − sinh(2r) sinh(2r) 0 cosh(2r) 0 0 − sinh(2r) 0 cosh(2r) . (6) The CV entanglement in the state σi,j(r) increases unboundedly as a function of r, and in the limit r → ∞ Eq. (6) approaches the (unnormalizable) Einstein-Podolski-Rosen (EPR) state [15], simultaneous eigenstate of relative position and total momentum of the two modes i and j. Concerning entanglement in general, the “positivity of partial transposition” (PPT) criterion states that a Gaussian CM γ is separable (with respect to a 1×N bipartition) if and only if the partially transposed CM γ̃ satisfies the uncertainty principle Eq. (2) [9, 16]. In phase space, partial transposition amounts to a mirror reflection of one quadrature associated to the single-mode partition. If {ν̃i} is the symplectic spectrum of the partially transposed CM γ̃, then a (N + 1)-mode Gaussian state with CM γ is separable if and only if ν̃i ≥ 1 ∀ i. A proper measure of CV entanglement is the logarithmic negativityEN [17], which is readily computed in terms of the symplectic spectrum ν̃i of γ̃ as EN = − i: ν̃i<1 log ν̃i . (7) Such an entanglement monotone [18] quantifies the extent to which the PPT condition ν̃i ≥ 1 is violated. For 1 × N Gaussian states, only the smallest symplectic eigenvalue ν̃− of the partially transposed CM can be smaller than one [10], thus simplifying the expression of EN : then the PPT criterion simply yields that γ is entangled as soon as ν̃− < 1, and infinite entanglement (accompanied by infinite energy in the state) is reached for ν̃− → 0+. For 1× 1 Gaussian states γi,j symmetric under mode permutations, the entanglement of formation EF is computable as well via the formula [19] EF (γi,j) = max{0, f(ν̃ − )} , (8) f(x) = (1 + x)2 (1 + x)2 − (1− x) (1− x)2 Being a monotonically decreasing function of the smallest symplectic eigenvalue ν̃i,j− of the partial transpose γ̃i,j of γi,j , the entanglement of formation is completely equivalent to the logarithmic negativity in this case. For a two-mode state, ν̃i,j can be computed from the symplectic invariants of the state [20], and experimentally estimated with measures of global and local purities [21] (the purity µ = Tr ρ2 of a Gaussian state ρ with CM γ is equal to µ = (Det γ)−1/2). 3. Gaussian valence bond states Let us review the basic definitions and notations for GVBS, as adopted in Ref. [4]. The so-called matrix product Gaussian states introduced in Ref. [3] are N -mode states obtained Optical implementation and entanglement distribution in Gaussian valence bond states 5Optical implementation and entanglement distribution in Gaussian valence bond states i i+1 Figure 1. Gaussian valence bond states. Γin is the state of N EPR bonds and γ is the three- mode building block. After the EPR measurements (depicted as curly brackets), the chain of modes γ collapses into a Gaussian valence bond state with global state Γout. See also Ref. [4]. by taking a fixed number, M , of infinitely entangled ancillary bonds (EPR pairs) shared by adjacent sites, and applying an arbitrary 2M → 1 Gaussian operation on each site i = 1, . . . , N . Such a construction, more properly definable as a “valence bond” picture for Gaussian states, can be better understood by resorting to the Jamiolkowski isomorphism between quantum operations and quantum states [22]. In this framework, one starts with a chain ofN Gaussian states of 2M +1 modes (the building blocks). The global Gaussian state of the chain is described by a CM Γ = i=1 γ [i]. As the interest in GVBS lies mainly in their connections with ground states of Hamiltonians invariant under translation [3], we can focus on pure (Detγ [i] = 1), translationally invariant (γ[i] ≡ γ ∀i) GVBS. Moreover, in this work we consider single-bonded GVBS, i.e. withM = 1. This is also physically motivated in view of experimental implementations of GVBS, as more than one EPR bond would result in a building block with five or more correlated modes, which appears technologically demanding. Under the considered prescriptions, the building block γ is a pure Gaussian state of three modes. As we aim to construct a translationally invariant state, it is convenient to consider a γ whose first two modes, which will be combined with two identical halves of consecutive EPR bonds (see Fig. 3), have the same reduced CM. This yields a pure, three-mode Gaussian building block with the property of being bisymmetric [23], that is with a CM invariant under permutation of the first two modes. This choice of the building block is further justified by the fact that, among all pure three-mode Gaussian states, bisymmetric states maximize the genuine tripartite entanglement [24]: no entanglement is thus wasted in the projection process. The 6 × 6 CM γ of the building block can be written as follows in terms of 2 × 2 submatrices, γs εss εsx εTss γs εsx εTsx ε sx γx . (9) The 4×4 CM of the first two modes (each of them having reduced CM γs) will be denoted by γss, and will be regarded as the input port of the building block. On the other hand, the CM γx of mode 3 will play the role of the output port. The intermodal correlations are encoded in the off-diagonal ε matrices. Without loss of generality, we can assume γ to be, up to local unitary operations, in the standard form [24] with γs = diag{s, s} , γx = diag{x, x} , (10) Optical implementation and entanglement distribution in Gaussian valence bond states 6 εss = diag{t+, t−} , εsx = diag{u+, u−} ; x2 − 1± 16s4 − 8(x2 + 1)s2 + (x2 − 1)2 x2 − 1 (x − 2s)2 − 1± (x+ 2s)2 − 1 The valence bond construction works as follows (see Fig. 3). The global CM Γ = i=1 γ acts as the projector from the state Γ in of the N ancillary EPR pairs, to the final N -mode GVBS Γout. This is realized by collapsing the state Γin, transposed in phase space, with the ‘input port’ Γss = i γss of Γ, so that the ‘output port’ Γx = i γx turns into the desired Γout. Here collapsing means that, at each site, the two two-mode states, each constituted by one mode (1 or 2) of γss and one half of the EPR bond between site i and its neighbor (i − 1 or i + 1, respectively), undergo an “EPR measurement” i.e. are projected onto the infinitely entangled EPR state [22, 3]. An EPR pair between modes i and j can be described, see Eq. (6), as a two-mode squeezed state σi,j(r) in the limit of infinite squeezing (r → ∞). The input state is then Γin = limr→∞ i σi,i+1(r), where we have set periodic boundary conditions so that N + 1 = 1 in labeling the sites. The projection corresponds mathematically to taking a Schur complement (see Refs. [4, 3, 22] for details), yielding an output pure GVBS of N modes on a ring with a CM out = Γx − ΓTsx(Γss + θΓ θ)−1Γsx , (11) where Γsx = i γsx, and θ = i diag{1, −1, 1, −1} represents transposition in phase space (q̂i → q̂i, p̂i → −p̂i). Within the building block picture, the valence bond construction can be in toto understood as a multiple CV entanglement swapping [25], as shown in Fig. 3: the GVBS is created as the entanglement in the bonds is swapped to the chain of output modes via teleportation [26] through the input port of the building blocks. It is thus clear that at a given initialization of the output port (i.e. at fixed x), changing the properties of the input port (i.e. varying s), which corresponds to implementing different Gaussian projections from the ancillary space to the physical one, will affect the structure and entanglement properties of the target GVBS. This link is explored in the following section. 4. Entanglement distribution In Ref. [4] the quantum correlations of GVBS of the form Eq. (11) have been studied, and related to the entanglement properties of the building block γ. Let us first recall the characterization of entanglement in the latter. As a consequence of the uncertainty principle Eq. (2), the CM Eq. (9) of the building block describes a physical state if [24] x ≥ 1 , s ≥ smin ≡ . (12) Let us keep the output parameter x fixed. Straightforward applications of the PPT separability conditions, and consequent calculations of the logarithmic negativity Eq. (7), reveal that the entanglement in the CM γss of the first two modes (input port) is monotonically increasing as a function of s, ranging from the case s = smin when γss is separable to the limit s → ∞ when the block γss is infinitely entangled. Accordingly, the entanglement between each of the first two modes γs of γ and the third one γx decreases with s. One can also show that the genuine tripartite entanglement in the building block increases with the difference s − smin [24]. The entanglement properties of the building block are summarized in Fig. 4. Optical implementation and entanglement distribution in Gaussian valence bond states 7 Figure 2. How a Gaussian valence bond state is created via continuous-variable entanglement swapping. At each step, Alice attempts to teleport her mode 0 (half of an EPR bond, depicted in yellow) to Bob, exploiting as an entangled resource two of the three modes of the building block (denoted at each step by 1 and 2). The curly bracket denotes homodyne detection, which together with classical communication and conditional displacement at Bob’s side achieves teleportation. The state will be approximately recovered in mode 2, owned by Bob. Since mode 0, at each step, is entangled with the respective half of an EPR bond, the process swaps entanglement from the ancillary chain of the EPR bonds to the modes in the building block. The picture has to be followed column-wise. For ease of clarity, we depict the process as constituted by two sequences: in the first sequence [frames (1) to (4)] modes 1 and 2 are the two input modes of the building block (depicted in blue); in the second sequence [frames (5) to (8)] modes 1 and 2 are respectively an input and an output mode of the building block. As a result of the multiple entanglement swapping [frame (9)] the chain of the output modes (depicted in red), initially in a product state, is transformed into a translationally invariant Gaussian valence bond state, possessing in general multipartite entanglement among all the modes (depicted in magenta). The main question addressed in Ref. [4] is how the initial entanglement in the building block γ redistributes in the Gaussian MPS Γout. The answer is that the more entanglement one prepares in the input port γss, the longer the range of pairwise quantum correlations in the output GVBS is, as pictorially shown in Fig. 4. In more detail, let us consider first a building block γ with s = smin = (x + 1)/2. In this case, a separability analysis shows that, for an arbitrary numberN of modes in the GVBS chain, the target state Γout exhibits bipartite entanglement only between nearest neighbor modes, for any value of x > 1 (for x = 1 we trivially obtain a product state). In fact, each reduced two-mode block γouti,j is separable for |i− j| > 1. With increasing s in the choice of the building block, one finds that in the target GVBS the correlations start to extend smoothly to distant modes. A series of thresholds sk can be Optical implementation and entanglement distribution in Gaussian valence bond states 8 Figure 3. Entanglement properties of the three-mode building block γ, Eq. (9), of the Gaussian valence bond construction, as functions of the standard form covariances x and d ≡ s − smin. (a) Bipartite entanglement, as quantified by the logarithmic negativity, between the first two input-port modes 1 and 2; (b) Bipartite entanglement, as quantified by the logarithmic negativity, between each of the first two modes and the output-port mode 3; (c) Genuine tripartite entanglement, as quantified by the residual Gaussian contangle [27, 24], among all the three modes. found such that for s > sk, two given modes i and j with |i − j| ≤ k are entangled. While trivially s1(x) = smin for anyN (notice that nearest neighbors are entangled also for s = s1), the entanglement boundaries for k > 1 are in general different functions of x, depending on the number of modes. We observe however a certain regularity in the process: sk(x,N) always increases with the integer k. Very remarkably, this means that the maximum range of bipartite entanglement between two modes, or equivalently the maximum distribution of multipartite entanglement, in a GVBS on a translationally invariant ring, is monotonically related to the amount of entanglement in the reduced two-mode input port of the building block [4]. Moreover, no complete transfer of entanglement to more distant modes occurs: closer sites remain still entangled even when correlations between farther pairs arise. The most interesting feature is perhaps obtained when infinite entanglement is fed in the input port (s → ∞): in this limit, the output GVBS turns out to be a fully symmetric, permutation-invariant, N -mode Gaussian state. This means that each individual mode is equally entangled with any other, no matter how distant they are [4]. These states, being thus Optical implementation and entanglement distribution in Gaussian valence bond states 9 Figure 4. Pictorial representation of the entanglement between a probe (green) mode and its neighbor (magenta) modes on an harmonic ring with an underlying valence bond structure. As soon as the parameter s (encoding entanglement in the input port of the valence bond building block) is increased, pairwise entanglement between the probe mode and its farther and farther neighbors gradually appears in the corresponding output Gaussian valence bond states. By translational invariance, each mode exhibits the same entanglement structure with its respective neighbors. In the limit s → ∞, every single mode becomes equally entangled with every other single mode on the ring, independently of their relative distance: the Gaussian valence bond state is in this case fully symmetric. built by a symmetric distribution of infinite pairwise entanglement among multiple modes, achieve maximum genuine multiparty entanglement among all Gaussian states (at a given energy) while keeping the strongest possible bipartite one in any pair, a property known as monogamous but promiscuous entanglement sharing [27]. Keeping Fig. 3 in mind, we can conclude that having the two input modes initially entangled in the building blocks, increases the efficiency of the entanglement-swapping mechanism, inducing correlations between distant modes on the GVBS chain, which enable to store and distribute joint information. In the asymptotic limit of an infinitely entangled input port of the building block, the entanglement range in the target GVBS states is engineered to be maximum, and communication between any two modes, independently of their distance, is enabled nonclassically. In the next sections, we investigate the possibility of producing GVBS with linear optics, and discuss with a specific example the usefulness of such resource states for multiparty CV quantum communication protocols such as telecloning [6] and teleportation networks [13]. 5. Optical implementation of Gaussian valence bond states The power of describing the production of GVBS in terms of physical states, the building blocks, rather than in terms of arbitrary non-unitary Gaussian maps, lies not only in the immediacy of the analytical treatment. From a practical point of view, the recipe of Fig. 3 can be directly implemented to produce GVBS experimentally in the domain of quantum optics. We first note that the EPR measurements are realized by the standard toolbox of a beamsplitter plus homodyne detection [22], as demonstrated in several CV teleportation experiments [28]. The next ingredient to produce aN -mode GVBS is constituted byN copies of the three- mode building block γ. We provide here an easy scheme (see also Refs. [6, 29]) to realize bisymmetric three-mode Gaussian states of the form Eq. (9). As shown in Fig. 5(a), one can start from three vacuum modes and first apply a twin-beam operation to modes 1 and 3, characterized by a squeezing r13, then apply another twin-beam operation to modes 1 and 2, Optical implementation and entanglement distribution in Gaussian valence bond states 10Optical implementation and entanglement distribution in Gaussian valence bond states 50:50 γγγγ0000 γγγγ0000 γγγγ0000 Figure 5. Optical production of bisymmetric three-mode Gaussian states, used as buildingFigure 5. Optical production of bisymmetric three-mode Gaussian states, used as building blocks for the valence bond construction. (a) Three initial vacuum modes are entangled through two sequential twin-beam boxes, the first (parametrized by a squeezing degree r13) acting on modes 1 and 3, and the second (parametrized by a squeezing degree r12) acting on the transformed mode 1 and mode 2. The output is a pure three-mode Gaussian state whose covariance matrix is equivalent, up to local unitary operations, to the standard form given in Eq. (9). (b) Detail of the entangling twin-beam transformation. One input mode is squeezed in a quadrature, say momentum, of a degree r (this transformation is denoted by stretching arrows→| |←); the other input mode is squeezed in the orthogonal quadrature, say position, of the same amount (this anti-squeezing transformation is denoted by the corresponding rotated symbol). Then the two squeezed modes are combined at a 50:50 beam-splitter. If the input modes are both in the vacuum state, the output is a pure two-mode squeezed Gaussian state, with entanglement proportional to the degree of squeezing r. parametrized by r12. The symplectic operation describing the twin-beam transformation (two- mode squeezing plus balanced beam splitter) is given by Eq. (5) and pictorially represented in Fig. 5(b). The output of this optical network is a pure, bisymmetric, three-mode Gaussian state with a CM γB = T12(r12)T13(r13)T 13(r13)T 12(r12) of the form Eq. (9), with γs = diag e−2r12 e4r12 cosh (2r13) + 1 e−2r12 cosh (2r13) + e γx = diag {cosh (2r13) , cosh (2r13)} , εss = diag e−2r12 e4r12 cosh (2r13)− 1 e−2r12 cosh (2r13) − e4r12 εsx = diag 2er12 cosh (r13) sinh (r13) , − 2e−r12 cosh (r13) sinh (r13) By means of local symplectic operations (unitary on the Hilbert space), like additional single- mode squeezings, the CM γB can be brought in the standard form of Eq. (10), from which Optical implementation and entanglement distribution in Gaussian valence bond states 11 one has r13 = arccos x+ 1√ , r12 = arccos −x3 + 2x2 + 4s2x− x For a given r13 (i.e. at fixed x), the quantity r12 is a monotonic function of the standard-form covariance s, so this squeezing parameter which enters in the production of the building block (see Fig. 5) directly regulates the entanglement distribution in the target GVBS, as discussed in Sec. 4. The only unfeasible part of the scheme seems constituted by the ancillary EPR pairs. But are infinitely entangled bonds truly necessary? In Ref. [4] the possibility is considered of using a Γin given by the direct sum of two-mode squeezed states of Eq. (6), but with finite r. Repeating the previous analysis to investigate the entanglement properties of the resulting GVBS with finitely entangled bonds, it is found that, at fixed (x, s), the entanglement in the various partitions is degraded as r decreases, as somehow expected. Crucially, this does not affect the connection between input entanglement and output correlation length. Numerical investigations show that, while the thresholds sk for the onset of entanglement between distant pairs are quantitatively modified – a bigger s is required at a given x to compensate the less entangled bonds – the overall structure stays untouched. This ensures that the possibility of engineering the entanglement structure in GVBS via the properties of the building block is robust against imperfect resources, definitely meaning that the presented scheme is feasible. Alternatively, one could from the beginning observe that the triples consisting of two projective measurements and one EPR pair can be replaced by a single projection onto the EPR state, applied at each site i between the input mode 2 of the building block and the consecutive input mode 1 of the building block of site i+1 [3]. The output of all the homodyne measurements will conditionally realize the target GVBS. 6. Telecloning with Gaussian valence bond resources The protocol of CV quantum telecloning [6] amongN parties is defined as a process in which one of them (Alice) owns an unknown coherent state, and wants to distribute her state to all the other N − 1 remote parties. The telecloning is achieved by a succession of standard two-party CV teleportations [26] between the sender Alice and each of the N − 1 remote receivers, exploiting each time the corresponding reduced two-mode state shared as resource by the selected pair of parties. The 1 → 2 CV telecloning of unknown coherent states has been recently demonstrated experimentally [7]. The no-cloning theorem [30] yields that the N − 1 remote clones can resemble the original input state only to a certain extent. The fidelity, which quantifies the success of a teleportation experiment, is defined as F ≡ 〈ψin|ρout|ψin〉, where “in” and “out” denote the input and the output state. F reaches unity only for a perfect state transfer, ρout = |ψin〉〈ψin|. Without using entanglement, by purely classical communication, an average fidelity of Fcl = 1/2 is the best that can be achieved if the alphabet of input states includes all coherent states with even weight [31]. The sufficient fidelity criterion states that, if teleportation is performed with F > Fcl, then the two parties exploited an entangled state [31]. The converse is generally false, i.e. some entangled resources may yield lower-than-classical fidelities. In Ref. [32] it has been shown, however, that if the fidelity is optimized over all possible local unitary operations performed on the shared Gaussian resource (which preserve entanglement by definition), then it becomes equivalent, both qualitatively and quantitatively, to the entanglement in the resource. Optical implementation and entanglement distribution in Gaussian valence bond states 12 Let us also recall that the fidelity of CV two-user teleportation [26] of arbitrary single- mode Gaussian states with CM γin (equal to the identity for coherent states) exploiting two- mode Gaussian resources with CM γab = γa εab εTab γb , can be computed [33] as F = 2√ , Σ ≡ 2γin + ξγaξ + γb + ξεab + εTabξ , (14) with ξ = diag{−1 , 1}. We can now consider the general setting of 1 → N − 1 telecloning, where N parties share a N -mode GVBS as an entangled resource, and one of them plays the role of Alice (the sender) distributing imperfect copies of unknown coherent states to all the N − 1 receivers. For any N , the fidelity can be easily computed from the reduced two-mode CMs via Eq. (14) and will depend, for translationally invariant states, on the relative distance between the two considered modes. In this work we focus on a practical example of a GVBS on a translationally invariant harmonic ring, with N = 6 modes. As shown in the previous section, these states can be produced with the current optical technology. They are completely characterized, up to local unitary operations, by a 12 × 12 CM analytically obtained from Eq. (11) by considering the building block in standard form Eq. (9), whose elements are algebraic functions of s and x here omitted for brevity (as no particular insight is gained from their explicit expressions). First of all we can construct the reduced CMs γouti,i+k of two modes with distance k, and evaluate for each k the respective symplectic eigenvalue ν̃i,i+k− of the corresponding partial transpose. The entanglement condition s > sk will correspond to the inequality ν̃ i,i+k − < 1. With this conditions one finds that s2(x) is the only acceptable solution to the equation: 72s 12(x2+1)s6+(−34x4+28x2−34)s4+(x6−5x4−5x2+1)s2+(x2−1)2(x4−6x2+1) = 0, while for the next-next-nearest neighbors threshold one has simply s3(x) = x. This enables us to classify the entanglement distribution and, more specifically, to observe the interaction scale in the GVBS Γout: as discussed in Sec. 4 and explicitly shown in Ref. [4], by increasing initial entanglement in γss one can gradually switch on pairwise quantum correlations between more and more distant sites. Accordingly, it is now interesting to test whether this entanglement is useful to achieve nonclassical telecloning towards distant receivers. In this specific instance, Alice will send two identical (approximate) clones to her nearest neighbors, two other identical clones (with in principle different fidelity than the previous case) to her next-nearest neighbors, and one final clone to the most distant site. The fidelities for the three transmissions can be computed from Eq. (14) and are plotted in Fig. 6(a). For s = smin, obviously, only the two nearest neighbor clones can be teleported with nonclassical fidelity, as the reduced states of more distant pairs are separable. With increasing s also the state transfer to more distant sites is enabled with nonclassical efficiency, but not in the whole region of the space of parameters s and x in which the corresponding two-mode resources are entangled. As mentioned before, one can optimize the telecloning fidelity considering resources prepared in a different way but whose CM can be brought by local unitary operations (single- mode symplectic transformations) in the standard form of Eq. (11). For GVBS resources, this local-unitary freedom can be transferred to the preparation of the building block. A more general γ locally equivalent to the standard form given in Eq. (10), can be realized by complementing the presented state engineering scheme for the three-mode building block as in Eq. (13) [see Fig. 5(a)], with additional single-mode rotations and squeezing transformations aimed at increasing the output fidelity in the target GVBS states, while keeping both the entanglement in the building block and consequently the entanglement in the final GVBS unchanged by definition. Optical implementation and entanglement distribution in Gaussian valence bond states 13 Figure 6. 1 → 5 quantum telecloning of unknown coherent states exploiting a six-mode translationally invariant Gaussian valence bond state as a shared resource. Alice owns mode i. Fidelities F for distributing clones to modes j such as k = |i − j| are plotted for k = 1 [(a),(d)]; k = 2 [(b),(e)]; and k = 3 [(c),(f)], as functions of the local invariants s and x of the building block. In the first row [(a)–(c)] the fidelities are achieved exploiting the non-optimized Gaussian valence bond resource in standard form. In the second row [(d)– (f)] fidelities optimized over local unitary operations on the resource are displayed, which are equivalent to the entanglement in the corresponding reduced two-mode states (see, as a comparison, Fig. 3 in Ref. [4]). Only nonclassical values of the fidelities (F > 0.5) are shown. The optimal telecloning fidelity, obtained in this way exploiting the results of Ref. [32], is plotted in Fig. 6(b) for the three teleportations between modes i and j with k = |i − j| = 1, 2, 3. In this case, one immediately recovers a non-classical fidelity as soon as the separability condition s ≤ sk is violated in the corresponding resources. Moreover, the optimal telecloning fidelity at a given k is itself a quantitative measure of the entanglement in the reduced two-mode resource, being equal to [32] Foptk = 1/(1 + ν̃ i,i+k − ) , (15) where ν̃i,i+k− is the smallest symplectic eigenvalue of the partially transposed CM in the corresponding bipartition. The optimal fidelity is thus completely equivalent to the entanglement of formation Eq. (8) and to the logarithmic negativity Eq. (7). In the limit s → ∞, as discussed in Sec. 4, the GVBS become fully permutation- invariant for anyN . Consequently, the (optimized and non-optimized) telecloning fidelity for distributing coherent states is equal for any pair of sender-receiver parties. These resources are thus useful for 1 → N−1 symmetric telecloning. However, due to the monogamy constraints on distribution of CV entanglement [27], this two-party fidelity will decrease with increasing N , vanishing in the limit N → ∞ where the resources become completely separable. In this Optical implementation and entanglement distribution in Gaussian valence bond states 14 respect, it is worth pointing out that the fully symmetric GVBS resources are more useful for teleportation networks [13, 34], where N − 2 parties first perform local measurements (momentum detections) on their single-mode portion of the entangled resource to concentrate as much entanglement as possible onto the two-mode state of Alice and Bob, who can accomplish non-classical teleportation (after the outcomes of the N − 2 measurements are classically communicated to Bob). In this case, the optimal fidelity of N -user teleportation network is an estimator of multipartite entanglement in the shared N -mode resource [32], which is indeed a GVBS obtained from an infinitely entangled building block. 7. Conclusion The valence bond picture is a valuable framework to study the structure of correlations in quantum states of harmonic lattices. In fact, the motivation for such a formalism is quite different from the finite-dimensional case, where valence bond/matrix product states are useful to efficiently approximate ground states of N -body systems – generally described by a number of parameters exponential in N – with polynomial resources [2]. In continuous variable systems, the key feature of GVBS lies in the understanding of their entanglement distribution as governed by the properties of simpler structures [4]. This has also experimental implications giving a robust recipe to engineer correlations in many-body Gaussian states from feasible operations on the building blocks. We have provided a simple scheme to produce bisymmetric three-mode building blocks with linear optics, and discussed the subsequent implementation of the valence bond construction. We have also investigated the usefulness of such GVBS as resources for nonclassical communication, like telecloning of unknown coherent states to distant receivers on a harmonic ring. It would be interesting to employ the valence bond picture to describe quantum computation with continuous-variable cluster states [35], and to devise efficient protocols for its optical implementation. Acknowledgments This work is supported by MIUR (Italy) and by the European Union through the Integrated Project RESQ (IST-2001-37559), QAP (IST-3-015848), SCALA (CT-015714), and SECOQC. References [1] S. L. Braunstein and P. van Loock, Rev. Mod. 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Adesso, A. Serafini, and F. Illuminati, New J. Phys. 9, 60 (2007). [30] W. K. Wootters and W. H. Zurek, Nature 299, 802 (1982). [31] S. L. Braunstein, C. A. Fuchs and H. J. Kimble, J. Mod. Opt. 47, 267 (2000); K. Hammerer, M. M. Wolf, E. S. Polzik, and J. I. Cirac, Phys. Rev. Lett. 94, 150503 (2005). [32] G. Adesso and F. Illuminati, Phys. Rev. Lett. 95, 150503 (2005). [33] J. Fiurás̆ek, Phys. Rev. A 66, 012304 (2002). [34] H. Yonezawa, T. Aoki, and A. Furusawa, Nature 431, 430 (2004). [35] N. C. Menicucci et al., Phys. Rev. Lett. 97, 110501 (2005); P. Van Loock, J. Opt. Soc. Am. B 24, 340 (2007). http://arxiv.org/abs/quant-ph/0703277 Introduction Continuous variable systems and Gaussian states Gaussian valence bond states Entanglement distribution Optical implementation of Gaussian valence bond states Telecloning with Gaussian valence bond resources Conclusion

Rich magnetic phase diagram in the Kagome-staircase compound Mn3V2O8 Rich magnetic phase diagram in the Kagome-staircase compound Mn3V2O8 E. Morosan,1 J. Fleitman,1 T. Klimczuk,2,3 and R. J. Cava1 1Department of Chemistry, Princeton University, Princeton NJ 08544 2Division of Thermal Physics, Los Alamos National Laboratories, Los Alamos NM 87545 3Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Poland Mn3V2O8 is a magnetic system in which S = 5/2 Mn2+ is found in the kagomé staircase lattice. Here we report the magnetic phase diagram for temperatures above 2 K and applied magnetic fields below 9 T, characterized by measurements of the magnetization and specific heat with field along the three unique lattice directions. At low applied magnetic fields, the system first orders magnetically below Tm1 ≈ 21 K, and then shows a second magnetic phase transition at Tm2 ≈ 15 K. In addition, a phase transition that is apparent in specific heat but not seen in magnetization is found for all three applied field orientations, converging towards Tm2 as H → 0. The magnetic behavior is highly anisotropic, with critical fields for magnetic phase boundaries much higher when the field is applied perpendicular to the Kagomé staircase plane than when applied in-plane. The field-temperature (H - T) phase diagrams are quite rich, with 7 distinct phases observed. Geometrically frustrated magnetic materials have recently emerged as the focus of intense study. Among these, compounds based on the kagomé net, a regular planar lattice made from corner sharing of equilateral triangles, are of particular interest due to the very high degeneracy of energetically equivalent magnetic ground states. Breaking the ideal triangular symmetry of the kagomé net typically favors one particular magnetically ordered state above others. For the particular case of the kagomé-staircase geometry, however, in which the symmetry breaking occurs via buckling of the kagomé plane (see inset to Fig. 1), an exquisitely close competition between different magnetically ordered states has been observed, resulting in complex temperature-applied field magnetic phase diagrams. The kagomé-staircase lattice is observed in the transition metal vanadates T3V2O8 (with T = Co, Ni, Cu, and Zn) [1-11], and the Ni [2-4,7-8,11] and Co [1-3,5-6] variants have been widely studied. Simultaneous long-range ferroelectric and magnetic order have been found in Ni3V2O8 [11], allowing its classification as a multiferroic compound. Orthorhombic symmetry Mn3V2O8 (MVO) is isostructural with Co3V2O8 (CVO) and Ni3V2O8 (NVO), but its physical properties have been only marginally studied [12]. The t2g3eg2, isotropic spin, S = 5/2 L = 0 configuration of Mn2+ in MVO presents an interesting contrast to the t2g5eg2 S = 3/2 Co2+ and t2g6eg2 S = 1 Ni2+ cases of CVO and NVO. Here we report the observation of rich anisotropic magnetic field-temperature (H - T) phase diagrams for MVO, as determined from magnetization and specific heat measurements on single crystals. Two distinct magnetic phase transitions, at 21 K and 15 K, are observed for fields applied in all three principal crystallographic directions. A phase transition that is apparent in specific heat but not seen in magnetization is found for all three applied field orientations, converging towards the 15 K transition as H → 0. The magnetic behavior is highly anisotropic, with critical fields for magnetic phase boundaries much higher when the magnetic field is applied perpendicular to the kagomé-staircase plane (H parallel to the b crystallographic axis) then when applied in-plane (H parallel to the a and c crystallographic axes). The field-temperature (H - T) phase diagrams are quite rich, with 7 distinct magnetic/structural phases observed. The magnetic phase diagrams are distinctly different from what is observed for CVO and NVO. Single crystals of MVO were grown out of a MoO3/V2O5/MnO flux as previously described [12]. The starting oxides (MnO 99% Aldrich, V2O5 99.6% Alfa Aesar, MoO3 99.95% Alfa Aesar) were packed in an alumina crucible, which was then heated in a vertical tube furnace under flowing Argon gas. Sacrificial MnO powder was placed in an alumina crucible above the MoO3/V2O5/MnO flux to create an oxygen partial pressure that would neither oxidize the Mn2+ nor reduce the V5+[ref 12]. The vertical furnace was heated to 1200 oC at 200 oC/hr, held at 1200 oC for 1 hour, cooled to 900 oC at 5 oC /hr, then cooled to room temperature at 300 oC/hr. After the heat treatment red-brown platelet crystals were extracted from the flux using a bath of 1 part glacial acetic acid (Fisher) and 3 parts deionized water. The crystals were found to be single phase by single crystal and powder X-ray diffraction, with the orthorhombic Cmca structure and lattice parameters a = 6.2672(3) Å, b = 11.7377(8) Å and c = 8.5044(5) Å. Field- and temperature-magnetization measurements were performed in a Quantum Design Physical Properties Measurement System (PPMS). The specific heat data were also collected in a PPMS, using a relaxation technique with fitting of the whole calorimeter (sample with sample platform and puck). The H = 0.5 T inverse magnetization data for MVO (Fig.1) indicates the presence of long range magnetic ordering below Tm1 ≈ 21 K. Previous low-field magnetization data [12] suggest the existence of an additional magnetic phase transition near 40 K, but the feature observed is most likely due to the presence of an Mn2V2O7 impurity phase. A high-temperature fit of the susceptibility (dotted line, Fig.1) to the Curie-Weiss law χ = χ0 + C/(T-θW) yields an effective moment µeff = 5.94 µB, in excellent agreement with the theoretical value µeff = 5.92 µB expected for high-spin S = 5/2 Mn2+. The Weiss temperature θW = -320 K indicates the dominance of antiferromagnetic exchange interactions. Given the kagomé staircase magnetic lattice (inset in Fig. 1), it is not surprising that |θW / Tm1| ≈ 15, characteristic of a strongly frustrated antiferromagnetic spin system. Deviations from Curie-Weiss behavior, typical of magnetically frustrated materials, are observed to begin on cooling at approximately 70 K. The easy magnetization axis lies close to the kagomé-staircase ac-plane, where the magnetization is largest (Fig.1). Upon further inspection of the behavior of the magnetization in different applied fields, two magnetic phase transitions can be identified in the M(T) data for all field orientations. Fig. 2a and c illustrate the field dependence of these transitions for H || a and H || c respectively. The competition between the antiferromagnetic spin coupling and the anisotropy associated with the kagomé staircase structure precludes the system from attaining a zero net magnetization ground state. This is suggested by the rapidly increasing magnetization as the system enters the high temperature, low field state (HT1) upon cooling below Tm1 ≈ 21 K. A net ferromagnetic component can probably be associated with the HT1 phase. Subsequent cooling of the sample gives rise to a sharp cusp followed by a local minimum around Tm2 =15 K, where a second magnetic phase transition, from HT1 to a low-T, low-H state (LT1) occurs. Increasing magnetic field (Fig.2a,c) has almost no effect on the long-range magnetic ordering temperature Tm1, but it broadens the cusp and slowly drives the second transition down in temperature. Above H = 0.04 T the H || a low temperature magnetization plateaus at a finite value, which strongly suggests a canted spin configuration even for the LT1 state, with a smaller ferromagnetic component along a than in the HT1 state. Very similar behavior occurs for the other in-plane orientation H || c (Fig.2c), with the two distinct transitions persisting up to slightly higher field H = 0.1 T. The insets in Fig. 2a,c represent examples of how the critical temperatures for the magnetic phase transitions at constant field are determined: as shown, the vertical arrows mark Tm1 and Tm2, H = 0.01 T, and correspond to local minima in the temperature-derivative of magnetization dM/dT. The two low-field phases that are observed in the magnetically ordered state are possibly a result of the ordering of the spins on one or both of the distinct Mn2+ ions (inset Fig. 1) in the ac-plane, similar to the transitions encountered in NVO [4]. Fewer magnetic phases are distinguishable at low fields in MVO, however, than are seen in either NVO or CVO. A more complex scenario is revealed in MVO at finite fields. Fig. 2b shows a selection of the H || b M(H) isotherms (full symbols), with the T = 2 K field-derivative dM/dH (open diamonds) as an example of how the critical field values were determined. At T = 2 K, the magnetization is low and linear with field for H < 2 T, which corresponds to the LT1 phase. This behavior is consistent with the antiferromagnetic spins slowly rotating from the easy axis in the ac plane, closer to the direction of the applied field H || b. A sharp step in magnetization around Hc1 = 2.1 T marks the transition from LT1 to LT2, possibly due to a spin-flop transition on one or both of the Mn2+ sites. Although the magnetization increases linearly with field above this transition, as expected for the spin-reorientation subsequent to a spin-flop, another transition occurs just below 3 T, where M(H) changes slope (full diamonds, Fig.2b) and the system enters the state LT3. The spin-flop transition yields a sharp peak in dM/dH (open diamonds, Fig.2b); the higher critical field value Hc2 is determined using an on-set criterion for dM/dH. Both transitions are marked by small vertical arrows in Fig.2b. As the temperature is raised, the initial slope of the M(H) curves increases in the magnetically ordered phase (Fig.2b) such that the magnetization jump at the spin-flop transition becomes indistinguishable. The two magnetic phase transitions move slightly down with field, and are hard to identify in the magnetization measurements above 16 K. Specific heat measurements complement the magnetization data, by confirming the magnetic phase lines, but also by revealing another phase transition that was not visible in the M(T,H) data. A selection of the specific heat curves, plotted as Cp/T vs. T, is shown in Fig. 3a, for H || b and applied fields up to 9 T. For H = 0 (full squares) a sharp peak associated with long range magnetic ordering is seen around Tm1 = 21 K, with a second peak at the lower phase transition temperature Tm2. After subtracting the lattice contribution to the specific heat as measured for the non-magnetic analogous compound Zn3V2O8 [2] (solid line, Fig.3b) one can estimate the magnetic specific heat Cm for MVO (open symbols, Fig.3b). The temperature dependence of the magnetic entropy can then be calculated and is shown in the inset in Fig.3b for H = 0 (open circles): only about 50% of the R ln6 entropy expected for a S = 5/2 state is accounted for between 2 and 40 K. This could be an indication that additional phase transitions may exist below 2 K. Another possible explanation, given the observed departure from Curie-Weiss behavior below ~ 70 K (Fig.1), is that more entropy is associated with short range order below 70 K. No additional entropy is recovered with the application of magnetic field, as the H = 9 T temperature-dependent entropy (crosses, inset Fig.3b) differs only slightly from the H = 0 data. However, as the field is turned on, very different behavior is observed for the two peaks in Cp (Fig.3a): the one just below 16 K is affected little in temperature by the increasing magnetic field, but the higher-temperature one moves down in field. Concurrently, a third, broader peak emerges above ~ 1.5 T and is driven higher in temperature with increasing field. It is likely that both phase transitions exist at finite fields even for H < 1.5 T, and converge at Tm1 for H → 0, but their proximity in temperature makes it impossible to discern two separate peaks. For H > Hbc1 the lower temperature peak is not associated with any phase transition observed in the magnetization data. Given its invisibility in the magnetization, and the relative insensitivity of the transition temperature to applied field, we speculate that this transition may have a structural component, though the fact that the amount of entropy in the transition is suppressed by the field indicates that there must be a magnetic component as well. Based on our extensive magnetization and specific heat measurements, we present the H – T magnetic phase diagrams for MVO, for magnetic fields applied along the unique structural directions, in Figs. 4 and 5. As the temperature is lowered in zero field, MVO orders magnetically at Tm1 = (20.7 ± 0.2) K, entering first a high temperature phase (HT1) and then a low temperature phase (LT1) at Tm2 = (15.2 ± 0.5) K. The response of the system to applied magnetic field is highly anisotropic. For H || a (Fig. 4a), in finite field, two distinct phase boundaries emerge at Tm2: one represents the lower temperature magnetic phase transition, which moves down in temperature as H increases, and the second is an almost vertical line, which is only visible in the specific heat data. The intermediate temperature phase delineated by these two phase boundaries is LT4, which extends in field up to about 0.04 T. An almost horizontal phase line cuts across the phase diagram at Hac1 ≈ 0.04 T. It separates the low field, low temperature (LT1) and a high temperature (HT1) phases from two different states (LT3 and HT3) at higher fields. For the other field orientation close to the plane (H || c, Fig. 4b), the low field phase diagram is similar to that for H || a, with the HT1, LT4 and LT1 phases extending up in field up to a much higher critical value Hcc1 = 0.3 T. In the T → 0 limit, a second magnetic phase transition occurs at Hcc2 = 2.6 T, and the critical field value is slowly reduced with temperature. The two almost horizontal phase lines at Hcc1 and Hcc2 separate a low temperature (LT2) and a high temperature (HT2) phase at intermediate field values from the high field states LT3 and HT3. When field is applied perpendicular to the kagomé planes (Fig. 5) the phase diagram is analogous to the in-plane ones. The most noticeable difference is that the critical field values are much higher: Hbc1 = 2.2 T and Hbc2 = 3.0 T respectively for T → 0. This is expected given the observed anisotropy, which constrains the magnetic moments to lie closer to the ac-plane: stronger fields are needed to pull the moments towards the “hard” axis b. In addition, the LT4 phase is missing, and the phase line that starts at Hbc1 at T → 0 converges at Tm2 in the H = 0 limit. As a consequence, the HT2 phase merges with HT1 just below the magnetic ordering at Tm1. The temperature-field magnetic phase diagram for Mn3V2O8 is quite different from those seen in Ni3V2O8 and Co3V2O8. In all three compounds, the competition between the crystalline anisotropy and the antiferromagnetic interactions in the kagomé staircase structure gives rise to strong geometric frustration. In NVO and CVO, differences in the magnetically ordered states have been found to involve differences in the ordering of the moments on the two kinds of magnetic ion sites, the so-called spine and crosstie sites. The same will no doubt prove true for MVO, with the present measurements revealing that the magnetic moments on the two distinct Mn2+ sites lie close to the ac-plane when in the H = 0 magnetically ordered states. For magnetic fields applied in-plane, the magnetic states in MVO are much more sensitive to applied field than they are in NVO and CVO, with fig. 4 showing for example that the LT1 and HT1 phases disappear in applied fields in the a direction as low as 0.04 T. The complexity of the anisotropic H – T phase diagrams in MVO appears to be derived from competition between nearly balanced magnetic interactions, leading to canted spin configurations or field-induced spin-flop transitions. An integration of the entropy observed under the H = 0 phase transitions between 2 and 40 K does not yield the expected Rln6 for Mn2+, suggesting that there may be more magnetic phase transitions below 2 K, or that additional entropy is associated with short-range order below 70 K. Detailed neutron scattering measurements are desirable in order to elucidate the nature of the different states observed in MVO, and also to clarify whether the almost field independent phase boundary at Tm2 is associated with a structural phase transition. Investigation of possible multiferroic phases will also be of considerable interest. Acknowledgements This research was supported by the US Department of Energy, Division of Basic Energy Sciences, grant DE-FG02- 98-ER45706. We thank G. Lawes for providing the specific heat data for Zn3V2O8. References 1. N. Krishnamachari, C. Calvo, Canad. J. Chem. 49 (1971) 1629 2. N. Rogado, G. Lawes, D. A. Huse, A. P. Ramirez, R. J. Cava Solid State Commun. 124 (2002) 229 3. G. Balakrishnan, O. A. Petrenko, M. R. Lees, D. M K Paul c J. Phys.: Condens. Matter 16 (2004) L347 4. M. Kenzelmann, A. B. Harris, A. Aharony, O. Entin-Wohlman, T. Yildirim, Q. Huang, S. Park, G. Lawes, C. Broholm, N. Rogado, R. J. Cava, K. H. Kim, G. Jorge, A. P. Ramirez Phys. Rev. B 74 14429 5. R. Szymczak, M. Baran, R. Diduszko, J. Fink-Finowicki, M. Gutowska, A. Szewczyk, H. Szymczak Phys. Rev. B 73 94425 6. Y. Chen, J. W. Lynn, Q. Huang, F. M. Woodward, T. Yildirim, G. Lawes, A. P. Ramirez, N. Rogado, R. J. Cava, A. Aharony, O. Entin-Wohlman, A. B. Harris Phys. Rev. B 74 (2006) 14430 7. R. P. Chaudhury, F. Yen, C. R. dela Cruz, B. Lorenz, Y. Q. Wang, Y. Y. Sun, C. W. Chu Phys. Rev. B 75 (2007) 12407 8. G. Lawes, M. Kenzelmann, N. Rogado, K. H. Kim, G. A. Jorge, R. J. Cava, A. Aharony, O. Entin-Wohlman, A. B. Harris, T. Yildirim, Q. Z. Huang, S. Park, C. Broholm, A. P. Ramirez Phys. Rev. Lett. 93 247201 9. N. Rogado, M. K. Haas, G. Lawes, D. A. Huse, A. P. Ramirez, R. J. Cava J. Phys.: Condens. Matter 15 (2003) 907 10. E. E. Sauerbrei, R. Faggiani, C. Calvo Acta Cryst. B 29 (1973) 2304 11. G. Lawes, A. B. Harris, T. Kimura, N. Rogado, R. J. Cava, A. Aharony, O. Entin- Wohlman, T. Yildirim, M. Kenzelmann, C. Broholm, A. P. Ramirez Phys. Rev. Lett. 95 (2005) 87205 12. X. Wang, Z. Liu, A. Ambrosini, A. Maignan, C. L. Stern, K. R. Poeppelmeier, V. P. Dravid Solid State Sciences 2 (2000) 99 Figure Captions Fig 1. Anisotropic inverse susceptibility data for H = 0.5 T (symbols) and linear fit of the high- temperature data (dotted line). Insert: kagomé staircase structure of the Mn2+ array in Mn3V2O8. Crystallographic axes are shown. Spine sites are shown in purple and crosstie sites are shown in pink. Fig. 2. (a) H || a M(T) data for H = 0.015 T, 0.025 T, 0.04 T, 0.06 T, 0.08 T, 0.1 T and 5.0 T. Inset: the H = 0.01 T dM/dT curve; small vertical arrows indicate the position of the minima in the derivative, from which the critical temperature values are determined. (b) H || b M(H) isotherms for T = 2 K, 15 K, 18 K, 20 K and 30 K (full symbols, left axis); the T = 2 K dM/dH curve (open symbols, right axis) illustrates how the critical field values Hc1 and Hc2, marked by vertical arrows, are determined. (c) H || c M(T) data for H = 0.02 T, 0.05 T, 0.5 T, 1.0 T, 2.0 T, 3.0 T and 5.0 T. Inset: the H = 0.01 T dM/dT curve; small vertical arrows indicate the position of the minima in the derivative, from which the critical temperature values are determined. Fig 3. (a) H || b Cp/T vs. T data for H = 0, 1.5 T, 2.0 T, 6.0 T and 9.0 T. (b) Cp/T data for MVO (full symbols) and Zn3V2O8 (solid line) (right axis) used to determine the magnetic specific heat Cm of MVO (open symbols, left axis) plotted as Cm/T. Inset: the temperature-dependence of the magnetic entropy Sm for H = 0 and 9 T. Fig 4. (a) H || a and (b) H || c H – T phase diagrams: points are determined from M(T) data (orange symbols), M(H) data (wine symbols) or Cp(T)|H data (open symbols). The solid lines are guides connecting the points determined experimentally; extrapolations of these phase boundaries in regions where measurements were missing or critical H and T values were difficult to determine are represented by dotted lines. Fig 5. H || b H – T phase diagrams: points are determined from M(T) data (orange symbols), M(H) data (wine symbols) or Cp(T)|H data (open symbols). The solid lines are guides connecting the points determined experimentally; extrapolations of these phase boundaries in regions where measurements were missing or critical H and T values were difficult to determine are represented by dotted lines. Fig.1. 0 50 100 150 200 250 H = 0.5 T H || c H || b 0 10 20 30 40 50 60 70 0 1 2 3 4 5 0 10 20 30 40 50 60 H || c 0.02T 0.05T 0.5 T H || b H || a 0.025T 0.015T 0.04T 0.06T 0.08T 0 10 20 30 H = 0.01 T 0 10 20 30 H = 0.01 T Fig. 2. Fig. 3. 8 12 16 20 24 28 32 36 40 0 10 20 30 40 0 10 20 30 40 10(a) (b)Mn H = 0 H = 1.5 T H = 2.0 T H = 6.0 T H = 9.0 T (J/m H = 0 H = 0 H = 9 T R ln2 R ln3 Fig.4. 0 4 8 12 16 20 24 28 T(K) PMLT3 HT3 HT2LT2 H || c LT1 HT1 H || a Fig.5. 0 4 8 12 16 20 24 H || b HT3LT3 LT1 HT1

Mn3V2O8 is a magnetic system in which S = 5/2 Mn2+ is found in the kagome staircase lattice. Here we report the magnetic phase diagram for temperatures above 2 K and applied magnetic fields below 9 T, characterized by measurements of the magnetization and specific heat with field along the three unique lattice directions. At low applied magnetic fields, the system first orders magnetically below Tm1 ~ 21 K, and then shows a second magnetic phase transition at Tm2 ~ 15 K. In addition, a phase transition that is apparent in specific heat but not seen in magnetization is found for all three applied field orientations, converging towards Tm2 as H -> 0. The magnetic behavior is highly anisotropic, with critical fields for magnetic phase boundaries much higher when the field is applied perpendicular to the Kagome staircase plane than when applied in-plane. The field-temperature (H - T) phase diagrams are quite rich, with 7 distinct phases observed.

Rich magnetic phase diagram in the Kagome-staircase compound Mn3V2O8 Rich magnetic phase diagram in the Kagome-staircase compound Mn3V2O8 E. Morosan,1 J. Fleitman,1 T. Klimczuk,2,3 and R. J. Cava1 1Department of Chemistry, Princeton University, Princeton NJ 08544 2Division of Thermal Physics, Los Alamos National Laboratories, Los Alamos NM 87545 3Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Poland Mn3V2O8 is a magnetic system in which S = 5/2 Mn2+ is found in the kagomé staircase lattice. Here we report the magnetic phase diagram for temperatures above 2 K and applied magnetic fields below 9 T, characterized by measurements of the magnetization and specific heat with field along the three unique lattice directions. At low applied magnetic fields, the system first orders magnetically below Tm1 ≈ 21 K, and then shows a second magnetic phase transition at Tm2 ≈ 15 K. In addition, a phase transition that is apparent in specific heat but not seen in magnetization is found for all three applied field orientations, converging towards Tm2 as H → 0. The magnetic behavior is highly anisotropic, with critical fields for magnetic phase boundaries much higher when the field is applied perpendicular to the Kagomé staircase plane than when applied in-plane. The field-temperature (H - T) phase diagrams are quite rich, with 7 distinct phases observed. Geometrically frustrated magnetic materials have recently emerged as the focus of intense study. Among these, compounds based on the kagomé net, a regular planar lattice made from corner sharing of equilateral triangles, are of particular interest due to the very high degeneracy of energetically equivalent magnetic ground states. Breaking the ideal triangular symmetry of the kagomé net typically favors one particular magnetically ordered state above others. For the particular case of the kagomé-staircase geometry, however, in which the symmetry breaking occurs via buckling of the kagomé plane (see inset to Fig. 1), an exquisitely close competition between different magnetically ordered states has been observed, resulting in complex temperature-applied field magnetic phase diagrams. The kagomé-staircase lattice is observed in the transition metal vanadates T3V2O8 (with T = Co, Ni, Cu, and Zn) [1-11], and the Ni [2-4,7-8,11] and Co [1-3,5-6] variants have been widely studied. Simultaneous long-range ferroelectric and magnetic order have been found in Ni3V2O8 [11], allowing its classification as a multiferroic compound. Orthorhombic symmetry Mn3V2O8 (MVO) is isostructural with Co3V2O8 (CVO) and Ni3V2O8 (NVO), but its physical properties have been only marginally studied [12]. The t2g3eg2, isotropic spin, S = 5/2 L = 0 configuration of Mn2+ in MVO presents an interesting contrast to the t2g5eg2 S = 3/2 Co2+ and t2g6eg2 S = 1 Ni2+ cases of CVO and NVO. Here we report the observation of rich anisotropic magnetic field-temperature (H - T) phase diagrams for MVO, as determined from magnetization and specific heat measurements on single crystals. Two distinct magnetic phase transitions, at 21 K and 15 K, are observed for fields applied in all three principal crystallographic directions. A phase transition that is apparent in specific heat but not seen in magnetization is found for all three applied field orientations, converging towards the 15 K transition as H → 0. The magnetic behavior is highly anisotropic, with critical fields for magnetic phase boundaries much higher when the magnetic field is applied perpendicular to the kagomé-staircase plane (H parallel to the b crystallographic axis) then when applied in-plane (H parallel to the a and c crystallographic axes). The field-temperature (H - T) phase diagrams are quite rich, with 7 distinct magnetic/structural phases observed. The magnetic phase diagrams are distinctly different from what is observed for CVO and NVO. Single crystals of MVO were grown out of a MoO3/V2O5/MnO flux as previously described [12]. The starting oxides (MnO 99% Aldrich, V2O5 99.6% Alfa Aesar, MoO3 99.95% Alfa Aesar) were packed in an alumina crucible, which was then heated in a vertical tube furnace under flowing Argon gas. Sacrificial MnO powder was placed in an alumina crucible above the MoO3/V2O5/MnO flux to create an oxygen partial pressure that would neither oxidize the Mn2+ nor reduce the V5+[ref 12]. The vertical furnace was heated to 1200 oC at 200 oC/hr, held at 1200 oC for 1 hour, cooled to 900 oC at 5 oC /hr, then cooled to room temperature at 300 oC/hr. After the heat treatment red-brown platelet crystals were extracted from the flux using a bath of 1 part glacial acetic acid (Fisher) and 3 parts deionized water. The crystals were found to be single phase by single crystal and powder X-ray diffraction, with the orthorhombic Cmca structure and lattice parameters a = 6.2672(3) Å, b = 11.7377(8) Å and c = 8.5044(5) Å. Field- and temperature-magnetization measurements were performed in a Quantum Design Physical Properties Measurement System (PPMS). The specific heat data were also collected in a PPMS, using a relaxation technique with fitting of the whole calorimeter (sample with sample platform and puck). The H = 0.5 T inverse magnetization data for MVO (Fig.1) indicates the presence of long range magnetic ordering below Tm1 ≈ 21 K. Previous low-field magnetization data [12] suggest the existence of an additional magnetic phase transition near 40 K, but the feature observed is most likely due to the presence of an Mn2V2O7 impurity phase. A high-temperature fit of the susceptibility (dotted line, Fig.1) to the Curie-Weiss law χ = χ0 + C/(T-θW) yields an effective moment µeff = 5.94 µB, in excellent agreement with the theoretical value µeff = 5.92 µB expected for high-spin S = 5/2 Mn2+. The Weiss temperature θW = -320 K indicates the dominance of antiferromagnetic exchange interactions. Given the kagomé staircase magnetic lattice (inset in Fig. 1), it is not surprising that |θW / Tm1| ≈ 15, characteristic of a strongly frustrated antiferromagnetic spin system. Deviations from Curie-Weiss behavior, typical of magnetically frustrated materials, are observed to begin on cooling at approximately 70 K. The easy magnetization axis lies close to the kagomé-staircase ac-plane, where the magnetization is largest (Fig.1). Upon further inspection of the behavior of the magnetization in different applied fields, two magnetic phase transitions can be identified in the M(T) data for all field orientations. Fig. 2a and c illustrate the field dependence of these transitions for H || a and H || c respectively. The competition between the antiferromagnetic spin coupling and the anisotropy associated with the kagomé staircase structure precludes the system from attaining a zero net magnetization ground state. This is suggested by the rapidly increasing magnetization as the system enters the high temperature, low field state (HT1) upon cooling below Tm1 ≈ 21 K. A net ferromagnetic component can probably be associated with the HT1 phase. Subsequent cooling of the sample gives rise to a sharp cusp followed by a local minimum around Tm2 =15 K, where a second magnetic phase transition, from HT1 to a low-T, low-H state (LT1) occurs. Increasing magnetic field (Fig.2a,c) has almost no effect on the long-range magnetic ordering temperature Tm1, but it broadens the cusp and slowly drives the second transition down in temperature. Above H = 0.04 T the H || a low temperature magnetization plateaus at a finite value, which strongly suggests a canted spin configuration even for the LT1 state, with a smaller ferromagnetic component along a than in the HT1 state. Very similar behavior occurs for the other in-plane orientation H || c (Fig.2c), with the two distinct transitions persisting up to slightly higher field H = 0.1 T. The insets in Fig. 2a,c represent examples of how the critical temperatures for the magnetic phase transitions at constant field are determined: as shown, the vertical arrows mark Tm1 and Tm2, H = 0.01 T, and correspond to local minima in the temperature-derivative of magnetization dM/dT. The two low-field phases that are observed in the magnetically ordered state are possibly a result of the ordering of the spins on one or both of the distinct Mn2+ ions (inset Fig. 1) in the ac-plane, similar to the transitions encountered in NVO [4]. Fewer magnetic phases are distinguishable at low fields in MVO, however, than are seen in either NVO or CVO. A more complex scenario is revealed in MVO at finite fields. Fig. 2b shows a selection of the H || b M(H) isotherms (full symbols), with the T = 2 K field-derivative dM/dH (open diamonds) as an example of how the critical field values were determined. At T = 2 K, the magnetization is low and linear with field for H < 2 T, which corresponds to the LT1 phase. This behavior is consistent with the antiferromagnetic spins slowly rotating from the easy axis in the ac plane, closer to the direction of the applied field H || b. A sharp step in magnetization around Hc1 = 2.1 T marks the transition from LT1 to LT2, possibly due to a spin-flop transition on one or both of the Mn2+ sites. Although the magnetization increases linearly with field above this transition, as expected for the spin-reorientation subsequent to a spin-flop, another transition occurs just below 3 T, where M(H) changes slope (full diamonds, Fig.2b) and the system enters the state LT3. The spin-flop transition yields a sharp peak in dM/dH (open diamonds, Fig.2b); the higher critical field value Hc2 is determined using an on-set criterion for dM/dH. Both transitions are marked by small vertical arrows in Fig.2b. As the temperature is raised, the initial slope of the M(H) curves increases in the magnetically ordered phase (Fig.2b) such that the magnetization jump at the spin-flop transition becomes indistinguishable. The two magnetic phase transitions move slightly down with field, and are hard to identify in the magnetization measurements above 16 K. Specific heat measurements complement the magnetization data, by confirming the magnetic phase lines, but also by revealing another phase transition that was not visible in the M(T,H) data. A selection of the specific heat curves, plotted as Cp/T vs. T, is shown in Fig. 3a, for H || b and applied fields up to 9 T. For H = 0 (full squares) a sharp peak associated with long range magnetic ordering is seen around Tm1 = 21 K, with a second peak at the lower phase transition temperature Tm2. After subtracting the lattice contribution to the specific heat as measured for the non-magnetic analogous compound Zn3V2O8 [2] (solid line, Fig.3b) one can estimate the magnetic specific heat Cm for MVO (open symbols, Fig.3b). The temperature dependence of the magnetic entropy can then be calculated and is shown in the inset in Fig.3b for H = 0 (open circles): only about 50% of the R ln6 entropy expected for a S = 5/2 state is accounted for between 2 and 40 K. This could be an indication that additional phase transitions may exist below 2 K. Another possible explanation, given the observed departure from Curie-Weiss behavior below ~ 70 K (Fig.1), is that more entropy is associated with short range order below 70 K. No additional entropy is recovered with the application of magnetic field, as the H = 9 T temperature-dependent entropy (crosses, inset Fig.3b) differs only slightly from the H = 0 data. However, as the field is turned on, very different behavior is observed for the two peaks in Cp (Fig.3a): the one just below 16 K is affected little in temperature by the increasing magnetic field, but the higher-temperature one moves down in field. Concurrently, a third, broader peak emerges above ~ 1.5 T and is driven higher in temperature with increasing field. It is likely that both phase transitions exist at finite fields even for H < 1.5 T, and converge at Tm1 for H → 0, but their proximity in temperature makes it impossible to discern two separate peaks. For H > Hbc1 the lower temperature peak is not associated with any phase transition observed in the magnetization data. Given its invisibility in the magnetization, and the relative insensitivity of the transition temperature to applied field, we speculate that this transition may have a structural component, though the fact that the amount of entropy in the transition is suppressed by the field indicates that there must be a magnetic component as well. Based on our extensive magnetization and specific heat measurements, we present the H – T magnetic phase diagrams for MVO, for magnetic fields applied along the unique structural directions, in Figs. 4 and 5. As the temperature is lowered in zero field, MVO orders magnetically at Tm1 = (20.7 ± 0.2) K, entering first a high temperature phase (HT1) and then a low temperature phase (LT1) at Tm2 = (15.2 ± 0.5) K. The response of the system to applied magnetic field is highly anisotropic. For H || a (Fig. 4a), in finite field, two distinct phase boundaries emerge at Tm2: one represents the lower temperature magnetic phase transition, which moves down in temperature as H increases, and the second is an almost vertical line, which is only visible in the specific heat data. The intermediate temperature phase delineated by these two phase boundaries is LT4, which extends in field up to about 0.04 T. An almost horizontal phase line cuts across the phase diagram at Hac1 ≈ 0.04 T. It separates the low field, low temperature (LT1) and a high temperature (HT1) phases from two different states (LT3 and HT3) at higher fields. For the other field orientation close to the plane (H || c, Fig. 4b), the low field phase diagram is similar to that for H || a, with the HT1, LT4 and LT1 phases extending up in field up to a much higher critical value Hcc1 = 0.3 T. In the T → 0 limit, a second magnetic phase transition occurs at Hcc2 = 2.6 T, and the critical field value is slowly reduced with temperature. The two almost horizontal phase lines at Hcc1 and Hcc2 separate a low temperature (LT2) and a high temperature (HT2) phase at intermediate field values from the high field states LT3 and HT3. When field is applied perpendicular to the kagomé planes (Fig. 5) the phase diagram is analogous to the in-plane ones. The most noticeable difference is that the critical field values are much higher: Hbc1 = 2.2 T and Hbc2 = 3.0 T respectively for T → 0. This is expected given the observed anisotropy, which constrains the magnetic moments to lie closer to the ac-plane: stronger fields are needed to pull the moments towards the “hard” axis b. In addition, the LT4 phase is missing, and the phase line that starts at Hbc1 at T → 0 converges at Tm2 in the H = 0 limit. As a consequence, the HT2 phase merges with HT1 just below the magnetic ordering at Tm1. The temperature-field magnetic phase diagram for Mn3V2O8 is quite different from those seen in Ni3V2O8 and Co3V2O8. In all three compounds, the competition between the crystalline anisotropy and the antiferromagnetic interactions in the kagomé staircase structure gives rise to strong geometric frustration. In NVO and CVO, differences in the magnetically ordered states have been found to involve differences in the ordering of the moments on the two kinds of magnetic ion sites, the so-called spine and crosstie sites. The same will no doubt prove true for MVO, with the present measurements revealing that the magnetic moments on the two distinct Mn2+ sites lie close to the ac-plane when in the H = 0 magnetically ordered states. For magnetic fields applied in-plane, the magnetic states in MVO are much more sensitive to applied field than they are in NVO and CVO, with fig. 4 showing for example that the LT1 and HT1 phases disappear in applied fields in the a direction as low as 0.04 T. The complexity of the anisotropic H – T phase diagrams in MVO appears to be derived from competition between nearly balanced magnetic interactions, leading to canted spin configurations or field-induced spin-flop transitions. An integration of the entropy observed under the H = 0 phase transitions between 2 and 40 K does not yield the expected Rln6 for Mn2+, suggesting that there may be more magnetic phase transitions below 2 K, or that additional entropy is associated with short-range order below 70 K. Detailed neutron scattering measurements are desirable in order to elucidate the nature of the different states observed in MVO, and also to clarify whether the almost field independent phase boundary at Tm2 is associated with a structural phase transition. Investigation of possible multiferroic phases will also be of considerable interest. Acknowledgements This research was supported by the US Department of Energy, Division of Basic Energy Sciences, grant DE-FG02- 98-ER45706. We thank G. Lawes for providing the specific heat data for Zn3V2O8. References 1. N. Krishnamachari, C. Calvo, Canad. J. Chem. 49 (1971) 1629 2. N. Rogado, G. Lawes, D. A. Huse, A. P. Ramirez, R. J. Cava Solid State Commun. 124 (2002) 229 3. G. Balakrishnan, O. A. Petrenko, M. R. Lees, D. M K Paul c J. Phys.: Condens. Matter 16 (2004) L347 4. M. Kenzelmann, A. B. Harris, A. Aharony, O. Entin-Wohlman, T. Yildirim, Q. Huang, S. Park, G. Lawes, C. Broholm, N. Rogado, R. J. Cava, K. H. Kim, G. Jorge, A. P. Ramirez Phys. Rev. B 74 14429 5. R. Szymczak, M. Baran, R. Diduszko, J. Fink-Finowicki, M. Gutowska, A. Szewczyk, H. Szymczak Phys. Rev. B 73 94425 6. Y. Chen, J. W. Lynn, Q. Huang, F. M. Woodward, T. Yildirim, G. Lawes, A. P. Ramirez, N. Rogado, R. J. Cava, A. Aharony, O. Entin-Wohlman, A. B. Harris Phys. Rev. B 74 (2006) 14430 7. R. P. Chaudhury, F. Yen, C. R. dela Cruz, B. Lorenz, Y. Q. Wang, Y. Y. Sun, C. W. Chu Phys. Rev. B 75 (2007) 12407 8. G. Lawes, M. Kenzelmann, N. Rogado, K. H. Kim, G. A. Jorge, R. J. Cava, A. Aharony, O. Entin-Wohlman, A. B. Harris, T. Yildirim, Q. Z. Huang, S. Park, C. Broholm, A. P. Ramirez Phys. Rev. Lett. 93 247201 9. N. Rogado, M. K. Haas, G. Lawes, D. A. Huse, A. P. Ramirez, R. J. Cava J. Phys.: Condens. Matter 15 (2003) 907 10. E. E. Sauerbrei, R. Faggiani, C. Calvo Acta Cryst. B 29 (1973) 2304 11. G. Lawes, A. B. Harris, T. Kimura, N. Rogado, R. J. Cava, A. Aharony, O. Entin- Wohlman, T. Yildirim, M. Kenzelmann, C. Broholm, A. P. Ramirez Phys. Rev. Lett. 95 (2005) 87205 12. X. Wang, Z. Liu, A. Ambrosini, A. Maignan, C. L. Stern, K. R. Poeppelmeier, V. P. Dravid Solid State Sciences 2 (2000) 99 Figure Captions Fig 1. Anisotropic inverse susceptibility data for H = 0.5 T (symbols) and linear fit of the high- temperature data (dotted line). Insert: kagomé staircase structure of the Mn2+ array in Mn3V2O8. Crystallographic axes are shown. Spine sites are shown in purple and crosstie sites are shown in pink. Fig. 2. (a) H || a M(T) data for H = 0.015 T, 0.025 T, 0.04 T, 0.06 T, 0.08 T, 0.1 T and 5.0 T. Inset: the H = 0.01 T dM/dT curve; small vertical arrows indicate the position of the minima in the derivative, from which the critical temperature values are determined. (b) H || b M(H) isotherms for T = 2 K, 15 K, 18 K, 20 K and 30 K (full symbols, left axis); the T = 2 K dM/dH curve (open symbols, right axis) illustrates how the critical field values Hc1 and Hc2, marked by vertical arrows, are determined. (c) H || c M(T) data for H = 0.02 T, 0.05 T, 0.5 T, 1.0 T, 2.0 T, 3.0 T and 5.0 T. Inset: the H = 0.01 T dM/dT curve; small vertical arrows indicate the position of the minima in the derivative, from which the critical temperature values are determined. Fig 3. (a) H || b Cp/T vs. T data for H = 0, 1.5 T, 2.0 T, 6.0 T and 9.0 T. (b) Cp/T data for MVO (full symbols) and Zn3V2O8 (solid line) (right axis) used to determine the magnetic specific heat Cm of MVO (open symbols, left axis) plotted as Cm/T. Inset: the temperature-dependence of the magnetic entropy Sm for H = 0 and 9 T. Fig 4. (a) H || a and (b) H || c H – T phase diagrams: points are determined from M(T) data (orange symbols), M(H) data (wine symbols) or Cp(T)|H data (open symbols). The solid lines are guides connecting the points determined experimentally; extrapolations of these phase boundaries in regions where measurements were missing or critical H and T values were difficult to determine are represented by dotted lines. Fig 5. H || b H – T phase diagrams: points are determined from M(T) data (orange symbols), M(H) data (wine symbols) or Cp(T)|H data (open symbols). The solid lines are guides connecting the points determined experimentally; extrapolations of these phase boundaries in regions where measurements were missing or critical H and T values were difficult to determine are represented by dotted lines. Fig.1. 0 50 100 150 200 250 H = 0.5 T H || c H || b 0 10 20 30 40 50 60 70 0 1 2 3 4 5 0 10 20 30 40 50 60 H || c 0.02T 0.05T 0.5 T H || b H || a 0.025T 0.015T 0.04T 0.06T 0.08T 0 10 20 30 H = 0.01 T 0 10 20 30 H = 0.01 T Fig. 2. Fig. 3. 8 12 16 20 24 28 32 36 40 0 10 20 30 40 0 10 20 30 40 10(a) (b)Mn H = 0 H = 1.5 T H = 2.0 T H = 6.0 T H = 9.0 T (J/m H = 0 H = 0 H = 9 T R ln2 R ln3 Fig.4. 0 4 8 12 16 20 24 28 T(K) PMLT3 HT3 HT2LT2 H || c LT1 HT1 H || a Fig.5. 0 4 8 12 16 20 24 H || b HT3LT3 LT1 HT1

L2-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS DAVID KYED Abstra t. We prove that a ompa t quantum group is oamen- able if and only if its orepresentation ring is amenable. We further propose a Følner ondition for ompa t quantum groups and prove it to be equivalent to oamenability. Using this Følner ondition, we prove that for a oamenable ompa t quantum group with tra- ial Haar state, the enveloping von Neumann algebra is dimension �at over the Hopf algebra of matrix oe� ients. This generalizes a theorem of Lü k from the group ase to the quantum group ase, and provides examples of ompa t quantum groups with vanishing -Betti numbers. Introdu tion The theory of L2-Betti numbers for dis rete groups is originally due to Atiyah and dates ba k to the seventies [Ati76℄. These L2-Betti num- bers are de�ned for those dis rete groups that permit a free, proper and o ompa t a tion on some ontra tible, Riemannian manifold X . If Γ is su h a group, the spa e of square integrable p-forms on X be omes a �nitely generated Hilbert module for the group von Neu- mann algebra L (Γ). As su h it has a Murray-von Neumann dimen- sion whi h turns out to be independent of the hoi e of X and is alled the p-th L2-Betti number of Γ, denoted β p (Γ). More re ently, Lü k [Lü 97, Lü 98a, Lü 98b℄ transported the notion of Murray-von Neu- mann dimension to the setting of �nitely generated proje tive (alge- brai ) L (Γ)-modules and extended thereafter the domain of de�nition to the lass of all modules. With this extended dimension fun tion, dimL (Γ)(−), it is possible to extend the notion of L2-Betti numbers to over all dis rete groups Γ by setting β(2)p (Γ) = dimL (Γ) Tor p (L (Γ),C). For more details on the relations between the di�erent de�nitions of L2- Betti numbers and the extended dimension fun tion we refer to Lü k's book [Lü 02℄. 2000 Mathemati s Subje t Classi� ation. 16W30,43A07, 46L89, 16E30. http://arxiv.org/abs/0704.1582v4 2 DAVID KYED All the ingredients in the homologi al algebrai de�nition above have fully developed analogues in the world of ompa t quantum groups, and using this di tionary the notion of L2-Betti numbers was generalized to the quantum group setting in [Kye08℄. Sin e this generalization is entral for the work in the present paper, we shall now explain it in greater detail. Consider a ompa t quantum group G = (A,∆) and assume that its Haar state h is a tra e. If we denote by A0 the unique dense Hopf ∗-algebra and by M the enveloping von Neumann algebra of A in the GNS representation arising from h, then the p-th L2-Betti number of G is de�ned as β(2)p (G) = dimM Tor p (M,C). Here C is onsidered an A0-module via the ounit ε : A0 → C and dimM(−) is Lü k's extended dimension fun tion arising from (the ex- tension of) the tra e-state h. This de�nition extends the lassi al one [Kye08, 1.3℄ in the sense that β(2)p (G) = β p (Γ) when G = (C∗red(Γ),∆red). The aim of this paper is to investigate the L2-Betti numbers of the lass of oamenable, ompa t quantum groups. In the lassi al ase we have that β p (Γ) = 0 for all p ≥ 1 whenever Γ is an amenable group. This an be seen as a spe ial ase of [Lü 98a, 5.1℄ where it is proved that the von Neumann algebra L (Γ) is dimension �at over CΓ, meaning dimL (Γ) Tor p (L (Γ), Z) = 0 (p ≥ 1) for any CΓ-module Z � provided, of ourse, that Γ is still assumed amenable. We generalize this result to the quantum group setting in Theorem 6.1. More pre isely, we prove that if G = (A,∆) is a ompa t, oamenable quantum group with tra ial Haar state and Z is any module for the algebra of matrix oe� ients A0 then dimM Tor p (M,Z) = 0. (p ≥ 1) Here M is again the enveloping von Neumann algebra in the GNS rep- resentation arising from the Haar state. In order to prove this result we need a Følner ondition for ompa t quantum groups. The lassi al Følner ondition for groups [Føl55℄ is a geometri al ondition, on the a tion of the group on itself, whi h is equivalent to amenability of the group. In order to obtain a quantum analogue of Følner's ondition a detailed study of the ring of orepresentations, asso iated to a ompa t quantum group, is needed. The ring of orepresentations is a spe ial -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 3 ase of a so- alled fusion algebra and we have therefore devoted a sub- stantial part of this paper to the study of abstra t fusion algebras and their amenability. Amenability for (�nitely generated) fusion algebras was introdu ed by Hiai and Izumi in [HI98℄ where they also gave two equivalent Følner-type onditions for fusion algebras. We generalize their results to the non-�nitely generated ase and prove that a om- pa t quantum group is oamenable if and only if its orepresentation ring is amenable. From this we obtain a Følner ondition for ompa t quantum groups whi h is equivalent to oamenability. Using this Følner ondition we prove our main result, Theorem 6.1, whi h implies that oamenable ompa t quantum groups have vanishing L2-Betti numbers in all positive degrees. Stru ture. The paper is organized as follows. In the �rst se tion we re- apitulate (parts of) Woronowi z's theory of ompa t quantum groups. The se ond and third se tion is devoted to the study of abstra t fu- sion algebras and amenability of su h. In the fourth se tion we dis uss oamenability of ompa t quantum groups and investigate the relation between oamenability of a ompa t quantum group and amenability of its orepresentation ring. The �fth se tion is an interlude in whi h the ne essary notation on erning von Neumann algebrai ompa t quan- tum groups and their dis rete duals is introdu ed. The sixth se tion is devoted to the proof of our main theorem (6.1) and the seventh, and �nal, se tion onsists of examples. A knowledgements. I wish to thank my supervisor Ryszard Nest for the many dis ussions about quantum groups and their ( o)amenability, and Andreas Thom for pointing out to me that the bi rossed produ t onstru tion ould be used to generate examples of quantum groups satisfying Følner's ondition. Notation. Throughout the paper, the symbol ⊙ will be used to denote algebrai tensor produ ts while the symbol ⊗̄ will be used to denote tensor produ ts in the ategory of Hilbert spa es or the ategory of von Neumann algebras. All tensor produ ts between C∗-algebras are assumed minimal/spatial and these will be denoted by the symbol ⊗. 1. Preliminaries on ompa t quantum groups In this se tion we brie�y re all Woronowi z's theory of ompa t quantum groups. Detailed treatments, and proofs of the results stated, an be found in [Wor98℄, [MVD98℄ and [KT99℄. 4 DAVID KYED A ompa t quantum group G is a pair (A,∆) where A is a unital C∗- algebra and ∆: A −→ A ⊗ A is a unital ∗-homomorphism from A to the minimal tensor produ t of A with itself satisfying: (id⊗∆)∆ = (∆⊗ id)∆ ( oasso iativity) ∆(A)(1⊗ A) = ∆(A)(A⊗ 1) = A⊗A (non-degenera y) For su h a ompa t quantum group G = (A,∆), there exists a unique state h : A → C, alled the Haar state, whi h is invariant in the sense (h⊗ id)∆(a) = (id⊗h)∆(a) = h(a)1, for all a ∈ A. Let H be a Hilbert spa e and let u ∈ M(K(H)⊗ A) be an invertible multiplier. Then u is alled a orepresentation if (id⊗∆)u = u(12)u(13), where we use the standard leg numbering onvention; for instan e u(12) = u⊗1. Intertwiners, dire t sums and equivalen es between orep- resentations as well as irredu ibility are de�ned in a straight forward manner. See e.g. [MVD98℄ for details. We shall denote by Mor(u, v) the set of intertwiners from u to v. It is a fa t that ea h irredu ible orepresentation is �nite dimensional and equivalent to a unitary o- representation. Moreover, every unitary orepresentation is unitarily equivalent to a dire t sum of irredu ible orepresentations. For two �nite dimensional unitary orepresentations u, v their tensor produ t is de�ned as u T©v = u(13)v(23). This is again a unitary orepresentation of G. The algebra A0 gener- ated by all matrix oe� ients arising from irredu ible orepresentations be omes a Hopf ∗-algebra (with the restri ted omultipli ation) whi h is dense in A. We denote its antipode by S and its ounit by ε. We also re all that the restri tion of the Haar state to the ∗-algebra A0 is always faithful. The quantum group G is alled a ompa t matrix quantum group if there exists a fundamental unitary orepresentation; i.e. a �nite dimensional, unitary orepresentation whose matrix oe� ients gener- ate A0 as a ∗-algebra. Ea h �nite dimensional, unitary orepresenta- tion u de�nes a ontragredient orepresentation uc on the dual Hilbert spa e; if u ∈ B(H) ⊙ A0 for some �nite dimensional Hilbert spa e H then uc ∈ B(H ′)⊙A0 is given by uc = (( · )′⊗S)u, where for T ∈ B(H) the operator T ′ ∈ B(H ′) is the natural dual (T ′(y′))(x) = y′(Tx). In general uc is not a unitary, but it is a orepresentation; i.e. it is invert- ible and satis�es (id⊗∆)uc = uc and is therefore equivalent to a unitary orepresentation. By hoosing an orthonormal basis e1, . . . , en -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 5 for H we get an identi� ation of B(H) ⊙ A0 with Mn(A0). If, under this identi� ation, u be omes the matrix (uij) then u is identi�ed with the matrix ū = (u∗ij), where we identify B(H ′)⊙A0 with Mn(A0) using the dual basis e′1, . . . , e n. From this it follows that u is equivalent to u. Note also that one has (u⊕ v)c = uc ⊕ vc and (u T©v)c = vc T©uc for unitary orepresentations u and v (see e.g. [Wor87℄). If u ∈ B(H)⊙A0 is a �nite dimensional orepresentation its hara ter is de�ned as χ(u) = (Tr⊗ id)u ∈ A0, where Tr is the unnormalized tra e on B(H). The hara ter map has the following properties. Proposition 1.1 ([Wor87℄). If u and v are �nite dimensional, unitary orepresentations then χ(u T©v) = χ(u)χ(v), χ(u⊕v) = χ(u)+χ(v) and χ(uc) = χ(u)∗. Moreover, if u and v are equivalent then χ(u) = χ(v). We end this se tion with the two basi examples of ompa t quantum groups arising from a tual groups. Example 1.2. If G is a ompa t, Hausdor� topologi al group then the Gelfand dual C(G) be omes a ompa t quantum group with omulti- pli ation ∆c : C(G) −→ C(G)⊗ C(G) = C(G×G) given by ∆c(f)(s, t) = f(st). The Haar state is in this ase given by integration against the Haar probability measure on G, and the �nite dimensional unitary orepre- sentations of C(G) are exa tly the �nite dimensional unitary represen- tations of G. Example 1.3. If Γ is a dis rete, ountable group then the redu ed group C∗-algebra C∗red(Γ) be omes a ompa t quantum group when endowed with omultipli ation given by ∆red(λγ) = λγ ⊗ λγ . Here λ denotes the left regular representation of Γ. In this ase, the Haar state is just the natural tra e on C∗red(Γ), and a omplete family of irredu ible, unitary orepresentations is given by the set {λγ | γ ∈ Γ}. Remark 1.4. All ompa t quantum groups to be onsidered in the following are assumed to have a separable underlying C∗-algebra. The quantum Peter-Weyl theorem [KT99, 3.2.3℄ then implies that the GNS spa e arising from the Haar state is separable and, in parti ular, that there are at most ountable many (pairwise inequivalent) irredu ible orepresentations. 6 DAVID KYED 2. Fusion Algebras In this se tion we introdu e the notion of fusion algebras and amen- ability of su h obje ts. This topi was treated by Hiai and Izumi in [HI98℄ and we will follow this referen e losely throughout this se - tion. Other referen es on the subje t are [Yam99℄, [HY00℄ and [Sun92℄. Throughout the se tion, N0 will denote the non-negative integers. De�nition 2.1 ([HI98℄). Let R be a unital ring and assume that R is free as Z-module with basis I. Then R is alled a fusion algebra if the unit e is an element of I and the following holds: (i) The abelian monoid N0[I] is stable under multipli ation. That is, for all ξ, η ∈ I the unique family (Nαξ,η)α∈I of integers satis- fying Nαξ,ηα, onsists of non-negative numbers. (ii) The ring R has a Z-linear, anti-multipli ative involution x 7→ x̄ preserving the basis I globally. (iii) Frobenius re ipro ity holds, i.e. for ξ, η, α ∈ I we have Nαξ,η = N α,η̄. (iv) There exists a Z-linear multipli ative fun tion d : R → [1,∞[ su h that d(ξ) = d(ξ̄) for all ξ ∈ I. This fun tion is alled the dimension fun tion. Note that the distinguished basis, involution and dimension fun tion are all in luded in the data de�ning a fusion algebra. Ea h fusion algebra omes with a natural tra e τ given by τ7−→ ke. We shall use this tra e later to de�ne a C∗-envelope of a fusion algebra. Note also that the multipli ativity of d implies d(ξ)d(η) Nαξ,η, for all ξ, η ∈ I. For an element r = α∈I kαα ∈ R, the set {α ∈ I | kα 6= 0} is alled the support of r and denoted supp(r). We shall also onsider the omplexi�ed fusion algebra C⊗ZZ[I] whi h will be denoted C[I] in the following. Note that this be omes a omplex ∗-algebra with the indu ed algebrai stru tures. -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 7 Example 2.2. For any dis rete group Γ the integral group ring Z[Γ] be- omes a fusion algebra when endowed with (the Z-linear extension of) inversion as involution and trivial dimension fun tion given by d(γ) = 1 for all γ ∈ Γ. The irredu ible representations of a ompa t group onstitute the basis in a fusion algebra where the tensor produ t of representations is the produ t. We shall not go into details with this onstru tion sin e it will be ontained in the following more general example. Example 2.3. If G = (A,∆) is a ompa t quantum group its irre- du ible orepresentations onstitute the basis of a fusion algebra with tensor produ t as multipli ation. Sin e this example will play a promi- nent role later, we shall now elaborate on the onstru tion. Denote by Irred(G) = (uα)α∈I a omplete family of representatives for the equivalen e lasses of irredu ible, unitary orepresentations of G. As explained in Se tion 1, for all uα, uβ ∈ Irred(G) there exists a �nite subset I0 ⊆ I and a family (Nγα,β)γ∈I0 of positive integers su h that uα T©uβ is equivalent to uγ ⊕ · · · ⊕ uγ ︸ ︷︷ ︸ times Thus, a produ t an be de�ned on the free Z-module Z[Irred(G)] by setting uα · uβ = and the trivial orepresentation e = 1A ∈ Irred(G) is a unit for this produ t. If we denote by uᾱ ∈ Irred(G) the unique representative equivalent to (uα)c, then the map uα 7→ uᾱ extends to a onjugation on the ring Z[Irred(G)] and sin e ea h uα is an element ofMnα(A) for some nα ∈ N we an also de�ne a dimension fun tion d : Z[Irred(G)] → [1,∞[ by d(uα) = nα. When endowed with this multipli ation, onjugation and dimension fun tion Z[Irred(G)] be omes a fusion algebra. The only thing that is not lear at this moment is that Frobenius re ipro ity holds. To see this, we �rst note that for any α ∈ I and any �nite dimensional orepresentation v we have (by S hur's Lemma [MVD98, 6.6℄) that uα o urs exa tly dimC Mor(u α, v) 8 DAVID KYED times in the de omposition of v. Moreover, we have for any two unitary orepresentations v and w that dimC Mor(v, w) = dimC((Vw ⊗ V ′v)w T#v dimCMor(v cc, w) = dimC((V v ⊗ Vw)v Here the right hand side denotes the linear dimension of the spa e of in- variant ve tors under the relevant oa tion. These formulas are proved in [Wor87, 3.4℄ for ompa t matrix quantum groups, but the same proof arries over to the ase where the ompa t quantum group in question does not ne essarily possess a fundamental orepresentation. Using the �rst formula, we get for α, β, γ ∈ I that α,β = dimC Mor(u γ, uα T©uβ) = dimC(Vα ⊗ Vβ ⊗ V ′γ)u T#uβ T#(uγ)c = dimC(Vγ ⊗ V ′β ⊗ V ′α)u T#(uβ)c T#(uα)c = dimC Mor(u α, uγ T©(uβ)c) The remaining identity in Frobenius re ipro ity follows similarly us- ing the se ond formula. The fusion algebra Z[Irred(G)] is alled the orepresentation ring (or fusion ring) of G and is denoted R(G). Re all that the hara ter of a orepresentation u ∈ Mn(A) is de�ned as χ(u) = i=1 uii. It follows from Proposition 1.1 that the Z-linear extension χ : Z[Irred(G)] −→ A0 is an inje tive homomorphism of ∗-rings. I.e. χ is additive and mul- tipli ative with χ(uᾱ) = (χ(uα))∗. This gives a link between the two ∗-algebras R(G) and A0 whi h will be of importan e later. Other interesting examples of fusion algebras arise from in lusions of II1-fa tors. See [HI98℄ for details. Remark 2.4. In the following we shall only onsider fusion algebras with an at most ountable basis. This will therefore be assumed with- out further noti e throughout the paper. Sin e we will primarily be interested in orepresentation rings of ompa t quantum groups, this is not very restri tive sin e the standing separability assumption (Re- mark 1.4) ensures that the orepresentation rings always have a ount- able basis. Consider again an abstra t fusion algebra R = Z[I]. For ξ, η ∈ I we de�ne the (weighted) onvolution of the orresponding Dira measures, -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 9 δξ and δη, as δξ ∗ δη = d(ξ)d(η) Nαξ,ηδα ∈ ℓ1(I). This extends linearly and ontinuously to a submultipli ative produ t on ℓ1(I). For f ∈ ℓ∞(I) and ξ ∈ I we de�ne λξ(f), ρξ(f) : I → C by λξ(f)(η) = f(α)(δξ̄ ∗ δη)(α) ρξ(f)(η) = f(α)(δη ∗ δξ)(α) Denote by σ the ounting measure on I s aled with d2; that is σ(ξ) = d(ξ)2. Combining Proposition 1.3, Remark 1.4 and Theorem 1.5 in [HI98℄ we get Proposition 2.5 ([HI98℄). For ea h f ∈ ℓ∞(I) we have λξ(f), ρξ(f) ∈ ℓ∞(I) and for ea h p ∈ N ∪ {∞} the maps λξ, ρξ : ℓ∞(I) → ℓ∞(I) re- stri t to bounded operators on ℓp(I, σ) denoted λp,ξ and ρp,ξ respe tively. By linear extension, we therefore obtain a map λp,− : Z[I] → B(ℓp(I, σ)) and this map respe ts the weighted onvolution produ t. Moreover, for p = 2 the operator U : ℓ2(I) → ℓ2(I, σ) given by U(δη) = 1d(η)δη is unitary and intertwines λ2,ξ with the operator lξ : δη 7−→ Nαξ,ηδα. Remark 2.6. Under the natural identi� ation of ℓ2(I) with the GNS spa e L2(C[I], τ), we see that πτ (ξ) = d(ξ)lξ. In parti ular the GNS representation onsists of bounded operators. Here τ is the natural tra e de�ned just after De�nition 2.1. 3. Amenability for Fusion Algebras The notion of amenability for fusion algebras was introdu ed in [HI98℄, but only in the slightly restri ted setting of �nitely generated fusion algebras; a fusion algebra R = Z[I] is alled �nitely generated if there exists a �nitely supported probability measure µ on I su h that supp(µ∗n) and µ(ξ̄) = µ(ξ) for all ξ ∈ I. That is, if the union of the supports of all powers of µ, with respe t to onvolution, is I and µ is invariant under the involution. The �rst ondition is referred to as non-degenera y of µ and the se ond ondition is referred to as symmetry of µ. 10 DAVID KYED In [HI98℄, amenability is de�ned, for a �nitely generated fusion al- gebra, by requiring that ‖λp,µ‖ = 1 for some 1 < p < ∞ and some �nitely supported, symmetri , non-degenerate probability measure µ. It is then proved that this is independent of the hoi e of µ and p, using the non-degenera y property of the measure. If we onsider a ompa t quantum group G = (A,∆) it is not di� ult to prove that its orepresentation ring R(G) is �nitely generated exa tly when G is a ompa t matrix quantum group. Sin e we are also interested in quan- tum groups without a fundamental orepresentation we will hoose the following de�nition of amenability. De�nition 3.1. A fusion algebra R = Z[I] is alled amenable if 1 ∈ σ(λ2,µ) for every �nitely supported, symmetri probability measure µ on I. Here σ(λ2,µ) denotes the spe trum of the operator λ2,µ. From Propo- sition 1.3 and Corollary 4.4 in [HI98℄ it follows that our de�nition agrees with the one in [HI98℄ on the lass of �nitely generated fusion algebras. The relation between amenability for fusion algebras and the lassi al notion of amenability for groups will be explained later. See e.g. Re- mark 3.8 and Corollary 4.7. De�nition 3.2. Let R = Z[I] be a fusion algebra. For two �nite subsets S, F ⊆ I we de�ne the boundary of F relative to S as the set ∂S(F ) = {α ∈ F | ∃ ξ ∈ S : supp(αξ) * F} ∪ {α ∈ F c | ∃ ξ ∈ S : supp(αξ) * F c}. Here, and in what follows, F c denotes the set I \ F . The modi�ed de�nition of amenability allows the following extension of [HI98, 4.6℄ from where we also adopt some notation. Theorem 3.3. Let R = Z[I] be a fusion algebra with dimension fun - tion d. Then the following are equivalent: (A) The fusion algebra is amenable. (FC1) For every �nitely supported, symmetri probability measure µ on I with e ∈ supp(µ) and every ε > 0 there exists a �nite subset F ⊆ I su h that ξ∈supp(χF ∗µ) d(ξ)2 < (1 + ε) d(ξ)2. (FC2) For every �nite, non-empty subset S ⊆ I and every ε > 0 there exists a �nite subset F ⊆ I su h that ∀ ξ ∈ S : ‖ρ1,ξ(χF )− χF‖1,σ < ε‖χF‖1,σ, -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 11 where ρ1,ξ ∈ B(ℓ1(I, σ)) is the operator from Proposition 2.5. (FC3) For every �nite, non-empty subset S ⊆ I and every ε > 0 there exists a �nite subset F ⊆ I su h that ξ∈∂S(F ) d(ξ)2 < ε d(ξ)2. The ondition (FC3) was not present in [HI98℄. It is to be onsidered as a fusion algebra analogue of the Følner ondition for groups as it is presented in [BP92, F.6℄. The strategy for the proof of Theorem 3.3 is to prove the following impli ations: (A) ⇔ (FC2) ⇒ (FC3) ⇒ (FC1) ⇒ (FC2). The proof of the impli ations (A) ⇔ (FC2) and (FC1) ⇒ (FC2) are small modi� ations of the orresponding proof in [HI98℄. We �rst set out to prove the ir le of impli ations (FC2) ⇒ (FC3) ⇒ (FC1) ⇒ (FC2). For the proof we will need the following simple lemma. Lemma 3.4. If Nαξ,η > 0 for some ξ, η, α ∈ I then d(α)d(η) ≥ d(ξ). Proof. By Frobenius re ipro ity, we have Nαξ,η = N α,η̄ > 0 and hen e d(α)d(η) = d(α)d(η̄) = α,η̄d(γ) ≥ N ξα,η̄d(ξ) ≥ d(ξ). Proof of (FC2) ⇒ (FC3). We �rst note that (FC2), by the triangle in- equality, implies the following ondition: For every �nite, non-empty set S ⊆ I and every ε > 0 there exists a �nite set F ⊆ I su h that ‖ρ1,χS(χF )− |S|χF‖1,σ < ε‖χF‖1,σ. (†) Here |S| denotes the ardinality of S. Let S and ε > 0 be given and hoose F su h that (†) is satis�ed. De�ne a map ϕ : I → R by 12 DAVID KYED ϕ(ξ) = ρ1,χS(χF )(ξ)− |S|χF (ξ). We note that ϕ(ξ) = χF (α)(δξ ∗ χS)(α) − |S|χF (ξ) (δξ ∗ δη)(α) − |S|χF (ξ) d(ξ)d(η) Nαξ,η − |S|χF (ξ). We now divide into four ases. (i) If ξ ∈ F ∩∂S(F )c then supp(ξη) ⊆ F for all η ∈ S and hen e we get the relation d(ξ)d(η) Nαξ,η = 1. This implies ϕ(ξ) = 0. (ii) If ξ ∈ F c ∩ ∂S(F )c we see that Nαξ,η = 0 for all α ∈ F and all η ∈ S and hen e ϕ(ξ) = 0. (iii) If ξ ∈ F c ∩ ∂S(F ) we have χF (ξ) = 0 and there exist α0 ∈ F and η0 ∈ S su h that Nα0ξ,η0 6= 0. Using Lemma 3.4, we now get ϕ(ξ) ≥ d(α0) d(ξ)d(η0) Nα0ξ,η0 ≥ d(η0)2 Nα0ξ,η0 ≥ d(η0)2 where M = max{d(η)2 | η ∈ S}. (iv) If ξ ∈ F ∩ ∂S(F ) we have ϕ(ξ) = d(ξ)d(η) Nαξ,η − |S| = (−1) d(ξ)d(η) Nαξ,η = (−1) d(ξ)d(η) Nαξ,η, and be ause ξ ∈ ∂S(F )∩ F there exist η0 ∈ S and α0 /∈ F su h that Nα0ξ,η0 6= 0. Using Lemma 3.4 again we on lude, as in (iii), that |ϕ(ξ)| ≥ 1 -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 13 We now get d(ξ)2 = ε‖χF‖1,σ > ‖ρ1,χS(χF )− |S|χF‖1,σ (by (†)) |ϕ(ξ)|d(ξ)2 ξ∈∂S(F ) |ϕ(ξ)|d(ξ)2 (by (i) and (ii)) ξ∈∂S(F ) d(ξ)2, (by (iii) and (iv)) and sin e ε was arbitrary the laim follows. � Proof of (FC3) ⇒ (FC1). Given a �nitely supported, symmetri pro- bability measure µ, with µ(e) > 0, and ε > 0 we put S = supp(µ) and hoose F ⊆ I su h that (FC3) is ful�lled with respe t to ε. We have (χF ∗ µ)(ξ) = α∈F,β∈S d(α)d(β) α,β , (χF ∗ µ)(ξ) = 0 ⇔ ∀α ∈ F ∀β ∈ S : N ξα,β = 0 ⇔ ∀α ∈ F ∀β ∈ S : Nα = 0 (Frobenius) ⇔ ∀α ∈ F ∀β ∈ S : Nαξ,β = 0 (S symmetri ) ⇔ ξ ∈ F c ∩ ∂S(F )c. (e ∈ S) Hen e supp(χF ∗ µ) = (F c ∩ ∂S(F )c)c = F ∪ ∂S(F ) and we get ξ∈supp(χF ∗µ) d(ξ)2 − d(ξ)2 = ξ∈F∪∂S(F ) d(ξ)2 − d(ξ)2 ξ∈∂S(F )∩F d(ξ)2 ξ∈∂S(F ) d(ξ)2 d(ξ)2. (by (FC3)) Proof of (FC1) ⇒ (FC2). Given ε > 0 and S ⊆ I we de�ne S̃ = S ∪ S̄ ∪{e} and µ = 1 χS̃. Choose F ⊆ I su h that µ and F satisfy (FC1) 14 DAVID KYED with respe t to . We aim to prove that (FC2) is satis�ed for all ξ ∈ S̃. For arbitrary ξ ∈ I we have ‖ρ1,ξ(χF )− χF‖1,σ = |ρ1,ξ(χF )(α)− χF (α)|d(α)2 d(α)d(ξ) α,ξ)− χF (α)|d(α)2 d(α)d(ξ) d(α)2 d(α)d(ξ) d(α)2 d(η)d(α) α,ξ + d(η)d(α) d(η)d(α) α,ξ +N d(η)d(α) +Nαη,ξ). (†) For ξ ∈ supp(µ) = S̃ and α /∈ F , it is easy to he k that (χF ∗µ)(α) > 0 if there exists an η ∈ F su h that Nα +Nαη,ξ > 0. Hen e the al ulation (†) implies that ‖ρ1,ξ(χF )− χF‖1,σ ≤ α∈supp(χF ∗µ)\F d(η)d(α) +Nαη,ξ) α∈supp(χF ∗µ)\F d(η)d(α) +Nαη,ξ) α∈supp(χF ∗µ)\F d(α)2 α∈supp(χF ∗µ) d(α)2 − d(α)2 < ε‖χF‖1,σ, where the last estimate follows from (FC1). Note that the ondition e ∈ supp(µ) was used to get the fourth step in the al ulation above. � We now set out to prove the remaining equivalen e in Theorem 3.3. -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 15 Proof of (A) ⇔ (FC2). At the end of this se tion four formulas are gathered; these will be used during the proof and referred to as (F1) - (F4). For the a tual proof we also need the following de�nitions. Con- sider a �nitely supported, symmetri probability measure µ on I and de�ne pµ : I × I → R by pµ(ξ, η) = (δξ ∗ µ)(η) = d(ξ)d(ω) Note that the fun tion pµ satis�es the reversibility ondition: σ(ξ)pµ(ξ, η) = σ(η)pµ(η, ξ). For a �nitely supported fun tion f ∈ c0(I) and r ∈ N we also de�ne ‖f‖Dµ(r) = σ(ξ)pµ(ξ, η)|f(ξ)− f(η)|r Although this is referred to as the generalized Diri hlet r-norm of f , one should keep in mind that the fun tion ‖·‖Dµ(r) is only a semi norm. We shall now onsider the following ondition: For all �nitely supported, symmetri , probability measures µ we have {‖f‖Dµ(r) ‖f‖r,σ | f ∈ c0(I) \ {0} = 0. (NWr) The reason for the name (NWr), whi h appeared in [HI98℄, is that the ondition is the negation of a so- alled Wirtinger inequality. See [HI98℄ for more details. To prove (A) ⇔ (FC2) we will a tually prove the following equivalen es (FC2) ⇔ (NW1) and ∀r : (NW1) ⇔ (NWr) and (A) ⇔ (NW2). For the latter of these equivalen es the following lemma will be useful. Lemma 3.5. For all f ∈ c0(I) we have ‖f‖2Dµ(2) = 〈f |f〉2,σ − 〈ρ2,µ(f)|f〉2,σ, where 〈·|·〉2,σ denotes the inner produ t on ℓ2(I, σ). Proof. This is proven by a dire t al ulation using the reversibility ondition and the formula (F4) from the end of this se tion. � Proof of (A)⇔ (NW2). Let µ be a �nitely supported, symmetri prob- ability measure on I. By [HI98, 1.3,1.5℄, we have that ρ2,µ is self-adjoint 16 DAVID KYED and ‖ρ2,µ‖ ≤ ‖µ‖1 = 1 so that 1− ρ2,µ ≥ 0. We now get 1 ∈ σ(λ2,µ) ⇔ 1 ∈ σ(ρ2,µ) ([HI98, 1.5℄) ⇔ 0 ∈ σ(1− ρ2,µ) ⇔ 0 ∈ σ( 1− ρ2,µ) ⇔ ∃xn ∈ (ℓ2(I, σ))1 : ‖( 1− ρ2,µ)xn‖2,σ −→ 0 ⇔ ∃fn ∈ (c0(I))1 : ‖( 1− ρ2,µ)fn‖2,σ −→ 0 ⇔ ∃fn ∈ (c0(I))1 : 〈(1− ρ2,µ)fn |fn〉2,σ −→ 0 ⇔ ∃fn ∈ (c0(I))1 : ‖fn‖Dµ(2) −→ 0 (Lem. 3.5) ⇔ inf {‖f‖Dµ(2) ‖f‖2,σ | f ∈ c0(I) \ {0} Hen e (A) ⇔ (NW2) as desired. � Proof of (NW1) ⇒ (FC2). Given ε > 0 and ξ1, . . . , ξn ∈ I, we hoose a �nitely supported, symmetri probability measure µ with ξ1, . . . , ξn ∈ supp(µ). De�ne min{µ(ξ) | ξ ∈ I}, and hoose, a ording to (NW1), an f ∈ c0(I) su h that ‖f‖Dµ(1) < ε′‖f‖1,σ. (∗) Sin e ‖|f |‖Dµ(1) ≤ ‖f‖Dµ(1) and ‖|f |‖1,σ = ‖f‖1,σ we may assume that f is positive. Sin e f an be approximated by a rational fun tion we may a tually assume that f has integer values. Put N = max{f(ξ) | ξ ∈ I} and de�ne, for k = 1, . . . , N , Fk = {ξ | f(ξ) ≥ k}. Then f = k=1 χFk and the following formulas hold. ‖f‖Dµ(1) = ‖χFk‖Dµ(1) and ‖f‖1,σ = ‖χFk‖1,σ. The �rst formula is proved by indu tion on the integer N and the se ond follows from a dire t al ulation using only the reversibility property of pµ. Be ause of (∗), there must therefore exist some j ∈ {1, . . . , N} su h that ‖χFj‖Dµ(1) < ε′‖χFj‖1,σ. (∗∗) -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 17 For the sake of simpli ity we denote this Fj by F in the following. We now get ‖χF‖Dµ(1) = σ(ξ)pµ(ξ, η)|χF (ξ)− χF (η)| ξ∈F,η/∈F σ(ξ)pµ(ξ, η) (reversibility) ξ∈F,η/∈F d(ξ)d(ω) ξ∈F,η/∈F d(ξ)d(η) ξ∈F,η/∈F d(ξ)d(η) ξ,ω +N ξ,ω̄) µ(ω)‖ρ1,ω(χF )− χF‖1,σ. (‡) Here the last equality follows from the omputation (†) in the proof of (FC1) ⇒ (FC2). The inequality (∗∗) therefore reads µ(ω)‖ρ1,ω(χF )− χF‖1,σ < ε′‖χF‖1,σ. For every ω ∈ I we therefore on lude, sin e ε′ = ε min(µ), that µ(ω)‖ρ1,ω(χF )− χF‖1,σ < min(µ)ε‖χF‖1,σ. Sin e ea h of the given ξi's are in supp(µ) we get for all i that ‖ρ1,ξi(χF )− χF‖1,σ < ε‖χF‖1,σ, as desired. � Proof of (FC2) ⇒ (NW1). Assume now (FC2) and let µ and ε be given. Choose F su h that ‖ρ1,ξ(χF )− χF‖1,σ < ε‖χF‖1,σ 18 DAVID KYED for all ξ ∈ supp(µ). Using the al ulation (‡), from the proof of opposite impli ation, we get ‖χF‖Dµ(1) = µ(ω)‖ρ1,ω(χF )− χF‖1,σ µ(ω)ε‖χF‖1,σ ‖χF‖1,σ < ε‖χF‖1,σ. For the proof of the statement (NW1) ⇔ (NWr) we will need the following lemma. Lemma 3.6 ([Ger88℄). For r ≥ 2 and f ∈ c0(I)+ we have ‖f r‖Dµ(1) ≤ 2r‖f‖r−1r,σ ‖f‖Dµ(r). Proof. First note that ‖f r‖Dµ(1) = σ(ξ)pµ(ξ, η)|f(ξ)r − f(η)r| σ(ξ)pµ(ξ, η)(f(ξ) r−1 + f(η)r−1)|f(ξ)− f(η)|, where the inequality follows from (F1). De�ne a measure ν on I × I by ν(ξ, η) = 1 σ(ξ)pµ(ξ, η) and onsider the fun tions ϕ, ψ : I × I → R given by ϕ(ξ, η) = f(ξ)r−1 + f(η)r−1 and ψ(ξ, η) = |f(ξ)− f(η)|. De�ne s > 1 by the equation 1 = 1. Then the inequality above an be written as ‖f r‖Dµ(1) ≤ r‖ϕψ‖1,ν and using Hölder's inequality we therefore get ‖f r‖Dµ(1) ≤ r‖ϕψ‖1,ν ≤ r‖ϕ‖s,ν‖ψ‖r,ν σ(ξ)pµ(ξ, η)(f(ξ) r−1 + f(η)r−1)s σ(ξ)pµ(ξ, η)|f(ξ)− f(η)|r -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 19 σ(ξ)pµ(ξ, η)(f(ξ) (r−1)s + f(η)(r−1)s) s‖f‖Dµ(r) σ(ξ)pµ(ξ, η)f(ξ) (r−1)s s‖f‖Dµ(r) pµ(ξ, η) f(ξ)(r−1)s s‖f‖Dµ(r) σ(ξ)f(ξ)(r−1)s s‖f‖Dµ(r) σ(ξ)f(ξ)r ] r−1 r ‖f‖Dµ(r) = 2r‖f‖r−1r,σ ‖f‖Dµ(r). Also the following observation will be useful. Observation 3.7. Under the assumptions of Lemma 3.6 we have ‖f‖Dµ(r) = σ(ξ)pµ(ξ, η)|f(ξ)− f(η)|r σ(ξ)pµ(ξ, η)|f(ξ)r − f(η)r| (by (F3)) = ‖f r‖ Dµ(1) Having these results, we are now able to prove (NW1) ⇔ (NWr). Proof of (NW1) ⇒ (NWr). Assume (NW1) and let µ and ε > 0 be given. Put ε′ = εr and hoose non-zero f ∈ c0(I)+ su h that ‖f‖Dµ(1) ‖f‖1,σ < ε′. Using Observation 3.7 we get f‖Dµ(r) f‖r,σ Dµ(1) < (ε′) r = ε. by (F2) by reversibility 20 DAVID KYED Proof of (NWr) ⇒ (NW1). Given µ and ε > 0 and put ε′ = 12rε. Then hoose non-zero f ∈ c0(I)+ with ‖f‖Dµ(r) ‖f‖r,σ < ε′. Using Lemma 3.6, we get ‖f r‖Dµ(1) ‖f r‖1,σ 2r‖f‖r−1r,σ ‖f‖Dµ(r) ‖f‖rr,σ < 2rε′ = ε. Gathering all the results just proven we get (A) ⇔ (FC2). � This on ludes the proof of Theorem 3.3. Remark 3.8. Consider a ountable, dis rete group Γ and the orre- sponding fusion algebra Z[Γ]. It is not di� ult to prove that Z[Γ] satis�es (FC3) from Theorem 3.3 if and only if Γ satis�es Følner's ondition (for groups) as presented in [BP92, F.6℄. Sin e a group is amenable if and only if it satis�es Følner's ondition, we see from this that Γ is amenable if and only if the orresponding fusion algebra Z[Γ] is amenable. 3.1. Formulas used in the proof of Theorem 3.3. We olle t here four formulas used in the proof of Theorem 3.3. Let r, s > 1 and assume = 1. Then for all z, w ∈ C, a, b ≥ 0 and n ∈ N we have |ar − br| ≤ r(ar−1 + br−1)|a− b| (F1) (a+ b)r ≤ 2r−1(ar + br) (F2) |a− b|n ≤ |an − bn| (F3) |z − w|2 + |w − z|2 = 2(|z|2 − zw̄) + 2(|w|2 − wz̄) (F4) Proof. The inequality (F1) an be proved using the mean value theorem on the fun tion f(x) = xr and the interval between a and b. To prove (F2), onsider a two-point set endowed with ounting measure. Using Hölder's inequality, we then get a + b = 1 · a + 1 · b ≤ (1s + 1s) 1s (ar + br) 1r . From this the desired inequality follows using the fa t that = r−1 The inequality (F3) follows using the binomial theorem. If, for instan e, a = b+ k for some k ≥ 0 we have (a− b)n = kn ≤ (b+ k)n − bn = an − bn. -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 21 The formula (F4) follows by splitting w and z into real and imaginary parts and al ulating both sides of the equation. � 4. Coamenable Compa t Quantum Groups In this se tion we introdu e the notion of oamenability for ompa t quantum groups and dis uss the relationship between oamenability of a ompa t quantum group and amenability of its orepresentation ring. The notion of ( o-)amenability has been treated in di�erent quantum group settings by numerous people. A number of referen es for this subje t are [BMT01℄, [Voi79℄, [Rua96℄, [Ban99a℄, [Ban99b℄, [ES92℄ and [BS93℄. For our purposes, the approa h of Bédos, Murphy and Tuset in [BMT01℄ is the most natural and we are therefore going to follow this referen e throughout this se tion. We will assume that the reader is familiar with the basi s on Woronowi z's theory of ompa t quantum groups. De�nitions, notation and some basi properties an be found in Se tion 1 and detailed treatments an be found in [Wor98℄, [MVD98℄ and [KT99℄. De�nition 4.1 ([BMT01℄). Let G = (A,∆) be a ompa t quantum group and let Ared be the image of A under the GNS representation πh arising from the Haar state h. Then G is said to be oamenable if the ounit ε : A0 → C extends ontinuously to Ared. Remark 4.2. It is well known that a dis rete group Γ is amenable if and only if the trivial representation of C∗full(Γ) fa torizes through C∗red(Γ). This amounts to saying that (C red(Γ),∆red) is oamenable if and only if Γ is amenable. Note also that the abelian ompa t quan- tum groups (C(G),∆c) are automati ally oamenable sin e the ounit is given by evaluation at the identity and therefore already globally de�ned and bounded. In the following theorem we olle t some fa ts on oamenable om- pa t quantum groups. For more oamenability riteria and a proof of the theorem below we refer to [BMT01℄. Theorem 4.3 ([BMT01℄). For a ompa t quantum group G = (A,∆) the following are equivalent. (i) G is oamenable. (ii) The Haar state h is faithful and the ounit is bounded with re- spe t to the norm on A. (iii) The natural map from the universal representation A to the redu ed representation Ared is an isomorphism. 22 DAVID KYED If G is a ompa t matrix quantum group with fundamental orepre- sentation u ∈ Mn(A) the above onditions are also equivalent to the following. (iv) The number n is in σ(πh(Re(χ(u))) where χ(u) = i=1 uii is the hara ter map from Se tion 2. Re all that σ(T ) denotes the spe trum of a given operator T . Thus, when we are dealing with a oamenable quantum group the Haar state is automati ally faithful and hen e the orresponding GNS representa- tion πh is faithful. We therefore an, and will, identify A and Ared. The ondition (iv) is Skandalis's quantum analogue of the so- alled Kesten ondition for groups (see [Kes59℄ and [Ban99a℄) whi h is proved by Bani a in [Ban99b℄. The next result is a generalization of the Kesten ondition to the ase where a fundamental orepresentation is not (ne- essarily) present. The proof draws inspiration from the orresponding proof in [BMT01℄. Theorem 4.4. Let G = (A,∆) be a ompa t quantum group. Then the following are equivalent: (i) G is oamenable. (ii) For any �nite dimensional, unitary orepresentation u ∈ Mnu(A) we have nu ∈ σ(πh(Re(χ(u)))). Proof. AssumeG to be oamenable and let a �nite dimensional, unitary orepresentation u ∈ Mnu(A) be given. Sin e the ounit extends to a hara ter ε : Ared → C and sin e ε(Re(χ(u))) = ε( uii + u ) = nu, we must have nu ∈ σ(πh(Re(χ(u)))). Assume onversely that the prop- erty (ii) is satis�ed and de�ne, for a �nite dimensional, unitary ore- presentation u, the set C(u) = {ϕ ∈ S (Ared) | ϕ(πh(Re(χ(u)))) = nu}. Here S (Ared) denotes the state spa e of Ared. It is lear that ea h C(u) is losed in the weak -topology and we now prove that the family F = {C(u) | u �nite dimensional, unitary orepresentation} has the �nite interse tion property. We �rst prove that ea h C(u) is non-empty. For given u, we put xij = uij − δij and x = ijxij . Then x is learly positive and a dire t al ulation reveals that x = 2(nu − Re(χ(u))). (†) -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 23 Hen e, nu ∈ σ(πh(Re(χ(u)))) if and only if there exists [KR83, 4.4.4℄ a ϕ ∈ S (Ared) with ϕ(πh(Re(χ(u)))) = nu. Thus, C(u) 6= ∅. Let now u(1), . . . , u(k) be given and put u = ⊕ki=1u(i). We aim at proving that C(u) ⊆ C(u(i)). Let ϕ ∈ C(u) be given and note that nu(k) = ϕ(πh(Re(χ(u)))) = u(i)∑ ϕ(πh(u jj ) + πh(u jj )). Sin e the matrix u is unitary, we have ‖πh(ust)‖ ≤ 1 for all s, t ∈ {1, . . . , nu} and hen e ϕ(πh(u jj ) + πh(u jj )) ∈ [−1, 1]. This for es ϕ(πh(u jj )+πh(u jj )) = 1 and hen e ϕ(πh(Re(χ(u (i))))) = nu(i). Thus ϕ is in ea h of the sets C(u (1)), . . . , C(u(k)) and we on lude that F has the �nite interse tion property. By ompa tness of S (Ared), we may therefore �nd a state ϕ su h that ϕ(πh(Re(χ(u)))) = nu for ev- ery unitary orepresentation u. Denote by H the GNS spa e asso iated with this ϕ, by ξ0 the natural y li ve tor and by π the orresponding GNS representation of Ared. Consider an arbitrary unitary orepresen- tation u and form as before the elements xij and x. Then the equation (†) shows that ϕ(x∗ijxij) = 0 and hen e π(xij)ξ0 = 0 and π(uij)ξ0 = δijξ0. From the Cau hy-S hwarz inequality we get |ϕ(xij)|2 ≤ ϕ(x∗ijxij)ϕ(1) = 0, and hen e ϕ(uij) = δij . We therefore have that π(uij)ξ0 = ϕ(uij)ξ0. Sin e the matrix oe� ients span A0 linearly we get π(a)ξ0 = ϕ(a)ξ0 for all a ∈ A0. By density of A0 in Ared it follows that π(a)ξ0 = ϕ(a)ξ0 for all a ∈ Ared. From this we see that H = π(Ared)ξ0 = Cξ0, and it follows that ϕ : Ared → C is a bounded ∗-homomorphism oin- iding with ε on A0. Thus, G is oamenable. 24 DAVID KYED The following result was mentioned, without proof, in [HI98, p.692℄ in the restri ted setting of ompa t matrix quantum groups whose Haar state is a tra e. Theorem 4.5. A ompa t quantum group G = (A,∆) is a oamenable if and only if the orepresentation ring R(G) is amenable. For the proof we will need the following lemma. For this, re all from Se tion 2 that the ∗-algebra C[Irred(G)] omes with a tra e τ given by u∈Irred(G) zuu 7−→ ze, where e ∈ Irred(G) denotes the identity in R(G). In what follows, we denote by C∗red(R(G)) the enveloping C -algebra of C[Irred(G)] on the GNS spa e L2(C[Irred(G)], τ) arising from τ . Lemma 4.6. The hara ter map χ : R(G) → A0 extends to an isome- tri ∗-homomorphism χ : C∗red(R(G)) → Ared. Proof. Put I = Irred(G). For an irredu ible, �nite dimensional, unitary orepresentation u we have h(uij) = 0 unless u is the trivial orepre- sentation and therefore the following diagram ommutes // A0 Hen e χ extends to an isometri embedding K = L2(C[I], τ) −֒→ L2(A0, h) = H. Denote by S the algebra χ(R(G)) and by S̄ the losure of πh(S) inside Ared. Sin e S is a ∗-algebra that maps K into itself it also maps K⊥ into itself and hen e πh(χ(a)) takes the form πh(χ(a)) 0 πh(χ(a)) ‖πh(χ(a))‖ = max{‖πh(χ(a)) ‖, ‖πh(χ(a)) ≥ ‖πh(χ(a)) = ‖πτ (a)‖. This proves that the map κ : πh(S) → πτ (C[I]) given by κ(πh(χ(a))) = πτ (a) is bounded and it therefore extends to a ontra tion κ̄ : S̄ → C∗red(R(G)). We now prove that κ̄ is inje tive. Sin e h is faithful on -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 25 Ared and τ is faithful on C red(R(G)) we get the following ommutative diagram πh(S) πτ (C[I])� C∗red(R(G))� L2(S̄, h) // L2(C∗red(R(G)), τ) One easily he ks that κ indu es an isometry L2(S̄, h) → L2(C∗red(G), τ) and it therefore follows that κ̄ is inje tive and hen e an isometry. Thus, for χ(a) ∈ S we have ‖πh(χ(a))‖ = ‖κ̄(πh(χ(a)))‖ = ‖πτ (a)‖, as desired. Proof of Theorem 4.5. Assume �rst that G is oamenable and put I = Irred(G). Consider a �nitely supported, symmetri probability measure µ on I. We aim to show that 1 ∈ σ(λ2,µ), where λ2,µ is the operator on ℓ2(I, σ) de�ned in Se tion 2. Write µ as ξ∈I tξδξ and re all (Lemma 4.6) that the hara ter map χ : C[I] → A0 extends to an inje tive ∗- homomorphism χ : C∗red(R(G)) → Ared. Using this, and Proposition 2.5, we get that σ(λ2,µ) = σ(lµ) tξlξ) πτ (ξ)) = σ(χ( πτ (ξ))) πh(ξii)). Sin e G is oamenable, the ounit extends to a hara ter ε : Ared → C and we have ξii)) = nξ = 1. 26 DAVID KYED Hen e 1 ∈ σ i=1 πh(ξii)) = σ(λ2,µ) and we on lude that R(G) is amenable. Assume, onversely, that R(G) is amenable. We aim at proving that G ful�lls the Kesten ondition from Theorem 4.4. Let therefore u ∈ Mn(A) be an arbitrary, �nite dimensional, unitary orepresenta- tion. Denote by (uα)α∈S ⊆ Irred(G) the irredu ible orepresentations o urring in the de omposition of u and by kα the multipli ity of uα in u. Now de�ne µu(uα) = if α ∈ S; 0 if α /∈ S. Putting µ = 1 µū we obtain a �nitely supported, symmetri probability measure and by assumption we have that 1 ∈ σ(λ2,µ). Using again that the hara ter map extends to an inje tive ∗-homomorphism χ : C∗red(R(G)) → Ared we obtain σ(λ2,µ) = σ λ2,uα + λ2,uᾱ luα + (Prop. 2.5) πτ (uα) + πτ (uᾱ) (Rem. 2.6) πh(χ(uα)) + πh(χ(uᾱ)) πh(χ(u)) + πh(χ(ū)) πh(Re(χ(u))) 1 ∈ σ(λ2,µ) if and only if n ∈ σ(Re(πh(χ(u)))), and the result now follows from Theorem 4.4. � In parti ular we (re-)obtain the following. Corollary 4.7. A dis rete group is amenable if and only if the group ring, onsidered as a fusion algebra, is amenable. Corollary 4.8 ([Ban99b℄). The quantum groups SUq(2) are oamen- able. -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 27 Proof. By Theorem 4.5, SUq(2) is oamenable if and only if R(SUq(2)) is amenable. But, R(SUq(2)) = R(SU(2)) (see e.g. [Wor88℄) and sin e (C(SU(2)),∆c) is a oamenable quantum group R(SU(2)) is amenable. As seen from Theorem 4.5, the answer to the question of whether a ompa t quantum group is oamenable or not an be determined using only information about its orepresentations � a fa t noted by Bani a in the setting of ompa t matrix quantum groups in [Ban99a℄ and [Ban99b℄. With this in mind, we now propose the following Følner ondition for quantum groups. De�nition 4.9. A ompa t quantum group G = (A,∆) is said to satis- fy Følner's ondition if for any �nite, non-empty subset S ⊆ Irred(G) and any ε > 0 there exists a �nite subset F ⊆ Irred(G) su h that u∈∂S(F ) n2u < ε Here nu denotes the dimension of the irredu ible orepresentation u and ∂S(F ) is the boundary of F relative to S as in De�nition 3.2. We immediately obtain the following. Corollary 4.10. A ompa t quantum group is oamenable if and only if it satis�es Følner's ondition. Proof. By Theorem 4.5, the ompa t quantum group G is oamenable if and only if R(G) is amenable. By Theorem 3.3, R(G) is amenable if and only if it satis�es (FC3) whi h is exa tly the same as saying that G satis�es Følner's ondition. � In Se tion 6 we will use this Følner ondition to dedu e a vanishing result on erning L2-Betti numbers of ompa t, oamenable quantum groups. 5. An Interlude In this se tion we gather various notation and minor results whi h will be used in the following se tion to prove our main result, Theorem 6.1. Some generalities on von Neumann algebrai quantum groups are stated without proofs; we refer to [KV03℄ for the details. Consider again a ompa t quantum group G = (A,∆) with tra ial Haar state h. Denote by {uα | α ∈ I} a omplete set of representatives for the equivalen e lasses of irredu ible, unitary orepresentations of G. Consider the dense Hopf ∗-algebra A0 = spanC{uαij | α ∈ I}, 28 DAVID KYED and its dis rete dual Hopf ∗-algebra Â0. Sin e h is tra ial, the dis rete quantum group Â0 is unimodular; i.e. the left and right invariant fun - tionals are the same. Denote by ϕ̂ the left and right invariant fun tional on Â0 normalized su h ϕ̂(h) = 1. For a ∈ A0 we denote by â ∈ A′0 the A0 ∋ x7−→h(ax) ∈ C. Then, by de�nition, we have Â0 = {â | a ∈ A0}. The algebra Â0 is ∗-isomorphi to Mnα(C), and be ause h is tra ial the isomorphism has a simple des ription; if we denote by Eαij the standard matrix units in Mnα(C) then the map Φ((̂uαij) ∗) = 1 Eαij , extends to a ∗-isomorphism [MVD98℄. Denote by λ the GNS represen- tation of A on H = L2(A0, h), by η the anoni al in lusion A0 ⊆ H and by M (or λ(M)) the enveloping von Neumann algebra λ(A0) The map η̂ : Â0 → H given by â 7→ η(a) makes (H, η̂) a GNS pair for (Â0, ϕ̂) and the orresponding GNS representation L is given by L(â)η(x) = η̂(âx̂). We denote by M̂ (or L(M̂)) the enveloping von Neumann algebra L(Â0) . This is a dis rete von Neumann algebrai quantum group and ϕ̂ gives rise to a left and right invariant, normal, semi�nite, faith- ful weight on M̂ . Ea h �nite subset E ⊆ I gives rise to a entral proje tion PE = Φ χE(α)1nα ∈ Â0, where 1nα denotes the unit in Mnα(C) and χE is the hara teristi fun tion for the set E. A dire t omputation shows that L(PE) is the orthogonal proje tion onto the �nite dimensional subspa e {uᾱij | 1 ≤ i, j ≤ nα, α ∈ E}. Re all from Example 2.3 that uβ̄ is the element in {uα | α ∈ I} whi h is equivalent to (uβ)c. Be ause h is tra ial, the left invariant weight ϕ̂ on Â0 has the parti ular simple form [VKV , p.47℄ nαTrnα -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 29 where Trnα is the non-normalized tra e on Mnα(C). In parti ular ϕ̂(PE) = n2α = ϕ̂(PĒ), for any �nite subset E ⊆ I. For any m ∈ M and any �nite subset E ⊆ I we have [VVD03, 2.10℄ that TrH(m ∗PEm) = h(m ∗m)ϕ̂(PE), where TrH denotes the standard tra e on B(H). Here, and in what fol- lows, we suppress the representations λ and L ofM and M̂ respe tively on H . The ommutant M ′ is the underlying von Neumann algebra of a ompa t, von Neumann algebrai quantum group whose Haar state is also given by the ve tor state h and whose dis rete dual is given by (M̂, ∆̂)op; this quantum group has M̂ as its underlying von Neumann algebra, but is endowed with omultipli ation ∆̂op = σ∆̂ where σ de- notes the �ip-automorphism on M̂⊗̄M̂ . Sin e (M̂, ∆̂) is unimodular we see that ϕ̂op = ϕ̂ and hen e the tra e-formula above extends in the following way. Lemma 5.1 ([VVD03℄). For any m ∈ M or m ∈ M ′ and any �nite subset E ⊆ I we have TrH(m∗PEm) = h(m∗m)ϕ̂(PE). With this lemma we on lude the interlude and move towards an appli ation of the quantum Følner ondition. 6. A Vanishing Result In this se tion we investigate the L2-Betti numbers of oamenable quantum groups. The notion of L2-Betti numbers for ompa t quantum groups was introdu ed in [Kye08℄ and we refer to that paper (and the introdu tion) for the de�nitions and basi results. Throughout the se tion, we will freely use Lü k's extended Murray-von Neumann dimension, but whenever expli it properties are used there will be a referen e. These referen es will be to the original work [Lü 97℄ and [Lü 98a℄, but for the reader who wants to learn the subje t Lü k's book [Lü 02℄ is probably a better general referen e. Consider again a ompa t quantum group G = (A,∆) with Haar state h and denote by M the enveloping von Neumann algebra in the GNS representation arising from h. As promised in the introdu tion, we will now prove the following theorem whi h should be onsidered as a quantum group analogue of Theorem 5.1 from [Lü 98a℄. 30 DAVID KYED Theorem 6.1. If G is oamenable and h is tra ial then for any left A0-module Z and any k ≥ 1 we have dimM Tor k (M,Z) = 0, where dimM(−) is Lü k's extended dimension fun tion arising from the extension of the tra e-state h. If M were �at as a module over A0 we would have Tor k (M,Z) = 0 for any Z and any k ≥ 1, and the property in Theorem 6.1 is therefore referred to as dimension �atness of the von Neumann algebra over the algebra of matrix oe� ients. The proof of Theorem 6.1, whi h is a generalization of the orresponding proof of [Lü 98a, 5.1℄, is divided into three parts. Part I onsists of redu tions while part II ontains the entral argument arried out in detail in a spe ial ase. Part III shows how to boost the argument from part II to the general ase. Throughout the proof, we will use freely the quantum group notation developed in the previous se tions without further referen e; in par- ti ular, {uα | α ∈ I} will denote a �xed, omplete set of pairwise inequivalent, irredu ible, unitary orepresentations of G. Proof of Theorem 6.1. Part I We begin with some redu tions. Let an arbitrary A0-module Z be given and hoose a free module F that surje ts onto Z. Then we have a short exa t sequen e 0 −→ K −→ F −→ Z −→ 0, and sin e F is free (in parti ular �at) the orresponding long exa t Tor-sequen e gives an isomorphism TorA0k+1(M,Z) ≃ Tor k (M,K) for k ≥ 1. It is therefore su� ient to prove the theorem for arbitrary Z and k = 1. Moreover, we may assume that Z is �nitely generated sin e Tor ommutes with dire t limits, every module is the dire ted union of its �nitely generated submodules and dimM(−) is well behaved with respe t to dire t limits [Lü 98a, 2.9℄. A tually, we an assume that Z is �nitely presented sin e any �nitely generated module Z is a dire t limit of �nitely presented modules. To see this, hoose a short exa t sequen e 0 −→ K −→ F −→ Z −→ 0, -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 31 with F �nitely generated and free. Denote by (Kj)j∈J the dire ted system of �nitely generated submodules in K. Then F/Kj is �nitely presented for ea h j ∈ J and Z = lim−→ F/Kj . Be ause of this and the dire t limit formula for the dimension fun tion [Lü 98a, 2.9℄ we may, and will, therefore assume that Z is �nitely presented. Choose a �nite presentation f−→ Am0 −→ Z −→ 0. Put H = L2(A, h), K = ker(f) ⊆ An0 ⊆ Hn and denote by f (2) : Hn → Hm the ontinuous extension of f . Then we have TorA01 (M,Z) = ker(idM ⊗f) and hen e dimM Tor 1 (M,Z) = dimM ker(idM ⊗f)− dimM M ⊗ = dimM ker(f (2))− dimM K where the se ond equality follows from [CS05, 2.11℄. See also [Lü 98a, p.158-159℄. So we need to prove that K = ker(f (2)). Part II We �rst treat the ase m = n = 1. Then the map f has the form Ra (right-multipli ation by a) for some a ∈ A0. If a = 0 we have = H = ker(f (2)) so we may assume a 6= 0. Sin e the uαij's onstitute a linear basis for A0, the element a ∈ A0 has a unique expansion i,j=1 tαiju ij, (t ij ∈ C) and we may therefore onsider the non-empty, �nite set S ⊆ I given S = {α ∈ I | ∃ 1 ≤ i, j ≤ nα : tαij 6= 0}. Denote by H0 the kernel of f and by q0 ∈ M ′ the proje tion onto it. Denote by q the proje tion onto H0 ∩ K⊥; we need to prove that this subspa e is trivial and sin e the ve tor-state h is faithful on M ′ this is equivalent to proving h(q) = 0. Let ε > 0 be given. Sin e G is 32 DAVID KYED assumed oamenable, the Følner ondition provides the existen e of a �nite, non-empty subset F ⊆ I su h that α∈∂S(F ) n2α < ε Here we identify a subset E ⊆ I with the orresponding set of orep- resentations {uα | α ∈ E}. To simplify notation further we will write ∂ instead of ∂S(F ) in the following and moreover we will suppress the GNS-representations λ : M → B(H) and L : M̂ → B(H) as in Se - tion 5. Sin e h is tra ial, Woronowi z's quantum Peter-Weyl Theorem [KT99, 3.2.3℄ takes a parti ular simple form and states that the set {√nαuαij | 1 ≤ i, j ≤ nα, α ∈ I} onstitutes an orthonormal basis for H . Hen e every x ∈ H has an ℓ2-expansion i,j=1 ij. (x ij ∈ C) Consider a ve tor x ∈ H and assume that P∂̄(x) = 0 su h that the ℓ2-expansion of x has the form i,j=1 x ij. For γ ∈ S and 1 ≤ p, q ≤ nγ we then have PF̄ (x) = α/∈∂,α∈F i,j=1 (x) = PF̄ i,j=1 Here R denotes the L2-extension of Ruγpq . Sin e u pq is ontained in the linear span of the matrix oe� ients of uα T©uγ and sin e α /∈ ∂ = ∂S(F ) and γ ∈ S we see that the two expressions above are equal. By linearity and ontinuity we obtain f (2)PF̄ (x) = PF̄f (2)(x). This holds for all x ∈ ker(P∂̄), so if x ∈ H0 ∩ ker(P∂̄) we have 0 = f (2)PF̄ (x) = f(PF̄ (x)), where the last equality is due to the fa t that rg(PF̄ ) ⊆ A0 ⊆ H . This proves that PF̄ (x) ∈ K = ker(f) and sin e q was de�ned as the proje tion onto H0∩K⊥ we get qPF̄ (x) = 0. Sin e this holds whenever x ∈ H0 = q0(H) and P∂̄(x) = 0 we get qPF̄ (q0 ∧ (1− P∂̄)) = 0. -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 33 Thus, the restri tion qPF̄ : H0 → H fa torizes through H0/H0∩ker(P∂̄) and we have dimC(qPF̄ (H0)) ≤ dimC(H0/H0 ∩ ker(P∂̄)) ≤ dimC(H/ ker(P∂̄)) = dimC(rg(P∂̄)) = ϕ̂(P∂). For any �nite rank operator T ∈ B(H) one has |TrH(T )| ≤ ‖T‖ dimC(T (H)) and using this and Lemma 5.1 we now get h(q)ϕ̂(PF ) = h(q)ϕ̂(PF̄ ) = TrH(qPF̄ q) ≤ ‖qPF̄ q‖ dimC(qPF̄ q(H)) ≤ dimC(qPF̄ (H0)) ≤ ϕ̂(P∂). h(q) ≤ ϕ̂(P∂) ϕ̂(PF ) and sin e ε > 0 was arbitrary we on lude that q = 0. Part III We now treat the general ase of a �nitely presented A0-module Z with �nite presentation f−→ Am0 −→ Z −→ 0. In this ase f is given by right multipli ation by an n × m matrix T = (tij) with entries in A0. Ea h tij has a unique linear expansion as tij = α,k,l t (i,j) α,k,lu kl and we put S = {α ∈ I | ∃ i, j, k, l, α : t(i,j)α,k,l 6= 0}. As in Part II, we may assume that T 6= 0 so that S 6= ∅. Denote by H0 the spa e ker(f (2)) ⊆ Hn, by q0 ∈ Mn(M ′) the proje tion onto H0 and by q ∈ Mn(M ′) the proje tion onto H0 ∩ K⊥. We need to show that q = 0. Denote by Trn the non-normalized tra e on Mn(C) and put hn = h ⊗ Trn : B(H) ⊗ Mn(C) → C. We aim at proving that 34 DAVID KYED hn(q) = 0, whi h su� es sin e h is faithful on M . For ea h x ∈ M̂ we denote by xn the diagonal operator on Hn whi h has x in ea h diagonal entry. Under the identi� ation B(H) ⊗ Mn(C) = B(Hn) we see that TrH ⊗Trn orresponds to TrHn , and Lemma 5.1 together with a dire t omputation therefore gives TrHn(A ∗P nEA) = hn(A ∗A)ϕ̂(PE), (†) for any �nite subset E ⊆ I and any A in Mn(M) or Mn(M ′). Let ε > 0 be given and hoose a ording to the Følner ondition a �nite subset F ⊆ I su h that α∈∂S(F ) n2α < and put ∂ = ∂S(F ) for simpli ity. By repeating the argument from the beginning of Part II we arrive at the equation (q0 ∧ (1− P n∂̄ )) = 0, whi h in turn yields dimC(qP (H0)) ≤ dimC(rg(P n∂̄ )) = n dimC(rg(P∂̄)) = nϕ̂(P∂). Using the tra e-formula (†) we on lude that hn(q)ϕ̂(PF ) = TrHn(qP F̄ q) ≤ ‖qP n q‖ dimC(qP nF̄ q(H)) ≤ dimC(qP nF̄ (H0)) ≤ nϕ̂(P∂). hn(q) ≤ n ϕ̂(P∂) ϕ̂(PF ) and sin e ε > 0 was arbitrary we on lude that hn(q) = 0 as desired. By putting Z = C in Theorem 6.1, we immediately obtain the fol- lowing orollary. Corollary 6.2. Let G = (A,∆) be a ompa t, oamenable quantum group with tra ial Haar state. Then β n (G) = 0 for all n ≥ 1. Here n (G) is the n-th L2-Betti number of G as de�ned in [Kye08℄. In parti ular we obtain the following extension of [Kye08, 3.3℄. Corollary 6.3. For an abelian, ompa t quantum group G we have n (G) = 0 for n ≥ 1. -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 35 Proof. Sin e G is abelian it is of the form (C(G),∆c) for some ompa t (se ond ountable) group G. Sin e the ounit, given by evaluation at the identity, is already globally de�ned and bounded it is lear that G is oamenable and the result now follows from Corollary 6.2. � We also obtain the lassi al result of Lü k. Corollary 6.4. [Lü 98a, 5.1℄ If Γ is an amenable, ountable, dis rete group then for all CΓ-modules Z and all n ≥ 1 we have dimL (Γ) Tor n (L (Γ), Z) = 0. In parti ular, β n (Γ) = 0 for n ≥ 1. Proof. Put G = (C∗red(Γ),∆red). Then G is oamenable if and only if Γ is amenable and the result now follows from Theorem 6.1 and Corollary 6.2 � Note, however, that this does not really give a new proof of Lü k's result sin e the proof of Theorem 6.1 oin ides with Lü k's proof of the statement in Corollary 6.4 when G = (C∗red(Γ),∆red). In [CS05℄, Connes and Shlyakhtenko introdu ed a notion of L2-Betti numbers for tra ial ∗-algebras. From the above results we also obtain vanishing of these Connes-Shlyakhtenko L2-Betti numbers for ertain Hopf ∗-algebras. More pre isely we get the following. Corollary 6.5. Let G = (A,∆) be a ompa t, oamenable quantum group with tra ial Haar state h. Then β n (A0, h) = 0 for all n ≥ 1, where β n (A0, h) is the n-th Connes-Shlyakhtenko L -Betti number of the ∗-algebra A0 with respe t to the tra e h. Proof. By [Kye08, 4.1℄ we have β n (G) = β n (A0, h) and the laim therefore follows from Corollary 6.2. � The knowledge of dimension �atness also gives genuine homologi al information about the ring extension A0 ⊆ M . More pre isely, the following holds. Corollary 6.6. If G = (A,∆) is ompa t and oamenable with tra ial Haar state then the indu tion fun torM⊙A0− is an exa t fun tor from the ategory of �nitely generated, proje tive A0-modules to the ategory of �nitely generated, proje tive M-modules. Proof. Let X and Y be �nitely generated, proje tive A0-modules and let f : X → Y be an inje tive homomorphism. Then 0 −→ X f−→ Y −→ Y/rg(f) −→ 0, 36 DAVID KYED is a proje tive resolution of Y/rg(f). Thus TorA01 (M,Y/rg(f)) = ker(idM ⊗f) and from Theorem 6.1 we on lude that dimM(ker(idM ⊗f)) = 0. Be ause idM ⊗f is a map of �nitely generated proje tive M-modules, it is not di� ult to prove that ker(idM ⊗f) = ker(idM ⊗f) where ker(idM ⊗f) is de�ned (see [Lü 98a℄) as the interse tion of all kernels arising from homomorphisms fromM⊙A0X toM vanishing on ker(idM ⊗f). By [Lü 98a, 0.6℄, we on lude from this that ker(idM ⊗f) is �nitely generated and proje tive. But, sin e the dimension fun tion is faithful on the ategory of �nitely generated, proje tive M-modules this for es ker(idM ⊗f) = {0} and the laim follows. � Corollary 6.6, in parti ular, implies the following result whi h was pointed out to us by A. Thom. Corollary 6.7. Let G = (A,∆) be ompa t and oamenable with tra ial Haar state and let x ∈ A0 be a non-zero element su h that there exists a non-zero m ∈ M with mx = 0. Then there exists a non-zero y ∈ A0 with yx = 0. Proof. This follows by using Corollary 6.6 on the map a 7−→ ax. � An analogous statement about produ ts in the opposite order follows by using the involution in M . So, formulated in ring theoreti al terms, we obtain the following: Any regular element in A0 stays regular in the over-ring M . 7. Examples A on rete example of a non- ommutative, non- o ommutative, o- amenable (matrix) quantum group with tra ial Haar state is the or- thogonal quantum group Ao(2) ≃ SU−1(2). It follows from [Ban99a, 5.1℄ that Ao(2) is oamenable. To see that the Haar state is tra ial, one observes that the orthogonality property of the anoni al fundamental orepresentation implies that the antipode has period two. 7.1. Examples arising from tensor produ ts. If G1 = (A1,∆1) and G2 = (A2,∆2) are ompa t quantum groups then the (minimal) tensor produ t A = A1 ⊗ A2 may be turned into a quantum group G by de�ning the omultipli ation ∆: A −→ A⊗ A to be ∆(a) = (id⊗σ ⊗ id)(∆1 ⊗∆2)(a), -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 37 where σ denotes the �ip-isomorphism from A1 ⊗ A2 to A2 ⊗ A1. The Haar state is the tensor produ t of the two Haar states and the ounit is the tensor produ t of the ounits. Using these fa ts, it is not di� ult to see [BMT01℄ that if both G1 and G2 are oamenable and have tra ial Haar states, then the same is true for G. See e.g. [KR86, 11.3.2℄. 7.2. Examples arising from bi rossed produ ts. Another way to obtain examples of ompa t, oamenable quantum groups is via bi- rossed produ ts. We therefore brie�y sket h the bi rossed produ t onstru tion following [VV03℄ losely. In [VV03℄, Vaes and Vainerman onsider the more general notion of o y le bi rossed produ ts, but sin e we will mainly be interested in the ase where the o y les are trivial we will restri t our attention to this ase in the following. The more general situation will be dis ussed brie�y in Remark 7.5. The bi rossed produ t onstru tion is de�ned using the language of von Neumann algebrai quantum groups. We will use this language freely in the following and refer to [KV03℄ for the ba kground material. Let (M1,∆1) and (M2,∆2) be lo ally ompa t (l. .) von Neumann algebrai quantum groups. Let τ : M1⊗̄M2 → M1⊗̄M2 be a faith- ful ∗-homomorphism and denote by σ : M1⊗̄M2 → M2⊗̄M1 the �ip- isomorphism. Then τ is alled a mat hing from M1 to M2 if the fol- lowing holds. • The map α : M2 −→ M1⊗̄M2 given by α(y) = τ(1 ⊗ y) is a (left) oa tion of (M1,∆1) on the von Neumann algebra M2. • De�ning β : M1 −→M1⊗̄M2 as β(x) = τ(x⊗ 1) the map σβ is a (left) oa tion of (M2,∆2) on the von Neumann algebra M1. • The oa tions satisfy the following two mat hing onditions: τ(13)(α⊗ 1)∆2 = (1⊗∆2)α (M1) τ(23)σ(23)(β ⊗ 1)∆1 = (∆1 ⊗ 1)β (M2) Here we use the standard leg numbering onvention (see e.g. [MVD98℄). If τ : M1⊗̄M2 → M1⊗̄M2 is a mat hing from M1 to M2 then it is easy to see that στσ−1 is a mat hing fromM2 to M1. We will therefore just refer to the pair (M1,M2) as a mat hed pair and to τ as a mat hing of the pair. Let (M1,∆1) and (M2,∆2) be su h a mat hed pair of l. . quantum groups and denote by τ the mat hing. We denote by Hi the GNS spa e ofMi with respe t to the left invariant weight ϕi and by Wi and Ŵi the natural multipli ative unitaries on Hi⊗̄Hi for Mi and M̂i respe tively. By H we denote H1⊗̄H2 and by Σ the �ip-unitary on 38 DAVID KYED H⊗̄H . We may now form two rossed produ ts: M =M1 ⋉α M2 = vNa{α(M2), M̂1 ⊗ 1} ⊆ B(H1⊗̄H2) M̃ =M2 ⋉σβ M1 = vNa{σβ(M1), M̂2 ⊗ 1} ⊆ B(H2⊗̄H1) Some of the main results in [VV03℄ are summarized in the following: Theorem 7.1 ([VV03℄). De�ne operators Ŵ = (β ⊗ 1⊗ 1)(W1 ⊗ 1)(1⊗ 1⊗ α)(1⊗ Ŵ2) and W = ΣŴ ∗Σ on H⊗̄H. Then W and Ŵ are multipli ative uni- taries and the map ∆: M → B(H⊗̄H) given by ∆(a) = W ∗(1⊗1⊗a)W de�nes a omultipli ation on M turning it into a l. . quantum group. Denoting by Σ12 the �ip-unitary fromH1⊗̄H2 to H2⊗̄H1, the dual quan- tum group M̂ be omes Σ∗12M̃Σ12 with omultipli ation implemented by Thus, up to a �ip the two rossed produ ts above are in duality. In [DQV02℄, Desmedt, Quaegebeur and Vaes studied ( o)amenability of bi rossed produ ts. Combining their Theorem 15 with [VV03, 2.17℄ we obtain the following: If (M1,M2) is a mat hed pair with M1 dis- rete and M2 ompa t then the bi rossed produ t M is ompa t, and M is oamenable if and only if both M2 and M̂1 are. Here a von Neu- mann algebrai ompa t quantum group is said to be oamenable if the orresponding C∗-algebrai quantum group is. Colle ting the results dis ussed above we obtain the following. Proposition 7.2. If (M1,M2) is a mat hed pair of l. . quantum groups in whi h M̂1 and M2 are ompa t and oamenable, then the bi rossed produ t M = M1 ⋉α M2 is oamenable and ompa t. So if the Haar state on M is tra ial the quantum group (M,∆) has vanishing L2-Betti numbers in all positive degrees. In order to produ e more on rete examples, we will now dis uss a spe ial ase of the bi rossed produ t onstru tion in whi h one of the oa tions omes from an a tual group a tion. This part of the theory is due to De Cannière [DC79℄ and is formulated using the language of Ka algebras. We remind the reader that a ompa t Ka algebra is nothing but a von Neumann algebrai , ompa t quantum group with tra ial Haar state. A dis rete, ountable group Γ a ts on a ompa t Ka algebra (M,∆, S, h) if the group a ts on the von Neumann algebra M and the a tion ommutes with both the oprodu t and the antipode. -BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 39 Denoting the a tion by ρ, this means that ∆(ργ(x)) = ργ ⊗ ργ(∆(x)), S(ργ(x)) = ργ(S(x)), for all γ ∈ Γ and all x ∈ M . In this situation, the a tion of Γ on M indu es a oa tion α : M −→ ℓ∞(Γ)⊗̄M . Denoting by H the Hilbert spa e on whi h M a ts and identifying ℓ2(Γ)⊗̄H with ℓ2(Γ, H), this oa tion is given by the formula α(x)(ξ)(γ) = ργ−1(x)(ξ(γ)), for ξ ∈ ℓ2(Γ, H). The rossed produ t, whi h is de�ned as Γ⋉ρ M = {α(M),L (Γ)⊗ 1}′′, be omes again a Ka algebra [DC79, Thm.1℄. One should note at this point that De Cannière works with the right rossed produ t a ting on H⊗̄ℓ2(Γ) where we work with the left rossed produ t a ting on ℓ2(Γ)⊗̄H . But, one an ome from one to the other by onjugation with the �ip-unitary and we may therefore freely transport all results from [DC79℄ to the setting of left rossed produ ts. We now prove that De Cannière's rossed produ t an also be onsidered as a bi rossed produ t. This is probably well known to experts in the �eld, but we were unable to �nd an expli it referen e. Proposition 7.3. De�ning τ : ℓ∞(Γ)⊗̄M −→ ℓ∞(Γ)⊗̄M by τ(δγ ⊗ x) = δγ ⊗ ργ−1(x) we obtain a mat hing with the above de�ned α as the orresponding oa tion of ℓ∞(Γ) on M and trivial oa tion of (M,∆) on ℓ∞(Γ). Proof. A dire t al ulation shows that α(x) = τ(1 ⊗ x) and β(f) = τ(f ⊗ 1) = f ⊗ 1. Therefore the two maps x 7→ τ(1 ⊗ x) and f 7→ στ(f ⊗ 1) are oa tions as required. We therefore just have to he k that the mat hing onditions are ful�lled. Denote the oprodu t on ℓ∞(Γ) by ∆1 and hoose f ∈ ℓ∞(Γ) su h that ∆1(f) ∈ ℓ∞(Γ)⊙ ℓ∞(Γ). Writing ∆1(f) as f(1) ⊗ f(2) we now get τ(23)σ(23)(β ⊗ 1)∆1f = τ(23)σ(23)(β ⊗ 1)(f(1) ⊗ f(2)) = τ(23)σ(23)(f(1) ⊗ 1⊗ f(2)) = τ(23)(f(1) ⊗ f(2) ⊗ 1) = f(1) ⊗ f(2) ⊗ 1 = (∆1 ⊗ 1)β(f), and hen e (M2) is satis�ed. An analogous, but slightly more umber- some, al ulation proves that (M1) is also satis�ed. � 40 DAVID KYED Thus, as von Neumann algebras, we have ℓ∞(Γ) ⋉α M = Γ ⋉ρ M . Using the fa t that β is trivial, one an prove that the elements λγ ⊗ 1 are group-like and it therefore follows from [DC79, 3.3℄ that also the omultipli ations agree. Hen e the two rossed produ t onstru tions are identi al as l. . quantum groups. In parti ular, the the bi rossed produ t ℓ∞(Γ) ⋉α M is a Ka algebra so if (M,∆) is ompa t then ℓ∞(Γ)⋉α M is also ompa t [VV03, 2.7℄ and the Haar state is tra ial. We therefore have the following. Proposition 7.4. If G = (M,∆, S, h) is a ompa t, oamenable Ka algebra and Γ is a ountable, dis rete, amenable group a ting on G then the rossed produ t Γ⋉M is again a ompa t, oamenable Ka algebra. Proof. That Γ⋉M is a Ka algebra follows from the dis ussion above and the oamenability of the rossed produ t follows from [DQV02, 15℄ sin e ℓ̂∞(Γ) = L (Γ) is oamenable if (and only if) Γ is amenable. � Remark 7.5. It is also possibly to onstru t examples using the more general notion of o y le rossed produ ts introdu ed in [VV03, 2.1℄. It is shown in [DQV02, 13℄ that weak amenability (i.e. the existen e of an invariant mean) is preserved under o y le bi rossed produ ts. In general it is not known whether or not weak amenability is equivalent to strong amenability, the latter being de�ned as the dual quantum group being oamenable in the sense of De�nition 4.1. But for dis- rete quantum groups this equivalen e has been proven by Tomatsu in [Tom06℄ and also by Blan hard and Vaes in unpublished work. 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Notes on amenability of ommutative fusion algebras. Positivity, 3(4):377�388, 1999. David Kyed, Mathematis hes Institut, Georg-August-Universität, Göttingen, Bunsenstraÿe 3-5, D-37073 Göttingen, Germany E-mail address : kyed�uni-math.gwdg.de URL: www.uni-math.gwdg.de/kyed www.wis.kuleuven.be/analyse/stefaan Introduction 1. Preliminaries on compact quantum groups 2. Fusion Algebras 3. Amenability for Fusion Algebras 3.1. Formulas used in the proof of Theorem ?? 4. Coamenable Compact Quantum Groups 5. An Interlude 6. A Vanishing Result 7. Examples 7.1. Examples arising from tensor products 7.2. Examples arising from bicrossed products References

We prove that a compact quantum group is coamenable if and only if its corepresentation ring is amenable. We further propose a Foelner condition for compact quantum groups and prove it to be equivalent to coamenability. Using this Foelner condition, we prove that for a coamenable compact quantum group with tracial Haar state, the enveloping von Neumann algebra is dimension flat over the Hopf algebra of matrix coefficients. This generalizes a theorem of Lueck from the group case to the quantum group case, and provides examples of compact quantum groups with vanishing L^2-Betti numbers.

Introduction 1. Preliminaries on compact quantum groups 2. Fusion Algebras 3. Amenability for Fusion Algebras 3.1. Formulas used in the proof of Theorem ?? 4. Coamenable Compact Quantum Groups 5. An Interlude 6. A Vanishing Result 7. Examples 7.1. Examples arising from tensor products 7.2. Examples arising from bicrossed products References

Mon. Not. R. Astron. Soc. 000, 1–5 (2006) Printed 11 November 2021 (MN LATEX style file v2.2) OPserver: interactive online-computations of opacities and radiative accelerations C. Mendoza,1,2⋆ M. J. Seaton,3 P. Buerger,4 A. Belloŕın,5 M. Meléndez,6† J. González,1,7 L. S. Rodŕıguez,8 F. Delahaye,9 E. Palacios,7 A. K. Pradhan10 and C. J. Zeippen9 1Centro de F́ısica, Instituto Venezolano de Investigaciones Cient́ıficas (IVIC), PO Box 21827, Caracas 1020A, Venezuela 2Centro Nacional de Cálculo Cient́ıfico Universidad de Los Andes (CeCalCULA), Mérida 5101, Venezuela 3Department of Physics and Astronomy, University College London, London WC1E 6BT, UK 4Ohio Supercomputer Center, Columbus, Ohio 43212, USA 5Escuela de F́ısica, Facultad de Ciencias, Universidad Central de Venezuela, PO Box 20513, Caracas 1020-A, Venezuela 6Departamento de F́ısica, Universidad Simón Boĺıvar, PO Box 89000, Caracas 1080-A, Venezuela 7Escuela de Computación, Facultad de Ciencia y Tecnoloǵıa, Universidad de Carabobo, Valencia, Venezuela 8Centro de Qúımica, Instituto Venezolano de Investigaciones Cient́ıficas (IVIC), P.O. Box 21827, Caracas 1020A, Venezuela 9LUTH, Observatoire de Paris, F-92195 Meudon, France 10Department of Astronomy, The Ohio State University, Columbus, Ohio 43210, USA Accepted. Received; in original form ABSTRACT Codes to compute mean opacities and radiative accelerations for arbitrary chemical mixtures using the Opacity Project recently revised data have been restructured in a client–server architecture and transcribed as a subroutine library. This implementation increases efficiency in stellar modelling where element stratification due to diffusion processes is depth dependent, and thus requires repeated fast opacity reestimates. Three user modes are provided to fit different computing environments, namely a web browser, a local workstation and a distributed grid. Key words: atomic processes – radiative transfer – stars: interior. 1 INTRODUCTION Astrophysical opacities from the Opacity Project (OP) have been updated by Badnell et al. (2005) to include inner- shell contributions and an improved frequency mesh. The complete data set of monochromatic opacities and a suite of codes to compute mean opacities and radiative accel- erations (OPCD 2.11) have also been publicly released by Seaton (2005) to make in-house calculations for arbitrary mixtures more versatile and expedient. Regarding data ac- curacy, there is excellent overall agreement between the OPAL (Iglesias & Rogers 1996) and OP results as dis- cussed by Seaton & Badnell (2004), Badnell et al. (2005) and Delahaye & Pinsonneault (2005). Rosseland mean opacities are sensitive to both the ⋆ E-mail: claudio@ivic.ve. † Present address: Institute for Astrophysics and Computational Sciences, Department of Physics, The Catholic University of America, Washington, DC 20064, and Exploration of the Uni- verse Division, Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA. 1 http://cdsweb.u-strasbg.fr/topbase/op.html basic atomic data used and the assumed abundances of the chemical elements. What had been a good agree- ment between theory and the helioseismological data was found to be less good using revised solar abundances from Asplund et al. (2005). The revised OP opacities have been instrumental in discussions of that problem (Antia & Basu 2005; Bahcall et al. 2005a,b,c; Bahcall & Serenelli 2005; Delahaye & Pinsonneault 2006). The modelling of stellar interiors, on the other hand, is being renewed with the solar experience. Present (WIRE, MOST, CoRoT) and future (Kepler) space probes and the well established solar methods are giving the field of asteroseismology remarkable growth and the guaran- tee of invaluable data (Metcalfe et al. 2004; Kurtz 2005; Christensen-Dalsgaard 2006). In future work on stellar mod- els it may be desirable to take account of revisions in abun- dances similar to those performed for the Sun. For some types of stars, models must take into ac- count microscopic diffusion processes, e.g. radiative levita- tion, gravitational settling and thermal diffusion, as they can affect the internal and thermal structures, the depth of the convection zone, pulsations and give rise to surface abun- dance anomalies (Seaton 1999; Delahaye & Pinsonneault c© 2006 RAS http://arxiv.org/abs/0704.1583v1 http://cdsweb.u-strasbg.fr/topbase/op.html 2 C. Mendoza et al. 2005; Bourge & Alecian 2006). As reviewed by Michaud (2004), such processes are relevant in the description of chemically peculiar stars, horizontal-branch stars, white dwarfs and neutron stars, and in globular cluster age de- terminations from Population II turnoff stars. Furthermore, in order to solve the outstanding discrepancy of the at- mospheric Li abundance in old stars with that predicted in big-bang nucleosynthesis, Richard et al. (2005) have pro- posed Li sinking deep into the star due to diffusion. This hypothesis has been recently confirmed in the observations by Korn et al. (2006). The OPCD 2.1 release includes data and codes to com- pute the radiative accelerations required for studies of dif- fusion processes. It should be noted that the radiative ac- celerations are summed over ionization stages and that data for the calculation of diffusion coefficients are calculated as- suming that the distribution over ionization stages of the diffusing ions is the same as that in the ambient plasma. The validity of this approximation is discussed by Gonzalez et al. (1995). In some cases, particularly when element stratification depends on stellar depth, calculations of mean opacities and radiative accelerations must be repeated at each depth point of the model and at each time step of the evolution, and thus the use of codes more efficient than those in OPCD 2.1 may be necessary. This becomes critical in the new dis- tributed computing grid environments where the network transfer of large volumes of data is a key issue. In the present work we have looked into these problems, and, as a solu- tion, report on the implementation of a general purpose, interactive server for astrophysical opacities referred to as OPserver. It has been installed at the Ohio Supercomputer Center, Columbus, Ohio, USA, from where it can be ac- cessed through a web page2 or a linkable subroutine library. It can also be downloaded locally to be run on a stand-alone basis but it will demand greater computational facilities. In Section 2 we discuss the computational strategy of the codes in OPCD 2.1 followed by a description of OPserver in Sec- tion 3. In Section 4 we include some tests as an indication of its performance with a final summary in Section 5. 2 OPCD CODES We highlight here some of the key features of the codes in OPCD 2.1. For a chemical mixture specified by abundance fractions fk, they essentially compute two types of data: Rosseland mean opacities (RMO) and radiative accelera- tions (RA). 2.1 Rosseland mean opacities For the frequency variable u = hν/kBT (1) where kB is the Boltzmann constant, RMO are given by the harmonic mean of the opacity cross section σ(u) of the mixture 2 http://opacities.osc.edu dv (2) where µ is the mean atomic weight. The σ(u) is a weighted sum of the monochromatic opacity cross sections for each of the chemical constituents σ(u) = fkσk(u) , (3) and is conveniently tabulated on the v-mesh v(u) = F (u) 1− exp(−u) du (4) where F (u) = 15u4 exp(−u) 4π4[1− exp(−u)]2 and v∞ = v(u → ∞). The rationale behind the v-mesh is that it enhances frequency resolution where F (u) is large (Badnell et al. 2005). 2.2 Radiative accelerations Similarly, the RA for a selected k element can be expressed grad = µκRγk F (6) where µk is its atomic weight and c the speed of light. The function F is given in terms of the effective temperature Teff and fractional depth r/R⋆ of the star by F = πB(Teff)(R⋆/r) B(T ) = 2(πkBT ) 15c2h3 . (8) The dimensionless parameter σmtak dv (9) depends on the cross section for momentum transfer to the k element k = σk(u)[1− exp(−u)]− ak(u) (10) where ak(u) is a correction to remove the contributions of electron scattering and momentum transfer to the electrons. Both σk(u) and ak(u), which are hereafter referred to as the mono data set (∼1 GB), are tabulated in equally spaced v intervals to facilitate accurate interpolation schemes. 2.3 Computational strategy The computational strategy adopted in the OPCD 2.1 re- lease is depicted in the flowcharts of Figure 1 where it may be seen that calculations of RMO and RA are carried in two stages. In a time consuming Stage 1, RMO and RA are computed with the mixv and accv codes, respectively, on a representative tabulation of the complete temperature– electron-density (T,Ne) plane. In mixv the chemical mixture is specified in the input file mixv.in as {X,Z,N,Zk, fk} (11) c© 2006 RAS, MNRAS 000, 1–5 http://opacities.osc.edu OPserver 3 http://vizier.u-strasbg.fr/topbase/opserver/fig1.eps Figure 1. Flowcharts for the computations of Rosseland mean opacities (RMO) and radiative accelerations (RA) with the codes in the OPCD 2.1 release. They are carried out in two stages: in a time consuming Stage 1, data are computed for the whole (T,Ne) plane followed by fast bicubic interpolations in Stage 2. The intermediate files mixv.xx and acc.xx enable communication between these two steps. where X and Z are the hydrogen and metal mass-fractions, N the number of elements, and Zk and fk are the metal nu- clear charges and fractional abundances. In accv, the input data (accv.in) are {N,Zk, fk, Zi, Nχ, χj} (12) where now k runs over the N elements of the mixture, and Zi and χj are respectively the nuclear charge and Nχ abun- dance multipliers of the test i element. Input data formats in either mixv.in or accv.in give the user flexible control over chemical mixture specifications. As shown in Figure 1, the intermediate output files mixv.xx (∼85 KB) containing {ρ, κR}(T,Ne) , (13) where ρ is the mass-density, and acc.xx (∼470 KB) with (T,Ne, χj) (14) are written to disk. They are then respectively read by the codes opfit and accfit in Stage 2 for fast bicubic interpolations of RMO and RA on stellar depth profiles {T, ρ, r/R⋆}(i) specified by the user in the opfit.in and accfit.in input files. The final output files are opfit.xx containing log κR, ∂ log κR ∂ log T ∂ log κR ∂ log ρ (i) (15) and accfit.xx with {log κR, log γ, log grad}(i, χj) . (16) In this computational approach, performance is mainly limited by the summation in equation (3) which implies disk reading the mono data set; for instance, in mixv it takes up to ∼90% of the total elapsed time. OPCD 2.1 also in- cludes other codes such as mx and ax which respectively compute RMO and RA for a star depth profile. The chemi- cal mixture can be fully varied at each depth point using the specifications in equations (11–12), the RMO and RA being obtained in a one-step process using bicubic interpolations without splines. These methods are thus suitable for cases with multi-mixture depth profiles (Seaton 2005). Further de- tails of all the OPCD codes are contained in the reference manual3. 3 OPSERVER In OPserver the computational capabilities of the codes in OPCD 2.1 are greatly enhanced by the following innovative adaptations. 3 http://opacities.osc.edu/publi/OPCD.pdf (i) The codes are restructured within a client–server net- work architecture whereby the time consuming Stage 1 is performed on a powerful processor while the fast Stage 2 is moved to the client, e.g. a web server or a user workstation. In this arrangement performance could be affected by the client–server transfer of the mixv.xx and acc.xx intermedi- ate files, but since they are never larger than 0.5 MB, it is not expected to be a deterrent with present-day bandwidths. In a local installation where both the client and server reside on the same machine, communication is managed through shared buffers in main memory; i.e. the corresponding data in mixv.xx and acc.xx are not written to disk. (ii) The codes are transcribed as a subroutine library— to be referred to hereafter as the OPlibrary—which can be linked by the user stellar modelling code for recurrent sub- routine calls that avoid data writing on disk. That is, the in- put data in the mixv.in, accv.in, opfit.in and accfit.in files and the output tables in the opfit.xx and accfit.xx files (see Figure 1) are now handled as subroutine param- eters while the intermediate mixv.xx and acc.xx files are passed via shared main-memory buffers. Chemical mixtures are again specified with the formats of equations (11–12) which allow full variation at each depth point in a single subroutine call. (iii) RMO/RA are computed with the complete mono data set always loaded in main memory thus avoiding lengthy and repeated disk readings. This is achieved by implementing OPserver on a dedicated server where mono is permanently resident in RAM, or in the case of a local installation, by disk-reading it once at the outset of a modelling calculation. (iv) When accessing the remote server, client data re- quests are addressed through the HTTP protocol, i.e. in terms of a Uniform Resource Locator (URL). This allows data fetching from the central facility through an interactive web page or a network access subroutine, the latter being particularly suitable for a stellar model code that is to be run in a distributed grid environment. (v) The do-loop that computes the summation of equa- tion (3) has been parallelized in OpenMP which provides a simple, scalable and portable scheme for shared-memory platforms. As shown in Figure 2, the current OPserver enterprise is implemented as a client–server model at the Ohio Supercom- puter Center (OSC). The web server communicates with the supercomputer via a socket interface. Earlier versions were developed on an SGI Origin2000 server with the PowerFor- tran parallelizing compiler. The current version runs on a Linux system with Fortran OpenMP directives. OPserver offers three user modes with full functionality except when otherwise indicated in the following description. Mode A In this mode OPserver is set up locally on a stand-alone basis (see Figure 2). The facilities of the OSC are c© 2006 RAS, MNRAS 000, 1–5 http://vizier.u-strasbg.fr/topbase/opserver/fig1.eps http://opacities.osc.edu/publi/OPCD.pdf 4 C. Mendoza et al. http://vizier.u-strasbg.fr/topbase/opserver/fig2.eps Figure 2. OPserver enterprise showing the web-server–supercomputer tandem at the Ohio Supercomputer Center (OSC) and the three available user modes. (A) The OPlibrary and monochromatic opacities (mono) are downloaded locally and linked to the user modelling code such that RMO/RA are computed locally. (B) The OPlibrary is downloaded locally and linked to the modelling code but RMO/RA are computed remotely at the OSC. (C) RMO/RA computations at the OSC through a web client. not used. A new OPCD release (OPCD 3.34) is downloaded, followed by (i) installation of both the OPlibrary and the mono data set and (ii) linking of the OPlibrary to the user modelling code. Computations of RMO/RA are preceded by the reading of the complete mono data set from disk and therefore requires at least 1 GB of RAM. Mode B In this mode, the OPlibrary is downloaded, in- stalled and linked to the user code, but Stage 1 is performed remotely at the OSC (see Figure 2). This option has been customized for stellar modelling in a distributed grid envi- ronment that would otherwise imply (i.e. Mode A) the net- work transfer, installation and disk-reading of the mono data set at runtime. It is also practical when local computer capa- bilities (RAM and/or disk space) are limited. The functions provided by the mx and ax codes have not been implemented. Mode C In this mode RMO/RA computations at the OSC are requested through an interactive web page5 which allows both Stage 1 and Stage 2 to be carried out re- motely or, alternatively, Stage 2 locally by downloading the mixv.xx/acc.xx intermediate files (see Figure. 1) with the browser for further processing with local opfit/accfit ex- ecutables. 4 TESTS OPserver benchmarks were initially carried out on an SGI Origin2000 multiprocessor at the OSC with an earlier re- lease of OPCD. For the standard S92 mixture (Seaton et al. 1994), the mixv code took up to 140 s to compute the mixv.xx file, of which 126 s were dedicated to disk-reading and 14 s to the actual computing of the mean opacities. OPserver took on average 12.0 ± 0.5 s to compute mixv.xx which was not written to disk unless requested. In Fig- ure 3 we show the acceleration obtained on the Origin2000 through parallelization where the calculation of mean opac- ities is reduced to 2 s with 8 processors. Further significant acceleration is prevented by data transfer overheads. On more recent workstations, the local performances of the codes in OPCD 2.1 and OPserver depend on processor speed and RAM and cache sizes. For instance, on a Pow- erMac G5 (PowerPC 970fx processor at 2.0 GHz, 1GB of RAM and L2 cache of 512 KB) the first time mixv is run it takes for a single S92 mixture 103.8 s to compute the RMO, but on subsequent runs the elapsed time is reduced to an av- erage of 28.2±0.2 s. Similarly, OPserver takes 103.3 s which is then reduced to 31.4 ± 0.4 s on subsequent runs. Once the mono data set is loaded in RAM by OPserver (Mode A), calculations of RMO for a single S92 mixture only take 5.29 ± 0.02 s and 6.13 ± 0.01 s for RA for the test element Ar. In Mode B, where Stage 1 is carried out remotely at the OSC and the mixv.xx and accv.xx files are transferred at 4 http://cdsweb.u-strasbg.fr/topbase/op.html 5 http://opacities.osc.edu the relatively low rate of 1.88 KB/s, computations of RMO and RA take 5.9±0.1 s and 9.3±0.3 s, respectively. The no- ticeable longer time taken for the latter is due to the transfer time taken for the larger accv.xx file. 5 SUMMARY Rosseland mean opacities and radiative accelerations can be computed from OP data in any one of the following ways. (i) Download the original OPCD 2.1 package as described by Seaton (2005) and perform all calculations locally. (ii) Mode A, download the upgraded OPCD 3.3 package, in- stall OPserver and perform all calculations locally by link- ing the subroutines in the OPlibrary. Calculations with OPserver are more efficient but require large local computer memory. (iii) Mode B, as Mode A but with Stage 1 performed re- motely at the OSC. Mode B is convenient if fast calculations are required but local computer memory is limited or when stellar modelling is to be carried out in a grid environment. (iv) Mode C, perform all calculations remotely at the OSC through an interactive web page whereby files are down- loaded locally with the browser. ACKNOWLEDGMENTS We acknowledge the invaluable assistance of Juan Luis Chaves and Gilberto Dı́az of CeCalCULA during the ini- tial stages of OPserver. We are also much indebted to the Ohio Supercomputer Center, Columbus, Ohio, USA, for hosting OPserver and for technical assistance; to the Centre de Données astronomiques de Strasbourg, France, for host- ing the OPCD releases; and to Drs Josslen Aray, Manuel Bautista, Juan Murgich and Fernando Ruette of IVIC for allowing us to test the OPserver installation on different platforms. FD would like to thank S. Rouchy for techni- cal support. AKP and FD have been partly supported by a grant from the US National Science Foundation. 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Codes to compute mean opacities and radiative accelerations for arbitrary chemical mixtures using the Opacity Project recently revised data have been restructured in a client--server architecture and transcribed as a subroutine library. This implementation increases efficiency in stellar modelling where element stratification due to diffusion processes is depth dependent, and thus requires repeated fast opacity reestimates. Three user modes are provided to fit different computing environments, namely a web browser, a local workstation and a distributed grid.

Introduction OPCD codes Rosseland mean opacities Radiative accelerations Computational strategy OPserver Tests Summary

arXiv:0704.1584v1 [math.ST] 12 Apr 2007 Can One Estimate The Unconditional Distribution of Post-Model-Selection Estimators? Hannes Leeb Department of Statistics, Yale University Benedikt M. Pötscher Department of Statistics, University of Vienna First version: April 2005 Revised version: February 2007 Abstract We consider the problem of estimating the unconditional distribution of a post-model-selection esti- mator. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model selection criterion like AIC or by a hypothesis testing procedure) and then estimating the parameters in the selected model (e.g., by least-squares or maximum likelihood), all based on the same data set. We show that it is impossible to estimate the unconditional distribution with reasonable accuracy even asymptotically. In particular, we show that no estimator for this distribution can be uniformly consistent (not even locally). This follows as a corollary to (lo- cal) minimax lower bounds on the performance of estimators for the distribution; performance is here measured by the probability that the estimation error exceeds a given threshold. These lower bounds are shown to approach 1/2 or even 1 in large samples, depending on the situation considered. Similar impossibility results are also obtained for the distribution of linear functions (e.g., predictors) of the post-model-selection estimator. AMS Mathematics Subject Classification 2000: 62F10, 62F12, 62J05, 62J07, 62C05. Keywords: Inference after model selection, Post-model-selection estimator, Pre-test estimator, Selection of re- gressors, Akaike’s information criterion AIC, Thresholding, Model uncertainty, Consistency, Uniform consistency, Lower risk bound. Research of the first author was supported by the Max Kade Foundation and by the Austrian National Science Foundation (FWF), Grant No. P13868-MAT. A preliminary draft of the material in this paper was already written in 1999. 1 Introduction and Overview In many statistical applications a data-based model selection step precedes the final parameter estimation and inference stage. For example, the specification of the model (choice of functional form, choice of regressors, http://arxiv.org/abs/0704.1584v1 number of lags, etc.) is often based on the data. In contrast, the traditional theory of statistical inference is concerned with the properties of estimators and inference procedures under the central assumption of an a priori given model. That is, it is assumed that the model is known to the researcher prior to the statistical analysis, except for the value of the true parameter vector. As a consequence, the actual statistical properties of estimators or inference procedures following a data-driven model selection step are not described by the traditional theory which assumes an a priori given model; in fact, they may differ substantially from the properties predicted by this theory, cf., e.g., Danilov and Magnus (2004), Dijkstra and Veldkamp (1988), Pötscher (1991, Section 3.3), or Rao and Wu (2001, Section 12). Ignoring the additional uncertainty originating from the data-driven model selection step and (inappropriately) applying traditional theory can hence result in very misleading conclusions. Investigations into the distributional properties of post-model-selection estimators, i.e., of estimators constructed after a data-driven model selection step, are relatively few and of recent vintage. Sen (1979) obtained the unconditional large-sample limit distribution of a post-model-selection estimator in an i.i.d. maximum likelihood framework, when selection is between two competing nested models. In Pötscher (1991) the asymptotic properties of a class of post-model-selection estimators (based on a sequence of hypothesis tests) were studied in a rather general setting covering non-linear models, dependent processes, and more than two competing models. In that paper, the large-sample limit distribution of the post-model-selection estimator was derived, both unconditional as well as conditional on having chosen a correct model, not necessarily the minimal one. See also Pötscher and Novak (1998) for further discussion and a simulation study, and Nickl (2003) for extensions. The finite-sample distribution of a post-model-selection estimator, both unconditional and conditional on having chosen a particular (possibly incorrect) model, was derived in Leeb and Pötscher (2003) in a normal linear regression framework; this paper also studied asymptotic approximations that are in a certain sense superior to the asymptotic distribution derived in Pötscher (1991). The distributions of corresponding linear predictors constructed after model selection were studied in Leeb (2005, 2006). Related work can also be found in Sen and Saleh (1987), Kabaila (1995), Pötscher (1995), Ahmed and Basu (2000), Kapetanios (2001), Hjort and Claeskens (2003), Dukić and Peña (2005), and Leeb and Pötscher (2005a). The latter paper provides a simple exposition of the problems of inference post model selection and may serve as an entry point to the present paper. It transpires from the papers mentioned above that the finite-sample distributions (as well as the large- sample limit distributions) of post-model-selection estimators typically depend on the unknown model pa- rameters, often in a complicated fashion. For inference purposes, e.g., for the construction of confidence sets, estimators of these distributions would be desirable. Consistent estimators of these distributions can typically be constructed quite easily, e.g., by suitably replacing unknown parameters in the large-sample limit distributions by estimators; cf. Section 2.2.1. However, the merits of such ‘plug-in’ estimators in small samples are questionable: It is known that the convergence of the finite-sample distributions to their large- sample limits is typically not uniform with respect to the underlying parameters (see Appendix B below and Corollary 5.5 in Leeb and Pötscher (2003)), and there is no reason to believe that this non-uniformity will disappear when unknown parameters in the large-sample limit are replaced by estimators. This observation is the main motivation for the present paper to investigate in general the performance of estimators of the distribution of a post-model-selection estimator, where the estimators of the distribution are not necessar- ily ‘plug-in’ estimators based on the limiting distribution. In particular, we ask whether estimators of the distribution function of post-model-selection estimators exist that do not suffer from the non-uniformity phenomenon mentioned above. As we show in this paper the answer in general is ‘No’. We also show that these negative results extend to the problem of estimating the distribution of linear functions (e.g., linear predictors) of post-model-selection estimators. Similar negative results apply also to the estimation of the mean squared error or bias of post-model-selection estimators; cf. Remark 4.7. To fix ideas consider for the moment the linear regression model Y = V χ+Wψ + u (1) where V and W , respectively, represent n× k and n× l non-stochastic regressor matrices (k ≥ 1, l ≥ 1), and the n× 1 disturbance vector u is normally distributed with mean zero and variance-covariance matrix σ2In. We also assume for the moment that (V : W )′(V : W )/n converges to a non-singular matrix as the sample size n goes to infinity and that limn→∞ V ′W/n 6= 0 (for a discussion of the case where this limit is zero see Example 1 in Section 2.2.2). Now suppose that the vector χ represents the parameters of interest, while the parameter vector ψ and the associated regressors in W have been entered into the model only to avoid possible misspecification. Suppose further that the necessity to include ψ or some of its components is then checked on the basis of the data, i.e., a model selection procedure is used to determine which components of ψ are to be retained in the model, the inclusion of χ not being disputed. The selected model is then used to obtain the final (post-model-selection) estimator χ̃ for χ. We are now interested in the unconditional finite-sample distribution of χ̃ (appropriately scaled and centered). Denote this k-dimensional cumulative distribution function (cdf) by Gn,θ,σ(t). As indicated in the notation, this distribution function depends on the true parameters θ = (χ′, ψ′)′ and σ. For the sake of definiteness of discussion assume for the moment that the model selection procedure used here is the particular ‘general-to-specific’ procedure described at the beginning of Section 2; we comment on other model selection procedures, including Akaike’s AIC and thresholding procedures, below. As mentioned above, it is not difficult to construct a consistent estimator of Gn,θ,σ(t) for any t, i.e., an estimator Ĝn(t) satisfying Pn,θ,σ (∣∣∣Ĝn(t)−Gn,θ,σ(t) ∣∣∣ > δ n→∞−→ 0 (2) for each δ > 0 and each θ, σ; see Section 2.2.1. However, it follows from the results in Section 2.2.2 that any estimator satisfying (2), i.e., any consistent estimator of Gn,θ,σ(t), necessarily also satisfies ||θ||<R Pn,θ,σ (∣∣∣Ĝn(t)−Gn,θ,σ(t) ∣∣∣ > δ n→∞−→ 1 (3) for suitable positive constants R and δ that do not depend on the estimator. That is, while the probability in (2) converges to zero for every given θ by consistency, relation (3) shows that it does not do so uniformly in θ. It follows that Ĝn(t) can never be uniformly consistent (not even when restricting consideration to uniform consistency over all compact subsets of the parameter space). Hence, a large sample size does not guarantee a small estimation error with high probability when estimating the distribution function of a post- model-selection estimator. In this sense, reliably assessing the precision of post-model-selection estimators is an intrinsically hard problem. Apart from (3), we also provide minimax lower bounds for arbitrary (not necessarily consistent) estimators of the conditional distribution function Gn,θ,σ(t). For example, we provide results that imply that lim inf Ĝn(t) ||θ||<R Pn,θ,σ (∣∣∣Ĝn(t)−Gn,θ,σ(t) ∣∣∣ > δ > 0 (4) holds for suitable positive constants R and δ, where the infimum extends over all estimators of Gn,θ,σ(t). The results in Section 2.2.2 in fact show that the balls ||θ|| < R in (3) and (4) can be replaced by suitable balls (not necessarily centered at the origin) shrinking at the rate n−1/2. This shows that the non-uniformity phenomenon described in (3)-(4) is a local, rather than a global, phenomenon. In Section 2.2.2 we further show that the non-uniformity phenomenon expressed in (3) and (4) typically also arises when the parameter of interest is not χ, but some other linear transformation of θ = (χ′, ψ′)′. As discussed in Remark 4.3, the results also hold for randomized estimators of the unconditional distribution function Gn,θ,σ(t). Hence no resampling procedure whatsoever can alleviate the problem. This explains the anecdotal evidence in the literature that resampling methods are often unsuccessful in approximating distributional properties of post- model-selection estimators (e.g., Dijkstra and Veldkamp (1988), or Freedman, Navidi, and Peters (1988)). See also the discussion on resampling in Section 6. The results outlined above are presented in Section 2.2 for the particular ‘general-to-specific’ model selection procedure described at the beginning of Section 2. Analogous results for a large class of model selection procedures, including Akaike’s AIC and thresholding procedures, are then given in Section 3, based on the results in Section 2.2. In fact, the non-uniformity phenomenon expressed in (3)-(4) is not specific to the model selection procedures discussed in Sections 2 and 3 of the present paper, but will occur for most (if not all) model selection procedures, including consistent ones; cf. Sections 5 and 6 for more discussion. Section 5 also shows that the results are – as is to be expected – by no means limited to the linear regression model. We focus on the unconditional distributions of post-model-selection estimators in the present paper. One can, however, also envisage a situation where one is more interested in the conditional distribution given the outcome of the model selection procedure. In line with the literature on conditional inference (see, e.g., Robinson (1979) or Lehmann and Casella (1998, p. 421)), one may argue that, given the outcome of the model selection step, the relevant object of interest is the conditional rather than the unconditional distribution of the post-model-selection estimator. In this case similar results can be obtained and are reported in Leeb and Pötscher (2006b). We note that on a technical level the results in Leeb and Pötscher (2006b) and in the present paper require separate treatment. The plan of the paper is as follows: Post-model-selection estimators based on a ‘general-to-specific’ model selection procedure are the subject of Section 2. After introducing the basic framework and some notation, like the family of models Mp from which the ‘general-to-specific’ model selection procedure p̂ selects, as well as the post-model-selection estimator θ̃, the unconditional cdf Gn,θ,σ(t) of (a linear function of) the post-model-selection estimator θ̃ is discussed in Section 2.1. Consistent estimators of Gn,θ,σ(t) are given in Section 2.2.1. The main results of the paper are contained in Section 2.2.2 and Section 3: In Section 2.2.2 we provide a detailed analysis of the non-uniformity phenomenon encountered in (3)-(4). In Section 3 the ‘impossibility’ result from Section 2.2.2 is extended to a large class of model selection procedures including Akaike’s AIC and to selection from a non-nested collection of models. Some remarks are collected in Section 4, while Section 5 discusses extensions and the scope of the results of the paper. Conclusions are drawn in Section 6. All proofs as well as some auxiliary results are collected into appendices. Finally a word on notation: The Euclidean norm is denoted by ‖·‖, and λmax(E) denotes the largest eigenvalue of a symmetric matrix E. A prime denotes transposition of a matrix. For vectors x and y the relation x ≤ y (x < y, respectively) denotes xi ≤ yi (xi < yi, respectively) for all i. As usual, Φ denotes the standard normal distribution function. 2 Results for Post-Model-Selection Estimators Based on a ‘General-to-Specific’ Model Selection Procedure Consider the linear regression model Y = Xθ + u, (5) where X is a non-stochastic n× P matrix with rank(X) = P and u ∼ N(0, σ2In), σ2 > 0. Here n denotes the sample size and we assume n > P ≥ 1. In addition, we assume that Q = limn→∞X ′X/n exists and is non-singular. In this section we shall – similar as in Pötscher (1991) – consider model selection from the collection of nested models MO ⊆ MO+1 ⊆ · · · ⊆ MP , where O is specified by the user, and where for 0 ≤ p ≤ P the model Mp is given by (θ1, . . . , θP ) ′ ∈ RP : θp+1 = · · · = θP = 0 [In Section 3 below also general non-nested families of models will be considered.] Clearly, the model Mp corresponds to the situation where only the first p regressors in (5) are included. For the most parsimonious model under consideration, i.e., forMO, we assume that O satisfies 0 ≤ O < P ; if O > 0, this model contains as free parameters only those components of the parameter vector θ that are not subject to model selection. [In the notation used in connection with (1) we then have χ = (θ1, . . . , θO) ′ and ψ = (θO+1, . . . , θP ) Furthermore, note that M0 = {(0, . . . , 0)′} and that MP = RP . We call Mp the regression model of order p. The following notation will prove useful. For matrices B and C of the same row-dimension, the column- wise concatenation of B and C is denoted by (B : C). If D is an m× P matrix, let D[p] denote the m× p matrix consisting of the first p columns of D. Similarly, let D[¬p] denote the m× (P − p) matrix consisting of the last P −p columns of D. If x is a P × 1 vector, we write in abuse of notation x[p] and x[¬p] for (x′[p])′ and (x′[¬p])′, respectively. [We shall use the above notation also in the ‘boundary’ cases p = 0 and p = P . It will always be clear from the context how expressions containing symbols like D[0], D[¬P ], x[0], or x[¬P ] are to be interpreted.] As usual, the i-th component of a vector x is denoted by xi, and the entry in the i-th row and j-th column of a matrix B is denoted by Bi,j . The restricted least-squares estimator of θ under the restriction θ[¬p] = 0, i.e., under θp+1 = · · · = θP = 0, will be denoted by θ̃(p), 0 ≤ p ≤ P (in case p = P the restriction being void). Note that θ̃(p) is given by the P × 1 vector θ̃(p) =  (X [p] ′X [p]) X [p]′Y (0, . . . , 0)′ where the expressions θ̃(0) and θ̃(P ), respectively, are to be interpreted as the zero-vector in RP and as the unrestricted least-squares estimator of θ. Given a parameter vector θ in RP , the order of θ (relative to the nested sequence of models Mp) is defined as p0(θ) = min {p : 0 ≤ p ≤ P, θ ∈Mp} . Hence, if θ is the true parameter vector, a model Mp is a correct model if and only if p ≥ p0(θ). We stress that p0(θ) is a property of a single parameter, and hence needs to be distinguished from the notion of the order of the model Mp introduced earlier, which is a property of the set of parameters Mp. A model selection procedure is now nothing else than a data-driven (measurable) rule p̂ that selects a value from {O, . . . , P} and thus selects a model from the list of candidate modelsMO, . . . ,MP . In this section we shall consider as an important leading case a ‘general-to-specific’ model selection procedure based on a sequence of hypothesis tests. [Results for a larger class of model selection procedures, including Akaike’s AIC, are provided in Section 3.] This procedure is given as follows: The sequence of hypotheses H 0 : p0(θ) < p is tested against the alternatives H 1 : p0(θ) = p in decreasing order starting at p = P . If, for some p > O, H is the first hypothesis in the process that is rejected, we set p̂ = p. If no rejection occurs until even HO+10 is not rejected, we set p̂ = O. Each hypothesis in this sequence is tested by a kind of t-test where the error variance is always estimated from the overall model (but see the discussion following Theorem 3.1 in Section 3 below for other choices of estimators of the error variance). More formally, we have p̂ = max {p : |Tp| ≥ cp, 0 ≤ p ≤ P} , (6) with cO = 0 in order to ensure a well-defined p̂ in the range {O,O + 1, . . . , P}. For O < p ≤ P , the critical values cp satisfy 0 < cp < ∞ and are independent of sample size (but see also Remark 4.2). The test-statistics are given by nθ̃p(p) σ̂ξn,p (0 < p ≤ P ) with the convention that T0 = 0. Furthermore, ξn,p = X [p]′X [p] (0 < p ≤ P ) denotes the nonnegative square root of the p-th diagonal element of the matrix indicated, and σ̂2 is given by σ̂2 = (n− P )−1(Y −Xθ̃(P ))′(Y −Xθ̃(P )). Note that under the hypothesis H 0 the statistic Tp is t-distributed with n − P degrees of freedom for 0 < p ≤ P . It is also easy to see that the so-defined model selection procedure p̂ is conservative: The probability of selecting an incorrect model, i.e., the probability of the event {p̂ < p0(θ)}, converges to zero as the sample size increases. In contrast, the probability of the event {p̂ = p}, for p satisfying max{p0(θ),O} ≤ p ≤ P , converges to a positive limit; cf., for example, Proposition 5.4 and equation (5.6) in Leeb (2006). The post-model-selection estimator θ̃ can now be defined as follows: On the event p̂ = p, θ̃ is given by the restricted least-squares estimator θ̃(p), i.e., θ̃(p)1(p̂ = p), (7) where 1(·) denotes the indicator function of the event shown in the argument. 2.1 The Distribution of the Post-Model-Selection Estimator We now introduce the distribution function of a linear transformation of θ̃ and summarize some of its properties that will be needed in the subsequent development. To this end, let A be a non-stochastic k × P matrix of rank k, 1 ≤ k ≤ P , and consider the cdf Gn,θ,σ(t) = Pn,θ,σ nA(θ̃ − θ) ≤ t (t ∈ Rk). (8) Here Pn,θ,σ(·) denotes the probability measure corresponding to a sample of size n from (5). Depending on the choice of the matrix A, several important scenarios are covered by (8): The cdf of n(θ̃ − θ) is obtained by setting A equal to the P × P identity matrix IP . In case O > 0, the cdf of those components of n(θ̃− θ) which correspond to the parameter of interest χ in (1) can be studied by setting A to the O×P matrix (IO : 0) as we then have Aθ = (θ1, . . . , θO)′ = χ. Finally, if A 6= 0 is an 1×P vector, we obtain the distribution of a linear predictor based on the post-model-selection estimator. See the examples at the end of Section 2.2.2 for more discussion. The cdf Gn,θ,σ and its properties have been analyzed in detail in Leeb and Pötscher (2003) and Leeb (2006). To be able to access these results we need some further notation. Note that on the event p̂ = p the expression A(θ̃ − θ) equals A(θ̃(p) − θ) in view of (7). The expected value of the restricted least-squares estimator θ̃(p) will be denoted by ηn(p) and is given by the P × 1 vector ηn(p) =  θ[p] + (X [p] ′X [p])−1X [p]′X [¬p]θ[¬p] (0, . . . , 0)′  (9) with the conventions that ηn(0) = (0, . . . , 0) ′ ∈ RP and that ηn(P ) = θ. Furthermore, let Φn,p denote the cdf of nA(θ̃(p) − ηn(p)), i.e., the cdf of nA times the restricted least-squares estimator based on model Mp centered at its mean. Hence, Φn,p is the cdf of a k-variate Gaussian random vector with mean zero and variance-covariance matrix σ2A[p](X [p]′X [p]/n)−1A[p]′ in case p > 0, and it is the cdf of point-mass at zero in Rk in case p = 0. If p > 0 and if the matrix A[p] has full row rank k, then Φn,p has a density with respect to Lebesgue measure, and we shall denote this density by φn,p. We note that ηn(p) depends on θ and that Φn,p depends on σ (in case p > 0), although these dependencies are not shown explicitly in the notation. For p > 0 we introduce bn,p = C n (A[p](X [p] ′X [p]/n)−1A[p]′)−, (10) ζ2n,p = ξ n,p − C(p) n (A[p](X [p] ′X [p]/n)−1A[p]′)−C(p)n , (11) with ζn,p ≥ 0. Here C n = A[p](X [p] ′X [p]/n)−1ep, where ep denotes the p-th standard basis vector in R and B− denotes a generalized inverse of a matrix B. [Observe that ζ2n,p is invariant under the choice of the generalized inverse. The same is not necessarily true for bn,p, but is true for bn,pz for all z in the column- space of A[p]. Also note that (12) below depends on bn,p only through bn,pz with z in the column-space of A[p].] We observe that the vector of covariances between Aθ̃(p) and θ̃p(p) is precisely given by σ 2n−1C (and hence does not depend on θ). Furthermore, observe that Aθ̃(p) and θ̃p(p) are uncorrelated if and only if ζ2n,p = ξ n,p if and only if bn,pz = 0 for all z in the column-space of A[p]; cf. Lemma A.2 in Leeb (2005). Finally, for a univariate Gaussian random variable N with zero mean and variance s2, s ≥ 0, we write ∆s(a, b) for P(|N− a| < b), a ∈ R∪{−∞,∞}, b ∈ R. Note that ∆s(·, ·) is symmetric around zero in its first argument, and that ∆s(−∞, b) = ∆s(∞, b) = 0 holds. In case s = 0, N is to be interpreted as being equal to zero, hence a 7→ ∆0(a, b) reduces to the indicator function of the interval (−b, b). We are now in a position to present the explicit formula for Gn,θ,σ(t) derived in Leeb (2006): Gn,θ,σ(t) = Φn,O(t− nA(ηn(O)− θ)) q=O+1 ∆σξn,q ( nηn,q(q), scqσξn,q)h(s)ds p=O+1 nA(ηn(p)−θ) [ ∫ ∞ (1−∆σζn,p( nηn,p(p) + bn,pz, scpσξn,p)) (12) q=p+1 ∆σξn,q( nηn,q(q), scqσξn,q)h(s)ds Φn,p(dz). In the above display, Φn,p(dz) denotes integration with respect to the measure induced by the normal cdf Φn,p on R k and h denotes the density of σ̂/σ, i.e., h is the density of (n−P )−1/2 times the square-root of a chi-square distributed random variable with n−P degrees of freedom. The finite-sample distribution of the post-model-selection estimator given in (12) is in general not normal, e.g., it can be bimodal; see Figure 2 in Leeb and Pötscher (2005a) or Figure 1 in Leeb (2006). [An exception where (12) is normal is the somewhat trivial case where C n = 0, i.e., where Aθ̃(p) and θ̃p(p) are uncorrelated, for p = O + 1, . . . , P ; see Leeb (2006, Section 3.3) for more discussion.] We note for later use that Gn,θ,σ(t) = p=O Gn,θ,σ(t|p)πn,θ,σ(p) where Gn,θ,σ(t|p) represents the cdf of nA(θ̃ − θ) conditional on the event {p̂ = p} and where πn,θ,σ(p) is the probability of this event under Pn,θ,σ. Note that πn,θ,σ(p) is always positive for O ≤ p ≤ P ; cf. Leeb (2006), Section 3.2. To describe the large-sample limit of Gn,θ,σ, some further notation is necessary. For p satisfying 0 < p ≤ P , partition the matrix Q = limn→∞X ′X/n as  Q[p : p] Q[p : ¬p] Q[¬p : p] Q[¬p : ¬p] where Q[p : p] is a p× p matrix. Let Φ∞,p be the cdf of a k-variate Gaussian random vector with mean zero and variance-covariance matrix σ2A[p]Q[p : p]−1A[p]′, 0 < p ≤ P , and let Φ∞,0 denote the cdf of point-mass at zero in Rk. Note that Φ∞,p has a Lebesgue density if p > 0 and the matrix A[p] has full row rank k; in this case, we denote the Lebesgue density of Φ∞,p by φ∞,p. Finally, for p = 1, . . . , P , define ξ2∞,p = (Q[p : p] −1)p,p, ζ2∞,p = ξ ∞,p − C(p)′∞ (A[p]Q[p : p]−1A[p]′)−C(p)∞ , (13) b∞,p = C ∞ (A[p]Q[p : p] −1A[p]′)−, where C ∞ = A[p]Q[p : p] −1ep, with ep denoting the p-th standard basis vector in R p; furthermore, take ζ∞,p and ξ∞,p as the nonnegative square roots of ζ ∞,p and ξ ∞,p, respectively. As the notation suggests, Φ∞,p is the large-sample limit of Φn,p, and C ∞ , ξ ∞,p, and ζ ∞,p are the limits of C n , ξ n,p, and ζ n,p, respectively; moreover, bn,pz converges to b∞,pz for each z in the column-space of A[p]. See Lemma A.2 in Leeb (2005). The next result describes the large-sample limit of the cdf under local alternatives to θ and is taken from Leeb (2006, Corollary 5.6). Recall that the total variation distance between two cdfs G and G∗ on Rk is defined as ||G−G∗||TV = supE |G(E)−G∗(E)|, where the supremum is taken over all Borel sets E. Clearly, the relation |G(t)−G∗(t)| ≤ ||G−G∗||TV holds for all t ∈ Rk. Thus, if G and G∗ are close with respect to the total variation distance, then G(t) is close to G∗(t), uniformly in t. Proposition 2.1 Suppose θ ∈ RP and γ ∈ RP and let σ(n) be a sequence of positive real numbers which converges to a (finite) limit σ > 0 as n → ∞. Then the cdf Gn,θ+γ/√n,σ(n) converges to a limit G∞,θ,σ,γ in total variation, i.e., ∣∣∣∣Gn,θ+γ/√n,σ(n) −G∞,θ,σ,γ n→∞−→ 0. (14) The large-sample limit cdf G∞,θ,σ,γ(t) is given by Φ∞,p∗(t− β (p∗)) q=p∗+1 ∆σξ∞,q (νq, cqσξ∞,q) p=p∗+1 z≤t−β(p) (1−∆σζ∞,p(νp + b∞,pz, cpσξ∞,p))Φ∞,p(dz) q=p+1 ∆σξ∞,q (νq, cqσξ∞,q) (15) where p∗ = max{p0(θ),O}. Here for 0 ≤ p ≤ P  Q[p : p] −1Q[p : ¬p]γ[¬p] −γ[¬p] with the convention that β(p) = −Aγ if p = 0 and that β(p) = (0, . . . , 0)′ if p = P . Furthermore, we have set νp = γp + (Q[p : p] −1Q[p : ¬p]γ[¬p])p for p > 0. [Note that β(p) = limn→∞ nA(ηn(p) − θ − γ/ for p ≥ p0(θ), and that νp = limn→∞ nηn,p(p) for p > p0(θ). Here ηn(p) is defined as in (9), but with θ + γ/ n replacing θ.] If p∗ > 0 and if the matrix A[p∗] has full row rank k, then the Lebesgue density φ∞,p of Φ∞,p exists for all p ≥ p∗ and hence the density of (15) exists and is given by φ∞,p∗(t− β (p∗)) q=p∗+1 ∆σξ∞,q (νq, cqσξ∞,q) p=p∗+1 (1−∆σζ∞,p(νp + b∞,p(t− β (p)), cpσξ∞,p))φ∞,p(t− β q=p+1 ∆σξ∞,q(νq, cqσξ∞,q). Like the finite-sample distribution, the limiting distribution of the post-model-selection estimator given in (15) is in general not normal. An exception is the case where C ∞ = 0 for p > p∗ in which case G∞,θ,σ,γ reduces to Φ∞,P ; see Remark A.6 in Appendix A. If γ = 0, we write G∞,θ,σ(t) as shorthand for G∞,θ,σ,0(t) in the following. 2.2 Estimators of the Finite-Sample Distribution For the purpose of inference after model selection the finite-sample distribution of the post-model-selection- estimator is an object of particular interest. As we have seen, it depends on unknown parameters in a complicated manner, and hence one will have to be satisfied with estimators of this cdf. As we shall see, it is not difficult to construct consistent estimators of Gn,θ,σ(t). However, despite this consistency result, we shall find in Section 2.2.2 that any estimator of Gn,θ,σ(t) typically performs unsatisfactory, in that the estimation error can not become small uniformly over (subsets of) the parameter space even as sample size goes to infinity. In particular, no uniformly consistent estimators exist, not even locally. 2.2.1 Consistent Estimators We construct a consistent estimator of Gn,θ,σ(t) by commencing from the asymptotic distribution. Spe- cializing to the case γ = 0 and σ(n) = σ in Proposition 2.1, the large-sample limit of Gn,θ,σ(t) is given G∞,θ,σ(t) = Φ∞,p∗(t) q=p∗+1 ∆σξ∞,q(0, cqσξ∞,q) p=p∗+1 (1 −∆σζ∞,p(b∞,pz, cpσξ∞,p))Φ∞,p(dz) q=p+1 ∆σξ∞,q(0, cqσξ∞,q) (16) with p∗ = max{p0(θ),O}. Note that G∞,θ,σ(t) depends on θ only through p∗. Let Φ̂n,p denote the cdf of a k- variate Gaussian random vector with mean zero and variance-covariance matrix σ̂2A[p](X [p]′X [p]/n)−1A[p]′, 0 < p ≤ P ; we also adopt the convention that Φ̂n,0 denotes the cdf of point-mass at zero in Rk. [We use the same convention for Φ̂n,p in case σ̂ = 0, which is a probability zero event.] An estimator Ǧn(t) of Gn,θ,σ(t) is now defined as follows: We first employ an auxiliary procedure p̄ that consistently estimates p0(θ) (e.g., p̄ could be obtained from BIC or from a ‘general-to-specific’ hypothesis testing procedure employing critical values that go to infinity but are o(n1/2) as n → ∞). The estimator Ǧn(t) is now given by the expression in (16) but with p∗, σ, b∞,p, ζ∞,p, ξ∞,p, and Φ∞,p replaced by max{p̄,O}, σ̂, bn,p, ζn,p, ξn,p, and Φ̂n,p, respectively. A little reflection shows that Ǧn is again a cdf. We have the following consistency results. Proposition 2.2 The estimator Ǧn is consistent (in the total variation distance) for Gn,θ,σ and G∞,θ,σ. That is, for every δ > 0 Pn,θ,σ (∣∣∣∣Ǧn(·)−Gn,θ,σ(·) ) n→∞−→ 0, (17) Pn,θ,σ (∣∣∣∣Ǧn(·)−G∞,θ,σ(·) ) n→∞−→ 0 (18) for all θ ∈ RP and all σ > 0. While the estimator constructed above on the basis of the formula for G∞,θ,σ is consistent, it can be expected to perform poorly in finite samples since convergence of Gn,θ,σ to G∞,θ,σ is typically not uniform in θ (cf. Appendix B), and since in case the true θ is ‘close’ to Mp0(θ)−1 the auxiliary decision procedure p̄ (although being consistent for p0(θ)) will then have difficulties making the correct decision in finite samples. In the next section we show that this poor performance is not particular to the estimator Ǧn constructed above, but is a genuine feature of the estimation problem under consideration. 2.2.2 Performance Limits and Impossibility Results We now provide lower bounds for the performance of estimators of the cdf Gn,θ,σ(t) of the post-model- selection estimator Aθ̃; that is, we give lower bounds on the worst-case probability that the estimation error exceeds a certain threshold. These lower bounds are large, being 1 or 1/2, depending on the situation considered; furthermore, they remain lower bounds even if one restricts attention only to certain subsets of the parameter space that shrink at the rate n−1/2. In this sense the ‘impossibility’ results are of a local nature. In particular, the lower bounds imply that no uniformly consistent estimator of the cdf Gn,θ,σ(t) exists, not even locally. In the following, the asymptotic ‘correlation’ between Aθ̃(p) and θ̃p(p) as measured by C limn→∞ C n will play an important rôle. [Recall that θ̃(p) denotes the least-squares estimator of θ based on model Mp and that Aθ is the parameter vector of interest. Furthermore, the vector of covariances be- tween Aθ̃(p) and θ̃p(p) is given by σ 2n−1C n with C n = A[p](X [p] ′X [p]/n)−1ep.] Note that C ∞ equals A[p]Q[p : p]−1ep, and hence does not depend on the unknown parameters θ or σ. In the important special case discussed in the Introduction, cf. (1), the matrix A equals the O×P matrix (IO : 0), and the condition ∞ 6= 0 reduces to the condition that the regressor corresponding to the p-th column of (V :W ) is asymp- totically correlated with at least one of the regressors corresponding to the columns of V . See Example 1 below for more discussion. In the result to follow we shall consider performance limits for estimators of Gn,θ,σ(t) at a fixed value of the argument t. An estimator of Gn,θ,σ(t) is now nothing else than a real-valued random variable Γn = Γn(Y,X). For mnemonic reasons we shall, however, use the symbol Ĝn(t) instead of Γn to denote an arbitrary estimator of Gn,θ,σ(t). This notation should not be taken as implying that the estimator is obtained by evaluating an estimated cdf at the argument t, or that it is a priori constrained to lie between zero and one. We shall use this notational convention mutatis mutandis also in subsequent sections. Regarding the non-uniformity phenomenon, we then have a dichotomy which is described in the following two results. Theorem 2.3 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C ∞ 6= 0, for some q sat- isfying O < q ≤ P , and let q∗ denote the largest q with this property. Then the following holds for every θ ∈ Mq∗−1, every σ, 0 < σ < ∞, and every t ∈ Rk: There exist δ0 > 0 and ρ0, 0 < ρ0 < ∞, such that any estimator Ĝn(t) of Gn,θ,σ(t) satisfying Pn,θ,σ (∣∣∣Ĝn(t)−Gn,θ,σ(t) ∣∣∣ > δ n→∞−→ 0 (19) for each δ > 0 (in particular, every estimator that is consistent) also satisfies ϑ∈Mq∗ ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣Ĝn(t)−Gn,ϑ,σ(t) ∣∣∣ > δ0 n→∞−→ 1. (20) The constants δ0 and ρ0 may be chosen in such a way that they depend only on t, Q, A, σ, and the critical values cp for O < p ≤ P . Moreover, lim inf Ĝn(t) ϑ∈Mq∗ ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣Ĝn(t)−Gn,ϑ,σ(t) ∣∣∣ > δ0 > 0 (21) lim inf Ĝn(t) ϑ∈Mq∗ ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣Ĝn(t)−Gn,ϑ,σ(t) ∣∣∣ > δ , (22) where the infima in (21) and (22) extend over all estimators Ĝn(t) of Gn,θ,σ(t). Remark 2.4 Assume that the conditions of the preceding theorem are satisfied. Suppose further that p⊙, O ≤ p⊙ < q∗, is such that either p⊙ > 0 and some row of A[p⊙] equals zero, or such that p⊙ = 0. Then there exist δ0 > 0 and 0 < ρ0 <∞ such that the left-hand side of (21) is not less than 1/2 for each θ ∈Mp⊙ . Theorem 2.3 a fortiori implies a corresponding ‘impossibility’ result for estimation of the functionGn,θ,σ(·) when the estimation error is measured in the total variation distance or the sup-norm; cf. also Section 5. It remains to consider the – quite exceptional – case where the assumption of Theorem 2.3 is not satisfied, i.e., where C ∞ = 0, for all q in the range O < q ≤ P . Under this ‘uncorrelatedness’ condition it is indeed possible to construct an estimator of Gn,θ,σ which is uniformly consistent: It is not difficult to see that the asymptotic distribution of Gn,θ,σ reduces to Φ∞,P under this ‘uncorrelatedness’ condition. Furthermore, the second half of Proposition B.1 in Appendix B shows that then the convergence of Gn,θ,σ to its large-sample limit is uniform w.r.t. θ, suggesting Φ̂n,P , an estimated version of Φ∞,P , as an estimator for Gn,θ,σ. Proposition 2.5 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically uncorrelated, i.e., C ∞ = 0, for all q satisfying O < q ≤ P . Then σ∗≤σ≤σ∗ Pn,θ,σ ∣∣∣Φ̂n,P −Gn,θ,σ n→∞−→ 0 (23) holds for each δ > 0, and for any constants σ∗ and σ ∗ satisfying 0 < σ∗ ≤ σ∗ <∞. Inspection of the proof of Proposition 2.5 shows that (23) continues to hold if the estimator Φ̂n,P is replaced by any of the estimators Φ̂n,p for O ≤ p ≤ P . We also note that in case O = 0 the assumption of Proposition 2.5 is never satisfied in view of Proposition 4.4 in Leeb and Pötscher (2006b), and hence Theorem 2.3 always applies in that case. Another consequence of Proposition 4.4 in Leeb and Pötscher (2006b) is that – under the ‘uncorrelatedness’ assumption of Proposition 2.5 – the restricted least squares estimators Aθ̃(q) for q ≥ O perform asymptotically as well as the unrestricted estimator Aθ̃(P ); this clearly shows that the case covered by Proposition 2.5 is highly exceptional. In summary we see that it is typically impossible to construct an estimator of Gn,θ,σ(t) which performs reasonably well even asymptotically. Whenever Theorem 2.3 applies, any estimator of Gn,θ,σ(t) suffers from a non-uniformity defect which is caused by parameters belonging to shrinking ‘tubes’ surrounding Mq∗−1. For the sake of completeness, we remark that outside a ‘tube’ of fixed positive radius that surrounds Mq∗−1 the non-uniformity need not be present: Let q ∗ be as in Theorem 2.3 and define the set U as U = {θ ∈ RP : |θq∗ | ≥ r} for some fixed r > 0. Then Φ̂n,P (t) is an estimator of Gn,θ,σ(t) that is uniformly consistent over θ ∈ U ; more generally, it can be shown that then the relation (23) holds if the supremum over θ on the left-hand side is restricted to θ ∈ U . We conclude this section by illustrating the above results with some important examples. Example 1: (The distribution of χ̃) Consider the model given in (1) with χ representing the parameter of interest. Using the general notation of Section 2, this corresponds to the case Aθ = (θ1, . . . , θO) ′ = χ with A representing the O × P matrix (IO : 0). Here k = O > 0. The cdf Gn,θ,σ then represents the cdf n (χ̃− χ). Assume first that limn→∞ V ′W/n 6= 0. Then C(q)∞ 6= 0 holds for some q > O. Consequently, the ‘impossibility’ results for the estimation of Gn,θ,σ given in Theorem 2.3 always apply. Next assume that limn→∞ V ′W/n = 0. Then C ∞ = 0 for every q > O. In this case Proposition 2.5 applies and a uniformly consistent estimator of Gn,θ,σ indeed exists. Summarizing we note that any estimator of Gn,θ,σ suffers from the non-uniformity phenomenon except in the special case where the columns of V and W are asymptotically orthogonal in the sense that limn→∞ V ′W/n = 0. But this is precisely the situation where inclusion or exclusion of the regressors in W has no effect on the distribution of the estimator χ̃ asymptotically; hence it is not surprising that also the model selection procedure does not have an effect on the estimation of the cdf of the post-model-selection estimator χ̃. This observation may tempt one to enforce orthogonality between the columns of V and W by either replacing the columns of V by their residuals from the projection on the column space ofW or vice versa. However, this is not helpful for the following reasons: In the first case one then in fact avoids model selection as all the restricted least-squares estimators for χ under consideration (and hence also the post-model selection estimator χ̃) in the reparameterized model coincide with the unrestricted least-squares estimator. In the second case the coefficients of the columns of V in the reparameterized model no longer coincide with the parameter of interest χ (and again are estimated by one and the same estimator regardless of inclusion/exclusion of columns of the transformed W -matrix). Example 2: (The distribution of θ̃) For A equal to IP , the cdf Gn,θ,σ is the cdf of n(θ̃ − θ). Here, Aθ̃(q) reduces to θ̃(q), and hence Aθ̃(q) and θ̃q(q) are perfectly correlated for every q > O. Consequently, the ‘impossibility’ result for estimation of Gn,θ,σ given in Theorem 2.3 applies. [In fact, the slightly stronger result mentioned in Remark 2.4 always applies here.] We therefore see that estimation of the distribution of the post-model-selection estimator of the entire parameter vector is always plagued by the non-uniformity phenomenon. Example 3: (The distribution of a linear predictor) Suppose A 6= 0 is a 1×P vector and one is interested in estimating the cdf Gn,θ,σ of the linear predictor Aθ̃. Then Theorem 2.3 and the discussion following Proposition 2.5 show that the non-uniformity phenomenon always arises in this estimation problem in case O = 0. In case O > 0, the non-uniformity problem is generically also present, except in the degenerate case where C ∞ = 0, for all q satisfying O < q ≤ P (in which case Proposition 4.4 in Leeb and Pötscher (2006b) shows that the least-squares predictors from all models Mp, O ≤ p ≤ P , perform asymptotically equally well). 3 Extensions to Other Model Selection Procedures Including AIC In this section we show that the ‘impossibility’ result obtained in the previous section for a ‘general-to- specific’ model selection procedure carries over to a large class of model selection procedures, including Akaike’s widely used AIC. Again consider the linear regression model (5) with the same assumptions on the regressors and the errors as in Section 2. Let {0, 1}P denote the set of all 0-1 sequences of length P . For each r ∈{0, 1}P let Mr denote the set {θ ∈ RP : θi(1 − ri) = 0 for 1 ≤ i ≤ P} where ri represents the i-th component of r. I.e., Mr describes a linear submodel with those parameters θi restricted to zero for which ri = 0. Now let R be a user-supplied subset of {0, 1}P . We consider model selection procedures that select from the set R, or equivalently from the set of models {Mr : r ∈ R}. Note that there is now no assumption that the candidate models are nested (for example, if R = {0, 1}P all possible submodels are candidates for selection). Also cases where the inclusion of a subset of regressors is undisputed on a priori grounds are obviously covered by this framework upon suitable choice of R. We shall assume throughout this section that R contains rfull = (1, . . . , 1) and also at least one element r∗ satisfying |r∗| = P − 1, where |r∗| represents the number of non-zero coordinates of r∗. Let r̂ be an arbitrary model selection procedure, i.e., r̂ = r̂(Y,X) is a random variable taking its values in R. We furthermore assume throughout this section that the model selection procedure r̂ satisfies the following mild condition: For every r∗ ∈ R with |r∗| = P − 1 there exists a positive finite constant c (possibly depending on r∗) such that for every θ ∈Mr∗ which has exactly P − 1 non-zero coordinates Pn,θ,σ ({r̂ = rfull}N{|Tr∗ | ≥ c}) = lim Pn,θ,σ ({r̂ = r∗}N{|Tr∗ | < c}) = 0 (24) holds for every 0 < σ < ∞. Here N denotes the symmetric difference operator and Tr∗ represents the usual t-statistic for testing the hypothesis θi(r∗) = 0 in the full model, where i(r∗) denotes the index of the unique coordinate of r∗ that equals zero. The above condition is quite natural for the following reason: For θ ∈ Mr∗ with exactly P − 1 non-zero coordinates, every reasonable model selection procedure will – with probability approaching unity – decide only betweenMr∗ andMrfull ; it is then quite natural that this decision will be based (at least asymptotically) on the likelihood ratio between these two models, which in turn boils down to the t-statistic. As will be shown below, condition (24) holds in particular for AIC-like procedures. Let A be a non-stochastic k×P matrix of full row rank k, 1 ≤ k ≤ P , as in Section 2.1. We then consider the cdf Kn,θ,σ(t) = Pn,θ,σ nA(θ̄ − θ) ≤ t (t ∈ Rk) (25) of a linear transformation of the post-model-selection estimator θ̄ obtained from the model selection procedure r̂, i.e., θ̃(r)1(r̂ = r) where the P × 1 vector θ̃(r) represents the restricted least-squares estimator obtained from model Mr, with the convention that θ̃(r) = 0 ∈ RP in case r = (0, . . . , 0). We then obtain the following result for estimation of Kn,θ,σ(t) at a fixed value of the argument t which parallels the corresponding ‘impossibility’ result in Theorem 2.3. Theorem 3.1 Let r∗ ∈ R satisfy |r∗| = P − 1, and let i(r∗) denote the index of the unique coordinate of r∗ that equals zero; furthermore, let c be the constant in (24) corresponding to r∗. Suppose that Aθ̃(rfull) and θ̃i(r∗)(rfull) are asymptotically correlated, i.e., AQ i(r∗) 6= 0, where e i(r∗) denotes the i(r∗)-th standard basis vector in RP . Then for every θ ∈ Mr∗ which has exactly P − 1 non-zero coordinates, for every σ, 0 < σ <∞, and for every t ∈ Rk the following holds: There exist δ0 > 0 and ρ0, 0 < ρ0 <∞, such that any estimator K̂n(t) of Kn,θ,σ(t) satisfying Pn,θ,σ (∣∣∣K̂n(t)−Kn,θ,σ(t) ∣∣∣ > δ n→∞−→ 0 (26) for each δ > 0 (in particular, every estimator that is consistent) also satisfies ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣K̂n(t)−Kn,ϑ,σ(t) ∣∣∣ > δ0 n→∞−→ 1 . (27) The constants δ0 and ρ0 may be chosen in such a way that they depend only on t, Q,A, σ, and c. Moreover, lim inf K̂n(t) ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣K̂n(t)−Kn,ϑ,σ(t) ∣∣∣ > δ0 > 0 (28) lim inf K̂n(t) ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣K̂n(t)−Kn,ϑ,σ(t) ∣∣∣ > δ ≥ 1/2 (29) hold, where the infima in (28) and (29) extend over all estimators K̂n(t) of Kn,θ,σ(t). The basic condition (24) on the model selection procedure employed in the above result will certainly hold for any hypothesis testing procedure that (i) asymptotically selects only correct models, (ii) employs a likelihood ratio test (or an asymptotically equivalent test) for testing Mrfull versus smaller models (at least versus the models Mr∗ with r∗ as in condition (24)), and (iii) uses a critical value for the likelihood ratio test that converges to a finite positive constant. In particular, this applies to usual thresholding procedures as well as to a variant of the ‘general-to-specific’ procedure discussed in Section 2 where the error variance in the construction of the test statistic for hypothesis H 0 is estimated from the fitted model Mp rather than from the overall model. We next verify condition (24) for AIC-like procedures. Let RSS(r) denote the residual sum of squares from the regression employing model Mr and set IC(r) = log (RSS(r)) + |r|Υn/n (30) where Υn ≥ 0 denotes a sequence of real numbers satisfying limn→∞ Υn = Υ and Υ is a positive real number. Of course, IC(r) = AIC(r) if Υn = 2. The model selection procedure r̂IC is then defined as a minimizer (more precisely, as a measurable selection from the set of minimizers) of IC(r) over R. It is well-known that the probability that r̂IC selects an incorrect model converges to zero. Hence, elementary calculations show that condition (24) is satisfied for c = Υ1/2. The analysis of post-model-selection estimators based on AIC-like model selection procedures given in this section proceeded by bringing this case under the umbrella of the results obtained in Section 2. Verification of condition (24) is the key that enables this approach. A complete analysis of post-model-selection estimators based on AIC-like model selection procedures, similar to the analysis in Section 2 for the ‘general-to-specific’ model selection procedure, is certainly possible but requires a direct and detailed analysis of the distribution of this post-model-selection estimator. [Even the mild condition that R contains rfull and also at least one element r∗ satisfying |r∗| = P − 1 can then be relaxed in such an analysis.] We furthermore note that in the special case where R = {rfull, r∗} and an AIC-like model selection procedure as in (30) is used, the results in the above theorem in fact hold for all θ ∈Mr∗ . 4 Remarks and Extensions Remark 4.1 Although not emphasized in the notation, all results in the paper also hold if the elements of the design matrix X depend on sample size. Furthermore, all results are expressed solely in terms of the distributions Pn,θ,σ(·) of Y , and hence they also apply if the elements of Y depend on sample size, including the case where the random vectors Y are defined on different probability spaces for different sample sizes. Remark 4.2 The model selection procedure considered in Section 2 is based on a sequence of tests which use critical values cp that do not depend on sample size and satisfy 0 < cp < ∞ for O < p ≤ P . If these critical values are allowed to depend on sample size such that they now satisfy cn,p → c∞,p as n → ∞ with 0 < c∞,p < ∞ for O < p ≤ P , the results in Leeb and Pötscher (2003) as well as in Leeb (2005, 2006) continue to hold; see Remark 6.2(i) in Leeb and Pötscher (2003) and Remark 6.1(ii) in Leeb (2005). As a consequence, the results in the present paper can also be extended to this case quite easily. Remark 4.3 The ‘impossibility’ results given in Theorems 2.3 and 3.1 (as well as the variants thereof discussed in the subsequent Remarks 4.4-4.7) also hold for the class of all randomized estimators (with P ∗n,θ,σ replacing Pn,θ,σ in those results, where P n,θ,σ denotes the distribution of the randomized sample). This follows immediately from Lemma 3.6 and the attending discussion in Leeb and Pötscher (2006a). Remark 4.4 a. Let ψn,θ,σ denote the expectation of θ̃ under Pn,θ,σ, and consider the cdf Hn,θ,σ(t) = Pn,θ,σ( nA(θ̃−ψn,θ,σ) ≤ t). Results for the cdf Hn,θ,σ quite similar to the results for Gn,θ,σ obtained in the present paper can be established. A similar remark applies to the post-model-selection estimator θ̄ considered in Section 3. b. In Leeb (2006) also the cdf G∗n,θ,σ is analyzed, which correspond to a (typically infeasible) model selection procedure that makes use of knowledge of σ. Results completely analogous to the ones in the present paper can also be obtained for this cdf. Remark 4.5 Results similar to the ones in Section 2.2.2 can also be obtained for estimation of the asymp- totic cdf G∞,θ,σ(t) (or of the asymptotic cdfs corresponding to the variants discussed in the previous remark). Since these results are of limited interest, we omit them. In particular, note that an ‘impossibility’ result for estimation of G∞,θ,σ(t) per se does not imply a corresponding ‘impossibility’ result for estimation of Gn,θ,σ(t), since Gn,θ,σ(t) does in general not converge uniformly to G∞,θ,σ(t) over the relevant subsets in the parameter space; cf. Appendix B. [An analogous remark applies to the model selection procedures considered in Section 3.] Remark 4.6 Let πn,θ,σ(p) denote the model selection probability Pn,θ,σ(p̂ = p), O ≤ p ≤ P corresponding to the model selection procedure discussed in Section 2. The finite-sample properties and the large-sample limit behavior of these quantities are thoroughly analyzed in Leeb (2006); cf. also Leeb and Pötscher (2003). For these model selection probabilities the following results can be established which we discuss here only briefly: a. The model selection probabilities πn,θ,σ(p) converge to well-defined large-sample limits which we denote by π∞,θ,σ(p). Similar as in Proposition B.1 in Appendix B, the convergence of πn,θ,σ(p) to π∞,θ,σ(p) is non-uniform w.r.t. θ. [For the case O = 0, this phenomenon is described in Corollary 5.6 of Leeb and Pötscher (2003).] b. The model selection probabilities πn,θ,σ(p) can be estimated consistently. However, uniformly consis- tent estimation is again not possible. A similar remark applies to the large-sample limits π∞,θ,σ(p). Remark 4.7 ‘Impossibility’ results similar to the ones given in Theorems 2.3 and 3.1 for the cdf can also be obtained for other characteristics of the distribution of a linear function of a post-model-selection estimator like the mean-squared error or the bias of nAθ̃. 5 On the Scope of the Impossibility Results The non-uniformity phenomenon described, e.g., in (20) of Theorem 2.3 is caused by a mechanism that can informally be described as follows. Under the assumptions of that theorem, one can find an appropriate θ and an appropriate sequence ϑn = θ + γ/ n exhibiting two crucial properties: a. The probability measures Pn,ϑn,σ corresponding to ϑn are ‘close’ to the measures Pn,θ,σ corresponding to θ, in the sense of contiguity. This entails that an estimator, that converges to some limit in probability under Pn,θ,σ, converges to the same limit also under Pn,ϑn,σ. b. For given t, the estimands Gn,ϑn,σ(t) corresponding to ϑn are ‘far away’ from the estimands Gn,θ,σ(t) corresponding to θ, in the sense that Gn,ϑn,σ(t) and Gn,θ,σ(t) converge to different limits, i.e., G∞,θ,σ,0(t) is different from G∞,θ,σ,γ(t). In view of Property a, an estimator Ĝn(t) satisfying Ĝn(t)−Gn,θ,σ(t) → 0 in probability under Pn,θ,σ, also satisfies Ĝn(t)−Gn,θ,σ(t) → 0 in probability under Pn,ϑn,σ. In view of Property b, such an estimator Ĝn(t) is hence ‘far away’ from the estimand Gn,ϑn,σ(t) with high probability under Pn,ϑn,σ. In other words, an estimator that is close to Gn,θ,σ(t) under Pn,θ,σ must be far away from Gn,ϑn,σ(t) under Pn,ϑn,σ. Formalized and refined, this argument leads to (20) and, as a consequence, to the non-existence of uniformly consistent estimators for Gn,θ,σ(t). [There are a number of technical details in this formalization process that need careful attention in order to obtain the results in their full strength as given in Sections 2 and 3.] The above informal argument that derives (20) from Properties a and b can be refined and formalized in a much more general and abstract framework, see Section 3 of Leeb and Pötscher (2006a) and the references therein. That paper also provides a general framework for deriving results like (21) and (22) of Theorem 2.3. The mechanism leading to such lower bounds is similar to the one outlined above, where for some of the results the concept of contiguity of the probability measures involved has to be replaced by closeness of these measures in total variation distance. We use the results in Section 3 of Leeb and Pötscher (2006a) to formally convert Properties a and b into the ‘impossibility’ results of the present paper; cf. Appendix C. Verifying the aforementioned Property a in the context of the present paper is straightforward because we consider a Gaussian linear model. What is technically more challenging and requires some work is the verification of Property b; this is done in Appendix A inter alia and rests on results of Leeb (2002, 2005, 2006). Two important observations on Properties a and b are in order: First, Property a is typically satisfied in general parametric models under standard regularity conditions; e.g., it is satisfied whenever the model is locally asymptotically normal. Second, Property b relies on limiting properties only and not on the finite- sample structure of the underlying statistical model. Now, the limit distributions of post-model-selection estimators in sufficiently regular parametric or semi-parametric models are typically the same as the limiting distributions of the corresponding post-model-selection estimators in a Gaussian linear model (see, e.g., Sen (1979), Pötscher (1991), Nickl (2003), or Hjort and Claeskens (2003)). Hence, establishing Property b for the Gaussian linear model then typically establishes the same result for a large class of general parametric or semi-parametric models.1 For example, Property b can be verified for a large class of pre-test estimators in sufficiently regular parametric models by arguing as in Appendix A and using the results of Nickl (2003) to reduce to the Gaussian linear case. Hence, the impossibility result given in Theorem 2.3 can be extended 1Some care has to be taken here. In the Gaussian linear case the finite-sample cdfs converge at every value of the argument t, cf. Propisition 2.1. In a general parametric model, sometimes the asymptotic results (e.g., Hjort and Claeskens (2003, Theorem 4.1)) only guarantee weak convergence. Hence, to ensure convergence of the relevant cdfs at a given argument t as required in Proberty b, additional considerations have to be employed. [This is, however, of no concern in the context discussed in the next but one paragraph in this section.] to more general parametric and semiparametric models with ease. The fact that we use a Gaussian linear model for the analysis in the present paper is a matter of convenience rather than a necessity. The non-uniformity results in Theorem 2.3 are for (conservative) ‘general-to-specific’ model selection from a nested family of models. Theorem 3.1 extends this to more general (conservative) model selection procedures (including AIC and related procedures) and to more general families of models. The proof of Theorem 3.1 proceeds by reducing the problem to one where only two nested models are considered, and then to appeal to the results of Theorem 2.3. The condition on the model selection procedures that enables this reduction is condition (24). It is apparent from the discussion in Section 3 that this condition is satisfied for many model selection procedures. Furthermore, for the same reasons as given in the preceding paragraph, also Theorem 3.1 can easily be extended to sufficiently regular parametric and semi-parametric models. The ‘impossibility’ results in the present paper are formulated for estimating Gn,θ,σ(t) for a given value of t. Suppose that we are now asking the question whether the cdf Gn,θ,σ(·) viewed as a function can be estimated uniformly consistently, where consistency is relative to a metric that metrizes weak convergence.2 Using a similar reasoning as above (which can again be made formal by using, e.g., Lemma 3.1 in Leeb and Pötscher (2006a)) the key step now is to show that the function G∞,θ,σ,0(·) is different from the function G∞,θ,σ,γ(·). Obviously, it is a much simpler problem to find a γ such that the functions G∞,θ,σ,0(·) and G∞,θ,σ,γ(·) differ, than to find a γ such that the values G∞,θ,σ,0(t) and G∞,θ,σ,γ(t) for a given t differ. Certainly, having solved the latter problem in Appendix A, this also provides an answer to the former. This then immediately delivers the desired ‘impossibility’ result. [We note that in some special cases simpler arguments than the ones used in Appendix A can be employed to solve the former problem: For example, in case A = I the functions G∞,θ,σ,0(·) and G∞,θ,σ,γ(·) can each be shown to be convex combinations of cdfs that are concentrated on subspaces of different dimensions. This can be exploited to establish without much difficulty that the functions G∞,θ,σ,0(·) and G∞,θ,σ,γ(·) differ. For purpose of comparison we note that for general A the distributions G∞,θ,σ,0 and G∞,θ,σ,γ can both be absolutely continuous w.r.t. Lebesgue measure, not allowing one to use this simple argument.] Again the discussion in this paragraph extends to more general parametric and semiparametric models without difficulty. The present paper, including the discussion in this section, has focussed on conservative model selection procedures. However, the discussion should make it clear that similar ‘impossibility’ results plague consistent model selection. Section 2.3 in Leeb and Pötscher (2006a) in fact gives such an ‘impossibility’ result in a simple case. We close with the following observations. Verification of Property b, whether it is for G∞,θ,σ,0(t) and G∞,θ,σ,γ(t) (for given t) or for G∞,θ,σ,0(·) and G∞,θ,σ,γ(·), shows that the post-model-selection estimator Aθ̃ is a so-called non-regular estimator for Aθ: Consider an estimator β̃ in a parametric model {Pn,β : β ∈ B} where the parameter space B is an open subset of Euclidean space Rd. Suppose β̃, properly scaled and centered, has a limit distribution under local alternatives, in the sense that n(β̃ − (β + γ/ n)) converges in law under Pn,β+γ/ n to a limit distribution L∞,β,γ(·) for every γ. The estimator β̃ is called regular if for every β the limit distribution L∞,β,γ(·) does not depend on γ; cf. van der Vaart (1998, Section 8.5). Suppose now that the model is, e.g., locally asymptotically normal (hence the contiguity property in Property a is satisfied). The informal argument outlined at the beginning of this section (and which is formalized in Lemma 3.1 of Leeb and Pötscher (2006a)) then in fact shows that the cdf of any non-regular estimator 2Or, in fact, any metric w.r.t. which the relevant cdfs converge. can not be estimated uniformly consistently (where consistency is relative to any metric that metrizes weak convergence). 6 Conclusions Despite the fact that we have shown that consistent estimators for the distribution of a post-model-selection estimator can be constructed with relative ease, we have also demonstrated that no estimator of this distri- bution can have satisfactory performance (locally) uniformly in the parameter space, even asymptotically. In particular, no (locally) uniformly consistent estimator of this distribution exists. Hence, the answer to the question posed in the title has to be negative. The results in the present paper also cover the case of linear functions (e.g., predictors) of the post-model-selection estimator. We would like to stress here that resampling procedures like, e.g., the bootstrap or subsampling, do not solve the problem at all. First note that standard bootstrap techniques will typically not even provide consistent estimators of the finite-sample distribution of the post-model-selection estimator, as the bootstrap can be shown to stay random in the limit (Kulperger and Ahmed (1992), Knight (1999, Example 3))3. Basically the only way one can coerce the bootstrap into delivering a consistent estimator is to resample from a model that has been selected by an auxiliary consistent model selection procedure. The consistent estimator constructed in Section 2.2.1 is in fact of this form. In contrast to the standard bootstrap, subsampling will typically deliver consistent estimators. However, the ‘impossibility’ results given in this paper apply to any estimator (including randomized estimators) of the cdf of a post-model-selection estimator. Hence, also any resampling based estimator suffers from the non-uniformity defects described in Theorems 2.3 and 3.1; cf. also Remark 4.3. The ‘impossibility’ results in Theorems 2.3 and 3.1 are derived in the framework of a normal linear regression model (and a fortiori these results continue to hold in any model which includes the normal linear regression model as a special case), but this is more a matter of convenience than anything else: As discussed in Section 5, similar results can be obtained in general statistical models allowing for nonlinearity or dependent data, e.g., as long as standard regularity conditions for maximum likelihood theory are satisfied. The results in the present paper are derived for a large class of conservative model selection procedures (i.e., procedures that select overparameterized models with positive probability asymptotically) including Akaike’s AIC and typical ‘general-to-specific’ hypothesis testing procedures. For consistent model selection procedures – like BIC or testing procedures with suitably diverging critical values cp (cf. Bauer, Pötscher, and Hackl (1988)) – the (pointwise) asymptotic distribution is always normal. [This is elementary, cf. Lemma 1 in Pötscher (1991).] However, as discussed at length in Leeb and Pötscher (2005a), this asymptotic nor- mality result paints a misleading picture of the finite sample distribution which can be far from a normal, the convergence of the finite-sample distribution to the asymptotic normal distribution not being uniform. ‘Impossibility’ results similar to the ones presented here can also be obtained for post-model-selection esti- mators based on consistent model selection procedures. These will be discussed in detail elsewhere. For a 3Brownstone (1990) claims the validity of a bootstrap procedure that is based on a conservative model selection procedure in a linear regression model. Kilian (1998) makes a similar claim in the context of autoregressive models selected by a conser- vative model selection procedure. Also Hansen (2003) contains such a claim for a stationary bootstrap procedure based on a conservative model selection procedure. The above discussion intimates that these claims are at least unsubstantiated. simple special case such an ‘impossibility’ result is given in Section 2.3 of Leeb and Pötscher (2006a). The ‘impossibility’ of estimating the distribution of the post-model-selection estimator does not per se preclude the possibility of conducting valid inference after model selection, a topic that deserves further study. However, it certainly makes this a more challenging task. A Auxiliary Lemmas Lemma A.1 Let Z be a random vector with values in Rk and let W be a univariate standard Gaussian random variable independent of Z. Furthermore, let C ∈ Rk and τ > 0. Then P(Z ≤ Cx)P(|W − x| < τ ) + P(Z ≤ CW, |W − x| ≥ τ) (31) is constant as a function of x ∈ R if and only if C = 0 or P(Z ≤ Cx) = 0 for each x ∈ R. Proof of Lemma A.1: Suppose C = 0 holds. Using independence of Z andW it is then easy to see that (31) reduces to P(Z ≤ 0), which is constant in x. If P(Z ≤ Cx) = 0 for every x ∈ R, then P(Z ≤ CW ) = 0, and hence (31) is again constant, namely equal to zero. To prove the converse, assume that (31) is constant in x ∈ R. Letting x→ ∞, we see that (31) must be equal to P(Z ≤ CW ). This entails that P(Z ≤ Cx)P(|W − x| < τ) = P(Z ≤ CW, |W − x| < τ ) holds for every x ∈ R. Write F (x) as shorthand for P(Z ≤ Cx), and let Φ(z) and φ(z) denote the cdf and density of W , respectively. Then the expression in the above display can be written as F (x)(Φ(x + τ )− Φ(x− τ)) = ∫ x+τ F (z)φ(z)dz. (x ∈ R) (32) We now further assume that C 6= 0 and that F (x) 6= 0 for at least one x ∈ R, and show that this leads to a contradiction. Consider first the case where all components of C are non-negative. Since F is not identically zero, it is then, up to a scale factor, the cdf of a random variable on the real line. But then (32) can not hold for all x ∈ R as shown in Example 7 in Leeb (2002) (cf. also equation (7) in that paper). The case where all components of C are non-positive follows similarly by applying the above argument to F (−x) and upon observing that both Φ(x+ τ )− Φ(x− τ) and φ(x) are symmetric around x = 0. Finally, consider the case where C has at least one positive and one negative component. In this case clearly limx→−∞ F (x) = limx→∞ F (x) = 0 holds. Since F (x) is continuous in view of (32), we see that F (x) attains its (positive) maximum at some point x1 ∈ R. Now note that (32) with x1 replacing x can be written as ∫ x1+τ (F (x1)− F (z))φ(z)dz = 0. This immediately entails that F (x) = F (x1) for each x ∈ [x1 − τ , x1 + τ ] (because F (x) is continuous and because of the definition of x1). Repeating this argument with x1−τ replacing x1 and proceeding inductively, we obtain that F (x) = F (x1) for each x satisfying x ≤ x1 + τ , a contradiction with limx→−∞ F (x) = 0. ✷ Lemma A.2 Let M and N be matrices of dimension k × p and k × q, respectively, such that the matrix (M : N) has rank k (k ≥ 1, p ≥ 1, q ≥ 1). Let t ∈ Rk, and let V be a random vector with values in Rp whose distribution assigns positive mass to every (non-empty) open subset of Rp (e.g., it possesses an almost everywhere positive Lebesgue density). Set f(x) = P(MV ≤ t + Nx), x ∈ Rq. If one of the rows of M consists of zeros only, then f is discontinuous at some point x0. More precisely, there exist x0 ∈ Rq, z ∈ Rq and a constant c > 0, such that f(x0 + δz) ≥ c and f(x0 − δz) = 0 hold for every sufficiently small δ > 0. Proof of Lemma A.2: The case where M is the zero-matrix is trivial. Otherwise, let I0 denote the set of indices i, 1 ≤ i ≤ k, for which the i-th row of M is zero. Let (M0 : N0) denote the matrix consisting of those rows of (M : N) whose index is in I0, and let (M1 : N1) denote the matrix consisting of the remaining rows of (M : N). Clearly, M0 is then the zero matrix. Furthermore, note that N0 has full row-rank. Moreover, let t0 denote the vector consisting of those components of t whose index is in I0 and let t1 denote the vector containing the remaining components. With this notation, f(x) can be written as P(0 ≤ t0 +N0x, M1V ≤ t1 +N1x). For vectors µ ∈ Rp and η ∈ Rq to be specified in a moment, set t∗ = t+Mµ+Nη, and let t∗0 and t∗1 be defined similarly to t0 and t1. Since the matrix (M : N) has full rank k, we can choose µ and η such that t∗0 = 0 and t 1 > 0. Choose z ∈ Rq such that N0z > 0, which is possible because N0 has full row-rank. Set x0 = η. Then for every ǫ ∈ R we have f(x0 + ǫz) = f(η + ǫz) = P(MV ≤ t+N(η + ǫz)) = P(0 ≤ t0 +N0(η + ǫz), M1V ≤ t1 +N1(η + ǫz)) = P(0 ≤ t∗0 + ǫN0z, M1(V + µ) ≤ t∗1 + ǫN1z) = P(0 ≤ ǫN0z, M1(V + µ) ≤ t∗1 + ǫN1z) Since t∗1 > 0, we can find a t 1 such that 0 < t 1 < t 1 + ǫN1z holds for every ǫ with |ǫ| small enough. If now ǫ > 0 then f(x0 + ǫz) = P(M1(V + µ) ≤ t∗1 + ǫN1z) ≥ P(M1(V + µ) ≤ t∗∗1 ). The r.h.s. in the above display is positive because t∗∗1 > 0 and because the distribution ofM1(V +µ) assigns positive mass to any neighborhood of the origin, since the same is true for the distribution of V + µ and since M1 maps neighborhoods of zero into neighborhoods of zero. Setting c = P(M1(V + µ) ≤ t∗∗1 )/2, we have f(x0 + ǫz) ≥ c > 0 for each sufficiently small ǫ > 0. Furthermore, for ǫ < 0 we have f(x0 + ǫz) = 0, since f(x0 + ǫz) ≤ P(0 ≤ ǫN0z) = 0 in view of N0z > 0. ✷ Lemma A.3 Let Z be a random vector with values in Rp, p ≥ 1, with a distribution that is absolutely continuous with respect to Lebesgue measure on Rp. Let B be a k×p matrix, k ≥ 1. Then the cdf P(BZ ≤ ·) of BZ, is discontinuous at t ∈ Rk if and only if P(BZ ≤ t) > 0 and if for some i0, 1 ≤ i0 ≤ k, the i0-th row of B and the i0-th component of t are both zero, i.e., Bi0,· = (0, . . . , 0) and ti0 = 0. Proof of Lemma A.3: To establish sufficiency of the above condition, let P(BZ ≤ t) > 0, ti0 = 0 and Bi0,· = (0, . . . , 0) for some i0, 1 ≤ i0 ≤ k. Then, of course, P(Bi0,·Z = 0) = 1. For tn = t−n−1ei0 , where ei0 denotes the i0-th unit vector in R k, we have P(BZ ≤ tn) ≤ P(Bi0,·Z ≤ tn,i0) = P(Bi0,·Z ≤ −1/n) = 0 for every n. Consequently, P(BZ ≤ t) is discontinuous at t. To establish necessity, we first show the following: If tn ∈ Rk is a sequence converging to t ∈ Rk as n→ ∞, then every accumulation point of the sequence P(BZ ≤ tn) has the form P(Bi1,·Z ≤ ti1 , . . . , Bim,·Z ≤ tim , Bim+1,·Z < tim+1 , . . . , Bik,·Z < tik) (33) for somem, 0 ≤ m ≤ k, and for some permutation (i1, . . . , ik) of (1, . . . , k). This can be seen as follows: Let α be an accumulation point of P(BZ ≤ tn). Then we may find a subsequence such that P(BZ ≤ tn) converges to α along this subsequence. From this subsequence we may even extract a further subsequence along which each component of the k × 1 vector tn converges to the corresponding component of t monotonously, that is, either from above or from below. Without loss of generality, we may also assume that those components which converge from below are strictly increasing. The resulting subsequence will be denoted by nj in the sequel. Assume that the components of tnj with indices i1, . . . , im converge from above, while the components with indices im+1, . . . , ik converge from below. Now P(BZ ≤ tnj ) = 1(−∞,tnj,s](zs)PBZ(dz), (34) where PBZ denotes the distribution of BZ. The integrand in (34) now converges to l=1 1(−∞,til ](zil) l=m+1 1(−∞,til )(zil) for all z ∈ R k as nj → ∞. The r.h.s. of (34) converges to the expression in (33) as nj → ∞ by the Dominated Convergence Theorem, while the l.h.s. of (34) converges to α by construction. This establishes the claim regarding (33). Now suppose that P(BZ ≤ t) is discontinuous at t; i.e., there exists a sequence tn converging to t as n → ∞, such that P(BZ ≤ tn) does not converge to P(BZ ≤ t) as n → ∞. From the sequence tn we can extract a subsequence tns along which P(BZ ≤ tns) converges to a limit different from P(BZ ≤ t) as ns → ∞. As shown above, the limit has to be of the form (33) and m < k has to hold. Consequently, the limit of P(BZ ≤ tns) is smaller than P(BZ ≤ t) = P(Bi,·Z ≤ ti, i = 1, . . . , k). The difference of P(BZ ≤ t) and the limit of P(BZ ≤ tns) is positive and because of (33) can be written as P(Bij ,·Z ≤ tij for each j = 1, . . . , k, Bij ,·Z = tij for some j = m+ 1, . . . , k) > 0. We thus see that P(Bij0 ,·Z = tij0 ) > 0 for some j0 satisfying m + 1 ≤ j0 ≤ k. As Z is absolutely continuous with respect to Lebesgue measure on Rp, this can only happen if Bij0 ,· = (0, . . . , 0) and tij0 = 0. Lemma A.4 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C ∞ 6= 0, for some q satisfying O < q ≤ P , and let q∗ denote the largest q with this property. Moreover let θ ∈ Mq∗−1, let σ satisfy 0 < σ < ∞, and let t ∈ Rk. Then G∞,θ,σ,γ(t) is non-constant as a function of γ ∈ Mq∗\Mq∗−1. More precisely, there exist δ0 > 0 and ρ0, 0 < ρ0 <∞, such that γ(1),γ(2)∈Mq∗ \Mq∗−1 ||γ(i)||<ρ0,i=1,2 ∣∣G∞,θ,σ,γ(1)(t)−G∞,θ,σ,γ(2)(t) ∣∣ > 2δ0 (35) holds. The constants δ0 and ρ0 can be chosen in such a way that they depend only on t, Q, A, σ, and the critical values cp for O < p ≤ P . Lemma A.5 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C ∞ 6= 0, for some q satisfying O < q ≤ P , and let q∗ denote the largest q with this property. Suppose further that for some p⊙ satisfying O ≤ p⊙ < q∗ either p⊙ = 0 holds or that p⊙ > 0 and A[p⊙] has a row of zeros. Then, for every θ ∈ Mp⊙ , every σ, 0 < σ < ∞, and every t ∈ Rk the quantity G∞,θ,σ,γ(t) is discontinuous as a function of γ ∈ Mq∗ . More precisely, for each s = O, . . . , p⊙, there exist vectors β∗ and γ∗ in Mq∗ and constants δ∗ > 0 and ǫ∗ > 0 such that ∣∣G∞,θ,σ,β∗+ǫγ∗(t)−G∞,θ,σ,β∗−ǫγ∗(t) ∣∣ ≥ δ∗ (36) holds for every θ satisfying max{p0(θ),O} = s and for every ǫ with 0 < ǫ < ǫ∗. The quantities δ∗, ǫ∗, β∗, and γ∗ can be chosen in such a way that – besides t, Q, A, σ, and the critical values cp for O < p ≤ P – they depend on θ only through max{p0(θ),O}. Before we prove the above lemmas, we provide a representation of G∞,θ,σ,γ(t) that will be useful in the following: For 0 < p ≤ P define Zp = r=1 ξ ∞ Wr, where C ∞ has been defined after (13) and the random variables Wr are independent normally distributed with mean zero and variances σ 2ξ2∞,r; for convenience, let Z0 denote the zero vector in R k. Observe that Zp, p > 0, is normally distributed with mean zero and variance-covariance matrix σ2A[p]Q[p : p]−1A[p]′ since it has been shown in the proof of Proposition 4.4 in Leeb and Pötscher (2006b) that the asymptotic variance-covariance matrix σ2A[p]Q[p : p]−1A[p]′ of nAθ̃(p) can be expressed as r=1 σ 2ξ−2∞,rC ∞ . Also the joint distribution of Zp and the set of variables Wr, 1 ≤ r ≤ P , is normal, with the covariance vector between Zp and Wr given by σ2C(r)∞ in case r ≤ p; otherwise Zp and Wr are independent. Define the constants νr = γr+(Q[r : r]−1Q[r : ¬r]γ[¬r])r for 0 < r ≤ P . It is now easy to see that for p ≥ p∗ = max{p0(θ),O} the quantity β(p) defined in Proposition 2.1 equals − r=p+1 ξ ∞ νr. [This is seen as follows: It was noted in Proposition 2.1 that β(p) = limn→∞ nA(ηn(p) − θ − γ/ n) for p ≥ p0(θ), when ηn(p) is defined as in (9), but with θ + γ/ replacing θ. Using the representation (20) of Leeb (2005) and taking limits, the result follows if we observe nηn,r(r) −→ νr for r > p ≥ p0(θ).] The cdf in (15) can now be written as Zp∗ ≤ t+ r=p∗+1 ξ−2∞,rC q=p∗+1 P(|Wq + νq| < cqσξ∞,q) p=p∗+1 Zp ≤ t+ r=p+1 ξ−2∞,rC ∞ νr, |Wp + νp| ≥ cpσξ∞,p q=p+1 P(|Wq + νq| < cqσξ∞,q). (37) That the terms corresponding to p = p∗ in (37) and (15) agree is obvious. Furthermore, for each p > p∗ the terms under the product sign in (37) and (15) coincide by definition of the function ∆s(a, b). It is also easy to see that the conditional distribution of Wp given Zp = z is Gaussian with mean b∞,pz and variance σ2ζ2∞,p. Consequently, the probability of the event {|Wp + νp| ≥ cpσξ∞,p} conditional on Zp = z is given by the integrand shown in (15). Since Zp has distribution Φ∞,p as noted above, it follows that (37) and (15) agree. Remark A.6 If C ∞ = 0 for p > p∗, then in view of the above discussion Zp∗ = Zp = ZP , and hence Φ∞,p∗ = Φ∞,p = Φ∞,P , holds for all p > p∗. Using the independence of Wr, r > p∗, from Zp∗ , inspection of (37) shows that G∞,θ,σ,γ reduces to Φ∞,P ; see also Leeb (2006, Remark 5.2). Proof of Lemma A.4: From (37) (or (15)) it follows that the map γ 7→ G∞,θ,σ,γ(t) depends only on t, Q, A, σ, the critical values cp for O < p ≤ P , as well as on θ; however, the dependence on θ is only through p∗ = max{p0(θ),O}. It hence suffices to find, for each possible value of p∗ in the range p∗ = O, . . . , q∗ − 1, constants 0 < ρ0 < ∞ and δ0 > 0 such that (35) is satisfied for some (and hence all) θ returning this particular value of p∗ = max{p0(θ),O}. For this in turn it is sufficient to show that for every θ ∈Mq∗−1 the quantity G∞,θ,σ,γ(t) is non-constant as a function of γ ∈Mq∗\Mq∗−1. Let θ ∈Mq∗−1 and assume that G∞,θ,σ,γ(t) is constant in γ ∈Mq∗\Mq∗−1. Observe that, by assumption, ∞ is non-zero while C ∞ = 0 for p > q ∗. For γ ∈ Mq∗ , we clearly have νq∗ = γq∗ and νr = 0 for r > q∗. Letting γq∗−1 → ∞ while γq∗ is held fixed, we see that νq∗−1 → ∞; hence, P(|Wq∗−1 + νq∗−1| < cq∗−1σξ∞,q∗−1) → 0. It follows that (37) converges to Zq∗−1 ≤ t+ ξ−2∞,q∗C(q ∞ γq∗ P(|Wq∗ + γq∗ | < cq∗σξ∞,q∗) q=q∗+1 P(|Wq| < cqσξ∞,q) Zq∗ ≤ t, |Wq∗ + γq∗ | ≥ cq∗σξ∞,q∗ q=q∗+1 P(|Wq| < cqσξ∞,q) (38) p=q∗+1 Zp ≤ t, |Wp| ≥ cpσξ∞,p q=p+1 P(|Wq| < cqσξ∞,q). By assumption, the expression in the above display is constant in γq∗ ∈ R\{0}. Dropping the terms that do not depend on γq∗ and observing that P(|Wq| < cqσξ∞,q) is never zero for q > q∗ > O, we see that Zq∗−1 ≤ t+ ξ−2∞,q∗C(q ∞ γq∗ P(|Wq∗ + γq∗ | < cq∗σξ∞,q∗) Zq∗ ≤ t, |Wq∗ + γq∗ | ≥ cq∗σξ∞,q∗ has to be constant in γq∗ ∈ R\{0}. We now show that the expression in (39) is in fact constant in γq∗ ∈ R: Observe first that P(|Wq∗ + γq∗ | < cq∗σξ∞,q∗) is positive and continuous in γq∗ ∈ R; also the probability Zq∗ ≤ t, |Wq∗ + γq∗ | ≥ cq∗σξ∞,q∗ is continuous in γq∗ ∈ R since Wq∗ , being normal with mean zero and positive variance, is absolutely continuously distributed. Concerning the remaining term in (39), we note that Zq∗−1 =MV where M = [ξ ∞ , . . . , ξ ∞,q∗−1C (q∗−1) ∞ ] and V = (W1, . . . ,Wq∗−1) ′. In case no row of M is identically zero, Lemma A.3 shows that also P Zq∗−1 ≤ t+ ξ−2∞,q∗C ∞ γq∗ is continuous in γq∗ ∈ R. Hence, in this case (39) is indeed constant for all γq∗ ∈ R. In case a row of M is identically zero, define N = ξ ∞,q∗C ∞ and rewrite the probability in question as P MV ≤ t+Nγq∗ . Note that (M : N) has full row-rank k, since (M : N)diag[ξ2∞,1, . . . , ξ ∞,q∗ ](M : N) ξ−2∞,rC ξ−2∞,rC ∞ = AQ −1A′ (40) by definition of q∗ and since the latter matrix is non-singular in view of rank A = k. Lemma A.2 then shows that there exists a γ q∗ ∈ R, z ∈ {−1, 1}, and a constant c > 0 such that P MV ≤ t+N(γ(0)q∗ − δz) and P MV ≤ t+N(γ(0)q∗ + δz) ≥ c holds for arbitrary small δ > 0. Observe that γ(0)q∗ − δz as well as q∗ − δz are non-zero for sufficiently small δ > 0. But then (39) – being constant for γq∗ ∈ R\{0} – gives the same value for γq∗ = γ q∗ − δz and γq∗ = γ q∗ + δz and all sufficiently small δ > 0. Letting δ go to zero in this equality and using the continuity properties for the second and third probability in (39) noted above we obtain that cP(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗) + P Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗ ≤ lim inf Zq∗−1 ≤ t+ ξ−2∞,q∗C(q q∗ + δz) P(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗) Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗ = lim inf Zq∗−1 ≤ t+ ξ−2∞,q∗C(q q∗ − δz) P(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗) Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗ Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗ which is impossible since c > 0 and P(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗) > 0. Hence we have shown that (39) is indeed constant for all γq∗ ∈ R. Now write Z,W , C, τ , and x for Zq∗−1−t, −Wq∗/σξ∞,q∗ , σξ ∞,q∗C ∞ , cq∗ , and γq∗/σξ∞,q∗ , respectively. Upon observing that Zq∗ equals Zq∗−1 + ξ ∞,q∗C ∞ Wq∗ , it is easy to see that (39) can be written as in (31). By our assumptions, this expression is constant in x = γq∗/σξ∞,q∗ ∈ R. Lemma A.1 then entails that either C = 0 or that P(Z ≤ Cx) = 0 for each x ∈ R. Since C equals σξ−1∞,q∗C ∞ , it is non-zero by assumption. Hence, Zq∗−1 ≤ t+ ξ−2∞,q∗C(q ∞ γq∗ must hold for every value of γq∗ . But the above probability is just the conditional probability that Zq∗ ≤ t given Wq∗ = −γq∗ . It follows that P(Zq∗ ≤ t) equals zero as well. By our assumption C ∞ = 0 for p > q and hence Zq∗ = ZP . We thus obtain P(ZP ≤ t) = 0, a contradiction with the fact that ZP is a Gaussian random variable on Rk with non-singular variance-covariance matrix σ2AQ−1A′. ✷ Inspection of the above proof shows that it can be simplified if the claim of non-constancy of G∞,θ,σ,γ(t) as a function of γ ∈Mq∗\Mq∗−1 in Lemma A.4 is weakened to non-constancy for γ ∈Mq∗ . The strong form of the lemma as given here is needed in the proof of Proposition B.1. Proof of Lemma A.5: Let p⊕ be the largest index p, O ≤ p ≤ P , for which A[p] has a row of zeroes, and set p⊕ = 0 if no such index exists. We first show that p⊕ satisfies p⊕ < q ∗. Suppose p⊕ ≥ q∗ would hold. Since Zp⊕ is a Gaussian random vector with mean zero and variance-covariance matrix σ 2A[p⊕]Q[p⊕ : p⊕] −1A[p⊕] at least one component of Zp⊕ is equal to zero with probability one. However, Zp⊕ equals ZP because of p⊕ ≥ q∗ and the definition of q∗. This leads to a contradiction since ZP has the non-singular variance- covariance matrix σ2AQ−1A′. Without loss of generality, we may hence assume that p⊙ = p⊕. In view of the discussion in the first paragraph of the proof of Lemma A.4, it suffices to establish, for each possible value s in the range O ≤ s ≤ p⊙, the result (36) for some θ with s = max{p0(θ),O} = p∗. Now fix such an s and θ (as well as, of course, t, Q, A, σ, and the critical values cp for O < p ≤ P ). Then (37) expresses the map γ 7→ G∞,θ,σ,γ(t) in terms of ν = (ν1, . . . , νP )′. It is easy to see that the correspondence between γ and ν is a linear bijection from RP onto itself, and that γ ∈ Mq∗ if and only if ν ∈ Mq∗ . It is hence sufficient to find a δ∗ > 0 and vectors ν and µ in Mq∗ such that (37) with ν + ǫµ in place of ν and (37) with ν − ǫµ in place of ν differ by at least δ∗ for sufficiently small ǫ > 0. Note that (37) is the sum of P − p∗ + 1 terms indexed by p = p∗, . . . , P . We shall now show that ν and µ can be chosen in such a way that, when replacing ν with ν + ǫµ and ν − ǫµ, respectively, (i) the resulting terms in (37) corresponding to p = p⊙ differ by some d > 0, while (ii) the difference of the other terms becomes arbitrarily small, provided that ǫ > 0 is sufficiently small. Consider first the case where s = p∗ = p⊙. Using the shorthand notation g(ν) = P Zp⊙ ≤ t+ r=p⊙+1 ξ−2∞,rC note that the p⊙-th term in (37) is given by g(ν) multiplied by a product of positive probabilities which are continuous in ν. To prove property (i) it thus suffices to find a constant c > 0, and vectors ν and µ in Mq∗ such that |g(ν + ǫµ)− g(ν − ǫµ)| ≥ c holds for each sufficiently small ǫ > 0. In the sub-case p⊙ = 0 choose c = 1, set ν = −[C(1)∞ , . . . , C(P )∞ ]′ ξ−2∞,rC µ = [C(1)∞ , . . . , C ξ−2∞,rC (1, . . . , 1)′, observing that the matrix to be inverted is indeed non-singular, since – as discussed after Lemma A.5 – it is up to a multiplicative factor σ2 identical to the variance-covariance matrix σ2AQ−1A′ of ZP . But then ν and µ satisfy r=p⊙+1 ξ−2∞,rC ∞ νr = −t and r=p⊙+1 ξ−2∞,rC ∞ µr = (1, . . . , 1) ′ if we note that by the definition of q∗ r=p⊙+1 ξ−2∞,rC ∞ νr = ξ−2∞,rC holds and that a similar relation holds with µ replacing ν. Since Zp⊙ = Z0 = 0 ∈ Rk, it is then obvious that g(ν + ǫµ) and g(ν − ǫµ) differ by 1 for each ǫ > 0. In the other sub-case p⊙ > 0, define M = [ξ ∞ , . . . , ξ ∞,p⊙C ∞ ], N = [ξ−2∞,p⊙+1C (p⊙+1) ∞ , . . . , ξ ∞,q∗C ∞ ], and V = (W1, . . . ,Wp⊙) ′. It is then easy to see that g(ν) equals f((νp⊙+1, . . . , νq∗) ′), with f defined as in Lemma A.2, and that M has a row of zeros. Furthermore, the matrix (M : N) has rank k by the same argument as in the proof of Lemma A.4; cf. (40). By Lemma A.2, we thus obtain vectors x0 and z, and a c > 0 such that |f(x0+ ǫz)−f(x0− ǫz)| ≥ c holds for each sufficiently small ǫ > 0. Setting (νp⊙+1, . . . , νq∗) ′ = x0, (µp⊙+1, . . . , µq∗) ′ = z, setting ν[¬q∗], and µ[¬q∗] each equal to zero, and setting ν[p⊙] and µ[p⊙] to arbitrary values, we see that g(ν ± ǫµ) has the desired properties. To complete the proof in case s = p∗ = p⊙, we need to establish property (ii) for which it suffices to show that, for p > p⊙, the p-th term in (37) depends continuously on ν. For p > q ∗, the p-th term does not depend on ν, because C ∞ = 0 for r = q∗, . . . , P . For p satisfying p⊙ < p ≤ q∗, it suffices to show that h(νp, . . . , νq∗) = P Zp ≤ t+ r=p+1 ∞ νr, |Wp + νp| ≥ cpσξ∞,p is a continuous function. Suppose that (ν p , . . . , ν q∗ ) converges to (νp, . . . , νq∗) as m→ ∞. For arbitrary α > 0, r=p+1 ξ ∞ νr and r=p+1 ξ r differ by less than α in each coordinate, provided that m is sufficiently large. This implies lim sup h(ν(m)p , . . . , ν ≤ lim sup P(Zp ≤ t+ r=p+1 ∞ νr + α(1, . . . , 1) ′, |Wp + ν(m)p | ≥ cpσξ∞,p) = P(Zp ≤ t+ r=p+1 ξ−2∞,rC ∞ νr + α(1, . . . , 1) ′, |Wp + νp| ≥ cpσξ∞,p), observing that the latter probability is obviously continuous in the single variable νp (since Wp has an ab- solutely continuous distribution). Letting α decrease to zero we obtain lim supm→∞ h(ν p , . . . , ν q∗ ) ≤ h(νp, . . . , νq∗). A similar argument establishes lim infm→∞ h(ν p , . . . , ν q∗ ) ≥ P(Zp < t + r=p+1 ξ ∞ νr, |Wp + νp| ≥ cpσξ∞,p). The proof of the continuity of h is then complete if we can show that P Zp ≤ ·, |Wp + νp| ≥ cpσξ∞,p is continuous or, equivalently, that P Zp ≤ · ∣∣|Wp + νp| ≥ cpσξ∞,p is a continuous cdf. Since p > p⊙, the variance-covariance matrix σ 2A[p]Q[p : p]−1A[p]′ of Zp does only have non-zero diagonal elements. Consequently, when representing Zp as B(W1, . . . ,Wp) ′, the matrix B cannot have rows that consist entirely of zeros. The conditional distribution of (W1, . . . ,Wp) ′ given the event {|Wp + νp| ≥ cpσξ∞,p} is clearly absolutely continuous w.r.t. p-dimensional Lebesgue measure. But then Lemma A.3 delivers the desired result. The case where s = p∗ < p⊙ is reduced to the previously discussed case as follows: It is easy to see that, for νp⊙ → ∞, the expression in (37) converges to a limit uniformly w.r.t. all νp with p 6= p⊙. Then observe that this limit is again of the form (37) but now with p⊙ taking the rôle of p∗. ✷ B Non-Uniformity of the Convergence of the Finite-Sample Cdf to the Large-Sample Limit Proposition B.1 a. Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C ∞ 6= 0, for some q satisfying O < q ≤ P , and let q∗ denote the largest q with this property. Then for every θ ∈Mq∗−1, every σ, 0 < σ <∞, and every t ∈ Rk there exists a ρ, 0 < ρ <∞, such that lim inf ϑ∈Mq∗ ||ϑ−θ||<ρ/ |Gn,ϑ,σ(t)−G∞,ϑ,σ(t)| > 0 (41) holds. The constant ρ may be chosen in such a way that it depends only on t, Q, A, σ, and the critical values cp for O < p ≤ P . b. Suppose that Aθ̃(q) and θ̃q(q) are asymptotically uncorrelated, i.e., C ∞ = 0, for all q satisfying O < q ≤ P . Then Gn,θ,σ converges to Φ∞,P in total variation uniformly in θ ∈ RP ; more precisely σ∗≤σ≤σ∗ ||Gn,θ,σ − Φ∞,P ||TV n→∞−→ 0 holds for any constants σ∗ and σ ∗ satisfying 0 < σ∗ ≤ σ∗ <∞. Under the assumptions of Proposition B.1(a), we see that convergence of Gn,θ,σ(t) to G∞,θ,σ(t) is non- uniform over shrinking ‘tubes’ aroundMq∗−1 that are contained in Mq∗ . [On the complement of a tube with a fixed positive radius, i.e., on the set U = {θ ∈ RP : |θq∗ | ≥ r} with fixed r > 0, convergence of Gn,θ,σ(t) to G∞,θ,σ(t) is in fact uniform (even with respect to the total variation distance), as can be shown. Note that for θ ∈ U the cdf G∞,θ,σ(t) reduces to the Gaussian cdf Φ∞,P (t), i.e., to the asymptotic distribution of the least-squares estimator based on the overall model; cf. Remark A.6.] A precursor to Proposition B.1(a) is Corollary 5.5 of Leeb and Pötscher (2003) which establishes (41) in the special case where O = 0 and where A is the P × P identity matrix. Proposition B.1(b) describes an exceptional case where convergence is uniform. [In this case G∞,θ,σ reduces to the Gaussian cdf Φ∞,P for all θ and Φ∞,P = Φ∞,p, O ≤ p ≤ P , holds; cf. Remark A.6.] Recall that under the assumptions of part (b) of Proposition B.1 we necessarily always have (i) O > 0, and (ii) rank A[O] = k; cf. Proposition 4.4 in Leeb and Pötscher (2006b). Proof of Proposition B.1: We first prove part (a). As noted at the beginning of the proof of Lemma A.4, the map γ 7→ G∞,θ,σ,γ(t) depends only on t, Q, A, σ, the critical values cp for O < p ≤ P , as well as on θ, but the dependence on θ is only through p∗ = max{p0(θ),O}. It hence suffices to find, for each possible value of p∗ in the range p∗ = O, . . . , q∗ − 1, a constant 0 < ρ < ∞ such that (41) is satisfied for some (and hence all) θ returning this particular value of p∗ = max{p0(θ),O}. For this in turn it is sufficient to show that given such a θ we can find a γ ∈Mq∗ such that lim inf |Gn,θ+γ/√n,σ(t)−G∞,θ+γ/√n,σ(t)| > 0 (42) holds. Note that (42) is equivalent to lim inf |G∞,θ,σ,γ(t)−G∞,θ+γ/√n,σ(t)| > 0 (43) in light of Proposition 2.1. To establish (43), we proceed as follows: For each γ ∈ Mq∗ with γq∗ 6= 0, G∞,θ+γ/ n,σ(t) in (15) reduces to Φ∞,q∗(t) as is easily seen from (37) since p0(θ+γ/ n) = q∗ which in turn follows from p0(θ) < q ∗ and γq∗ 6= 0. Furthermore, Lemma A.4 entails that G∞,θ,σ,γ(t) is non-constant in γ ∈Mq∗\Mq∗−1. But this shows that (43) must hold. To prove part (b), we write ||Gn,θ,σ − Φ∞,P ||TV = ∣∣∣∣∣∣ ∣∣∣∣∣∣ Gn,θ,σ(·|p)πn,θ,σ(p)− Φ∞,P (·) ∣∣∣∣∣∣ ∣∣∣∣∣∣ ||Gn,θ,σ(·|p)− Φ∞,P (·)||TV πn,θ,σ(p), where the conditional cdfs Gn,θ,σ(·|p) and the model selection probabilities πn,θ,σ(p) have been introduced after (12). By the ‘uncorrelatedness’ assumption, we have that Φ∞,p = Φ∞,P for all p in the rangeO ≤ p ≤ P ; cf. Remark A.6. We hence obtain σ∗≤σ≤σ∗ ||Gn,θ,σ − Φ∞,P ||TV ≤ σ∗≤σ≤σ∗ ||Gn,θ,σ(·|p)− Φ∞,p(·)||TV πn,θ,σ(p). (44) Now for every p with O ≤ p ≤ P and for every ρ, 0 < ρ <∞, we can write σ∗≤σ≤σ∗ ||Gn,θ,σ(·|p)− Φ∞,p(·)||TV πn,θ,σ(p) ≤ max ‖θ[¬p]‖<ρ/ σ∗≤σ≤σ∗ ||Gn,θ,σ(·|p)− Φ∞,p(·)||TV , sup ‖θ[¬p]‖≥ρ/ σ∗≤σ≤σ∗ πn,θ,σ(p) . (45) In case p = P , we use here the convention that the second term in the maximum is absent and that the first supremum in the first term in the maximum extends over all of RP . Letting first n and then ρ go to infinity in (45), we may apply Lemmas C.2 and C.3 in Leeb and Pötscher (2005b) to conclude that the l.h.s. of (45), and hence the l.h.s. of (44), goes to zero as n→ ∞. ✷ C Proofs for Sections 2.1 to 2.2.2 In the proofs below it will be convenient to show the dependence of Φn,p and Φ∞,p on σ in the notation. Thus, in the following we shall write Φn,p,σ and Φ∞,p,σ, respectively, for the cdf of a k-variate Gaussian random vector with mean zero and variance-covariancematrix σ2A[p](X [p]′X [p]/n)−1A[p]′ and σ2A[p]Q[p : p]−1A[p]′, respectively. For convenience, let Φn,0,σ and Φ∞,0,σ denote the cdf of point-mass at zero in R The following lemma is elementary to prove, if we recall that bn,pz converges to b∞,pz as n → ∞ for every z ∈ ImA[p], the column space of A[p]. Lemma C.1 Suppose p > O. Define Rn,p(z, σ) = 1 − ∆σζn,p(bn,pz, cpσξn,p) and R∞,p(z, σ) = 1 − ∆σζ∞,p(b∞,pz, cpσξ∞,p) for z ∈ ImA[p], 0 < σ <∞. Let σ (n) converge to σ, 0 < σ < ∞. If ζ∞,p 6= 0, then Rn,p(z, σ (n)) converges to R∞,p(z, σ) for every z ∈ ImA[p]; if ζ∞,p = 0, then convergence holds for every z ∈ ImA[p], except possibly for z ∈ ImA[p] satisfying |b∞,pz| = cpσξ∞,p. [This exceptional subset of ImA[p] has rank(A[p])-dimensional Lebesgue measure zero since cpσξ∞,p > 0.] The following observation is useful in the proof of Proposition 2.2 below: Since the proposition depends on Y only through its distribution (cf. Remark 4.1), we may assume without loss of generality that the errors in (5) are given by ut = σεt, t ∈ N, with i.i.d. εt that are standard normal. In particular, all random variables involved are then defined on the same probability space. Proof of Proposition 2.2: Since Pn,θ,σ(p̄ = p0(θ)) → 1 by consistency, we may replace max{p̄,O} by p∗ = max{p0(θ),O} in the formula for Ǧn for the remainder of the proof. Furthermore, since σ̂ → σ in Pn,θ,σ-probability, each subsequence contains a further subsequence along which σ̂ → σ almost surely (with respect to the probability measure on the common probability space supporting all random variables involved), and we restrict ourselves to such a further subsequence for the moment. In particular, we write {σ̂ → σ} for the event that σ̂ converges to σ along the subsequence under consideration; clearly, the event {σ̂ → σ} has probability one. Also note that we can assume without loss of generality that σ̂ > 0 holds on this event (at least from some data-dependent n onwards), since σ > 0 holds. But then obviously q=p∗+1 ∆σ̂ξn,q (0, cqσ̂ξn,q) converges to q=p∗+1 ∆σξ∞,q (0, cqσξ∞,q), and Φ̂n,p∗(t) converges to Φ∞,p∗,σ(t) in total variation by Lemma A.3 of Leeb (2005) in case p∗ > 0, and trivially so in case p∗ = 0. This proves that the first term in the formula for Ǧn converges to the corresponding term in the formula for G∞,θ,σ in total variation. Next, consider the term in Ǧn that carries the index p > p∗. By Lemma A.3 in Leeb (2005), Φ̂n,p = Φn,p,σ̂ has a density dΦn,p,σ̂/dΦ∞,p,σ with respect to Φ∞,p,σ, which converges to 1 except on a set that has measure zero under Φ∞,p,σ. By Scheffé’s Lemma (Billingsley (1995), Theorem 16.12), dΦn,p,σ̂/dΦ∞,p,σ converges to 1 also in the L1(Φ∞,p,σ)-sense. By Lemma C.1, Rn,p(z, σ̂) converges to R∞,p(z, σ) except possibly on a set that has measure zero under Φ∞,p,σ. (Recall that Φ∞,p,σ is concentrated on ImA[p] and is not degenerate there.) Observing that |Rn,p(z, σ̂)| is uniformly bounded by 1, we obtain that Rn,p(z, σ̂) converges to R∞,p(z, σ) also in the L1(Φ∞,p,σ)-sense. Hence, ∥∥∥∥Rn,p(z, σ̂) dΦn,p,σ̂ dΦ∞,p,σ (z)−R∞,p(z, σ) ∥∥∥∥Rn,p(z, σ̂) dΦn,p,σ̂ dΦ∞,p,σ (z)−Rn,p(z, σ̂) ∥∥∥∥+ ‖Rn,p(z, σ̂)−R∞,p(z, σ)‖ (46) dΦn,p,σ̂ dΦ∞,p,σ (z)− 1 ∥∥∥∥+ ‖Rn,p(z, σ̂)−R∞,p(z, σ)‖ n→∞−→ 0 where ‖·‖ denotes the L1(Φ∞,p,σ)-norm. Since q=p+1 ∆σ̂ξn,q (0, cqσ̂ξn,q) obviously converges to∏P q=p+1 ∆σξ∞,q (0, cqσξ∞,q), the relation (46) shows that the term in Ǧn carrying the index p converges to the corresponding term in G∞,θ,σ in the total variation sense. This proves (18) along the subsequence under consideration. However, since any subsequence contains such a further subsequence, this establishes (18). Since Gn,θ,σ converges to G∞,θ,σ in total variation by Proposition 2.1, the claim in (17) also follows.✷ Before we prove the main result we observe that the total variation distance between Pn,θ,σ and Pn,ϑ,σ satisfies ||Pn,θ,σ − Pn,ϑ,σ||TV ≤ 2Φ(‖θ − ϑ‖λ max(X ′X)/2σ) − 1; furthermore, if θ(n) and ϑ(n) sat- ∥∥∥θ(n) − ϑ(n) ∥∥∥ = O(n−1/2), the sequence Pn,ϑ(n),σ is contiguous with respect to the sequence Pn,θ(n),σ (and vice versa). This follows exactly in the same way as Lemma A.1 in Leeb and Pötscher (2006a). Proof of Theorem 2.3: We first prove (20) and (21). For this purpose we make use of Lemma 3.1 in Leeb and Pötscher (2006a) with α = θ ∈ Mq∗−1, B = Mq∗ , Bn = {ϑ ∈ Mq∗ : ‖ϑ− θ‖ < ρ0n−1/2}, β = ϑ, ϕn(β) = Gn,ϑ,σ(t), ϕ̂n = Ĝn(t), where ρ0, 0 < ρ0 < ∞, will be chosen shortly (and σ is held fixed). The contiguity assumption of this lemma (as well as the mutual contiguity assumption used in the corrigendum to Leeb and Pötscher (2006a)) is satisfied in view of the preparatory remark above. It hence remains only to show that there exists a value of ρ0, 0 < ρ0 < ∞, such that δ in Lemma 3.1 of Leeb and Pötscher (2006a) (which represents the limit inferior of the oscillation of ϕn(·) over Bn) is positive. Applying Lemma 3.5(i) of Leeb and Pötscher (2006a) with ζn = ρ0n −1/2 and the set G0 equal to the set G, it remains, in light of Proposition 2.1, to show that there exists a ρ0, 0 < ρ0 < ∞, such that G∞,θ,σ,γ(t) as a function of γ is non-constant on the set {γ ∈ Mq∗ : ‖γ‖ < ρ0}. In view of Lemma 3.1 of Leeb and Pötscher (2006a), the corresponding δ0 can then be chosen as any positive number less than one-half of the oscillation of G∞,θ,σ,γ(t) over this set. That such a ρ0 indeed exists follows now from Lemma A.4 in Appendix A, where it is also shown that ρ0 and δ0 can be chosen such that they depend only on t, Q,A, σ, and cp for O < p ≤ P . This completes the proof of (20) and (21). To prove (22) we use Corollary 3.4 in Leeb and Pötscher (2006a) with the same identification of notation as above, with ζn = ρ0n −1/2, and with V = Mq∗ (viewed as a vector space isomorphic to ). The asymptotic uniform equicontinuity condition in that corollary is then satisfied in view of ||Pn,θ,σ − Pn,ϑ,σ||TV ≤ 2Φ(‖θ − ϑ‖λ max(X ′X)/2σ) − 1. Given that the positivity of δ∗ has already be es- tablished in the previous paragraph, applying Corollary 3.4(i) in Leeb and Pötscher (2006a) then establishes (22). ✷ Proof of Remark 2.4: The proof is similar to the proof of (22) just given, except for using Corol- lary 3.4(ii) and Lemma 3.5(ii) in Leeb and Pötscher (2006a) instead of Corollary 3.4(i) and Lemma 3.5(i) from that paper. Furthermore, Lemma A.5 in Appendix A instead of Lemma A.4 is used. ✷ Proof of Proposition 2.5: In view of Proposition B.1(b) and the fact that Φ̂n,P (·) = Φn,P,σ̂(·) holds (in case σ̂ > 0), it suffices to show that σ∗≤σ≤σ∗ ||Φn,P,σ(·)− Φ∞,P,σ(·)||TV n→∞−→ 0 (47) σ∗≤σ≤σ∗ Pn,θ,σ ||Φn,P,σ̂(·)− Φn,P,σ(·)||TV > δ ) n→∞−→ 0 (48) hold for each δ > 0, and for any constants σ∗ and σ ∗ satisfying 0 < σ∗ ≤ σ∗ <∞. [Note that the probability in (48) does in fact not depend on θ.] But this has already been established in the proof of Proposition 4.3 of Leeb and Pötscher (2005b). ✷ D Proofs for Section 3 Proof of Theorem 3.1: After rearranging the elements of θ (and hence the regressors) if necessary and then correspondingly rearranging the rows of the matrix A, we may assume without loss of generality that r∗ = (1, . . . , 1, 0), and hence that i(r∗) = P . That is, Mr∗ = MP−1 and Mrfull = MP . Furthermore, note that after this arrangement C ∞ 6= 0. Let p̂ be the model selection procedure introduced in Section 2 with O = P − 1, cP = c, and cO = 0. Let θ̃ be the corresponding post-model-selection estimator and let Gn,θ,σ(t) be as defined in Section 2.1. Condition (24) now implies: For every θ ∈ MP−1 which has exactly P − 1 non-zero coordinates Pn,θ,σ ({r̂ = rfull}N{p̂ = P}) = lim Pn,θ,σ ({r̂ = r∗}N{p̂ = P − 1}) = 0 (49) holds for every 0 < σ < ∞. 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We consider the problem of estimating the unconditional distribution of a post-model-selection estimator. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model selection criterion like AIC or by a hypothesis testing procedure) and then estimating the parameters in the selected model (e.g., by least-squares or maximum likelihood), all based on the same data set. We show that it is impossible to estimate the unconditional distribution with reasonable accuracy even asymptotically. In particular, we show that no estimator for this distribution can be uniformly consistent (not even locally). This follows as a corollary to (local) minimax lower bounds on the performance of estimators for the distribution; performance is here measured by the probability that the estimation error exceeds a given threshold. These lower bounds are shown to approach 1/2 or even 1 in large samples, depending on the situation considered. Similar impossibility results are also obtained for the distribution of linear functions (e.g., predictors) of the post-model-selection estimator.

Introduction and Overview In many statistical applications a data-based model selection step precedes the final parameter estimation and inference stage. For example, the specification of the model (choice of functional form, choice of regressors, http://arxiv.org/abs/0704.1584v1 number of lags, etc.) is often based on the data. In contrast, the traditional theory of statistical inference is concerned with the properties of estimators and inference procedures under the central assumption of an a priori given model. That is, it is assumed that the model is known to the researcher prior to the statistical analysis, except for the value of the true parameter vector. As a consequence, the actual statistical properties of estimators or inference procedures following a data-driven model selection step are not described by the traditional theory which assumes an a priori given model; in fact, they may differ substantially from the properties predicted by this theory, cf., e.g., Danilov and Magnus (2004), Dijkstra and Veldkamp (1988), Pötscher (1991, Section 3.3), or Rao and Wu (2001, Section 12). Ignoring the additional uncertainty originating from the data-driven model selection step and (inappropriately) applying traditional theory can hence result in very misleading conclusions. Investigations into the distributional properties of post-model-selection estimators, i.e., of estimators constructed after a data-driven model selection step, are relatively few and of recent vintage. Sen (1979) obtained the unconditional large-sample limit distribution of a post-model-selection estimator in an i.i.d. maximum likelihood framework, when selection is between two competing nested models. In Pötscher (1991) the asymptotic properties of a class of post-model-selection estimators (based on a sequence of hypothesis tests) were studied in a rather general setting covering non-linear models, dependent processes, and more than two competing models. In that paper, the large-sample limit distribution of the post-model-selection estimator was derived, both unconditional as well as conditional on having chosen a correct model, not necessarily the minimal one. See also Pötscher and Novak (1998) for further discussion and a simulation study, and Nickl (2003) for extensions. The finite-sample distribution of a post-model-selection estimator, both unconditional and conditional on having chosen a particular (possibly incorrect) model, was derived in Leeb and Pötscher (2003) in a normal linear regression framework; this paper also studied asymptotic approximations that are in a certain sense superior to the asymptotic distribution derived in Pötscher (1991). The distributions of corresponding linear predictors constructed after model selection were studied in Leeb (2005, 2006). Related work can also be found in Sen and Saleh (1987), Kabaila (1995), Pötscher (1995), Ahmed and Basu (2000), Kapetanios (2001), Hjort and Claeskens (2003), Dukić and Peña (2005), and Leeb and Pötscher (2005a). The latter paper provides a simple exposition of the problems of inference post model selection and may serve as an entry point to the present paper. It transpires from the papers mentioned above that the finite-sample distributions (as well as the large- sample limit distributions) of post-model-selection estimators typically depend on the unknown model pa- rameters, often in a complicated fashion. For inference purposes, e.g., for the construction of confidence sets, estimators of these distributions would be desirable. Consistent estimators of these distributions can typically be constructed quite easily, e.g., by suitably replacing unknown parameters in the large-sample limit distributions by estimators; cf. Section 2.2.1. However, the merits of such ‘plug-in’ estimators in small samples are questionable: It is known that the convergence of the finite-sample distributions to their large- sample limits is typically not uniform with respect to the underlying parameters (see Appendix B below and Corollary 5.5 in Leeb and Pötscher (2003)), and there is no reason to believe that this non-uniformity will disappear when unknown parameters in the large-sample limit are replaced by estimators. This observation is the main motivation for the present paper to investigate in general the performance of estimators of the distribution of a post-model-selection estimator, where the estimators of the distribution are not necessar- ily ‘plug-in’ estimators based on the limiting distribution. In particular, we ask whether estimators of the distribution function of post-model-selection estimators exist that do not suffer from the non-uniformity phenomenon mentioned above. As we show in this paper the answer in general is ‘No’. We also show that these negative results extend to the problem of estimating the distribution of linear functions (e.g., linear predictors) of post-model-selection estimators. Similar negative results apply also to the estimation of the mean squared error or bias of post-model-selection estimators; cf. Remark 4.7. To fix ideas consider for the moment the linear regression model Y = V χ+Wψ + u (1) where V and W , respectively, represent n× k and n× l non-stochastic regressor matrices (k ≥ 1, l ≥ 1), and the n× 1 disturbance vector u is normally distributed with mean zero and variance-covariance matrix σ2In. We also assume for the moment that (V : W )′(V : W )/n converges to a non-singular matrix as the sample size n goes to infinity and that limn→∞ V ′W/n 6= 0 (for a discussion of the case where this limit is zero see Example 1 in Section 2.2.2). Now suppose that the vector χ represents the parameters of interest, while the parameter vector ψ and the associated regressors in W have been entered into the model only to avoid possible misspecification. Suppose further that the necessity to include ψ or some of its components is then checked on the basis of the data, i.e., a model selection procedure is used to determine which components of ψ are to be retained in the model, the inclusion of χ not being disputed. The selected model is then used to obtain the final (post-model-selection) estimator χ̃ for χ. We are now interested in the unconditional finite-sample distribution of χ̃ (appropriately scaled and centered). Denote this k-dimensional cumulative distribution function (cdf) by Gn,θ,σ(t). As indicated in the notation, this distribution function depends on the true parameters θ = (χ′, ψ′)′ and σ. For the sake of definiteness of discussion assume for the moment that the model selection procedure used here is the particular ‘general-to-specific’ procedure described at the beginning of Section 2; we comment on other model selection procedures, including Akaike’s AIC and thresholding procedures, below. As mentioned above, it is not difficult to construct a consistent estimator of Gn,θ,σ(t) for any t, i.e., an estimator Ĝn(t) satisfying Pn,θ,σ (∣∣∣Ĝn(t)−Gn,θ,σ(t) ∣∣∣ > δ n→∞−→ 0 (2) for each δ > 0 and each θ, σ; see Section 2.2.1. However, it follows from the results in Section 2.2.2 that any estimator satisfying (2), i.e., any consistent estimator of Gn,θ,σ(t), necessarily also satisfies ||θ||<R Pn,θ,σ (∣∣∣Ĝn(t)−Gn,θ,σ(t) ∣∣∣ > δ n→∞−→ 1 (3) for suitable positive constants R and δ that do not depend on the estimator. That is, while the probability in (2) converges to zero for every given θ by consistency, relation (3) shows that it does not do so uniformly in θ. It follows that Ĝn(t) can never be uniformly consistent (not even when restricting consideration to uniform consistency over all compact subsets of the parameter space). Hence, a large sample size does not guarantee a small estimation error with high probability when estimating the distribution function of a post- model-selection estimator. In this sense, reliably assessing the precision of post-model-selection estimators is an intrinsically hard problem. Apart from (3), we also provide minimax lower bounds for arbitrary (not necessarily consistent) estimators of the conditional distribution function Gn,θ,σ(t). For example, we provide results that imply that lim inf Ĝn(t) ||θ||<R Pn,θ,σ (∣∣∣Ĝn(t)−Gn,θ,σ(t) ∣∣∣ > δ > 0 (4) holds for suitable positive constants R and δ, where the infimum extends over all estimators of Gn,θ,σ(t). The results in Section 2.2.2 in fact show that the balls ||θ|| < R in (3) and (4) can be replaced by suitable balls (not necessarily centered at the origin) shrinking at the rate n−1/2. This shows that the non-uniformity phenomenon described in (3)-(4) is a local, rather than a global, phenomenon. In Section 2.2.2 we further show that the non-uniformity phenomenon expressed in (3) and (4) typically also arises when the parameter of interest is not χ, but some other linear transformation of θ = (χ′, ψ′)′. As discussed in Remark 4.3, the results also hold for randomized estimators of the unconditional distribution function Gn,θ,σ(t). Hence no resampling procedure whatsoever can alleviate the problem. This explains the anecdotal evidence in the literature that resampling methods are often unsuccessful in approximating distributional properties of post- model-selection estimators (e.g., Dijkstra and Veldkamp (1988), or Freedman, Navidi, and Peters (1988)). See also the discussion on resampling in Section 6. The results outlined above are presented in Section 2.2 for the particular ‘general-to-specific’ model selection procedure described at the beginning of Section 2. Analogous results for a large class of model selection procedures, including Akaike’s AIC and thresholding procedures, are then given in Section 3, based on the results in Section 2.2. In fact, the non-uniformity phenomenon expressed in (3)-(4) is not specific to the model selection procedures discussed in Sections 2 and 3 of the present paper, but will occur for most (if not all) model selection procedures, including consistent ones; cf. Sections 5 and 6 for more discussion. Section 5 also shows that the results are – as is to be expected – by no means limited to the linear regression model. We focus on the unconditional distributions of post-model-selection estimators in the present paper. One can, however, also envisage a situation where one is more interested in the conditional distribution given the outcome of the model selection procedure. In line with the literature on conditional inference (see, e.g., Robinson (1979) or Lehmann and Casella (1998, p. 421)), one may argue that, given the outcome of the model selection step, the relevant object of interest is the conditional rather than the unconditional distribution of the post-model-selection estimator. In this case similar results can be obtained and are reported in Leeb and Pötscher (2006b). We note that on a technical level the results in Leeb and Pötscher (2006b) and in the present paper require separate treatment. The plan of the paper is as follows: Post-model-selection estimators based on a ‘general-to-specific’ model selection procedure are the subject of Section 2. After introducing the basic framework and some notation, like the family of models Mp from which the ‘general-to-specific’ model selection procedure p̂ selects, as well as the post-model-selection estimator θ̃, the unconditional cdf Gn,θ,σ(t) of (a linear function of) the post-model-selection estimator θ̃ is discussed in Section 2.1. Consistent estimators of Gn,θ,σ(t) are given in Section 2.2.1. The main results of the paper are contained in Section 2.2.2 and Section 3: In Section 2.2.2 we provide a detailed analysis of the non-uniformity phenomenon encountered in (3)-(4). In Section 3 the ‘impossibility’ result from Section 2.2.2 is extended to a large class of model selection procedures including Akaike’s AIC and to selection from a non-nested collection of models. Some remarks are collected in Section 4, while Section 5 discusses extensions and the scope of the results of the paper. Conclusions are drawn in Section 6. All proofs as well as some auxiliary results are collected into appendices. Finally a word on notation: The Euclidean norm is denoted by ‖·‖, and λmax(E) denotes the largest eigenvalue of a symmetric matrix E. A prime denotes transposition of a matrix. For vectors x and y the relation x ≤ y (x < y, respectively) denotes xi ≤ yi (xi < yi, respectively) for all i. As usual, Φ denotes the standard normal distribution function. 2 Results for Post-Model-Selection Estimators Based on a ‘General-to-Specific’ Model Selection Procedure Consider the linear regression model Y = Xθ + u, (5) where X is a non-stochastic n× P matrix with rank(X) = P and u ∼ N(0, σ2In), σ2 > 0. Here n denotes the sample size and we assume n > P ≥ 1. In addition, we assume that Q = limn→∞X ′X/n exists and is non-singular. In this section we shall – similar as in Pötscher (1991) – consider model selection from the collection of nested models MO ⊆ MO+1 ⊆ · · · ⊆ MP , where O is specified by the user, and where for 0 ≤ p ≤ P the model Mp is given by (θ1, . . . , θP ) ′ ∈ RP : θp+1 = · · · = θP = 0 [In Section 3 below also general non-nested families of models will be considered.] Clearly, the model Mp corresponds to the situation where only the first p regressors in (5) are included. For the most parsimonious model under consideration, i.e., forMO, we assume that O satisfies 0 ≤ O < P ; if O > 0, this model contains as free parameters only those components of the parameter vector θ that are not subject to model selection. [In the notation used in connection with (1) we then have χ = (θ1, . . . , θO) ′ and ψ = (θO+1, . . . , θP ) Furthermore, note that M0 = {(0, . . . , 0)′} and that MP = RP . We call Mp the regression model of order p. The following notation will prove useful. For matrices B and C of the same row-dimension, the column- wise concatenation of B and C is denoted by (B : C). If D is an m× P matrix, let D[p] denote the m× p matrix consisting of the first p columns of D. Similarly, let D[¬p] denote the m× (P − p) matrix consisting of the last P −p columns of D. If x is a P × 1 vector, we write in abuse of notation x[p] and x[¬p] for (x′[p])′ and (x′[¬p])′, respectively. [We shall use the above notation also in the ‘boundary’ cases p = 0 and p = P . It will always be clear from the context how expressions containing symbols like D[0], D[¬P ], x[0], or x[¬P ] are to be interpreted.] As usual, the i-th component of a vector x is denoted by xi, and the entry in the i-th row and j-th column of a matrix B is denoted by Bi,j . The restricted least-squares estimator of θ under the restriction θ[¬p] = 0, i.e., under θp+1 = · · · = θP = 0, will be denoted by θ̃(p), 0 ≤ p ≤ P (in case p = P the restriction being void). Note that θ̃(p) is given by the P × 1 vector θ̃(p) =  (X [p] ′X [p]) X [p]′Y (0, . . . , 0)′ where the expressions θ̃(0) and θ̃(P ), respectively, are to be interpreted as the zero-vector in RP and as the unrestricted least-squares estimator of θ. Given a parameter vector θ in RP , the order of θ (relative to the nested sequence of models Mp) is defined as p0(θ) = min {p : 0 ≤ p ≤ P, θ ∈Mp} . Hence, if θ is the true parameter vector, a model Mp is a correct model if and only if p ≥ p0(θ). We stress that p0(θ) is a property of a single parameter, and hence needs to be distinguished from the notion of the order of the model Mp introduced earlier, which is a property of the set of parameters Mp. A model selection procedure is now nothing else than a data-driven (measurable) rule p̂ that selects a value from {O, . . . , P} and thus selects a model from the list of candidate modelsMO, . . . ,MP . In this section we shall consider as an important leading case a ‘general-to-specific’ model selection procedure based on a sequence of hypothesis tests. [Results for a larger class of model selection procedures, including Akaike’s AIC, are provided in Section 3.] This procedure is given as follows: The sequence of hypotheses H 0 : p0(θ) < p is tested against the alternatives H 1 : p0(θ) = p in decreasing order starting at p = P . If, for some p > O, H is the first hypothesis in the process that is rejected, we set p̂ = p. If no rejection occurs until even HO+10 is not rejected, we set p̂ = O. Each hypothesis in this sequence is tested by a kind of t-test where the error variance is always estimated from the overall model (but see the discussion following Theorem 3.1 in Section 3 below for other choices of estimators of the error variance). More formally, we have p̂ = max {p : |Tp| ≥ cp, 0 ≤ p ≤ P} , (6) with cO = 0 in order to ensure a well-defined p̂ in the range {O,O + 1, . . . , P}. For O < p ≤ P , the critical values cp satisfy 0 < cp < ∞ and are independent of sample size (but see also Remark 4.2). The test-statistics are given by nθ̃p(p) σ̂ξn,p (0 < p ≤ P ) with the convention that T0 = 0. Furthermore, ξn,p = X [p]′X [p] (0 < p ≤ P ) denotes the nonnegative square root of the p-th diagonal element of the matrix indicated, and σ̂2 is given by σ̂2 = (n− P )−1(Y −Xθ̃(P ))′(Y −Xθ̃(P )). Note that under the hypothesis H 0 the statistic Tp is t-distributed with n − P degrees of freedom for 0 < p ≤ P . It is also easy to see that the so-defined model selection procedure p̂ is conservative: The probability of selecting an incorrect model, i.e., the probability of the event {p̂ < p0(θ)}, converges to zero as the sample size increases. In contrast, the probability of the event {p̂ = p}, for p satisfying max{p0(θ),O} ≤ p ≤ P , converges to a positive limit; cf., for example, Proposition 5.4 and equation (5.6) in Leeb (2006). The post-model-selection estimator θ̃ can now be defined as follows: On the event p̂ = p, θ̃ is given by the restricted least-squares estimator θ̃(p), i.e., θ̃(p)1(p̂ = p), (7) where 1(·) denotes the indicator function of the event shown in the argument. 2.1 The Distribution of the Post-Model-Selection Estimator We now introduce the distribution function of a linear transformation of θ̃ and summarize some of its properties that will be needed in the subsequent development. To this end, let A be a non-stochastic k × P matrix of rank k, 1 ≤ k ≤ P , and consider the cdf Gn,θ,σ(t) = Pn,θ,σ nA(θ̃ − θ) ≤ t (t ∈ Rk). (8) Here Pn,θ,σ(·) denotes the probability measure corresponding to a sample of size n from (5). Depending on the choice of the matrix A, several important scenarios are covered by (8): The cdf of n(θ̃ − θ) is obtained by setting A equal to the P × P identity matrix IP . In case O > 0, the cdf of those components of n(θ̃− θ) which correspond to the parameter of interest χ in (1) can be studied by setting A to the O×P matrix (IO : 0) as we then have Aθ = (θ1, . . . , θO)′ = χ. Finally, if A 6= 0 is an 1×P vector, we obtain the distribution of a linear predictor based on the post-model-selection estimator. See the examples at the end of Section 2.2.2 for more discussion. The cdf Gn,θ,σ and its properties have been analyzed in detail in Leeb and Pötscher (2003) and Leeb (2006). To be able to access these results we need some further notation. Note that on the event p̂ = p the expression A(θ̃ − θ) equals A(θ̃(p) − θ) in view of (7). The expected value of the restricted least-squares estimator θ̃(p) will be denoted by ηn(p) and is given by the P × 1 vector ηn(p) =  θ[p] + (X [p] ′X [p])−1X [p]′X [¬p]θ[¬p] (0, . . . , 0)′  (9) with the conventions that ηn(0) = (0, . . . , 0) ′ ∈ RP and that ηn(P ) = θ. Furthermore, let Φn,p denote the cdf of nA(θ̃(p) − ηn(p)), i.e., the cdf of nA times the restricted least-squares estimator based on model Mp centered at its mean. Hence, Φn,p is the cdf of a k-variate Gaussian random vector with mean zero and variance-covariance matrix σ2A[p](X [p]′X [p]/n)−1A[p]′ in case p > 0, and it is the cdf of point-mass at zero in Rk in case p = 0. If p > 0 and if the matrix A[p] has full row rank k, then Φn,p has a density with respect to Lebesgue measure, and we shall denote this density by φn,p. We note that ηn(p) depends on θ and that Φn,p depends on σ (in case p > 0), although these dependencies are not shown explicitly in the notation. For p > 0 we introduce bn,p = C n (A[p](X [p] ′X [p]/n)−1A[p]′)−, (10) ζ2n,p = ξ n,p − C(p) n (A[p](X [p] ′X [p]/n)−1A[p]′)−C(p)n , (11) with ζn,p ≥ 0. Here C n = A[p](X [p] ′X [p]/n)−1ep, where ep denotes the p-th standard basis vector in R and B− denotes a generalized inverse of a matrix B. [Observe that ζ2n,p is invariant under the choice of the generalized inverse. The same is not necessarily true for bn,p, but is true for bn,pz for all z in the column- space of A[p]. Also note that (12) below depends on bn,p only through bn,pz with z in the column-space of A[p].] We observe that the vector of covariances between Aθ̃(p) and θ̃p(p) is precisely given by σ 2n−1C (and hence does not depend on θ). Furthermore, observe that Aθ̃(p) and θ̃p(p) are uncorrelated if and only if ζ2n,p = ξ n,p if and only if bn,pz = 0 for all z in the column-space of A[p]; cf. Lemma A.2 in Leeb (2005). Finally, for a univariate Gaussian random variable N with zero mean and variance s2, s ≥ 0, we write ∆s(a, b) for P(|N− a| < b), a ∈ R∪{−∞,∞}, b ∈ R. Note that ∆s(·, ·) is symmetric around zero in its first argument, and that ∆s(−∞, b) = ∆s(∞, b) = 0 holds. In case s = 0, N is to be interpreted as being equal to zero, hence a 7→ ∆0(a, b) reduces to the indicator function of the interval (−b, b). We are now in a position to present the explicit formula for Gn,θ,σ(t) derived in Leeb (2006): Gn,θ,σ(t) = Φn,O(t− nA(ηn(O)− θ)) q=O+1 ∆σξn,q ( nηn,q(q), scqσξn,q)h(s)ds p=O+1 nA(ηn(p)−θ) [ ∫ ∞ (1−∆σζn,p( nηn,p(p) + bn,pz, scpσξn,p)) (12) q=p+1 ∆σξn,q( nηn,q(q), scqσξn,q)h(s)ds Φn,p(dz). In the above display, Φn,p(dz) denotes integration with respect to the measure induced by the normal cdf Φn,p on R k and h denotes the density of σ̂/σ, i.e., h is the density of (n−P )−1/2 times the square-root of a chi-square distributed random variable with n−P degrees of freedom. The finite-sample distribution of the post-model-selection estimator given in (12) is in general not normal, e.g., it can be bimodal; see Figure 2 in Leeb and Pötscher (2005a) or Figure 1 in Leeb (2006). [An exception where (12) is normal is the somewhat trivial case where C n = 0, i.e., where Aθ̃(p) and θ̃p(p) are uncorrelated, for p = O + 1, . . . , P ; see Leeb (2006, Section 3.3) for more discussion.] We note for later use that Gn,θ,σ(t) = p=O Gn,θ,σ(t|p)πn,θ,σ(p) where Gn,θ,σ(t|p) represents the cdf of nA(θ̃ − θ) conditional on the event {p̂ = p} and where πn,θ,σ(p) is the probability of this event under Pn,θ,σ. Note that πn,θ,σ(p) is always positive for O ≤ p ≤ P ; cf. Leeb (2006), Section 3.2. To describe the large-sample limit of Gn,θ,σ, some further notation is necessary. For p satisfying 0 < p ≤ P , partition the matrix Q = limn→∞X ′X/n as  Q[p : p] Q[p : ¬p] Q[¬p : p] Q[¬p : ¬p] where Q[p : p] is a p× p matrix. Let Φ∞,p be the cdf of a k-variate Gaussian random vector with mean zero and variance-covariance matrix σ2A[p]Q[p : p]−1A[p]′, 0 < p ≤ P , and let Φ∞,0 denote the cdf of point-mass at zero in Rk. Note that Φ∞,p has a Lebesgue density if p > 0 and the matrix A[p] has full row rank k; in this case, we denote the Lebesgue density of Φ∞,p by φ∞,p. Finally, for p = 1, . . . , P , define ξ2∞,p = (Q[p : p] −1)p,p, ζ2∞,p = ξ ∞,p − C(p)′∞ (A[p]Q[p : p]−1A[p]′)−C(p)∞ , (13) b∞,p = C ∞ (A[p]Q[p : p] −1A[p]′)−, where C ∞ = A[p]Q[p : p] −1ep, with ep denoting the p-th standard basis vector in R p; furthermore, take ζ∞,p and ξ∞,p as the nonnegative square roots of ζ ∞,p and ξ ∞,p, respectively. As the notation suggests, Φ∞,p is the large-sample limit of Φn,p, and C ∞ , ξ ∞,p, and ζ ∞,p are the limits of C n , ξ n,p, and ζ n,p, respectively; moreover, bn,pz converges to b∞,pz for each z in the column-space of A[p]. See Lemma A.2 in Leeb (2005). The next result describes the large-sample limit of the cdf under local alternatives to θ and is taken from Leeb (2006, Corollary 5.6). Recall that the total variation distance between two cdfs G and G∗ on Rk is defined as ||G−G∗||TV = supE |G(E)−G∗(E)|, where the supremum is taken over all Borel sets E. Clearly, the relation |G(t)−G∗(t)| ≤ ||G−G∗||TV holds for all t ∈ Rk. Thus, if G and G∗ are close with respect to the total variation distance, then G(t) is close to G∗(t), uniformly in t. Proposition 2.1 Suppose θ ∈ RP and γ ∈ RP and let σ(n) be a sequence of positive real numbers which converges to a (finite) limit σ > 0 as n → ∞. Then the cdf Gn,θ+γ/√n,σ(n) converges to a limit G∞,θ,σ,γ in total variation, i.e., ∣∣∣∣Gn,θ+γ/√n,σ(n) −G∞,θ,σ,γ n→∞−→ 0. (14) The large-sample limit cdf G∞,θ,σ,γ(t) is given by Φ∞,p∗(t− β (p∗)) q=p∗+1 ∆σξ∞,q (νq, cqσξ∞,q) p=p∗+1 z≤t−β(p) (1−∆σζ∞,p(νp + b∞,pz, cpσξ∞,p))Φ∞,p(dz) q=p+1 ∆σξ∞,q (νq, cqσξ∞,q) (15) where p∗ = max{p0(θ),O}. Here for 0 ≤ p ≤ P  Q[p : p] −1Q[p : ¬p]γ[¬p] −γ[¬p] with the convention that β(p) = −Aγ if p = 0 and that β(p) = (0, . . . , 0)′ if p = P . Furthermore, we have set νp = γp + (Q[p : p] −1Q[p : ¬p]γ[¬p])p for p > 0. [Note that β(p) = limn→∞ nA(ηn(p) − θ − γ/ for p ≥ p0(θ), and that νp = limn→∞ nηn,p(p) for p > p0(θ). Here ηn(p) is defined as in (9), but with θ + γ/ n replacing θ.] If p∗ > 0 and if the matrix A[p∗] has full row rank k, then the Lebesgue density φ∞,p of Φ∞,p exists for all p ≥ p∗ and hence the density of (15) exists and is given by φ∞,p∗(t− β (p∗)) q=p∗+1 ∆σξ∞,q (νq, cqσξ∞,q) p=p∗+1 (1−∆σζ∞,p(νp + b∞,p(t− β (p)), cpσξ∞,p))φ∞,p(t− β q=p+1 ∆σξ∞,q(νq, cqσξ∞,q). Like the finite-sample distribution, the limiting distribution of the post-model-selection estimator given in (15) is in general not normal. An exception is the case where C ∞ = 0 for p > p∗ in which case G∞,θ,σ,γ reduces to Φ∞,P ; see Remark A.6 in Appendix A. If γ = 0, we write G∞,θ,σ(t) as shorthand for G∞,θ,σ,0(t) in the following. 2.2 Estimators of the Finite-Sample Distribution For the purpose of inference after model selection the finite-sample distribution of the post-model-selection- estimator is an object of particular interest. As we have seen, it depends on unknown parameters in a complicated manner, and hence one will have to be satisfied with estimators of this cdf. As we shall see, it is not difficult to construct consistent estimators of Gn,θ,σ(t). However, despite this consistency result, we shall find in Section 2.2.2 that any estimator of Gn,θ,σ(t) typically performs unsatisfactory, in that the estimation error can not become small uniformly over (subsets of) the parameter space even as sample size goes to infinity. In particular, no uniformly consistent estimators exist, not even locally. 2.2.1 Consistent Estimators We construct a consistent estimator of Gn,θ,σ(t) by commencing from the asymptotic distribution. Spe- cializing to the case γ = 0 and σ(n) = σ in Proposition 2.1, the large-sample limit of Gn,θ,σ(t) is given G∞,θ,σ(t) = Φ∞,p∗(t) q=p∗+1 ∆σξ∞,q(0, cqσξ∞,q) p=p∗+1 (1 −∆σζ∞,p(b∞,pz, cpσξ∞,p))Φ∞,p(dz) q=p+1 ∆σξ∞,q(0, cqσξ∞,q) (16) with p∗ = max{p0(θ),O}. Note that G∞,θ,σ(t) depends on θ only through p∗. Let Φ̂n,p denote the cdf of a k- variate Gaussian random vector with mean zero and variance-covariance matrix σ̂2A[p](X [p]′X [p]/n)−1A[p]′, 0 < p ≤ P ; we also adopt the convention that Φ̂n,0 denotes the cdf of point-mass at zero in Rk. [We use the same convention for Φ̂n,p in case σ̂ = 0, which is a probability zero event.] An estimator Ǧn(t) of Gn,θ,σ(t) is now defined as follows: We first employ an auxiliary procedure p̄ that consistently estimates p0(θ) (e.g., p̄ could be obtained from BIC or from a ‘general-to-specific’ hypothesis testing procedure employing critical values that go to infinity but are o(n1/2) as n → ∞). The estimator Ǧn(t) is now given by the expression in (16) but with p∗, σ, b∞,p, ζ∞,p, ξ∞,p, and Φ∞,p replaced by max{p̄,O}, σ̂, bn,p, ζn,p, ξn,p, and Φ̂n,p, respectively. A little reflection shows that Ǧn is again a cdf. We have the following consistency results. Proposition 2.2 The estimator Ǧn is consistent (in the total variation distance) for Gn,θ,σ and G∞,θ,σ. That is, for every δ > 0 Pn,θ,σ (∣∣∣∣Ǧn(·)−Gn,θ,σ(·) ) n→∞−→ 0, (17) Pn,θ,σ (∣∣∣∣Ǧn(·)−G∞,θ,σ(·) ) n→∞−→ 0 (18) for all θ ∈ RP and all σ > 0. While the estimator constructed above on the basis of the formula for G∞,θ,σ is consistent, it can be expected to perform poorly in finite samples since convergence of Gn,θ,σ to G∞,θ,σ is typically not uniform in θ (cf. Appendix B), and since in case the true θ is ‘close’ to Mp0(θ)−1 the auxiliary decision procedure p̄ (although being consistent for p0(θ)) will then have difficulties making the correct decision in finite samples. In the next section we show that this poor performance is not particular to the estimator Ǧn constructed above, but is a genuine feature of the estimation problem under consideration. 2.2.2 Performance Limits and Impossibility Results We now provide lower bounds for the performance of estimators of the cdf Gn,θ,σ(t) of the post-model- selection estimator Aθ̃; that is, we give lower bounds on the worst-case probability that the estimation error exceeds a certain threshold. These lower bounds are large, being 1 or 1/2, depending on the situation considered; furthermore, they remain lower bounds even if one restricts attention only to certain subsets of the parameter space that shrink at the rate n−1/2. In this sense the ‘impossibility’ results are of a local nature. In particular, the lower bounds imply that no uniformly consistent estimator of the cdf Gn,θ,σ(t) exists, not even locally. In the following, the asymptotic ‘correlation’ between Aθ̃(p) and θ̃p(p) as measured by C limn→∞ C n will play an important rôle. [Recall that θ̃(p) denotes the least-squares estimator of θ based on model Mp and that Aθ is the parameter vector of interest. Furthermore, the vector of covariances be- tween Aθ̃(p) and θ̃p(p) is given by σ 2n−1C n with C n = A[p](X [p] ′X [p]/n)−1ep.] Note that C ∞ equals A[p]Q[p : p]−1ep, and hence does not depend on the unknown parameters θ or σ. In the important special case discussed in the Introduction, cf. (1), the matrix A equals the O×P matrix (IO : 0), and the condition ∞ 6= 0 reduces to the condition that the regressor corresponding to the p-th column of (V :W ) is asymp- totically correlated with at least one of the regressors corresponding to the columns of V . See Example 1 below for more discussion. In the result to follow we shall consider performance limits for estimators of Gn,θ,σ(t) at a fixed value of the argument t. An estimator of Gn,θ,σ(t) is now nothing else than a real-valued random variable Γn = Γn(Y,X). For mnemonic reasons we shall, however, use the symbol Ĝn(t) instead of Γn to denote an arbitrary estimator of Gn,θ,σ(t). This notation should not be taken as implying that the estimator is obtained by evaluating an estimated cdf at the argument t, or that it is a priori constrained to lie between zero and one. We shall use this notational convention mutatis mutandis also in subsequent sections. Regarding the non-uniformity phenomenon, we then have a dichotomy which is described in the following two results. Theorem 2.3 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C ∞ 6= 0, for some q sat- isfying O < q ≤ P , and let q∗ denote the largest q with this property. Then the following holds for every θ ∈ Mq∗−1, every σ, 0 < σ < ∞, and every t ∈ Rk: There exist δ0 > 0 and ρ0, 0 < ρ0 < ∞, such that any estimator Ĝn(t) of Gn,θ,σ(t) satisfying Pn,θ,σ (∣∣∣Ĝn(t)−Gn,θ,σ(t) ∣∣∣ > δ n→∞−→ 0 (19) for each δ > 0 (in particular, every estimator that is consistent) also satisfies ϑ∈Mq∗ ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣Ĝn(t)−Gn,ϑ,σ(t) ∣∣∣ > δ0 n→∞−→ 1. (20) The constants δ0 and ρ0 may be chosen in such a way that they depend only on t, Q, A, σ, and the critical values cp for O < p ≤ P . Moreover, lim inf Ĝn(t) ϑ∈Mq∗ ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣Ĝn(t)−Gn,ϑ,σ(t) ∣∣∣ > δ0 > 0 (21) lim inf Ĝn(t) ϑ∈Mq∗ ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣Ĝn(t)−Gn,ϑ,σ(t) ∣∣∣ > δ , (22) where the infima in (21) and (22) extend over all estimators Ĝn(t) of Gn,θ,σ(t). Remark 2.4 Assume that the conditions of the preceding theorem are satisfied. Suppose further that p⊙, O ≤ p⊙ < q∗, is such that either p⊙ > 0 and some row of A[p⊙] equals zero, or such that p⊙ = 0. Then there exist δ0 > 0 and 0 < ρ0 <∞ such that the left-hand side of (21) is not less than 1/2 for each θ ∈Mp⊙ . Theorem 2.3 a fortiori implies a corresponding ‘impossibility’ result for estimation of the functionGn,θ,σ(·) when the estimation error is measured in the total variation distance or the sup-norm; cf. also Section 5. It remains to consider the – quite exceptional – case where the assumption of Theorem 2.3 is not satisfied, i.e., where C ∞ = 0, for all q in the range O < q ≤ P . Under this ‘uncorrelatedness’ condition it is indeed possible to construct an estimator of Gn,θ,σ which is uniformly consistent: It is not difficult to see that the asymptotic distribution of Gn,θ,σ reduces to Φ∞,P under this ‘uncorrelatedness’ condition. Furthermore, the second half of Proposition B.1 in Appendix B shows that then the convergence of Gn,θ,σ to its large-sample limit is uniform w.r.t. θ, suggesting Φ̂n,P , an estimated version of Φ∞,P , as an estimator for Gn,θ,σ. Proposition 2.5 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically uncorrelated, i.e., C ∞ = 0, for all q satisfying O < q ≤ P . Then σ∗≤σ≤σ∗ Pn,θ,σ ∣∣∣Φ̂n,P −Gn,θ,σ n→∞−→ 0 (23) holds for each δ > 0, and for any constants σ∗ and σ ∗ satisfying 0 < σ∗ ≤ σ∗ <∞. Inspection of the proof of Proposition 2.5 shows that (23) continues to hold if the estimator Φ̂n,P is replaced by any of the estimators Φ̂n,p for O ≤ p ≤ P . We also note that in case O = 0 the assumption of Proposition 2.5 is never satisfied in view of Proposition 4.4 in Leeb and Pötscher (2006b), and hence Theorem 2.3 always applies in that case. Another consequence of Proposition 4.4 in Leeb and Pötscher (2006b) is that – under the ‘uncorrelatedness’ assumption of Proposition 2.5 – the restricted least squares estimators Aθ̃(q) for q ≥ O perform asymptotically as well as the unrestricted estimator Aθ̃(P ); this clearly shows that the case covered by Proposition 2.5 is highly exceptional. In summary we see that it is typically impossible to construct an estimator of Gn,θ,σ(t) which performs reasonably well even asymptotically. Whenever Theorem 2.3 applies, any estimator of Gn,θ,σ(t) suffers from a non-uniformity defect which is caused by parameters belonging to shrinking ‘tubes’ surrounding Mq∗−1. For the sake of completeness, we remark that outside a ‘tube’ of fixed positive radius that surrounds Mq∗−1 the non-uniformity need not be present: Let q ∗ be as in Theorem 2.3 and define the set U as U = {θ ∈ RP : |θq∗ | ≥ r} for some fixed r > 0. Then Φ̂n,P (t) is an estimator of Gn,θ,σ(t) that is uniformly consistent over θ ∈ U ; more generally, it can be shown that then the relation (23) holds if the supremum over θ on the left-hand side is restricted to θ ∈ U . We conclude this section by illustrating the above results with some important examples. Example 1: (The distribution of χ̃) Consider the model given in (1) with χ representing the parameter of interest. Using the general notation of Section 2, this corresponds to the case Aθ = (θ1, . . . , θO) ′ = χ with A representing the O × P matrix (IO : 0). Here k = O > 0. The cdf Gn,θ,σ then represents the cdf n (χ̃− χ). Assume first that limn→∞ V ′W/n 6= 0. Then C(q)∞ 6= 0 holds for some q > O. Consequently, the ‘impossibility’ results for the estimation of Gn,θ,σ given in Theorem 2.3 always apply. Next assume that limn→∞ V ′W/n = 0. Then C ∞ = 0 for every q > O. In this case Proposition 2.5 applies and a uniformly consistent estimator of Gn,θ,σ indeed exists. Summarizing we note that any estimator of Gn,θ,σ suffers from the non-uniformity phenomenon except in the special case where the columns of V and W are asymptotically orthogonal in the sense that limn→∞ V ′W/n = 0. But this is precisely the situation where inclusion or exclusion of the regressors in W has no effect on the distribution of the estimator χ̃ asymptotically; hence it is not surprising that also the model selection procedure does not have an effect on the estimation of the cdf of the post-model-selection estimator χ̃. This observation may tempt one to enforce orthogonality between the columns of V and W by either replacing the columns of V by their residuals from the projection on the column space ofW or vice versa. However, this is not helpful for the following reasons: In the first case one then in fact avoids model selection as all the restricted least-squares estimators for χ under consideration (and hence also the post-model selection estimator χ̃) in the reparameterized model coincide with the unrestricted least-squares estimator. In the second case the coefficients of the columns of V in the reparameterized model no longer coincide with the parameter of interest χ (and again are estimated by one and the same estimator regardless of inclusion/exclusion of columns of the transformed W -matrix). Example 2: (The distribution of θ̃) For A equal to IP , the cdf Gn,θ,σ is the cdf of n(θ̃ − θ). Here, Aθ̃(q) reduces to θ̃(q), and hence Aθ̃(q) and θ̃q(q) are perfectly correlated for every q > O. Consequently, the ‘impossibility’ result for estimation of Gn,θ,σ given in Theorem 2.3 applies. [In fact, the slightly stronger result mentioned in Remark 2.4 always applies here.] We therefore see that estimation of the distribution of the post-model-selection estimator of the entire parameter vector is always plagued by the non-uniformity phenomenon. Example 3: (The distribution of a linear predictor) Suppose A 6= 0 is a 1×P vector and one is interested in estimating the cdf Gn,θ,σ of the linear predictor Aθ̃. Then Theorem 2.3 and the discussion following Proposition 2.5 show that the non-uniformity phenomenon always arises in this estimation problem in case O = 0. In case O > 0, the non-uniformity problem is generically also present, except in the degenerate case where C ∞ = 0, for all q satisfying O < q ≤ P (in which case Proposition 4.4 in Leeb and Pötscher (2006b) shows that the least-squares predictors from all models Mp, O ≤ p ≤ P , perform asymptotically equally well). 3 Extensions to Other Model Selection Procedures Including AIC In this section we show that the ‘impossibility’ result obtained in the previous section for a ‘general-to- specific’ model selection procedure carries over to a large class of model selection procedures, including Akaike’s widely used AIC. Again consider the linear regression model (5) with the same assumptions on the regressors and the errors as in Section 2. Let {0, 1}P denote the set of all 0-1 sequences of length P . For each r ∈{0, 1}P let Mr denote the set {θ ∈ RP : θi(1 − ri) = 0 for 1 ≤ i ≤ P} where ri represents the i-th component of r. I.e., Mr describes a linear submodel with those parameters θi restricted to zero for which ri = 0. Now let R be a user-supplied subset of {0, 1}P . We consider model selection procedures that select from the set R, or equivalently from the set of models {Mr : r ∈ R}. Note that there is now no assumption that the candidate models are nested (for example, if R = {0, 1}P all possible submodels are candidates for selection). Also cases where the inclusion of a subset of regressors is undisputed on a priori grounds are obviously covered by this framework upon suitable choice of R. We shall assume throughout this section that R contains rfull = (1, . . . , 1) and also at least one element r∗ satisfying |r∗| = P − 1, where |r∗| represents the number of non-zero coordinates of r∗. Let r̂ be an arbitrary model selection procedure, i.e., r̂ = r̂(Y,X) is a random variable taking its values in R. We furthermore assume throughout this section that the model selection procedure r̂ satisfies the following mild condition: For every r∗ ∈ R with |r∗| = P − 1 there exists a positive finite constant c (possibly depending on r∗) such that for every θ ∈Mr∗ which has exactly P − 1 non-zero coordinates Pn,θ,σ ({r̂ = rfull}N{|Tr∗ | ≥ c}) = lim Pn,θ,σ ({r̂ = r∗}N{|Tr∗ | < c}) = 0 (24) holds for every 0 < σ < ∞. Here N denotes the symmetric difference operator and Tr∗ represents the usual t-statistic for testing the hypothesis θi(r∗) = 0 in the full model, where i(r∗) denotes the index of the unique coordinate of r∗ that equals zero. The above condition is quite natural for the following reason: For θ ∈ Mr∗ with exactly P − 1 non-zero coordinates, every reasonable model selection procedure will – with probability approaching unity – decide only betweenMr∗ andMrfull ; it is then quite natural that this decision will be based (at least asymptotically) on the likelihood ratio between these two models, which in turn boils down to the t-statistic. As will be shown below, condition (24) holds in particular for AIC-like procedures. Let A be a non-stochastic k×P matrix of full row rank k, 1 ≤ k ≤ P , as in Section 2.1. We then consider the cdf Kn,θ,σ(t) = Pn,θ,σ nA(θ̄ − θ) ≤ t (t ∈ Rk) (25) of a linear transformation of the post-model-selection estimator θ̄ obtained from the model selection procedure r̂, i.e., θ̃(r)1(r̂ = r) where the P × 1 vector θ̃(r) represents the restricted least-squares estimator obtained from model Mr, with the convention that θ̃(r) = 0 ∈ RP in case r = (0, . . . , 0). We then obtain the following result for estimation of Kn,θ,σ(t) at a fixed value of the argument t which parallels the corresponding ‘impossibility’ result in Theorem 2.3. Theorem 3.1 Let r∗ ∈ R satisfy |r∗| = P − 1, and let i(r∗) denote the index of the unique coordinate of r∗ that equals zero; furthermore, let c be the constant in (24) corresponding to r∗. Suppose that Aθ̃(rfull) and θ̃i(r∗)(rfull) are asymptotically correlated, i.e., AQ i(r∗) 6= 0, where e i(r∗) denotes the i(r∗)-th standard basis vector in RP . Then for every θ ∈ Mr∗ which has exactly P − 1 non-zero coordinates, for every σ, 0 < σ <∞, and for every t ∈ Rk the following holds: There exist δ0 > 0 and ρ0, 0 < ρ0 <∞, such that any estimator K̂n(t) of Kn,θ,σ(t) satisfying Pn,θ,σ (∣∣∣K̂n(t)−Kn,θ,σ(t) ∣∣∣ > δ n→∞−→ 0 (26) for each δ > 0 (in particular, every estimator that is consistent) also satisfies ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣K̂n(t)−Kn,ϑ,σ(t) ∣∣∣ > δ0 n→∞−→ 1 . (27) The constants δ0 and ρ0 may be chosen in such a way that they depend only on t, Q,A, σ, and c. Moreover, lim inf K̂n(t) ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣K̂n(t)−Kn,ϑ,σ(t) ∣∣∣ > δ0 > 0 (28) lim inf K̂n(t) ||ϑ−θ||<ρ0/ Pn,ϑ,σ (∣∣∣K̂n(t)−Kn,ϑ,σ(t) ∣∣∣ > δ ≥ 1/2 (29) hold, where the infima in (28) and (29) extend over all estimators K̂n(t) of Kn,θ,σ(t). The basic condition (24) on the model selection procedure employed in the above result will certainly hold for any hypothesis testing procedure that (i) asymptotically selects only correct models, (ii) employs a likelihood ratio test (or an asymptotically equivalent test) for testing Mrfull versus smaller models (at least versus the models Mr∗ with r∗ as in condition (24)), and (iii) uses a critical value for the likelihood ratio test that converges to a finite positive constant. In particular, this applies to usual thresholding procedures as well as to a variant of the ‘general-to-specific’ procedure discussed in Section 2 where the error variance in the construction of the test statistic for hypothesis H 0 is estimated from the fitted model Mp rather than from the overall model. We next verify condition (24) for AIC-like procedures. Let RSS(r) denote the residual sum of squares from the regression employing model Mr and set IC(r) = log (RSS(r)) + |r|Υn/n (30) where Υn ≥ 0 denotes a sequence of real numbers satisfying limn→∞ Υn = Υ and Υ is a positive real number. Of course, IC(r) = AIC(r) if Υn = 2. The model selection procedure r̂IC is then defined as a minimizer (more precisely, as a measurable selection from the set of minimizers) of IC(r) over R. It is well-known that the probability that r̂IC selects an incorrect model converges to zero. Hence, elementary calculations show that condition (24) is satisfied for c = Υ1/2. The analysis of post-model-selection estimators based on AIC-like model selection procedures given in this section proceeded by bringing this case under the umbrella of the results obtained in Section 2. Verification of condition (24) is the key that enables this approach. A complete analysis of post-model-selection estimators based on AIC-like model selection procedures, similar to the analysis in Section 2 for the ‘general-to-specific’ model selection procedure, is certainly possible but requires a direct and detailed analysis of the distribution of this post-model-selection estimator. [Even the mild condition that R contains rfull and also at least one element r∗ satisfying |r∗| = P − 1 can then be relaxed in such an analysis.] We furthermore note that in the special case where R = {rfull, r∗} and an AIC-like model selection procedure as in (30) is used, the results in the above theorem in fact hold for all θ ∈Mr∗ . 4 Remarks and Extensions Remark 4.1 Although not emphasized in the notation, all results in the paper also hold if the elements of the design matrix X depend on sample size. Furthermore, all results are expressed solely in terms of the distributions Pn,θ,σ(·) of Y , and hence they also apply if the elements of Y depend on sample size, including the case where the random vectors Y are defined on different probability spaces for different sample sizes. Remark 4.2 The model selection procedure considered in Section 2 is based on a sequence of tests which use critical values cp that do not depend on sample size and satisfy 0 < cp < ∞ for O < p ≤ P . If these critical values are allowed to depend on sample size such that they now satisfy cn,p → c∞,p as n → ∞ with 0 < c∞,p < ∞ for O < p ≤ P , the results in Leeb and Pötscher (2003) as well as in Leeb (2005, 2006) continue to hold; see Remark 6.2(i) in Leeb and Pötscher (2003) and Remark 6.1(ii) in Leeb (2005). As a consequence, the results in the present paper can also be extended to this case quite easily. Remark 4.3 The ‘impossibility’ results given in Theorems 2.3 and 3.1 (as well as the variants thereof discussed in the subsequent Remarks 4.4-4.7) also hold for the class of all randomized estimators (with P ∗n,θ,σ replacing Pn,θ,σ in those results, where P n,θ,σ denotes the distribution of the randomized sample). This follows immediately from Lemma 3.6 and the attending discussion in Leeb and Pötscher (2006a). Remark 4.4 a. Let ψn,θ,σ denote the expectation of θ̃ under Pn,θ,σ, and consider the cdf Hn,θ,σ(t) = Pn,θ,σ( nA(θ̃−ψn,θ,σ) ≤ t). Results for the cdf Hn,θ,σ quite similar to the results for Gn,θ,σ obtained in the present paper can be established. A similar remark applies to the post-model-selection estimator θ̄ considered in Section 3. b. In Leeb (2006) also the cdf G∗n,θ,σ is analyzed, which correspond to a (typically infeasible) model selection procedure that makes use of knowledge of σ. Results completely analogous to the ones in the present paper can also be obtained for this cdf. Remark 4.5 Results similar to the ones in Section 2.2.2 can also be obtained for estimation of the asymp- totic cdf G∞,θ,σ(t) (or of the asymptotic cdfs corresponding to the variants discussed in the previous remark). Since these results are of limited interest, we omit them. In particular, note that an ‘impossibility’ result for estimation of G∞,θ,σ(t) per se does not imply a corresponding ‘impossibility’ result for estimation of Gn,θ,σ(t), since Gn,θ,σ(t) does in general not converge uniformly to G∞,θ,σ(t) over the relevant subsets in the parameter space; cf. Appendix B. [An analogous remark applies to the model selection procedures considered in Section 3.] Remark 4.6 Let πn,θ,σ(p) denote the model selection probability Pn,θ,σ(p̂ = p), O ≤ p ≤ P corresponding to the model selection procedure discussed in Section 2. The finite-sample properties and the large-sample limit behavior of these quantities are thoroughly analyzed in Leeb (2006); cf. also Leeb and Pötscher (2003). For these model selection probabilities the following results can be established which we discuss here only briefly: a. The model selection probabilities πn,θ,σ(p) converge to well-defined large-sample limits which we denote by π∞,θ,σ(p). Similar as in Proposition B.1 in Appendix B, the convergence of πn,θ,σ(p) to π∞,θ,σ(p) is non-uniform w.r.t. θ. [For the case O = 0, this phenomenon is described in Corollary 5.6 of Leeb and Pötscher (2003).] b. The model selection probabilities πn,θ,σ(p) can be estimated consistently. However, uniformly consis- tent estimation is again not possible. A similar remark applies to the large-sample limits π∞,θ,σ(p). Remark 4.7 ‘Impossibility’ results similar to the ones given in Theorems 2.3 and 3.1 for the cdf can also be obtained for other characteristics of the distribution of a linear function of a post-model-selection estimator like the mean-squared error or the bias of nAθ̃. 5 On the Scope of the Impossibility Results The non-uniformity phenomenon described, e.g., in (20) of Theorem 2.3 is caused by a mechanism that can informally be described as follows. Under the assumptions of that theorem, one can find an appropriate θ and an appropriate sequence ϑn = θ + γ/ n exhibiting two crucial properties: a. The probability measures Pn,ϑn,σ corresponding to ϑn are ‘close’ to the measures Pn,θ,σ corresponding to θ, in the sense of contiguity. This entails that an estimator, that converges to some limit in probability under Pn,θ,σ, converges to the same limit also under Pn,ϑn,σ. b. For given t, the estimands Gn,ϑn,σ(t) corresponding to ϑn are ‘far away’ from the estimands Gn,θ,σ(t) corresponding to θ, in the sense that Gn,ϑn,σ(t) and Gn,θ,σ(t) converge to different limits, i.e., G∞,θ,σ,0(t) is different from G∞,θ,σ,γ(t). In view of Property a, an estimator Ĝn(t) satisfying Ĝn(t)−Gn,θ,σ(t) → 0 in probability under Pn,θ,σ, also satisfies Ĝn(t)−Gn,θ,σ(t) → 0 in probability under Pn,ϑn,σ. In view of Property b, such an estimator Ĝn(t) is hence ‘far away’ from the estimand Gn,ϑn,σ(t) with high probability under Pn,ϑn,σ. In other words, an estimator that is close to Gn,θ,σ(t) under Pn,θ,σ must be far away from Gn,ϑn,σ(t) under Pn,ϑn,σ. Formalized and refined, this argument leads to (20) and, as a consequence, to the non-existence of uniformly consistent estimators for Gn,θ,σ(t). [There are a number of technical details in this formalization process that need careful attention in order to obtain the results in their full strength as given in Sections 2 and 3.] The above informal argument that derives (20) from Properties a and b can be refined and formalized in a much more general and abstract framework, see Section 3 of Leeb and Pötscher (2006a) and the references therein. That paper also provides a general framework for deriving results like (21) and (22) of Theorem 2.3. The mechanism leading to such lower bounds is similar to the one outlined above, where for some of the results the concept of contiguity of the probability measures involved has to be replaced by closeness of these measures in total variation distance. We use the results in Section 3 of Leeb and Pötscher (2006a) to formally convert Properties a and b into the ‘impossibility’ results of the present paper; cf. Appendix C. Verifying the aforementioned Property a in the context of the present paper is straightforward because we consider a Gaussian linear model. What is technically more challenging and requires some work is the verification of Property b; this is done in Appendix A inter alia and rests on results of Leeb (2002, 2005, 2006). Two important observations on Properties a and b are in order: First, Property a is typically satisfied in general parametric models under standard regularity conditions; e.g., it is satisfied whenever the model is locally asymptotically normal. Second, Property b relies on limiting properties only and not on the finite- sample structure of the underlying statistical model. Now, the limit distributions of post-model-selection estimators in sufficiently regular parametric or semi-parametric models are typically the same as the limiting distributions of the corresponding post-model-selection estimators in a Gaussian linear model (see, e.g., Sen (1979), Pötscher (1991), Nickl (2003), or Hjort and Claeskens (2003)). Hence, establishing Property b for the Gaussian linear model then typically establishes the same result for a large class of general parametric or semi-parametric models.1 For example, Property b can be verified for a large class of pre-test estimators in sufficiently regular parametric models by arguing as in Appendix A and using the results of Nickl (2003) to reduce to the Gaussian linear case. Hence, the impossibility result given in Theorem 2.3 can be extended 1Some care has to be taken here. In the Gaussian linear case the finite-sample cdfs converge at every value of the argument t, cf. Propisition 2.1. In a general parametric model, sometimes the asymptotic results (e.g., Hjort and Claeskens (2003, Theorem 4.1)) only guarantee weak convergence. Hence, to ensure convergence of the relevant cdfs at a given argument t as required in Proberty b, additional considerations have to be employed. [This is, however, of no concern in the context discussed in the next but one paragraph in this section.] to more general parametric and semiparametric models with ease. The fact that we use a Gaussian linear model for the analysis in the present paper is a matter of convenience rather than a necessity. The non-uniformity results in Theorem 2.3 are for (conservative) ‘general-to-specific’ model selection from a nested family of models. Theorem 3.1 extends this to more general (conservative) model selection procedures (including AIC and related procedures) and to more general families of models. The proof of Theorem 3.1 proceeds by reducing the problem to one where only two nested models are considered, and then to appeal to the results of Theorem 2.3. The condition on the model selection procedures that enables this reduction is condition (24). It is apparent from the discussion in Section 3 that this condition is satisfied for many model selection procedures. Furthermore, for the same reasons as given in the preceding paragraph, also Theorem 3.1 can easily be extended to sufficiently regular parametric and semi-parametric models. The ‘impossibility’ results in the present paper are formulated for estimating Gn,θ,σ(t) for a given value of t. Suppose that we are now asking the question whether the cdf Gn,θ,σ(·) viewed as a function can be estimated uniformly consistently, where consistency is relative to a metric that metrizes weak convergence.2 Using a similar reasoning as above (which can again be made formal by using, e.g., Lemma 3.1 in Leeb and Pötscher (2006a)) the key step now is to show that the function G∞,θ,σ,0(·) is different from the function G∞,θ,σ,γ(·). Obviously, it is a much simpler problem to find a γ such that the functions G∞,θ,σ,0(·) and G∞,θ,σ,γ(·) differ, than to find a γ such that the values G∞,θ,σ,0(t) and G∞,θ,σ,γ(t) for a given t differ. Certainly, having solved the latter problem in Appendix A, this also provides an answer to the former. This then immediately delivers the desired ‘impossibility’ result. [We note that in some special cases simpler arguments than the ones used in Appendix A can be employed to solve the former problem: For example, in case A = I the functions G∞,θ,σ,0(·) and G∞,θ,σ,γ(·) can each be shown to be convex combinations of cdfs that are concentrated on subspaces of different dimensions. This can be exploited to establish without much difficulty that the functions G∞,θ,σ,0(·) and G∞,θ,σ,γ(·) differ. For purpose of comparison we note that for general A the distributions G∞,θ,σ,0 and G∞,θ,σ,γ can both be absolutely continuous w.r.t. Lebesgue measure, not allowing one to use this simple argument.] Again the discussion in this paragraph extends to more general parametric and semiparametric models without difficulty. The present paper, including the discussion in this section, has focussed on conservative model selection procedures. However, the discussion should make it clear that similar ‘impossibility’ results plague consistent model selection. Section 2.3 in Leeb and Pötscher (2006a) in fact gives such an ‘impossibility’ result in a simple case. We close with the following observations. Verification of Property b, whether it is for G∞,θ,σ,0(t) and G∞,θ,σ,γ(t) (for given t) or for G∞,θ,σ,0(·) and G∞,θ,σ,γ(·), shows that the post-model-selection estimator Aθ̃ is a so-called non-regular estimator for Aθ: Consider an estimator β̃ in a parametric model {Pn,β : β ∈ B} where the parameter space B is an open subset of Euclidean space Rd. Suppose β̃, properly scaled and centered, has a limit distribution under local alternatives, in the sense that n(β̃ − (β + γ/ n)) converges in law under Pn,β+γ/ n to a limit distribution L∞,β,γ(·) for every γ. The estimator β̃ is called regular if for every β the limit distribution L∞,β,γ(·) does not depend on γ; cf. van der Vaart (1998, Section 8.5). Suppose now that the model is, e.g., locally asymptotically normal (hence the contiguity property in Property a is satisfied). The informal argument outlined at the beginning of this section (and which is formalized in Lemma 3.1 of Leeb and Pötscher (2006a)) then in fact shows that the cdf of any non-regular estimator 2Or, in fact, any metric w.r.t. which the relevant cdfs converge. can not be estimated uniformly consistently (where consistency is relative to any metric that metrizes weak convergence). 6 Conclusions Despite the fact that we have shown that consistent estimators for the distribution of a post-model-selection estimator can be constructed with relative ease, we have also demonstrated that no estimator of this distri- bution can have satisfactory performance (locally) uniformly in the parameter space, even asymptotically. In particular, no (locally) uniformly consistent estimator of this distribution exists. Hence, the answer to the question posed in the title has to be negative. The results in the present paper also cover the case of linear functions (e.g., predictors) of the post-model-selection estimator. We would like to stress here that resampling procedures like, e.g., the bootstrap or subsampling, do not solve the problem at all. First note that standard bootstrap techniques will typically not even provide consistent estimators of the finite-sample distribution of the post-model-selection estimator, as the bootstrap can be shown to stay random in the limit (Kulperger and Ahmed (1992), Knight (1999, Example 3))3. Basically the only way one can coerce the bootstrap into delivering a consistent estimator is to resample from a model that has been selected by an auxiliary consistent model selection procedure. The consistent estimator constructed in Section 2.2.1 is in fact of this form. In contrast to the standard bootstrap, subsampling will typically deliver consistent estimators. However, the ‘impossibility’ results given in this paper apply to any estimator (including randomized estimators) of the cdf of a post-model-selection estimator. Hence, also any resampling based estimator suffers from the non-uniformity defects described in Theorems 2.3 and 3.1; cf. also Remark 4.3. The ‘impossibility’ results in Theorems 2.3 and 3.1 are derived in the framework of a normal linear regression model (and a fortiori these results continue to hold in any model which includes the normal linear regression model as a special case), but this is more a matter of convenience than anything else: As discussed in Section 5, similar results can be obtained in general statistical models allowing for nonlinearity or dependent data, e.g., as long as standard regularity conditions for maximum likelihood theory are satisfied. The results in the present paper are derived for a large class of conservative model selection procedures (i.e., procedures that select overparameterized models with positive probability asymptotically) including Akaike’s AIC and typical ‘general-to-specific’ hypothesis testing procedures. For consistent model selection procedures – like BIC or testing procedures with suitably diverging critical values cp (cf. Bauer, Pötscher, and Hackl (1988)) – the (pointwise) asymptotic distribution is always normal. [This is elementary, cf. Lemma 1 in Pötscher (1991).] However, as discussed at length in Leeb and Pötscher (2005a), this asymptotic nor- mality result paints a misleading picture of the finite sample distribution which can be far from a normal, the convergence of the finite-sample distribution to the asymptotic normal distribution not being uniform. ‘Impossibility’ results similar to the ones presented here can also be obtained for post-model-selection esti- mators based on consistent model selection procedures. These will be discussed in detail elsewhere. For a 3Brownstone (1990) claims the validity of a bootstrap procedure that is based on a conservative model selection procedure in a linear regression model. Kilian (1998) makes a similar claim in the context of autoregressive models selected by a conser- vative model selection procedure. Also Hansen (2003) contains such a claim for a stationary bootstrap procedure based on a conservative model selection procedure. The above discussion intimates that these claims are at least unsubstantiated. simple special case such an ‘impossibility’ result is given in Section 2.3 of Leeb and Pötscher (2006a). The ‘impossibility’ of estimating the distribution of the post-model-selection estimator does not per se preclude the possibility of conducting valid inference after model selection, a topic that deserves further study. However, it certainly makes this a more challenging task. A Auxiliary Lemmas Lemma A.1 Let Z be a random vector with values in Rk and let W be a univariate standard Gaussian random variable independent of Z. Furthermore, let C ∈ Rk and τ > 0. Then P(Z ≤ Cx)P(|W − x| < τ ) + P(Z ≤ CW, |W − x| ≥ τ) (31) is constant as a function of x ∈ R if and only if C = 0 or P(Z ≤ Cx) = 0 for each x ∈ R. Proof of Lemma A.1: Suppose C = 0 holds. Using independence of Z andW it is then easy to see that (31) reduces to P(Z ≤ 0), which is constant in x. If P(Z ≤ Cx) = 0 for every x ∈ R, then P(Z ≤ CW ) = 0, and hence (31) is again constant, namely equal to zero. To prove the converse, assume that (31) is constant in x ∈ R. Letting x→ ∞, we see that (31) must be equal to P(Z ≤ CW ). This entails that P(Z ≤ Cx)P(|W − x| < τ) = P(Z ≤ CW, |W − x| < τ ) holds for every x ∈ R. Write F (x) as shorthand for P(Z ≤ Cx), and let Φ(z) and φ(z) denote the cdf and density of W , respectively. Then the expression in the above display can be written as F (x)(Φ(x + τ )− Φ(x− τ)) = ∫ x+τ F (z)φ(z)dz. (x ∈ R) (32) We now further assume that C 6= 0 and that F (x) 6= 0 for at least one x ∈ R, and show that this leads to a contradiction. Consider first the case where all components of C are non-negative. Since F is not identically zero, it is then, up to a scale factor, the cdf of a random variable on the real line. But then (32) can not hold for all x ∈ R as shown in Example 7 in Leeb (2002) (cf. also equation (7) in that paper). The case where all components of C are non-positive follows similarly by applying the above argument to F (−x) and upon observing that both Φ(x+ τ )− Φ(x− τ) and φ(x) are symmetric around x = 0. Finally, consider the case where C has at least one positive and one negative component. In this case clearly limx→−∞ F (x) = limx→∞ F (x) = 0 holds. Since F (x) is continuous in view of (32), we see that F (x) attains its (positive) maximum at some point x1 ∈ R. Now note that (32) with x1 replacing x can be written as ∫ x1+τ (F (x1)− F (z))φ(z)dz = 0. This immediately entails that F (x) = F (x1) for each x ∈ [x1 − τ , x1 + τ ] (because F (x) is continuous and because of the definition of x1). Repeating this argument with x1−τ replacing x1 and proceeding inductively, we obtain that F (x) = F (x1) for each x satisfying x ≤ x1 + τ , a contradiction with limx→−∞ F (x) = 0. ✷ Lemma A.2 Let M and N be matrices of dimension k × p and k × q, respectively, such that the matrix (M : N) has rank k (k ≥ 1, p ≥ 1, q ≥ 1). Let t ∈ Rk, and let V be a random vector with values in Rp whose distribution assigns positive mass to every (non-empty) open subset of Rp (e.g., it possesses an almost everywhere positive Lebesgue density). Set f(x) = P(MV ≤ t + Nx), x ∈ Rq. If one of the rows of M consists of zeros only, then f is discontinuous at some point x0. More precisely, there exist x0 ∈ Rq, z ∈ Rq and a constant c > 0, such that f(x0 + δz) ≥ c and f(x0 − δz) = 0 hold for every sufficiently small δ > 0. Proof of Lemma A.2: The case where M is the zero-matrix is trivial. Otherwise, let I0 denote the set of indices i, 1 ≤ i ≤ k, for which the i-th row of M is zero. Let (M0 : N0) denote the matrix consisting of those rows of (M : N) whose index is in I0, and let (M1 : N1) denote the matrix consisting of the remaining rows of (M : N). Clearly, M0 is then the zero matrix. Furthermore, note that N0 has full row-rank. Moreover, let t0 denote the vector consisting of those components of t whose index is in I0 and let t1 denote the vector containing the remaining components. With this notation, f(x) can be written as P(0 ≤ t0 +N0x, M1V ≤ t1 +N1x). For vectors µ ∈ Rp and η ∈ Rq to be specified in a moment, set t∗ = t+Mµ+Nη, and let t∗0 and t∗1 be defined similarly to t0 and t1. Since the matrix (M : N) has full rank k, we can choose µ and η such that t∗0 = 0 and t 1 > 0. Choose z ∈ Rq such that N0z > 0, which is possible because N0 has full row-rank. Set x0 = η. Then for every ǫ ∈ R we have f(x0 + ǫz) = f(η + ǫz) = P(MV ≤ t+N(η + ǫz)) = P(0 ≤ t0 +N0(η + ǫz), M1V ≤ t1 +N1(η + ǫz)) = P(0 ≤ t∗0 + ǫN0z, M1(V + µ) ≤ t∗1 + ǫN1z) = P(0 ≤ ǫN0z, M1(V + µ) ≤ t∗1 + ǫN1z) Since t∗1 > 0, we can find a t 1 such that 0 < t 1 < t 1 + ǫN1z holds for every ǫ with |ǫ| small enough. If now ǫ > 0 then f(x0 + ǫz) = P(M1(V + µ) ≤ t∗1 + ǫN1z) ≥ P(M1(V + µ) ≤ t∗∗1 ). The r.h.s. in the above display is positive because t∗∗1 > 0 and because the distribution ofM1(V +µ) assigns positive mass to any neighborhood of the origin, since the same is true for the distribution of V + µ and since M1 maps neighborhoods of zero into neighborhoods of zero. Setting c = P(M1(V + µ) ≤ t∗∗1 )/2, we have f(x0 + ǫz) ≥ c > 0 for each sufficiently small ǫ > 0. Furthermore, for ǫ < 0 we have f(x0 + ǫz) = 0, since f(x0 + ǫz) ≤ P(0 ≤ ǫN0z) = 0 in view of N0z > 0. ✷ Lemma A.3 Let Z be a random vector with values in Rp, p ≥ 1, with a distribution that is absolutely continuous with respect to Lebesgue measure on Rp. Let B be a k×p matrix, k ≥ 1. Then the cdf P(BZ ≤ ·) of BZ, is discontinuous at t ∈ Rk if and only if P(BZ ≤ t) > 0 and if for some i0, 1 ≤ i0 ≤ k, the i0-th row of B and the i0-th component of t are both zero, i.e., Bi0,· = (0, . . . , 0) and ti0 = 0. Proof of Lemma A.3: To establish sufficiency of the above condition, let P(BZ ≤ t) > 0, ti0 = 0 and Bi0,· = (0, . . . , 0) for some i0, 1 ≤ i0 ≤ k. Then, of course, P(Bi0,·Z = 0) = 1. For tn = t−n−1ei0 , where ei0 denotes the i0-th unit vector in R k, we have P(BZ ≤ tn) ≤ P(Bi0,·Z ≤ tn,i0) = P(Bi0,·Z ≤ −1/n) = 0 for every n. Consequently, P(BZ ≤ t) is discontinuous at t. To establish necessity, we first show the following: If tn ∈ Rk is a sequence converging to t ∈ Rk as n→ ∞, then every accumulation point of the sequence P(BZ ≤ tn) has the form P(Bi1,·Z ≤ ti1 , . . . , Bim,·Z ≤ tim , Bim+1,·Z < tim+1 , . . . , Bik,·Z < tik) (33) for somem, 0 ≤ m ≤ k, and for some permutation (i1, . . . , ik) of (1, . . . , k). This can be seen as follows: Let α be an accumulation point of P(BZ ≤ tn). Then we may find a subsequence such that P(BZ ≤ tn) converges to α along this subsequence. From this subsequence we may even extract a further subsequence along which each component of the k × 1 vector tn converges to the corresponding component of t monotonously, that is, either from above or from below. Without loss of generality, we may also assume that those components which converge from below are strictly increasing. The resulting subsequence will be denoted by nj in the sequel. Assume that the components of tnj with indices i1, . . . , im converge from above, while the components with indices im+1, . . . , ik converge from below. Now P(BZ ≤ tnj ) = 1(−∞,tnj,s](zs)PBZ(dz), (34) where PBZ denotes the distribution of BZ. The integrand in (34) now converges to l=1 1(−∞,til ](zil) l=m+1 1(−∞,til )(zil) for all z ∈ R k as nj → ∞. The r.h.s. of (34) converges to the expression in (33) as nj → ∞ by the Dominated Convergence Theorem, while the l.h.s. of (34) converges to α by construction. This establishes the claim regarding (33). Now suppose that P(BZ ≤ t) is discontinuous at t; i.e., there exists a sequence tn converging to t as n → ∞, such that P(BZ ≤ tn) does not converge to P(BZ ≤ t) as n → ∞. From the sequence tn we can extract a subsequence tns along which P(BZ ≤ tns) converges to a limit different from P(BZ ≤ t) as ns → ∞. As shown above, the limit has to be of the form (33) and m < k has to hold. Consequently, the limit of P(BZ ≤ tns) is smaller than P(BZ ≤ t) = P(Bi,·Z ≤ ti, i = 1, . . . , k). The difference of P(BZ ≤ t) and the limit of P(BZ ≤ tns) is positive and because of (33) can be written as P(Bij ,·Z ≤ tij for each j = 1, . . . , k, Bij ,·Z = tij for some j = m+ 1, . . . , k) > 0. We thus see that P(Bij0 ,·Z = tij0 ) > 0 for some j0 satisfying m + 1 ≤ j0 ≤ k. As Z is absolutely continuous with respect to Lebesgue measure on Rp, this can only happen if Bij0 ,· = (0, . . . , 0) and tij0 = 0. Lemma A.4 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C ∞ 6= 0, for some q satisfying O < q ≤ P , and let q∗ denote the largest q with this property. Moreover let θ ∈ Mq∗−1, let σ satisfy 0 < σ < ∞, and let t ∈ Rk. Then G∞,θ,σ,γ(t) is non-constant as a function of γ ∈ Mq∗\Mq∗−1. More precisely, there exist δ0 > 0 and ρ0, 0 < ρ0 <∞, such that γ(1),γ(2)∈Mq∗ \Mq∗−1 ||γ(i)||<ρ0,i=1,2 ∣∣G∞,θ,σ,γ(1)(t)−G∞,θ,σ,γ(2)(t) ∣∣ > 2δ0 (35) holds. The constants δ0 and ρ0 can be chosen in such a way that they depend only on t, Q, A, σ, and the critical values cp for O < p ≤ P . Lemma A.5 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C ∞ 6= 0, for some q satisfying O < q ≤ P , and let q∗ denote the largest q with this property. Suppose further that for some p⊙ satisfying O ≤ p⊙ < q∗ either p⊙ = 0 holds or that p⊙ > 0 and A[p⊙] has a row of zeros. Then, for every θ ∈ Mp⊙ , every σ, 0 < σ < ∞, and every t ∈ Rk the quantity G∞,θ,σ,γ(t) is discontinuous as a function of γ ∈ Mq∗ . More precisely, for each s = O, . . . , p⊙, there exist vectors β∗ and γ∗ in Mq∗ and constants δ∗ > 0 and ǫ∗ > 0 such that ∣∣G∞,θ,σ,β∗+ǫγ∗(t)−G∞,θ,σ,β∗−ǫγ∗(t) ∣∣ ≥ δ∗ (36) holds for every θ satisfying max{p0(θ),O} = s and for every ǫ with 0 < ǫ < ǫ∗. The quantities δ∗, ǫ∗, β∗, and γ∗ can be chosen in such a way that – besides t, Q, A, σ, and the critical values cp for O < p ≤ P – they depend on θ only through max{p0(θ),O}. Before we prove the above lemmas, we provide a representation of G∞,θ,σ,γ(t) that will be useful in the following: For 0 < p ≤ P define Zp = r=1 ξ ∞ Wr, where C ∞ has been defined after (13) and the random variables Wr are independent normally distributed with mean zero and variances σ 2ξ2∞,r; for convenience, let Z0 denote the zero vector in R k. Observe that Zp, p > 0, is normally distributed with mean zero and variance-covariance matrix σ2A[p]Q[p : p]−1A[p]′ since it has been shown in the proof of Proposition 4.4 in Leeb and Pötscher (2006b) that the asymptotic variance-covariance matrix σ2A[p]Q[p : p]−1A[p]′ of nAθ̃(p) can be expressed as r=1 σ 2ξ−2∞,rC ∞ . Also the joint distribution of Zp and the set of variables Wr, 1 ≤ r ≤ P , is normal, with the covariance vector between Zp and Wr given by σ2C(r)∞ in case r ≤ p; otherwise Zp and Wr are independent. Define the constants νr = γr+(Q[r : r]−1Q[r : ¬r]γ[¬r])r for 0 < r ≤ P . It is now easy to see that for p ≥ p∗ = max{p0(θ),O} the quantity β(p) defined in Proposition 2.1 equals − r=p+1 ξ ∞ νr. [This is seen as follows: It was noted in Proposition 2.1 that β(p) = limn→∞ nA(ηn(p) − θ − γ/ n) for p ≥ p0(θ), when ηn(p) is defined as in (9), but with θ + γ/ replacing θ. Using the representation (20) of Leeb (2005) and taking limits, the result follows if we observe nηn,r(r) −→ νr for r > p ≥ p0(θ).] The cdf in (15) can now be written as Zp∗ ≤ t+ r=p∗+1 ξ−2∞,rC q=p∗+1 P(|Wq + νq| < cqσξ∞,q) p=p∗+1 Zp ≤ t+ r=p+1 ξ−2∞,rC ∞ νr, |Wp + νp| ≥ cpσξ∞,p q=p+1 P(|Wq + νq| < cqσξ∞,q). (37) That the terms corresponding to p = p∗ in (37) and (15) agree is obvious. Furthermore, for each p > p∗ the terms under the product sign in (37) and (15) coincide by definition of the function ∆s(a, b). It is also easy to see that the conditional distribution of Wp given Zp = z is Gaussian with mean b∞,pz and variance σ2ζ2∞,p. Consequently, the probability of the event {|Wp + νp| ≥ cpσξ∞,p} conditional on Zp = z is given by the integrand shown in (15). Since Zp has distribution Φ∞,p as noted above, it follows that (37) and (15) agree. Remark A.6 If C ∞ = 0 for p > p∗, then in view of the above discussion Zp∗ = Zp = ZP , and hence Φ∞,p∗ = Φ∞,p = Φ∞,P , holds for all p > p∗. Using the independence of Wr, r > p∗, from Zp∗ , inspection of (37) shows that G∞,θ,σ,γ reduces to Φ∞,P ; see also Leeb (2006, Remark 5.2). Proof of Lemma A.4: From (37) (or (15)) it follows that the map γ 7→ G∞,θ,σ,γ(t) depends only on t, Q, A, σ, the critical values cp for O < p ≤ P , as well as on θ; however, the dependence on θ is only through p∗ = max{p0(θ),O}. It hence suffices to find, for each possible value of p∗ in the range p∗ = O, . . . , q∗ − 1, constants 0 < ρ0 < ∞ and δ0 > 0 such that (35) is satisfied for some (and hence all) θ returning this particular value of p∗ = max{p0(θ),O}. For this in turn it is sufficient to show that for every θ ∈Mq∗−1 the quantity G∞,θ,σ,γ(t) is non-constant as a function of γ ∈Mq∗\Mq∗−1. Let θ ∈Mq∗−1 and assume that G∞,θ,σ,γ(t) is constant in γ ∈Mq∗\Mq∗−1. Observe that, by assumption, ∞ is non-zero while C ∞ = 0 for p > q ∗. For γ ∈ Mq∗ , we clearly have νq∗ = γq∗ and νr = 0 for r > q∗. Letting γq∗−1 → ∞ while γq∗ is held fixed, we see that νq∗−1 → ∞; hence, P(|Wq∗−1 + νq∗−1| < cq∗−1σξ∞,q∗−1) → 0. It follows that (37) converges to Zq∗−1 ≤ t+ ξ−2∞,q∗C(q ∞ γq∗ P(|Wq∗ + γq∗ | < cq∗σξ∞,q∗) q=q∗+1 P(|Wq| < cqσξ∞,q) Zq∗ ≤ t, |Wq∗ + γq∗ | ≥ cq∗σξ∞,q∗ q=q∗+1 P(|Wq| < cqσξ∞,q) (38) p=q∗+1 Zp ≤ t, |Wp| ≥ cpσξ∞,p q=p+1 P(|Wq| < cqσξ∞,q). By assumption, the expression in the above display is constant in γq∗ ∈ R\{0}. Dropping the terms that do not depend on γq∗ and observing that P(|Wq| < cqσξ∞,q) is never zero for q > q∗ > O, we see that Zq∗−1 ≤ t+ ξ−2∞,q∗C(q ∞ γq∗ P(|Wq∗ + γq∗ | < cq∗σξ∞,q∗) Zq∗ ≤ t, |Wq∗ + γq∗ | ≥ cq∗σξ∞,q∗ has to be constant in γq∗ ∈ R\{0}. We now show that the expression in (39) is in fact constant in γq∗ ∈ R: Observe first that P(|Wq∗ + γq∗ | < cq∗σξ∞,q∗) is positive and continuous in γq∗ ∈ R; also the probability Zq∗ ≤ t, |Wq∗ + γq∗ | ≥ cq∗σξ∞,q∗ is continuous in γq∗ ∈ R since Wq∗ , being normal with mean zero and positive variance, is absolutely continuously distributed. Concerning the remaining term in (39), we note that Zq∗−1 =MV where M = [ξ ∞ , . . . , ξ ∞,q∗−1C (q∗−1) ∞ ] and V = (W1, . . . ,Wq∗−1) ′. In case no row of M is identically zero, Lemma A.3 shows that also P Zq∗−1 ≤ t+ ξ−2∞,q∗C ∞ γq∗ is continuous in γq∗ ∈ R. Hence, in this case (39) is indeed constant for all γq∗ ∈ R. In case a row of M is identically zero, define N = ξ ∞,q∗C ∞ and rewrite the probability in question as P MV ≤ t+Nγq∗ . Note that (M : N) has full row-rank k, since (M : N)diag[ξ2∞,1, . . . , ξ ∞,q∗ ](M : N) ξ−2∞,rC ξ−2∞,rC ∞ = AQ −1A′ (40) by definition of q∗ and since the latter matrix is non-singular in view of rank A = k. Lemma A.2 then shows that there exists a γ q∗ ∈ R, z ∈ {−1, 1}, and a constant c > 0 such that P MV ≤ t+N(γ(0)q∗ − δz) and P MV ≤ t+N(γ(0)q∗ + δz) ≥ c holds for arbitrary small δ > 0. Observe that γ(0)q∗ − δz as well as q∗ − δz are non-zero for sufficiently small δ > 0. But then (39) – being constant for γq∗ ∈ R\{0} – gives the same value for γq∗ = γ q∗ − δz and γq∗ = γ q∗ + δz and all sufficiently small δ > 0. Letting δ go to zero in this equality and using the continuity properties for the second and third probability in (39) noted above we obtain that cP(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗) + P Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗ ≤ lim inf Zq∗−1 ≤ t+ ξ−2∞,q∗C(q q∗ + δz) P(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗) Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗ = lim inf Zq∗−1 ≤ t+ ξ−2∞,q∗C(q q∗ − δz) P(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗) Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗ Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗ which is impossible since c > 0 and P(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗) > 0. Hence we have shown that (39) is indeed constant for all γq∗ ∈ R. Now write Z,W , C, τ , and x for Zq∗−1−t, −Wq∗/σξ∞,q∗ , σξ ∞,q∗C ∞ , cq∗ , and γq∗/σξ∞,q∗ , respectively. Upon observing that Zq∗ equals Zq∗−1 + ξ ∞,q∗C ∞ Wq∗ , it is easy to see that (39) can be written as in (31). By our assumptions, this expression is constant in x = γq∗/σξ∞,q∗ ∈ R. Lemma A.1 then entails that either C = 0 or that P(Z ≤ Cx) = 0 for each x ∈ R. Since C equals σξ−1∞,q∗C ∞ , it is non-zero by assumption. Hence, Zq∗−1 ≤ t+ ξ−2∞,q∗C(q ∞ γq∗ must hold for every value of γq∗ . But the above probability is just the conditional probability that Zq∗ ≤ t given Wq∗ = −γq∗ . It follows that P(Zq∗ ≤ t) equals zero as well. By our assumption C ∞ = 0 for p > q and hence Zq∗ = ZP . We thus obtain P(ZP ≤ t) = 0, a contradiction with the fact that ZP is a Gaussian random variable on Rk with non-singular variance-covariance matrix σ2AQ−1A′. ✷ Inspection of the above proof shows that it can be simplified if the claim of non-constancy of G∞,θ,σ,γ(t) as a function of γ ∈Mq∗\Mq∗−1 in Lemma A.4 is weakened to non-constancy for γ ∈Mq∗ . The strong form of the lemma as given here is needed in the proof of Proposition B.1. Proof of Lemma A.5: Let p⊕ be the largest index p, O ≤ p ≤ P , for which A[p] has a row of zeroes, and set p⊕ = 0 if no such index exists. We first show that p⊕ satisfies p⊕ < q ∗. Suppose p⊕ ≥ q∗ would hold. Since Zp⊕ is a Gaussian random vector with mean zero and variance-covariance matrix σ 2A[p⊕]Q[p⊕ : p⊕] −1A[p⊕] at least one component of Zp⊕ is equal to zero with probability one. However, Zp⊕ equals ZP because of p⊕ ≥ q∗ and the definition of q∗. This leads to a contradiction since ZP has the non-singular variance- covariance matrix σ2AQ−1A′. Without loss of generality, we may hence assume that p⊙ = p⊕. In view of the discussion in the first paragraph of the proof of Lemma A.4, it suffices to establish, for each possible value s in the range O ≤ s ≤ p⊙, the result (36) for some θ with s = max{p0(θ),O} = p∗. Now fix such an s and θ (as well as, of course, t, Q, A, σ, and the critical values cp for O < p ≤ P ). Then (37) expresses the map γ 7→ G∞,θ,σ,γ(t) in terms of ν = (ν1, . . . , νP )′. It is easy to see that the correspondence between γ and ν is a linear bijection from RP onto itself, and that γ ∈ Mq∗ if and only if ν ∈ Mq∗ . It is hence sufficient to find a δ∗ > 0 and vectors ν and µ in Mq∗ such that (37) with ν + ǫµ in place of ν and (37) with ν − ǫµ in place of ν differ by at least δ∗ for sufficiently small ǫ > 0. Note that (37) is the sum of P − p∗ + 1 terms indexed by p = p∗, . . . , P . We shall now show that ν and µ can be chosen in such a way that, when replacing ν with ν + ǫµ and ν − ǫµ, respectively, (i) the resulting terms in (37) corresponding to p = p⊙ differ by some d > 0, while (ii) the difference of the other terms becomes arbitrarily small, provided that ǫ > 0 is sufficiently small. Consider first the case where s = p∗ = p⊙. Using the shorthand notation g(ν) = P Zp⊙ ≤ t+ r=p⊙+1 ξ−2∞,rC note that the p⊙-th term in (37) is given by g(ν) multiplied by a product of positive probabilities which are continuous in ν. To prove property (i) it thus suffices to find a constant c > 0, and vectors ν and µ in Mq∗ such that |g(ν + ǫµ)− g(ν − ǫµ)| ≥ c holds for each sufficiently small ǫ > 0. In the sub-case p⊙ = 0 choose c = 1, set ν = −[C(1)∞ , . . . , C(P )∞ ]′ ξ−2∞,rC µ = [C(1)∞ , . . . , C ξ−2∞,rC (1, . . . , 1)′, observing that the matrix to be inverted is indeed non-singular, since – as discussed after Lemma A.5 – it is up to a multiplicative factor σ2 identical to the variance-covariance matrix σ2AQ−1A′ of ZP . But then ν and µ satisfy r=p⊙+1 ξ−2∞,rC ∞ νr = −t and r=p⊙+1 ξ−2∞,rC ∞ µr = (1, . . . , 1) ′ if we note that by the definition of q∗ r=p⊙+1 ξ−2∞,rC ∞ νr = ξ−2∞,rC holds and that a similar relation holds with µ replacing ν. Since Zp⊙ = Z0 = 0 ∈ Rk, it is then obvious that g(ν + ǫµ) and g(ν − ǫµ) differ by 1 for each ǫ > 0. In the other sub-case p⊙ > 0, define M = [ξ ∞ , . . . , ξ ∞,p⊙C ∞ ], N = [ξ−2∞,p⊙+1C (p⊙+1) ∞ , . . . , ξ ∞,q∗C ∞ ], and V = (W1, . . . ,Wp⊙) ′. It is then easy to see that g(ν) equals f((νp⊙+1, . . . , νq∗) ′), with f defined as in Lemma A.2, and that M has a row of zeros. Furthermore, the matrix (M : N) has rank k by the same argument as in the proof of Lemma A.4; cf. (40). By Lemma A.2, we thus obtain vectors x0 and z, and a c > 0 such that |f(x0+ ǫz)−f(x0− ǫz)| ≥ c holds for each sufficiently small ǫ > 0. Setting (νp⊙+1, . . . , νq∗) ′ = x0, (µp⊙+1, . . . , µq∗) ′ = z, setting ν[¬q∗], and µ[¬q∗] each equal to zero, and setting ν[p⊙] and µ[p⊙] to arbitrary values, we see that g(ν ± ǫµ) has the desired properties. To complete the proof in case s = p∗ = p⊙, we need to establish property (ii) for which it suffices to show that, for p > p⊙, the p-th term in (37) depends continuously on ν. For p > q ∗, the p-th term does not depend on ν, because C ∞ = 0 for r = q∗, . . . , P . For p satisfying p⊙ < p ≤ q∗, it suffices to show that h(νp, . . . , νq∗) = P Zp ≤ t+ r=p+1 ∞ νr, |Wp + νp| ≥ cpσξ∞,p is a continuous function. Suppose that (ν p , . . . , ν q∗ ) converges to (νp, . . . , νq∗) as m→ ∞. For arbitrary α > 0, r=p+1 ξ ∞ νr and r=p+1 ξ r differ by less than α in each coordinate, provided that m is sufficiently large. This implies lim sup h(ν(m)p , . . . , ν ≤ lim sup P(Zp ≤ t+ r=p+1 ∞ νr + α(1, . . . , 1) ′, |Wp + ν(m)p | ≥ cpσξ∞,p) = P(Zp ≤ t+ r=p+1 ξ−2∞,rC ∞ νr + α(1, . . . , 1) ′, |Wp + νp| ≥ cpσξ∞,p), observing that the latter probability is obviously continuous in the single variable νp (since Wp has an ab- solutely continuous distribution). Letting α decrease to zero we obtain lim supm→∞ h(ν p , . . . , ν q∗ ) ≤ h(νp, . . . , νq∗). A similar argument establishes lim infm→∞ h(ν p , . . . , ν q∗ ) ≥ P(Zp < t + r=p+1 ξ ∞ νr, |Wp + νp| ≥ cpσξ∞,p). The proof of the continuity of h is then complete if we can show that P Zp ≤ ·, |Wp + νp| ≥ cpσξ∞,p is continuous or, equivalently, that P Zp ≤ · ∣∣|Wp + νp| ≥ cpσξ∞,p is a continuous cdf. Since p > p⊙, the variance-covariance matrix σ 2A[p]Q[p : p]−1A[p]′ of Zp does only have non-zero diagonal elements. Consequently, when representing Zp as B(W1, . . . ,Wp) ′, the matrix B cannot have rows that consist entirely of zeros. The conditional distribution of (W1, . . . ,Wp) ′ given the event {|Wp + νp| ≥ cpσξ∞,p} is clearly absolutely continuous w.r.t. p-dimensional Lebesgue measure. But then Lemma A.3 delivers the desired result. The case where s = p∗ < p⊙ is reduced to the previously discussed case as follows: It is easy to see that, for νp⊙ → ∞, the expression in (37) converges to a limit uniformly w.r.t. all νp with p 6= p⊙. Then observe that this limit is again of the form (37) but now with p⊙ taking the rôle of p∗. ✷ B Non-Uniformity of the Convergence of the Finite-Sample Cdf to the Large-Sample Limit Proposition B.1 a. Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C ∞ 6= 0, for some q satisfying O < q ≤ P , and let q∗ denote the largest q with this property. Then for every θ ∈Mq∗−1, every σ, 0 < σ <∞, and every t ∈ Rk there exists a ρ, 0 < ρ <∞, such that lim inf ϑ∈Mq∗ ||ϑ−θ||<ρ/ |Gn,ϑ,σ(t)−G∞,ϑ,σ(t)| > 0 (41) holds. The constant ρ may be chosen in such a way that it depends only on t, Q, A, σ, and the critical values cp for O < p ≤ P . b. Suppose that Aθ̃(q) and θ̃q(q) are asymptotically uncorrelated, i.e., C ∞ = 0, for all q satisfying O < q ≤ P . Then Gn,θ,σ converges to Φ∞,P in total variation uniformly in θ ∈ RP ; more precisely σ∗≤σ≤σ∗ ||Gn,θ,σ − Φ∞,P ||TV n→∞−→ 0 holds for any constants σ∗ and σ ∗ satisfying 0 < σ∗ ≤ σ∗ <∞. Under the assumptions of Proposition B.1(a), we see that convergence of Gn,θ,σ(t) to G∞,θ,σ(t) is non- uniform over shrinking ‘tubes’ aroundMq∗−1 that are contained in Mq∗ . [On the complement of a tube with a fixed positive radius, i.e., on the set U = {θ ∈ RP : |θq∗ | ≥ r} with fixed r > 0, convergence of Gn,θ,σ(t) to G∞,θ,σ(t) is in fact uniform (even with respect to the total variation distance), as can be shown. Note that for θ ∈ U the cdf G∞,θ,σ(t) reduces to the Gaussian cdf Φ∞,P (t), i.e., to the asymptotic distribution of the least-squares estimator based on the overall model; cf. Remark A.6.] A precursor to Proposition B.1(a) is Corollary 5.5 of Leeb and Pötscher (2003) which establishes (41) in the special case where O = 0 and where A is the P × P identity matrix. Proposition B.1(b) describes an exceptional case where convergence is uniform. [In this case G∞,θ,σ reduces to the Gaussian cdf Φ∞,P for all θ and Φ∞,P = Φ∞,p, O ≤ p ≤ P , holds; cf. Remark A.6.] Recall that under the assumptions of part (b) of Proposition B.1 we necessarily always have (i) O > 0, and (ii) rank A[O] = k; cf. Proposition 4.4 in Leeb and Pötscher (2006b). Proof of Proposition B.1: We first prove part (a). As noted at the beginning of the proof of Lemma A.4, the map γ 7→ G∞,θ,σ,γ(t) depends only on t, Q, A, σ, the critical values cp for O < p ≤ P , as well as on θ, but the dependence on θ is only through p∗ = max{p0(θ),O}. It hence suffices to find, for each possible value of p∗ in the range p∗ = O, . . . , q∗ − 1, a constant 0 < ρ < ∞ such that (41) is satisfied for some (and hence all) θ returning this particular value of p∗ = max{p0(θ),O}. For this in turn it is sufficient to show that given such a θ we can find a γ ∈Mq∗ such that lim inf |Gn,θ+γ/√n,σ(t)−G∞,θ+γ/√n,σ(t)| > 0 (42) holds. Note that (42) is equivalent to lim inf |G∞,θ,σ,γ(t)−G∞,θ+γ/√n,σ(t)| > 0 (43) in light of Proposition 2.1. To establish (43), we proceed as follows: For each γ ∈ Mq∗ with γq∗ 6= 0, G∞,θ+γ/ n,σ(t) in (15) reduces to Φ∞,q∗(t) as is easily seen from (37) since p0(θ+γ/ n) = q∗ which in turn follows from p0(θ) < q ∗ and γq∗ 6= 0. Furthermore, Lemma A.4 entails that G∞,θ,σ,γ(t) is non-constant in γ ∈Mq∗\Mq∗−1. But this shows that (43) must hold. To prove part (b), we write ||Gn,θ,σ − Φ∞,P ||TV = ∣∣∣∣∣∣ ∣∣∣∣∣∣ Gn,θ,σ(·|p)πn,θ,σ(p)− Φ∞,P (·) ∣∣∣∣∣∣ ∣∣∣∣∣∣ ||Gn,θ,σ(·|p)− Φ∞,P (·)||TV πn,θ,σ(p), where the conditional cdfs Gn,θ,σ(·|p) and the model selection probabilities πn,θ,σ(p) have been introduced after (12). By the ‘uncorrelatedness’ assumption, we have that Φ∞,p = Φ∞,P for all p in the rangeO ≤ p ≤ P ; cf. Remark A.6. We hence obtain σ∗≤σ≤σ∗ ||Gn,θ,σ − Φ∞,P ||TV ≤ σ∗≤σ≤σ∗ ||Gn,θ,σ(·|p)− Φ∞,p(·)||TV πn,θ,σ(p). (44) Now for every p with O ≤ p ≤ P and for every ρ, 0 < ρ <∞, we can write σ∗≤σ≤σ∗ ||Gn,θ,σ(·|p)− Φ∞,p(·)||TV πn,θ,σ(p) ≤ max ‖θ[¬p]‖<ρ/ σ∗≤σ≤σ∗ ||Gn,θ,σ(·|p)− Φ∞,p(·)||TV , sup ‖θ[¬p]‖≥ρ/ σ∗≤σ≤σ∗ πn,θ,σ(p) . (45) In case p = P , we use here the convention that the second term in the maximum is absent and that the first supremum in the first term in the maximum extends over all of RP . Letting first n and then ρ go to infinity in (45), we may apply Lemmas C.2 and C.3 in Leeb and Pötscher (2005b) to conclude that the l.h.s. of (45), and hence the l.h.s. of (44), goes to zero as n→ ∞. ✷ C Proofs for Sections 2.1 to 2.2.2 In the proofs below it will be convenient to show the dependence of Φn,p and Φ∞,p on σ in the notation. Thus, in the following we shall write Φn,p,σ and Φ∞,p,σ, respectively, for the cdf of a k-variate Gaussian random vector with mean zero and variance-covariancematrix σ2A[p](X [p]′X [p]/n)−1A[p]′ and σ2A[p]Q[p : p]−1A[p]′, respectively. For convenience, let Φn,0,σ and Φ∞,0,σ denote the cdf of point-mass at zero in R The following lemma is elementary to prove, if we recall that bn,pz converges to b∞,pz as n → ∞ for every z ∈ ImA[p], the column space of A[p]. Lemma C.1 Suppose p > O. Define Rn,p(z, σ) = 1 − ∆σζn,p(bn,pz, cpσξn,p) and R∞,p(z, σ) = 1 − ∆σζ∞,p(b∞,pz, cpσξ∞,p) for z ∈ ImA[p], 0 < σ <∞. Let σ (n) converge to σ, 0 < σ < ∞. If ζ∞,p 6= 0, then Rn,p(z, σ (n)) converges to R∞,p(z, σ) for every z ∈ ImA[p]; if ζ∞,p = 0, then convergence holds for every z ∈ ImA[p], except possibly for z ∈ ImA[p] satisfying |b∞,pz| = cpσξ∞,p. [This exceptional subset of ImA[p] has rank(A[p])-dimensional Lebesgue measure zero since cpσξ∞,p > 0.] The following observation is useful in the proof of Proposition 2.2 below: Since the proposition depends on Y only through its distribution (cf. Remark 4.1), we may assume without loss of generality that the errors in (5) are given by ut = σεt, t ∈ N, with i.i.d. εt that are standard normal. In particular, all random variables involved are then defined on the same probability space. Proof of Proposition 2.2: Since Pn,θ,σ(p̄ = p0(θ)) → 1 by consistency, we may replace max{p̄,O} by p∗ = max{p0(θ),O} in the formula for Ǧn for the remainder of the proof. Furthermore, since σ̂ → σ in Pn,θ,σ-probability, each subsequence contains a further subsequence along which σ̂ → σ almost surely (with respect to the probability measure on the common probability space supporting all random variables involved), and we restrict ourselves to such a further subsequence for the moment. In particular, we write {σ̂ → σ} for the event that σ̂ converges to σ along the subsequence under consideration; clearly, the event {σ̂ → σ} has probability one. Also note that we can assume without loss of generality that σ̂ > 0 holds on this event (at least from some data-dependent n onwards), since σ > 0 holds. But then obviously q=p∗+1 ∆σ̂ξn,q (0, cqσ̂ξn,q) converges to q=p∗+1 ∆σξ∞,q (0, cqσξ∞,q), and Φ̂n,p∗(t) converges to Φ∞,p∗,σ(t) in total variation by Lemma A.3 of Leeb (2005) in case p∗ > 0, and trivially so in case p∗ = 0. This proves that the first term in the formula for Ǧn converges to the corresponding term in the formula for G∞,θ,σ in total variation. Next, consider the term in Ǧn that carries the index p > p∗. By Lemma A.3 in Leeb (2005), Φ̂n,p = Φn,p,σ̂ has a density dΦn,p,σ̂/dΦ∞,p,σ with respect to Φ∞,p,σ, which converges to 1 except on a set that has measure zero under Φ∞,p,σ. By Scheffé’s Lemma (Billingsley (1995), Theorem 16.12), dΦn,p,σ̂/dΦ∞,p,σ converges to 1 also in the L1(Φ∞,p,σ)-sense. By Lemma C.1, Rn,p(z, σ̂) converges to R∞,p(z, σ) except possibly on a set that has measure zero under Φ∞,p,σ. (Recall that Φ∞,p,σ is concentrated on ImA[p] and is not degenerate there.) Observing that |Rn,p(z, σ̂)| is uniformly bounded by 1, we obtain that Rn,p(z, σ̂) converges to R∞,p(z, σ) also in the L1(Φ∞,p,σ)-sense. Hence, ∥∥∥∥Rn,p(z, σ̂) dΦn,p,σ̂ dΦ∞,p,σ (z)−R∞,p(z, σ) ∥∥∥∥Rn,p(z, σ̂) dΦn,p,σ̂ dΦ∞,p,σ (z)−Rn,p(z, σ̂) ∥∥∥∥+ ‖Rn,p(z, σ̂)−R∞,p(z, σ)‖ (46) dΦn,p,σ̂ dΦ∞,p,σ (z)− 1 ∥∥∥∥+ ‖Rn,p(z, σ̂)−R∞,p(z, σ)‖ n→∞−→ 0 where ‖·‖ denotes the L1(Φ∞,p,σ)-norm. Since q=p+1 ∆σ̂ξn,q (0, cqσ̂ξn,q) obviously converges to∏P q=p+1 ∆σξ∞,q (0, cqσξ∞,q), the relation (46) shows that the term in Ǧn carrying the index p converges to the corresponding term in G∞,θ,σ in the total variation sense. This proves (18) along the subsequence under consideration. However, since any subsequence contains such a further subsequence, this establishes (18). Since Gn,θ,σ converges to G∞,θ,σ in total variation by Proposition 2.1, the claim in (17) also follows.✷ Before we prove the main result we observe that the total variation distance between Pn,θ,σ and Pn,ϑ,σ satisfies ||Pn,θ,σ − Pn,ϑ,σ||TV ≤ 2Φ(‖θ − ϑ‖λ max(X ′X)/2σ) − 1; furthermore, if θ(n) and ϑ(n) sat- ∥∥∥θ(n) − ϑ(n) ∥∥∥ = O(n−1/2), the sequence Pn,ϑ(n),σ is contiguous with respect to the sequence Pn,θ(n),σ (and vice versa). This follows exactly in the same way as Lemma A.1 in Leeb and Pötscher (2006a). Proof of Theorem 2.3: We first prove (20) and (21). For this purpose we make use of Lemma 3.1 in Leeb and Pötscher (2006a) with α = θ ∈ Mq∗−1, B = Mq∗ , Bn = {ϑ ∈ Mq∗ : ‖ϑ− θ‖ < ρ0n−1/2}, β = ϑ, ϕn(β) = Gn,ϑ,σ(t), ϕ̂n = Ĝn(t), where ρ0, 0 < ρ0 < ∞, will be chosen shortly (and σ is held fixed). The contiguity assumption of this lemma (as well as the mutual contiguity assumption used in the corrigendum to Leeb and Pötscher (2006a)) is satisfied in view of the preparatory remark above. It hence remains only to show that there exists a value of ρ0, 0 < ρ0 < ∞, such that δ in Lemma 3.1 of Leeb and Pötscher (2006a) (which represents the limit inferior of the oscillation of ϕn(·) over Bn) is positive. Applying Lemma 3.5(i) of Leeb and Pötscher (2006a) with ζn = ρ0n −1/2 and the set G0 equal to the set G, it remains, in light of Proposition 2.1, to show that there exists a ρ0, 0 < ρ0 < ∞, such that G∞,θ,σ,γ(t) as a function of γ is non-constant on the set {γ ∈ Mq∗ : ‖γ‖ < ρ0}. In view of Lemma 3.1 of Leeb and Pötscher (2006a), the corresponding δ0 can then be chosen as any positive number less than one-half of the oscillation of G∞,θ,σ,γ(t) over this set. That such a ρ0 indeed exists follows now from Lemma A.4 in Appendix A, where it is also shown that ρ0 and δ0 can be chosen such that they depend only on t, Q,A, σ, and cp for O < p ≤ P . This completes the proof of (20) and (21). To prove (22) we use Corollary 3.4 in Leeb and Pötscher (2006a) with the same identification of notation as above, with ζn = ρ0n −1/2, and with V = Mq∗ (viewed as a vector space isomorphic to ). The asymptotic uniform equicontinuity condition in that corollary is then satisfied in view of ||Pn,θ,σ − Pn,ϑ,σ||TV ≤ 2Φ(‖θ − ϑ‖λ max(X ′X)/2σ) − 1. Given that the positivity of δ∗ has already be es- tablished in the previous paragraph, applying Corollary 3.4(i) in Leeb and Pötscher (2006a) then establishes (22). ✷ Proof of Remark 2.4: The proof is similar to the proof of (22) just given, except for using Corol- lary 3.4(ii) and Lemma 3.5(ii) in Leeb and Pötscher (2006a) instead of Corollary 3.4(i) and Lemma 3.5(i) from that paper. Furthermore, Lemma A.5 in Appendix A instead of Lemma A.4 is used. ✷ Proof of Proposition 2.5: In view of Proposition B.1(b) and the fact that Φ̂n,P (·) = Φn,P,σ̂(·) holds (in case σ̂ > 0), it suffices to show that σ∗≤σ≤σ∗ ||Φn,P,σ(·)− Φ∞,P,σ(·)||TV n→∞−→ 0 (47) σ∗≤σ≤σ∗ Pn,θ,σ ||Φn,P,σ̂(·)− Φn,P,σ(·)||TV > δ ) n→∞−→ 0 (48) hold for each δ > 0, and for any constants σ∗ and σ ∗ satisfying 0 < σ∗ ≤ σ∗ <∞. [Note that the probability in (48) does in fact not depend on θ.] But this has already been established in the proof of Proposition 4.3 of Leeb and Pötscher (2005b). ✷ D Proofs for Section 3 Proof of Theorem 3.1: After rearranging the elements of θ (and hence the regressors) if necessary and then correspondingly rearranging the rows of the matrix A, we may assume without loss of generality that r∗ = (1, . . . , 1, 0), and hence that i(r∗) = P . That is, Mr∗ = MP−1 and Mrfull = MP . Furthermore, note that after this arrangement C ∞ 6= 0. Let p̂ be the model selection procedure introduced in Section 2 with O = P − 1, cP = c, and cO = 0. Let θ̃ be the corresponding post-model-selection estimator and let Gn,θ,σ(t) be as defined in Section 2.1. Condition (24) now implies: For every θ ∈ MP−1 which has exactly P − 1 non-zero coordinates Pn,θ,σ ({r̂ = rfull}N{p̂ = P}) = lim Pn,θ,σ ({r̂ = r∗}N{p̂ = P − 1}) = 0 (49) holds for every 0 < σ < ∞. Since the sequences Pn,ϑ(n),σ and Pn,θ,σ are contiguous for ϑ (n) satisfying∥∥∥θ − ϑ(n) ∥∥∥ = O(n−1/2) as remarked prior to the proof of Theorem 2.3 in Appendix C, it follows that condition (49) continues to hold with Pn,ϑ(n),σ replacing Pn,θ,σ. This implies that for every sequence of positive real numbers sn with sn = O(n −1/2), for every σ, 0 < σ < ∞, and for every θ ∈ MP−1 which has exactly P − 1 non-zero coordinates ||ϑ−θ||<sn ‖Kn,ϑ,σ −Gn,ϑ,σ‖TV → 0 (50) holds as n → ∞. From (50) we conclude that the limit of Kn,θ+γ/√n,σ (with respect to total variation distance) exists and coincides with G∞,θ,σ,γ . Repeating the proof of Theorem 2.3 with q ∗ = P , with Kn,ϑ,σ(t) replacing Gn,ϑ,σ(t), and with K̂n(t) replacing Ĝn(t) gives the desired result. ✷ E References Ahmed, S. E. & A. K. Basu (2000): Least squares, preliminary test and Stein-type estimation in general vector AR(p) models. Statistica Neerlandica 54, 47–66. Bauer, P., Pötscher, B. M. & P. 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Fixed Phase Quantum Search Algorithm Ahmed Younes∗ Department of Math. & Comp. Science Faculty of Science Alexandria University Alexandria, Egypt October 24, 2018 Abstract Building quantum devices using fixed operators is a must to simplify the hardware con- struction. Quantum search engine is not an exception. In this paper, a fixed phase quantum search algorithm that searches for M matches in an unstructured search space of size N will be presented. Selecting phase shifts of 1.91684π in the standard amplitude amplification will make the technique perform better so as to get probability of success at least 99.58% in better than any know fixed operator quantum search algorithms. The algorithm will be able to handle either a single match or multiple matches in the search space. The algorithm will find a match in O whether the number of matches is known or not in advance. 1 Introduction In 1996, Lov Grover [10] presented an algorithm that quantum mechanically searches an unstruc- tured list assuming that a unique match exists in the list with quadratic speed-up over classical algorithms. To be able to define the target problem of this paper, we have to organize the ef- forts done by others in that field. The unstructured search problem targeted by Grover’s original algorithm is deviated in the literature to the following four major problems: • Unstructured list with a unique match. • Unstructured list with one or more matches, where the number of matches is known • Unstructured list with one or more matches, where the number of matches is unknown. • Unstructured list with strictly multiple matches. The efforts done in all the above cases, similar to Grover’s original work, used quantum paral- lelism by preparing superposition that represents all the items in the list. The superposition could be uniform or arbitrary. The techniques used in most of the cases to amplify the amplitude(s) of the required state(s) have been generalized to an amplitude amplification technique that iterates ∗ayounes2@yahoo.com http://arxiv.org/abs/0704.1585v2 the operation URs (φ)U †Rt (ϕ), on U |s〉 where U is unitary operator, Rs (φ) = I− (1− eiφ) |s〉 〈s|, Rt (ϕ) = I − (1− eiϕ) |t〉 〈t|, |s〉 is the initial state of the system, |t〉 represents the target state(s) and I is the identity operator. Grover’s original algorithm replaces U be W , where W is the Walsh-Hadamard transform, pre- pares the superpositionW |0〉 (uniform superposition) and iteratesWRs (π)WRt (π) for O where N is the size of the list, which was shown be optimal to get the highest probability with the minimum number of iterations [23], such that there is only one match in the search space. In [11, 15, 9, 17, 1], Grover’s algorithm is generalized by showing that U can be replaced by almost any arbitrary superposition and the phase shifts φ and ϕ can be generalized to deal with the arbitrary superposition and/or to increase the probability of success even with a factor increase in the number of iterations to still run in O( N). These give a larger class of algorithms for amplitude amplification using variable operators from which Grover’s algorithm was shown to be a special case. In another direction, work has been done trying to generalize Grover’s algorithm with a uniform superposition for known number of multiple matches in the search space [3, 8, 7, 6], where it was shown that the required number of iterations is approximately π/4 N/M for small M/N , where M is the number of matches. The required number of iterations will increase for M > N/2, i.e. the problem will be harder where it might be excepted to be easier [19]. Another work has been done for known number of multiple matches with arbitrary superposition and phase shifts [18, 2, 4, 14, 16] where the same problem for multiple matches occurs. In [5, 18, 4], a hybrid algorithm was presented to deal with this problem by applying Grover’s fixed operators algorithm for π/4 N/M times then apply one more step using specific φ and ϕ according to the knowledge of the number of matches M to get the solution with probability close to certainty. Using this algorithm will increase the hardware cost since we have to build one more Rs and Rt for each particular M . For the sake of practicality, the operators should be fixed for any given M and are able to handle the problem with high probability whether or notM is known in advance. In [21, 22], Younes et al presented an algorithm that exploits entanglement and partial diffusion operator to perform the search and can perform in case of either a single match or multiple matches where the number of matches is known or not [22] covering the whole possible range, i.e. 1 ≤ M ≤ N . Grover described this algorithm as the best quantum search algorithm [12]. It can be shown that we can get the same probability of success of [21] using amplitude amplification with phase shifts φ = ϕ = π/2, although the amplitude amplification mechanism will be different. The mechanism used to manipulate the amplitudes could be useful in many applications, for example, superposition preparation and error-correction. For unknown number of matches, an algorithm for estimating the number of matches (quantum counting algorithm) was presented [5, 18]. In [3], another algorithm was presented to find a match even if the number of matches is unknown which will be able to work if M lies within the range 1 ≤M ≤ 3N/4 [22]. For strictly multiple matches, Younes et al [20] presented an algorithm which works very ef- ficiently only in case of multiple matches within the search space that splits the solution states over more states, inverts the sign of half of them (phase shift of -1) and keeps the other half unchanged every iteration. This will keep the mean of the amplitudes to a minimum for multiple matches. The same result was rediscovered by Grover using amplitude amplification with phase shifts φ = ϕ = π/3 [13], in both algorithms the behavior will be similar to the classical algorithms in the worst case. In this paper, we will propose a fixed phase quantum search algorithm that runs inO This algorithm is able to handle the range 1 ≤ M ≤ N for both known and unknown number of matches more reliably than known fixed operator quantum search algorithms that target this case. The plan of the paper is as follows: Section 2 introduces the general definition of the target unstructured search problem. Section 3 presents the algorithm for both known and unknown number of matches. The paper will end up with a general conclusion in Section 4. 2 Unstructured Search Problem Consider an unstructured list L of N items. For simplicity and without loss of generality we will assume that N = 2n for some positive integer n. Suppose the items in the list are labeled with the integers {0, 1, ..., N − 1}, and consider a function (oracle) f which maps an item i ∈ L to either 0 or 1 according to some properties this item should satisfy, i.e. f : L → {0, 1}. The problem is to find any i ∈ L such that f(i) = 1 assuming that such i exists in the list. In conventional computers, solving this problem needs O (N/M) calls to the oracle (query),where M is the number of items that satisfy the oracle. 3 Fixed Phase Algorithm 3.1 Known Number of Matches Assume that the system is initially in state |s〉 = |0〉. Assume that denotes a sum over i which are desired matches, and denotes a sum over i which are undesired items in the list. So, Applying U |s〉 we get, ∣ψ(0) = U |s〉 = 1√ ′ |i〉+ 1√ ′′ |i〉, (1) where U =W and the superscript in ∣ψ(0) represents the iteration number. Let M be the number of matches, sin(θ) = M/N and 0 < θ ≤ π/2, then the system can be re-written as follows, ∣ψ(0) = sin(θ) |ψ1〉+ cos(θ) |ψ0〉 , (2) where |ψ1〉 = |t〉 represents the matches subspace and |ψ0〉 represents the non-matches subspace. Let D = URs (φ)U †Rt (ϕ), Rs (φ) = I − (1 − eiφ) |s〉 〈s|, Rt (ϕ) = I − (1 − eiϕ) |t〉 〈t| and set φ = ϕ as the best choice [14]. Applying D on ∣ψ(0) we get, ∣ψ(1) ∣ψ(0) = a1 |ψ1〉+ b1 |ψ0〉 , (3) such that, a1 = sin(θ)(2 cos (δ) e iφ + 1), (4) b1 = e iφ cos(θ)(2 cos (δ) + 1), (5) where cos (δ) = 2 sin2(θ) sin2(φ )− 1. Let q represents the required number of iterations to get a match with the highest possible probability. After q applications of D on ∣ψ(0) we get, ∣ψ(q) ∣ψ(0) = aq |ψ1〉+ bq |ψ0〉 , (6) such that, aq = sin(θ) eiqφUq (y) + e i(q−1)φUq−1 (y) , (7) bq = cos(θ)e i(q−1)φ (Uq (y) + Uq−1 (y)) , (8) where y = cos(δ) and Uq is the Chebyshev polynomial of the second kind defined as follows, Uq (y) = sin ((q + 1) δ) sin (δ) . (9) Let P qs represents the probability of success to get a match after q iterations and P ns is the probability not to get a match after applying measurement, so P qs = |aq| and P qns = |bq| that P qs +P ns = 1. To calculate the required number of iterations q to get a match with certainty, one the following two approaches might be followed: • Analytically. The usual approach used in the literature when the number of matches M is known in advance is to equate P qs to 1 or P ns to 0 and then find an algebraic formula that represents the required number of iterations, as well as, the phase shifts φ and ϕ in terms on M . Using this approach is not possible for the case that the phase shifts should be fixed for an arbitrary M such that 1 ≤M ≤ N as shown in the following theorem. Theorem 3.1 (No Certainty Principle) Let D be an amplitude amplification operator such that D = URs (φ)U †Rt (ϕ), where U is unitary operator, Rs (φ) = I − (1 − eiφ) |s〉 〈s|, Rt (ϕ) = I − (1 − eiϕ) |t〉 〈t|, |s〉 is the initial state of the system, |t〉 represents the target state(s) and I is the identity operator. Let D performs on a system initially set to U |s〉. If the phase shifts φ and ϕ should be fixed, then iterating D an arbitrary number of times will not find a match with certainty for an arbitrary known number of matches M such that 1 ≤M ≤ N . Proof To prove this theorem, we will use the usual approach, i.e. start with P qs = 1 or P qns = 0 and calculate the required number of iterations q. Since P qs = |aq| and from Eqn.7, we can re-write P qs as follows setting φ = ϕ as the best choice [14], P qs = sin2 (θ) sin2 (δ) (1− cos (δ) cos ((2q + 1) δ) + 2 cos (φ) sin ((q + 1) δ) sin (qδ)) . (10) Setting P qs = 1 and using simple trigonometric identities we get, q = , i.e. the required number of iterations is independent of M , φ and ϕ, and represents an impossible value for a required number of iterations. • Direct Search. The alternative approach used in this paper is to empirically assume an algebraic form for the required number of iterations that satisfy the quadratic speed-up of the known quantum search algorithms and use a computer program to search for the best phase shift φ that satisfy the condition, max (min (P qs (φ))) such that 0 ≤ φ ≤ 2π and, 1 ≤ M ≤ N. (11) i.e. find the value of φ that maximize the minimum value of P qs over the range 1 ≤M ≤ N . 0 0.2 0.4 0.6 0.8 1 0.991 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999 Probability of Success Probability lower bound Figure 1: The probability of success the proposed algorithm after the required number of iterations. Assume that q = sin(θ) . Using this form for q, a computer program has been written using C language to find the best φ with precision 10−15 that satisfy the conditions shown in Eqn. 11. The program shows that using φ = 6.021930660106538 ≈ 1.91684π, the minimum probability of success will be at least 99.58% compared with 87.88 % for Younes et al [22] and 50% for the original Grover’s algorithm [3] as shown in Fig. 2. To prove these results, using φ = 1.91684π, the lower bound for the probability of success is as follows as shown in Fig. 1. P qs = sin2(θ) sin2(δ) (1− cos (δ) cos ((2q + 1) δ) + 2 cos (φ) sin ((q + 1) δ) sin (qδ)) sin2(θ) sin2(δ) (1− cos (δ) cos ((2q + 1) δ) + cos (φ) cos (δ)− cos (φ) cos ((2q + 1) δ)) ≥ sin sin2(δ) (1 + cos2 (δ) + 2 cos (φ) cos (δ)) ≥ 0.9958. where, cos (δ) = 2 sin2(θ) sin2(φ )− 1, 0 < θ ≤ π/2, and cos ((2q + 1) δ) ≤ −cos(δ). 3.2 Unknown Number of Matches In case we do not know the number of matches M in advance, we can apply the algorithm shown in [3] for 1 ≤M ≤ N by replacing Grover’s step with the proposed algorithm. The algorithm can be summarized as follows, 1- Initialize m = 1 and λ = 8/7. (where λ can take any value between 1 and 4/3) 2- Pick an integer j between 0 and m− 1 in a uniform random manner. 3- Run j iterations of the proposed algorithm on the state ∣ψ(0) ∣ψ(j) ∣ψ(0) . (13) 0 0.2 0.4 0.6 0.8 1 Grover’s Younes et al[21] Fixed Phase Figure 2: The probability of success of Grover’s algorithm, Younes et al algorithm [21] and the proposed algorithm after the required number of iterations. 4- Measure the register ∣ψ(j) and assume i is the output. 5- If f(i) = 1, then we found a solution and exit. 6- Set m = min and go to step 2. where m represents the range of random numbers (step 2), j represents the random number of iterations (step3), and λ is a factor used to increase the range of random numbers after each trial (step 6). For the sake of simplicity and to be able to compare the performance of this algorithm with that shown in [3], we will try to follow the same style of analysis used in [3]. Before we construct the analysis, we need the following lemmas. Lemma 3.2 For any positive integer m and real numbers θ, δ such that cos (δ) = c sin2(θ) − 1, 0 < θ ≤ π/2 where c = 2 sin2(φ ) is a constant, sin2 ((q + 1) δ) + sin2 (qδ) = m− cos (δ) sin (2mδ) 2 sin (δ) Proof By mathematical induction. Lemma 3.3 For any positive integer m and real numbers θ, δ such that cos (δ) = c sin2(θ) − 1, 0 < θ ≤ π/2 where c = 2 sin2(φ ) is a constant, sin ((q + 1) δ) sin (qδ) = cos (δ)− sin (2mδ) 4 sin (δ) Proof By mathematical induction. Lemma 3.4 AssumeM is the unknown number of matches such that 1 ≤M ≤ N . Let θ, δ be real numbers such that cos (δ) = 2 sin2(θ) sin2(φ )− 1, sin2(θ) =M/N , φ = 1.91684π and 0 < θ ≤ π/2. Let m be any positive integer. Let q be any integer picked in a uniform random manner between 0 and m− 1. Measuring the register after applying q iterations of the proposed algorithm starting from the initial state, the probability Pm of finding a solution is as follows, c (1− cos (δ)) 1 + cos (δ) cos (φ)− (cos (δ) + cos (φ)) sin (2mδ) 2m sin (δ) where c = 2 sin2(φ ), then Pm ≥ 1/4 for m ≥ 1/ sin (δ) and small M/N . Proof The average probability of success when applying q iterations of the proposed algorithm when 0 ≤ q ≤ m is picked in a uniform random manner is as follows, sin2(θ) m sin2(δ) sin2 ((q + 1) δ) + sin2 (qδ) + 2 cos (φ) sin ((q + 1) δ) sin (qδ) sin2(θ) m sin2(δ) m− cos(δ) sin(2mδ) 2 sin(δ) + cos (φ) cos (δ)− cos(φ) sin(2mδ) 2 sin(δ) c(1−cos(δ)) 1 + cos (δ) cos (φ)− (cos(δ)+cos(φ)) sin(2mδ) 2m sin(δ) If m ≥ 1/ sin (δ) and M ≪ N then cos (δ) ≈ −1, so, 1− cos (φ)− (cos (φ)− 1) sin (2mδ) 1− cos (φ)− (1− cos (φ)) = 0.25 where −1 ≤ sin (2mδ) ≤ 1 for 0 < θ ≤ π/2. We calculate the total expected number of iterations as done in Theorem 3 in [3]. Assume that mq ≥ 1/ sin (δ), and vq = ⌈logλmq⌉. Notice that, mq = O for 1 ≤M ≤ N , then: 1- The total expected number of iterations to reach the critical stage, i.e. when m ≥ mq: λv−1 ≤ 1 2 (λ− 1) mq = 3.5mq. (14) 2- The total expected number of iterations after reaching the critical stage: λvq+u = 2 (1− 0.75λ) mq = 3.5mq. (15) The total expected number of iterations whether we reach to the critical stage or not is 7mq which is in O( N/M) for 1 ≤M ≤ N . When this algorithm employed Grover’s algorithm, and based on the conditionmG ≥ 1/ sin (2θG) = for M ≤ 3N/4,the total expected number of iterations is approximately 8mG for 1 ≤ M ≤ 3N/4. Employing the proposed algorithm instead, and based on the condition 0 0.2 0.4 0.6 0.8 1 s Fixed Phase Younes et al[22] Grover’s Figure 3: The actual behavior of the functions representing the total expected number of iterations for Grover’s algorithm, Younes et al algorithm [22] and the proposed algorithm taking λ = 8/7, where the number of iterations is the flooring of the values (step function). mq ≥ 1/ sin (δ) = O ,the total expected number of iterations is approximately 7mq for 1 ≤M ≤ N , i.e. the algorithm will be able to handle the whole range, since mq will be able to act as a lower bound for q over 1 ≤ M ≤ N . Fig. 3 compares between the total expected number of iterations for Grover’s algorithm, Younes et al algorithm [22] and the Fixed Phase algorithm taking λ = 8/7. 4 Conclusion To be able to build a practical search engine, the engine should be constructed from fixed operators that can handle the whole possible range of the search problem, i.e. whether a single match or multiple matches exist in the search space. It should also be able to handle the case where the number of matches is unknown. The engine should perform with the highest possible probability after performing the required number of iterations. In this paper, a fixed phase quantum search algorithm is presented. It was shown that selecting the phase shifts to 1.91684π could enhance the searching process so as to get a solution with probability at least 99.58%. The algorithm still achieves the quadratic speed up of Grover’s original algorithm. It was shown that Younes et al algorithm [22] might perform better in case the number of matches is unknown, although the presented algorithm might scale similar with an acceptable delay. i.e. both run in O . In that sense, the Fixed Phase algorithm can act efficiently in all the possible classes of the unstructured search problem. References [1] E. Biham and D. Dan Kenigsberg. Grover’s quantum search algorithm for an arbitrary initial mixed state. Physical Review A, 66:062301, 2002. [2] D. Biron, O. Biham, E. Biham, M. Grassl, and D. A. Lidar. Generalized Grover search algorithm for arbitrary initial amplitude distribution. arXiv e-Print quant-ph/9801066, 1998. [3] M. Boyer, G. Brassard, P. Høyer, and A. Tapp. Tight bounds on quantum searching. Fortschritte der Physik, 46:493, 1998. [4] G. Brassard, P. Høyer, M. Mosca, , and A. Tapp. Quantum amplitude amplification and estimation. arXiv e-Print quant-ph/0005055, 2000. [5] G. Brassard, P. Høyer, and A. Tapp. Quantum counting. arXiv e-Print quant-ph/9805082, 1998. [6] G. Chen and S. Fulling. Generalization of Grover’s algorithm to multiobject search in quantum computing, part II: General unitary transformation. arXiv e-Print quant-ph/0007124, 2000. [7] G. Chen, S. Fulling, and J. Chen. Generalization of Grover’s algorithm to multiobject search in quantum computing, part I: Continuous time and discrete time. arXiv e-Print quant- ph/0007123, 2000. [8] G. Chen, S. Fulling, and M. Scully. Grover’s algorithm for multiobject search in quantum computing. arXiv e-Print quant-ph/9909040, 1999. [9] A. Galindo and M. A. Martin-Delgado. Family of Grover’s quantum-searching algorithms. Physical Review A, 62:062303, 2000. [10] L. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, pages 212–219, 1996. [11] L. Grover. Quantum computers can search rapidly by using almost any transformation. Physical Review Letters, 80(19):4329–4332, 1998. [12] L. Grover. A different kind of quantum search. arXiv e-Print quant-ph/0503205, 2005. [13] L. Grover. Fixed-point quantum search. Phys. Rev. Lett., 95(15):150501, 2005. [14] P. Høyer. Arbitrary phases in quantum amplitude amplification. Physical Review A, 62:052304, 2000. [15] R. Jozsa. Searching in Grover’s algorithm. arXiv e-Print quant-ph/9901021, 1999. [16] C. Li, C. Hwang, J. Hsieh, and K. Wang. A general phase matching condition for quantum searching algorithm. arXiv e-Print quant-ph/0108086, 2001. [17] G. L. Long. Grover algorithm with zero theoretical failure rate. arXiv e-Print quant- ph/0106071, 2001. [18] M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector anal- ysis. In Proceedings of Randomized Algorithms, Workshop of Mathematical Foundations of Computer Science, pages 90–100, 1998. [19] M. Nielsen and I. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, United Kingdom, 2000. [20] A. Younes, J. Rowe, and J. Miller. A hybrid quantum search engine: A fast quantum algo- rithm for multiple matches. In Proceedings of the 2nd International Computer Engineering Conference, 2003. [21] A. Younes, J. Rowe, and J. Miller. Quantum search algorithm with more reliable behaviour using partial diffusion. In Proceedings of the 7th International Conference on Quantum Com- munication, Measurement and Computing, 2004. [22] A. Younes, J. Rowe, and J. Miller. Quantum searching via entangelment and partial diffusion. Technical Report CSR-04-9, University of Birmingham, School of Computer Science, arXiv e-Print quant-ph/0406207, June 2004. [23] C. Zalka. Grover’s quantum searching algorithm is optimal. Physical Review A, 60(4):2746– 2751, 1999. Introduction Unstructured Search Problem Fixed Phase Algorithm Known Number of Matches Unknown Number of Matches Conclusion

Building quantum devices using fixed operators is a must to simplify the hardware construction. Quantum search engine is not an exception. In this paper, a fixed phase quantum search algorithm that searches for M matches in an unstructured search space of size N will be presented. Selecting phase shifts of 1.91684\pi in the standard amplitude amplification will make the technique perform better so as to get probability of success at least 99.58% in O(sqrt(N/M)) better than any know fixed operator quantum search algorithms. The algorithm will be able to handle either a single match or multiple matches in the search space. The algorithm will find a match in O(sqrt(N/M)) whether the number of matches is known or not in advance.

Introduction In 1996, Lov Grover [10] presented an algorithm that quantum mechanically searches an unstruc- tured list assuming that a unique match exists in the list with quadratic speed-up over classical algorithms. To be able to define the target problem of this paper, we have to organize the ef- forts done by others in that field. The unstructured search problem targeted by Grover’s original algorithm is deviated in the literature to the following four major problems: • Unstructured list with a unique match. • Unstructured list with one or more matches, where the number of matches is known • Unstructured list with one or more matches, where the number of matches is unknown. • Unstructured list with strictly multiple matches. The efforts done in all the above cases, similar to Grover’s original work, used quantum paral- lelism by preparing superposition that represents all the items in the list. The superposition could be uniform or arbitrary. The techniques used in most of the cases to amplify the amplitude(s) of the required state(s) have been generalized to an amplitude amplification technique that iterates ∗ayounes2@yahoo.com http://arxiv.org/abs/0704.1585v2 the operation URs (φ)U †Rt (ϕ), on U |s〉 where U is unitary operator, Rs (φ) = I− (1− eiφ) |s〉 〈s|, Rt (ϕ) = I − (1− eiϕ) |t〉 〈t|, |s〉 is the initial state of the system, |t〉 represents the target state(s) and I is the identity operator. Grover’s original algorithm replaces U be W , where W is the Walsh-Hadamard transform, pre- pares the superpositionW |0〉 (uniform superposition) and iteratesWRs (π)WRt (π) for O where N is the size of the list, which was shown be optimal to get the highest probability with the minimum number of iterations [23], such that there is only one match in the search space. In [11, 15, 9, 17, 1], Grover’s algorithm is generalized by showing that U can be replaced by almost any arbitrary superposition and the phase shifts φ and ϕ can be generalized to deal with the arbitrary superposition and/or to increase the probability of success even with a factor increase in the number of iterations to still run in O( N). These give a larger class of algorithms for amplitude amplification using variable operators from which Grover’s algorithm was shown to be a special case. In another direction, work has been done trying to generalize Grover’s algorithm with a uniform superposition for known number of multiple matches in the search space [3, 8, 7, 6], where it was shown that the required number of iterations is approximately π/4 N/M for small M/N , where M is the number of matches. The required number of iterations will increase for M > N/2, i.e. the problem will be harder where it might be excepted to be easier [19]. Another work has been done for known number of multiple matches with arbitrary superposition and phase shifts [18, 2, 4, 14, 16] where the same problem for multiple matches occurs. In [5, 18, 4], a hybrid algorithm was presented to deal with this problem by applying Grover’s fixed operators algorithm for π/4 N/M times then apply one more step using specific φ and ϕ according to the knowledge of the number of matches M to get the solution with probability close to certainty. Using this algorithm will increase the hardware cost since we have to build one more Rs and Rt for each particular M . For the sake of practicality, the operators should be fixed for any given M and are able to handle the problem with high probability whether or notM is known in advance. In [21, 22], Younes et al presented an algorithm that exploits entanglement and partial diffusion operator to perform the search and can perform in case of either a single match or multiple matches where the number of matches is known or not [22] covering the whole possible range, i.e. 1 ≤ M ≤ N . Grover described this algorithm as the best quantum search algorithm [12]. It can be shown that we can get the same probability of success of [21] using amplitude amplification with phase shifts φ = ϕ = π/2, although the amplitude amplification mechanism will be different. The mechanism used to manipulate the amplitudes could be useful in many applications, for example, superposition preparation and error-correction. For unknown number of matches, an algorithm for estimating the number of matches (quantum counting algorithm) was presented [5, 18]. In [3], another algorithm was presented to find a match even if the number of matches is unknown which will be able to work if M lies within the range 1 ≤M ≤ 3N/4 [22]. For strictly multiple matches, Younes et al [20] presented an algorithm which works very ef- ficiently only in case of multiple matches within the search space that splits the solution states over more states, inverts the sign of half of them (phase shift of -1) and keeps the other half unchanged every iteration. This will keep the mean of the amplitudes to a minimum for multiple matches. The same result was rediscovered by Grover using amplitude amplification with phase shifts φ = ϕ = π/3 [13], in both algorithms the behavior will be similar to the classical algorithms in the worst case. In this paper, we will propose a fixed phase quantum search algorithm that runs inO This algorithm is able to handle the range 1 ≤ M ≤ N for both known and unknown number of matches more reliably than known fixed operator quantum search algorithms that target this case. The plan of the paper is as follows: Section 2 introduces the general definition of the target unstructured search problem. Section 3 presents the algorithm for both known and unknown number of matches. The paper will end up with a general conclusion in Section 4. 2 Unstructured Search Problem Consider an unstructured list L of N items. For simplicity and without loss of generality we will assume that N = 2n for some positive integer n. Suppose the items in the list are labeled with the integers {0, 1, ..., N − 1}, and consider a function (oracle) f which maps an item i ∈ L to either 0 or 1 according to some properties this item should satisfy, i.e. f : L → {0, 1}. The problem is to find any i ∈ L such that f(i) = 1 assuming that such i exists in the list. In conventional computers, solving this problem needs O (N/M) calls to the oracle (query),where M is the number of items that satisfy the oracle. 3 Fixed Phase Algorithm 3.1 Known Number of Matches Assume that the system is initially in state |s〉 = |0〉. Assume that denotes a sum over i which are desired matches, and denotes a sum over i which are undesired items in the list. So, Applying U |s〉 we get, ∣ψ(0) = U |s〉 = 1√ ′ |i〉+ 1√ ′′ |i〉, (1) where U =W and the superscript in ∣ψ(0) represents the iteration number. Let M be the number of matches, sin(θ) = M/N and 0 < θ ≤ π/2, then the system can be re-written as follows, ∣ψ(0) = sin(θ) |ψ1〉+ cos(θ) |ψ0〉 , (2) where |ψ1〉 = |t〉 represents the matches subspace and |ψ0〉 represents the non-matches subspace. Let D = URs (φ)U †Rt (ϕ), Rs (φ) = I − (1 − eiφ) |s〉 〈s|, Rt (ϕ) = I − (1 − eiϕ) |t〉 〈t| and set φ = ϕ as the best choice [14]. Applying D on ∣ψ(0) we get, ∣ψ(1) ∣ψ(0) = a1 |ψ1〉+ b1 |ψ0〉 , (3) such that, a1 = sin(θ)(2 cos (δ) e iφ + 1), (4) b1 = e iφ cos(θ)(2 cos (δ) + 1), (5) where cos (δ) = 2 sin2(θ) sin2(φ )− 1. Let q represents the required number of iterations to get a match with the highest possible probability. After q applications of D on ∣ψ(0) we get, ∣ψ(q) ∣ψ(0) = aq |ψ1〉+ bq |ψ0〉 , (6) such that, aq = sin(θ) eiqφUq (y) + e i(q−1)φUq−1 (y) , (7) bq = cos(θ)e i(q−1)φ (Uq (y) + Uq−1 (y)) , (8) where y = cos(δ) and Uq is the Chebyshev polynomial of the second kind defined as follows, Uq (y) = sin ((q + 1) δ) sin (δ) . (9) Let P qs represents the probability of success to get a match after q iterations and P ns is the probability not to get a match after applying measurement, so P qs = |aq| and P qns = |bq| that P qs +P ns = 1. To calculate the required number of iterations q to get a match with certainty, one the following two approaches might be followed: • Analytically. The usual approach used in the literature when the number of matches M is known in advance is to equate P qs to 1 or P ns to 0 and then find an algebraic formula that represents the required number of iterations, as well as, the phase shifts φ and ϕ in terms on M . Using this approach is not possible for the case that the phase shifts should be fixed for an arbitrary M such that 1 ≤M ≤ N as shown in the following theorem. Theorem 3.1 (No Certainty Principle) Let D be an amplitude amplification operator such that D = URs (φ)U †Rt (ϕ), where U is unitary operator, Rs (φ) = I − (1 − eiφ) |s〉 〈s|, Rt (ϕ) = I − (1 − eiϕ) |t〉 〈t|, |s〉 is the initial state of the system, |t〉 represents the target state(s) and I is the identity operator. Let D performs on a system initially set to U |s〉. If the phase shifts φ and ϕ should be fixed, then iterating D an arbitrary number of times will not find a match with certainty for an arbitrary known number of matches M such that 1 ≤M ≤ N . Proof To prove this theorem, we will use the usual approach, i.e. start with P qs = 1 or P qns = 0 and calculate the required number of iterations q. Since P qs = |aq| and from Eqn.7, we can re-write P qs as follows setting φ = ϕ as the best choice [14], P qs = sin2 (θ) sin2 (δ) (1− cos (δ) cos ((2q + 1) δ) + 2 cos (φ) sin ((q + 1) δ) sin (qδ)) . (10) Setting P qs = 1 and using simple trigonometric identities we get, q = , i.e. the required number of iterations is independent of M , φ and ϕ, and represents an impossible value for a required number of iterations. • Direct Search. The alternative approach used in this paper is to empirically assume an algebraic form for the required number of iterations that satisfy the quadratic speed-up of the known quantum search algorithms and use a computer program to search for the best phase shift φ that satisfy the condition, max (min (P qs (φ))) such that 0 ≤ φ ≤ 2π and, 1 ≤ M ≤ N. (11) i.e. find the value of φ that maximize the minimum value of P qs over the range 1 ≤M ≤ N . 0 0.2 0.4 0.6 0.8 1 0.991 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999 Probability of Success Probability lower bound Figure 1: The probability of success the proposed algorithm after the required number of iterations. Assume that q = sin(θ) . Using this form for q, a computer program has been written using C language to find the best φ with precision 10−15 that satisfy the conditions shown in Eqn. 11. The program shows that using φ = 6.021930660106538 ≈ 1.91684π, the minimum probability of success will be at least 99.58% compared with 87.88 % for Younes et al [22] and 50% for the original Grover’s algorithm [3] as shown in Fig. 2. To prove these results, using φ = 1.91684π, the lower bound for the probability of success is as follows as shown in Fig. 1. P qs = sin2(θ) sin2(δ) (1− cos (δ) cos ((2q + 1) δ) + 2 cos (φ) sin ((q + 1) δ) sin (qδ)) sin2(θ) sin2(δ) (1− cos (δ) cos ((2q + 1) δ) + cos (φ) cos (δ)− cos (φ) cos ((2q + 1) δ)) ≥ sin sin2(δ) (1 + cos2 (δ) + 2 cos (φ) cos (δ)) ≥ 0.9958. where, cos (δ) = 2 sin2(θ) sin2(φ )− 1, 0 < θ ≤ π/2, and cos ((2q + 1) δ) ≤ −cos(δ). 3.2 Unknown Number of Matches In case we do not know the number of matches M in advance, we can apply the algorithm shown in [3] for 1 ≤M ≤ N by replacing Grover’s step with the proposed algorithm. The algorithm can be summarized as follows, 1- Initialize m = 1 and λ = 8/7. (where λ can take any value between 1 and 4/3) 2- Pick an integer j between 0 and m− 1 in a uniform random manner. 3- Run j iterations of the proposed algorithm on the state ∣ψ(0) ∣ψ(j) ∣ψ(0) . (13) 0 0.2 0.4 0.6 0.8 1 Grover’s Younes et al[21] Fixed Phase Figure 2: The probability of success of Grover’s algorithm, Younes et al algorithm [21] and the proposed algorithm after the required number of iterations. 4- Measure the register ∣ψ(j) and assume i is the output. 5- If f(i) = 1, then we found a solution and exit. 6- Set m = min and go to step 2. where m represents the range of random numbers (step 2), j represents the random number of iterations (step3), and λ is a factor used to increase the range of random numbers after each trial (step 6). For the sake of simplicity and to be able to compare the performance of this algorithm with that shown in [3], we will try to follow the same style of analysis used in [3]. Before we construct the analysis, we need the following lemmas. Lemma 3.2 For any positive integer m and real numbers θ, δ such that cos (δ) = c sin2(θ) − 1, 0 < θ ≤ π/2 where c = 2 sin2(φ ) is a constant, sin2 ((q + 1) δ) + sin2 (qδ) = m− cos (δ) sin (2mδ) 2 sin (δ) Proof By mathematical induction. Lemma 3.3 For any positive integer m and real numbers θ, δ such that cos (δ) = c sin2(θ) − 1, 0 < θ ≤ π/2 where c = 2 sin2(φ ) is a constant, sin ((q + 1) δ) sin (qδ) = cos (δ)− sin (2mδ) 4 sin (δ) Proof By mathematical induction. Lemma 3.4 AssumeM is the unknown number of matches such that 1 ≤M ≤ N . Let θ, δ be real numbers such that cos (δ) = 2 sin2(θ) sin2(φ )− 1, sin2(θ) =M/N , φ = 1.91684π and 0 < θ ≤ π/2. Let m be any positive integer. Let q be any integer picked in a uniform random manner between 0 and m− 1. Measuring the register after applying q iterations of the proposed algorithm starting from the initial state, the probability Pm of finding a solution is as follows, c (1− cos (δ)) 1 + cos (δ) cos (φ)− (cos (δ) + cos (φ)) sin (2mδ) 2m sin (δ) where c = 2 sin2(φ ), then Pm ≥ 1/4 for m ≥ 1/ sin (δ) and small M/N . Proof The average probability of success when applying q iterations of the proposed algorithm when 0 ≤ q ≤ m is picked in a uniform random manner is as follows, sin2(θ) m sin2(δ) sin2 ((q + 1) δ) + sin2 (qδ) + 2 cos (φ) sin ((q + 1) δ) sin (qδ) sin2(θ) m sin2(δ) m− cos(δ) sin(2mδ) 2 sin(δ) + cos (φ) cos (δ)− cos(φ) sin(2mδ) 2 sin(δ) c(1−cos(δ)) 1 + cos (δ) cos (φ)− (cos(δ)+cos(φ)) sin(2mδ) 2m sin(δ) If m ≥ 1/ sin (δ) and M ≪ N then cos (δ) ≈ −1, so, 1− cos (φ)− (cos (φ)− 1) sin (2mδ) 1− cos (φ)− (1− cos (φ)) = 0.25 where −1 ≤ sin (2mδ) ≤ 1 for 0 < θ ≤ π/2. We calculate the total expected number of iterations as done in Theorem 3 in [3]. Assume that mq ≥ 1/ sin (δ), and vq = ⌈logλmq⌉. Notice that, mq = O for 1 ≤M ≤ N , then: 1- The total expected number of iterations to reach the critical stage, i.e. when m ≥ mq: λv−1 ≤ 1 2 (λ− 1) mq = 3.5mq. (14) 2- The total expected number of iterations after reaching the critical stage: λvq+u = 2 (1− 0.75λ) mq = 3.5mq. (15) The total expected number of iterations whether we reach to the critical stage or not is 7mq which is in O( N/M) for 1 ≤M ≤ N . When this algorithm employed Grover’s algorithm, and based on the conditionmG ≥ 1/ sin (2θG) = for M ≤ 3N/4,the total expected number of iterations is approximately 8mG for 1 ≤ M ≤ 3N/4. Employing the proposed algorithm instead, and based on the condition 0 0.2 0.4 0.6 0.8 1 s Fixed Phase Younes et al[22] Grover’s Figure 3: The actual behavior of the functions representing the total expected number of iterations for Grover’s algorithm, Younes et al algorithm [22] and the proposed algorithm taking λ = 8/7, where the number of iterations is the flooring of the values (step function). mq ≥ 1/ sin (δ) = O ,the total expected number of iterations is approximately 7mq for 1 ≤M ≤ N , i.e. the algorithm will be able to handle the whole range, since mq will be able to act as a lower bound for q over 1 ≤ M ≤ N . Fig. 3 compares between the total expected number of iterations for Grover’s algorithm, Younes et al algorithm [22] and the Fixed Phase algorithm taking λ = 8/7. 4 Conclusion To be able to build a practical search engine, the engine should be constructed from fixed operators that can handle the whole possible range of the search problem, i.e. whether a single match or multiple matches exist in the search space. It should also be able to handle the case where the number of matches is unknown. The engine should perform with the highest possible probability after performing the required number of iterations. In this paper, a fixed phase quantum search algorithm is presented. It was shown that selecting the phase shifts to 1.91684π could enhance the searching process so as to get a solution with probability at least 99.58%. The algorithm still achieves the quadratic speed up of Grover’s original algorithm. It was shown that Younes et al algorithm [22] might perform better in case the number of matches is unknown, although the presented algorithm might scale similar with an acceptable delay. i.e. both run in O . In that sense, the Fixed Phase algorithm can act efficiently in all the possible classes of the unstructured search problem. References [1] E. Biham and D. Dan Kenigsberg. Grover’s quantum search algorithm for an arbitrary initial mixed state. Physical Review A, 66:062301, 2002. [2] D. Biron, O. Biham, E. Biham, M. Grassl, and D. A. Lidar. Generalized Grover search algorithm for arbitrary initial amplitude distribution. arXiv e-Print quant-ph/9801066, 1998. [3] M. Boyer, G. Brassard, P. Høyer, and A. Tapp. Tight bounds on quantum searching. Fortschritte der Physik, 46:493, 1998. [4] G. Brassard, P. Høyer, M. Mosca, , and A. Tapp. Quantum amplitude amplification and estimation. arXiv e-Print quant-ph/0005055, 2000. [5] G. Brassard, P. Høyer, and A. Tapp. Quantum counting. arXiv e-Print quant-ph/9805082, 1998. [6] G. Chen and S. Fulling. Generalization of Grover’s algorithm to multiobject search in quantum computing, part II: General unitary transformation. arXiv e-Print quant-ph/0007124, 2000. [7] G. Chen, S. Fulling, and J. Chen. Generalization of Grover’s algorithm to multiobject search in quantum computing, part I: Continuous time and discrete time. arXiv e-Print quant- ph/0007123, 2000. [8] G. Chen, S. Fulling, and M. Scully. Grover’s algorithm for multiobject search in quantum computing. arXiv e-Print quant-ph/9909040, 1999. [9] A. Galindo and M. A. Martin-Delgado. Family of Grover’s quantum-searching algorithms. Physical Review A, 62:062303, 2000. [10] L. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, pages 212–219, 1996. [11] L. Grover. Quantum computers can search rapidly by using almost any transformation. Physical Review Letters, 80(19):4329–4332, 1998. [12] L. Grover. A different kind of quantum search. arXiv e-Print quant-ph/0503205, 2005. [13] L. Grover. Fixed-point quantum search. Phys. Rev. Lett., 95(15):150501, 2005. [14] P. Høyer. Arbitrary phases in quantum amplitude amplification. Physical Review A, 62:052304, 2000. [15] R. Jozsa. Searching in Grover’s algorithm. arXiv e-Print quant-ph/9901021, 1999. [16] C. Li, C. Hwang, J. Hsieh, and K. Wang. A general phase matching condition for quantum searching algorithm. arXiv e-Print quant-ph/0108086, 2001. [17] G. L. Long. Grover algorithm with zero theoretical failure rate. arXiv e-Print quant- ph/0106071, 2001. [18] M. Mosca. Quantum searching, counting and amplitude amplification by eigenvector anal- ysis. In Proceedings of Randomized Algorithms, Workshop of Mathematical Foundations of Computer Science, pages 90–100, 1998. [19] M. Nielsen and I. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, United Kingdom, 2000. [20] A. Younes, J. Rowe, and J. Miller. A hybrid quantum search engine: A fast quantum algo- rithm for multiple matches. In Proceedings of the 2nd International Computer Engineering Conference, 2003. [21] A. Younes, J. Rowe, and J. Miller. Quantum search algorithm with more reliable behaviour using partial diffusion. In Proceedings of the 7th International Conference on Quantum Com- munication, Measurement and Computing, 2004. [22] A. Younes, J. Rowe, and J. Miller. Quantum searching via entangelment and partial diffusion. Technical Report CSR-04-9, University of Birmingham, School of Computer Science, arXiv e-Print quant-ph/0406207, June 2004. [23] C. Zalka. Grover’s quantum searching algorithm is optimal. Physical Review A, 60(4):2746– 2751, 1999. Introduction Unstructured Search Problem Fixed Phase Algorithm Known Number of Matches Unknown Number of Matches Conclusion

Core excitation in the elastic scattering and breakup of 11Be on protons N. C. Summers1, 2, 3 and F. M. Nunes3, 4 Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996 Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854 National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824 Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824 (Dated: October 25, 2018) The elastic scattering and breakup of 11Be from a proton target at intermediate energies is studied. We explore the role of core excitation in the reaction mechanism. Comparison with the data suggests that there is still missing physics in the description. PACS numbers: 24.10.Eq, 25.40.Cm, 25.60.Gc, 27.20.+n I. INTRODUCTION Nuclear reactions offer the most diverse methods to study nuclei at the limit of stability. Understanding re- action mechanisms in nuclear processes involving nuclei near the driplines is of great importance, particularly at this time, when there is such a high demand for accuracy on the structure information to be extracted from the data. Reaction and structure models are undoubtedly entangled, therefore improving reaction models often im- plies incorporating more detailed structure models in the description [1]. It is generally accepted that, in reactions with loosely bound nuclei, the coupling to the continuum needs to be considered. Continuum effects are very much enhanced in breakup but can also have imprints on other reac- tion channels, for example elastic and inelastic scatter- ing. One framework that explicitly includes continuum effects is the Continuum Discretized Coupled Channel (CDCC) method [2]. A large amount of work has been devoted to the analysis of experiments within this frame- work [3, 4, 5, 6, 7] and in general results are very good. The eXtended Continuum Discretized Coupled Chan- nel (XCDCC) method [8, 9] was recently developed. It brings together a coupled channel description of the pro- jectile with a coupled channel model of the reaction, en- abling the description of interference between the multi- channel components of the projectile as well as dynam- ical excitation of the core within the projectile, during the reaction [9]. The model has been applied to the breakup of 11Be→10Be+n and 17C→16C+n on 9Be at ≈ 60 MeV/nucleon [8, 9]. The projectiles are described within a two-body core + n multi-channel model, where the core can be in the ground state but also in an ex- cited state. This model produces breakup cross sections to specific final states of the core, given a coupled chan- nel Hamiltonian for the projectile. Results presented in Ref. [8, 9] show that core excitation effects in the total cross section to the ground state of the core are small, but become very large when considering the total popula- tion of the core’s excited state. Other differences can be seen in angular and energy distributions but at present no such data is available. The effect of core excitation needs to be studied in other regimes, for very light and very heavy targets, as well as a variety of energies. In this work we concentrate on the proton target. In this case the process is nuclear driven, and recoil effects are very important. Several reactions of 11Be on protons have been measured in a number of facilities, namely elastic scattering at 50 MeV/nucleon [10], quasi-elastic and breakup at 64 MeV/nucleon [11] as well as transfer at 35 MeV/nucleon [12]. 11Be proton elastic scattering at 49.3 MeV/nucleon was performed in GANIL [10], at the same time as the elastic scattering of the core 10Be at 59.4 MeV/nucleon. Even though the outcoming 11Be measurements corre- spond to quasi-elastic, these are essentially elastic as the contribution from the first excited state is negligible. Standard optical potentials (either density folding as in JLM [13] or global optical potentials coming from elas- tic fits as in CH89 [14]) could reproduce the 10Be elastic reasonably well, requiring small renormalizations of the real and imaginary parts of the interaction (λV = 0.9 and λW = 1.1 for CH89) [10] . Larger renormalizations were required in order to reproduce the distribution of 11Be (λV = 0.7 or λW = 1.3 for CH89) [10]. It is clear from Ref. [10] that the global optical poten- tial overestimates the elastic cross section for 11Be. In Ref. [15] the elastic scattering of 11Be on 12C was success- fully described using a 10Be+n two-body model, incor- porating breakup effects. As the 10Be-target interaction was fixed by the 10Be elastic scattering data, the large modification in the 11Be+12C elastic data was described, without renormalization, purely through breakup effects. Due to the loosely bound nature of the last neutron in 11Be this loss of flux from the elastic channel can be at- tributed to breakup into 10Be+n. It is thus possible that the large renormalizations for 11Be scattering on protons [10] are also due to breakup effects. In Ref. [11], elastic data is only described after large renormalizations of both the real and imaginary part of the 10Be+p interaction (λV = 0.75 and λW = 1.8), much larger than those used in Ref. [10]. These same renormal- izations can no longer describe the 10Be+p elastic data from Ref. [10], and are inconsistent with few-body reac- tion theory. We will re-examine the elastic scattering of http://arxiv.org/abs/0704.1586v1 10/11Be+p to see if one can consistently describe both sets of data using the same interaction for 10Be+p, by including continuum and core excitation. In addition to elastic and inelastic measurements, breakup data from NSCL exist at 63.7 MeV/nucleon [11]. This breakup data is integrated into two wide energy bins due to statistics. The lower energy bin covers the 1.78 MeV resonance and a reasonable angular distribu- tion is obtained, which underpredicts the cross section [11]. The higher energy bin covers resonances that are thought to be built on excited core states. The cal- culations presented in Ref. [11] failed to reproduce the shape of this higher energy bin, and the authors suggest that the source of the disagreement may be due to an active core during the reaction. Now that it is possi- ble to include core excitation in the reaction mechanism [9] we will re-examine the breakup data using a consis- tent 10Be+p interaction, and including systematically the coupling to the 2+ state in 10Be. Transfer reactions have also been performed with the 11Be beam, at 35.3 MeV/u in GANIL [12] with the aim of extracting spectroscopic factors for the ground state. While the reaction mechanisms proved to be more com- plicated than the 1-step DWBA theory, results for (p,d) show evidence for a significant core excited component. The inverse reaction, 10Be(d,p)11Be, has also been stud- ied in GANIL [16], the main interest being the resonance structure of 11Be. This illustrates how transfer is being used beyond the standard application of spectroscopy of bound states, underlining the need to better understand the transfer mechanism and its coupling to the contin- All these different data offer a good testing ground for theory. A comprehensive theoretical study [17] focusing on 10Be(d,p) show inconsistencies of the extracted spec- troscopic factors for data at different energies. Optical potential uncertainties and core excitation effects could be at the heart of the problem. In this work we perform calculations including elastic, inelastic, and breakup channels of 11Be on protons at in- termediate energies. We explore explicitly the effect of the inclusion of core excitation in the reaction mecha- nism. Comparison to elastic and breakup data will be presented here. The analysis of the inelastic channel is presented in [18] and we leave a detailed study of the transfer channel for a future publication. In section II we provide the details of the calculations. In section III we present the results: first for the elastic channel (III A), then for the breakup (III B). Finally, in section IV, we draw our conclusions and provide an outlook into the future. II. DETAILS OF THE CALCULATIONS The calculations for breakup of loosely bound systems on protons have a rather different convergence require- ment as compared to the breakup on heavier systems. energy V RV aV W RW aW 40 60.84 1.000 0.7 23.16 0.600 0.6 60 31.64 1.145 0.69 8.78 1.134 0.69 TABLE I: 10Be-proton Woods-Saxon potential parameters. All energies are in MeV and lengths in fm. The model space needs to span large excitation energies, while the radial dependence can be reduced significantly. For the CDCC calculations at 40 MeV/nucleon, the continuum was discretized upto 35 MeV, with 10 bins upto 10 MeV for s-, p-, and d-waves, and 8 bins from 10–35 MeV. We include 12 bins from 0–35 MeV for all other partial waves up to lmax = 4. The same binning scheme was used for the XCDCC calculations, except that the higher bin density upto 10 MeV was only used for channels with outgoing ground state core components. Partial waves up to lmax = 4 were used for the coupled channels projectile states. For the 60 MeV/nucleon CDCC calculations, a slightly different binning scheme was adopted to match the ex- perimental energy bin integrations. From 0–0.5 MeV, 2 bins were used; over the observed energy bins from 0.5– 3.0 and 3.0–5.5 MeV, 3 bins were used in each case; and from 5.5–30 MeV, 6 bins. For the XCDCC calculations where the outgoing channel had excited core states, only 1 bin was used from 0–0.5 MeV, 1 bin for each observed energy range, and 5 bins above. The radial integrals for the bins were calculated upto 40 fm in steps of 0.1 fm. The radial equations in the CDCC method were calculated for 30 partial waves with the lower radial cutoff for the integrals set to 4 fm inside the point Coulomb radius, and matched to the asymp- totic Coulomb functions at 150 fm. The 11Be bound state potential parameters are taken from Ref. [19], using the Be12-pure interaction for the CDCC calculations and the Be12-b for the XCDCC cal- culations. The 10Be-proton interaction is fitted to the proton elastic data available at the two energies. A good fit could be obtained from a renormalized CH89 interac- tion [14]. The parameters are given in Table I. For the cases including 10Be excitation, the OM potentials were deformed with the same β2 deformations as used in the 11Be bound state. The coupling matrix elements to the excited state in 10Be assume a rotational model with the deformation fitted to the experimental B(E2) strength [20]. The deformation length is in good agreement with that obtained from inelastic scattering ot the 2+ state in 10Be [21], and the optical potential used here reproduces the angular distribution of the inelastic scattering well. 0 20 40 60 80 (deg) Be OM Be CF Be CDCC Be XCDCC FIG. 1: (Color online) 10/11Be elastic scattering on a proton target at ∼40 MeV/nucleon, using an optical model fit to the 10Be elastic and various 11Be reaction models for the 11Be data. The experimental data are from GANIL [22]. 0 20 40 60 (deg) Be OM Be CDCC Be XCDCC FIG. 2: (Color online) 10/11Be elastic scattering on a proton target at ∼60 MeV/nucleon, using an optical model fit to the 10Be elastic and various 11Be reaction models for the 11Be data. The experimental data are from Ref. [10] (10Be) and Ref. [11] (11Be). III. RESULTS A. Elastic channel The elastic scattering is the first test on the reaction model. In Fig. 1 we show the 10Be and 11Be elastic data and theoretical calculations at ∼40 MeV/nucleon. The optical model for the 10Be (dashed/black line) is fitted to the 10Be elastic data (open circles). The cluster folding model (dotted/red line) folds the 10Be+p and n-p interac- tions over the 11Be ground state wave function to produce the 11Be+p potential. Also shown in Fig. 1 is the effect of the 11Be continuum within CDCC (dot-dashed/blue line) and core excitation within XCDCC (solid line). Even though there is significant improvement over the simple optical model when including breakup, results still over- estimate the 11Be elastic cross section at larger angles, and no improvement is found by including excited core contributions. Calculations were repeated for a higher energy, around 60 MeV/nucleon, where both 10Be and 11Be elastic data exist. Once again, when the 10Be data is fitted with an optical model, and the 11Be elastic is described within the CDCC approach, the cross section is over-estimated (see Fig. 2). Note that the data at this energy does not span a large angular range, but it is evident that the pattern of over-predicting the 11Be cross section remains. Other reaction calculations have been performed in an effort to describe this data [23], which consisted of a transfer to the continuum approach in which the breakup continuum was described using the deuteron basis. This also failed to describe the data when the 10Be potential was fixed to the elastic data. The same pattern of over- predicting the 11Be elastic was also seen at a lower energy of ∼40 MeV/nucleon [23]. As pointed out earlier, in [10, 11] large renormaliza- tion factors were needed to reproduce the elastic cross section. By including more relevant reaction channels, one might account for a part of the renormalization re- quired, corresponding to flux that is being removed from the elastic channel. This suggests that there are still channels coupled to the elastic that have not been con- sidered. Preliminary calculations including the deuteron transfer channel along with the breakup in the 11Be basis show improvement at small angles, but the disagreement still remains at large angles. Due to large non-orthonality corrections, CDCC calculations including the deuteron transfer coupling turn out to be numerically challenging. They will be discussed in a later publication. B. Breakup: comparison with data at ∼60 MeV/nucleon Breakup data was also obtained at 63.7 MeV/nucleon [11], summed into two energy bins. The first covers the energy range 0.5–3.0 MeV, which spans the 1.78 MeV res- onance, predominantly a d-wave neutron coupled to the ground state of the core. The second energy bin is over the energy range 3.0–5.5 MeV, which spans a resonance at 3.89 MeV, thought to be predominantly an s-wave neutron coupled to a 10Be(2+) core [11]. In Ref. [11], CDCC results were presented which underestimated the cross section for the lower energy bin, but did not repro- duce the higher energy bin. It was suggested that since the higher energy bin spanned a resonance with a pos- sible excited core component, the disagreement could be due to the spectator core approximation in the standard CDCC theory. Since XCDCC can handle excited core components, this data is re-examined. The breakup angular distribution data and the asso- ciated theory prediction for the lower energy bin (0.5– 3.0 MeV) and the higher energy bin (3.0–5.5 MeV) are presented in Figs. 3 and 4 (the equivalent of Figures 3b and 3c of Ref. [11]). Firstly, the CDCC calculations of Ref. [11] were redone, with a higher CDCC bin density. 0 20 40 (deg) XCDCC Be+n)p @ 63.7 MeV/nucleon 0.5-3.0 MeV FIG. 3: 11Be breakup at 60 MeV/nucleon with the relative energy between breakup fragments in the range 0.5–3.0 MeV. Experimental data are from Ref. [11]. 0 20 40 (deg) XCDCC ) x10 Be+n)p @ 63.7 MeV/nucleon 3.0-5.5 MeV FIG. 4: (Color online) 11Be breakup at 60 MeV/nucleon with the relative energy between breakup fragments in the range 3.0–5.5 MeV. Experimental data are from Ref. [11]. We find that converged CDCC calculations, for the lower energy bin, do in fact agree well with the data (dashed line in Fig. 3), contrary to what is presented in Ref. [11]. Results do not change significantly when core excitation is included with XCDCC (solid line in Fig. 3), as could be expected expected due to the resonance structure in this energy region. One can conclude that a single-particle description for the first d5/2 resonance is adequate. The data for the higher energy bin is not well described within the single particle CDCC model (dashed line in Fig. 4). To see if this discrepancy can be explained by ex- cited core contributions, we include the excited 10Be(2+) components in the reaction mechanism, within XCDCC (solid line). As shown in Fig. 4, core excitation lowers the cross section but does not significantly change the shape of the distribution. It becomes clear that core excitation does not help to reproduce the shape of the higher en- ergy angular distribution. The main reason for this is that for the 11Be coupled channel model of [20], most of the breakup ends up in the 10Be ground state. Fig. 4 also shows the breakup cross section to the 0+ and 2+ states of 10Be (red/dotted line and the dot-dashed/blue line respectively). We see that whereas the ground state distribution has the original shape of the CDCC calcu- lation, the shape of the distribution to the excited state reproduces the data (to illustrate this fact, we show the breakup cross section to 10Be(2+) multiplied by 10 by the dashed/blue line). The reason for this maybe that the large number of resonances in this region are not re- produced well by our particle-rotor model for 11Be. The only resonance that appears in this model is the 3/2+, for which the width is not narrow enough to attract signifi- cant cross section. Some suggest that more exotic struc- tures are responsible for resonances in this region [24]. Without exotic resonances built on excited core compo- nents in our structure model, the breakup cross section is still dominated by ground state 10Be fragments. IV. CONCLUDING REMARKS A consistent analysis of reactions involving the halo nucleus 11Be on protons, at two intermediate energies (∼40 and ∼60 MeV/nucleon) are performed and com- pared with data. An optical model approach, based on a cluster folding potential constructed from the 10Be+p potential fitted to the appropriate elastic data, is un- able to describe the 11Be elastic data. The inclusion of breakup effects improve the description, but theoretical predictions still overestimate the elastic cross section at larger scattering angles. The inclusion of core excitation does not affect the elastic distribution significantly. Note however that these results include no artificial renormal- ization of the optical potential. Elastic scattering ex- periments with radioactive beams at large facilities have repeatedly been undermined. The fact that the best re- action models are still unable to fully describe the mecha- nisms for the 11Be case, shows the need for a more varied and better elastic scattering experimental program. In this work we also study the breakup channel explic- itly. Core excitation in the description of the continuum, within XCDCC, produces a slight modification of the dis- tribution. These breakup calculations are compared to the data at 63.7 MeV/nucleon, for two energy bins 0.5– 3.0 MeV and 3.0–5.5 MeV. For the lower energy bin, the shape of the angular distribution is well reproduced by the models. The same cannot be said for the higher en- ergy bin. The XCDCC calculations predict breakup states to specific states of the core 10Be. This level of detail is still not available in the data, but it could be helpful in- formation, even at an integrated level, to identify possible causes for the remaining disagreement with the data. Another important point is related to the basis used to describe the breakup states. As discussed in Ref. [25], within CDCC, one can describe the three body final state continuum 10Be+n+p in the 11Be continuum basis or in the deuteron continuum basis. In this work we used the 11Be basis. Work in Ref. [25] shows that in practice the two choices do not provide the same result. Efforts are underway to tackle this problem within a Faddeev frame- work [26]. These results may have important implications to the theory-experiment mismatch. Acknowledgements We thank the high performance computing center (HPCC) at MSU for the use of their facilities. This work is supported by NSCL, Michigan State University, the National Science Foundation through grant PHY- 0555893, by NNSA through US DOE Cooperative Agree- ment DEFC03-03NA00143 at Rutgers University, and by the US DOE under contract No. DE-FG02-96ER40983 at the University of Tennessee. [1] J. S. Al-Khalili and F. M. Nunes, J. Phys. G 29, R89 (2003). [2] Y. Sakuragi, M. Yahiro, and M. Kamimura, Prog. Theor. Phys. Suppl. 89, 136 (1986). [3] F. M. Nunes and I. J. Thompson, Phys. Rev. C 59, 2652 (1999). [4] J. A. Tostevin, D. Bazin, B. A. Brown, T. Glasmacher, P. G. Hansen, V. Maddalena, A. Navin, and B. M. Sher- rill, Phys. Rev. C 66, 024607 (2002). [5] A. M. Moro, R. Crespo, F. M. Nunes, and I. J. Thomp- son, Phys. Rev. C 67, 047602 (2003). [6] T. Matsumoto, E. Hiyama, K. Ogata, Y. Iseri, M. Kamimura, S. Chiba, and M. Yahiro, Phys. Rev. C 70, 061601(R) (2004). [7] K. Ogata, S. Hashimoto, Y. Iseri, M. Kamimura, and M. Yahiro, Phys. Rev. C 73, 024605 (2006). [8] N. C. Summers, F. M. Nunes, and I. J. Thompson, Phys. Rev. C 73, 031603(R) (2006). [9] N. C. Summers, F. M. Nunes, and I. J. Thompson, Phys. Rev. C 74, 014606 (2006). [10] M. D. Cortina-Gil et al., Phys. Lett. B 401, 9 (1997). [11] A. Shrivastava et al., Phys. Lett. B 596, 54 (2004). [12] J. S. Winfield et al., Nucl. Phys. A683, 48 (2001). [13] J.-P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rev. C 15, 10 (1977). [14] W. W. Daehnick, J. D. Childs, and Z. Vrcelj, Phys. Rev. C 21, 2253 (1980). [15] R. C. Johnson, J. S. Al-Khalili, and J. A. Tostevin, Phys. Rev. Lett. 79, 2771 (1997). [16] F. Delaunay et al., in preparation. [17] N. Keeley, N. Alamanos and V. Lapoux, Phys. Rev. C 69, 064604 (2004). [18] N. C. Summers, S. D. Pain et al., submitted to Phys. Lett. B. [19] F. M. Nunes, J. A. Christley, I. J. Thompson, R. C. John- son and V. D. Efros, Nucl. Phys. A609, 43 (1996). [20] F. M. Nunes, I. J. Thompson, and R. C. Johnson, Nucl. Phys. A596, 171 (1996). [21] H. Iwasaki et al., Phys. Lett. B 481, 7 (2000). [22] V. Lapoux et al., submitted to Phys. Lett. B. [23] R. Crespo, J. C. Fernandes, V. Lapoux, A. M. Moro, F. M. Nunes, private communication (2003). [24] F. Cappuzzello et al., Phys. Lett. B 516, 21 (2001). [25] A. M. Moro and F. M. Nunes, Nucl. Phys. A767, 138 (2006). [26] A.Deltuva, A.M. Moro, F.M. Nunes and A.C. Fonseca, DNP meeting, Nashville 24-27 October 2006.

The elastic scattering and breakup of $^{11}$Be from a proton target at intermediate energies is studied. We explore the role of core excitation in the reaction mechanism. Comparison with the data suggests that there is still missing physics in the description.

Core excitation in the elastic scattering and breakup of 11Be on protons N. C. Summers1, 2, 3 and F. M. Nunes3, 4 Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996 Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854 National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824 Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824 (Dated: October 25, 2018) The elastic scattering and breakup of 11Be from a proton target at intermediate energies is studied. We explore the role of core excitation in the reaction mechanism. Comparison with the data suggests that there is still missing physics in the description. PACS numbers: 24.10.Eq, 25.40.Cm, 25.60.Gc, 27.20.+n I. INTRODUCTION Nuclear reactions offer the most diverse methods to study nuclei at the limit of stability. Understanding re- action mechanisms in nuclear processes involving nuclei near the driplines is of great importance, particularly at this time, when there is such a high demand for accuracy on the structure information to be extracted from the data. Reaction and structure models are undoubtedly entangled, therefore improving reaction models often im- plies incorporating more detailed structure models in the description [1]. It is generally accepted that, in reactions with loosely bound nuclei, the coupling to the continuum needs to be considered. Continuum effects are very much enhanced in breakup but can also have imprints on other reac- tion channels, for example elastic and inelastic scatter- ing. One framework that explicitly includes continuum effects is the Continuum Discretized Coupled Channel (CDCC) method [2]. A large amount of work has been devoted to the analysis of experiments within this frame- work [3, 4, 5, 6, 7] and in general results are very good. The eXtended Continuum Discretized Coupled Chan- nel (XCDCC) method [8, 9] was recently developed. It brings together a coupled channel description of the pro- jectile with a coupled channel model of the reaction, en- abling the description of interference between the multi- channel components of the projectile as well as dynam- ical excitation of the core within the projectile, during the reaction [9]. The model has been applied to the breakup of 11Be→10Be+n and 17C→16C+n on 9Be at ≈ 60 MeV/nucleon [8, 9]. The projectiles are described within a two-body core + n multi-channel model, where the core can be in the ground state but also in an ex- cited state. This model produces breakup cross sections to specific final states of the core, given a coupled chan- nel Hamiltonian for the projectile. Results presented in Ref. [8, 9] show that core excitation effects in the total cross section to the ground state of the core are small, but become very large when considering the total popula- tion of the core’s excited state. Other differences can be seen in angular and energy distributions but at present no such data is available. The effect of core excitation needs to be studied in other regimes, for very light and very heavy targets, as well as a variety of energies. In this work we concentrate on the proton target. In this case the process is nuclear driven, and recoil effects are very important. Several reactions of 11Be on protons have been measured in a number of facilities, namely elastic scattering at 50 MeV/nucleon [10], quasi-elastic and breakup at 64 MeV/nucleon [11] as well as transfer at 35 MeV/nucleon [12]. 11Be proton elastic scattering at 49.3 MeV/nucleon was performed in GANIL [10], at the same time as the elastic scattering of the core 10Be at 59.4 MeV/nucleon. Even though the outcoming 11Be measurements corre- spond to quasi-elastic, these are essentially elastic as the contribution from the first excited state is negligible. Standard optical potentials (either density folding as in JLM [13] or global optical potentials coming from elas- tic fits as in CH89 [14]) could reproduce the 10Be elastic reasonably well, requiring small renormalizations of the real and imaginary parts of the interaction (λV = 0.9 and λW = 1.1 for CH89) [10] . Larger renormalizations were required in order to reproduce the distribution of 11Be (λV = 0.7 or λW = 1.3 for CH89) [10]. It is clear from Ref. [10] that the global optical poten- tial overestimates the elastic cross section for 11Be. In Ref. [15] the elastic scattering of 11Be on 12C was success- fully described using a 10Be+n two-body model, incor- porating breakup effects. As the 10Be-target interaction was fixed by the 10Be elastic scattering data, the large modification in the 11Be+12C elastic data was described, without renormalization, purely through breakup effects. Due to the loosely bound nature of the last neutron in 11Be this loss of flux from the elastic channel can be at- tributed to breakup into 10Be+n. It is thus possible that the large renormalizations for 11Be scattering on protons [10] are also due to breakup effects. In Ref. [11], elastic data is only described after large renormalizations of both the real and imaginary part of the 10Be+p interaction (λV = 0.75 and λW = 1.8), much larger than those used in Ref. [10]. These same renormal- izations can no longer describe the 10Be+p elastic data from Ref. [10], and are inconsistent with few-body reac- tion theory. We will re-examine the elastic scattering of http://arxiv.org/abs/0704.1586v1 10/11Be+p to see if one can consistently describe both sets of data using the same interaction for 10Be+p, by including continuum and core excitation. In addition to elastic and inelastic measurements, breakup data from NSCL exist at 63.7 MeV/nucleon [11]. This breakup data is integrated into two wide energy bins due to statistics. The lower energy bin covers the 1.78 MeV resonance and a reasonable angular distribu- tion is obtained, which underpredicts the cross section [11]. The higher energy bin covers resonances that are thought to be built on excited core states. The cal- culations presented in Ref. [11] failed to reproduce the shape of this higher energy bin, and the authors suggest that the source of the disagreement may be due to an active core during the reaction. Now that it is possi- ble to include core excitation in the reaction mechanism [9] we will re-examine the breakup data using a consis- tent 10Be+p interaction, and including systematically the coupling to the 2+ state in 10Be. Transfer reactions have also been performed with the 11Be beam, at 35.3 MeV/u in GANIL [12] with the aim of extracting spectroscopic factors for the ground state. While the reaction mechanisms proved to be more com- plicated than the 1-step DWBA theory, results for (p,d) show evidence for a significant core excited component. The inverse reaction, 10Be(d,p)11Be, has also been stud- ied in GANIL [16], the main interest being the resonance structure of 11Be. This illustrates how transfer is being used beyond the standard application of spectroscopy of bound states, underlining the need to better understand the transfer mechanism and its coupling to the contin- All these different data offer a good testing ground for theory. A comprehensive theoretical study [17] focusing on 10Be(d,p) show inconsistencies of the extracted spec- troscopic factors for data at different energies. Optical potential uncertainties and core excitation effects could be at the heart of the problem. In this work we perform calculations including elastic, inelastic, and breakup channels of 11Be on protons at in- termediate energies. We explore explicitly the effect of the inclusion of core excitation in the reaction mecha- nism. Comparison to elastic and breakup data will be presented here. The analysis of the inelastic channel is presented in [18] and we leave a detailed study of the transfer channel for a future publication. In section II we provide the details of the calculations. In section III we present the results: first for the elastic channel (III A), then for the breakup (III B). Finally, in section IV, we draw our conclusions and provide an outlook into the future. II. DETAILS OF THE CALCULATIONS The calculations for breakup of loosely bound systems on protons have a rather different convergence require- ment as compared to the breakup on heavier systems. energy V RV aV W RW aW 40 60.84 1.000 0.7 23.16 0.600 0.6 60 31.64 1.145 0.69 8.78 1.134 0.69 TABLE I: 10Be-proton Woods-Saxon potential parameters. All energies are in MeV and lengths in fm. The model space needs to span large excitation energies, while the radial dependence can be reduced significantly. For the CDCC calculations at 40 MeV/nucleon, the continuum was discretized upto 35 MeV, with 10 bins upto 10 MeV for s-, p-, and d-waves, and 8 bins from 10–35 MeV. We include 12 bins from 0–35 MeV for all other partial waves up to lmax = 4. The same binning scheme was used for the XCDCC calculations, except that the higher bin density upto 10 MeV was only used for channels with outgoing ground state core components. Partial waves up to lmax = 4 were used for the coupled channels projectile states. For the 60 MeV/nucleon CDCC calculations, a slightly different binning scheme was adopted to match the ex- perimental energy bin integrations. From 0–0.5 MeV, 2 bins were used; over the observed energy bins from 0.5– 3.0 and 3.0–5.5 MeV, 3 bins were used in each case; and from 5.5–30 MeV, 6 bins. For the XCDCC calculations where the outgoing channel had excited core states, only 1 bin was used from 0–0.5 MeV, 1 bin for each observed energy range, and 5 bins above. The radial integrals for the bins were calculated upto 40 fm in steps of 0.1 fm. The radial equations in the CDCC method were calculated for 30 partial waves with the lower radial cutoff for the integrals set to 4 fm inside the point Coulomb radius, and matched to the asymp- totic Coulomb functions at 150 fm. The 11Be bound state potential parameters are taken from Ref. [19], using the Be12-pure interaction for the CDCC calculations and the Be12-b for the XCDCC cal- culations. The 10Be-proton interaction is fitted to the proton elastic data available at the two energies. A good fit could be obtained from a renormalized CH89 interac- tion [14]. The parameters are given in Table I. For the cases including 10Be excitation, the OM potentials were deformed with the same β2 deformations as used in the 11Be bound state. The coupling matrix elements to the excited state in 10Be assume a rotational model with the deformation fitted to the experimental B(E2) strength [20]. The deformation length is in good agreement with that obtained from inelastic scattering ot the 2+ state in 10Be [21], and the optical potential used here reproduces the angular distribution of the inelastic scattering well. 0 20 40 60 80 (deg) Be OM Be CF Be CDCC Be XCDCC FIG. 1: (Color online) 10/11Be elastic scattering on a proton target at ∼40 MeV/nucleon, using an optical model fit to the 10Be elastic and various 11Be reaction models for the 11Be data. The experimental data are from GANIL [22]. 0 20 40 60 (deg) Be OM Be CDCC Be XCDCC FIG. 2: (Color online) 10/11Be elastic scattering on a proton target at ∼60 MeV/nucleon, using an optical model fit to the 10Be elastic and various 11Be reaction models for the 11Be data. The experimental data are from Ref. [10] (10Be) and Ref. [11] (11Be). III. RESULTS A. Elastic channel The elastic scattering is the first test on the reaction model. In Fig. 1 we show the 10Be and 11Be elastic data and theoretical calculations at ∼40 MeV/nucleon. The optical model for the 10Be (dashed/black line) is fitted to the 10Be elastic data (open circles). The cluster folding model (dotted/red line) folds the 10Be+p and n-p interac- tions over the 11Be ground state wave function to produce the 11Be+p potential. Also shown in Fig. 1 is the effect of the 11Be continuum within CDCC (dot-dashed/blue line) and core excitation within XCDCC (solid line). Even though there is significant improvement over the simple optical model when including breakup, results still over- estimate the 11Be elastic cross section at larger angles, and no improvement is found by including excited core contributions. Calculations were repeated for a higher energy, around 60 MeV/nucleon, where both 10Be and 11Be elastic data exist. Once again, when the 10Be data is fitted with an optical model, and the 11Be elastic is described within the CDCC approach, the cross section is over-estimated (see Fig. 2). Note that the data at this energy does not span a large angular range, but it is evident that the pattern of over-predicting the 11Be cross section remains. Other reaction calculations have been performed in an effort to describe this data [23], which consisted of a transfer to the continuum approach in which the breakup continuum was described using the deuteron basis. This also failed to describe the data when the 10Be potential was fixed to the elastic data. The same pattern of over- predicting the 11Be elastic was also seen at a lower energy of ∼40 MeV/nucleon [23]. As pointed out earlier, in [10, 11] large renormaliza- tion factors were needed to reproduce the elastic cross section. By including more relevant reaction channels, one might account for a part of the renormalization re- quired, corresponding to flux that is being removed from the elastic channel. This suggests that there are still channels coupled to the elastic that have not been con- sidered. Preliminary calculations including the deuteron transfer channel along with the breakup in the 11Be basis show improvement at small angles, but the disagreement still remains at large angles. Due to large non-orthonality corrections, CDCC calculations including the deuteron transfer coupling turn out to be numerically challenging. They will be discussed in a later publication. B. Breakup: comparison with data at ∼60 MeV/nucleon Breakup data was also obtained at 63.7 MeV/nucleon [11], summed into two energy bins. The first covers the energy range 0.5–3.0 MeV, which spans the 1.78 MeV res- onance, predominantly a d-wave neutron coupled to the ground state of the core. The second energy bin is over the energy range 3.0–5.5 MeV, which spans a resonance at 3.89 MeV, thought to be predominantly an s-wave neutron coupled to a 10Be(2+) core [11]. In Ref. [11], CDCC results were presented which underestimated the cross section for the lower energy bin, but did not repro- duce the higher energy bin. It was suggested that since the higher energy bin spanned a resonance with a pos- sible excited core component, the disagreement could be due to the spectator core approximation in the standard CDCC theory. Since XCDCC can handle excited core components, this data is re-examined. The breakup angular distribution data and the asso- ciated theory prediction for the lower energy bin (0.5– 3.0 MeV) and the higher energy bin (3.0–5.5 MeV) are presented in Figs. 3 and 4 (the equivalent of Figures 3b and 3c of Ref. [11]). Firstly, the CDCC calculations of Ref. [11] were redone, with a higher CDCC bin density. 0 20 40 (deg) XCDCC Be+n)p @ 63.7 MeV/nucleon 0.5-3.0 MeV FIG. 3: 11Be breakup at 60 MeV/nucleon with the relative energy between breakup fragments in the range 0.5–3.0 MeV. Experimental data are from Ref. [11]. 0 20 40 (deg) XCDCC ) x10 Be+n)p @ 63.7 MeV/nucleon 3.0-5.5 MeV FIG. 4: (Color online) 11Be breakup at 60 MeV/nucleon with the relative energy between breakup fragments in the range 3.0–5.5 MeV. Experimental data are from Ref. [11]. We find that converged CDCC calculations, for the lower energy bin, do in fact agree well with the data (dashed line in Fig. 3), contrary to what is presented in Ref. [11]. Results do not change significantly when core excitation is included with XCDCC (solid line in Fig. 3), as could be expected expected due to the resonance structure in this energy region. One can conclude that a single-particle description for the first d5/2 resonance is adequate. The data for the higher energy bin is not well described within the single particle CDCC model (dashed line in Fig. 4). To see if this discrepancy can be explained by ex- cited core contributions, we include the excited 10Be(2+) components in the reaction mechanism, within XCDCC (solid line). As shown in Fig. 4, core excitation lowers the cross section but does not significantly change the shape of the distribution. It becomes clear that core excitation does not help to reproduce the shape of the higher en- ergy angular distribution. The main reason for this is that for the 11Be coupled channel model of [20], most of the breakup ends up in the 10Be ground state. Fig. 4 also shows the breakup cross section to the 0+ and 2+ states of 10Be (red/dotted line and the dot-dashed/blue line respectively). We see that whereas the ground state distribution has the original shape of the CDCC calcu- lation, the shape of the distribution to the excited state reproduces the data (to illustrate this fact, we show the breakup cross section to 10Be(2+) multiplied by 10 by the dashed/blue line). The reason for this maybe that the large number of resonances in this region are not re- produced well by our particle-rotor model for 11Be. The only resonance that appears in this model is the 3/2+, for which the width is not narrow enough to attract signifi- cant cross section. Some suggest that more exotic struc- tures are responsible for resonances in this region [24]. Without exotic resonances built on excited core compo- nents in our structure model, the breakup cross section is still dominated by ground state 10Be fragments. IV. CONCLUDING REMARKS A consistent analysis of reactions involving the halo nucleus 11Be on protons, at two intermediate energies (∼40 and ∼60 MeV/nucleon) are performed and com- pared with data. An optical model approach, based on a cluster folding potential constructed from the 10Be+p potential fitted to the appropriate elastic data, is un- able to describe the 11Be elastic data. The inclusion of breakup effects improve the description, but theoretical predictions still overestimate the elastic cross section at larger scattering angles. The inclusion of core excitation does not affect the elastic distribution significantly. Note however that these results include no artificial renormal- ization of the optical potential. Elastic scattering ex- periments with radioactive beams at large facilities have repeatedly been undermined. The fact that the best re- action models are still unable to fully describe the mecha- nisms for the 11Be case, shows the need for a more varied and better elastic scattering experimental program. In this work we also study the breakup channel explic- itly. Core excitation in the description of the continuum, within XCDCC, produces a slight modification of the dis- tribution. These breakup calculations are compared to the data at 63.7 MeV/nucleon, for two energy bins 0.5– 3.0 MeV and 3.0–5.5 MeV. For the lower energy bin, the shape of the angular distribution is well reproduced by the models. The same cannot be said for the higher en- ergy bin. The XCDCC calculations predict breakup states to specific states of the core 10Be. This level of detail is still not available in the data, but it could be helpful in- formation, even at an integrated level, to identify possible causes for the remaining disagreement with the data. Another important point is related to the basis used to describe the breakup states. As discussed in Ref. [25], within CDCC, one can describe the three body final state continuum 10Be+n+p in the 11Be continuum basis or in the deuteron continuum basis. In this work we used the 11Be basis. Work in Ref. [25] shows that in practice the two choices do not provide the same result. Efforts are underway to tackle this problem within a Faddeev frame- work [26]. These results may have important implications to the theory-experiment mismatch. Acknowledgements We thank the high performance computing center (HPCC) at MSU for the use of their facilities. This work is supported by NSCL, Michigan State University, the National Science Foundation through grant PHY- 0555893, by NNSA through US DOE Cooperative Agree- ment DEFC03-03NA00143 at Rutgers University, and by the US DOE under contract No. DE-FG02-96ER40983 at the University of Tennessee. [1] J. S. Al-Khalili and F. M. Nunes, J. Phys. G 29, R89 (2003). [2] Y. Sakuragi, M. Yahiro, and M. Kamimura, Prog. Theor. Phys. Suppl. 89, 136 (1986). [3] F. M. Nunes and I. J. Thompson, Phys. Rev. C 59, 2652 (1999). [4] J. A. Tostevin, D. Bazin, B. A. Brown, T. Glasmacher, P. G. Hansen, V. Maddalena, A. Navin, and B. M. Sher- rill, Phys. Rev. C 66, 024607 (2002). [5] A. M. Moro, R. Crespo, F. M. Nunes, and I. J. Thomp- son, Phys. Rev. C 67, 047602 (2003). [6] T. Matsumoto, E. Hiyama, K. Ogata, Y. Iseri, M. Kamimura, S. Chiba, and M. Yahiro, Phys. Rev. C 70, 061601(R) (2004). [7] K. Ogata, S. Hashimoto, Y. Iseri, M. Kamimura, and M. Yahiro, Phys. Rev. C 73, 024605 (2006). [8] N. C. Summers, F. M. Nunes, and I. J. Thompson, Phys. Rev. C 73, 031603(R) (2006). [9] N. C. Summers, F. M. Nunes, and I. J. Thompson, Phys. Rev. C 74, 014606 (2006). [10] M. D. Cortina-Gil et al., Phys. Lett. B 401, 9 (1997). [11] A. Shrivastava et al., Phys. Lett. B 596, 54 (2004). [12] J. S. Winfield et al., Nucl. Phys. A683, 48 (2001). [13] J.-P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rev. C 15, 10 (1977). [14] W. W. Daehnick, J. D. Childs, and Z. Vrcelj, Phys. Rev. C 21, 2253 (1980). [15] R. C. Johnson, J. S. Al-Khalili, and J. A. Tostevin, Phys. Rev. Lett. 79, 2771 (1997). [16] F. Delaunay et al., in preparation. [17] N. Keeley, N. Alamanos and V. Lapoux, Phys. Rev. C 69, 064604 (2004). [18] N. C. Summers, S. D. Pain et al., submitted to Phys. Lett. B. [19] F. M. Nunes, J. A. Christley, I. J. Thompson, R. C. John- son and V. D. Efros, Nucl. Phys. A609, 43 (1996). [20] F. M. Nunes, I. J. Thompson, and R. C. Johnson, Nucl. Phys. A596, 171 (1996). [21] H. Iwasaki et al., Phys. Lett. B 481, 7 (2000). [22] V. Lapoux et al., submitted to Phys. Lett. B. [23] R. Crespo, J. C. Fernandes, V. Lapoux, A. M. Moro, F. M. Nunes, private communication (2003). [24] F. Cappuzzello et al., Phys. Lett. B 516, 21 (2001). [25] A. M. Moro and F. M. Nunes, Nucl. Phys. A767, 138 (2006). [26] A.Deltuva, A.M. Moro, F.M. Nunes and A.C. Fonseca, DNP meeting, Nashville 24-27 October 2006.

Possible X-ray diagnostic for jet/disk dominance in Type 1 AGN Barbara J. Mattson⋆, Kimberly A. Weaver NASA/Goddard Space Flight Center, Astrophysics Science Division, Greenbelt, MD, 20771 Christopher S. Reynolds Department of Astronomy, University of Maryland, College Park, MD, 20742 ABSTRACT Using Rossi X-ray Timing Explorer Seyfert 1 and 1.2 data spanning 9 years, we study correlations between X-ray spectral features. The sample consists of 350 time-resolved spectra from 12 Seyfert 1 and 1.2 galaxies. Each spectrum is fitted to a model with an intrinsic powerlaw X-ray spectrum produced close to the central black hole that is reprocessed and absorbed by material around the black hole. To test the robustness of our results, we performed Monte Carlo simulations of the spectral sample. We find a complex relationship between the iron line equivalent width (EW ) and the underlying power law index (Γ). The data reveal a correlation between Γ and EW which turns over at Γ . 2, but finds a weak anti-correlation for steeper photon indices. We propose that this relationship is driven by dilution of a disk spectrum (which includes the narrow iron line) by a beamed jet component and, hence, could be used as a diagnostic of jet-dominance. In addition, our sample shows a strong correlation between R and Γ, but we find that it is likely the result of modeling degeneracies. We also see the X-ray Baldwin effect (an anti-correlation between the 2-10 keV X- ray luminosity and EW ) for the sample as a whole, but not for the individual galaxies and galaxy types. Subject headings: galaxies: Seyfert, X-rays: galaxies also Department of Astronomy, University of Maryland, College Park, MD and Adnet Systems, Inc., Rockville, MD http://arxiv.org/abs/0704.1587v1 – 2 – 1. Introduction Time-resolved X-ray spectroscopy studies of active galactic nuclei (AGN) offer the op- portunity to investigate emission regions near the central black hole. In fact, X-ray spec- troscopy offers the clearest view of processes occurring very close to the black hole itself, probing matter to its final plunge into the black hole. Armed with such information, we can unlock the structure of the innermost regions of AGN. Typical X-ray spectra of AGN show an underlying powerlaw produced near the central black hole with signatures of reprocessed photons often present. These reprocessed photons show up as an Fe Kα line at ∼6.4 keV and a “reflection hump” which starts to dominate near 10 keV and is produced by the combined effects of photoelectric absorption and Compton downscattering in optically-thin cold matter irradiated by the hard X-ray continuum. The Fe Kα line has been observed in both type 1 (unabsorbed) and type 2 (absorbed) Seyfert galaxies. It has been attributed to either the broad line region, the accretion disk, the molec- ular torus of unification models (Antonucci 1993), or some combination of these. Signatures of reflection have also been observed in both Seyfert 1 and 2 galaxies. If the unification models are correct, we should see similar spectral correlations be- tween Seyfert 1 and 2 galaxies, with any differences easily attributable to our viewing angle. Regardless of the accuracy of the reflection models, we expect changes in the un- derlying continuum to drive changes in the reprocessing features. However, results from X-ray spectral studies of AGN have so far produced puzzling results. Samples of Seyfert 1 observations from ASCA (Weaver, Gelbord & Yaqoob 2001) and Rossi X-ray Timing Ex- plorer (Markowitz, Edelson, & Vaughan 2003) have shown no obvious relationship between changes in the continuum and iron line. Several galaxies have shown an anticorrelation between reflection and/or iron line equivalent width and the source flux; e.g. NGC 5548 (Chiang et al. 2000), MCG −6-30-15 (Papadakis et al. 2002), NGC 4051 (Papadakis et al. 2002; Wang et al. 1999), NGC 5506 (Papadakis et al. 2002; Lamer, Uttley & McHardy 2000). Recent data from Suzaku on MCG −6-30-15, on the other hand, show that the iron line and reflection remain relatively constant while the powerlaw is highly variable (Miniutti et al. 2006). Zdziarski, Lubiński & Smith (1999) found that Seyfert galaxies and X-ray binaries show a correlation between the continuum slope and reflection fraction, so those with soft intrinsic spectra show stronger reflection than those with hard spectra. However, other stud- ies have found either a shallower relationship than Zdziarski et al. (Perola et al. 2002) or an anticorrelation (Papadakis et al. 2002; Lamer, Uttley & McHardy 2000). Here we present the first results of a larger study of the X-ray spectral properties of Seyfert galaxies observed by the Rossi X-ray Timing Explorer (RXTE ). Our full study consists of observations of 30 galaxies. In this letter, we focus on the spectral results from – 3 – the subset of 12 Seyfert 1 and 1.2 galaxies. In § 2 we present our method of data analysis, including our sample selection criteria (§ 2.1), a description of our data pipeline (§ 2.2), and results of our spectral analysis (§ 2.3). We discuss the implications of our results in § 3 and detail our conclusions in § 4. 2. Data Analysis 2.1. The Sample The RXTE public archive1 represents one of the largest collections of X-ray data for AGN, with pointed observations of over 100 AGN spanning 10 years. The RXTE bandpass allows the study of absorption and iron line properties of AGN spectra, as well as a glimpse at the Compton reflection hump. We use data from the RXTE proportional counter array (PCA), which is sensitive to energies from 2 to 60 keV and consists of five Proportional Counter Units (PCUs). Most of the sources in our sample do not show significant counts in the RXTE Hard Energy X-ray Timing Experiment (HEXTE), so we do not include HEXTE data in this study. To focus this study, we choose only Seyfert galaxies for which the RXTE public archive contained a minimum of two pointings separated by at least two weeks. We further required the total observed time be > 40 ks. These selection criteria led to a sample of 40 Seyfert galaxies. For the analysis presented here, we examine the 18 Seyfert 1 and 1.2 galaxies. Six galaxies were eliminated after they were put through our data pipeline (see § 2.2 for more), so the final sample presented here consists of 12 galaxies, listed in Table 1. Because the data come from the public archive, the sample is not uniform from galaxy to galaxy or even from observation to observation; however, we use the Standard 2 data, which provides a standard data mode for these diverse observations. 2.2. Data Pipeline To ensure consistent data reduction of the large volume of data, we developed a data pipeline. The Standard 2 data for each observation was reduced using a combination of FTOOLs and the Pythonr scripting language. The pipeline produces time-resolved spectra, 1Hosted by the High Energy Astrophysics Science Archive and Research Center (HEASARC; http://heasarc.gsfc.nasa.gov/) http://heasarc.gsfc.nasa.gov/ – 4 – each with a minimum of 125,000 net photons, which are extracted using standard PCA selection criteria and background models (Jahoda et al. 2006). Sources which did not have sufficient net photons for even one spectrum were eliminated from the final sample (Table 1 shows the final sample with the 6 eliminated sources listed in the table notes). Each spectrum includes 1% systematic errors. We are confident in the instrument response and background models up to energies of ∼25 keV, so we ignore channels with higher energies. 2.3. Spectral Fitting and Results The data pipeline produced 350 spectra for the 12 galaxies in our sample. Each spectrum was fitted from 3 to 25 keV with an absorbed Compton reflection model plus a Gaussian iron line. In xspec, the PEXRAV (Magdziarz & Zdziarski 1995) model simulates the effects of an exponentially cut-off powerlaw reflected by neutral matter and has seven model parameters: photon index of the intrinsic underlying power-law (Γ), the cutoff energy of the power law in keV (Ec), the relative amount of reflection (R), the redshift (z), the abundance of heavy elements in solar units (Z), the disk inclination angle (i), and the photon flux of the power law at 1 keV in the observer’s frame (A). The relative amount of reflection is normalized to 1 for the case of an isotropic source above a disk of neutral material (Ω = 2π). Adding a Gaussian line (energy in keV (EFe), physical width (σ) in keV, and normalization in units of photons cm−2 s−1) and an absorbing column (NH , in cm −2) yields a total of 12 parameters. We fixed the following values in PEXRAV: Ec = 500 keV, Z = 1.0, and cos i = 0.95. This inclination represents an almost face-on disk; however, since we are seeking trends in the spectral parameters, rather than absolute values, the precise value is not important to this study. In addition, z is fixed at the appropriate value from the NASA Extragalactic Database for each galaxy2. After fitting all spectra to this model, we derived the mean Gaussian width for each source (Table 1), then held σ fixed for a second fit to the model. Our final model has free parameters: Γ, R, A, EFe, iron line normalization and NH . To prevent xspec from pursuing unphysical values of the parameters, we set the following hard limits: 0 ≤ Γ ≤ 5, 0 ≤ R ≤ 5, 5.5 ≤ EFe ≤ 7.5 keV, and 0 ≤ σ ≤ 1.5 keV (for the free-σ fits). Looking at the iron line equivalent width (EW ) and Γ, we find a complex relationship with a “hump” peaking near Γ ∼ 2.0 (Figure 1a). The EW -Γ plot shows a correlation for Γ . 2.0 and an anti-correlation for Γ & 2.0, with a peak near Γ ∼ 2.0 with EW ∼ 250 eV. We also find a strong correlation between R and Γ (Figure 2a), with a best-fit line of 2http://nedwww.ipac.caltech.edu/ http://nedwww.ipac.caltech.edu/ – 5 – R = −0.87 + 0.54 Γ (χ2 = 506/349 = 1.46). We performed a Monte Carlo simulation to determine if our results were an artifact of modeling degeneracies. Each spectrum in the Monte Carlo sample was simulated with NH=10 22 cm−2, Γ=2.0, R=1.0, EFe=6.4 keV, and σFe=0.23 keV. The flux and exposure times were randomly varied for each spectrum. The flux was varied by randomly choosing A from a uniform distribution between 0.004 and 0.06 photons keV−1 cm−2 s−1. The exposure time was randomly generated from a uniform distribution between 300 and 11000s. The ranges for A and the exposure time represent the range of A and exposure for the spectra in the full sample. We generated 200 spectra: 100 simulated using an RXTE Epoch 3 response, 50 using an Epoch 4 response, and 50 using an Epoch 5 response, roughly corresponding to our RXTE sample. Each spectrum was then fitted to the same model as our full sample. The R over Γ plot (Figure 2b) clearly shows a strong correlation with a best-fit line of R = −7.3 + 4.1 Γ (χ2 = 28.96/159 = 0.182), which strongly suggests that the observed R-Γ correlation is a result of modeling degeneracies. The correlation shows a much steeper relationship than the Seyfert 1 data, due to the large number of Seyfert 1 spectra showing R ∼ 0. EW and Γ, however, do not suffer the same degeneracies, which is clear from the Monte Carlo results (Figure 1b). Based on the lack of correlation in our Monte Carlo results, we are confident that the shape of the EW -Γ plot for the data sample is real. To further examine the EW -Γ relationship, we reproduced the EW -Γ plot to show the contribution from each galaxy (Figure 3). The radio-loud galaxies form the rising leg, with the quasar, 3C 273, anchoring the low Γ-low EW portion of the plot. The Seyfert 1 (radio quiet) and 1.2 galaxies tend to congregate at the peak and the falling leg of the plot. The one narrow-line Seyfert 1 diverges from the main cluster of points. Finally, we examined EW as a function of the intrinsic 2-10 keV X-ray luminosity (Lx), using H0 = 70 km s −1 Mpc−1. We fitted the data for each galaxy, each type, and the sample as a whole to linear and powerlaw models. The data were well-fit for either model. For consistency with other publications, we report here the powerlaw results. For the sample as a whole, we see an anticorrelation, i.e. the X-ray Baldwin effect (Iwasawa & Taniguchi 1993), with EW ∝ L−0.14±0.01x . When examining galaxy types, however, the anticorrelation does not always hold up (Table 1). We find an anticorrelation in the radio loud galaxies and the Seyfert 1.2s, but a marginal correlation for the quasar and radio quiet Seyfert 1s. – 6 – 3. Discussion 3.1. EW -Γ Relationship The simulations of George & Fabian (1991) for the observed spectrum from an X-ray source illuminating a half-slab showed that the spectra should include a “Compton hump” and an iron line. They found that the iron line EW should decrease as the spectrum softens. This is easy to understand, since as the spectrum softens (Γ increases), there are fewer photons with energies above the iron photoionization threshold. Our results show that the relationship between EW and Γ is not quite so simple. We find a correlation between EW and Γ when Γ . 2 and an anticorrelation when Γ & 2. Other researchers have found a correlation for Seyfert 1 samples (Perola et al. 2002; Lubiński & Zdziarski 2001), but the galaxies in their samples primarily fell in the Γ . 2 region. Page et al. (2004) also find that their data suggest a slight correlation for a sample of radio loud and radio quiet Type 1 A close examination of our EW -Γ plot shows that the data for different galaxy types progresses across the plot. The plot is anchored at the low-Γ, low-EW end by the quasar, 3C 273, in our sample. The rising arm of the plot, Γ ∼ 1.5− 2.0 and EW ∼ 0 − 300 eV, is primarily formed by radio loud Seyfert 1 galaxies. The radio-quiet Seyfert 1 galaxies cluster near the Γ ∼ 2.0, EW ∼ 300 eV peak of the hump, and the radio-quiet Seyfert 1.2 galaxies form the falling arm of the plot for Γ > 2.0. Physically, the most obvious difference between these sources is the presence or absence of a strong jet. We propose that this relationship is driven by the degree of jet-dominance of the source. The iron line features are associated with the X-ray emission from the disk. Since the disk is essentially isotropic, it will excite an observable iron line from matter out of our line-of-sight. On the other hand, the jet is beamed away from the obvious configurations of matter in the system and, more importantly, is beamed toward us in the quasar and radio- loud sources. Both of these jet-related phenomena reduce the observed equivalent width of any iron line emission associated with the jet continuum. In order for the Γ to increase as the jet-dominance decreases, the jet in these sources must have a hard X-ray component, which implies that the radio-loud Seyferts in our sample are to be associated with low-peaked BL Lac objects (LBLs). BL Lac objects show two broad peaks in their spectral energy density plots (Giommi & Padovani 1994), with the lower-energy peak due to synchrotron emission and the higher-energy peak due to inverse Compton emission. BL Lacs are divided into two classes, depending on where the peaks occur: high-peaked BL Lacs (HBLs) and LBLs. The X-ray continuum in the HBLs is rather soft, since we are seeing the synchrotron spectrum cutting off in these sources. LBLs, on the other hand, tend to – 7 – have a harder X-ray continua, since we are observing well into the inverse Compton part of the spectrum (Donato, Sambruna, & Gliozzi 2005). We also note that much of the falling arm of the EW -Γ relationship is formed by MCG −6-30-15. Recent observations of MCG −6-30-15 by Suzaku have shown that the reflection component, including the iron line, remains relatively constant Miniutti et al. (2006). We would expect, then, that as Γ increases, the EW should decrease, which is exactly what we see in our data. 3.2. R-Γ Relationship Significant degeneracies between the photon index, absorbing column, and reflection fraction can easily lead to false conclusions about spectral correlations. These degeneracies occur as these three parameters trade off against each other in the modeling process, an effect that is especially strong in the RXTE bandpass. Our R-Γ plot shows a strong correlation which is mimicked in our Monte Carlo results. The few points that lie under the main concentration are likely to be outliers, and not indicative of a subclass of galaxy. These points all come from spectra that have been fitted to have NH = 0, and are primarily radio-loud galaxies. We conclude that the observed R-Γ correlation in our sample cannot be trusted as a real correlation. 3.3. EW -Lx Relationship Looking at the EW -Lx relationship, we do see the X-ray Baldwin effect for our sample as a whole, with a slighly shallower anticorrelation than reported elsewhere. We find EW ∝ L−0.14x , whereas Iwasawa & Taniguchi (1993) and Jiang, Wang & Wang (2006) find EW ∝ L−0.20x and Page et al. (2004) find EW ∝ L −0.17 x . 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C. 1991, MNRAS, 249, 352 Giommi, P., Padovani, P. 1994, MNRAS, 268, 51 Iwasawa, K. Taniguchi, Y. 1993, ApJ, 413, L15 Jahoda, K., Markwardt, C. B., Radeva, Y., Rots, A. H., Stark, M. J., Swank, J. H., Strohmayer, T. E., Zhang, W., 2006, ApJS, 163, 401 Jiang, P., Wang, J. X., Wang, T. G. 2006, ApJ, 644, 725 – 9 – Lamer, G., Uttley, P., McHardy, I. M. 2000, MNRAS, 319, 949 Lubiński, P., Zdziarski, A. A. 2001, MNRAS, 323, L37 Magdziarz, P. & Zdziarski, A. A. 1995, MNRAS, 273, 837 Markowitz, A., Edelson, R., Vaughan, S. 2003, ApJ, 598, 935 Miniutti, et al. 2006, PASJ, accepted (astro-ph/0609521) Page, K. L., O’Brien, P. T., Reeves, J. N. Turner, M. J. L. 2004, MNRAS, 347, 316 Papadakis, I. E., Petrucci, P. O., Maraschi, L., McHardy, I. M., Uttley, P., Haardt, F. 2002, ApJ, 573, 92 Perola, G. C., Matt, G., Cappi, M., Fiore, G., Guainazzi, M., Maraschi, L., Petrucci, P. O., Piro, L. 2002, A&A, 389, 802 Wang, J. X., Zhou, Y. Y., Xu, H. G., Wang, T. G. 1999, ApJ, 516, L65 Weaver, K. A., Gelbord, J., Yaqoob, T. 2001, ApJ, 550, 261 Zdziarski, A. A., Lubiński, P., Smith, D. A. 1999, MNRAS, 303, L11 This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0609521 – 10 – Table 1. Sample of RXTE -observed Seyfert 1 and 1.2 galaxiesa Galaxy Seyfert Fitted Average EW/Lx correlation Typeb Spectrac σFeKα d α WV/Num. All -0.14+0.01 −0.01 700/350 Quasars +0.09+0.20 −0.25 105/81 3C 273 1 81 0.329 +0.09+0.20 −0.25 105/81 Broadline Seyfert 1s -0.24+0.14 −0.15 48.0/66 3C 111 1 4 0.239 +0.70+2.60 −1.52 0.654/4 3C 120f 1 40 0.261 -0.70+0.63 −0.61 20.9/39 3C 382 1 5 0.328 -0.80+1.69 −1.70 2.54/5 3C 390.3 1 17 0.203 -0.51+0.44 −0.41 2.70/17 Seyfert 1s (Radio quiet) 0.01+0.300.30 23.6/31 Ark 120 1 15 0.197 -0.66+0.58 −0.57 6.62/15 Fairall 9 1 16 0.155 +0.41+0.44 −0.44 11.1/16 Seyfert 1.2s -0.08+0.03 −0.03 192/169 IC 4329A 1.2 41 0.214 -0.55+0.36 −0.37 27.5/41 MCG -6-30-15 1.2 75 0.292 -0.65+0.34 −0.33 89.2/75 Mkn 509 1.2 16 0.102 -0.52+0.91 −0.99 7.57/16 NGC 7469 1.2 37 0.145 -0.58+0.30 −0.31 17.7/37 Narrow Line Seyfert 1 8.80+20.80 −6.08 0.196/3 TON S180 1.2 3 0.379 8.80+20.80 −6.08 0.196/3 aThe following sources were eliminated after running the data pipeline described in the text, due to having no spectra with at least 125,000 net photons: Mkn 110, PG 0804+761, PG 1211+143, Mkn 79, Mkn 335, and PG 0052+251. bSeyfert type based on the NASA Extragalactic Database cTotal number of spectra extracted using our data pipeline (§ 2.2). dThe average physical width of the Fe Kα line for all spectra from a source when fitted to the absorbed powerlaw model with Compton reflection and Gaussian iron line (§ 2.3). – 11 – eResults of fitting the X-ray luminosity over EW plot to a powerlaw model; e.g. EW ∝ Lαx , where Lx is the 2-10 keV X-ray luminosity in ergs s −1 and EW is the iron line equivalent width in eV. fOne 3C 120 spectrum shows a flare, where Lx jumps by ∼ 6×. The number quoted above excludes this point from the sample. If we include the flare, we find EW ∝ L 0.07(+0.18/−0.25) – 12 – (a) (b) Fig. 1.— Iron line equivalent width in eV (EW ) versus powerlaw photon index (Γ) for the Seyfert 1/1.2 sample (a) and for the Monte Carlo simulations (b). – 13 – (a) (b) Fig. 2.— Reflection fraction (R) versus powerlaw photon index (Γ) for the Seyfert 1/1.2 sample (a) and for the Monte Carlo simulations (b). In both plots, the line shows the best-fit linear model for the Monte Carlo simulations. – 14 – Fig. 3.— The iron line equivalent width in eV versus the powerlaw photon index. This plot is similar to the left panel in Figure 1, but with each galaxy plotted with a separate symbol. The open circles are 3C 111, open squares are 3C120, pluses (+) are 3C273, open triangles are 3C 382, open diamonds 3C 390.3, open stars Akn 120, open crosses Fairall 9, filled circles IC 4329A, filled squares MCG −6-30-15, filled triangles Mkn 509, filled stars NGC 7469, and asterisks (*) TON S180. Introduction Data Analysis The Sample Data Pipeline Spectral Fitting and Results Discussion EW- Relationship R- Relationship EW-Lx Relationship Conclusions

Using Rossi X-ray Timing Explorer Seyfert 1 and 1.2 data spanning 9 years, we study correlations between X-ray spectral features. The sample consists of 350 time-resolved spectra from 12 Seyfert 1 and 1.2 galaxies. Each spectrum is fitted to a model with an intrinsic powerlaw X-ray spectrum produced close to the central black hole that is reprocessed and absorbed by material around the black hole. To test the robustness of our results, we performed Monte Carlo simulations of the spectral sample. We find a complex relationship between the iron line equivalent width (EW) and the underlying power law index (Gamma). The data reveal a correlation between Gamma and EW which turns over at Gamma <~ 2, but finds a weak anti-correlation for steeper photon indices. We propose that this relationship is driven by dilution of a disk spectrum (which includes the narrow iron line) by a beamed jet component and, hence, could be used as a diagnostic of jet-dominance. In addition, our sample shows a strong correlation between the reflection fraction (R) and Gamma, but we find that it is likely the result of modeling degeneracies. We also see the X-ray Baldwin effect (an anti-correlation between the 2-10 keV X-ray luminosity and EW) for the sample as a whole, but not for the individual galaxies and galaxy types.

Introduction Time-resolved X-ray spectroscopy studies of active galactic nuclei (AGN) offer the op- portunity to investigate emission regions near the central black hole. In fact, X-ray spec- troscopy offers the clearest view of processes occurring very close to the black hole itself, probing matter to its final plunge into the black hole. Armed with such information, we can unlock the structure of the innermost regions of AGN. Typical X-ray spectra of AGN show an underlying powerlaw produced near the central black hole with signatures of reprocessed photons often present. These reprocessed photons show up as an Fe Kα line at ∼6.4 keV and a “reflection hump” which starts to dominate near 10 keV and is produced by the combined effects of photoelectric absorption and Compton downscattering in optically-thin cold matter irradiated by the hard X-ray continuum. The Fe Kα line has been observed in both type 1 (unabsorbed) and type 2 (absorbed) Seyfert galaxies. It has been attributed to either the broad line region, the accretion disk, the molec- ular torus of unification models (Antonucci 1993), or some combination of these. Signatures of reflection have also been observed in both Seyfert 1 and 2 galaxies. If the unification models are correct, we should see similar spectral correlations be- tween Seyfert 1 and 2 galaxies, with any differences easily attributable to our viewing angle. Regardless of the accuracy of the reflection models, we expect changes in the un- derlying continuum to drive changes in the reprocessing features. However, results from X-ray spectral studies of AGN have so far produced puzzling results. Samples of Seyfert 1 observations from ASCA (Weaver, Gelbord & Yaqoob 2001) and Rossi X-ray Timing Ex- plorer (Markowitz, Edelson, & Vaughan 2003) have shown no obvious relationship between changes in the continuum and iron line. Several galaxies have shown an anticorrelation between reflection and/or iron line equivalent width and the source flux; e.g. NGC 5548 (Chiang et al. 2000), MCG −6-30-15 (Papadakis et al. 2002), NGC 4051 (Papadakis et al. 2002; Wang et al. 1999), NGC 5506 (Papadakis et al. 2002; Lamer, Uttley & McHardy 2000). Recent data from Suzaku on MCG −6-30-15, on the other hand, show that the iron line and reflection remain relatively constant while the powerlaw is highly variable (Miniutti et al. 2006). Zdziarski, Lubiński & Smith (1999) found that Seyfert galaxies and X-ray binaries show a correlation between the continuum slope and reflection fraction, so those with soft intrinsic spectra show stronger reflection than those with hard spectra. However, other stud- ies have found either a shallower relationship than Zdziarski et al. (Perola et al. 2002) or an anticorrelation (Papadakis et al. 2002; Lamer, Uttley & McHardy 2000). Here we present the first results of a larger study of the X-ray spectral properties of Seyfert galaxies observed by the Rossi X-ray Timing Explorer (RXTE ). Our full study consists of observations of 30 galaxies. In this letter, we focus on the spectral results from – 3 – the subset of 12 Seyfert 1 and 1.2 galaxies. In § 2 we present our method of data analysis, including our sample selection criteria (§ 2.1), a description of our data pipeline (§ 2.2), and results of our spectral analysis (§ 2.3). We discuss the implications of our results in § 3 and detail our conclusions in § 4. 2. Data Analysis 2.1. The Sample The RXTE public archive1 represents one of the largest collections of X-ray data for AGN, with pointed observations of over 100 AGN spanning 10 years. The RXTE bandpass allows the study of absorption and iron line properties of AGN spectra, as well as a glimpse at the Compton reflection hump. We use data from the RXTE proportional counter array (PCA), which is sensitive to energies from 2 to 60 keV and consists of five Proportional Counter Units (PCUs). Most of the sources in our sample do not show significant counts in the RXTE Hard Energy X-ray Timing Experiment (HEXTE), so we do not include HEXTE data in this study. To focus this study, we choose only Seyfert galaxies for which the RXTE public archive contained a minimum of two pointings separated by at least two weeks. We further required the total observed time be > 40 ks. These selection criteria led to a sample of 40 Seyfert galaxies. For the analysis presented here, we examine the 18 Seyfert 1 and 1.2 galaxies. Six galaxies were eliminated after they were put through our data pipeline (see § 2.2 for more), so the final sample presented here consists of 12 galaxies, listed in Table 1. Because the data come from the public archive, the sample is not uniform from galaxy to galaxy or even from observation to observation; however, we use the Standard 2 data, which provides a standard data mode for these diverse observations. 2.2. Data Pipeline To ensure consistent data reduction of the large volume of data, we developed a data pipeline. The Standard 2 data for each observation was reduced using a combination of FTOOLs and the Pythonr scripting language. The pipeline produces time-resolved spectra, 1Hosted by the High Energy Astrophysics Science Archive and Research Center (HEASARC; http://heasarc.gsfc.nasa.gov/) http://heasarc.gsfc.nasa.gov/ – 4 – each with a minimum of 125,000 net photons, which are extracted using standard PCA selection criteria and background models (Jahoda et al. 2006). Sources which did not have sufficient net photons for even one spectrum were eliminated from the final sample (Table 1 shows the final sample with the 6 eliminated sources listed in the table notes). Each spectrum includes 1% systematic errors. We are confident in the instrument response and background models up to energies of ∼25 keV, so we ignore channels with higher energies. 2.3. Spectral Fitting and Results The data pipeline produced 350 spectra for the 12 galaxies in our sample. Each spectrum was fitted from 3 to 25 keV with an absorbed Compton reflection model plus a Gaussian iron line. In xspec, the PEXRAV (Magdziarz & Zdziarski 1995) model simulates the effects of an exponentially cut-off powerlaw reflected by neutral matter and has seven model parameters: photon index of the intrinsic underlying power-law (Γ), the cutoff energy of the power law in keV (Ec), the relative amount of reflection (R), the redshift (z), the abundance of heavy elements in solar units (Z), the disk inclination angle (i), and the photon flux of the power law at 1 keV in the observer’s frame (A). The relative amount of reflection is normalized to 1 for the case of an isotropic source above a disk of neutral material (Ω = 2π). Adding a Gaussian line (energy in keV (EFe), physical width (σ) in keV, and normalization in units of photons cm−2 s−1) and an absorbing column (NH , in cm −2) yields a total of 12 parameters. We fixed the following values in PEXRAV: Ec = 500 keV, Z = 1.0, and cos i = 0.95. This inclination represents an almost face-on disk; however, since we are seeking trends in the spectral parameters, rather than absolute values, the precise value is not important to this study. In addition, z is fixed at the appropriate value from the NASA Extragalactic Database for each galaxy2. After fitting all spectra to this model, we derived the mean Gaussian width for each source (Table 1), then held σ fixed for a second fit to the model. Our final model has free parameters: Γ, R, A, EFe, iron line normalization and NH . To prevent xspec from pursuing unphysical values of the parameters, we set the following hard limits: 0 ≤ Γ ≤ 5, 0 ≤ R ≤ 5, 5.5 ≤ EFe ≤ 7.5 keV, and 0 ≤ σ ≤ 1.5 keV (for the free-σ fits). Looking at the iron line equivalent width (EW ) and Γ, we find a complex relationship with a “hump” peaking near Γ ∼ 2.0 (Figure 1a). The EW -Γ plot shows a correlation for Γ . 2.0 and an anti-correlation for Γ & 2.0, with a peak near Γ ∼ 2.0 with EW ∼ 250 eV. We also find a strong correlation between R and Γ (Figure 2a), with a best-fit line of 2http://nedwww.ipac.caltech.edu/ http://nedwww.ipac.caltech.edu/ – 5 – R = −0.87 + 0.54 Γ (χ2 = 506/349 = 1.46). We performed a Monte Carlo simulation to determine if our results were an artifact of modeling degeneracies. Each spectrum in the Monte Carlo sample was simulated with NH=10 22 cm−2, Γ=2.0, R=1.0, EFe=6.4 keV, and σFe=0.23 keV. The flux and exposure times were randomly varied for each spectrum. The flux was varied by randomly choosing A from a uniform distribution between 0.004 and 0.06 photons keV−1 cm−2 s−1. The exposure time was randomly generated from a uniform distribution between 300 and 11000s. The ranges for A and the exposure time represent the range of A and exposure for the spectra in the full sample. We generated 200 spectra: 100 simulated using an RXTE Epoch 3 response, 50 using an Epoch 4 response, and 50 using an Epoch 5 response, roughly corresponding to our RXTE sample. Each spectrum was then fitted to the same model as our full sample. The R over Γ plot (Figure 2b) clearly shows a strong correlation with a best-fit line of R = −7.3 + 4.1 Γ (χ2 = 28.96/159 = 0.182), which strongly suggests that the observed R-Γ correlation is a result of modeling degeneracies. The correlation shows a much steeper relationship than the Seyfert 1 data, due to the large number of Seyfert 1 spectra showing R ∼ 0. EW and Γ, however, do not suffer the same degeneracies, which is clear from the Monte Carlo results (Figure 1b). Based on the lack of correlation in our Monte Carlo results, we are confident that the shape of the EW -Γ plot for the data sample is real. To further examine the EW -Γ relationship, we reproduced the EW -Γ plot to show the contribution from each galaxy (Figure 3). The radio-loud galaxies form the rising leg, with the quasar, 3C 273, anchoring the low Γ-low EW portion of the plot. The Seyfert 1 (radio quiet) and 1.2 galaxies tend to congregate at the peak and the falling leg of the plot. The one narrow-line Seyfert 1 diverges from the main cluster of points. Finally, we examined EW as a function of the intrinsic 2-10 keV X-ray luminosity (Lx), using H0 = 70 km s −1 Mpc−1. We fitted the data for each galaxy, each type, and the sample as a whole to linear and powerlaw models. The data were well-fit for either model. For consistency with other publications, we report here the powerlaw results. For the sample as a whole, we see an anticorrelation, i.e. the X-ray Baldwin effect (Iwasawa & Taniguchi 1993), with EW ∝ L−0.14±0.01x . When examining galaxy types, however, the anticorrelation does not always hold up (Table 1). We find an anticorrelation in the radio loud galaxies and the Seyfert 1.2s, but a marginal correlation for the quasar and radio quiet Seyfert 1s. – 6 – 3. Discussion 3.1. EW -Γ Relationship The simulations of George & Fabian (1991) for the observed spectrum from an X-ray source illuminating a half-slab showed that the spectra should include a “Compton hump” and an iron line. They found that the iron line EW should decrease as the spectrum softens. This is easy to understand, since as the spectrum softens (Γ increases), there are fewer photons with energies above the iron photoionization threshold. Our results show that the relationship between EW and Γ is not quite so simple. We find a correlation between EW and Γ when Γ . 2 and an anticorrelation when Γ & 2. Other researchers have found a correlation for Seyfert 1 samples (Perola et al. 2002; Lubiński & Zdziarski 2001), but the galaxies in their samples primarily fell in the Γ . 2 region. Page et al. (2004) also find that their data suggest a slight correlation for a sample of radio loud and radio quiet Type 1 A close examination of our EW -Γ plot shows that the data for different galaxy types progresses across the plot. The plot is anchored at the low-Γ, low-EW end by the quasar, 3C 273, in our sample. The rising arm of the plot, Γ ∼ 1.5− 2.0 and EW ∼ 0 − 300 eV, is primarily formed by radio loud Seyfert 1 galaxies. The radio-quiet Seyfert 1 galaxies cluster near the Γ ∼ 2.0, EW ∼ 300 eV peak of the hump, and the radio-quiet Seyfert 1.2 galaxies form the falling arm of the plot for Γ > 2.0. Physically, the most obvious difference between these sources is the presence or absence of a strong jet. We propose that this relationship is driven by the degree of jet-dominance of the source. The iron line features are associated with the X-ray emission from the disk. Since the disk is essentially isotropic, it will excite an observable iron line from matter out of our line-of-sight. On the other hand, the jet is beamed away from the obvious configurations of matter in the system and, more importantly, is beamed toward us in the quasar and radio- loud sources. Both of these jet-related phenomena reduce the observed equivalent width of any iron line emission associated with the jet continuum. In order for the Γ to increase as the jet-dominance decreases, the jet in these sources must have a hard X-ray component, which implies that the radio-loud Seyferts in our sample are to be associated with low-peaked BL Lac objects (LBLs). BL Lac objects show two broad peaks in their spectral energy density plots (Giommi & Padovani 1994), with the lower-energy peak due to synchrotron emission and the higher-energy peak due to inverse Compton emission. BL Lacs are divided into two classes, depending on where the peaks occur: high-peaked BL Lacs (HBLs) and LBLs. The X-ray continuum in the HBLs is rather soft, since we are seeing the synchrotron spectrum cutting off in these sources. LBLs, on the other hand, tend to – 7 – have a harder X-ray continua, since we are observing well into the inverse Compton part of the spectrum (Donato, Sambruna, & Gliozzi 2005). We also note that much of the falling arm of the EW -Γ relationship is formed by MCG −6-30-15. Recent observations of MCG −6-30-15 by Suzaku have shown that the reflection component, including the iron line, remains relatively constant Miniutti et al. (2006). We would expect, then, that as Γ increases, the EW should decrease, which is exactly what we see in our data. 3.2. R-Γ Relationship Significant degeneracies between the photon index, absorbing column, and reflection fraction can easily lead to false conclusions about spectral correlations. These degeneracies occur as these three parameters trade off against each other in the modeling process, an effect that is especially strong in the RXTE bandpass. Our R-Γ plot shows a strong correlation which is mimicked in our Monte Carlo results. The few points that lie under the main concentration are likely to be outliers, and not indicative of a subclass of galaxy. These points all come from spectra that have been fitted to have NH = 0, and are primarily radio-loud galaxies. We conclude that the observed R-Γ correlation in our sample cannot be trusted as a real correlation. 3.3. EW -Lx Relationship Looking at the EW -Lx relationship, we do see the X-ray Baldwin effect for our sample as a whole, with a slighly shallower anticorrelation than reported elsewhere. We find EW ∝ L−0.14x , whereas Iwasawa & Taniguchi (1993) and Jiang, Wang & Wang (2006) find EW ∝ L−0.20x and Page et al. (2004) find EW ∝ L −0.17 x . However, when Jiang, Wang & Wang (2006) exclude the radio loud galaxies from their sample, they find EW ∝ L−0.10x . We find, though, that when we examine our data on a galaxy-by-galaxy or type-by-type basis, the effect is not consistent from source to source. At this point, we cannot determine if these variations are real or are simply due to the small number of spectra for some of our galaxies and types. – 8 – 4. Conclusions We have examined time-resolved spectra of 12 Seyfert 1 and 1.2 galaxies observed by RXTE over seven years. We find a complex relationship between the iron line equivalent width and the continuum slope, with a correlation for Γ . 2 that turns over to an anticor- relation for Γ & 2. We propose that this relationship is a possible diagnostic for jet- versus disk-dominated sources, where jet-dominated sources show a correlation between EW and Γ, and disk-dominated sources show an anticorrelation. We also see a strong correlation between Γ and R which is likely an artifact of modeling degeneracies caused by the interplay of Γ, R, and nH in the RXTE bandpass. Finally, we observe the X-ray Baldwin effect for the sample as a whole, but not for each galaxy and galaxy type individually. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center. This research has also made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. CSR gratefully acknowledges support from the National Science Foundation under grants AST0205990 and AST0607428. REFERENCES Antonucci, R. 1993, ARA&A, 31, 473 Chiang, J. 2000, ApJ, 528, 292 Donato, D., Sambruna, R. M., Gliozzi, M. 2005, å, 433, 1163 George, I. M., Fabian, A. C. 1991, MNRAS, 249, 352 Giommi, P., Padovani, P. 1994, MNRAS, 268, 51 Iwasawa, K. Taniguchi, Y. 1993, ApJ, 413, L15 Jahoda, K., Markwardt, C. B., Radeva, Y., Rots, A. H., Stark, M. J., Swank, J. H., Strohmayer, T. E., Zhang, W., 2006, ApJS, 163, 401 Jiang, P., Wang, J. X., Wang, T. G. 2006, ApJ, 644, 725 – 9 – Lamer, G., Uttley, P., McHardy, I. M. 2000, MNRAS, 319, 949 Lubiński, P., Zdziarski, A. A. 2001, MNRAS, 323, L37 Magdziarz, P. & Zdziarski, A. A. 1995, MNRAS, 273, 837 Markowitz, A., Edelson, R., Vaughan, S. 2003, ApJ, 598, 935 Miniutti, et al. 2006, PASJ, accepted (astro-ph/0609521) Page, K. L., O’Brien, P. T., Reeves, J. N. Turner, M. J. L. 2004, MNRAS, 347, 316 Papadakis, I. E., Petrucci, P. O., Maraschi, L., McHardy, I. M., Uttley, P., Haardt, F. 2002, ApJ, 573, 92 Perola, G. C., Matt, G., Cappi, M., Fiore, G., Guainazzi, M., Maraschi, L., Petrucci, P. O., Piro, L. 2002, A&A, 389, 802 Wang, J. X., Zhou, Y. Y., Xu, H. G., Wang, T. G. 1999, ApJ, 516, L65 Weaver, K. A., Gelbord, J., Yaqoob, T. 2001, ApJ, 550, 261 Zdziarski, A. A., Lubiński, P., Smith, D. A. 1999, MNRAS, 303, L11 This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0609521 – 10 – Table 1. Sample of RXTE -observed Seyfert 1 and 1.2 galaxiesa Galaxy Seyfert Fitted Average EW/Lx correlation Typeb Spectrac σFeKα d α WV/Num. All -0.14+0.01 −0.01 700/350 Quasars +0.09+0.20 −0.25 105/81 3C 273 1 81 0.329 +0.09+0.20 −0.25 105/81 Broadline Seyfert 1s -0.24+0.14 −0.15 48.0/66 3C 111 1 4 0.239 +0.70+2.60 −1.52 0.654/4 3C 120f 1 40 0.261 -0.70+0.63 −0.61 20.9/39 3C 382 1 5 0.328 -0.80+1.69 −1.70 2.54/5 3C 390.3 1 17 0.203 -0.51+0.44 −0.41 2.70/17 Seyfert 1s (Radio quiet) 0.01+0.300.30 23.6/31 Ark 120 1 15 0.197 -0.66+0.58 −0.57 6.62/15 Fairall 9 1 16 0.155 +0.41+0.44 −0.44 11.1/16 Seyfert 1.2s -0.08+0.03 −0.03 192/169 IC 4329A 1.2 41 0.214 -0.55+0.36 −0.37 27.5/41 MCG -6-30-15 1.2 75 0.292 -0.65+0.34 −0.33 89.2/75 Mkn 509 1.2 16 0.102 -0.52+0.91 −0.99 7.57/16 NGC 7469 1.2 37 0.145 -0.58+0.30 −0.31 17.7/37 Narrow Line Seyfert 1 8.80+20.80 −6.08 0.196/3 TON S180 1.2 3 0.379 8.80+20.80 −6.08 0.196/3 aThe following sources were eliminated after running the data pipeline described in the text, due to having no spectra with at least 125,000 net photons: Mkn 110, PG 0804+761, PG 1211+143, Mkn 79, Mkn 335, and PG 0052+251. bSeyfert type based on the NASA Extragalactic Database cTotal number of spectra extracted using our data pipeline (§ 2.2). dThe average physical width of the Fe Kα line for all spectra from a source when fitted to the absorbed powerlaw model with Compton reflection and Gaussian iron line (§ 2.3). – 11 – eResults of fitting the X-ray luminosity over EW plot to a powerlaw model; e.g. EW ∝ Lαx , where Lx is the 2-10 keV X-ray luminosity in ergs s −1 and EW is the iron line equivalent width in eV. fOne 3C 120 spectrum shows a flare, where Lx jumps by ∼ 6×. The number quoted above excludes this point from the sample. If we include the flare, we find EW ∝ L 0.07(+0.18/−0.25) – 12 – (a) (b) Fig. 1.— Iron line equivalent width in eV (EW ) versus powerlaw photon index (Γ) for the Seyfert 1/1.2 sample (a) and for the Monte Carlo simulations (b). – 13 – (a) (b) Fig. 2.— Reflection fraction (R) versus powerlaw photon index (Γ) for the Seyfert 1/1.2 sample (a) and for the Monte Carlo simulations (b). In both plots, the line shows the best-fit linear model for the Monte Carlo simulations. – 14 – Fig. 3.— The iron line equivalent width in eV versus the powerlaw photon index. This plot is similar to the left panel in Figure 1, but with each galaxy plotted with a separate symbol. The open circles are 3C 111, open squares are 3C120, pluses (+) are 3C273, open triangles are 3C 382, open diamonds 3C 390.3, open stars Akn 120, open crosses Fairall 9, filled circles IC 4329A, filled squares MCG −6-30-15, filled triangles Mkn 509, filled stars NGC 7469, and asterisks (*) TON S180. Introduction Data Analysis The Sample Data Pipeline Spectral Fitting and Results Discussion EW- Relationship R- Relationship EW-Lx Relationship Conclusions

On algebraic automorphisms and their rational invariants Philippe Bonnet Mathematisches Institut, Universität Basel Rheinsprung 21, 4051 Basel, Switzerland e-mail: Philippe.bonnet@unibas.ch Abstract Let X be an affine irreducible variety over an algebraically closed field k of char- acteristic zero. Given an automorphism Φ, we denote by k(X)Φ its field of invariants, i.e the set of rational functions f on X such that f ◦Φ = f . Let n(Φ) be the transcen- dence degree of k(X)Φ over k. In this paper, we study the class of automorphisms Φ of X for which n(Φ) = dimX − 1. More precisely, we show that under some condi- tions on X, every such automorphism is of the form Φ = ϕg, where ϕ is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(Φ) = 1. 1 Introduction Let k be an algebraically closed field of characteristic zero. Let X be an affine irreducible variety of dimension n over k. We denote by O(X) its ring of regular functions, and by k(X) its field of rational functions. Given an algebraic automorphism Φ of X , denote by Φ∗ the field automorphism induced by Φ on k(X), i.e. Φ∗(f) = f ◦Φ for any f ∈ k(X). An element f of k(X) is invariant for Φ (or simply invariant) if Φ∗(f) = f . Invariant rational functions form a field denoted k(X)Φ, and we set: n(Φ) = trdegk k(X) In this paper, we are going to study the class of automorphisms of X for which n(Φ) = n − 1. There are natural candidates for such automorphisms, such as exponentials of locally nilpotent derivations (see [M] or [Da]). More generally, one can construct such automorphisms by means of algebraic group actions as follows. Let G be a linear algebraic group over k. An algebraic action of G on X is a regular map: ϕ : G×X −→ X http://arxiv.org/abs/0704.1588v1 of affine varieties, such that ϕ(g.g′, x) = ϕ(g, ϕ(g′, x)) for any (g, g′, x) in G × G × X . Given an element g of G, denote by ϕg the map x 7→ ϕ(g, x). Then ϕg clearly defines an automorphism of X . Let k(X)G be the field of invariants of G, i.e. the set of rational functions f on X such that f ◦ ϕg = f for any g ∈ G. If G is an algebraic group of dimension 1, acting faithfully on X , and if g is an element of G of infinite order, then one can prove by Rosenlicht’s Theorem (see [Ro]) that: n(ϕg) = trdegk k(X) G = n− 1 We are going to see that, under some mild conditions on X , there are no other automor- phisms with n(Φ) = n−1 than those constructed above. In what follows, denote by O(X)ν the normalization of O(X), and by G(X) the group of invertible elements of O(X)ν . Theorem 1.1 Let X be an affine irreducible variety of dimension n over k, such that char(k) = 0 and G(X)∗ = k∗. Let Φ be an algebraic automorphism of X such that n(Φ) = n − 1. Then there exist an abelian linear algebraic group G of dimension 1, and an algebraic action ϕ of G on X such that Φ = ϕg for some g ∈ G of infinite order. Note that the structure of G is fairly simple. Since every connected linear algebraic group of dimension 1 is either isomorphic to Ga(k) = (k,+) or Gm(k) = (k ∗,×) (see [Hum], p. 131), there exists a finite abelian group H such that G is either equal to H × Ga(k) or H × Gm(k). Moreover, the assumption on the group G(X) is essential. Indeed, consider the automorphism Φ of k∗ × k given by Φ(x, y) = (x, xy). Obviously, its field of invariants is equal to k(x). However, it is easy to check that Φ cannot have the form given in the conclusion of Theorem 1.1. This theorem is analogous to a result given by Van den Essen and Peretz (see [V-P]). More precisely, they establish a criterion to decide if an automorphism Φ is the exponential of a locally nilpotent derivation, based on the invariants and on the form of Φ. A similar result has been developed by Daigle (see [Da]). We apply these results to the group of automorphisms of the plane. First, we obtain a classification of the automorphisms Φ of k2 for which n(Φ) = 1. Second, we derive a criterion on automorphisms of k2 to have no nonconstant rational invariants. Corollary 1.2 Let Φ be an algebraic automorphism of k2. If n(Φ) = 1, then Φ is conjugate to one of the following forms: • Φ1(x, y) = (a nx, amby), where (n,m) 6= (0, 0), a, b ∈ k, b is a root of unity but a is • Φ2(x, y) = (ax, by + P (x)), where P belongs to k[t]− {0}, a, b ∈ k are roots of unity. Corollary 1.3 Let Φ be an algebraic automorphism of k2. Assume that Φ has a unique fixpoint p and that dΦp is unipotent. Then n(Φ) = 0. We then apply Corollary 1.3 to an automorphism of C3 recently discovered by Pierre-Marie Poloni and Lucy Moser (see [M-P]). We may wonder whether Theorem 1.1 still holds if the ground field k is not algebraically closed or has positive characteristic. The answer is not known for the moment. In fact, two obstructions appear in the proof of Theorem 1.1 when k is arbitrary. First, the group Gm(k) needs to be divisible (see Lemma 4.2), which is not always the case if k is not algebraically closed. Second, the proof uses the fact that every Ga(k)-action on X can be reconstructed from a locally nilpotent derivation on O(X) (see subsection 4.1), which is no longer true if k has positive characteristic. This phenomenom is due to the existence of differents forms for the affine line (see [Ru]). Note that, in case Theorem 1.1 holds and k is not algebraically closed, the algebraic group G needs not be isomorphic to H ×Ga(k) or H ×Gm(k), where H is finite. Indeed consider the unit circle X in the plane R 2, given by the equation x2 + y2 = 1. Let Φ be a rotation in R2 with center at the origin and angle θ 6∈ 2πQ. Then Φ defines an algebraic automorphism of X with n(Φ) = 0, and the subgroup spanned by Φ is dense in SO2(R). But SO2(R) is not isomorphic to either Ga(R) or Gm(R), even though it is a connected linear algebraic group of dimension 1. We may also wonder what happens to the automorphisms Φ of X for which n(Φ) = dimX − 2. More precisely, does there exist an action ϕ of a linear algebraic group G on X , of dimension 2, such that Φ = ϕg for a given g ∈ G? The answer is no. Indeed consider the automorphism Φ of k2 given by Φ = f ◦ g, where f(x, y) = (x + y2, y) and g(x, y) = (x, y + x2). Let d(n) denote the maximum of the homogeneous degrees of the coordinate functions of the iterate Φn. If there existed an action ϕ of a linear algebraic group G such that Φ = ϕg, then the function d would be bounded, which is impossible since d(n) = 4n. A similar argument on the length of the iterates also yields the result. But if we restrict to some specific varieties X , for instance X = k3, one may ask the following question: If n(Φ) = 1, is Φ birationally conjugate to an automorphism that leaves the first coordinate of k3 invariant? The answer is still unknown. 2 Reduction to an affine curve C Let X be an affine irreducible variety of dimension n over k. Let Φ be an algebraic automorphism of X such that n(Φ) = n − 1. In this section, we are going to construct an irreducible affine curve on which Φ acts naturally. This will allow us to use some well-known results on automorphisms of curves. We set: K = {f ∈ k(X)|∃m > 0, f ◦ Φm = f ◦ Φ ◦ ... ◦ Φ = f} It is straightforward that K is a subfield of k(X) containing both k and k(X)Φ. We begin with some properties of this field. Lemma 2.1 K has transcendence degree (n − 1) over k, and is algebraically closed in k(X). In particular, the automorphism Φ of X has infinite order. Proof: First we show that K has transcendence degree (n − 1) over k. Since K contains the field k(X)Φ, whose transcendence degree is (n − 1), we only need to show that the extension K/k(X)Φ is algebraic, or in other words that every element of K is algebraic over k(X)Φ. Let f be any element of K. By definition, there exists an integer m > 0 such that f ◦ Φm = f . Let P (t) be the polynomial of k(X)[t] defined as: P (t) = (t− f ◦ Φi) By construction, the coefficients of this polynomial are all invariant for Φ, and P (t) belongs to k(X)Φ[t]. Moreover P (f) = 0, f is algebraic over k(X)Φ and the first assertion follows. Second we show that K is algebraically closed in k(X). Let f be an element of k(X) that is algebraic over K. We need to prove that f belongs to K. By the first assertion of the lemma, f is algebraic over k(X)Φ. Let P (t) = a0 + a1t + ... + apt p be a nonzero minimal polynomial of f over k(X)Φ. Since P (f) = 0 and all ai are invariant, we have P (f ◦ Φ) = P (f) ◦ Φ = 0. In particular, all elements of the form f ◦ Φi, with i ∈ N, are roots of P . Since P has finitely many roots, there exist two distinct integers m′ < m′′ such that f ◦ Φm = f ◦ Φm . In particular, f ◦ Φm ′′−m′ = f and f belongs to K. Now if Φ were an automorphism of finite order, then K would be equal to k(X). But this is impossible since K and k(X) have different transcendence degrees. Lemma 2.2 There exists an integer m > 0 such that K = k(X)Φ Proof: By definition, k(X) is a field of finite type over k. Since K is contained in k(X), K has also finite type over k. Let f1, ..., fr be some elements of k(X) such that K = k(f1, ..., fr). Let m1, ..., mr be some positive integers such that fi ◦ Φ mi = fi, and set m = m1...mr. By construction, all fi are invariant for Φ m. In particular, K is invariant for Φm and K ⊆ k(X)Φ . Since k(X)Φ ⊆ K, the result follows. Let L be the algebraic closure of k(X), and let A be the K-subalgebra of L spanned by O(X). By construction, A is an integral K-algebra of finite type of dimension 1. Let m be an integer satisfying the conditions of lemma 2.2. The automorphism Ψ∗ = (Φm)∗ of O(X) stabilizes A, hence it defines a K-automorphism of A, of infinite order (see lemma 2.1). Let B be the integral closure of A. Then B is also an integral K-algebra of finite type, of dimension 1, and the K-automorphism Ψ∗ extends uniquely to B. If K stands for the algebraic closure of K, we set: C = B ⊗K K By construction, C = Spec(C) is an affine curve over the algebraically closed field K. Moreover the automorphism Ψ∗ acts on C via the operation: Ψ∗ : C −→ C, x⊗ y 7−→ Ψ∗(x)⊗ y This makes sense since Ψ∗ fixes the field K. Therefore Ψ∗ induces an algebraic automor- phism of the curve C. Since K is algebraically closed in k(X) by lemma 2.1, C is integral (see [Z-S], Chap. VII, §11, Theorem 38). But by construction, B and K are normal rings. Since C is a domain and char(K) = 0, C is also integrally closed by a result of Bourbaki (see [Bou], p. 29). So C is a normal domain and C is a smooth irreducible curve. Lemma 2.3 Let C be the K-algebra constructed above. Then either C = K[t] or C = K[t, 1/t]. Proof: By lemma 2.1, the automorphism Φ of X has infinite order. Since the fraction field of B is equal to k(X), Ψ∗ has infinite order on B. But B ⊗ 1 ⊂ C, so Ψ∗ has infinite order on C. In particular, Ψ acts like an automorphism of infinite order on C. Since C is affine, it has genus zero (see [Ro2]). Since K is algebraically closed, the curve C is rational (see [Che], p. 23 ). Since C is smooth, it is isomorphic to P1(K) − E, where E is a finite set. Moreover, Ψ acts like an automorphism of P1(K) that stabilizes P1(K) − E. Up to replacing Ψ by one of its iterates, we may assume that Ψ fixes every point of E. But an automorphism of P1(K) that fixes at least three points is the identity, which is impossible. Therefore E consists of at most two points, and C is either isomorphic to K or to K particular, either C = K[t] or C = K[t, 1/t]. 3 Normal forms for the automorphism Ψ Let C and Ψ∗ be the K-algebra and the K-automorphism constructed in the previous section. In this section, we are going to give normal forms for the couple (C,Ψ∗), in case the group G(X) is trivial, i.e. G(X) = k∗. We begin with a few lemmas. Lemma 3.1 Let X be an irreducible affine variety over k. Let Ψ be an automorphism of X. Let α, f be some elements of k(X)∗ such that (Ψ∗)n(f) = αnf for any n ∈ Z. Then α belongs to G(X). Proof: Given an element h of k(X)∗ and a prime divisor D on the normalization Xν , we consider h as a rational function on Xν , and denote by ordD(h) the multiplicity of h along D. This makes sense since the variety Xν is normal. Fix any prime divisor D on X . Since (Ψ∗)n(f) = αnf for any n ∈ Z, we obtain: ordD((Ψ ∗)n(f)) = nordD(α) + ordD(f) Since Ψ is an algebraic automorphism of X , it extends uniquely to an algebraic automor- phism of Xν , which is still denoted Ψ. Moreover, this extension maps every prime divisor to another prime divisor, does not change the multiplicity and maps distinct prime divisors into distinct ones. If div(f) = i niDi, where all Di are prime, then we have: div((Ψ∗)n(f)) = ∗)n(Di) where all (Ψ∗)n(Di) are prime and distinct. So the multiplicity of (Ψ ∗)n(f) along D is equal to zero if D is none of the (Ψ∗)n(Di), and equal to ni if D = (Ψ ∗)n(Di). In all cases, if R = max{|ni|}, then we find that |ordD((Ψ ∗)n(f))| ≤ R and |ordD(f)| ≤ R, and this implies for any integer n: |nordD(α)| ≤ 2R In particular we find ordD(α) = 0. Since this holds for any prime divisor D, the support of div(α) in Xν is empty and div(α) = 0. Since Xν is normal, α is an invertible element of O(X)ν , hence it belongs to G(X). Lemma 3.2 Let K be a field of characteristic zero and K its algebraic closure. Let C be either equal to K[t] or to K[t, 1/t]. Let Ψ∗ be a K-automorphism of C such that Ψ∗(t) = at, where a belongs to K. Let σ1 be a K-automorphism of C, commuting with Ψ ∗, such that σ1(K) = K. Then σ1(a) is either equal to a or to 1/a. Proof: We distinguish two cases depending on the ring C. First assume that C = K[t]. Since σ1 is a K-automorphism of C that maps K to itself, we have K[t] = K[σ1(t)]. In particular σ1(t) = λt + µ, where λ, µ belong to K and λ 6= 0. Since Ψ ∗ and σ1 commute, we obtain: Ψ∗ ◦ σ1(t) = λat + µ = σ1 ◦Ψ ∗(t) = σ1(a)(λt + µ) In particular, we have σ1(a) = a and the lemma follows in this case. Second assume that C = K[t, 1/t]. Since σ1 is a K-automorphism of C, we find: σ1(t)σ1(1/t) = σ1(t.1/t) = σ1(1) = 1 Therefore σ1(t) is an invertible element of C, and has the form σ1(t) = a1t n1 , where a1 ∈ K and n1 is an integer. Since σ1 is a K-automorphism of C that maps K to K, we have K[t, 1/t] = K[σ1(t), 1/σ1(t)]. In particular |n1| = 1 and either σ1(t) = a1t or σ1(t) = a1/t. If σ1(t) = a1t, the relation Ψ ∗ ◦ σ1(t) = σ1 ◦ Ψ ∗(t) yields σ(a) = a. If σ1(t) = a1/t, then the same relation yields σ(a) = 1/a. Lemma 3.3 Let X be an irreducible affine variety of dimension n over k, such that G(X) = k∗. Let Φ be an automorphism of X such that n(Φ) = (n− 1). Let Ψ∗ be the au- tomorphism of C constructed in the previous section. If either C = K[t] or C = K[t, 1/t], and if Ψ∗(t) = at, then a belongs to k∗. Proof: We are going to prove by contradiction that a belongs to k∗. So assume that a 6∈ k∗. Let σ be any element of Gal(K/K), and denote by σ1 the K-automorphism of C defined as follows: ∀(x, y) ∈ B ×K, σ1(x⊗ y) = x⊗ σ1(y) Since Ψ∗ ◦σ1(x⊗ y) = Ψ ∗(x)⊗σ1(y) = σ1 ◦Ψ ∗(x⊗ y) for any element x⊗ y of B⊗KK, Ψ and σ1 commute. Moreover if we identify K with 1⊗K, then σ1(K) = K by construction. By lemma 3.2, we obtain: ∀σ ∈ Gal(K/K), σ(a) = a or σ(a) = In particular, the element (ai + a−i) is invariant under the action of Gal(K/K) for any i, and so it belongs to K because char(K) = 0. Now let f be an element of B −K. Since f belongs to C, we can express f as follows: Choose an f ∈ B−K such that the difference (s−r) is minimal. We claim that (s−r) = 0, i.e. f = fst s. Indeed, assume that s > r. Since f is an element of B, the following expressions: Ψ∗(f) + (Ψ∗)−1(f)− (as + a−s)f = i=r fi(a i + a−i − as − a−s)ti Ψ∗(f) + (Ψ∗)−1(f)− (ar + a−r)f = i=r+1 i + a−i − ar − a−r)ti also belong to B. By minimality of (s− r), these expressions belong to K. In other words, i+ a−i− as− a−s) = 0 (resp. fi(a i+ a−i− ar− a−r) = 0) for any i 6= 0, s (resp. for any i 6= 0, r). Since k is algebraically closed and a 6∈ k∗ by assumption, (ai + a−i − as − a−s) (resp. (ai+a−i−ar−a−r)) is nonzero for any i 6= s (resp. for any i 6= r). Therefore fi = 0 for any i 6= 0, and f belongs to K, a contradiction. Therefore s = r and f = fst s. Since f belongs to B, it also belongs to k(X). Since Ψ is an automorphism of X , the element as = Ψ∗(f)/f belongs to k(X). Moreover (Ψ∗)n(f) = ansf for any n ∈ Z. By lemma 3.1, as belongs to G(X) = k∗. Since k is algebraically closed, a belongs to k∗, hence a contradiction, and the result follows. Proposition 3.4 Let X be an irreducible affine variety of dimension n over k, such that G(X) = k∗. Let Φ be an automorphism of X such that n(Φ) = (n− 1). Let C and Ψ∗ be the K-algebra and the K-automorphism constructed in the previous section. Then up to conjugation, one of the following three cases occurs: • C = K[t] and Ψ∗(t) = t + 1, • C = K[t] and Ψ∗(t) = at, where a ∈ k∗ is not a root of unity, • C = K[t, 1/t] and Ψ∗(t) = at, where a ∈ k∗ is not a root of unity. Proof: By lemma 2.3, we know that either C = K[t] or C = K[t, 1/t]. We are going to study both cases. First case: C = K[t]. The automorphism Ψ∗ maps t to at + b, where a ∈ K and b ∈ K. If a = 1, then b 6= 0 and up to replacing t with t/b, we may assume that Ψ∗(t) = t+ 1. If a 6= 1, then up to replacing t with t− c for a suitable c, we may assume that Ψ∗(t) = at. But then lemma 3.3 implies that a belongs to k∗. Since Ψ∗ has infinite order, a cannot be a root of unity. Second case: C = K[t, 1/t]. Since Ψ∗(t)Ψ∗(1/t) = Ψ∗(1) = 1, Ψ∗(t) is an invertible element of C. So Ψ∗(t) = atn, where a ∈ K and n 6= 0. Since Ψ∗ is an automorphism, n is either equal to 1 or to −1. But if n were equal to −1, then a simple computation shows that (Ψ∗)2 would be the identity, which is impossible. So Ψ∗(t) = at, where a ∈ K . By lemma 3.3, a belongs to k∗. As before, a cannot be a root of unity. 4 Proof of the main theorem In this section, we are going to establish Theorem 1.1. We will split its proof in two steps depending on the form of the automorphism Ψ∗ given in Proposition 3.4. But before, we begin with a few lemmas. Lemma 4.1 Let Φ be an automorphism of an affine irreducible variety X. Let G be a linear algebraic group and ψ be an algebraic G-action on X. Let h be an element of G such that the group <h> spanned by h is Zariski dense in G. If Φ and ψh commute, then Φ and ψg commute for any g in G. Proof: It suffices to check that Φ∗ and ψ∗g commute for any g ∈ G. For any k-algebra automorphisms α, β of O(X), denote by [α, β] their commutator, i.e. [α, β] = α ◦β ◦α−1 ◦ β−1. For any f ∈ O(X), set: λ(g, f)(x) = [Φ∗, ψ∗g ](f)(x)− f(x) Since G is a linear algebraic group acting algebraically on the affine variety X , λ(g, f)(x) is a regular function on G×X . Since Φ∗ and ψ∗h commute, the automorphisms Φ ∗ and ψ∗hn commute for any integer n. So the regular function λ(g, f)(x) vanishes on <h> ×X . Since < h > is dense in G by assumption, < h > ×X is dense in G × X and λ(g, f)(x) vanishes identically on G × X . In particular, [Φ∗, ψ∗g ](f) = f for any g ∈ G. Since this holds for any element f of O(X), the bracket [Φ∗, ψ∗g ] coincides with the identity on O(X) for any g ∈ G, and the result follows. Lemma 4.2 Let Φ be an automorphism of an affine irreducible variety X. Let G be a linear algebraic group and ψ be an algebraic G-action on X. Let h be an element of G such that the group <h> spanned by h is Zariski dense in G. Assume there exists a nonzero integer r such that Φr = ψh, and that G is divisible. Then there exists an algebraic action ϕ of G′ = Z/rZ×G such that Φ = ϕg′ for some g ′ in G′. Proof: Fix an element b in G such that br = h, and set ∆ = Φ◦ψb−1 . This is possible since G is divisible. By construction, ∆ is an automorphism of X . Since Φr = ψh, Φ and ψh commute. By lemma 4.1, Φ and ψg commute for any g ∈ G. In particular, we have: ∆r = (Φr) ◦ ψb−r = (Φ r) ◦ ψh−1 = Id So ∆ is finite, Φ = ∆ ◦ ψb and ∆ commutes with ψg for any g ∈ G. The group G ′ then acts on X via the map ϕ defined by: ϕ(i,g)(x) = ∆ i ◦ ψg(x) Moreover we have Φ = ϕg′ for g ′ = (1, b). The proof of Theorem 1.1 will then go as follows. In the following subsections, we are going to exhibit an algebraic action ψ of Ga(k) (resp. Gm(k)) on X , such that Ψ = Φ m = ψh for some h. In both cases, the group G we will consider will be linear algebraic of dimension 1, and divisible. Moreover the element h will span a Zariski dense set because h 6= 0 (resp. h is not a root of unity). With these conditions, Theorem 1.1 will become a direct application of Lemma 4.2. 4.1 The case Ψ∗(t) = t+ 1 Assume that C = K[t] and Ψ∗(t) = t+1. We are going to construct a nontrivial algebraic Ga(k)-action ψ on X such that Ψ = ψ1. Since O(X) ⊂ C, every element f of O(X) can be written as f = P (t), where P belongs to K[t]. We set r = degt P (t). Since Ψ ∗ stabilizes O(X), the expression: (Ψi)∗(f) = P (t+ i) = P (j)(t) belongs to O(X) for any integer i. Since the matrix M = (ij/j!)0≤i,j≤r is invertible in Mr+1(Q), the polynomial P (j)(t) belongs to O(X) for any j ≤ r. So the K-derivation D = ∂/∂t on C stabilizes the k-algebra O(X). Since Dr+1(f) = 0, the operator D, considered as a k-derivation on O(X), is locally nilpotent (see [Van]). Therefore the exponential map: exp uD : O(X) −→ O(X)[u], f 7−→ Dj(f) is a well-defined k-algebra morphism. But exp uD defines also a K-algebra morphism from C to C[u]. Since exp uD(t) = t + u, expD coincides with Ψ∗ on C. Since C contains the ring O(X), we have expD = Ψ∗ on O(X). So the exponential map induces an algebraic Ga(k)-action ψ on X such that Ψ = ψ1 (see [Van]). 4.2 The case Ψ∗(t) = at Assume that Ψ∗(t) = at and a is not a root of unity. We are going to construct a nontrivial algebraic Gm(k)-action ψ on X such that Ψ = ψa. First note that either C = K[t] or C = K[t, 1/t]. Let f be any element of O(X). Since O(X) ⊂ C, we can write f as: f = P (t) = where the fit i belong a priori to C. Since Ψ∗ stabilizes O(X), the expression: (Ψj)∗(f) = P (ajt) = ajifit belongs to O(X) for any integer j. Since a belongs to k∗ and is not a root of unity, the Vandermonde matrix M = (aij)0≤i,j≤s−r is invertible in Ms−r+1(k). So the elements fit all belong to O(X) for any integer i. Consider the map: ψ∗ : O(X) −→ O(X)[v, 1/v], f 7−→ Then ψ∗ is a well-defined k-algebra morphism, which induces a regular map ψ from k∗×X to X . Moreover we have ψv ◦ ψv′ = ψvv′ on X for any v, v ′ ∈ k∗. So ψ defines an algebraic Gm(k)-action on X such that Ψ = ψa. 5 Proof of Corollary 1.2 Let Φ be an automorphism of the affine plane k2, such that n(Φ) = 1. By Theorem 1.1, there exists an algebraic action ϕ of an abelian linear algebraic group G of dimension 1 such that Φ = ϕg. We will distinguish the cases G = Z/rZ×Gm(k) and G = Z/rZ×Ga(k). First case: G = Z/rZ×Gm(k). Then G is linearly reductive and ϕ is conjugate to a representation in Gl2(k) (see [Ka] or [Kr]). Since G consists solely of semisimple elements, ϕ is even diagonalizable. In par- ticular, there exists a system (x, y) of polynomial coordinates, some integers n,m and some r-roots of unity a, b such that: ϕ(i,u)(x, y) = (a iunx, biumy) Note that, since the action is faithful, the couple (n,m) is distinct from (0, 0). Since k is algebraically closed, we can even reduce Φ = ϕg to the first form given in Corollary 1.2. Second case: G = Z/rZ×Ga(k). Let ψ and ∆ be respectively the Ga(k)-action and finite automorphism constructed in Lemma 4.2. By Rentschler’s theorem (see [Re]), there exists a system (x, y) of polynomial coordinates and an element P of k[t] such that: ψu(x, y) = (x, y + uP (x)) For any f ∈ k[x, y], set degψ(f) = degu exp uD(f). It is well-known that this defines a degree function on k[x, y] (see [Da]). Since ψ and ∆ commute, ∆∗ preserves the space En of polynomials of degree ≤ n with respect to degψ. In particular, ∆ ∗ preserves E0 = k[x]. So ∆∗ induces a finite automorphism of k[x], hence ∆∗(x) = ax + b, where a is a root of unity. Since ∆ is finite, either a 6= 1 or a = 1 and b = 0. In any case, up to replacing x by x − µ for a suitable constant µ, we may assume that ∆∗(x) = ax. Moreover ∆∗ preserves the space E1 = k[x]{1, y}. With the same arguments as before, we obtain that ∆∗(y) = cy+ d(x), where c is a root of unity and d(x) belongs to k[x]. Composing ∆ with ψ1/m then yields the second form given in Corollary 1.2. 6 Proof of Corollary 1.3 Let Φ be an algebraic automorphism of k2. We assume that Φ has a unique fixpoint p and that dΦp is unipotent. We are going to prove that n(Φ) = 0. First we check that n(Φ) cannot be equal to 2. Assume that n(Φ) = 2. Then k(x, y)Φ has transcendence degree 2, and the extension k(x, y)/k(x, y)Φ is algebraic, hence finite. Moreover Φ∗ acts like an element of the Galois group of this extension. In particular, Φ∗ is finite. By a result of Kambayashi (see [Ka]), Φ can be written as h◦A◦h−1, where A is an element of Gl2(k) of finite order and h belongs to Aut(k 2). Since Φ has a unique fixpoint p, we have h(0, 0) = p. In particular, dΦp is conjugate to A in Gl2(k). Since dΦp is unipotent and A is finite, A is the identity. Therefore Φ is also the identity, which contradicts the fact that it has a unique fixpoint. Second we check that n(Φ) cannot be equal to 1. Assume that n(Φ) = 1. By the previous corollary, up to conjugacy, we may assume that Φ has one of the following forms: • Φ1(x, y) = (a nx, amby), where (n,m) 6= (0, 0), b is a root of unity but a is not, • Φ2(x, y) = (ax, by + P (x)), where P belongs to k[t]− {0} and a, b are roots of unity. Assume that Φ is an automorphism of type Φ1. Then dΦp is a diagonal matrix of Gl2(k), distinct from the identity. But this is impossible since dΦp is unipotent. So assume that Φ is an automorphism of type Φ2. Then dΦp is a linear map of the form (u, v) 7→ (au, bv+du), with d ∈ k. Since dΦp is unipotent, we have a = b = 1. So (α, β) is a fixpoint if and only if P (α) = 0. In particular, the set of fixpoints is either empty or a finite union of parallel lines. But this is impossible since there is only one fixpoint by assumption. Therefore n(Φ) = 0. 7 An application of Corollary 1.3 In this section, we are going to see how Corollary 1.3 can be applied to the determination of invariants for automorphisms of C3. Set Q(x, y, z) = x2y − z2 − xz3 and consider the following automorphism (see [M-P]): Φ : C3 −→ C3,  7−→ y(1− xz) + Q z − Q We are going to show that: C(x, y, z)Φ = C(x) and C[x, y, z]Φ = C[x] Let k be the algebraic closure of C(x). Since Φ∗(x) = x, the morphism Φ∗ induces an automorphism of k[y, z], which we denote by Ψ∗. The automorphism Ψ has clearly (0, 0) as a fixpoint, and its differential at this point is unipotent, distinct from the identity. Indeed, it is given by the matrix: dΨ(0,0) = 1 −x3/2 Moreover, the set of fixpoints of Ψ is reduced to the origin. Indeed, if (α, β) is a point of k2 fixed by Ψ, then xQ = 0 and 4β4 − 4xαβ +Q2 = 0. Since x belongs to k∗, we have: Q = x2α− β2 − xβ3 = 0 and β4 − xαβ = 0 If β = 0, then α = 0 and we find the origin. If β 6= 0, then dividing by β and multiplying by −x yields the relation: x2α− xβ3 = 0 This implies β2 = 0 and β = 0, hence a contradiction. By Corollary 1.3, the field of invari- ants of Ψ has transcendence degree zero. So the field of invariants of Φ has transcendence degree ≤ 1 over C. Since this field contains C(x) and that C(x) is algebraically closed in C(x, y, z), we obtain that C(x, y, z)Φ = C(x). As a consequence, the ring of invariants of Φ is equal to C[x]. References [Bou] N.Bourbaki Eléments de Mathématiques: Algèbre Commutative, chapitres 5-6, Her- mann Paris 1964. [Che] C.Chevalley Introduction to the theory of algebraic functions of one variable, Math- ematical Surveys no VI, American Mathematical Society, New York 1951. [Da] D.Daigle On some properties of locally nilpotent derivations, J. Pure Appl. Algebra 114 (1997), n03, 221-230. [Hum] J.Humphreys Linear algebraic groups, Graduate texts in mathematics 21, Springer Verlag Berlin 1981. [Ka] T.Kambayashi Automorphism group of a polynomial ring and algebraic group action on an affine space, J.Algebra 60 (1979), no 2, 439-451. [Kr] H.Kraft Challenging problems on affine n-space, Séminaire Bourbaki 802, 1994-95. [M] L.Makar-Limanov On the hypersurface x + x2y + z2 + t3 = 0 in C4 or a C3-like threefold which is not C3, Israel J. Math. 96 (1996), part B, 419-429. [M-P] L.Moser-Jauslin, P-M.Poloni Embeddings of a family of Danielewsky hypersurfaces and certain (C,+)-actions on C3, preprint. [Re] R.Rentschler Opérations du groupe additif sur le plan affine, C.R.A.S 267 (1968) 384-387. [Ro] M.Rosenlicht A remark on quotient spaces, An. Acad. Brasil. Cienc. 35 (1963), 487- [Ro2] M.Rosenlicht Automorphisms of function fields, Trans. Amer. Math. Soc. 79 (1955), 1-11. [Ru] P.Russell Forms of the affine line and its additive group, Pacific J. Math. 32 (1970) 527-539. [S-B] G.Schwarz, M.Brion Théorie des invariants et Géométrie des variétés quotients, Collection Travaux en cours 61, Hermann Paris 2000. [Van] A.Van den Essen Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics no190, Birkhäuser Verlag, Basel 2000. [V-P] A.Van den Essen, R. Peretz Polynomial automorphisms and invariants, J. Algebra 269 (2003), n01, 317-328. [Z-S] O.Zariski, P.Samuel Commutative Algebra, Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New- York 1960. Introduction Reduction to an affine curve C Normal forms for the automorphism Proof of the main theorem The case *(t)=t+1 The case *(t)=at Proof of Corollary ?? Proof of Corollary ?? An application of Corollary ??

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism F, we denote by k(X)^F its field of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let n(F) be the transcendence degree of k(X)^F over k. In this paper, we study the class of automorphisms F of X for which n(F)= dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form F=A_g, where A is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(F)=1.

Introduction Let k be an algebraically closed field of characteristic zero. Let X be an affine irreducible variety of dimension n over k. We denote by O(X) its ring of regular functions, and by k(X) its field of rational functions. Given an algebraic automorphism Φ of X , denote by Φ∗ the field automorphism induced by Φ on k(X), i.e. Φ∗(f) = f ◦Φ for any f ∈ k(X). An element f of k(X) is invariant for Φ (or simply invariant) if Φ∗(f) = f . Invariant rational functions form a field denoted k(X)Φ, and we set: n(Φ) = trdegk k(X) In this paper, we are going to study the class of automorphisms of X for which n(Φ) = n − 1. There are natural candidates for such automorphisms, such as exponentials of locally nilpotent derivations (see [M] or [Da]). More generally, one can construct such automorphisms by means of algebraic group actions as follows. Let G be a linear algebraic group over k. An algebraic action of G on X is a regular map: ϕ : G×X −→ X http://arxiv.org/abs/0704.1588v1 of affine varieties, such that ϕ(g.g′, x) = ϕ(g, ϕ(g′, x)) for any (g, g′, x) in G × G × X . Given an element g of G, denote by ϕg the map x 7→ ϕ(g, x). Then ϕg clearly defines an automorphism of X . Let k(X)G be the field of invariants of G, i.e. the set of rational functions f on X such that f ◦ ϕg = f for any g ∈ G. If G is an algebraic group of dimension 1, acting faithfully on X , and if g is an element of G of infinite order, then one can prove by Rosenlicht’s Theorem (see [Ro]) that: n(ϕg) = trdegk k(X) G = n− 1 We are going to see that, under some mild conditions on X , there are no other automor- phisms with n(Φ) = n−1 than those constructed above. In what follows, denote by O(X)ν the normalization of O(X), and by G(X) the group of invertible elements of O(X)ν . Theorem 1.1 Let X be an affine irreducible variety of dimension n over k, such that char(k) = 0 and G(X)∗ = k∗. Let Φ be an algebraic automorphism of X such that n(Φ) = n − 1. Then there exist an abelian linear algebraic group G of dimension 1, and an algebraic action ϕ of G on X such that Φ = ϕg for some g ∈ G of infinite order. Note that the structure of G is fairly simple. Since every connected linear algebraic group of dimension 1 is either isomorphic to Ga(k) = (k,+) or Gm(k) = (k ∗,×) (see [Hum], p. 131), there exists a finite abelian group H such that G is either equal to H × Ga(k) or H × Gm(k). Moreover, the assumption on the group G(X) is essential. Indeed, consider the automorphism Φ of k∗ × k given by Φ(x, y) = (x, xy). Obviously, its field of invariants is equal to k(x). However, it is easy to check that Φ cannot have the form given in the conclusion of Theorem 1.1. This theorem is analogous to a result given by Van den Essen and Peretz (see [V-P]). More precisely, they establish a criterion to decide if an automorphism Φ is the exponential of a locally nilpotent derivation, based on the invariants and on the form of Φ. A similar result has been developed by Daigle (see [Da]). We apply these results to the group of automorphisms of the plane. First, we obtain a classification of the automorphisms Φ of k2 for which n(Φ) = 1. Second, we derive a criterion on automorphisms of k2 to have no nonconstant rational invariants. Corollary 1.2 Let Φ be an algebraic automorphism of k2. If n(Φ) = 1, then Φ is conjugate to one of the following forms: • Φ1(x, y) = (a nx, amby), where (n,m) 6= (0, 0), a, b ∈ k, b is a root of unity but a is • Φ2(x, y) = (ax, by + P (x)), where P belongs to k[t]− {0}, a, b ∈ k are roots of unity. Corollary 1.3 Let Φ be an algebraic automorphism of k2. Assume that Φ has a unique fixpoint p and that dΦp is unipotent. Then n(Φ) = 0. We then apply Corollary 1.3 to an automorphism of C3 recently discovered by Pierre-Marie Poloni and Lucy Moser (see [M-P]). We may wonder whether Theorem 1.1 still holds if the ground field k is not algebraically closed or has positive characteristic. The answer is not known for the moment. In fact, two obstructions appear in the proof of Theorem 1.1 when k is arbitrary. First, the group Gm(k) needs to be divisible (see Lemma 4.2), which is not always the case if k is not algebraically closed. Second, the proof uses the fact that every Ga(k)-action on X can be reconstructed from a locally nilpotent derivation on O(X) (see subsection 4.1), which is no longer true if k has positive characteristic. This phenomenom is due to the existence of differents forms for the affine line (see [Ru]). Note that, in case Theorem 1.1 holds and k is not algebraically closed, the algebraic group G needs not be isomorphic to H ×Ga(k) or H ×Gm(k), where H is finite. Indeed consider the unit circle X in the plane R 2, given by the equation x2 + y2 = 1. Let Φ be a rotation in R2 with center at the origin and angle θ 6∈ 2πQ. Then Φ defines an algebraic automorphism of X with n(Φ) = 0, and the subgroup spanned by Φ is dense in SO2(R). But SO2(R) is not isomorphic to either Ga(R) or Gm(R), even though it is a connected linear algebraic group of dimension 1. We may also wonder what happens to the automorphisms Φ of X for which n(Φ) = dimX − 2. More precisely, does there exist an action ϕ of a linear algebraic group G on X , of dimension 2, such that Φ = ϕg for a given g ∈ G? The answer is no. Indeed consider the automorphism Φ of k2 given by Φ = f ◦ g, where f(x, y) = (x + y2, y) and g(x, y) = (x, y + x2). Let d(n) denote the maximum of the homogeneous degrees of the coordinate functions of the iterate Φn. If there existed an action ϕ of a linear algebraic group G such that Φ = ϕg, then the function d would be bounded, which is impossible since d(n) = 4n. A similar argument on the length of the iterates also yields the result. But if we restrict to some specific varieties X , for instance X = k3, one may ask the following question: If n(Φ) = 1, is Φ birationally conjugate to an automorphism that leaves the first coordinate of k3 invariant? The answer is still unknown. 2 Reduction to an affine curve C Let X be an affine irreducible variety of dimension n over k. Let Φ be an algebraic automorphism of X such that n(Φ) = n − 1. In this section, we are going to construct an irreducible affine curve on which Φ acts naturally. This will allow us to use some well-known results on automorphisms of curves. We set: K = {f ∈ k(X)|∃m > 0, f ◦ Φm = f ◦ Φ ◦ ... ◦ Φ = f} It is straightforward that K is a subfield of k(X) containing both k and k(X)Φ. We begin with some properties of this field. Lemma 2.1 K has transcendence degree (n − 1) over k, and is algebraically closed in k(X). In particular, the automorphism Φ of X has infinite order. Proof: First we show that K has transcendence degree (n − 1) over k. Since K contains the field k(X)Φ, whose transcendence degree is (n − 1), we only need to show that the extension K/k(X)Φ is algebraic, or in other words that every element of K is algebraic over k(X)Φ. Let f be any element of K. By definition, there exists an integer m > 0 such that f ◦ Φm = f . Let P (t) be the polynomial of k(X)[t] defined as: P (t) = (t− f ◦ Φi) By construction, the coefficients of this polynomial are all invariant for Φ, and P (t) belongs to k(X)Φ[t]. Moreover P (f) = 0, f is algebraic over k(X)Φ and the first assertion follows. Second we show that K is algebraically closed in k(X). Let f be an element of k(X) that is algebraic over K. We need to prove that f belongs to K. By the first assertion of the lemma, f is algebraic over k(X)Φ. Let P (t) = a0 + a1t + ... + apt p be a nonzero minimal polynomial of f over k(X)Φ. Since P (f) = 0 and all ai are invariant, we have P (f ◦ Φ) = P (f) ◦ Φ = 0. In particular, all elements of the form f ◦ Φi, with i ∈ N, are roots of P . Since P has finitely many roots, there exist two distinct integers m′ < m′′ such that f ◦ Φm = f ◦ Φm . In particular, f ◦ Φm ′′−m′ = f and f belongs to K. Now if Φ were an automorphism of finite order, then K would be equal to k(X). But this is impossible since K and k(X) have different transcendence degrees. Lemma 2.2 There exists an integer m > 0 such that K = k(X)Φ Proof: By definition, k(X) is a field of finite type over k. Since K is contained in k(X), K has also finite type over k. Let f1, ..., fr be some elements of k(X) such that K = k(f1, ..., fr). Let m1, ..., mr be some positive integers such that fi ◦ Φ mi = fi, and set m = m1...mr. By construction, all fi are invariant for Φ m. In particular, K is invariant for Φm and K ⊆ k(X)Φ . Since k(X)Φ ⊆ K, the result follows. Let L be the algebraic closure of k(X), and let A be the K-subalgebra of L spanned by O(X). By construction, A is an integral K-algebra of finite type of dimension 1. Let m be an integer satisfying the conditions of lemma 2.2. The automorphism Ψ∗ = (Φm)∗ of O(X) stabilizes A, hence it defines a K-automorphism of A, of infinite order (see lemma 2.1). Let B be the integral closure of A. Then B is also an integral K-algebra of finite type, of dimension 1, and the K-automorphism Ψ∗ extends uniquely to B. If K stands for the algebraic closure of K, we set: C = B ⊗K K By construction, C = Spec(C) is an affine curve over the algebraically closed field K. Moreover the automorphism Ψ∗ acts on C via the operation: Ψ∗ : C −→ C, x⊗ y 7−→ Ψ∗(x)⊗ y This makes sense since Ψ∗ fixes the field K. Therefore Ψ∗ induces an algebraic automor- phism of the curve C. Since K is algebraically closed in k(X) by lemma 2.1, C is integral (see [Z-S], Chap. VII, §11, Theorem 38). But by construction, B and K are normal rings. Since C is a domain and char(K) = 0, C is also integrally closed by a result of Bourbaki (see [Bou], p. 29). So C is a normal domain and C is a smooth irreducible curve. Lemma 2.3 Let C be the K-algebra constructed above. Then either C = K[t] or C = K[t, 1/t]. Proof: By lemma 2.1, the automorphism Φ of X has infinite order. Since the fraction field of B is equal to k(X), Ψ∗ has infinite order on B. But B ⊗ 1 ⊂ C, so Ψ∗ has infinite order on C. In particular, Ψ acts like an automorphism of infinite order on C. Since C is affine, it has genus zero (see [Ro2]). Since K is algebraically closed, the curve C is rational (see [Che], p. 23 ). Since C is smooth, it is isomorphic to P1(K) − E, where E is a finite set. Moreover, Ψ acts like an automorphism of P1(K) that stabilizes P1(K) − E. Up to replacing Ψ by one of its iterates, we may assume that Ψ fixes every point of E. But an automorphism of P1(K) that fixes at least three points is the identity, which is impossible. Therefore E consists of at most two points, and C is either isomorphic to K or to K particular, either C = K[t] or C = K[t, 1/t]. 3 Normal forms for the automorphism Ψ Let C and Ψ∗ be the K-algebra and the K-automorphism constructed in the previous section. In this section, we are going to give normal forms for the couple (C,Ψ∗), in case the group G(X) is trivial, i.e. G(X) = k∗. We begin with a few lemmas. Lemma 3.1 Let X be an irreducible affine variety over k. Let Ψ be an automorphism of X. Let α, f be some elements of k(X)∗ such that (Ψ∗)n(f) = αnf for any n ∈ Z. Then α belongs to G(X). Proof: Given an element h of k(X)∗ and a prime divisor D on the normalization Xν , we consider h as a rational function on Xν , and denote by ordD(h) the multiplicity of h along D. This makes sense since the variety Xν is normal. Fix any prime divisor D on X . Since (Ψ∗)n(f) = αnf for any n ∈ Z, we obtain: ordD((Ψ ∗)n(f)) = nordD(α) + ordD(f) Since Ψ is an algebraic automorphism of X , it extends uniquely to an algebraic automor- phism of Xν , which is still denoted Ψ. Moreover, this extension maps every prime divisor to another prime divisor, does not change the multiplicity and maps distinct prime divisors into distinct ones. If div(f) = i niDi, where all Di are prime, then we have: div((Ψ∗)n(f)) = ∗)n(Di) where all (Ψ∗)n(Di) are prime and distinct. So the multiplicity of (Ψ ∗)n(f) along D is equal to zero if D is none of the (Ψ∗)n(Di), and equal to ni if D = (Ψ ∗)n(Di). In all cases, if R = max{|ni|}, then we find that |ordD((Ψ ∗)n(f))| ≤ R and |ordD(f)| ≤ R, and this implies for any integer n: |nordD(α)| ≤ 2R In particular we find ordD(α) = 0. Since this holds for any prime divisor D, the support of div(α) in Xν is empty and div(α) = 0. Since Xν is normal, α is an invertible element of O(X)ν , hence it belongs to G(X). Lemma 3.2 Let K be a field of characteristic zero and K its algebraic closure. Let C be either equal to K[t] or to K[t, 1/t]. Let Ψ∗ be a K-automorphism of C such that Ψ∗(t) = at, where a belongs to K. Let σ1 be a K-automorphism of C, commuting with Ψ ∗, such that σ1(K) = K. Then σ1(a) is either equal to a or to 1/a. Proof: We distinguish two cases depending on the ring C. First assume that C = K[t]. Since σ1 is a K-automorphism of C that maps K to itself, we have K[t] = K[σ1(t)]. In particular σ1(t) = λt + µ, where λ, µ belong to K and λ 6= 0. Since Ψ ∗ and σ1 commute, we obtain: Ψ∗ ◦ σ1(t) = λat + µ = σ1 ◦Ψ ∗(t) = σ1(a)(λt + µ) In particular, we have σ1(a) = a and the lemma follows in this case. Second assume that C = K[t, 1/t]. Since σ1 is a K-automorphism of C, we find: σ1(t)σ1(1/t) = σ1(t.1/t) = σ1(1) = 1 Therefore σ1(t) is an invertible element of C, and has the form σ1(t) = a1t n1 , where a1 ∈ K and n1 is an integer. Since σ1 is a K-automorphism of C that maps K to K, we have K[t, 1/t] = K[σ1(t), 1/σ1(t)]. In particular |n1| = 1 and either σ1(t) = a1t or σ1(t) = a1/t. If σ1(t) = a1t, the relation Ψ ∗ ◦ σ1(t) = σ1 ◦ Ψ ∗(t) yields σ(a) = a. If σ1(t) = a1/t, then the same relation yields σ(a) = 1/a. Lemma 3.3 Let X be an irreducible affine variety of dimension n over k, such that G(X) = k∗. Let Φ be an automorphism of X such that n(Φ) = (n− 1). Let Ψ∗ be the au- tomorphism of C constructed in the previous section. If either C = K[t] or C = K[t, 1/t], and if Ψ∗(t) = at, then a belongs to k∗. Proof: We are going to prove by contradiction that a belongs to k∗. So assume that a 6∈ k∗. Let σ be any element of Gal(K/K), and denote by σ1 the K-automorphism of C defined as follows: ∀(x, y) ∈ B ×K, σ1(x⊗ y) = x⊗ σ1(y) Since Ψ∗ ◦σ1(x⊗ y) = Ψ ∗(x)⊗σ1(y) = σ1 ◦Ψ ∗(x⊗ y) for any element x⊗ y of B⊗KK, Ψ and σ1 commute. Moreover if we identify K with 1⊗K, then σ1(K) = K by construction. By lemma 3.2, we obtain: ∀σ ∈ Gal(K/K), σ(a) = a or σ(a) = In particular, the element (ai + a−i) is invariant under the action of Gal(K/K) for any i, and so it belongs to K because char(K) = 0. Now let f be an element of B −K. Since f belongs to C, we can express f as follows: Choose an f ∈ B−K such that the difference (s−r) is minimal. We claim that (s−r) = 0, i.e. f = fst s. Indeed, assume that s > r. Since f is an element of B, the following expressions: Ψ∗(f) + (Ψ∗)−1(f)− (as + a−s)f = i=r fi(a i + a−i − as − a−s)ti Ψ∗(f) + (Ψ∗)−1(f)− (ar + a−r)f = i=r+1 i + a−i − ar − a−r)ti also belong to B. By minimality of (s− r), these expressions belong to K. In other words, i+ a−i− as− a−s) = 0 (resp. fi(a i+ a−i− ar− a−r) = 0) for any i 6= 0, s (resp. for any i 6= 0, r). Since k is algebraically closed and a 6∈ k∗ by assumption, (ai + a−i − as − a−s) (resp. (ai+a−i−ar−a−r)) is nonzero for any i 6= s (resp. for any i 6= r). Therefore fi = 0 for any i 6= 0, and f belongs to K, a contradiction. Therefore s = r and f = fst s. Since f belongs to B, it also belongs to k(X). Since Ψ is an automorphism of X , the element as = Ψ∗(f)/f belongs to k(X). Moreover (Ψ∗)n(f) = ansf for any n ∈ Z. By lemma 3.1, as belongs to G(X) = k∗. Since k is algebraically closed, a belongs to k∗, hence a contradiction, and the result follows. Proposition 3.4 Let X be an irreducible affine variety of dimension n over k, such that G(X) = k∗. Let Φ be an automorphism of X such that n(Φ) = (n− 1). Let C and Ψ∗ be the K-algebra and the K-automorphism constructed in the previous section. Then up to conjugation, one of the following three cases occurs: • C = K[t] and Ψ∗(t) = t + 1, • C = K[t] and Ψ∗(t) = at, where a ∈ k∗ is not a root of unity, • C = K[t, 1/t] and Ψ∗(t) = at, where a ∈ k∗ is not a root of unity. Proof: By lemma 2.3, we know that either C = K[t] or C = K[t, 1/t]. We are going to study both cases. First case: C = K[t]. The automorphism Ψ∗ maps t to at + b, where a ∈ K and b ∈ K. If a = 1, then b 6= 0 and up to replacing t with t/b, we may assume that Ψ∗(t) = t+ 1. If a 6= 1, then up to replacing t with t− c for a suitable c, we may assume that Ψ∗(t) = at. But then lemma 3.3 implies that a belongs to k∗. Since Ψ∗ has infinite order, a cannot be a root of unity. Second case: C = K[t, 1/t]. Since Ψ∗(t)Ψ∗(1/t) = Ψ∗(1) = 1, Ψ∗(t) is an invertible element of C. So Ψ∗(t) = atn, where a ∈ K and n 6= 0. Since Ψ∗ is an automorphism, n is either equal to 1 or to −1. But if n were equal to −1, then a simple computation shows that (Ψ∗)2 would be the identity, which is impossible. So Ψ∗(t) = at, where a ∈ K . By lemma 3.3, a belongs to k∗. As before, a cannot be a root of unity. 4 Proof of the main theorem In this section, we are going to establish Theorem 1.1. We will split its proof in two steps depending on the form of the automorphism Ψ∗ given in Proposition 3.4. But before, we begin with a few lemmas. Lemma 4.1 Let Φ be an automorphism of an affine irreducible variety X. Let G be a linear algebraic group and ψ be an algebraic G-action on X. Let h be an element of G such that the group <h> spanned by h is Zariski dense in G. If Φ and ψh commute, then Φ and ψg commute for any g in G. Proof: It suffices to check that Φ∗ and ψ∗g commute for any g ∈ G. For any k-algebra automorphisms α, β of O(X), denote by [α, β] their commutator, i.e. [α, β] = α ◦β ◦α−1 ◦ β−1. For any f ∈ O(X), set: λ(g, f)(x) = [Φ∗, ψ∗g ](f)(x)− f(x) Since G is a linear algebraic group acting algebraically on the affine variety X , λ(g, f)(x) is a regular function on G×X . Since Φ∗ and ψ∗h commute, the automorphisms Φ ∗ and ψ∗hn commute for any integer n. So the regular function λ(g, f)(x) vanishes on <h> ×X . Since < h > is dense in G by assumption, < h > ×X is dense in G × X and λ(g, f)(x) vanishes identically on G × X . In particular, [Φ∗, ψ∗g ](f) = f for any g ∈ G. Since this holds for any element f of O(X), the bracket [Φ∗, ψ∗g ] coincides with the identity on O(X) for any g ∈ G, and the result follows. Lemma 4.2 Let Φ be an automorphism of an affine irreducible variety X. Let G be a linear algebraic group and ψ be an algebraic G-action on X. Let h be an element of G such that the group <h> spanned by h is Zariski dense in G. Assume there exists a nonzero integer r such that Φr = ψh, and that G is divisible. Then there exists an algebraic action ϕ of G′ = Z/rZ×G such that Φ = ϕg′ for some g ′ in G′. Proof: Fix an element b in G such that br = h, and set ∆ = Φ◦ψb−1 . This is possible since G is divisible. By construction, ∆ is an automorphism of X . Since Φr = ψh, Φ and ψh commute. By lemma 4.1, Φ and ψg commute for any g ∈ G. In particular, we have: ∆r = (Φr) ◦ ψb−r = (Φ r) ◦ ψh−1 = Id So ∆ is finite, Φ = ∆ ◦ ψb and ∆ commutes with ψg for any g ∈ G. The group G ′ then acts on X via the map ϕ defined by: ϕ(i,g)(x) = ∆ i ◦ ψg(x) Moreover we have Φ = ϕg′ for g ′ = (1, b). The proof of Theorem 1.1 will then go as follows. In the following subsections, we are going to exhibit an algebraic action ψ of Ga(k) (resp. Gm(k)) on X , such that Ψ = Φ m = ψh for some h. In both cases, the group G we will consider will be linear algebraic of dimension 1, and divisible. Moreover the element h will span a Zariski dense set because h 6= 0 (resp. h is not a root of unity). With these conditions, Theorem 1.1 will become a direct application of Lemma 4.2. 4.1 The case Ψ∗(t) = t+ 1 Assume that C = K[t] and Ψ∗(t) = t+1. We are going to construct a nontrivial algebraic Ga(k)-action ψ on X such that Ψ = ψ1. Since O(X) ⊂ C, every element f of O(X) can be written as f = P (t), where P belongs to K[t]. We set r = degt P (t). Since Ψ ∗ stabilizes O(X), the expression: (Ψi)∗(f) = P (t+ i) = P (j)(t) belongs to O(X) for any integer i. Since the matrix M = (ij/j!)0≤i,j≤r is invertible in Mr+1(Q), the polynomial P (j)(t) belongs to O(X) for any j ≤ r. So the K-derivation D = ∂/∂t on C stabilizes the k-algebra O(X). Since Dr+1(f) = 0, the operator D, considered as a k-derivation on O(X), is locally nilpotent (see [Van]). Therefore the exponential map: exp uD : O(X) −→ O(X)[u], f 7−→ Dj(f) is a well-defined k-algebra morphism. But exp uD defines also a K-algebra morphism from C to C[u]. Since exp uD(t) = t + u, expD coincides with Ψ∗ on C. Since C contains the ring O(X), we have expD = Ψ∗ on O(X). So the exponential map induces an algebraic Ga(k)-action ψ on X such that Ψ = ψ1 (see [Van]). 4.2 The case Ψ∗(t) = at Assume that Ψ∗(t) = at and a is not a root of unity. We are going to construct a nontrivial algebraic Gm(k)-action ψ on X such that Ψ = ψa. First note that either C = K[t] or C = K[t, 1/t]. Let f be any element of O(X). Since O(X) ⊂ C, we can write f as: f = P (t) = where the fit i belong a priori to C. Since Ψ∗ stabilizes O(X), the expression: (Ψj)∗(f) = P (ajt) = ajifit belongs to O(X) for any integer j. Since a belongs to k∗ and is not a root of unity, the Vandermonde matrix M = (aij)0≤i,j≤s−r is invertible in Ms−r+1(k). So the elements fit all belong to O(X) for any integer i. Consider the map: ψ∗ : O(X) −→ O(X)[v, 1/v], f 7−→ Then ψ∗ is a well-defined k-algebra morphism, which induces a regular map ψ from k∗×X to X . Moreover we have ψv ◦ ψv′ = ψvv′ on X for any v, v ′ ∈ k∗. So ψ defines an algebraic Gm(k)-action on X such that Ψ = ψa. 5 Proof of Corollary 1.2 Let Φ be an automorphism of the affine plane k2, such that n(Φ) = 1. By Theorem 1.1, there exists an algebraic action ϕ of an abelian linear algebraic group G of dimension 1 such that Φ = ϕg. We will distinguish the cases G = Z/rZ×Gm(k) and G = Z/rZ×Ga(k). First case: G = Z/rZ×Gm(k). Then G is linearly reductive and ϕ is conjugate to a representation in Gl2(k) (see [Ka] or [Kr]). Since G consists solely of semisimple elements, ϕ is even diagonalizable. In par- ticular, there exists a system (x, y) of polynomial coordinates, some integers n,m and some r-roots of unity a, b such that: ϕ(i,u)(x, y) = (a iunx, biumy) Note that, since the action is faithful, the couple (n,m) is distinct from (0, 0). Since k is algebraically closed, we can even reduce Φ = ϕg to the first form given in Corollary 1.2. Second case: G = Z/rZ×Ga(k). Let ψ and ∆ be respectively the Ga(k)-action and finite automorphism constructed in Lemma 4.2. By Rentschler’s theorem (see [Re]), there exists a system (x, y) of polynomial coordinates and an element P of k[t] such that: ψu(x, y) = (x, y + uP (x)) For any f ∈ k[x, y], set degψ(f) = degu exp uD(f). It is well-known that this defines a degree function on k[x, y] (see [Da]). Since ψ and ∆ commute, ∆∗ preserves the space En of polynomials of degree ≤ n with respect to degψ. In particular, ∆ ∗ preserves E0 = k[x]. So ∆∗ induces a finite automorphism of k[x], hence ∆∗(x) = ax + b, where a is a root of unity. Since ∆ is finite, either a 6= 1 or a = 1 and b = 0. In any case, up to replacing x by x − µ for a suitable constant µ, we may assume that ∆∗(x) = ax. Moreover ∆∗ preserves the space E1 = k[x]{1, y}. With the same arguments as before, we obtain that ∆∗(y) = cy+ d(x), where c is a root of unity and d(x) belongs to k[x]. Composing ∆ with ψ1/m then yields the second form given in Corollary 1.2. 6 Proof of Corollary 1.3 Let Φ be an algebraic automorphism of k2. We assume that Φ has a unique fixpoint p and that dΦp is unipotent. We are going to prove that n(Φ) = 0. First we check that n(Φ) cannot be equal to 2. Assume that n(Φ) = 2. Then k(x, y)Φ has transcendence degree 2, and the extension k(x, y)/k(x, y)Φ is algebraic, hence finite. Moreover Φ∗ acts like an element of the Galois group of this extension. In particular, Φ∗ is finite. By a result of Kambayashi (see [Ka]), Φ can be written as h◦A◦h−1, where A is an element of Gl2(k) of finite order and h belongs to Aut(k 2). Since Φ has a unique fixpoint p, we have h(0, 0) = p. In particular, dΦp is conjugate to A in Gl2(k). Since dΦp is unipotent and A is finite, A is the identity. Therefore Φ is also the identity, which contradicts the fact that it has a unique fixpoint. Second we check that n(Φ) cannot be equal to 1. Assume that n(Φ) = 1. By the previous corollary, up to conjugacy, we may assume that Φ has one of the following forms: • Φ1(x, y) = (a nx, amby), where (n,m) 6= (0, 0), b is a root of unity but a is not, • Φ2(x, y) = (ax, by + P (x)), where P belongs to k[t]− {0} and a, b are roots of unity. Assume that Φ is an automorphism of type Φ1. Then dΦp is a diagonal matrix of Gl2(k), distinct from the identity. But this is impossible since dΦp is unipotent. So assume that Φ is an automorphism of type Φ2. Then dΦp is a linear map of the form (u, v) 7→ (au, bv+du), with d ∈ k. Since dΦp is unipotent, we have a = b = 1. So (α, β) is a fixpoint if and only if P (α) = 0. In particular, the set of fixpoints is either empty or a finite union of parallel lines. But this is impossible since there is only one fixpoint by assumption. Therefore n(Φ) = 0. 7 An application of Corollary 1.3 In this section, we are going to see how Corollary 1.3 can be applied to the determination of invariants for automorphisms of C3. Set Q(x, y, z) = x2y − z2 − xz3 and consider the following automorphism (see [M-P]): Φ : C3 −→ C3,  7−→ y(1− xz) + Q z − Q We are going to show that: C(x, y, z)Φ = C(x) and C[x, y, z]Φ = C[x] Let k be the algebraic closure of C(x). Since Φ∗(x) = x, the morphism Φ∗ induces an automorphism of k[y, z], which we denote by Ψ∗. The automorphism Ψ has clearly (0, 0) as a fixpoint, and its differential at this point is unipotent, distinct from the identity. Indeed, it is given by the matrix: dΨ(0,0) = 1 −x3/2 Moreover, the set of fixpoints of Ψ is reduced to the origin. Indeed, if (α, β) is a point of k2 fixed by Ψ, then xQ = 0 and 4β4 − 4xαβ +Q2 = 0. Since x belongs to k∗, we have: Q = x2α− β2 − xβ3 = 0 and β4 − xαβ = 0 If β = 0, then α = 0 and we find the origin. If β 6= 0, then dividing by β and multiplying by −x yields the relation: x2α− xβ3 = 0 This implies β2 = 0 and β = 0, hence a contradiction. By Corollary 1.3, the field of invari- ants of Ψ has transcendence degree zero. So the field of invariants of Φ has transcendence degree ≤ 1 over C. Since this field contains C(x) and that C(x) is algebraically closed in C(x, y, z), we obtain that C(x, y, z)Φ = C(x). As a consequence, the ring of invariants of Φ is equal to C[x]. References [Bou] N.Bourbaki Eléments de Mathématiques: Algèbre Commutative, chapitres 5-6, Her- mann Paris 1964. [Che] C.Chevalley Introduction to the theory of algebraic functions of one variable, Math- ematical Surveys no VI, American Mathematical Society, New York 1951. [Da] D.Daigle On some properties of locally nilpotent derivations, J. Pure Appl. Algebra 114 (1997), n03, 221-230. [Hum] J.Humphreys Linear algebraic groups, Graduate texts in mathematics 21, Springer Verlag Berlin 1981. [Ka] T.Kambayashi Automorphism group of a polynomial ring and algebraic group action on an affine space, J.Algebra 60 (1979), no 2, 439-451. [Kr] H.Kraft Challenging problems on affine n-space, Séminaire Bourbaki 802, 1994-95. [M] L.Makar-Limanov On the hypersurface x + x2y + z2 + t3 = 0 in C4 or a C3-like threefold which is not C3, Israel J. Math. 96 (1996), part B, 419-429. [M-P] L.Moser-Jauslin, P-M.Poloni Embeddings of a family of Danielewsky hypersurfaces and certain (C,+)-actions on C3, preprint. [Re] R.Rentschler Opérations du groupe additif sur le plan affine, C.R.A.S 267 (1968) 384-387. [Ro] M.Rosenlicht A remark on quotient spaces, An. Acad. Brasil. Cienc. 35 (1963), 487- [Ro2] M.Rosenlicht Automorphisms of function fields, Trans. Amer. Math. Soc. 79 (1955), 1-11. [Ru] P.Russell Forms of the affine line and its additive group, Pacific J. Math. 32 (1970) 527-539. [S-B] G.Schwarz, M.Brion Théorie des invariants et Géométrie des variétés quotients, Collection Travaux en cours 61, Hermann Paris 2000. [Van] A.Van den Essen Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics no190, Birkhäuser Verlag, Basel 2000. [V-P] A.Van den Essen, R. Peretz Polynomial automorphisms and invariants, J. Algebra 269 (2003), n01, 317-328. [Z-S] O.Zariski, P.Samuel Commutative Algebra, Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New- York 1960. Introduction Reduction to an affine curve C Normal forms for the automorphism Proof of the main theorem The case *(t)=t+1 The case *(t)=at Proof of Corollary ?? Proof of Corollary ?? An application of Corollary ??

Improving immunization strategies Lazaros K. Gallos1, Fredrik Liljeros2, Panos Argyrakis1, Armin Bunde3, and Shlomo Havlin4 Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece Department of Sociology, Stockholm University 106 91 Stockholm, Sweden Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany and Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel (Dated: November 20, 2018) We introduce an immunization method where the percentage of required vaccinations for immunity are close to the optimal value of a targeted immunization scheme of highest degree nodes. Our strategy retains the advantage of being purely local, without the need of knowledge on the global network structure or identification of the highest degree nodes. The method consists of selecting a random node and asking for a neighbor that has more links than himself or more than a given threshold and immunizing him. We compare this method to other efficient strategies on three real social networks and on a scale-free network model, and find it to be significantly more effective. PACS numbers: 89.75.Hc, 87.23.Ge Immunization of large populations through vaccination is an extremely important issue with obvious implications for the public health [1, 2, 3]. The eradication of Small Pox through a global mass vaccination campaign during the second part of the 20th century represents, for exam- ple, a landmark in the history of the medical sciences [4]. Global or national mass vaccination may however not al- ways be possible. The number of vaccinated people may need to be minimized due to severe side effects of vacci- nation such as for Small Pox, or temporary shortage of vaccine that could be the case for a pandemic influenza. The cost for a vaccine may also be an important limiting factor. Improving efficiency of immunization is thus an urgent task. Recently [5], developments in the study of population connectivities helped researchers in the field to present new ideas on immunization, based on the heterogeneity in the number of contacts between individuals. A number of strategies have been proposed for lowering the required minimum fraction fc of the population to be immunized. The problem can be mapped to the well-known percola- tion problem where nodes are immunized (removed) up to a concentration fc, above which the spanning clus- ter does not survive. Random immunization of nodes has been shown incapable of protecting the population when the contacts distribution is wide, since the perco- lation threshold is close to fc = 1, i.e. practically all nodes need to be immunized [6, 7, 8]. The best known strategy today is believed to be targeted immunization, where the highest connected nodes in the system are im- munized in decreasing order of their degree. In this case fc is less than 10% [7, 9, 10]. For all practical applica- tions, though, this approach is unrealistic because it is a ‘global’ strategy and requires a complete knowledge of the high degree nodes, which is in many cases impossi- ble. An effective strategy, called acquaintance immuniza- tion, was recently introduced [11] that combines both ef- ficiency and somewhat greater ease of applicability. Ac- cording to this scheme a random individual is selected who then points to one of his random acquaintances and this node is the one to be immunized. This method is more efficient compared to random immunization (fc is of the order of 20-25%) but less efficient than targeted immunization. In this paper we introduce an immunization method, which is practically as efficient as the accepted as opti- mum strategy, but at the same time depends on local in- formation only. The method consists in selecting random individuals and asking them to direct us to their friend who is more connected than they are and this acuain- tance is immunized. If such a friend does not exist we continue with another random selection. Alternatively, in a second variation of the method we ask the randomly chosen individual to point us to a random neighbor that has a number of neighbors larger than e.g. k = 5 (or an equally small and easily countable threshold value). If they point to such an individual it is immunized, oth- erwise we select another individual. Similar results are obtained if the chosen individual is asked to estimate his own number of contacts, rather than of his random neigh- bor. Although this procedure is simpler, the selection of a neighbor can also eliminate the bias that may be in- troduced due to selfish people, lying about their contacts in order to receive the vaccine themselves. The method is proposed for social networks, but it is expected that it can be even more efficient for technological networks, such as e.g. the Internet, where the number of links for a given node is exactly known to the local network ad- ministrator, and need not be estimated. Our method is local because the decision for immu- nization of a given node is taken without the need to know the connectivity of other nodes. This is in contrast with global strategies where immunization of a node has to be decided only after we have gathered information for the entire network. This means that for immunization of e.g. a city or a country in a global method we have to http://arxiv.org/abs/0704.1589v1 send special teams to collect this information and trans- mit it to a central place. This central authority decides then which nodes should be immunized and transmits back the outcome to the local authorities which then go on with vaccinations. For a local method, there is no need to collect or compare data from other areas of the network. Based on the answer of each individual the de- cision is made immediately on whether a node should be immunized or not. We study the proposed method on real social networks with a fat tail in their degree distribution, as well as on a random scale-free model network. We also compare this method with several other immunization strategies, including such that partial knowledge on the global net- work of contacts is available and we demonstrate the ad- vantage of the proposed method via the improvement in The social networks used in this study represent dif- ferent interactions among the members of an online com- munity, as described in Ref. [12]. These interactions in- clude a) exchange of messages, b) signing of guestbooks, c) flirt requests, and d) established friendships. The first three networks are directed but we consider only their undirected projection, by transforming arcs into edges. No significant difference is observed in the results for the undirected network and the projections of the di- rected networks. The size of the networks is of the order N = 104. The percentage of immunized nodes is denoted with f , while the percentage of nodes suveyed is denoted with p. The four strategies that we employ are summa- rized below. Strategy I: Immunize a node with probabil- ity proportional to kα, where k is the number of connec- tions and α tunes the probability of preferentially select- ing high-connectivity or low-connectivity nodes. Large positive values of α tend towards mainly selecting the hubs (α → ∞ is equivalent to targeted immunization), the value α = 0 represents the random immunization model, while negative α values lead to selecting the lower- connected nodes [13]. This parameter can be interpreted as a measure of the extent of our knowledge on the struc- ture. Strategy II: Select a node with probability propor- tional to kα and immunize a random acquaintance of this node. The value α = 0 corresponds to the acquaintance immunization scheme [11]. Strategy III: Select a random node and immunize one of its acquaintances i, with prob- ability proportional to kαi , where ki represents the degree of the neighbor. Strategy IV: Select a random node and ask for an acquaintance, which is immunized if a cer- tain condition is met. We study two variations: a) The selected node points randomly to a node which is more connected than himself. If there are no such neighbors no node is immunized. b) The selected node is asked to choose a random neighbor with degree larger than a threshold value kcut then this acquaintance is immunized. Equivalently, we can ask the node to estimate its own de- gree. If it is larger than a threshold value we immunize -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (b) Guestbook(a) Messages (c) Flirt (d) γ=2.5 II II IV IV FIG. 1: Critical immunized fraction fc of the population as a function of α for (a)-(c) Real-life social networks, and (d) scale-free network model with γ = 2.5. Four different strate- gies are used as described in the text and indicated in the plot. The two symbols correspond to the critical fraction for the strategies of the enhanced acquaintance immuniza- tion method (the open circle corresponds to asking for an acquintance with threshold kcut = 7, while the filled circle corresponds to asking for a better connected node). the node, otherwise we ignore it. These two variations are similar when kcut = 〈k〉. We call strategy IV “enhanced acquaintance immunization” (EAI) method. In Figs. 1a-c we present the results of fc for the four de- scribed strategies applied to three of the social networks, as defined by different types of interactions. All networks follow similar patterns for a given strategy. In strategy I we can see the abrupt decrease of fc when increasing α from α ≤ 0 (random immunization) with fc = 1 to α = ∞ (targeted immunization) with fc ≪ 1. Strategy II presents an improvement over the first strategy for val- ues α . 1. The critical value fc presents a minimum at α ≃ 1, indicating that identification of large hubs actu- ally deteriorates the results, since the neighbors of large hubs, which are chosen to be immunized, are with higher probability low degree nodes for dissasortative networks, similarly with the acquaintance immunization method [11]. Strategy III leads to monotonic decrease in fc and prevails from the first two methods when we have limited global network knowledge, i.e. in the range α ∈ [0, 1]. However, in Strategy III we find that when α = ∞ (i.e. we always immunize the most connected neighbor) it may be impossible to destroy the spanning cluster, because al- most all selected nodes point to the same hubs. Finally, the enhanced acquaintance immunization strategy seems to be the most efficient method, although it assumes no 1 2 3 4 5 6 7 1 2 3 4 5 6 7 (a) Messages (b) Guestbook (c) Flirt (d) γ=2.5 FIG. 2: Critical immunized fraction fc of the population as a function of the threshold value kcut for the enhanced ac- quaintance immunization strategy applied to (a)-(c) social in- teraction networks, (d) random model scale-free network with γ = 2.5 (of size N = 105 nodes). Filled symbols correspond to immunizing a random neighbor of the selected node if its de- gree is ≥ kcut and open symbols to immunizing the selected node itself. The upper horizontal dotted line is the result for acquaintance immunization, the dashed line in the middle corresponds to immunizing a more connected acquaintance, while the lower line refers to targeted immunization. knowledge of the underlying structure (the method is in- dependent of α). The value of fc is lower than an attack with α = 3 and very close to the results of the targeted immunization. To gain more insight into the different immunization methods we also performed numerical simulations on a model network. We consider each member of a popula- tion represented by a node, while the acquaintances of a person with other people form links. It is well established that many social networks follow a broad distribution in the degree of a node, such as the power-law distribution P (k) ∼ k−γ , where the exponent γ is usually found to be between 2 < γ < 4 [5, 14, 15, 16]. The above real networks are scale-free with γ ≃ 2.4[12]. The results in Fig. 1d correspond to the four strategies in such a model network (created with the configuration random model [17]) with exponent γ = 2.5, which is close to the re- ported exponent γ ≃ 2.4 of the real networks used. All strategies in this plot follow closely the results for the real networks. The two ‘transition’ points for the first three strategies are located at α = 0 and α = 1. At α = 0, strategies II and III coincide. In the range α ∈ [0, 1] strategy III is 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 P∞(f) P∞(f) 1 2 3 4 5 6 7 1 2 3 4 5 6 7 (b) Guestbook(a) Messages (c) Flirt (d) γ=2.5 FIG. 3: Size of epidemics, measured via the fraction of nodes belonging to the largest cluster over the number of not- immunized nodes P∞(f), as a function of the fraction f of im- munized nodes. In each plot, from top to bottom, the curves correspond to acquaintance immunization, EAI redirecting to a better connected node, the EAI with kcut = 7, and targeted immunization. (a)-(c): real networks, and (d): random scale- free network with γ = 2.5. Insets: Ratios for f1/fc of the critical immnunized fraction fc over the critical fraction f1 for acquaintance immunization (kcut = 1) and pc/p1, i.e. the number of people surveyed, as a function of kcut for the EAI method. more efficient, indicating that in this range it is preferable to let the nodes choose their neighbors according to their connectivity, rather than selecting nodes with probabil- ity proportional to kα and following random links. The value α ≃ 1 is the optimum value for strategy II. In prac- tice, the process is equivalent to selecting a random link and immunizing one of the two nodes attached to the given link (provided the uncorrelated network hypothe- sis holds). It is also interesting to note that up to the value α = 1 the acquaintance immunization strategy is superior to direct immunization of the initially selected nodes, but close to this value the two methods yield a similar value for fc. When α > 1 the direct immuniza- tion method becomes more efficient than acquaintance immunization. The enhanced acquaintance immunization is, however, found to be superior to all the above methods. The value of fc for a given kcut value is of course independent of α, meaning that it works equally well when there is no fur- ther information on the network structure, i.e. global knowledge does not offer any significant advantage over completely random selections. Thus, the strategy is local and easy to implement. The choice of kcut, though, influ- ences fc and can further reduce the fc value when more accurate knowledge on the network structure is available. The gain of this method for kcut = 7 when compared to the original acquaintance immunization method is about a factor of 4, which is for practical purposes a significant improvement. This striking variation is evident in Fig. 2, where the critical percentage decays from fc ≃ 0.26 at kcut = 1 (acquaintance immunization) to fc ≃ 0.06 at kcut = 7. For kcut = 7 the strategy works comparably well to the targeted immunization. The fraction fc, how- ever, remains very low even when the cutoff value kcut decreases to values close to, but less than 7. This stabil- ity over the value of kcut offers greater flexibility since the method seems tolerant to mistakes of lower degree nodes being pointed at for immunization, without siginificant loss in the efficiency (even at a value of kcut = 4 the critical fraction fc remains lower than 10%). The results are different when we immunize directly the initially se- lected random node (without asking for an acquaintance) and only at kcut = 7 the two methods seem to coincide (Fig. 2). There exists, though, a critical degree above which this strategy no longer works, simply because the number of nodes with degree larger than this value is smaller than the critical number needed for complete im- munization. Thus, it seems preferable to remain con- servative on the estimation of kcut and choose a smaller value over a larger one. A considerable advantage is gained, even when the question is posed in a much simpler way, i.e. we ask a random node to direct us to a friend who is better con- nected than his and immunize him. This simple approach already offers a significant improvement over the original acquaintance method, as is evident in Fig. 2, although it is not as efficient as when asking for a friend whose degree exceeds the cutoff value. Since it is, however, much easier for an individual to estimate an acquaintance who is bet- ter connected than himself, and practically everyone can understand and correctly answer this simple question, we consider this method as a useful strategy which is easy to apply in real-life situations. In order to assess the size of the epidemics in the im- munization process we measure the size of the spanning cluster (epidemics size) as a function of the immunized nodes f . In Figs. 3a-c we present the fraction of nodes belonging to the spanning cluster over the total number of non-immunized nodes for the real networks described above and compare the targeted immunization with the enhanced acquaintance immunization and the original ac- quaintance immunization methods. The results for the model scale-free networks (Fig. 3d) are averages over 100 different realizations of networks with exponent γ = 2.5. In all cases the critical fraction for the targeted immu- nization and the EAI with the cutoff value are similar, while acquaintance immunization leads to considerably higher values of fc. Again, the EAI with an estimation of a better connected friend yields a result between these two extremes. However, during the removal process the targeted immunization yields the faster decomposition of the spanning cluster, since it first removes the most con- nected nodes in the system. The results for all the ac- quaintance immunization methods depend on when these largest hubs will be selected and the averaging conceals the fact that during one realization the size of the largest cluster drops abruptly when the largest hubs are selected. Despite this, the proposed methods follow closely the re- sults of targeted immunization, while retaining the ad- vantage of being local. In the insets of Fig. 3 we can see that compared to the acquaintance immunization method (which is the EAI method with kcut = 1) in general we need to survey more nodes for their acquaintances as kcut increases, but this is a small change compared to the improvement in the number of required immunizations presented in the same plots. A work with similar scope was performed by Holme [18]. Among other methods, an immunization scheme was introduced, where a random node points to one of its highest degree neighbors or to its most connected neigh- bor. This corresponds to strategy III of the current work with α → ∞ (where we encounter the problem of select- ing always the same nodes as described above) and the first variation of Strategy IV. The results in that paper are consistent with the ones presented above for these limiting cases. In summary, we introduced and compared various im- munization strategies on real and model networks. We have shown that the fraction of immunized nodes can be significantly reduced to the almost optimum level of intentional immunization using a completely local infor- mation strategy. This simple process is enough to ensure that the immunization threshold is significantly lowered, as compared to other local methods. We thank R. Cohen and D. Brewer for useful discus- sions. This work was supported by a European research NEST/PATHFINDER project DYSONET 012911, by a project of the Greek GGET in conjunction with ESF and by the Israel Science Foundation. [1] R.M. Andserson and R.M. May, Infectious Diseases of Humans: Dynamics and Control (Oxford University Press, Oxford, U.K., 1992). [2] T. Britton, J. R. Stat. Soc. B 63, 705 (2001). [3] F. Ball, D. Mollison, and G. Scalia-Tomba, Ann. Appl. Prob. 7, 46 (1997). [4] H. Bazin, The Eradication of Small Pox: Edward Jenner and the First and Only Eradication of a Human Infec- tious Disease (Academic Press, London, 2000). [5] R. Albert and A.-L. Barabasi, Rev. Mod. Phys. 74, 47 (2002). [6] R. Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). [7] D.S. Callaway et al., Phys. Rev. Lett. 85, 5468 (2000). [8] R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001). [9] R. Albert et al., Nature 406, 378 (2000). [10] R. Cohen et al., Phys. Rev. Lett. 86, 3682 (2001). [11] R. Cohen et al., Phys. Rev. Lett. 91, 247901 (2003). [12] P. Holme et al., Soc. Networks 26, 155 (2004). [13] L.K. Gallos et al., Phys. Rev. Lett. 94, 188701 (2005). [14] S.N. Dorogovtsev and J.F.F. Mendes, Evolution of Net- works: From Biological nets to the Internet and WWW (Oxford University Press, 2003). [15] R. Pastor-Satorras and A. Vespignani, Evolution and Structure of the Internet, (Cambridge Univ. Press, 2004). [16] M.E.J. Newman, SIAM Review 45, 167 (2003). [17] M. Molloy and B. Reed, Random Struct. Algor. 6, 161 (1995); Comb. Probab. Comput. 7, 295 (1998). [18] P. Holme, Europhys. Lett. 68, 908 (2004).

We introduce an immunization method where the percentage of required vaccinations for immunity are close to the optimal value of a targeted immunization scheme of highest degree nodes. Our strategy retains the advantage of being purely local, without the need of knowledge on the global network structure or identification of the highest degree nodes. The method consists of selecting a random node and asking for a neighbor that has more links than himself or more than a given threshold and immunizing him. We compare this method to other efficient strategies on three real social networks and on a scale-free network model, and find it to be significantly more effective.

Improving immunization strategies Lazaros K. Gallos1, Fredrik Liljeros2, Panos Argyrakis1, Armin Bunde3, and Shlomo Havlin4 Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece Department of Sociology, Stockholm University 106 91 Stockholm, Sweden Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany and Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel (Dated: November 20, 2018) We introduce an immunization method where the percentage of required vaccinations for immunity are close to the optimal value of a targeted immunization scheme of highest degree nodes. Our strategy retains the advantage of being purely local, without the need of knowledge on the global network structure or identification of the highest degree nodes. The method consists of selecting a random node and asking for a neighbor that has more links than himself or more than a given threshold and immunizing him. We compare this method to other efficient strategies on three real social networks and on a scale-free network model, and find it to be significantly more effective. PACS numbers: 89.75.Hc, 87.23.Ge Immunization of large populations through vaccination is an extremely important issue with obvious implications for the public health [1, 2, 3]. The eradication of Small Pox through a global mass vaccination campaign during the second part of the 20th century represents, for exam- ple, a landmark in the history of the medical sciences [4]. Global or national mass vaccination may however not al- ways be possible. The number of vaccinated people may need to be minimized due to severe side effects of vacci- nation such as for Small Pox, or temporary shortage of vaccine that could be the case for a pandemic influenza. The cost for a vaccine may also be an important limiting factor. Improving efficiency of immunization is thus an urgent task. Recently [5], developments in the study of population connectivities helped researchers in the field to present new ideas on immunization, based on the heterogeneity in the number of contacts between individuals. A number of strategies have been proposed for lowering the required minimum fraction fc of the population to be immunized. The problem can be mapped to the well-known percola- tion problem where nodes are immunized (removed) up to a concentration fc, above which the spanning clus- ter does not survive. Random immunization of nodes has been shown incapable of protecting the population when the contacts distribution is wide, since the perco- lation threshold is close to fc = 1, i.e. practically all nodes need to be immunized [6, 7, 8]. The best known strategy today is believed to be targeted immunization, where the highest connected nodes in the system are im- munized in decreasing order of their degree. In this case fc is less than 10% [7, 9, 10]. For all practical applica- tions, though, this approach is unrealistic because it is a ‘global’ strategy and requires a complete knowledge of the high degree nodes, which is in many cases impossi- ble. An effective strategy, called acquaintance immuniza- tion, was recently introduced [11] that combines both ef- ficiency and somewhat greater ease of applicability. Ac- cording to this scheme a random individual is selected who then points to one of his random acquaintances and this node is the one to be immunized. This method is more efficient compared to random immunization (fc is of the order of 20-25%) but less efficient than targeted immunization. In this paper we introduce an immunization method, which is practically as efficient as the accepted as opti- mum strategy, but at the same time depends on local in- formation only. The method consists in selecting random individuals and asking them to direct us to their friend who is more connected than they are and this acuain- tance is immunized. If such a friend does not exist we continue with another random selection. Alternatively, in a second variation of the method we ask the randomly chosen individual to point us to a random neighbor that has a number of neighbors larger than e.g. k = 5 (or an equally small and easily countable threshold value). If they point to such an individual it is immunized, oth- erwise we select another individual. Similar results are obtained if the chosen individual is asked to estimate his own number of contacts, rather than of his random neigh- bor. Although this procedure is simpler, the selection of a neighbor can also eliminate the bias that may be in- troduced due to selfish people, lying about their contacts in order to receive the vaccine themselves. The method is proposed for social networks, but it is expected that it can be even more efficient for technological networks, such as e.g. the Internet, where the number of links for a given node is exactly known to the local network ad- ministrator, and need not be estimated. Our method is local because the decision for immu- nization of a given node is taken without the need to know the connectivity of other nodes. This is in contrast with global strategies where immunization of a node has to be decided only after we have gathered information for the entire network. This means that for immunization of e.g. a city or a country in a global method we have to http://arxiv.org/abs/0704.1589v1 send special teams to collect this information and trans- mit it to a central place. This central authority decides then which nodes should be immunized and transmits back the outcome to the local authorities which then go on with vaccinations. For a local method, there is no need to collect or compare data from other areas of the network. Based on the answer of each individual the de- cision is made immediately on whether a node should be immunized or not. We study the proposed method on real social networks with a fat tail in their degree distribution, as well as on a random scale-free model network. We also compare this method with several other immunization strategies, including such that partial knowledge on the global net- work of contacts is available and we demonstrate the ad- vantage of the proposed method via the improvement in The social networks used in this study represent dif- ferent interactions among the members of an online com- munity, as described in Ref. [12]. These interactions in- clude a) exchange of messages, b) signing of guestbooks, c) flirt requests, and d) established friendships. The first three networks are directed but we consider only their undirected projection, by transforming arcs into edges. No significant difference is observed in the results for the undirected network and the projections of the di- rected networks. The size of the networks is of the order N = 104. The percentage of immunized nodes is denoted with f , while the percentage of nodes suveyed is denoted with p. The four strategies that we employ are summa- rized below. Strategy I: Immunize a node with probabil- ity proportional to kα, where k is the number of connec- tions and α tunes the probability of preferentially select- ing high-connectivity or low-connectivity nodes. Large positive values of α tend towards mainly selecting the hubs (α → ∞ is equivalent to targeted immunization), the value α = 0 represents the random immunization model, while negative α values lead to selecting the lower- connected nodes [13]. This parameter can be interpreted as a measure of the extent of our knowledge on the struc- ture. Strategy II: Select a node with probability propor- tional to kα and immunize a random acquaintance of this node. The value α = 0 corresponds to the acquaintance immunization scheme [11]. Strategy III: Select a random node and immunize one of its acquaintances i, with prob- ability proportional to kαi , where ki represents the degree of the neighbor. Strategy IV: Select a random node and ask for an acquaintance, which is immunized if a cer- tain condition is met. We study two variations: a) The selected node points randomly to a node which is more connected than himself. If there are no such neighbors no node is immunized. b) The selected node is asked to choose a random neighbor with degree larger than a threshold value kcut then this acquaintance is immunized. Equivalently, we can ask the node to estimate its own de- gree. If it is larger than a threshold value we immunize -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (b) Guestbook(a) Messages (c) Flirt (d) γ=2.5 II II IV IV FIG. 1: Critical immunized fraction fc of the population as a function of α for (a)-(c) Real-life social networks, and (d) scale-free network model with γ = 2.5. Four different strate- gies are used as described in the text and indicated in the plot. The two symbols correspond to the critical fraction for the strategies of the enhanced acquaintance immuniza- tion method (the open circle corresponds to asking for an acquintance with threshold kcut = 7, while the filled circle corresponds to asking for a better connected node). the node, otherwise we ignore it. These two variations are similar when kcut = 〈k〉. We call strategy IV “enhanced acquaintance immunization” (EAI) method. In Figs. 1a-c we present the results of fc for the four de- scribed strategies applied to three of the social networks, as defined by different types of interactions. All networks follow similar patterns for a given strategy. In strategy I we can see the abrupt decrease of fc when increasing α from α ≤ 0 (random immunization) with fc = 1 to α = ∞ (targeted immunization) with fc ≪ 1. Strategy II presents an improvement over the first strategy for val- ues α . 1. The critical value fc presents a minimum at α ≃ 1, indicating that identification of large hubs actu- ally deteriorates the results, since the neighbors of large hubs, which are chosen to be immunized, are with higher probability low degree nodes for dissasortative networks, similarly with the acquaintance immunization method [11]. Strategy III leads to monotonic decrease in fc and prevails from the first two methods when we have limited global network knowledge, i.e. in the range α ∈ [0, 1]. However, in Strategy III we find that when α = ∞ (i.e. we always immunize the most connected neighbor) it may be impossible to destroy the spanning cluster, because al- most all selected nodes point to the same hubs. Finally, the enhanced acquaintance immunization strategy seems to be the most efficient method, although it assumes no 1 2 3 4 5 6 7 1 2 3 4 5 6 7 (a) Messages (b) Guestbook (c) Flirt (d) γ=2.5 FIG. 2: Critical immunized fraction fc of the population as a function of the threshold value kcut for the enhanced ac- quaintance immunization strategy applied to (a)-(c) social in- teraction networks, (d) random model scale-free network with γ = 2.5 (of size N = 105 nodes). Filled symbols correspond to immunizing a random neighbor of the selected node if its de- gree is ≥ kcut and open symbols to immunizing the selected node itself. The upper horizontal dotted line is the result for acquaintance immunization, the dashed line in the middle corresponds to immunizing a more connected acquaintance, while the lower line refers to targeted immunization. knowledge of the underlying structure (the method is in- dependent of α). The value of fc is lower than an attack with α = 3 and very close to the results of the targeted immunization. To gain more insight into the different immunization methods we also performed numerical simulations on a model network. We consider each member of a popula- tion represented by a node, while the acquaintances of a person with other people form links. It is well established that many social networks follow a broad distribution in the degree of a node, such as the power-law distribution P (k) ∼ k−γ , where the exponent γ is usually found to be between 2 < γ < 4 [5, 14, 15, 16]. The above real networks are scale-free with γ ≃ 2.4[12]. The results in Fig. 1d correspond to the four strategies in such a model network (created with the configuration random model [17]) with exponent γ = 2.5, which is close to the re- ported exponent γ ≃ 2.4 of the real networks used. All strategies in this plot follow closely the results for the real networks. The two ‘transition’ points for the first three strategies are located at α = 0 and α = 1. At α = 0, strategies II and III coincide. In the range α ∈ [0, 1] strategy III is 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 P∞(f) P∞(f) 1 2 3 4 5 6 7 1 2 3 4 5 6 7 (b) Guestbook(a) Messages (c) Flirt (d) γ=2.5 FIG. 3: Size of epidemics, measured via the fraction of nodes belonging to the largest cluster over the number of not- immunized nodes P∞(f), as a function of the fraction f of im- munized nodes. In each plot, from top to bottom, the curves correspond to acquaintance immunization, EAI redirecting to a better connected node, the EAI with kcut = 7, and targeted immunization. (a)-(c): real networks, and (d): random scale- free network with γ = 2.5. Insets: Ratios for f1/fc of the critical immnunized fraction fc over the critical fraction f1 for acquaintance immunization (kcut = 1) and pc/p1, i.e. the number of people surveyed, as a function of kcut for the EAI method. more efficient, indicating that in this range it is preferable to let the nodes choose their neighbors according to their connectivity, rather than selecting nodes with probabil- ity proportional to kα and following random links. The value α ≃ 1 is the optimum value for strategy II. In prac- tice, the process is equivalent to selecting a random link and immunizing one of the two nodes attached to the given link (provided the uncorrelated network hypothe- sis holds). It is also interesting to note that up to the value α = 1 the acquaintance immunization strategy is superior to direct immunization of the initially selected nodes, but close to this value the two methods yield a similar value for fc. When α > 1 the direct immuniza- tion method becomes more efficient than acquaintance immunization. The enhanced acquaintance immunization is, however, found to be superior to all the above methods. The value of fc for a given kcut value is of course independent of α, meaning that it works equally well when there is no fur- ther information on the network structure, i.e. global knowledge does not offer any significant advantage over completely random selections. Thus, the strategy is local and easy to implement. The choice of kcut, though, influ- ences fc and can further reduce the fc value when more accurate knowledge on the network structure is available. The gain of this method for kcut = 7 when compared to the original acquaintance immunization method is about a factor of 4, which is for practical purposes a significant improvement. This striking variation is evident in Fig. 2, where the critical percentage decays from fc ≃ 0.26 at kcut = 1 (acquaintance immunization) to fc ≃ 0.06 at kcut = 7. For kcut = 7 the strategy works comparably well to the targeted immunization. The fraction fc, how- ever, remains very low even when the cutoff value kcut decreases to values close to, but less than 7. This stabil- ity over the value of kcut offers greater flexibility since the method seems tolerant to mistakes of lower degree nodes being pointed at for immunization, without siginificant loss in the efficiency (even at a value of kcut = 4 the critical fraction fc remains lower than 10%). The results are different when we immunize directly the initially se- lected random node (without asking for an acquaintance) and only at kcut = 7 the two methods seem to coincide (Fig. 2). There exists, though, a critical degree above which this strategy no longer works, simply because the number of nodes with degree larger than this value is smaller than the critical number needed for complete im- munization. Thus, it seems preferable to remain con- servative on the estimation of kcut and choose a smaller value over a larger one. A considerable advantage is gained, even when the question is posed in a much simpler way, i.e. we ask a random node to direct us to a friend who is better con- nected than his and immunize him. This simple approach already offers a significant improvement over the original acquaintance method, as is evident in Fig. 2, although it is not as efficient as when asking for a friend whose degree exceeds the cutoff value. Since it is, however, much easier for an individual to estimate an acquaintance who is bet- ter connected than himself, and practically everyone can understand and correctly answer this simple question, we consider this method as a useful strategy which is easy to apply in real-life situations. In order to assess the size of the epidemics in the im- munization process we measure the size of the spanning cluster (epidemics size) as a function of the immunized nodes f . In Figs. 3a-c we present the fraction of nodes belonging to the spanning cluster over the total number of non-immunized nodes for the real networks described above and compare the targeted immunization with the enhanced acquaintance immunization and the original ac- quaintance immunization methods. The results for the model scale-free networks (Fig. 3d) are averages over 100 different realizations of networks with exponent γ = 2.5. In all cases the critical fraction for the targeted immu- nization and the EAI with the cutoff value are similar, while acquaintance immunization leads to considerably higher values of fc. Again, the EAI with an estimation of a better connected friend yields a result between these two extremes. However, during the removal process the targeted immunization yields the faster decomposition of the spanning cluster, since it first removes the most con- nected nodes in the system. The results for all the ac- quaintance immunization methods depend on when these largest hubs will be selected and the averaging conceals the fact that during one realization the size of the largest cluster drops abruptly when the largest hubs are selected. Despite this, the proposed methods follow closely the re- sults of targeted immunization, while retaining the ad- vantage of being local. In the insets of Fig. 3 we can see that compared to the acquaintance immunization method (which is the EAI method with kcut = 1) in general we need to survey more nodes for their acquaintances as kcut increases, but this is a small change compared to the improvement in the number of required immunizations presented in the same plots. A work with similar scope was performed by Holme [18]. Among other methods, an immunization scheme was introduced, where a random node points to one of its highest degree neighbors or to its most connected neigh- bor. This corresponds to strategy III of the current work with α → ∞ (where we encounter the problem of select- ing always the same nodes as described above) and the first variation of Strategy IV. The results in that paper are consistent with the ones presented above for these limiting cases. In summary, we introduced and compared various im- munization strategies on real and model networks. We have shown that the fraction of immunized nodes can be significantly reduced to the almost optimum level of intentional immunization using a completely local infor- mation strategy. This simple process is enough to ensure that the immunization threshold is significantly lowered, as compared to other local methods. We thank R. Cohen and D. Brewer for useful discus- sions. This work was supported by a European research NEST/PATHFINDER project DYSONET 012911, by a project of the Greek GGET in conjunction with ESF and by the Israel Science Foundation. [1] R.M. Andserson and R.M. May, Infectious Diseases of Humans: Dynamics and Control (Oxford University Press, Oxford, U.K., 1992). [2] T. Britton, J. R. Stat. Soc. B 63, 705 (2001). [3] F. Ball, D. Mollison, and G. Scalia-Tomba, Ann. Appl. Prob. 7, 46 (1997). [4] H. Bazin, The Eradication of Small Pox: Edward Jenner and the First and Only Eradication of a Human Infec- tious Disease (Academic Press, London, 2000). [5] R. Albert and A.-L. Barabasi, Rev. Mod. Phys. 74, 47 (2002). [6] R. Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). [7] D.S. Callaway et al., Phys. Rev. Lett. 85, 5468 (2000). [8] R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001). [9] R. Albert et al., Nature 406, 378 (2000). [10] R. Cohen et al., Phys. Rev. Lett. 86, 3682 (2001). [11] R. Cohen et al., Phys. Rev. Lett. 91, 247901 (2003). [12] P. Holme et al., Soc. Networks 26, 155 (2004). [13] L.K. Gallos et al., Phys. Rev. Lett. 94, 188701 (2005). [14] S.N. Dorogovtsev and J.F.F. Mendes, Evolution of Net- works: From Biological nets to the Internet and WWW (Oxford University Press, 2003). [15] R. Pastor-Satorras and A. Vespignani, Evolution and Structure of the Internet, (Cambridge Univ. Press, 2004). [16] M.E.J. Newman, SIAM Review 45, 167 (2003). [17] M. Molloy and B. Reed, Random Struct. Algor. 6, 161 (1995); Comb. Probab. Comput. 7, 295 (1998). [18] P. Holme, Europhys. Lett. 68, 908 (2004).

April 2007 Constraints on the Very Early Universe from Thermal WIMP Dark Matter Manuel Dreesa, ∗, Hoernisa Iminniyaza,b, † and Mitsuru Kakizakia, ‡ aPhysikalisches Institut der Universität Bonn, Nussallee 12, 53115 Bonn, Germany bPhysics Dept., Univ. of Xinjiang, 830046 Urumqi, P.R. China Abstract We investigate the relic density nχ of non–relativistic long–lived or sta- ble particles χ in non–standard cosmological scenarios. We calculate the relic abundance starting from arbitrary initial temperatures of the radiation– dominated epoch, and derive the lower bound on the initial temperature T0 ≥ mχ/23, assuming that thermally produced χ particles account for the dark matter energy density in the universe; this bound holds for all χ annihi- lation cross sections. We also investigate cosmological scenarios with modified expansion rate. Even in this case an approximate formula similar to the stan- dard one is capable of predicting the final relic abundance correctly. Choosing the χ annihilation cross section such that the observed cold dark matter abun- dance is reproduced in standard cosmology, we constrain possible modifications of the expansion rate at T ∼ mχ/20, well before Big Bang Nucleosynthesis. ∗drees@th.physik.uni-bonn.de †hoernisa@th.physik.uni-bonn.de ‡kakizaki@th.physik.uni-bonn.de http://arxiv.org/abs/0704.1590v2 1 Introduction One of the most notable recent developments in cosmology is the precise determina- tion of cosmological parameters from observations of the large–scale structure of the universe, most notably by the Wilkinson Microwave Anisotropy Probe (WMAP). In particular, the accurate determination of the non–baryonic cold Dark Matter (DM) density [1], 0.08 < ΩCDMh 2 < 0.12 (95% C.L.) , (1) has great influence on particle physics models which possess dark matter candidates [2, 3]. The requirement that the predicted DM density falls in the range (1) is a powerful tool for discriminating between various models and for constraining the parameter space of surviving models. Many dark matter candidate particles have been proposed. In particular long– lived or stable weakly interacting massive particles (WIMPs) with weak–scale masses are excellent candidates. In standard cosmology WIMPs decoupled from the thermal background during the radiation–dominated epoch after inflation. In this framework convenient and accurate analytic approximate solutions for the relic abundance have been derived [4, 5]. One of the best motivated candidates for WIMPs is the lightest neutralino in supersymmetric (SUSY) models. Assuming that the neutralino is the lightest supersymmetric particle (LSP) stabilized due to R–parity, its relic abundance has been extensively discussed [3]. Similar analyses have also been performed for other WIMPs whose existence is postulated in other extensions of the Standard Model (SM) of particle physics. In many cases the cosmologically favored parameter space of WIMP models can be directly tested at the CERN Large Hadron Collider (LHC) in a few years [6]. The same parameter space often also leads to rates of WIMP interactions with matter within the sensitivity of near–future direct DM detection experiments. This discussion shows that we are now entering an interesting time where the standard cosmological scenario can be examined by experiments at high–energy col- liders as well as DM searches [7]. In this respect we should emphasize that the relic abundance of thermally produced WIMPs depends not only on their annihila- tion cross section, which can be determined by particle physics experiments, but in general is also very sensitive to cosmological parameters during the era of WIMP production and annihilation. Of particular importance are the initial temperature T0 at which WIMPs began to be thermally produced, and the expansion rate of the universe H . In the standard cosmological scenario, the expansion rate is uniquely determined through the Friedmann equation of general relativity. In this scenario the density of WIMPs with massmχ followed its equilibrium value until the freeze–out temperature TF ≃ mχ/20. Below TF , interactions of WIMPs are decoupled, and thus the present density is independent of T0 as long as T0 > TF . It should be noted that in non–standard scenarios the relic density can be larger or smaller than the value in the standard scenario. One example is the case where T0 is smaller than or comparable to TF , which can be realized in inflationary models with low reheat temperature. Since in many models the inflationary energy scale must be much higher than mχ in order to correctly predict the density perturbations [8], the standard assumption T0 > TF is not unreasonable. On the other hand, the constraint on the reheat temperature from Big Bang Nucleosynthesis (BBN) is as low as T0 >∼ MeV [9, 10]. From the purely phenomenological viewpoint, it is therefore also interesting to investigate the production of WIMPs in low reheat temperature scenarios [9, 11, 12, 13, 14]. The standard scenario also assumes that entropy per comoving volume is con- served for all temperatures T ≤ TF . Late entropy production can dilute the predicted relic density [15, 16]. The reason is that the usual calculation actually predicts the ratio of the WIMP number density to the entropy density. On the other hand, if late decays of a heavier particle non–thermally produce WIMPs in addition to the usual thermal production mechanism, the resulting increase of the WIMP density competes with the dilution caused by the decay of this particle into radiation, which increases the entropy density [17, 18, 19, 20, 13, 21]. Another example of a non–standard cosmology changing the WIMP relic density is a modified expansion rate of the universe. This might be induced by an anisotropic expansion [16], by a modification of general relativity [16, 22], by additional contri- butions to the total energy density from quintessence [23], by branes in a warped geometry [24], or by a superstring dilaton [25]. These examples show that, once the WIMP annihilation cross section is fixed, with the help of precise measurements of the cold dark matter density we can probe the very early stage of the universe at temperatures of O(mχ/20) ∼ 10 GeV. This is reminiscent of constraining the early evolution of the universe at T = O(100) keV ∗Note that TF can be formally defined in the standard way even if T0 < TF . In this case WIMPs never were in full equilibrium, and correspondingly never “froze out”. using the primordial abundances of the light elements produced by BBN. The goal of this paper is to investigate to what extent the constraint (1) on the WIMP relic abundance might allow us to derive quantitative constraints on modifications of standard cosmology. So far the history of the universe has been established by cosmological observations as far back as the BBN era. In this paper we try to derive bounds on cosmological parameters relevant to the era before BBN. Rather than studying specific extensions of the standard cosmological scenario, we simply parameterize deviations from the standard scenario, and attempt to derive constraints on these new parameters. Since we only have the single constraint (1), for the most part we only allow a single quantity to differ from its standard value. We expect that varying two quantities simultaneously will allow to get the right relic density for almost any WIMP annihilation cross section. This has been shown explicitly in [13] for the case that both late entropy production and non–thermal WIMP production are considered, even if both originate from the late decay of a single scalar field. We first analyze the dependence of the WIMP abundance on the initial temper- ature T0 of the conventional radiation–dominated epoch. We showed in [14] that for fixed T0 the predicted WIMP relic density reaches a maximum as the annihilation cross section is varied from very small to very large values. A small annihilation cross section corresponds to a large TF > T0; in this case the relic density increases with the annihilation cross section, since WIMP production from the thermal plasma is more important than WIMP annihilation. On the other hand, increasing this cross section reduces TF ; once TF < T0 a further increase of the cross section leads to smaller relic densities since in this case WIMPs continue to annihilate even after the temperature is too low for WIMP production. Here we turn this argument around, and derive the lower bound on T0 ≥ mχ/23 under the assumption that all WIMPs are produced thermally. Note that we do not need to know the WIMP annihilation cross section to derive this bound. We then examine the dependence of the predicted WIMP relic density on the expansion rate in the epoch prior to BBN, where we allow the Hubble parameter to depart from the standard value. The standard method of calculating the thermal relic density [2, 4] is found to be still applicable in this case. Our working hypothesis here is that the standard prediction for the Hubble expansion rate is essentially correct, i.e. that the true expansion rate differs by at most a factor of a few from the standard prediction. We then simply employ a generic Taylor expansion for the temperature dependence of this modification factor; note that the success of standard BBN indicates that this factor cannot deviate by more than ∼ 20% from unity at low temperatures, T <∼ 1 MeV. Similarly, we assume that the WIMP annihilation cross section has been determined (from experiments at particle colliders) to have the value required in standard cosmology. Our approach is thus quite different from that taken in [7], where present upper bounds on the fluxes of WIMP annihilation products are used to place upper bounds on the Hubble expansion rate during WIMP decoupling. The advantage of their approach is that no prior assumption on the WIMP annihilation cross section needs to be made, whereas we assume a cross section that reproduces the correct relic density in the standard scenario. On the other hand, the bounds derived in refs.[7] are still quite weak, allowing the Hubble parameter to exceed its standard prediction by a factor >∼ 30; moreover, no lower bound on H can be derived in this fashion. The remainder of this paper is organized as follows: In Sec. 2 we will briefly review the calculation of the WIMP relic abundance assuming a conventional radiation– dominated universe, and derive the lower bound on the initial temperature T0. In Sec. 3 we discuss the case where the pre–BBN expansion rate is allowed to depart from the standard one. Using approximate analytic formulae for the predicted WIMP relic density for this scenario, we derive constraints on the early expansion parameter. Finally, Sec. 4 is devoted to summary and conclusions. 2 Relic Abundance in the Radiation–Dominated Universe We start the discussion of the relic density nχ of stable or long–lived particles χ by reviewing the structure of the Boltzmann equation which describes their creation and annihilation. The goal of this Section is to find the lowest possible initial temperature of the radiation–dominated universe, assuming that the present relic abundance of cold dark matter is entirely due to thermally produced χ particles. As usual, we will assume that χ is self–conjugate†, χ = χ̄, and that some symme- try, for example R–parity, forbids decays of χ into SM particles; the same symmetry then also forbids single production of χ from the thermal background. However, †The case χ 6= χ̄ differs in a non–trivial way only in the presence of a χ− χ̄ asymmetry, i.e. if nχ 6= nχ̄. the creation and annihilation of χ pairs remains allowed. The time evolution of the number density nχ of particles χ in the expanding universe is then described by the Boltzmann equation [2], + 3Hnχ = −〈σv〉(n2χ − n2χ,eq) , (2) where nχ,eq is the equilibrium number density of χ, and 〈σv〉 is the thermally averaged annihilation cross section multiplied with the relative velocity of the two annihilating χ particles. Finally, the Hubble parameter H = Ṙ/R is the expansion rate of the universe, R being the scale factor in the Friedmann–Robertson–Walker metric. The first (second) term on the right–hand side of Eq.(2) describes the decrease (increase) of the number density due to annihilation into (production from) lighter particles. Eq.(2) assumes that χ is in kinetic equilibrium with standard model particles. It is useful to rewrite Eq.(2) in terms of the scaled inverse temperature x = mχ/T as well as the dimensionless quantities Yχ = nχ/s and Yχ,eq = nχ,eq/s. The entropy density is given by s = (2π2/45)g∗sT 3, where g∗s = i=bosons i=fermions . (3) Here gi denotes the number of intrinsic degrees of freedom for particle species i (e.g. due to spin and color), and Ti is the temperature of species i. Assuming that the universe expands adiabatically, the entropy per comoving volume, sR3, remains constant, which implies + 3Hs = 0 . (4) The time dependence of the temperature is then given by . (5) Therefore the Boltzmann equation (2) can be written as = −〈σv〉s (Y 2χ − Y 2χ,eq) . (6) Thermal production of WIMPs takes place during the radiation–dominated epoch, when the expansion rate is given by , (7) with MPl = 2.4× 1018 GeV being the reduced Planck mass and i=bosons i=fermions . (8) In the following we use Hst to denote the standard expansion rate (7). If the post– inflationary reheat temperature was sufficiently high, WIMPs reached full thermal equilibrium. This remains true for temperatures well below mχ. We can therefore use the non–relativistic expression for the χ equilibrium number density, nχ,eq = gχ e−mχ/T . (9) In the absence of non–thermal production mechanisms, nχ ≤ nχ,eq at early times. The annihilation rate Γ = nχ〈σv〉 then depends exponentially on T , and thus drops more rapidly with decreasing temperature than the expansion rate Hst of Eq.(7) does. When the annihilation rate falls below the expansion rate, the number density of WIMPs ceases to follow its equilibrium value and is frozen out. For T ≪ mχ the annihilation cross section can often (but not always [5]) be approximated by a non–relativistic expansion in powers of v2. Its thermal average is then given by 〈σv〉 = a + b〈v2〉+O(〈v4〉) = a+ 6b . (10) In this standard scenario, the following approximate formula has been shown [4, 2, 5] to accurately reproduce the exact (numerically calculated) relic density: Yχ,∞ ≡ Yχ(x → ∞) ≃ 1.3 mχMPl g∗(xF )(a/xF + 3b/x , (11) with xF = mχ/TF , TF being the decoupling temperature. For WIMPs, xF ≃ 22. Here we assume g∗ ≃ g∗s and dg∗/dx ≃ 0. It is useful to express the χ mass density as Ωχ = ρχ/ρc, ρc = 3H Pl being the critical density of the universe. The present relic mass density is then given by ρχ = mχnχ,∞ = mχs0Yχ,∞; here s0 ≃ 2900 cm−3 is the present entropy density. Eq.(11) then leads to 2 = 2.7× 1010 Yχ,∞ 100 GeV ≃ 8.5× 10 −11 xF GeV g∗(xF )(a+ 3b/xF ) , (12) where h ≃ 0.7 is the scaled Hubble constant in units of 100 km sec−1 Mpc−1. We defer further discussions of this expression to Sec. 3, where scenarios with modified expansion rate are analyzed. Note that in the standard scenario leading to Eq.(12), the present χ relic density is inversely proportional to its annihilation cross section and has no dependence on the reheat temperature. Recall that this result depends on the assumption that the highest temperature in the post–inflationary radiation dominated epoch, which we denote by T0, exceeded TF significantly. On the other hand, if T0 was too low to fully thermalize WIMPs, the final result for Ωχ will depend on T0. In particular, if WIMPs were thermally produced in a com- pletely out–of–equilibrium manner starting from vanishing initial abundance during the radiation–dominated era, such that WIMP annihilation remains negligible, the present relic abundance is given by [14] Y0(x → ∞) ≃ 0.014 g2χg−3/2∗ mχMPle−2x0x0 . (13) Note that the final abundance depends exponentially on T0, and increases with increasing cross section. In in–between cases where WIMPs are not completely thermalized but WIMP annihilation can no longer be neglected, we have shown [14] that re–summing the first correction term δ enables us to reproduce the full temperature dependence of the density of WIMPs: 1− δ/Y0 ≡ Y1,r . (14) Here δ < 0 describes the annihilation of WIMPs produced according to Eq.(13): δ(x → ∞) ≃ −2.5 × 10−4 g4χg−5/2∗ m3χM3Ple−4x0x0 . (15) Since δ is proportional to the third power of the cross section, the re–summed ex- pression Y1,r is inversely proportional to the cross section for large cross section. In ref.[14] we have shown that this feature allows the approximation (14) to be smoothly matched to the standard result (12). Not surprisingly, as long as we only consider thermal χ production, decreasing T0 can only reduce the final χ relic density. With the help of these results, we can explore the dependence of the χ relic density on T0 as well as on the annihilation cross section. Some results are shown in Fig. 1, where we take (a) a 6= 0, b = 0, and (b) a = 0, b 6= 0. We choose Yχ(x0) = 0, mχ = 100 GeV, gχ = 2 and g∗ = 90. The results depicted in this Figure can be understood as follows. For small T0, i.e. large x0, Eq.(13) is valid, leading to a very strong dependence of Ωχh 2 on x0. 10−10 10−9 10−8 10−7 a (GeV−2 ) b = 0 10−310−4 10−9 10−8 10−7 10−6 b (GeV−2 ) a = 0 10−310−4 Figure 1: Contour plots of the present relic abundance Ωχh2. Here we take (a) a 6= 0, b = 0, and (b) a = 0, b 6= 0. We choose Yχ(x0) = 0, mχ = 100 GeV, gχ = 2, g∗ = 90. The shaded region corresponds to the WMAP bound on the cold dark matter abundance, 0.08 < ΩCDMh 2 < 0.12 (95% C.L.). Recall that in this case the relic density is proportional to the cross section. In this regime one can reproduce the relic density (1) with quite small annihilation cross section, a + 6b/x0 <∼ 10−9 GeV−2, for some narrow range of initial temperature, x0 <∼ 22.5. Note that this allows much smaller annihilation cross sections than the standard result, at the cost of a very strong dependence of the final result on the initial temperature T0. In this Section we set out to derive a lower bound on T0. In this regard the region of parameter space described by Eq.(13) is not optimal. Increasing the χ annihilation cross section at first allows to obtain the correct relic density for larger x0, i.e. smaller T0. However, the correction δ then quickly increases in size; as noted earlier, once |δ| > Y0 a further increase of the cross section will lead to a decrease of the final relic density. The lower bound on T0 is therefore saturated if Ωχh 2 as a function of the cross section reaches a maximum. From Fig.1 we read off T0 ≥ mχ/23 , (16) if we require Ωχh 2 to fall in the range (1). We just saw that in the regime where this bound is saturated, the final relic den- sity is (to first order) independent of the annihilation cross section, ∂(Ωχh 2)/∂〈σv〉 = 0. If T0 is slightly above the absolute lower bound (16), the correct relic density can therefore be obtained for a rather wide range of cross sections. For example, if x0 = 22.5, the entire range 3 × 10−10 GeV−2 <∼ a <∼ 2 × 10−9 GeV −2 is allowed. Of course, the correct relic density can also be obtained in the standard scenario of (arbitrarily) high T0, if a + 3b/22 falls within ∼ 20% of 2× 10−9 GeV−2. 3 Relic Abundance for Modified Expansion Rate In this section we discuss the calculation of the WIMP relic density nχ in modi- fied cosmological scenarios where the expansion parameter of the pre–BBN universe differed from the standard value Hst of Eq.(7). For the most part we will assume that WIMPs have been in full thermal equilibrium. Various cosmological models predict a non–standard early expansion history [22, 23, 24, 25]. Here we analyze to what extent the relic density of WIMP Dark Matter can be used to constrain the Hubble parameter during the epoch of WIMP decoupling. As long as we assume large T0 we can use a modification of the standard treatment [4, 2] to estimate the relic density for given annihilation cross section and expansion rate. We will show that the resulting approximate solutions again accurately reproduce the numerically evaluated relic abundance. Let us introduce the modification parameter A(x), which parameterizes the ratio of the standard value Hst(x) to the assumed H(x): A(x) = Hst(x) . (17) Note that A > 1 means that the expansion rate is smaller than in standard cos- mology. Allowing for this modified expansion rate, the Boltzmann equation (6) is altered to G(x)mχMPl 〈σv〉A(x) Y 2χ − Y 2χ,eq , (18) where G(x) = . (19) Following refs.[4, 2], we can obtain an approximate solution of this equation by considering the differential equation for ∆ = Yχ − Yχ,eq. For temperatures higher than the decoupling temperature, Yχ tracks Yχ,eq very closely and the ∆ 2-term can be ignored: ≃ −dYeq − 4π√ mχMPl G(x)〈σv〉A(x) (2Yχ,eq∆) . (20) Here dYχ,eq/dx ≃ −Yχ,eq for x ≫ 1. In order to keep |∆| small, the derivative d∆/dx must also be small, which implies ∆ ≃ x 90)mχMPlG(x)〈σv〉A(x) . (21) This solution is used down to the freeze–out temperature TF , defined via ∆(xF ) = ξYχ,eq(xF ) , (22) where ξ is a constant of order of unity. This leads to the following expression: xF = ln ξmχMPlgχ 〈σv〉A(x) xg∗(x) , (23) which can e.g. be solved iteratively. In our numerical calculations we will choose 2− 1 [4, 2]. On the other hand, for low temperatures (T < TF ), the production term ∝ Y 2χ,eq in Eq.(18) can be ignored. In this limit, Yχ ≃ ∆, and the solution of Eq.(18) is given ∆(xF ) ∆(x → ∞) mχMPlI(xF ) , (24) where the annihilation integral is defined as I(xF ) = G(x)〈σv〉A(x) . (25) Assuming ∆(x → ∞) ≪ ∆(xF ), the final relic abundance is Yχ,∞ ≡ Yχ(x → ∞) = 90)mχMPlI(xF ) . (26) Plugging in numerical values for the Planck mass and for today’s entropy density, the present relic density can thus be written as 8.5× 10−11 I(xF ) GeV . (27) The constraint (1) therefore corresponds to the allowed range for the annihilation integral 7.1× 10−10 GeV−2 < I(xF ) < 1.1× 10−9 GeV−2 . (28) The standard formula (12) for the final relic density is recovered if A(x) is set to unity and G(x) is replaced by the constant g∗(xF ). The further discussion is simplified if we use the normalized temperature z = T/mχ ≡ 1/x, rather than x. Phenomenologically A(z) can be any function subject to the condition that A(z) approaches unity at late times in order not to contradict the successful predictions of BBN. We need to know A(z) only for the interval from around the freeze-out to BBN: zBBN ∼ 10−5 − 10−4 <∼ z <∼ zF ∼ 1/20. This suggests a parameterization of A(z) in terms of a power series in (z − zF,st): A(z) = A(zF,st) + (z − zF,st)A′(zF,st) + (z − zF,st)2A′′(zF,st) , (29) where zF,st is the normalized freeze–out temperature in the standard scenario and a prime denotes a derivative with respect to z. The ansatz (29) should be of quite general validity, so long as the modification of the expansion rate is relatively modest. This suits our purpose, since we wish to find out what constraints can be derived on the expansion history if standard cosmology leads to the correct WIMP relic density. We further introduce the variable k = A(z → 0) = A(zF,st)− zF,stA′(zF,st) + z2F,stA ′′(zF,st) , (30) which describes the modification parameter at late times. Since zBBN is almost zero, we treat k as the modification parameter at the era of BBN in this paper.‡ Deviations from k = 1 are conveniently discussed in terms of the equivalent number of light neutrino degrees of freedom Nν . BBN permits that the number of neutrinos differs from the standard model value Nν = 3 by δNν = 1.5 or so [26]. We therefore take the uncertainty of k to be 20%. In the following we treat A(zF,st), A ′(zF,st) and k as free parameters; A′′(zF,st) is then a derived quantity. Note that we allow A(z) to cross unity, i.e. to switch from an expansion that is faster than in standard cosmology to a slower expansion or vice versa. This is illustrated in Fig. 2, which shows examples of possible evolutions of A(z) as function of z for zF = 0.05. Here we take k = 1.2 (left frame) and k = 0.8 (right). In each case we consider scenarios with A(zF ) = 1.3 (slower expansion at TF than in standard cosmology) as well as A(zF ) = 0.7 (faster expansion); moreover, we allow the change of A at z = zF to be either positive or negative. However, we insist that H remains positive at all times, i.e. A(z) must not cross zero. This excludes scenarios with very large positive A′(zF,st), which would lead to A < 0 at some z < zF . Similarly, ‡Presumably the Hubble expansion rate has to approach the standard rate even more closely for T < TBBN. However, since all WIMP annihilation effectively ceased well before the onset of BBN, this epoch plays no role in our analysis. 0 0.01 0.02 0.03 0.04 0.05 z = T/mχ k = 1.2 A=1.3, A’=−3 A=1.3, A’= 9 A=0.7, A’=−3 A=0.7, A’= 9 0 0.01 0.02 0.03 0.04 0.05 z = T/mχ k = 0.8A=1.3, A’=−3 A=1.3, A’= 9 A=0.7, A’=−3 A=0.7, A’= 9 Figure 2: Examples of possible evolutions of the modification parameter A(z) as function of z for zF = 0.05. Here we take k = 1.2 (left frame) and k = 0.8 (right). In each frame we choose A(zF ) = 1.3, A ′(zF ) = −3 (thick line), A(zF ) = 1.3, A′(zF ) = 9 (dashed), A(zF ) = 0.7, A′(zF ) = −3 (dotted), A(zF ) = 0.7, A′(zF ) = 9 (dot–dashed). demanding that our ansatz (29) remains valid for some range of temperatures above TF excludes scenarios with very large negative A ′(zF,st). We will come back to this point shortly. Eq.(23) shows that zF 6= zF,st (xF 6= xF,st) if A(zF ) 6= 1. This is illustrated by Fig. 3, which shows the difference between xF and xF,st in the (A(zF,st), A ′(zF,st)) plane. Here we take parameters such that Ωχh 2 = 0.099 in the standard cosmology, which is recovered for A(zF,st) = 1, A ′(zF,st) = 0. Due to the logarithmic dependence on A, xF (or zF ) differs by at most a few percent from its standard value if A(zF,st) is O(1). Since TF only depends on the expansion rate at TF , it is essentially insensitive to the derivative A′(zF,st). In our treatment the modification of the expansion parameter affects the WIMP relic density mostly via the annihilation integral (25). In terms of the normalized temperature z, the latter can be rewritten as I(zF ) = dz G(z)〈σv〉A(z) . (31) One advantage of the expansion (29) is that this integral can be evaluated analyti- cally: I(zF ) ≃ G(zF ) k(azF + 3bz F ) + (A ′(zF,st)− zF,stA′′(zF,st)) z2F + 2bz A′′(zF,st) z3F + . (32) 0.6 0.8 1 1.2 1.4 A(zF,st ) xF − x F,st a = 2.0*10−9 GeV −2 b = 0 k = 1 xF,st = 22.0 −0.4 −0.2 0 0.2 0.4 Figure 3: Contour plot of xF −xF,st in the (A(zF,st), A′(zF,st)) plane. Here we take a = 2.0×10−9 GeV−2, b = 0, mχ = 100 GeV, gχ = 2, g∗ = 90 (constant) and k = 1. This parameter set produces xF,st = 22.0 and Ωχh 2 = 0.099 for the standard approximation. 0.6 0.8 1 1.2 1.4 A(zF,st ) 2(approx) / Ωχh 2(exact) 0.997 0.998 0.999 1.000 a = 2.0*10−9 GeV−2 b = 0 k = 1 0.6 0.8 1 1.2 1.4 A(zF,st) 2(approx) / Ωχh 2(exact) 0.996 0.998 1.000 1.002 1.004 a = 0 b = 1.5*10−8 GeV−2 k = 1 Figure 4: Ratio of the analytic result of the relic density to the exact value in the (A(zF,st), A′(zF,st)) plane for a = 2.0× 10−9 GeV−2, b = 0 (left frame) and for a = 0, b = 1.5× 10−8 GeV−2 (right). The other parameters are as in Fig. 3. Here we have assumed that G(z) varies only slowly. Before proceeding, we first have to convince ourselves that the analytic treatment developed in this Section still works for A 6= 1. This is demonstrated by Fig. 4, which shows the ratio of the analytic solution obtained from Eqs. (27) and (32) to the exact one, obtained by numerically integrating the Boltzmann equation (18), assuming constant g∗. We see that our analytical treatment is accurate to better than 1%, and can thus safely be employed in the subsequent analysis. 0.6 0.8 1 1.2 1.4 A(zF,st ) a = 2.0*10−9 GeV −2 b = 0 k = 1 0.6 0.8 1 1.2 1.4 A(zF,st ) a = 0 b = 1.5*10−8 GeV−2 k = 1 0.6 0.8 1 1.2 1.4 A(zF,st ) a = 2.0*10−9 GeV−2 b = 0 k = 1.2 0.6 0.8 1 1.2 1.4 A(zF,st ) a = 2.0*10−9 GeV−2 b = 0 k = 0.8 Figure 5: Contour plots of the relic abundance in the (A(zF,st), A′(zF,st)) plane. Here we choose (a) a = 2.0×10−9 GeV−2, b = 0, k = 1; (b) a = 0, b = 1.5×10−8 GeV−2, k = 1; (c) a = 2.0×10−9 GeV−2, b = 0, k = 1.2; (d) a = 2.0× 10−9 GeV−2, b = 0, k = 0.8. The other parameters are as in Fig. 3. We are now ready to analyze the impact of the modified expansion rate on the WIMP relic density. In Fig. 5, we show contour plots of Ωχh 2 in the (A(zF,st), A′(zF,st)) plane. Recall that large (small) values of A correspond to a small (large) expansion rate. Since a smaller expansion rate allows the WIMPs more time to annihilate, A > 1 leads to a reduced WIMP relic density, whereas A < 1 means larger relic density, if the cross section is kept fixed. However, unlike the freeze–out temperature, the annihilation integral is sensitive to A(z) for all z ≤ zF . Note that A′(zF,st) > 0 implies A(z) < A(zF,st) for z < zF,st ≃ zF . A positive first derivative, A′(zF,st) > 0, can therefore to some extent compensate for A(zF,st) > 1; analogously, a negative first derivative can compensate for A(zF,st) < 1. This explains the slopes of the curves in Fig. 5. Recall also that A′(zF,st) = 0 does not imply a constant modification factor A(z); rather, the term ∝ A′′(zF,st) in Eq.(29) makes sure that A approaches k as z → 0. This explains why a change of A by some given percentage leads to a smaller relative change of Ωχh 2, as can be seen in the Figure. This also illustrates the importance of ensuring appropriate (near–standard) expansion rate in the BBN era. Finally, since the expansion rate at late times is given by Hst/k, bigger (smaller) values of k imply that the χ relic density is reduced (enhanced). Fig. 5 shows that we need additional physical constraints if we want to derive bounds on A(zF,st) and/or A ′(zF,st). Once the annihilation cross section is known, the requirement (1) will single out a region in the space spanned by our three new parameters (including k) which describe the non–standard evolution of the universe, but this region is not bounded. Such additional constraints can be derived from the requirement that the Hubble parameter should remain positive throughout the epoch we are considering. As noted earlier, requiring H > 0 for all T < TF,st leads to an upper bound on A′(zF,st); explicitly, A′(zF,st) < A(zF,st) + kA(zF,st) zF,st . (33) On the other hand, a lower bound on A′(zF,st) is obtained from the condition that the modified Hubble parameter is positive between the highest temperature Ti where the ansatz (29) holds and TF,st: A′(zF,st) > − zi − zF,st 2− zi zF,st A(zF,st) + k zF,st , (34) for (1− zF,st/zi)2k < A(zF,st), and A′(zF,st) > A(zF,st)− kA(zF,st) zF,st , (35) for A(zF,st) < (1− zF,st/zi)2k, where zi = Ti/mχ. Evidently the lower bound on A′(zF,st) depends on zi, i.e. on the maximal tem- perature where we assume our ansatz (29) to be valid. In ref.[14] we have shown that in standard cosmology (A ≡ 1) essentially full thermalization is already achieved for xi <∼ xF − 0.5, even if nχ(xi) = 0. However, it seems reasonable to demand that H should remain positive at least up to xi = xF−(a few). In Fig. 6 we therefore show 0 1 2 3 4 5 6 7 A(zF,st ) a = 2.0*10−9 GeV−2 b = 0 k = 1 xF,st − x i = 4 xF,st − x i = 10 0 1 2 3 4 5 6 7 A(zF,st ) a = 0 b = 1.5*10−8 GeV−2 k = 1 xF,st − x i = 4 xF,st − x i = 10 Figure 6: Contour plots of the relic abundance Ωχh2 in the (A(zF,st), A′(zF,st)) plane. The dashed line corresponds to the upper bound on A′(zF,st). The dotted lines correspond to the lower bounds calculated for xF,st − xi = 4, 10. We take a = 2.0 × 10−9 GeV−2, b = 0 (left frame) and a = 0, b = 1.5× 10−8 GeV−2 (right frame). The other parameters are as in Fig 3. the physical constraints on the modification parameter A(z) for xF,st − xi = 4, 10 and k = 1. The dashed and dotted lines correspond to the upper and lower bounds on A′(zF,st), described by Eq.(33) and Eqs.(34), (35), respectively. We see that when xF,st − xi = 4 the allowed region is 0.4 <∼ A(zF,st) <∼ 6.5 with −60 <∼ A′(zF,st) <∼ 400 for b = 0 (left frame), and 0.4 <∼ A(zF,st) <∼ 4.5 with −60 <∼ A′(zF,st) <∼ 300 for a = 0 (right frame). When xF,st − xi = 10, the lower bounds are altered to 0.6 <∼ A(zF,st), −10 <∼ A′(zF,st) for b = 0 (left frame), and 0.6 <∼ A(zF,st), −20 <∼ A′(zF,st) for a = 0 (right frame). Note that the lower bounds on A(zF,st), which depend only weakly on xi so long as it is not very close to xF , are almost the same in both cases, which also lead to very similar relic densities in standard cosmology. However, the two upper bounds differ significantly. The reason is that large values of A(zF,st), i.e. a strongly suppressed Hubble expansion, require some degree of finetuning: One also has to take large positive A′(zF,st), so that A becomes smaller than one for some range of z values below zF , leading to an annihilation integral of similar size as in standard cosmology. Since the b−terms show different zF dependence in the annihilation in- tegral (32), the required tuning between A(zF,st) and A ′(zF,st) is somewhat different than for the a−terms, leading to a steeper slope of the allowed region. This allowed region therefore saturates the upper bound (33) on the slope for somewhat smaller A(zF,st). The effect of this tuning can be seen by analyzing the special case where A′′(zF,st) = 0 0.5 1 1.5 2 A(zF,st ) a = 2.0*10−9 GeV−2 b = 0 4 0.08 0 0.5 1 1.5 2 A(zF,st ) a = 0 b = 1.5*10−8 GeV−2 Figure 7: Contour plots of the relic abundance Ωχh2 in the (A(zF,st), k) plane for A′′(zF,st) = 0. The dotted lines correspond to the lower bounds of A(zF,st), calculated for xF,st − xi = 4, 10. We take a = 2.0 × 10−9 GeV−2, b = 0 (left frame) and a = 0, b = 1.5 × 10−8 GeV−2 (right frame). The other parameters are as in Fig 3. 0. The modification parameter then reads A(z) = A(zF,st)− k zF,st z + k . (36) Note that A is now a monotonic function of z, making large cancellations in the annihilation integral impossible. Imposing that A(z) remains positive for zF,st ≤ z ≤ zi leads to the lower limit A(zF,st) > zF,st k . (37) There is no upper bound, since A(z) is now automatically positive for all z ∈ [0, zF,st] if A(zF,st) and A(0) ≡ k are both positive. Fig. 7 shows constraints on the relic abundance in the (A(zF,st), k) plane for A ′′(zF,st) = 0. The dotted lines correspond to the lower bounds (37) on A(zF,st) for xF,st − xi = 4, 10. As noted earlier, k is constrained by the BBN bound. This leads to the bounds 0.5 <∼ A(zF,st) <∼ 1.8 for b = 0 (left frame), and 0.65 <∼ A(zF,st) <∼ 1.6 for a = 0 (right frame), when xF,st − xi = 10. Evidently the constraints now only depend weakly on whether the a− or b−term dominates in the annihilation cross section. As the initial temperature is lowered, the impact of the constraint (37) disappears. So far we have assumed in this Section that the reheat temperature was high enough for WIMPs to have attained full thermal equilibrium. If this was not the case, the initial temperature as well as the suppression parameter affects the final 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 A(zF,st ) a = 2.0*10−9 GeV−2 b = 0 k = 1, A’’(zF,st ) = 0 0.080.12 Figure 8: Contour plot of the relic abundance Ωχh2 in the (A(zF,st), x0) plane. Here we choose a = 2.0 × 10−9 GeV−2, b = 0, k = 1, A′′(zF,st) = 0. The other parameters are as in Fig. 3. The shaded region corresponds to the WMAP bound on the cold dark matter abundance, 0.08 < ΩCDMh 2 < 0.12 (95% C.L.). relic abundance. Here we show that the lower bound on the reheat temperature derived in the previous Section survives even in scenarios with altered expansion history as long as WIMPs were only produced thermally. This can be understood from the observation that the Boltzmann equation with modified expansion rate is obtained by replacing 〈σv〉 in the radiation–dominated case by 〈σv〉A. Increasing (decreasing) A therefore has the same effect as an in- crease (decrease) of the annihilation cross section. Since the lower bound on T0 was independent of σ (more exactly: we quoted the absolute minimum, for the optimal choice of σ), we expect it to survive even if A(z) 6= 1 is introduced. This is borne out by Fig. 8, which shows the relic abundance Ωχh 2 in the (A(zF,st), x0) plane for the simplified case A ′′(zF,st) = 0; similar results can be obtained for the more general ansatz (29). The shaded region corresponds to the bound (1) on the cold dark matter abundance. As expected, this figure looks similar to Fig. 1 if the annihilation cross section in Fig. 1 is replaced by A(zF,st). The maximal value of x0 consistent with the WMAP data remains around 23 even in these scenarios with modified expansion rate. Fig. 8 also shows that A(zF,st) ≪ 1 is allowed for some narrow range of initial temperature T0 < TF . This is analogous to the low cross section branch in Fig. 1. 4 Summary and Conclusions In this paper we have investigated the relic abundance of WIMPs χ, which are non- relativistic long–lived or stable particles, in non–standard cosmological scenarios. One motivation for studying such scenarios is that they allow to reproduce the ob- served Dark Matter density for a large range of WIMP annihilation cross sections. Our motivation was the opposite: we wanted to quantify the constraints that can be obtained on parameters describing the early universe, under the assumption that thermally produced WIMPs form all Dark Matter. Wherever necessary, we fixed particle physics quantities such that standard cosmology yields the correct relic den- sity. Specifically, we first considered scenarios with low post–inflationary reheat tem- perature, while keeping all other features of standard cosmology (known particle content and Hubble expansion parameter during WIMP decoupling; no late entropy production; no non–thermal WIMP production channels). If the temperature was so low that WIMPs could not achieve full thermal equilibrium, the dependence of the abundance on the mass and annihilation cross section of the WIMPs is completely different from that in the standard thermal WIMP scenario. In particular, if the maximal temperature T0 is much less than the decoupling temperature TF , nχ re- mains exponentially suppressed. By applying the observed cosmological amount of cold dark matter to the predicted WIMP abundance, we therefore found the lower bound of the initial temperature T0 >∼ mχ/23. One might naively think that this bound could be evaded by choosing a sufficiently large WIMP production (or anni- hilation) cross section. However, increasing this cross section also reduces TF . For sufficiently large cross section one therefore has TF ≤ T0 again; in this regime the relic density drops with increasing cross section. Our lower bound is the minimal T0 required for any cross section; once the latter is known, the bound on T0 might be slightly sharpened. As a by–product, we also noted that the final relic density depends only weakly on the annihilation cross section if T0 is slightly above this lower bound. We also investigated the effect of a non–standard expansion rate of the universe on the WIMP relic abundance. In general the abundance of thermal relics depends on the ratio of the annihilation cross section to the expansion rate; the latter is determined unambiguously in standard cosmology. We found that even for non– standard Hubble parameter the relic abundance can be calculated accurately in terms of an annihilation integral, very similar to the case of standard cosmology. We assumed that the WIMP annihilation cross section is such that the standard scenario yields the observed relic density, and parameterized the modification of the Hubble parameter as a quadratic function of the temperature. In this analysis it is crucial to make sure that at low temperatures the Hubble parameter approaches its standard value to within ∼ 20%, as required for the success of Big Bang Nucleosynthesis (BBN). Keeping the annihilation cross section fixed and allowing a 20% variation in the relic density, roughly corresponding to the present “2σ” band, we found that the expansion of the universe at T = TF might have been more than two times faster, or more than six times slower, than in standard cosmology. These large variations of H(TF ) can only be realized by finetuning of the parameters describing H(T < TF ). However, even if we forbid such finetuning by choosing a linear parameterization for the modification of the expansion rate, a 20% variation of Ωχh 2 allows a difference between H(TF ) and its standard expectation of more than 50%. This relatively weak sensitivity of the predicted Ωχh 2 on H(TF ) is due to the fact that the relic density depends on all H(T < TF ); as stressed above, we have to require that H(T ≪ TF ) approaches its standard value to within ∼ 20%. The fact that determining Ωχh2 will yield relatively poor bounds on H(TF ) remains true even if the annihilation cross section is such that a non–standard behavior of H(T ) is required for obtaining the correct χ relic density. Finally, we showed that the absolute lower bound on the temperature required for thermal χ production is unaltered by allowing H(T ) to differ from its standard value. Of course, in order to draw the conclusions derived in this article, we need to convince ourselves that WIMPs do indeed form (nearly) all Dark Matter. This requires not only the detection of WIMPs, e.g. in direct search experiments; we also need to show that their density is in accord with the local Dark Matter density derived from astronomical observations. To that end, the cross sections appearing in the calculation of the detection rate need to be known independently. This can only be done in the framework of a definite theory, using data from collider experiments. For example, in order to determine the cross section for the direct detection of supersymmetric WIMPs, one needs to know the parameters of the supersymmetric neutralino, Higgs and squark sectors [3]. We also saw that inferences about H(TF ) can only be made if the WIMP annihilation cross section is known. This again requires highly non–trivial analyses of collider data [27], as well as a consistent overall theory. 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D63, 035008 (2001), hep–ph/0007202; B. C. Allanach, G. Bélanger, F. Boudjema and A. Pukhov, JHEP 0412, 020 (2004), hep–ph/0410091; M. M. Nojiri, G. Polesello and D. R. Tovey, JHEP, 0603, 063 (2006), hep–ph/0512204; E. A. Baltz, M. Battaglia, M. E. Peskin and T. Wizansky, Phys. Rev. D74, 103521 (2006), hep-ph/0602187. http://arxiv.org/abs/astro--ph/0408033 http://arxiv.org/abs/astro--ph/0302554 http://arxiv.org/abs/hep--ph/0007202 http://arxiv.org/abs/hep--ph/0410091 http://arxiv.org/abs/hep--ph/0512204 http://arxiv.org/abs/hep-ph/0602187 Introduction Relic Abundance in the Radiation–Dominated Universe Relic Abundance for Modified Expansion Rate Summary and Conclusions

We investigate the relic density n_\chi of non-relativistic long-lived or stable particles \chi in non-standard cosmological scenarios. We calculate the relic abundance starting from arbitrary initial temperatures of the radiation-dominated epoch, and derive the lower bound on the initial temperature T_0 \geq m_\chi/23, assuming that thermally produced \chi particles account for the dark matter energy density in the universe; this bound holds for all \chi annihilation cross sections. We also investigate cosmological scenarios with modified expansion rate. Even in this case an approximate formula similar to the standard one is capable of predicting the final relic abundance correctly. Choosing the \chi annihilation cross section such that the observed cold dark matter abundance is reproduced in standard cosmology, we constrain possible modifications of the expansion rate at T \sim m_\chi/20, well before Big Bang nucleosynthesis.

Introduction One of the most notable recent developments in cosmology is the precise determina- tion of cosmological parameters from observations of the large–scale structure of the universe, most notably by the Wilkinson Microwave Anisotropy Probe (WMAP). In particular, the accurate determination of the non–baryonic cold Dark Matter (DM) density [1], 0.08 < ΩCDMh 2 < 0.12 (95% C.L.) , (1) has great influence on particle physics models which possess dark matter candidates [2, 3]. The requirement that the predicted DM density falls in the range (1) is a powerful tool for discriminating between various models and for constraining the parameter space of surviving models. Many dark matter candidate particles have been proposed. In particular long– lived or stable weakly interacting massive particles (WIMPs) with weak–scale masses are excellent candidates. In standard cosmology WIMPs decoupled from the thermal background during the radiation–dominated epoch after inflation. In this framework convenient and accurate analytic approximate solutions for the relic abundance have been derived [4, 5]. One of the best motivated candidates for WIMPs is the lightest neutralino in supersymmetric (SUSY) models. Assuming that the neutralino is the lightest supersymmetric particle (LSP) stabilized due to R–parity, its relic abundance has been extensively discussed [3]. Similar analyses have also been performed for other WIMPs whose existence is postulated in other extensions of the Standard Model (SM) of particle physics. In many cases the cosmologically favored parameter space of WIMP models can be directly tested at the CERN Large Hadron Collider (LHC) in a few years [6]. The same parameter space often also leads to rates of WIMP interactions with matter within the sensitivity of near–future direct DM detection experiments. This discussion shows that we are now entering an interesting time where the standard cosmological scenario can be examined by experiments at high–energy col- liders as well as DM searches [7]. In this respect we should emphasize that the relic abundance of thermally produced WIMPs depends not only on their annihila- tion cross section, which can be determined by particle physics experiments, but in general is also very sensitive to cosmological parameters during the era of WIMP production and annihilation. Of particular importance are the initial temperature T0 at which WIMPs began to be thermally produced, and the expansion rate of the universe H . In the standard cosmological scenario, the expansion rate is uniquely determined through the Friedmann equation of general relativity. In this scenario the density of WIMPs with massmχ followed its equilibrium value until the freeze–out temperature TF ≃ mχ/20. Below TF , interactions of WIMPs are decoupled, and thus the present density is independent of T0 as long as T0 > TF . It should be noted that in non–standard scenarios the relic density can be larger or smaller than the value in the standard scenario. One example is the case where T0 is smaller than or comparable to TF , which can be realized in inflationary models with low reheat temperature. Since in many models the inflationary energy scale must be much higher than mχ in order to correctly predict the density perturbations [8], the standard assumption T0 > TF is not unreasonable. On the other hand, the constraint on the reheat temperature from Big Bang Nucleosynthesis (BBN) is as low as T0 >∼ MeV [9, 10]. From the purely phenomenological viewpoint, it is therefore also interesting to investigate the production of WIMPs in low reheat temperature scenarios [9, 11, 12, 13, 14]. The standard scenario also assumes that entropy per comoving volume is con- served for all temperatures T ≤ TF . Late entropy production can dilute the predicted relic density [15, 16]. The reason is that the usual calculation actually predicts the ratio of the WIMP number density to the entropy density. On the other hand, if late decays of a heavier particle non–thermally produce WIMPs in addition to the usual thermal production mechanism, the resulting increase of the WIMP density competes with the dilution caused by the decay of this particle into radiation, which increases the entropy density [17, 18, 19, 20, 13, 21]. Another example of a non–standard cosmology changing the WIMP relic density is a modified expansion rate of the universe. This might be induced by an anisotropic expansion [16], by a modification of general relativity [16, 22], by additional contri- butions to the total energy density from quintessence [23], by branes in a warped geometry [24], or by a superstring dilaton [25]. These examples show that, once the WIMP annihilation cross section is fixed, with the help of precise measurements of the cold dark matter density we can probe the very early stage of the universe at temperatures of O(mχ/20) ∼ 10 GeV. This is reminiscent of constraining the early evolution of the universe at T = O(100) keV ∗Note that TF can be formally defined in the standard way even if T0 < TF . In this case WIMPs never were in full equilibrium, and correspondingly never “froze out”. using the primordial abundances of the light elements produced by BBN. The goal of this paper is to investigate to what extent the constraint (1) on the WIMP relic abundance might allow us to derive quantitative constraints on modifications of standard cosmology. So far the history of the universe has been established by cosmological observations as far back as the BBN era. In this paper we try to derive bounds on cosmological parameters relevant to the era before BBN. Rather than studying specific extensions of the standard cosmological scenario, we simply parameterize deviations from the standard scenario, and attempt to derive constraints on these new parameters. Since we only have the single constraint (1), for the most part we only allow a single quantity to differ from its standard value. We expect that varying two quantities simultaneously will allow to get the right relic density for almost any WIMP annihilation cross section. This has been shown explicitly in [13] for the case that both late entropy production and non–thermal WIMP production are considered, even if both originate from the late decay of a single scalar field. We first analyze the dependence of the WIMP abundance on the initial temper- ature T0 of the conventional radiation–dominated epoch. We showed in [14] that for fixed T0 the predicted WIMP relic density reaches a maximum as the annihilation cross section is varied from very small to very large values. A small annihilation cross section corresponds to a large TF > T0; in this case the relic density increases with the annihilation cross section, since WIMP production from the thermal plasma is more important than WIMP annihilation. On the other hand, increasing this cross section reduces TF ; once TF < T0 a further increase of the cross section leads to smaller relic densities since in this case WIMPs continue to annihilate even after the temperature is too low for WIMP production. Here we turn this argument around, and derive the lower bound on T0 ≥ mχ/23 under the assumption that all WIMPs are produced thermally. Note that we do not need to know the WIMP annihilation cross section to derive this bound. We then examine the dependence of the predicted WIMP relic density on the expansion rate in the epoch prior to BBN, where we allow the Hubble parameter to depart from the standard value. The standard method of calculating the thermal relic density [2, 4] is found to be still applicable in this case. Our working hypothesis here is that the standard prediction for the Hubble expansion rate is essentially correct, i.e. that the true expansion rate differs by at most a factor of a few from the standard prediction. We then simply employ a generic Taylor expansion for the temperature dependence of this modification factor; note that the success of standard BBN indicates that this factor cannot deviate by more than ∼ 20% from unity at low temperatures, T <∼ 1 MeV. Similarly, we assume that the WIMP annihilation cross section has been determined (from experiments at particle colliders) to have the value required in standard cosmology. Our approach is thus quite different from that taken in [7], where present upper bounds on the fluxes of WIMP annihilation products are used to place upper bounds on the Hubble expansion rate during WIMP decoupling. The advantage of their approach is that no prior assumption on the WIMP annihilation cross section needs to be made, whereas we assume a cross section that reproduces the correct relic density in the standard scenario. On the other hand, the bounds derived in refs.[7] are still quite weak, allowing the Hubble parameter to exceed its standard prediction by a factor >∼ 30; moreover, no lower bound on H can be derived in this fashion. The remainder of this paper is organized as follows: In Sec. 2 we will briefly review the calculation of the WIMP relic abundance assuming a conventional radiation– dominated universe, and derive the lower bound on the initial temperature T0. In Sec. 3 we discuss the case where the pre–BBN expansion rate is allowed to depart from the standard one. Using approximate analytic formulae for the predicted WIMP relic density for this scenario, we derive constraints on the early expansion parameter. Finally, Sec. 4 is devoted to summary and conclusions. 2 Relic Abundance in the Radiation–Dominated Universe We start the discussion of the relic density nχ of stable or long–lived particles χ by reviewing the structure of the Boltzmann equation which describes their creation and annihilation. The goal of this Section is to find the lowest possible initial temperature of the radiation–dominated universe, assuming that the present relic abundance of cold dark matter is entirely due to thermally produced χ particles. As usual, we will assume that χ is self–conjugate†, χ = χ̄, and that some symme- try, for example R–parity, forbids decays of χ into SM particles; the same symmetry then also forbids single production of χ from the thermal background. However, †The case χ 6= χ̄ differs in a non–trivial way only in the presence of a χ− χ̄ asymmetry, i.e. if nχ 6= nχ̄. the creation and annihilation of χ pairs remains allowed. The time evolution of the number density nχ of particles χ in the expanding universe is then described by the Boltzmann equation [2], + 3Hnχ = −〈σv〉(n2χ − n2χ,eq) , (2) where nχ,eq is the equilibrium number density of χ, and 〈σv〉 is the thermally averaged annihilation cross section multiplied with the relative velocity of the two annihilating χ particles. Finally, the Hubble parameter H = Ṙ/R is the expansion rate of the universe, R being the scale factor in the Friedmann–Robertson–Walker metric. The first (second) term on the right–hand side of Eq.(2) describes the decrease (increase) of the number density due to annihilation into (production from) lighter particles. Eq.(2) assumes that χ is in kinetic equilibrium with standard model particles. It is useful to rewrite Eq.(2) in terms of the scaled inverse temperature x = mχ/T as well as the dimensionless quantities Yχ = nχ/s and Yχ,eq = nχ,eq/s. The entropy density is given by s = (2π2/45)g∗sT 3, where g∗s = i=bosons i=fermions . (3) Here gi denotes the number of intrinsic degrees of freedom for particle species i (e.g. due to spin and color), and Ti is the temperature of species i. Assuming that the universe expands adiabatically, the entropy per comoving volume, sR3, remains constant, which implies + 3Hs = 0 . (4) The time dependence of the temperature is then given by . (5) Therefore the Boltzmann equation (2) can be written as = −〈σv〉s (Y 2χ − Y 2χ,eq) . (6) Thermal production of WIMPs takes place during the radiation–dominated epoch, when the expansion rate is given by , (7) with MPl = 2.4× 1018 GeV being the reduced Planck mass and i=bosons i=fermions . (8) In the following we use Hst to denote the standard expansion rate (7). If the post– inflationary reheat temperature was sufficiently high, WIMPs reached full thermal equilibrium. This remains true for temperatures well below mχ. We can therefore use the non–relativistic expression for the χ equilibrium number density, nχ,eq = gχ e−mχ/T . (9) In the absence of non–thermal production mechanisms, nχ ≤ nχ,eq at early times. The annihilation rate Γ = nχ〈σv〉 then depends exponentially on T , and thus drops more rapidly with decreasing temperature than the expansion rate Hst of Eq.(7) does. When the annihilation rate falls below the expansion rate, the number density of WIMPs ceases to follow its equilibrium value and is frozen out. For T ≪ mχ the annihilation cross section can often (but not always [5]) be approximated by a non–relativistic expansion in powers of v2. Its thermal average is then given by 〈σv〉 = a + b〈v2〉+O(〈v4〉) = a+ 6b . (10) In this standard scenario, the following approximate formula has been shown [4, 2, 5] to accurately reproduce the exact (numerically calculated) relic density: Yχ,∞ ≡ Yχ(x → ∞) ≃ 1.3 mχMPl g∗(xF )(a/xF + 3b/x , (11) with xF = mχ/TF , TF being the decoupling temperature. For WIMPs, xF ≃ 22. Here we assume g∗ ≃ g∗s and dg∗/dx ≃ 0. It is useful to express the χ mass density as Ωχ = ρχ/ρc, ρc = 3H Pl being the critical density of the universe. The present relic mass density is then given by ρχ = mχnχ,∞ = mχs0Yχ,∞; here s0 ≃ 2900 cm−3 is the present entropy density. Eq.(11) then leads to 2 = 2.7× 1010 Yχ,∞ 100 GeV ≃ 8.5× 10 −11 xF GeV g∗(xF )(a+ 3b/xF ) , (12) where h ≃ 0.7 is the scaled Hubble constant in units of 100 km sec−1 Mpc−1. We defer further discussions of this expression to Sec. 3, where scenarios with modified expansion rate are analyzed. Note that in the standard scenario leading to Eq.(12), the present χ relic density is inversely proportional to its annihilation cross section and has no dependence on the reheat temperature. Recall that this result depends on the assumption that the highest temperature in the post–inflationary radiation dominated epoch, which we denote by T0, exceeded TF significantly. On the other hand, if T0 was too low to fully thermalize WIMPs, the final result for Ωχ will depend on T0. In particular, if WIMPs were thermally produced in a com- pletely out–of–equilibrium manner starting from vanishing initial abundance during the radiation–dominated era, such that WIMP annihilation remains negligible, the present relic abundance is given by [14] Y0(x → ∞) ≃ 0.014 g2χg−3/2∗ mχMPle−2x0x0 . (13) Note that the final abundance depends exponentially on T0, and increases with increasing cross section. In in–between cases where WIMPs are not completely thermalized but WIMP annihilation can no longer be neglected, we have shown [14] that re–summing the first correction term δ enables us to reproduce the full temperature dependence of the density of WIMPs: 1− δ/Y0 ≡ Y1,r . (14) Here δ < 0 describes the annihilation of WIMPs produced according to Eq.(13): δ(x → ∞) ≃ −2.5 × 10−4 g4χg−5/2∗ m3χM3Ple−4x0x0 . (15) Since δ is proportional to the third power of the cross section, the re–summed ex- pression Y1,r is inversely proportional to the cross section for large cross section. In ref.[14] we have shown that this feature allows the approximation (14) to be smoothly matched to the standard result (12). Not surprisingly, as long as we only consider thermal χ production, decreasing T0 can only reduce the final χ relic density. With the help of these results, we can explore the dependence of the χ relic density on T0 as well as on the annihilation cross section. Some results are shown in Fig. 1, where we take (a) a 6= 0, b = 0, and (b) a = 0, b 6= 0. We choose Yχ(x0) = 0, mχ = 100 GeV, gχ = 2 and g∗ = 90. The results depicted in this Figure can be understood as follows. For small T0, i.e. large x0, Eq.(13) is valid, leading to a very strong dependence of Ωχh 2 on x0. 10−10 10−9 10−8 10−7 a (GeV−2 ) b = 0 10−310−4 10−9 10−8 10−7 10−6 b (GeV−2 ) a = 0 10−310−4 Figure 1: Contour plots of the present relic abundance Ωχh2. Here we take (a) a 6= 0, b = 0, and (b) a = 0, b 6= 0. We choose Yχ(x0) = 0, mχ = 100 GeV, gχ = 2, g∗ = 90. The shaded region corresponds to the WMAP bound on the cold dark matter abundance, 0.08 < ΩCDMh 2 < 0.12 (95% C.L.). Recall that in this case the relic density is proportional to the cross section. In this regime one can reproduce the relic density (1) with quite small annihilation cross section, a + 6b/x0 <∼ 10−9 GeV−2, for some narrow range of initial temperature, x0 <∼ 22.5. Note that this allows much smaller annihilation cross sections than the standard result, at the cost of a very strong dependence of the final result on the initial temperature T0. In this Section we set out to derive a lower bound on T0. In this regard the region of parameter space described by Eq.(13) is not optimal. Increasing the χ annihilation cross section at first allows to obtain the correct relic density for larger x0, i.e. smaller T0. However, the correction δ then quickly increases in size; as noted earlier, once |δ| > Y0 a further increase of the cross section will lead to a decrease of the final relic density. The lower bound on T0 is therefore saturated if Ωχh 2 as a function of the cross section reaches a maximum. From Fig.1 we read off T0 ≥ mχ/23 , (16) if we require Ωχh 2 to fall in the range (1). We just saw that in the regime where this bound is saturated, the final relic den- sity is (to first order) independent of the annihilation cross section, ∂(Ωχh 2)/∂〈σv〉 = 0. If T0 is slightly above the absolute lower bound (16), the correct relic density can therefore be obtained for a rather wide range of cross sections. For example, if x0 = 22.5, the entire range 3 × 10−10 GeV−2 <∼ a <∼ 2 × 10−9 GeV −2 is allowed. Of course, the correct relic density can also be obtained in the standard scenario of (arbitrarily) high T0, if a + 3b/22 falls within ∼ 20% of 2× 10−9 GeV−2. 3 Relic Abundance for Modified Expansion Rate In this section we discuss the calculation of the WIMP relic density nχ in modi- fied cosmological scenarios where the expansion parameter of the pre–BBN universe differed from the standard value Hst of Eq.(7). For the most part we will assume that WIMPs have been in full thermal equilibrium. Various cosmological models predict a non–standard early expansion history [22, 23, 24, 25]. Here we analyze to what extent the relic density of WIMP Dark Matter can be used to constrain the Hubble parameter during the epoch of WIMP decoupling. As long as we assume large T0 we can use a modification of the standard treatment [4, 2] to estimate the relic density for given annihilation cross section and expansion rate. We will show that the resulting approximate solutions again accurately reproduce the numerically evaluated relic abundance. Let us introduce the modification parameter A(x), which parameterizes the ratio of the standard value Hst(x) to the assumed H(x): A(x) = Hst(x) . (17) Note that A > 1 means that the expansion rate is smaller than in standard cos- mology. Allowing for this modified expansion rate, the Boltzmann equation (6) is altered to G(x)mχMPl 〈σv〉A(x) Y 2χ − Y 2χ,eq , (18) where G(x) = . (19) Following refs.[4, 2], we can obtain an approximate solution of this equation by considering the differential equation for ∆ = Yχ − Yχ,eq. For temperatures higher than the decoupling temperature, Yχ tracks Yχ,eq very closely and the ∆ 2-term can be ignored: ≃ −dYeq − 4π√ mχMPl G(x)〈σv〉A(x) (2Yχ,eq∆) . (20) Here dYχ,eq/dx ≃ −Yχ,eq for x ≫ 1. In order to keep |∆| small, the derivative d∆/dx must also be small, which implies ∆ ≃ x 90)mχMPlG(x)〈σv〉A(x) . (21) This solution is used down to the freeze–out temperature TF , defined via ∆(xF ) = ξYχ,eq(xF ) , (22) where ξ is a constant of order of unity. This leads to the following expression: xF = ln ξmχMPlgχ 〈σv〉A(x) xg∗(x) , (23) which can e.g. be solved iteratively. In our numerical calculations we will choose 2− 1 [4, 2]. On the other hand, for low temperatures (T < TF ), the production term ∝ Y 2χ,eq in Eq.(18) can be ignored. In this limit, Yχ ≃ ∆, and the solution of Eq.(18) is given ∆(xF ) ∆(x → ∞) mχMPlI(xF ) , (24) where the annihilation integral is defined as I(xF ) = G(x)〈σv〉A(x) . (25) Assuming ∆(x → ∞) ≪ ∆(xF ), the final relic abundance is Yχ,∞ ≡ Yχ(x → ∞) = 90)mχMPlI(xF ) . (26) Plugging in numerical values for the Planck mass and for today’s entropy density, the present relic density can thus be written as 8.5× 10−11 I(xF ) GeV . (27) The constraint (1) therefore corresponds to the allowed range for the annihilation integral 7.1× 10−10 GeV−2 < I(xF ) < 1.1× 10−9 GeV−2 . (28) The standard formula (12) for the final relic density is recovered if A(x) is set to unity and G(x) is replaced by the constant g∗(xF ). The further discussion is simplified if we use the normalized temperature z = T/mχ ≡ 1/x, rather than x. Phenomenologically A(z) can be any function subject to the condition that A(z) approaches unity at late times in order not to contradict the successful predictions of BBN. We need to know A(z) only for the interval from around the freeze-out to BBN: zBBN ∼ 10−5 − 10−4 <∼ z <∼ zF ∼ 1/20. This suggests a parameterization of A(z) in terms of a power series in (z − zF,st): A(z) = A(zF,st) + (z − zF,st)A′(zF,st) + (z − zF,st)2A′′(zF,st) , (29) where zF,st is the normalized freeze–out temperature in the standard scenario and a prime denotes a derivative with respect to z. The ansatz (29) should be of quite general validity, so long as the modification of the expansion rate is relatively modest. This suits our purpose, since we wish to find out what constraints can be derived on the expansion history if standard cosmology leads to the correct WIMP relic density. We further introduce the variable k = A(z → 0) = A(zF,st)− zF,stA′(zF,st) + z2F,stA ′′(zF,st) , (30) which describes the modification parameter at late times. Since zBBN is almost zero, we treat k as the modification parameter at the era of BBN in this paper.‡ Deviations from k = 1 are conveniently discussed in terms of the equivalent number of light neutrino degrees of freedom Nν . BBN permits that the number of neutrinos differs from the standard model value Nν = 3 by δNν = 1.5 or so [26]. We therefore take the uncertainty of k to be 20%. In the following we treat A(zF,st), A ′(zF,st) and k as free parameters; A′′(zF,st) is then a derived quantity. Note that we allow A(z) to cross unity, i.e. to switch from an expansion that is faster than in standard cosmology to a slower expansion or vice versa. This is illustrated in Fig. 2, which shows examples of possible evolutions of A(z) as function of z for zF = 0.05. Here we take k = 1.2 (left frame) and k = 0.8 (right). In each case we consider scenarios with A(zF ) = 1.3 (slower expansion at TF than in standard cosmology) as well as A(zF ) = 0.7 (faster expansion); moreover, we allow the change of A at z = zF to be either positive or negative. However, we insist that H remains positive at all times, i.e. A(z) must not cross zero. This excludes scenarios with very large positive A′(zF,st), which would lead to A < 0 at some z < zF . Similarly, ‡Presumably the Hubble expansion rate has to approach the standard rate even more closely for T < TBBN. However, since all WIMP annihilation effectively ceased well before the onset of BBN, this epoch plays no role in our analysis. 0 0.01 0.02 0.03 0.04 0.05 z = T/mχ k = 1.2 A=1.3, A’=−3 A=1.3, A’= 9 A=0.7, A’=−3 A=0.7, A’= 9 0 0.01 0.02 0.03 0.04 0.05 z = T/mχ k = 0.8A=1.3, A’=−3 A=1.3, A’= 9 A=0.7, A’=−3 A=0.7, A’= 9 Figure 2: Examples of possible evolutions of the modification parameter A(z) as function of z for zF = 0.05. Here we take k = 1.2 (left frame) and k = 0.8 (right). In each frame we choose A(zF ) = 1.3, A ′(zF ) = −3 (thick line), A(zF ) = 1.3, A′(zF ) = 9 (dashed), A(zF ) = 0.7, A′(zF ) = −3 (dotted), A(zF ) = 0.7, A′(zF ) = 9 (dot–dashed). demanding that our ansatz (29) remains valid for some range of temperatures above TF excludes scenarios with very large negative A ′(zF,st). We will come back to this point shortly. Eq.(23) shows that zF 6= zF,st (xF 6= xF,st) if A(zF ) 6= 1. This is illustrated by Fig. 3, which shows the difference between xF and xF,st in the (A(zF,st), A ′(zF,st)) plane. Here we take parameters such that Ωχh 2 = 0.099 in the standard cosmology, which is recovered for A(zF,st) = 1, A ′(zF,st) = 0. Due to the logarithmic dependence on A, xF (or zF ) differs by at most a few percent from its standard value if A(zF,st) is O(1). Since TF only depends on the expansion rate at TF , it is essentially insensitive to the derivative A′(zF,st). In our treatment the modification of the expansion parameter affects the WIMP relic density mostly via the annihilation integral (25). In terms of the normalized temperature z, the latter can be rewritten as I(zF ) = dz G(z)〈σv〉A(z) . (31) One advantage of the expansion (29) is that this integral can be evaluated analyti- cally: I(zF ) ≃ G(zF ) k(azF + 3bz F ) + (A ′(zF,st)− zF,stA′′(zF,st)) z2F + 2bz A′′(zF,st) z3F + . (32) 0.6 0.8 1 1.2 1.4 A(zF,st ) xF − x F,st a = 2.0*10−9 GeV −2 b = 0 k = 1 xF,st = 22.0 −0.4 −0.2 0 0.2 0.4 Figure 3: Contour plot of xF −xF,st in the (A(zF,st), A′(zF,st)) plane. Here we take a = 2.0×10−9 GeV−2, b = 0, mχ = 100 GeV, gχ = 2, g∗ = 90 (constant) and k = 1. This parameter set produces xF,st = 22.0 and Ωχh 2 = 0.099 for the standard approximation. 0.6 0.8 1 1.2 1.4 A(zF,st ) 2(approx) / Ωχh 2(exact) 0.997 0.998 0.999 1.000 a = 2.0*10−9 GeV−2 b = 0 k = 1 0.6 0.8 1 1.2 1.4 A(zF,st) 2(approx) / Ωχh 2(exact) 0.996 0.998 1.000 1.002 1.004 a = 0 b = 1.5*10−8 GeV−2 k = 1 Figure 4: Ratio of the analytic result of the relic density to the exact value in the (A(zF,st), A′(zF,st)) plane for a = 2.0× 10−9 GeV−2, b = 0 (left frame) and for a = 0, b = 1.5× 10−8 GeV−2 (right). The other parameters are as in Fig. 3. Here we have assumed that G(z) varies only slowly. Before proceeding, we first have to convince ourselves that the analytic treatment developed in this Section still works for A 6= 1. This is demonstrated by Fig. 4, which shows the ratio of the analytic solution obtained from Eqs. (27) and (32) to the exact one, obtained by numerically integrating the Boltzmann equation (18), assuming constant g∗. We see that our analytical treatment is accurate to better than 1%, and can thus safely be employed in the subsequent analysis. 0.6 0.8 1 1.2 1.4 A(zF,st ) a = 2.0*10−9 GeV −2 b = 0 k = 1 0.6 0.8 1 1.2 1.4 A(zF,st ) a = 0 b = 1.5*10−8 GeV−2 k = 1 0.6 0.8 1 1.2 1.4 A(zF,st ) a = 2.0*10−9 GeV−2 b = 0 k = 1.2 0.6 0.8 1 1.2 1.4 A(zF,st ) a = 2.0*10−9 GeV−2 b = 0 k = 0.8 Figure 5: Contour plots of the relic abundance in the (A(zF,st), A′(zF,st)) plane. Here we choose (a) a = 2.0×10−9 GeV−2, b = 0, k = 1; (b) a = 0, b = 1.5×10−8 GeV−2, k = 1; (c) a = 2.0×10−9 GeV−2, b = 0, k = 1.2; (d) a = 2.0× 10−9 GeV−2, b = 0, k = 0.8. The other parameters are as in Fig. 3. We are now ready to analyze the impact of the modified expansion rate on the WIMP relic density. In Fig. 5, we show contour plots of Ωχh 2 in the (A(zF,st), A′(zF,st)) plane. Recall that large (small) values of A correspond to a small (large) expansion rate. Since a smaller expansion rate allows the WIMPs more time to annihilate, A > 1 leads to a reduced WIMP relic density, whereas A < 1 means larger relic density, if the cross section is kept fixed. However, unlike the freeze–out temperature, the annihilation integral is sensitive to A(z) for all z ≤ zF . Note that A′(zF,st) > 0 implies A(z) < A(zF,st) for z < zF,st ≃ zF . A positive first derivative, A′(zF,st) > 0, can therefore to some extent compensate for A(zF,st) > 1; analogously, a negative first derivative can compensate for A(zF,st) < 1. This explains the slopes of the curves in Fig. 5. Recall also that A′(zF,st) = 0 does not imply a constant modification factor A(z); rather, the term ∝ A′′(zF,st) in Eq.(29) makes sure that A approaches k as z → 0. This explains why a change of A by some given percentage leads to a smaller relative change of Ωχh 2, as can be seen in the Figure. This also illustrates the importance of ensuring appropriate (near–standard) expansion rate in the BBN era. Finally, since the expansion rate at late times is given by Hst/k, bigger (smaller) values of k imply that the χ relic density is reduced (enhanced). Fig. 5 shows that we need additional physical constraints if we want to derive bounds on A(zF,st) and/or A ′(zF,st). Once the annihilation cross section is known, the requirement (1) will single out a region in the space spanned by our three new parameters (including k) which describe the non–standard evolution of the universe, but this region is not bounded. Such additional constraints can be derived from the requirement that the Hubble parameter should remain positive throughout the epoch we are considering. As noted earlier, requiring H > 0 for all T < TF,st leads to an upper bound on A′(zF,st); explicitly, A′(zF,st) < A(zF,st) + kA(zF,st) zF,st . (33) On the other hand, a lower bound on A′(zF,st) is obtained from the condition that the modified Hubble parameter is positive between the highest temperature Ti where the ansatz (29) holds and TF,st: A′(zF,st) > − zi − zF,st 2− zi zF,st A(zF,st) + k zF,st , (34) for (1− zF,st/zi)2k < A(zF,st), and A′(zF,st) > A(zF,st)− kA(zF,st) zF,st , (35) for A(zF,st) < (1− zF,st/zi)2k, where zi = Ti/mχ. Evidently the lower bound on A′(zF,st) depends on zi, i.e. on the maximal tem- perature where we assume our ansatz (29) to be valid. In ref.[14] we have shown that in standard cosmology (A ≡ 1) essentially full thermalization is already achieved for xi <∼ xF − 0.5, even if nχ(xi) = 0. However, it seems reasonable to demand that H should remain positive at least up to xi = xF−(a few). In Fig. 6 we therefore show 0 1 2 3 4 5 6 7 A(zF,st ) a = 2.0*10−9 GeV−2 b = 0 k = 1 xF,st − x i = 4 xF,st − x i = 10 0 1 2 3 4 5 6 7 A(zF,st ) a = 0 b = 1.5*10−8 GeV−2 k = 1 xF,st − x i = 4 xF,st − x i = 10 Figure 6: Contour plots of the relic abundance Ωχh2 in the (A(zF,st), A′(zF,st)) plane. The dashed line corresponds to the upper bound on A′(zF,st). The dotted lines correspond to the lower bounds calculated for xF,st − xi = 4, 10. We take a = 2.0 × 10−9 GeV−2, b = 0 (left frame) and a = 0, b = 1.5× 10−8 GeV−2 (right frame). The other parameters are as in Fig 3. the physical constraints on the modification parameter A(z) for xF,st − xi = 4, 10 and k = 1. The dashed and dotted lines correspond to the upper and lower bounds on A′(zF,st), described by Eq.(33) and Eqs.(34), (35), respectively. We see that when xF,st − xi = 4 the allowed region is 0.4 <∼ A(zF,st) <∼ 6.5 with −60 <∼ A′(zF,st) <∼ 400 for b = 0 (left frame), and 0.4 <∼ A(zF,st) <∼ 4.5 with −60 <∼ A′(zF,st) <∼ 300 for a = 0 (right frame). When xF,st − xi = 10, the lower bounds are altered to 0.6 <∼ A(zF,st), −10 <∼ A′(zF,st) for b = 0 (left frame), and 0.6 <∼ A(zF,st), −20 <∼ A′(zF,st) for a = 0 (right frame). Note that the lower bounds on A(zF,st), which depend only weakly on xi so long as it is not very close to xF , are almost the same in both cases, which also lead to very similar relic densities in standard cosmology. However, the two upper bounds differ significantly. The reason is that large values of A(zF,st), i.e. a strongly suppressed Hubble expansion, require some degree of finetuning: One also has to take large positive A′(zF,st), so that A becomes smaller than one for some range of z values below zF , leading to an annihilation integral of similar size as in standard cosmology. Since the b−terms show different zF dependence in the annihilation in- tegral (32), the required tuning between A(zF,st) and A ′(zF,st) is somewhat different than for the a−terms, leading to a steeper slope of the allowed region. This allowed region therefore saturates the upper bound (33) on the slope for somewhat smaller A(zF,st). The effect of this tuning can be seen by analyzing the special case where A′′(zF,st) = 0 0.5 1 1.5 2 A(zF,st ) a = 2.0*10−9 GeV−2 b = 0 4 0.08 0 0.5 1 1.5 2 A(zF,st ) a = 0 b = 1.5*10−8 GeV−2 Figure 7: Contour plots of the relic abundance Ωχh2 in the (A(zF,st), k) plane for A′′(zF,st) = 0. The dotted lines correspond to the lower bounds of A(zF,st), calculated for xF,st − xi = 4, 10. We take a = 2.0 × 10−9 GeV−2, b = 0 (left frame) and a = 0, b = 1.5 × 10−8 GeV−2 (right frame). The other parameters are as in Fig 3. 0. The modification parameter then reads A(z) = A(zF,st)− k zF,st z + k . (36) Note that A is now a monotonic function of z, making large cancellations in the annihilation integral impossible. Imposing that A(z) remains positive for zF,st ≤ z ≤ zi leads to the lower limit A(zF,st) > zF,st k . (37) There is no upper bound, since A(z) is now automatically positive for all z ∈ [0, zF,st] if A(zF,st) and A(0) ≡ k are both positive. Fig. 7 shows constraints on the relic abundance in the (A(zF,st), k) plane for A ′′(zF,st) = 0. The dotted lines correspond to the lower bounds (37) on A(zF,st) for xF,st − xi = 4, 10. As noted earlier, k is constrained by the BBN bound. This leads to the bounds 0.5 <∼ A(zF,st) <∼ 1.8 for b = 0 (left frame), and 0.65 <∼ A(zF,st) <∼ 1.6 for a = 0 (right frame), when xF,st − xi = 10. Evidently the constraints now only depend weakly on whether the a− or b−term dominates in the annihilation cross section. As the initial temperature is lowered, the impact of the constraint (37) disappears. So far we have assumed in this Section that the reheat temperature was high enough for WIMPs to have attained full thermal equilibrium. If this was not the case, the initial temperature as well as the suppression parameter affects the final 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 A(zF,st ) a = 2.0*10−9 GeV−2 b = 0 k = 1, A’’(zF,st ) = 0 0.080.12 Figure 8: Contour plot of the relic abundance Ωχh2 in the (A(zF,st), x0) plane. Here we choose a = 2.0 × 10−9 GeV−2, b = 0, k = 1, A′′(zF,st) = 0. The other parameters are as in Fig. 3. The shaded region corresponds to the WMAP bound on the cold dark matter abundance, 0.08 < ΩCDMh 2 < 0.12 (95% C.L.). relic abundance. Here we show that the lower bound on the reheat temperature derived in the previous Section survives even in scenarios with altered expansion history as long as WIMPs were only produced thermally. This can be understood from the observation that the Boltzmann equation with modified expansion rate is obtained by replacing 〈σv〉 in the radiation–dominated case by 〈σv〉A. Increasing (decreasing) A therefore has the same effect as an in- crease (decrease) of the annihilation cross section. Since the lower bound on T0 was independent of σ (more exactly: we quoted the absolute minimum, for the optimal choice of σ), we expect it to survive even if A(z) 6= 1 is introduced. This is borne out by Fig. 8, which shows the relic abundance Ωχh 2 in the (A(zF,st), x0) plane for the simplified case A ′′(zF,st) = 0; similar results can be obtained for the more general ansatz (29). The shaded region corresponds to the bound (1) on the cold dark matter abundance. As expected, this figure looks similar to Fig. 1 if the annihilation cross section in Fig. 1 is replaced by A(zF,st). The maximal value of x0 consistent with the WMAP data remains around 23 even in these scenarios with modified expansion rate. Fig. 8 also shows that A(zF,st) ≪ 1 is allowed for some narrow range of initial temperature T0 < TF . This is analogous to the low cross section branch in Fig. 1. 4 Summary and Conclusions In this paper we have investigated the relic abundance of WIMPs χ, which are non- relativistic long–lived or stable particles, in non–standard cosmological scenarios. One motivation for studying such scenarios is that they allow to reproduce the ob- served Dark Matter density for a large range of WIMP annihilation cross sections. Our motivation was the opposite: we wanted to quantify the constraints that can be obtained on parameters describing the early universe, under the assumption that thermally produced WIMPs form all Dark Matter. Wherever necessary, we fixed particle physics quantities such that standard cosmology yields the correct relic den- sity. Specifically, we first considered scenarios with low post–inflationary reheat tem- perature, while keeping all other features of standard cosmology (known particle content and Hubble expansion parameter during WIMP decoupling; no late entropy production; no non–thermal WIMP production channels). If the temperature was so low that WIMPs could not achieve full thermal equilibrium, the dependence of the abundance on the mass and annihilation cross section of the WIMPs is completely different from that in the standard thermal WIMP scenario. In particular, if the maximal temperature T0 is much less than the decoupling temperature TF , nχ re- mains exponentially suppressed. By applying the observed cosmological amount of cold dark matter to the predicted WIMP abundance, we therefore found the lower bound of the initial temperature T0 >∼ mχ/23. One might naively think that this bound could be evaded by choosing a sufficiently large WIMP production (or anni- hilation) cross section. However, increasing this cross section also reduces TF . For sufficiently large cross section one therefore has TF ≤ T0 again; in this regime the relic density drops with increasing cross section. Our lower bound is the minimal T0 required for any cross section; once the latter is known, the bound on T0 might be slightly sharpened. As a by–product, we also noted that the final relic density depends only weakly on the annihilation cross section if T0 is slightly above this lower bound. We also investigated the effect of a non–standard expansion rate of the universe on the WIMP relic abundance. In general the abundance of thermal relics depends on the ratio of the annihilation cross section to the expansion rate; the latter is determined unambiguously in standard cosmology. We found that even for non– standard Hubble parameter the relic abundance can be calculated accurately in terms of an annihilation integral, very similar to the case of standard cosmology. We assumed that the WIMP annihilation cross section is such that the standard scenario yields the observed relic density, and parameterized the modification of the Hubble parameter as a quadratic function of the temperature. In this analysis it is crucial to make sure that at low temperatures the Hubble parameter approaches its standard value to within ∼ 20%, as required for the success of Big Bang Nucleosynthesis (BBN). Keeping the annihilation cross section fixed and allowing a 20% variation in the relic density, roughly corresponding to the present “2σ” band, we found that the expansion of the universe at T = TF might have been more than two times faster, or more than six times slower, than in standard cosmology. These large variations of H(TF ) can only be realized by finetuning of the parameters describing H(T < TF ). However, even if we forbid such finetuning by choosing a linear parameterization for the modification of the expansion rate, a 20% variation of Ωχh 2 allows a difference between H(TF ) and its standard expectation of more than 50%. This relatively weak sensitivity of the predicted Ωχh 2 on H(TF ) is due to the fact that the relic density depends on all H(T < TF ); as stressed above, we have to require that H(T ≪ TF ) approaches its standard value to within ∼ 20%. The fact that determining Ωχh2 will yield relatively poor bounds on H(TF ) remains true even if the annihilation cross section is such that a non–standard behavior of H(T ) is required for obtaining the correct χ relic density. Finally, we showed that the absolute lower bound on the temperature required for thermal χ production is unaltered by allowing H(T ) to differ from its standard value. Of course, in order to draw the conclusions derived in this article, we need to convince ourselves that WIMPs do indeed form (nearly) all Dark Matter. This requires not only the detection of WIMPs, e.g. in direct search experiments; we also need to show that their density is in accord with the local Dark Matter density derived from astronomical observations. To that end, the cross sections appearing in the calculation of the detection rate need to be known independently. This can only be done in the framework of a definite theory, using data from collider experiments. For example, in order to determine the cross section for the direct detection of supersymmetric WIMPs, one needs to know the parameters of the supersymmetric neutralino, Higgs and squark sectors [3]. We also saw that inferences about H(TF ) can only be made if the WIMP annihilation cross section is known. This again requires highly non–trivial analyses of collider data [27], as well as a consistent overall theory. 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FTUAM 07-07 IFT-UAM/CSIC 07-18 Flavour-Dependent Type II Leptogenesis S. Antuscha Departamento de F́ısica Teórica C-XI and Instituto de F́ısica Teórica C-XVI, Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain Abstract We reanalyse leptogenesis via the out-of-equilibrium decay of the lightest right- handed neutrino in type II seesaw scenarios, taking into account flavour-dependent effects. In the type II seesaw mechanism, in addition to the type I seesaw con- tribution, an additional direct mass term for the light neutrinos is present. We consider type II seesaw scenarios where this additional contribution arises from the vacuum expectation value of a Higgs triplet, and furthermore an effective model-independent approach. We investigate bounds on the flavour-specific de- cay asymmetries, on the mass of the lightest right-handed neutrino and on the reheat temperature of the early universe, and compare them to the corresponding bounds in the type I seesaw framework. We show that while flavour-dependent thermal type II leptogenesis becomes more efficient for larger mass scale of the light neutrinos, and the bounds become relaxed, the type I seesaw scenario for leptogenesis becomes more constrained. We also argue that in general, flavour- dependent effects cannot be ignored when dealing with leptogenesis in type II seesaw models. aE-mail: antusch@delta.ft.uam.es http://arxiv.org/abs/0704.1591v3 1 Introduction Leptogenesis [1] is one of the most attractive and minimal mechanisms for explaining the observed baryon asymmetry of the universe nB/nγ ≈ (6.0965 ± 0.2055) × 10−10 [2]. A lepton asymmetry is dynamically generated and then converted into a baryon asym- metry due to (B + L)-violating sphaleron interactions [3] which exist in the Standard Model (SM) and its minimal supersymmetric extension, the MSSM. Leptogenesis can be implemented within the type I seesaw scenario [4], consisting of the SM (MSSM) plus three right-handed Majorana neutrinos (and their superpartners) with a hierarchical spectrum. In thermal leptogenesis [5], the lightest of the right-handed neutrinos is pro- duced by thermal scattering after inflation, and subsequently decays out-of-equilibrium in a lepton number and CP-violating way, thus satisfying Sakharov’s constraints [6]. In models with a left-right symmetric particle content like minimal left-right sym- metric models, Pati-Salam models or Grand Unified Theories (GUTs) based on SO(10), the type I seesaw mechanism is typically generalised to a type II seesaw [7], where an additional direct mass term mIILL for the light neutrinos is present. From a model independent perspective, the type II mass term can be considered as an additional con- tribution to the lowest dimensional effective neutrino mass operator. In most explicit models, the type II contribution stems from seesaw suppressed induced vevs of SU(2)L- triplet Higgs fields. One motivation for considering the type II seesaw is that it allows to construct unified flavour models for partially degenerate neutrinos in an elegant way, e.g. via a type II upgrade [8], which is otherwise difficult to achieve in type I models. For leptogenesis in type II seesaw scenarios with SU(2)L-triplet Higgs fields, there are in general two possibilities to generate the baryon asymmetry: via decays of the lightest right-handed neutrinos or via decays of the SU(2)L-triplets [9,10,11,12]. In the first case, there are additional one-loop diagrams where virtual triplets are running in the loop [9, 13, 14, 15, 16]. In the following, we focus on this possibility, and assume hierarchical right-handed neutrino masses (and that the triplets are heavier than ν1R). In this limit, to a good approximation the decay asymmetry depends mainly on the low energy neutrino mass matrix mνLL = m LL + m LL and on the Yukawa couplings to the lightest right-handed neutrino and its mass [16]. It has been shown that type II leptogenesis imposes constraints on the seesaw parameters, which, in the flavour- independent approximation, differ substantially from the constraints in the type I case. For instance, the bound on the decay asymmetry increases with increasing neutrino mass scale [16], in contrast to the type I case where it decreases. As a consequence, the lower bound on the mass of the lightest right-handed neutrino from leptogenesis decreases for increasing neutrino mass scale [16]. One interesting application of type II leptogenesis is the possibility to improve consistency of classes of unified flavour models with respect to thermal leptogenesis [17]. Finally, since the type II contribution typically does not effect washout, there is no bound on the absolute neutrino mass scale from type II leptogenesis, as has been pointed out in [15]. For further applications and realisations of type II leptogenesis in specific models of fermion masses and mixings, see e.g. [18]. In recent years, the impact of flavour in thermal leptogenesis has merited increasing attention [19] - [38]. In fact, the one-flavour approximation is only rigorously correct when the interactions mediated by the charged lepton Yukawa couplings are out of equilibrium. Below a given temperature (e.g. O(1012GeV) in the SM and (1+tan2 β)× O(1012GeV) in the MSSM), the tau Yukawa coupling comes into equilibrium (later followed by the couplings of the muon and electron). Flavour effects are then physical and become manifest, not only at the level of the generated CP asymmetries, but also regarding the washout processes that destroy the asymmetries created for each flavour. In the full computation, the asymmetries in each distinguishable flavour are differently washed out, and appear with distinct weights in the final baryon asymmetry. Flavour-dependent leptogenesis in the type I seesaw scenario has recently been ad- dressed in detail by several authors. In particular, flavour-dependent effects in lepto- genesis have been studied, and shown to be relevant, in the two right-handed neutrino models [24] as well as in classes of neutrino mass models with three right-handed neu- trinos [26]. The quantum oscillations/correlations of the asymmetries in lepton flavour space have been included in [22, 32, 33, 35] and the treatment has been generalised to the MSSM [26,29]. Effects of reheating, and constraints on the seesaw parameters from upper bounds on the reheat temperature, have been investigated in [29]. Leptogenesis bounds on the reheat temperature [29] and on the mass of the lightest right-handed neutrino [29, 36] have also been considered including flavour-dependent effects. Strong connections between the low-energy CP phases of the UMNS matrix and CP violation for flavour-dependent leptogenesis have been shown to emerge in certain classes of neutrino mass models [26] or under the hypothesis of no CP violation sources associated with the right-handed neutrino sector (real R) [25,27,28,31]. Possible effects regarding the decays of the heavier right-handed neutrinos for leptogenesis have been discussed in this con- text in [21, 34], and flavour-dependent effects for resonant leptogenesis were addressed in [38]. Regarding the masses of the light neutrinos, assuming hierarchical right-handed neutrinos and considering experimentally allowed light neutrino masses (below about 0.4 eV), there is no longer a bound on the neutrino mass scale from thermal leptogenesis if flavour-dependent effects are included [24]. In view of the importance of flavour-dependent effects on leptogenesis in the type I seesaw case, it is pertinent to investigate their effects on type II leptogenesis. In this paper, we therefore reanalyse leptogenesis via the out-of-equilibrium decay of the lightest right-handed neutrino in type II seesaw scenarios, taking into account flavour-dependent effects. We investigate bounds on the decay asymmetries, on the mass of the lightest right-handed neutrino and on the reheat temperature of the early universe, and discuss how increasing the neutrino mass scale affects thermal leptogenesis in the type I and type II seesaw frameworks. 2 Type I and type II seesaw mechanisms Motivated by left-right symmetric unified theories, we consider two generic possibilities for explaining the smallness of neutrino masses: via heavy SM (MSSM) singlet fermions (i.e. right-handed neutrinos) [4] and via heavy SU(2)L-triplet Higgs fields [7]. In both cases, the effective dimension five operator for Majorana neutrino masses in the SM or p2≪M2 Figure 1: Generation of the dimension 5 neutrino mass operator in the type I seesaw mechanism. the MSSM, respectively, κgf (LC · φ) (Lf · φ) + h.c. , (1a) κ = − κgf (L̂ g · Ĥu) (L̂f · Ĥu) + h.c. , (1b) is generated from integrating out the heavy fields. This is illustrated in figures 1 and 2. In equation (1), the dots indicate the SU(2)L-invariant product, (L̂ f · Ĥu) = L̂fa(iτ2)ab(Ĥu)b, with τA (A ∈ {1, 2, 3}) being the Pauli matrices. Superfields are marked by hats. After electroweak symmetry breaking, the operators of equation (1) lead to Majorana mass terms for the light neutrinos, Lν = −12m LLνLν L , with m LL = − (κ)∗ . (2) In the type I seesaw mechanism, it is assumed that only the singlet (right-handed) neutrinos νRi contribute to the neutrino masses. With Yν being the neutrino Yukawa matrix in left-right convention,1 MRR the mass matrix of the right-handed neutrinos and vu = 〈φ0〉 (= 〈H0u〉) the vacuum expectation value of the Higgs field which couples to the right-handed neutrinos, the effective mass matrix of the light neutrinos is given by the conventional type I seesaw formula mILL = −v2u Yν M−1RR Y ν . (3) In the type II seesaw mechanism, the contributions to the neutrino mass matrix from both, right-handed neutrinos νRi and Higgs triplet(s) ∆L, are considered. The additional contribution to the neutrino masses from ∆L can be understood in two ways: as another contribution to the effective neutrino mass operator in the low energy effective theory or, equivalently, as a direct mass term after the Higgs triplet obtains an induced small vev after electroweak symmetry breaking (c.f. figure 2). The neutrino mass matrix in the type II seesaw mechanism has the form mνLL = m LL +m LL = m LL − v2uYνM−1RRY ν , (4) 1The neutrino Yukawa matrix corresponds to −(Yν)fi(Lf · φ) νiR in the Lagrangian of the SM and, analogously, to (Yν)fi(L̂ f · Ĥu) ν̂Ci in the superpotential of the MSSM (see [16] for further details). p2≪M2 Figure 2: Extra diagram generating the dimension 5 neutrino mass operator in the type II seesaw mechanism from a SU(2)L-triplet Higgs field. where mIILL is the additional term from the Higgs triplet(s). In left-right symmetric unified theories, the generic size of both seesaw contributionsmILL andm LL isO(v2u/vB−L) where vB−L is the B-L breaking scale (i.e. the mass scale of the right-handed neutrinos and of the Higgs triplet(s)). 3 Baryogenesis via flavour-dependent leptogenesis Flavour-dependent effects can have a strong impact in baryogenesis via thermal lep- togenesis [19] - [38]. The effects are manifest not only in the flavour-dependent CP asymmetries, but also in the flavour-dependence of scattering processes in the thermal bath, which can destroy a previously produced asymmetry. The relevance of the flavour-dependent effects depends on the temperatures at which thermal leptogenesis takes place, and thus on which interactions mediated by the charged lepton Yukawa couplings are in thermal equilibrium. For example, in the MSSM, for temperatures between circa (1+tan2 β)×105GeV and (1+tan2 β)×109GeV, the µ and τ Yukawa couplings are in thermal equilibrium and all flavours in the Boltzmann equations are to be treated separately. For tan β = 30, this applies for temperatures below about 1012 GeV and above 108GeV, a temperature range which is of most interest for thermal leptogenesis in the MSSM. In the SM, in the temperature range between circa 109 GeV and 1012 GeV, only the τ Yukawa coupling is in equilibrium and is treated separately in the Boltzmann equations, whereas µ and e flavours are indistinguishable. A discussion of the temperature regimes in the SM and MSSM, where flavour is important, can be found, e.g., in [26]. We now briefly review the estimation of the produced baryon asymmetry in flavour- dependent leptogenesis.2 For definiteness, we focus on the temperature range where all flavours are to be treated separately. In the following discussion of thermal type II leptogenesis, we will assume that the mass M∆L of the triplet(s) is much larger than MR1. In this limit, the flavour-dependent efficiencies calculated in the type I seesaw scenario can also be used in the type II framework. The out-of-equilibrium decays of the heavy right-handed (s)neutrinos ν1R and ν̃ R give rise to flavour-dependent asymmetries in the (s)lepton sector, which are then partly transformed via sphaleron conversion into 2For a discussion of approximations which typically enter these estimates, and which also apply to our discussion, see e.g. section 3.1.3 in [29]. a baryon asymmetry YB. 3 The final baryon asymmetry can be calculated as Y SMB = Y SM∆f , (5) Y MSSMB = Ŷ MSSM∆f , (6) where Ŷ∆f ≡ YB/3 − YLf are the total (particle and sparticle) B/3 − Lf asymmetries, with YLf the lepton number densities in the flavour f = e, µ, τ . The asymmetries Ŷ and Y SM∆f , which are conserved by sphalerons and by the other SM (MSSM) interactions, are then usually calculated by solving a set of coupled Boltzmann equations, describing the evolution of the number densities as a function of temperature. It is convenient to parameterise the produced asymmetries in terms of flavour-specific efficiency factors ηf and decay asymmetries ε1,f as Y SM∆f = η f ε1,f Y , (7) Ŷ MSSM∆f = η (ε1,f + ε1, ef) Y (εe1,f + εe1, ef) Y . (8) and Y are the number densities of the neutrino and sneutrino for T ≫ M1 if they were in thermal equilibrium, normalised with respect to the entropy density. In the Boltzmann approximation, they are given by Y ≈ Y eq ≈ 45/(π4g∗). g∗ is the effective number of degrees of freedom, which amounts 106.75 in the SM and 228.75 in the MSSM. ε1,f , ε1, ef , εe1,f and εe1, ef are the decay asymmetries for the decay of neutrino into Higgs and lepton, neutrino into Higgsino and slepton, sneutrino into Higgsino and lepton, and sneutrino into Higgs and slepton, respectively, defined by ε1,f = − Γν1 f (Γν1RLf + Γν1RLf 1, ef f (Γν1 εe1,f = Γeν∗1 f (Γeν∗1R Lf + Γeν1RLf , εe1, ef = f(Γeν1 . (9) The flavour-dependent efficiency factors ηf in the SM and in the MSSM are defined by Eqs. (7) and (8), respectively. As stated above, we assume that the mass M∆L of the triplet(s) is much larger than MR1. In this limit, the efficiencies for flavour-dependent thermal leptogenesis in the type I and type II frameworks are mainly determined by the properties of ν1R, which means in particular that the flavour-dependent efficiencies 3In the following, Y will always be used for quantities which are normalised to the entropy density s. The quantities normalised with respect to the photon density can be obtained using the relation s/nγ ≈ 7.04k. -2 -1 0 1 2 ÈAff K f È ������������������������������� È Aff K f È ������������������������������� È Aff K f È ������������������������������� È Aff K f È = 100 Figure 3: Flavour-dependent efficiency factor η(AffKf ,K) in the MSSM as a function of AffKf , for fixed values of K/|AffKf | = 2, 5 and 100, obtained from solving the flavour-dependent Boltzmann equations in the MSSM with zero initial abundance of right-handed (s)neutrinos (figure from [26]). A is a matrix which appears in the Boltzmann equations (see [19, 24] for A in the SM and [26] for the MSSM case), and which has diagonal elements |Aff | of O(1). The small off-diagonal entries of A have been neglected, which is a good approximation in most cases. In general, however, they have to be included. More relevant than the differences in the flavour-dependent efficiency factors for different K/|AffKf | is that the total baryon asymmetry is the sum of each individual lepton asymmetries, which is weighted by the corresponding efficiency factors. calculated in the type I seesaw scenario can also be used in the type II framework. In the definition of the efficiency factor, the equilibrium number densities serve as a nor- malization: A thermal population νR1 (and ν̃R1) decaying completely out of equilibrium (without washout effects) would lead to ηf = 1. The efficiency factors can be computed by means of the flavour-dependent Boltzmann equations, which can be found for the SM in [19,22,23,24] and for the MSSM in [26,29]. In general, the flavour-dependent efficiencies depend strongly on the washout parameters m̃1,f for each flavour, and on the total washout parameter m̃1, which are defined as m̃1,f = v2u |(Yν)f1|2 , m̃1 = m̃1,f . (10) Alternatively, one may use the quantities Kf , K, which are related to m̃1,f , m̃1 by m̃1,f , K = Kf , (11) with m∗SM ≈ 1.08 × 10−3 eV and m∗MSSM ≈ sin2(β) × 1.58 × 10−3 eV. Figure 3 shows the flavour-specific efficiency factor ηf in the MSSM. Maximal efficiency for a specific flavour corresponds to Kf ≈ 1 (m̃1,f ≈ m∗). The most relevant difference between the flavour-independent approximation and the correct flavour-dependent treatment is the fact that in the latter, the total baryon asymmetry is the sum of each individual lepton asymmetries, which is weighted by the corresponding efficiency factor. Therefore, upon summing over the lepton asymmetries, the total baryon number is generically not proportional to the sum over the CP asymme- tries, ε1 = f ε1,f , as in the flavour-independent approximation where the lepton flavour is neglected in the Boltzmann equations. In other words, in the flavour-independent ap- proximation the total baryon asymmetry is a function of f ε1,f × ηind ( g Kg). In the correct flavour treatment the baryon asymmetry is (approximately) a function of∑ f ε1,fη (AffKf , K). From this, it is already clear that flavour-dependent effects can have important consequences also in type II leptogenesis. The most important quantities for computing the produced baryon asymmetry are thus the decay asymmetries ε1,f and the efficiency factors ηf (which depend mainly on m̃1,f and m̃1 (or Kf and K)). While the efficiency factors can be computed similarly to the type I seesaw case, important differences between leptogenesis in type I and type II seesaw scenarios arise concerning the decay asymmetries as well as concerning the connection between leptogenesis and seesaw parameters. 4 Decay asymmetries 4.1 Right-handed neutrinos plus triplets Regarding the decay asymmetry in the type II seesaw mechanism, where the direct mass term for the neutrinos stems from the induced vev of a Higgs triplet, there are new contributions from 1-loop diagrams where virtual SU(2)L-triplet scalar fields (or their superpartners) are exchanged in the loop. The relevant diagrams for the decay ν1R → LfaHub in the limit M1 ≪ MR2,MR2,M∆ are shown in figure 4. Compared to the type I seesaw framework, the new contributions are the diagrams (c) and (f). The calculation of the corresponding decay asymmetries for each lepton flavour yields 1,f = j 6=1 Im [(Y †)1f(Y ν Yν)1j(Y T )jf ] ν Yν)11 1− (1 + xj) ln xj + 1 , (12a) 1,f = j 6=1 Im [(Y †)1f(Y ν Yν)1j(Y T )jf ] ν Yν)11 1− xj , (12b) 1,f = − g Im [(Y ν )f1(Y ν )g1(m LL)fg] ν Yν)11 −1 + y ln y + 1 , (12c) 1,f = j 6=1 Im [(Y †)1f(Y ν Yν)1j(Y T )jf ] ν Yν)11 −1 + xj ln xj + 1 , (12d) 1,f = j 6=1 Im [(Y †)1f(Y ν Yν)1j(Y T )jf ] ν Yν)11 1− xj , (12e) 1,f = − g Im [(Y ν )f1(Y ν )g1(m LL)fg] ν Yν)11 1− (1 + y) ln y + 1 , (12f) ∆̃1, ∆̃2 Figure 4: Loop diagrams in the MSSM which contribute to the decay ν1 → LfaHub for the case of a type II seesaw mechanism where the direct mass term for the neutrinos stems from the induced vev of a Higgs triplet. In diagram (f), ∆̃1 and ∆̃2 are the mass eigenstates corresponding to the superpartners of the SU(2)L-triplet scalar fields ∆ and ∆̄. The SM diagrams are the ones where no superpartners (marked by a tilde) are involved and where Hu is renamed to the SM Higgs φ. where y := M2∆/M R1 and xj := M R1 for j 6= 1 and where we assume hierarchical right-handed neutrino masses and M∆ ≫ MR1.4 The MSSM results for the type II contributions have been derived in [16]. In the SM, the results in [16] correct the previous result of [15] by a factor of −3/2. In equation (12) they have been generalised to the flavour-dependent case. The results for the contributions to the decay asymmetries from the triplet in the SM and from the triplet superfield in the MSSM are SM,II 1,f = ε 1,f , (13a) MSSM,II 1,f = ε 1,f + ε 1,f . (13b) In the MSSM, we furthermore obtain MSSM,II 1,f = ε MSSM,II 1, ef MSSM,II MSSM,II e1, ef . (14) The results corresponding to the diagrams (a), (b), (d) and (e) which contribute to εI1 in the type I seesaw in the SM and in the MSSM, have been presented first in [39]. The results for the type I contribution to the decay asymmetries in the SM and in the 4Integrating out the heavy particles ν2 ,∆ (and their superpartners) in figure 4 leads to an effective approach involving the dimension 5 neutrino mass operator (c.f. figures 1, 2 and 5), as will be discussed in section 4.2. We note that there are additional diagrams not shown in figure 4 (since they are generically suppressed for M1 ≪ MR2,MR2,M∆) which are related to the dimension 6 operator containing two lepton doublets, two Higgs doublets and a derivative. MSSM are 1,f = ε 1,f + ε 1,f , (15a) MSSM,I 1,f = ε 1,f + ε 1,f + ε 1,f + ε 1,f . (15b) Again, in the MSSM, the remaining decay asymmetries are equal to ε MSSM,I 1,f : MSSM,I 1,f = ε MSSM,I 1, ef MSSM,I MSSM,I e1, ef . (16) Finally, the total decay asymmetries from the decay of ν1R in the type II seesaw, where the direct mass term for the neutrinos stems from the induced vev of a Higgs triplet, are given by εSM1,f = ε 1,f + ε SM,II 1,f , (17) εMSSM1,f = ε MSSM,I 1,f + ε MSSM,II 1,f . (18) It is interesting to note that the type I results can be brought to a form which contains the neutrino mass matrix using j 6=1 Im [(Y †)1f (Y ν Yν)1j(Y T )jf ] 8π (Y ν Yν)11 = −MR1 g Im [(Y ν )f1(Y ν )g1(m LL)fg] 8π (Y ν Yν)11 .(19) In the limit y ≫ 1 and xj ≫ 1 for all j 6= 1, which corresponds to a large gap between the mass MR1 and the masses MR2, MR3 and M∆, we obtain the simple results for the flavour-specific decay asymmetries εSM1,f and ε 1,f [16] εSM1,f = g Im [(Y ν )f1(Y ν )g1(m LL +m LL)fg] ν Yν)11 , (20a) εMSSM1,f = g Im [(Y ν )f1(Y ν )g1(m LL +m LL)fg] ν Yν)11 . (20b) In the presence of such a mass gap, the calculation can also be performed in an effective approach after integrating out the two heavy right-handed neutrinos and the heavy triplet, as we now discuss. 4.2 Effective approach to leptogenesis Let us now explicitly use the assumption that the lepton asymmetry is generated via the decay of the lightest right-handed neutrino and that all other additional particles, in particular the ones which generate the type II contribution, are much heavier than MR1. Furthermore, we assume that we can neglect their population in the early universe, e.g. that their masses are much larger than the reheat temperature TRH and that they are Figure 5: Loop diagrams contributing to the decay asymmetry via the decay ν1 → LfaHub in the MSSM with a (lightest) right-handed neutrino ν1 and a neutrino mass matrix determined by κ′ [16]. Further contributions to the generated baryon asymmetry stem from the decay of ν1 into slepton and Higgsino and from the decays of the sneutrino ν̃1 . With Hu renamed to the SM Higgs, the first diagram contributes in the extended SM. not produced non-thermally in a large amount. Under these assumptions we can apply an effective approach to leptogenesis, which is independent of the mechanism which generates the additional (type II) contribution to the neutrino mass matrix [16]. For this minimal effective approach, it is convenient to isolate the type I contribution from the lightest right-handed neutrino as follows: (mνLL)fg = − 2(Yν)f1M R1 (Y ν )1g + (κ ′∗)fg . (21) κ′ includes type I contributions from the heavier right-handed neutrinos, plus any ad- ditional (type II) contributions from heavier particles. Examples for realisations of the neutrino mass operator can be found, e.g., in [40]. At MR1, the minimal effective field theory extension of the SM (MSSM) for lepto- genesis includes the effective neutrino mass operator κ′ plus one right-handed neutrino ν1R with mass MR1 and Yukawa couplings (Yν)f1 to the lepton doublets L f , defined as −(Yν)f1(Lf · φ) ν1R in the Lagrangian of the SM and, analogously, as (Yν)f1(L̂f · Ĥu) ν̂C1 in the superpotential of the MSSM. The contributions to the decay asymmetries in the effective approach stem from the interference of the diagram(s) for the tree-level decay of νR1 (and ν̃R1) with the loop diagrams containing the effective operator, shown in figure 5. In the SM, we obtain the simple result [16] for the flavour-specific effective decay asymmetries (corresponding to diagram (a) of figure 5) εSM1,f = g Im [(Y ν )f1(Y ν )g1(m LL)fg] ν Yν)11 . (22) For the supersymmetric case, diagram (a) and diagram (b) contribute to εMSSM1,f and we obtain [16]: εMSSM1,f = g Im [(Y ν )f1(Y ν )g1(m LL)fg] ν Yν)11 . (23) Explicit calculation furthermore yields εMSSM1,f = ε 1, ef = εMSSM = εMSSM e1, ef . (24) The results are independent of the details of the realisation of the neutrino mass operator κ′. Note that, since the diagrams where the lightest right-handed neutrino runs in the loop do not contribute to leptogenesis, we have written mνLL = −v2u(κ)∗/2 instead of m′νLL := −v2u(κ′)∗/2 in the formulae in equations (22) - (23). The decay asymmetries are directly related to the neutrino mass matrix mνLL. For neutrino masses via the type I seesaw mechanism, the results are in agreement with the known results [39], in the limit MR2,MR3 ≫ MR1. The results obtained in the effective approach are also in agreement with our full theory calculation in the type II scenarios with SU(2)L-triplets in equation (12) [16], in the limit M∆ ≫ MR1. 5 Type II bounds on decay asymmetries and on MR1 In the limit MR2,MR3,M∆ ≫ MR1 (or alternatively in the effective approach), upper bounds for the total decay asymmetries in type II leptgenesis, i.e. for the sums |εSM1 | = 1,f | and |εMSSM1 | = | 1,f |, have been derived in [16]. For the flavour-specific decay asymmetries εSM1,f and ε 1,f , the bounds can readily be obtained as |εSM1,f | ≤ mνmax , |εMSSM1,f | ≤ mνmax . (25) They are thus identical to the bounds for the total asymmetries. In particular, they also increase with increasing mass scale of the light neutrinos. Note that, compared to the low energy value, the neutrino masses at the scale MR1 are enlarged by renormalization group running by ≈ +20% in the MSSM and ≈ +30% in the SM, which raises the bounds on the decay asymmetries by the same values (see e.g. figure 4 of [41]). A situation where an almost maximal baryon asymmetry is generated by thermal leptogenesis can be realised, for example, if the total decay asymmetry nearly saturates its upper bound and if, in addition, the washout parameters m̃1,f for all three flavours approximately take its optimal value. Classes of type II seesaw models, where this can be accommodated, have been considered in [8, 42, 17]. In these so-called “type- II-upgraded” seesaw models, the type II contribution to the neutrino mass matrix is proportional to the unit matrix (enforced e.g. by an SO(3) flavour symmetry or by one of its non-Abelian discrete subgroups). From equation (20), one can readily see that if the type II contribution (∝ 1) dominates the neutrino mass matrix mνLL, and if (Yν)f1 are approximately equal for all flavours f = 1, 2, 3 and chosen such that the resulting m̃1,f are approximately equal to m ∗, we have realised ηf ≈ ηmax for all flavours and simultaneously nearly saturated the bound for the total decay asymmetry.5 5We further note that the bound for one of the flavour-specific decay asymmetries can be nearly saturated in this scenario if, for instance, (Yν)21 ≈ (Yν)31 ≈ 0. 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL -0.75 -0.25 type I seesaw type II seesaw Figure 6: Bound on the decay asymmetry ε1,f in type II leptogenesis (solid blue line) and type I lepto- genesis (dotted red line) as a function of the mass of the lightest neutrino mν := min (mν1 ,mν3 ,mν3) in type I and type II seesaw scenarios (see also [29]). The washout parameter |Aff |m̃1,f is fixed to m∗ (close to optimal), and the asymmetry is normalised to ε max,0 = 3MR1 (∆m )1/2/(16π v ), where ≈ 2.5× 10−3 eV2 is the atmospheric neutrino mass squared difference. We have considered the MSSM with tanβ = 30 as an explicit example. Assuming a maximal efficiency factor ηmax for all flavours in a given scenario, and taking an upper bound for the masses of the light neutrinos mνmax as well as the observed value nB/nγ ≈ (6.0965 ± 0.2055) × 10−10 [2] for the baryon asymmetry, equation (25) can be transformed into lower type II bounds for the mass of the lightest right-handed neutrino [16]: MSMR1 ≥ mνmax nB/nγ 0.99 · 10−2 ηmax , MMSSMR1 ≥ mνmax nB/nγ 0.92 · 10−2 ηmax . (26) The bound on MR1 is lower for a larger neutrino mass scale. The situation in the type II framework differs from the type I seesaw case: In the latter, the flavour-specific decay asymmetries are constrained by [24] |εI,SM1,f | ≤ mνmax m̃1,f , |εI,MSSM1,f | ≤ mνmax m̃1,f . (27) Note that compared to the type II bounds, there is an extra factor of (m̃1,f/m̃1) which depends on the washout parameters. As we shall now discuss, this factor implies that it is not possible to have a maximal decay asymmetry ε1,f and an optimal washout parameter m̃1,f simultaneously. Let us recall first that in the type I seesaw, in contrast to the type II case, the flavour-independent washout parameter has the lower bound [43] m̃1 ≥ mνmin , (28) with mνmin = min (mν1 , mν3 , mν3). On the contrary, in the type I and type II seesaw, the flavour-dependent washout parameters m̃1,f are generically not constrained. Note that in 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL type I seesaw type II seesaw Figure 7: Lower bound on MR1 in type II leptogenesis (solid blue line) and type I leptogenesis (dotted red line) as a function of the mass of the lightest neutrinomν := min (mν1 ,mν3 ,mν3). For definiteness, the MSSM with tanβ = 30 has been considered as an example. the flavour-independent approximation, Eq. (28) leads to a dramatically more restrictive bound on ε1 = f ε1,f [44] for quasi-degenerate light neutrino masses, and finally even to a bound on the neutrino mass scale [43]. This can be understood from the fact that for m̃1 ≫ m∗ in the flavour-independent approximation, washout effects strongly reduce the efficiency of thermal leptogenesis. Similarly, in the flavour-dependent treatment, m̃1,f ≫ m∗ would lead to a strongly reduced efficiency for this specific flavour. This strong washout for quasi-degenerate light neutrinos can be avoided in flavour-dependent type I leptogenesis, and m̃1,f ≈ m∗ can realise a nearly optimal scenario regarding washout (c.f. figure 3). However, we see from equation (27) that the decay asymmetries in this case are reduced by a factor of (m∗/mνmin) 1/2 when compared to the optimal value, leading to a reduced baryon asymmetry. On the other hand, realizing nearly optimal ε1,f requires m̃1,f ≈ m̃1 ≥ mνmin, leading to large washout effects for quasi-degenerate light neutrinos and even to a more strongly suppressed generation of baryon asymmetry (c.f. figure 3). As a consequence, increasing the neutrino mass scale increases the lower bound on MR1 (also in the presence of flavour-dependent effects), in contrast to the type II seesaw case. Comparing the type II and type I seesaw cases, in the latter the baryon asymmetry is suppressed for quasi-degenerate light neutrino masses either by a factor (m∗/mνmin) in the decay asymmetries or by a non-optimal washout parameter much larger than m∗ (or Kf ≫ 1, c.f. figure 3). The bounds on the decay asymmetries in type I and type II leptogenesis are compared in figure 6, where m̃1,f has been fixed to m ∗, close to its optimal value. From figure 6 we see that in the type I case the maximal baryon asymme- try is obtained for hierarchical neutrino masses, whereas in the type II case, increasing the neutrino mass scale increases the produced baryon asymmetry and therefore allows to relax the bound on MR1, as shown in figure 7. In addition, for the same reason, increasing the neutrino mass scale also relaxes the lower bound on the reheat tempera- ture TRH from the requirement of successful type II leptogenesis. Including reheating in the flavour-dependent Boltzmann equations as in Ref. [29] (for the flavour-independent 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL 2´109 4´109 6´109 8´109 type I seesaw 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL 2´109 4´109 6´109 8´109 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL 5´108 1´109 1.5´109 2´109 type II seesaw 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL 5´108 1´109 1.5´109 2´109 Figure 8: Lower bound on the reheat temperature TRH in type I leptogenesis (left panel) and in type II leptogenesis (right panel) as a function of the mass of the lightest neutrino mν = min (mν1 ,mν3 ,mν3), in the MSSM with tanβ = 30. In the grey regions, values of TRH are incompatible with thermal leptogenesis for the corresponding mν case, see [45]), we obtain the mνmin-dependent lower bounds on TRH in type I and type II scenarios shown in figure 8. While the bound decreases in type II leptogenesis by about an order of magnitude when the neutrino mass scale increases to 0.4 eV, it in- creases in the type I seesaw case. In the presence of upper bounds on TRH, this can lead to constraints on the neutrino mass scale , i.e. on mνmin = min (mν1 , mν3, mν3). For instance, with an upper bound TRH ≤ 5× 109 GeV, values of mνmin in the approximate range [0.01 eV, 0.32 eV] would be incompatible with leptogenesis in the type I seesaw framework (c.f. figure 8). 6 Summary, discussion and conclusions We have analysed flavour-dependent leptogenesis via the out-of-equilibrium decay of the lightest right-handed neutrino in type II seesaw scenarios, where, in addition to the type I seesaw, an additional direct mass term for the light neutrinos is present. We have considered type II seesaw scenarios where this additional contribution stems from the vacuum expectation value of a Higgs triplet, and furthermore an effective approach, which is independent of the mechanism which generates the additional (type II) contribution to neutrino masses. We have taken into account flavour-dependent effects, which are relevant if thermal leptogenesis takes place at temperatures below circa 1012 GeV in the SM and below circa (1 + tan2 β) × 1012 GeV in the MSSM. As in type I leptogenesis, in the flavour-dependent regime the decays of the right-handed (s)neutrinos generate asymmetries in in each distinguishable flavour (proportional to the flavour-specific decay asymmetries ε1,f), which are differently washed out by scattering processes in the thermal bath, and thus appear with distinct weights (efficiency factors ηf ) in the final baryon asymmetry. The most important quantities for computing the produced baryon asymmetry are the decay asymmetries ε1,f and the efficiency factors ηf (which mainly depend on washout parameters m̃1,f and m̃1 = f m̃1,f). With respect to the flavour-specific efficiency factors ηf , in the limit that the mass M∆L of the triplet is much larger than MR1 (and MR1 ≪ MR2,MR3), they can be estimated from the same Boltzmann equa- tions as in the type I seesaw framework. Regarding the decay asymmetries ε1,f , in the type II seesaw case there are additional contributions where virtual Higgs triplets (and their superpartners) run in the 1-loop diagrams. Here, we have generalised the results of [16] to the flavour-dependent case. The most important effects of flavour in leptoge- nesis are a consequence of the fact that in the flavour-independent approximation the total baryon asymmetry is a function of f ε1,f × ηind ( g m̃1,g), whereas in the cor- rect flavour-dependent treatment the baryon asymmetry is (approximately) a function f ε1,fη (Affm̃1,f , m̃1). We have then investigated the bounds on the flavour-specific decay asymmetries ε1,f . In the type I seesaw case, it is known that the bound on the flavour-specific asymmetries εI1,f is substantially relaxed [24] compared to the bound on ε 1,f [44] in the case of a quasi-degenerate spectrum of light neutrinos. For experimentally allowed light neutrino masses below about 0.4 eV, there is no longer a bound on the neutrino mass scale from the requirement of successful thermal leptogenesis. In the type II seesaw case, we have derived the bound on the flavour-specific decay asymmetries ε1,f = ε 1,f + ε 1,f , which turns out to be identical to the bound on the total decay asymmetry ε1 = f ε1,f . We have compared the bound on the flavour-specific decay asymmetries in type I and type II scenarios, and found that while the type II bound increases with the neutrino mass scale, the type I bound decreases (for experimentally allowed light neutrino masses below about 0.4 eV). The relaxed bound on ε1,f (figure 6) leads to a lower bound on the mass of the lightest right-handed neutrino MR1 in the type II seesaw scenario (figure 7), which decreases when the neutrino mass scale increases. Furthermore, it leads to a relaxed lower bound on the reheat temperature TRH of the early universe (figure 8), which helps to improve consistency of thermal leptogenesis with upper bounds on TRH in some supergravity models. This is in contrast to the type I seesaw scenario, where the lower bound on TRH from thermal leptogenesis increases with increasing neutrino mass scale. Constraints on TRH can therefore imply constraints on the mass scale of the light neutrinos also in flavour-dependent type I leptogenesis, although a general bound is absent. We have furthermore argued that these relaxed bounds on ε1,f MR1 and TRH in the type II case can be nearly saturated in an elegant way in classes of so-called “type- II-upgraded” seesaw models [8], where the type II contribution to the neutrino mass matrix is proportional to the unit matrix (enforced e.g. by an SO(3) flavour symmetry or by one of its non-Abelian subgroups). One interesting application of these type II seesaw scenarios is that the consistency of thermal leptogenesis with unified theories of flavour is improved compared to the type I seesaw case. This effect, investigated in the flavour-independent approximation in [17], is also present analogously in the flavour- dependent treatment of leptogenesis. The reason is that if the type II contribution (∝ 1) dominates, the decay asymmetries ε1,f become approximately equal and the estimate for the produced baryon asymmetry is similar to the flavour-independent case. Nevertheless, an accurate analysis of leptogenesis in this scenario requires careful inclusion of the flavour-dependent effects. In many applications and realisations of type II leptogenesis in specific models of fermion masses and mixings (see e.g. [18]), flavour-dependent effects may substantially change the results and they therefore have to be taken into account. In summary, type II leptogenesis provides a well-motivated generalisation of the conventional scenario of leptogenesis in the type I seesaw framework. We have argued that flavour-dependent effects have to be included in type II leptogenesis, and can change predictions of existing models as well as open up new possibilities for for successful models of leptogenesis. 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Ratz, Nucl. Phys. B674 (2003), 401– [42] S. Antusch and S. F. King, Phys. Lett. B 591 (2004) 104 [arXiv:hep-ph/0403053]. [43] W. Buchmüller, P. Di Bari, and M. Plümacher, Nucl. Phys. B665 (2003), 445–468. [44] S. Davidson and A. Ibarra, Phys. Lett. B535 (2002), 25–32. [45] G. F. Giudice, A. Notari, M. Raidal, A. Riotto, and A. Strumia, Nucl. Phys. B 685, 89 (2004). http://arxiv.org/abs/hep-ph/0403053 Introduction Type I and type II seesaw mechanisms Baryogenesis via flavour-dependent leptogenesis Decay asymmetries Right-handed neutrinos plus triplets Effective approach to leptogenesis Type II bounds on decay asymmetries and on bold0mu mumu MR1MR13.91663pt plus 1.95831pt minus 1.30554ptMR1MR1MR1MR1 Summary, discussion and conclusions

We reanalyse leptogenesis via the out-of-equilibrium decay of the lightest right-handed neutrino in type II seesaw scenarios, taking into account flavour-dependent effects. In the type II seesaw mechanism, in addition to the type I seesaw contribution, an additional direct mass term for the light neutrinos is present. We consider type II seesaw scenarios where this additional contribution arises from the vacuum expectation value of a Higgs triplet, and furthermore an effective model-independent approach. We investigate bounds on the flavour-specific decay asymmetries, on the mass of the lightest right-handed neutrino and on the reheat temperature of the early universe, and compare them to the corresponding bounds in the type I seesaw framework. We show that while flavour-dependent thermal type II leptogenesis becomes more efficient for larger mass scale of the light neutrinos, and the bounds become relaxed, the type I seesaw scenario for leptogenesis becomes more constrained. We also argue that in general, flavour-dependent effects cannot be ignored when dealing with leptogenesis in type II seesaw models.

Introduction Leptogenesis [1] is one of the most attractive and minimal mechanisms for explaining the observed baryon asymmetry of the universe nB/nγ ≈ (6.0965 ± 0.2055) × 10−10 [2]. A lepton asymmetry is dynamically generated and then converted into a baryon asym- metry due to (B + L)-violating sphaleron interactions [3] which exist in the Standard Model (SM) and its minimal supersymmetric extension, the MSSM. Leptogenesis can be implemented within the type I seesaw scenario [4], consisting of the SM (MSSM) plus three right-handed Majorana neutrinos (and their superpartners) with a hierarchical spectrum. In thermal leptogenesis [5], the lightest of the right-handed neutrinos is pro- duced by thermal scattering after inflation, and subsequently decays out-of-equilibrium in a lepton number and CP-violating way, thus satisfying Sakharov’s constraints [6]. In models with a left-right symmetric particle content like minimal left-right sym- metric models, Pati-Salam models or Grand Unified Theories (GUTs) based on SO(10), the type I seesaw mechanism is typically generalised to a type II seesaw [7], where an additional direct mass term mIILL for the light neutrinos is present. From a model independent perspective, the type II mass term can be considered as an additional con- tribution to the lowest dimensional effective neutrino mass operator. In most explicit models, the type II contribution stems from seesaw suppressed induced vevs of SU(2)L- triplet Higgs fields. One motivation for considering the type II seesaw is that it allows to construct unified flavour models for partially degenerate neutrinos in an elegant way, e.g. via a type II upgrade [8], which is otherwise difficult to achieve in type I models. For leptogenesis in type II seesaw scenarios with SU(2)L-triplet Higgs fields, there are in general two possibilities to generate the baryon asymmetry: via decays of the lightest right-handed neutrinos or via decays of the SU(2)L-triplets [9,10,11,12]. In the first case, there are additional one-loop diagrams where virtual triplets are running in the loop [9, 13, 14, 15, 16]. In the following, we focus on this possibility, and assume hierarchical right-handed neutrino masses (and that the triplets are heavier than ν1R). In this limit, to a good approximation the decay asymmetry depends mainly on the low energy neutrino mass matrix mνLL = m LL + m LL and on the Yukawa couplings to the lightest right-handed neutrino and its mass [16]. It has been shown that type II leptogenesis imposes constraints on the seesaw parameters, which, in the flavour- independent approximation, differ substantially from the constraints in the type I case. For instance, the bound on the decay asymmetry increases with increasing neutrino mass scale [16], in contrast to the type I case where it decreases. As a consequence, the lower bound on the mass of the lightest right-handed neutrino from leptogenesis decreases for increasing neutrino mass scale [16]. One interesting application of type II leptogenesis is the possibility to improve consistency of classes of unified flavour models with respect to thermal leptogenesis [17]. Finally, since the type II contribution typically does not effect washout, there is no bound on the absolute neutrino mass scale from type II leptogenesis, as has been pointed out in [15]. For further applications and realisations of type II leptogenesis in specific models of fermion masses and mixings, see e.g. [18]. In recent years, the impact of flavour in thermal leptogenesis has merited increasing attention [19] - [38]. In fact, the one-flavour approximation is only rigorously correct when the interactions mediated by the charged lepton Yukawa couplings are out of equilibrium. Below a given temperature (e.g. O(1012GeV) in the SM and (1+tan2 β)× O(1012GeV) in the MSSM), the tau Yukawa coupling comes into equilibrium (later followed by the couplings of the muon and electron). Flavour effects are then physical and become manifest, not only at the level of the generated CP asymmetries, but also regarding the washout processes that destroy the asymmetries created for each flavour. In the full computation, the asymmetries in each distinguishable flavour are differently washed out, and appear with distinct weights in the final baryon asymmetry. Flavour-dependent leptogenesis in the type I seesaw scenario has recently been ad- dressed in detail by several authors. In particular, flavour-dependent effects in lepto- genesis have been studied, and shown to be relevant, in the two right-handed neutrino models [24] as well as in classes of neutrino mass models with three right-handed neu- trinos [26]. The quantum oscillations/correlations of the asymmetries in lepton flavour space have been included in [22, 32, 33, 35] and the treatment has been generalised to the MSSM [26,29]. Effects of reheating, and constraints on the seesaw parameters from upper bounds on the reheat temperature, have been investigated in [29]. Leptogenesis bounds on the reheat temperature [29] and on the mass of the lightest right-handed neutrino [29, 36] have also been considered including flavour-dependent effects. Strong connections between the low-energy CP phases of the UMNS matrix and CP violation for flavour-dependent leptogenesis have been shown to emerge in certain classes of neutrino mass models [26] or under the hypothesis of no CP violation sources associated with the right-handed neutrino sector (real R) [25,27,28,31]. Possible effects regarding the decays of the heavier right-handed neutrinos for leptogenesis have been discussed in this con- text in [21, 34], and flavour-dependent effects for resonant leptogenesis were addressed in [38]. Regarding the masses of the light neutrinos, assuming hierarchical right-handed neutrinos and considering experimentally allowed light neutrino masses (below about 0.4 eV), there is no longer a bound on the neutrino mass scale from thermal leptogenesis if flavour-dependent effects are included [24]. In view of the importance of flavour-dependent effects on leptogenesis in the type I seesaw case, it is pertinent to investigate their effects on type II leptogenesis. In this paper, we therefore reanalyse leptogenesis via the out-of-equilibrium decay of the lightest right-handed neutrino in type II seesaw scenarios, taking into account flavour-dependent effects. We investigate bounds on the decay asymmetries, on the mass of the lightest right-handed neutrino and on the reheat temperature of the early universe, and discuss how increasing the neutrino mass scale affects thermal leptogenesis in the type I and type II seesaw frameworks. 2 Type I and type II seesaw mechanisms Motivated by left-right symmetric unified theories, we consider two generic possibilities for explaining the smallness of neutrino masses: via heavy SM (MSSM) singlet fermions (i.e. right-handed neutrinos) [4] and via heavy SU(2)L-triplet Higgs fields [7]. In both cases, the effective dimension five operator for Majorana neutrino masses in the SM or p2≪M2 Figure 1: Generation of the dimension 5 neutrino mass operator in the type I seesaw mechanism. the MSSM, respectively, κgf (LC · φ) (Lf · φ) + h.c. , (1a) κ = − κgf (L̂ g · Ĥu) (L̂f · Ĥu) + h.c. , (1b) is generated from integrating out the heavy fields. This is illustrated in figures 1 and 2. In equation (1), the dots indicate the SU(2)L-invariant product, (L̂ f · Ĥu) = L̂fa(iτ2)ab(Ĥu)b, with τA (A ∈ {1, 2, 3}) being the Pauli matrices. Superfields are marked by hats. After electroweak symmetry breaking, the operators of equation (1) lead to Majorana mass terms for the light neutrinos, Lν = −12m LLνLν L , with m LL = − (κ)∗ . (2) In the type I seesaw mechanism, it is assumed that only the singlet (right-handed) neutrinos νRi contribute to the neutrino masses. With Yν being the neutrino Yukawa matrix in left-right convention,1 MRR the mass matrix of the right-handed neutrinos and vu = 〈φ0〉 (= 〈H0u〉) the vacuum expectation value of the Higgs field which couples to the right-handed neutrinos, the effective mass matrix of the light neutrinos is given by the conventional type I seesaw formula mILL = −v2u Yν M−1RR Y ν . (3) In the type II seesaw mechanism, the contributions to the neutrino mass matrix from both, right-handed neutrinos νRi and Higgs triplet(s) ∆L, are considered. The additional contribution to the neutrino masses from ∆L can be understood in two ways: as another contribution to the effective neutrino mass operator in the low energy effective theory or, equivalently, as a direct mass term after the Higgs triplet obtains an induced small vev after electroweak symmetry breaking (c.f. figure 2). The neutrino mass matrix in the type II seesaw mechanism has the form mνLL = m LL +m LL = m LL − v2uYνM−1RRY ν , (4) 1The neutrino Yukawa matrix corresponds to −(Yν)fi(Lf · φ) νiR in the Lagrangian of the SM and, analogously, to (Yν)fi(L̂ f · Ĥu) ν̂Ci in the superpotential of the MSSM (see [16] for further details). p2≪M2 Figure 2: Extra diagram generating the dimension 5 neutrino mass operator in the type II seesaw mechanism from a SU(2)L-triplet Higgs field. where mIILL is the additional term from the Higgs triplet(s). In left-right symmetric unified theories, the generic size of both seesaw contributionsmILL andm LL isO(v2u/vB−L) where vB−L is the B-L breaking scale (i.e. the mass scale of the right-handed neutrinos and of the Higgs triplet(s)). 3 Baryogenesis via flavour-dependent leptogenesis Flavour-dependent effects can have a strong impact in baryogenesis via thermal lep- togenesis [19] - [38]. The effects are manifest not only in the flavour-dependent CP asymmetries, but also in the flavour-dependence of scattering processes in the thermal bath, which can destroy a previously produced asymmetry. The relevance of the flavour-dependent effects depends on the temperatures at which thermal leptogenesis takes place, and thus on which interactions mediated by the charged lepton Yukawa couplings are in thermal equilibrium. For example, in the MSSM, for temperatures between circa (1+tan2 β)×105GeV and (1+tan2 β)×109GeV, the µ and τ Yukawa couplings are in thermal equilibrium and all flavours in the Boltzmann equations are to be treated separately. For tan β = 30, this applies for temperatures below about 1012 GeV and above 108GeV, a temperature range which is of most interest for thermal leptogenesis in the MSSM. In the SM, in the temperature range between circa 109 GeV and 1012 GeV, only the τ Yukawa coupling is in equilibrium and is treated separately in the Boltzmann equations, whereas µ and e flavours are indistinguishable. A discussion of the temperature regimes in the SM and MSSM, where flavour is important, can be found, e.g., in [26]. We now briefly review the estimation of the produced baryon asymmetry in flavour- dependent leptogenesis.2 For definiteness, we focus on the temperature range where all flavours are to be treated separately. In the following discussion of thermal type II leptogenesis, we will assume that the mass M∆L of the triplet(s) is much larger than MR1. In this limit, the flavour-dependent efficiencies calculated in the type I seesaw scenario can also be used in the type II framework. The out-of-equilibrium decays of the heavy right-handed (s)neutrinos ν1R and ν̃ R give rise to flavour-dependent asymmetries in the (s)lepton sector, which are then partly transformed via sphaleron conversion into 2For a discussion of approximations which typically enter these estimates, and which also apply to our discussion, see e.g. section 3.1.3 in [29]. a baryon asymmetry YB. 3 The final baryon asymmetry can be calculated as Y SMB = Y SM∆f , (5) Y MSSMB = Ŷ MSSM∆f , (6) where Ŷ∆f ≡ YB/3 − YLf are the total (particle and sparticle) B/3 − Lf asymmetries, with YLf the lepton number densities in the flavour f = e, µ, τ . The asymmetries Ŷ and Y SM∆f , which are conserved by sphalerons and by the other SM (MSSM) interactions, are then usually calculated by solving a set of coupled Boltzmann equations, describing the evolution of the number densities as a function of temperature. It is convenient to parameterise the produced asymmetries in terms of flavour-specific efficiency factors ηf and decay asymmetries ε1,f as Y SM∆f = η f ε1,f Y , (7) Ŷ MSSM∆f = η (ε1,f + ε1, ef) Y (εe1,f + εe1, ef) Y . (8) and Y are the number densities of the neutrino and sneutrino for T ≫ M1 if they were in thermal equilibrium, normalised with respect to the entropy density. In the Boltzmann approximation, they are given by Y ≈ Y eq ≈ 45/(π4g∗). g∗ is the effective number of degrees of freedom, which amounts 106.75 in the SM and 228.75 in the MSSM. ε1,f , ε1, ef , εe1,f and εe1, ef are the decay asymmetries for the decay of neutrino into Higgs and lepton, neutrino into Higgsino and slepton, sneutrino into Higgsino and lepton, and sneutrino into Higgs and slepton, respectively, defined by ε1,f = − Γν1 f (Γν1RLf + Γν1RLf 1, ef f (Γν1 εe1,f = Γeν∗1 f (Γeν∗1R Lf + Γeν1RLf , εe1, ef = f(Γeν1 . (9) The flavour-dependent efficiency factors ηf in the SM and in the MSSM are defined by Eqs. (7) and (8), respectively. As stated above, we assume that the mass M∆L of the triplet(s) is much larger than MR1. In this limit, the efficiencies for flavour-dependent thermal leptogenesis in the type I and type II frameworks are mainly determined by the properties of ν1R, which means in particular that the flavour-dependent efficiencies 3In the following, Y will always be used for quantities which are normalised to the entropy density s. The quantities normalised with respect to the photon density can be obtained using the relation s/nγ ≈ 7.04k. -2 -1 0 1 2 ÈAff K f È ������������������������������� È Aff K f È ������������������������������� È Aff K f È ������������������������������� È Aff K f È = 100 Figure 3: Flavour-dependent efficiency factor η(AffKf ,K) in the MSSM as a function of AffKf , for fixed values of K/|AffKf | = 2, 5 and 100, obtained from solving the flavour-dependent Boltzmann equations in the MSSM with zero initial abundance of right-handed (s)neutrinos (figure from [26]). A is a matrix which appears in the Boltzmann equations (see [19, 24] for A in the SM and [26] for the MSSM case), and which has diagonal elements |Aff | of O(1). The small off-diagonal entries of A have been neglected, which is a good approximation in most cases. In general, however, they have to be included. More relevant than the differences in the flavour-dependent efficiency factors for different K/|AffKf | is that the total baryon asymmetry is the sum of each individual lepton asymmetries, which is weighted by the corresponding efficiency factors. calculated in the type I seesaw scenario can also be used in the type II framework. In the definition of the efficiency factor, the equilibrium number densities serve as a nor- malization: A thermal population νR1 (and ν̃R1) decaying completely out of equilibrium (without washout effects) would lead to ηf = 1. The efficiency factors can be computed by means of the flavour-dependent Boltzmann equations, which can be found for the SM in [19,22,23,24] and for the MSSM in [26,29]. In general, the flavour-dependent efficiencies depend strongly on the washout parameters m̃1,f for each flavour, and on the total washout parameter m̃1, which are defined as m̃1,f = v2u |(Yν)f1|2 , m̃1 = m̃1,f . (10) Alternatively, one may use the quantities Kf , K, which are related to m̃1,f , m̃1 by m̃1,f , K = Kf , (11) with m∗SM ≈ 1.08 × 10−3 eV and m∗MSSM ≈ sin2(β) × 1.58 × 10−3 eV. Figure 3 shows the flavour-specific efficiency factor ηf in the MSSM. Maximal efficiency for a specific flavour corresponds to Kf ≈ 1 (m̃1,f ≈ m∗). The most relevant difference between the flavour-independent approximation and the correct flavour-dependent treatment is the fact that in the latter, the total baryon asymmetry is the sum of each individual lepton asymmetries, which is weighted by the corresponding efficiency factor. Therefore, upon summing over the lepton asymmetries, the total baryon number is generically not proportional to the sum over the CP asymme- tries, ε1 = f ε1,f , as in the flavour-independent approximation where the lepton flavour is neglected in the Boltzmann equations. In other words, in the flavour-independent ap- proximation the total baryon asymmetry is a function of f ε1,f × ηind ( g Kg). In the correct flavour treatment the baryon asymmetry is (approximately) a function of∑ f ε1,fη (AffKf , K). From this, it is already clear that flavour-dependent effects can have important consequences also in type II leptogenesis. The most important quantities for computing the produced baryon asymmetry are thus the decay asymmetries ε1,f and the efficiency factors ηf (which depend mainly on m̃1,f and m̃1 (or Kf and K)). While the efficiency factors can be computed similarly to the type I seesaw case, important differences between leptogenesis in type I and type II seesaw scenarios arise concerning the decay asymmetries as well as concerning the connection between leptogenesis and seesaw parameters. 4 Decay asymmetries 4.1 Right-handed neutrinos plus triplets Regarding the decay asymmetry in the type II seesaw mechanism, where the direct mass term for the neutrinos stems from the induced vev of a Higgs triplet, there are new contributions from 1-loop diagrams where virtual SU(2)L-triplet scalar fields (or their superpartners) are exchanged in the loop. The relevant diagrams for the decay ν1R → LfaHub in the limit M1 ≪ MR2,MR2,M∆ are shown in figure 4. Compared to the type I seesaw framework, the new contributions are the diagrams (c) and (f). The calculation of the corresponding decay asymmetries for each lepton flavour yields 1,f = j 6=1 Im [(Y †)1f(Y ν Yν)1j(Y T )jf ] ν Yν)11 1− (1 + xj) ln xj + 1 , (12a) 1,f = j 6=1 Im [(Y †)1f(Y ν Yν)1j(Y T )jf ] ν Yν)11 1− xj , (12b) 1,f = − g Im [(Y ν )f1(Y ν )g1(m LL)fg] ν Yν)11 −1 + y ln y + 1 , (12c) 1,f = j 6=1 Im [(Y †)1f(Y ν Yν)1j(Y T )jf ] ν Yν)11 −1 + xj ln xj + 1 , (12d) 1,f = j 6=1 Im [(Y †)1f(Y ν Yν)1j(Y T )jf ] ν Yν)11 1− xj , (12e) 1,f = − g Im [(Y ν )f1(Y ν )g1(m LL)fg] ν Yν)11 1− (1 + y) ln y + 1 , (12f) ∆̃1, ∆̃2 Figure 4: Loop diagrams in the MSSM which contribute to the decay ν1 → LfaHub for the case of a type II seesaw mechanism where the direct mass term for the neutrinos stems from the induced vev of a Higgs triplet. In diagram (f), ∆̃1 and ∆̃2 are the mass eigenstates corresponding to the superpartners of the SU(2)L-triplet scalar fields ∆ and ∆̄. The SM diagrams are the ones where no superpartners (marked by a tilde) are involved and where Hu is renamed to the SM Higgs φ. where y := M2∆/M R1 and xj := M R1 for j 6= 1 and where we assume hierarchical right-handed neutrino masses and M∆ ≫ MR1.4 The MSSM results for the type II contributions have been derived in [16]. In the SM, the results in [16] correct the previous result of [15] by a factor of −3/2. In equation (12) they have been generalised to the flavour-dependent case. The results for the contributions to the decay asymmetries from the triplet in the SM and from the triplet superfield in the MSSM are SM,II 1,f = ε 1,f , (13a) MSSM,II 1,f = ε 1,f + ε 1,f . (13b) In the MSSM, we furthermore obtain MSSM,II 1,f = ε MSSM,II 1, ef MSSM,II MSSM,II e1, ef . (14) The results corresponding to the diagrams (a), (b), (d) and (e) which contribute to εI1 in the type I seesaw in the SM and in the MSSM, have been presented first in [39]. The results for the type I contribution to the decay asymmetries in the SM and in the 4Integrating out the heavy particles ν2 ,∆ (and their superpartners) in figure 4 leads to an effective approach involving the dimension 5 neutrino mass operator (c.f. figures 1, 2 and 5), as will be discussed in section 4.2. We note that there are additional diagrams not shown in figure 4 (since they are generically suppressed for M1 ≪ MR2,MR2,M∆) which are related to the dimension 6 operator containing two lepton doublets, two Higgs doublets and a derivative. MSSM are 1,f = ε 1,f + ε 1,f , (15a) MSSM,I 1,f = ε 1,f + ε 1,f + ε 1,f + ε 1,f . (15b) Again, in the MSSM, the remaining decay asymmetries are equal to ε MSSM,I 1,f : MSSM,I 1,f = ε MSSM,I 1, ef MSSM,I MSSM,I e1, ef . (16) Finally, the total decay asymmetries from the decay of ν1R in the type II seesaw, where the direct mass term for the neutrinos stems from the induced vev of a Higgs triplet, are given by εSM1,f = ε 1,f + ε SM,II 1,f , (17) εMSSM1,f = ε MSSM,I 1,f + ε MSSM,II 1,f . (18) It is interesting to note that the type I results can be brought to a form which contains the neutrino mass matrix using j 6=1 Im [(Y †)1f (Y ν Yν)1j(Y T )jf ] 8π (Y ν Yν)11 = −MR1 g Im [(Y ν )f1(Y ν )g1(m LL)fg] 8π (Y ν Yν)11 .(19) In the limit y ≫ 1 and xj ≫ 1 for all j 6= 1, which corresponds to a large gap between the mass MR1 and the masses MR2, MR3 and M∆, we obtain the simple results for the flavour-specific decay asymmetries εSM1,f and ε 1,f [16] εSM1,f = g Im [(Y ν )f1(Y ν )g1(m LL +m LL)fg] ν Yν)11 , (20a) εMSSM1,f = g Im [(Y ν )f1(Y ν )g1(m LL +m LL)fg] ν Yν)11 . (20b) In the presence of such a mass gap, the calculation can also be performed in an effective approach after integrating out the two heavy right-handed neutrinos and the heavy triplet, as we now discuss. 4.2 Effective approach to leptogenesis Let us now explicitly use the assumption that the lepton asymmetry is generated via the decay of the lightest right-handed neutrino and that all other additional particles, in particular the ones which generate the type II contribution, are much heavier than MR1. Furthermore, we assume that we can neglect their population in the early universe, e.g. that their masses are much larger than the reheat temperature TRH and that they are Figure 5: Loop diagrams contributing to the decay asymmetry via the decay ν1 → LfaHub in the MSSM with a (lightest) right-handed neutrino ν1 and a neutrino mass matrix determined by κ′ [16]. Further contributions to the generated baryon asymmetry stem from the decay of ν1 into slepton and Higgsino and from the decays of the sneutrino ν̃1 . With Hu renamed to the SM Higgs, the first diagram contributes in the extended SM. not produced non-thermally in a large amount. Under these assumptions we can apply an effective approach to leptogenesis, which is independent of the mechanism which generates the additional (type II) contribution to the neutrino mass matrix [16]. For this minimal effective approach, it is convenient to isolate the type I contribution from the lightest right-handed neutrino as follows: (mνLL)fg = − 2(Yν)f1M R1 (Y ν )1g + (κ ′∗)fg . (21) κ′ includes type I contributions from the heavier right-handed neutrinos, plus any ad- ditional (type II) contributions from heavier particles. Examples for realisations of the neutrino mass operator can be found, e.g., in [40]. At MR1, the minimal effective field theory extension of the SM (MSSM) for lepto- genesis includes the effective neutrino mass operator κ′ plus one right-handed neutrino ν1R with mass MR1 and Yukawa couplings (Yν)f1 to the lepton doublets L f , defined as −(Yν)f1(Lf · φ) ν1R in the Lagrangian of the SM and, analogously, as (Yν)f1(L̂f · Ĥu) ν̂C1 in the superpotential of the MSSM. The contributions to the decay asymmetries in the effective approach stem from the interference of the diagram(s) for the tree-level decay of νR1 (and ν̃R1) with the loop diagrams containing the effective operator, shown in figure 5. In the SM, we obtain the simple result [16] for the flavour-specific effective decay asymmetries (corresponding to diagram (a) of figure 5) εSM1,f = g Im [(Y ν )f1(Y ν )g1(m LL)fg] ν Yν)11 . (22) For the supersymmetric case, diagram (a) and diagram (b) contribute to εMSSM1,f and we obtain [16]: εMSSM1,f = g Im [(Y ν )f1(Y ν )g1(m LL)fg] ν Yν)11 . (23) Explicit calculation furthermore yields εMSSM1,f = ε 1, ef = εMSSM = εMSSM e1, ef . (24) The results are independent of the details of the realisation of the neutrino mass operator κ′. Note that, since the diagrams where the lightest right-handed neutrino runs in the loop do not contribute to leptogenesis, we have written mνLL = −v2u(κ)∗/2 instead of m′νLL := −v2u(κ′)∗/2 in the formulae in equations (22) - (23). The decay asymmetries are directly related to the neutrino mass matrix mνLL. For neutrino masses via the type I seesaw mechanism, the results are in agreement with the known results [39], in the limit MR2,MR3 ≫ MR1. The results obtained in the effective approach are also in agreement with our full theory calculation in the type II scenarios with SU(2)L-triplets in equation (12) [16], in the limit M∆ ≫ MR1. 5 Type II bounds on decay asymmetries and on MR1 In the limit MR2,MR3,M∆ ≫ MR1 (or alternatively in the effective approach), upper bounds for the total decay asymmetries in type II leptgenesis, i.e. for the sums |εSM1 | = 1,f | and |εMSSM1 | = | 1,f |, have been derived in [16]. For the flavour-specific decay asymmetries εSM1,f and ε 1,f , the bounds can readily be obtained as |εSM1,f | ≤ mνmax , |εMSSM1,f | ≤ mνmax . (25) They are thus identical to the bounds for the total asymmetries. In particular, they also increase with increasing mass scale of the light neutrinos. Note that, compared to the low energy value, the neutrino masses at the scale MR1 are enlarged by renormalization group running by ≈ +20% in the MSSM and ≈ +30% in the SM, which raises the bounds on the decay asymmetries by the same values (see e.g. figure 4 of [41]). A situation where an almost maximal baryon asymmetry is generated by thermal leptogenesis can be realised, for example, if the total decay asymmetry nearly saturates its upper bound and if, in addition, the washout parameters m̃1,f for all three flavours approximately take its optimal value. Classes of type II seesaw models, where this can be accommodated, have been considered in [8, 42, 17]. In these so-called “type- II-upgraded” seesaw models, the type II contribution to the neutrino mass matrix is proportional to the unit matrix (enforced e.g. by an SO(3) flavour symmetry or by one of its non-Abelian discrete subgroups). From equation (20), one can readily see that if the type II contribution (∝ 1) dominates the neutrino mass matrix mνLL, and if (Yν)f1 are approximately equal for all flavours f = 1, 2, 3 and chosen such that the resulting m̃1,f are approximately equal to m ∗, we have realised ηf ≈ ηmax for all flavours and simultaneously nearly saturated the bound for the total decay asymmetry.5 5We further note that the bound for one of the flavour-specific decay asymmetries can be nearly saturated in this scenario if, for instance, (Yν)21 ≈ (Yν)31 ≈ 0. 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL -0.75 -0.25 type I seesaw type II seesaw Figure 6: Bound on the decay asymmetry ε1,f in type II leptogenesis (solid blue line) and type I lepto- genesis (dotted red line) as a function of the mass of the lightest neutrino mν := min (mν1 ,mν3 ,mν3) in type I and type II seesaw scenarios (see also [29]). The washout parameter |Aff |m̃1,f is fixed to m∗ (close to optimal), and the asymmetry is normalised to ε max,0 = 3MR1 (∆m )1/2/(16π v ), where ≈ 2.5× 10−3 eV2 is the atmospheric neutrino mass squared difference. We have considered the MSSM with tanβ = 30 as an explicit example. Assuming a maximal efficiency factor ηmax for all flavours in a given scenario, and taking an upper bound for the masses of the light neutrinos mνmax as well as the observed value nB/nγ ≈ (6.0965 ± 0.2055) × 10−10 [2] for the baryon asymmetry, equation (25) can be transformed into lower type II bounds for the mass of the lightest right-handed neutrino [16]: MSMR1 ≥ mνmax nB/nγ 0.99 · 10−2 ηmax , MMSSMR1 ≥ mνmax nB/nγ 0.92 · 10−2 ηmax . (26) The bound on MR1 is lower for a larger neutrino mass scale. The situation in the type II framework differs from the type I seesaw case: In the latter, the flavour-specific decay asymmetries are constrained by [24] |εI,SM1,f | ≤ mνmax m̃1,f , |εI,MSSM1,f | ≤ mνmax m̃1,f . (27) Note that compared to the type II bounds, there is an extra factor of (m̃1,f/m̃1) which depends on the washout parameters. As we shall now discuss, this factor implies that it is not possible to have a maximal decay asymmetry ε1,f and an optimal washout parameter m̃1,f simultaneously. Let us recall first that in the type I seesaw, in contrast to the type II case, the flavour-independent washout parameter has the lower bound [43] m̃1 ≥ mνmin , (28) with mνmin = min (mν1 , mν3 , mν3). On the contrary, in the type I and type II seesaw, the flavour-dependent washout parameters m̃1,f are generically not constrained. Note that in 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL type I seesaw type II seesaw Figure 7: Lower bound on MR1 in type II leptogenesis (solid blue line) and type I leptogenesis (dotted red line) as a function of the mass of the lightest neutrinomν := min (mν1 ,mν3 ,mν3). For definiteness, the MSSM with tanβ = 30 has been considered as an example. the flavour-independent approximation, Eq. (28) leads to a dramatically more restrictive bound on ε1 = f ε1,f [44] for quasi-degenerate light neutrino masses, and finally even to a bound on the neutrino mass scale [43]. This can be understood from the fact that for m̃1 ≫ m∗ in the flavour-independent approximation, washout effects strongly reduce the efficiency of thermal leptogenesis. Similarly, in the flavour-dependent treatment, m̃1,f ≫ m∗ would lead to a strongly reduced efficiency for this specific flavour. This strong washout for quasi-degenerate light neutrinos can be avoided in flavour-dependent type I leptogenesis, and m̃1,f ≈ m∗ can realise a nearly optimal scenario regarding washout (c.f. figure 3). However, we see from equation (27) that the decay asymmetries in this case are reduced by a factor of (m∗/mνmin) 1/2 when compared to the optimal value, leading to a reduced baryon asymmetry. On the other hand, realizing nearly optimal ε1,f requires m̃1,f ≈ m̃1 ≥ mνmin, leading to large washout effects for quasi-degenerate light neutrinos and even to a more strongly suppressed generation of baryon asymmetry (c.f. figure 3). As a consequence, increasing the neutrino mass scale increases the lower bound on MR1 (also in the presence of flavour-dependent effects), in contrast to the type II seesaw case. Comparing the type II and type I seesaw cases, in the latter the baryon asymmetry is suppressed for quasi-degenerate light neutrino masses either by a factor (m∗/mνmin) in the decay asymmetries or by a non-optimal washout parameter much larger than m∗ (or Kf ≫ 1, c.f. figure 3). The bounds on the decay asymmetries in type I and type II leptogenesis are compared in figure 6, where m̃1,f has been fixed to m ∗, close to its optimal value. From figure 6 we see that in the type I case the maximal baryon asymme- try is obtained for hierarchical neutrino masses, whereas in the type II case, increasing the neutrino mass scale increases the produced baryon asymmetry and therefore allows to relax the bound on MR1, as shown in figure 7. In addition, for the same reason, increasing the neutrino mass scale also relaxes the lower bound on the reheat tempera- ture TRH from the requirement of successful type II leptogenesis. Including reheating in the flavour-dependent Boltzmann equations as in Ref. [29] (for the flavour-independent 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL 2´109 4´109 6´109 8´109 type I seesaw 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL 2´109 4´109 6´109 8´109 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL 5´108 1´109 1.5´109 2´109 type II seesaw 0 0.1 0.2 0.3 0.4 log10 Hm min �eVL 5´108 1´109 1.5´109 2´109 Figure 8: Lower bound on the reheat temperature TRH in type I leptogenesis (left panel) and in type II leptogenesis (right panel) as a function of the mass of the lightest neutrino mν = min (mν1 ,mν3 ,mν3), in the MSSM with tanβ = 30. In the grey regions, values of TRH are incompatible with thermal leptogenesis for the corresponding mν case, see [45]), we obtain the mνmin-dependent lower bounds on TRH in type I and type II scenarios shown in figure 8. While the bound decreases in type II leptogenesis by about an order of magnitude when the neutrino mass scale increases to 0.4 eV, it in- creases in the type I seesaw case. In the presence of upper bounds on TRH, this can lead to constraints on the neutrino mass scale , i.e. on mνmin = min (mν1 , mν3, mν3). For instance, with an upper bound TRH ≤ 5× 109 GeV, values of mνmin in the approximate range [0.01 eV, 0.32 eV] would be incompatible with leptogenesis in the type I seesaw framework (c.f. figure 8). 6 Summary, discussion and conclusions We have analysed flavour-dependent leptogenesis via the out-of-equilibrium decay of the lightest right-handed neutrino in type II seesaw scenarios, where, in addition to the type I seesaw, an additional direct mass term for the light neutrinos is present. We have considered type II seesaw scenarios where this additional contribution stems from the vacuum expectation value of a Higgs triplet, and furthermore an effective approach, which is independent of the mechanism which generates the additional (type II) contribution to neutrino masses. We have taken into account flavour-dependent effects, which are relevant if thermal leptogenesis takes place at temperatures below circa 1012 GeV in the SM and below circa (1 + tan2 β) × 1012 GeV in the MSSM. As in type I leptogenesis, in the flavour-dependent regime the decays of the right-handed (s)neutrinos generate asymmetries in in each distinguishable flavour (proportional to the flavour-specific decay asymmetries ε1,f), which are differently washed out by scattering processes in the thermal bath, and thus appear with distinct weights (efficiency factors ηf ) in the final baryon asymmetry. The most important quantities for computing the produced baryon asymmetry are the decay asymmetries ε1,f and the efficiency factors ηf (which mainly depend on washout parameters m̃1,f and m̃1 = f m̃1,f). With respect to the flavour-specific efficiency factors ηf , in the limit that the mass M∆L of the triplet is much larger than MR1 (and MR1 ≪ MR2,MR3), they can be estimated from the same Boltzmann equa- tions as in the type I seesaw framework. Regarding the decay asymmetries ε1,f , in the type II seesaw case there are additional contributions where virtual Higgs triplets (and their superpartners) run in the 1-loop diagrams. Here, we have generalised the results of [16] to the flavour-dependent case. The most important effects of flavour in leptoge- nesis are a consequence of the fact that in the flavour-independent approximation the total baryon asymmetry is a function of f ε1,f × ηind ( g m̃1,g), whereas in the cor- rect flavour-dependent treatment the baryon asymmetry is (approximately) a function f ε1,fη (Affm̃1,f , m̃1). We have then investigated the bounds on the flavour-specific decay asymmetries ε1,f . In the type I seesaw case, it is known that the bound on the flavour-specific asymmetries εI1,f is substantially relaxed [24] compared to the bound on ε 1,f [44] in the case of a quasi-degenerate spectrum of light neutrinos. For experimentally allowed light neutrino masses below about 0.4 eV, there is no longer a bound on the neutrino mass scale from the requirement of successful thermal leptogenesis. In the type II seesaw case, we have derived the bound on the flavour-specific decay asymmetries ε1,f = ε 1,f + ε 1,f , which turns out to be identical to the bound on the total decay asymmetry ε1 = f ε1,f . We have compared the bound on the flavour-specific decay asymmetries in type I and type II scenarios, and found that while the type II bound increases with the neutrino mass scale, the type I bound decreases (for experimentally allowed light neutrino masses below about 0.4 eV). The relaxed bound on ε1,f (figure 6) leads to a lower bound on the mass of the lightest right-handed neutrino MR1 in the type II seesaw scenario (figure 7), which decreases when the neutrino mass scale increases. Furthermore, it leads to a relaxed lower bound on the reheat temperature TRH of the early universe (figure 8), which helps to improve consistency of thermal leptogenesis with upper bounds on TRH in some supergravity models. This is in contrast to the type I seesaw scenario, where the lower bound on TRH from thermal leptogenesis increases with increasing neutrino mass scale. Constraints on TRH can therefore imply constraints on the mass scale of the light neutrinos also in flavour-dependent type I leptogenesis, although a general bound is absent. We have furthermore argued that these relaxed bounds on ε1,f MR1 and TRH in the type II case can be nearly saturated in an elegant way in classes of so-called “type- II-upgraded” seesaw models [8], where the type II contribution to the neutrino mass matrix is proportional to the unit matrix (enforced e.g. by an SO(3) flavour symmetry or by one of its non-Abelian subgroups). One interesting application of these type II seesaw scenarios is that the consistency of thermal leptogenesis with unified theories of flavour is improved compared to the type I seesaw case. This effect, investigated in the flavour-independent approximation in [17], is also present analogously in the flavour- dependent treatment of leptogenesis. The reason is that if the type II contribution (∝ 1) dominates, the decay asymmetries ε1,f become approximately equal and the estimate for the produced baryon asymmetry is similar to the flavour-independent case. Nevertheless, an accurate analysis of leptogenesis in this scenario requires careful inclusion of the flavour-dependent effects. In many applications and realisations of type II leptogenesis in specific models of fermion masses and mixings (see e.g. [18]), flavour-dependent effects may substantially change the results and they therefore have to be taken into account. In summary, type II leptogenesis provides a well-motivated generalisation of the conventional scenario of leptogenesis in the type I seesaw framework. We have argued that flavour-dependent effects have to be included in type II leptogenesis, and can change predictions of existing models as well as open up new possibilities for for successful models of leptogenesis. 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Buras, S. Jager, S. Uhlig and A. Weiler, arXiv:hep-ph/0609067. [39] L. Covi, E. Roulet, and F. Vissani, Phys. Lett. B384 (1996), 169–174. [40] E. Ma, Phys. Rev. Lett. 81 (1998), 1171–1174. http://arxiv.org/abs/hep-ph/0308276 http://arxiv.org/abs/hep-ph/0505076 http://arxiv.org/abs/hep-ph/0512160 http://arxiv.org/abs/hep-ph/0601083 http://arxiv.org/abs/hep-ph/0601084 http://arxiv.org/abs/hep-ph/0605281 http://arxiv.org/abs/hep-ph/0607330 http://arxiv.org/abs/hep-ph/0609038 http://arxiv.org/abs/hep-ph/0609125 http://arxiv.org/abs/hep-ph/0609297 http://arxiv.org/abs/hep-ph/0611232 http://arxiv.org/abs/hep-ph/0612262 http://arxiv.org/abs/hep-ph/0611338 http://arxiv.org/abs/hep-ph/0611337 http://arxiv.org/abs/hep-ph/0611357 http://arxiv.org/abs/hep-ph/0612187 http://arxiv.org/abs/hep-ph/0703175 http://arxiv.org/abs/hep-ph/0703084 http://arxiv.org/abs/hep-ph/0703183 http://arxiv.org/abs/hep-ph/0506107 http://arxiv.org/abs/hep-ph/0609067 [41] S. Antusch, J. Kersten, M. Lindner, and M. Ratz, Nucl. Phys. B674 (2003), 401– [42] S. Antusch and S. F. King, Phys. Lett. B 591 (2004) 104 [arXiv:hep-ph/0403053]. [43] W. Buchmüller, P. Di Bari, and M. Plümacher, Nucl. Phys. B665 (2003), 445–468. [44] S. Davidson and A. Ibarra, Phys. Lett. B535 (2002), 25–32. [45] G. F. Giudice, A. Notari, M. Raidal, A. Riotto, and A. Strumia, Nucl. Phys. B 685, 89 (2004). http://arxiv.org/abs/hep-ph/0403053 Introduction Type I and type II seesaw mechanisms Baryogenesis via flavour-dependent leptogenesis Decay asymmetries Right-handed neutrinos plus triplets Effective approach to leptogenesis Type II bounds on decay asymmetries and on bold0mu mumu MR1MR13.91663pt plus 1.95831pt minus 1.30554ptMR1MR1MR1MR1 Summary, discussion and conclusions

The Physics of Chromospheric Plasmas ASP Conference Series, Vol. 368, 2007 Petr Heinzel, Ivan Dorotovič and Robert J. Rutten, eds. Multi-wavelength Analysis of a Quiet Solar Region G. Tsiropoula1, K. Tziotziou1, J. Giannikakis1, P. Young2, U. Schühle3 and P. Heinzel4 1Institute for Space Applications and Remote Sensing, Athens, Greece 2CCLRC Rutherford Appleton Laboratory, United Kingdom 3MPI für Sonnensystemforschung, Katlenburg-Lindau, Germany 4Astronomical Institute AS, Ondřejov, Czech Republic Abstract. We present observations of a solar quiet region obtained by the ground-based Dutch Open Telescope (DOT), and by instruments on the space- craft SOHO and TRACE. The observations were obtained during a coordinated observing campaign on October 2005. The aim of this work is to present the rich diversity of fine-scale structures that are found at the network boundaries and their appearance in different instruments and different spectral lines that span the photosphere to the corona. Detailed studies of these structures are crucial to understanding their dynamics in different solar layers, as well as the role such structures play in the mass balance and heating of the solar atmosphere. 1. Introduction In the quiet regions of the solar surface the magnetic field is mainly concen- trated at the boundaries of the network cells. Over the past decade, apart from the well-known mottles and spicules, several other structures residing at the network boundaries such as explosive events, blinkers, network flares, upflow events have been mentioned in the literature. However, their interpretation, inter-relationship and their relation to the underlying photospheric magnetic concentrations remain ambiguous, because the same feature has a different ap- pearance when observed in different spectral lines and by different instruments. For most of the events mentioned above magnetic reconnection is suggested as the driving mechanism. This is not surprising, since it is now well established from investigations of high resolution magnetograms, that new bipolar elements emerge continuously inside the cell interiors and are, subsequently, swept at the network boundaries by the supergranular flow (Wang et al. 2006; Schrijver et al. 1997). Interactions of the magnetic fields have as a result either the enhance- ment of the flux concentration in the case of same polarities or its cancellation in the case of opposite polarities. Observations support the idea that flux cancel- lation most likely invokes magnetic reconnection. In this context, the study and comprehension of the dynamical behaviour of the different fine-scale structures is crucial to the understanding of the dynamics of the solar atmosphere. In this work we present observations of a solar quiet region and some of the properties of several different structures appearing at the network boundaries and observed in different wavelengths by the different instruments involved in a coordinated campaign. http://arxiv.org/abs/0704.1592v1 172 Tsiropoula et al. Figure 1. Left : C IV TRACE image. Right : MDI magnetogram. The white rectangle inside the images marks the DOT’s field-of-view. 2. Observations and Data Reduction In October 2005 we ran a 12 days observational campaign. The aim of that campaign was the collection of multi-wavelength observations both from the ground and space that could be used for the study of the dynamical behaviour of mottles/spicules and other fine structures, observed in different layers of the solar atmosphere. Three ground-based telescopes were involved in that campaign: DOT on La Palma, THEMIS on Tenerife and SOLIS at Kitt Peak. From space telescopes two spacecraft were involved: SOHO (with CDS, SUMER, and MDI) and TRACE. The analysed data were obtained on October 14 and consist of time se- quences of observations of a quiet region found at the solar disk center recorded by different instruments. Sequences recorded by the DOT were obtained be- tween 10:15:43 – 10:30:42 UT and consist of 26 speckle reconstructed images taken simultaneously at a cadence of 35 s with a pixel size of 0.071′′ in 5 wave- lengths along the Hα line profile (i.e. at −0.7 Å, −0.35 Å, line centre, 0.35 Å and 0.7 Å), in the G band with a 10 Å filter, in the Ca IIH line taken with a narrow band filter and in the blue and red continuum. TRACE obtained high cadence filter images at 1550 Å, 1600 Å and 1700 Å. SUMER obtained raster scans and sit-and-stare observations from 8:15 to 10:30 UT. CDS obtained sit- and-stare observations from 6:44 to 10:46 UT and six 154′′ × 240′′ raster scans (each one having a duration of 30min) from 10:46 to 13:52 UT. Both SOHO instruments (i.e., CDS and SUMER) observed in several spectral lines spanning the upper solar atmosphere. Using the standard software the raw measurements were corrected for flat field, cosmic rays and other instrumental effects. A single Gaussian with a linear background and Poisson statistics were used for fitting each spectral line profile. MDI obtained high cadence images at its high resolu- tion mode. Multi-wavelength Analysis of a Quiet Region 173 Figure 2. DOT images of a rosette region. Left : Hα -0.7 Å (first row), Hα+0.7 Å (second row). Right : Ca IIH (first row), G band (second row). Extensive work went into collecting, scaling and co-aligning the various data sets to a common coordinate system (see Fig. 1 showing the coalignment of TRACE, MDI, and DOT images). 3. Analysis and Results 3.1. DOT observations DOT’s field-of-view (FOV) is 80′′ in the X direction and 63′′ in the Y direction. For the present study we selected a smaller region which contains a rosette with several mottles pointing to a common center (see Fig. 2) and is found at the middle upper part of the DOT’s FOV. In the G-band image (Fig. 2, right, second row) isolated bright points show up in the regions of strong magnetic field as can be seen in the MDI image. These seem to be passively advected with the general granular flow field in the intergranular lanes. While we have not conducted an exhaustive study of bright point lifetimes we find that bright points can be visible from some minutes up to almost the entire length of the time series. In the contemporal Ca IIH images (Fig. 2, right, first row bright points are less sharp due to strong scattering in this line and possibly due to increasing flux tube with height. Reversed granulation caused by convection reversal is obvious in this image. In the Hα − 0.7 Å (Fig. 2, left, first row) the dark streaks are part of the elongated Hα mottles seen better at Hα line center. Some mottle endings appear 174 Tsiropoula et al. Figure 3. Cross correlation function vs time for Left : the intensity at Hα− 0.7 Å, Middle: the intensity at Hα line center. Right : the velocity at Hα − 0.7 Å. extra dark in the blue wing image through Doppler blueshift. Near the mottle endings one can see bright points. These are sharper in the G-band but stand out much clearer in the Hα wing. Thus Hα wing represents a promising proxy magnetometer to locate and track isolated intermittent magnetic elements. This is because the Hα wing has a strong photospheric contribution, as it is shown by Leenaarts et al. (2006) who arrived to this conclusion by using radiative transfer calculations and convective simulations. In the Hα+0.7 Å (Fig. 2, left, second row) dark streaks around the rosette’s center are signatures of redshifts. Blueshifts in the outer endings and redshifts in the inner endings of mottles provide evidence for the presence of bi-directional flows along these structures. An important parameter for the study of the dynamics of mottles is their velocity. For its determination, when filtergrams at two wavelengths of equal intensity at the blue and the red side of the line are available, a technique based on the subtraction of images can be used. In this technique, by using the well known representation of the line intensity profile and assuming a Gaussian wavelength dependence of the optical thickness, we can define the parameter ΣI − 2I0λ , (1) where ∆I = I(−∆λ) − I(+∆λ), ΣI = I(−∆λ) + I(+∆λ) and I0(∆λ) is the reference profile emitted by the background. DS is called Doppler signal, has the same sign as the velocity and can be used for a qualitative description of the velocity field (for a description of the method see Tsiropoula 2000). When an optical depth less than one is assumed then quantitative values of the velocity can be obtained from the relation: 1 +DS , (2) since in that case the velocity depends only on DS (obtained from the obser- vations) and the Doppler width, ∆λD (obtained from the literature). By using these relations we have constructed 2-D intensity and velocity images for the whole time series. We found out that it is difficult to follow each one mottle for more than two or three frames and that the general appearance of the region Multi-wavelength Analysis of a Quiet Region 175 Figure 4. CDS raster image obtained at the OV 629.7 Å line with an over- plotted SUMER image at NeVIII 770 Å. seems to change quite rapidly with time. For a quantitative estimate of the temporal changes we computed the value of the cross correlation (CC) function over the 2-D FOV both for the intensity and velocity. Figure 3 shows the inten- sity CC curve at Hα− 0.7 Å (left) and Hα line center (middle) and the velocity CC curve at Hα± 0.7 Å (right). The decay of the CC curve is a measure of the lifetime of the structures. The e-folding time for the left curve is found equal to 2min, the middle curve equal to 5min and the right curve is of the order of the cadence. 3.2. CDS and SUMER observations In Fig. 4 we show the CDS raster image obtained in the OV 629.7 Å line with the SUMER image at NeVIII 770 Å overplotted. Although there is a time difference of 3 hours between the two images the network is constant enough to allowed the coalignement of the two images. In CDS intensity maps several brightenings are observed which are called blinkers (Harrison et al. 1999). These events are best observed in transition region lines and show an intensity increase of 60 - 80%. Most of them have a repetitive character and reappear at the same position several times. In Fig. 5 (left, up) we show an integrated (over the spectral line with the background included) intensity image in the Ne VIII 770 Å line produced by sit- and-stare observations of a network region. The image is produced by binning over 6 spectra in order to improve the signal-to-noise ratio. The Doppler shift map was derived by applying a single Gaussian fitting (Fig. 5, left, bottom). The Doppler shift map and the spectral line profiles were visually inspected for any non-Gaussian profiles with enhancements in both the blue and red wings that are the main characteristics of the presence of bi-directional jets. Large numbers of such profiles were found at the network boundaries (Fig. 5, right). 176 Tsiropoula et al. Figure 5. SUMER sit-and-stare observations in the NeVIII line. Left : in- tensities (up), Doppler velocities (bottom) (network boundaries are bright in the intensity image). Right : non-Gaussian profiles in the positions marked by “x” inside the images in the left. 4. Conclusions In this work we present observations of a quiet solar region obtained by different instruments in different spectral lines. Network boundaries are found to be the locus of several structures which have different appearances when observed by different instruments e.g., blinkers (when observed with CDS), mottles (when observed with DOT), jets (when observed with SUMER). Their interrelationship is to be further explored. Regarding flows no-clear pattern is found in blinkers, while bi-directional flows are found in jets. In dark mottles downward velocities are found at their footpoints and upwards velocities at their upper parts and very fast changes in their appearance. The network shows a remarkable constancy when observed in low resolution images. However when seen in high resolution images several fine structures are observed which change so fast that it is very hard to follow. Acknowledgments. K. Tziotziou acknowledges support by Marie Curie European Reintegration Grant MERG-CT-2004-021626. This work has been partly supported by a Greek-Czech programme of cooperation. References Harrison, A., Lang, J., Brooks, D.H., & Innes, D.E. 1999, A&A, 351, 1115 Leenarts, J., Rutten, R.J., Sütterlin, P., Carlsson, M., & Uitenbroek H. 2006, A&A, 449, 1209 Schrijver, C.J., Title, A.M., Van Ballegooijen, A.A., Hagenaar, H.J., & Shine, R.A. 1997, ApJ, 487, 424 Tsiropoula, G. 2000, New Astronomy, 5, 1 Wang, H., Tang, F., Zirin, H., & Wang, J. 1996, Solar Phys. , 165, 223

We present observations of a solar quiet region obtained by the ground-based Dutch Open Telescope (DOT), and by instruments on the spacecraft SOHO and TRACE. The observations were obtained during a coordinated observing campaign on October 2005. The aim of this work is to present the rich diversity of fine-scale structures that are found at the network boundaries and their appearance in different instruments and different spectral lines that span the photosphere to the corona. Detailed studies of these structures are crucial to understanding their dynamics in different solar layers, as well as the role such structures play in the mass balance and heating of the solar atmosphere.

Introduction In the quiet regions of the solar surface the magnetic field is mainly concen- trated at the boundaries of the network cells. Over the past decade, apart from the well-known mottles and spicules, several other structures residing at the network boundaries such as explosive events, blinkers, network flares, upflow events have been mentioned in the literature. However, their interpretation, inter-relationship and their relation to the underlying photospheric magnetic concentrations remain ambiguous, because the same feature has a different ap- pearance when observed in different spectral lines and by different instruments. For most of the events mentioned above magnetic reconnection is suggested as the driving mechanism. This is not surprising, since it is now well established from investigations of high resolution magnetograms, that new bipolar elements emerge continuously inside the cell interiors and are, subsequently, swept at the network boundaries by the supergranular flow (Wang et al. 2006; Schrijver et al. 1997). Interactions of the magnetic fields have as a result either the enhance- ment of the flux concentration in the case of same polarities or its cancellation in the case of opposite polarities. Observations support the idea that flux cancel- lation most likely invokes magnetic reconnection. In this context, the study and comprehension of the dynamical behaviour of the different fine-scale structures is crucial to the understanding of the dynamics of the solar atmosphere. In this work we present observations of a solar quiet region and some of the properties of several different structures appearing at the network boundaries and observed in different wavelengths by the different instruments involved in a coordinated campaign. http://arxiv.org/abs/0704.1592v1 172 Tsiropoula et al. Figure 1. Left : C IV TRACE image. Right : MDI magnetogram. The white rectangle inside the images marks the DOT’s field-of-view. 2. Observations and Data Reduction In October 2005 we ran a 12 days observational campaign. The aim of that campaign was the collection of multi-wavelength observations both from the ground and space that could be used for the study of the dynamical behaviour of mottles/spicules and other fine structures, observed in different layers of the solar atmosphere. Three ground-based telescopes were involved in that campaign: DOT on La Palma, THEMIS on Tenerife and SOLIS at Kitt Peak. From space telescopes two spacecraft were involved: SOHO (with CDS, SUMER, and MDI) and TRACE. The analysed data were obtained on October 14 and consist of time se- quences of observations of a quiet region found at the solar disk center recorded by different instruments. Sequences recorded by the DOT were obtained be- tween 10:15:43 – 10:30:42 UT and consist of 26 speckle reconstructed images taken simultaneously at a cadence of 35 s with a pixel size of 0.071′′ in 5 wave- lengths along the Hα line profile (i.e. at −0.7 Å, −0.35 Å, line centre, 0.35 Å and 0.7 Å), in the G band with a 10 Å filter, in the Ca IIH line taken with a narrow band filter and in the blue and red continuum. TRACE obtained high cadence filter images at 1550 Å, 1600 Å and 1700 Å. SUMER obtained raster scans and sit-and-stare observations from 8:15 to 10:30 UT. CDS obtained sit- and-stare observations from 6:44 to 10:46 UT and six 154′′ × 240′′ raster scans (each one having a duration of 30min) from 10:46 to 13:52 UT. Both SOHO instruments (i.e., CDS and SUMER) observed in several spectral lines spanning the upper solar atmosphere. Using the standard software the raw measurements were corrected for flat field, cosmic rays and other instrumental effects. A single Gaussian with a linear background and Poisson statistics were used for fitting each spectral line profile. MDI obtained high cadence images at its high resolu- tion mode. Multi-wavelength Analysis of a Quiet Region 173 Figure 2. DOT images of a rosette region. Left : Hα -0.7 Å (first row), Hα+0.7 Å (second row). Right : Ca IIH (first row), G band (second row). Extensive work went into collecting, scaling and co-aligning the various data sets to a common coordinate system (see Fig. 1 showing the coalignment of TRACE, MDI, and DOT images). 3. Analysis and Results 3.1. DOT observations DOT’s field-of-view (FOV) is 80′′ in the X direction and 63′′ in the Y direction. For the present study we selected a smaller region which contains a rosette with several mottles pointing to a common center (see Fig. 2) and is found at the middle upper part of the DOT’s FOV. In the G-band image (Fig. 2, right, second row) isolated bright points show up in the regions of strong magnetic field as can be seen in the MDI image. These seem to be passively advected with the general granular flow field in the intergranular lanes. While we have not conducted an exhaustive study of bright point lifetimes we find that bright points can be visible from some minutes up to almost the entire length of the time series. In the contemporal Ca IIH images (Fig. 2, right, first row bright points are less sharp due to strong scattering in this line and possibly due to increasing flux tube with height. Reversed granulation caused by convection reversal is obvious in this image. In the Hα − 0.7 Å (Fig. 2, left, first row) the dark streaks are part of the elongated Hα mottles seen better at Hα line center. Some mottle endings appear 174 Tsiropoula et al. Figure 3. Cross correlation function vs time for Left : the intensity at Hα− 0.7 Å, Middle: the intensity at Hα line center. Right : the velocity at Hα − 0.7 Å. extra dark in the blue wing image through Doppler blueshift. Near the mottle endings one can see bright points. These are sharper in the G-band but stand out much clearer in the Hα wing. Thus Hα wing represents a promising proxy magnetometer to locate and track isolated intermittent magnetic elements. This is because the Hα wing has a strong photospheric contribution, as it is shown by Leenaarts et al. (2006) who arrived to this conclusion by using radiative transfer calculations and convective simulations. In the Hα+0.7 Å (Fig. 2, left, second row) dark streaks around the rosette’s center are signatures of redshifts. Blueshifts in the outer endings and redshifts in the inner endings of mottles provide evidence for the presence of bi-directional flows along these structures. An important parameter for the study of the dynamics of mottles is their velocity. For its determination, when filtergrams at two wavelengths of equal intensity at the blue and the red side of the line are available, a technique based on the subtraction of images can be used. In this technique, by using the well known representation of the line intensity profile and assuming a Gaussian wavelength dependence of the optical thickness, we can define the parameter ΣI − 2I0λ , (1) where ∆I = I(−∆λ) − I(+∆λ), ΣI = I(−∆λ) + I(+∆λ) and I0(∆λ) is the reference profile emitted by the background. DS is called Doppler signal, has the same sign as the velocity and can be used for a qualitative description of the velocity field (for a description of the method see Tsiropoula 2000). When an optical depth less than one is assumed then quantitative values of the velocity can be obtained from the relation: 1 +DS , (2) since in that case the velocity depends only on DS (obtained from the obser- vations) and the Doppler width, ∆λD (obtained from the literature). By using these relations we have constructed 2-D intensity and velocity images for the whole time series. We found out that it is difficult to follow each one mottle for more than two or three frames and that the general appearance of the region Multi-wavelength Analysis of a Quiet Region 175 Figure 4. CDS raster image obtained at the OV 629.7 Å line with an over- plotted SUMER image at NeVIII 770 Å. seems to change quite rapidly with time. For a quantitative estimate of the temporal changes we computed the value of the cross correlation (CC) function over the 2-D FOV both for the intensity and velocity. Figure 3 shows the inten- sity CC curve at Hα− 0.7 Å (left) and Hα line center (middle) and the velocity CC curve at Hα± 0.7 Å (right). The decay of the CC curve is a measure of the lifetime of the structures. The e-folding time for the left curve is found equal to 2min, the middle curve equal to 5min and the right curve is of the order of the cadence. 3.2. CDS and SUMER observations In Fig. 4 we show the CDS raster image obtained in the OV 629.7 Å line with the SUMER image at NeVIII 770 Å overplotted. Although there is a time difference of 3 hours between the two images the network is constant enough to allowed the coalignement of the two images. In CDS intensity maps several brightenings are observed which are called blinkers (Harrison et al. 1999). These events are best observed in transition region lines and show an intensity increase of 60 - 80%. Most of them have a repetitive character and reappear at the same position several times. In Fig. 5 (left, up) we show an integrated (over the spectral line with the background included) intensity image in the Ne VIII 770 Å line produced by sit- and-stare observations of a network region. The image is produced by binning over 6 spectra in order to improve the signal-to-noise ratio. The Doppler shift map was derived by applying a single Gaussian fitting (Fig. 5, left, bottom). The Doppler shift map and the spectral line profiles were visually inspected for any non-Gaussian profiles with enhancements in both the blue and red wings that are the main characteristics of the presence of bi-directional jets. Large numbers of such profiles were found at the network boundaries (Fig. 5, right). 176 Tsiropoula et al. Figure 5. SUMER sit-and-stare observations in the NeVIII line. Left : in- tensities (up), Doppler velocities (bottom) (network boundaries are bright in the intensity image). Right : non-Gaussian profiles in the positions marked by “x” inside the images in the left. 4. Conclusions In this work we present observations of a quiet solar region obtained by different instruments in different spectral lines. Network boundaries are found to be the locus of several structures which have different appearances when observed by different instruments e.g., blinkers (when observed with CDS), mottles (when observed with DOT), jets (when observed with SUMER). Their interrelationship is to be further explored. Regarding flows no-clear pattern is found in blinkers, while bi-directional flows are found in jets. In dark mottles downward velocities are found at their footpoints and upwards velocities at their upper parts and very fast changes in their appearance. The network shows a remarkable constancy when observed in low resolution images. However when seen in high resolution images several fine structures are observed which change so fast that it is very hard to follow. Acknowledgments. K. Tziotziou acknowledges support by Marie Curie European Reintegration Grant MERG-CT-2004-021626. This work has been partly supported by a Greek-Czech programme of cooperation. References Harrison, A., Lang, J., Brooks, D.H., & Innes, D.E. 1999, A&A, 351, 1115 Leenarts, J., Rutten, R.J., Sütterlin, P., Carlsson, M., & Uitenbroek H. 2006, A&A, 449, 1209 Schrijver, C.J., Title, A.M., Van Ballegooijen, A.A., Hagenaar, H.J., & Shine, R.A. 1997, ApJ, 487, 424 Tsiropoula, G. 2000, New Astronomy, 5, 1 Wang, H., Tang, F., Zirin, H., & Wang, J. 1996, Solar Phys. , 165, 223

Orbital currents in the Colle-Salvetti correlation energy functional and the degeneracy problem S. Pittalis1, S. Kurth1, S. Sharma1,2 and E.K.U. Gross1 1 Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany and 2 Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, D-14195 Berlin, Germany. (Dated: October 30, 2018) Abstract Popular density functionals for the exchange-correlation energy typically fail to reproduce the degeneracy of different ground states of open-shell atoms. As a remedy, functionals which explic- itly depend on the current density have been suggested. We present an analysis of this problem by investigating functionals that explicitly depend on the Kohn-Sham orbitals. Going beyond the exact-exchange approximation by adding correlation in the form of the Colle-Salvetti functional we show how current-dependent terms enter the Colle-Salvetti expression and their relevance is evaluated. A very good description of the degeneracy of ground-states for atoms of the first and second row of the periodic table is obtained. http://arxiv.org/abs/0704.1593v2 I. INTRODUCTION Common approximations to the exchange-correlation functional of density functional theory (DFT) [1, 2] and spin-DFT (SDFT) [3] often fail to reproduce the degeneracy of dif- ferent ground states. An illustrative example are ground states of open-shell atoms where one usually and erroneously obtains different total energies for states with zero and non- vanishing current density. The local spin density approximation (LSDA) gives rather small splittings (of the order of 1 kcal/mole), but generalized gradient approximations (GGAs) and meta-GGAs can introduce splittings of 10 kcal/mol [4, 5, 6, 7, 8, 9, 10, 11]. These spurious energy splittings would vanish if the exact exchange-correlation func- tional could be used. The exchange-correlation energy functional can be represented in terms of the exchange-correlation hole function. Considering current-carrying states, Dobson showed how the expression for the exchange-hole curvature has to be changed by including current-dependent terms [12]. Later, Becke found the same kind of terms in the short-range behavior of the exchange-correlation hole, and observed that they also enter the spin-like correlation-hole function [13]. For open-shell atoms, inclusion of these current-dependent terms results in spurious energy splittings of less than 1 kcal/mole [9]. Along these lines, Maximoff at al. [10] worked out a correction for the system-averaged exchange hole of the Perdew-Burke-Ernzerhof (PBE) GGA [14], which improves the corresponding spurious splittings. Alternatively, Tao and Perdew [11, 15] proposed a scheme for the extension of existing functionals using ideas of current density functional theory (CDFT) [16, 17] which, again, improves the description of the degener- The performance of the exact-exchange (EXX) energy functional - which, by definition, describes the exchange hole correctly - has been evaluated for the spurious splittings in DFT and SDFT [18]. In the EXX-DFT (i.e., spin-restricted calculations using one and the same Kohn-Sham potential for spin-up and spin-down orbitals) the degeneracy is well re- produced to within 0.6 kcal/mole but, surprisingly, in EXX-SDFT (i.e., spin-unrestricted calculations using two Kohn-Sham potentials, one for spin-up and one for spin-down or- bitals) spurious splittings up to 3 kcal/mole are obtained. In particular, current-carrying states always have higher total energies than states without current. This observation motivated the applications of the optimized-effective-potential (OEP) method [19, 20, 21] generalized to current-spin-density functional theory (CSDFT) to these current-carrying states [22]. As expected, EXX-CSDFT total energies for current-carrying states are lower than those of EXX-SDFT. However, this lowering is too small to give a substantial im- provement of the spurious energy splittings. These studies lead to the conclusion that correlation is needed for any further improvement. The construction of a correlation energy functional compatible with EXX is a difficult task [23, 24, 25], but for spherical atoms it was found that EXX combined with the Colle- Salvetti (CS) functional for correlation [26, 27, 28] leads to very accurate total energies [29]. The CS functional has been used to derive the popular Lee-Yang-Parr (LYP) func- tional [30], which is most commonly used together with Becke’s exchange functional [31] (BLYP) and in hybrid schemes such as B3LYP [32, 33]. On the other hand, the CS correla- tion energy functional also has its limitations [34, 35, 36]. In particular, while short-range correlations are well described [35], very important long-range correlations are missing. These correlations often cannot be ignored in molecules and solids, but are negligible in atoms. This fact, together with the encouraging results for spherical atoms [29], indicates that it is appropriate to employ the CS functional to analyze the degeneracy problem for open-shell atoms beyond EXX. Furthermore, the expression of the CS functional also al- lows a reconsideration of the relevance of the orbital currents as ingredient of correlation functionals. Although the general density functional formalism to deal with degenerate ground states includes densities which can only be obtained by a weighted sum of several deter- minantal densities [37, 38], as in many previous investigations [8, 9, 10, 11] we only con- sider densities which may be represented by a single Slater determinant of Kohn-Sham orbitals. II. THEORY Going beyond the EXX approximation, we here consider the correlation-energy func- tional of Colle and Salvetti [26]. This expression relies on the assumption that the corre- lated two-body reduced density matrix may be approximated by the Hartree-Fock (HF) two-body reduced density matrix ρHF2 (r1, r2), multiplied by a Jastrow-type correlation factor. After a series of approximations, the following expression is obtained for the cor- relation energy Ec = −4a ρHF2 (r, r) 1 + bρ− 3 (r) , r− s 3 (r) 1 + dρ− 3 (r) where ρHF2 (r, s) is expressed in terms of the average and relative coordinates r = (r1+r2) and s = r1 − r2. Here, ρ(r) is the electron density and the constants a = 0.049, b = 0.132, c = 0.2533, d = 0.349 are determined by a fitting procedure using the Hartree-Fock (HF) orbitals for the Helium atom. Following Lee, Yang and Parr, this expression can be restated as a formula involving only the total charge-density, the charge-density of each Hartree-Fock orbital and their gradient and Laplacian [30]. In this derivation, the single-particle orbitals are tacitly as- sumed to be real. We denote the resulting expression as CSLYP. In the following, we relax this restriction and consider complex orbitals. We then proceed in analogy to the inclusion of current-dependent terms in the Fermi-hole curvature [9, 12] and in the ex- tension of the electron-localization-function (ELF) [39] for time-dependent states [40]. As a consequence, in addition to the term already present in CSLYP expression, the current densities of the single-particle orbitals appear in the final formula. In order to obtain this expression, which in the following will be denoted as JCSLYP, we rewrite the Laplacian of the Hartree-Fock (HF) two-body reduced density matrix in Eq.(1) in terms of the original particle coordinates ∇21 + ∇22 − ∇1 · ∇2 2 (r1, r2)|r1=r2 . (2) where 2 (r1, r2) = ρ(r1)ρ(r2)− 1,σ (r1, r2)ρ 1,σ (r1, r2) . (3) Here, 1,σ (r1, r2) = ψk,σ(r1)ψ k,σ(r2), (4) is the first-order HF density matrix (for a single Slater determinant) expressed in terms of the single-particle orbitals ψk,σ(r). The corresponding spin-density is simply given by ρσ(r) = ρ 1,σ (r, r). (5) Allowing the single-particle orbitals ψk,σ(r) to be complex, a given orbital not only gives the contribution ρk,σ(r) = |ψk,σ(r)| 2 to the density, but also the contribution jp k,σ(r) = ψk,σ(r)∇ψ k,σ(r) to the paramagnetic current density which is given by jp,σ(r) = jp k,σ(r) . (6) After some straightforward algebra, the Laplacian of the second-order HF reduced den- sity matrix takes the final form ρ(r)∇2ρ(r)− (∇ρ(r)) ρσ(r)∇ ρσ(r) (7) ρσ(r) (∇ρk,σ(r)) ρk,σ(r) + J(r) where J(r) = ρσ(r) j2pσ(r) ρσ(r) j2p k,σ(r) ρk,σ(r) contains all the current-dependent terms. Alternatively, Eq. (7) may also be expressed in terms of the non-interacting kinetic energy density τσ(r) = |∇ψk,σ(r)| (∇ρk,σ(r)) ρk,σ(r) j2p k,σ(r) ρk,σ(r) ρ(r)∇2ρ(r)− (∇ρ(r)) ρσ(r)∇ ρσ(r) 2ρσ(r)τσ(r)− jpσ(r) . (10) Comparison of Eq.(7) and Eq.(10) shows that J(r), as defined in Eq.(8), also contains current-dependent terms coming from the kinetic energy density. Thus, this would also suggest to reconsider the gradient expansion of τ for current-carrying states. Impor- tant consequences maybe expected for all approximated exchange-correlation functional involving the kinetic energy density as ingredient, such as the CS functional, and meta- GGAs. This issue will be specifically considered in a future work. In the next section, we assess the performance of the CS functional, and, in particular, the relevance of J(r), in reproducing the degeneracy of atomic states. III. RESULTS AND DISCUSSION We consider ground states of open-shell atoms having densities that can be repre- sented by a single Slater determinant of Kohn-Sham orbitals. Due to the symmetry of the problem, these Slater determinants are eigenstates of the z-component of both spin and orbital angular momentum. The single-particle orbitals are generally complex-valued and states with different total magnetic quantum numbers, ML, correspond to differ- ent current densities. By means of an accurate exchange-correlation functional the same total energies would be obtained. In the previous section, we have shown how current- dependent terms enter the expression of the CS functional when complex-valued orbitals are considered. Here, we evaluate the performance of EXX plus the CS functional in re- producing the degeneracy and study the effect of the current-dependent terms, in SDFT and DFT calculations. We consider atoms of the first and second row of the periodic table: these are the ref- erence cases for which a vast amount of numerical data is available [8, 9, 10, 11, 18, 22]. In analogy to the procedure where Hartree-Fock orbitals are used as input to the CS for- mula, we have evaluated the correlation energies in a post-hoc fashion using Kohn-Sham (KS) orbitals. We expand the KS orbitals in Slater-type basis functions (QZ4P of Ref. [42]) for the radial part, multiplied with spherical harmonics for the angular part. We obtain the KS orbitals from self-consistent EXX-only calculations employing the approximation of Krieger, Li, and Iafrate (KLI) [41], which has been shown to be extremely good at least for small systems [21]. In principle a functional should be evaluated with KS orbitals obtained from self-consistent calculations, and this is certainly possible for the CS func- tional [29]. However, thanks to the variational nature of DFT, it is common experience to observe only minor quantitative differences between post-hoc and self-consistent evalu- ation of total energies. This is the reason why the functionals designed for solving the degeneracy problem are typically evaluated in a post-hoc manner [9, 10, 11]. Table I shows the spurious energy splittings (difference in the total energies) between Kohn-Sham Slater determinants with total magnetic quantum number |ML| = 1 and ML = 0, from our SDFT and DFT calculations. The deviation of these total energies from the exact values is plotted in Fig. (1), where again the spurious energy splittings are visi- SDFT (DFT) Atom ∆JCSLY P ∆CSLY P B 0.8 (-0.3) 2.4 (1.4) C 0.9 (-0.1) -3.2 (-4.3) O -0.6 (-1.9) 0.9 (-0.4) F -0.1 (-1.5) -3.5 (-5.1) Al 0.4 (-0.5) 1.1 (0.2) Si 0.5 (-0.4) -1.2 (-2.2) S 0.1 (-1.6) 1.1 (-0.7) Cl 0.7 (-1.3) -1.1 (-3.2) me 0.3 (-1.0) -0.4 (-1.8) mae 0.5 (1.0) 1.8 (2.2) TABLE I: Spurios energy splittings, ∆ = E(|ML| = 1) − E(ML = 0) in kcal/mol for open-shell atoms, computed in SDFT (DFT results in parenthesis for comparison). Correlation energy has been added to the KLI-EXX energies including (JCSLYP) and neglecting (CSLYP) the current terms of Eq. (7). The last row shows the mean error (me) and mean absolute (mae) of the spurious splittings. ble. These results highlight the importance of including J(r) in Eq. (7). In particular, it is remarkable to observe that SDFT splittings are within 0.9 kcal/mol (with a mean error of 0.3 kcal/mol and a mean absolute error of 0.5 kcal/mol), and the corresponding DFT spu- rious energy splittings are less than 1.9 kcal/mol (with mean errors of 1.0 kcal/mol). It is worthwhile to note that in several cases inclusion of correlation leads to current-carrying states (|ML| = 1) with lower total energy than zero-current states (ML = 0). These results are in contrast to EXX-only [18] cases where: (a) the zero-current states are always lowest in energy and (b) the spurious energy splittings are always smaller in DFT than in SDFT. Going beyond EXX by including correlation in the form of CS functional accurate to- tal energies can be obtained within the OEP method [29]. Figure (1) and Table II show the deviations from exact total energies for the states with different magnetic quantum B C O F Al Si S Cl CSLYP M = 0 JCSLYP M CSLYP |M |= 1 JCSLYP |M B C O F Al Si S Cl SDFT DFT FIG. 1: Deviation from exact total energies for SDFT and DFT calculations employing the CS functional, including (JCSLYP) and not including (CSLYP) the current-dependent term J in Eq. (7). States with different magnetic quantum numbers ML are plotted. Exact total energies are taken from Ref. [29] and references therein. numbers, i.e. different current-carrying states. This further emphasizes the importance of proper inclusion of J(r) in Eq. (7). IV. CONCLUSIONS We have shown that going beyond the exact-exchange approximation by including correlation energy in the form of the Colle-Salvetti functional leads to a very good de- scription of the degeneracy of open-shell atoms in both SDFT and DFT calculations. Comparing DFT and SDFT results for the first and second row of the periodic table, on average we observe a reduction of the spurious energy splittings and better total ener- gies for SDFT. Furthermore, we have also shown how current-dependent terms enter the expression of the Colle-Salvetti functional. If these terms are neglected, the degeneracy is not well described and the total energies are also less accurate. Thus, this analysis re- confirms the advantage of properly including the orbital current dependent terms as an JCSLYP (CSLYP) SDFT DFT SDFT DFT |ML| 0 0 1 1 me 2.3 (16.1) 4.7 (18.5) 2.6 (15.7) 3.7 (16.8) mae 4.8 (16.4) 5.4 (18.5) 4.7 (15.7) 4.7 (16.8) TABLE II: Mean error (me) and mean absolute error (mae) in the total energies for the CS functional, including (JCSLYP) and not including (CSLYP results in parenthesis) the current- dependent term J in Eq. (7), in kcal/mol. Exact total energies are taken from Ref. 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Popular density functionals for the exchange-correlation energy typically fail to reproduce the degeneracy of different ground states of open-shell atoms. As a remedy, functionals which explicitly depend on the current density have been suggested. We present an analysis of this problem by investigating functionals that explicitly depend on the Kohn-Sham orbitals. Going beyond the exact-exchange approximation by adding correlation in the form of the Colle-Salvetti functional we show how current-dependent terms enter the Colle-Salvetti expression and their relevance is evaluated. A very good description of the degeneracy of ground-states for atoms of the first and second row of the periodic table is obtained.

Introduction Theory Results and discussion Conclusions Acknowledgements References

Cooper pairs in atomic nuclei G. G. Dussel1, S. Pittel2, J. Dukelsky3, and P. Sarriguren3 Departamento de Fisica J. J. Giambiagi, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, DE 19716 USA Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain (Dated: Received October 26, 2018) We consider the development of Cooper pairs in a self-consistent Hartree Fock mean field for the even Sm isotopes. Results are presented at the level of a BCS treatment, a number-projected BCS treatment and an exact treatment using the Richardson ansatz. While projected BCS captures much of the pairing correlation energy that is absent from BCS, it still misses a sizable correlation energy, typically of order 1 MeV . Furthermore, because it does not average over the properties of the fermion pairs, the exact Richardson solution permits a more meaningful definition of the Cooper wave function and of the fraction of pairs that are collective. PACS numbers: 21.60.-n, 03.75.Ss, 02.30.Ik, 74.20.Fg The first breakthrough in the derivation of a micro- scopic theory of superconductivity was the demonstra- tion by Cooper [1] in 1956 that bound pairs could be produced in the vicinity of the Fermi surface for an ar- bitrarily small attractive interaction. This was followed soon thereafter by the development of the BCS theory [2], in which superconductivity was described as the con- densation of a set of correlated pairs averaged over the whole system. Soon after the BCS paper, Bohr, Mottel- son and Pines [3] suggested that a similar phenomenon could explain the large gaps in the spectra of even-even nuclei. Since then, the BCS theory has been widely used to describe superconductivity in condensed matter and nuclear systems. Moreover, the concept of Cooper pairs as strongly overlapping objects that go through a con- densation process at the superconducting transition is central in the interpretation of the superconducting phe- nomenon. However, it is not easy to define the Cooper pair wave function from the mean field BCS theory, and most frequently it has been related to the pair correlator. By using the exact solution of the BCS Hamiltonian given by Richardson in the sixties [4], it was recently shown [5] that the Cooper pair wave function in a super- conducting medium has a precise definition. The unique form of its wave function transforms from a Cooper reso- nance in the weak coupling BCS region to a quasi bound pair in the Bose-Einstein condensed (BEC) phase. More- over, the Richardson solution gives a clear prescription for evaluating the fraction of correlated pairs as compared with Yang’s definition [6], providing a more accurate de- scription of the condensation phenomenon. The subject of Cooper pairing in atomic nuclei has come under renewed focus recently in the context of the mean-field Hartree-Fock-Bogolyubov approach [7, 8]. In this work, we also explore the role of Cooper pairs in mean-field treatments of atomic nuclei, comparing the traditional number-nonconserving BCS approach with a projected BCS approach and the exact Richardson treat- ment. We show that substantial differences in correlation energies arise when pairing is treated exactly for the same pairing strength, and that interesting differences emerge in some conceptual properties of the paired system. We begin by detailing the differences between the three approaches, focusing on a pairing Hamiltonian with con- stant strength G acting in a space of doubly-degenerate time-reversed states (k, k̄), ck −G ′ck′ , (1) where ǫk are the single-particle energies for the doubly- degenerate orbits k, k̄. Cooper studied the problem of adding a pair of fermions with an attractive pairing interaction on top of an inert Fermi sea (FS). He showed that the pair eigen- state is |ΨCooper〉 = 2ǫk − E |FS〉 , (2) where E is the energy eigenvalue. It turns out that E is negative for any attractive value of G, implying that the Cooper pair is bound and that the FS is unstable against the formation of bound pairs. Cooper suggested [1] that a theory considering a collection of bound pairs on top of an effective FS could explain superconductivity. The BCS approach follows a somewhat different path, defining instead a variational wave function as a coherent state of pairs properly averaged over the whole system, |ΨBCS〉 = e Γ† |0〉 , (3) where Γ† = is the coherent pair. The BCS wave function breaks particle-number conservation. Though errors due to the nonconservation of particle number are negligible in the thermodynamic limit, they can be important in finite systems such as atomic nuclei. Indeed, Bohr, Mottelson and Pines [3] noted already in 1958 the importance of taking into account finite size ef- fects in its application to nuclei. To accommodate these http://arxiv.org/abs/0704.1594v1 effects, the number-projected BCS formalism (PBCS) [9] assumes a condensed state of pairs of the form |ΨPBCS〉 = |0〉 , (4) where M is the number of pairs and Γ† has the same form as in BCS. We would like to emphasize here that Γ† should not be confused with the operator that creates a Cooper pair since its structure contains an average over the correlated pairs close to the Fermi energy and the free fermions deep inside the Fermi sphere. The Richardson ansatz [4] for the exact solution of the pairing Hamiltonian (1) follows closely Cooper’s original idea. For a system with 2M particles, it involves (in the ν = 0 sector) a product of M distinct pairs of the form |Φ >= Γ†α| 0〉 , Γ 2ǫk − eα . (5) The eα, called pair energies in analogy with the Cooper wave function (2), are in general complex parameters, which are obtained by solving the set of coupled non- linear Richardson equations 2ǫk − eα β( 6=α)=1,M eβ − eα = 0 . (6) The energy eigenvalues are obtained by summing the lowest M pair energies of each independent solution The key point to note upon inspection of the Richard- son pair (5) is that a pair energy close to a particular 2ǫk, i.e. close to the energy of an unperturbed pair, is dominated by this particular configuration and thus de- fines an uncorrelated pair. In contrast, a pair energy lying sufficiently far away in the complex plane produces a correlated Cooper pair. As mentioned before, the BCS coherent pairs, with amplitudes zk = vk/uk, cannot be interpreted as Cooper pairs since they mix correlated and uncorrelated pairs over the whole system. Indeed, it has been shown [5] that only in the extreme BEC limit are all pairs bound and condensed, and amenable to description by the two ap- proaches. Usually the structure of the Cooper pair is as- signed to the pair correlator 〈BCS| c |BCS〉 = ukvk. However, if the BCS state represents a fraction of cor- related pairs within a Fermi sea of free uncorrelated fermions, the pair correlator cannot guarantee that it picks up the two fermions from the same pair. The pair correlator is another averaged property over the set of correlated pairs. In what follows we explore the structure of pairing cor- relations in the even Sm isotopes, from 144Sm through 158Sm. The results are based on a series of self-consistent deformed Hartree Fock+BCS calculations. The calcula- tions make use of the density-dependent Skyrme force, SLy4, and treat pairing correlations using a pairing force with constant strength G. The calculations are carried out in an axially sym- metric harmonic oscillator space of 11 major shells (286 doubly-degenerate single-particle states). This basis in- volves oscillator parameters b0 and axis ratio q, opti- mized in order to minimize the energy in the given space. The strength of the pairing force for protons and neu- trons is chosen in such a way as to reproduce the ex- perimental pairing gaps in 154Sm (∆n = 0.98 MeV , ∆p = 0.94 MeV ), extracted from the binding energies in neighboring nuclei. We obtain Gn = 0.106 MeV and Gp = 0.117 MeV . Once we have fitted this reference strength, we determine the pairing strengths appropri- ate to the 142−158Sm isotopic chain by assuming a 1/A dependence. These calculations provide an excellent de- scription of the properties of the even Sm isotopes. We then use the results at self-consistency to define the HF mean field and consider the alternative number- conserving PBCS and exact Richardson approach to treat the pairing correlations within this mean field. We ig- nore the issue of whether the mean field should be self- consistently modified in these other approaches. In this way we are able to directly compare the three approaches to pairing with the same pairing Hamiltonian, which is the focus of this investigation. As is well known that the numerical solution of the Richardson equations (6) involves instabilities due to sin- gularities arising at some critical values of the pairing strength G. There have been two recent works that study these critical regions of parameter space [10] and propose ways to overcome the singularities [11]. While these methods alleviate the numerical divergences, thus allowing for an interpolation method to cross the criti- cal regions, some problems still persist and we have thus chosen to use a different approach. Since the singulari- ties arise as crossings of real pair energies eα with the unperturbed single-pair energies 2ǫk in the denomina- tors of (6), we start the numerical procedure at strong coupling (G = 1 MeV ) with complex single-particle en- ergies, obtained by adding a small arbitrary imaginary component. In this way, the singularities are avoided in the evolution of the system from strong coupling almost to the G = 0 limit. To obtain the exact solution at the physical value of G, we then let the imaginary parts go to zero starting with the solution already obtained for that G value. The method seems to work for any distribution of single-particle energies. A principal focus of our investigation is on the pairing correlation energy, defined as EC = 〈Φcorr|H | Φcorr〉 − 〈Φuncorr|H | Φuncorr〉 , (7) where |Φcorr〉 is the correlated ground-state wave func- tion and |Φuncorr〉 is the uncorrelated Hartree Fock Slater determinant obtained by filling all levels up to the Fermi energy. This quantity reflects the additional energy that derives from the inclusion of pairing. Table 1 summarizes our results for the pairing corre- lation energy in table 1 for all the even Sm isotopes un- der consideration. Note that the calculations include the semi-magic nucleus 144Sm, for which the BCS calculation leads to a normal solution with no pairing correlation en- ergy. In contrast, the projected BCS calculation leads to substantial pairing correlations in the ground state. That number projection is critical in mean-field treatments of semi-magic nuclei is well known from other calculations [12]. The exact treatment of pairing leads to a further lowering of the energy of the ground state of the system, by 0.3 MeV . In the calculations other than 144Sm, the effect on the pairing correlation energy of the exact solution is more pronounced. While PBCS gives a significant lowering of the energy of the system due to number projection, it misses about 1 MeV of the full correlation energy of an exact treatment. Considering the extensive recent efforts to carry out systematic microscopic calculations of nu- clear masses using mean-field methods [13], we feel that this effect may be quite meaningful. It is not clear that a renormalization of the strength of the pairing interaction can accommodate these important corrections. The results obtained for the Sm isotopes are consis- tent with studies performed in ultrasmall superconduct- ing grains [14]. The quantum phase transition from a superconducting to a normal metal predicted by BCS and PBCS completely disappears after fully including the pairing fluctuations by means of the exact solution of the BCS model. Moreover, the PBCS wave function displays a strange behavior in the transitional region as compared with the smooth behavior of the exact wave function [15]. Table I: Pairing correlation energies associated with the BCS, PBCS and exact Richardson treatments of pairing for the even Sm isotopes. All energies are given in MeV Mass EC(Exact) EC(PBCS) EC(BCS) 142 -4.146 -3.096 -1.107 144 -2.960 -2.677 0. 146 -4.340 -3.140 -1.384 148 -4.221 -3.014 -1.075 150 -3.761 -2.932 -0.386 152 -3.922 -2.957 -0.637 154 -3.678 -2.859 -0.390 156 -3.716 -2.832 -0.515 158 -3.832 -2.824 -0.717 A second important feature of Cooper pairing is the condensate fraction, namely the fraction of pairs of the whole system that are correlated. Analysis of the off- diagonal long-range order (ODLRO) that characterizes superconductors and superfluids led Yang [6] to a defini- tion of the condensate fraction, λ, in terms of the single macroscopic eigenvalue of the two-body density matrix. For a homogeneous system of two spin fermion species in the thermodynamic limit, λ is given by d3r1d r2 |〈ψ↓ (r1)ψ↑ (r2)〉| = k . (8) This definition is not appropriate for finite Fermi sys- tems, however, where several eigenvalues of the two-body density matrix are of the same order. We modify it, therefore, by excluding from the two-body density ma- trix the amplitude of finding two uncorrelated fermions. More specifically, our prescription for finite systems is to evaluate the matrix elements of the operator M(1−M/L) k,k′=1 ck̄′ck〉 − 〈c ck〉〈c ck̄′〉 , where L is the total number of doubly-degenerate, canon- ically conjugate pair states k, k̄. In BCS approximation, the modified Yang prescription leads to a condensate fraction λBCS = M(1−M/L) . (10) We have calculated this quantity for the BCS solutions obtained for 154Sm as a function of the pairing strength G and plot the results as the smooth curve in figure 1. An alternative prescription for the condensate fraction from the exact Richardson solution was proposed in [5] and shown to more properly reflect the properties of a su- perfluid system as it undergoes the crossover from BCS to BEC. In particular, this new prescription gives a fully condensed state at the change of sign of the chemical potential where the whole system becomes bound. This prescription, however, requires knowledge of the proper- ties of the precise Cooper pairs in the problem, not an average over the whole system as provided by the BCS or PBCS approximations (3,4). The Richardson ansatz (5) is ideally suited for this as it provides an exact wave function for each individual Cooper pair. One has to sim- ply distinguish which pairs are correlated and which are not. As previously discussed, a correlated pair is char- acterized by a pair energy eα that is far enough away in the complex plane from any particular 2ǫk. We therefore propose the following practical definition for the conden- sate fraction. It is the fraction of pair energies which in the complex energy plane lie further from any unperturbed single-pair energy, 2ǫk, than the mean single-particle level spacing. We now return to a discussion of the condensate frac- tion, as plotted in figure 1 for 154Sm as a function of the pairing strength G. In addition to the results based on the pair correlator, as discussed earlier, we also plot (in the sawtooth curve) the results that derive from the exact Richardson solution using the prescription just de- scribed. To illustrate how these latter results emerge, we show in figure 2 the associated pair energies for four 0.0 0.2 0.4 0.6 0.8 G (MeV) Exact FIG. 1: The modified Yang prescription for the BCS treat- ment of pairing (smooth curve) and the alternative prescrip- tion discussed in the text (sawtooth curve) for the exact Richardson treatment. G0 = 0.106 MeV denotes the physical value of the pairing strength and ǫ1 = µ denotes the strength at which the whole system binds. values of G in 154Sm, ranging from the physical value of G = 0.106 MeV to a fairly strong pairing strength of G = 0.4 MeV . In 154Sm the mean level spacing be- tween the Hartree Fock single-particle levels is roughly 0.5 MeV , both around the Fermi surface and far from it. For G = 0.106 MeV , most of the pair energies lie very near the real axis and quite close to at least one un- perturbed single-pair energy, 2ǫk. Two of them (which form a complex conjugate pair) extend about 1 MeV in the complex plane, while another two are marginally collective, lying roughly 0.5 MeV from the closest 2ǫk. The two most collective pairs, denoted C1 in the figure, each have a real energy of −15.55MeV , which is roughly twice the energy of the single-particle levels just below the Fermi surface. This suggests that the first pairs that become collective are indeed those built out of the va- lence orbits. As G increases, we see a gradual increase in the number of collective pairs, which form an arc in the complex plane. As can be seen from figure 1, by a pairing strength of roughly 0.5 MeV all of the pairs of the system are correlated giving a condensate fraction of 1, even though the BEC regime has not yet been reached. The BEC limit is realized when the chemical potential µ crosses the lowest single-particle energy ǫ1 at G = 0.788 for 154Sm. At this point all pairs are bound. However, the revised Yang prescription (9) fails to predict a com- plete condensate at this point, in the same way as it fails to do so in the homogeneous case [5]. The Richardson prescription for Cooper pairs also gives rise to a different interpretation of their internal structure. In figure 3, we compare the square of the wave function for the most correlated Cooper pairs in 154Sm, i.e. those whose pair energies lie farthest from any un- -80 -60 -40 -20 0 20 40 60 80 -1.0 -0.5 0.0 0.5 1.0 -40 -20 0 20 40 -20 -15 -10 -5 0 5 10 15 20 Imaginary Part G=0.4 2C2C1 C1 G=0.106 Imaginary Part G=0.3 G=0.2 FIG. 2: Pair energies (in MeV ) for the exact Cooper pairs that emerge from four calculations of the 154Sm isotope. G = 0.106 MeV is the physical value of the pairing strength. In that panel, we denote the most collective pairs as Ci, for subsequent notational purposes. 38 40 42 44 46 48 50 52 FIG. 3: Square of the wave function of the most collective Cooper pairs in 154Sm (denoted C1, C2, C3, C4, and C5) and the pair correlator (BCS) versus the single-particle levels. perturbed single-pair energy, with the square of the pair correlator wave function obtained from the BCS calcula- tion. All wave functions are plotted versus the order of the single-particle states to make clear the relevant mix- ing of configurations in each pair. The pair labels in the figure (C1 through C5) refer to corresponding labels in the upper left panel of figure 2. C1 refers to the two most collective pairs, namely those that are farthest from any unperturbed single-particle pair. Being complex conju- gate pairs, both have exactly the same absolute square of their wave function and thus we only show one in the figure. C2 refers to the next two most collective pairs, which as noted earlier are marginally collective according to our prescription. C3 refers to the next two most col- lective pairs after C2, which according to the prescription given above involve perturbative mixing of configurations and are not collective. C4 and C5, the following pairs in descending collective order, have real pair energies and involve almost pure single-particle configurations. ¿From the figure, we see that the pair correlator wave function is quite spread over several single-particle con- figurations and is peaked between at the 47th single- particle level, just beyond the Fermi energy (154Sm has 46 neutron pairs). In contrast, the most highly corre- lated Cooper pair wave function C1 is somewhat narrower (less collective) and is peaked slightly within the Fermi sphere. The less-collective Cooper pairs, C2 through C3, are peaked progressively further inside the Fermi sphere and are progressively narrower. From this figure, we con- clude that the size of even the most collective Cooper pairs in coordinate space will be larger than the size of the pair correlator, as was already demonstrated in the weak coupling BCS regime of cold atomic gases [5]. Re- cent investigations [7, 8] on the size of the pair correlator in spherical nuclei have concluded that it is unexpectedly small in the nuclear surface (2− 3 fm). The present cal- culations would suggest that the actual size of the few highly collective Cooper pairs is larger than the typical size of the pair correlations in the nuclear medium. Fur- thermore, as is also evident from the figure, less bound pairs get progressively closer to a particular 2ǫk and the corresponding Cooper pair wave function is less collec- tive, i.e. more narrow in energy space, and peaked at this particular configuration. In this work, we have studied the role of Cooper pair- ing in atomic nuclei, focusing on a realistic description of the even Sm isotopes. We assume that the mean field is given by the self-consistent HF solution from coupled HF+BCS calculations, and then consider how the effects of pairing on that mean field would be modified at several levels of improved treatment. We consider both the pro- jected BCS approximation and an exact treatment based on Richardson’s solution of the pairing problem. Sev- eral important points emerged. On the one hand, even though PBCS approximation gives a significant gain in binding energy over ordinary BCS, it still fails to capture a sizable component, typically of order 1 MeV . This might have important implications in efforts to derive nuclear masses from a microscopic approach. Second, we discussed a new and improved prescription for identify- ing the fraction of the pairs in a nucleus that are collec- tive, which can only be realized when the properties of the various Cooper pairs in the problem are treated sep- arately. This new prescription suggests that a slightly larger number of pairs are collective when compared to the more usual prescription based on Yang’s definition of the condensate fraction. Furthermore, it suggests that the few collective Cooper pairs that arise in real nuclei, being individually less collective than the pair correlator, would be spatially more spread out. The Richardson solution, as generalized in ref. [16], can be obtained for integrable pairing hamiltonians only. It is possible, however, to use the Richardson ansatz (5) in a variational treatment of general non-integrable pair- ing hamiltonians. The pair energies would play the role of variational parameters within a generalized Pfaffian pairing wave function [17], making it possible to treat pair correlations in a more precise manner for realistic nuclear systems. We acknowledge fruitful discussions with N. Sand- ulescu, P. Schuck and W. Nazarewicz. This work was supported in part by the Spanish DGI under grants FIS2005-00640 and FIS2006-12783-C03-01, in part by the US National Science Foundation under grant # 0553127, and in part by UBACYT X-053, Carrera del Investigador Cient́ıfico and PIP-5287 (CONICET-Argentina). One of the authors (SP) wishes to acknowledge the generous sup- port and hospitality of the CSIC in Madrid where much of his contribution to the work was carried out [1] Leon N. Cooper, Phys. Rev. 104, 1189 (1956). [2] J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). [3] A. Bohr, B. R. Mottelson and D. Pines, Phys. Rev. 110, 936 (1958). [4] R. W. Richardson, Phys. Rev. Lett. 3, 277 (1963); Phys. Rev. 141, 949 (1966). [5] G. Ortiz and J. Dukelsky, Phys. Rev. A 72, 043611 (2005). [6] C. N. Yang, Rev. Mod. Phys. 34, 694 (1962). [7] Masayuki Matsuo, Kazuhito Mizuyama and Yasuyoshi Seizawa, Phys. Rev. C 71, 064326 (2005). [8] N. Pillet, N. Sandulescu and P. Schuck, LANL preprint # nucl-th/0701086. [9] K. Dietrich, H. J. Mang and J. H. Pradal, Phys. Rev. 135, B22 (1964). [10] F. Dominguez, C. Esebbag and J. Dukelsky, J. Phys. A: Math. Gen. 39, 11349 (2006). [11] S. Rombouts, D. Van Neck and J. Dukelsky, Phys. Rev. C 69, 061303 (2004). [12] M.V. Stoitsov, J. Dobaczewski, R. Kirchner, W. Nazarewicz and J. Terasaki, LANL preprint # nucl-th/0610061. [13] D. Lunney, J.M. Pearson, C. Thibault Rev. Mod. Phys. 75, 1021 (2003). [14] J. Dukelsky and G. Sierra, Phys. Rev. Lett. 83, 172 (1999). [15] J. Dukelsky and G. Sierra, Phys. Rev. B 61, 12302 (2000). [16] J. Dukelsky, C. Esebbag and P. Schuck, Phys. Rev. Lett. 87, 66403 (2001). [17] M. Bajdich, L. Mitas, G. Drobn, L. K. Wagner, and K. E. Schmidt, Phys. Rev. Lett. 96, 130201 (2006). http://arxiv.org/abs/nucl-th/0701086 http://arxiv.org/abs/nucl-th/0610061

We consider the development of Cooper pairs in a self-consistent Hartree Fock mean field for the even Sm isotopes. Results are presented at the level of a BCS treatment, a number-projected BCS treatment and an exact treatment using the Richardson ansatz. While projected BCS captures much of the pairing correlation energy that is absent from BCS, it still misses a sizable correlation energy, typically of order $1 MeV$. Furthermore, because it does not average over the properties of the fermion pairs, the exact Richardson solution permits a more meaningful definition of the Cooper wave function and of the fraction of pairs that are collective.

Cooper pairs in atomic nuclei G. G. Dussel1, S. Pittel2, J. Dukelsky3, and P. Sarriguren3 Departamento de Fisica J. J. Giambiagi, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, DE 19716 USA Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain (Dated: Received October 26, 2018) We consider the development of Cooper pairs in a self-consistent Hartree Fock mean field for the even Sm isotopes. Results are presented at the level of a BCS treatment, a number-projected BCS treatment and an exact treatment using the Richardson ansatz. While projected BCS captures much of the pairing correlation energy that is absent from BCS, it still misses a sizable correlation energy, typically of order 1 MeV . Furthermore, because it does not average over the properties of the fermion pairs, the exact Richardson solution permits a more meaningful definition of the Cooper wave function and of the fraction of pairs that are collective. PACS numbers: 21.60.-n, 03.75.Ss, 02.30.Ik, 74.20.Fg The first breakthrough in the derivation of a micro- scopic theory of superconductivity was the demonstra- tion by Cooper [1] in 1956 that bound pairs could be produced in the vicinity of the Fermi surface for an ar- bitrarily small attractive interaction. This was followed soon thereafter by the development of the BCS theory [2], in which superconductivity was described as the con- densation of a set of correlated pairs averaged over the whole system. Soon after the BCS paper, Bohr, Mottel- son and Pines [3] suggested that a similar phenomenon could explain the large gaps in the spectra of even-even nuclei. Since then, the BCS theory has been widely used to describe superconductivity in condensed matter and nuclear systems. Moreover, the concept of Cooper pairs as strongly overlapping objects that go through a con- densation process at the superconducting transition is central in the interpretation of the superconducting phe- nomenon. However, it is not easy to define the Cooper pair wave function from the mean field BCS theory, and most frequently it has been related to the pair correlator. By using the exact solution of the BCS Hamiltonian given by Richardson in the sixties [4], it was recently shown [5] that the Cooper pair wave function in a super- conducting medium has a precise definition. The unique form of its wave function transforms from a Cooper reso- nance in the weak coupling BCS region to a quasi bound pair in the Bose-Einstein condensed (BEC) phase. More- over, the Richardson solution gives a clear prescription for evaluating the fraction of correlated pairs as compared with Yang’s definition [6], providing a more accurate de- scription of the condensation phenomenon. The subject of Cooper pairing in atomic nuclei has come under renewed focus recently in the context of the mean-field Hartree-Fock-Bogolyubov approach [7, 8]. In this work, we also explore the role of Cooper pairs in mean-field treatments of atomic nuclei, comparing the traditional number-nonconserving BCS approach with a projected BCS approach and the exact Richardson treat- ment. We show that substantial differences in correlation energies arise when pairing is treated exactly for the same pairing strength, and that interesting differences emerge in some conceptual properties of the paired system. We begin by detailing the differences between the three approaches, focusing on a pairing Hamiltonian with con- stant strength G acting in a space of doubly-degenerate time-reversed states (k, k̄), ck −G ′ck′ , (1) where ǫk are the single-particle energies for the doubly- degenerate orbits k, k̄. Cooper studied the problem of adding a pair of fermions with an attractive pairing interaction on top of an inert Fermi sea (FS). He showed that the pair eigen- state is |ΨCooper〉 = 2ǫk − E |FS〉 , (2) where E is the energy eigenvalue. It turns out that E is negative for any attractive value of G, implying that the Cooper pair is bound and that the FS is unstable against the formation of bound pairs. Cooper suggested [1] that a theory considering a collection of bound pairs on top of an effective FS could explain superconductivity. The BCS approach follows a somewhat different path, defining instead a variational wave function as a coherent state of pairs properly averaged over the whole system, |ΨBCS〉 = e Γ† |0〉 , (3) where Γ† = is the coherent pair. The BCS wave function breaks particle-number conservation. Though errors due to the nonconservation of particle number are negligible in the thermodynamic limit, they can be important in finite systems such as atomic nuclei. Indeed, Bohr, Mottelson and Pines [3] noted already in 1958 the importance of taking into account finite size ef- fects in its application to nuclei. To accommodate these http://arxiv.org/abs/0704.1594v1 effects, the number-projected BCS formalism (PBCS) [9] assumes a condensed state of pairs of the form |ΨPBCS〉 = |0〉 , (4) where M is the number of pairs and Γ† has the same form as in BCS. We would like to emphasize here that Γ† should not be confused with the operator that creates a Cooper pair since its structure contains an average over the correlated pairs close to the Fermi energy and the free fermions deep inside the Fermi sphere. The Richardson ansatz [4] for the exact solution of the pairing Hamiltonian (1) follows closely Cooper’s original idea. For a system with 2M particles, it involves (in the ν = 0 sector) a product of M distinct pairs of the form |Φ >= Γ†α| 0〉 , Γ 2ǫk − eα . (5) The eα, called pair energies in analogy with the Cooper wave function (2), are in general complex parameters, which are obtained by solving the set of coupled non- linear Richardson equations 2ǫk − eα β( 6=α)=1,M eβ − eα = 0 . (6) The energy eigenvalues are obtained by summing the lowest M pair energies of each independent solution The key point to note upon inspection of the Richard- son pair (5) is that a pair energy close to a particular 2ǫk, i.e. close to the energy of an unperturbed pair, is dominated by this particular configuration and thus de- fines an uncorrelated pair. In contrast, a pair energy lying sufficiently far away in the complex plane produces a correlated Cooper pair. As mentioned before, the BCS coherent pairs, with amplitudes zk = vk/uk, cannot be interpreted as Cooper pairs since they mix correlated and uncorrelated pairs over the whole system. Indeed, it has been shown [5] that only in the extreme BEC limit are all pairs bound and condensed, and amenable to description by the two ap- proaches. Usually the structure of the Cooper pair is as- signed to the pair correlator 〈BCS| c |BCS〉 = ukvk. However, if the BCS state represents a fraction of cor- related pairs within a Fermi sea of free uncorrelated fermions, the pair correlator cannot guarantee that it picks up the two fermions from the same pair. The pair correlator is another averaged property over the set of correlated pairs. In what follows we explore the structure of pairing cor- relations in the even Sm isotopes, from 144Sm through 158Sm. The results are based on a series of self-consistent deformed Hartree Fock+BCS calculations. The calcula- tions make use of the density-dependent Skyrme force, SLy4, and treat pairing correlations using a pairing force with constant strength G. The calculations are carried out in an axially sym- metric harmonic oscillator space of 11 major shells (286 doubly-degenerate single-particle states). This basis in- volves oscillator parameters b0 and axis ratio q, opti- mized in order to minimize the energy in the given space. The strength of the pairing force for protons and neu- trons is chosen in such a way as to reproduce the ex- perimental pairing gaps in 154Sm (∆n = 0.98 MeV , ∆p = 0.94 MeV ), extracted from the binding energies in neighboring nuclei. We obtain Gn = 0.106 MeV and Gp = 0.117 MeV . Once we have fitted this reference strength, we determine the pairing strengths appropri- ate to the 142−158Sm isotopic chain by assuming a 1/A dependence. These calculations provide an excellent de- scription of the properties of the even Sm isotopes. We then use the results at self-consistency to define the HF mean field and consider the alternative number- conserving PBCS and exact Richardson approach to treat the pairing correlations within this mean field. We ig- nore the issue of whether the mean field should be self- consistently modified in these other approaches. In this way we are able to directly compare the three approaches to pairing with the same pairing Hamiltonian, which is the focus of this investigation. As is well known that the numerical solution of the Richardson equations (6) involves instabilities due to sin- gularities arising at some critical values of the pairing strength G. There have been two recent works that study these critical regions of parameter space [10] and propose ways to overcome the singularities [11]. While these methods alleviate the numerical divergences, thus allowing for an interpolation method to cross the criti- cal regions, some problems still persist and we have thus chosen to use a different approach. Since the singulari- ties arise as crossings of real pair energies eα with the unperturbed single-pair energies 2ǫk in the denomina- tors of (6), we start the numerical procedure at strong coupling (G = 1 MeV ) with complex single-particle en- ergies, obtained by adding a small arbitrary imaginary component. In this way, the singularities are avoided in the evolution of the system from strong coupling almost to the G = 0 limit. To obtain the exact solution at the physical value of G, we then let the imaginary parts go to zero starting with the solution already obtained for that G value. The method seems to work for any distribution of single-particle energies. A principal focus of our investigation is on the pairing correlation energy, defined as EC = 〈Φcorr|H | Φcorr〉 − 〈Φuncorr|H | Φuncorr〉 , (7) where |Φcorr〉 is the correlated ground-state wave func- tion and |Φuncorr〉 is the uncorrelated Hartree Fock Slater determinant obtained by filling all levels up to the Fermi energy. This quantity reflects the additional energy that derives from the inclusion of pairing. Table 1 summarizes our results for the pairing corre- lation energy in table 1 for all the even Sm isotopes un- der consideration. Note that the calculations include the semi-magic nucleus 144Sm, for which the BCS calculation leads to a normal solution with no pairing correlation en- ergy. In contrast, the projected BCS calculation leads to substantial pairing correlations in the ground state. That number projection is critical in mean-field treatments of semi-magic nuclei is well known from other calculations [12]. The exact treatment of pairing leads to a further lowering of the energy of the ground state of the system, by 0.3 MeV . In the calculations other than 144Sm, the effect on the pairing correlation energy of the exact solution is more pronounced. While PBCS gives a significant lowering of the energy of the system due to number projection, it misses about 1 MeV of the full correlation energy of an exact treatment. Considering the extensive recent efforts to carry out systematic microscopic calculations of nu- clear masses using mean-field methods [13], we feel that this effect may be quite meaningful. It is not clear that a renormalization of the strength of the pairing interaction can accommodate these important corrections. The results obtained for the Sm isotopes are consis- tent with studies performed in ultrasmall superconduct- ing grains [14]. The quantum phase transition from a superconducting to a normal metal predicted by BCS and PBCS completely disappears after fully including the pairing fluctuations by means of the exact solution of the BCS model. Moreover, the PBCS wave function displays a strange behavior in the transitional region as compared with the smooth behavior of the exact wave function [15]. Table I: Pairing correlation energies associated with the BCS, PBCS and exact Richardson treatments of pairing for the even Sm isotopes. All energies are given in MeV Mass EC(Exact) EC(PBCS) EC(BCS) 142 -4.146 -3.096 -1.107 144 -2.960 -2.677 0. 146 -4.340 -3.140 -1.384 148 -4.221 -3.014 -1.075 150 -3.761 -2.932 -0.386 152 -3.922 -2.957 -0.637 154 -3.678 -2.859 -0.390 156 -3.716 -2.832 -0.515 158 -3.832 -2.824 -0.717 A second important feature of Cooper pairing is the condensate fraction, namely the fraction of pairs of the whole system that are correlated. Analysis of the off- diagonal long-range order (ODLRO) that characterizes superconductors and superfluids led Yang [6] to a defini- tion of the condensate fraction, λ, in terms of the single macroscopic eigenvalue of the two-body density matrix. For a homogeneous system of two spin fermion species in the thermodynamic limit, λ is given by d3r1d r2 |〈ψ↓ (r1)ψ↑ (r2)〉| = k . (8) This definition is not appropriate for finite Fermi sys- tems, however, where several eigenvalues of the two-body density matrix are of the same order. We modify it, therefore, by excluding from the two-body density ma- trix the amplitude of finding two uncorrelated fermions. More specifically, our prescription for finite systems is to evaluate the matrix elements of the operator M(1−M/L) k,k′=1 ck̄′ck〉 − 〈c ck〉〈c ck̄′〉 , where L is the total number of doubly-degenerate, canon- ically conjugate pair states k, k̄. In BCS approximation, the modified Yang prescription leads to a condensate fraction λBCS = M(1−M/L) . (10) We have calculated this quantity for the BCS solutions obtained for 154Sm as a function of the pairing strength G and plot the results as the smooth curve in figure 1. An alternative prescription for the condensate fraction from the exact Richardson solution was proposed in [5] and shown to more properly reflect the properties of a su- perfluid system as it undergoes the crossover from BCS to BEC. In particular, this new prescription gives a fully condensed state at the change of sign of the chemical potential where the whole system becomes bound. This prescription, however, requires knowledge of the proper- ties of the precise Cooper pairs in the problem, not an average over the whole system as provided by the BCS or PBCS approximations (3,4). The Richardson ansatz (5) is ideally suited for this as it provides an exact wave function for each individual Cooper pair. One has to sim- ply distinguish which pairs are correlated and which are not. As previously discussed, a correlated pair is char- acterized by a pair energy eα that is far enough away in the complex plane from any particular 2ǫk. We therefore propose the following practical definition for the conden- sate fraction. It is the fraction of pair energies which in the complex energy plane lie further from any unperturbed single-pair energy, 2ǫk, than the mean single-particle level spacing. We now return to a discussion of the condensate frac- tion, as plotted in figure 1 for 154Sm as a function of the pairing strength G. In addition to the results based on the pair correlator, as discussed earlier, we also plot (in the sawtooth curve) the results that derive from the exact Richardson solution using the prescription just de- scribed. To illustrate how these latter results emerge, we show in figure 2 the associated pair energies for four 0.0 0.2 0.4 0.6 0.8 G (MeV) Exact FIG. 1: The modified Yang prescription for the BCS treat- ment of pairing (smooth curve) and the alternative prescrip- tion discussed in the text (sawtooth curve) for the exact Richardson treatment. G0 = 0.106 MeV denotes the physical value of the pairing strength and ǫ1 = µ denotes the strength at which the whole system binds. values of G in 154Sm, ranging from the physical value of G = 0.106 MeV to a fairly strong pairing strength of G = 0.4 MeV . In 154Sm the mean level spacing be- tween the Hartree Fock single-particle levels is roughly 0.5 MeV , both around the Fermi surface and far from it. For G = 0.106 MeV , most of the pair energies lie very near the real axis and quite close to at least one un- perturbed single-pair energy, 2ǫk. Two of them (which form a complex conjugate pair) extend about 1 MeV in the complex plane, while another two are marginally collective, lying roughly 0.5 MeV from the closest 2ǫk. The two most collective pairs, denoted C1 in the figure, each have a real energy of −15.55MeV , which is roughly twice the energy of the single-particle levels just below the Fermi surface. This suggests that the first pairs that become collective are indeed those built out of the va- lence orbits. As G increases, we see a gradual increase in the number of collective pairs, which form an arc in the complex plane. As can be seen from figure 1, by a pairing strength of roughly 0.5 MeV all of the pairs of the system are correlated giving a condensate fraction of 1, even though the BEC regime has not yet been reached. The BEC limit is realized when the chemical potential µ crosses the lowest single-particle energy ǫ1 at G = 0.788 for 154Sm. At this point all pairs are bound. However, the revised Yang prescription (9) fails to predict a com- plete condensate at this point, in the same way as it fails to do so in the homogeneous case [5]. The Richardson prescription for Cooper pairs also gives rise to a different interpretation of their internal structure. In figure 3, we compare the square of the wave function for the most correlated Cooper pairs in 154Sm, i.e. those whose pair energies lie farthest from any un- -80 -60 -40 -20 0 20 40 60 80 -1.0 -0.5 0.0 0.5 1.0 -40 -20 0 20 40 -20 -15 -10 -5 0 5 10 15 20 Imaginary Part G=0.4 2C2C1 C1 G=0.106 Imaginary Part G=0.3 G=0.2 FIG. 2: Pair energies (in MeV ) for the exact Cooper pairs that emerge from four calculations of the 154Sm isotope. G = 0.106 MeV is the physical value of the pairing strength. In that panel, we denote the most collective pairs as Ci, for subsequent notational purposes. 38 40 42 44 46 48 50 52 FIG. 3: Square of the wave function of the most collective Cooper pairs in 154Sm (denoted C1, C2, C3, C4, and C5) and the pair correlator (BCS) versus the single-particle levels. perturbed single-pair energy, with the square of the pair correlator wave function obtained from the BCS calcula- tion. All wave functions are plotted versus the order of the single-particle states to make clear the relevant mix- ing of configurations in each pair. The pair labels in the figure (C1 through C5) refer to corresponding labels in the upper left panel of figure 2. C1 refers to the two most collective pairs, namely those that are farthest from any unperturbed single-particle pair. Being complex conju- gate pairs, both have exactly the same absolute square of their wave function and thus we only show one in the figure. C2 refers to the next two most collective pairs, which as noted earlier are marginally collective according to our prescription. C3 refers to the next two most col- lective pairs after C2, which according to the prescription given above involve perturbative mixing of configurations and are not collective. C4 and C5, the following pairs in descending collective order, have real pair energies and involve almost pure single-particle configurations. ¿From the figure, we see that the pair correlator wave function is quite spread over several single-particle con- figurations and is peaked between at the 47th single- particle level, just beyond the Fermi energy (154Sm has 46 neutron pairs). In contrast, the most highly corre- lated Cooper pair wave function C1 is somewhat narrower (less collective) and is peaked slightly within the Fermi sphere. The less-collective Cooper pairs, C2 through C3, are peaked progressively further inside the Fermi sphere and are progressively narrower. From this figure, we con- clude that the size of even the most collective Cooper pairs in coordinate space will be larger than the size of the pair correlator, as was already demonstrated in the weak coupling BCS regime of cold atomic gases [5]. Re- cent investigations [7, 8] on the size of the pair correlator in spherical nuclei have concluded that it is unexpectedly small in the nuclear surface (2− 3 fm). The present cal- culations would suggest that the actual size of the few highly collective Cooper pairs is larger than the typical size of the pair correlations in the nuclear medium. Fur- thermore, as is also evident from the figure, less bound pairs get progressively closer to a particular 2ǫk and the corresponding Cooper pair wave function is less collec- tive, i.e. more narrow in energy space, and peaked at this particular configuration. In this work, we have studied the role of Cooper pair- ing in atomic nuclei, focusing on a realistic description of the even Sm isotopes. We assume that the mean field is given by the self-consistent HF solution from coupled HF+BCS calculations, and then consider how the effects of pairing on that mean field would be modified at several levels of improved treatment. We consider both the pro- jected BCS approximation and an exact treatment based on Richardson’s solution of the pairing problem. Sev- eral important points emerged. On the one hand, even though PBCS approximation gives a significant gain in binding energy over ordinary BCS, it still fails to capture a sizable component, typically of order 1 MeV . This might have important implications in efforts to derive nuclear masses from a microscopic approach. Second, we discussed a new and improved prescription for identify- ing the fraction of the pairs in a nucleus that are collec- tive, which can only be realized when the properties of the various Cooper pairs in the problem are treated sep- arately. This new prescription suggests that a slightly larger number of pairs are collective when compared to the more usual prescription based on Yang’s definition of the condensate fraction. Furthermore, it suggests that the few collective Cooper pairs that arise in real nuclei, being individually less collective than the pair correlator, would be spatially more spread out. The Richardson solution, as generalized in ref. [16], can be obtained for integrable pairing hamiltonians only. It is possible, however, to use the Richardson ansatz (5) in a variational treatment of general non-integrable pair- ing hamiltonians. The pair energies would play the role of variational parameters within a generalized Pfaffian pairing wave function [17], making it possible to treat pair correlations in a more precise manner for realistic nuclear systems. We acknowledge fruitful discussions with N. Sand- ulescu, P. Schuck and W. Nazarewicz. This work was supported in part by the Spanish DGI under grants FIS2005-00640 and FIS2006-12783-C03-01, in part by the US National Science Foundation under grant # 0553127, and in part by UBACYT X-053, Carrera del Investigador Cient́ıfico and PIP-5287 (CONICET-Argentina). One of the authors (SP) wishes to acknowledge the generous sup- port and hospitality of the CSIC in Madrid where much of his contribution to the work was carried out [1] Leon N. Cooper, Phys. Rev. 104, 1189 (1956). [2] J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). [3] A. Bohr, B. R. Mottelson and D. Pines, Phys. Rev. 110, 936 (1958). [4] R. W. Richardson, Phys. Rev. Lett. 3, 277 (1963); Phys. Rev. 141, 949 (1966). [5] G. Ortiz and J. Dukelsky, Phys. Rev. A 72, 043611 (2005). [6] C. N. Yang, Rev. Mod. Phys. 34, 694 (1962). [7] Masayuki Matsuo, Kazuhito Mizuyama and Yasuyoshi Seizawa, Phys. Rev. C 71, 064326 (2005). [8] N. Pillet, N. Sandulescu and P. Schuck, LANL preprint # nucl-th/0701086. [9] K. Dietrich, H. J. Mang and J. H. Pradal, Phys. Rev. 135, B22 (1964). [10] F. Dominguez, C. Esebbag and J. Dukelsky, J. Phys. A: Math. Gen. 39, 11349 (2006). [11] S. Rombouts, D. Van Neck and J. Dukelsky, Phys. Rev. C 69, 061303 (2004). [12] M.V. Stoitsov, J. Dobaczewski, R. Kirchner, W. Nazarewicz and J. Terasaki, LANL preprint # nucl-th/0610061. [13] D. Lunney, J.M. Pearson, C. Thibault Rev. Mod. Phys. 75, 1021 (2003). [14] J. Dukelsky and G. Sierra, Phys. Rev. Lett. 83, 172 (1999). [15] J. Dukelsky and G. Sierra, Phys. Rev. B 61, 12302 (2000). [16] J. Dukelsky, C. Esebbag and P. Schuck, Phys. Rev. Lett. 87, 66403 (2001). [17] M. Bajdich, L. Mitas, G. Drobn, L. K. Wagner, and K. E. Schmidt, Phys. Rev. Lett. 96, 130201 (2006). http://arxiv.org/abs/nucl-th/0701086 http://arxiv.org/abs/nucl-th/0610061

An adaptive numerical method for the Vlasov equation based on a multiresolution analysis N. Besse1, F. Filbet2, M. Gutnic2, I. Paun2, E. Sonnendrücker2 1 C.E.A, BP 12, 91680 Bruyères-le-Châtel, France, nicolas.besse@cea.fr 2 IRMA, Université Louis Pasteur, 67084 Strasbourg cedex, France, filbet,gutnic,ipaun,sonnen@math.u-strasbg.fr 1 Introduction Plasmas, which are gases of charged particles, and charged particle beams can be described by a distribution function f(t, x, v) dependent on time t, on position x and on velocity v. The function f represents the probability of presence of a particle at position (x, v) in phase space at time t. It satisfies the so-called Vlasov equation + v · ∇xf + F (t, x, v) · ∇vf = 0. (1) The force field F (t, x, v) consists of applied and self-consistent electric and magnetic fields: (Eself + Eapp + v × (Bself +Bapp)), wherem represents the mass of a particle and q its charge. The self-consistent part of the force field is solution of Maxwell’s equations +∇×B = µ0j, ∇ · E = +∇×E = 0, ∇ ·B = 0. The coupling with the Vlasov equation results from the source terms ρ and j such that: ρ(t, x) = q f(t, x, v) dv, j = q f(t, x, v)v dv. We then obtain the nonlinear Vlasov-Maxwell equations. In some cases, when the field are slowly varying the magnetic field becomes negligible and the Maxwell equations can be replaced by the Poisson equation where: Eself (t, x) = −∇xφ(t, x), −ε0∆xφ = ρ. (2) The numerical resolution of the Vlasov equation is usually performed by particle methods (PIC) which consist in approximating the plasma by a http://arxiv.org/abs/0704.1595v1 2 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker finite number of particles. The trajectories of these particles are computed from the characteristic curves given by the Vlasov equation, whereas self- consistent fields are computed on a mesh of the physical space. This method allows to obtain satisfying results with a few number of particles. However, it is well known that, in some cases, the numerical noise inherent to the particle method becomes too important to have an accurate description of the distribution function in phase space. Moreover, the numerical noise only decreases in N , when the number of particles N is increased. To remedy to this problem, methods discretizing the Vlasov equation on a mesh of phase space have been proposed. A review of the main methods for the resolution of the Vlasov equation is given in these proceedings [5]. The major drawback of methods using a uniform and fixed mesh is that their numerical cost is high, which makes them rather inefficient when the dimension of phase-space grows. For this reason we are investigating here a method using an adaptive mesh. The adaptive method is overlayed to a classical semi-Lagrangian method which is based on the conservation of the distribution function along characteristics. Indeed, this method uses two steps to update the value of the distribution function at a given mesh point. The first one consists in following the characteristic ending at this mesh point backward in time, and the second one in interpolating its value there from the old values at the surrounding mesh points. Using the conservation of the distribution function along the characteristics this will yield its new value at the given mesh point. This idea was originally introduced by Cheng and Knorr [2] along with a time splitting technique enabling to compute exactly the origin of the characteristics at each fractional step. In the original method, the interpolation was performed using cubic splines. This method has since been used extensively by plasma physicists (see for example [4, 6] and the ref- erences therein). It has then been generalized to the frame of semi-Lagrangian methods by E. Sonnendrücker et al. [8]. This method has also been used to investigate problems linked to the propagation of strongly nonlinear heavy ion beams [9]. In the present work, we have chosen to introduce a phase-space mesh which can be refined or derefined adaptively in time. For this purpose, we use a technique based on multiresolution analysis which is in the same spirit as the methods developed in particular by S. Bertoluzza [1], A. Cohen et al. [3] and M. Griebel and F. Koster [7]. We represent the distribution function on a wavelet basis at different scales. We can then compress it by eliminat- ing coefficients which are small and accordingly remove the associated mesh points. Another specific feature of our method is that we use an advection in physical and velocity space forward in time to predict the useful grid points for the next time step, rather than restrict ourselves to the neighboring points. This enables us to use a much larger time step, as in the semi-Lagrangian method the time step is not limited by a Courant condition. Once the new mesh is predicted, the semi-Lagrangian methodology is used to compute the An adaptive numerical method for the Vlasov equation 3 new values of the distribution function at the predicted mesh points, using an interpolation based on the wavelet decomposition of the old distribution function. The mesh is then refined again by performing a wavelet transform, and eliminating the points associated to small coefficients. This paper is organized as follows. In section 2, we recall the tools of multiresolution analysis which will be needed for our method, precizing what kind of wavelets seem to be the most appropriate in our case. Then, we describe in section 3 the algorithm used in our method, first for the non adaptive mesh case and then for the adaptive mesh case. Finally we present a few preliminary numerical results. 2 Multiresolution analysis The semi-Lagrangian method consists mainly of two steps, an advection step and an interpolation step. The interpolation part is performed using for ex- ample a Lagrange interpolating polynomial on a uniform grid. Thus interpo- lating wavelets provide a natural way to extend this procedure to an adaptive grid in the way we shall now shortly describe. For simplicity, we shall restrict our description to the 1D case of the whole real line. It is straightforward to extend it to periodic boundary conditions and it can also be extended to an interval with Dirichlet boundary conditions. The extension to higher dimension is performed using a tensor product of wavelets and will be addressed at the end of the section. For any value of j ∈ Z, we consider a uniform grid Gj of step 2−j . The grid points are located at x k = k2 −j . This defines an infinite sequence of grids that we denote by (Gj)j∈Z, and j will be called the level of the grid. In order to go from one level to the next or the previous, we define a pro- jection operator and a prediction operator. Consider two grid levels Gj and Gj+1 and discrete values (of a function) denoted by (c k)k∈Z and (c k )k∈Z. Even though we use the same index k for the grid points in the two cases, there are of course twice as many points in any given interval on Gj+1 as on Gj . Using the terminology in [3], we then define the projection operator j+1 : Gj+1 → Gj , 2k 7→ c which is merely a restriction operator, as well as the prediction operator j : Gj → Gj+1, such that c 2k = c 2k+1 = P2N+1(x 2k+1), 4 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker where P2N+1 stands for the Lagrange interpolation polynomial of odd degree 2N + 1 centered at the point (x 2k+1). Using the just defined prediction operator, we can construct on Gj a subspace of L2(R) that we shall denote by Vj , a basis of which being given by (ϕ k)k∈Z such that ϕ k′ ) = δkk′ where δkk′ is the Kronecker symbol. The value of ϕ k at any point of the real line is then obtained by applying, possibly an infinite number of times, the prediction operator. In the wavelets terminology the ϕ k are called scaling functions. We shall also denote by ϕ = ϕ0 . Let us notice that k(x) = ϕ(2 jx− k). It can be easily verified that the scaling functions satisfy the following prop- erties: – Compact support: the support of ϕ is included in [−2N − 1, 2N + 1]. – Interpolation: by construction ϕ(x) is interpolating in the sense that ϕ(0) = 1 and ϕ(k) = 0 if k 6= 0. – Polynomial representation: all polynomials of degree less or equal to 2N+1 can be expressed exactly as linear combinations of the ϕ – Change of scale: the ϕ at a given scale can be expressed as a linear combi- nation of the ϕ at the scale immediately below: ϕ(x) = −2N−1 hlϕ(2x− l). Moreover the sequence of spaces (Vj)j∈Z defines a multiresolution analysis of L2(R), i.e. it satisfies the following properties: – . . . ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ . . . ⊂ Vn ⊂ . . . ⊂ L2(R). – ∩Vj = {0}, ∪Vj = L2(R). – f ∈ Vj ↔ f(2 ·)Vj+1. – ∃ϕ (scaling function) such that {ϕ(x− k)}k∈Z is a basis of V0 and {ϕjk = 2j/2ϕ(2j x− k)}k∈Z is a basis of Vj . As Vj ⊂ Vj+1, there exists a supplementary of Vj in Vj+1 that we shall call the detail space and denote by Wj : Vj+1 = Vj ⊕Wj . The construction of Wj can be made in the following way: an element of Vj+1 is characterized by the sequence(c k )k∈Z and by construction we have k = c 2k . Thus, if we define d k = c 2k+1 − P2N+1(x 2k+1), where P2N+1 is the Lagrange interpolation polynomial by which the value of an element of Vj at the point (x 2k+1) can be computed, d k represents exactly the difference between the value in Vj+1 and the value predicted in Vj . Finally, any element An adaptive numerical method for the Vlasov equation 5 of Vj+1 can be characterized by the two sequences (c k)k of values in Vj and (d k)k of details in Wj . Moreover this strategy for constructing Wj is particularly interesting for adaptive refinement as d k will be small at places where the prediction from Vj is good and large elsewhere, which gives us a natural refinement criterion. Besides, there exists a function ψ, called wavelet such that {ψjk = 2j/2ψ(2j x− k)}k∈Z is a basis of Wj . In practise, for adaptive refinement we set the coarsest level j0 and the finest level j1, j0 < j1, and we decompose the space corresponding to the finest level on all the levels in between: Vj1 = Vj0 ⊕Wj0 ⊕Wj0+1 ⊕ · · · ⊕Wj1−1. A function f ∈ Vj1 can then be decomposed as follows f(x) = l (x) + l (x), where the (c l )l are the coefficients on the coarse mesh and the (d l )l the details at the different level in between. 2k1+2,2k2+1 k1,k2 2k1,2k2+1 2k1+1,2k2+1 2k1+1,2k2+2 k1,k2+1 k1+1,k2+1 2k1+1,2k2 k1+1,k2 Fig. 1. Mesh refinement in 2D. In two dimensions, the prediction operator which defines the multireso- lution analysis is constructed by tensor product from the 1D operator. In practise three different cases must be considered (see figure 1 for notations): 1. Refinement in x (corresponding to points c 2k1+1,2k2 and c 2k1+1,2k2+2 ): we use the 1D prediction operator in x for fixed k2. 2. Refinement in v (corresponding to points c 2k1,2k2+1 and c 2k1+2,2k2+1 ): we use the 1D prediction operator in v for fixed k1. 3. Refinement in v (corresponding to point c 2k1+1,2k2+1 ): we first use the 1D prediction operator in v for fixed k1 to determine the points which are necessary for applying the 1D prediction operator in x for fixed k2 which we then apply. 6 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker The corresponding wavelet bases are respectively of type ψ(x)ϕ(v), ϕ(x)ψ(v) and ψ(x)ψ(v) where ϕ and ψ are respectively the scaling function and the 1D wavelet. We then obtain a 2D wavelet decomposition of the following form: f(x, v) = k1,k2 k1,k2 (v) + row,j k1,k2 col,j k1,k2 (v) + d mid,j k1,k2 . (3) 3 The algorithms We want to numerically solve the Vlasov equation (1) given an initial value of the distribution function f0. We start by describing the method based on an interpolation using the wavelet decomposition of f in the non adaptive case. Then we overlay an adaptive algorithm to this method. For those two algorithms, we first pick the resolution levels for the phase- space meshes, from the coarsest j0 to the finest j1. Although these levels could be different in x and v, we consider here for the sake of conciseness and clarity that they are identical. We also compute our scaling function on a very fine grid so that we can obtain with enough precision its value at any point. 3.1 The non adaptive algorithm We are working in this case on the finest level corresponding to j1 keeping all the points. Initialization: We decompose the initial condition in the wavelet basis by computing the coefficients ck1,k2 of the decomposition in Vj0 for the coarse mesh, and then adding the details d k1,k2 in the detail spaces Wj for all the other levels j = j0, . . . , j1 − 1. We then compute the initial electric field. Time iterations: – Advection in x: We start by computing for each mesh point the origin of the corresponding characteristic exactly, the displacement being vj∆t. As we do not necessarily land on a mesh point, we compute the values of the distribution function at the intermediate time level, denoted by f∗, at the origin of the characteristics by interpolation from fn. We use for this the wavelet decomposition (3) applied to fn from which we can compute fn at any point in phase space. An adaptive numerical method for the Vlasov equation 7 – Computation of the electric field: We compute the charge density by integrating f∗ with respect to v, then the electric field by solving the Poisson equation (this step vanishes for the linear case of the rotating cylinder where the advection field is exactly known). – Advection in v: We start by computing exactly the origin of the char- acteristic for each mesh point, the displacement being E(tn, xi)∆t. As we do not necessarily land on a mesh point, we compute the values of the distribution function at the intermediate time level, denoted by fn+1, at the origin of the characteristics by interpolation from f∗. We use for this the wavelet decomposition of f∗ given by (3) used at the previous step. 3.2 The adaptive algorithm In the initialization phase, we first compute the wavelet decomposition of the initial condition f0, and then proceed by compressing it, i.e. eliminating the details which are smaller than a threshold that we impose. We then construct an adaptive mesh which, from all the possible points at all the levels between our coarsest and finest, contains only those of the coarsest and those corresponding to details which are above the threshold. We denote by G̃ this mesh. – Prediction in x: We predict the positions of points where the details should be important at the next time split step by advancing in x the characteristics originating from the points of the mesh G̃. For this we use an explicit Euler scheme for the numerical integration of the characteristics. Then we retain the grid points, at one level finer as the starting point, surrounding the end point the characteristic. – Construction of mesh Ĝ: From the predicted mesh G̃, we construct the mesh Ĝ where the values of the distribution at the next time step shall be computed. This mesh Ĝ contains exactly the points necessary for computing the wavelet transform of f∗ at the points of G̃. – Advection in x: As in the non adaptive case. – Wavelet transform of f∗: We compute the ck and dk coefficients at the points of G̃ from the values of f∗ at the points of Ĝ. – Compression:We eliminate the points of G̃ where the details dk are lower than the fixed threshold. – Computation of the electric field: As in the non adaptive case. – Prediction in v: As for the prediction in x. – Construction of mesh Ĝ: As previously. This mesh Ĝ contains exactly the points necessary for computing the wavelet transform of fn+1 at the points of G̃ determined in the prediction in v step. – Advection in v: As in the non adaptive case. – Wavelet transform of fn+1: We compute the ck and dk at the points of G̃ from the values of fn+1 at the points of Ĝ. – Compression:We eliminate the points of G̃ where the details dk are lower than the fixed threshold. 8 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker 4 Numerical results We show here our first results obtained with the adaptive method. We con- sider first a linear problem, namely the test case of the rotating cylinder introduced by Zalesak [10] to test advection schemes. Then we consider a classical nonlinear Vlasov-Poisson test case, namely the two stream instabil- 4.1 The slit rotating cylinder We consider the following initial condition: f(0, x, v) = x2 + v2 < 0.5 and if x < 0 or |v| > 0.125, 0 else. The computational domain is [−0.5, 0.5]× [−0.5, 0.5]. The advection field is (v,−x), which corresponds to the Vlasov equation with an applied electric field Eapp(x, t) = −x and without self-consistent field. Figure 2 represents the evolution of the rotating cylinder on a half turn with a coarse mesh of 16× 16 points and 4 adaptive refinement levels. We notice that the cylinder is well represented and that the mesh points concentrate along the discontinuities. 4.2 The two-stream instability We consider two streams symmetric with respect to v = 0 and represented by the initial distribution function f(0, x, v) = v2 exp(−v2/2)(1 + α cos(k0 x)), with α = 0.25, k0 = 0.5, and L = 2 π/k0. We use a maximum of Nx = 128 points in the x direction, and Nv = 128 points in the v direction with vmax = 7, and a time step∆t = 1/8. The solution varies first very slowly and then fine scales are generated. Between times of around t ≃ 20 ω−1p and t ≃ 40 ω−1p , the instability increases rapidly and a hole appears in the middle of the computational domain. After t = 45 ω−1p until the end of the simulation, particles inside the hole are trapped. On figure 3 we show a snapshot of the distribution function at times t = 5 ω−1p and t = 30 ω p for a coarse mesh of 16 × 16 points and 3 levels of refinement. The adaptive method reproduces well the results obtained in the non adaptive case. 5 Conclusion In this paper we have described a new method for the numerical resolution of the Vlasov equation using an adaptive mesh of phase-space. The adaptive al- gorithm is based on a multiresolution analysis. It performs qualitatively well. An adaptive numerical method for the Vlasov equation 9 Fig. 2. Rotating cylinder: evolution for a coarse mesh of 24×24 points and 4 adap- tive refinement levels. Snapshots of the cylinder and the corresponding adaptive mesh: (upper) after one time step, (lower) after 1/2 turn. Fig. 3. Two stream instability for a coarse mesh of 24 × 24, and 3 adaptive refine- ment levels, (left) at time t = 5ω−1p , (right) at time t = 30ω 10 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker However, there is a large overhead due to the handling of the adaptive mesh which has not been optimized yet. The performance of the code needs to be improved before we can recommend this technique for actual computations. We are currently working on optimizing the code and trying different kinds of wavelets, as well as obtaining error estimates for the adaptive method. References 1. S. Bertoluzza, An adaptive collocation method based on interpolating wavelets. Multiscale wavelet methods for partial differential equations, pp. 109–135, Wavelet Anal. Appl., 6, Academic Press, San Diego, CA, 1997. 2. C.Z. Cheng, G. Knorr, The integration of the Vlasov equation in configuration space. J. Comput. Phys., 22 (1976), pp. 330–348. 3. A. Cohen, S.M. Kaber, S. Mueller and M. Postel, Fully adaptive multiresolu- tion finite volume schemes for conservation laws, to appear in Mathematics of Computation. 4. M.R. Feix, P. Bertrand, A. Ghizzo, Eulerian codes for the Vlasov equation, Series on Advances in Mathematics for Applied Sciences, 22, Kinetic Theory and Computing (1994), pp. 45–81. 5. F. Filbet, E. Sonnendrücker, Numerical methods for the Vlasov equation, these proceedings. 6. A. Ghizzo, P. Bertrand, M. Shoucri, T.W. Johnston, E. Filjakow, M.R. Feix, A Vlasov code for the numerical simulation of stimulated Raman scattering, J. Comput. Phys., 90 (1990), no. 2, pp. 431–457. 7. M. Griebel, F. Koster, Adaptive wavelet solvers for the unsteady incompressible Navier-Stokes equations, Advances in Mathematical Fluid Mechanics J. Malek and J. Necas and M. Rokyta eds., Springer Verlag, (2000). 8. E. Sonnendrücker, J. Roche, P. Bertrand, A. Ghizzo, The Semi-Lagrangian Method for the Numerical Resolution of Vlasov Equations, J. Comput. Phys., 149 (1999), no. 2, pp. 201–220. 9. E. Sonnendrücker, J.J. Barnard, A. Friedman, D.P. Grote, S.M. Lund, Sim- ulation of heavy ion beams with a semi-Lagrangian Vlasov solver, Nuclear Instruments and Methods in Physics Research, Section A, 464, no. 1-3, (2001), pp. 653–661. 10. S.T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31 (1979), no. 3, pp. 335–362. An adaptive numerical method for the Vlasov equation based on a multiresolution analysis N. Besse, F. Filbet, M. Gutnic, I. Paun , E. Sonnendrücker

In this paper, we present very first results for the adaptive solution on a grid of the phase space of the Vlasov equation arising in particles accelarator and plasma physics. The numerical algorithm is based on a semi-Lagrangian method while adaptivity is obtained using multiresolution analysis.

Introduction Plasmas, which are gases of charged particles, and charged particle beams can be described by a distribution function f(t, x, v) dependent on time t, on position x and on velocity v. The function f represents the probability of presence of a particle at position (x, v) in phase space at time t. It satisfies the so-called Vlasov equation + v · ∇xf + F (t, x, v) · ∇vf = 0. (1) The force field F (t, x, v) consists of applied and self-consistent electric and magnetic fields: (Eself + Eapp + v × (Bself +Bapp)), wherem represents the mass of a particle and q its charge. The self-consistent part of the force field is solution of Maxwell’s equations +∇×B = µ0j, ∇ · E = +∇×E = 0, ∇ ·B = 0. The coupling with the Vlasov equation results from the source terms ρ and j such that: ρ(t, x) = q f(t, x, v) dv, j = q f(t, x, v)v dv. We then obtain the nonlinear Vlasov-Maxwell equations. In some cases, when the field are slowly varying the magnetic field becomes negligible and the Maxwell equations can be replaced by the Poisson equation where: Eself (t, x) = −∇xφ(t, x), −ε0∆xφ = ρ. (2) The numerical resolution of the Vlasov equation is usually performed by particle methods (PIC) which consist in approximating the plasma by a http://arxiv.org/abs/0704.1595v1 2 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker finite number of particles. The trajectories of these particles are computed from the characteristic curves given by the Vlasov equation, whereas self- consistent fields are computed on a mesh of the physical space. This method allows to obtain satisfying results with a few number of particles. However, it is well known that, in some cases, the numerical noise inherent to the particle method becomes too important to have an accurate description of the distribution function in phase space. Moreover, the numerical noise only decreases in N , when the number of particles N is increased. To remedy to this problem, methods discretizing the Vlasov equation on a mesh of phase space have been proposed. A review of the main methods for the resolution of the Vlasov equation is given in these proceedings [5]. The major drawback of methods using a uniform and fixed mesh is that their numerical cost is high, which makes them rather inefficient when the dimension of phase-space grows. For this reason we are investigating here a method using an adaptive mesh. The adaptive method is overlayed to a classical semi-Lagrangian method which is based on the conservation of the distribution function along characteristics. Indeed, this method uses two steps to update the value of the distribution function at a given mesh point. The first one consists in following the characteristic ending at this mesh point backward in time, and the second one in interpolating its value there from the old values at the surrounding mesh points. Using the conservation of the distribution function along the characteristics this will yield its new value at the given mesh point. This idea was originally introduced by Cheng and Knorr [2] along with a time splitting technique enabling to compute exactly the origin of the characteristics at each fractional step. In the original method, the interpolation was performed using cubic splines. This method has since been used extensively by plasma physicists (see for example [4, 6] and the ref- erences therein). It has then been generalized to the frame of semi-Lagrangian methods by E. Sonnendrücker et al. [8]. This method has also been used to investigate problems linked to the propagation of strongly nonlinear heavy ion beams [9]. In the present work, we have chosen to introduce a phase-space mesh which can be refined or derefined adaptively in time. For this purpose, we use a technique based on multiresolution analysis which is in the same spirit as the methods developed in particular by S. Bertoluzza [1], A. Cohen et al. [3] and M. Griebel and F. Koster [7]. We represent the distribution function on a wavelet basis at different scales. We can then compress it by eliminat- ing coefficients which are small and accordingly remove the associated mesh points. Another specific feature of our method is that we use an advection in physical and velocity space forward in time to predict the useful grid points for the next time step, rather than restrict ourselves to the neighboring points. This enables us to use a much larger time step, as in the semi-Lagrangian method the time step is not limited by a Courant condition. Once the new mesh is predicted, the semi-Lagrangian methodology is used to compute the An adaptive numerical method for the Vlasov equation 3 new values of the distribution function at the predicted mesh points, using an interpolation based on the wavelet decomposition of the old distribution function. The mesh is then refined again by performing a wavelet transform, and eliminating the points associated to small coefficients. This paper is organized as follows. In section 2, we recall the tools of multiresolution analysis which will be needed for our method, precizing what kind of wavelets seem to be the most appropriate in our case. Then, we describe in section 3 the algorithm used in our method, first for the non adaptive mesh case and then for the adaptive mesh case. Finally we present a few preliminary numerical results. 2 Multiresolution analysis The semi-Lagrangian method consists mainly of two steps, an advection step and an interpolation step. The interpolation part is performed using for ex- ample a Lagrange interpolating polynomial on a uniform grid. Thus interpo- lating wavelets provide a natural way to extend this procedure to an adaptive grid in the way we shall now shortly describe. For simplicity, we shall restrict our description to the 1D case of the whole real line. It is straightforward to extend it to periodic boundary conditions and it can also be extended to an interval with Dirichlet boundary conditions. The extension to higher dimension is performed using a tensor product of wavelets and will be addressed at the end of the section. For any value of j ∈ Z, we consider a uniform grid Gj of step 2−j . The grid points are located at x k = k2 −j . This defines an infinite sequence of grids that we denote by (Gj)j∈Z, and j will be called the level of the grid. In order to go from one level to the next or the previous, we define a pro- jection operator and a prediction operator. Consider two grid levels Gj and Gj+1 and discrete values (of a function) denoted by (c k)k∈Z and (c k )k∈Z. Even though we use the same index k for the grid points in the two cases, there are of course twice as many points in any given interval on Gj+1 as on Gj . Using the terminology in [3], we then define the projection operator j+1 : Gj+1 → Gj , 2k 7→ c which is merely a restriction operator, as well as the prediction operator j : Gj → Gj+1, such that c 2k = c 2k+1 = P2N+1(x 2k+1), 4 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker where P2N+1 stands for the Lagrange interpolation polynomial of odd degree 2N + 1 centered at the point (x 2k+1). Using the just defined prediction operator, we can construct on Gj a subspace of L2(R) that we shall denote by Vj , a basis of which being given by (ϕ k)k∈Z such that ϕ k′ ) = δkk′ where δkk′ is the Kronecker symbol. The value of ϕ k at any point of the real line is then obtained by applying, possibly an infinite number of times, the prediction operator. In the wavelets terminology the ϕ k are called scaling functions. We shall also denote by ϕ = ϕ0 . Let us notice that k(x) = ϕ(2 jx− k). It can be easily verified that the scaling functions satisfy the following prop- erties: – Compact support: the support of ϕ is included in [−2N − 1, 2N + 1]. – Interpolation: by construction ϕ(x) is interpolating in the sense that ϕ(0) = 1 and ϕ(k) = 0 if k 6= 0. – Polynomial representation: all polynomials of degree less or equal to 2N+1 can be expressed exactly as linear combinations of the ϕ – Change of scale: the ϕ at a given scale can be expressed as a linear combi- nation of the ϕ at the scale immediately below: ϕ(x) = −2N−1 hlϕ(2x− l). Moreover the sequence of spaces (Vj)j∈Z defines a multiresolution analysis of L2(R), i.e. it satisfies the following properties: – . . . ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ . . . ⊂ Vn ⊂ . . . ⊂ L2(R). – ∩Vj = {0}, ∪Vj = L2(R). – f ∈ Vj ↔ f(2 ·)Vj+1. – ∃ϕ (scaling function) such that {ϕ(x− k)}k∈Z is a basis of V0 and {ϕjk = 2j/2ϕ(2j x− k)}k∈Z is a basis of Vj . As Vj ⊂ Vj+1, there exists a supplementary of Vj in Vj+1 that we shall call the detail space and denote by Wj : Vj+1 = Vj ⊕Wj . The construction of Wj can be made in the following way: an element of Vj+1 is characterized by the sequence(c k )k∈Z and by construction we have k = c 2k . Thus, if we define d k = c 2k+1 − P2N+1(x 2k+1), where P2N+1 is the Lagrange interpolation polynomial by which the value of an element of Vj at the point (x 2k+1) can be computed, d k represents exactly the difference between the value in Vj+1 and the value predicted in Vj . Finally, any element An adaptive numerical method for the Vlasov equation 5 of Vj+1 can be characterized by the two sequences (c k)k of values in Vj and (d k)k of details in Wj . Moreover this strategy for constructing Wj is particularly interesting for adaptive refinement as d k will be small at places where the prediction from Vj is good and large elsewhere, which gives us a natural refinement criterion. Besides, there exists a function ψ, called wavelet such that {ψjk = 2j/2ψ(2j x− k)}k∈Z is a basis of Wj . In practise, for adaptive refinement we set the coarsest level j0 and the finest level j1, j0 < j1, and we decompose the space corresponding to the finest level on all the levels in between: Vj1 = Vj0 ⊕Wj0 ⊕Wj0+1 ⊕ · · · ⊕Wj1−1. A function f ∈ Vj1 can then be decomposed as follows f(x) = l (x) + l (x), where the (c l )l are the coefficients on the coarse mesh and the (d l )l the details at the different level in between. 2k1+2,2k2+1 k1,k2 2k1,2k2+1 2k1+1,2k2+1 2k1+1,2k2+2 k1,k2+1 k1+1,k2+1 2k1+1,2k2 k1+1,k2 Fig. 1. Mesh refinement in 2D. In two dimensions, the prediction operator which defines the multireso- lution analysis is constructed by tensor product from the 1D operator. In practise three different cases must be considered (see figure 1 for notations): 1. Refinement in x (corresponding to points c 2k1+1,2k2 and c 2k1+1,2k2+2 ): we use the 1D prediction operator in x for fixed k2. 2. Refinement in v (corresponding to points c 2k1,2k2+1 and c 2k1+2,2k2+1 ): we use the 1D prediction operator in v for fixed k1. 3. Refinement in v (corresponding to point c 2k1+1,2k2+1 ): we first use the 1D prediction operator in v for fixed k1 to determine the points which are necessary for applying the 1D prediction operator in x for fixed k2 which we then apply. 6 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker The corresponding wavelet bases are respectively of type ψ(x)ϕ(v), ϕ(x)ψ(v) and ψ(x)ψ(v) where ϕ and ψ are respectively the scaling function and the 1D wavelet. We then obtain a 2D wavelet decomposition of the following form: f(x, v) = k1,k2 k1,k2 (v) + row,j k1,k2 col,j k1,k2 (v) + d mid,j k1,k2 . (3) 3 The algorithms We want to numerically solve the Vlasov equation (1) given an initial value of the distribution function f0. We start by describing the method based on an interpolation using the wavelet decomposition of f in the non adaptive case. Then we overlay an adaptive algorithm to this method. For those two algorithms, we first pick the resolution levels for the phase- space meshes, from the coarsest j0 to the finest j1. Although these levels could be different in x and v, we consider here for the sake of conciseness and clarity that they are identical. We also compute our scaling function on a very fine grid so that we can obtain with enough precision its value at any point. 3.1 The non adaptive algorithm We are working in this case on the finest level corresponding to j1 keeping all the points. Initialization: We decompose the initial condition in the wavelet basis by computing the coefficients ck1,k2 of the decomposition in Vj0 for the coarse mesh, and then adding the details d k1,k2 in the detail spaces Wj for all the other levels j = j0, . . . , j1 − 1. We then compute the initial electric field. Time iterations: – Advection in x: We start by computing for each mesh point the origin of the corresponding characteristic exactly, the displacement being vj∆t. As we do not necessarily land on a mesh point, we compute the values of the distribution function at the intermediate time level, denoted by f∗, at the origin of the characteristics by interpolation from fn. We use for this the wavelet decomposition (3) applied to fn from which we can compute fn at any point in phase space. An adaptive numerical method for the Vlasov equation 7 – Computation of the electric field: We compute the charge density by integrating f∗ with respect to v, then the electric field by solving the Poisson equation (this step vanishes for the linear case of the rotating cylinder where the advection field is exactly known). – Advection in v: We start by computing exactly the origin of the char- acteristic for each mesh point, the displacement being E(tn, xi)∆t. As we do not necessarily land on a mesh point, we compute the values of the distribution function at the intermediate time level, denoted by fn+1, at the origin of the characteristics by interpolation from f∗. We use for this the wavelet decomposition of f∗ given by (3) used at the previous step. 3.2 The adaptive algorithm In the initialization phase, we first compute the wavelet decomposition of the initial condition f0, and then proceed by compressing it, i.e. eliminating the details which are smaller than a threshold that we impose. We then construct an adaptive mesh which, from all the possible points at all the levels between our coarsest and finest, contains only those of the coarsest and those corresponding to details which are above the threshold. We denote by G̃ this mesh. – Prediction in x: We predict the positions of points where the details should be important at the next time split step by advancing in x the characteristics originating from the points of the mesh G̃. For this we use an explicit Euler scheme for the numerical integration of the characteristics. Then we retain the grid points, at one level finer as the starting point, surrounding the end point the characteristic. – Construction of mesh Ĝ: From the predicted mesh G̃, we construct the mesh Ĝ where the values of the distribution at the next time step shall be computed. This mesh Ĝ contains exactly the points necessary for computing the wavelet transform of f∗ at the points of G̃. – Advection in x: As in the non adaptive case. – Wavelet transform of f∗: We compute the ck and dk coefficients at the points of G̃ from the values of f∗ at the points of Ĝ. – Compression:We eliminate the points of G̃ where the details dk are lower than the fixed threshold. – Computation of the electric field: As in the non adaptive case. – Prediction in v: As for the prediction in x. – Construction of mesh Ĝ: As previously. This mesh Ĝ contains exactly the points necessary for computing the wavelet transform of fn+1 at the points of G̃ determined in the prediction in v step. – Advection in v: As in the non adaptive case. – Wavelet transform of fn+1: We compute the ck and dk at the points of G̃ from the values of fn+1 at the points of Ĝ. – Compression:We eliminate the points of G̃ where the details dk are lower than the fixed threshold. 8 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker 4 Numerical results We show here our first results obtained with the adaptive method. We con- sider first a linear problem, namely the test case of the rotating cylinder introduced by Zalesak [10] to test advection schemes. Then we consider a classical nonlinear Vlasov-Poisson test case, namely the two stream instabil- 4.1 The slit rotating cylinder We consider the following initial condition: f(0, x, v) = x2 + v2 < 0.5 and if x < 0 or |v| > 0.125, 0 else. The computational domain is [−0.5, 0.5]× [−0.5, 0.5]. The advection field is (v,−x), which corresponds to the Vlasov equation with an applied electric field Eapp(x, t) = −x and without self-consistent field. Figure 2 represents the evolution of the rotating cylinder on a half turn with a coarse mesh of 16× 16 points and 4 adaptive refinement levels. We notice that the cylinder is well represented and that the mesh points concentrate along the discontinuities. 4.2 The two-stream instability We consider two streams symmetric with respect to v = 0 and represented by the initial distribution function f(0, x, v) = v2 exp(−v2/2)(1 + α cos(k0 x)), with α = 0.25, k0 = 0.5, and L = 2 π/k0. We use a maximum of Nx = 128 points in the x direction, and Nv = 128 points in the v direction with vmax = 7, and a time step∆t = 1/8. The solution varies first very slowly and then fine scales are generated. Between times of around t ≃ 20 ω−1p and t ≃ 40 ω−1p , the instability increases rapidly and a hole appears in the middle of the computational domain. After t = 45 ω−1p until the end of the simulation, particles inside the hole are trapped. On figure 3 we show a snapshot of the distribution function at times t = 5 ω−1p and t = 30 ω p for a coarse mesh of 16 × 16 points and 3 levels of refinement. The adaptive method reproduces well the results obtained in the non adaptive case. 5 Conclusion In this paper we have described a new method for the numerical resolution of the Vlasov equation using an adaptive mesh of phase-space. The adaptive al- gorithm is based on a multiresolution analysis. It performs qualitatively well. An adaptive numerical method for the Vlasov equation 9 Fig. 2. Rotating cylinder: evolution for a coarse mesh of 24×24 points and 4 adap- tive refinement levels. Snapshots of the cylinder and the corresponding adaptive mesh: (upper) after one time step, (lower) after 1/2 turn. Fig. 3. Two stream instability for a coarse mesh of 24 × 24, and 3 adaptive refine- ment levels, (left) at time t = 5ω−1p , (right) at time t = 30ω 10 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker However, there is a large overhead due to the handling of the adaptive mesh which has not been optimized yet. The performance of the code needs to be improved before we can recommend this technique for actual computations. We are currently working on optimizing the code and trying different kinds of wavelets, as well as obtaining error estimates for the adaptive method. References 1. S. Bertoluzza, An adaptive collocation method based on interpolating wavelets. Multiscale wavelet methods for partial differential equations, pp. 109–135, Wavelet Anal. Appl., 6, Academic Press, San Diego, CA, 1997. 2. C.Z. Cheng, G. Knorr, The integration of the Vlasov equation in configuration space. J. Comput. Phys., 22 (1976), pp. 330–348. 3. A. Cohen, S.M. Kaber, S. Mueller and M. Postel, Fully adaptive multiresolu- tion finite volume schemes for conservation laws, to appear in Mathematics of Computation. 4. M.R. Feix, P. Bertrand, A. Ghizzo, Eulerian codes for the Vlasov equation, Series on Advances in Mathematics for Applied Sciences, 22, Kinetic Theory and Computing (1994), pp. 45–81. 5. F. Filbet, E. Sonnendrücker, Numerical methods for the Vlasov equation, these proceedings. 6. A. Ghizzo, P. Bertrand, M. Shoucri, T.W. Johnston, E. Filjakow, M.R. Feix, A Vlasov code for the numerical simulation of stimulated Raman scattering, J. Comput. Phys., 90 (1990), no. 2, pp. 431–457. 7. M. Griebel, F. Koster, Adaptive wavelet solvers for the unsteady incompressible Navier-Stokes equations, Advances in Mathematical Fluid Mechanics J. Malek and J. Necas and M. Rokyta eds., Springer Verlag, (2000). 8. E. Sonnendrücker, J. Roche, P. Bertrand, A. Ghizzo, The Semi-Lagrangian Method for the Numerical Resolution of Vlasov Equations, J. Comput. Phys., 149 (1999), no. 2, pp. 201–220. 9. E. Sonnendrücker, J.J. Barnard, A. Friedman, D.P. Grote, S.M. Lund, Sim- ulation of heavy ion beams with a semi-Lagrangian Vlasov solver, Nuclear Instruments and Methods in Physics Research, Section A, 464, no. 1-3, (2001), pp. 653–661. 10. S.T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31 (1979), no. 3, pp. 335–362. An adaptive numerical method for the Vlasov equation based on a multiresolution analysis N. Besse, F. Filbet, M. Gutnic, I. Paun , E. Sonnendrücker

Turbulence and the Navier-Stokes Equations R. M. Kiehn Emeritus Professor of Physics, University of Houston Retired to Mazan, France http://www.cartan.pair.com Abstract: The concept of continuous topological evolution, based upon Car- tan’s methods of exterior differential systems, is used to develop a topological theory of non-equilibrium thermodynamics, within which there exist processes that exhibit continuous topological change and thermodynamic irreversibility. The technique furnishes a universal, topological foundation for the partial differ- ential equations of hydrodynamics and electrodynamics; the technique does not depend upon a metric, connection or a variational principle. Certain topological classes of solutions to the Navier-Stokes equations are shown to be equivalent to thermodynamically irreversible processes. Prologue THE POINT OF DEPARTURE This presentation summarizes a portion of 40 years of research interests1 in applied physics from a perspective of continuous topological evolution. The motivation for the past and present effort continues to be based on the recognition that topological evolution (not geometrical evolution) is required if non-equilibrium thermodynamic systems and irre- versible turbulent processes are to be understood without the use of statistics. This essay is written (by an applied physicist) as an alternative response to the (more mathematical) challenge of the Clay Institute regarding the properties of the Navier-Stokes equations and their relationship to hydrodynamic turbulence. To replicate a statement made by the Clay Institute: ”The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.” 1This work is summarized in a series of reference monographs [40], [41], [42], [43], [44] which have been constructed and updated from numerous publications. These volumes contain many examples and proofs of the basic concepts. http://arxiv.org/abs/0704.1596v1 http://www.cartan.pair.com The point of departure starts with a topological (not statistical) formulation of Thermo- dynamics, which furnishes a universal foundation for the Partial Differential Equations of classical hydrodynamics and electrodynamics [40]. The topology that is of significance is defined in terms of Cartan’s topological structure [34], which can be constructed from an ex- terior differential 1-form, A, defined on a pre-geometric domain of base variables. The topo- logical method extends the classical geometrical approach to the study of non-equilibrium thermodynamic systems. Claim 1 The topological method permits the conclusion that among the solutions to the Navier-Stokes equations there are C2 smooth, thermodynamically irreversible processes which permit description of topological change and the decay of turbulence. In addition, the method permits examples to be constructed showing the difference be- tween certain piecewise-linear processes which are reversible, but which are different from certain smooth processes which are irreversible. Such concepts (of smoothness) seem to be of direct interest to the challenge of the Clay Institute, and are to be associated with the fact that there are differences between piecewise linear, smooth, and topological manifolds (see p. 106, [38]). However, the methods of topological thermodynamics go well beyond these types of questions. In particular, the methods permit non-statistical engineering design criteria to be developed for non-equilibrium systems. The theory of Topological Thermodynamics, based upon Continuous Topological Evolution [35] of Cartan’s topological structure, can explain why topologically coherent, compact structures, far from equilibrium, will emerge as long-lived artifacts of thermodynamically irreversible, turbulent, continuous processes. I want to present the idea that: Theorem 2 The Pfaff Topological Dimension (PTD) of a Thermodynamic System can change dynamically and continuously via irreversible dissipative processes from a non-equilibrium turbulent state of PTD = 4 to an excited “topologically stationary, but excited,” state of PTD = 3, which is still far from equilibrium! The PTD=3 state admits an extremal Hamiltonian evolutionary process which, if dominant, produces a relatively long lifetime. There exist C2 smooth processes that can describe the topological evolution from an Open non-equilibrium turbulent domain of Pfaff Topological Dimension 4 to Closed, but non-equilibrium, domains of Pfaff Topological Dimension 3, and ultimately to equilibrium domains of Pfaff dimension 2 or less. The Topological domains of Pfaff Topological Di- mension 3 emerge via thermodynamically irreversible, dissipative processes as topologically coherent, deformable defects, embedded in the turbulent environment of Pfaff Topological Dimension 4. Now I am well aware of the fact that Thermodynamics (much less Topological Thermo- dynamics) is a topic often treated with apprehension. In addition, I must confess, that as undergraduates at MIT we used to call the required physics course in Thermodynamics, The Hour of Mystery! Let me present a few quotations (taken from Uffink, [39]) that describe the apprehensive views of several very famous scientists: Any mathematician knows it is impossible to understand an elementary course in thermodynamics ....... V. Arnold 1990. It is always emphasized that thermodynamics is concerned with reversible pro- cesses and equilibrium states, and that it can have nothing to do with irreversible processes or systems out of equilibrium ......Bridgman 1941 No one knows what entropy really is, so in a debate (if you use the term entropy) you will always have an advantage ...... Von Neumann (1971) On the other hand Uffink states: Einstein, ..., remained convinced throughout his life that thermodynamics is the only universal physical theory that will never be overthrown. I wish to demonstrate that from the point of view of Continuous Topological Evolution (which is based upon Cartan’s theory of exterior differential forms) many of the mysteries of non-equilibrium thermodynamics, irreversible processes, and turbulent flows, can be resolved. In addition, the non-equilibrium methods can lead to many new processes and patentable devices and concepts. There are many intuitive, yet disputed, definitions of what is meant by turbulence, but the one property of turbulence that everyone agrees upon is that turbulent evolution in a fluid is a thermodynamic irreversible process. Isolated or equilibrium thermodynamics can be defined on a 4D space-time variety in terms of a connected Cartan topology of Pfaff Topological Dimension 2 or less. Non-equilibrium thermodynamics can be constructed in terms of disconnected Cartan topology of Pfaff Topological Dimension of 3 or more. As irreversibility requires a change in topology, the point of departure for this article will be to use the thermodynamic theory of continuous topological evolution in 4D space-time. It will be demonstrated, by example, that the non-equilibrium component of the Cartan topology can support topological change, thermodynamic irreversible processes and turbulent solu- tions to the Navier-Stokes equations, while the equilibrium topological component cannot. In addition, it will be demonstrated that complex isotropic macroscopic Spinors are the source of topological fluctuations and irreversible processes in the topological dynamics of non-equilibrium systems. This, perhaps surprising, fact has been ignored by almost all re- searchers in classical hydrodynamics who use classic real vector analysis and symmetries to produce conservation laws, which do not require Spinor components. The flaw in such sym- metrical based theories is that they describe evolutionary processes that are time reversible. Time irreversibility requires topological change. EXTERIOR DIFFERENTIAL FORMS overcomes the LIMITATIONS of REAL VECTOR ANALYSIS During the period 1965-1992 it became apparent that new theoretical foundations were needed to describe non-equilibrium systems and continuous irreversible processes - which require topological (not geometrical) evolution. I selected Cartan’s methods of exterior differential topology to encode Continuous Topological Evolution. The reason for this choice is that many years of teaching experience indicated that such methods were rapidly learned by all research scientists and engineers. In short: 1. Vector and Tensor analysis is not adequate to study the evolution of topology. The tensor constraint of diffeomorphic equivalences implies that the topology of the initial state must be equal to the topology of the final state. Turbulence is a thermodynamic, irreversible process which can not be described by tensor fields alone. 2. However, Cartan’s methods of exterior differential systems and the topological perspec- tive of Continuous Topological Evolution (not geometrical evolution) CAN be used to construct a theory of non-equilibrium thermodynamic systems and irreversible pro- cesses. 3. Bottom Line: Exterior differential forms carry topological information and can be used to describe topological change induced by processes. Real ”Vector” direction- fields alone cannot describe processes that cause topological change; but Spinor direc- tionfields can. A cornerstone of classic Vector (tensor) analysis is the constraint of functional equivalence with respect to diffeomorphisms. However, diffeomorphisms are a subset of a homeomor- phisms, and homeomorphisms preserve topology. Hence to study topological change, Vector (tensor) analysis is not adequate. In topological thermodynamics, processes are defined in terms of directionfields which may or may not be tensors. The ubiquitous concepts of 1-1 dif- feomorphic equivalence, and non-zero congruences, for the eigen directionfields of symmetric matrices do not apply to the eigen directionfields of antisymmetric matrices. The eigen di- rection fields of antisymmetric matrices (which are equivalent to Cartan’s isotropic Spinors) may be used to define components of a thermodynamic process, but such Spinors have a null congruence (zero valued quadratic form), admit chirality, and are not 1-1. Where classic geometric evolution is described in terms of symmetries and conservation laws, topological evolution is described in terms of antisymmetries. Cartan’s theory of exterior differential forms is built over completely antisymmetric struc- tures, and therefore is the method of choice for studying topological evolution. The exterior differential defines limit sets; the Lie differential defines continuous topological evolution. The concept of Spinors arise naturally in theories using Cartan’s methods of exterior dif- ferential forms; i.e., Spinors are not added to the theory ad hoc. The Cartan theory of extended differential forms can be used to study topological change. The word extended is used to emphasize the fact that differential forms are functionally well defined with respect a larger class of transformations than those used to define tensors. Extended differential forms behave as scalars with respect to C1 maps which do not have an inverse, much less an inverse Jacobian. Both the inverse map and the inverse Jacobian are required by a diffeomorphism. The exterior differential form on the final state of such C1 non-invertible maps permits the functional form of the differential form on the initial state to be functionally well defined in a retrodictive, pullback sense - not just at a point, but over a neighborhood. Theorem 3 Tensor fields can be neither retrodicted nor predicted in functional form by maps that are not diffeomorphisms [14]. CONTINUOUS TOPOLOGICAL EVOLUTION Objectives of CTE The objectives of the theory of Continuous Topological Evolution are to: 1. Establish the long sought for connection between irreversible thermodynamic processes and dynamical systems – without statistics! 2. Demonstrate the connection between thermodynamic irreversibility and Pfaff Topolog- ical Dimension equal to 4. The result suggests that “2-D Turbulence is a myth” for it is a thermodynamic system of Pfaff Topological Dimension equal to 3 [21]. 3. Demonstrate that topological thermodynamics leads to universal topological equiva- lences between Electromagnetism, Hydrodynamics, Cosmology, and Topological Quan- tum Mechanics. 4. Demonstrate that Cartan’s methods of exterior differential forms permits important topological concepts to be displayed in a useful, engineering format. New Concepts deduced from CTE The theory of Continuous Topological Evolution introduces several new important concepts that are not apparent in a geometric equilibrium analysis. 1. Continuous Topological Evolution is the dynamical equivalent of the FIRST LAW OF THERMODYNAMICS. 2. The Pfaff Topological Dimension, PTD, is a topological property associated with any Cartan exterior differential 1-form, A. The PTD can change via topologically contin- uous processes. 3. Topological Torsion is a 3-form (on any 4D geometrical domain) that can be used to describe irreversible processes. As a 4D non-equilibrium direction field it is completely determined by the coefficient functions that encode the thermodynamic system. Other process directionfields are determined by the system topology based upon the 1-form of Action, A, and the refinement based on the topology of the 1-form of work, W . 4. Closed thermodynamic topological defects of Pfaff Topological Dimension 3 can emerge from Open thermodynamic systems of Pfaff Topological Dimension 4 by means of irre- versible dissipative processes that represent topological evolution and change. When the topologically coherent defect structures emerge, their evolution can be dominated by a Hamiltonian component (modulo topological fluctuations), which maintains the topological deformation invariance, and yields hydrodynamic wakes [20] and other Soli- ton structures. These objects are of Pfaff Topological Dimension 3 and are far from equilibrium. They behave as if they were ”stationary excited” states above the equi- librium ground state. Falaco Solitons are an easily reproduced hydrodynamic example that came to my attention in 1986 [41] [33] . PRESENTATION OUTLINE The essay is constructed in several sections: Section 1. Topological Thermodynamics In Section 1, the concepts of topological thermodynamics in a space-time variety are reviewed (briefly) in terms of Cartan’s method of exterior differential forms. A thermodynamic system is encoded in terms of a 1-form of Action, A. Thermodynamic processes are encoded in terms of the Lie differential with respect to a directionfield, V , acting on the 1-form, A, to produce a 1-form, Q. The process directionfield can have Vector and Spinor components. The definition of the Lie differential is a statement of cohomology and defines Q as the composite of a 1-form, W , and a perfect differential, dU . The formula abstractly represents a dynamical version of the First Law of Thermodynamics. The existence of a 1-form on a 4D space-time variety generates a Cartan topology. If the Pfaff Topological (not geometrical) Dimension of the 1-form of Action, A, is 2 or less, then the thermodynamic system is an isolated or equilibrium system on the 4D variety. If the Pfaff Topological Dimension of A is greater than 3, then the system is a non-equilibrium system on the 4D variety. Examples of systems of Pfaff Topological Dimension 4 which admit processes which are thermodynamically irreversible are given in the reference monographs (see footnote page 1). Section 2. Applications In Section 2, the abstract formalism will be given a specific real- ization appropriate for fluids in general. First, an electromagnetic format will be described because my teaching experience has demonstrated that the concepts of non-equilibrium phenomena are more readily recognized in an electromagnetic format. Then it will be demonstrated how the PDE’s representing the Hamiltonian version of the hydrodynamic Lagrange-Euler equations arise from the constraint that the work 1-form, W , should vanish (Pfaff Topological Dimension of W = 0). The Bernoulli flow will be obtained by constrain- ing the thermodynamic Work 1-form to be exact, W = dΘ (Pfaff Topological Dimension 1), and the Helmholtz flow will follow from the constraint that the thermodynamic Work 1-form be closed, but not necessarily exact, dW = 0. Such reversible dynamical processes belong to the connected component of the Work 1-form; irreversible processes belong to the disconnected component of the Work 1-form. Section 3. The Navier-Stokes system In Section 3, the topological constraints of isolated equilibrium systems will be relaxed to produce more general PDE’s defining the topological evolution of the system relative to an applied process. These relaxed topolog- ical constraints will include the partial differential equations known as the Navier-Stokes equations. The method used will be to augment the topology induced by the 1-form of Action, A, by studying the topological refinements induced by the 1-form of Work, W . It will be demonstrated that when the Pfaff Topological Dimension of A and W and Q are 4, there exist C2 solutions (processes) to the Navier-Stokes equations which are thermodynam- ically irreversible (the most significant property of turbulent flow). An interesting result is the set of conditions on solutions of the Navier-Stokes equations that produce an adiabatic irreversible flow. Those topological refinements of the Work 1-form, required to include the Navier-Stokes equations, can be related directly to the concept of macroscopic Spinors. Macroscopic, complex Spinor solutions occur naturally in terms of the eigendirection fields of (real) anti- symmetric matrices with non-zero eigenvalues, whenever the thermodynamic Work 1-form is not zero. Spinors can also be associated with topological fluctuations of position and velocity about kinematic perfection generated by 1-parameter groups. These topological fluctuations are presumed to be representations of pressure and temperature. Section 4. Closed States of Topological Coherence embedded as deformable defects in Turbulent Domains One of the key interests of the Clay problem has to do with the smoothness of the solutions to the Navier-Stokes equations. In Section 4, the problem will be attacked from the point of view of thermodynamics. First, the properties of the different species of topological defects will be discussed. These defects are non- equilibrium closed domains (of PTD = 3) which can emerge by C2 smooth irreversible process in open domains (of PTD = 4), as excited states far from equilibrium, yet with long relative lifetimes. Falaco Solitons are an easily reproduced experimental example of such topological defects, and are discussed in detail in [41]. The properties of two different species of PTD = 3 defect domains will be given in detail. In addition, an analytic solution of a thermodynamically irreversible process that causes the defect domain to emerge will be displayed. Finally, an example will be given where by combinations of Spinor solutions produce piece- wise linear processes. These piecewise linear processes are thermodynamically reversible, while the Spinor solutions of which they are composed are not. Section 5. Topological Fluctuations and Spinors In Section 5, a few concluding remarks will be made about the ongoing research concerning topological fluctuations, as generated by Spinors. The methods of fiber bundle theory are used extend the 4D thermo- dynamic domain. Such topological fluctuations can be associated with fluid pressure and temperature. 1 Topological Thermodynamics 1.1 The Axioms of Topological Thermodynamics The topological methods used herein are based upon Cartan’s theory of exterior differential forms. The thermodynamic view assumes that the physical systems to be studied can be encoded in terms of a 1-form of Action Potentials (per unit source, or, per mole), A, on a four-dimensional variety of ordered independent variables, {ξ1, ξ2, ξ3, ξ4}. The variety supports a differential volume element Ω4 = dξ 1ˆdξ2ˆdξ3ˆdξ4. This statement implies that the differentials of the µ = 4 base variables are functionally independent. No metric, no connection, no constraint of gauge symmetry is imposed upon the four-dimensional pre- geometric variety. Topological constraints can be expressed in terms of exterior differential systems placed upon this set of base variables [1]. In order to make the equations more suggestive to the reader, the symbolism for the variety of independent variables will be changed to the format {x, y, z, t}, but be aware that no constraints of metric or connection are imposed upon this variety, at this, thermodynamic, level. For instance, it is NOT assumed that the variety is spatially Euclidean. With this notation, the Axioms of Topological Thermodynamics can be summarized as: Axiom 1. Thermodynamic physical systems can be encoded in terms of a 1- form of covariant Action Potentials, Aµ(x, y, z, t...), on a four-dimensional ab- stract variety of ordered independent variables, {x, y, z, t}. The variety supports differential volume element Ω4 = dxˆdyˆdzˆdt. Axiom 2. Thermodynamic processes are assumed to be encoded, to within a factor, ρ(x, y, z, t...), in terms of a contravariant Vector and/or complex Spinor directionfields, symbolized as V4(x, y, z, t). Axiom 3. Continuous Topological Evolution of the thermodynamic system can be encoded in terms of Cartan’s magic formula (see p. 122 in [10]). The Lie differential with respect to the process, ρV4, when applied to an exterior differen- tial 1-form of Action, A = Aµdx µ, is equivalent, abstractly, to the first law of thermodynamics. Cartan’s Magic Formula L(ρV4)A = i(ρV4)dA+ d(i(ρV4)A), (1) First Law : W + dU = Q, (2) Inexact Heat 1-form Q = W + dU = L(ρV4)A, (3) Inexact Work 1-form W = i(ρV4)dA, (4) Internal Energy U = i(ρV4)A. (5) Axiom 4. Equivalence classes of systems and continuous processes can be de- fined in terms of the Pfaff Topological Dimension and topological structure gen- erated by of the 1-forms of Action, A, Work, W , and Heat, Q. Axiom 5. If QˆdQ 6= 0, then the thermodynamic process is irreversible. 1.2 Cartan’s Magic Formula ≈ First Law of Thermodynamics The Lie differential (not Lie derivative) is the fundamental generator of Continuous Topo- logical Evolution. When acting on an exterior differential 1-form of Action, A = Aµdx Cartan’s magic (algebraic) formula is equivalent abstractly to the first law of thermodynam- L(ρV4)A = i(ρV4)dA+ d(i(ρV4)A), (6) = W + dU = Q. (7) In effect, Cartan’s magic formula leads to a topological basis of thermodynamics, where the thermodynamic Work, W , thermodynamic Heat, Q, and the thermodynamic internal energy, U , are defined dynamically in terms of Continuous Topological Evolution. In effect, the First Law is a statement of Continuous Topological Evolution in terms of deRham cohomology theory; the difference between two non-exact differential forms is equal to an exact differential, Q−W = dU . My recognition (some 30 years ago) of this correspondence between the Lie differential and the First Law of thermodynamics has been the corner stone of my research efforts in applied topology. It is important to realize that the Cartan formula is to be interpreted algebraically. Many textbook presentations of the Cartan-Lie differential formula presume a dynamic constraint, such that the vector field V4(x, y, z, t) be the generator of a single parameter group. If true, then the topological constraint of Kinematic Perfection cn be established as an exterior differential system of the format: Kinematic Perfection : dxk −Vkdt⇒ 0. (8) The topological constraint of Kinematic Perfection, in effect, defines (or presumes) a limit process. This constraint leads to the concept of the Lie derivative2 of the 1-form A. The evolution then is represented by the infinitesimal propagation of the 1-form, A, down the 2Professor Zbigniew Oziewicz told me that Slebodzinsky was the first to formulate the idea of the Lie derivative in his thesis (in Polish). flow lines generated by the 1-parameter group. Cartan called this set of flow lines ”the tube of trajectories”. However, such a topological, kinematic constraint is not imposed in the presentation found in this essay; the directionfield, V4, may have multiple parameters. This observation leads to the important concept of topological fluctuations (about Kinematic Perfection), such as given by the expressions: Topological : Fluctuations (dxk −Vkdt) = ∆xk 6= 0, ( ∼ Pressure) (9) (dV k −Akdt) = (∆Vk) 6= 0, ( ∼ Temperature) (10) d(∆xk) = −(dVk −Akdt)ˆdt = −(∆Vk)ˆdt, (11) In this context it is interesting to note that in Felix Klein’s discussions [4] of the development of calculus, he says ”The primary thing for him (Leibniz) was not the differential quotient (the deriva- tive) thought of as a limit. The differential, dx, of the variable x had for him (Leibniz) actual existence...” The Leibniz concept is followed throughout this presentation. It is important for the reader to remember that the concept of a differential form is different from the concept of a derivative, where a (topological) limit has been defined, thereby constraining the topological evolution. The topological methods to be described below go beyond the notion of processes which are confined to equilibrium systems of kinematic perfection. Non-equilibrium systems and processes which are thermodynamically irreversible, as well as many other classical ther- modynamic ideas, can be formulated in precise mathematical terms using the topological structure and refinements generated by the three thermodynamic 1-forms, A, W, and Q. 1.3 The Pfaff Sequence and the Pfaff Topological Dimension 1.3.1 The Pfaff Topological Dimension of the System 1-form, A It is important to realize that the Pfaff Topological Dimension of the system 1-form of Action, A, determines whether the thermodynamic system is Open, Closed, Isolated or Equilibrium. Also, it is important to realize that the Pfaff Topological Dimension of the thermodynamic Work 1-form, W , determines a specific category of reversible and/or irreversible processes. It is therefore of some importance to understand the meaning of the Pfaff Topological Di- mension of a 1-form. Given the functional format of a general 1-form, A, on a 4D variety it is an easy step to compute the Pfaff Sequence, using one exterior differential operation, and several algebraic exterior products. For a differential 1-form, A, defined on a geometric domain of 4 base variables, the Pfaff Sequence is defined as: Pfaff Sequence {A, dA,AˆdA, dAˆdA...} (12) It is possible that over some domains, as the elements of the sequence are computed, one of the elements (and subsequent elements) of the Pfaff Sequence will vanish. The number of non-zero elements in the Pfaff Sequence (PS) defines the Pfaff Topological Dimension (PTD) of the specified 1-form3. Modulo singularities, the Pfaff Topological Dimension determines the minimum number M of N functions of base variables (N ≥ M) required to define the topological properties of the connected component of the 1-form A. The Pfaff Topological Dimension of the 1-form of Action, A, can be put into correspon- dence with the four classic topological structures of thermodynamics. Equilibrium, Isolated, Closed, and Open systems. The classic thermodynamic interpretation is that the first two structures do not exchange mass (mole numbers) or radiation with their environment. The Closed structure can exchange radiation with its environment but not mass (mole numbers). The Open structure can exchange both mass and radiation with its environment. The fol- lowing table summarizes these properties. For reference purposes, I have given the various elements of the Pfaff sequence specific names: Topological p-form name element Nulls PTD Thermodynamic system Action A dA = 0 1 Equilibrium Vorticity dA AˆdA = 0 2 Isolated Torsion AˆdA dAˆdA = 0 3 Closed Parity dAˆdA − 4 Open Table 1 Applications of the Pfaff Topological Dimension. The four thermodynamic systems can be placed into two disconnected topological cate- gories. If the Pfaff Topological Dimension of A is 2 or less, the first category is determined by the closure (or differential ideal) of the 1-form of Action, A∪ dA. This Cartan topology is a connected topology. In the case that the Pfaff Topological Dimension is greater than 2, the Cartan topology is based on the union of two closures, {A ∪ dA ∪ AˆdA ∪ dAˆdA}, and is a disconnected topology. 3The Pfaff Topological dimension has been called the ”class” of a 1-form in the old literature. I prefer the more suggestive name. It is a topological fact that there exists a (topologically) continuous C2 process from a disconnected topology to a connected topology, but there does not exist a C2 continuous process from a connected topology to a disconnected topology. This fact implies that topological change can occur continuously by a ”pasting” processes representing the decay of turbulence by ”condensations” from non-equilibrium to equilibrium systems. On the other hand, the creation of Turbulence involves a discontinuous (non C2) process of ”cutting” into parts. This warning was given long ago [19] to prove that computer analyses that smoothly match value and slope will not replicate the creation of turbulence, but can faithfully replicate the decay of turbulence. 1.3.2 The Pfaff Topological Dimension of the Thermodynamic Work 1-form, W The topological structure of the thermodynamic Work 1-form, W , can be used to refine the topology of the physical system; recall that the physical system is encoded by the Action 1-form, A. Claim 4 The PDE’s that represent the system dynamics are determined by the Pfaff Topo- logical Dimension of the 1-form of Work, W , and the 1-form of Action, A, that encodes the physical system. The Pfaff Topological Dimension of the thermodynamic Work 1-form depends upon both the physical system, A, and the process, V4. In particular if the Pfaff Dimension of the thermodynamic Work 1-form is zero, (W = 0), then system dynamics is generated by an extremal vector field which admits a Hamiltonian realization. However, such extremal direction fields can occur only when the Pfaff Topological Dimension of the system encoded by A is odd, and equal or less than the geometric dimension of the base variables. For example, if the geometric dimension is 3, and the Pfaff Topological Dimension of A is 3, then there exists a unique extremal field on the Contact manifold defined by dA. This unique directionfield is the unique eigen directionfield of the 3x3 antisymmetric matrix (created by the 2-form F = dA) with eigenvalue equal to zero. If the geometric dimension is 4, and the Pfaff Topological Dimension of A is 3, then there exists a two extremal fields on the geometric manifold. These directionfields are those generated as the eigen directionfields of the 4x4 antisymmetric matrix (created by the 2-form F = dA) with eigenvalue equal to zero. If the geometric dimension is 4, and the Pfaff Topological Dimension of A is 4, then there do not exist extremal fields on the Symplectic manifold defined by dA. All of the eigen directionfields of the 4x4 antisymmetric matrix (created by the 2-form F = dA) are complex isotropic spinors with pure imaginary eigenvalues not equal to zero. 1.4 Topological Torsion and other Continuous Processes. 1.4.1 Reversible Processes Physical Processes are determined by directionfields4 with the symbol, V4, to within an arbitrary function, ρ. There are several classes of direction fields that are defined as follows Associated Class:i(ρV4)A = 0, (13) Extremal Class:i(ρV4)dA = 0, (14) Characteristic Class:i(ρV4)A = 0, (15) and : i(ρV4)dA = 0, (16) Helmholtz Class: d(i(ρV4)dA) = 0, (17) Extremal Vectors (relative to the 1-form of Action, A) produce zero thermodynamic work, W = i(ρV4)dA = 0, and admit a Hamiltonian representation. Associated Vectors (relative to the 1-form of Action, A) can be adiabatic if the process remains orthogonal to the 1- form, A. Helmholtz processes (which include Hamiltonian processes, Bernoulli processes and Stokes flow) conserve the 2-form of Topological vorticity, dA. All such processes are thermodynamically reversible. Many examples of these systems are detailed in the reference monographs (see footnote on page 1). 1.4.2 Irreversible Processes There is one directionfield that is uniquely defined by the coefficient functions of the 1-form, A, that encodes the thermodynamic system on a 4D geometric variety. This vector exists only in non-equilibrium systems, for which the Pfaff Topological Dimension of A is 3 or 4. This 4 vector is defined herein as the topological Torsion vector, T4. To within a factor, this directionfield5 has the four coefficients of the 3-form AˆdA, with the following properties: 4Which include both vector and spinor fields. 5A direction field is defined by the components of a vector field which establish the ”line of action” of the vector in a projective sense. An arbitrary factor times the direction field defines the same projective line of action, just reparameterized. In metric based situations, the arbitrary factor can be interpreted as a renormalization factor. Properties of :Topological Torsion T4 on Ω4 (18) i(T4)Ω4 = i(T4)dxˆdyˆdzˆdt = AˆdA, (19) W = i(T4)dA = σ A, (20) dW = dσˆA+ σdA = dQ (21) U = i(T4)A = 0, T4 is associative (22) i(T4)dU = 0 (23) i(T4)Q = 0 T4 is adiabatic (24) L(T4)A = σ A, T4 is homogeneous (25) L(T4)dA = dσˆA+ σdA = dQ, (26) QˆdQ = L(T4)AˆL(T4)dA = σ 2AˆdA 6= 0, (27) dAˆdA = d(AˆdA) = d{(i(T4)Ω4} = (div4T4)Ω4, (28) L(T4)Ω4 = d{(i(T4)Ω4} = (2σ)Ω4, (29) If the Pfaff Topological Dimension of A is 4 (an Open thermodynamic system), then T4 has a non-zero 4 divergence, (2σ), representing an expansion or a contraction of the 4D volume element Ω4. The Heat 1-form, Q, generated by the process, T4, is NOT integrable. Q is of Pfaff Topological Dimension greater that 2, when σ 6= 0. Furthermore the T4 process is locally adiabatic as the change of internal energy in the direction of the process path is zero. Therefore, in the Pfaff Topological Dimension 4 case, where dAˆdA 6= 0, the T4 direction field represents an irreversible, adiabatic process. When σ is zero and dσ = 0, but AˆdA 6= 0, the Pfaff Topological Dimension of the system is 3 (a Closed thermodynamic system). In this case, the T4 direction field becomes a characteristic vector field which is both extremal and associative, and induces a Hamilton- Jacobi representation (the ground state of the system for which dQ = 0). For any process and any system, equation ( 27) can be used as a test for irreversibility. It seems a pity, that the concept of the Topological Torsion vector and its association with non-equilibrium systems, where it can be used to establish design criteria to minimize energy dissipation, has been ignored by the engineering community. 1.4.3 The Spinor class It is rather remarkable (and only fully appreciated by me in February, 2005) that there is a large class of direction fields useful to the topological dynamics of thermodynamic systems (given herein the symbol ρS4) that do not behave as vectors (with respect to rotations). They are isotropic complex vectors of zero length, defined as Spinors by E. Cartan [2], but which are most easily recognized as the eigen directionfields relative to the antisymmetric matrix, [F ], generated by the component of the 2-form F = dA: The Spinor Class [F ] ◦ |ρS4〉 = λ |ρS4〉 6= 0, (30) 〈ρS4| ◦ |ρS4〉 = 0, λ 6= 0 (31) In the language of exterior differential forms, if the Work 1-form is not zero, the process must contain Spinor components: W = i(ρS4)dA 6= 0 (32) As mentioned above, Spinors have metric properties, behave as vectors with respect to transitive maps, but do not behave as vectors with respect to rotations (see p. 3, [2]). Spinors generate harmonic forms and are related to conjugate pairs of minimal surfaces. The notation that a Spinor is a complex isotropic directionfield is preferred over the names ”complex isotropic vector”, or ”null vector” that appear in the literature. As shown below, the familiar formats of Hamiltonian mechanical systems exclude the concept of Spinor process directionfields, for the processes permitted are restricted to be represented by direction fields of the extremal class, which have zero eigenvalues. Remark 5 Spinors are normal consequences of antisymmetric matrices, and, as topological artifacts, they are not restricted to physical microscopic or quantum constraints. According to the topological thermodynamic arguments, Spinors are implicitly involved in all processes for which the 1-form of thermodynamic Work is not zero. Spinors play a role in topological fluctuations and irreversible processes. The thermodynamic Work 1-form, W , is generated by a completely antisymmetric 2- form, F , and therefore, if not zero, must have Spinor components. In the odd dimensional Contact manifold case there is one eigen Vector, with eigenvalue zero, which generates the extremal processes that can be associated with a Hamiltonian representation. The other two eigendirection fields are Spinors. In the even dimensional Symplectic manifold case, any non-zero component of work requires that the evolutionary directionfields must contain Spinor components. All eigen directionfields on symplectic spaces are Spinors. The fundamental problem of Spinor components is that there can be more than one Spinor direction field that generates the same geometric path. For example, there can be Spinors of left or right handed polarizations and Spinors of expansion or contraction that produce the same optical (null congruence) path. This result does not fit with the classic arguments of mechanics, which require unique initial data to yield unique paths. Furthermore, the concept of Spinor processes can annihilate the concept of time reversal symmetry, inherent in classical hydrodynamics. The requirement of uniqueness is not a requirement of non- equilibrium thermodynamics, where Spinor ”entanglement” has to be taken into account. 1.5 Emergent Topological Defects Suppose an evolutionary process starts in a domain of Pfaff Topological Dimension 4, for which a process in the direction of the Topological Torsion vector, T4 , is known to represent an irreversible process. Examples can demonstrate that the irreversible process can proceed to a domain of the geometric variety for which the dissipation coefficient, σ, becomes zero. Physical examples [41] such as the skidding bowling ball proceed with irreversible dissipation (PTD = 6) until the ”no-slip” condition is reached (PTD = 5). In fluid systems the topological defects can emerge as long lived states far from equilibrium. The process is most simply visualized as a ”condensation” from a turbulent gas, such as the creation of a star in the model which presumes the universe is a very dilute, turbulent van der Waals gas near its critical point. The red spot of Jupiter, a hurricane, the ionized plasma ring in a nuclear explosion, Falaco Solitons, the wake behind an aircraft are all exhibitions of the emergence process to long lived topological structures far from equilibrium. It is most remarkable that the emergence of these experimental defect structures occurs in finite time. The idea is that a subdomain of the original system of Pfaff Topological Dimension 4 can evolve continuously with a change of topology to a region of Pfaff Topological Dimension 3. The emergent subdomain of Pfaff Topological Dimension 3 is a topological defect, with topological coherence, and often with an extended lifetime (as a soliton structure with a dominant Hamiltonian evolutionary path), embedded in the Pfaff dimension 4 turbulent background. The Topological Torsion vector in a region of Pfaff Topological Dimension 3 is an extremal vector direction field in systems of Pfaff Topological Dimension 3; it then has a zero 4D divergence, and leaves the volume element invariant. Moreover the existence of an extremal direction field implies that the 1-form of Action can be given a Hamiltonian representation, k + H(P, q, t)dt. In the domain of Pfaff dimension 3 for the Action, A, the subse- quent continuous evolution of the system, A, relative to the process T4, can proceed in an energy conserving, Hamiltonian manner, representing a ”stationary” or ”excited” state far from equilibrium (the ground state). This argument is based on the assumption that the Hamiltonian component of the direction field is dominant, and any Spinor components in the PTD = 3 domain, representing topological fluctuations, can be ignored. These excited states, far from equilibrium, can be interpreted as the evolutionary topological defects that emerge and self-organize due to irreversible processes in the turbulent dissipative system of Pfaff dimension 4. The descriptive words of self-organized states far from equilibrium have been abstracted from the intuition and conjectures of I. Prigogine [9]. The methods of Continuous Topolog- ical Evolution correct the Prigogine conjecture that ”dissipative structures” can be caused by dissipative processes and fluctuations. The long-lived excited state structures created by irreversible processes are non-equilibrium, deformable topological defects almost void of irreversible dissipation. The topological theory presented herein presents for the first time a solid, formal, mathematical justification (with examples) for the Prigogine conjec- tures. Precise definitions of equilibrium and non-equilibrium systems, as well as reversible and irreversible processes can be made in terms of the topological features of Cartan’s ex- terior calculus. Using Cartan’s methods of exterior differential systems, thermodynamic irreversibility and the arrow of time can be well defined in a topological sense, a technique that goes beyond (and without) statistical analysis [23]. Thermodynamic irreversibility and the arrow of time requires that the evolutionary process produce topological change. 2 Applications 2.1 An Electromagnetic format The thermodynamic identification of the terms in Cartan’s magic formula are not whimsical. To establish an initial level of credence in the terminology, consider the 1-form of Action, A, where the component functions are the symbols representing the familiar vector and scalar potentials in electromagnetic theory. The coefficient functions have arguments over the four independent variables {x, y, z, t}, A = Aµ(x, y, z, t)dx µ = A ◦ dr− φ dt. (33) Construct the 2-form of field intensities as the exterior differential of the 1-form of Action, F = dA = (∂Ak/∂x j − ∂Aj/∂xk)dxjˆdxk (34) = Fjkdx jˆdxk = +Bzdxˆdy...+ Exdxˆdt... . (35) The engineering variables are defined as electric and magnetic field intensities: E = −∂A/∂t− grad φ, B = curl A. (36) Relative to the ordered set of base variables, {x, y, z, t}, define a process directionfield, ρV4, as a 4-vector with components, [V, 1], with a scaling factor, ρ. ρ[V4] = ρ[V, 1]. (37) Note that this direction field can be used to construct a useful 3-form of (matter) current, C, in terms of the 4-volume element, Ω4 = dxˆdyˆdzˆdt : C = i(ρV4)dxˆdyˆdzˆdt = i(C4)Ω4. (38) The process 3-form, C, is not necessarily the same as electromagnetic charge current density 3-form of electromagnetic theory, J . The 4-divergence of C, need not be zero: dC 6= 0. Using the above expressions, the evaluation of the thermodynamic work 1-form in terms of 3-vector engineering components becomes: The thermodynamic Work 1-form: W = i(ρV4)dA = i(ρV4)F, (39) ⇒ −ρ{E+V ×B} ◦ dr+ ρ{V ◦ E}dt (40) = −ρ{fLorentz} ◦ dr+ ρ{V ◦ E}dt. (41) The Lorentz force = −{fLorentz} ◦ dr (spatial component) (42) The dissipative power = +{V ◦ E}dt (time component). (43) For those with experience in electromagnetism, note that the construction yields the format, automatically and naturally, for the ”Lorentz force” as a derivation consequence, without further ad hoc assumptions. The dot product of a 3 component force, fLorentz, and a differential spatial displacement, dr, defines the elementary classic concept of ”differential work”. The 4-component thermodynamic Work 1-form,W , includes the spatial component and a differential time component, Pdt, with a coefficient which is recognized to be the ”dissipative power”, P = {V ◦ E}. The thermodynamic Work 1-form, W , is not necessarily a perfect differential, and therefore can be path dependent. Closed cycles of Work need not be zero. Next compute the Internal Energy term, relative to the process defined as ρV4: Internal Energy: U = i(ρV4)A = ρ(V ◦A− φ). (44) The result is to be recognized as the ”interaction” energy density in electromagnetic plasma systems. It is apparent that the internal energy, U , corresponds to the interaction energy of the physical system; that is, U is the internal stress energy of system deformation. Therefore, the electromagnetic terminology can be used to demonstrate the premise that Cartan’s magic formula is not just another way to state that the first law of thermodynamics makes practical sense. The topological methods permit the long sought for integration of mechanical and thermodynamic concepts, without the constraints of equilibrium systems, and/or statistical analysis. It is remarkable that although the symbols are different, the same basic constructions and conclusions apply to many classical physical systems. The correspondence so established between the Cartan magic formula acting on a 1-form of Action, and the first law of ther- modynamics is taken both literally and seriously in this essay. The methods yield explicit constructions for testing when a process acting on a physical system is irreversible. The methods permit irreversible adiabatic processes to be distinguished from reversible adiabatic processes, analytically. Adiabatic processes need not be ”slow” or quasi-static. Given any 1-form, A, W, and/or Q, the concept of Pfaff Topological Dimension (for each of the three 1-forms, A, W , Q) permits separation of processes and systems into equiva- lence classes. For example, dynamical processes can be classified in terms of the topological Pfaff dimension of the thermodynamic Work 1-form, W . All extremal Hamiltonian systems have a thermodynamic Work 1-form, W , of topological Pfaff dimension of 1, (dW = 0). Hamiltonian systems can describe reversible processes in non-equilibrium systems for which the topological Pfaff dimension is 3. Such systems are topological defects whose topology is preserved by the Hamiltonian dynamics, but all processes which preserve topology are re- versible. In non-equilibrium systems, topological fluctuations can be associated with Spinors of the 2-form, F = dA. Even if the dominant component of the process is Hamiltonian, Spinor fluctuations can cause the system (ultimately) to decay. 2.1.1 Topological 3-forms and 4-forms in EM format Construct the elements of the Pfaff Sequence for the EM notation, {A, F = dA,AˆF, FˆF}, (45) and note that the algebraic expressions of Topological Torsion, AˆF , can be evaluated in terms of 4-component engineering variables T4 as: Topological Torsion vector (46) AˆF = i(T4)Ω4 = i(T4)dxˆdyˆdzˆdt (47) T4 = [T, h] = −[E×A+Bφ,A ◦B]. (48) The exterior 3-form, AˆF , with physical units of (~/unit mole)2, is not found (usually) in classical discussions of electromagnetism6. If T4 is used as to define the direction field of a process, then L(T4)A = σA, i(T4)A = 0. (49) where 2σ = {div4(T4)} = 2(E ◦B). (50) The important (universal) result is that if the acceleration associated with the direction field, E, is parallel to the vorticity associated with the direction field, B, then according to the equations starting with eq. (18) et. seq. the process is dissipatively irreversible. This result establishes the design criteria for engineering applications to minimize dissipation from turbulent processes. The Topological Torsion vector has had almost no utilization in applications of classical electromagnetic theory. 2.1.2 Topological Torsion quanta The 4-form of Topological Parity, FˆF , can be evaluated in terms of 4-component engineering variables as: Topological Parity d(AˆF ) = FˆF = {div4(T4)}Ω4 = {2E ◦B} Ω4. (51) 6The unit mole number is charge, e, in EM theory. This 4-form is also known as the second Poincare Invariant of Electromagnetic Theory. The fact that FˆF need not be zero implies that the Pfaff Topological Dimension of the 1-form of Action, A, must be 4, and therefore A represents a non-equilibrium Open thermo- dynamic system. Similarly, if FˆF = 0, but AˆF 6= 0, then the Pfaff Topological Dimension of the 1-form of Action, A, must be 3, and the physical system is a non-equilibrium Closed thermodynamic system. When FˆF = 0, the corresponding three-dimensional integral of the closed 3-form, AˆF , when integrated over a closed 3D-cycle, becomes a deRham period integral, defined as the Torsion quantum. In other words, the closed integral of the (closed) 3-form of Topological Torsion becomes a deformation (Hopf) invariant with integral values proportional to the integers. Torsion quantum = 3D cycle AˆF. (52) On the other hand, topological evolution and transitions between ”quantized” states of Torsion require that the respective Parity 4-form is are not zero. As, L(T4)Ω4 = d{i(T4)Ω4} = (2σ) Ω4 = 2(E ◦B) Ω4 6= 0, (53) it is apparent that the evolution of the differential volume element, Ω4, depends upon the existence and colinearity of both the electric field, E, and the magnetic field, B. It is here that contact is made with the phenomenological concept of ”4D bulk” viscosity = 2σ. It is tempting to identify σ2 with the concept of entropy production. Note that the Topological Torsion directionfield appears only in non-equilibrium systems. These results are universal and can be used in hydrodynamic systems discussed in that which follows. 2.2 A Hydrodynamic format 2.2.1 The Topological Continuum vs. the Geometrical Continuum In many treatments of fluid mechanics the (geometrical) continuum hypothesis is invoked from the start. The idea is ”matter” occupies all points of the space of interest, and that properties of the fluid can be represented by piecewise continuous functions of space and time, as long as length and time scales are not too small. The problem is that at very small scales, one has been led to believe the molecular or atomic structure of particles will become evident, and the ”macroscopic” theory will breakdown. However, these problems of scale are geometric issues, important to many applications, but not pertinent to a topological perspective, where shape and size are unimportant. Suppose that the dynamics can be formulated in terms of topological concepts which are independent from sizes and shapes. Then such a theory of a Topological Continuum would be valid at all scales. Such is the goal of this monograph. Remark 6 However, one instance where ”scale” many have topological importance is as- sociated with the example of a surface with a ”teeny” hole. If the hole, no matter what its size, has a twisted ear (Moebius band) then the whole surface is non-orientable, no mat- ter how ”small” the hole. Could it be that the world of the quantum is, in effect, that of non-orientable defects embedded in an otherwise orientable manifold that originally had no such defects. Note further the strong correspondence with Fermions with non-oriented (half- integer) multiplet ribbons, and Bosons with oriented (integer) multiplet ribbons of both right and left twists [44]. As will be developed below, the fundamental equations of exterior differential systems can lead to field equations in terms of systems of Partial Differential Equations (PDE’s). The format of the fundamental theory will be in terms of objects (exterior differential forms) which, although composed of algebraic constructions of tensorial7 things, are in a sense scalars (or pseudo scalars) that are homogeneous with respect to concepts of scale. The theory then developed is applicable to hydrodynamics at all scales, from the microworld to the cosmological arena. The ”breakdown” of the continuum model is not relevant. The topological system may consist of many disconnected parts when the system is not in thermodynamic equilibrium or isolation, and the parts can have topological obstructions or defects, some of which can be used to construct period integrals that are topologically ”quantized”. Hence the ”quantization” of the micro-scaled geometric systems can have it genesis in the non-equilibrium theory of thermodynamics. However, from the topological perspective, the rational topological quantum values can also occur at all scales. 2.2.2 Topological Hydrodynamics The axioms of Topological Thermodynamics are summarized in Section 1.1. For hydrody- namics (or electrodynamics the axioms are essentially the same. Just exchange the word, hydrodynamics (or electrodynamics) for the word, thermodynamics, in the formats of Section By 1969 it had become evident to me that electromagnetism (without geometric con- straints), when written in terms of differential forms, was a topological theory, and that the concept of dissipation and irreversible processes required more than that offered by Hamil- tonian mechanics. At that time I was interested in possible interactions of the gravitational field and the polarizations of an electromagnetic signal. One of the first ideas discovered 7Relative to diffeomorphisms. about topological electrodynamics was that there existed an intrinsic transport theorem [11] that introduced the concept of what is now called Topological Spin, AˆG, into electromag- netic theory [43]. As a transport theorem not recognized by classical electromagnetism, the first publication was as a letter to Physics of Fluids. That started my interest in a topological formulation of fluids. It was not until 1974 that the Lie differential acting on exterior differential forms was es- tablished as the key to the problem of intrinsically describing dissipation and the production of topological defects in physical systems; but methods of visualization of such topological defects in classical electrodynamics were not known [12]. It was hoped that something in the more visible fluid mechanics arena would lend credence to the concepts of topological defects. The first formulations of the PDE’s of fluid dynamics in terms of differential forms and Cartan’s Magic formula followed quickly [13]. In 1976 it was argued that topological evolution was at the cause of turbulence in fluid dynamics, and the notion of what is now called Topological Torsion, AˆF, became recognized as an important concept. It was apparent that streamline flow imposed the constraint that AˆF = 0 on the equations of hydrodynamics. Turbulent flow, being the antithesis of streamline flow, must admit AˆF 6= 0. In 1977 it was recognized that topological defect structures could become ”quantized” in terms of deRham period integrals [15], forming a possible link between topology and both macroscopic and microscopic quantum physics. The research effort then turned back to a study of topological electrodynamics in terms of the dual polarized ring laser, where it was experimentally determined that the speed of an electromagnetic signal outbound could be different from the speed of an electromagnetic signal inbound: a topological result not within the realm of classical theory. Then in 1986 the long sought for creation and visualization of topological defects in fluids [16] became evident. The creation of Falaco Solitons in a swimming pool was the experiment that established credence in the ideas of what had, by that time, become a theory of continuous topological evolution. It was at the Cambridge conference in 1989 [17] that the notions of topological evolution, hydrodynamics and thermodynamics were put together in a rudimentary form, but it was a year later at the Permb conference in 1990 [18] that the ideas were well established. The Permb presentation also suggested that the ambiguous (at that time) notion of coherent structures in fluids could be made precise in terms of topological coherence. A number of conference presentations followed in which the ideas of continuous thermodynamic irreversible topological evolution in hydrodynamics were described [19], but the idea that the topological methods of thermodynamics could be used to distinguish non-equilibrium processes and non-equilibrium systems and irreversible processes with out the use of statistics slowly came into being in the period 1985-2005 [22]. These efforts have been summarized in [40], and a collection of the old publications appears in [45]. 2.3 Classical Hydrodynamic Theory There are two classical techniques for describing the evolutionary motion of a fluid: the Lagrangian method and the Eulerian method. Both methods treat the fluid relative to a Euclidean 3D manifold, with time as a parameter. The first (Lagrangian) technique treats a fluid as a collection of ”particles, or parcels” and the flow is computed in terms of ”initial” data {a, b, c, τ} imposed upon solutions to Newtonian equations of motion for ”particles, or parcels”. The solution functions describe a map from an initial state {a, b, c; τ} to a final state {x, y, z, t}. This method is related to solutions of kinematic equations, and is contravariant in the sense of an immersion to velocities (the tangent space). The kinematic basis for the Lagrangian motion draws heavily from the Frenet-Serret analysis of a point moving along a space curve. The second technique treats a fluid as ”field”, and is representative of a ”wave” point of view of a Hamiltonian system. The functions that define the field (the covariant momenta) depend on ”final data” {x, y, z, t}, and are covariant in the sense of a submersion. Each method has its preimage in the form of an exterior differential 1-form of Action. The primitive classical Lagrangian Action concept is written in the form AL = L(x k, V m, t)dt, (54) and the primitive Eulerian Action is written as AE = pkdx k + Hdt. (55) As both Actions supposedly describe the same fluid, are they equivalent? That is, does AE ⇔ AL ? (56) Note that AL is composed from only two functions (L and t) such that at most AL is of Pfaff Topological Dimension 2. On the other hand, the Pfaff Topological Dimension of AE (as written) could be as high as 8, if all functions and differentials are presumed to be independent. So the two formulations are NOT equivalent, unless constraints reduce the topological dimension of AE to 2, or if additions are made to the 1-form AL to increase its Pfaff dimension. 2.3.1 The Lagrange-Hilbert Action The classic addition to AL is of the form, pk(dx k − V kdt), where the pk are presumed to be Lagrange multipliers of the fluctuations in kinematic velocity. The result is defined as the Cartan - Hilbert 1-form of Action: ACH = L(x k, V k, t)dt+ pk(dx k − V kdt). (57) Note that at first glance it appears that there are 10=3N+1 independent geometric variables {xk, V k, pk, t} in the formula for ACH , but if the Pfaff Sequence is constructed, the Pfaff Topological Dimension turns out to be 8. So with this addition of Lagrange multipliers to AL, the topological dimensions of the two actions are the same. However, note that by rearranging variables, ACH = pkdx k + (L(xk, V m, t)− pkV k)dt, (58) = pkdx k + Hdt = AE . (59) For a fluid the Eulerian ”momenta” per unit parcel of mass is usually defined as pk/m = vk, (60) such that the Eulerian Action per unit mass becomes AE ⇒ vkdxk + Hdt. (61) The bottom line is that the Lagrangian and Eulerian point of view can be made compatible if fluctuations in Kinematic Perfection are allowed. Recall that the development of elasticity theory (and its emphasis on symmetrical tensors) also spawned the development of hydrodynamics. Much of the theory of classical hydro- dynamics was phrased geometrically in the language of vector analysis. The development followed the phenomenological concepts of an extended Newtonian theory of elasticity. The classical theory was developed from ”balance” equations for a bounded sample, or parcel, of matter (mass), which express assumptions (defined as the conservation of mass, momenta and energy) in terms of integrals over the bounded sample, or parcel, of matter. The classical integrals are performed usually over three-dimensional volumes (and not over 4D space-time). The classical Cauchy result (for the momentum equations) is ρ{∂v/∂t + v ◦ ∇v} = ∇◦T+ ρf , (62) ρ{∂v/∂t + grad(v ◦ v/2)− v×curlv} = ∇◦T+ ρf , (63) Constitutive assumptions are then made for the 3D stress tensor T , such that (in matrix format) [T] = (−P + λ(∇ ◦ v) [I] + ν{[∇v] + [∇v]T }, (64) where P is the Pressure, ν the affine ”shear” viscosity and λ the ”expansion” viscosity. The antisymmetric components {[∇v]− [∇v]T } have been ignored. It will be demonstrated how the Axioms of Hydrodynamics yield topological information about the classic Cauchy development, and goes beyond the symmetrical formulations by recognizing that antisymmetries can introduce complex spinor contributions to the dynamics. 2.4 Euler flows and Hamiltonian fluids Consider the Action 1-form per unit source (in thermodynamics, the unit source is mole number, or sometimes mass), constructed from a covariant 3D velocity field, v = vk(x, y, z, t), and a scalar potential function, φ: A = v ◦ dr−φdt = vk(x, y, z, t)dxk − φdt. (65) Compute the exterior differential dA and define the following (3D vector) functions as, ω = curl v and a = +{∂v/∂t + grad(φ)}, (66) such that, F = dA = {∂Ak/∂xj − ∂Aj/∂xk}dxjˆdxk = Fjkdxjˆdxk (67) = ωzdxˆdy + ωxdyˆdz + ωydzˆdx− axdxˆdt− aydyˆdt− azdzˆdt. These vector fields always satisfy the Poincare-Faraday induction equations, dF = ddA = 0, or, curl (−a) + ∂ω/∂t = 0, div ω = 0. (68) The Eulerian Fluid Consider a process created by the contravariant vector directionfield, V4 = [V x,Vy,Vz, 1] and use Cartan’s magic formula, L(ρV4)A = i(ρV4)dA+ d(i(ρV4)A) =W + dU = Q, (69) to compute the thermodynamic Work 1-form,W . The expressions for Work,W, and internal energy, U, become: W = i(ρV4)dA = −ρ{−∂v/∂t − grad(φ) +V × ω} ◦ dr − ρV ◦ {∂v/∂t + grad(φ)}dt, (70) = ρ{a−V × ω} ◦ dr−ρV ◦ {a)}dt (71) U = i(V4)A = ρ(V · v−φ). (72) At first, topologically constrain the thermodynamic Work 1-form to be of the Bernoulli class in terms of the exterior differential system: W = −dP (73) ρ{+a−V × ω} ◦ dr−ρV ◦ {a)}dt = −grad P ◦ dr−∂P/∂t dt (74) Assume that formally V = v, and the potential is equal to φ = v · v/2. Compare the coefficients of dr to deduce the classic equations of motion for the Eulerian fluid. {∂v/∂t + grad(v · v/2)− v× ω} = −grad(P )/ρ. (75) This formula should be compared to the derivation of the Lorentz force term for Work in electromagnetic systems. The functional format of the hydrodynamic 1-form of Action, A, is the same as that specified above for the electromagnetic system. All that is changed is the notation. In essence, the two topological theories are equivalent to the extent that there is a correspondence between functions: A ⇔ v, φ ⇔ v · v/2, (76) E ⇔ −a, B ⇔ ω. (77) All the results of the preceding section using an electromagnetic format can be translated into the hydrodynamic format. Note that the Bernoulli ”pressure”, P , is an evolutionary invariant along a trajectory, L(ρV4)P = i(V4)dP = i(V4)i(V4)dA = 0. (78) If the Pressure is barotropic, then the Bernoulli function becomes, dΘ = dP/ρ. The function Θ can be amalgamated with the potential, φ, such that the thermodynamic Work 1-form becomes equal to zero. The system then admits an extremal Hamiltonian direction field such that the thermodynamic Work 1-form is zero. W = i(VH)dA = 0. (79) For a process defined in terms of an extremal directionfield, the First Law indicates that the 1-form of Heat, Q, is exact, dQ = 0, and equal to the change in internal energy, Q = dU . Any Hamiltonian process is reversible, as QˆdQ = 0. The time-like component of the exterior differential system W + dP = 0 leads to the equation, ∂P/∂t = −ρv ◦ {∂v/∂t + grad(v · v/2) = ρ(v ◦ a). (80) It is apparent that if the velocity, v, and the acceleration, a, are orthogonal, then the time rate of change of the Bernoulli pressure is zero. It also follows that the ”Master” equation is valid, with the only difference being that curl v is defined as ω, the vorticity of the hydrodynamic flow. The master equation becomes, curl(v× ω) = ∂ω/∂t, (81) and this equation is to be recognized as the equivalent of Helmholtz’ equation for the con- servation of vorticity. When the Pfaff Topological Dimension of the Work 1-form is 1, it is possible to show that the ”Master” leads to a diffusion equation. In the hydrodynamic sense, conservation of vorticity implies uniform continuity. In other words, the Eulerian flow is not only Hamiltonian, it is also uniformly continuous, and satisfies both the master equation and the conservation of vorticity constraints. In addition, it may be demonstrated that such systems are at most of Pfaff dimension 3, and admit a relative integral invariant which generalizes the hydrodynamic concept of invariant helicity. In the electromagnetic topology, the Hamiltonian constraint is equivalent to the statement that the Lorentz force vanishes, a condition that has been used to define the ”ideal” plasma or ”force-free” plasma state [46]. 3 The Navier-Stokes fluid 3.1 The classic Navier-Stokes equations It can be demonstrated that the ”ideal fluid” has a Hamiltonian representation, for which the dynamics preserves a ”Hamiltonian” energy. This result is in disagreement with experiment in that it is observed that motions of ”non-ideal” fluids exhibit decay to a stationary state. The Lagrange Euler equations must be modified to accommodate dissipation of kinetic energy and angular momentum. In fact, the ideal fluid constraint of zero affine shear stresses should be replaced by dissipative terms related to both affine shears and a new phenomena of rotational and expansion shears which have a fixed point. The classical phenomenological outcome is the Navier-Stokes PDE’s, ∂v/∂t + grad(v · v/2)− v× curl v = −gradϕ− (1/ρ) gradP + ν∆v − (µB + ν) grad div v, (82) where µB is the ”bulk” viscosity coefficient and ν is the ”shear” viscosity coefficient. If the fluid is ”incompressible” then the last term, which includes corrections due to bulk viscosity, vanishes; the incompressible constraint requires that div v ⇒ 0. In that which follows, the basic momentum equation (82) will be deduced from the per- spective of Continuous Topological Evolution. Different topological equivalence classes of thermodynamic processes depend upon the Pfaff Topological Dimension of the Work 1-form. The different classes of thermodynamic processes are related to the velocity field in a Hy- drodynamic system. The phenomenological (geometrical) derivation of the equations of hydrodynamics will be replaced by determining the format of the PDE’s that agree with the constraint required to satisfy the various PTD equivalence classes the Work 1-form. The 1-form of Work (for barotropic flows as in eq. (75)) will be of Pfaff Topological Dimension 1. The Pfaff Topological Dimension of the 1-form of Work for the Navier-Stokes fluid can be as high as 4, and is required to be 4 if the flow is fully turbulent. Various intermediate classes of the work 1-form are of interest, as well. In particular, the Pfaff Topological Dimension of the work 1-form must be 3 for a baroclinic system, a result that admits frontal systems with propagating tangential discontinuities as found in weather systems. 3.2 The Navier-Stokes equations embedded in a non-equilibrium thermodynamic system In this subsection, the topological refinement due to the Pfaff Topological Dimension of the Work 1-form will be employed to demonstrate that processes in a non-equilibrium thermody- namic system can be put into correspondence with solutions of the Navier-Stokes equations. In order to go beyond extremal (Hamiltonian) processes, it is necessary that the Pfaff Topological Dimension of the Work 1-form must be greater than 1. Recall that for any process, the Work done is transverse to the process trajectory, (i(ρV4)W = (i(ρV4)(i(ρV4)dA = 0. (83) Hence, if the PTD of the Work 1-form, W , is to be greater than 1, it must have the format, W = i(ρV4)dA = −dP +̟j(dxj − vjdt) = −dP +̟j∆xj , (84) where the ”Bernoulli function”, P , if it exists, must be a first integral (a process invariant), L(ρV4)P = (i(ρV4)dP = 0. (85) It is also important to remember that such non-zero contributions to the work 1-form are due to the complex, isotropic Cartan Spinors, which are the eigen direction fields of the 2-form, F . The coefficients, ̟j, of the topological fluctuations, ∆x j , act in the manner of Lagrange multipliers, and mimic the concept of system forces. If ̟j/ρ is defined (arbitrarily 8) as υ curl curl v then the spatial components of the thermodynamic Work 1-form, W , are con- strained to yield the partial differential equations for a constant density Navier-Stokes fluid: {∂v/∂t + grad(v · v/2)− v × ω} = −grad(P )/ρ+ υ curl curl v. (86) Density variations can be included by adding a term λdiv(V) to the potential {v · v/2} to yield: ∂v/∂t + grad{v.v/2} − v × curl v = −gradP/ρ (87) + λ{grad(div v)} (88) + υ{curl curl v}. (89) Classically, v can be identified with the geometric kinematic shear viscosity, and λ = µB −υ. The coefficient µB can be identified with the topological (space-time) bulk viscosity. 8This is one of many formal choices, but the choice demonstrates that the Navier-Stokes equations reside within the domain of non-equilibrium thermodynamics. QED 3.3 The Topological Torsion process for the Navier-Stokes fluid The Navier-Stokes constraint implies that the thermodynamic Work 1-form need not be closed. Then there are thermodynamic processes represented by solutions to the Navier- Stokes equations that are thermodynamically irreversible. In this subsection, the Topological Torsion vector will be expressed in terms of the solutions of the Navier-Stokes equations. From the work in section 2, the 1-form of Action will generate a 3-form of Topological Torsion, AˆdA = i(T4)Ω4, and leads to the 4 Vector of Topological Torsion (written in hydrodynamic notation): T4 = [−a × v + {v.v/2} curl v, (v ◦ curl v)], (90) = [−a × v + {v.v/2} ω,(v ◦ ω)] = [T, h]. (91) Use the Navier-Stokes equations (82) to solve for a, a = [grad{v.v/2}+ ∂v/∂t] (92) = v × curl v − gradP/ρ + λ{grad(div v)}+ υ{curl curl v}, (93) and then substitute this result into the expression for T4, to yield: T = [hv − (v ◦ v/2)curl v− v × (gradP lρ) + λ{v× grad(div v)} − υ{v× (curl curl v)}], (94) h = v · curl v, (95) Note that T4 exists even for Euler flows, where υ = 0, if the flow is baroclinic. The measurement of the components of the Torsion vector, T4, have been completely ignored by experimentalists in hydrodynamics. By a similar substitution, the topological parity 4-form, FˆF, becomes expressible in terms of engineering quantities as, K = {2(−a ◦ ω)}Ω4 = {2(σ)} ⇔ 2(E ◦B) σ = {gradP/ρ ◦ curl v (96) − λ{ grad(div v) ◦ curl v} − υ{curl v ◦ (curl curl v)}}Ω4. (97) The coefficient σ is a measure of the space-time bulk dissipation coefficient (not λ), and it is the square of this number which must not be zero if the process is irreversible (see eq (27)). Recall that a turbulent dissipative irreversible flow is defined when the Pfaff dimension of the Action 1-form is equal to 4, which implies that K 6= 0. From the expression for σ, it is apparent that if the 3D vector of vorticity is of Pfaff dimension 2, such that ω◦curl ω = 0, then the last term vanishes, and there is no irreversible dissipation due to shear viscosity, υ (a result useful in the theory of wakes). Other useful situations and design criteria for dissipation, or the lack thereof, can be gleaned from the formula. If the vector field is harmonic, then an irreversible process requires that, σ = (−a ◦ ω) = {(gradP/ρ− µBgrad(div v)) ◦ curl v} 6= 0. (98) (Recall that harmonic vector fields are generators of minimal surfaces.) For fluids where (µB) ⇒ 0, if the pressure gradient is orthogonal to the vorticity and the flow field is harmonic, then there is no irreversible dissipation as σ = 0, and the flow is not turbulent. Note that for many fluids the bulk viscosity is much greater than the shear viscosity. When σ = 0, no topological torsion defects are created; the acceleration, a, and the vorticity, ω, of the Navier-Stokes fluid are colinear. Theorem 7 It is thereby demonstrated that solutions to the Navier-Stokes equations corre- spond to processes of a non-equilibrium thermodynamic system of PTD(A) = 4, and Work 1-forms of PTD(W ) > 2. Such processes include Spinor direction fields generated by the Topological Torsion vector. The Topological Torsion vector generates processes, and hence solutions to the Navier-Stokes equations, that are thermodynamically irreversible.. These results should be compared to those generated by Lamb and Eckart [3] for the fluid dissipation function, which is defined by the requirement that the dissipative flow has a (geometric) entropy production rate greater than or equal to zero. More examples can be found in, ”Wakes, Coherent Structures, and Turbulence” [42]. 4 Closed States of Topological Coherence embedded as deformable defects in Turbulent Domains In this section 4, the problem of C2 smoothness will be attacked from the point of view of topological thermodynamics. First, two distinct examples will be given demonstrating two different emergent PTD = 3 states, that emerge from different 4D rotations (see p. 108, [38]). Then, an example demonstrating the decay of a PTD = 4 state into a PTD = 3 state will be given in detail. 4.1 Examples of PTD = 3 domains and their Emergence In section 3, it was demonstrated that there are solutions (thermodynamic processes) to the Navier Stokes equations in non-equilibrium thermodynamic domains. The properties of those PTD = 3 domains which emerge by C2 irreversible solutions from domains of PTD = 4 are of particular interest. From section 2, it is apparent that the key feature of PTD = 3 domains is that the electric E field (acceleration field a in hydrodynamics) must be orthogonal to the magnetic B field (vorticity field ω in hydrodynamics). There are 8 cases to consider (including chirality), Pfaff Topological Dimension 3 E = 0, ±B 6= 0, (99) B = 0, ±E 6= 0, (100) E ◦B = 0, with chirality choices, ±E = ±B 6= 0, (101) of which two will be discussed in detail. The Finite Helicity case (both E and B finite) PTD = 3 Start with the 4D ther- modynamic domain, and first consider the 1-form of Action, A, with the format9: A = Ax(z)dx+ Ay(z)dy − φ(z)dt, (102) and its induced 2-form, F = dA, F = dA = (∂Ax(z)/∂z)dzˆdx + (∂Ay(z)/∂z)dzˆdy − (∂φ(z)/∂z)dzˆdt, (103) = Bx(z)dzˆdx−By(z)dzˆdy + Ez(z)dzˆdt. (104) 9The +E, +B chirality has been selected. The 3-form of Topological Torsion 3-form becomes i(T4)Ω4 = AˆF where (105) T4(z) = [EzAy + φBx, −EzAx + φBy, 0, AxBx + AyBy] (106) with div4(T4(z)) = 2(E ◦B) = 0, A ◦B 6= 0. (107) The Zero Helicity case (both E and B finite) PTD = 3 Start with the 4D thermo- dynamic domain, and consider the 1-form of Action, A, with the format: A = Ax(x, y)dx+ Ay(x, y)dy − φ(x, y)dt, (108) and its induced 2-form, F = dA, F = dA = {(∂Ay(x, y)/∂x)− (∂Ax(x, y)/∂x)dxˆdy} − (∂φ(x, y)/∂x)dxˆdt− (∂φ(x, y)/∂y)dyˆdt, (109) = Bz(x, y)dxˆdy + Ex(x, y)dxˆdt+ Ey(x, y)dyˆdt. (110) The 3-form of Topological Torsion 3-form becomes, i(T4)Ω4 = AˆF where (111) T4(x, y) = [0, 0, (ExAy − EyAx) + φBz, 0] (112) with div4(T4(x, y)) = 2(E ◦B) = 0, A ◦B = 0. (113) This case of zero helicity (A ◦B = 0), has the Topological Torsion vector, T4(x, y), colinear with the B field. Zero Helicity case: PTD = 4 decays to PTD = 3 The two distinct cases, modulo chirality, are suggestive of the idea (see p. 108 [38]) that the rotation group of a 4D domain is not simple. The example,immediately above, is particularly useful because the algebra of the decay from Pfaff dimension 4 to 3 is transparent. Start with the 4D thermodynamic domain, and consider the 1-form of Action, A, with the format: A = Ax(x, y)dx+ Ay(x, y)dy − φ(x, y, z, t)dt, (114) and its induced 2-form, F = dA, F = dA = {(∂Ay(x, y)/∂x)− (∂Ax(x, y)/∂x)dxˆdy} − (∂φ(x, y, z)/∂x)dxˆdt− (∂φ(x, y, z)/∂y)dyˆdt− (∂φ(x, y, z)/∂z)dzˆdt, (115) = Bz(x, y)dxˆdy (116) + Ex(x, y, z, t)dxˆdt+ Ey(x, y, z, t)dyˆdt+ Ez(x, y, z, t)dyˆdt. (117) The 3-form of Topological Torsion 3-form becomes, i(T4)Ω4 = AˆF with PTD(A) = 4 (118) T4(x, y, z, t) = [−EzAy,+EzAx, (ExAy − EyAx) + φBz, 0] (119) with div4(T4(x, y, z, t)) = 2{Ez(x, y, z, t)Bz(x, y)} 6= 0. (120) In this case, the helicity (A ◦B = 0) is still zero, but now the Topological Torsion vector, T4(x, y, z, t), has three spatial components. Moreover, the Process generated by T4(x, y, z) is thermodynamically irreversible, as (E ◦B) 6= 0. The example 1-form is of PTD = 4. To demonstrate the emergence of the PTD = 3 state, suppose the potential function in this example has the format, φ = ψ(x, y) + ϕ(z)e−αt (121) Ez(z, t) = −(∂ϕ(z)/∂z)e−αt = Ez(z)e−αt. (122) Then the irreversible dissipation function decays as {Ez(z)Bz}e−αt. By addition of Spinor fluctuation terms to represent the very small components of irreversible dissipation at late times, the PTD = 3 solution, T4(x, y) = [0, 0, (ExAy − EyAx) + φBz, 0] (123) becomes dominant, and represents a long lived ”stationary” state far from equilibrium, modulo the small Spinor decay terms10. 4.2 Piecewise Linear Vector Processes vs. C2 Spinor processes It will be demonstrated on thermodynamic spaces of Pfaff Topological Dimension 3, that there exist piecewise continuous processes (solutions to the Navier-Stokes equations) which are thermodynamically reversible. These Vector processes can be fabricated by combinations of Spinor processes, each of which is irreversible. This topological result demonstrates, by example, the difference between piecewise linear 3-manifolds and smooth complex manifolds. It appears that the key feature of the irreversible processes is that they have a fixed point of ”rotation or expansion”. Consider those abstract physical systems that are represented by 1-forms, A, of Pfaff Topological Dimension 3. The concept implies that the topological features can be described 10The experimental fact that the defect structures emerge in finite time is still an open topological problem, although some geomtric success has been achieved through Ricci flows. in terms of 3 functions (of perhaps many geometrical coordinates and parameters) and their differentials. For example, if one presumes the fundamental independent base variables are the set {x, y, z}, with an exterior differential oriented volume element consisting of a product11 of exact 1-forms Ω3 = +dxˆdyˆdz, (then a local) Darboux representation for a physical system could have the appearance, A = xdy + dz. (124) The objective is to use the features of Cartan’s magic formula to compute the possible evolutionary features of such a system. The evolutionary dynamics is essentially the first law of thermodynamics: LρVA = i(ρV)dA+ di(ρV)A) = W + dU = Q. (125) The elements of the Pfaff sequence for this Action become, A = xdy + dz., (126) dA = dxˆdy, (127) AˆdA = dxˆdyˆdz, (128) dAˆdA = 0. (129) Note that for this example the coefficient of the 3-form of Topological Torsion is not zero, and depends upon the Enstrophy (square of the Vorticity) of the fluid flow. 4.3 The Vector Processes Relative to the position vector R = [x, y, z] of ordered topological coordinates {x, y, z}, consider the 3 abstract, linearly independent, orthogonal (supposedly) vector direction fields: , (130) , (131) . (132) 11More abstract systems could be constructed from differential forms which are not exact. These direction fields can be used to define a class of (real) Vector processes, but these real vectors do not exhibit the complex Spinor class of eigendirection fields for the 2-form, dA. The Spinor eigendirection fields are missing from this basis frame. The important fact is that thermodynamic processes defined in terms of a real basis frame (and its connection) are incomplete, as such processes ignore the complex spinor direction fields. For each of the real direction fields, deform the (assumed) process by an arbitrary function, ρ. Then construct the terms that make up the First Law of topological thermodynamics. First construct the contractions to form the internal energy for each process, UVx = i(ρVx)A = 0, dUVx = 0, (133) UVy = i(ρVy)A = ρx, dUVy = d(ρx), (134) UE = i(ρE)A = ρ, dUE = dρ. (135) The extremal vector E is the unique eigenvector with eigenvalue zero relative to the maximal rank antisymmetric matrix generated by the 2-form, dA. The associated vector Vx (relative to the 1-form of Action, A, is orthogonal to the y, z plane. Recall that any associated vector represents a local adiabatic process, as the Heat flow is transverse to the process. The linearly independent thermodynamic Work 1-forms for evolution in the direction of the 3 basis vectors are determined to be, WVx = i(ρVx)dA = +ρdy, (136) WVy = i(ρVy)dA = −ρdx, (137) WE = i(ρE)dA = 0. (138) From Cartan’s Magic Formula representing the First Law as a description of topological evolution, L(V)A = i(ρV)dA+ d(i(ρV)A) ≡ Q, (139) it becomes apparent that, QVx = −ρdy, dQVx = −dρˆdy, (140) QVy = +xdρ, dQVy = −dρˆdx, (141) QE = dρ dQE = 0, (142) All processes in the extremal direction satisfy the conditions that QEˆdQE = 0. Hence, all extremal processes are reversible. It is also true that evolutionary processes in the direction of the other basis vectors, separately, are reversible, as the 3-form QˆdQ vanishes forVx, Vy, or E. Hence all such piecewise continuous, transitive, processes are thermodynamically reversible. Note further that the ”rotation” induced by the antisymmetric matrix [dA] acting on Vx yields Vy and the 4th power of the matrix yields the identity rotation, [dA] ◦ |Vx〉 = |Vy〉 , (143) 2 ◦ |Vx〉 = − |Vx〉 , (144) 4 ◦ |Vx〉 = + |Vx〉 . (145) This concept is a signature of Spinor phenomena. 4.4 The Spinor Processes Now consider processes defined in terms of the Spinors. The eigendirection fields of the antisymmetric matrix representation of F = dA, [F ] = 0 1 0 −1 0 0 0 0 0 , (146) are given by the equations: EigenSpinor1 |Sp1〉 = Eigenvalue = + -1, (147) EigenSpinor2 |Sp2〉 = Eigenvalue = - -1 (148) EigenVector1 |E〉 = Eigenvalue = 0 (149) Now consider the processes defined by ρ times the Spinor eigendirection fields. Compute the change in internal energy, dU , the Work, W and the Heat, Q, for each Spinor eigendi- rection field: = i(ρSp1)A = −1ρx d(UρSp −1d(ρx), (150) = i(ρSp2)A = − −1ρx d(UρSp ) = − −1d(ρx), (151) UρE = i(ρE)A = ρ, d(UρE) = dρ, . (152) = i(ρSp1)dA = ρ(dy − -1dx), (153) = i(ρSp2)dA = +ρ(dy + -1dx) (154) WρV1 = i(ρV1)dA = 0, . (155) = Li(ρSp )A = ρ(dy − -1dx) + -1d(ρx), (156) = Li(ρSp )A = ρ(dy + -1dx)− -1d(ρx), (157) QρV1 = Li(ρV1)A = dρ. (158) 4.5 Irreversible Spinor processes Next compute the 3-forms of QˆdQ for each direction field, including the spinors: QρV1ˆdQρV1 = 0, (159) ˆdQρSp -1ρdρˆdxˆdy, (160) ˆdQρSp -1ρdρˆdxˆdy. (161) It is apparent that evolution in the direction of the Spinor fields can be irreversible in a thermodynamic sense, if dρˆdxˆdy is not zero. This is not true for the ”piecewise linear” combinations of the complex Spinors that produce the real vectors, V and V⊥. Evolution in the direction of ”smooth” combinations of the base vectors may not satisfy the reversibility conditions, QˆdQ = 0, when the combination involves a fixed point in the x, y plane. For example, it is possible to consider smooth rotations (polarization chirality) in the x, y plane: Vrotation right = V⊥ + -1V = Sp1, (162) QˆdQ = − -1ρdρˆdxˆdy. (163) Vrotation left = V − -1V⊥= Sp2, (164) QˆdQ = + -1ρdρˆdxˆdy. (165) The non-zero value ofQˆdQ for the continuous rotations are related to the non-zero Godbillon- Vey class [7]. A key feature of the rotations is that they have a fixed point in the plane; the motions are not transitive. If the physical system admits an equation of state of the form, θ = θ(x, y, ρ) = 0, then the rotation or expansion processes are not irreversible. Note that the (supposedly) Vector processes of the preceding subsection are combinations of the Spinor processes, Vx = (a · Sp1 + b · Sp2)/2 (166) Vy = − -1(a · Sp1− b · Sp2)/2. (167) Almost always, a process defined in terms a linear combinations of the Spinor direction fields will generate a Heat 1-form, Q, that does not satisfy the Frobenius integrability theorem, and therefore all such processes are thermodynamically irreversible: QˆdQ 6= 0. However, with the requirement that a2 is precisely the same as b2, then either piecewise linear process is reversible, for QˆdQ = 0. If the coefficients, and therefore the Spinor contributions, have slight fluctuations, the cancellation of the complex terms is not precise. Then either of the (now approximately) piecewise continuous process will NOT be reversible due to Spinor fluctuations. Remark 8 The facts that piecewise (sequential) C1 transitive evolution along a set of di- rection fields in odd (3) dimensions can be thermodynamically reversible, QˆdQ = 0, while (smooth) C2 evolution processes composed from complex Spinors can be thermodynamically irreversible, QˆdQ 6= 0, is a remarkable result which appears to have a relationship to Nash’s theorem on C1 embedding. Physically, the results are related to tangential discontinuities such as hydrodynamic wakes. For systems of Pfaff dimension 4, all of the eigendirection fields are Spinors. The Spinors occur as two conjugate pairs. If the conjugate variables are taken to be x,y and z,t then the z,t spinor pair can be interpreted in terms of a chirality of expansion or contraction, where the x,y pair can be interpreted as a chirality of polarization. In this sense it may be said that thermodynamic time irreversibility is an artifact of dimension 4. It is remarkable that a rotation and an expansion can be combined (eliminating the fixed point) to produce a thermodynamically reversible process. Ian Stewart points out that there are three types of manifold structure: piecewise linear, smooth, topological. Theorems on piecewise-linear manifolds may not be true on smooth manifolds. The work above seems to describe such an effect. Piecewise continuous processes are reversible, where smooth continuous processes are not (see page 106, [38])! 5 Epilogue: Topological Fluctuations and Spinors This Section 5 goes beyond the original objective of demonstrating that the Navier-Stokes equations, based upon continuous topological evolution, can describe the irreversible decay of turbulence, but not its creation. However, the key features of process irreversibility and turbulence are entwined with the concept of Topological Torsion and Spinors. Hence this epilogue calls attention to the fact that the Cartan topological methods permit the analysis of Spinor entanglement, as well as the analysis of fluctuations about kinematic perfection. This research area is in its infancy, and extends the thermodynamic approach to the realm of fiber bundles. A few of the introductory ideas are presented below. Remark 9 These concepts go beyond the scope of this essay which has the objective of presenting the important topological ideas in a manner palatable (if not recognizable) to the engineering community of hydrodynamics. 5.1 The Cartan-Hilbert Action 1-form To start, consider those physical systems that can be described by a function L(q,v,t) and a 1-form of Action given by Cartan-Hilbert format, A = L(qk,vk,t)dt + pk·(dqk − vkdt). (168) The classic Lagrange function, L(qk,vk,t)dt, is extended to include fluctuations in the kinematic variables, (dqk−vkdt) 6= 0. It is no longer assumed that the equation of Kinematic Perfection is satisfied. Fluctuations of the topological constraint of Kinematic Perfection are permitted; Topological Fluctuations in position: ∆q = (dqk − vkdt) 6= 0. (169) As the fluctuations are 1-forms, it is some interest to compute their Pfaff Topological Di- mension. The first step in the construction of the Pfaff Sequence is to compute the exterior differential of the fluctuation 1-form: Fluctuation 2-form: d(∆q) = −(dvk − akdt)ˆdt (170) = −∆vˆdt, (171) Topological Fluctuations in velocity:∆v =(dvk − akdt) 6= 0. (172) It is apparent that the Pfaff Topological Dimension of the fluctuations is at most 3, as ∆qˆ∆vˆdt 6= 0, and has a Heisenberg component, When dealing with fluctuations in this prologue, the geometric dimension of independent base variables will not be constrained to the 4 independent base variables of the Thermody- namic model. At first glance it appears that the domain of definition is a (3n+1)-dimensional variety of independent base variables, {pk,qk,vk,t}. Do not make the assumption that the pk are constrained to be canonically defined. Instead, consider pk to be a (set of) Lagrange multiplier(s) to be determined later. Also, do not assume at this stage that v is a kinematic velocity function, such that (dqk−vkdt) ⇒ 0. The classical idea is to assert that topological fluctuations in position are related to pressure, and topological fluctuations in velocity are related to temperature. For the given Action, construct the Pfaff Sequence (12) in order to determine the Pfaff dimension or class [8] of the Cartan-Hilbert 1-form of Action. The top Pfaffian is defined as the non-zero p-form of largest degree p in the sequence. The top Pfaffian for the Cartan- Hilbert Action is given by the formula, Top Pfaffian is 2n+2 (dA)n+1 = (n+ 1)!{Σnk=1(∂L/∂vk − pk)dvk}ˆΩ2n+1, (173) Ω2n+1 = dp1ˆ...dpnˆdq 1ˆ..dqnˆdt. (174) The formula is a bit surprising in that it indicates that the Pfaff Topological Dimension of the Cartan-Hilbert 1-form is 2n+2, and not the geometrical dimension 3n + 1. For n = 3 ”degrees of freedom”, the top Pfaffian indicates that the Pfaff Topological Dimension of the 2-form, dA is 2n + 2 = 8. The value 3n + 1 = 10 might be expected as the 1-form was defined initially on a space of 3n+ 1 ”independent” base variables. The implication is that there exists an irreducible number of independent variables equal to 2n + 2 = 8 which completely characterize the differential topology of the first order system described by the Cartan-Hilbert Action. It follows that the exact 2-form, dA, satisfies the equations (dA)n+1 6= 0, but Aˆ(dA)n+1 = 0. (175) Remark 10 The idea that the 2-form, dA, is a symplectic generator of even maximal rank, 2n+2, implies that ALL eigendirection fields of the 2-form, F = dA, are complex isotropic Spinors, and all processes on such domains have Spinor components. The format of the top Pfaffian requires that the bracketed factor in the expression above, {Σnk=1(∂L/∂vk − pk)dvk}, can be represented (to within a factor) by a perfect differential, dS = (n+ 1)!{Σnk=1(∂L/∂vk − pk)dvk}. (176) The result is also true for any closed addition γ added to A; e.g., the result is ”gauge invariant”. Addition of a closed 1-form does not change the Pfaff dimension from even to odd. On the other hand the result is not renormalizable, for multiplication of the Action 1-form by a function can change the algebraic Pfaff dimension from even to odd. On the 2n+2 domain, the components of (2n+1)-form T = Aˆ(dA)n generate what has been defined herein as the Topological Torsion vector, to within a factor equal to the Torsion Current. The coefficients of the (2n+1)-form are components of a contravariant vector density Tm defined as the Topological Torsion vector, the same concept as defined previously on a 4D thermodynamic domain, but now extended to (2n+2)-dimensions. This vector is orthogonal (transversal) to the 2n+2 components of the covector, Am. In other words, AˆT = Aˆ(Aˆ(dA)n) = 0 ⇒ i(T)(A) = TmAm = 0. (177) This result demonstrates that the extended Topological Torsion vector represents an adia- batic process. This topological result does not depend upon geometric ideas such as metric. It was demonstrated above that, on a space of 4 independent variables, evolution in the direction of the Topological Torsion vector is irreversible in a thermodynamic sense, subject to the symplectic condition of non-zero divergence, d(AˆdA) 6= 0. The same concept holds on dimension 2n+2. The 2n+2 symplectic domain so constructed can not be compact without boundary for it has a volume element which is exact. By Stokes theorem, if the boundary is empty, then the surface integral is zero, which would require that the volume element vanishes; but that is in contradiction to the assumption that the volume element is finite. For the 2n+2 domain to be symplectic, the top Pfaffian can never vanish. The domain is therefore orientable, but has two components, of opposite orientation. Examination of the constraint that the symplectic space be of dimension 2n+2 implies that the Lagrange multipliers, pk, cannot be used to define momenta in the classical ”conjugate or canonical” manner. Define the non-canonical components of the momentum, ℏkj, as, non-canonical momentum: ℏkj = (pj − ∂L/∂vj), (178) such that the top Pfaffian can be written as, (dA)n+1 = (n+ 1)!{Σnj=1ℏkjdvj}ˆΩ2n+1, (179) Ω2n+1 = dp1ˆ...dpnˆdq 1ˆ..dqnˆdt. (180) For the Cartan-Hilbert Action to be of Pfaff Topological Dimension 2n+2, the factor {Σnj=1ℏkjdvj} 6= 0. It is important to note, however, that as (dA)n+1 is a volume ele- ment of geometric dimension 2n+2, the 1-form Σnj=1ℏkjdv j is exact (to within a factor, say T (qk, t, pk,Sv)); hence, Σnj=1ℏkjdv j = TdSv. (181) Tentatively, this 1-form, dSv, will be defined as the Topological Entropy production relative to topological fluctuations of momentum, kinematic differential position and velocity. 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Moulden, Editors, The University of Tennessee Space Institute, Tullahoma, TN 37388 USA. [20] Kiehn, R. M. (1993), Instability Patterns, Wakes and Topological Limit Sets, in ”Eddy Structure Identification in Free Turbulent Shear Flows”, J.P.Bonnet and M.N. Glauser, (eds), Kluwer Academic Publishers, 363. [21] Kiehn, R. M. (2000), 2D turbulence is a Myth, (invited speaker EGS XXIV General Assembly IUTAM, the Hague, 1999 (http://www22.pair.com/csdc/pdf/hague6.pdf) [22] Kiehn, R.M. (2002), The Photon Spin and other Topological Features of Classical Elec- tromagnetism, in ”Gravitation and Cosmology, From the Hubble Radius to the Planck Scale”, Amoroso, R., et al., eds., Kluwer, Dordrecht, Netherlands, 197-206. Vigier 3 conference in 2000. (http://www22.pair.com/csdc/pdf/vig2000.pdf) http://www22.pair.com/csdc/pdf/periods.pdf http://www22.pair.com/csdc/pdf/falaco85o.pdf http://www22.pair.com/csdc/pdf/falaco97.pdf http://www22.pair.com/pdf/csdc/camb89.pdf http://www22.pair.com/csdc/pdf/hague6.pdf http://www22.pair.com/csdc/pdf/vig2000.pdf [23] Kiehn, R. M. (2003), Thermodynamic Irreversibility and the Arrow of Time, in ”The Nature of Time: Geometry, Physics and Perception”, R. Bucher et al. (eds.), Kluwer, Dordrecht, Netherlands, 243-250. (http://www22.pair.com/csdc/pdf/arwfinal.pdf) [24] Kiehn, R. M. (2005), Propagating topological singularities: the photon, in ”The Na- ture of Light: What Is a Photon?”; Chandrasekhar Roychoudhuri, Katherine Creath; Eds,Proc. SPIE 5866, 192-206. [25] Kiehn, R. M., ”Thermodynamics and quantum cosmology – Continuous topological evolution of topologically coherent defects”, arXiv:gr-qc/0603072 [26] Kiehn, R. M.,”A topological theory of the Physical Vacuum”, arXiv:gr-qc/0602118 [27] Kiehn, R. M., ”Instability patterns, wakes and topological limit sets”,arXiv:physics/0102005 [28] Kiehn, R. M., ”Topology and Turbulence”, arXiv:physics/0102003 [29] Kiehn, R. M., ”Some closed form solutions to the Navier Stokes equations”, rmkarXiv:physics/0102002 [30] Kiehn, R. M., ”Topological-Torsion and Topological-Spin as coherent structures in plas- mas:, rmkarXiv:physics/0102001 [31] Kiehn, R. M., ”Curvature and torsion of implicit hypersurfaces and the origin of charge”, rmkarXiv:gr-qc/0101109 [32] Kiehn, R. M., ”Chirality and helicity in terms of topological spin and topological tor- sion”, rmkarXiv:physics/0101101 [33] Kiehn, R. M., ”Falaco Solitons, Cosmic Strings in a Swimming Pool”, rmkarXiv:gr- qc/0101098 [34] Kiehn, R. M. and Baldwin, P. ”Cartan’s topological structure”, rmkarXiv:math- ph/0101033 [35] Kiehn, R. M., ”Continuous topological evolution”, rmkarXiv:math-ph/0101032 [36] Kiehn, R. M., ”Electromagnetic Waves in the Vacuum with Torsion and Spin”, rmkarXiv:physics/9802033 [37] W. Slebodzinsky, (1970), ”Exterior Forms and their Applications”, PWN, Warsaw. http://www22.pair.com/csdc/pdf/arwfinal.pdf http://arxiv.org/abs/gr-qc/0603072 http://arxiv.org/abs/gr-qc/0602118 http://arxiv.org/abs/physics/0102005 http://arxiv.org/abs/physics/0102003 [38] Stewart, I. (1988), The Problems of Mathematics, Oxford, NY.. [39] Uffink, J. arXiv.org/cond-mat/0005327 [40] Kiehn, R. M. (2007), Non-Equilibrium Thermodynamics, 2nd Edition, ”Non-Equilibrium Systems and Irreversible Processes Vol 1”, see see (http://www.lulu.com/kiehn). [41] Kiehn, R. M. (2007), Cosmology, Falaco Solitons and the Arrow of Time, 2nd Edition, ”Non-Equilibrium Systems and Irreversible Processes Vol 2”, see (http://www.lulu.com/kiehn). [42] Kiehn, R. M. (2007), Wakes, Coherent Structures and Turbulence, 2nd Edi- tion, ”Non-Equilibrium Systems and Irreversible Processes Vol 3”, s see (http://www.lulu.com/kiehn). [43] Kiehn, R. M. (2007), Plasmas and non-equilibrium Electrodynamics, 2nd Edition, ”Non-Equilibrium Systems and Irreversible Processes Vol 4”, see (http://www.lulu.com/kiehn). [44] Kiehn, R. M. (2007), Exterior Differential Forms and Differential Topol- ogy, 1st Edition ”Topological Torsion and Macroscopic Spinors Vol 5”, see (http://www.lulu.com/kiehn). [45] Kiehn, R. M. (2004), Selected Publications ”Non-Equilibrium Systems and Irreversible Processes Vol 7”, see (http://www.lulu.com/kiehn). [46] Wesson, P. S. (2000) ”Space Time Matter - Modern Kaluza Klein theory”, World Sci- entific, Singapore. http://www.lulu.com/kiehn http://www.lulu.com/kiehn http://www.lulu.com/kiehn http://www.lulu.com/kiehn http://www.lulu.com/kiehn http://www.lulu.com/kiehn Topological Thermodynamics The Axioms of Topological Thermodynamics Cartan's Magic Formula First Law of Thermodynamics The Pfaff Sequence and the Pfaff Topological Dimension The Pfaff Topological Dimension of the System 1-form, A The Pfaff Topological Dimension of the Thermodynamic Work 1-form, W Topological Torsion and other Continuous Processes. Reversible Processes Irreversible Processes The Spinor class Emergent Topological Defects Applications An Electromagnetic format Topological 3-forms and 4-forms in EM format Topological Torsion quanta A Hydrodynamic format The Topological Continuum vs. the Geometrical Continuum Topological Hydrodynamics Classical Hydrodynamic Theory The Lagrange-Hilbert Action Euler flows and Hamiltonian fluids The Navier-Stokes fluid The classic Navier-Stokes equations The Navier-Stokes equations embedded in a non-equilibrium thermodynamic system The Topological Torsion process for the Navier-Stokes fluid Closed States of Topological Coherence embedded as deformable defects in Turbulent Domains Examples of PTD = 3 domains and their Emergence Piecewise Linear Vector Processes vs. C2 Spinor processes The Vector Processes The Spinor Processes Irreversible Spinor processes Epilogue: Topological Fluctuations and Spinors The Cartan-Hilbert Action 1-form

The concept of continuous topological evolution, based upon Cartan's methods of exterior differential systems, is used to develop a topological theory of non-equilibrium thermodynamics, within which there exist processes that exhibit continuous topological change and thermodynamic irreversibility. The technique furnishes a universal, topological foundation for the partial differential equations of hydrodynamics and electrodynamics; the technique does not depend upon a metric, connection or a variational principle. Certain topological classes of solutions to the Navier-Stokes equations are shown to be equivalent to thermodynamically irreversible processes.

Introduction to Mechanics and Symmetry”, Springer-Verlag, 122. [11] Kiehn, R.M. and Pierce, J. F. (1969), An Intrinsic Transport Theorem, Phys. Fluids, 12, #9, 1971. (http://www22.pair.com/csdc/pdf/inttrans.pdf) [12] Kiehn, R.M. (1974), Extensions of Hamilton’s Principle to Include Dissipative Systems, J. Math Phys. 15, 9. [13] Kiehn, R.M. (1975), Intrinsic hydrodynamics with applications to space-time fluids, Int. J. of Eng. Sci. 13, 941. (http://www22.pair.com/csdc/pdf/inthydro.pdf) arXiv/math- ph/0101032 [14] Kiehn, R.M. (1976), Retrodictive Determinism, Int. J. of Eng. Sci. 14, 749. (http://www22.pair.com/csdc/pdf/retrodic.pdf) http://www22.pair.com/csdc/pdf/inttrans.pdf http://www22.pair.com/csdc/pdf/inthydro.pdf http://www22.pair.com/csdc/pdf/retrodic.pdf [15] Kiehn, R.M. (1977), Periods on manifolds, quantization and gauge, J. Math. Phy. 18, 614. (http://www22.pair.com/csdc/pdf/periods.pdf) [16] Kiehn, R. M., (1987), The Falaco Effect as a topological defect was first noticed by the present author in the swimming pool of an old MIT friend, during a visit in Rio de Janeiro, at the time of Halley’s comet, March 1986. The concept was presented at the Austin Meeting of Dynamic Days in Austin, January 1987, and caused some interest among the resident topologists. The easily reproduced ex- periment added to the credence of topological defects in fluids. It is now perceived that this topological phenomena is universal, and will appear at all levels from the microscopic to the galactic. (http://www22.pair.com/csdc/pdf/falaco85o.pdf), arXiv.org/gr-qc/0101098 (http://www22.pair.com/csdc/pdf/falaco97.pdf and (http://www22.pair.com/csdc/pdf/topturb.pdf) [17] Kiehn, R.M. (1990), Topological Torsion, Pfaff Dimension and Coherent Structures, in: ”Topological Fluid Mechanics”, H. K. Moffatt and T. S. Tsinober eds, Cambridge University Press, 449-458. (http://www22.pair.com/pdf/csdc/camb89.pdf) [18] Kiehn, R. M. (1991), Compact Dissipative Flow Structures with Topological Coherence Embedded in Eulerian Environments, in: ”Non-linear Dynamics of Structures”, edited by R.Z. Sagdeev, U. Frisch, F. Hussain, S. S. Moiseev and N. S. Erokhin, 139-164, World Scientific Press, Singapore. [19] Kiehn, R. M. (1992), Topological Defects, Coherent Structures and Turbulence in Terms of Cartan’s Theory of Differential Topology, in ”Developments in Theoretical and Ap- plied Mathematics, Proceedings of the SECTAM XVI conference”, B. N. Antar, R. Engels, A.A. Prinaris and T. H. Moulden, Editors, The University of Tennessee Space Institute, Tullahoma, TN 37388 USA. [20] Kiehn, R. M. (1993), Instability Patterns, Wakes and Topological Limit Sets, in ”Eddy Structure Identification in Free Turbulent Shear Flows”, J.P.Bonnet and M.N. Glauser, (eds), Kluwer Academic Publishers, 363. [21] Kiehn, R. M. (2000), 2D turbulence is a Myth, (invited speaker EGS XXIV General Assembly IUTAM, the Hague, 1999 (http://www22.pair.com/csdc/pdf/hague6.pdf) [22] Kiehn, R.M. (2002), The Photon Spin and other Topological Features of Classical Elec- tromagnetism, in ”Gravitation and Cosmology, From the Hubble Radius to the Planck Scale”, Amoroso, R., et al., eds., Kluwer, Dordrecht, Netherlands, 197-206. Vigier 3 conference in 2000. (http://www22.pair.com/csdc/pdf/vig2000.pdf) http://www22.pair.com/csdc/pdf/periods.pdf http://www22.pair.com/csdc/pdf/falaco85o.pdf http://www22.pair.com/csdc/pdf/falaco97.pdf http://www22.pair.com/pdf/csdc/camb89.pdf http://www22.pair.com/csdc/pdf/hague6.pdf http://www22.pair.com/csdc/pdf/vig2000.pdf [23] Kiehn, R. M. (2003), Thermodynamic Irreversibility and the Arrow of Time, in ”The Nature of Time: Geometry, Physics and Perception”, R. Bucher et al. (eds.), Kluwer, Dordrecht, Netherlands, 243-250. (http://www22.pair.com/csdc/pdf/arwfinal.pdf) [24] Kiehn, R. M. (2005), Propagating topological singularities: the photon, in ”The Na- ture of Light: What Is a Photon?”; Chandrasekhar Roychoudhuri, Katherine Creath; Eds,Proc. SPIE 5866, 192-206. [25] Kiehn, R. M., ”Thermodynamics and quantum cosmology – Continuous topological evolution of topologically coherent defects”, arXiv:gr-qc/0603072 [26] Kiehn, R. M.,”A topological theory of the Physical Vacuum”, arXiv:gr-qc/0602118 [27] Kiehn, R. M., ”Instability patterns, wakes and topological limit sets”,arXiv:physics/0102005 [28] Kiehn, R. M., ”Topology and Turbulence”, arXiv:physics/0102003 [29] Kiehn, R. M., ”Some closed form solutions to the Navier Stokes equations”, rmkarXiv:physics/0102002 [30] Kiehn, R. M., ”Topological-Torsion and Topological-Spin as coherent structures in plas- mas:, rmkarXiv:physics/0102001 [31] Kiehn, R. M., ”Curvature and torsion of implicit hypersurfaces and the origin of charge”, rmkarXiv:gr-qc/0101109 [32] Kiehn, R. M., ”Chirality and helicity in terms of topological spin and topological tor- sion”, rmkarXiv:physics/0101101 [33] Kiehn, R. M., ”Falaco Solitons, Cosmic Strings in a Swimming Pool”, rmkarXiv:gr- qc/0101098 [34] Kiehn, R. M. and Baldwin, P. ”Cartan’s topological structure”, rmkarXiv:math- ph/0101033 [35] Kiehn, R. M., ”Continuous topological evolution”, rmkarXiv:math-ph/0101032 [36] Kiehn, R. M., ”Electromagnetic Waves in the Vacuum with Torsion and Spin”, rmkarXiv:physics/9802033 [37] W. Slebodzinsky, (1970), ”Exterior Forms and their Applications”, PWN, Warsaw. http://www22.pair.com/csdc/pdf/arwfinal.pdf http://arxiv.org/abs/gr-qc/0603072 http://arxiv.org/abs/gr-qc/0602118 http://arxiv.org/abs/physics/0102005 http://arxiv.org/abs/physics/0102003 [38] Stewart, I. (1988), The Problems of Mathematics, Oxford, NY.. [39] Uffink, J. arXiv.org/cond-mat/0005327 [40] Kiehn, R. M. (2007), Non-Equilibrium Thermodynamics, 2nd Edition, ”Non-Equilibrium Systems and Irreversible Processes Vol 1”, see see (http://www.lulu.com/kiehn). [41] Kiehn, R. M. (2007), Cosmology, Falaco Solitons and the Arrow of Time, 2nd Edition, ”Non-Equilibrium Systems and Irreversible Processes Vol 2”, see (http://www.lulu.com/kiehn). [42] Kiehn, R. M. (2007), Wakes, Coherent Structures and Turbulence, 2nd Edi- tion, ”Non-Equilibrium Systems and Irreversible Processes Vol 3”, s see (http://www.lulu.com/kiehn). [43] Kiehn, R. M. (2007), Plasmas and non-equilibrium Electrodynamics, 2nd Edition, ”Non-Equilibrium Systems and Irreversible Processes Vol 4”, see (http://www.lulu.com/kiehn). [44] Kiehn, R. M. (2007), Exterior Differential Forms and Differential Topol- ogy, 1st Edition ”Topological Torsion and Macroscopic Spinors Vol 5”, see (http://www.lulu.com/kiehn). [45] Kiehn, R. M. (2004), Selected Publications ”Non-Equilibrium Systems and Irreversible Processes Vol 7”, see (http://www.lulu.com/kiehn). [46] Wesson, P. S. (2000) ”Space Time Matter - Modern Kaluza Klein theory”, World Sci- entific, Singapore. http://www.lulu.com/kiehn http://www.lulu.com/kiehn http://www.lulu.com/kiehn http://www.lulu.com/kiehn http://www.lulu.com/kiehn http://www.lulu.com/kiehn Topological Thermodynamics The Axioms of Topological Thermodynamics Cartan's Magic Formula First Law of Thermodynamics The Pfaff Sequence and the Pfaff Topological Dimension The Pfaff Topological Dimension of the System 1-form, A The Pfaff Topological Dimension of the Thermodynamic Work 1-form, W Topological Torsion and other Continuous Processes. Reversible Processes Irreversible Processes The Spinor class Emergent Topological Defects Applications An Electromagnetic format Topological 3-forms and 4-forms in EM format Topological Torsion quanta A Hydrodynamic format The Topological Continuum vs. the Geometrical Continuum Topological Hydrodynamics Classical Hydrodynamic Theory The Lagrange-Hilbert Action Euler flows and Hamiltonian fluids The Navier-Stokes fluid The classic Navier-Stokes equations The Navier-Stokes equations embedded in a non-equilibrium thermodynamic system The Topological Torsion process for the Navier-Stokes fluid Closed States of Topological Coherence embedded as deformable defects in Turbulent Domains Examples of PTD = 3 domains and their Emergence Piecewise Linear Vector Processes vs. C2 Spinor processes The Vector Processes The Spinor Processes Irreversible Spinor processes Epilogue: Topological Fluctuations and Spinors The Cartan-Hilbert Action 1-form

Numerical estimation of critical parameters using the Bond entropy Rafael A. Molina1 and Peter Schmitteckert2 Instituto de Estructura de la Materia - CSIC, Serrano 123, 28006, Madrid, Spain Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany Using a model of spinless fermions in a lattice with nearest neighbor and next-nearest neighbor interaction we show that the entropy of the reduced two site density matrix (the bond entropy) can be used as an extremely accurate and easy to calculate numerical indicator for the critical parameters of the quantum phase transition when the basic ordering pattern has a two-site periodicity. The actual behavior of the bond entropy depends on the particular characteristics of the transition under study. For the Kosterlitz-Thouless type phase transition from a Luttinger liquid phase to a charge density wave state the bond entropy has a local maximum while in the transition from the Luttinger liquid to the phase separated state the derivative of the bond entropy has a divergence due to the cancelation of the third eigenvalue of the two-site reduced density matrix. PACS numbers: 03.67.Mn,75.10.Jm Keywords: I. INTRODUCTION A Quantum Phase Transitions (QPT) is a qualitative change in the ground state of a quantum system as some parameter is varied [1, 2]. Contrary to classical phase transitions, QPTs occur at zero temperature and are due to the effect of quantum fluctuations and not of ther- mal fluctuations. The previous definition is very gen- eral, however, the abrupt change in the structure of the ground state that define the phase transition can have different consequences depending on the different cases. The ground state energy may become non-analytic when approaching the critical parameter. The energy gap be- tween the ground state and the first excited state may go to zero in the critical point. The correlations at the crit- ical point may decay as power laws instead of exponen- tially indicating a diverging correlation length. It is pos- sible to find quantum systems which have some of these indications of the QPT and not others [3, 4]. For this rea- son, alternative ways for the classification of QPTs and for the numerical investigation of the critical parameters in a QPT can be very helpful. In recent years quantum information concepts have started to be applied to the study of QPTs. One cen- tral concept in quantum information theory is the con- cept of entanglement [5]. Two quantum systems in a pure state are entangled if their state cannot be writ- ten as the product of two separate pure states for each of the quantum systems. Entanglement measures quan- tum correlations and as correlations are typically maxi- mum at the critical points of QPTs it was realized that some entanglement measures may have a singularity or a maximum at the critical point. The amount of en- tanglement has been shown to be a very sensitive quan- tity to the value of the critical parameter governing the phase transition [6, 7, 8, 9, 10]. In particular, concur- rence [11] has been used to investigate spin models and this quantity shows an extreme or singular behavior at the corresponding critical points [7]. The block-block entanglement between two parts of the system has also been used, establishing connections with conformal field theory [8, 12, 13]. In a recent work Gu et al. analyzed the local en- tanglement and its relationship with phase transitions in the one-dimensional and two-dimensional XXZ spin models [14]. The local entanglement was measured with the Von-Neumann entropy of the two-site density matrix. It is more convenient numerically than the block-block entanglement as the size of the density matrix needed for the latter quantity grows exponentially with the size of the block. Using Bethe ansatz results for the one- dimensional XXZ model Gu and coworkers showed that the local information obtained from the entanglement en- tropy of the two-site density matrix is enough to study the phase transitions that occur in this model. Consider- ing blocks larger than the characteristic length scales of the system (that in the critical point diverge) was shown to be unnecessary as the entanglement between a block of two spins and the rest of the system is sufficient to re- veal the most important information about the system. Their results hint to the possibility of using these prop- erties for the numerical study of phase transitions with small systems. It is the purpose of this work to study numerically the local entanglement as a function of the size of the system. In particular, we will concentrate in the crit- ical points of the phase transitions. We will study a one-dimensional model of spinless fermions with nearest- neighbor interactions that can be transformed through the Jordan-Wigner transformation to the XXZ model with spins at each site Sj = 1/2 [15]. For repulsive inter- action V1 = 2.0 the ground state performs a Kosterlitz- Thouless type phase transition from a Luttinger Liquid to a charge density wave, which corresponds to an antifer- romagnetically ordered state in the spin picture. For at- tractive interaction V1 = −2.0 there is a phase transition to a phase separated state, which corresponds to a ferro- magnet in the spin picture. We will show that the bond http://arxiv.org/abs/0704.1597v1 entropy Sbond enables us to determine the critical points of both quantum phase transitions with an astonishing accuracy, albeit they present very different characteris- tics and symmetries. We will show how these differences are reflected in the behavior of the bond entropy. Finally we include next-nearest-neighbor interaction to test the generality of our finding. The effect of longer ranged in- teraction was studied before in the context of multiple umklapp scattering [16]. In this case the interaction can now lead to phase transitions at fillings different from 1/2 and to ordering patterns with increased unit cell. II. BOND-ENTROPY FOR THE NEAREST-NEIGHBOR INTERACTION MODEL We shall consider the one-dimensional spinless fermions model with next-neighbor interactions. Ĥ = −t i ĉi−1 + ĉ i−1ĉi V1(n̂i− )(n̂i−1− where the operators appearing in the formula are the usual fermionic creation, annihilation, and number oper- ator at site i, L is the total number of sites, t is the hop- ping matrix element between neighboring sites, and V1 is the nearest-neighbor interaction strength. An important property of this model is that it can be transformed into the XXZ spin S = 1/2 model through the Jordan-Wigner transformation [15, 17]. Ŝ−j = exp cj , (2) Ŝzj = n̂j − 1/2. (3) For an even number of particles a phase term appears in the boundary condtion when we apply the Jordan- Wigner transformation to Hamiltonian (1). As we are not interested in this even-odd effect we will consider periodic (anti-periodic) boundary conditions, c0 ≡ cM (c0 ≡ −cM ), for N odd (even) and both models will be equivalent in our examples. Although we will mainly consider the spinless fermion model we will make com- ments regarding the equivalent behavior of both models when we believe it will be useful to clarify some situation (specially in the “ferromagnetic” phase). The Hamilto- nian (1) commutes with the total N̂ = n̂i operator so the total number of particles is a conserved quantity and we will consider subspaces with a definite number of fermions, equivalent to consider subspaces with a definite value of Sz in the XXZ model. This model has an interesting phase diagram depend- ing on the value of the ratio of the interaction parameter and the hopping term V1/t and also on the number of particles N . Without loss of generality we can consider t = 1 and consider the phases as we change V1. For V1 < −2 the equivalent spin system is ferromagnetic and the ground state is fully spin polarized. When we cross the first transition point Vca = −2 the ground state of the system can be shown to be non-degenerate and with spin S = 0 [18]. Only in the half-filled case, N = L/2, there is another transition point at Vcb = 2. For V1 > 2 the system is a charge density wave type insulator, the transition is of the Kosterlitz-Thouless type and the or- der parameter depends exponentially on the difference V − Vcb making an accurate numerical determination of the transition point notoriously hard. We will define the bond-entropy as the Von Neumann entropy of the reduced density matrix of two-neighboring sites ρ̂ii+1. As a result of the conservation of N the re- duced density matrix can be written as a 4 × 4 matrix with three sectors of N = 0, N = 1, and N = 2. In the two-site basis |00〉, |01〉, |10〉, |11〉 it can be represented ρ̂ii+1 = u− 0 0 0 0 ω z 0 0 z∗ ω 0 0 0 0 u+ . (4) For this particular model using its invariance under translations it can be shown that we can write this matrix elements in terms of certain correlation functions [19, 20], u− = 1+ 〈n̂in̂i+1〉 − 〈n̂i〉 − 〈n̂i+1〉 . (5) u+ = 〈n̂in̂i+1〉+ 〈n̂i〉 − 〈n̂i+1〉 . (6) 〈n̂i〉+ 〈n̂i+1〉 − 〈n̂in̂i+1〉 . (7) i ĉi+1 i+1ĉi . (8) We define the bond entropy Si as the Von Neu- mann entropy of the two-site density matrix ρ̂i i+1. In the model under study, the invariance under translations also implies that Si does not depend on the site i. Sbond = − λj lnλj , (9) where λj are the four eigenvalues of the reduced two-site density matrix ρ̂i i+1. III. NUMERICAL RESULTS In this section we will show the numerical results for the bond entropy as a function of the size of the system using the DMRG algorithm [21]. In Figure 1 we show the results of Sbond at half fill- ing for different number of sites L. The behavior of the -2 -1 0 1 2 3 4 5 L = 10 N = 5 L = 14 N = 7 L = 18 N = 9 L = 22 N = 11 L = 26 N = 13 L = 30 N = 15 FIG. 1: (Color online) Bond entropy Sbond as a function of the interaction for different sizes of the spinless fermions ring L at half filling N = L/2. The results for L = 26 and L = 30 are hardly distinguishable. We have a maximum of Sbond at V1 = 2 with extremely good approximation, see next figure, that marks the CDW insulator transition. The slope of the entropy diverges at V1 = −2 marking the appearance of the ferromagnetic transition. bond entropy around the two critical points is very dif- ferent, reflecting the different changes in the symmetries and correlations of the ground state. The slope of Sbond diverges around Vca = −2 (we will explain that in more detail in the next section), while Sbond is continuous but has a local maximum in the proximity of the second crit- ical point Vcb = 2. In both cases one can understand the behavior of Sbond from the behavior of the correlation functions in the different phases [14]. More importantly, one can estimate with extraordinary precision the value of the critical parameter from very small system sizes even in the case of the transition to the CDW phase. In addition, Sbond has a minimum for V1 = 0. In Figure 2 we see a zoom of the last figure 1 for one particular case with parameter L = 30, N = 15 in the region around V1 = 2. When we calculate numerically the position of the maximum with DMRG we obtain the value of V1 with a precision better than 10 −5. Of course, in order to take full advantage of these proper- ties of Sbond we need a very accurate algorithm such as DMRG. We used at least 500 states per block for the L ≤ 30 sites leading to an discarded entropy below 10−9, and 1400 states per block for the 36 site system (see be- low), including the five lowest lying states in order to treat the degeneracies correctly, leading to a discarded entropy typically below 10−10, and up to 2 · 10−8 close to the phase transitions and applied always eleven finite lattice sweeps. Close to the CDW-I – Luttinger liquid transition (see below) we used at least 2050 states per block and only two low lying states to check our results. This resulted in a discarded entropy below 10−12. We’d like to note that despite the large number of states the DMRG runs are much cheaper as compared to the cal- culations in [16], since no resolvent has to be computed, e.g. the largest run took about a hundred CPU minutes. 0.9519 0.951902 0.951904 0.951906 0.951908 0.95191 0.951912 0.951914 1.95 1.96 1.97 1.98 1.99 2 2.01 2.02 2.03 2.04 2.05 FIG. 2: (Color online) Behavior of Sbond as a function of V1 in the neighborhood of the phase transition at V1 = 2 in the case L = 30, N = 15. We can numerically pinpoint the maximum of the curve at V1 = 2 with a precision better than 10 The function f(x) is the second order polynomial fit used to obtain the maximum of the curve. -2 0 2 4 L = 26 N = 13 L = 26 N = 9 L = 22 N = 9 FIG. 3: (Color online) Behavior of Sbond as a function of V for L = 26, N = 13 (half-filling) and L = 26, N = 9, and L = 22, N = 9 (outside half-filling). We observe the same behavior in the ferromagnetic transition around Vca = −2 but a complete different one in the transition around Vcb = 2. In Figure 3 we see some examples comparing results at half-filling and outside half-filling. The qualitative be- havior of the bond entropy is exactly the same around the first critical point Vca = −2, but the maximum of the bond entropy around Vcb = 2 disappears as soon as we move outside half-filling reflecting the absence of the CDW transition for fillings different from 1/2. IV. FERROMAGNETISM AND THE TWO-SITE DENSITY MATRIX As we have mentioned before the slope of Sbond di- verges at V1 = Vca. In Fig. 4 we show the results for the value of the third eigenvalue of the two-site density matrix λ3 for different sizes. We can reach a very high numerical precision in the determination of the ferromag- -0.0001 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 -2.05 -2.04 -2.03 -2.02 -2.01 -2 -1.99 -1.98 -1.97 -1.96 -1.95 FIG. 4: (Color online) Behavior of the third eigenvalue of the two-site density matrix λ3 as a function of V in the neighbor- hood of the phase transition at V1 = −2 for different sizes at half filling. λ3 vanishes at V1 = −2 with very high precision. netic critical point studying the cancellation of the third eigenvalue of the two-site density matrix. In the thermodynamic limit at V1 = −2, ω = z = 1/4 in Eq. (4). One can immediately see that the equality of both matrix elements implies that one of the eigenvalues of the density matrix is zero, which leads to the singular- ity of Sbond. If one tries a direct numerical examination of the values of the correlation functions independently one does not get a very accurate estimation of the critical parameter. However, the examination of the particular combination appearing in the Von Neumann entropy of the two-site density matrix allows a very accurate calcu- lation even with very small system sizes due to the fact that although ω and z converge slowly to the thermo- dynamic limit value 1/4, their difference converges very quickly to zero at the critical value of the interaction. V. NEXT NEAREST-NEIGHBOR INTERACTION MODEL In order to test the generality of our conclusions and to obtain a critical parameter of a phase transition in a model not solvable with Bethe ansatz we add next- nearest neighbors interaction, Ĥ = −t i ĉi−1 + ĉ i−1ĉi n̂i − n̂i−1 − n̂i − n̂i−2 − , (10) where V2 is the strength of the interaction between sites separated by two lattice spacings. This Hamiltonian has been used to study the physics of materials that exhibit 0 0,25 0,5 0,75 1 0,951 0,952 0,953 FIG. 5: Bond entropy as a function of V2 in the line V2 = 5− 2V1 for L = 36 and N = 18. The maximum can be used to estimate the critical point at V2,c = 0.280 with very high precision. multiple phase transitions. Usually one considers V2 < V1 as one expects the interaction to reduce with distance. However, there can be exceptions if the nearest-neighbor interaction is supressed by the lattice geometry. The phase diagram of the model represented by the Hamiltonian (10) has been studied as a function of V1 and V2 by Schmitteckert and Werner [16]. In this paper the authors used DMRG to calculate the ground state curva- ture. The phase diagram depends on the filling, commen- surability effects are extremely important due to the mul- tiple umklapp scattering. If we concentrate on half-filling and repulsive interactions we have a charge density wave (CDW) phase in which the ground state is twofold de- generate with ordering pattern (•◦•◦) and (◦•◦•). Here ◦ denotes a vacant and • denotes an occupied site. In phase CDW II the ground state is fourfold degenerate with or- dering pattern (••◦◦), (◦••◦), (◦◦••), and (•◦◦•). We will follow reference [16] and study the critical parameters along the line V2 = 5 − 2V . For example, studying sys- tems of sizes up to L = 60 they obtained a critical point for the transition between the CDW I phase and the Lut- tinger Liquid phase as (V1,c, V2,c) = (2.4±0.05, 0.2±0.1)}. In Fig. (5) we show results for the ground state bond entropy as a function of V2 for along the previous men- tioned line for L = 36 and N = 18. Even from the small size used we can accurately determine the critical V2 as V2,c = 0.280, which is within the error bars previously given by Schmitteckert and Werner [16]. The determina- tion of the critical parameter for the transition between CDW I and the Luttinger liquid is done with much less numerical work as compared to the finite size analysis of excitation gaps and the ground state curvate in [16]. Notably, the finite size corrections are smaller, e.g. the results of the same analysis with L = 18 already gives a critical parameter of V2,c = 0.277. In Fig. 6 we show the numerical results for the value of the next-neighbor inter- action V2 in which we have a local maximum of Sbond ( along the same line as before) as a function of the inverse 0 0,025 0,05 0,075 0,275 0 0,04 0,08 0,945 FIG. 6: Finite size scaling for the position of the maximum of the bond entropy V2,c as a function of 1/L. The continuous line is a second order polynomial fit. It is used to extrapolate the value for L = ∞. In the inset we show the value of the local maximum of Sbond as a function of 1/L, a second order polynomial also fits very well the numerical results. of the total length of the system 1/L. We have calculated numerically the bond entropy at each size with an inter- val of 0.001 in V2 in the region around the maximum of Sbond, except in the case of L = 36 where we have used an interval of = 0.0002. The values of V2 in the maxi- mum where obtained through a second-order polynomial fit of the numerical results for Sbond. The actual value of the interval used was not very critical as the fits were very good. With another second-order polynomial fit we can extrapolate the calculated values to obtain the result for the thermodynamic limit V2,c = 0.2814± 0.0001. In the inset of the figure we can see the numerical results used in the extrapolation of the value of the maximum of the bond entropy in the critical point. The extrapolated value being Smax = 0.95385± 0.00001. In this case the bond entropy and the two-site den- sity matrix does not give very useful information about the phase transitions to charge density wave phases with ordering patterns with basic sizes bigger than two. We obtain no clear signature in the bond entropy for the quantum phase transition to CDW II. One may have to study entropies of density matrices of blocks with at least the size of the basic ordered block of the phases we are looking at. Also, one could try to study the bond entropy for the excited states. The numerical determination of critical parameters in this case is out of the scope of this study of the bond entropy and will be subject of future work. We note that in the case of CDW II the ground state is fourfold degenerate in the thermodynamic limit. However, the degeneracy of the two lowest lying states is lifted by finite size effects. VI. CONCLUSIONS The bond entropy defined as the Von Neumann en- tropy of the two-site density matrix can be a very effec- tive tool for the study of phase transitions and critical parameters. Its behavior depends on the correlations in the ground state of the system. We have studied the bond entropy for a model of spin- less fermions with nearest-neighbor interactions and pe- riodic boundary conditions. The size dependence of its behavior near the two critical points in the model has been studied in detail, showing an amazing precision in the estimation of the critical parameter. We have also studied a model with next-nearest neighbor interactions. We could determine the critical point of the phase tran- sition from the Luttinger liquid to the CDW I state with an ordering pattern of period two. If the fundamental block contains only two sites we show that the bond en- tropy displays a clear signature of the quantum phase transitons and allows for the determination of the crit- ical parameters. The bond entropy of the ground state could not be used for the transition to CDW II with or- dering pattern of period four. In this case we may have to turn to a block entropy of higher size. In general, we can say that the bond entropy can be used as a numerical indicator for phase transitions but the actual behavior of the bond entropy is not universal and will depend on the QPT under study. Our results should open the way to the numerical study of phase transitions with small sized systems. Acknowledgments RAM wishes to acknowledge useful discussions with J. Dukelsky. He also acknowledges finantial support at the Instituto de Estructura de la Materia-CSIC by an I3P contract funded by the European Social Fund. This work is supported in part by Spanish Government Grant FIS2006-12783-C03-01. [1] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999). [2] S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Rev. Mod. Phys. 69, 315 (1997). [3] F.D.M. Haldane, Phys. Lett. 93A, 464 (1983); F. Ver- straete, M. Popp, J.I. Cirac, Phys. Rev. Lett. 92, 027901 (2004). [4] F. Verstraete, M.A. Mart́ın Delgado, J.I. Cirac, Phys. Rev. Lett. 92, 087201 (2004). [5] F. Mintert, A. R. R. Carvalho, M. Kuś, A. Buchleitner, Phys. Rep. 415, 207 (2005). [6] T. J. Osborne, M. A. Nielsen, Phys. Rev. A 66, 032110 (2002). [7] A. Osterloh, L. Amico, G. Falci, R. Fazio, Nature 416, 608 (2002). [8] G. Vidal, J.I. Latorre, E. Rico, A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003). [9] J. Vidal, G. Palacios, R. Mosseri, Phys. Rev. A 69, 022107 (2004). [10] J. Vidal, R. Mosseri, J. Dukelsky, Phys. Rev. A 69, 054101 (2004). [11] S. Hill, W. K.Wootters, Phys. Rev. Lett. 78, 5022 (1997); W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [12] V. E. Korepin, Phys. Rev. Lett. 92, 096402 (2004). [13] J.P. Keating, F. Mezzadri, Phys. Rev. Lett. 94, 050501 (2005). [14] Shi-Jian Gu, Guang-Shan Tian, Hai-Qing Lin, New J. Phys. 8, 61 (2006). [15] P. Jordan, E. Wigner, Z. Phys. 47, 631 (1928). [16] P. Schmitteckert, R. Werner, Phys. Rev. B 69, 195115 (2004). [17] N. Nagaosa, Quantum Field Theory in Strongly Corre- lated Electronic Systems (Springer-Verlag, Berlin, 1999). [18] E. Lieb, D. Mattis, J. Math. Phys. 3, 749 (1962). [19] X. Wang, P. Zanardi, Phys. Lett. A 301, 1 (2002). [20] S. J. Gu, G. S. Tian, H. Q. Lin, Phys. Rev. A 71, 052322 (2005). [21] S. R. White, Phys. Rev. Lett. 69, 2863 (1992).

Using a model of spinless fermions in a lattice with nearest neighbor and next-nearest neighbor interaction we show that the entropy of the reduced two site density matrix (the bond entropy) can be used as an extremely accurate and easy to calculate numerical indicator for the critical parameters of the quantum phase transition when the basic ordering pattern has a two-site periodicity. The actual behavior of the bond entropy depends on the particular characteristics of the transition under study. For the Kosterlitz-Thouless type phase transition from a Luttinger liquid phase to a charge density wave state the bond entropy has a local maximum while in the transition from the Luttinger liquid to the phase separated state the derivative of the bond entropy has a divergence due to the cancelation of the third eigenvalue of the two-site reduced density matrix.

Numerical estimation of critical parameters using the Bond entropy Rafael A. Molina1 and Peter Schmitteckert2 Instituto de Estructura de la Materia - CSIC, Serrano 123, 28006, Madrid, Spain Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany Using a model of spinless fermions in a lattice with nearest neighbor and next-nearest neighbor interaction we show that the entropy of the reduced two site density matrix (the bond entropy) can be used as an extremely accurate and easy to calculate numerical indicator for the critical parameters of the quantum phase transition when the basic ordering pattern has a two-site periodicity. The actual behavior of the bond entropy depends on the particular characteristics of the transition under study. For the Kosterlitz-Thouless type phase transition from a Luttinger liquid phase to a charge density wave state the bond entropy has a local maximum while in the transition from the Luttinger liquid to the phase separated state the derivative of the bond entropy has a divergence due to the cancelation of the third eigenvalue of the two-site reduced density matrix. PACS numbers: 03.67.Mn,75.10.Jm Keywords: I. INTRODUCTION A Quantum Phase Transitions (QPT) is a qualitative change in the ground state of a quantum system as some parameter is varied [1, 2]. Contrary to classical phase transitions, QPTs occur at zero temperature and are due to the effect of quantum fluctuations and not of ther- mal fluctuations. The previous definition is very gen- eral, however, the abrupt change in the structure of the ground state that define the phase transition can have different consequences depending on the different cases. The ground state energy may become non-analytic when approaching the critical parameter. The energy gap be- tween the ground state and the first excited state may go to zero in the critical point. The correlations at the crit- ical point may decay as power laws instead of exponen- tially indicating a diverging correlation length. It is pos- sible to find quantum systems which have some of these indications of the QPT and not others [3, 4]. For this rea- son, alternative ways for the classification of QPTs and for the numerical investigation of the critical parameters in a QPT can be very helpful. In recent years quantum information concepts have started to be applied to the study of QPTs. One cen- tral concept in quantum information theory is the con- cept of entanglement [5]. Two quantum systems in a pure state are entangled if their state cannot be writ- ten as the product of two separate pure states for each of the quantum systems. Entanglement measures quan- tum correlations and as correlations are typically maxi- mum at the critical points of QPTs it was realized that some entanglement measures may have a singularity or a maximum at the critical point. The amount of en- tanglement has been shown to be a very sensitive quan- tity to the value of the critical parameter governing the phase transition [6, 7, 8, 9, 10]. In particular, concur- rence [11] has been used to investigate spin models and this quantity shows an extreme or singular behavior at the corresponding critical points [7]. The block-block entanglement between two parts of the system has also been used, establishing connections with conformal field theory [8, 12, 13]. In a recent work Gu et al. analyzed the local en- tanglement and its relationship with phase transitions in the one-dimensional and two-dimensional XXZ spin models [14]. The local entanglement was measured with the Von-Neumann entropy of the two-site density matrix. It is more convenient numerically than the block-block entanglement as the size of the density matrix needed for the latter quantity grows exponentially with the size of the block. Using Bethe ansatz results for the one- dimensional XXZ model Gu and coworkers showed that the local information obtained from the entanglement en- tropy of the two-site density matrix is enough to study the phase transitions that occur in this model. Consider- ing blocks larger than the characteristic length scales of the system (that in the critical point diverge) was shown to be unnecessary as the entanglement between a block of two spins and the rest of the system is sufficient to re- veal the most important information about the system. Their results hint to the possibility of using these prop- erties for the numerical study of phase transitions with small systems. It is the purpose of this work to study numerically the local entanglement as a function of the size of the system. In particular, we will concentrate in the crit- ical points of the phase transitions. We will study a one-dimensional model of spinless fermions with nearest- neighbor interactions that can be transformed through the Jordan-Wigner transformation to the XXZ model with spins at each site Sj = 1/2 [15]. For repulsive inter- action V1 = 2.0 the ground state performs a Kosterlitz- Thouless type phase transition from a Luttinger Liquid to a charge density wave, which corresponds to an antifer- romagnetically ordered state in the spin picture. For at- tractive interaction V1 = −2.0 there is a phase transition to a phase separated state, which corresponds to a ferro- magnet in the spin picture. We will show that the bond http://arxiv.org/abs/0704.1597v1 entropy Sbond enables us to determine the critical points of both quantum phase transitions with an astonishing accuracy, albeit they present very different characteris- tics and symmetries. We will show how these differences are reflected in the behavior of the bond entropy. Finally we include next-nearest-neighbor interaction to test the generality of our finding. The effect of longer ranged in- teraction was studied before in the context of multiple umklapp scattering [16]. In this case the interaction can now lead to phase transitions at fillings different from 1/2 and to ordering patterns with increased unit cell. II. BOND-ENTROPY FOR THE NEAREST-NEIGHBOR INTERACTION MODEL We shall consider the one-dimensional spinless fermions model with next-neighbor interactions. Ĥ = −t i ĉi−1 + ĉ i−1ĉi V1(n̂i− )(n̂i−1− where the operators appearing in the formula are the usual fermionic creation, annihilation, and number oper- ator at site i, L is the total number of sites, t is the hop- ping matrix element between neighboring sites, and V1 is the nearest-neighbor interaction strength. An important property of this model is that it can be transformed into the XXZ spin S = 1/2 model through the Jordan-Wigner transformation [15, 17]. Ŝ−j = exp cj , (2) Ŝzj = n̂j − 1/2. (3) For an even number of particles a phase term appears in the boundary condtion when we apply the Jordan- Wigner transformation to Hamiltonian (1). As we are not interested in this even-odd effect we will consider periodic (anti-periodic) boundary conditions, c0 ≡ cM (c0 ≡ −cM ), for N odd (even) and both models will be equivalent in our examples. Although we will mainly consider the spinless fermion model we will make com- ments regarding the equivalent behavior of both models when we believe it will be useful to clarify some situation (specially in the “ferromagnetic” phase). The Hamilto- nian (1) commutes with the total N̂ = n̂i operator so the total number of particles is a conserved quantity and we will consider subspaces with a definite number of fermions, equivalent to consider subspaces with a definite value of Sz in the XXZ model. This model has an interesting phase diagram depend- ing on the value of the ratio of the interaction parameter and the hopping term V1/t and also on the number of particles N . Without loss of generality we can consider t = 1 and consider the phases as we change V1. For V1 < −2 the equivalent spin system is ferromagnetic and the ground state is fully spin polarized. When we cross the first transition point Vca = −2 the ground state of the system can be shown to be non-degenerate and with spin S = 0 [18]. Only in the half-filled case, N = L/2, there is another transition point at Vcb = 2. For V1 > 2 the system is a charge density wave type insulator, the transition is of the Kosterlitz-Thouless type and the or- der parameter depends exponentially on the difference V − Vcb making an accurate numerical determination of the transition point notoriously hard. We will define the bond-entropy as the Von Neumann entropy of the reduced density matrix of two-neighboring sites ρ̂ii+1. As a result of the conservation of N the re- duced density matrix can be written as a 4 × 4 matrix with three sectors of N = 0, N = 1, and N = 2. In the two-site basis |00〉, |01〉, |10〉, |11〉 it can be represented ρ̂ii+1 = u− 0 0 0 0 ω z 0 0 z∗ ω 0 0 0 0 u+ . (4) For this particular model using its invariance under translations it can be shown that we can write this matrix elements in terms of certain correlation functions [19, 20], u− = 1+ 〈n̂in̂i+1〉 − 〈n̂i〉 − 〈n̂i+1〉 . (5) u+ = 〈n̂in̂i+1〉+ 〈n̂i〉 − 〈n̂i+1〉 . (6) 〈n̂i〉+ 〈n̂i+1〉 − 〈n̂in̂i+1〉 . (7) i ĉi+1 i+1ĉi . (8) We define the bond entropy Si as the Von Neu- mann entropy of the two-site density matrix ρ̂i i+1. In the model under study, the invariance under translations also implies that Si does not depend on the site i. Sbond = − λj lnλj , (9) where λj are the four eigenvalues of the reduced two-site density matrix ρ̂i i+1. III. NUMERICAL RESULTS In this section we will show the numerical results for the bond entropy as a function of the size of the system using the DMRG algorithm [21]. In Figure 1 we show the results of Sbond at half fill- ing for different number of sites L. The behavior of the -2 -1 0 1 2 3 4 5 L = 10 N = 5 L = 14 N = 7 L = 18 N = 9 L = 22 N = 11 L = 26 N = 13 L = 30 N = 15 FIG. 1: (Color online) Bond entropy Sbond as a function of the interaction for different sizes of the spinless fermions ring L at half filling N = L/2. The results for L = 26 and L = 30 are hardly distinguishable. We have a maximum of Sbond at V1 = 2 with extremely good approximation, see next figure, that marks the CDW insulator transition. The slope of the entropy diverges at V1 = −2 marking the appearance of the ferromagnetic transition. bond entropy around the two critical points is very dif- ferent, reflecting the different changes in the symmetries and correlations of the ground state. The slope of Sbond diverges around Vca = −2 (we will explain that in more detail in the next section), while Sbond is continuous but has a local maximum in the proximity of the second crit- ical point Vcb = 2. In both cases one can understand the behavior of Sbond from the behavior of the correlation functions in the different phases [14]. More importantly, one can estimate with extraordinary precision the value of the critical parameter from very small system sizes even in the case of the transition to the CDW phase. In addition, Sbond has a minimum for V1 = 0. In Figure 2 we see a zoom of the last figure 1 for one particular case with parameter L = 30, N = 15 in the region around V1 = 2. When we calculate numerically the position of the maximum with DMRG we obtain the value of V1 with a precision better than 10 −5. Of course, in order to take full advantage of these proper- ties of Sbond we need a very accurate algorithm such as DMRG. We used at least 500 states per block for the L ≤ 30 sites leading to an discarded entropy below 10−9, and 1400 states per block for the 36 site system (see be- low), including the five lowest lying states in order to treat the degeneracies correctly, leading to a discarded entropy typically below 10−10, and up to 2 · 10−8 close to the phase transitions and applied always eleven finite lattice sweeps. Close to the CDW-I – Luttinger liquid transition (see below) we used at least 2050 states per block and only two low lying states to check our results. This resulted in a discarded entropy below 10−12. We’d like to note that despite the large number of states the DMRG runs are much cheaper as compared to the cal- culations in [16], since no resolvent has to be computed, e.g. the largest run took about a hundred CPU minutes. 0.9519 0.951902 0.951904 0.951906 0.951908 0.95191 0.951912 0.951914 1.95 1.96 1.97 1.98 1.99 2 2.01 2.02 2.03 2.04 2.05 FIG. 2: (Color online) Behavior of Sbond as a function of V1 in the neighborhood of the phase transition at V1 = 2 in the case L = 30, N = 15. We can numerically pinpoint the maximum of the curve at V1 = 2 with a precision better than 10 The function f(x) is the second order polynomial fit used to obtain the maximum of the curve. -2 0 2 4 L = 26 N = 13 L = 26 N = 9 L = 22 N = 9 FIG. 3: (Color online) Behavior of Sbond as a function of V for L = 26, N = 13 (half-filling) and L = 26, N = 9, and L = 22, N = 9 (outside half-filling). We observe the same behavior in the ferromagnetic transition around Vca = −2 but a complete different one in the transition around Vcb = 2. In Figure 3 we see some examples comparing results at half-filling and outside half-filling. The qualitative be- havior of the bond entropy is exactly the same around the first critical point Vca = −2, but the maximum of the bond entropy around Vcb = 2 disappears as soon as we move outside half-filling reflecting the absence of the CDW transition for fillings different from 1/2. IV. FERROMAGNETISM AND THE TWO-SITE DENSITY MATRIX As we have mentioned before the slope of Sbond di- verges at V1 = Vca. In Fig. 4 we show the results for the value of the third eigenvalue of the two-site density matrix λ3 for different sizes. We can reach a very high numerical precision in the determination of the ferromag- -0.0001 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 -2.05 -2.04 -2.03 -2.02 -2.01 -2 -1.99 -1.98 -1.97 -1.96 -1.95 FIG. 4: (Color online) Behavior of the third eigenvalue of the two-site density matrix λ3 as a function of V in the neighbor- hood of the phase transition at V1 = −2 for different sizes at half filling. λ3 vanishes at V1 = −2 with very high precision. netic critical point studying the cancellation of the third eigenvalue of the two-site density matrix. In the thermodynamic limit at V1 = −2, ω = z = 1/4 in Eq. (4). One can immediately see that the equality of both matrix elements implies that one of the eigenvalues of the density matrix is zero, which leads to the singular- ity of Sbond. If one tries a direct numerical examination of the values of the correlation functions independently one does not get a very accurate estimation of the critical parameter. However, the examination of the particular combination appearing in the Von Neumann entropy of the two-site density matrix allows a very accurate calcu- lation even with very small system sizes due to the fact that although ω and z converge slowly to the thermo- dynamic limit value 1/4, their difference converges very quickly to zero at the critical value of the interaction. V. NEXT NEAREST-NEIGHBOR INTERACTION MODEL In order to test the generality of our conclusions and to obtain a critical parameter of a phase transition in a model not solvable with Bethe ansatz we add next- nearest neighbors interaction, Ĥ = −t i ĉi−1 + ĉ i−1ĉi n̂i − n̂i−1 − n̂i − n̂i−2 − , (10) where V2 is the strength of the interaction between sites separated by two lattice spacings. This Hamiltonian has been used to study the physics of materials that exhibit 0 0,25 0,5 0,75 1 0,951 0,952 0,953 FIG. 5: Bond entropy as a function of V2 in the line V2 = 5− 2V1 for L = 36 and N = 18. The maximum can be used to estimate the critical point at V2,c = 0.280 with very high precision. multiple phase transitions. Usually one considers V2 < V1 as one expects the interaction to reduce with distance. However, there can be exceptions if the nearest-neighbor interaction is supressed by the lattice geometry. The phase diagram of the model represented by the Hamiltonian (10) has been studied as a function of V1 and V2 by Schmitteckert and Werner [16]. In this paper the authors used DMRG to calculate the ground state curva- ture. The phase diagram depends on the filling, commen- surability effects are extremely important due to the mul- tiple umklapp scattering. If we concentrate on half-filling and repulsive interactions we have a charge density wave (CDW) phase in which the ground state is twofold de- generate with ordering pattern (•◦•◦) and (◦•◦•). Here ◦ denotes a vacant and • denotes an occupied site. In phase CDW II the ground state is fourfold degenerate with or- dering pattern (••◦◦), (◦••◦), (◦◦••), and (•◦◦•). We will follow reference [16] and study the critical parameters along the line V2 = 5 − 2V . For example, studying sys- tems of sizes up to L = 60 they obtained a critical point for the transition between the CDW I phase and the Lut- tinger Liquid phase as (V1,c, V2,c) = (2.4±0.05, 0.2±0.1)}. In Fig. (5) we show results for the ground state bond entropy as a function of V2 for along the previous men- tioned line for L = 36 and N = 18. Even from the small size used we can accurately determine the critical V2 as V2,c = 0.280, which is within the error bars previously given by Schmitteckert and Werner [16]. The determina- tion of the critical parameter for the transition between CDW I and the Luttinger liquid is done with much less numerical work as compared to the finite size analysis of excitation gaps and the ground state curvate in [16]. Notably, the finite size corrections are smaller, e.g. the results of the same analysis with L = 18 already gives a critical parameter of V2,c = 0.277. In Fig. 6 we show the numerical results for the value of the next-neighbor inter- action V2 in which we have a local maximum of Sbond ( along the same line as before) as a function of the inverse 0 0,025 0,05 0,075 0,275 0 0,04 0,08 0,945 FIG. 6: Finite size scaling for the position of the maximum of the bond entropy V2,c as a function of 1/L. The continuous line is a second order polynomial fit. It is used to extrapolate the value for L = ∞. In the inset we show the value of the local maximum of Sbond as a function of 1/L, a second order polynomial also fits very well the numerical results. of the total length of the system 1/L. We have calculated numerically the bond entropy at each size with an inter- val of 0.001 in V2 in the region around the maximum of Sbond, except in the case of L = 36 where we have used an interval of = 0.0002. The values of V2 in the maxi- mum where obtained through a second-order polynomial fit of the numerical results for Sbond. The actual value of the interval used was not very critical as the fits were very good. With another second-order polynomial fit we can extrapolate the calculated values to obtain the result for the thermodynamic limit V2,c = 0.2814± 0.0001. In the inset of the figure we can see the numerical results used in the extrapolation of the value of the maximum of the bond entropy in the critical point. The extrapolated value being Smax = 0.95385± 0.00001. In this case the bond entropy and the two-site den- sity matrix does not give very useful information about the phase transitions to charge density wave phases with ordering patterns with basic sizes bigger than two. We obtain no clear signature in the bond entropy for the quantum phase transition to CDW II. One may have to study entropies of density matrices of blocks with at least the size of the basic ordered block of the phases we are looking at. Also, one could try to study the bond entropy for the excited states. The numerical determination of critical parameters in this case is out of the scope of this study of the bond entropy and will be subject of future work. We note that in the case of CDW II the ground state is fourfold degenerate in the thermodynamic limit. However, the degeneracy of the two lowest lying states is lifted by finite size effects. VI. CONCLUSIONS The bond entropy defined as the Von Neumann en- tropy of the two-site density matrix can be a very effec- tive tool for the study of phase transitions and critical parameters. Its behavior depends on the correlations in the ground state of the system. We have studied the bond entropy for a model of spin- less fermions with nearest-neighbor interactions and pe- riodic boundary conditions. The size dependence of its behavior near the two critical points in the model has been studied in detail, showing an amazing precision in the estimation of the critical parameter. We have also studied a model with next-nearest neighbor interactions. We could determine the critical point of the phase tran- sition from the Luttinger liquid to the CDW I state with an ordering pattern of period two. If the fundamental block contains only two sites we show that the bond en- tropy displays a clear signature of the quantum phase transitons and allows for the determination of the crit- ical parameters. The bond entropy of the ground state could not be used for the transition to CDW II with or- dering pattern of period four. In this case we may have to turn to a block entropy of higher size. In general, we can say that the bond entropy can be used as a numerical indicator for phase transitions but the actual behavior of the bond entropy is not universal and will depend on the QPT under study. Our results should open the way to the numerical study of phase transitions with small sized systems. Acknowledgments RAM wishes to acknowledge useful discussions with J. Dukelsky. He also acknowledges finantial support at the Instituto de Estructura de la Materia-CSIC by an I3P contract funded by the European Social Fund. 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Mon. Not. R. Astron. Soc. 000, 1–14 (2008) Printed 10 November 2018 (MN LATEX style file v2.2) Radiative transitions of the helium atom in highly magnetized neutron star atmospheres Z. Medin1, D. Lai1, and A. Y. Potekhin1,2 1Center for Radiophysics and Space Research, Department of Astronomy, Cornell University, Ithaca, NY 14853 2Ioffe Physico-Technical Institute, Politekhnicheskaya 26, 194021 Saint-Petersburg, Russia Accepted 2007 September 26. Received 2007 May 21; in original form 2007 April 12 ABSTRACT Recent observations of thermally emitting isolated neutron stars revealed spectral fea- tures that could be interpreted as radiative transitions of He in a magnetized neutron star atmosphere. We present Hartree–Fock calculations of the polarization-dependent photoionization cross sections of the He atom in strong magnetic fields ranging from 1012 G to 1014 G. Convenient fitting formulae for the cross sections are given as well as related oscillator strengths for various bound-bound transitions. The effects of finite nucleus mass on the radiative absorption cross sections are examined using perturbation theory. Key words: atomic processes – magnetic fields – stars: atmospheres – stars: neutron 1 INTRODUCTION An important advance in neutron star astrophysics in the last few years has been the detection and detailed studies of surface emission from a large number of isolated neutron stars (NSs), including radio pulsars, magnetars, and radio- quiet NSs (e.g., Kaspi et al. 2006; Harding & Lai 2006). This was made possible by X-ray telescopes such as Chan- dra and XMM-Newton. Such studies can potentially provide invaluable information on the physical properties and evolu- tion of NSs (e.g., equation of state at super-nuclear densities, cooling history, surface magnetic field and composition). Of great interest are the radio-quiet, thermally emitting NSs (e.g., Haberl 2006): they share the common property that their spectra appear to be entirely thermal, indicating that the emission arises directly from the NS surfaces, uncon- taminated by magnetospheric emission. The true nature of these sources, however, is unclear at present: they could be young cooling NSs, or NSs kept hot by accretion from the ISM, or magnetar descendants. While some of these NSs (e.g., RX J1856.5−3754) have featureless X-ray spectrum remarkably well described by blackbody (e.g., Burwitz et al 2003) or by emission from a condensed surface covered by a thin atmosphere (Ho et al. 2007), a single or multiple ab- sorption features at E ≃ 0.2–1 keV have been detected from several sources (see van Kerkwijk & Kaplan 2007): e.g., 1E 1207.4−5209 (0.7 and 1.4 keV, possibly also 2.1, 2.8 keV; Sanwal et al. 2002; De Luca et al. 2004; Mori et al. 2005), RX J1308.6+2127 (0.2–0.3 keV; Haberl et al. 2003), RX J1605.3+3249 (0.45 keV; van Kerkwijk et al. 2004), RX J0720.4−3125 (0.27 keV; Haberl et al. 2006), and possibly RBS 1774 (∼ 0.7 keV; Zane et al. 2005). The identifications of these features, however, remain uncertain, with sugges- tions ranging from proton cyclotron lines to atomic transi- tions of H, He, or mid-Z atoms in a strong magnetic field (see Sanwal et al. 2002; Ho & Lai 2004; Pavlov & Bezchastnov 2005; Mori & Ho 2007). Clearly, understanding these ab- sorption lines is very important as it would lead to direct measurement of the NS surface magnetic fields and composi- tions, shedding light on the nature of these objects. Multiple lines also have the potential of constraining the mass-radius relation of NSs (through measurement of gravitational red- shift). Since the thermal radiation from a NS is mediated by its atmosphere (if T is sufficiently high so that the surface does not condense into a solid; see, e.g., van Adelsberg et al. 2005; Medin & Lai 2006, 2007), detailed modelling of ra- diative transfer in magnetized NS atmospheres is impor- tant. The atmosphere composition of the NS is unknown a priori. Because of the efficient gravitational separation of light and heavy elements, a pure H atmosphere is ex- pected even if a small amount of fallback or accretion oc- curs after NS formation. A pure He atmosphere results if H is completely burnt out, and a heavy-element (e.g., Fe) atmosphere may be possible if no fallback/accretion oc- curs. The atmosphere composition may also be affected by (slow) diffusive nuclear burning in the outer NS envelope (Chang, Arras & Bildsten 2004), as well as by the bombard- ment on the surface by fast particles from NS magneto- spheres (e.g., Beloborodov & Thompson 2007). Fully ion- ized atmosphere models in various magnetic field regimes have been extensively studied (e.g., Shibanov et al. 1992; Zane et al. 2001; Ho & Lai 2001), including the effect of vac- uum polarization (see Ho & Lai 2003; Lai & Ho 2002, 2003; c© 2008 RAS http://arxiv.org/abs/0704.1598v2 2 Z. Medin, D. Lai, and A. Y. Potekhin van Adelsberg & Lai 2006). Because a strong magnetic field greatly increases the binding energies of atoms, molecules, and other bound species (for a review, see Lai 2001), these bound states may have appreciable abundances in the NS atmosphere, as guessed by Cohen, Lodenquai, & Ruderman (1970) and confirmed by calculations of Lai & Salpeter (1997) and Potekhin, Chabrier & Shibanov (1999). Early considerations of partially ionized and strongly magnetized atmospheres (e.g., Rajagopal, Romani & Miller 1997) relied on oversimplified treatments of atomic physics and plasma thermodynamics (ionization equilibrium, equation of state, and nonideal plasma effects). Recently, a thermodynami- cally consistent equation of state and opacities for mag- netized (B = 1012 − 1015 G), partially ionized H plasma have been obtained (Potekhin & Chabrier 2003, 2004), and the effect of bound atoms on the dielectric tensor of the plasma has also been studied (Potekhin et al. 2004). These improvements have been incorporated into partially ion- ized, magnetic NS atmosphere models (Ho et al. 2003, 2007; Potekhin et al. 2004, 2006). Mid-Z element atmospheres for B ∼ 1012 − 1013 G were recently studied by Mori & Ho (2007). In this paper we focus on He atoms and their radiative transitions in magnetic NS atmospheres. It is well known that for B ≫ Z2B0, where Z is the charge number of the nucleus and B0 = e 3m2e/h̄ 3c = 2.35 × 109 G, the binding energy of an atom is significantly increased over its zero-field value. In this strong-field regime the electrons are confined to the ground Landau level, and one may apply the adiabatic approximation, in which electron motions along and across the field are assumed to be decoupled from each other (see Sect. 2.1). Using this approximation in combination with the Hartree–Fock method (“1DHF approximation”), several groups calcu- lated binding energies for the helium atom (Pröschel et al. 1982; Thurner et al. 1993) and also for some other atoms and molecules (Neuhauser, Langanke & Koonin 1986; Neuhauser, Koonin & Langanke 1987; Miller & Neuhauser 1991; Lai, Salpeter & Shapiro 1992). Mori & Hailey (2002) developed a “multiconfigurational perturbative hybrid Hartree–Fock” approach, which is a perturba- tive improvement of the 1DHF method. Other methods of calculation include Thomas–Fermi-like models (e.g., Abrahams & Shapiro 1991), the density functional the- ory (e.g., Relovsky & Ruder 1996; Medin & Lai 2006), variational methods (e.g., Müller 1984; Vincke & Baye 1989; Jones et al. 1999; Turbiner & Guevara 2006), and 2D Hartree–Fock mesh calculations (Ivanov 1994; Ivanov & Schmelcher 2000) which do not directly employ the adiabatic approximation. In strong magnetic fields, the finite nuclear mass and centre-of-mass motion affect the atomic structure in a non- trivial way (e.g., Lai 2001; see Sect. 5). The stronger B is, the more important the effects of finite nuclear mass are. Apart from the H atom, these effects have been cal- culated only for the He atom which rests as a whole, but has a moving nucleus (Al-Hujaj & Schmelcher 2003a,b), and for the He+ ion (Bezchastnov, Pavlov & Ventura 1998; Pavlov & Bezchastnov 2005). There were relatively few publications devoted to ra- diative transitions of non-hydrogenic atoms in strong mag- netic fields. Several authors (Miller & Neuhauser 1991; Thurner et al. 1993; Jones et al. 1999; Mori & Hailey 2002; Al-Hujaj & Schmelcher 2003b) calculated oscillator strengths for bound-bound transitions; Miller & Neuhauser (1991) presented also a few integrated bound-free oscilla- tor strengths. Rajagopal et al. (1997) calculated opacities of strongly magnetized iron, using photoionization cross sections obtained by M. C. Miller (unpublished). To the best of our knowledge, there were no published calculations of polarization-dependent photoionization cross sections for the He atom in the strong-field regime, as well as the calcu- lations of the atomic motion effect on the photoabsorption coefficients for He in this regime. Moreover, the subtle effect of exchange interaction involving free electrons and the pos- sible role of two-electron transitions (see Sect. 3.2) were not discussed before. In this paper we perform detailed calculations of radia- tive transitions of the He atom using the 1DHF approx- imation. The total error introduced into our calculations by the use of these two approximations, the Hartree-Fock method and the adiabatic approximation, is of order 1% or less, as can be seen by the following considerations: The Hartree-Fock method is approximate because electron corre- lations are neglected. Due to their mutual repulsion, any pair of electrons tend to be more distant from each other than the Hartree-Fock wave function would indicate. In zero-field, this correlation effect is especially pronounced for the spin- singlet states of electrons for which the spatial wave function is symmetrical. In strong magnetic fields (B ≫ B0), the elec- tron spins (in the ground state) are all aligned antiparallel to the magnetic field, and the multielectron spatial wave function is antisymmetric with respect to the interchange of two electrons. Thus the error in the Hartree-Fock approach is expected to be less than the 1% accuracy characteristic of zero-field Hartree-Fock calculations (Neuhauser et al. 1987; Schmelcher, Ivanov & Becken 1999; for B = 0 see Scrinzi 1998). The adiabatic approximation is also very accurate at B ≫ Z2B0. Indeed, a comparision of the ground-state en- ergy values calculated here to those of Ivanov (1994) (who did not use the adiabatic approximation) shows an agree- ment to within 1% for B = 1012 G and to within 0.1% for B = 1013 G. The paper is organized as follows. Section 2 describes our calculations of the bound states and continuum states of the He atom, and section 3 contains relevant equations for radiative transitions. We present our numerical results and fitting formulae in section 4 and examine the effects of finite nucleus mass on the photoabsorption cross sections in section 5. 2 BOUND STATES AND SINGLY-IONIZED STATES OF HELIUM ATOMS IN STRONG MAGNETIC FIELDS 2.1 Bound states of the helium atom To define the notation, we briefly describe 1DHF calcula- tions for He atoms in strong magnetic fields. Each electron in the atom is described by a one-electron wave function (orbital). If the magnetic field is sufficiently strong (e.g., B ≫ 1010 G for He ground state), the motion of an electron perpendicular to the magnetic field lines is mainly governed c© 2008 RAS, MNRAS 000, 1–14 Radiative transitions of the helium atom 3 by the Lorentz force, which is, on the average, stronger than the Coulomb force. In this case, the adiabatic approximation can be employed – i.e., the wave function can be separated into a transverse (perpendicular to the external magnetic field) component and a longitudinal (along the magnetic field) component: φmν(r) = fmν(z)Wm(r⊥) . (1) Here Wm is the ground-state Landau wave function (e.g., Landau & Lifshitz 1977) given by Wm(r⊥) = , (2) where (ρ, ϕ) are the polar coordinates of r⊥, ρ0 = (h̄c/eB)1/2 is the magnetic length and fmν is the longitudi- nal wave function which can be calculated numerically. The quantum number m (> 0 for the considered ground Landau state) specifies the negative of the z-projection of the elec- tron orbital angular momentum. We restrict our considera- tion to electrons in the ground Landau level; for these elec- trons, m specifies also the (transverse) distance of the guid- ing centre of the electron from the ion, ρm = (2m+1) 1/2ρ0. The quantum number ν specifies the number of nodes in the longitudinal wave function. The spins of the electrons are taken to be aligned anti-parallel with the magnetic field, and so do not enter into any of our equations. In addition, we assume that the ion is completely stationary (the ‘infinite ion mass’ approximation). In general, the latter assumption is not necessary for the applicability of the adiabatic ap- proximation (see, e.g., Potekhin 1994). The accuracy of the infinite ion mass approximation will be discussed in Sect. 5. Note that we use non-relativistic quantum mechanics in our calculations, even when h̄ωBe & mec 2 or B & BQ = 2 = 4.414 × 1013 G. This is valid for two reasons: (i) The free-electron energy in relativistic theory is 1 + 2nL )]1/2 . (3) For electrons in the ground Landau level (nL = 0), Eq. (3) reduces to E ≃ mec2 + p2z/(2me) for pzc ≪ mec2; the elec- tron remains non-relativistic in the z direction as long as the electron energy is much less than mec 2; (ii) it is well known (e.g., Sokolov & Ternov 1986) that Eq. (2) describes the transverse motion of an electron with nL = 0 at any field strength, and thus Eq. (2) is valid in the relativistic theory. Our calculations assume that the longitudinal motion of the electron is non-relativistic. This is valid for helium at all field strengths considered in this paper. Thus relativistic correc- tions to our calculated electron wave functions, binding ener- gies, and transition cross sections are all small. Our approx- imation is justified in part by Chen & Goldman (1992), who find that the relativistic corrections to the binding energy of the hydrogen atom are of order ∆E/E ∼ 10−5.5−10−4.5 for the range of field strengths we are considering in this work (B = 1012 − 1014 G). A bound state of the He atom, in which one electron occupies the (m1ν1) orbital, and the other occupies the (m2ν2) orbital, is denoted by |m1ν1,m2ν2〉 = |Wm1fm1ν1 ,Wm2fm2ν2〉 (clearly, |m1ν1,m2ν2〉 = |m2ν2,m1ν1〉). The two-electron wave function is Ψm1ν1,m2ν2(r1, r2) = Wm1(r1⊥)fm1ν1(z1) ×Wm2(r2⊥)fm2ν2(z2) −Wm2(r1⊥)fm2ν2(z1)Wm1(r2⊥)fm1ν1(z2) . (4) The one-electron wave functions are found using Hartree–Fock theory, by varying the total energy with re- spect to the wave functions. The total energy is given by (see, e.g., Neuhauser et al. 1987): E = EK + EeZ + Edir + Eexc , (5) where dz |f ′mν (z)|2 , (6) EeZ = −Ze2 dz |fmν (z)|2Vm(z) , (7) Edir = mν,m′ν′ ′ |fmν (z)|2 |fm′ν′(z′)|2 ×Dmm′(z − z′) , (8) Eexc = − mν,m′ν′ m′ν′(z)fmν(z) ×f∗mν(z′)fm′ν′(z′)Emm′(z − z′) ; (9) Vm(z) = |Wm(r⊥)|2 , (10) Dmm′ (z − z′) = dr⊥dr |Wm(r⊥)|2|Wm′(r′⊥)|2 |r′ − r| , (11) Emm′(z − z′) = dr⊥dr |r′ − r| ×W ∗m′(r⊥)Wm(r⊥)W ⊥)Wm′(r ⊥) . (12) Variation of Eq. (5) with respect to fmν(z) yields − Ze2Vm(z) ′ |fm′ν′(z′)|2Dmm′ (z − z′)− εmν fmν (z) mν (z )fm′ν′(z )Emm′(z − z′)fm′ν′(z) . In these equations, asterisks denote complex conjugates, and f ′mν(z) ≡ dfmν/dz. The wave functions fmν(z) must satisfy appropriate boundary conditions, i.e., fmν → 0 as z → ±∞, and must have the required symmetry [fmν(z) = ±fmν(−z)] and the required number of nodes (ν). The equations are solved iteratively until self-consistency is reached for each wave function fmν and energy εmν . The total energy of the bound He state |m1ν1,m2ν2〉 can then be found, using either Eq. (5) or εmν − Edir − Eexc . (14) c© 2008 RAS, MNRAS 000, 1–14 4 Z. Medin, D. Lai, and A. Y. Potekhin 2.2 Continuum states of the helium atom The He state in which one electron occupies the bound (m3ν3) orbital, and other occupies the continuum state (m4k) is denoted by |m3ν3,m4k〉 = |Wm3fm3ν3 ,Wm4fm4k〉. The corresponding two-electron wave function is Ψm3ν3,m4k(r1, r2) = [Wm3(r1⊥)fm3ν3(z1) ×Wm4(r2⊥)fm4k(z2) −Wm4(r1⊥)fm4k(z1)Wm3(r2⊥)fm3ν3(z2)] . (15) Here fm4k(z) is the longitudinal wave function of the con- tinuum electron, and k is the z-wavenumber of the electron at |z| → ∞ (far away from the He nucleus). We can use Hartree–Fock theory to solve for the ion- ized He states as we did for the bound He states. Since the continuum electron wave function fm4k(z) is non-localized in z, while the bound electron wave function fm3ν3(z) is lo- calized around z = 0, it is a good approximation to neglect the continuum electron’s influence on the bound electron. We therefore solve for the bound electron orbital using the equation − Ze2Vm3(z) fm3ν3(z) = εm3ν3fm3ν3(z) . (16) The continuum electron, however, is influenced by the bound electron, and its longitudinal wave function is determined − Ze2Vm4(z) ′ |fm3ν3(z )|2Dm3m4(z − z )− εf fm4k(z) )fm3ν3(z )Em3m4(z − z )fm3ν4(z) . where εf = εm4k = h̄ 2k2/(2me). Here, the bound electron orbital |m3ν3〉 satisfies the same boundary conditions as dis- cussed in Sect. 2.1. The shape of the free electron wave func- tion is determined by the energy of the incoming photon and the direction the electron is emitted from the ion. We will discuss this boundary condition in the next section. The to- tal energy of the ionized He state |m3ν3,m4k〉 is simply E = εm3ν3 + εf . (18) Note that the correction terms Edir and Eexc that appear in Eq. (14) do not also appear in Eq. (18). The direct and exchange energies depend on the local overlap of the elec- tron wave functions, but the non-localized nature of the free electron ensures that these terms are zero for the continuum states. 3 RADIATIVE TRANSITIONS We will be considering transitions of helium atoms from two initial states: the ground state, |00, 10〉, and the first excited state, |00, 20〉. In the approximation of an infinitely massive, pointlike nucleus, the Hamiltonian of the He atom in electromagnetic field is (see, e.g., Landau & Lifshitz 1977) j=1,2 Atot(rj) j=1,2 |r1 − r2| , (19) where pj = −ih̄∇j is the canonical momentum operator, acting on the jth electron, rj is the jth electron radius vec- tor, measured from the nucleus, andAtot(r) is the vector po- tential of the field. In our case, Atot(r) = AB(r) +Aem(r), where AB(r) and Aem(r) are vector potentials of the sta- tionary magnetic field and electromagnetic wave, respec- tively. The interaction operator is Hint = H − H0, where H0 is obtained from H by setting Aem(r) = 0. The un- perturbed Hamiltonian H0 is responsible for the stationary states of He, discussed in Sect. 2. The vector potential and the wave functions may be subject to gauge transformations; the wave functions presented in Sect. 2 correspond to the cylindrical gauge AB(r) = B × r. Neglecting non-linear (quadratic in Aem) term, we have Hint ≈ j=1,2 [πj ·Aem(rj) +Aem(rj) · πj ], (20) where π = p + AB(r). (21) is the non-perturbed kinetic momentum operator: π = meṙ = me(i/h̄)[H0 r − rH0]. For a monochromatic wave of the form Aem(r) ∝ ǫ eiq·r , where ǫ is the unit polarization vector, applying the Fermi’s Golden Rule and assuming the transverse polarization (ǫ · q = 0), one obtains the following general formula for the cross section of absorption of radiation from a given initial state |a〉 (see, e.g., Armstrong & Nicholls 1972): σ(ω, ǫ) = ∣ǫ · 〈b|eiq·rj|a〉 δ(ω − ωba), (22) where |b〉 is the final state, ω = qc is the photon frequency, ωba = (Eb − Ea)/h̄, and j is the electric current operator. In our case, j = (−e/me)(π1 + π2). We shall calculate the cross sections in the dipole ap- proximation – i.e., drop eiq·r from Eq. (22). This approxima- tion is sufficiently accurate for calculation of the total cross section as long as h̄ω ≪ mec2 (cf., e.g., Potekhin & Pavlov 1993, 1997 for the case of H atom). In the dipole approxi- mation, Eq. (22) can be written as σ(ω, ǫ) = 2π2e2 fbaδ(ω − ωba), (23) where fba = h̄ωbame |〈b|ǫ · π|a〉|2 = 2meωba |〈b|ǫ · r|a〉|2 (24) is the oscillator strength. In the second equality we have passed from the ‘velocity form’ to the ‘length form’ of the matrix element (cf., e.g., Chandrasekhar 1945). These rep- resentations are identical for the exact wave functions, but it is not so for approximate ones. In the adiabatic ap- proximation, the length representation [i.e., the right-hand side of Eq. (24)] is preferable (see Potekhin & Pavlov 1993; Potekhin, Pavlov, & Ventura 1997). To evaluate the matrix element, we decompose the unit polarization vector ǫ into three cyclic components, ǫ = ǫ−ê+ + ǫ+ê− + ǫ0ê0, (25) c© 2008 RAS, MNRAS 000, 1–14 Radiative transitions of the helium atom 5 with ê0 = êz along the external magnetic field direction (the z-axis), ê± = (êx ± iêy)/ 2, and ǫα = êα · ǫ (with α = ±, 0). Then we can write the cross section as the sum of three components, σ(ω, ǫ) = σ+(ω)|ǫ+|2 + σ−(ω)|ǫ−|2 + σ0(ω)|ǫ0|2, (26) where σα has the same form as Eq. (23), with the corre- sponding oscillator strength given by 2meωbaρ |Mba|2 = |Mba|2, (27) Mba = 〈b|ê∗α · r̄|a〉, (28) where r̄ = r/ρ0 and ωc = eB/(mec) is the electron cyclotron frequency. 3.1 Bound-bound transitions Consider the electronic transition |a〉 = |mν,m2ν2〉 = |Wmfmν ,Wm2fm2ν2〉 −→ |b〉 = |m′ν′,m2ν2〉 = |Wm′gm′ν′ ,Wm2gm2ν2〉. (29) The selection rules for allowed transitions and the related matrix elements are σ0 : ∆m = 0, ∆ν = odd, Mba = 〈gmν′ |z̄|fmν 〉〈gm2ν2 |fm2ν2〉, (30) σ+ : ∆m = 1, ∆ν = even, Mba = m+ 1 〈gm′ν′ |fmν 〉〈gm2ν2 |fm2ν2〉, (31) σ− : ∆m = −1, ∆ν = even, Mba = m 〈gm′ν′ |fmν〉〈gm2ν2 |fm2ν2〉, (32) where ∆m = m′ −m, ∆ν = ν′ − ν. The oscillator strengths for bound-bound transitions from the states |00, 10〉 and |00, 20〉 are given in Table 1. The selection rules (30) – (32) are exact in the dipole approximation. The selection rules in m follow from the con- servation of the z-projection of total (for the photon and two electrons) angular momentum. Technically, in the adiabatic approximation, they follow from the properties of the Lan- dau functions (e.g., Potekhin & Pavlov 1993). The selection rules in ν follow from the fact that the functions gm′ν′ and fmν have the same parity for even ν ′−ν and opposite parity for odd ν′ − ν. In addition to these selection rules, there are approxi- mate selection rules which rely on the approximate orthog- onality of functions gm′ν′ and fmν (for general ν 6= ν′). Be- cause of this approximate orthogonality, which holds better the larger B is, we have 〈gm′ν′ |fmν 〉〈gm2ν2 |fm2ν2〉 = δν,ν′ + ε, (33) where |ε| ≪ 1 and ε → 0 as ∆ν → ±∞. Therefore, the oscil- lator strengths for transitions with α = ± and ∆ν = 2, 4, . . . are small compared to those with ∆ν = 0. The latter oscilla- tor strengths can be approximated, according to Eqs. (27), (31), (32) and (33), by ba ≈ 2(m+ 1)ωba/ωc, f ba ≈ 2mωba/ωc (34) (α = ∆m = ±1, ν′ = ν). The same approximate orthogonality leads to the smallness of matrix elements for transitions of the type |mν,m2ν2〉 −→ |m′ν′,m2ν′2〉 with ν′2 6= ν2 for α = ± and the smallness of cross terms in the matrix elements of the form 〈gm2ν2 |fmν〉〈gm′ν′ |fm2ν2〉 when m′ = m2 (i.e., the so-called “one-electron jump rule”); we have therefore excluded such terms from the selection rule equations above [Eqs. (30) – (32)]. 3.2 Photoionization The bound-free absorption cross section for the transition from the bound state |b〉 to the continuum state |f〉 is given by Eq. (22) with obvious substitutions |a〉 → |b〉, |b〉 → |f〉, → (Lz/2π) dk, (35) where Lz is the normalization length of the continuum elec- tron [ ∫ Lz/2 −Lz/2 dz |gmk(z)|2 = 1] and k is the wave number of the outgoing electron (Sect. 2.2). Therefore we have σbf(ω, ǫ) = 2πe2Lz mech̄ 2ωfbk ∣〈fk|eiq·rǫ · π|b〉 ∣〈f−k|eiq·rǫ · π|b〉 , (36) where k = 2meεf/h̄ and |f±k〉 represents the final state where the free electron has wave number ±k (here and here- after we assume k > 0). The asymptotic conditions for these outgoing free electrons are (cf., e.g., Potekhin et al. 1997) gmk(z) ∼ exp[iϕk(z)] at z → ±∞, where ϕk(z) = |kz|+ (ka0)−1 ln |kz| and a0 = h̄2/mee2 is the Bohr radius. Since we do not care about direction of the outgoing elec- tron, we can use for calculations a basis of symmetric and antisymmetric wave functions of the continuum – that is, in Eq. (36) we can replace 〈fk| and 〈f−k| by 〈feven| and 〈fodd|. The symmetric state |feven〉 is determined by the free electron boundary condition g′mk,even(0) = 0 and the antisymmetric state |fodd〉 is determined by gmk,odd(0) = 0. Since the coefficients in Eq. (17) are real, gmk,even(z) and gmk,odd(z) can be chosen real. At z → ±∞, they behave as gmk,(even,odd)(z) ∼ sin[ϕ(z) + constant] (where the value of constant depends on all quantum numbers, including k). We still have the normalization ∫ Lz/2 −Lz/2 dz |gmk,(even,odd)(z)|2 = Similar to bound-bound transitions, we can decompose the bound-free cross section into three components, Eq. (26). Thus, using the dipole approximation and the length form of the matrix elements, as discussed above, we have for (α = ±, 0)-components of the bound-free cross section σbf,α(ω) = × |〈f |ê∗α · r̄|b〉| , (37) where |f〉 = |feven〉 or |f〉 = |fodd〉 depending on the parity of the initial state and according to the selection rules, and σTh = (8π/3) (e 2/mec 2)2 is the Thomson cross section. The selection rules and related matrix elements for the bound- free transitions |b〉 = |mν,m2ν2〉 = |Wmfmν ,Wm2fm2ν2〉 c© 2008 RAS, MNRAS 000, 1–14 6 Z. Medin, D. Lai, and A. Y. Potekhin −→ |f〉 = |m′k,m2ν2〉 = |Wm′gm′k,Wm2gm2ν2〉 (38) are similar to those for the bound-bound transitions [see Eqs. (30) – (32)]: σ0 : ∆m = 0, ∆ν = odd, Mfb = 〈gmk|z̄|fmν 〉〈gm2ν2 |fm2ν2〉, (39) σ+ : ∆m = 1, ∆ν = even, Mfb = m+ 1 (〈gm′k|fmν 〉〈gm2ν2 |fm2ν2〉 −δm′ν,m2ν2〈gm2ν2 |fmν 〉〈gm′k|fm2ν2〉) , (40) σ− : ∆m = −1, ∆ν = even, Mfb = m (〈gm′k|fmν〉〈gm2ν2 |fm2ν2〉 −δm′ν,m2ν2〈gm2ν2 |fmν 〉〈gm′k|fm2ν2〉) , (41) In this case, the condition ∆ν = odd means that gm′k and fmν must have opposite parity, and the condition ∆ν = even means that gm′k and fmν must have the same parity. The oscillator strengths for bound-free transitions from the states |00, 10〉 and |00, 20〉 are given in Table 2. Note that in Eqs. (40) and (41), the second term in the matrix element (of the form 〈gm2ν2 |fmν〉〈gm′k|fm2ν2〉) corresponds to transitions of both electrons. This appears to violate the “one-electron jump rule” and other approxi- mate selection rules discussed in Sect. 3.1 [see Eq. (33)]. In fact, these approximate rules are not directly relevant for bound-free transitions, since the matrix elements involving a continuuum state are always small: 〈gm′k|fmν 〉 → 0 as the normalization length Lz → ∞. Rather, we use a different set of selection rules to determine which of these ‘small’ matrix elements are smaller than the rest. The first is that 〈gm′k|fmν 〉〈gm2ν2 |fm2ν2〉 ≫ 〈gm′k|fmν 〉〈gm2ν′2 |fm2ν2〉, when ν′2 6= ν2. This selection rule is similar to the bound- bound transition case as 〈gm2ν′2 |fm2ν2〉 involves a bound electron transition, not a free electron transition. The second approximate selection rule that applies here is more compli- cated: terms of the form 〈gm′ν |fmν 〉〈gm2k|fm2ν2〉 are small, unless m′ = m2 and ν2 = ν. This exception for m ′ = m2 and ν2 = ν is due to the exchange term in the differential equation for the free electron wave function [Eq. (17)], which strongly (anti)correlates the two final wave functions |gm′ν〉 and |gm2k〉. If m′ = m2 and ν = ν2, then since 〈gm′ν |fm2ν2〉 is not small (in fact, it is of order 1), 〈gm2k|fm2ν2〉 will not be small but will be of the same order as other terms involv- ing the free electron wave function. In particular, the second selection rule means, e.g., that the matrix element for the transition from |00, 10〉 to |00, 0k〉 is M00,10→00,0k = 〈g0k|f10〉〈g00|f00〉 − 〈g00|f10〉〈g0k|f00〉, (43) where the second term is non-negligible, but that the matrix element for the transition from |00, 10〉 to |0k, 20〉, which is M00,10→0k,20 = 〈g20|f10〉〈g0k|f00〉, (44) is small compared to the other matrix elements and can be ignored (see Fig. 1). We make one final comment here about the effect of exchange interaction on the free electron state. If the ex- change term [the right-hand side of Eq. (17)] is neglected in the calculation of the free electron wave function, then the cross terms (i.e., those involving two-electron transitions) in the matrix elements of Eqs. (40) and (41) are small and can be neglected. One then obtains approximate photoionization cross sections which are within a factor of two of the true val- ues in most cases and much better for σ0 transitions. If the exchange term is included in Eq. (17) but the cross terms in the matrix elements are ignored, significant errors in the σ± photoionization cross sections will result. To obtain reliable cross sections for all cases, both the exchange effect on the free electron and the contribution of two-electron transitions must be included. 4 RESULTS Tables 1 and 2 give results for transitions of helium atoms from the ground state (|00, 10〉) and the first excited state (|00, 20〉). Table 1 gives results (photon energies and oscilla- tor strengths) for all possible bound-bound transitions with ∆ν 6 1, for the field strengths B12 = 1, 5, 10, 50, 100, where B12 = B/(10 12 G). Transitions |a〉 → |b〉 for α = − are not listed separately, being equivalent to transitions |b〉 → |a〉 for α = +. One can check that the oscillator strengths fba presented in Table 1 for α = + are well described by the approximation (34). Table 2 gives results (threshold photon energies and cross section fitting formulas, see below) for all possible bound-free transitions. Figure 1 shows partial cross section curves for all bound-free transitions from the ground state of helium for B12 = 1. The transition |00, 10〉 → |0k, 20〉 is an example of a ‘weak’ transition, whose oscillator strength is small because of the approximate orthogonality of one- electron wave functions, as discussed at the end of Sect. 3.1. It is included in this figure to confirm the accuracy of our assumption. Figures 2 and 3 show total cross section curves for a photon polarized along the magnetic field, for B12 = 1 and 100 respectively. Figures 4 and 5 show total cross sec- tions for the circular polarizations, α = ±, for B12 = 1. Finally, Figs. 6 and 7 show total cross sections for α = ± and B12 = 100. 4.1 Fitting Formula The high-energy cross section scaling relations from Potekhin & Pavlov (1993), which were derived for hydro- gen photoionization in strong magnetic fields, also hold for helium: σbf,0 ∝ )2mi+9/2 σbf,± ∝ )2mi+7/2 , (46) where mi is the m value of the initial electron that tran- sitions to the free state. In addition, we use similar fitting formulae for our numerical cross sections: σbf,0 ≃ (1 +Ay)2.5(1 + B( 1 + y − 1))4(mi+1) σTh (47) σbf,± ≃ C(1 + y) (1 +Ay)2.5(1 + B( 1 + y − 1))4(mi+1) σTh (48) where y = εf/h̄ωthr and h̄ωthr is the threshold photon en- ergy for photoionization. These formulas have been fit to the c© 2008 RAS, MNRAS 000, 1–14 Radiative transitions of the helium atom 7 1 10 100 1000 10000 εf (eV) 00,10->00,1k σbf,0 1 10 100 1000 10000 εf (eV) σbf,0 00,10->0k,10 1000 10000 1 10 100 1000 10000 εf (eV) 00,10->00,2k σbf,+ 1000 10000 1 10 100 1000 10000 εf (eV) σbf,+ 00,10->10,1k 0.001 0.01 1 10 100 εf (eV) 00,10->0k,20 σbf,+ 1000 10000 1 10 100 1000 10000 εf (eV) σbf,- 00,10->00,0k Figure 1. Partial cross sections σ(0,+,−) versus final ionized electron energy for photoionization of the ground state helium atom ((m1,m2) = (1, 0)). The field strength is 10 12 G. The transition |00, 10〉 → |0k,20〉 in the bottom left panel is an example of a ‘weak’ transition. We have ignored these transitions in our calculations of the total cross sections. cross section curves with respect to the free electron energy εf in approximately the 1 – 10 4 eV range (the curves are fit up to 105 eV for strong magnetic fields B12 = 50 − 100, in order to obtain the appropriate high-energy factor). The data points to be fit are weighted proportional to their cross section values plus a slight weight toward low-energy values, according to the formula (error in σ) ∝ σ εf0.25. Results for the three fitting parameters, A, B, and C, are given in Table 2 for various partial cross sections over a range of magnetic field strengths. For photoionization in strong magnetic fields (B12 & 50) the cross section curves we generate for the σ+ and σ− transitions have a slight deficiency at low electron energies, such that the curves peak at εf ≃ 10 eV, rather than at threshold as expected. These peaks do not represent a real effect, but rather re- flect the limits on the accuracy of our code (the overlap of the wave function of the transitioning electron pre- and post-ionization is extremely small under these conditions). Because the cross section values are not correct at low ener- gies, our fits are not as accurate for these curves. In Table 2 we have marked with a ‘∗’ those transitions which are most inaccurately fit by our fitting formula, determined by cross section curves with low-energy dips greater than 5% of the threshold cross section value. c© 2008 RAS, MNRAS 000, 1–14 8 Z. Medin, D. Lai, and A. Y. Potekhin Table 1. Bound-bound transitions |a〉 → |b〉: The photon energy h̄ωba = Eb − Ea (in eV) and the oscillator strength fα for different polarization components α [see Eq. (27)]. All transitions ∆ν 6 1 from the initial states |00, 10〉 and |00, 20〉 are listed, for several magnetic field strengths B12 = B/(10 12 G). The last two columns list the transition energies h̄ω∗ and oscillator strengths , corrected for the finite mass of the nucleus, according to Sect. 5.1. B12 σ |a〉 → |b〉 h̄ωba fba h̄ω 1 0 |00, 10〉 → |00, 11〉 147.5 0.234 – – → |10, 01〉 271.8 0.124 – – + → |00, 20〉 43.11 0.0147 44.70 0.0153 0 |00, 20〉 → |00, 21〉 104.4 0.312 – – → |20, 01〉 277.7 0.115 – – + → |00, 30〉 18.01 0.00930 19.60 0.0101 → |20, 10〉 100.7 0.0170 102.3 0.0172 5 0 |00, 10〉 → |00, 11〉 256.2 0.127 – – → |10, 01〉 444.8 0.0603 – – + → |00, 20〉 66.95 0.00459 74.89 0.00512 0 |00, 20〉 → |00, 21〉 189.2 0.176 – – → |20, 01〉 455.0 0.0537 – – + → |00, 30〉 28.94 0.00299 36.88 0.00381 → |20, 10〉 151.1 0.00512 159.0 0.00539 10 0 |00, 10〉 → |00, 11〉 318.9 0.0974 – – → |10, 01〉 540.8 0.0457 – – + → |00, 20〉 79.54 0.00273 95.42 0.00327 0 |00, 20〉 → |00, 21〉 239.4 0.136 – – → |20, 01〉 553.3 0.0405 – – + → |00, 30〉 34.84 0.00179 50.72 0.00261 → |20, 10〉 177.0 0.00301 192.9 0.00328 50 0 |00, 10〉 → |00, 11〉 510.9 0.0557 – – → |10, 01〉 822.2 0.0266 – – + → |00, 20〉 114.2 7.85e−4 193.6 0.00133 0 |00, 20〉 → |00, 21〉 396.7 0.0776 – – → |20, 01〉 841.1 0.0235 – – + → |00, 30〉 51.92 5.37e−4 131.3 0.00136 → |20, 10〉 246.5 8.41e−4 325.9 0.00111 100 0 |00, 10〉 → |00, 11〉 616.4 0.0452 – – → |10, 01〉 971.4 0.0221 – – + → |00, 20〉 131.4 4.52e−4 290.2 9.98e−4 0 |00, 20〉 → |00, 21〉 485.0 0.0626 – – → |20, 01〉 993.4 0.0195 – – + → |00, 30〉 60.57 3.13e−4 219.4 0.00114 → |20, 10〉 280.7 4.80e−4 439.5 7.51e−4 5 FINITE NUCLEUS MASS EFFECTS So far we have used the infinite ion mass approximation. In this section we shall evaluate the validity range of this approximation and suggest possible corrections. It is convenient to use the coordinate system which con- tains the centre-of-mass coordinate Rcm and the relative coordinates {rj} of the electrons with respect to the nu- cleus. Using a suitable canonical transformation, the Hamil- tonian H of an arbitrary atom or ion can be separated into three terms (Vincke & Baye 1988; Baye & Vincke 1990; Schmelcher & Cederbaum 1991): H1 which describes the motion of a free pseudo-particle with net charge Q and total mass M of the ion (atom), the coupling term H2 between the collective and internal motion, and H3 which describes the internal relative motion of the electrons and the nu- cleus. H1 and H2 are proportional to M −1, so they van- ish in the infinite mass approximation. It is important to note, however, that H3 (the only non-zero term in the in- finite mass approximation) also contains a term that de- pends on M−10 , where M0 ≈ M is the mass of the nucleus. Thus, there are two kinds of non-trivial finite-mass effects: the effects due to H1 + H2, which can be interpreted as caused by the electric field induced in the co-moving refer- ence frame, and the effects due to H3, which arise irrespec- tive of the atomic motion. Both kinds of effects have been included in calculations only for the H atom (Potekhin 1994; Potekhin & Pavlov 1997, and references therein) and He+ c© 2008 RAS, MNRAS 000, 1–14 Radiative transitions of the helium atom 9 Table 2. Bound-free transitions |b〉 → |f〉: The threshold photon energy h̄ωthr (in eV) and the fitting parameters A, B, and C used in the cross section fitting formulas [Eq. (48)]. All transitions from the initial states |00, 10〉 and |00, 20〉 are listed, for several magnetic field strengths B12 = B/(10 12 G). B12 σ |b〉 → |f〉 mi h̄ωthr A B C 1 0 |00, 10〉 → |00, 1k〉 1 159.2 0.96 0.093 1.43e6 → |10, 0k〉 0 283.2 0.89 0.20 8.83e5 + → |00, 2k〉 1 159.2 0.70 0.061 7.95e2 → |10, 1k〉 0 283.2 0.86 0.094 1.30e3 – → |00, 0k〉 1 159.2 0.62 0.030 8.89e2 0 |00, 20〉 → |00, 2k〉 2 116.0 1.00 0.062 1.78e6 → |20, 0k〉 0 289.2 0.88 0.22 8.71e5 + → |00, 3k〉 2 116.0 0.66 0.038 3.94e2 → |20, 1k〉 0 289.2 0.54 0.14 6.48e2 – → |00, 1k〉 2 116.0 0.62 0.029 5.82e2 5 0 |00, 10〉 → |00, 1k〉 1 268.2 0.86 0.061 8.39e5 → |10, 0k〉 0 456.4 0.69 0.16 4.60e5 + → |00, 2k〉 1 268.2 0.68 0.036 1.14e2 → |10, 1k〉 0 456.4 0.83 0.057 1.93e2 – → |00, 0k〉 1 268.2 0.60 0.020 1.36e2 0 |00, 20〉 → |00, 2k〉 2 201.2 0.92 0.039 1.11e6 → |20, 0k〉 0 466.5 0.65 0.18 4.39e5 + → |00, 3k〉 2 201.2 0.65 0.021 5.95e1 → |20, 1k〉 0 466.5 0.54 0.084 9.13e1 – → |00, 1k〉 2 201.2 0.61 0.015 7.82e1 10 0 |00, 10〉 → |00, 1k〉 1 331.1 0.82 0.051 6.58e5 → |10, 0k〉 0 552.5 0.63 0.15 3.51e5 + → |00, 2k〉 1 331.1 0.67 0.029 4.94e1 → |10, 1k〉 0 552.5 0.81 0.046 8.43e1 – → |00, 0k〉 1 331.1 0.59 0.016 6.00e1 0 |00, 20〉 → |00, 2k〉 2 251.6 0.88 0.033 8.77e5 → |20, 0k〉 0 564.9 0.59 0.16 3.31e5 + → |00, 3k〉 2 251.6 0.64 0.017 2.64e1 → |20, 1k〉 0 564.9 0.53 0.069 3.97e1 – → |00, 1k〉 2 251.6 0.61 0.012 3.25e1 50 0 |00, 10〉 → |00, 1k〉 1 523.3 0.73 0.034 3.74e5 → |10, 0k〉 0 834.2 0.54 0.11 1.96e5 + → |00, 2k〉 1 523.3 0.63 0.020 7.15e0 → |10, 1k〉 0 834.2 0.77 0.033 1.22e1 – → |00, 0k〉 1 523.3 0.57 0.012 8.94e0 0 |00, 20〉 → |00, 2k〉 2 409.1 0.79 0.021 5.02e5 → |20, 0k〉 0 853.0 0.50 0.13 1.83e5 + → |00, 3k〉 2 409.1 0.62 0.0104 4.04e0 → |20, 1k〉 0 853.0 *0.52 0.052 5.88e0 – → |00, 1k〉 2 409.1 0.59 0.0058 4.13e0 100 0 |00, 10〉 → |00, 1k〉 1 628.8 0.69 0.029 2.96e5 → |10, 0k〉 0 983.4 0.51 0.101 1.56e5 + → |00, 2k〉 1 628.8 0.62 0.019 3.12e0 → |10, 1k〉 0 983.4 0.75 0.031 5.33e0 – → |00, 0k〉 1 628.8 0.56 0.012 3.94e0 0 |00, 20〉 → |00, 2k〉 2 498.0 0.75 0.018 3.96e5 → |20, 0k〉 0 1008 0.47 0.12 1.45e5 + → |00, 3k〉 2 498.0 0.60 0.0092 1.81e0 → |20, 1k〉 0 1008 *0.50 0.050 2.60e0 – → |00, 1k〉 2 498.0 0.58 0.0042 1.69e0 c© 2008 RAS, MNRAS 000, 1–14 10 Z. Medin, D. Lai, and A. Y. Potekhin Figure 2. Total cross section σ0 versus photon energy for helium photoionization, from initial states (m1,m2) = (1, 0) (solid lines) and (2, 0) (dashed lines). The field strength is 1012 G. The dotted lines extending from each cross section curve represent the effect of magnetic broadening on these cross sections, as approximated in Eq. (55), for T = 104.5 K (steeper lines) and 106 K (flatter lines). Figure 3. Total cross section σ0 versus photon energy for helium photoionization, from initial states (m1,m2) = (1, 0) (solid lines) and (2, 0) (dashed lines). The field strength is 1014 G. The dotted lines extending from each cross section curve represent the effect of magnetic broadening on these cross sections, as approximated in Eq. (55), for T = 105.5 K (steeper lines) and 106 K (flatter lines). Figure 4. Total cross section σ+ versus photon energy for helium photoionization, from initial states (m1,m2) = (1, 0) (solid lines) and (2, 0) (dashed lines). The field strength is 1012 G. The dotted lines extending from each cross section curve represent the effect of magnetic broadening on these cross sections, as approximated in Eq. (55), for T = 106 K. Figure 5. The same as in Fig. 4, but for σ−. ion (Bezchastnov et al. 1998; Pavlov & Bezchastnov 2005). For the He atom, only the second kind of effects have been studied (Al-Hujaj & Schmelcher 2003a,b). c© 2008 RAS, MNRAS 000, 1–14 Radiative transitions of the helium atom 11 Figure 6. The same as in Fig. 4, but for B = 1014 G. Figure 7. The same as in Fig. 6, but for σ−. 5.1 Non-moving helium atom The state of motion of an atom can be described by pseu- domomentum K , which is a conserved vector since Q = 0 (e.g., Vincke & Baye 1988; Schmelcher & Cederbaum 1991). Let us consider first the non-moving helium atom: K = 0. According to Al-Hujaj & Schmelcher (2003a), there are trivial normal mass corrections, which consist in the ap- pearance of reduced masses me/(1 ± me/M0) in H3, and non-trivial specific mass corrections, which originate from the mass polarization operator. The normal mass corrections for the total energy E of the He state |m1ν1,m2ν2〉 can be described as follows: E(M0, B) = E(∞, (1 +me/M0)2B) 1 +me/M0 + h̄Ωc mj , (49) where Ωc = (me/M0)ωc (for He, h̄Ωc = 1.588B12 eV). The first term on the right-hand side describes the reduced mass transformation. The second term represents the energy shift due to conservation of the total z component of the angu- lar momentum. Because of this shift, the states with suffi- ciently large values of m1 +m2 become unbound (autoion- izing, in analogy with the case of the H atom considered by Potekhin et al. 1997). This shift is also important for radia- tive transitions which change (m1 + m2) by ∆m 6= 0: the transition energy h̄ωba is changed by h̄Ωc∆m. The dipole matrix elements Mba are only slightly affected by the normal mass corrections, but the oscillator strengths are changed with changing ωba according to Eq. (27). The energy shift also leads to the splitting of the photoionization thresh- old by the same quantity h̄Ωc∆m, with ∆m = 0,±1 de- pending on the polarization (in the dipole approximation). Clearly, these corrections must be taken into account, unless Ωc ≪ ωba or ∆m = 0, as illustrated in the last two columns of Table 1. The specific mass corrections are more difficult to evalu- ate, but they can be neglected in the considered B range. In- deed, calculations by Al-Hujaj & Schmelcher (2003a) show that these corrections do not exceed 0.003 eV at B 6 104B0. 5.2 Moving helium atom Eigenenergies and wave functions of a moving atom depend on its pseudomomentum K perpendicular to the magnetic field. This dependence can be described by Hamiltonian components (e.g., Schmelcher & Cederbaum 1991) H1 +H2 = K · (B × rj), (50) where is the sum over all electrons. The dependence on Kz is trivial, but the dependence on the perpendicular component K⊥ is not. The energies depend on the abso- lute value K⊥. For calculation of radiative transitions, it is important to take into account that the pseudomomen- tum of the atom in the initial and final state differ due to recoil: K ′ = K + h̄q. Effectively the recoil adds a term ∝ q into the interaction operator (cf. Potekhin et al. 1997; Potekhin & Pavlov 1997). The recoil should be neglected in the dipole approximation. The atomic energy E depends on K⊥ differently for different quantum states of the atom. In a real neutron star atmosphere, one should integrate the binding energies and cross sections over the K⊥-distribution of the atoms, in or- der to obtain the opacities.1 Such integration leads to the specific magnetic broadening of spectral lines and ioniza- tion edges. Under the conditions typical for neutron star at- mospheres, the magnetic broadening turns out to be much 1 For the hydrogen atom, this has been done by Pavlov & Potekhin (1995) for bound-bound transitions and by Potekhin & Pavlov (1997) for bound-free transitions. c© 2008 RAS, MNRAS 000, 1–14 12 Z. Medin, D. Lai, and A. Y. Potekhin larger than the conventional Doppler and collisional broad- enings (Pavlov & Potekhin 1995). At present the binding energies and cross sections of a moving helium atom have not been calculated. However, we can approximately estimate the magnetic broadening for T ≪ |(∆E)min|/kB, where (∆E)min is the energy difference from a considered atomic level to the nearest level admixed by the perturbation due to atomic motion, and kB is the Boltzmann constant. In this case, the K⊥-dependence of E can be approximated by the formula E(K⊥) = E(0) + , (51) where E(0) is the energy in the infinite mass approxima- tion and M⊥ = K⊥(∂E/∂K⊥) −1 is an effective ‘transverse’ mass, whose value (M⊥ > M) depends on the quantum state considered (e.g., Vincke & Baye 1988; Pavlov & Mészáros 1993). Generally, at every value of K⊥ one has a different cross section σ(ω,K⊥). Assuming the equilibrium (Maxwell– Boltzmann) distribution of atomic velocities, the K⊥- averaged cross section can be written as σ(ω) = E(0)− E(K⊥) σ(ω,K⊥) dE(K⊥) , (52) where Emin = −h̄ω. The transitions that were dipole-forbidden for an atom at rest due to the conservation of the total z-projection of an- gular momentum become allowed for a moving atom. There- fore, the selection rule ∆m = α [Eqs. (30)–(32)] does not strictly hold, and we must write σ(ω,K⊥) = σm′(ω,K⊥), (53) where the sum of partial cross sections is over all final quan- tum numbers m′ (with m′ > 0 and m′ 6= m2 for ∆ν = 0) which are energetically allowed. For bound-bound transi- tions, this results in the splitting of an absorption line at a frequency ωba in a multiplet at frequencies ωba + δmΩc + −M−1⊥ )K ⊥/2h̄, where δm ≡ m′−m−α and M⊥,m′ is the transverse mass of final states. For photoionization, we have the analogous splitting of the threshold. In particular, there appear bound-free transitions at frequencies ω < ωthr – they correspond to δm < K2⊥/(2M⊥h̄Ωc). Here, ωthr is the threshold in the infinite ion mass approximation, and one should keep in mind that the considered perturbation theory is valid for K2⊥/2M⊥ ≪ |(∆E)min| < h̄ωthr. Accord- ing to Eq. (53), σ(ω,K⊥) is notched at ω < ωthr, with the cogs at partial thresholds ωthr + δmΩc − K2⊥/(2M⊥h̄) (cf. Fig. 2 in Potekhin & Pavlov 1997). Let us approximately evaluate the resulting envelope of the notched photoionization cross section (53), assum- ing that the ‘longitudinal’ matrix elements [〈. . .〉 con- structions in Eqs. (30)–(32)] do not depend on K⊥. The ‘transverse’ matrix elements can be evaluated following Potekhin & Pavlov (1997): in the perturbation approxima- tion, they are proportional to |ξ||δm|e−|ξ| 2/2, where |ξ|2 = 0/(2h̄ 2). Then σ(ω < ωthr,K⊥) ≈ σ(ωthr, 0) exp ωthr − ω − h̄(ωthr − ω) , (54) where θ(x) is the step function. A comparison of this approx- imation with numerical calculations for the hydrogen atom (Potekhin & Pavlov 1997) shows that it gives the correct qualitative behaviour of σ(ω,K⊥). For a quantitative agree- ment, one should multiply the exponential argument by a numerical factor ∼ 0.5–2, depending on the state and po- larization. This numerical correction is likely due to the ne- glected K⊥-dependence of the longitudinal matrix elements. We assume that this approximation can be used also for the helium atom. Using Eq. (52), we obtain σ(ω) ≈ σ(ωthr) exp ωthr − ω − h̄(ωthr − ω) for ω < ωthr. Here the transverse mass M⊥ can be evaluated by treating the coupling Hamiltonian H2 as a perturbation, as was done by Pavlov & Mészáros (1993) for the H atom. Following this approach, retaining only the main perturba- tion terms according to the approximate orthogonality rela- tion (33) and neglecting the difference between M and M0, we obtain an estimate b(∆m=α) ba/(2ωba) 1 + ωba/Ωc , (56) where |a〉 is the considered bound state (|00, 10〉 or |00, 20〉 for the examples in Figs. 2–7) and |b〉 are the final bound states to which α = ± transitions |a〉 → |b〉 are allowed. According to Eq. (34), the numerator in Eq. (56) is close to m+ 1 for α = + and to m for α = −. For the transitions from the ground state with po- larization α = −, which are strictly forbidden in the infinite ion mass approximation, using the same approx- imations as above we obtain the estimate σ−(ω) ∝ σ+(ω)h̄ΩckBT/(kBT + h̄Ωc) Examples of the photoionization envelope approxima- tion, as described in Eq. (55) above, are shown in Figs. 2–7. In Figs. 6 and 7 (for B = 1014 G), in addition to the mag- netic broadening, we see a significant shift of the maximum, which originates from the last term in Eq. (49). Such shift is negligible in Figs. 4 and 5 because of the relatively small Ωc value for B = 10 12 G. Finally, let us note that the Doppler and collisional broadening of spectral features in a strong magnetic field can be estimated, following Pavlov & Mészáros (1993), Pavlov & Potekhin (1995) and Rajagopal et al. (1997). The Doppler spectral broadening profile is φD(ω) = − (ω − ω0) , (57) ∆ωD = ]−1/2 , (58) where θB is the angle between the wave vector and B. The collisional broadening is given by φcoll(ω) = Λcoll (ω − ω0)2 + (Λcoll/2)2 , (59) c© 2008 RAS, MNRAS 000, 1–14 Radiative transitions of the helium atom 13 h̄Λcoll = 4.8nea0r = 41.5 1024 cm−3 eV, (60) where ne is the electron number density and reff is an effective electron-atom interaction radius, which is about the quantum-mechanical size of the atom. The convolution of the Doppler, collisional and magnetic broadening pro- files gives the total shape of the cross section. For bound- free transitions, the Doppler and collisional factors can be neglected, but for the bound-bound transitions they give the correct blue wings of the spectral features (cf. Pavlov & Potekhin 1995). 6 CONCLUSION We have presented detailed numerical results and fitting for- mulae for the dominant radiative transitions (both bound- bound and bound-free) of He atoms in strong magnetic fields in the range of 1012 − 1014 G. These field strengths may be most appropriate for the identification of spectral lines observed in thermally emitting isolated neutron stars (see Sect. 1). While most of our calculations are based on the infinite- nucleus-mass approximation, we have examined the effects of finite nucleus mass and atomic motion on the opacities. 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Y., Arras P., 2005, ApJ, 628, 902 van Kerkwijk M. H., Kaplan D. L., Durant M., Kulkarni S. R., Paerels F., 2004, ApJ, 608, 432 van Kerkwijk M. H., Kaplan D. L., 2007, Ap&SS, 308, 191 Vincke M., Baye D., 1988, J. Phys. B., 21, 2407 Vincke M., Baye D., 1989, J. Phys. B., 22, 2089 Zane S., Turolla R., Stella L., Treves A., 2001, ApJ, 560, Zane S., Cropper M., Turolla R., Zampieri L., Chieregato M., Drake J. J., Treves A., 2005, ApJ, 627, 397 c© 2008 RAS, MNRAS 000, 1–14 Introduction Bound states and singly-ionized states of helium atoms in strong magnetic fields Bound states of the helium atom Continuum states of the helium atom Radiative transitions Bound-bound transitions Photoionization Results Fitting Formula Finite nucleus mass effects Non-moving helium atom Moving helium atom Conclusion

Recent observations of thermally emitting isolated neutron stars revealed spectral features that could be interpreted as radiative transitions of He in a magnetized neutron star atmosphere. We present Hartree-Fock calculations of the polarization-dependent photoionization cross sections of the He atom in strong magnetic fields ranging from 10^12 G to 10^14 G. Convenient fitting formulae for the cross sections are given as well as related oscillator strengths for various bound-bound transitions. The effects of finite nucleus mass on the radiative absorption cross sections are examined using perturbation theory.

Introduction Bound states and singly-ionized states of helium atoms in strong magnetic fields Bound states of the helium atom Continuum states of the helium atom Radiative transitions Bound-bound transitions Photoionization Results Fitting Formula Finite nucleus mass effects Non-moving helium atom Moving helium atom Conclusion

Astronomy & Astrophysics manuscript no. VVDS˙mf˙Pozzetti˙accepted c© ESO 2018 November 12, 2018 The VIMOS VLT Deep Survey ⋆ The Assembly History of the Stellar Mass in Galaxies: from the Young to the Old Universe L. Pozzetti1 , M. Bolzonella1 , F. Lamareille1,9,6, G. Zamorani1, P. Franzetti2, O. Le Fèvre3, A. Iovino4, S. Temporin4, O. Ilbert5, S. Arnouts3, S. Charlot6,7, J. Brinchmann8, E. Zucca1, L. Tresse3, M. Scodeggio2, L. Guzzo4, D. Bottini2, B. Garilli2, V. Le Brun3, D. Maccagni2, J.P. Picat9, R. Scaramella10,11, G. Vettolani10, A. Zanichelli10, C. Adami3, S. Bardelli1, A. Cappi1, P. Ciliegi1, T. Contini9, S. Foucaud12, I. Gavignaud13, H.J. McCracken7,14, B. Marano15, C. Marinoni16, A. Mazure3, B. Meneux2,4, R. Merighi1, S. Paltani17,18, R. Pellò9, A. Pollo3,19, M. Radovich20, M. Bondi10, A. Bongiorno15, O. Cucciati4,21, S. de la Torre3, L. Gregorini22,10, Y. Mellier7,14, P. Merluzzi20, D. Vergani2, and C.J. Walcher3 (Affiliations can be found after the references) Received 04 04 2007; accepted 18 08 2007 ABSTRACT We present a detailed analysis of the Galaxy Stellar Mass Function (GSMF) of galaxies up to z = 2.5 as obtained from the VIMOS VLT Deep Survey (VVDS). Our survey offers the possibility to investigate it using two different samples: (1) an optical (I-selected 17.5 < IAB < 24) main spectroscopic sample of about 6500 galaxies over 1750 arcmin 2 and (2) a near-IR (K-selected KAB < 22.34 & KAB < 22.84) sample of about 10200 galaxies, with photometric redshifts accurately calibrated on the VVDS spectroscopic sample, over 610 arcmin2. We apply and compare two different methods to estimate the stellar mass Mstars from broad-band photometry based on different assumptions on the galaxy star-formation history. We find that the accuracy of the photometric stellar mass is overall satisfactory, and show that the addition of secondary bursts to a continuous star formation history produces systematically higher (up to 40%) stellar masses. We derive the cosmic evolution of the GSMF, the galaxy number density and the stellar mass density in different mass ranges. At low redshift (z ≃ 0.2) we find a substantial population of low-mass galaxies (< 109M⊙) composed by faint blue galaxies (MI −MK ≃ 0.3). In general the stellar mass function evolves slowly up to z ∼ 0.9 and more significantly above this redshift, in particular for low mass systems. Conversely, a massive population is present up to z = 2.5 and have extremely red colours (MI−MK ≃ 0.7−0.8). We find a decline with redshift of the overall number density of galaxies for all masses (59± 5% for Mstars > 10 8M⊙ at z = 1), and a mild mass-dependent average evolution (‘mass-downsizing’). In particular our data are consistent with mild/negligible (< 30%) evolution up to z ∼ 0.7 for massive galaxies (> 6× 1010M⊙). For less massive systems the no-evolution scenario is excluded. Specifically, a large fraction (≥ 50%) of massive galaxies have been already assembled and converted most of their gas into stars at z ∼ 1, ruling out the ‘dry mergers’ as the major mechanism of their assembly history below z ≃ 1. This fraction decreases to ∼ 33% at z ∼ 2. Low-mass systems have decreased continuously in number density (by a factor up to 4.1 ± 0.9) from the present age to z = 2, consistently with a prolonged mass assembly also at z < 1. The evolution of the stellar mass density is relatively slow with redshift, with a decrease of a factor 2.3 ± 0.1 at z = 1 and about 4.5± 0.3 at z = 2.5. Key words. galaxies: evolution – galaxies: luminosity function, mass function – galaxies: statistics – surveys 1. Introduction One of the main and still open question of modern cos- mology is how and when galaxies formed and in particular when they assembled their stellar mass. There are growing but still controversial evidences in near-IR (NIR) surveys that luminous and rather massive old galaxies were quite common already at z ∼ 1 (Pozzetti et al. 2003, Fontana et al. 2004, Saracco et al. 2004, 2005, Caputi et al. 2006a) and up to z ∼ 2 (Cimatti et al. 2004, Glazebrook et al Send offprint requests to: Lucia Pozzetti e-mail: lucia.pozzetti@oabo.inaf.it ⋆ based on data obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, program 070.A-9007(A), and on data obtained at the Canada-France- Hawaii Telescope, operated by the CNRS of France, CNRC in Canada and the University of Hawaii 2004). These surveys indicate that a significant fraction of early-type massive galaxies were already in place at least up to z ∼ 1. Therefore they should have formed their stars and assembled their stellar mass at higher redshifts. As in the local universe, at z ≃ 1.5 these galaxies still dominate the near-IR luminosity function and stellar mass density of the universe (Pozzetti et al. 2003, Fontana et al. 2004, Strazzullo et al. 2006). These results favour a high-z mass assembly, in particular for massive galaxies, in apparent contradiction with the hierarchical scenario of galaxy for- mation, applied to both dark and baryonic matter, which predicts that galaxies form through merging at later cosmic time. In these models massive galaxies, in particular, assem- bled most of their stellar mass via merging only at z < 1 (De Lucia et al. 2006). From several observations it seems that baryonic matter has a mass-dependent assembly his- http://arxiv.org/abs/0704.1600v2 2 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History tory, from massive to small objects, (i.e. the ‘downsizing’ scenario in star formation, firstly defined by Cowie et al. 1996, is valid also for mass assembly), opposite to the dark matter (DM) halos assembly. The continuous merging of DM halos in the hierarchical models, indeed, should result in an ‘upsizing’ in mass assembly, with the most massive galaxies being the last to be fully assembled. If we trust the hierarchical ΛCDM universe, the source of this discrep- ancy between observations and simple basic models could be due to the difficult physical treatment of the baryonic component, such as the star formation history/timescale, feedback, dust content, AGN feedback or to a missing in- gredient in the hierarchical models of galaxy formation (for the inclusion of AGN feedback, see Bower et al. 2006, Kang et al. 2006, De Lucia & Blaizot 2007, Menci et al. 2006, Monaco et al. 2006; and see Neistein et al. 2006 for the de- scription of a natural downsizing in star formation in the hierarchical galaxy formation models and a recent review by Renzini 2007). Considering optically selected surveys, a strong number density evolution of early type galaxies has been recently reported from the COMBO17 and DEEP2 surveys (Bell et al. 2004, Faber et al. 2005), with a corresponding increase by a factor 2 of their stellar mass since z ∼ 1, possibly due to so called ‘dry-mergers’ (even if the observational results on major merging and dry-merging are still contra- dictory, see Bell et al. 2006, van Dokkum 2005, Lin et al. 2004 and Renzini 2007 for a summary). This is at variance with results from the VIMOS-VLT Deep Survey (VVDS, Le Fèvre et al. 2003b), conducted at greater depth and us- ing spectroscopic redshifts in a large contiguous area. From the VVDS, Zucca et al. (2006) found that the B-band lu- minosity function of early type galaxies is consistent with passive evolution up to z ∼ 1.1, while the number of bright (MBAB < −20) early type galaxies has decreased only by ∼ 40% from z ∼ 0.3 to z ∼ 1.1. Similarly, Brown et al. (2007), in the NOAO Deep Wide Field survey over ∼ 10 deg2, found that the B-band luminosity density of L∗ galax- ies increases by only 36 ± 13% from z = 0 to z = 1 and conclude that mergers do not produce rapid growth of lu- minous red galaxy stellar masses between z = 1 and the present day. The VVDS is very well suited for this kind of stud- ies, thanks to its depth and wide area, covered by multi- wavelength photometry and deep spectroscopy. The simple 17.5 < IAB < 24 VVDS magnitude limit selection is signif- icantly fainter than other complete spectroscopic surveys and allows the determination of the faint and low mass population with unprecedented accuracy. Most of the pre- vious existing surveys are instead very small and/or not deep enough, or based only on photometric redshifts. Given the still controversial results based on morphol- ogy or colour-selected early-type galaxies (see Franzetti et al. 2007 for a discussion on colour-selected contamina- tion), we prefer to study the total galaxy population us- ing the stellar mass content. Here we present results on the cosmic evolution of the Galaxy Stellar Mass Function (GSMF) and mass density to z = 2.5 in the deep VVDS spectroscopic survey, limited to 17.5 < IAB < 24, over ∼ 1750 arcmin2 and based on about 6500 galaxies with secure spectroscopic redshifts and multiband (from UV to near-IR) photometry. In addition, we derive the GSMF also for a K-selected sample based on about 6600 galax- ies (KAB < 22.34) in an area of 442 arcmin 2 and about 3600 galaxies in a deeper (KAB < 22.84) smaller area of 168 arcmin2, making use of photometric redshifts, accu- rately calibrated on the VVDS spectroscopic sample, and spectroscopic redshifts when available. Throughout the paper we adopt the cosmology Ωm = 0.3 and ΩΛ = 0.7, with h70 = H0/70 km s −1 Mpc−1. Magnitudes are given in the AB system and the suffix AB will be dropped from now on. 2. The First Epoch VVDS Sample The VVDS is an ongoing program aiming to map the evo- lution of galaxies, large scale structures and AGN through redshift measurements of ∼ 105 objects, obtained with the VIsible Multi-Object Spectrograph (VIMOS, Le Fèvre et al. 2003a), mounted on the ESO Very Large Telescope (UT3), in combination with a multi-wavelength dataset from radio to X-rays. The VVDS is described in detail in Le Fèvre et al. (2005). Here we summarize only the main characteristics of the survey. The VVDS is made of a wide part, with spectroscopy in the range 17.5 ≤ I ≤ 22.5 on 4 fields (∼ 2 × 2 deg2 each), and a deep part, with spectroscopy in the range 17.5 ≤ I ≤ 24 on the field 0226-04 (F02 hereafter). Multicolour photometry is available for each field (Le Fèvre et al. 2004). In particular, the B, V , R, I photometry for the 0226-04 deep field, covering ∼ 1 deg2, has been ob- tained at CFHT and is described in detail in McCracken et al. (2003). The photometric depth reached in this field is 26.5, 26.2, 25.9, 25.0 (50% completeness for point-like sources), respectively in the B, V , R, I bands. Moreover, U < 25.4 (50% completeness) photometry obtained with the WFI at the ESO-2.2m telescope (Radovich et al. 2004) and Ks band (hereafter K) photometry with NTT+SOFI at the depth (50% completeness) of 23.34 (Temporin et al. in preparation) are available for wide sub-areas of this field. Moreover, an area of about 170 arcmin2 has been covered by deeper J and K band observations with NTT+SOFI at the depth (50% completeness) of 24.15 and 23.84, re- spectively (Iovino et al. 2005). The deep F02 field has been observed also by the CFHT Legacy Survey (CFHTLS1) in several optical bands (u∗, g′, r′, i′, z′) at very faint depth (u∗ = 26.4, g′ = 26.3, r′ = 26.1, i′ = 25.9, z′ = 24.9, 50% completeness). Spectroscopic observations of a randomly selected sub- sample of objects in an area of ∼ 0.5 deg2, with an average sampling rate of about 25%, were performed in the F02 field with VIMOS at the VLT. Spectroscopic data were reduced with the VIMOS Interactive Pipeline Graphical Interface (VIPGI, Scodeggio et al. 2005, Zanichelli et al. 2005) and redshift mea- surements were performed with an automatic package 1 Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 3 Fig. 1. Upper panel: Comparison of photometric and spec- troscopic redshifts in theK-selected sample for objects with highly reliable (confidence level > 97% , i.e. about 1400 galaxies with flag=3, 4) spectroscopic redshifts. The accu- racy obtained is σ∆z = 0.02(1 + z) 2 (shown as solid lines) with only 3.7% of outliers, defined as the objects outside the region limited by the 2 dotted lines (zphoto = zspec ± 0.15(1 + zspec)) in the figure. Lower panel: Spectroscopic (solid line) and photometric (dotted line) redshift distribu- tion for the same comparison sample. (KBRED) and then visually checked. Each redshift mea- surement was assigned a quality flag, ranging from 0 (failed measurement) to 4 (100% confidence level); flag 9 indicates spectra with a single emission line, for which multiple red- shift solutions are possible. Further details on the quality flags are given in Le Fèvre et al. (2005). The analysis presented in this paper is based on the first epoch VVDS deep sample, which has been obtained from the first spectroscopic observations (fall 2002) on the field VVDS-02h, which cover 1750 arcmin2. 2.1. The I-selected Spectroscopic Sample In this study we use the F02-VVDS deep spectroscopic sam- ple, purely magnitude limited (17.5 ≤ I ≤ 24), in combi- nation with the multi-wavelength optical/near-IR dataset. From the total sample of 8281 objects with measured red- shift, we removed the spectroscopically confirmed stars and broad line AGN, as well the galaxies with low quality red- shift flag (i.e. flag 1), remaining with 6419 galaxy spec- tra with secure spectroscopic measurement (flags 2, 3, 4, 9), corresponding to a confidence level higher than 80%. Galaxies with redshift flags 0 and 1 are taken into account statistically (see Section 4 and Ilbert et al. 2005 and Zucca et al. 2006 for details). This spectroscopic sample has a median redshift of ∼ 0.76. Compared to previous optically selected samples, the VVDS has not only the advantage of having an unprecedented high fraction of spectroscopic redshifts (compared, for example, to the purely photomet- ric redshifts as in COMBO17, Wolf et al. 2003 and Borch et al. 2006 for the MF), but also of being purely magnitude selected (17.5 < I < 24), differently, for example, from the DEEP2 (Bundy et al. 2006 for the MF) survey, which has a colour-colour selection. Moreover, the VVDS covers an area from 10 to 40 times wider than the GOODS-MUSIC field (Fontana et al. 2006) and the FORS Deep Field (FDF, Drory et al. 2005), respectively. 2.2. The K-selected Photometric Sample A wide part of the VVDS-02h field (about 623 arcmin2) has been observed also in the near-IR (Iovino et al. 2005, Temporin et al. in preparation). This allows us to build a K-selected sample with a total area of 610 arcmin2 (after excluding low-S/N borders): 442 arcmin2 are 90% complete to K < 22.34, while 168 arcmin2 are 90% complete to K < 22.84 (equivalent to KVega = 21). This sample consists of 11221 objects, of which 2882 have VVDS spectroscopy. In particular, the deep sample (K < 22.84) consists of 3821 objects, of which 749 have VVDS spectroscopy, and 596 of them are galaxies with a secure spectroscopic identification (flags 2, 3, 4, 9). This latter deep sample is more than one magnitude deeper than the samples from the K20 spectroscopic survey (Cimatti et al. 2002) and the MUNICS survey (Drory et al. 2001). Additionally, the total K-selected sample covers an area more than 10 times wider than the K20 and the GOODS- CDFS sample used by Drory et al. (2005) and 4 times wider than the GOODS-MUSIC field (Fontana et al. 2006). Since the spectroscopic sampling of the K-selected sam- ple is less than satisfactory, we take advantage of the high quality photometric redshifts (zphoto). The method and the calibration are presented and discussed in Ilbert et al. (2006). The comparison sample contains 3241 ac- curate spectroscopic redshifts (confidence level > 97% , i.e. flag=3, 4) up to I = 24 obtaining a global accu- racy of σ∆z/(1+z) = 0.037 with only 3.7% of outliers. Also in the K-selected photometric sample the agreement be- tween photometric and highly reliable spectroscopic red- shifts (about 1400) is excellent (Figure 1). We note, how- ever, a non-negligible number of catastrophic solutions with zphoto ∼ 1.2 and zspec ∼ 1 which could introduce a bias at high-redshift (see also discussion in Section 2.3). Even if we cannot rely on a wide spectroscopic comparison sample at high-z, the number of galaxies with zphoto > 1.2 is simi- lar or only slightly higher (about 20%) than the number of galaxies with zspec > 1.2 (63 vs. 51, see Figure 1) and have very similar fluxes and colors. For this reason we do not expect that our results on the mass function and the mass density will be strongly biased by the effect of catas- trophic redshifts (see Section 4). Furthermore, at high-z the dispersion between photometric and spectroscopic redshifts increases, but not drammatically, to σ∆z/(1+z) ≃ 0.05, 0.06 4 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History at z > 1.2, 1.4. Over the whole redshift range it can be represented with σ∆z ≃ 0.02(1 + z) 2 (shown in Figure 1). For the whole K-selected sample, the median errors on photometric redshifts, based on χ2 statistics, are σzphoto = 0.06 (0.04 at z < 1, increasing to 0.14 at z > 1.5). As ex- pected, there is also an increase of σzphoto for the faintest objects, but this increase is only about 0.02 in the faintest magnitude bin. As previously noted by Ilbert et al. (2006) the statistical errors are consistent with σ∆z and could be used as an indication of their accuracy. We will discuss in the following sections the effects of these uncertainties and conclude that they do not affect significantly our conclu- sions. We note, moreover, that the K-selected sample se- lects a different population, in particular of Extremely Red Objects (EROs) at zphoto > 1 (see Section 2.3 and Fig. 3), compared to the I-selected sample used to calibrate the de- rived photometric redshift. Actually, photometric redshifts greater than 0.8-1.0 for the EROs population have been confirmed spectroscopically with very low contamination of low-z objects (Cimatti et al. 2002). Moreover, the near-IR bands are crucial to constrain photometric redshifts in the redshift desert since the J-band is sensitive to the Balmer break up to z = 2.5. Indeed Ilbert et al. (2006) obtain for the deep sample at K < 23 the most reliable photometric redshifts on this sub-sample with only 2.1% of outliers and σ∆z/(1+z) = 0.035 (see their figure 13). In this paper we therefore use photometric redshifts for the whole K-selected photometric sample and the highly reliable spectroscopic redshifts when available. In order to select galaxies from the total K-selected photometric sample, we have used a number of photomet- ric methods to remove candidate stars, as described be- low. Some of the possible criteria to select stars are: (i) the CLASS STAR parameter given by SExtractor (Bertin & Arnouts 1996), providing the “stellarity-index” for each object, reliable up to I ≃ 21; (ii) the FLUX RADIUS K parameter, computed by SExtractor from the K band im- ages, which gives an estimate of the radius containing half of the flux for each object; this can be considered a good criterion to isolate point-like sources up to K ≃ 19 (see Iovino et al. 2005); (iii) the BzK criterion, proposed by Daddi et al. (2004a), with stars characterized by colours z − K < 0.3(B − z) − 0.5; (iv) the χ2 of the SED fitting carried out during the photometric redshift estimate (Ilbert et al. 2006), with template SEDs of both stars and galaxies. To efficiently remove stars in the whole magnitude range of our sample, avoiding as much as possible to lose galaxies, we decided to use the intersection of the first three crite- ria. We therefore selected as stars the objects fulfilling all the constraints (i) CLASS STAR ≥ 0.95 for I < 22.5 or CLASS STAR ≥ 0.90 if I > 22.5, (ii) FLUX RADIUS K < 3.4 and (iii) z − K < 0.3(B − z) − 0.5. When it was not possible to apply criterion (iii), because of non detec- tion either in the B or z filters, we used criterion (iv) in its place. Furthermore, we added to the sample of candidate stars also the objects with K < 16 and FLUX RADIUS K < 4, to be sure to exclude from the galaxy sample these sat- urated point-like objects. The final sample consists of 653 candidate stars, which we have removed from the sample in the following analysis. Comparing to the spectroscopic subsample (we remind that stars were not excluded from the spectroscopic targets of VVDS), we found about 87% of efficiency to photometrically select stars, i.e. only 28 out Fig. 2. Redshift distributions for the K-selected photomet- ric sample (filled histogram) and for the I-selected spectro- scopic sample (empty histogram). of the 214 spectroscopic stars have not been selected in this way, and only 3 (1.4%) with highly reliable spectroscopic flag (3, 4), whereas 21 spectroscopic extragalactic objects (less than 1%) fall inside the candidate star sample. Three of them are broad line AGN and the others are all compact objects, most of them with redshift flags 1 or 2 and only one with flag 3. This latter object has not been eliminated from the galaxy sample. We have furthermore removed from the galaxy sample the spectroscopically confirmed AGNs and the three secure spectroscopic stars which were not removed with the photometric criteria. The final K-selected sample consists of 10160 galaxies with either photometric redshifts or highly reliable spec- troscopic redshifts, when available, in the range between 0 and 2.5: 6720 galaxies in the shallow K < 22.34 area of 442 arcmin2 and 3440 galaxies in the deeper area (K < 22.84) of 168 arcmin2. 2.3. Comparison of the Two Samples As shown in Fig. 2, the redshift distribution in the K- selected sample peaks at higher redshift than in the I- selected spectroscopic sample, with the two median red- shifts being 0.91 and 0.76, respectively. Even if we cannot rely on a wide spectroscopic comparison sample at high-z, we have better investigated the reliability of the high-z tail in the K-selected sample in term of fraction and colors. We found indeed that at K < 22 the fraction of objects with z > 1, 1.5 (35, 13% respectively) is similar to previous spec- troscopic (K20 survey, see Cimatti et al. 2002) or photomet- ric studies (Somerville et al. 2004). Moreover, we have used the BzK color-color diagnostic proposed and calibrated on a spectroscopic sample to cull galaxies at 1.4 < z < 2.5 (Daddi et al. 2004). We found that most (92%) of the galax- L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 5 Fig. 3. Colour-magnitude diagram and colour distribution at different redshifts for the K-selected photometric sam- ple (filled squares and histogram) and for the I-selected spectroscopic sample (open circles and empty histogram) ies at 1.5 < zphoto < 2.5 lie in the high-z region of the BzK diagram. We conclude that our K-selected sample shows no indi- cation of significant bias in its high-redshift tail. The global (I −K) colour distributions of the two samples are similar to each other up to z ∼ 1.2 (see right panels in Fig. 3), but they are significantly different at higher redshift. At z > 1.2 the I-selected sample misses many red galaxies fainter than the I limit, most of them being Extremely Red Objects (EROs: defined as objects with colours I−K > 2.6), which are instead included in the K sample (∼ 81% of EROs in the deep K < 22.84 sample have I > 24). For this rea- son the K-selected sample is more adequate to study the massive tail of the GSMF at high-z. Vice versa, the K- selected sample misses at all redshifts a number of faint blue galaxies, which are included in the I-selected spectro- scopic sample (see left panels in Fig. 3). These faint blue galaxies are important in the estimate of the low-mass tail of the GSMF. 3. Estimate of the Stellar Masses The rest-frame near-IR light has been widely used as a tracer of the galaxy stellar mass, in particular for lo- cal galaxies (e.g. Gavazzi et al. 1996; Madau, Pozzetti & Dickinson 1998, Bell & de Jong 2001). However, an accurate estimate of the galaxy stellar mass at high z, where galax- ies are observed at widely different evolutionary stages, is more uncertain because of the variation of the Mstars/LK ratio as a function of age and other parameters of the stel- lar population, such as the star formation history and the metallicity. The use of multiband imaging from UV to near- IR bands is a way to take into account the contribution to the observed light of both the old and the young stellar populations in order to obtain a more reliable estimate of the stellar mass. However, even stellar masses estimated using the fit to the multicolour spectral energy distribution (SED) are model dependent (e.g. changing with different assumptions on the initial mass function, IMF) and subject to various degeneracies (age – metallicity – extinction). In order to re- duce such degeneracies we have used a large grid of stellar population synthesis models, covering a wide range of pa- rameters, in particular in star formation histories (SFH). Indeed, in the case of real galaxies the possibly complex star-formation histories and the presence of major and/or minor bursts of star formation can affect the derived mass estimate (see Fontana et al. 2004). We have applied and compared two different methods to estimate the stellar masses from the observed magnitudes (using 12 photometric bands from u∗ to K), that are based on different assumptions on the star-formation history. For both of them we have adopted the Bruzual & Charlot (2003; BC03 hereafter) code for spectral synthesis models, in its more recent rendition, using its low resolution version with the “Padova 1994” tracks. Different models have also been considered, e.g. Maraston 2005, and Pégase models (Fioc and Rocca-Volmerange 1997). The results obtained with these models are compared with those obtained with the BC03 models at the end of Section 3.1. Since most of previous studies at high-z assumed models with exponentially decreasing SFHs, we have used the same simple smooth SFHs (see Section 3.1) in order to compare our results with those of previous surveys. In addition, to further test the uncertainties in mass determination, we have used models with complex SFHs (see Section 3.2), in which secondary bursts have been added to exponentially decreasing SFHs. These models have been widely used in studies of SDSS galaxies (see Kauffmann et al. 2003 and Salim et al. 2005 for further details). Table 1 summarizes the model parameters used in the 2 methods described in the following sections. In our analysis we have adopted the Chabrier IMF (Chabrier et al. 2003), with lower and upper cutoffs of 0.1 and 100 M⊙. Indeed, all empirical determinations of the IMF indicate that its slope flattens below ∼ 0.5 M⊙ (Kroupa 2001, Gould et al. 1996, Zoccali et al. 2000) and a similar flattening is required to reproduce the observed Mstars/LB ratio in local elliptical galaxies (see e.g. Renzini 2005). As discussed extensively by Bell et al. (2003), the Salpeter IMF (Salpeter 1955) is too rich in low mass stars to satisfy dynamical constraints (Kauffmann et al. 2003, Kranz et al. 2003). Moreover, di Serego Alighieri et al. (2005) show a rather good agreement between dynamical masses and stellar masses estimated with the Chabrier IMF at z ∼ 1. Specifically, this is true at least for high-mass el- liptical galaxies, less affected than lower-mass galaxies by uncertainties in the estimate of their dynamical mass due to possibly substantial rotational contribution to the observed velocity dispersion. At fixed age the masses obtained with the Chabrier IMF are smaller by a factor ∼ 1.7, roughly independent of the age of the population, than those derived with the classical Salpeter IMF, used in several previous works that we shall compare with (e.g. Brinchmann & Ellis 2000; Cole et al. 2001; Dickinson et al. 2003; Fontana et al. 2004). We have checked this statement in our sample, finding a systematic median offset of a factor 1.7 and a very small dispersion 6 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History (σ = 0.082 dex) in the masses derived with the two different IMFs. Since this ratio is approximately constant for a wide range of star formation histories (SFH), the uncertainty in the IMF does not introduce a fundamental limitation with respect to the results we will discuss in the following Sections. Even if the absolute value of the mass estimate is uncertain, the use of Salpeter or Chabrier IMFs does not introduce any significant difference in the relative evolution with redshift of the mass function and mass density. One possible limitation of our approach to derive stellar masses in our sample is the contamination by narrow-line AGN (broad line AGN have been already excluded, see Section 2.1). From the available spectroscopic diagnostic, in the I-selected spectroscopic sample at the mean red- shift z ≃ 0.7, we found that the contamination due to type II AGN is less than 10%. Recently, several studies (Papovich et al. 2006, Kriek et al. 2007, Daddi et al. 2007) suggest that the fraction of type II AGN increases with redshift and stellar mass. According to Kriek et al. (2007) at 2 < z < 2.7 and K < 21.5 the fraction is about 20% for massive (2.9 × 1011M⊙ for a Salpeter IMF) galaxies. To derive the contribution of type II AGN to the massive tail of the MF is beyond the scope of this paper. However we note, as shown also by the above studies, that for most of these objects the optical light is dominated by the in- tegrated stellar emission. Therefore, both our photometric redshift and mass estimates are likely to be approximately correct also for them. 3.1. Smooth SFHs Consistently with previous studies, we have used synthetic models with smooth SFH models (exponentially decreas- ing SFH with time scale τ : SFR(t) ∝ exp(−t/τ)) and a best-fit technique to derive stellar masses from multicolour photometry. To this purpose we have developed the code HyperZmass, a modified version of the public photomet- ric redshift code HyperZ (Bolzonella et al. 2000): like the public version, HyperZmass uses the SED fitting technique, computing the best fit SED by minimizing the χ2 between observed and model fluxes. We used models built with the Bruzual & Charlot (2003) synthetic library. When the red- shift is known, either spectroscopic or photometric, the best fit SED and its normalization provide an estimate of the stellar mass contained in the observed galaxy. In particu- lar, we estimate the stellar mass content of the galaxies, derived by BC03 code, by integrating the star formation history over the galaxy age and subtracting from it the “Return fraction” (R) due to mass loss during the stellar evolution. For a Chabrier IMF, this fraction is already as high as ∼ 40% at an age of the order of 1 Gyr and ap- proaches asymptotically about 50% at older ages. The parameters used to define the library of synthetic models are listed in Table 1. Similar parameters have been used in Fontana et al. (2004). The Calzetti (2000) extinction law has been used. Following that paper, we have excluded from the grid some models which may be not physical (e.g. those implying large dust extinctions, AV > 0.6, in absence of a significant star-formation rate, Age/τ > 4, see Table 1). To better match the ages of early-type galaxies in the local universe and following SDSS studies, we also removed models with τ < 1 Gyr and with star formation starting at z < 1. Table 1. Parameters Used for the Library of Template Method Smooth SFHs Complex SFHs IMF Chabrier Chabrier SFR τ (Gyr) [0.1,∞]a [1,∞] log(Age)b (yr) [8, 10.2] [8, 10] burst age (yr) − [0, 1010] burst fraction − [0, 0.9] Metallicities Z⊙ [0.1Z⊙, 2Z⊙] Extinction Calzetti law Charlot&Fall model (n = 0.7, µ ∈ [0.1, 1]) Dust content AcV ∈ [0, 2.4] τV ∈ [0, 6] a τ < 1 if star formation starts at z < 1. b At each redshift, galaxies are forced to have ages smaller than the Hubble time at that redshift. c AV < 0.6 if Age/τ > 4. We find that the “formal” typical 1σ statistical errors (defined as the 68% range as derived from the χ2 statis- tics) on the estimated masses, not taking into account the error on the estimate of the photometric redshift for the K-selected sample, are of the order of 0.04 dex for the K-selected sample and 0.05 for the I-selected sample. A more reliable estimate of the errors has been obtained us- ing HyperZ to simulate catalogs to the same depth of our sample (see Bolzonella et al. 2000). Using all 12 photo- metric bands (from u∗ to K), available for a subset of our data, and realistic photometric errors, the recovered stellar masses reproduce the input masses with no significant offset and a dispersion of 0.12 dex up to z ∼ 3. For comparison, using only the optical bands (from u∗ to z′) the disper- sion increases to ∼ 0.49 dex at z > 1. The best fit masses obtained from input simulations built using randomly all available metallicities and analyzed only with solar metal- licity models are not significantly shifted from the input masses, but the dispersion increases from 0.12 dex to 0.21 dex. These dispersions, computed using a 4σ clipping, pro- vide an estimate of the minimum, intrinsic uncertainties of this method at our depth. For the K-selected photometric sample further uncertainties in the fitting technique are due to the photometric redshift accuracy (σ∆z ≃ 0.02(1 + z) up to z = 2.5) which corresponds on average to about 0.12 dex of uncertainty in mass, being larger at low redshift (∼ 0.2 dex at z < 0.4) than at high-z (∼ 0.10 dex at z = 2). Although in principle the best-fitting technique pro- vides estimates also for age, metallicity, dust content and SFH timescale, our simulations show that on average all these quantities are much more affected by degeneracies and therefore less constrained than the stellar mass. In addition, we have compared our derived masses with those obtained by using different population synthesis mod- els, such as Pégase and Maraston (2005) models. In partic- ular, Maraston (2005) models include the thermally pulsing asymptotic giant branch (TP-AGB) phase, calibrated with L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 7 local stellar populations. This stellar phase is the dominant source of bolometric and near-IR energy for a simple stellar population in the age range 0.2 to 2 Gyr. We have tested the differences with BC03 models using the I-selected spec- troscopic sample and we found only a small but systematic shift (∼ −0.14 dex and a similar dispersion) up to z ∼ 1.2 both with and without the use of near-IR photometry. On the contrary, masses derived using Pégase models and sim- ilar SFHs have instead no significant offset. At higher redshifts the differences between our esti- mated masses and those obtained with Maraston models in the K-selected spectroscopic subsample are slightly smaller and even smaller in the K-selected photometric sample (∼ −0.11,−0.08, respectively). This differences are smaller than that found by Maraston et al. (2006), ∼ −0.2, in their SED fitting (from B up to Spitzer IRAC and MIPS bands) of a few high redshift passive galaxies with typical ages in the range 0.5 – 2.0 Gyr, selected in the Hubble Ultra Deep Field (HUDF). This difference between our results and those of Maraston et al. could be due to a combina- tion of effects, such as the absence in our photometric data of mid-IR Spitzer photometry, which at these redshifts is sampling the rest frame near-IR part of the SED, mostly in- fluenced by the TP-AGB phase, and also to the wide range of complex stellar populations in our sample, in which the effect of the TP-AGB phase may be diluted by the SFH. 3.2. Complex SFHs Real galaxies could have undergone a more complex SFH, in particular with the possible presence of bursts of star formation on the top of a smooth SFH. Thus, we have com- puted masses also following a different approach, which has been intensively used in previous studies of SDSS galaxies (e.g. Kauffmann et al. 2003, Brinchmann et al. 2004, Salim et al. 2005, Gallazzi et al. 2005). In this approach we pa- rameterize each SF history in terms of two components: an underlying continuous model, with an exponentially de- clining SF law (SFR(t) ∝ exp(−t/τ)), and random bursts superimposed on it. We assume that random bursts occur with equal probability at all times up to galaxy age. They are parameterized in terms of the ratio between the mass of stars formed in the burst and the total mass of stars formed by the continuous model over the age. This ratio is taken to be distributed between 0.0 and 0.9. During a burst, stars are assumed to form at a constant rate for a time dis- tributed uniformly in the range 30 – 300 Myr. The burst probability is set so that 50% of the galaxies in the library have experienced a burst in the past 2 Gyr. Attenuation by dust is described by a two-component model (see Charlot & Fall 2000), defined by two parameters: the effective V - band absorption optical depth τV affecting stars younger than 10 Myr and arising from giant molecular clouds and the diffuse ISM, and the fraction µ of it contributed by the diffuse ISM, that also affects older stars. We take τV to be distributed between 0 and 6 with a broad peak around 1 and µ to be distributed between 0.1 and 1 with a broad peak around 0.3. Finally, our model galaxies have metallic- ities uniformly distributed between 0.1 and 2 Z⊙. The model spectra are computed at the galaxy redshift and in each of them we measure the k-shifted model magni- tudes for each VVDS photometric band. We also force the age of all models in a specific redshift range to be smaller than the Hubble time at that redshift. The model SEDs Fig. 4. Effect of NIR photometry in the mass determina- tion: ratio between masses estimated without and with NIR photometry vs. mass determined without NIR photometry. The data have been splitted into different redshift ranges. Left: masses determined using smooth SFHs. Right: The same, but using complex SFHs are then scaled to each observed SED with a least squares method and the same scaling factor is applied to the model stellar mass. We compare the observed to the model fluxes in each photometric band and the χ2 goodness of fit of each model determines the weight (∝ exp[−χ2/2]) to be assigned to the physical parameters of that model when building the probability distributions for each parameter of any given galaxy. The probability distribution function (PDF) of a given physical parameter is thus obtained from the distribution of the weights of all models in the library at the specified redshift. We characterize the PDF using its median and the 16 – 84 percentile range (equivalent to ±1σ range for Gaussian distributions), and also record the χ2 of the best-fitting model. Similarly to what has been done for the models with smooth SFH (see Section 3.1), also in this case the stel- lar mass content of galaxies is derived by subtracting the return fraction R from the total formed stellar mass. We find that the average “formal” 1σ error (defined as half of the 16 – 84 percentile range) on the estimated masses is of the order of 0.09 dex for the K-selected sample. The av- erage error increases with redshift from ∼ 0.06 dex at low redshift to ∼ 0.11 dex at 1 < z < 2 and decreases with in- creasing mass from ∼ 0.08 dex for logM < 10 to ∼ 0.05 for logM > 10 at z ≃ 0.7. In the K-selected sample the photo- metric redshift accuracy induced a further uncertainty on the mass of the order of 30% up to z > 1.5. In the I-selected sample, where near-IR photometry is not always available, the typical error on the mass is larger and is of the order of ∼ 0.13 dex. 8 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 3.3. Effect of NIR Photometry For about half of the sources in the I-selected sample only optical photometry is available. We have therefore used the results obtained for theK-selected spectroscopic subsample to better understand the reliability of the mass estimates in the whole I-selected sample and quantify potential system- atic effects. We found that the mass estimates derived using only optical bands are on average in rather good agreement with those obtained using also NIR bands up to z ∼ 1.2. In absence of NIR bands the galaxy stellar masses tend to be only slightly overestimated, with a median shift < 0.1 dex; this is due to the fact that already at z = 0.4, for example, the z′-band (the reddest band used in the fit in absence of NIR) samples the R-band rest-frame and therefore the SED fitting is less reliable for the estimate of the stellar masses. There is however a significant fraction of the galaxies for which the ratio between the two masses is higher than a factor of three (see upper panels of Fig.4). This fraction of galaxies with significantly discrepant mass estimates is ∼ 5% for the models with smooth SFH and ∼ 9 % for the models with complex SFH. At higher redshifts, where our reddest optical band, i.e. the z-band, is sampling the rest-frame spectrum bluewards of the 4000 Å break, the comparison of the two sets of mass estimates (i.e. with and without near-IR photometry) is significantly worse. Not only the median shift increases significantly, but also the ratio of the two sets of masses is significantly correlated with the mass derived without using NIR photometry (see lower panels of Fig. 4). For this reason, we have decided to use the whole I-selected VVDS spectroscopic sample only up to z ∼ 1.2, whereas at higher redshifts we use as reference the K-selected photometric sample. As shown in the upper panels of Fig.4, the ratio between the masses computed without and with NIR photometry has a non-negligible dispersion also for z < 1.2, with the masses computed without NIR photometry being higher on average. In order to statistically correct for this effect, we have performed the following Monte Carlo simulation. For each galaxy without near-IR photometry in the I-selected spectroscopic sample we have applied a correction factor to its estimated mass. This correction factor has been de- rived randomly from the observed distribution, at the mass of each galaxy, of the ratios of the masses with and with- out NIR photometry. The effect on the mass function of using these “statistically corrected” masses is shown and discussed in Sect. 4. 3.4. Comparison of the Masses Obtained with the Two Methods In this section we compare the mass estimates we obtained using the two different methods described above for the VVDS galaxies. Since the ratio of the two estimates is al- most independent of the mass, in Fig. 5 we show the his- tograms of this ratio, integrated over all masses, for two dif- ferent redshift bins. In the upper panel (z < 1.2), both data from the I-selected and the K-selected samples are shown; in the lower panel only data from the K-selected sample are shown, since the I-selected sample is not used to derive the mass function in this redshift range. Gaussian curves, representing the bulk of the population, are drawn on the Fig. 5. Histograms of the ratio of the masses derived with the smooth and the complex star formation histories. In the upper panel (z < 1.2) both data from the I-selected and the K-selected samples are shown; in the lower panel (z > 1.2) only data from the K-selected sample are shown, since the I-selected sample is not used to derive the mass function in this redshift range. top of each histogram. The parameters of each Gaussian are reported in the figure. The global comparison of the two sets of masses is rather satisfactory, even if it shows a systematic shift, larger for theK-selected sample, between the two sets of masses, with the ‘smooth SFH’ masses being on average smaller than the ‘complex SFH’ masses. The values of σ of these Gaussians are similar, of the order of 0.13 dex. However, in two of the three cases (i.e. the I-selected sample at low redshift and the K-selected sample at high redshift) the distributions of the ratio of the masses appear to be asymmetric, with tails which are not well represented by a Gaussian distribution. The fractions of these “outliers” are given in Fig. 5. This tail is particularly significant for the I-selected sample, for which there are galaxies with the ratio between the two masses higher than 3 and in a few cases reaching a value of We have analyzed the effect of the different parameters used in the two methods. The impact of different extinction curves on the mass estimates has already been investigated by Papovich et al. (2001), Dickinson et al. (2003), Fontana et al. (2004), and found to be small. We have repeated the same exercise for the two different dust attenuation models adopted, finding that for a given SFH the mass estimated with different extinction laws are similar, with an average shift of 0.02 dex. Analyzing in some detail the properties of the galaxies which are in the extended tail of large mass ratios for the I- selected sample at low redshift (see the upper panel in Fig. 5), we found that, even if many of them do not have near- IR photometry and therefore their mass is more uncertain L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 9 (see previous section), on average they are characterized by blue colours in the bluer bands (B − I) and red colours in the redder bands (I − z or I − K). Because of this, they have been fitted typically with low values of the popula- tion age by HyperZmass with a smooth SFH, while the estimate obtained with a complex SFH corresponds to an older, and therefore more massive, population plus a more recent burst with a mass fraction in the burst of the order of fburst < 0.15. Vice versa the low ratio outliers in the K- selected sample at high redshift have been fitted typically with moderately higher dust content by HyperZmass than with complex SFHs. Finally, we conclude that the main dif- ferences between the two methods to determine the masses are largely due to different assumed SFHs and, in particu- lar, to the secondary burst component allowed in the model with complex SFHs. To summarize, we have explored in detail two differ- ent methods and a wide parameter space (see Table 1) to estimate the stellar mass content in galaxies in order to better understand the uncertainties in the photomet- ric stellar mass determinations. Indeed, a good estimate of the intrinsic errors may be critical for the GSMF measure- ment and interpretation. We found that, within a given assumption on the SFH, the accuracy of the photomet- ric stellar mass is overall satisfactory, with intrinsic un- certainties in the fitting technique of the order of ∼ 30%, in agreement with similar results in the K20 (Fontana et al. 2004), HDFN (Dickinson et al. 2003) and HDFS (Fontana et al. 2003) at the same redshifts. These errors are smaller than the estimates at higher redshift (z ≃ 3) (Papovich et al. 2001, Shapley et al. 2001), since at our av- erage redshifts (z ≃ 0.7−1) we can rely on a better sampling of the rest-frame near-IR part of the spectrum, while are similar to the uncertainties estimated at high-z using IRAC data by Shapley et al. (2005). For theK-selected photomet- ric sample the uncertainties in the stellar mass due to the photometric redshift errors are on average of the order of 30% up to z > 1.5, giving a total fitting uncertainty up to 45% at z > 1.5. Finally, systematic shifts, mainly due to different assumptions on the SFHs, can be as large as ∼ 40% over the entire redshift range 0.05 < z < 2.5 when NIR photometry is available. The uncertainty in the de- rived masses is obviously higher also at low redshift when NIR photometry is not available and in this case it becomes extremely large at z > 1.2 (see Fig.4). For what concerns the absolute value of the mass, its uncertainty is mainly due to the assumptions on the IMF and it is within a factor of 2 for the typical IMFs usually adopted in the literature. In the following sections we discuss in some detail the effects of the two methods on the derivation of the GSMF. 3.5. Massive Galaxies at z > 1 Figure 6 shows the stellar masses for the 2 samples (I and K-selected) derived using the smooth SFHs. It is inter- esting to note the presence of numerous massive objects (logM > 11) at all redshifts and up to z = 2.5. High- z massive galaxies have been already observed in previous surveys (Fontana et al. 2004, Saracco et al. 2005, Cimatti et al. 2004, Glazebrook et al 2004, Fontana et al. 2006, Trujillo et al. 2006). Here the relatively wide area (the K-selected sample is more than 10 times wider and from 0.5 to 1 mag- nitude deeper than the K20 survey) allows to better sam- Fig. 6. Stellar Mass as a function of redshift for the I- selected spectroscopic (left) and for the K-selected photo- metric (right) samples for smooth SFHs. ple the massive tail of the population. We note that mas- sive galaxies have typically redder optical-NIR rest-frame colours (〈MI −MK〉 ≃ 0.7) compared to the whole popu- lation (〈MI −MK〉 ≃ 0.5), consistently with the idea that massive galaxies host the oldest stellar population. Further analysis of the stellar population properties and spectral features of massive galaxies, as well as of red objects will be presented in forthcoming papers (Lamareille et al. in preparation, Vergani et al. 2007, Temporin et al. in prepa- ration). 4. Mass Function Estimate Once the stellar mass has been estimated for each galaxy in the sample, the derivation of the corresponding Galaxy Stellar Mass Function (MF) follows the traditional tech- niques used for the computation of the luminosity function. Here we apply both the classical non-parametric 1/Vmax formalism (Schmidt 1968, Felten 1976) and the parametric STY (Sandage, Tammann & Yahil 1979) method to esti- mate best-fit Schechter (1976) parameters (α,M∗stars, φ In the case of the K-selected sample, in order to take into account the two different magnitude limits, we perform a “Coherent Analysis of independent samples” as described by Avni & Bahcall (1980). Ilbert et al. (2004) have shown that the estimate of the faint end of the global luminosity function can be biased, because, due to different k-corrections, different galaxy types have different absolute magnitude limits for the same apparent magnitude limit. The same bias is present also for the low mass end of the mass function. This is due to the fact that, because of the existing dispersion in the mass-to- light ratio of different galaxy types, at small masses the ob- jects with the largest mass-to-light ratio are not included in a magnitude limited sample (see Appendix B in Fontana et 10 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History Fig. 7. Effect on the I-selected MF of the use of the statistically corrected masses, which take into account the effect of near-IR photometry on the mass determination (see text for details). Complex SFHs have been used to derive masses. Empty circles represent the MF obtained using the original uncorrected masses, filled squares show the MF obtained using the statistically corrected masses. For comparison the STY Schechter MF is shown for the subsample where near-IR photometry is available. The vertical dashed lines represent the completeness limit of the sample as defined in the text. al. 2004 for an extensive discussion). For this reason, when computing the global mass functions with the STY method, in order to fully avoid this bias, we should use in each red- shift range only galaxies above the stellar mass limit where all the SEDs are potentially observable. In both the K- and I-selected samples these limits derive from the conversion between photometry and stellar mass for early-type galax- ies and are very restrictive. However, recent results show that the faint-end and the low-mass end of the luminosity and mass function is dominated by late-type galaxies up to z ∼ 2 (Fontana et al. 2004, Zucca et al. 2006, Bundy et al. 2006). Therefore for the STY estimate we will use as lower limit of the mass range the minimum mass above which late-type SEDs (defined by rest-frame optical/NIR colours MI −MK < 0.4) are potentially observable (see Fig. 9). We have estimated the MF for both the deep I-selected (17.5 < I < 24) spectroscopic sample and the photomet- ric K-selected (K < 22.34 & K < 22.84) sample. For each sample the MF has been estimated using masses com- puted with both methods described in Section 2.3. In the case of the spectroscopic sample, in order to correct for both the non-targeted sources in spectroscopy and those for which the spectroscopic measurement failed, we use a statistical weight wi, associated with each galaxy i with a secure redshift measurement (see Ilbert et al. 2005 for details). This weight is the inverse of the product of the Target Sampling Rate times the Spectroscopic Success Rate. Accurate weights have been derived by Ilbert et al. (2005) for all objects with secure spectroscopic redshifts, tak- ing into account all the parameters involved (magnitudes, galaxy size and redshift). For the K-selected sample, we have tested the effect of catastrophic photometric redshifts (see discussion in Sec. 2.2) on the evolution of the mass function and mass den- sity. We have used the I-selected spectroscopic sample, re- placing spectroscopic redshifts with photometric redshifts. The two MFs (with either spectroscopic or photometric red- shifts) are very similar in the whole mass and redshift range (0.05 < z < 2.5) analyzed and even at z > 1.2, where we note a not negligible number of catastrophic photometric redshifts (see discussion in Sec. 2.2). There is no evidence of a strong bias in the normalization and in the shape of the MF; also the massive tails of the MFs are similar, within the statistical errors. We conclude that the catastrophic L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 11 Fig. 8.K-selected MF derived from the 2 subsamples, deep (filled circles) and shallow (empty circles), separately (smooth SFHs have been used to derive masses). For comparison the STY Schechter MF for the globalK-selected sample is shown. The vertical dashed lines represent the completeness limit of the 2 K-selected subsamples. solutions at high photometric redshifts (i.e. masses) do not strongly affect our results. 4.1. The VVDS Galaxy Stellar Mass Function The resulting stellar mass functions of the VVDS sample are derived in the following redshift ranges: (a) 0.05 < z < 1.2 for the I-selected sample, because at higher redshift the mass estimate becomes very uncertain (see figure 4) and (b) 0.05 < z < 2.5 for the K-selected sample. We have furthermore divided the 2 samples into different redshift bins in order to sample evolution with similar numbers of sources in each bin. For the galaxies in the I-selected sample not covered by near-IR data, we have used the statistically corrected masses derived through a Monte Carlo simulation, to take into account the effect of the near-IR photometry in the mass determination (see Sec. 3.3). Figure 7 shows the ef- fect in the MF for complex SFHs. The high mass tail is significantly reduced if we use statistically corrected masses when near-IR is not available. Consistent MFs have been 12 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History Fig. 9. Galaxy Stellar Mass Functions in the I-selected (squares) and K-selected (triangles) using both methods to estimate the stellar masses (empty symbols for smooth SFHs and filled for complex SFHs). The STY Schechter fits for the 2 methods limit the hatched regions (horizontal hatched for the K-selected and vertical hatched for the I-selected samples). Vertical hatched regions represents the completeness limit of the 2 samples. The local MFs by Cole et al. (2001), both original and “rescaled” version (Fontana et al. 2004), and by Bell et al. (2003) are reported in each panel as dotted lines. L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 13 Table 2. STY parameters in the different redshift ranges sample method z range mean redshift α logM∗stars(h 70 M⊙) φ ∗(10−3h370Mpc I smooth 0.05 - 0.4 0.27 -1.26+0.01 −0.02 11 1.90 +0.08 −0.16 I smooth 0.4 - 0.7 0.58 -1.23+0.04 −0.04 11 +0.08 −0.08 1.72 +0.33 −0.28 I smooth 0.7 - 0.9 0.8 -1.23+0.12 −0.11 10.88 +0.15 −0.13 1.6 +0.55 −0.45 I smooth 0.9 - 1.2 1.05 -1.09+0.19 −0.17 10.85 +0.14 −0.14 1.3 +0.43 −0.46 I complex 0.05 - 0.4 0.27 -1.28+0.02 −0.01 11.15 1.75 +0.16 −0.08 I complex 0.4 - 0.7 0.58 -1.22+0.04 −0.04 11.15 +0.08 −0.08 1.58 +0.30 −0.26 I complex 0.7 - 0.9 0.81 -1.04+0.08 −0.07 10.83 +0.07 −0.07 3.02 +0.57 −0.51 I complex 0.9 - 1.2 1.04 -1.16+0.1 −0.09 10.89 +0.08 −0.07 1.80 +0.44 −0.39 K smooth 0.05 - 0.4 0.26 -1.38+0.02 −0.01 10.93 1.29 +0.10 −0.05 K smooth 0.4 - 0.7 0.57 -1.14+0.04 −0.04 10.93 +0.06 −0.06 1.83 +0.27 −0.24 K smooth 0.7 - 0.9 0.81 -1.01+0.07 −0.08 10.67 +0.07 −0.05 2.6 +0.38 −0.44 K smooth 0.9 - 1.2 1.05 -1.1+0.07 −0.08 10.78 +0.06 −0.05 1.83 +0.28 −0.30 K smooth 1.2 - 1.6 1.4 -1.15+0.12 −0.12 10.72 +0.07 −0.06 1.48 +0.30 −0.30 K smooth 1.6 - 2.5 1.96 -1.15 10.96+0.01 −0.02 0.9 +0.30 −0.30 K complex 0.05 - 0.4 0.26 -1.39+0.01 −0.02 11.12 1.17 +0.05 −0.09 K complex 0.4 - 0.7 0.57 -1.16+0.04 −0.04 11.12 +0.06 −0.06 1.58 +0.24 −0.22 K complex 0.7 - 0.9 0.81 -1.16+0.07 −0.07 10.98 +0.07 −0.07 1.74 +0.36 −0.30 K complex 0.9 - 1.2 1.05 -1.2+0.07 −0.06 11.07 +0.06 −0.06 1.34 +0.26 −0.21 K complex 1.2 - 1.6 1.4 -1.17+0.12 −0.12 10.93 +0.07 −0.06 1.39 +0.29 −0.28 K complex 1.6 - 2.5 1.96 -1.17 10.97+0.01 −0.02 1.25 +0.09 −0.04 obtained in the sub-area of I-selected sample where near-IR photometry is available (see Fig. 7). For the K-selected sample, we have analyzed the ef- fect of the cosmic variance on small areas, deriving the MF for the two K-selected subsamples, deep and shallow, sepa- rately (see Figure 8). We find a significantly lower MF (by a factor 1.8 and 1.6) in the redshift range 0.4 < z < 0.7 and 0.7 < z < 0.9 in the deep K-selected sample (K < 22.84 over 168 arcmin2) compared to the shallow K-band sam- ple (K < 22.34 over 442 arcmin2). The significance of such differences in the MF, is of ∼ 2 − 3σ in each mass bin at 0.4 < z < 0.7 and ∼ 1 − 2σ at 0.7 < z < 0.9. Globally, i.e. for the total number densities over the complete mass range, the differences are significant at about 5-3 σ level in the two redshift ranges, respectively. This problem leads to a clear warning on the results based on small fields, as covered by most of the previous existing surveys. In Figure 9 we show the MFs derived using the I- selected spectroscopic sample and the K-selected photo- metric sample for both methods (smooth and complex SFHs) to derive the masses. The resulting mass functions are quite well fitted by Schechter functions. The best-fit Schechter parameters are summarized in Table 2, with the uncertainties derived from the projection of the 68% confi- dence ellipse. Since in the lowest and highest redshift bins (z ≃ 0.2 and 2) the values of M∗stars and the low-mass-end slope (α), respectively, are poorly constrained, they have been fixed to the values measured in the following and pre- vious redshift bins respectively. We note, first, that the overall agreement between the MF derived with the different methods for masses determi- nation is fairly satisfactory, albeit complex SFHs estimates provide typically larger masses. The systematic shift be- tween the 2 methods (Section 3.4) is reflected in most of the redshift bins also in the characteristic mass (M∗stars) of the MF while the best fit slopes (α) and φ∗ Schechter parameters agree within the errors between the different methods in most of the redshift bins. We note, furthermore, an overall agreement between the 2 samples (I- and K-selected), and in most of the redshift bins the Schechter parameters agree within the errors, even if some differences exist. More in detail, the I-selected sam- ple in the range 0.4 < z < 0.9 has a higher low-mass (< 9.5 dex) end and a slightly steeper MF (α ∼ −1.23) than the K-selected one (α > −1.15). These differences are probably due to the population of blue K-faint galaxies, that are missed in the K-sample, as discussed in Section 3. These galaxies have, indeed, median colours in the I- selected sample that are bluer than in the K-selected sam- ple (I−K ≃ 0.45 compared to ≃ 0.89). A similar behaviour has been noted in the local MF derived using an optically selected sample (g band) compared to the local MF from the near-IR (2MASS) sample (Bell et al. 2003). On the contrary, at even lower masses (< 8.5 dex) at z < 0.4 the K-selected MF is slightly steeper than the I-selected one, but no significant differences in the colour of the two pop- ulations is found. 4.2. Comparison with Previous Surveys In general, previous efforts to derive MF have relied on smaller or more limited samples, or often based mainly on photometric redshifts (Drory et al. 2004, 2005). We have compared our MF determination with literature re- sults based on different surveys (K20, COMBO17, MUSIC, DEEP2, FDF+CDFS), rescaled to Chabrier IMF (see Figure 10). Our MFs rely on a higher statistics at inter- mediate to high-mass ranges, and therefore present lower statistical errors. At z < 0.2 we sample unprecedented mass ranges, more than one order of magnitude lower than previ- ous surveys, while at z > 0.4 the FDF and MUSIC surveys reach lower mass limits even if on significantly smaller area. 14 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History Fig. 10. Comparison between the I-selected and the K-selected MFs in the VVDS (hatched vertical and horizontal STY regions, respectively, see caption Figure 9) and the literature data (K20, COMBO17, MUSIC, DEEP2, FDF+CDFS; for each the band of selection is indicated in the parenthesis). The vertical hatched regions represent the completeness limits of the VVDS samples. Our MFs are in fairly good agreement with previous studies over the whole mass range up to z ∼ 1.2. However, some differences exist, in particular at the massive end, which is more sensitive to the different selections, methods, statis- tics and to cosmic variance due to large scale structures: for example, in the MUSIC-GOODS survey there are two sig- nificant overdensities at z ∼ 0.7. The MFs from COMBO17 (Borch et al. 2006) and also from DEEP2 (Bundy et al. 2006) are systematically higher than previous surveys at the massive-end, in particular in the range 0.7 < z < 1.2. The MFs in the FDF+CDFS are instead systematically lower than ours at the massive-end and higher than our extrapolation to masses lower than our completeness limit. At z > 1.2 our MF is systematically higher than previ- ous studies. Given the area sampled (more than a factor 4 wider compared to FDF+CDFS and to MUSIC) and the consistency at these redshifts of our MF in the 2 K- selected separated areas (see Figure 8), we are confident in our results. However at high-z the uncertainties on the stellar masses estimate increase (up to 0.16 dex including L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 15 also the photometric redshift errors) and could produce a partially spurios excess in the number densities of galax- ies, in particular in the massive tail of the MF. This ef- fect is discussed in Kitzbichler & White (2007) which take into account in the hierarchical formation Millenium simu- lation the effect of the dispersion in the mass determination (0.25 dex, i.e. 78%). We have performed a similar analysis, taking into account the uncertainty on the mass, due to the fitting technique (∼ 30%) and to the uncertainty of the photometric redshifts (both its dependence on redshift and magnitudes as described in previous sections, i.e. up to σz ≃ 0.2 at z = 2 and K > 21.5). We found that the effect on the MF is always small, and only the very massive tail (M > 2 × 1011M⊙) is systematically overestimated (up to 0.2 dex). This effect can not completely explain the excess found compared to previous surveys, which are affected in a similar way by the same bias. 4.3. The Evolution of the Galaxy Stellar Mass Function The VVDS allows us to follow the evolution of the MF within a single sample over a wide redshift range. Difficulties in the interpretation of the evolution are, in- deed, due to the comparison with the local MFs, which have been determined with different methods and sample selection. For example, no local MF has been derived using complex SFHs for mass determinations. In our analysis we use, as reference, the local MF by Bell et al. (2003) and Cole et al. (2001) rescaled to Chabrier IMF. In particular, the Cole et al. (2001) local mass function, derived with smooth SFHs but with formation redshift fixed at z = 20, has been rescaled to smooth SFHs method with free formation red- shift by Fontana et al. (2004). The first important result is that, thanks to our very deep samples, both I- and K-selected, the low-mass end of the MF is even better determined than in the local sam- ple up to z < 0.4, probing for the first time masses down to about 3 × 107M⊙. The low mass-end is rather steep (−1.38 < α < −1.26), and could even be described by a double Schechter function, and is steeper than the local es- timates (α = −1.18±0.03 Cole et al. 2001, α = −1.10±0.02 Bell et al. 2003), possibly due to the fact they are not prob- ing masses smaller than 109 M⊙ (more than one order of magnitude more massive than in our sample). As evident from Figure 9, we find a substantial population of low-mass (< 109M⊙) galaxies at low redshifts (z < 0.4). This popu- lation is composed by faint blue galaxies with similar prop- erties in the 2 samples (I- and K-selected): I,K ≃ 22− 23, MI ,MK ≃ −16,−17 with median MI − MK ≃ 0.3, and median z ≃ 0.1− 0.2. This is a very strong result from our survey which can rely on a wider area and a deeper sam- ple than previous surveys at low redshifts. At z > 0.4 the low-mass slope is on the contrary always consistent with the local values. Even if we are not probing masses smaller than 108M⊙ at z > 0.4, we found that the MF remains quite flat (−1.23 < α < −1.04) at all redshifts, similar to that of Fontana et al. (2006) which probe lower masses (see figure 10). From a visual inspection of Figure 9, we see that up to z ∼ 0.9 there is only a weak evolution of the MF, as sug- gested by previous results (Fontana et al. 2004, 2006, Drory et al. 2005), while at higher redshifts there appears to begin a decrease in the normalization of the MF, even if a massive tail remains present up to z = 2.5. At intermediate masses Fig. 11. Cosmological evolution of the galaxy number den- sity as a function of redshift, as observed from the VVDS in various mass ranges (> 108M⊙, > 10 9.77M⊙ and > 1010.77M⊙ from top to bottom). Observed data from Vmax (shown as lower limit in the top panel) have been corrected, when necessary, for incompleteness integrating the mass function using the best fit Schechter parameters. VVDS data (big filled circles), averaged over the I- and the K- selected samples and the 2 methods to derive the mass, are plotted along with their statistical errors (solid error bars) and the scatter between the 2 different samples and meth- ods (dotted error bars). The solid lines show the best-fit power laws ∝ (1 + z)β, while the dashed lines correspond to the no-evolution solution normalized at z = 0. Results from previous surveys (small points and dot-dashed lines) are also shown. (9.5 < logM < 10.5), our VVDS MF is very well defined and shows a clear evolution, i.e. the number density de- creases with increasing redshifts compared to both the first VVDS redshift bin and the local MF. This evolution is quite mild up to z ≃ 0.9, while it becomes faster at higher z. At larger masses the high mass end of the MF (> 1011 M⊙) shows a small evolution up to z ≃ 2.5. However, its evolu- tion is extremely dependent on the assumed local MF and on the uncertainties in the mass determination, which pro- duce a larger dispersion between the different methods and samples compared to the intermediate-mass range. In order to quantify the MF evolution, and its mass dependency independently from the local MF, in the next section we derive number densities for different mass limits. 16 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History Table 3. Number Density and Stellar Mass Density zinf zsup Log(ρN ) Scatter Error Log(ρstars)Scatter Error log(Mstars)> 8 0.05 0.40 -1.40 0.04 0.01 8.45 0.09 0.01 0.40 0.70 -1.63 0.08 0.02 8.34 0.08 0.02 0.70 0.90 -1.70 0.08 0.04 8.22 0.11 0.01 0.90 1.20 -1.78 0.13 0.05 8.14 0.12 0.01 1.20 1.60 -1.81 0.05 0.11 8.04 0.14 0.02 1.60 2.50 -1.90 0.13 0.01 8.05 0.11 0.01 log(Mstars)> 9.77 0.05 0.40 -2.20 0.06 0.01 0.40 0.70 -2.29 0.03 0.01 0.70 0.90 -2.33 0.04 0.01 0.90 1.20 -2.45 0.05 0.01 1.20 1.60 -2.54 0.02 0.01 1.60 2.50 -2.65 0.01 0.01 log(Mstars)> 10.77 0.05 0.40 -3.07 0.18 0.05 8.00 0.22 0.04 0.40 0.70 -3.04 0.10 0.02 8.04 0.14 0.02 0.70 0.90 -3.22 0.16 0.02 7.80 0.19 0.02 0.90 1.20 -3.28 0.15 0.02 7.76 0.18 0.02 1.20 1.60 -3.42 0.23 0.02 7.59 0.28 0.02 1.60 2.50 -3.35 0.11 0.01 7.70 0.11 0.01 4.4. Galaxy Number Density Here we derive the number density of galaxies as a function of redshift, using different lower limits in mass (Mmin). We have estimated the number density from the observed data (from Vmax), as well as from the incompleteness-corrected MFs, i.e. integrating the best-fit Schechter functions over the considered mass range. The corrections due to faint galaxies dominate for Mmin = 10 8M⊙, while they are neg- ligible for the other mass limits considered. A formal uncer- tainty in this procedure was estimated by considering the Vmax statistical errors and the range of acceptable Schechter parameters values. In Figure 11 we plot our VVDS determi- nations, averaged over the I- and K-selected samples and the 2 methods for mass determination (listed in Table 3), along with their statistical errors (always less than 10%) and the scatter between the 2 different samples and meth- ods (ranging between 10 and 45% and due mainly to the dif- ferent methods rather than to the different samples). With the two methods we find similar trends with redshift of the number densities of galaxies, but with a systematic shift which is significant only for the highest mass limit (for com- plex SFHs the galaxy number densities are ∼ 50% higher than for smooth SFHs). The effect of photometric redshift and mass uncertainty on the number densities is always small (< 15%) for the mass range shown in Figure 11, ex- cept for the very massive galaxies (> 2 × 1011 M⊙, not shown in the figure because of the small number of galaxies in this mass range) where the intrinsic values could be up to a factor ∼ 2 lower (see discussion in section 4.2). The decrease in number density with redshift for all the adopted mass limits is evident. We have compared VVDS results to previous surveys and with different local determinations. For the total num- ber density (108 < Mstars < 10 13M⊙) VVDS data are very well consistent with the evolutionary STY fit determined by Fontana et al. (2006) in the GOODS-MUSIC survey. At in- Fig. 12. Cosmological evolution of the stellar mass density as a function of redshift as observed from the VVDS for 2 mass ranges: integrated over the whole range 108M⊙ ≤ Mstars ≤ 10 13M⊙ (upper panels) and for massive galax- ies (> 1010.77M⊙) (lower panels). Symbols and lines as in Figure 11. termediate masses (> 109.77M⊙, corresponding to 10 for Salpeter IMF) our VVDS data have a better determina- tion and smaller uncertainties than previous ones and are consistent with most of them at z < 1.2 and in the upper en- velope at higher z. For the high mass range (> 1010.77M⊙, corresponding to 1011M⊙ for Salpeter IMF) we are quite consistent with previous results, and even if our VVDS have lower errors than previous ones, the dispersion within the various VVDS measurements reflect the uncertainties for massive galaxies. If we represent the average number density evolution by a power law ρN ∝ (1 + z) β , we find that β(Mstars > 108) = −1.28 ± 0.15, β(Mstars > 10 9.77) = −1.26 ± 0.10, and β(Mstars > 10 10.77) = −1.01 ± 0.05 (the errors on β represent the uncertainties due to the 2 different methods). We find on average a similar evolution for the 2 methods analyzed and a slightly milder evolution with increasing mass limit (‘downsizing’ in mass assembly). The average evolution from z = 0 to z = 1 is a factor 2.4 ± 0.3 and 2.0± 0.1 from low to high-mass galaxies, respectively, and increases to a factor 4.0 ± 0.9 and 3.0 ± 0.2, respectively, at z = 2. We note moreover that for the highest mass limit (Mstars > 10 10.77) at low redshift (z < 0.7) the number density observed is consistent with no-evolution (fixing the L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 17 value to z = 0 we found an evolution < 30%), excluded instead for the intermediate and low-mass limit. At high-z (z > 1.5) for intermediate and high-mass range we note a flattening in the number density from the VVDS data, sim- ilar to Drory et al. (2005), but higher than Fontana et al. (2006). This flattening is due to the high mass tail observed in the range 1.6 < z < 2.5. This population shows ex- tremely red colours (MI −MK ≃ 0.8) and could be related to the appearance of a population of massive star forming dusty galaxies, observed in previous surveys (Fontana et al. 2004, Daddi et al. 2004b). The small excess induced by uncertainties on the mass and photometric redshifts (see discussion in section 4.2) can not completely explain the difference with Fontana et al. (2006) survey, which is af- fected by the same bias in a similar way. The mass dependent evolution (“mass downsizing”) is very debated and the results from different surveys are still controversial. Deep surveys, such as the FDF & CDFS (an- alyzed by Drory et al. 2005) find an evolution consistent with ours (a decrease of about a factor 2.5 – 4 at z = 1− 2 in the number density of galaxies> 1010.77 M⊙). Fontana et al. (2006) suggest a similar mild evolution up to z = 1, for massive galaxies (> 1010.77M⊙) and a stronger evolution at z > 1.5, reaching a factor about 10 at z = 3. Similarly, Cimatti et al. (2006) show that the number density of lu- minous (massive, Mstars > 10 11M⊙) early-type galaxies is nearly constant up to z ∼ 0.8, while Bundy et al. (2006) find a slight decrease, consistent with no evolution, only for even more massive system (> 3×1011M⊙) and a more sig- nificant decline for Mstars < 3× 10 11M⊙. Vice versa, data from the MUNICS survey (Drory et al. 2004) show a faster evolution of massive galaxies, even faster than for the less massive systems (see also Figure 4 in Drory et al. 2005). To summarize, our accurate results show that the MF evolves mildly up to z ≃ 1 (about a factor 2.5 in the total number density) and that a high-mass tail is still present up to z = 2.5. Moreover, we find that massive systems show an evolution that is on average milder (< 50% at z < 1) than intermediate and low-mass galaxies and con- sistent with a mild/negligible evolution (< 30%) up to z ∼ 0.8. Conversely, a no-evolution scenario in the same redshift range is definitely excluded for intermediate- and low-mass galaxies. This behaviour suggests that the assembly of the stellar mass in objects with mass smaller than the localM∗stars was quite significant between z = 2 and z = 0. Qualitatively, this behaviour is expected for galaxies with SFHs prolonged over cosmic time, which therefore continue to grow in terms of stellar mass after z ∼ 1. Conversely, our results fur- ther strengthen the fact that the number density of mas- sive galaxies is roughly constant up to z ≃ 0.8, consistently with a SFH peaked at higher redshifts, with the conversion of most of their gas into stars happening at z > 1.5−2, rul- ing out the ‘dry mergers’ as the major mechanism of their assembly history, below z < 1. 5. Mass Density Various attempts to reconstruct the cosmic evolution of the stellar mass density have been previously made, mainly us- ing NIR-selected samples (Dickinson et al. 2003, Fontana et al. 2004, Drory et al. 2005). Our survey offers the possibil- ity to investigate it using the MF derived from two different optical- and NIR-selected samples, taking advantage of our depth, and relatively wide area covered. Furthermore, the different methods analyzed here to derive the stellar mass content give us a direct measure of the uncertainties in- volved. We have estimated the stellar mass density from the observed data, as well as from the incompleteness-corrected MFs. Up to z < 1 the corrections due to faint galaxies are relatively small. A formal uncertainty in this procedure was estimated by considering the Vmax statistical errors and the range of acceptable Schechter parameters values. Figure 12 shows our results (averaged over the I- andK- selected samples and the 2 methods with a typical scatter of about 30-50% and statistical errors always less than 5% see Table 3), for the total mass density and for the density in massive galaxies (> 1010.77), along with their represen- tative power laws (ρstars ∝ (1+z) β), and compared to liter- ature data (see references in the figure). For the total mass density, even if the results from our survey cover a range of values with some significant differences between the two different methods (up to ∼ 40%), the general behaviour and evolutionary trend is well defined by β = −1.19± 0.05. We find that the evolution of the stellar mass density is relatively slow with redshift, with a decrease of a factor 2.3 ± 0.1 up to z ≃ 1, up to a factor 4.5 ± 0.3 at z = 2.5. The agreement of average total mass density with previous surveys is reasonably good, and the range covered by VVDS data reflect the different selection techniques and methods used in different surveys. The average total mass density evolution is milder than in the MUSIC sample (Fontana et al. 2006) already at z > 0.5. Our evolutionary trend is con- sistent with the upper envelope of previous surveys, even if our highest-redshift value is uncertain because the low-mass slope is poorly constrained. For comparison the analysis of VVDS data using a IRAC-selected sample (see Arnouts et al. 2007) finds similar values for the mass density, except that the highest redshift point is lower than ours. Given the present uncertainties on the low-mass slope of the GSMF, the total mass density at z ≃ 2 remains poorly constrained. The mass density of high-mass objects (> 1011M⊙ with Salpeter IMF) varies by a factor up to 1.8 within the 2 methods adopted, but the evolutionary trend is similar (β = −1.13± 0.01) and consistent with a decrease of about a factor 2.18 ± 0.02 to z = 1 and 3.44 ± 0.04 to z = 2.0. Moreover at low redshift (z < 0.7) the VVDS observed data are consistent with a mild/negligible evolution (< 30%), as indicated by the number density of massive galaxies (see previous Section). Our data are roughly consistent with Fontana et al. (2006) up to z = 1.5 (even if the slope of the evolutionary trend is shallower), while at z > 1.5 the VVDS mass-density of massive galaxies is significantly higher than that in Fontana et al. (2006), reflecting the excess in MF at high-z noted in the VVDS MF compared to previous ones (see Section 4.2). This results, therefore, in a flatter evolu- tionary trend over the total redshift range. Given the wider area and completeness for high-mass objects, our samples guarantee a higher statistical accuracy and confidence level than before. However some caveat remains due to the effect of photometric redshift and mass uncertainty on the mass densities, which is anyhow always small (< 15%) except for the massive galaxies (> 6 × 1010 M⊙) where the intrinsic values could be up to 20-30% lower than our estimates (see discussion in Section 4.2). 18 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 6. Summary and Discussion We have investigated the evolution of the Galaxy Stellar Mass Function up to to z = 2.5 using the VVDS survey covered by deep VIMOS spectroscopy (17.5 < I < 24) and multiband photometry (from U to K-band). For our analysis we have used two different samples: (1) the opti- cal (I-selected, 17.5 < I < 24) main spectroscopic sam- ple, based on about 6500 secure redshifts over about 1750 arcmin2, and (2) a near-IR sample (K-selected, K < 22.84 & K < 22.34), in a sub-area of about 610 arcmin2 and based on about 10200 galaxies with accurate photomet- ric and spectroscopic redshifts. For the first time we have probed masses down to a very low limit, in particular at low-z (down to ∼ 3 × 107M⊙ at z ∼ 0.2), while the rela- tively wide area has allowed us to determine the MF with much higher statistical accuracy than previous samples. In order to better understand uncertainties we have ap- plied and compared two methods to estimate the stellar mass content in galaxies from multiband SED fitting. The 2 methods differ in the explored parameter space (metal- licity, dust law and content) and are based on different as- sumptions on previous star formation history. The main results from the stellar mass estimate can be summarized as follows: – The agreement between the 2 methods is fairly good even if masses estimated with ‘complex SFHs’ are systematically higher than ‘smooth SFHs’ masses. For the K-selected sample the mean difference is 〈d logMstars〉 ≃ 0.12 dex, and the dispersion is σ = 0.13. The differences are mainly due to the secondary burst component (complex SFHs) compared to smooth SFHs. – We found that mass estimates using only optical bands are in rather good agreement with those using also NIR bands up to z ∼ 1.2. We have used this information to statistically correct masses for objects without near-IR photometry. At higher redshifts the shift and dispersion dramatically increase and the mass estimates become unreliable if near-IR photometry is not available. We have, thus, derived the MF using the VVDS I- selected sample and extended it up to z = 2.5 thanks to the K-selected sample. From a detailed analysis of the MF, galaxy number density and mass density, in different mass ranges, through cosmic time, we found evidences for: – a substantial population of low-mass galaxies (< 109M⊙) at z ≃ 0.2 composed by faint (I,K ≃ 22, 23) blue galaxies with median MI−MK ≃ 0.3, and absolute magnitudes MI ,MK ≃ −16,−17; – a slow evolution of the stellar mass function with red- shift up to z ∼ 0.9 and a faster evolution at higher-z, in particular for less massive systems. A massive popu- lation is present up to z = 2.5 and have extremely red colours (MI −MK ≃ 0.7− 0.8). – at z > 0.4 the low-mass slope of the GSMF does not evolve significantly and remains quite flat (−1.23 < α < −1.04). – the number density shows, on average, a mild differen- tial evolution with mass, which is slower with increas- ing mass limit. Such evolution can be described by a power law ∝ (1 + z)β(>M). Within the VVDS redshift range we found that β(> 108M⊙) = −1.28 ± 0.15, β(> 109.77M⊙) = −1.26± 0.10 and β(> 10 10.77M⊙) = −1.01± 0.05. For massive galaxies at low redshift (z < 0.7) the evolution is consistent with mild/negligible- evolution (< 30%), which is excluded for low-mass sys- tems. – the evolution of the stellar mass density is relatively slow with redshift, with a decrease of about a factor 2.3±0.1 to z ≃ 1, while at z ≃ 2.5 the decrease amounts to a fac- tor up to 4.5± 0.3, milder than in previous surveys. For massive galaxies the evolution at low redshift (z < 0.7) is consistent with a mild/negligible evolution(< 30%), and shows a flattening compared to previous results at z > 1.5 due to a population with extremely red colours. Our results provide new clues on the controversial ques- tion of when galaxy formed and assembled their stellar mass. Most of the massive galaxies seem to be in place up to z = 1 and have, therefore, formed their stellar mass at high redshift (z > 1), rather than assembled it mainly through continuous galaxy merging of small galaxies at z < 1. On the contrary, less massive systems have assembled their mass (through merging or prolonged star formation his- tory) later in cosmic time. In agreement with our results, a substantial population of high-z (z ∼ 2−3) dusty and mas- sive objects have been discovered in near-IR surveys (Daddi et al. 2004b) and detected by Spitzer in the far-IR (Daddi et al. 2005, Caputi et al. 2006b). This population could be related to the initial phase of massive galaxy formation during their strong star forming and dusty phase. Finally, our results are not completely accounted for by most of theoretical models of galaxy formation (see Fontana et al. 2004, 2006 and Caputi et al. 2006a for a detailed com- parison with models). For instance, models by De Lucia et al. (2006) predict that the most massive galaxies generally form their stars earlier, but assemble them later, mainly at z < 1 via merging, than the less massive galaxies (i.e. ’downsizing in star formation but ’upsizing’ in mass assem- bly, see Renzini 2007 for a recent discussion). Furthermore, the stronger decrease with redshift of the low-mass popu- lation, with a low-mass end of the GSMF which remains substantially flat up to high redshift, is not reproduced by most of the theoretical galaxy assembly models, which tend, indeed, to overpredict the low-mass end of the MF (see Fontana et al. 2006). Understanding the mass assembly of less massive ob- jects and disentangling merging processes from prolonged star formation history is more complicated. In this respect for a better comprehension of galaxy formation the VVDS will allow us to further investigate the evolution of the stel- lar mass function up to high-z also for different galaxy types (spectral and morphological) and in different environments. For example Arnouts et al. (2007) study the mass density evolution of different galaxy population. Further analysis of galaxy mass dependent evolution, using stellar popula- tion properties, as well as observed spectral features, will be presented in forthcoming papers (Lamareille et al., in preparation, Vergani et al. 2007). Furthermore, it will be possible to push the study of the galaxy stellar mass func- tion at higher redshifts using SPITZER mid-IR observa- tions. While most of present studies (Dickinson et al. 2003, Drory et al. 2005) do not use rest-frame near-IR photome- try to estimate stellar masses, our VVDS-SWIRE collabo- ration will allow to combine the deep VVDS spectroscopic sample with SPITZER-IRAC photometry. L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 19 Acknowledgements. This research has been developed within the framework of the VVDS consortium. This work has been partially supported by the CNRS-INSU and its Programme National de Cosmologie (France), and by Italian Ministry (MIUR) grants COFIN2000 (MM02037133) and COFIN2003 (num.2003020150). The VLT-VIMOS observations have been carried out on guaranteed time (GTO) allocated by the European Southern Observatory (ESO) to the VIRMOS consortium, under a contractual agreement between the Centre National de la Recherche Scientifique of France, heading a consortium of French and Italian institutes, and ESO, to design, manufacture and test the VIMOS instrument. We are in debt with E. Bell, S. Salimbeni, and E. Fontana for providing the data from their survey in electronic format, and to C. Maraston for her galaxy evolution models in BC format. References Avni, Y, Bahcall, J. 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A., et al. 2000, ApJ, 530, 418 Zucca, E., Ilbert, O., Bardelli, S., et al. 2006, A&A, 455, 879 1 INAF-Osservatorio Astronomico di Bologna - Via Ranzani,1, I-40127, Bologna, Italy 2 INAF-IASF - via Bassini 15, I-20133, Milano, Italy 3 Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS-Université de Provence, BP8, 13376 Marseille Cedex 12, France 4 INAF-Osservatorio Astronomico di Brera - Via Brera 28, Milan, Italy http://arxiv.org/abs/astro-ph/0506044 http://arxiv.org/abs/astro-ph/0611724 http://arxiv.org/abs/astro-ph/0311475 http://arxiv.org/abs/astro-ph/0410295 http://arxiv.org/abs/astro-ph/0702148 20 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 5 Institute for Astronomy, 2680 Woodlawn Dr., University of Hawaii, Honolulu, Hawaii, 96822 6 Max Planck Institut fur Astrophysik, 85741, Garching, Germany 7 Institut d’Astrophysique de Paris, UMR 7095, 98 bis Bvd Arago, 75014 Paris, France 8 Centro de Astrofsica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal 9 Laboratoire d’Astrophysique de Toulouse/Tabres (UMR5572), CNRS, Université Paul Sabatier - Toulouse III, Observatoire Midi-Pyriénées, 14 av. E. Belin, F-31400 Toulouse, France 10 INAF-IRA - Via Gobetti,101, I-40129, Bologna, Italy 11 INAF-Osservatorio Astronomico di Roma - Via di Frascati 33, I-00040, Monte Porzio Catone, Italy 12 School of Physics & Astronomy, University of Nottingham, University Park, Nottingham, NG72RD, UK 13 Astrophysical Institute Potsdam, An der Sternwarte 16, D- 14482 Potsdam, Germany 14 Observatoire de Paris, LERMA, 61 Avenue de l’Observatoire, 75014 Paris, France 15 Università di Bologna, Dipartimento di Astronomia - Via Ranzani,1, I-40127, Bologna, Italy 16 Centre de Physique Théorique, UMR 6207 CNRS-Université de Provence, F-13288 Marseille France 17 Integral Science Data Centre, ch. d’Écogia 16, CH-1290 Versoix 18 Geneva Observatory, ch. des Maillettes 51, CH-1290 Sauverny, Switzerland 19 Astronomical Observatory of the Jagiellonian University, ul Orla 171, 30-244 Kraków, Poland 20 INAF-Osservatorio Astronomico di Capodimonte - Via Moiariello 16, I-80131, Napoli, Italy 21 Universitá di Milano-Bicocca, Dipartimento di Fisica - Piazza delle Scienze, 3, I-20126 Milano, Italy 22 Università di Bologna, Dipartimento di Fisica - Via Irnerio 46, I-40126, Bologna, Italy Introduction The First Epoch VVDS Sample The I-selected Spectroscopic Sample The K-selected Photometric Sample Comparison of the Two Samples Estimate of the Stellar Masses Smooth SFHs Complex SFHs Effect of NIR Photometry Comparison of the Masses Obtained with the Two Methods Massive Galaxies at z>1 Mass Function Estimate The VVDS Galaxy Stellar Mass Function Comparison with Previous Surveys The Evolution of the Galaxy Stellar Mass Function Galaxy Number Density Mass Density Summary and Discussion

We present a detailed analysis of the Galaxy Stellar Mass Function of galaxies up to z=2.5 as obtained from the VVDS. We estimate the stellar mass from broad-band photometry using 2 different assumptions on the galaxy star formation history and show that the addition of secondary bursts to a continuous star formation history produces systematically higher (up to 40%) stellar masses. At low redshift (z=0.2) we find a substantial population of low-mass galaxies (<10^9 Msun) composed by faint blue galaxies (M_I-M_K=0.3). In general the stellar mass function evolves slowly up to z=0.9 and more significantly above this redshift. Conversely, a massive tail is present up to z=2.5 and have extremely red colours (M_I-M_K=0.7-0.8). We find a decline with redshift of the overall number density of galaxies for all masses (59+-5% for M>10^8 Msun at z=1), and a mild mass-dependent average evolution (`mass-downsizing'). In particular our data are consistent with mild/negligible (<30%) evolution up to z=0.7 for massive galaxies (>6x10^10 Msun). For less massive systems the no-evolution scenario is excluded. A large fraction (>=50%) of massive galaxies have been already assembled and converted most of their gas into stars at z=1, ruling out the `dry mergers' as the major mechanism of their assembly history below z=1. This fraction decreases to 33% at z=2. Low-mass systems have decreased continuously in number and mass density (by a factor up to 4) from the present age to z=2, consistently with a prolonged mass assembly also at z<1.

Introduction One of the main and still open question of modern cos- mology is how and when galaxies formed and in particular when they assembled their stellar mass. There are growing but still controversial evidences in near-IR (NIR) surveys that luminous and rather massive old galaxies were quite common already at z ∼ 1 (Pozzetti et al. 2003, Fontana et al. 2004, Saracco et al. 2004, 2005, Caputi et al. 2006a) and up to z ∼ 2 (Cimatti et al. 2004, Glazebrook et al Send offprint requests to: Lucia Pozzetti e-mail: lucia.pozzetti@oabo.inaf.it ⋆ based on data obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, program 070.A-9007(A), and on data obtained at the Canada-France- Hawaii Telescope, operated by the CNRS of France, CNRC in Canada and the University of Hawaii 2004). These surveys indicate that a significant fraction of early-type massive galaxies were already in place at least up to z ∼ 1. Therefore they should have formed their stars and assembled their stellar mass at higher redshifts. As in the local universe, at z ≃ 1.5 these galaxies still dominate the near-IR luminosity function and stellar mass density of the universe (Pozzetti et al. 2003, Fontana et al. 2004, Strazzullo et al. 2006). These results favour a high-z mass assembly, in particular for massive galaxies, in apparent contradiction with the hierarchical scenario of galaxy for- mation, applied to both dark and baryonic matter, which predicts that galaxies form through merging at later cosmic time. In these models massive galaxies, in particular, assem- bled most of their stellar mass via merging only at z < 1 (De Lucia et al. 2006). From several observations it seems that baryonic matter has a mass-dependent assembly his- http://arxiv.org/abs/0704.1600v2 2 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History tory, from massive to small objects, (i.e. the ‘downsizing’ scenario in star formation, firstly defined by Cowie et al. 1996, is valid also for mass assembly), opposite to the dark matter (DM) halos assembly. The continuous merging of DM halos in the hierarchical models, indeed, should result in an ‘upsizing’ in mass assembly, with the most massive galaxies being the last to be fully assembled. If we trust the hierarchical ΛCDM universe, the source of this discrep- ancy between observations and simple basic models could be due to the difficult physical treatment of the baryonic component, such as the star formation history/timescale, feedback, dust content, AGN feedback or to a missing in- gredient in the hierarchical models of galaxy formation (for the inclusion of AGN feedback, see Bower et al. 2006, Kang et al. 2006, De Lucia & Blaizot 2007, Menci et al. 2006, Monaco et al. 2006; and see Neistein et al. 2006 for the de- scription of a natural downsizing in star formation in the hierarchical galaxy formation models and a recent review by Renzini 2007). Considering optically selected surveys, a strong number density evolution of early type galaxies has been recently reported from the COMBO17 and DEEP2 surveys (Bell et al. 2004, Faber et al. 2005), with a corresponding increase by a factor 2 of their stellar mass since z ∼ 1, possibly due to so called ‘dry-mergers’ (even if the observational results on major merging and dry-merging are still contra- dictory, see Bell et al. 2006, van Dokkum 2005, Lin et al. 2004 and Renzini 2007 for a summary). This is at variance with results from the VIMOS-VLT Deep Survey (VVDS, Le Fèvre et al. 2003b), conducted at greater depth and us- ing spectroscopic redshifts in a large contiguous area. From the VVDS, Zucca et al. (2006) found that the B-band lu- minosity function of early type galaxies is consistent with passive evolution up to z ∼ 1.1, while the number of bright (MBAB < −20) early type galaxies has decreased only by ∼ 40% from z ∼ 0.3 to z ∼ 1.1. Similarly, Brown et al. (2007), in the NOAO Deep Wide Field survey over ∼ 10 deg2, found that the B-band luminosity density of L∗ galax- ies increases by only 36 ± 13% from z = 0 to z = 1 and conclude that mergers do not produce rapid growth of lu- minous red galaxy stellar masses between z = 1 and the present day. The VVDS is very well suited for this kind of stud- ies, thanks to its depth and wide area, covered by multi- wavelength photometry and deep spectroscopy. The simple 17.5 < IAB < 24 VVDS magnitude limit selection is signif- icantly fainter than other complete spectroscopic surveys and allows the determination of the faint and low mass population with unprecedented accuracy. Most of the pre- vious existing surveys are instead very small and/or not deep enough, or based only on photometric redshifts. Given the still controversial results based on morphol- ogy or colour-selected early-type galaxies (see Franzetti et al. 2007 for a discussion on colour-selected contamina- tion), we prefer to study the total galaxy population us- ing the stellar mass content. Here we present results on the cosmic evolution of the Galaxy Stellar Mass Function (GSMF) and mass density to z = 2.5 in the deep VVDS spectroscopic survey, limited to 17.5 < IAB < 24, over ∼ 1750 arcmin2 and based on about 6500 galaxies with secure spectroscopic redshifts and multiband (from UV to near-IR) photometry. In addition, we derive the GSMF also for a K-selected sample based on about 6600 galax- ies (KAB < 22.34) in an area of 442 arcmin 2 and about 3600 galaxies in a deeper (KAB < 22.84) smaller area of 168 arcmin2, making use of photometric redshifts, accu- rately calibrated on the VVDS spectroscopic sample, and spectroscopic redshifts when available. Throughout the paper we adopt the cosmology Ωm = 0.3 and ΩΛ = 0.7, with h70 = H0/70 km s −1 Mpc−1. Magnitudes are given in the AB system and the suffix AB will be dropped from now on. 2. The First Epoch VVDS Sample The VVDS is an ongoing program aiming to map the evo- lution of galaxies, large scale structures and AGN through redshift measurements of ∼ 105 objects, obtained with the VIsible Multi-Object Spectrograph (VIMOS, Le Fèvre et al. 2003a), mounted on the ESO Very Large Telescope (UT3), in combination with a multi-wavelength dataset from radio to X-rays. The VVDS is described in detail in Le Fèvre et al. (2005). Here we summarize only the main characteristics of the survey. The VVDS is made of a wide part, with spectroscopy in the range 17.5 ≤ I ≤ 22.5 on 4 fields (∼ 2 × 2 deg2 each), and a deep part, with spectroscopy in the range 17.5 ≤ I ≤ 24 on the field 0226-04 (F02 hereafter). Multicolour photometry is available for each field (Le Fèvre et al. 2004). In particular, the B, V , R, I photometry for the 0226-04 deep field, covering ∼ 1 deg2, has been ob- tained at CFHT and is described in detail in McCracken et al. (2003). The photometric depth reached in this field is 26.5, 26.2, 25.9, 25.0 (50% completeness for point-like sources), respectively in the B, V , R, I bands. Moreover, U < 25.4 (50% completeness) photometry obtained with the WFI at the ESO-2.2m telescope (Radovich et al. 2004) and Ks band (hereafter K) photometry with NTT+SOFI at the depth (50% completeness) of 23.34 (Temporin et al. in preparation) are available for wide sub-areas of this field. Moreover, an area of about 170 arcmin2 has been covered by deeper J and K band observations with NTT+SOFI at the depth (50% completeness) of 24.15 and 23.84, re- spectively (Iovino et al. 2005). The deep F02 field has been observed also by the CFHT Legacy Survey (CFHTLS1) in several optical bands (u∗, g′, r′, i′, z′) at very faint depth (u∗ = 26.4, g′ = 26.3, r′ = 26.1, i′ = 25.9, z′ = 24.9, 50% completeness). Spectroscopic observations of a randomly selected sub- sample of objects in an area of ∼ 0.5 deg2, with an average sampling rate of about 25%, were performed in the F02 field with VIMOS at the VLT. Spectroscopic data were reduced with the VIMOS Interactive Pipeline Graphical Interface (VIPGI, Scodeggio et al. 2005, Zanichelli et al. 2005) and redshift mea- surements were performed with an automatic package 1 Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 3 Fig. 1. Upper panel: Comparison of photometric and spec- troscopic redshifts in theK-selected sample for objects with highly reliable (confidence level > 97% , i.e. about 1400 galaxies with flag=3, 4) spectroscopic redshifts. The accu- racy obtained is σ∆z = 0.02(1 + z) 2 (shown as solid lines) with only 3.7% of outliers, defined as the objects outside the region limited by the 2 dotted lines (zphoto = zspec ± 0.15(1 + zspec)) in the figure. Lower panel: Spectroscopic (solid line) and photometric (dotted line) redshift distribu- tion for the same comparison sample. (KBRED) and then visually checked. Each redshift mea- surement was assigned a quality flag, ranging from 0 (failed measurement) to 4 (100% confidence level); flag 9 indicates spectra with a single emission line, for which multiple red- shift solutions are possible. Further details on the quality flags are given in Le Fèvre et al. (2005). The analysis presented in this paper is based on the first epoch VVDS deep sample, which has been obtained from the first spectroscopic observations (fall 2002) on the field VVDS-02h, which cover 1750 arcmin2. 2.1. The I-selected Spectroscopic Sample In this study we use the F02-VVDS deep spectroscopic sam- ple, purely magnitude limited (17.5 ≤ I ≤ 24), in combi- nation with the multi-wavelength optical/near-IR dataset. From the total sample of 8281 objects with measured red- shift, we removed the spectroscopically confirmed stars and broad line AGN, as well the galaxies with low quality red- shift flag (i.e. flag 1), remaining with 6419 galaxy spec- tra with secure spectroscopic measurement (flags 2, 3, 4, 9), corresponding to a confidence level higher than 80%. Galaxies with redshift flags 0 and 1 are taken into account statistically (see Section 4 and Ilbert et al. 2005 and Zucca et al. 2006 for details). This spectroscopic sample has a median redshift of ∼ 0.76. Compared to previous optically selected samples, the VVDS has not only the advantage of having an unprecedented high fraction of spectroscopic redshifts (compared, for example, to the purely photomet- ric redshifts as in COMBO17, Wolf et al. 2003 and Borch et al. 2006 for the MF), but also of being purely magnitude selected (17.5 < I < 24), differently, for example, from the DEEP2 (Bundy et al. 2006 for the MF) survey, which has a colour-colour selection. Moreover, the VVDS covers an area from 10 to 40 times wider than the GOODS-MUSIC field (Fontana et al. 2006) and the FORS Deep Field (FDF, Drory et al. 2005), respectively. 2.2. The K-selected Photometric Sample A wide part of the VVDS-02h field (about 623 arcmin2) has been observed also in the near-IR (Iovino et al. 2005, Temporin et al. in preparation). This allows us to build a K-selected sample with a total area of 610 arcmin2 (after excluding low-S/N borders): 442 arcmin2 are 90% complete to K < 22.34, while 168 arcmin2 are 90% complete to K < 22.84 (equivalent to KVega = 21). This sample consists of 11221 objects, of which 2882 have VVDS spectroscopy. In particular, the deep sample (K < 22.84) consists of 3821 objects, of which 749 have VVDS spectroscopy, and 596 of them are galaxies with a secure spectroscopic identification (flags 2, 3, 4, 9). This latter deep sample is more than one magnitude deeper than the samples from the K20 spectroscopic survey (Cimatti et al. 2002) and the MUNICS survey (Drory et al. 2001). Additionally, the total K-selected sample covers an area more than 10 times wider than the K20 and the GOODS- CDFS sample used by Drory et al. (2005) and 4 times wider than the GOODS-MUSIC field (Fontana et al. 2006). Since the spectroscopic sampling of the K-selected sam- ple is less than satisfactory, we take advantage of the high quality photometric redshifts (zphoto). The method and the calibration are presented and discussed in Ilbert et al. (2006). The comparison sample contains 3241 ac- curate spectroscopic redshifts (confidence level > 97% , i.e. flag=3, 4) up to I = 24 obtaining a global accu- racy of σ∆z/(1+z) = 0.037 with only 3.7% of outliers. Also in the K-selected photometric sample the agreement be- tween photometric and highly reliable spectroscopic red- shifts (about 1400) is excellent (Figure 1). We note, how- ever, a non-negligible number of catastrophic solutions with zphoto ∼ 1.2 and zspec ∼ 1 which could introduce a bias at high-redshift (see also discussion in Section 2.3). Even if we cannot rely on a wide spectroscopic comparison sample at high-z, the number of galaxies with zphoto > 1.2 is simi- lar or only slightly higher (about 20%) than the number of galaxies with zspec > 1.2 (63 vs. 51, see Figure 1) and have very similar fluxes and colors. For this reason we do not expect that our results on the mass function and the mass density will be strongly biased by the effect of catas- trophic redshifts (see Section 4). Furthermore, at high-z the dispersion between photometric and spectroscopic redshifts increases, but not drammatically, to σ∆z/(1+z) ≃ 0.05, 0.06 4 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History at z > 1.2, 1.4. Over the whole redshift range it can be represented with σ∆z ≃ 0.02(1 + z) 2 (shown in Figure 1). For the whole K-selected sample, the median errors on photometric redshifts, based on χ2 statistics, are σzphoto = 0.06 (0.04 at z < 1, increasing to 0.14 at z > 1.5). As ex- pected, there is also an increase of σzphoto for the faintest objects, but this increase is only about 0.02 in the faintest magnitude bin. As previously noted by Ilbert et al. (2006) the statistical errors are consistent with σ∆z and could be used as an indication of their accuracy. We will discuss in the following sections the effects of these uncertainties and conclude that they do not affect significantly our conclu- sions. We note, moreover, that the K-selected sample se- lects a different population, in particular of Extremely Red Objects (EROs) at zphoto > 1 (see Section 2.3 and Fig. 3), compared to the I-selected sample used to calibrate the de- rived photometric redshift. Actually, photometric redshifts greater than 0.8-1.0 for the EROs population have been confirmed spectroscopically with very low contamination of low-z objects (Cimatti et al. 2002). Moreover, the near-IR bands are crucial to constrain photometric redshifts in the redshift desert since the J-band is sensitive to the Balmer break up to z = 2.5. Indeed Ilbert et al. (2006) obtain for the deep sample at K < 23 the most reliable photometric redshifts on this sub-sample with only 2.1% of outliers and σ∆z/(1+z) = 0.035 (see their figure 13). In this paper we therefore use photometric redshifts for the whole K-selected photometric sample and the highly reliable spectroscopic redshifts when available. In order to select galaxies from the total K-selected photometric sample, we have used a number of photomet- ric methods to remove candidate stars, as described be- low. Some of the possible criteria to select stars are: (i) the CLASS STAR parameter given by SExtractor (Bertin & Arnouts 1996), providing the “stellarity-index” for each object, reliable up to I ≃ 21; (ii) the FLUX RADIUS K parameter, computed by SExtractor from the K band im- ages, which gives an estimate of the radius containing half of the flux for each object; this can be considered a good criterion to isolate point-like sources up to K ≃ 19 (see Iovino et al. 2005); (iii) the BzK criterion, proposed by Daddi et al. (2004a), with stars characterized by colours z − K < 0.3(B − z) − 0.5; (iv) the χ2 of the SED fitting carried out during the photometric redshift estimate (Ilbert et al. 2006), with template SEDs of both stars and galaxies. To efficiently remove stars in the whole magnitude range of our sample, avoiding as much as possible to lose galaxies, we decided to use the intersection of the first three crite- ria. We therefore selected as stars the objects fulfilling all the constraints (i) CLASS STAR ≥ 0.95 for I < 22.5 or CLASS STAR ≥ 0.90 if I > 22.5, (ii) FLUX RADIUS K < 3.4 and (iii) z − K < 0.3(B − z) − 0.5. When it was not possible to apply criterion (iii), because of non detec- tion either in the B or z filters, we used criterion (iv) in its place. Furthermore, we added to the sample of candidate stars also the objects with K < 16 and FLUX RADIUS K < 4, to be sure to exclude from the galaxy sample these sat- urated point-like objects. The final sample consists of 653 candidate stars, which we have removed from the sample in the following analysis. Comparing to the spectroscopic subsample (we remind that stars were not excluded from the spectroscopic targets of VVDS), we found about 87% of efficiency to photometrically select stars, i.e. only 28 out Fig. 2. Redshift distributions for the K-selected photomet- ric sample (filled histogram) and for the I-selected spectro- scopic sample (empty histogram). of the 214 spectroscopic stars have not been selected in this way, and only 3 (1.4%) with highly reliable spectroscopic flag (3, 4), whereas 21 spectroscopic extragalactic objects (less than 1%) fall inside the candidate star sample. Three of them are broad line AGN and the others are all compact objects, most of them with redshift flags 1 or 2 and only one with flag 3. This latter object has not been eliminated from the galaxy sample. We have furthermore removed from the galaxy sample the spectroscopically confirmed AGNs and the three secure spectroscopic stars which were not removed with the photometric criteria. The final K-selected sample consists of 10160 galaxies with either photometric redshifts or highly reliable spec- troscopic redshifts, when available, in the range between 0 and 2.5: 6720 galaxies in the shallow K < 22.34 area of 442 arcmin2 and 3440 galaxies in the deeper area (K < 22.84) of 168 arcmin2. 2.3. Comparison of the Two Samples As shown in Fig. 2, the redshift distribution in the K- selected sample peaks at higher redshift than in the I- selected spectroscopic sample, with the two median red- shifts being 0.91 and 0.76, respectively. Even if we cannot rely on a wide spectroscopic comparison sample at high-z, we have better investigated the reliability of the high-z tail in the K-selected sample in term of fraction and colors. We found indeed that at K < 22 the fraction of objects with z > 1, 1.5 (35, 13% respectively) is similar to previous spec- troscopic (K20 survey, see Cimatti et al. 2002) or photomet- ric studies (Somerville et al. 2004). Moreover, we have used the BzK color-color diagnostic proposed and calibrated on a spectroscopic sample to cull galaxies at 1.4 < z < 2.5 (Daddi et al. 2004). We found that most (92%) of the galax- L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 5 Fig. 3. Colour-magnitude diagram and colour distribution at different redshifts for the K-selected photometric sam- ple (filled squares and histogram) and for the I-selected spectroscopic sample (open circles and empty histogram) ies at 1.5 < zphoto < 2.5 lie in the high-z region of the BzK diagram. We conclude that our K-selected sample shows no indi- cation of significant bias in its high-redshift tail. The global (I −K) colour distributions of the two samples are similar to each other up to z ∼ 1.2 (see right panels in Fig. 3), but they are significantly different at higher redshift. At z > 1.2 the I-selected sample misses many red galaxies fainter than the I limit, most of them being Extremely Red Objects (EROs: defined as objects with colours I−K > 2.6), which are instead included in the K sample (∼ 81% of EROs in the deep K < 22.84 sample have I > 24). For this rea- son the K-selected sample is more adequate to study the massive tail of the GSMF at high-z. Vice versa, the K- selected sample misses at all redshifts a number of faint blue galaxies, which are included in the I-selected spectro- scopic sample (see left panels in Fig. 3). These faint blue galaxies are important in the estimate of the low-mass tail of the GSMF. 3. Estimate of the Stellar Masses The rest-frame near-IR light has been widely used as a tracer of the galaxy stellar mass, in particular for lo- cal galaxies (e.g. Gavazzi et al. 1996; Madau, Pozzetti & Dickinson 1998, Bell & de Jong 2001). However, an accurate estimate of the galaxy stellar mass at high z, where galax- ies are observed at widely different evolutionary stages, is more uncertain because of the variation of the Mstars/LK ratio as a function of age and other parameters of the stel- lar population, such as the star formation history and the metallicity. The use of multiband imaging from UV to near- IR bands is a way to take into account the contribution to the observed light of both the old and the young stellar populations in order to obtain a more reliable estimate of the stellar mass. However, even stellar masses estimated using the fit to the multicolour spectral energy distribution (SED) are model dependent (e.g. changing with different assumptions on the initial mass function, IMF) and subject to various degeneracies (age – metallicity – extinction). In order to re- duce such degeneracies we have used a large grid of stellar population synthesis models, covering a wide range of pa- rameters, in particular in star formation histories (SFH). Indeed, in the case of real galaxies the possibly complex star-formation histories and the presence of major and/or minor bursts of star formation can affect the derived mass estimate (see Fontana et al. 2004). We have applied and compared two different methods to estimate the stellar masses from the observed magnitudes (using 12 photometric bands from u∗ to K), that are based on different assumptions on the star-formation history. For both of them we have adopted the Bruzual & Charlot (2003; BC03 hereafter) code for spectral synthesis models, in its more recent rendition, using its low resolution version with the “Padova 1994” tracks. Different models have also been considered, e.g. Maraston 2005, and Pégase models (Fioc and Rocca-Volmerange 1997). The results obtained with these models are compared with those obtained with the BC03 models at the end of Section 3.1. Since most of previous studies at high-z assumed models with exponentially decreasing SFHs, we have used the same simple smooth SFHs (see Section 3.1) in order to compare our results with those of previous surveys. In addition, to further test the uncertainties in mass determination, we have used models with complex SFHs (see Section 3.2), in which secondary bursts have been added to exponentially decreasing SFHs. These models have been widely used in studies of SDSS galaxies (see Kauffmann et al. 2003 and Salim et al. 2005 for further details). Table 1 summarizes the model parameters used in the 2 methods described in the following sections. In our analysis we have adopted the Chabrier IMF (Chabrier et al. 2003), with lower and upper cutoffs of 0.1 and 100 M⊙. Indeed, all empirical determinations of the IMF indicate that its slope flattens below ∼ 0.5 M⊙ (Kroupa 2001, Gould et al. 1996, Zoccali et al. 2000) and a similar flattening is required to reproduce the observed Mstars/LB ratio in local elliptical galaxies (see e.g. Renzini 2005). As discussed extensively by Bell et al. (2003), the Salpeter IMF (Salpeter 1955) is too rich in low mass stars to satisfy dynamical constraints (Kauffmann et al. 2003, Kranz et al. 2003). Moreover, di Serego Alighieri et al. (2005) show a rather good agreement between dynamical masses and stellar masses estimated with the Chabrier IMF at z ∼ 1. Specifically, this is true at least for high-mass el- liptical galaxies, less affected than lower-mass galaxies by uncertainties in the estimate of their dynamical mass due to possibly substantial rotational contribution to the observed velocity dispersion. At fixed age the masses obtained with the Chabrier IMF are smaller by a factor ∼ 1.7, roughly independent of the age of the population, than those derived with the classical Salpeter IMF, used in several previous works that we shall compare with (e.g. Brinchmann & Ellis 2000; Cole et al. 2001; Dickinson et al. 2003; Fontana et al. 2004). We have checked this statement in our sample, finding a systematic median offset of a factor 1.7 and a very small dispersion 6 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History (σ = 0.082 dex) in the masses derived with the two different IMFs. Since this ratio is approximately constant for a wide range of star formation histories (SFH), the uncertainty in the IMF does not introduce a fundamental limitation with respect to the results we will discuss in the following Sections. Even if the absolute value of the mass estimate is uncertain, the use of Salpeter or Chabrier IMFs does not introduce any significant difference in the relative evolution with redshift of the mass function and mass density. One possible limitation of our approach to derive stellar masses in our sample is the contamination by narrow-line AGN (broad line AGN have been already excluded, see Section 2.1). From the available spectroscopic diagnostic, in the I-selected spectroscopic sample at the mean red- shift z ≃ 0.7, we found that the contamination due to type II AGN is less than 10%. Recently, several studies (Papovich et al. 2006, Kriek et al. 2007, Daddi et al. 2007) suggest that the fraction of type II AGN increases with redshift and stellar mass. According to Kriek et al. (2007) at 2 < z < 2.7 and K < 21.5 the fraction is about 20% for massive (2.9 × 1011M⊙ for a Salpeter IMF) galaxies. To derive the contribution of type II AGN to the massive tail of the MF is beyond the scope of this paper. However we note, as shown also by the above studies, that for most of these objects the optical light is dominated by the in- tegrated stellar emission. Therefore, both our photometric redshift and mass estimates are likely to be approximately correct also for them. 3.1. Smooth SFHs Consistently with previous studies, we have used synthetic models with smooth SFH models (exponentially decreas- ing SFH with time scale τ : SFR(t) ∝ exp(−t/τ)) and a best-fit technique to derive stellar masses from multicolour photometry. To this purpose we have developed the code HyperZmass, a modified version of the public photomet- ric redshift code HyperZ (Bolzonella et al. 2000): like the public version, HyperZmass uses the SED fitting technique, computing the best fit SED by minimizing the χ2 between observed and model fluxes. We used models built with the Bruzual & Charlot (2003) synthetic library. When the red- shift is known, either spectroscopic or photometric, the best fit SED and its normalization provide an estimate of the stellar mass contained in the observed galaxy. In particu- lar, we estimate the stellar mass content of the galaxies, derived by BC03 code, by integrating the star formation history over the galaxy age and subtracting from it the “Return fraction” (R) due to mass loss during the stellar evolution. For a Chabrier IMF, this fraction is already as high as ∼ 40% at an age of the order of 1 Gyr and ap- proaches asymptotically about 50% at older ages. The parameters used to define the library of synthetic models are listed in Table 1. Similar parameters have been used in Fontana et al. (2004). The Calzetti (2000) extinction law has been used. Following that paper, we have excluded from the grid some models which may be not physical (e.g. those implying large dust extinctions, AV > 0.6, in absence of a significant star-formation rate, Age/τ > 4, see Table 1). To better match the ages of early-type galaxies in the local universe and following SDSS studies, we also removed models with τ < 1 Gyr and with star formation starting at z < 1. Table 1. Parameters Used for the Library of Template Method Smooth SFHs Complex SFHs IMF Chabrier Chabrier SFR τ (Gyr) [0.1,∞]a [1,∞] log(Age)b (yr) [8, 10.2] [8, 10] burst age (yr) − [0, 1010] burst fraction − [0, 0.9] Metallicities Z⊙ [0.1Z⊙, 2Z⊙] Extinction Calzetti law Charlot&Fall model (n = 0.7, µ ∈ [0.1, 1]) Dust content AcV ∈ [0, 2.4] τV ∈ [0, 6] a τ < 1 if star formation starts at z < 1. b At each redshift, galaxies are forced to have ages smaller than the Hubble time at that redshift. c AV < 0.6 if Age/τ > 4. We find that the “formal” typical 1σ statistical errors (defined as the 68% range as derived from the χ2 statis- tics) on the estimated masses, not taking into account the error on the estimate of the photometric redshift for the K-selected sample, are of the order of 0.04 dex for the K-selected sample and 0.05 for the I-selected sample. A more reliable estimate of the errors has been obtained us- ing HyperZ to simulate catalogs to the same depth of our sample (see Bolzonella et al. 2000). Using all 12 photo- metric bands (from u∗ to K), available for a subset of our data, and realistic photometric errors, the recovered stellar masses reproduce the input masses with no significant offset and a dispersion of 0.12 dex up to z ∼ 3. For comparison, using only the optical bands (from u∗ to z′) the disper- sion increases to ∼ 0.49 dex at z > 1. The best fit masses obtained from input simulations built using randomly all available metallicities and analyzed only with solar metal- licity models are not significantly shifted from the input masses, but the dispersion increases from 0.12 dex to 0.21 dex. These dispersions, computed using a 4σ clipping, pro- vide an estimate of the minimum, intrinsic uncertainties of this method at our depth. For the K-selected photometric sample further uncertainties in the fitting technique are due to the photometric redshift accuracy (σ∆z ≃ 0.02(1 + z) up to z = 2.5) which corresponds on average to about 0.12 dex of uncertainty in mass, being larger at low redshift (∼ 0.2 dex at z < 0.4) than at high-z (∼ 0.10 dex at z = 2). Although in principle the best-fitting technique pro- vides estimates also for age, metallicity, dust content and SFH timescale, our simulations show that on average all these quantities are much more affected by degeneracies and therefore less constrained than the stellar mass. In addition, we have compared our derived masses with those obtained by using different population synthesis mod- els, such as Pégase and Maraston (2005) models. In partic- ular, Maraston (2005) models include the thermally pulsing asymptotic giant branch (TP-AGB) phase, calibrated with L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 7 local stellar populations. This stellar phase is the dominant source of bolometric and near-IR energy for a simple stellar population in the age range 0.2 to 2 Gyr. We have tested the differences with BC03 models using the I-selected spec- troscopic sample and we found only a small but systematic shift (∼ −0.14 dex and a similar dispersion) up to z ∼ 1.2 both with and without the use of near-IR photometry. On the contrary, masses derived using Pégase models and sim- ilar SFHs have instead no significant offset. At higher redshifts the differences between our esti- mated masses and those obtained with Maraston models in the K-selected spectroscopic subsample are slightly smaller and even smaller in the K-selected photometric sample (∼ −0.11,−0.08, respectively). This differences are smaller than that found by Maraston et al. (2006), ∼ −0.2, in their SED fitting (from B up to Spitzer IRAC and MIPS bands) of a few high redshift passive galaxies with typical ages in the range 0.5 – 2.0 Gyr, selected in the Hubble Ultra Deep Field (HUDF). This difference between our results and those of Maraston et al. could be due to a combina- tion of effects, such as the absence in our photometric data of mid-IR Spitzer photometry, which at these redshifts is sampling the rest frame near-IR part of the SED, mostly in- fluenced by the TP-AGB phase, and also to the wide range of complex stellar populations in our sample, in which the effect of the TP-AGB phase may be diluted by the SFH. 3.2. Complex SFHs Real galaxies could have undergone a more complex SFH, in particular with the possible presence of bursts of star formation on the top of a smooth SFH. Thus, we have com- puted masses also following a different approach, which has been intensively used in previous studies of SDSS galaxies (e.g. Kauffmann et al. 2003, Brinchmann et al. 2004, Salim et al. 2005, Gallazzi et al. 2005). In this approach we pa- rameterize each SF history in terms of two components: an underlying continuous model, with an exponentially de- clining SF law (SFR(t) ∝ exp(−t/τ)), and random bursts superimposed on it. We assume that random bursts occur with equal probability at all times up to galaxy age. They are parameterized in terms of the ratio between the mass of stars formed in the burst and the total mass of stars formed by the continuous model over the age. This ratio is taken to be distributed between 0.0 and 0.9. During a burst, stars are assumed to form at a constant rate for a time dis- tributed uniformly in the range 30 – 300 Myr. The burst probability is set so that 50% of the galaxies in the library have experienced a burst in the past 2 Gyr. Attenuation by dust is described by a two-component model (see Charlot & Fall 2000), defined by two parameters: the effective V - band absorption optical depth τV affecting stars younger than 10 Myr and arising from giant molecular clouds and the diffuse ISM, and the fraction µ of it contributed by the diffuse ISM, that also affects older stars. We take τV to be distributed between 0 and 6 with a broad peak around 1 and µ to be distributed between 0.1 and 1 with a broad peak around 0.3. Finally, our model galaxies have metallic- ities uniformly distributed between 0.1 and 2 Z⊙. The model spectra are computed at the galaxy redshift and in each of them we measure the k-shifted model magni- tudes for each VVDS photometric band. We also force the age of all models in a specific redshift range to be smaller than the Hubble time at that redshift. The model SEDs Fig. 4. Effect of NIR photometry in the mass determina- tion: ratio between masses estimated without and with NIR photometry vs. mass determined without NIR photometry. The data have been splitted into different redshift ranges. Left: masses determined using smooth SFHs. Right: The same, but using complex SFHs are then scaled to each observed SED with a least squares method and the same scaling factor is applied to the model stellar mass. We compare the observed to the model fluxes in each photometric band and the χ2 goodness of fit of each model determines the weight (∝ exp[−χ2/2]) to be assigned to the physical parameters of that model when building the probability distributions for each parameter of any given galaxy. The probability distribution function (PDF) of a given physical parameter is thus obtained from the distribution of the weights of all models in the library at the specified redshift. We characterize the PDF using its median and the 16 – 84 percentile range (equivalent to ±1σ range for Gaussian distributions), and also record the χ2 of the best-fitting model. Similarly to what has been done for the models with smooth SFH (see Section 3.1), also in this case the stel- lar mass content of galaxies is derived by subtracting the return fraction R from the total formed stellar mass. We find that the average “formal” 1σ error (defined as half of the 16 – 84 percentile range) on the estimated masses is of the order of 0.09 dex for the K-selected sample. The av- erage error increases with redshift from ∼ 0.06 dex at low redshift to ∼ 0.11 dex at 1 < z < 2 and decreases with in- creasing mass from ∼ 0.08 dex for logM < 10 to ∼ 0.05 for logM > 10 at z ≃ 0.7. In the K-selected sample the photo- metric redshift accuracy induced a further uncertainty on the mass of the order of 30% up to z > 1.5. In the I-selected sample, where near-IR photometry is not always available, the typical error on the mass is larger and is of the order of ∼ 0.13 dex. 8 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 3.3. Effect of NIR Photometry For about half of the sources in the I-selected sample only optical photometry is available. We have therefore used the results obtained for theK-selected spectroscopic subsample to better understand the reliability of the mass estimates in the whole I-selected sample and quantify potential system- atic effects. We found that the mass estimates derived using only optical bands are on average in rather good agreement with those obtained using also NIR bands up to z ∼ 1.2. In absence of NIR bands the galaxy stellar masses tend to be only slightly overestimated, with a median shift < 0.1 dex; this is due to the fact that already at z = 0.4, for example, the z′-band (the reddest band used in the fit in absence of NIR) samples the R-band rest-frame and therefore the SED fitting is less reliable for the estimate of the stellar masses. There is however a significant fraction of the galaxies for which the ratio between the two masses is higher than a factor of three (see upper panels of Fig.4). This fraction of galaxies with significantly discrepant mass estimates is ∼ 5% for the models with smooth SFH and ∼ 9 % for the models with complex SFH. At higher redshifts, where our reddest optical band, i.e. the z-band, is sampling the rest-frame spectrum bluewards of the 4000 Å break, the comparison of the two sets of mass estimates (i.e. with and without near-IR photometry) is significantly worse. Not only the median shift increases significantly, but also the ratio of the two sets of masses is significantly correlated with the mass derived without using NIR photometry (see lower panels of Fig. 4). For this reason, we have decided to use the whole I-selected VVDS spectroscopic sample only up to z ∼ 1.2, whereas at higher redshifts we use as reference the K-selected photometric sample. As shown in the upper panels of Fig.4, the ratio between the masses computed without and with NIR photometry has a non-negligible dispersion also for z < 1.2, with the masses computed without NIR photometry being higher on average. In order to statistically correct for this effect, we have performed the following Monte Carlo simulation. For each galaxy without near-IR photometry in the I-selected spectroscopic sample we have applied a correction factor to its estimated mass. This correction factor has been de- rived randomly from the observed distribution, at the mass of each galaxy, of the ratios of the masses with and with- out NIR photometry. The effect on the mass function of using these “statistically corrected” masses is shown and discussed in Sect. 4. 3.4. Comparison of the Masses Obtained with the Two Methods In this section we compare the mass estimates we obtained using the two different methods described above for the VVDS galaxies. Since the ratio of the two estimates is al- most independent of the mass, in Fig. 5 we show the his- tograms of this ratio, integrated over all masses, for two dif- ferent redshift bins. In the upper panel (z < 1.2), both data from the I-selected and the K-selected samples are shown; in the lower panel only data from the K-selected sample are shown, since the I-selected sample is not used to derive the mass function in this redshift range. Gaussian curves, representing the bulk of the population, are drawn on the Fig. 5. Histograms of the ratio of the masses derived with the smooth and the complex star formation histories. In the upper panel (z < 1.2) both data from the I-selected and the K-selected samples are shown; in the lower panel (z > 1.2) only data from the K-selected sample are shown, since the I-selected sample is not used to derive the mass function in this redshift range. top of each histogram. The parameters of each Gaussian are reported in the figure. The global comparison of the two sets of masses is rather satisfactory, even if it shows a systematic shift, larger for theK-selected sample, between the two sets of masses, with the ‘smooth SFH’ masses being on average smaller than the ‘complex SFH’ masses. The values of σ of these Gaussians are similar, of the order of 0.13 dex. However, in two of the three cases (i.e. the I-selected sample at low redshift and the K-selected sample at high redshift) the distributions of the ratio of the masses appear to be asymmetric, with tails which are not well represented by a Gaussian distribution. The fractions of these “outliers” are given in Fig. 5. This tail is particularly significant for the I-selected sample, for which there are galaxies with the ratio between the two masses higher than 3 and in a few cases reaching a value of We have analyzed the effect of the different parameters used in the two methods. The impact of different extinction curves on the mass estimates has already been investigated by Papovich et al. (2001), Dickinson et al. (2003), Fontana et al. (2004), and found to be small. We have repeated the same exercise for the two different dust attenuation models adopted, finding that for a given SFH the mass estimated with different extinction laws are similar, with an average shift of 0.02 dex. Analyzing in some detail the properties of the galaxies which are in the extended tail of large mass ratios for the I- selected sample at low redshift (see the upper panel in Fig. 5), we found that, even if many of them do not have near- IR photometry and therefore their mass is more uncertain L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 9 (see previous section), on average they are characterized by blue colours in the bluer bands (B − I) and red colours in the redder bands (I − z or I − K). Because of this, they have been fitted typically with low values of the popula- tion age by HyperZmass with a smooth SFH, while the estimate obtained with a complex SFH corresponds to an older, and therefore more massive, population plus a more recent burst with a mass fraction in the burst of the order of fburst < 0.15. Vice versa the low ratio outliers in the K- selected sample at high redshift have been fitted typically with moderately higher dust content by HyperZmass than with complex SFHs. Finally, we conclude that the main dif- ferences between the two methods to determine the masses are largely due to different assumed SFHs and, in particu- lar, to the secondary burst component allowed in the model with complex SFHs. To summarize, we have explored in detail two differ- ent methods and a wide parameter space (see Table 1) to estimate the stellar mass content in galaxies in order to better understand the uncertainties in the photomet- ric stellar mass determinations. Indeed, a good estimate of the intrinsic errors may be critical for the GSMF measure- ment and interpretation. We found that, within a given assumption on the SFH, the accuracy of the photomet- ric stellar mass is overall satisfactory, with intrinsic un- certainties in the fitting technique of the order of ∼ 30%, in agreement with similar results in the K20 (Fontana et al. 2004), HDFN (Dickinson et al. 2003) and HDFS (Fontana et al. 2003) at the same redshifts. These errors are smaller than the estimates at higher redshift (z ≃ 3) (Papovich et al. 2001, Shapley et al. 2001), since at our av- erage redshifts (z ≃ 0.7−1) we can rely on a better sampling of the rest-frame near-IR part of the spectrum, while are similar to the uncertainties estimated at high-z using IRAC data by Shapley et al. (2005). For theK-selected photomet- ric sample the uncertainties in the stellar mass due to the photometric redshift errors are on average of the order of 30% up to z > 1.5, giving a total fitting uncertainty up to 45% at z > 1.5. Finally, systematic shifts, mainly due to different assumptions on the SFHs, can be as large as ∼ 40% over the entire redshift range 0.05 < z < 2.5 when NIR photometry is available. The uncertainty in the de- rived masses is obviously higher also at low redshift when NIR photometry is not available and in this case it becomes extremely large at z > 1.2 (see Fig.4). For what concerns the absolute value of the mass, its uncertainty is mainly due to the assumptions on the IMF and it is within a factor of 2 for the typical IMFs usually adopted in the literature. In the following sections we discuss in some detail the effects of the two methods on the derivation of the GSMF. 3.5. Massive Galaxies at z > 1 Figure 6 shows the stellar masses for the 2 samples (I and K-selected) derived using the smooth SFHs. It is inter- esting to note the presence of numerous massive objects (logM > 11) at all redshifts and up to z = 2.5. High- z massive galaxies have been already observed in previous surveys (Fontana et al. 2004, Saracco et al. 2005, Cimatti et al. 2004, Glazebrook et al 2004, Fontana et al. 2006, Trujillo et al. 2006). Here the relatively wide area (the K-selected sample is more than 10 times wider and from 0.5 to 1 mag- nitude deeper than the K20 survey) allows to better sam- Fig. 6. Stellar Mass as a function of redshift for the I- selected spectroscopic (left) and for the K-selected photo- metric (right) samples for smooth SFHs. ple the massive tail of the population. We note that mas- sive galaxies have typically redder optical-NIR rest-frame colours (〈MI −MK〉 ≃ 0.7) compared to the whole popu- lation (〈MI −MK〉 ≃ 0.5), consistently with the idea that massive galaxies host the oldest stellar population. Further analysis of the stellar population properties and spectral features of massive galaxies, as well as of red objects will be presented in forthcoming papers (Lamareille et al. in preparation, Vergani et al. 2007, Temporin et al. in prepa- ration). 4. Mass Function Estimate Once the stellar mass has been estimated for each galaxy in the sample, the derivation of the corresponding Galaxy Stellar Mass Function (MF) follows the traditional tech- niques used for the computation of the luminosity function. Here we apply both the classical non-parametric 1/Vmax formalism (Schmidt 1968, Felten 1976) and the parametric STY (Sandage, Tammann & Yahil 1979) method to esti- mate best-fit Schechter (1976) parameters (α,M∗stars, φ In the case of the K-selected sample, in order to take into account the two different magnitude limits, we perform a “Coherent Analysis of independent samples” as described by Avni & Bahcall (1980). Ilbert et al. (2004) have shown that the estimate of the faint end of the global luminosity function can be biased, because, due to different k-corrections, different galaxy types have different absolute magnitude limits for the same apparent magnitude limit. The same bias is present also for the low mass end of the mass function. This is due to the fact that, because of the existing dispersion in the mass-to- light ratio of different galaxy types, at small masses the ob- jects with the largest mass-to-light ratio are not included in a magnitude limited sample (see Appendix B in Fontana et 10 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History Fig. 7. Effect on the I-selected MF of the use of the statistically corrected masses, which take into account the effect of near-IR photometry on the mass determination (see text for details). Complex SFHs have been used to derive masses. Empty circles represent the MF obtained using the original uncorrected masses, filled squares show the MF obtained using the statistically corrected masses. For comparison the STY Schechter MF is shown for the subsample where near-IR photometry is available. The vertical dashed lines represent the completeness limit of the sample as defined in the text. al. 2004 for an extensive discussion). For this reason, when computing the global mass functions with the STY method, in order to fully avoid this bias, we should use in each red- shift range only galaxies above the stellar mass limit where all the SEDs are potentially observable. In both the K- and I-selected samples these limits derive from the conversion between photometry and stellar mass for early-type galax- ies and are very restrictive. However, recent results show that the faint-end and the low-mass end of the luminosity and mass function is dominated by late-type galaxies up to z ∼ 2 (Fontana et al. 2004, Zucca et al. 2006, Bundy et al. 2006). Therefore for the STY estimate we will use as lower limit of the mass range the minimum mass above which late-type SEDs (defined by rest-frame optical/NIR colours MI −MK < 0.4) are potentially observable (see Fig. 9). We have estimated the MF for both the deep I-selected (17.5 < I < 24) spectroscopic sample and the photomet- ric K-selected (K < 22.34 & K < 22.84) sample. For each sample the MF has been estimated using masses com- puted with both methods described in Section 2.3. In the case of the spectroscopic sample, in order to correct for both the non-targeted sources in spectroscopy and those for which the spectroscopic measurement failed, we use a statistical weight wi, associated with each galaxy i with a secure redshift measurement (see Ilbert et al. 2005 for details). This weight is the inverse of the product of the Target Sampling Rate times the Spectroscopic Success Rate. Accurate weights have been derived by Ilbert et al. (2005) for all objects with secure spectroscopic redshifts, tak- ing into account all the parameters involved (magnitudes, galaxy size and redshift). For the K-selected sample, we have tested the effect of catastrophic photometric redshifts (see discussion in Sec. 2.2) on the evolution of the mass function and mass den- sity. We have used the I-selected spectroscopic sample, re- placing spectroscopic redshifts with photometric redshifts. The two MFs (with either spectroscopic or photometric red- shifts) are very similar in the whole mass and redshift range (0.05 < z < 2.5) analyzed and even at z > 1.2, where we note a not negligible number of catastrophic photometric redshifts (see discussion in Sec. 2.2). There is no evidence of a strong bias in the normalization and in the shape of the MF; also the massive tails of the MFs are similar, within the statistical errors. We conclude that the catastrophic L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 11 Fig. 8.K-selected MF derived from the 2 subsamples, deep (filled circles) and shallow (empty circles), separately (smooth SFHs have been used to derive masses). For comparison the STY Schechter MF for the globalK-selected sample is shown. The vertical dashed lines represent the completeness limit of the 2 K-selected subsamples. solutions at high photometric redshifts (i.e. masses) do not strongly affect our results. 4.1. The VVDS Galaxy Stellar Mass Function The resulting stellar mass functions of the VVDS sample are derived in the following redshift ranges: (a) 0.05 < z < 1.2 for the I-selected sample, because at higher redshift the mass estimate becomes very uncertain (see figure 4) and (b) 0.05 < z < 2.5 for the K-selected sample. We have furthermore divided the 2 samples into different redshift bins in order to sample evolution with similar numbers of sources in each bin. For the galaxies in the I-selected sample not covered by near-IR data, we have used the statistically corrected masses derived through a Monte Carlo simulation, to take into account the effect of the near-IR photometry in the mass determination (see Sec. 3.3). Figure 7 shows the ef- fect in the MF for complex SFHs. The high mass tail is significantly reduced if we use statistically corrected masses when near-IR is not available. Consistent MFs have been 12 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History Fig. 9. Galaxy Stellar Mass Functions in the I-selected (squares) and K-selected (triangles) using both methods to estimate the stellar masses (empty symbols for smooth SFHs and filled for complex SFHs). The STY Schechter fits for the 2 methods limit the hatched regions (horizontal hatched for the K-selected and vertical hatched for the I-selected samples). Vertical hatched regions represents the completeness limit of the 2 samples. The local MFs by Cole et al. (2001), both original and “rescaled” version (Fontana et al. 2004), and by Bell et al. (2003) are reported in each panel as dotted lines. L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 13 Table 2. STY parameters in the different redshift ranges sample method z range mean redshift α logM∗stars(h 70 M⊙) φ ∗(10−3h370Mpc I smooth 0.05 - 0.4 0.27 -1.26+0.01 −0.02 11 1.90 +0.08 −0.16 I smooth 0.4 - 0.7 0.58 -1.23+0.04 −0.04 11 +0.08 −0.08 1.72 +0.33 −0.28 I smooth 0.7 - 0.9 0.8 -1.23+0.12 −0.11 10.88 +0.15 −0.13 1.6 +0.55 −0.45 I smooth 0.9 - 1.2 1.05 -1.09+0.19 −0.17 10.85 +0.14 −0.14 1.3 +0.43 −0.46 I complex 0.05 - 0.4 0.27 -1.28+0.02 −0.01 11.15 1.75 +0.16 −0.08 I complex 0.4 - 0.7 0.58 -1.22+0.04 −0.04 11.15 +0.08 −0.08 1.58 +0.30 −0.26 I complex 0.7 - 0.9 0.81 -1.04+0.08 −0.07 10.83 +0.07 −0.07 3.02 +0.57 −0.51 I complex 0.9 - 1.2 1.04 -1.16+0.1 −0.09 10.89 +0.08 −0.07 1.80 +0.44 −0.39 K smooth 0.05 - 0.4 0.26 -1.38+0.02 −0.01 10.93 1.29 +0.10 −0.05 K smooth 0.4 - 0.7 0.57 -1.14+0.04 −0.04 10.93 +0.06 −0.06 1.83 +0.27 −0.24 K smooth 0.7 - 0.9 0.81 -1.01+0.07 −0.08 10.67 +0.07 −0.05 2.6 +0.38 −0.44 K smooth 0.9 - 1.2 1.05 -1.1+0.07 −0.08 10.78 +0.06 −0.05 1.83 +0.28 −0.30 K smooth 1.2 - 1.6 1.4 -1.15+0.12 −0.12 10.72 +0.07 −0.06 1.48 +0.30 −0.30 K smooth 1.6 - 2.5 1.96 -1.15 10.96+0.01 −0.02 0.9 +0.30 −0.30 K complex 0.05 - 0.4 0.26 -1.39+0.01 −0.02 11.12 1.17 +0.05 −0.09 K complex 0.4 - 0.7 0.57 -1.16+0.04 −0.04 11.12 +0.06 −0.06 1.58 +0.24 −0.22 K complex 0.7 - 0.9 0.81 -1.16+0.07 −0.07 10.98 +0.07 −0.07 1.74 +0.36 −0.30 K complex 0.9 - 1.2 1.05 -1.2+0.07 −0.06 11.07 +0.06 −0.06 1.34 +0.26 −0.21 K complex 1.2 - 1.6 1.4 -1.17+0.12 −0.12 10.93 +0.07 −0.06 1.39 +0.29 −0.28 K complex 1.6 - 2.5 1.96 -1.17 10.97+0.01 −0.02 1.25 +0.09 −0.04 obtained in the sub-area of I-selected sample where near-IR photometry is available (see Fig. 7). For the K-selected sample, we have analyzed the ef- fect of the cosmic variance on small areas, deriving the MF for the two K-selected subsamples, deep and shallow, sepa- rately (see Figure 8). We find a significantly lower MF (by a factor 1.8 and 1.6) in the redshift range 0.4 < z < 0.7 and 0.7 < z < 0.9 in the deep K-selected sample (K < 22.84 over 168 arcmin2) compared to the shallow K-band sam- ple (K < 22.34 over 442 arcmin2). The significance of such differences in the MF, is of ∼ 2 − 3σ in each mass bin at 0.4 < z < 0.7 and ∼ 1 − 2σ at 0.7 < z < 0.9. Globally, i.e. for the total number densities over the complete mass range, the differences are significant at about 5-3 σ level in the two redshift ranges, respectively. This problem leads to a clear warning on the results based on small fields, as covered by most of the previous existing surveys. In Figure 9 we show the MFs derived using the I- selected spectroscopic sample and the K-selected photo- metric sample for both methods (smooth and complex SFHs) to derive the masses. The resulting mass functions are quite well fitted by Schechter functions. The best-fit Schechter parameters are summarized in Table 2, with the uncertainties derived from the projection of the 68% confi- dence ellipse. Since in the lowest and highest redshift bins (z ≃ 0.2 and 2) the values of M∗stars and the low-mass-end slope (α), respectively, are poorly constrained, they have been fixed to the values measured in the following and pre- vious redshift bins respectively. We note, first, that the overall agreement between the MF derived with the different methods for masses determi- nation is fairly satisfactory, albeit complex SFHs estimates provide typically larger masses. The systematic shift be- tween the 2 methods (Section 3.4) is reflected in most of the redshift bins also in the characteristic mass (M∗stars) of the MF while the best fit slopes (α) and φ∗ Schechter parameters agree within the errors between the different methods in most of the redshift bins. We note, furthermore, an overall agreement between the 2 samples (I- and K-selected), and in most of the redshift bins the Schechter parameters agree within the errors, even if some differences exist. More in detail, the I-selected sam- ple in the range 0.4 < z < 0.9 has a higher low-mass (< 9.5 dex) end and a slightly steeper MF (α ∼ −1.23) than the K-selected one (α > −1.15). These differences are probably due to the population of blue K-faint galaxies, that are missed in the K-sample, as discussed in Section 3. These galaxies have, indeed, median colours in the I- selected sample that are bluer than in the K-selected sam- ple (I−K ≃ 0.45 compared to ≃ 0.89). A similar behaviour has been noted in the local MF derived using an optically selected sample (g band) compared to the local MF from the near-IR (2MASS) sample (Bell et al. 2003). On the contrary, at even lower masses (< 8.5 dex) at z < 0.4 the K-selected MF is slightly steeper than the I-selected one, but no significant differences in the colour of the two pop- ulations is found. 4.2. Comparison with Previous Surveys In general, previous efforts to derive MF have relied on smaller or more limited samples, or often based mainly on photometric redshifts (Drory et al. 2004, 2005). We have compared our MF determination with literature re- sults based on different surveys (K20, COMBO17, MUSIC, DEEP2, FDF+CDFS), rescaled to Chabrier IMF (see Figure 10). Our MFs rely on a higher statistics at inter- mediate to high-mass ranges, and therefore present lower statistical errors. At z < 0.2 we sample unprecedented mass ranges, more than one order of magnitude lower than previ- ous surveys, while at z > 0.4 the FDF and MUSIC surveys reach lower mass limits even if on significantly smaller area. 14 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History Fig. 10. Comparison between the I-selected and the K-selected MFs in the VVDS (hatched vertical and horizontal STY regions, respectively, see caption Figure 9) and the literature data (K20, COMBO17, MUSIC, DEEP2, FDF+CDFS; for each the band of selection is indicated in the parenthesis). The vertical hatched regions represent the completeness limits of the VVDS samples. Our MFs are in fairly good agreement with previous studies over the whole mass range up to z ∼ 1.2. However, some differences exist, in particular at the massive end, which is more sensitive to the different selections, methods, statis- tics and to cosmic variance due to large scale structures: for example, in the MUSIC-GOODS survey there are two sig- nificant overdensities at z ∼ 0.7. The MFs from COMBO17 (Borch et al. 2006) and also from DEEP2 (Bundy et al. 2006) are systematically higher than previous surveys at the massive-end, in particular in the range 0.7 < z < 1.2. The MFs in the FDF+CDFS are instead systematically lower than ours at the massive-end and higher than our extrapolation to masses lower than our completeness limit. At z > 1.2 our MF is systematically higher than previ- ous studies. Given the area sampled (more than a factor 4 wider compared to FDF+CDFS and to MUSIC) and the consistency at these redshifts of our MF in the 2 K- selected separated areas (see Figure 8), we are confident in our results. However at high-z the uncertainties on the stellar masses estimate increase (up to 0.16 dex including L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 15 also the photometric redshift errors) and could produce a partially spurios excess in the number densities of galax- ies, in particular in the massive tail of the MF. This ef- fect is discussed in Kitzbichler & White (2007) which take into account in the hierarchical formation Millenium simu- lation the effect of the dispersion in the mass determination (0.25 dex, i.e. 78%). We have performed a similar analysis, taking into account the uncertainty on the mass, due to the fitting technique (∼ 30%) and to the uncertainty of the photometric redshifts (both its dependence on redshift and magnitudes as described in previous sections, i.e. up to σz ≃ 0.2 at z = 2 and K > 21.5). We found that the effect on the MF is always small, and only the very massive tail (M > 2 × 1011M⊙) is systematically overestimated (up to 0.2 dex). This effect can not completely explain the excess found compared to previous surveys, which are affected in a similar way by the same bias. 4.3. The Evolution of the Galaxy Stellar Mass Function The VVDS allows us to follow the evolution of the MF within a single sample over a wide redshift range. Difficulties in the interpretation of the evolution are, in- deed, due to the comparison with the local MFs, which have been determined with different methods and sample selection. For example, no local MF has been derived using complex SFHs for mass determinations. In our analysis we use, as reference, the local MF by Bell et al. (2003) and Cole et al. (2001) rescaled to Chabrier IMF. In particular, the Cole et al. (2001) local mass function, derived with smooth SFHs but with formation redshift fixed at z = 20, has been rescaled to smooth SFHs method with free formation red- shift by Fontana et al. (2004). The first important result is that, thanks to our very deep samples, both I- and K-selected, the low-mass end of the MF is even better determined than in the local sam- ple up to z < 0.4, probing for the first time masses down to about 3 × 107M⊙. The low mass-end is rather steep (−1.38 < α < −1.26), and could even be described by a double Schechter function, and is steeper than the local es- timates (α = −1.18±0.03 Cole et al. 2001, α = −1.10±0.02 Bell et al. 2003), possibly due to the fact they are not prob- ing masses smaller than 109 M⊙ (more than one order of magnitude more massive than in our sample). As evident from Figure 9, we find a substantial population of low-mass (< 109M⊙) galaxies at low redshifts (z < 0.4). This popu- lation is composed by faint blue galaxies with similar prop- erties in the 2 samples (I- and K-selected): I,K ≃ 22− 23, MI ,MK ≃ −16,−17 with median MI − MK ≃ 0.3, and median z ≃ 0.1− 0.2. This is a very strong result from our survey which can rely on a wider area and a deeper sam- ple than previous surveys at low redshifts. At z > 0.4 the low-mass slope is on the contrary always consistent with the local values. Even if we are not probing masses smaller than 108M⊙ at z > 0.4, we found that the MF remains quite flat (−1.23 < α < −1.04) at all redshifts, similar to that of Fontana et al. (2006) which probe lower masses (see figure 10). From a visual inspection of Figure 9, we see that up to z ∼ 0.9 there is only a weak evolution of the MF, as sug- gested by previous results (Fontana et al. 2004, 2006, Drory et al. 2005), while at higher redshifts there appears to begin a decrease in the normalization of the MF, even if a massive tail remains present up to z = 2.5. At intermediate masses Fig. 11. Cosmological evolution of the galaxy number den- sity as a function of redshift, as observed from the VVDS in various mass ranges (> 108M⊙, > 10 9.77M⊙ and > 1010.77M⊙ from top to bottom). Observed data from Vmax (shown as lower limit in the top panel) have been corrected, when necessary, for incompleteness integrating the mass function using the best fit Schechter parameters. VVDS data (big filled circles), averaged over the I- and the K- selected samples and the 2 methods to derive the mass, are plotted along with their statistical errors (solid error bars) and the scatter between the 2 different samples and meth- ods (dotted error bars). The solid lines show the best-fit power laws ∝ (1 + z)β, while the dashed lines correspond to the no-evolution solution normalized at z = 0. Results from previous surveys (small points and dot-dashed lines) are also shown. (9.5 < logM < 10.5), our VVDS MF is very well defined and shows a clear evolution, i.e. the number density de- creases with increasing redshifts compared to both the first VVDS redshift bin and the local MF. This evolution is quite mild up to z ≃ 0.9, while it becomes faster at higher z. At larger masses the high mass end of the MF (> 1011 M⊙) shows a small evolution up to z ≃ 2.5. However, its evolu- tion is extremely dependent on the assumed local MF and on the uncertainties in the mass determination, which pro- duce a larger dispersion between the different methods and samples compared to the intermediate-mass range. In order to quantify the MF evolution, and its mass dependency independently from the local MF, in the next section we derive number densities for different mass limits. 16 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History Table 3. Number Density and Stellar Mass Density zinf zsup Log(ρN ) Scatter Error Log(ρstars)Scatter Error log(Mstars)> 8 0.05 0.40 -1.40 0.04 0.01 8.45 0.09 0.01 0.40 0.70 -1.63 0.08 0.02 8.34 0.08 0.02 0.70 0.90 -1.70 0.08 0.04 8.22 0.11 0.01 0.90 1.20 -1.78 0.13 0.05 8.14 0.12 0.01 1.20 1.60 -1.81 0.05 0.11 8.04 0.14 0.02 1.60 2.50 -1.90 0.13 0.01 8.05 0.11 0.01 log(Mstars)> 9.77 0.05 0.40 -2.20 0.06 0.01 0.40 0.70 -2.29 0.03 0.01 0.70 0.90 -2.33 0.04 0.01 0.90 1.20 -2.45 0.05 0.01 1.20 1.60 -2.54 0.02 0.01 1.60 2.50 -2.65 0.01 0.01 log(Mstars)> 10.77 0.05 0.40 -3.07 0.18 0.05 8.00 0.22 0.04 0.40 0.70 -3.04 0.10 0.02 8.04 0.14 0.02 0.70 0.90 -3.22 0.16 0.02 7.80 0.19 0.02 0.90 1.20 -3.28 0.15 0.02 7.76 0.18 0.02 1.20 1.60 -3.42 0.23 0.02 7.59 0.28 0.02 1.60 2.50 -3.35 0.11 0.01 7.70 0.11 0.01 4.4. Galaxy Number Density Here we derive the number density of galaxies as a function of redshift, using different lower limits in mass (Mmin). We have estimated the number density from the observed data (from Vmax), as well as from the incompleteness-corrected MFs, i.e. integrating the best-fit Schechter functions over the considered mass range. The corrections due to faint galaxies dominate for Mmin = 10 8M⊙, while they are neg- ligible for the other mass limits considered. A formal uncer- tainty in this procedure was estimated by considering the Vmax statistical errors and the range of acceptable Schechter parameters values. In Figure 11 we plot our VVDS determi- nations, averaged over the I- and K-selected samples and the 2 methods for mass determination (listed in Table 3), along with their statistical errors (always less than 10%) and the scatter between the 2 different samples and meth- ods (ranging between 10 and 45% and due mainly to the dif- ferent methods rather than to the different samples). With the two methods we find similar trends with redshift of the number densities of galaxies, but with a systematic shift which is significant only for the highest mass limit (for com- plex SFHs the galaxy number densities are ∼ 50% higher than for smooth SFHs). The effect of photometric redshift and mass uncertainty on the number densities is always small (< 15%) for the mass range shown in Figure 11, ex- cept for the very massive galaxies (> 2 × 1011 M⊙, not shown in the figure because of the small number of galaxies in this mass range) where the intrinsic values could be up to a factor ∼ 2 lower (see discussion in section 4.2). The decrease in number density with redshift for all the adopted mass limits is evident. We have compared VVDS results to previous surveys and with different local determinations. For the total num- ber density (108 < Mstars < 10 13M⊙) VVDS data are very well consistent with the evolutionary STY fit determined by Fontana et al. (2006) in the GOODS-MUSIC survey. At in- Fig. 12. Cosmological evolution of the stellar mass density as a function of redshift as observed from the VVDS for 2 mass ranges: integrated over the whole range 108M⊙ ≤ Mstars ≤ 10 13M⊙ (upper panels) and for massive galax- ies (> 1010.77M⊙) (lower panels). Symbols and lines as in Figure 11. termediate masses (> 109.77M⊙, corresponding to 10 for Salpeter IMF) our VVDS data have a better determina- tion and smaller uncertainties than previous ones and are consistent with most of them at z < 1.2 and in the upper en- velope at higher z. For the high mass range (> 1010.77M⊙, corresponding to 1011M⊙ for Salpeter IMF) we are quite consistent with previous results, and even if our VVDS have lower errors than previous ones, the dispersion within the various VVDS measurements reflect the uncertainties for massive galaxies. If we represent the average number density evolution by a power law ρN ∝ (1 + z) β , we find that β(Mstars > 108) = −1.28 ± 0.15, β(Mstars > 10 9.77) = −1.26 ± 0.10, and β(Mstars > 10 10.77) = −1.01 ± 0.05 (the errors on β represent the uncertainties due to the 2 different methods). We find on average a similar evolution for the 2 methods analyzed and a slightly milder evolution with increasing mass limit (‘downsizing’ in mass assembly). The average evolution from z = 0 to z = 1 is a factor 2.4 ± 0.3 and 2.0± 0.1 from low to high-mass galaxies, respectively, and increases to a factor 4.0 ± 0.9 and 3.0 ± 0.2, respectively, at z = 2. We note moreover that for the highest mass limit (Mstars > 10 10.77) at low redshift (z < 0.7) the number density observed is consistent with no-evolution (fixing the L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 17 value to z = 0 we found an evolution < 30%), excluded instead for the intermediate and low-mass limit. At high-z (z > 1.5) for intermediate and high-mass range we note a flattening in the number density from the VVDS data, sim- ilar to Drory et al. (2005), but higher than Fontana et al. (2006). This flattening is due to the high mass tail observed in the range 1.6 < z < 2.5. This population shows ex- tremely red colours (MI −MK ≃ 0.8) and could be related to the appearance of a population of massive star forming dusty galaxies, observed in previous surveys (Fontana et al. 2004, Daddi et al. 2004b). The small excess induced by uncertainties on the mass and photometric redshifts (see discussion in section 4.2) can not completely explain the difference with Fontana et al. (2006) survey, which is af- fected by the same bias in a similar way. The mass dependent evolution (“mass downsizing”) is very debated and the results from different surveys are still controversial. Deep surveys, such as the FDF & CDFS (an- alyzed by Drory et al. 2005) find an evolution consistent with ours (a decrease of about a factor 2.5 – 4 at z = 1− 2 in the number density of galaxies> 1010.77 M⊙). Fontana et al. (2006) suggest a similar mild evolution up to z = 1, for massive galaxies (> 1010.77M⊙) and a stronger evolution at z > 1.5, reaching a factor about 10 at z = 3. Similarly, Cimatti et al. (2006) show that the number density of lu- minous (massive, Mstars > 10 11M⊙) early-type galaxies is nearly constant up to z ∼ 0.8, while Bundy et al. (2006) find a slight decrease, consistent with no evolution, only for even more massive system (> 3×1011M⊙) and a more sig- nificant decline for Mstars < 3× 10 11M⊙. Vice versa, data from the MUNICS survey (Drory et al. 2004) show a faster evolution of massive galaxies, even faster than for the less massive systems (see also Figure 4 in Drory et al. 2005). To summarize, our accurate results show that the MF evolves mildly up to z ≃ 1 (about a factor 2.5 in the total number density) and that a high-mass tail is still present up to z = 2.5. Moreover, we find that massive systems show an evolution that is on average milder (< 50% at z < 1) than intermediate and low-mass galaxies and con- sistent with a mild/negligible evolution (< 30%) up to z ∼ 0.8. Conversely, a no-evolution scenario in the same redshift range is definitely excluded for intermediate- and low-mass galaxies. This behaviour suggests that the assembly of the stellar mass in objects with mass smaller than the localM∗stars was quite significant between z = 2 and z = 0. Qualitatively, this behaviour is expected for galaxies with SFHs prolonged over cosmic time, which therefore continue to grow in terms of stellar mass after z ∼ 1. Conversely, our results fur- ther strengthen the fact that the number density of mas- sive galaxies is roughly constant up to z ≃ 0.8, consistently with a SFH peaked at higher redshifts, with the conversion of most of their gas into stars happening at z > 1.5−2, rul- ing out the ‘dry mergers’ as the major mechanism of their assembly history, below z < 1. 5. Mass Density Various attempts to reconstruct the cosmic evolution of the stellar mass density have been previously made, mainly us- ing NIR-selected samples (Dickinson et al. 2003, Fontana et al. 2004, Drory et al. 2005). Our survey offers the possibil- ity to investigate it using the MF derived from two different optical- and NIR-selected samples, taking advantage of our depth, and relatively wide area covered. Furthermore, the different methods analyzed here to derive the stellar mass content give us a direct measure of the uncertainties in- volved. We have estimated the stellar mass density from the observed data, as well as from the incompleteness-corrected MFs. Up to z < 1 the corrections due to faint galaxies are relatively small. A formal uncertainty in this procedure was estimated by considering the Vmax statistical errors and the range of acceptable Schechter parameters values. Figure 12 shows our results (averaged over the I- andK- selected samples and the 2 methods with a typical scatter of about 30-50% and statistical errors always less than 5% see Table 3), for the total mass density and for the density in massive galaxies (> 1010.77), along with their represen- tative power laws (ρstars ∝ (1+z) β), and compared to liter- ature data (see references in the figure). For the total mass density, even if the results from our survey cover a range of values with some significant differences between the two different methods (up to ∼ 40%), the general behaviour and evolutionary trend is well defined by β = −1.19± 0.05. We find that the evolution of the stellar mass density is relatively slow with redshift, with a decrease of a factor 2.3 ± 0.1 up to z ≃ 1, up to a factor 4.5 ± 0.3 at z = 2.5. The agreement of average total mass density with previous surveys is reasonably good, and the range covered by VVDS data reflect the different selection techniques and methods used in different surveys. The average total mass density evolution is milder than in the MUSIC sample (Fontana et al. 2006) already at z > 0.5. Our evolutionary trend is con- sistent with the upper envelope of previous surveys, even if our highest-redshift value is uncertain because the low-mass slope is poorly constrained. For comparison the analysis of VVDS data using a IRAC-selected sample (see Arnouts et al. 2007) finds similar values for the mass density, except that the highest redshift point is lower than ours. Given the present uncertainties on the low-mass slope of the GSMF, the total mass density at z ≃ 2 remains poorly constrained. The mass density of high-mass objects (> 1011M⊙ with Salpeter IMF) varies by a factor up to 1.8 within the 2 methods adopted, but the evolutionary trend is similar (β = −1.13± 0.01) and consistent with a decrease of about a factor 2.18 ± 0.02 to z = 1 and 3.44 ± 0.04 to z = 2.0. Moreover at low redshift (z < 0.7) the VVDS observed data are consistent with a mild/negligible evolution (< 30%), as indicated by the number density of massive galaxies (see previous Section). Our data are roughly consistent with Fontana et al. (2006) up to z = 1.5 (even if the slope of the evolutionary trend is shallower), while at z > 1.5 the VVDS mass-density of massive galaxies is significantly higher than that in Fontana et al. (2006), reflecting the excess in MF at high-z noted in the VVDS MF compared to previous ones (see Section 4.2). This results, therefore, in a flatter evolu- tionary trend over the total redshift range. Given the wider area and completeness for high-mass objects, our samples guarantee a higher statistical accuracy and confidence level than before. However some caveat remains due to the effect of photometric redshift and mass uncertainty on the mass densities, which is anyhow always small (< 15%) except for the massive galaxies (> 6 × 1010 M⊙) where the intrinsic values could be up to 20-30% lower than our estimates (see discussion in Section 4.2). 18 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 6. Summary and Discussion We have investigated the evolution of the Galaxy Stellar Mass Function up to to z = 2.5 using the VVDS survey covered by deep VIMOS spectroscopy (17.5 < I < 24) and multiband photometry (from U to K-band). For our analysis we have used two different samples: (1) the opti- cal (I-selected, 17.5 < I < 24) main spectroscopic sam- ple, based on about 6500 secure redshifts over about 1750 arcmin2, and (2) a near-IR sample (K-selected, K < 22.84 & K < 22.34), in a sub-area of about 610 arcmin2 and based on about 10200 galaxies with accurate photomet- ric and spectroscopic redshifts. For the first time we have probed masses down to a very low limit, in particular at low-z (down to ∼ 3 × 107M⊙ at z ∼ 0.2), while the rela- tively wide area has allowed us to determine the MF with much higher statistical accuracy than previous samples. In order to better understand uncertainties we have ap- plied and compared two methods to estimate the stellar mass content in galaxies from multiband SED fitting. The 2 methods differ in the explored parameter space (metal- licity, dust law and content) and are based on different as- sumptions on previous star formation history. The main results from the stellar mass estimate can be summarized as follows: – The agreement between the 2 methods is fairly good even if masses estimated with ‘complex SFHs’ are systematically higher than ‘smooth SFHs’ masses. For the K-selected sample the mean difference is 〈d logMstars〉 ≃ 0.12 dex, and the dispersion is σ = 0.13. The differences are mainly due to the secondary burst component (complex SFHs) compared to smooth SFHs. – We found that mass estimates using only optical bands are in rather good agreement with those using also NIR bands up to z ∼ 1.2. We have used this information to statistically correct masses for objects without near-IR photometry. At higher redshifts the shift and dispersion dramatically increase and the mass estimates become unreliable if near-IR photometry is not available. We have, thus, derived the MF using the VVDS I- selected sample and extended it up to z = 2.5 thanks to the K-selected sample. From a detailed analysis of the MF, galaxy number density and mass density, in different mass ranges, through cosmic time, we found evidences for: – a substantial population of low-mass galaxies (< 109M⊙) at z ≃ 0.2 composed by faint (I,K ≃ 22, 23) blue galaxies with median MI−MK ≃ 0.3, and absolute magnitudes MI ,MK ≃ −16,−17; – a slow evolution of the stellar mass function with red- shift up to z ∼ 0.9 and a faster evolution at higher-z, in particular for less massive systems. A massive popu- lation is present up to z = 2.5 and have extremely red colours (MI −MK ≃ 0.7− 0.8). – at z > 0.4 the low-mass slope of the GSMF does not evolve significantly and remains quite flat (−1.23 < α < −1.04). – the number density shows, on average, a mild differen- tial evolution with mass, which is slower with increas- ing mass limit. Such evolution can be described by a power law ∝ (1 + z)β(>M). Within the VVDS redshift range we found that β(> 108M⊙) = −1.28 ± 0.15, β(> 109.77M⊙) = −1.26± 0.10 and β(> 10 10.77M⊙) = −1.01± 0.05. For massive galaxies at low redshift (z < 0.7) the evolution is consistent with mild/negligible- evolution (< 30%), which is excluded for low-mass sys- tems. – the evolution of the stellar mass density is relatively slow with redshift, with a decrease of about a factor 2.3±0.1 to z ≃ 1, while at z ≃ 2.5 the decrease amounts to a fac- tor up to 4.5± 0.3, milder than in previous surveys. For massive galaxies the evolution at low redshift (z < 0.7) is consistent with a mild/negligible evolution(< 30%), and shows a flattening compared to previous results at z > 1.5 due to a population with extremely red colours. Our results provide new clues on the controversial ques- tion of when galaxy formed and assembled their stellar mass. Most of the massive galaxies seem to be in place up to z = 1 and have, therefore, formed their stellar mass at high redshift (z > 1), rather than assembled it mainly through continuous galaxy merging of small galaxies at z < 1. On the contrary, less massive systems have assembled their mass (through merging or prolonged star formation his- tory) later in cosmic time. In agreement with our results, a substantial population of high-z (z ∼ 2−3) dusty and mas- sive objects have been discovered in near-IR surveys (Daddi et al. 2004b) and detected by Spitzer in the far-IR (Daddi et al. 2005, Caputi et al. 2006b). This population could be related to the initial phase of massive galaxy formation during their strong star forming and dusty phase. Finally, our results are not completely accounted for by most of theoretical models of galaxy formation (see Fontana et al. 2004, 2006 and Caputi et al. 2006a for a detailed com- parison with models). For instance, models by De Lucia et al. (2006) predict that the most massive galaxies generally form their stars earlier, but assemble them later, mainly at z < 1 via merging, than the less massive galaxies (i.e. ’downsizing in star formation but ’upsizing’ in mass assem- bly, see Renzini 2007 for a recent discussion). Furthermore, the stronger decrease with redshift of the low-mass popu- lation, with a low-mass end of the GSMF which remains substantially flat up to high redshift, is not reproduced by most of the theoretical galaxy assembly models, which tend, indeed, to overpredict the low-mass end of the MF (see Fontana et al. 2006). Understanding the mass assembly of less massive ob- jects and disentangling merging processes from prolonged star formation history is more complicated. In this respect for a better comprehension of galaxy formation the VVDS will allow us to further investigate the evolution of the stel- lar mass function up to high-z also for different galaxy types (spectral and morphological) and in different environments. For example Arnouts et al. (2007) study the mass density evolution of different galaxy population. Further analysis of galaxy mass dependent evolution, using stellar popula- tion properties, as well as observed spectral features, will be presented in forthcoming papers (Lamareille et al., in preparation, Vergani et al. 2007). Furthermore, it will be possible to push the study of the galaxy stellar mass func- tion at higher redshifts using SPITZER mid-IR observa- tions. While most of present studies (Dickinson et al. 2003, Drory et al. 2005) do not use rest-frame near-IR photome- try to estimate stellar masses, our VVDS-SWIRE collabo- ration will allow to combine the deep VVDS spectroscopic sample with SPITZER-IRAC photometry. L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 19 Acknowledgements. This research has been developed within the framework of the VVDS consortium. This work has been partially supported by the CNRS-INSU and its Programme National de Cosmologie (France), and by Italian Ministry (MIUR) grants COFIN2000 (MM02037133) and COFIN2003 (num.2003020150). The VLT-VIMOS observations have been carried out on guaranteed time (GTO) allocated by the European Southern Observatory (ESO) to the VIRMOS consortium, under a contractual agreement between the Centre National de la Recherche Scientifique of France, heading a consortium of French and Italian institutes, and ESO, to design, manufacture and test the VIMOS instrument. We are in debt with E. Bell, S. Salimbeni, and E. Fontana for providing the data from their survey in electronic format, and to C. Maraston for her galaxy evolution models in BC format. References Avni, Y, Bahcall, J. 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A., et al. 2000, ApJ, 530, 418 Zucca, E., Ilbert, O., Bardelli, S., et al. 2006, A&A, 455, 879 1 INAF-Osservatorio Astronomico di Bologna - Via Ranzani,1, I-40127, Bologna, Italy 2 INAF-IASF - via Bassini 15, I-20133, Milano, Italy 3 Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS-Université de Provence, BP8, 13376 Marseille Cedex 12, France 4 INAF-Osservatorio Astronomico di Brera - Via Brera 28, Milan, Italy http://arxiv.org/abs/astro-ph/0506044 http://arxiv.org/abs/astro-ph/0611724 http://arxiv.org/abs/astro-ph/0311475 http://arxiv.org/abs/astro-ph/0410295 http://arxiv.org/abs/astro-ph/0702148 20 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 5 Institute for Astronomy, 2680 Woodlawn Dr., University of Hawaii, Honolulu, Hawaii, 96822 6 Max Planck Institut fur Astrophysik, 85741, Garching, Germany 7 Institut d’Astrophysique de Paris, UMR 7095, 98 bis Bvd Arago, 75014 Paris, France 8 Centro de Astrofsica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal 9 Laboratoire d’Astrophysique de Toulouse/Tabres (UMR5572), CNRS, Université Paul Sabatier - Toulouse III, Observatoire Midi-Pyriénées, 14 av. E. Belin, F-31400 Toulouse, France 10 INAF-IRA - Via Gobetti,101, I-40129, Bologna, Italy 11 INAF-Osservatorio Astronomico di Roma - Via di Frascati 33, I-00040, Monte Porzio Catone, Italy 12 School of Physics & Astronomy, University of Nottingham, University Park, Nottingham, NG72RD, UK 13 Astrophysical Institute Potsdam, An der Sternwarte 16, D- 14482 Potsdam, Germany 14 Observatoire de Paris, LERMA, 61 Avenue de l’Observatoire, 75014 Paris, France 15 Università di Bologna, Dipartimento di Astronomia - Via Ranzani,1, I-40127, Bologna, Italy 16 Centre de Physique Théorique, UMR 6207 CNRS-Université de Provence, F-13288 Marseille France 17 Integral Science Data Centre, ch. d’Écogia 16, CH-1290 Versoix 18 Geneva Observatory, ch. des Maillettes 51, CH-1290 Sauverny, Switzerland 19 Astronomical Observatory of the Jagiellonian University, ul Orla 171, 30-244 Kraków, Poland 20 INAF-Osservatorio Astronomico di Capodimonte - Via Moiariello 16, I-80131, Napoli, Italy 21 Universitá di Milano-Bicocca, Dipartimento di Fisica - Piazza delle Scienze, 3, I-20126 Milano, Italy 22 Università di Bologna, Dipartimento di Fisica - Via Irnerio 46, I-40126, Bologna, Italy Introduction The First Epoch VVDS Sample The I-selected Spectroscopic Sample The K-selected Photometric Sample Comparison of the Two Samples Estimate of the Stellar Masses Smooth SFHs Complex SFHs Effect of NIR Photometry Comparison of the Masses Obtained with the Two Methods Massive Galaxies at z>1 Mass Function Estimate The VVDS Galaxy Stellar Mass Function Comparison with Previous Surveys The Evolution of the Galaxy Stellar Mass Function Galaxy Number Density Mass Density Summary and Discussion

Año 8, No.27, Ene – Mar. 2007 What does Hirsch index evolution explain us? A case study: Turkish Journal of Chemistry Metin Orbay, Orhan Karamustafaoğlu and Feda Öner Amasya University (Turkey) morbay@omu.edu.tr, orseka@yahoo.com, oner3@yahoo.com Abstract The evolution of Turkish Journal of Chemistry’s (TURK J CHEM) Hirsch index (h-index) over the period 1995-2005 is studied and determined in the case of the self and without self-citations. It is seen that the effect of Hirsch index of TURK J CHEM has a highly positive trend during the last five years. It proves that TURK J CHEM is improving itself both in quantity and quality since h-index reflects peer review, and peer review reflects research quality of a journal. Key words Bibliographic citations, citation index, h-index, Hirsch number, Scientometric analysis, Bibliometry Resumen Presenta la evolución del índice Hirsch (h-index) de la Turkish Journal of Chemistry’s (TURK J CHEM) durante el período 1995-2005; el caso es estudiado y precisado tanto con auto-citaciones como sin auto-citaciones. El indice Hirsch permite apreciar una alta tendencia positiva de los indicadores de la TURK J CHEM durante los últimos cinco años. Se comprueba así que la TURK J CHEM va mejorando tanto en cantidad como en calidad de contenidos dado que el h-index refleja la revisión por pares, y la revisión por pares refleja la calidad de las investigaciones de una revista. Palabras clave Citas bibliográficas, índice de citación, indice-h, NúmeroHirsch, Cienciometría, Análisis cienciométrico, Bibliometría Introduction Hitherto, several citation-based indicators have been used to measure research performance (e.g. the number of citations to each of the q most cited papers, the total number of citations, the citations per paper, the number of highly cited published papers). There are valid reservations about using above mentioned indicators to measure performance because some papers are cited for reasons that are unrelated to the quality or utility of a study (see: Kelly & Jennions, 2006; Miller, 2007 and references therein). essay Orbay, Karamustafaoğlu &, Öner - What does Hirsch index evolution explain us? Recently, taking into account above citation-based indicators with advantages and disadvantages, Jorge E. Hirsch has suggested a new indicator called h-index, which means that one single index for the assessment of the research performance of an individual scientist. According to the definition by Hirsch, “A scientist has index h if his/her N papers have at least h citations each, and the other (N-h) papers have fewer than h citations each” (Hirsch, 2005). Hirsch’s article has generated considerable interest and almost immediately provoked reactions in the scientific community (Ball, 2005; Braun, Glanzel & Schubert, 2005; Glanzel & Persson, 2005; Glanzel, 2006a; Egghe & Rousseau, 2006; Egghe, 2006; Cronin & Meho, 2006; Burrell, 2007; Rousseau, 2007a). The h-index has generally well received by the research group. Of course, the h-index has also a number of disadvantages as point out by some authors (Kelly & Jennions, 2006; Van Raan, 2006). After all these beneficial arguments, W. Glanzel has summarized some pros and cons of h-index in his excellent recent paper (Glanzel, 2006b). After a short time, the h-index definition has been adapted into journals and article citations, as a h-type index-equal to h if you have published h papers, each of which has at least h citations (Braun, Glanzel & Schubert, 2006). T. Braun and co-workers stressed that the h-type index for journals would advantageously supplement journal impact factor (IF), the total number of citations divided by the number of articles (Garfield, 1955), at least two aspects: respectively, i. It is robust in the sense that it is insensitive to an accidental excess of uncited articles, and to one or several highly cited articles, ii. It combines the effect of “quantity” and “quality” in a rather specific. Naturally, the journal h-index would not be calculated for a “lifetime contributions”, as defined by Hirsch for the scientific output of a researcher, but for a definite period-in the simplest case for a given year. Using this procedure, R. Rousseau studied the evolution of the Journal of American Society of Information Sciences’ Hirsch index and introduced relative h-index (Rousseau, 2007b). In this opinion article, the evolution of Hirsch index of Turkish Journal of Chemistry (TURK J CHEM) over period 1995-2005 is studied and determined in the case of the self and elimination self-citation (or without self citation) of the journal. Method and results As is well known, Web of Science database offers a very simple way to determine the annual h-index of a journal. Retrieving all source items of a given journal from a given period and shorting them by the number of “times cited”, it is easy to find the h-index of the journal for the given year (http://isiknowledge.com). In this study, we conduct a case study for h-index of TURK J CHEM over period 1995- 2005. Meanwhile, we consider a fixed moment in time when citations are collected from Web of Science (http://isiknowledge.com, retrieved date 24.12.2006). The h-index of TURK J CHEM over the period 1995-2005 is determined in the case of the self and without self-citations, as shown in Figure 1. Año 8, No.27, Ene – Mar. 2007 h-index h-index(w ithout self citations) Figure 1. h-index of TURK J CHEM. However, besides the period over which a volume can collect citations, also the number of articles published in that volume influences the h-index. For this reason, the h-index must be divided by the number of articles published, leading to a normalized (or relative) h-index (Rousseau, 2007b). In this case, the results are shown with self and without self citations for the journal in Figure 2.a. and Figure 2.b., respectively. Figure 2a. Normalized h index with self citiations of TURK J CHEM. Orbay, Karamustafaoğlu &, Öner - What does Hirsch index evolution explain us? Figure 2b. Normalized h index with eliminated self citations of TURK J CHEM. As can be clearly seen from Figure 2, using the normalized h-index leads to a linear increase when going backward in time (or decrease when going forward in time). The Pearson correlation coefficients of the regression lines of this journal are 0.686 for normalized h-index (continuous line in Figure 2.a) with self citations and 0.689 for normalized h-index without self-citations (dot line in Figure 2.b), which are moderate, but statistically significant (1% level). It is not surprising that these two correlation coefficients are very close to each other because of the fact that the self-citations over this period are limited by approximately 20%. On the other hand, we encounter that this value is high in other twenty randomly selected journals published in the same field. It is obvious from the Figure 1 and Figure 2 that h-index and normalized h-index are extremely different trend between 1995-2000 and 2000-2005 periods. So, we focus on two periods. In the former period, the Pearson correlation coefficients are 0.336 for normalized h-index with self citations and 0.192 for without self-citation, which are very low, but statistically significant (1% level). It is not surprising that TURK J CHEM has started to be scanned in Web of Science newly in this period. For this reason, it can be thought that a few researchers were aware of this journal. On the other hand, in the latter period, the Pearson correlation coefficients are 0.858 for normalized h-index with self citations and 0.941 for without self-citation, which are very high, and statistically significant (1% level). From these interesting results, we conclude that a lot of published papers in this journal have been very high impact with respect to “quantity” (number of publications) and “quality” (citation rate), recently. Furthermore, it is known that TURK J CHEM has started open access in the latter period. Thus, we think that open access contributes h-index of this journal. References Ball, P. (2005). Index aims for ranking of scientists, Nature, 436, 900. Braun, T., Glanzel W. & Schubert, A. (2005). A Hirsch-type index for journals, The Scientist, 19(22), 8. Año 8, No.27, Ene – Mar. 2007 Braun, T., Glanzel, W. & Schubert, A. (2006). A Hirsch-type index for journals, Scientometrics, 69(1), 169-173. Burrell, Q. L. (2007). Hirsch's h-index: A stochastic model, Journal of Informetrics, 1(1), 16-25. Cronin, B. & Meho, L. (2006). Using the h-index to rank influential information scientists, Journal of the American Society for Information Science and Technology, 57(9), 1275-1278. Egghe, L. (2006). How to improve the h-index: letter, The Scientist, 20(3), 121. Egghe, L. & Rousseau, R. (2006). An informetric model for the Hirch-index, Scientometrics, 69(1), 121-129. Garfield, E. (1955). Citation indexes to science: a new dimension in documentation through association of ideas. Science, 122, 108-111. Glanzel W. (2006a). On the h-index- A mathematical approach to a new measure of publication activity and citation impact, Scientometrics, 67(2), 315-321. Glanzel, W. (2006b). On the opportunities and limitations of the h-index, Science Focus, 1, 10. Glanzel W. & Persson, O. (2005). H-index for prize medalists, International Society for Scientometrics and Informetrics, 1, 15-18. Hirsch, J. E. (2005). An index to quantity an individual’s scientific research output, Proceedings of the National Academy of Sciences of the United States of America, 102(46), 16569-16572. http://isiknowledge.com, Retrieved date 24.12.2006. Kelly, C. D. & Jennions, M. D. (2006). The h index and career assessment by numbers, TRENDS in Ecology and Evolution, 21, 167-170. Miller, C. W. (2007). Superiority of the h-index over the impact factor for physics, American Journal of Physics. Retrieved date 13.01.2007 from http://arxiv.org/abs/physics/0608183. Rousseau, R. (2007a). The influence of missing publications on the Hirsch index, Journal of Informetrics, 1(1), 2-7. Rousseau, R. (2007b). A case study: evolution of JASIS’ Hirsch index, Science Focus. Retrieved date 07.01.2007 from http://eprints.rclis.org/archive/00005430/01/Evolution_of_h_JASIS_rev.pdf Van Raan, A. F. J. (2006). Comparison of the Hirsch-index with standard bibliometric indicators and with peer judgment for 147 chemistry research groups, Scientometrics, 67(3), 491-502.

The evolution of Turkish Journal of Chemistry (Turk J. Chem) Hirsch index (h-index) over the period 1995-2005 is studied and determined in the case of the self and without self-citations. It is seen that the effect of Hirsch index of Turk J. Chem has a highly positive trend during the last five years. It proves that Turk J. Chem is improving itself both in quantity and quality since h-index reflects peer review, and peer review reflects research quality of a journal.

Introduction Hitherto, several citation-based indicators have been used to measure research performance (e.g. the number of citations to each of the q most cited papers, the total number of citations, the citations per paper, the number of highly cited published papers). There are valid reservations about using above mentioned indicators to measure performance because some papers are cited for reasons that are unrelated to the quality or utility of a study (see: Kelly & Jennions, 2006; Miller, 2007 and references therein). essay Orbay, Karamustafaoğlu &, Öner - What does Hirsch index evolution explain us? Recently, taking into account above citation-based indicators with advantages and disadvantages, Jorge E. Hirsch has suggested a new indicator called h-index, which means that one single index for the assessment of the research performance of an individual scientist. According to the definition by Hirsch, “A scientist has index h if his/her N papers have at least h citations each, and the other (N-h) papers have fewer than h citations each” (Hirsch, 2005). Hirsch’s article has generated considerable interest and almost immediately provoked reactions in the scientific community (Ball, 2005; Braun, Glanzel & Schubert, 2005; Glanzel & Persson, 2005; Glanzel, 2006a; Egghe & Rousseau, 2006; Egghe, 2006; Cronin & Meho, 2006; Burrell, 2007; Rousseau, 2007a). The h-index has generally well received by the research group. Of course, the h-index has also a number of disadvantages as point out by some authors (Kelly & Jennions, 2006; Van Raan, 2006). After all these beneficial arguments, W. Glanzel has summarized some pros and cons of h-index in his excellent recent paper (Glanzel, 2006b). After a short time, the h-index definition has been adapted into journals and article citations, as a h-type index-equal to h if you have published h papers, each of which has at least h citations (Braun, Glanzel & Schubert, 2006). T. Braun and co-workers stressed that the h-type index for journals would advantageously supplement journal impact factor (IF), the total number of citations divided by the number of articles (Garfield, 1955), at least two aspects: respectively, i. It is robust in the sense that it is insensitive to an accidental excess of uncited articles, and to one or several highly cited articles, ii. It combines the effect of “quantity” and “quality” in a rather specific. Naturally, the journal h-index would not be calculated for a “lifetime contributions”, as defined by Hirsch for the scientific output of a researcher, but for a definite period-in the simplest case for a given year. Using this procedure, R. Rousseau studied the evolution of the Journal of American Society of Information Sciences’ Hirsch index and introduced relative h-index (Rousseau, 2007b). In this opinion article, the evolution of Hirsch index of Turkish Journal of Chemistry (TURK J CHEM) over period 1995-2005 is studied and determined in the case of the self and elimination self-citation (or without self citation) of the journal. Method and results As is well known, Web of Science database offers a very simple way to determine the annual h-index of a journal. Retrieving all source items of a given journal from a given period and shorting them by the number of “times cited”, it is easy to find the h-index of the journal for the given year (http://isiknowledge.com). In this study, we conduct a case study for h-index of TURK J CHEM over period 1995- 2005. Meanwhile, we consider a fixed moment in time when citations are collected from Web of Science (http://isiknowledge.com, retrieved date 24.12.2006). The h-index of TURK J CHEM over the period 1995-2005 is determined in the case of the self and without self-citations, as shown in Figure 1. Año 8, No.27, Ene – Mar. 2007 h-index h-index(w ithout self citations) Figure 1. h-index of TURK J CHEM. However, besides the period over which a volume can collect citations, also the number of articles published in that volume influences the h-index. For this reason, the h-index must be divided by the number of articles published, leading to a normalized (or relative) h-index (Rousseau, 2007b). In this case, the results are shown with self and without self citations for the journal in Figure 2.a. and Figure 2.b., respectively. Figure 2a. Normalized h index with self citiations of TURK J CHEM. Orbay, Karamustafaoğlu &, Öner - What does Hirsch index evolution explain us? Figure 2b. Normalized h index with eliminated self citations of TURK J CHEM. As can be clearly seen from Figure 2, using the normalized h-index leads to a linear increase when going backward in time (or decrease when going forward in time). The Pearson correlation coefficients of the regression lines of this journal are 0.686 for normalized h-index (continuous line in Figure 2.a) with self citations and 0.689 for normalized h-index without self-citations (dot line in Figure 2.b), which are moderate, but statistically significant (1% level). It is not surprising that these two correlation coefficients are very close to each other because of the fact that the self-citations over this period are limited by approximately 20%. On the other hand, we encounter that this value is high in other twenty randomly selected journals published in the same field. It is obvious from the Figure 1 and Figure 2 that h-index and normalized h-index are extremely different trend between 1995-2000 and 2000-2005 periods. So, we focus on two periods. In the former period, the Pearson correlation coefficients are 0.336 for normalized h-index with self citations and 0.192 for without self-citation, which are very low, but statistically significant (1% level). It is not surprising that TURK J CHEM has started to be scanned in Web of Science newly in this period. For this reason, it can be thought that a few researchers were aware of this journal. On the other hand, in the latter period, the Pearson correlation coefficients are 0.858 for normalized h-index with self citations and 0.941 for without self-citation, which are very high, and statistically significant (1% level). From these interesting results, we conclude that a lot of published papers in this journal have been very high impact with respect to “quantity” (number of publications) and “quality” (citation rate), recently. Furthermore, it is known that TURK J CHEM has started open access in the latter period. Thus, we think that open access contributes h-index of this journal. References Ball, P. (2005). Index aims for ranking of scientists, Nature, 436, 900. Braun, T., Glanzel W. & Schubert, A. (2005). A Hirsch-type index for journals, The Scientist, 19(22), 8. Año 8, No.27, Ene – Mar. 2007 Braun, T., Glanzel, W. & Schubert, A. (2006). A Hirsch-type index for journals, Scientometrics, 69(1), 169-173. Burrell, Q. L. (2007). Hirsch's h-index: A stochastic model, Journal of Informetrics, 1(1), 16-25. Cronin, B. & Meho, L. (2006). Using the h-index to rank influential information scientists, Journal of the American Society for Information Science and Technology, 57(9), 1275-1278. Egghe, L. (2006). How to improve the h-index: letter, The Scientist, 20(3), 121. Egghe, L. & Rousseau, R. (2006). An informetric model for the Hirch-index, Scientometrics, 69(1), 121-129. Garfield, E. (1955). Citation indexes to science: a new dimension in documentation through association of ideas. Science, 122, 108-111. Glanzel W. (2006a). On the h-index- A mathematical approach to a new measure of publication activity and citation impact, Scientometrics, 67(2), 315-321. Glanzel, W. (2006b). On the opportunities and limitations of the h-index, Science Focus, 1, 10. Glanzel W. & Persson, O. (2005). H-index for prize medalists, International Society for Scientometrics and Informetrics, 1, 15-18. Hirsch, J. E. (2005). An index to quantity an individual’s scientific research output, Proceedings of the National Academy of Sciences of the United States of America, 102(46), 16569-16572. http://isiknowledge.com, Retrieved date 24.12.2006. Kelly, C. D. & Jennions, M. D. (2006). The h index and career assessment by numbers, TRENDS in Ecology and Evolution, 21, 167-170. Miller, C. W. (2007). Superiority of the h-index over the impact factor for physics, American Journal of Physics. Retrieved date 13.01.2007 from http://arxiv.org/abs/physics/0608183. Rousseau, R. (2007a). The influence of missing publications on the Hirsch index, Journal of Informetrics, 1(1), 2-7. Rousseau, R. (2007b). A case study: evolution of JASIS’ Hirsch index, Science Focus. Retrieved date 07.01.2007 from http://eprints.rclis.org/archive/00005430/01/Evolution_of_h_JASIS_rev.pdf Van Raan, A. F. J. (2006). Comparison of the Hirsch-index with standard bibliometric indicators and with peer judgment for 147 chemistry research groups, Scientometrics, 67(3), 491-502.

Measuring energy dependent polarization in soft gamma-rays using Compton scattering in PoGOLite M. Axelsson a,∗, O. Engdeg̊ard b,a, F. Ryde b,a, S. Larsson a, M. Pearce b, L. Hjalmarsdotter c,a, M. Kiss b, C. Marini Bettolo b, M. Arimoto d, C.-I. Björnsson a, P. Carlson b, Y. Fukazawa e, T. Kamae f,g, Y. Kanai d, J. Kataoka d, N. Kawai d, W. Klamra b, G. Madejski f,g, T. Mizuno e, J. Ng f , H. Tajima f,g, T. Takahashi h, T. Tanaka e, M. Ueno d, G. Varner i, K. Yamamoto e aStockholm Observatory, AlbaNova, SE-106 91 Stockholm, Sweden bPhysics Department, Royal Institute of Technology, AlbaNova, SE-106 91 Stockholm, Sweden cObservatory, PO Box 14, FIN-00014 University of Helsinki, Finland dTokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan eHiroshima University, Physics Department, Higashi-Hiroshima 739-8526, Japan fStanford Linear Accelerator Center, 2575 Sand Hill Road, Menlo Park, CA 94025, USA gKavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA hInstitute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara 229-8510, Japan iDepartment of Physics and Astronomy, University of Hawaii, 2505 Correa Road, Honolulu, HI 96822, USA Abstract Linear polarization in X- and gamma-rays is an important diagnostic of many astrophysical sources, foremost giving information about their geometry, magnetic fields, and radiation mechanisms. However, very few X-ray polarization measurements have been made, and then only mono-energetic detections, whilst several objects are assumed to have energy dependent polarization signatures. In this paper we investigate whether detection of energy dependent polarization from cosmic sources is possible using the Compton technique, in particular with the proposed PoGOLite balloon-experiment, in the 25–100 keV range. We use Geant4 simulations of a PoGOLite model and input photon spectra based on Cygnus X-1 and accreting magnetic pulsars (100mCrab). Effective observing times of 6 and 35 hours were simulated, corresponding to a standard and a long duration flight respectively. Both smooth and sharp energy variations of the polarization are investigated and compared to constant polarization signals using chi-square statistics. We can reject constant polarization, with energy, for the Cygnus X-1 spectrum (in the hard state), if the reflected component is assumed to be completely polarized, whereas the distinction cannot be made for weaker polarization. For the accreting pulsar, constant polarization can be rejected in the case of polarization in a narrow energy band with at least 50% polarization, and similarly for a negative step distribution from 30% to 0% polarization. Key words: Polarization, X-rays, Gamma-rays, Compton technique, PoGOLite, Geant4, Simulations PACS: 95.55.Ka, 95.55.Qf, 95.75.Hi, 98.70.Qy 1. Introduction In the areas of spectral and temporal studies, X-ray and gamma-ray astronomers have been given a wealth of data on a wide range of objects. Polarization has long been pre- dicted to play a crucial role in determining physical and geo- metrical parameters in many astrophysical sources, thereby discriminating among current models. However, there have so far been very few measurements of polarization at these ∗ Corresponding author. E-mail: magnusa@astro.su.se energies. In light of this, the possibility to detect energy dependent polarization has hardly been discussed at all in the literature. In this paper, we present the results from simulations of a dedicated soft gamma-ray polarimeter us- ing Compton scattering, and study the response when the degree of polarization varies with the energy of the emit- ted photons. While energy-dependent polarization is ex- pected from many sources, its detection requires an instru- ment of sufficiently good energy response. The Compton polarimeter presented in this paper utilizes plastic scintilla- tors, which are relatively inefficient for energy depositions Preprint submitted to Elsevier 15 September 2021 http://arxiv.org/abs/0704.1603v2 below a few keV. Thus, simulations are necessary to deter- mine how sensitive the instrument is and how large varia- tions must be for detection. We begin by describing the organisation of the pa- per. The remainder of this section is devoted to giving a background of polarimetry in the X/γ-ray regime, and an overview of the scientific motivation for such measure- ments. In Section 2 we focus on polarimetry using Comp- ton scattering and describe an instrument design based on this technique. We then present the set-up of our simula- tion of the instrument in Section 3, and the results of the simulations in Section 4. Finally, in Sections 5 and 6, we discuss and summarise our results. 1.1. Measurement of polarization The aim of any polarimetric measurement is to deter- mine the degree and direction of polarization of incident radiation. When combined with the traditionally measured quantities of energy and time, polarimetry has the poten- tial to double the parameter space available. As such, it can be a powerful tool to discriminate between physical models proposed for a given source. Historically, polarimetry has proven very successful at optical and radio wavelengths. In these bands, it has been extensively used to probe both radiation physics and ge- ometry of sources (see, e.g., [1]). In the X-ray regime, how- ever, the results are more meagre. Early rocket observa- tions measured X-ray polarization from the Crab Nebula [2]. This result was later confirmed by the Orbiting Solar Observatory 8 (OSO-8, measuring a polarization degree of 19.2% ± 1.0%, [3,4]), the only satellite mission carrying a dedicated polarimeter to date. As the design was based on Bragg reflection on graphite crystals, the energies probed were constrained to 2.6 keV and 5.2 keV. A number of new polarimetric instruments, designed to work in the X/γ-ray regime, have recently been proposed. These include POLAR (10–300 keV, [5]), GRAPE (50–300 keV, [6]), PHENEX (40–300 keV, [7]), CIPHER (10 keV – 1MeV, [8]), and POLARIX (1.5–10 keV, [9]). In this paper we present PoGOLite, a Compton polarimeter currently under construction [10]. 1.2. Expected objects of interest The lack of polarimetric measurements in X-rays is not due to a lack of potential targets. Indeed, from a theoreti- cal point of view there are many sources that are expected to display detectable degrees of polarization. Over the past decades, there have been publications discussing the po- tential for polarization in sources such as X-ray binary (XRB) systems, active galactic nuclei (AGN), accretion and rotation powered pulsars as well as cataclysmic vari- ables (CVs); see e.g., [11,12,13,14,15]. Other work has fo- cused on the processes producing polarized radiation, either the radiative processes themselves (e.g., synchrotron and Fig. 1. Likely geometry in the hard state of Cyg X-1. Mass being accreted forms an accretion disc around the compact object. In the inner regions, there is a hot inner flow/corona. Soft seed photons from the disc may be Comptonized in the hot flow. A fraction of the resulting hard photons can then be reflected off the disc, giving a net polarization. non-thermal bremsstrahlung, [16,17]), or processes such as reflection/asymmetric scattering (e.g., [18,19,20]), strong- field gravity [21,22] and vacuum birefringence in strong magnetic fields [23]. In most sources, polarization is not expected to re- main constant with energy. An example is radiation from strongly magnetized plasmas where the polarization may change dramatically near the cyclotron resonance energy. It is therefore important to understand not only what degree of polarization is needed for detection, but also how sensitive a given instrument will be to the changes of polarization with energy. To study such effects we have chosen to simulate two example sources: Cygnus X-1 and an accreting magnetic neutron star. 1.2.1. Cygnus X-1 Cygnus X-1 is a high-mass XRB where the compact ob- ject is believed to be a black hole. The source exhibits two main spectral states, commonly referred to as hard and soft. Most of the time is spent in the hard state. Several mod- els have been proposed to explain the observed states and transitions. The two main components of such models are usually a geometrically thin, optically thick accretion disc and a hot inner flow or corona [24]. A schematic picture of a likely geometry in the hard state is shown in Fig. 1. Soft X-rays are produced in the accretion disc, and may then be Comptonized in the hot inner flow/corona. A fraction of the hard radiation can be reflected off the accretion disc before reaching the observer. Polarization from this system may arise through several processes. In this paper, we will focus on the polarization introduced by the reflection (for more details, see, e.g., [19,25,26]). In Cygnus X-1, this con- tribution is strongest in the energy range of ∼ 20–100keV. The polarization degree is expected to vary with energy, following the relative strength of the reflection component. 1.2.2. Accreting magnetic neutron stars In many high-mass XRBs the accreting object is a highly magnetic neutron star. The strong magnetic field, ∼ 1013 gauss at the surface, directs the accretion flow to- wards the magnetic poles of the star. Most of the accretion energy is released just above the polar cap where the emis- sion and propagation of radiation is directly connected to the magnetic field as well as the local properties of the plasma. For a number of sources cyclotron spectral features have been observed in hard X-rays, and from these, mag- netic field strengths have been deduced. The X-rays are expected to be polarized and the degree, angle and energy dependence of the polarization will depend on the physical conditions in the emission region [27]. Measurements of the detailed polarization properties would therefore provide a new and very powerful probe of the radiating plasma near the surface of the neutron star. 2. The Compton technique Apart from the special case of Bragg reflection, all three main physical processes of photon-matter interaction in the X/γ-ray regime may be used in polarimetry: photoab- sorption, Compton scattering and pair production. Each of these preserves information on the polarization of the in- coming radiation. For photon energies between ∼ 100 keV and 1MeV, Compton scattering is the dominant process. In this section, we will briefly outline the theoretical basis for a polarimeter based on Compton scattering, and present a design for a dedicated polarimeter based on this technique. 2.1. Basic principle The differential cross section for Compton scattering is given by: − 2 sin2 θ cos2 φ , (1) where re is the classical electron radius, E0 and E are the photon frequency before and after scattering, θ is the an- gle between incident and scattered direction, and φ is the azimuthal scattering angle relative to the plane of polar- ization. When projected on a plane, the angle of scattering will thus be modulated as cos2 φ. To measure the scattering angles, it is necessary to de- tect both the site of scattering and that of photoabsorp- tion. If more than two scattering sites are identified, the relative energy depositions can be used to help distinguish between Compton scattering and photoelectric absorption sites. Some form of segmentation of the detector is neces- sary to provide spatial resolution, required to determine the positions of the signals. Fig. 2. The design of the PoGOLite instrument. The side anticoinci- dence shield has been partially cut away for clarity. The total length of the instrument will be ∼ 100 cm. 2.2. PoGOLite The Polarized Gamma-ray Observer - Light weight ver- sion (PoGOLite) is a balloon-borne polarimeter, planned for launch with a stratospheric balloon in 2009. Figure 2 shows the design of the instrument. The instrument con- sists of 217 phoswich detector cells (PDCs) arranged in a hexagonal pattern. Each PDC is made up of a hollow slow scintillator tube, a fast scintillator detector, a bottom bismuth germanate (BGO) crystal, and a photomultiplier tube (PMT). Signals from the different optical components are distinguished using a pulse shape discrimination tech- nique based on the different scintillation decay times of the materials [28]. The configuration is surrounded by an an- ticoincidence shield made of BGO crystals. Together with the bottom BGO crystals, this allows side and back enter- ing photons and cosmic rays to be rejected. The hollow slow scintillator tube acts as an active col- limator. Photons or charged particles entering the instru- ment off-axis will be registered in the slow scintillator and can be rejected. The desired events are from photons that enter cleanly through the slow scintillator and scatter in the fast scintillator. After scattering, the photon may be absorbed in one of the neighbouring fast scintillator cells, allowing the azimuthal scattering angle to be determined. The well-type design of PoGOLite allows for efficient background rejection [29,30], and gives a field of view of ∼ 5 deg2. This allows the instrument to be accurately pointed at specific sources. As both the initial Compton scattering and subsequent photon absorption occur in the same material (the plastic fast scintillator), the effective energy range is determined by the cross-sections for both these processes, as well as the background. PoGOLite will have an energy range of ∼ 25–100keV, which is lower than the range where Compton scattering dominates. A more detailed description of the instrument may be found in [28,31]. The capability of PoGOLite to measure the energy de- Azimuth angle Fig. 3. Simplified sketch of the distribution of scattering angles, used to determine the modulation factor. The maximum (Cmax), minimum (Cmin) and average (T = [Cmax + Cmin]/2) values of the distribution are indicated. pendence of polarization is limited both by the signal-to- background ratio and the energy resolution. Due to redis- tribution, some of the higher energy photons will produce events at lower energies. The flux and polarization in the low energy band will therefore be affected by the spectrum at higher energies but not vice-versa. The energy response has been carefully simulated using Geant4 [32]. Figure 3 shows a hypothetical distribution of azimuthal scattering angles. The maximum (Cmax) and minimum (Cmin) values of the distribution and the average (T = [Cmax+Cmin]/2) can be used to define a modulation factor: Cmax − Cmin Cmax + Cmin Cmax − Cmin . (2) The modulation factor is determined by fitting the follow- ing function to the distribution of azimuthal scattering an- f(x) = T (1 +M cos(2x+ 2α)) , (3) with angle x (a function variable, not a fitting parameter), average T , modulation factorM , and polarization angle α. In this work, the modulation factor is the discriminator between different polarizationmodels. If the response of the instrument to a 100% polarized source is known, the mod- ulation factor can be used to determine the polarization of the incoming photon beam [33]. 3. Simulations In this section we will describe the setup of our simula- tions. The source models used as input are also presented, as well as the background considered. 3.1. Geant4 Geant4 1 is a multi-purpose software package for sim- ulating particles travelling through and interacting with 1 http://geant4.cern.ch matter, using Monte Carlo techniques [34]. The standard Geant4 package was earlier found [35] to have incorrect im- plementations concerning photon polarization in Compton and Rayleigh scattering; the Geant4 version used here is a corrected version of 4.8.0.p01. 3.2. Simulation setup The Geant4 implementation includes the essential parts of PoGOLite: 217 PDCs with slow and fast plastic scintil- lators and bottom BGO crystals together with a BGO side shield. The model has no PMTs, and uses a solid BGO side shield instead of discrete pentagonal bars (cf. Fig. 2). Lay- ers of tin (50µm) and lead (50µm) surrounding each slow scintillator and the BaSO4 coating (200µm) of the BGO crystals are included. The mechanical support structure is not represented. During the simulation, separate photons are generated with random energies from a spectral model. An event is triggered by a hit in two or three of the fast plastic scintil- lators. The following is saved as output data for each event: information about the original gamma momentum, the ID- number of the cells that had an interaction (ranging from 1 to 217) and the energy deposited in each cell. These data are preprocessed to simulate the resolution of the PMTs, as described in Sect. 4.1. 3.3. Source Models As stated in Sect. 1.2, two sources were considered: Cygnus X-1 (in the hard state) and an accreting neutron star. Below we describe the model used for the incident radiation and polarization in each case. As shown in [10], PoGOLite is expected to detect polarization in both these sources; what we are investigating is the sensitivity to changes in polarization degree with energy. 3.3.1. Cygnus X-1 For our simulations of Cygnus X-1 we used an input spec- trum of a power-law, with photon index α = −1.2 and an exponential cutoff at energy Ecut = 120 keV. It was nor- malised to match the observed spectrum of Cygnus X-1. The spectrum of the reflection was approximated by the logarithmic quadratic curve EFE = 10 −c(logE−log a)(logE−log b), (4) with a = 24, b = 98 and c = 1.89. Figure 4 shows the observed radiation of Cygnus X-1, and our model of the total spectrum as well as that assumed for the reflection component. In our simulations, the polarization is assumed to arise due to the reflection component. Two scenarios were tested: 100% and 20% polarization for the reflection component, with unpolarized direct emission. This corresponds to a to- tal average polarization around 17% and 3% respectively. Fig. 4. Observed radiation spectrum and input model used in the simulations of observations of Cygnus X-1. Gray lines: Typical ra- diation spectrum of Cygnus X-1 in the hard state. Black lines: As- sumed input spectrum: a cut-off power law with index α = −1.2 and cut-off Ecut = 120 keV, normalized to match the measured flux. The reflection component is shown, and the energy range of PoGOLite is indicated by vertical lines. The energy dependent polarization Π(E) used as input was set to the relative strength of the reflection component com- pared to the total flux, scaled down in the case of 20% po- larization. Simulations were performed for effective observ- ing times of 6 hours and 35 hours. These times are chosen as realistic estimates for short and long duration balloon flights, respectively. 3.3.2. Accreting Magnetic Neutron Star In the case of the neutron star, we study the observability of energy dependent effects by simulations of three different idealized polarized spectra: – Polarization in a narrow band. – Polarization only at low energies. – Polarization only at high energies. The neutron star spectrum was in all cases approxi- mated with an exponentially cut-off power law, with index α = −1.1 and energy cut-off at Ecut = 70keV. It was normalized to correspond to a 100mCrab source. Assuming a cyclotron energy Ec at 50 keV, we use three toy models of the polarization energy dependence Π(E), with Πmax ≡ p%: – A Gaussian peak centred at 50 keV, Gp, modelling a rise in polarization from 0% to maximum p%, using the Gaussian curve Π(E) = pe −(E−50)2 2σ2 % (5) with E measured in keV and σ = 5 keV. – Two step functions, Sp and S−p, with polarization 0% if E < 50 keV p% if E ≥ 50 keV for Sp, and 10 20 30 40 50 60 70 80 90 100 E (keV) Fig. 5. Example of assumed energy dependence of the polarization fraction in the case of an accreting magnetic neutron star. The figure shows a Gaussian curve with Πmax = 20% (G20), and a positive step with Πmax = 10% (S10). p% if E < 50 keV 0% if E ≥ 50 keV for S−p. Simply put, p is the jump in polarization that occurs at E = 50 keV. Figure 5 illustrates examples of G20 and S10, the Gaussian and positive steps with maxima 20% and 10% respectively. In the simulations, the values p = {10, 20, 30, 40, 50}% were used, each assuming an observa- tion time of 35 hours. 3.4. Background Balloon-borne gamma-ray polarimetry measurements are subject to several significant sources of background. Through the use of the well-type phoswich detector tech- nique, the PoGOLite instrument has been designed to reduce these backgrounds, allowing 10% polarization of a 100 mCrab source to be measured in one 6 hour balloon observation in the 25–100keV energy range. The basic phoswhich design was used in the WELCOME series of balloon-borne observations and allowed effective back- ground suppression [36,37,38,39,40,41,42]. The concept was subsequently improved and effectively used in a satel- lite instrument, the Suzaku Hard X-ray Detector (HXD) [29,43,44,45]. The background to PoGOLite measurements can arise from charged cosmic rays, neutrons (atmospheric and in- strumental) and gamma-rays (primary and atmospheric). The background from charged cosmic rays (predominantly protons, ∼90%, and helium nuclei, ∼10%) is rejected by the BGO anticoincidence shields and slow plastic collima- tors. Cosmic rays are minimum ionizing particles and can be identified through their relatively large energy deposits. The background presented by atmospheric neutrons and neutrons produced in the PoGOLite instrument and sur- rounding structures is currently being studied in detail [46]. Fig. 6. Estimated contribution to the gamma-ray background in the PoGOLite energy range (∼ 25–100 keV). The radiation from the two considered sources, Cygnus X-1 and an accreting neutron star, are also shown. For the purposes of the study presented in this paper, par- ticular attention has been paid to what is expected to be the dominant background: primary and atmospheric gamma- rays. The gamma-ray background rate is estimated from a model derived from measurements taken in Texas with the GLAST Balloon Flight Engineering Model [47]. The primary gamma-ray component originates outside the atmosphere, i.e., above PoGOLite. The angular distri- bution of the radiation is uniform within the hemisphere above PoGOLite. The energy spectrum is modeled by a doubly-broken power-law with breaks at 50 keV and 1 MeV [48]. Secondary gamma-rays are created in the Earth’s at- mosphere through bremsstrahlung interactions of charged cosmic-rays. Two separate components are considered, one directed upwards and one downwards. The upward flux is dependent on the zenith angle [49], and the energy spec- trum consists of a doubly-broken power-law with breaks at 10 MeV and 1 GeV, and a 511 keV line from electron- positron annihilation. The downward component is simi- lar, but with breaks at 1 MeV and 1 GeV. Energies up to 100 GeV were generated for all components. These models are based on data from satellite- and balloon-borne instru- ments ([50] and [51], and references therein). Figure 6 shows the estimated gamma-ray backgrounds compared to the accreting pulsar and Cygnus X-1 models. The total gamma-ray background is at the 10 mCrab level. 4. Analysis and Results 4.1. Data Processing In the first data processing step, the resolution of the scintillator-PMT assembly is simulated by fluctuating the number of photo-electrons generated in the scintillating materials. It is assumed that when the energyE is deposited in a cell, the average number of photo-electrons generated is En, with n set to 0.5 photo-electrons per keV. Now, we fluctuate En to (En)fluct by applying a Gaussian spread. If En ≤ 10, we do it in two steps: First we subject it to a Poissonian spread, and thereafter a Gaussian spread with variance Enσ2, with σ set to 0.4. If En > 10, only a Gaus- sian spread with variance En is used. If (En)fluct < 0, then it is set to 0. Finally, we take Emes = (En)fluct as the energy actually measured by the PMT in the cell of interest. We reject all fast scintillator interactions with Emes below a certain measurement threshold (2 keV). For the analysis described in this paper, only events with two or three hits in the fast scintillators are retained (the veto logic is not considered at this stage). At PoGOLite energies, more than 80% of the events are from photons interacting in no more than three detector cells [52]. For two-site events, the chronological order of the two cells does not matter for angle calculation, as the distribu- tion is periodic over the angle π. In the case of three hits, we calculate the scattering angle by ignoring the hit with the lowest energy measured, assuming that a low-energy interaction does not affect direction much, and derive an angle from the positions of the two cells with highest en- ergy deposits. Most photons do not scatter very far; about half will only go from one cell to its neighbour. As the range of possible scattering angles resulting in detection in a given adjacent cell is large, this causes strong peaks in each of the six direc- tions corresponding to the neighbouring cells. The PoGO- Lite instrument will rotate about its axis, causing the range for a given cell to smoothly vary and thereby creating a continuous distribution over angles. In the simulations, the uncertainty of the angle determination is instead approxi- mated by introducing a Gaussian spread to the measured scattering angles. In the last steps of data processing we take into account the mass of air in the atmosphere above the balloon, filter- ing out roughly half of our incident source radiation, as- suming the atmospheric overburden 4 g/cm2 at 40 km alti- tude. We also apply the veto logic, rejecting all events with detection in any slow plastic scintillator or BGO crystal. 4.2. A χ2 measure Tomeasure the polarization energy dependence, one can- not simply calculate the polarization at certain energies and construct Π(E), since the photon energy Eγ always will be unknown due to the response of the instrument. Instead we calculate the modulation factor at different measured en- ergies, obtaining a curve M(Emes). This curve can then be compared with theoretic curves resulting from other mod- els, possibly from the same family of curves, enabling us to reject complete families of energy dependencies. One such family, which we will be concerned with here, is the set of constant polarizations. Fig. 7. Example of the measured distribution of events as a function of scattering angle (histogram). In this example, the simulated results of a six hour observation of Cygnus X-1 are shown, assuming a completely polarized reflection component. The data are from the measured energy range of 30–35 keV. A sinusoidal function is fit to the data (solid line, cf. Eq. 3) and a modulation factor is calculated. For a given source model (polarization energy depen- dence) A, the modulation factor M (Eq. 2) was fitted at different measured energies, yielding a curve MA = MA(Emes). Figure 7 shows an example of a modulation curve in the 30–35keV band, generated for a six hour observation of Cygnus X-1 in the case of a completely polarized reflection component. The resulting modulation factor is 2.69± 0.30. The modulation factor is in this way calculated for each energy band. The result is a curve showing how the modulation factor varies with energy, which can then be compared to the corresponding curves for various models of polarization. To test the results against constant polarization, we also generated curves MΠ with constant levels of polarization Π. Since these latter curves should not be thought of as measured, but fluctuation-free theoretical constructs, they were generated by much longer simulations than the obser- vational curves. A measure of how much two curves differ is defined as χ2Π = (MA,Emes −MΠ,Emes) σ2Emes , (9) with σi as the sum of the two errors in fitting MA,Emes and MΠ,Emes . When this is calculated for all reasonable values of Π, we can reject the hypothesis of constant polarization if the minimum of χ2Π is high enough. For 16 degrees of freedom, corresponding to data points up to 100 keV, the 95% certainty level requires χ2 > 26.3. 4.3. Cygnus X-1 Table 1 summarises the results for the simulations of Cygnus X-1. For a 100% polarized reflection component, the energy dependence is detected both after 6 hours of observation and after 35 hours. Figure 8 shows the expected modulation factors and χ2 values for a 35h observation. In Reflection 100% Reflection 20% Obs. time 6h 35h 6h 35h Significance 99.4% >99.99% 19.1% 51.2% Table 1 The significance in rejecting constant polarization models, shown for different reflection polarization strengths and observation times in the case of Cygnus X-1. Gaussian peaks Πmax (%) 10 20 30 40 50 Significance 24.3% 24.3% 35.3% 79.9% 99.7% Positive steps Πmax (%) 10 20 30 40 50 Significance 46.3% 70.0% 96.8% 91.8% 83.5% Negative steps Πmax (%) 10 20 30 40 50 Significance 51.0% 99.3% 95.8% 99.6% >99.99% Table 2 The significance in rejecting constant polarization models, shown for different polarization shapes and maxima in the case of a 100mCrab neutron star. the case of 20% polarization of the reflection component, shown in Fig. 9, constant polarization cannot be ruled out at any higher significance level. 4.4. Magnetic NS The results for the simulations of the accreting neutron star are summarised in Table 2. For a 100mCrab source, the background will start to become significant already in the higher end of the PoGOLite energy range. To be con- servative, only M(E) data points up to 60 keV were used for the χ2 analysis in order to limit the errors due to un- certainties in the background flux. This is equivalent to 8 degrees of freedom, demanding χ2 > 15.5 for 95% certainty in rejecting constant polarization. 4.4.1. Gaussian peaks In Fig. 10 we seeM(E) curves forG50 and a few constant polarizations (left panel), clearly illustrating their different characteristics. The resulting χ2 curve (right panel) con- firms that these models are significantly (99% level) differ- ent. However, this was not the case for any lower value of Πmax. 4.4.2. Positive steps Figure 11 shows the result for S50 when compared to cases of constant polarization. In this case, the difference in characteristics between the constant models and the energy dependent model is not large enough, and constant polar- ization cannot be ruled out. The only value of Πmax yielding a significant (95% level) difference was Πmax = 30%. What is interesting to note is that a higher Πmax, which implies 20 30 40 50 60 70 80 90 100 Modulation factors, 35h E (keV) Cygx1 P(E), ref=100 P=15% P=17% Fig. 8. Results from simulations of a 35h observation of Cygnus X-1, with the reflection component assumed to be 100% polarized. Left panel: Expected modulation factor M(E), together with models of constant polarization at 15% and 17%. The modulation is fitted in intervals of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed line marks the limit for 99.9% significance in rejecting constant polarization. Constant polarization is rejected with high significance. 20 30 40 50 60 70 80 90 100 Modulation factors, 35h E (keV) Cygx1 P(E), ref=20 Fig. 9. Same as Fig. 8, but with the reflection component assumed to be 20% polarized. Left panel: Expected modulation factor M(E), together with models of constant polarization at 2% and 5%. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed line marks the limit for 95% significance in rejecting constant polarization. Constant polarization cannot be rejected with high significance. 20 30 40 50 60 70 80 Modulation factors, 35h E (keV) NS, G50 Const. 0% Const. 10% Const. 20% Fig. 10. Results from simulations of a 35h observation of an accreting magnetic neutron star, assuming a Gaussian polarization curve. Left panel: Expected modulation factor M(E), together with models of constant polarization at 0%, 10% and 20%. The modulation is fitted in intervals of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed line marks the limit for 95% significance in rejecting constant polarization. Constant polarization is rejected with high significance. 20 30 40 50 60 70 80 Modulation factors, 35h E (keV) NS, S50 Const. 20% Const. 30% Const. 40% Fig. 11. Results from simulations of a 35h observation of an accreting magnetic neutron star, assuming a positive step with Πmax = 50%. Left panel: Expected modulation factor M(E), together with models of constant polarization at 20%, 30% and 40%. The modulation is fitted in intervals of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed line marks the limit for 95% significance in rejecting constant polarization. Constant polarization cannot be rejected with high significance. a higher total polarization, does not necessarily make the energy dependence easier to measure. 4.4.3. Negative steps The result of the S−50 model is shown in Fig. 12, where the χ2 plot (right panel) allows us to reject constant polar- ization with a certainty much higher than 99.99%. Other models yielding significant detections were S−20 (99%), S−30 (95%), and S−40 (99%). One reason these models are so easily measurable stems from the fact that, due to the shape of the neutron star spectrum, data and statistics are much poorer at higher energies. If the polarized part only lies at high energies, the total polarization will be much lower or even undetectable. 5. Discussion Although no measurements have been made of polariza- tion in the energy range covered by PoGOLite, some theo- retical models predict changes in polarization with energy. The results of the simulations clearly show that it is possible for the PoGOLite instrument to detect the energy depen- dence of polarization for several of the investigated cases. The highest significance is found for Cygnus X-1 assuming a fully polarized reflection component and a neutron star in the case of a negative step; in the remaining cases, con- stant polarization can not be rejected. Our results there- fore show that PoGOLite has the potential to discriminate among these models. As described above, in Cygnus X-1 the hard X-rays are believed to originate from Comptonization of soft seed pho- tons in a predominantly thermal electron distribution. Al- though this process involves Compton scattering – which could introduce a net polarization – multiple scatterings are required, making our assumption of the direct compo- nent being unpolarized a reasonable one. The degree of po- larization of the reflected component is however more un- certain. Our idealized case of 100% polarization is certainly an overestimation. Calculations [19] show that the degree of polarization in the reflected component varies with in- clination, with a maximum of ∼ 30% expected at high in- clination. The inclination of the Cygnus X-1 system is not well known, but estimates put it at 30◦–50◦ [53], making our assumption of 20% polarization in the reflected compo- nent reasonable. The relative size of the reflected compo- nent compared to the direct emission is in turn dependent on both inclination, where the dependence is the opposite one, and system geometry. We note that in other sources the reflected component may be much stronger, or even dominate the radiation spectrum (e.g., Cygnus X-3, [54]). Another issue which may complicate measurements of energy dependent polarization is the behaviour of the angle of polarization. In our simulations, we have implicitly as- sumed that the angle does not vary with energy. However, for both black hole and neutron star systems, this assump- tion may be an oversimplification. It is certainly true that emission originating from the region close to a black hole will be affected by the strong gravity, affecting the polar- ization angle [21]. It is not clear how large this effect would be on the reflected component in, for instance, Cygnus X-1, but results from both temporal and spectral analysis show that the accretion disc – assumed to be the reflector – is pre- sumably truncated at a large distance (Rin & 30Rg, [55,56]) from the black hole in the hard state. We therefore do not expect this effect to alter the outcome of our simulations. Our results from simulating Cygnus X-1 indicate that long observations with PoGOLite are required to search for energy dependence of polarization. The first flights of the instrument will likely be shorter flights, covering several targets. While these observations should be long enough to detect polarization down to the level of a few per cent, we do not expect to detect any changes in polarizationwith en- ergy. However, long duration flights spanning several days are also planned, and such flights would provide the obser- 20 30 40 50 60 70 80 Modulation factors, 35h E (keV) NS, S−50 Const. 20% Const. 30% Const. 40% Fig. 12. Same as in Fig. 11, but using the negative step. In this case, constant polarization is rejected with high significance. vation time needed to search for variations of polarization degree with energy. A point to note from the neutron star simulations is the result that it is easier to rule out constant polarization in the case of a negative step than for a positive step. As noted in Sect. 4, the energy response of the instrument is such that the energy of the incoming photon cannot be uniquely determined. The result will be a redistribution of energy, with a possibility for higher energy photons to be detected with lower energies. The reverse is however not true – a low energy photon will not be detected as having a higher energy. In the case of a positive step, some polarized high energy photons will be detected at lower energies. This will give a false polarization signal at lower energies, and act to “smooth” the detected energy dependance of polarization. For a negative step, the polarization is introduced at lower energies and will not “spread” to higher energies. The low energy polarization will be somewhat diluted by redistribu- tion of high energy photons but the polarization contrast will still be higher than in the positive case. By excluding energies above 60 keV in the χ2 analysis of accreting X-ray pulsars we have been fairly conservative in our estimate of significances. The restriction of the energy range was motivated by a potentially high sensitivity to systematic errors in the background level at high energies. In this analysis however, we have not taken advantage of the fact that these sources are pulsating. By analysing the polarization of the pulsed flux, rather than the total flux, it should be possible to include all points up to 100 keV and thereby increase the sensitivity. On the other hand the polarization direction will probably change over the pulsa- tion period which will have the opposite effect of reducing the sensitivity. How important this effect is depends on the precise source geometry, radiation beaming and our view- ing angle. As PoGOLite’s field-of-view is rather large (∼ 5 deg2), the pointing errors with respect to the axis of rotation must be small to avoid introducing systematic errors in the po- larization measurements. The attitude control system used for PoGOLite will assure accurate pointing to within a few arcminutes, keeping the systematic error below 1% [10]. Al- though this figure refers to the whole energy band, we do not expect any such effect to change the results presented here. A comprehensive study of systematic effects is beyond the scope of this paper, but will be crucial once PoGOLite is in operation. The performance of the PoGOLite instrument has been extensively evaluated, both with laboratory-based tests [57], accelerator-based tests [28], and simulations [32]. These tests show that it will be able to detect low (∼ 10%) degrees of polarization even for 100 mCrab sources. What has not previously been tested is its sensitivity to a polar- ization degree that varies with energy. Despite the rela- tively modest inherent energy resolution, our results show that PoGOLite has the capacity to detect changes in po- larization degree with energy. The simulations show that significant results can be obtained in a 35h observation, attainable in the long duration flights already planned for PoGOLite. We stress that the design is not optimized for such detections, and future instruments will in all likeli- hood develop this technique further. 6. Conclusions The Compton technique applied to an array of plastic scintillators is an effective method to measure broad en- ergy band polarization, which is demonstrated with the proposed PoGOLite mission, using Geant4 simulations. In particular, energy dependence can be detected. However, in our model of polarization from X-ray binaries, we require the reflection to contribute a large fraction of the observed flux and/or have high degree of polarization for energy de- pendence to be detected. Similarly, for accreting magnetic neutron stars, sharp energy variations in the polarization are needed for a clear detection. This is made easier if the lower energies contain most of the polarization. Acknowledgments The authors gratefully acknowledge support from the Knut and Alice Wallenberg Foundation, the Swedish Na- tional Space Board, the Swedish Research Council, the Kavli Institute for Particle Astrophysics and Cosmology (KIPAC) at Stanford University through an Enterprise Fund, and the Ministry of Education, Science, Sports and Culture (Japan) Grant-in-Aid in Science No.18340052. J.K. and N.K. acknowledge support by JSPS.KAKENHI (16340055).J.K. was also supported by a grant for the in- ternational mission research, which was provided by the In- stitute for Space and Astronautical Science (ISAS/JAXA). T.M. acknowledges support by Grants-in-Aid for Young Scientists (B) from Japan Society for the Promotion of Science (No. 18740154). References [1] J. Tinbergen, Astronomical Polarimetry, Cambridge University Press (2005) [2] R. Novick, M. C. Weisskopf, R. Berthelsdorf, R. Linke, & R. S. Wolff, Astrophys. J. Lett. 174 (1972), L1 [3] M. C. Weisskopf et al, Astrophys. J. 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Soc. 288 (1997), 958 [57] M. Kiss, Construction and laboratory tests of the PoGO-Lite prototype, Master’s thesis (2006) http://www.particle.kth.se/pogolite http://arxiv.org/abs/astro-ph/0609576 http://arxiv.org/abs/astro-ph/0501579 http://www.particle.kth.se/pogolite http://heasarc.gsfc.nasa.gov/docs/astroe/prop_tools/suzaku_td/ http://www.particle.kth.se/pogolite Introduction Measurement of polarization Expected objects of interest The Compton technique Basic principle PoGOLite Simulations Geant4 Simulation setup Source Models Background Analysis and Results Data Processing A 2 measure Cygnus X-1 Magnetic NS Discussion Conclusions References

Linear polarization in X- and gamma-rays is an important diagnostic of many astrophysical sources, foremost giving information about their geometry, magnetic fields, and radiation mechanisms. However, very few X-ray polarization measurements have been made, and then only mono-energetic detections, whilst several objects are assumed to have energy dependent polarization signatures. In this paper we investigate whether detection of energy dependent polarization from cosmic sources is possible using the Compton technique, in particular with the proposed PoGOLite balloon-experiment, in the 25-100 keV range. We use Geant4 simulations of a PoGOLite model and input photon spectra based on Cygnus X-1 and accreting magnetic pulsars (100 mCrab). Effective observing times of 6 and 35 hours were simulated, corresponding to a standard and a long duration flight respectively. Both smooth and sharp energy variations of the polarization are investigated and compared to constant polarization signals using chi-square statistics. We can reject constant polarization, with energy, for the Cygnus X-1 spectrum (in the hard state), if the reflected component is assumed to be completely polarized, whereas the distinction cannot be made for weaker polarization. For the accreting pulsar, constant polarization can be rejected in the case of polarization in a narrow energy band with at least 50% polarization, and similarly for a negative step distribution from 30% to 0% polarization.

Introduction In the areas of spectral and temporal studies, X-ray and gamma-ray astronomers have been given a wealth of data on a wide range of objects. Polarization has long been pre- dicted to play a crucial role in determining physical and geo- metrical parameters in many astrophysical sources, thereby discriminating among current models. However, there have so far been very few measurements of polarization at these ∗ Corresponding author. E-mail: magnusa@astro.su.se energies. In light of this, the possibility to detect energy dependent polarization has hardly been discussed at all in the literature. In this paper, we present the results from simulations of a dedicated soft gamma-ray polarimeter us- ing Compton scattering, and study the response when the degree of polarization varies with the energy of the emit- ted photons. While energy-dependent polarization is ex- pected from many sources, its detection requires an instru- ment of sufficiently good energy response. The Compton polarimeter presented in this paper utilizes plastic scintilla- tors, which are relatively inefficient for energy depositions Preprint submitted to Elsevier 15 September 2021 http://arxiv.org/abs/0704.1603v2 below a few keV. Thus, simulations are necessary to deter- mine how sensitive the instrument is and how large varia- tions must be for detection. We begin by describing the organisation of the pa- per. The remainder of this section is devoted to giving a background of polarimetry in the X/γ-ray regime, and an overview of the scientific motivation for such measure- ments. In Section 2 we focus on polarimetry using Comp- ton scattering and describe an instrument design based on this technique. We then present the set-up of our simula- tion of the instrument in Section 3, and the results of the simulations in Section 4. Finally, in Sections 5 and 6, we discuss and summarise our results. 1.1. Measurement of polarization The aim of any polarimetric measurement is to deter- mine the degree and direction of polarization of incident radiation. When combined with the traditionally measured quantities of energy and time, polarimetry has the poten- tial to double the parameter space available. As such, it can be a powerful tool to discriminate between physical models proposed for a given source. Historically, polarimetry has proven very successful at optical and radio wavelengths. In these bands, it has been extensively used to probe both radiation physics and ge- ometry of sources (see, e.g., [1]). In the X-ray regime, how- ever, the results are more meagre. Early rocket observa- tions measured X-ray polarization from the Crab Nebula [2]. This result was later confirmed by the Orbiting Solar Observatory 8 (OSO-8, measuring a polarization degree of 19.2% ± 1.0%, [3,4]), the only satellite mission carrying a dedicated polarimeter to date. As the design was based on Bragg reflection on graphite crystals, the energies probed were constrained to 2.6 keV and 5.2 keV. A number of new polarimetric instruments, designed to work in the X/γ-ray regime, have recently been proposed. These include POLAR (10–300 keV, [5]), GRAPE (50–300 keV, [6]), PHENEX (40–300 keV, [7]), CIPHER (10 keV – 1MeV, [8]), and POLARIX (1.5–10 keV, [9]). In this paper we present PoGOLite, a Compton polarimeter currently under construction [10]. 1.2. Expected objects of interest The lack of polarimetric measurements in X-rays is not due to a lack of potential targets. Indeed, from a theoreti- cal point of view there are many sources that are expected to display detectable degrees of polarization. Over the past decades, there have been publications discussing the po- tential for polarization in sources such as X-ray binary (XRB) systems, active galactic nuclei (AGN), accretion and rotation powered pulsars as well as cataclysmic vari- ables (CVs); see e.g., [11,12,13,14,15]. Other work has fo- cused on the processes producing polarized radiation, either the radiative processes themselves (e.g., synchrotron and Fig. 1. Likely geometry in the hard state of Cyg X-1. Mass being accreted forms an accretion disc around the compact object. In the inner regions, there is a hot inner flow/corona. Soft seed photons from the disc may be Comptonized in the hot flow. A fraction of the resulting hard photons can then be reflected off the disc, giving a net polarization. non-thermal bremsstrahlung, [16,17]), or processes such as reflection/asymmetric scattering (e.g., [18,19,20]), strong- field gravity [21,22] and vacuum birefringence in strong magnetic fields [23]. In most sources, polarization is not expected to re- main constant with energy. An example is radiation from strongly magnetized plasmas where the polarization may change dramatically near the cyclotron resonance energy. It is therefore important to understand not only what degree of polarization is needed for detection, but also how sensitive a given instrument will be to the changes of polarization with energy. To study such effects we have chosen to simulate two example sources: Cygnus X-1 and an accreting magnetic neutron star. 1.2.1. Cygnus X-1 Cygnus X-1 is a high-mass XRB where the compact ob- ject is believed to be a black hole. The source exhibits two main spectral states, commonly referred to as hard and soft. Most of the time is spent in the hard state. Several mod- els have been proposed to explain the observed states and transitions. The two main components of such models are usually a geometrically thin, optically thick accretion disc and a hot inner flow or corona [24]. A schematic picture of a likely geometry in the hard state is shown in Fig. 1. Soft X-rays are produced in the accretion disc, and may then be Comptonized in the hot inner flow/corona. A fraction of the hard radiation can be reflected off the accretion disc before reaching the observer. Polarization from this system may arise through several processes. In this paper, we will focus on the polarization introduced by the reflection (for more details, see, e.g., [19,25,26]). In Cygnus X-1, this con- tribution is strongest in the energy range of ∼ 20–100keV. The polarization degree is expected to vary with energy, following the relative strength of the reflection component. 1.2.2. Accreting magnetic neutron stars In many high-mass XRBs the accreting object is a highly magnetic neutron star. The strong magnetic field, ∼ 1013 gauss at the surface, directs the accretion flow to- wards the magnetic poles of the star. Most of the accretion energy is released just above the polar cap where the emis- sion and propagation of radiation is directly connected to the magnetic field as well as the local properties of the plasma. For a number of sources cyclotron spectral features have been observed in hard X-rays, and from these, mag- netic field strengths have been deduced. The X-rays are expected to be polarized and the degree, angle and energy dependence of the polarization will depend on the physical conditions in the emission region [27]. Measurements of the detailed polarization properties would therefore provide a new and very powerful probe of the radiating plasma near the surface of the neutron star. 2. The Compton technique Apart from the special case of Bragg reflection, all three main physical processes of photon-matter interaction in the X/γ-ray regime may be used in polarimetry: photoab- sorption, Compton scattering and pair production. Each of these preserves information on the polarization of the in- coming radiation. For photon energies between ∼ 100 keV and 1MeV, Compton scattering is the dominant process. In this section, we will briefly outline the theoretical basis for a polarimeter based on Compton scattering, and present a design for a dedicated polarimeter based on this technique. 2.1. Basic principle The differential cross section for Compton scattering is given by: − 2 sin2 θ cos2 φ , (1) where re is the classical electron radius, E0 and E are the photon frequency before and after scattering, θ is the an- gle between incident and scattered direction, and φ is the azimuthal scattering angle relative to the plane of polar- ization. When projected on a plane, the angle of scattering will thus be modulated as cos2 φ. To measure the scattering angles, it is necessary to de- tect both the site of scattering and that of photoabsorp- tion. If more than two scattering sites are identified, the relative energy depositions can be used to help distinguish between Compton scattering and photoelectric absorption sites. Some form of segmentation of the detector is neces- sary to provide spatial resolution, required to determine the positions of the signals. Fig. 2. The design of the PoGOLite instrument. The side anticoinci- dence shield has been partially cut away for clarity. The total length of the instrument will be ∼ 100 cm. 2.2. PoGOLite The Polarized Gamma-ray Observer - Light weight ver- sion (PoGOLite) is a balloon-borne polarimeter, planned for launch with a stratospheric balloon in 2009. Figure 2 shows the design of the instrument. The instrument con- sists of 217 phoswich detector cells (PDCs) arranged in a hexagonal pattern. Each PDC is made up of a hollow slow scintillator tube, a fast scintillator detector, a bottom bismuth germanate (BGO) crystal, and a photomultiplier tube (PMT). Signals from the different optical components are distinguished using a pulse shape discrimination tech- nique based on the different scintillation decay times of the materials [28]. The configuration is surrounded by an an- ticoincidence shield made of BGO crystals. Together with the bottom BGO crystals, this allows side and back enter- ing photons and cosmic rays to be rejected. The hollow slow scintillator tube acts as an active col- limator. Photons or charged particles entering the instru- ment off-axis will be registered in the slow scintillator and can be rejected. The desired events are from photons that enter cleanly through the slow scintillator and scatter in the fast scintillator. After scattering, the photon may be absorbed in one of the neighbouring fast scintillator cells, allowing the azimuthal scattering angle to be determined. The well-type design of PoGOLite allows for efficient background rejection [29,30], and gives a field of view of ∼ 5 deg2. This allows the instrument to be accurately pointed at specific sources. As both the initial Compton scattering and subsequent photon absorption occur in the same material (the plastic fast scintillator), the effective energy range is determined by the cross-sections for both these processes, as well as the background. PoGOLite will have an energy range of ∼ 25–100keV, which is lower than the range where Compton scattering dominates. A more detailed description of the instrument may be found in [28,31]. The capability of PoGOLite to measure the energy de- Azimuth angle Fig. 3. Simplified sketch of the distribution of scattering angles, used to determine the modulation factor. The maximum (Cmax), minimum (Cmin) and average (T = [Cmax + Cmin]/2) values of the distribution are indicated. pendence of polarization is limited both by the signal-to- background ratio and the energy resolution. Due to redis- tribution, some of the higher energy photons will produce events at lower energies. The flux and polarization in the low energy band will therefore be affected by the spectrum at higher energies but not vice-versa. The energy response has been carefully simulated using Geant4 [32]. Figure 3 shows a hypothetical distribution of azimuthal scattering angles. The maximum (Cmax) and minimum (Cmin) values of the distribution and the average (T = [Cmax+Cmin]/2) can be used to define a modulation factor: Cmax − Cmin Cmax + Cmin Cmax − Cmin . (2) The modulation factor is determined by fitting the follow- ing function to the distribution of azimuthal scattering an- f(x) = T (1 +M cos(2x+ 2α)) , (3) with angle x (a function variable, not a fitting parameter), average T , modulation factorM , and polarization angle α. In this work, the modulation factor is the discriminator between different polarizationmodels. If the response of the instrument to a 100% polarized source is known, the mod- ulation factor can be used to determine the polarization of the incoming photon beam [33]. 3. Simulations In this section we will describe the setup of our simula- tions. The source models used as input are also presented, as well as the background considered. 3.1. Geant4 Geant4 1 is a multi-purpose software package for sim- ulating particles travelling through and interacting with 1 http://geant4.cern.ch matter, using Monte Carlo techniques [34]. The standard Geant4 package was earlier found [35] to have incorrect im- plementations concerning photon polarization in Compton and Rayleigh scattering; the Geant4 version used here is a corrected version of 4.8.0.p01. 3.2. Simulation setup The Geant4 implementation includes the essential parts of PoGOLite: 217 PDCs with slow and fast plastic scintil- lators and bottom BGO crystals together with a BGO side shield. The model has no PMTs, and uses a solid BGO side shield instead of discrete pentagonal bars (cf. Fig. 2). Lay- ers of tin (50µm) and lead (50µm) surrounding each slow scintillator and the BaSO4 coating (200µm) of the BGO crystals are included. The mechanical support structure is not represented. During the simulation, separate photons are generated with random energies from a spectral model. An event is triggered by a hit in two or three of the fast plastic scintil- lators. The following is saved as output data for each event: information about the original gamma momentum, the ID- number of the cells that had an interaction (ranging from 1 to 217) and the energy deposited in each cell. These data are preprocessed to simulate the resolution of the PMTs, as described in Sect. 4.1. 3.3. Source Models As stated in Sect. 1.2, two sources were considered: Cygnus X-1 (in the hard state) and an accreting neutron star. Below we describe the model used for the incident radiation and polarization in each case. As shown in [10], PoGOLite is expected to detect polarization in both these sources; what we are investigating is the sensitivity to changes in polarization degree with energy. 3.3.1. Cygnus X-1 For our simulations of Cygnus X-1 we used an input spec- trum of a power-law, with photon index α = −1.2 and an exponential cutoff at energy Ecut = 120 keV. It was nor- malised to match the observed spectrum of Cygnus X-1. The spectrum of the reflection was approximated by the logarithmic quadratic curve EFE = 10 −c(logE−log a)(logE−log b), (4) with a = 24, b = 98 and c = 1.89. Figure 4 shows the observed radiation of Cygnus X-1, and our model of the total spectrum as well as that assumed for the reflection component. In our simulations, the polarization is assumed to arise due to the reflection component. Two scenarios were tested: 100% and 20% polarization for the reflection component, with unpolarized direct emission. This corresponds to a to- tal average polarization around 17% and 3% respectively. Fig. 4. Observed radiation spectrum and input model used in the simulations of observations of Cygnus X-1. Gray lines: Typical ra- diation spectrum of Cygnus X-1 in the hard state. Black lines: As- sumed input spectrum: a cut-off power law with index α = −1.2 and cut-off Ecut = 120 keV, normalized to match the measured flux. The reflection component is shown, and the energy range of PoGOLite is indicated by vertical lines. The energy dependent polarization Π(E) used as input was set to the relative strength of the reflection component com- pared to the total flux, scaled down in the case of 20% po- larization. Simulations were performed for effective observ- ing times of 6 hours and 35 hours. These times are chosen as realistic estimates for short and long duration balloon flights, respectively. 3.3.2. Accreting Magnetic Neutron Star In the case of the neutron star, we study the observability of energy dependent effects by simulations of three different idealized polarized spectra: – Polarization in a narrow band. – Polarization only at low energies. – Polarization only at high energies. The neutron star spectrum was in all cases approxi- mated with an exponentially cut-off power law, with index α = −1.1 and energy cut-off at Ecut = 70keV. It was normalized to correspond to a 100mCrab source. Assuming a cyclotron energy Ec at 50 keV, we use three toy models of the polarization energy dependence Π(E), with Πmax ≡ p%: – A Gaussian peak centred at 50 keV, Gp, modelling a rise in polarization from 0% to maximum p%, using the Gaussian curve Π(E) = pe −(E−50)2 2σ2 % (5) with E measured in keV and σ = 5 keV. – Two step functions, Sp and S−p, with polarization 0% if E < 50 keV p% if E ≥ 50 keV for Sp, and 10 20 30 40 50 60 70 80 90 100 E (keV) Fig. 5. Example of assumed energy dependence of the polarization fraction in the case of an accreting magnetic neutron star. The figure shows a Gaussian curve with Πmax = 20% (G20), and a positive step with Πmax = 10% (S10). p% if E < 50 keV 0% if E ≥ 50 keV for S−p. Simply put, p is the jump in polarization that occurs at E = 50 keV. Figure 5 illustrates examples of G20 and S10, the Gaussian and positive steps with maxima 20% and 10% respectively. In the simulations, the values p = {10, 20, 30, 40, 50}% were used, each assuming an observa- tion time of 35 hours. 3.4. Background Balloon-borne gamma-ray polarimetry measurements are subject to several significant sources of background. Through the use of the well-type phoswich detector tech- nique, the PoGOLite instrument has been designed to reduce these backgrounds, allowing 10% polarization of a 100 mCrab source to be measured in one 6 hour balloon observation in the 25–100keV energy range. The basic phoswhich design was used in the WELCOME series of balloon-borne observations and allowed effective back- ground suppression [36,37,38,39,40,41,42]. The concept was subsequently improved and effectively used in a satel- lite instrument, the Suzaku Hard X-ray Detector (HXD) [29,43,44,45]. The background to PoGOLite measurements can arise from charged cosmic rays, neutrons (atmospheric and in- strumental) and gamma-rays (primary and atmospheric). The background from charged cosmic rays (predominantly protons, ∼90%, and helium nuclei, ∼10%) is rejected by the BGO anticoincidence shields and slow plastic collima- tors. Cosmic rays are minimum ionizing particles and can be identified through their relatively large energy deposits. The background presented by atmospheric neutrons and neutrons produced in the PoGOLite instrument and sur- rounding structures is currently being studied in detail [46]. Fig. 6. Estimated contribution to the gamma-ray background in the PoGOLite energy range (∼ 25–100 keV). The radiation from the two considered sources, Cygnus X-1 and an accreting neutron star, are also shown. For the purposes of the study presented in this paper, par- ticular attention has been paid to what is expected to be the dominant background: primary and atmospheric gamma- rays. The gamma-ray background rate is estimated from a model derived from measurements taken in Texas with the GLAST Balloon Flight Engineering Model [47]. The primary gamma-ray component originates outside the atmosphere, i.e., above PoGOLite. The angular distri- bution of the radiation is uniform within the hemisphere above PoGOLite. The energy spectrum is modeled by a doubly-broken power-law with breaks at 50 keV and 1 MeV [48]. Secondary gamma-rays are created in the Earth’s at- mosphere through bremsstrahlung interactions of charged cosmic-rays. Two separate components are considered, one directed upwards and one downwards. The upward flux is dependent on the zenith angle [49], and the energy spec- trum consists of a doubly-broken power-law with breaks at 10 MeV and 1 GeV, and a 511 keV line from electron- positron annihilation. The downward component is simi- lar, but with breaks at 1 MeV and 1 GeV. Energies up to 100 GeV were generated for all components. These models are based on data from satellite- and balloon-borne instru- ments ([50] and [51], and references therein). Figure 6 shows the estimated gamma-ray backgrounds compared to the accreting pulsar and Cygnus X-1 models. The total gamma-ray background is at the 10 mCrab level. 4. Analysis and Results 4.1. Data Processing In the first data processing step, the resolution of the scintillator-PMT assembly is simulated by fluctuating the number of photo-electrons generated in the scintillating materials. It is assumed that when the energyE is deposited in a cell, the average number of photo-electrons generated is En, with n set to 0.5 photo-electrons per keV. Now, we fluctuate En to (En)fluct by applying a Gaussian spread. If En ≤ 10, we do it in two steps: First we subject it to a Poissonian spread, and thereafter a Gaussian spread with variance Enσ2, with σ set to 0.4. If En > 10, only a Gaus- sian spread with variance En is used. If (En)fluct < 0, then it is set to 0. Finally, we take Emes = (En)fluct as the energy actually measured by the PMT in the cell of interest. We reject all fast scintillator interactions with Emes below a certain measurement threshold (2 keV). For the analysis described in this paper, only events with two or three hits in the fast scintillators are retained (the veto logic is not considered at this stage). At PoGOLite energies, more than 80% of the events are from photons interacting in no more than three detector cells [52]. For two-site events, the chronological order of the two cells does not matter for angle calculation, as the distribu- tion is periodic over the angle π. In the case of three hits, we calculate the scattering angle by ignoring the hit with the lowest energy measured, assuming that a low-energy interaction does not affect direction much, and derive an angle from the positions of the two cells with highest en- ergy deposits. Most photons do not scatter very far; about half will only go from one cell to its neighbour. As the range of possible scattering angles resulting in detection in a given adjacent cell is large, this causes strong peaks in each of the six direc- tions corresponding to the neighbouring cells. The PoGO- Lite instrument will rotate about its axis, causing the range for a given cell to smoothly vary and thereby creating a continuous distribution over angles. In the simulations, the uncertainty of the angle determination is instead approxi- mated by introducing a Gaussian spread to the measured scattering angles. In the last steps of data processing we take into account the mass of air in the atmosphere above the balloon, filter- ing out roughly half of our incident source radiation, as- suming the atmospheric overburden 4 g/cm2 at 40 km alti- tude. We also apply the veto logic, rejecting all events with detection in any slow plastic scintillator or BGO crystal. 4.2. A χ2 measure Tomeasure the polarization energy dependence, one can- not simply calculate the polarization at certain energies and construct Π(E), since the photon energy Eγ always will be unknown due to the response of the instrument. Instead we calculate the modulation factor at different measured en- ergies, obtaining a curve M(Emes). This curve can then be compared with theoretic curves resulting from other mod- els, possibly from the same family of curves, enabling us to reject complete families of energy dependencies. One such family, which we will be concerned with here, is the set of constant polarizations. Fig. 7. Example of the measured distribution of events as a function of scattering angle (histogram). In this example, the simulated results of a six hour observation of Cygnus X-1 are shown, assuming a completely polarized reflection component. The data are from the measured energy range of 30–35 keV. A sinusoidal function is fit to the data (solid line, cf. Eq. 3) and a modulation factor is calculated. For a given source model (polarization energy depen- dence) A, the modulation factor M (Eq. 2) was fitted at different measured energies, yielding a curve MA = MA(Emes). Figure 7 shows an example of a modulation curve in the 30–35keV band, generated for a six hour observation of Cygnus X-1 in the case of a completely polarized reflection component. The resulting modulation factor is 2.69± 0.30. The modulation factor is in this way calculated for each energy band. The result is a curve showing how the modulation factor varies with energy, which can then be compared to the corresponding curves for various models of polarization. To test the results against constant polarization, we also generated curves MΠ with constant levels of polarization Π. Since these latter curves should not be thought of as measured, but fluctuation-free theoretical constructs, they were generated by much longer simulations than the obser- vational curves. A measure of how much two curves differ is defined as χ2Π = (MA,Emes −MΠ,Emes) σ2Emes , (9) with σi as the sum of the two errors in fitting MA,Emes and MΠ,Emes . When this is calculated for all reasonable values of Π, we can reject the hypothesis of constant polarization if the minimum of χ2Π is high enough. For 16 degrees of freedom, corresponding to data points up to 100 keV, the 95% certainty level requires χ2 > 26.3. 4.3. Cygnus X-1 Table 1 summarises the results for the simulations of Cygnus X-1. For a 100% polarized reflection component, the energy dependence is detected both after 6 hours of observation and after 35 hours. Figure 8 shows the expected modulation factors and χ2 values for a 35h observation. In Reflection 100% Reflection 20% Obs. time 6h 35h 6h 35h Significance 99.4% >99.99% 19.1% 51.2% Table 1 The significance in rejecting constant polarization models, shown for different reflection polarization strengths and observation times in the case of Cygnus X-1. Gaussian peaks Πmax (%) 10 20 30 40 50 Significance 24.3% 24.3% 35.3% 79.9% 99.7% Positive steps Πmax (%) 10 20 30 40 50 Significance 46.3% 70.0% 96.8% 91.8% 83.5% Negative steps Πmax (%) 10 20 30 40 50 Significance 51.0% 99.3% 95.8% 99.6% >99.99% Table 2 The significance in rejecting constant polarization models, shown for different polarization shapes and maxima in the case of a 100mCrab neutron star. the case of 20% polarization of the reflection component, shown in Fig. 9, constant polarization cannot be ruled out at any higher significance level. 4.4. Magnetic NS The results for the simulations of the accreting neutron star are summarised in Table 2. For a 100mCrab source, the background will start to become significant already in the higher end of the PoGOLite energy range. To be con- servative, only M(E) data points up to 60 keV were used for the χ2 analysis in order to limit the errors due to un- certainties in the background flux. This is equivalent to 8 degrees of freedom, demanding χ2 > 15.5 for 95% certainty in rejecting constant polarization. 4.4.1. Gaussian peaks In Fig. 10 we seeM(E) curves forG50 and a few constant polarizations (left panel), clearly illustrating their different characteristics. The resulting χ2 curve (right panel) con- firms that these models are significantly (99% level) differ- ent. However, this was not the case for any lower value of Πmax. 4.4.2. Positive steps Figure 11 shows the result for S50 when compared to cases of constant polarization. In this case, the difference in characteristics between the constant models and the energy dependent model is not large enough, and constant polar- ization cannot be ruled out. The only value of Πmax yielding a significant (95% level) difference was Πmax = 30%. What is interesting to note is that a higher Πmax, which implies 20 30 40 50 60 70 80 90 100 Modulation factors, 35h E (keV) Cygx1 P(E), ref=100 P=15% P=17% Fig. 8. Results from simulations of a 35h observation of Cygnus X-1, with the reflection component assumed to be 100% polarized. Left panel: Expected modulation factor M(E), together with models of constant polarization at 15% and 17%. The modulation is fitted in intervals of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed line marks the limit for 99.9% significance in rejecting constant polarization. Constant polarization is rejected with high significance. 20 30 40 50 60 70 80 90 100 Modulation factors, 35h E (keV) Cygx1 P(E), ref=20 Fig. 9. Same as Fig. 8, but with the reflection component assumed to be 20% polarized. Left panel: Expected modulation factor M(E), together with models of constant polarization at 2% and 5%. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed line marks the limit for 95% significance in rejecting constant polarization. Constant polarization cannot be rejected with high significance. 20 30 40 50 60 70 80 Modulation factors, 35h E (keV) NS, G50 Const. 0% Const. 10% Const. 20% Fig. 10. Results from simulations of a 35h observation of an accreting magnetic neutron star, assuming a Gaussian polarization curve. Left panel: Expected modulation factor M(E), together with models of constant polarization at 0%, 10% and 20%. The modulation is fitted in intervals of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed line marks the limit for 95% significance in rejecting constant polarization. Constant polarization is rejected with high significance. 20 30 40 50 60 70 80 Modulation factors, 35h E (keV) NS, S50 Const. 20% Const. 30% Const. 40% Fig. 11. Results from simulations of a 35h observation of an accreting magnetic neutron star, assuming a positive step with Πmax = 50%. Left panel: Expected modulation factor M(E), together with models of constant polarization at 20%, 30% and 40%. The modulation is fitted in intervals of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed line marks the limit for 95% significance in rejecting constant polarization. Constant polarization cannot be rejected with high significance. a higher total polarization, does not necessarily make the energy dependence easier to measure. 4.4.3. Negative steps The result of the S−50 model is shown in Fig. 12, where the χ2 plot (right panel) allows us to reject constant polar- ization with a certainty much higher than 99.99%. Other models yielding significant detections were S−20 (99%), S−30 (95%), and S−40 (99%). One reason these models are so easily measurable stems from the fact that, due to the shape of the neutron star spectrum, data and statistics are much poorer at higher energies. If the polarized part only lies at high energies, the total polarization will be much lower or even undetectable. 5. Discussion Although no measurements have been made of polariza- tion in the energy range covered by PoGOLite, some theo- retical models predict changes in polarization with energy. The results of the simulations clearly show that it is possible for the PoGOLite instrument to detect the energy depen- dence of polarization for several of the investigated cases. The highest significance is found for Cygnus X-1 assuming a fully polarized reflection component and a neutron star in the case of a negative step; in the remaining cases, con- stant polarization can not be rejected. Our results there- fore show that PoGOLite has the potential to discriminate among these models. As described above, in Cygnus X-1 the hard X-rays are believed to originate from Comptonization of soft seed pho- tons in a predominantly thermal electron distribution. Al- though this process involves Compton scattering – which could introduce a net polarization – multiple scatterings are required, making our assumption of the direct compo- nent being unpolarized a reasonable one. The degree of po- larization of the reflected component is however more un- certain. Our idealized case of 100% polarization is certainly an overestimation. Calculations [19] show that the degree of polarization in the reflected component varies with in- clination, with a maximum of ∼ 30% expected at high in- clination. The inclination of the Cygnus X-1 system is not well known, but estimates put it at 30◦–50◦ [53], making our assumption of 20% polarization in the reflected compo- nent reasonable. The relative size of the reflected compo- nent compared to the direct emission is in turn dependent on both inclination, where the dependence is the opposite one, and system geometry. We note that in other sources the reflected component may be much stronger, or even dominate the radiation spectrum (e.g., Cygnus X-3, [54]). Another issue which may complicate measurements of energy dependent polarization is the behaviour of the angle of polarization. In our simulations, we have implicitly as- sumed that the angle does not vary with energy. However, for both black hole and neutron star systems, this assump- tion may be an oversimplification. It is certainly true that emission originating from the region close to a black hole will be affected by the strong gravity, affecting the polar- ization angle [21]. It is not clear how large this effect would be on the reflected component in, for instance, Cygnus X-1, but results from both temporal and spectral analysis show that the accretion disc – assumed to be the reflector – is pre- sumably truncated at a large distance (Rin & 30Rg, [55,56]) from the black hole in the hard state. We therefore do not expect this effect to alter the outcome of our simulations. Our results from simulating Cygnus X-1 indicate that long observations with PoGOLite are required to search for energy dependence of polarization. The first flights of the instrument will likely be shorter flights, covering several targets. While these observations should be long enough to detect polarization down to the level of a few per cent, we do not expect to detect any changes in polarizationwith en- ergy. However, long duration flights spanning several days are also planned, and such flights would provide the obser- 20 30 40 50 60 70 80 Modulation factors, 35h E (keV) NS, S−50 Const. 20% Const. 30% Const. 40% Fig. 12. Same as in Fig. 11, but using the negative step. In this case, constant polarization is rejected with high significance. vation time needed to search for variations of polarization degree with energy. A point to note from the neutron star simulations is the result that it is easier to rule out constant polarization in the case of a negative step than for a positive step. As noted in Sect. 4, the energy response of the instrument is such that the energy of the incoming photon cannot be uniquely determined. The result will be a redistribution of energy, with a possibility for higher energy photons to be detected with lower energies. The reverse is however not true – a low energy photon will not be detected as having a higher energy. In the case of a positive step, some polarized high energy photons will be detected at lower energies. This will give a false polarization signal at lower energies, and act to “smooth” the detected energy dependance of polarization. For a negative step, the polarization is introduced at lower energies and will not “spread” to higher energies. The low energy polarization will be somewhat diluted by redistribu- tion of high energy photons but the polarization contrast will still be higher than in the positive case. By excluding energies above 60 keV in the χ2 analysis of accreting X-ray pulsars we have been fairly conservative in our estimate of significances. The restriction of the energy range was motivated by a potentially high sensitivity to systematic errors in the background level at high energies. In this analysis however, we have not taken advantage of the fact that these sources are pulsating. By analysing the polarization of the pulsed flux, rather than the total flux, it should be possible to include all points up to 100 keV and thereby increase the sensitivity. On the other hand the polarization direction will probably change over the pulsa- tion period which will have the opposite effect of reducing the sensitivity. How important this effect is depends on the precise source geometry, radiation beaming and our view- ing angle. As PoGOLite’s field-of-view is rather large (∼ 5 deg2), the pointing errors with respect to the axis of rotation must be small to avoid introducing systematic errors in the po- larization measurements. The attitude control system used for PoGOLite will assure accurate pointing to within a few arcminutes, keeping the systematic error below 1% [10]. Al- though this figure refers to the whole energy band, we do not expect any such effect to change the results presented here. A comprehensive study of systematic effects is beyond the scope of this paper, but will be crucial once PoGOLite is in operation. The performance of the PoGOLite instrument has been extensively evaluated, both with laboratory-based tests [57], accelerator-based tests [28], and simulations [32]. These tests show that it will be able to detect low (∼ 10%) degrees of polarization even for 100 mCrab sources. What has not previously been tested is its sensitivity to a polar- ization degree that varies with energy. Despite the rela- tively modest inherent energy resolution, our results show that PoGOLite has the capacity to detect changes in po- larization degree with energy. The simulations show that significant results can be obtained in a 35h observation, attainable in the long duration flights already planned for PoGOLite. We stress that the design is not optimized for such detections, and future instruments will in all likeli- hood develop this technique further. 6. Conclusions The Compton technique applied to an array of plastic scintillators is an effective method to measure broad en- ergy band polarization, which is demonstrated with the proposed PoGOLite mission, using Geant4 simulations. In particular, energy dependence can be detected. However, in our model of polarization from X-ray binaries, we require the reflection to contribute a large fraction of the observed flux and/or have high degree of polarization for energy de- pendence to be detected. Similarly, for accreting magnetic neutron stars, sharp energy variations in the polarization are needed for a clear detection. This is made easier if the lower energies contain most of the polarization. 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Soc. 288 (1997), 958 [57] M. Kiss, Construction and laboratory tests of the PoGO-Lite prototype, Master’s thesis (2006) http://www.particle.kth.se/pogolite http://arxiv.org/abs/astro-ph/0609576 http://arxiv.org/abs/astro-ph/0501579 http://www.particle.kth.se/pogolite http://heasarc.gsfc.nasa.gov/docs/astroe/prop_tools/suzaku_td/ http://www.particle.kth.se/pogolite Introduction Measurement of polarization Expected objects of interest The Compton technique Basic principle PoGOLite Simulations Geant4 Simulation setup Source Models Background Analysis and Results Data Processing A 2 measure Cygnus X-1 Magnetic NS Discussion Conclusions References

EFI-07-08 Baryon Number-Induced Chern-Simons Couplings of Vector and Axial-Vector Mesons in Holographic QCD Sophia K. Domokos and Jeffrey A. Harvey Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago Illinois 60637, USA (Dated: April 2007) We show that holographic models of QCD predict the presence of a Chern-Simons coupling between vector and axial-vector mesons at finite baryon density. In the AdS/CFT dictionary, the coefficient of this coupling is proportional to the baryon number density, and is fixed uniquely in the five-dimensional holographic dual by anomalies in the flavor currents. For the lightest mesons, the coupling mixes transverse ρ and a1 polarization states. At sufficiently large baryon number densities, it produces an instability, which causes the ρ and a1 mesons to condense in a state breaking both rotational and translational invariance. INTRODUCTION Models which use the gravity/gauge correspondence to treat strongly-coupled QCD as a five-dimensional theory of gravity have progressed dramatically in recent years [1, 2, 3]. Particularly at high energies, these theories dif- fer significantly from QCD – yet those models which in- corporate light quarks [4] and chiral symmetry-breaking of the form observed in QCD [5] do capture much of the important low-energy structure of the theory, and give rise to a spectrum of mesons whose masses, decay con- stants, and couplings match those of QCD to within 20%. The gravity/gauge approach includes both top-down models of QCD arising from D-brane constructions in string theory [5], and bottom-up phenomenological mod- els, which attempt to capture the essential dynamics us- ing a simple choice of five-dimensional metric (AdS5) and a minimal field content consisting of a scalar X and gauge fields AaLµ, and A Rµ [6, 7]. These fields are holographically dual to the quark bilinear q̄αqβ , and to the SU(Nf )L × SU(Nf)R flavor currents q̄LγµtaqL and µtaqR of QCD, respectively. These holographic models can be used to study QCD at finite baryon density [8, 9]. In this paper we focus on a novel effect, in which a Chern-Simons term leads to mixing between vector and axial-vector mesons. We will use the model introduced in [6, 7] and for the most part follow the conventions and notation of [6]. THE MODEL We work in a slice of AdS5 with metric ds2 = −dz2 + dxµdxµ , 0 < z ≤ zm . (1) The fifth coordinate, z, is dual to the energy scale of QCD. We generate confinement by imposing an IR cutoff zm, and specifying the IR boundary conditions on the fields. The UV behavior, meanwhile, is governed by z → In AdS/CFT calculations, boundary contributions to the action must be treated with care. In the full AdS space, the only boundary is in the UV (at z = 0). UV- divergent contributions to the action and to other quanti- ties are canceled by counterterms. For details see [10, 11]. In the model at hand, the IR boundary at z = zm may contribute to the action. We follow the approach of [6, 7] by (1) dropping IR boundary terms, and (2) taking pa- rameters normally fixed by IR boundary conditions on the classical solution as input parameters of the model. We generalize the gauge symmetry to U(Nf )L × U(Nf )R and add a Chern-Simons term which gives the correct holographic description of the QCD flavor anoma- lies [3]. The Chern-Simons term does not depend on the metric and on general grounds will be present in any holo- graphic dual description of QCD. The U(1) axial symme- try in QCD is anomalous, but in the spirit of the large Nc approximation we treat it as an exact symmetry of QCD with massless quarks. Including the anomaly would not affect our conclusions. The Lagrangian is thus d4xdz |DX |2 + 3|X |2 − 1 (F 2L + F +SCS . The Chern-Simons term is given by SCS = [ω5(AL)− ω5(AR)] (3) where dω5 = TrF 3, Nc is the number of colors, and AL,R = ÂL,Rt̂ + A a where ta are the generators of SU(Nf )L,R normalized so that Tr t atb = δab/2 and t̂ = 1/ 2Nf is the generator of the U(1) subalgebra of U(Nf). In what follows, we take Nf = 2 so that a = 1, 2, 3. We will often work with the vector and axial- vector fields V = (AL +AR)/2 and A = (AL −AR)/2. CLASSICAL BACKGROUND We expand around a nontrivial solution to the classical equations of motion for the scalar X . Following [6, 7] we http://arxiv.org/abs/0704.1604v1 find the scalar background X0(z) = ≡ v(z) 1 (4) where the coefficient M of the non-normalizable term is proportional to the quark mass matrix, and Σ is the q̄q expectation value. We take bothM and Σ to be diagonal: M ≡ mq1 and Σ ≡ σ1. As shown in [6, 7], we can fix the five-dimensional coupling g5 by comparison with the vector current two-point function in QCD at large Euclidean momentum. This leads to the identification g25 = . (5) The model is thus defined by three parameters: zm, mq and σ. Note that including the U(1) gauge fields and Chern-Simons coupling does not mandate the addition of any new parameters. We use zm = 1/(346 MeV), mq = 2.3 MeV and σ = (308 MeV) 3, which correspond to values found through a global fit to seven observables (Model B) in [6]. A background with a static, constant quark density is described by a classical solution to the equation of motion for the time component of the U(1) vector gauge field V̂µ, which is dual to the quark number current. Solving the V̂0 equation of motion at zero four-momentum yields V̂0(z) = A+ Bz2 . (6) By the general philosophy of AdS/CFT, the coefficient of the non-normalizable term, A, is proportional to the coefficient with which the operator dual to V̂0 enters the gauge theory Lagrangian. Since V̂µ is dual to the quark number current, A must be proportional to the quark chemical potential. Meanwhile, the coefficient of the nor- malizable term, B, is proportional to the expectation value of the operator dual to V̂0: the quark number den- sity. We now obtain the normalizations of A and B. The action evaluated for the background Eq. (6) is given by a boundary term: V̂0∂zV̂0|z=0 = d4x . (7) At finite temperature and baryon number, the Euclidean action is equal to the grand canonical potential. Using Eq. (5), this implies that nqµq (8) with nq the quark number density and µq the quark chemical potential. To fix A we separate U(Nf )L,R into U(1)L,R and SU(Nf )L,R components and note that the Chern-Simons term contains the coupling d4xdzǫMNPQ(ÂL0 TrF PQ−ÂR0 TrFRMNFRPQ) where the indices M,N,P,Q run over 1, 2, 3, z and the trace is over SU(Nf ). Defining the SU(Nf)L,R instanton numbers by nL,R = d3xdzǫMNPQ TrF PQ (10) and taking  constant, this reduces to the coupling nL − ÂR0 nR . (11) Using the connection between instantons and Skyrmion configurations of the pion field carrying non-zero baryon number [12, 13, 14, 15, 16], we can interpret an instanton with nL = −nR = Nb as a state with baryon number Nb. Eq. (11) then fixes A = µb/Nc = µq with µq the quark chemical potential; Eq. (8) fixes B = 24π2nq/Nc. QUADRATIC ACTION In vacuum, the spectrum of the theory consists of towers of scalar, vector, pseudoscalar, and axial-vector mesons given by mode-expanding the five-dimensional fields along the holographic (z) direction, and integrat- ing over z. In this section, we identify the spectrum of excitations and their dispersion relations at non-zero baryon density by expanding the action to quadratic or- der around the background given by Eqns. (4),(6). We focus on the π mesons and the isospin triplet vector ρ and axial-vector a1 mesons, ignoring contributions from heavier mesons, and from the scalar σ which arises from fluctuations in the magnitude of X . Couplings similar to those for the ρ− a1 mesons exist for the isoscalar ω and f1 mesons. For simplicity, we omit these as well. Pions arise as Nambu-Goldstone modes associated with the breaking of U(Nf )L × U(Nf )R to U(Nf )V . We write X(x, z) = X0(z) exp(i2π ata) and expand to quadratic order in πa. The four-dimensional pion field is obtained by writing πa(x, z) = πa(x)ψπ(z). Similarly, the ρa and a1 mesons appear by writing V aµ (x, z) = g5ρ µ(x)ψρ(z), A µ(x, z) = g5a µ(x)ψa(z). The wave functions ψπ(z), ψρ(z), and ψa(z) are solutions of the quadratic equations of motion for fields with four- momentum q2 = m2 and with boundary conditions ψ(0) = ∂zψ(zm) = 0. For details see [6, 7]. Making the above substitutions and expanding to quadratic order yields the four-dimensional action a∂µπa − 1 aπa − 1 (ρaµν) (aaµν) +µǫijk (ρai ∂ja k + a i ∂jρ , (12) with ρµν , aµν the field strengths for ρµ, aµ. The Chern- Simons term with coefficient µ mixes the ρ and a1 mesons. It arises from reduction of a term of the form dV̂ TrAdV in the expansion of Eq. (3). As usual, to obtain Eq. (12) one must remove the mix- ing between aaµ and ∂µπ a by performing the transforma- tion aaµ → aaµ+ξ∂µπa and then rescaling the pion field to obtain a canonical kinetic energy term [17]. This leads to a pion contribution to the Chern-Simons term. A total spatial derivative, it does not contribute to the equations of motion and may be dropped. Since the ρ has JPC = 1−− and the a1 has J 1++, the Chern-Simons coupling is even under P and odd under C. This is indeed consistent with a background having non-zero baryon number, which preserves P and violates C: the coupling is rotationally invariant, but not Lorentz invariant due to the preferred rest frame of the baryons. We can deduce the existence of the Chern-Simons cou- pling in four-dimensional terms as follows. The reduction of the five-dimensional Chern-Simons term [5, 22] gives rise to the usual gauged WZW action [18, 19, 20], as well as a set of couplings which arise from inexact bulk terms. These include a ρ − a1 − ω coupling which, in the pres- ence of a coherent ω field in nuclear matter, gives rise to a coupling of the form given in Eq. (12). The ρ− a1 − ω coupling has been considered previously in a general dis- cussion of chiral effective Lagrangians [25], and is implicit in the formulae of [23]. Related terms appear in [21, 22]. In AdS/QCD, different forms of the gauged WZW action can be obtained by the addition of UV counterterms [24], but these will not cancel the Chern-Simons coupling and lead to explicit breaking of chiral symmetry beyond that given by the quark mass term in Eq. (4). The mass of the ρ meson is given by mρ = 2.405/zm, while ma1 must be determined from a numerical solution of the equation of motion. Model B of [6] finds mρ = 832 MeV, ma1 = 1200 MeV which should be compared to the experimental values mρ = 775.8 ± 0.5 MeV and ma1 = 1230 ± 40 MeV [26]. The parameter µ in the Chern-Simons coupling is given by µ = 18π2nbz mI (13) where I is the dimensionless overlap integral dzzψρ(z)ψa1(z) . (14) Numerical evaluation of the integral gives I = 0.54. A typical baryon density in nuclear matter, n0b ≃ 0.16/(fermi)3 , gives µ ≃ 1.05 GeV . (15) PHENOMENOLOGICAL APPLICATIONS We now outline two potentially observable conse- quences of the Chern-Simons coupling between the ρ and a1. Details will appear elsewhere. Mixing of transverse ρ and a1 states We consider plane-wave solutions to the equations of motion resulting from Eq. (12), dropping the pion fields and focusing on the ρ and a1 dispersion relation and po- larization vectors. Without loss of generality, we consider propagation along x3: ρµ(x) = ǫ µ(q)e −iq·x, aµ(x) = ǫ µ(q)e −iq·x (16) with q = (q0, 0, 0, q3). For convenience, we suppress the SU(2) indices in the following. The components ρ0, ρ3, a0, and a3 have standard dispersion relations, unaffected by the Chern-Simons coupling. The transverse compo- nents ρ1, ρ2, a1, and a2 mix through a derivative cou- pling. The equations of motion yield the dispersion rela- tion for the transverse polarizations (m2ρ +m (m2a1 −m2ρ)2 + 16µ2q The lower sign in Eq. (17) gives a state which is pure ρ as q3 → 0. At non-zero q3, it is a mixture of transverse ρ and a1 states with orthogonal polarization vectors: iM2(q3) = − iM 2(q3) where we have defined ∆2 = m2a1 − m ρ and M2(q3) = ∆4 + 16µ2q2 − ∆2)/2. The upper sign in Eq. (17) gives a pure a1 state for q3 = 0, while for non-zero q3, = − iM 2(q3) iM2(q3) . (19) For µ greater than some momentum-dependent critical value, the dispersion relation Eq. (17) leads to tachyonic modes (modes having dq0/dq3 > 1). For very large mo- menta, this critical value becomes µcrit = (m2ρ +m )/2 ≃ 1.09 GeV . (20) For a range of µ below µcrit the dispersion relation with the lower sign in Eq. (17) exhibits interesting anomalous behavior, the analysis of which is beyond the scope of this letter. It would be interesting to explore signatures of these mixed polarization states in the quark-gluon plasma and in nuclear matter. Vector Meson Condensation To identify the tachyonic instability which occurs for µ > µcrit we start with the energy density corresponding to Eq. (12) for the diagonal component of the ρ and a fields, aa = aδa3, ρa = ρδa3. Completing the square and dropping the terms involving the electric components of the field strengths, which play no role in the instability, we find H = 1 (m2a − µ2)~a · ~a+ (m2ρ − µ2)~ρ · ~ρ ( ~Ba − µ~ρ)2 + ( ~Bρ − µ~a)2 (21) where ~Bρ = ~∇× ~ρ , ~Ba = ~∇× ~a. Applying the ansatz ~a = v cos(µx3)x̂2, ~ρ = v sin(µx3)x̂1 , (22) the last two terms in Eq. (21) vanish, while the average of the first two terms over x3 is negative for µ 2 > µ2 leading to an instability to v 6= 0. Understanding the sta- bilization of the configuration Eq. (22) requires general- izing H to include higher order terms. Note that Eq. (22) breaks both rotational and translational symmetry, ex- hibiting a structure similar to the smectic phase of liquid crystals which includes an interesting set of topological defects. The critical value Eq. (20) is remarkably close to the estimate Eq. (15) for µ at ordinary nuclear densities. If this model is accurate then there should be a conden- sate of vector and axial-vector mesons in nuclear matter with baryon densities at or slightly above n0b . In ordi- nary nuclei, there are finite size effects as well as other corrections to the ρ and a1 interactions which will have to be included to determine whether this condensate oc- curs. Neutron stars are more likely to produce such a condensate, as they are thought to contain matter at a density somewhat greater than n0b . The interplay be- tween this condensate and other conjectured effects in nuclear matter, such as pion condensation and color su- perconductivity, deserves further study. ACKNOWLEDGEMENTS We thank O. Lunin and J. Rosner for helpful conversa- tions. JH thanks the Galileo Galilei Institute in Arcetri, Florence for hospitality while this work was being com- pleted. The work of SD and JH was supported in part by NSF Grant No. PHY-00506630 and NSF Grant 0529954. Any opinions, findings, and conclusions or recommenda- tions expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. [1] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). [2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109]. [3] E. Witten, Adv. Theor. Math. 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We show that holographic models of QCD predict the presence of a Chern-Simons coupling between vector and axial-vector mesons at finite baryon density. In the AdS/CFT dictionary, the coefficient of this coupling is proportional to the baryon number density, and is fixed uniquely in the five-dimensional holographic dual by anomalies in the flavor currents. For the lightest mesons, the coupling mixes transverse $\rho$ and $a_1$ polarization states. At sufficiently large baryon number densities, it produces an instability, which causes the $\rho$ and $a_1$ mesons to condense in a state breaking both rotational and translational invariance.

EFI-07-08 Baryon Number-Induced Chern-Simons Couplings of Vector and Axial-Vector Mesons in Holographic QCD Sophia K. Domokos and Jeffrey A. Harvey Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago Illinois 60637, USA (Dated: April 2007) We show that holographic models of QCD predict the presence of a Chern-Simons coupling between vector and axial-vector mesons at finite baryon density. In the AdS/CFT dictionary, the coefficient of this coupling is proportional to the baryon number density, and is fixed uniquely in the five-dimensional holographic dual by anomalies in the flavor currents. For the lightest mesons, the coupling mixes transverse ρ and a1 polarization states. At sufficiently large baryon number densities, it produces an instability, which causes the ρ and a1 mesons to condense in a state breaking both rotational and translational invariance. INTRODUCTION Models which use the gravity/gauge correspondence to treat strongly-coupled QCD as a five-dimensional theory of gravity have progressed dramatically in recent years [1, 2, 3]. Particularly at high energies, these theories dif- fer significantly from QCD – yet those models which in- corporate light quarks [4] and chiral symmetry-breaking of the form observed in QCD [5] do capture much of the important low-energy structure of the theory, and give rise to a spectrum of mesons whose masses, decay con- stants, and couplings match those of QCD to within 20%. The gravity/gauge approach includes both top-down models of QCD arising from D-brane constructions in string theory [5], and bottom-up phenomenological mod- els, which attempt to capture the essential dynamics us- ing a simple choice of five-dimensional metric (AdS5) and a minimal field content consisting of a scalar X and gauge fields AaLµ, and A Rµ [6, 7]. These fields are holographically dual to the quark bilinear q̄αqβ , and to the SU(Nf )L × SU(Nf)R flavor currents q̄LγµtaqL and µtaqR of QCD, respectively. These holographic models can be used to study QCD at finite baryon density [8, 9]. In this paper we focus on a novel effect, in which a Chern-Simons term leads to mixing between vector and axial-vector mesons. We will use the model introduced in [6, 7] and for the most part follow the conventions and notation of [6]. THE MODEL We work in a slice of AdS5 with metric ds2 = −dz2 + dxµdxµ , 0 < z ≤ zm . (1) The fifth coordinate, z, is dual to the energy scale of QCD. We generate confinement by imposing an IR cutoff zm, and specifying the IR boundary conditions on the fields. The UV behavior, meanwhile, is governed by z → In AdS/CFT calculations, boundary contributions to the action must be treated with care. In the full AdS space, the only boundary is in the UV (at z = 0). UV- divergent contributions to the action and to other quanti- ties are canceled by counterterms. For details see [10, 11]. In the model at hand, the IR boundary at z = zm may contribute to the action. We follow the approach of [6, 7] by (1) dropping IR boundary terms, and (2) taking pa- rameters normally fixed by IR boundary conditions on the classical solution as input parameters of the model. We generalize the gauge symmetry to U(Nf )L × U(Nf )R and add a Chern-Simons term which gives the correct holographic description of the QCD flavor anoma- lies [3]. The Chern-Simons term does not depend on the metric and on general grounds will be present in any holo- graphic dual description of QCD. The U(1) axial symme- try in QCD is anomalous, but in the spirit of the large Nc approximation we treat it as an exact symmetry of QCD with massless quarks. Including the anomaly would not affect our conclusions. The Lagrangian is thus d4xdz |DX |2 + 3|X |2 − 1 (F 2L + F +SCS . The Chern-Simons term is given by SCS = [ω5(AL)− ω5(AR)] (3) where dω5 = TrF 3, Nc is the number of colors, and AL,R = ÂL,Rt̂ + A a where ta are the generators of SU(Nf )L,R normalized so that Tr t atb = δab/2 and t̂ = 1/ 2Nf is the generator of the U(1) subalgebra of U(Nf). In what follows, we take Nf = 2 so that a = 1, 2, 3. We will often work with the vector and axial- vector fields V = (AL +AR)/2 and A = (AL −AR)/2. CLASSICAL BACKGROUND We expand around a nontrivial solution to the classical equations of motion for the scalar X . Following [6, 7] we http://arxiv.org/abs/0704.1604v1 find the scalar background X0(z) = ≡ v(z) 1 (4) where the coefficient M of the non-normalizable term is proportional to the quark mass matrix, and Σ is the q̄q expectation value. We take bothM and Σ to be diagonal: M ≡ mq1 and Σ ≡ σ1. As shown in [6, 7], we can fix the five-dimensional coupling g5 by comparison with the vector current two-point function in QCD at large Euclidean momentum. This leads to the identification g25 = . (5) The model is thus defined by three parameters: zm, mq and σ. Note that including the U(1) gauge fields and Chern-Simons coupling does not mandate the addition of any new parameters. We use zm = 1/(346 MeV), mq = 2.3 MeV and σ = (308 MeV) 3, which correspond to values found through a global fit to seven observables (Model B) in [6]. A background with a static, constant quark density is described by a classical solution to the equation of motion for the time component of the U(1) vector gauge field V̂µ, which is dual to the quark number current. Solving the V̂0 equation of motion at zero four-momentum yields V̂0(z) = A+ Bz2 . (6) By the general philosophy of AdS/CFT, the coefficient of the non-normalizable term, A, is proportional to the coefficient with which the operator dual to V̂0 enters the gauge theory Lagrangian. Since V̂µ is dual to the quark number current, A must be proportional to the quark chemical potential. Meanwhile, the coefficient of the nor- malizable term, B, is proportional to the expectation value of the operator dual to V̂0: the quark number den- sity. We now obtain the normalizations of A and B. The action evaluated for the background Eq. (6) is given by a boundary term: V̂0∂zV̂0|z=0 = d4x . (7) At finite temperature and baryon number, the Euclidean action is equal to the grand canonical potential. Using Eq. (5), this implies that nqµq (8) with nq the quark number density and µq the quark chemical potential. To fix A we separate U(Nf )L,R into U(1)L,R and SU(Nf )L,R components and note that the Chern-Simons term contains the coupling d4xdzǫMNPQ(ÂL0 TrF PQ−ÂR0 TrFRMNFRPQ) where the indices M,N,P,Q run over 1, 2, 3, z and the trace is over SU(Nf ). Defining the SU(Nf)L,R instanton numbers by nL,R = d3xdzǫMNPQ TrF PQ (10) and taking  constant, this reduces to the coupling nL − ÂR0 nR . (11) Using the connection between instantons and Skyrmion configurations of the pion field carrying non-zero baryon number [12, 13, 14, 15, 16], we can interpret an instanton with nL = −nR = Nb as a state with baryon number Nb. Eq. (11) then fixes A = µb/Nc = µq with µq the quark chemical potential; Eq. (8) fixes B = 24π2nq/Nc. QUADRATIC ACTION In vacuum, the spectrum of the theory consists of towers of scalar, vector, pseudoscalar, and axial-vector mesons given by mode-expanding the five-dimensional fields along the holographic (z) direction, and integrat- ing over z. In this section, we identify the spectrum of excitations and their dispersion relations at non-zero baryon density by expanding the action to quadratic or- der around the background given by Eqns. (4),(6). We focus on the π mesons and the isospin triplet vector ρ and axial-vector a1 mesons, ignoring contributions from heavier mesons, and from the scalar σ which arises from fluctuations in the magnitude of X . Couplings similar to those for the ρ− a1 mesons exist for the isoscalar ω and f1 mesons. For simplicity, we omit these as well. Pions arise as Nambu-Goldstone modes associated with the breaking of U(Nf )L × U(Nf )R to U(Nf )V . We write X(x, z) = X0(z) exp(i2π ata) and expand to quadratic order in πa. The four-dimensional pion field is obtained by writing πa(x, z) = πa(x)ψπ(z). Similarly, the ρa and a1 mesons appear by writing V aµ (x, z) = g5ρ µ(x)ψρ(z), A µ(x, z) = g5a µ(x)ψa(z). The wave functions ψπ(z), ψρ(z), and ψa(z) are solutions of the quadratic equations of motion for fields with four- momentum q2 = m2 and with boundary conditions ψ(0) = ∂zψ(zm) = 0. For details see [6, 7]. Making the above substitutions and expanding to quadratic order yields the four-dimensional action a∂µπa − 1 aπa − 1 (ρaµν) (aaµν) +µǫijk (ρai ∂ja k + a i ∂jρ , (12) with ρµν , aµν the field strengths for ρµ, aµ. The Chern- Simons term with coefficient µ mixes the ρ and a1 mesons. It arises from reduction of a term of the form dV̂ TrAdV in the expansion of Eq. (3). As usual, to obtain Eq. (12) one must remove the mix- ing between aaµ and ∂µπ a by performing the transforma- tion aaµ → aaµ+ξ∂µπa and then rescaling the pion field to obtain a canonical kinetic energy term [17]. This leads to a pion contribution to the Chern-Simons term. A total spatial derivative, it does not contribute to the equations of motion and may be dropped. Since the ρ has JPC = 1−− and the a1 has J 1++, the Chern-Simons coupling is even under P and odd under C. This is indeed consistent with a background having non-zero baryon number, which preserves P and violates C: the coupling is rotationally invariant, but not Lorentz invariant due to the preferred rest frame of the baryons. We can deduce the existence of the Chern-Simons cou- pling in four-dimensional terms as follows. The reduction of the five-dimensional Chern-Simons term [5, 22] gives rise to the usual gauged WZW action [18, 19, 20], as well as a set of couplings which arise from inexact bulk terms. These include a ρ − a1 − ω coupling which, in the pres- ence of a coherent ω field in nuclear matter, gives rise to a coupling of the form given in Eq. (12). The ρ− a1 − ω coupling has been considered previously in a general dis- cussion of chiral effective Lagrangians [25], and is implicit in the formulae of [23]. Related terms appear in [21, 22]. In AdS/QCD, different forms of the gauged WZW action can be obtained by the addition of UV counterterms [24], but these will not cancel the Chern-Simons coupling and lead to explicit breaking of chiral symmetry beyond that given by the quark mass term in Eq. (4). The mass of the ρ meson is given by mρ = 2.405/zm, while ma1 must be determined from a numerical solution of the equation of motion. Model B of [6] finds mρ = 832 MeV, ma1 = 1200 MeV which should be compared to the experimental values mρ = 775.8 ± 0.5 MeV and ma1 = 1230 ± 40 MeV [26]. The parameter µ in the Chern-Simons coupling is given by µ = 18π2nbz mI (13) where I is the dimensionless overlap integral dzzψρ(z)ψa1(z) . (14) Numerical evaluation of the integral gives I = 0.54. A typical baryon density in nuclear matter, n0b ≃ 0.16/(fermi)3 , gives µ ≃ 1.05 GeV . (15) PHENOMENOLOGICAL APPLICATIONS We now outline two potentially observable conse- quences of the Chern-Simons coupling between the ρ and a1. Details will appear elsewhere. Mixing of transverse ρ and a1 states We consider plane-wave solutions to the equations of motion resulting from Eq. (12), dropping the pion fields and focusing on the ρ and a1 dispersion relation and po- larization vectors. Without loss of generality, we consider propagation along x3: ρµ(x) = ǫ µ(q)e −iq·x, aµ(x) = ǫ µ(q)e −iq·x (16) with q = (q0, 0, 0, q3). For convenience, we suppress the SU(2) indices in the following. The components ρ0, ρ3, a0, and a3 have standard dispersion relations, unaffected by the Chern-Simons coupling. The transverse compo- nents ρ1, ρ2, a1, and a2 mix through a derivative cou- pling. The equations of motion yield the dispersion rela- tion for the transverse polarizations (m2ρ +m (m2a1 −m2ρ)2 + 16µ2q The lower sign in Eq. (17) gives a state which is pure ρ as q3 → 0. At non-zero q3, it is a mixture of transverse ρ and a1 states with orthogonal polarization vectors: iM2(q3) = − iM 2(q3) where we have defined ∆2 = m2a1 − m ρ and M2(q3) = ∆4 + 16µ2q2 − ∆2)/2. The upper sign in Eq. (17) gives a pure a1 state for q3 = 0, while for non-zero q3, = − iM 2(q3) iM2(q3) . (19) For µ greater than some momentum-dependent critical value, the dispersion relation Eq. (17) leads to tachyonic modes (modes having dq0/dq3 > 1). For very large mo- menta, this critical value becomes µcrit = (m2ρ +m )/2 ≃ 1.09 GeV . (20) For a range of µ below µcrit the dispersion relation with the lower sign in Eq. (17) exhibits interesting anomalous behavior, the analysis of which is beyond the scope of this letter. It would be interesting to explore signatures of these mixed polarization states in the quark-gluon plasma and in nuclear matter. Vector Meson Condensation To identify the tachyonic instability which occurs for µ > µcrit we start with the energy density corresponding to Eq. (12) for the diagonal component of the ρ and a fields, aa = aδa3, ρa = ρδa3. Completing the square and dropping the terms involving the electric components of the field strengths, which play no role in the instability, we find H = 1 (m2a − µ2)~a · ~a+ (m2ρ − µ2)~ρ · ~ρ ( ~Ba − µ~ρ)2 + ( ~Bρ − µ~a)2 (21) where ~Bρ = ~∇× ~ρ , ~Ba = ~∇× ~a. Applying the ansatz ~a = v cos(µx3)x̂2, ~ρ = v sin(µx3)x̂1 , (22) the last two terms in Eq. (21) vanish, while the average of the first two terms over x3 is negative for µ 2 > µ2 leading to an instability to v 6= 0. Understanding the sta- bilization of the configuration Eq. (22) requires general- izing H to include higher order terms. Note that Eq. (22) breaks both rotational and translational symmetry, ex- hibiting a structure similar to the smectic phase of liquid crystals which includes an interesting set of topological defects. The critical value Eq. (20) is remarkably close to the estimate Eq. (15) for µ at ordinary nuclear densities. If this model is accurate then there should be a conden- sate of vector and axial-vector mesons in nuclear matter with baryon densities at or slightly above n0b . In ordi- nary nuclei, there are finite size effects as well as other corrections to the ρ and a1 interactions which will have to be included to determine whether this condensate oc- curs. 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Total Quantum Zeno Effect beyond Zeno Time D. Mundarain1, M. Orszag2 and J. Stephany1 Departmento de F́ısica, Universidad Simón Boĺıvar, Apartado Postal 89000, Caracas 1080A, Venezuela Facultad de F́ısica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile In this work we show that is possible to obtain Total Quantum Zeno Effect in an unstable systems for times larger than the correlation time of the bath. The effect is observed for some particular systems in which one can chose appropriate observables which frequent measurements freeze the system into the initial state. For a two level system in a squeezed bath one can show that there are two bath dependent observables displaying Total Zeno Effect when the system is initialized in some particular states. We show also that these states are intelligent states of two conjugate observables associated to the electromagnetic fluctuations of the bath. I. INTRODUCTION An interesting consequence of the fact that frequent measurements can modify the dynamics of quantum sys- tems is known as Quantum Zeno Effect (QZE) [1, 2, 3, 4, 5] . In general QZE is related to suppression of in- duced transitions in interacting systems or reduction of the decay rate in unstable systems. Also, the opposite effect of enhancing the decay process by frequent mea- surements has been predicted and is known as Anti-Zeno Effect (AZE). The experimental observation of QZE in the early days was restricted to oscillating quantum sys- tems [6] but recently, both QZE and AZE were success- fully observed in irreversible decaying processes.[7, 8, 9]. Quantum theory of measurements predicts reduction of the decay rate in unstable systems when the time be- tween successive measurements is smaller than the Zeno Time which is known to be smaller than the correlation time of the bath. This effect is universal in the sense that it does not depend on the measured observable whenever the time between measurements is very small. This ob- servation does not preclude the manifestation of Zeno Effect for times larger than the correlation time for some well selected observables in a particular bath. In this work we show that is possible for a two-level system inter- acting with a squeezed bath to select a couple observables whose measurements beyond the correlation time for ad- equately prepared systems lead to the total suppression of transitions, i.e Total Zeno Effect. This work is organized as follows: In section (II) we discuss some general facts and review some results ob- tained in reference [10] which are needed for our discus- sion. In Section (III) we define the system we deal with and identify the observables and the corresponding initial states which are shown to display Total Zeno Effect. In section (V) we show that the initial states which show To- tal Zeno Effect are intelligent spin states, i.e states that saturate the Heisenberg Uncertainty Relation for two fic- titious spin operators. Finally, we discuss the results in Section (VI). II. TOTAL ZENO EFFECT IN UNSTABLE SYSTEMS Consider a closed system with Hamiltonian H and an observable A with discrete spectrum. If the initial state of the system is the eigenstate |an〉 of A with eigenvalue an, the probability of survival in a sequence of S measure- ments, that is the probability that in all measurements one gets the same result an, is Pn(∆t, S) = where ∆2nH = 〈an|H 2|an〉 − 〈an|H |an〉2 (2) and ∆t is the time between consecutive measurements. In the limit of continuous monitoring ( S → ∞,∆t → 0 and S∆t → t ), Pn → 1 and the system is freezed in the initial state. In an unstable system and for times larger than the correlation time of the bath, the irreversible evolution of the system can be described in terms of the Liouville operator L{ρ} by using the master equation; = L{ρ} . (3) In this case the survival probability in a sequence of S measurements is: Pn(∆t, S) = (1 + ∆t 〈an|L{|an〉〈an|}|an〉)S (4) Then, the survival probability in the limit of contin- uous monitoring is time dependent and is easy to show that it is given by Pn(t) = exp {〈an|L{|an〉〈an|}|an〉t} . (5) In fact for non zero bath correlation time (τD 6= 0) one cannot take the continous monitoring limit and the equa- tion (5) is an aproximation since ∆t cannot be strictly zero and at the same time be larger than τD. In that case this expression is valid only when the time between con- secutive measurements is small enough but greater than http://arxiv.org/abs/0704.1605v1 the correlation time of the bath. For mathematical sim- plicity in what follows we consider the zero correlation time limit and then one is allowed to take the limit of continuous monitoring. ¿From equation (5) one observes that the Total Zeno Effect is possible when 〈an|L{|an〉〈an|}|an〉 = 0 . (6) Then, for times larger than the correlation time, the pos- sibility of having Total Zeno Effect depends on the dy- namics of the system ( determined by the interaction with the baths), on the observable to be measured and on the particular eigenstate of the observable chosen as the initial state of the system. If equation (6) is satisfied, then equation (5) must be corrected, taking the next non-zero contribution in the expansion of ρ(∆t). In that case the eq. (4) becomes: Pn(∆t, s) = 1 + 〈an|L{L{|an〉〈an|}}|an〉∆t2/2 Then the survival probility for continous monitoring is Pn(t) = exp{ 〈an|L{L{|an〉〈an|}}|an〉∆t t} (8) In general L is proportional to γ, the decay constant for vacuum. Then as one can see a decay rate proportional to γ2∆t appears. and the decay time is ∝ 1 , which is in general a number much larger than the typical evolution time of the system since ∆t ≪ γ. This observation is particularly important for system in which one cannot take the zero limit in ∆t, i.e when one has a bath with a non zero correlation time. Notice that as the spectrum of the bath gets broader, τD becomes smaller, and one is able to choose a smaller ∆t, approaching in this way the ideal situation and the Total Zeno Effect. III. TOTAL ZENO OBSERVABLES In the interaction picture the Liouville operator for a two level system in a broadband squeezed vacuum has the following structure [11], L{ρ} = 1 γ (N + 1) 2σρσ† − σ†σρ− ρσ†σ 2σ†ρσ − σσ†ρ− ρσσ† −γMeiφσ†ρσ† − γMe−iφσρσ (9) where γ is the vacuum decay constant and N,M = N(N + 1) and ψ are the parameters of the squeezed bath. Here σ and σ† are the ladder operators for a two level system, (σx − iσy) σ† = (σx + iσy) (10) with σx, σy and σz the Pauli matrices. Let us introduce the Bloch representation of the two level density matrix (1 + ~ρ · ~σ) (11) Using this representation and the master equation one can obtain the following set of differential equation for the components of the Bloch vector (ρx, ρy, ρz): ρ̇x = −γ (N + 1/2 +M cos(ψ)) ρx + γM sin(ψ)ρy ρ̇y = −γ (N + 1/2−M cos(ψ)) ρy + γM sin(ψ)ρx ρ̇z = −γ (2N + 1)ρz − γ (12) which has the following solutions: ρx(t) = ρx(0) sin 2(ψ/2) + ρy(0) sin(ψ/2) cos(ψ/2) e−γ(N+1/2−M) t ρx(0) cos 2(ψ/2)− ρy(0) sin(ψ/2) cos(ψ/2) e−γ(N+1/2+M) t (13) ρy(t) = ρy(0) cos 2(ψ/2) + ρx(0) sin(ψ/2) cos(ψ/2) e−γ(N+1/2−M) t ρy(0) sin 2(ψ/2)− ρx(0) sin(φ/2) cos(ψ/2) e−γ(N+1/2+M) t (14) ρz(t) = ρz(0)e −γ(2N+1)t + 2N + 1 e−γ(2N+1)t − 1 These equations describe the behavior of the system when there are no measurements. Consider now the hermitian operator σµ associated to the fictitious spin component in the direction of the uni- tary vector µ̂ = (cos(φ) sin(θ), sin(φ) sin(θ), cos(θ)) de- fined by the angles θ and φ, σµ = ~σ · µ̂ = σx cos(φ) sin(θ)+σy sin(φ) sin(θ)+σz cos(θ) The eigenstates of σµ are, |+〉µ = cos(θ/2) |+〉+ sin(θ/2) exp (iφ) |−〉 (17) |−〉µ = − sin(θ/2) |+〉+ cos(θ/2) exp (iφ) |−〉 (18) If the system is initialized in the state |+〉µ the survival probability at time t is P+µ (t) = exp {F (θ, φ) t } (19) where F (θ, φ) = µ〈+| L { |+〉µ µ〈+| } |+〉µ . (20) In this case the function F (θ, φ) has the structure F (θ, φ) = −1 γ (N + 1) ρz(0) + ρ z(0) + ρ2x(0) + ρ2y(0) (ρz(0)− ρ2z(0)− ρ2x(0)− ρ2y(0) γMρx(0)(cos(ψ)ρx(0)− sin(ψ)ρy(0)) γMρy(0)(sin(ψ)ρx(0) + cos(ψ)ρy(0)) (21) where now ~ρ(0) = µ̂ is a function of the angles.. In figure (1) we show F (φ, θ) for N = 1 and ψ = 0 as function of φ and θ. The maxima correspond to F (φ, θ) = 0. For arbitrary values of N and ψ there are two maxima corresponding to the following angles: φM1 = π − ψ and cos(θM ) = − 1 2 (N +M + 1/2) φM2 = π − ψ + π and cos(θM ) = − 1 2 (N +M + 1/2) Theta FIG. 1: F (φ, θ) for N = 1 and ψ = 0 These preferential directions given by the vectors µ̂1 = (cos(φM1 ) sin(θ M ), sin(φM1 ) sin(θ M ), cos(θM )) and µ̂2 = (cos(φM2 ) sin(θ M ), sin(φM2 ) sin(θ M ), cos(θM ))) define the operators σµ1 and σµ2 which show Total Zeno Effect if the initial state of the system is the eigenstate |+〉µ1 or respectively |+〉µ2 , then each preferential observable has only one eigenstate displaying Total Zeno Effect. These eigenstates are: |+〉µ1 = |+〉+ i exp{−iψ }|−〉 (24) |+〉µ2 = |+〉 − i exp{−iψ }|−〉 (25) The other eigenstates of the observables do not dis- play Total Zeno Effect. As final remark is important to observe that in the previous calculations we have ever chosen the state |+〉µ in order to optimize the function F (φ, θ). In fact one can select the state |−〉µ but the final observables displaying Total Zeno Effect will be in the same preferential directions indicated above. IV. MASTER EQUATION AND MEASUREMENTS Besides of the Total Zeno effect obtained in the cases specified previously it is also very interesting to discuss the effect of measurements for other choices of the initial state, the states which do not display Total Zeno Effect. To be specific let us consider measurements of the ob- servable σµ = ~σ · µ̂. The modified master equation with the measurement of σµ is given by [10]: = Pµ L {ρ} Pµ + (1− Pµ) L {ρ} (1− Pµ) (26) where Pµ = |+〉µ µ〈+| (27) and L{ρ} is given by (9). This equation can be solved using the Bloch representation of the density matrix. In this case we can write the density operator in terms of a second set of rotated Pauli matrices that includes the Pauli observable which we are measuring : (1 + ρµσµ + ρασα + ρβσβ) (28) where σα and σβ are two Pauli matrices projected in two orthogonal direction to the vector µ̂. During the process of measurement one obtains always eigenvectors of σµ observable, these eigenvectors have the property of being zero valued for the other two observables. Then during the measurement process the quantities ρα and ρβ are equal to zero because these quantities coprrespond to the mean values of the respectives observables. Then in this case the density matrix can be written in term of one parameter which corresponds to the mean value of the observable that is bein measured; (1 + ρµσµ) (29) ρµ = 〈σµ〉 = Tr {ρσµ} (30) Then the master equation is reduced to the following differential equation: ρ̇µ = Tr {ρ̇σµ} = Tr {(Pµ L {ρ} Pµ + (1− Pµ) L {ρ} (1− Pµ)) σµ} = Tr {L {ρ}σµ} (31) This equation could induced to think that the evolution with and withouth measurements are equal, but we must remember that the density matrix in the right hand side of (??) is the density with measurements. Substituting the form of the density matrix during the measuring pro- cess one can obtain a real differential equation for ρµ: ρ̇µ = α+ βρµ (32) where Tr {L {1}σµ} (33) Tr {L {σµ}σµ} (34) In our case and measuring σµ1 one obtains α = 2 γ (N −M + 1/2) (35) β = −α = −2 γ (N −M + 1/2) (36) The solution to the differential equation is ρµ(t) = 1 + (ρµ(o) − 1) e−αt (37) one can observe the Total Zeno Effect when ρµ(o) = 1 which correspond to having as initial state |+〉µ1 . In figure (2) we show the evolution of 〈σµ1〉, that is the mean value of observable σµ1 , when the system is initial- ized in the state |+〉µ1 . without measurements (master equation (9)) and with frequent monitoring of σµ1 (mas- ter equation (26)). Consistently with our discussion of frequent measurements, the system is freezed in the state |+〉µ1 (Total Zeno Effect). In figure (3) we show the time evolution of 〈σµ1〉 when the initial state is |−〉µ1 without measurements and with measurements of the same observable as in previous case. One observes that with measurements the system evolves from |−〉µ1 to |+〉µ1 . In general for any initial state the system under frequent measurements evolves to |+〉µ1 which is the stationary state of Eq. ( 26) whenever we do measurements in σµ1 . Analogous effects are observed if one measures σµ2 . In contrast, for measurements in other directions different from those defined by µ̂1 or µ̂2 , the system evolves to states which are not eigenstates of the measured observables. < σ 1 > (t) FIG. 2: 〈σµ1(t)〉 for N = 1 and ψ = 0. Solid circles: no mea- surements. Empty circles: with measurements. One measures σµ1 and the initial state is |+〉µ1 .5 0 .5 1 < σ 1 > (t) .5 0 .5 1 < σ 1 > (t) FIG. 3: 〈σµ1(t)〉 for N = 1 y ψ = 0. Solid circles: no mea- surements. Empty circles: with measurements. One measures σµ1 and the initial state is |−〉µ1 V. INTELLIGENT STATES Aragone et al [12] considered well defined angular mo- mentum states that satisfy the equality (∆Jx∆Jy) | 〈Jz〉 |2 in the uncertainty relation. They are called Intelligent States in the literature. The difference with the coherent or squeezed states, associated to harmonic oscillators, is that these Intelligent States are not Mini- mum Uncertainty States (MUS), since the uncertainty is a function of the state itself. In this section we show that the states |+〉µ1 and |+〉µ2 are intelligent states of two observables associated to the bath fluctuations. The master equation (9) can be writ- ten in an explicit Lindblad form 2SρS† − ρS†S − S†Sρ using only one Lindblad operator S, N + 1σ − N exp {iψ}σ† (39) S = cosh(r)σ − sinh(r) exp {iψ}σ† (40) Obviously any eigenstate of S satisfies the condi- tion (6). It is very easy to show that the S opera- tor has two eigenvectors |λ±〉 with eigenvalues λ± = M exp{iψ/2}. It is also easy to observe that these two states are exactly the same states founded in the previous section, |λ+〉 = |+〉µ1 and |λ−〉 = |+〉µ2 . Consider now the standard fictitious angular momen- tum operators for the two level system are {Jx = σx/2, Jy = σy/2, Jz = σz/2} and also two rotated op- erators J1 and J2 which are consistent with the electro- magnetic bath fluctuations in phase space (see fig. 2 in ref [10]) and which satisfy the same Heisenberg uncer- tainty relation that Jx and Jy . They are, J1 = exp{iψ/2Jz}Jx exp{−iψ/2Jz} = cos(ψ/2)Jx − sin(ψ/2)Jy (41) J2 = exp{iψ/2Jz}Jy exp{−iψ/2Jz} = sin(ψ/2)Jx + cos(ψ/2)Jy (42) These two operators are associated respectively with the major and minor axes of the ellipse which represents the fluctuations of bath. In terms of J1 y J2 we have J− = σ = (Jx − iJy) = exp{iψ/2}(J1 − iJ2) , (43) J+ = σ † = (Jx + iJy) = exp{−iψ/2}(J1 + iJ2) . (44) Then S can be written in the following form: S = exp{iψ/2} (cosh(r) − sinh(r)) (J1 − iαJ2) (45) cosh(r) + sinh(r) cosh(r) − sinh(r) = exp{2r} (46) Following Rashid et al ( [13]) we define a non hermitian operator J−(α) J−(α) = (J1 − iαJ2) (1 − α2)1/2 so that S = exp{iψ/2} (cosh(r) − sinh(r)) (1− α2)1/2 J−(α) After some algebra one obtains that S = 2λ+ J−(α) (49) ¿From this equation one can observe that the eigen- states of S are also eigenstates of J−(α) with eigenvalues ±1/2. It is known that the eigenstates of J−(α) are in- telligent states of J1 and J2, i.e they satisfy the equality condition in the Heisenberg uncertainty relation for these observables: ∆2J1∆ 2J2 = |〈Jz〉|2 VI. DISCUSSION We have shown that Total Zeno Effect is obtained for two particular observables σµ1 or σµ2 , for which the az- imuthal phases in the fictitious spin representation de- pend on the phase of the squeezing parameter of the bath and the polar phases depend on the squeeze amplitude. In this sense, the parameters of the squeezed bath specify some definite atomic directions. When performing frequent measurements on σµ1 , starting from the initial state |+〉µ1 , the system freezes at the initial state as opposed to the usual decay when no measurements are done. On the other hand, if the system is initially prepared in the state |−〉µ1 , the fre- quent measurements on σµ1 will makes it evolve from the state |−〉µ1 to |+〉µ1 . More generally, when perform- ing the measurements on σµ1 , any initial state evolves to the same state |+〉µ1 which is the steady state of the master equation (26) in this situation. The above discussion could appear at a first sight sur- prising. However, taking a more familiar case of a two- level atom in contact with a thermal bath at zero temper- ature, if one starts from any initial state, the atom will necessarily decay to the ground state. This is because the time evolution of 〈σz〉 is the same with or without measurements of σz . In both cases the system goes to the ground state, which is an eigenstate of the measured observable σz. In the limit N,M → 0, σµ1 → −σz , and the state |+〉µ1 → |−〉z, which agrees with the known results. Finally, we found that the two eigentates of the two preferential observables displaying QZE are also eigen- states of S operator and consequently intelligent states of J1, J2 which are rotated versions of Jx, Jy obsevables. A. Acknowledgements Two of the authors(D.M. and J.S.) were supported by Did-Usb Grant Gid-30 and by Fonacit Grant No G- 2001000712. M.O was supported by Fondecyt # 1051062 and Nu- cleo Milenio ICM(P02-049) [1] B.Misra and E.C.G.Sudarshan, J.Math.Phys(N.Y), 18, 756 (1977) [2] A.Perez and A.Ron, Phys.Rev.A, 42, 5720( 1990) [3] L.S.Schulman, Phys.Rev A, 57, 1590 (1998) [4] A.D.Panov, Ann.Phys(N.Y), 249, 5720 (1990) [5] A.G.Kofman and G.Kuritzki, Nature(London), 405, 546 (2000) [6] W.M.Itano, D.J.Heinzen,J.J.Bollinger and D.Wineland, Phys.Rev.A, 41, 2295 (1990) [7] S.R.Wilkinson,C.F.Bharucha,M.C.Fischer,K.W.Madison, P.R.Morrow,Q.Miu,B.Sudaram and M.G.Raizen, Na- ture(London), 387, 575 (1997) [8] M.C.Fischer,B.Gutierrez-Medina and G.Raizen, Phys.Rev.Lett, 87, 040402 (2001) [9] P.E.Toschek and C.Wunderlich, Eur.Phys.J.D, 14, 387 (2001) [10] D. Mundarain and J. Stephany, Phys. Rev. A, 73, 042113 (2005). [11] C.W.Gardiner, Phys. Rev. Lett., 56, 1917 (1986). [12] C.Aragone,E.Chalbaud and S.Salamo, J.Math.Phys,17 ,1963(1976) [13] M.A.Rashid, J.Math.Phys,19,1391(1978)

In this work we show that is possible to obtain Total Quantum Zeno Effect in an unstable systems for times larger than the correlation time of the bath. The effect is observed for some particular systems in which one can chose appropriate observables which frequent measurements freeze the system into the initial state. For a two level system in a squeezed bath one can show that there are two bath dependent observables displaying Total Zeno Effect when the system is initialized in some particular states. We show also that these states are intelligent states of two conjugate observables associated to the electromagnetic fluctuations of the bath.

Total Quantum Zeno Effect beyond Zeno Time D. Mundarain1, M. Orszag2 and J. Stephany1 Departmento de F́ısica, Universidad Simón Boĺıvar, Apartado Postal 89000, Caracas 1080A, Venezuela Facultad de F́ısica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile In this work we show that is possible to obtain Total Quantum Zeno Effect in an unstable systems for times larger than the correlation time of the bath. The effect is observed for some particular systems in which one can chose appropriate observables which frequent measurements freeze the system into the initial state. For a two level system in a squeezed bath one can show that there are two bath dependent observables displaying Total Zeno Effect when the system is initialized in some particular states. We show also that these states are intelligent states of two conjugate observables associated to the electromagnetic fluctuations of the bath. I. INTRODUCTION An interesting consequence of the fact that frequent measurements can modify the dynamics of quantum sys- tems is known as Quantum Zeno Effect (QZE) [1, 2, 3, 4, 5] . In general QZE is related to suppression of in- duced transitions in interacting systems or reduction of the decay rate in unstable systems. Also, the opposite effect of enhancing the decay process by frequent mea- surements has been predicted and is known as Anti-Zeno Effect (AZE). The experimental observation of QZE in the early days was restricted to oscillating quantum sys- tems [6] but recently, both QZE and AZE were success- fully observed in irreversible decaying processes.[7, 8, 9]. Quantum theory of measurements predicts reduction of the decay rate in unstable systems when the time be- tween successive measurements is smaller than the Zeno Time which is known to be smaller than the correlation time of the bath. This effect is universal in the sense that it does not depend on the measured observable whenever the time between measurements is very small. This ob- servation does not preclude the manifestation of Zeno Effect for times larger than the correlation time for some well selected observables in a particular bath. In this work we show that is possible for a two-level system inter- acting with a squeezed bath to select a couple observables whose measurements beyond the correlation time for ad- equately prepared systems lead to the total suppression of transitions, i.e Total Zeno Effect. This work is organized as follows: In section (II) we discuss some general facts and review some results ob- tained in reference [10] which are needed for our discus- sion. In Section (III) we define the system we deal with and identify the observables and the corresponding initial states which are shown to display Total Zeno Effect. In section (V) we show that the initial states which show To- tal Zeno Effect are intelligent spin states, i.e states that saturate the Heisenberg Uncertainty Relation for two fic- titious spin operators. Finally, we discuss the results in Section (VI). II. TOTAL ZENO EFFECT IN UNSTABLE SYSTEMS Consider a closed system with Hamiltonian H and an observable A with discrete spectrum. If the initial state of the system is the eigenstate |an〉 of A with eigenvalue an, the probability of survival in a sequence of S measure- ments, that is the probability that in all measurements one gets the same result an, is Pn(∆t, S) = where ∆2nH = 〈an|H 2|an〉 − 〈an|H |an〉2 (2) and ∆t is the time between consecutive measurements. In the limit of continuous monitoring ( S → ∞,∆t → 0 and S∆t → t ), Pn → 1 and the system is freezed in the initial state. In an unstable system and for times larger than the correlation time of the bath, the irreversible evolution of the system can be described in terms of the Liouville operator L{ρ} by using the master equation; = L{ρ} . (3) In this case the survival probability in a sequence of S measurements is: Pn(∆t, S) = (1 + ∆t 〈an|L{|an〉〈an|}|an〉)S (4) Then, the survival probability in the limit of contin- uous monitoring is time dependent and is easy to show that it is given by Pn(t) = exp {〈an|L{|an〉〈an|}|an〉t} . (5) In fact for non zero bath correlation time (τD 6= 0) one cannot take the continous monitoring limit and the equa- tion (5) is an aproximation since ∆t cannot be strictly zero and at the same time be larger than τD. In that case this expression is valid only when the time between con- secutive measurements is small enough but greater than http://arxiv.org/abs/0704.1605v1 the correlation time of the bath. For mathematical sim- plicity in what follows we consider the zero correlation time limit and then one is allowed to take the limit of continuous monitoring. ¿From equation (5) one observes that the Total Zeno Effect is possible when 〈an|L{|an〉〈an|}|an〉 = 0 . (6) Then, for times larger than the correlation time, the pos- sibility of having Total Zeno Effect depends on the dy- namics of the system ( determined by the interaction with the baths), on the observable to be measured and on the particular eigenstate of the observable chosen as the initial state of the system. If equation (6) is satisfied, then equation (5) must be corrected, taking the next non-zero contribution in the expansion of ρ(∆t). In that case the eq. (4) becomes: Pn(∆t, s) = 1 + 〈an|L{L{|an〉〈an|}}|an〉∆t2/2 Then the survival probility for continous monitoring is Pn(t) = exp{ 〈an|L{L{|an〉〈an|}}|an〉∆t t} (8) In general L is proportional to γ, the decay constant for vacuum. Then as one can see a decay rate proportional to γ2∆t appears. and the decay time is ∝ 1 , which is in general a number much larger than the typical evolution time of the system since ∆t ≪ γ. This observation is particularly important for system in which one cannot take the zero limit in ∆t, i.e when one has a bath with a non zero correlation time. Notice that as the spectrum of the bath gets broader, τD becomes smaller, and one is able to choose a smaller ∆t, approaching in this way the ideal situation and the Total Zeno Effect. III. TOTAL ZENO OBSERVABLES In the interaction picture the Liouville operator for a two level system in a broadband squeezed vacuum has the following structure [11], L{ρ} = 1 γ (N + 1) 2σρσ† − σ†σρ− ρσ†σ 2σ†ρσ − σσ†ρ− ρσσ† −γMeiφσ†ρσ† − γMe−iφσρσ (9) where γ is the vacuum decay constant and N,M = N(N + 1) and ψ are the parameters of the squeezed bath. Here σ and σ† are the ladder operators for a two level system, (σx − iσy) σ† = (σx + iσy) (10) with σx, σy and σz the Pauli matrices. Let us introduce the Bloch representation of the two level density matrix (1 + ~ρ · ~σ) (11) Using this representation and the master equation one can obtain the following set of differential equation for the components of the Bloch vector (ρx, ρy, ρz): ρ̇x = −γ (N + 1/2 +M cos(ψ)) ρx + γM sin(ψ)ρy ρ̇y = −γ (N + 1/2−M cos(ψ)) ρy + γM sin(ψ)ρx ρ̇z = −γ (2N + 1)ρz − γ (12) which has the following solutions: ρx(t) = ρx(0) sin 2(ψ/2) + ρy(0) sin(ψ/2) cos(ψ/2) e−γ(N+1/2−M) t ρx(0) cos 2(ψ/2)− ρy(0) sin(ψ/2) cos(ψ/2) e−γ(N+1/2+M) t (13) ρy(t) = ρy(0) cos 2(ψ/2) + ρx(0) sin(ψ/2) cos(ψ/2) e−γ(N+1/2−M) t ρy(0) sin 2(ψ/2)− ρx(0) sin(φ/2) cos(ψ/2) e−γ(N+1/2+M) t (14) ρz(t) = ρz(0)e −γ(2N+1)t + 2N + 1 e−γ(2N+1)t − 1 These equations describe the behavior of the system when there are no measurements. Consider now the hermitian operator σµ associated to the fictitious spin component in the direction of the uni- tary vector µ̂ = (cos(φ) sin(θ), sin(φ) sin(θ), cos(θ)) de- fined by the angles θ and φ, σµ = ~σ · µ̂ = σx cos(φ) sin(θ)+σy sin(φ) sin(θ)+σz cos(θ) The eigenstates of σµ are, |+〉µ = cos(θ/2) |+〉+ sin(θ/2) exp (iφ) |−〉 (17) |−〉µ = − sin(θ/2) |+〉+ cos(θ/2) exp (iφ) |−〉 (18) If the system is initialized in the state |+〉µ the survival probability at time t is P+µ (t) = exp {F (θ, φ) t } (19) where F (θ, φ) = µ〈+| L { |+〉µ µ〈+| } |+〉µ . (20) In this case the function F (θ, φ) has the structure F (θ, φ) = −1 γ (N + 1) ρz(0) + ρ z(0) + ρ2x(0) + ρ2y(0) (ρz(0)− ρ2z(0)− ρ2x(0)− ρ2y(0) γMρx(0)(cos(ψ)ρx(0)− sin(ψ)ρy(0)) γMρy(0)(sin(ψ)ρx(0) + cos(ψ)ρy(0)) (21) where now ~ρ(0) = µ̂ is a function of the angles.. In figure (1) we show F (φ, θ) for N = 1 and ψ = 0 as function of φ and θ. The maxima correspond to F (φ, θ) = 0. For arbitrary values of N and ψ there are two maxima corresponding to the following angles: φM1 = π − ψ and cos(θM ) = − 1 2 (N +M + 1/2) φM2 = π − ψ + π and cos(θM ) = − 1 2 (N +M + 1/2) Theta FIG. 1: F (φ, θ) for N = 1 and ψ = 0 These preferential directions given by the vectors µ̂1 = (cos(φM1 ) sin(θ M ), sin(φM1 ) sin(θ M ), cos(θM )) and µ̂2 = (cos(φM2 ) sin(θ M ), sin(φM2 ) sin(θ M ), cos(θM ))) define the operators σµ1 and σµ2 which show Total Zeno Effect if the initial state of the system is the eigenstate |+〉µ1 or respectively |+〉µ2 , then each preferential observable has only one eigenstate displaying Total Zeno Effect. These eigenstates are: |+〉µ1 = |+〉+ i exp{−iψ }|−〉 (24) |+〉µ2 = |+〉 − i exp{−iψ }|−〉 (25) The other eigenstates of the observables do not dis- play Total Zeno Effect. As final remark is important to observe that in the previous calculations we have ever chosen the state |+〉µ in order to optimize the function F (φ, θ). In fact one can select the state |−〉µ but the final observables displaying Total Zeno Effect will be in the same preferential directions indicated above. IV. MASTER EQUATION AND MEASUREMENTS Besides of the Total Zeno effect obtained in the cases specified previously it is also very interesting to discuss the effect of measurements for other choices of the initial state, the states which do not display Total Zeno Effect. To be specific let us consider measurements of the ob- servable σµ = ~σ · µ̂. The modified master equation with the measurement of σµ is given by [10]: = Pµ L {ρ} Pµ + (1− Pµ) L {ρ} (1− Pµ) (26) where Pµ = |+〉µ µ〈+| (27) and L{ρ} is given by (9). This equation can be solved using the Bloch representation of the density matrix. In this case we can write the density operator in terms of a second set of rotated Pauli matrices that includes the Pauli observable which we are measuring : (1 + ρµσµ + ρασα + ρβσβ) (28) where σα and σβ are two Pauli matrices projected in two orthogonal direction to the vector µ̂. During the process of measurement one obtains always eigenvectors of σµ observable, these eigenvectors have the property of being zero valued for the other two observables. Then during the measurement process the quantities ρα and ρβ are equal to zero because these quantities coprrespond to the mean values of the respectives observables. Then in this case the density matrix can be written in term of one parameter which corresponds to the mean value of the observable that is bein measured; (1 + ρµσµ) (29) ρµ = 〈σµ〉 = Tr {ρσµ} (30) Then the master equation is reduced to the following differential equation: ρ̇µ = Tr {ρ̇σµ} = Tr {(Pµ L {ρ} Pµ + (1− Pµ) L {ρ} (1− Pµ)) σµ} = Tr {L {ρ}σµ} (31) This equation could induced to think that the evolution with and withouth measurements are equal, but we must remember that the density matrix in the right hand side of (??) is the density with measurements. Substituting the form of the density matrix during the measuring pro- cess one can obtain a real differential equation for ρµ: ρ̇µ = α+ βρµ (32) where Tr {L {1}σµ} (33) Tr {L {σµ}σµ} (34) In our case and measuring σµ1 one obtains α = 2 γ (N −M + 1/2) (35) β = −α = −2 γ (N −M + 1/2) (36) The solution to the differential equation is ρµ(t) = 1 + (ρµ(o) − 1) e−αt (37) one can observe the Total Zeno Effect when ρµ(o) = 1 which correspond to having as initial state |+〉µ1 . In figure (2) we show the evolution of 〈σµ1〉, that is the mean value of observable σµ1 , when the system is initial- ized in the state |+〉µ1 . without measurements (master equation (9)) and with frequent monitoring of σµ1 (mas- ter equation (26)). Consistently with our discussion of frequent measurements, the system is freezed in the state |+〉µ1 (Total Zeno Effect). In figure (3) we show the time evolution of 〈σµ1〉 when the initial state is |−〉µ1 without measurements and with measurements of the same observable as in previous case. One observes that with measurements the system evolves from |−〉µ1 to |+〉µ1 . In general for any initial state the system under frequent measurements evolves to |+〉µ1 which is the stationary state of Eq. ( 26) whenever we do measurements in σµ1 . Analogous effects are observed if one measures σµ2 . In contrast, for measurements in other directions different from those defined by µ̂1 or µ̂2 , the system evolves to states which are not eigenstates of the measured observables. < σ 1 > (t) FIG. 2: 〈σµ1(t)〉 for N = 1 and ψ = 0. Solid circles: no mea- surements. Empty circles: with measurements. One measures σµ1 and the initial state is |+〉µ1 .5 0 .5 1 < σ 1 > (t) .5 0 .5 1 < σ 1 > (t) FIG. 3: 〈σµ1(t)〉 for N = 1 y ψ = 0. Solid circles: no mea- surements. Empty circles: with measurements. One measures σµ1 and the initial state is |−〉µ1 V. INTELLIGENT STATES Aragone et al [12] considered well defined angular mo- mentum states that satisfy the equality (∆Jx∆Jy) | 〈Jz〉 |2 in the uncertainty relation. They are called Intelligent States in the literature. The difference with the coherent or squeezed states, associated to harmonic oscillators, is that these Intelligent States are not Mini- mum Uncertainty States (MUS), since the uncertainty is a function of the state itself. In this section we show that the states |+〉µ1 and |+〉µ2 are intelligent states of two observables associated to the bath fluctuations. The master equation (9) can be writ- ten in an explicit Lindblad form 2SρS† − ρS†S − S†Sρ using only one Lindblad operator S, N + 1σ − N exp {iψ}σ† (39) S = cosh(r)σ − sinh(r) exp {iψ}σ† (40) Obviously any eigenstate of S satisfies the condi- tion (6). It is very easy to show that the S opera- tor has two eigenvectors |λ±〉 with eigenvalues λ± = M exp{iψ/2}. It is also easy to observe that these two states are exactly the same states founded in the previous section, |λ+〉 = |+〉µ1 and |λ−〉 = |+〉µ2 . Consider now the standard fictitious angular momen- tum operators for the two level system are {Jx = σx/2, Jy = σy/2, Jz = σz/2} and also two rotated op- erators J1 and J2 which are consistent with the electro- magnetic bath fluctuations in phase space (see fig. 2 in ref [10]) and which satisfy the same Heisenberg uncer- tainty relation that Jx and Jy . They are, J1 = exp{iψ/2Jz}Jx exp{−iψ/2Jz} = cos(ψ/2)Jx − sin(ψ/2)Jy (41) J2 = exp{iψ/2Jz}Jy exp{−iψ/2Jz} = sin(ψ/2)Jx + cos(ψ/2)Jy (42) These two operators are associated respectively with the major and minor axes of the ellipse which represents the fluctuations of bath. In terms of J1 y J2 we have J− = σ = (Jx − iJy) = exp{iψ/2}(J1 − iJ2) , (43) J+ = σ † = (Jx + iJy) = exp{−iψ/2}(J1 + iJ2) . (44) Then S can be written in the following form: S = exp{iψ/2} (cosh(r) − sinh(r)) (J1 − iαJ2) (45) cosh(r) + sinh(r) cosh(r) − sinh(r) = exp{2r} (46) Following Rashid et al ( [13]) we define a non hermitian operator J−(α) J−(α) = (J1 − iαJ2) (1 − α2)1/2 so that S = exp{iψ/2} (cosh(r) − sinh(r)) (1− α2)1/2 J−(α) After some algebra one obtains that S = 2λ+ J−(α) (49) ¿From this equation one can observe that the eigen- states of S are also eigenstates of J−(α) with eigenvalues ±1/2. It is known that the eigenstates of J−(α) are in- telligent states of J1 and J2, i.e they satisfy the equality condition in the Heisenberg uncertainty relation for these observables: ∆2J1∆ 2J2 = |〈Jz〉|2 VI. DISCUSSION We have shown that Total Zeno Effect is obtained for two particular observables σµ1 or σµ2 , for which the az- imuthal phases in the fictitious spin representation de- pend on the phase of the squeezing parameter of the bath and the polar phases depend on the squeeze amplitude. In this sense, the parameters of the squeezed bath specify some definite atomic directions. When performing frequent measurements on σµ1 , starting from the initial state |+〉µ1 , the system freezes at the initial state as opposed to the usual decay when no measurements are done. On the other hand, if the system is initially prepared in the state |−〉µ1 , the fre- quent measurements on σµ1 will makes it evolve from the state |−〉µ1 to |+〉µ1 . More generally, when perform- ing the measurements on σµ1 , any initial state evolves to the same state |+〉µ1 which is the steady state of the master equation (26) in this situation. The above discussion could appear at a first sight sur- prising. However, taking a more familiar case of a two- level atom in contact with a thermal bath at zero temper- ature, if one starts from any initial state, the atom will necessarily decay to the ground state. This is because the time evolution of 〈σz〉 is the same with or without measurements of σz . In both cases the system goes to the ground state, which is an eigenstate of the measured observable σz. In the limit N,M → 0, σµ1 → −σz , and the state |+〉µ1 → |−〉z, which agrees with the known results. Finally, we found that the two eigentates of the two preferential observables displaying QZE are also eigen- states of S operator and consequently intelligent states of J1, J2 which are rotated versions of Jx, Jy obsevables. A. Acknowledgements Two of the authors(D.M. and J.S.) were supported by Did-Usb Grant Gid-30 and by Fonacit Grant No G- 2001000712. M.O was supported by Fondecyt # 1051062 and Nu- cleo Milenio ICM(P02-049) [1] B.Misra and E.C.G.Sudarshan, J.Math.Phys(N.Y), 18, 756 (1977) [2] A.Perez and A.Ron, Phys.Rev.A, 42, 5720( 1990) [3] L.S.Schulman, Phys.Rev A, 57, 1590 (1998) [4] A.D.Panov, Ann.Phys(N.Y), 249, 5720 (1990) [5] A.G.Kofman and G.Kuritzki, Nature(London), 405, 546 (2000) [6] W.M.Itano, D.J.Heinzen,J.J.Bollinger and D.Wineland, Phys.Rev.A, 41, 2295 (1990) [7] S.R.Wilkinson,C.F.Bharucha,M.C.Fischer,K.W.Madison, P.R.Morrow,Q.Miu,B.Sudaram and M.G.Raizen, Na- ture(London), 387, 575 (1997) [8] M.C.Fischer,B.Gutierrez-Medina and G.Raizen, Phys.Rev.Lett, 87, 040402 (2001) [9] P.E.Toschek and C.Wunderlich, Eur.Phys.J.D, 14, 387 (2001) [10] D. Mundarain and J. Stephany, Phys. Rev. A, 73, 042113 (2005). [11] C.W.Gardiner, Phys. Rev. Lett., 56, 1917 (1986). [12] C.Aragone,E.Chalbaud and S.Salamo, J.Math.Phys,17 ,1963(1976) [13] M.A.Rashid, J.Math.Phys,19,1391(1978)

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 1 November 2018 (MN LATEX style file v2.2) Asteroseismic Signatures of Stellar Magnetic Activity Cycles T. S. Metcalfe1,2, W. A. Dziembowski3,4, P. G. Judge1, M. Snow5 1 High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO 80307-3000 USA 2 Scientific Computing Division, National Center for Atmospheric Research, Boulder, CO 80307-3000 USA 3 Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warsaw, Poland 4 Copernicus Astronomical Centre, Bartycka 18, 00-716 Warsaw, Poland 5 Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado 80309-0392 USA 1 November 2018 ABSTRACT Observations of stellar activity cycles provide an opportunity to study magnetic dynamos under many different physical conditions. Space-based asteroseismology missions will soon yield useful constraints on the interior conditions that nurture such magnetic cycles, and will be sensitive enough to detect shifts in the oscillation frequencies due to the magnetic varia- tions. We derive a method for predicting these shifts from changes in the Mg II activity index by scaling from solar data. We demonstrate this technique on the solar-type subgiant β Hyi, using archival International Ultraviolet Explorer spectra and two epochs of ground-based as- teroseismic observations. We find qualitative evidence of the expected frequency shifts and predict the optimal timing for future asteroseismic observations of this star. Key words: stars: activity – stars: individual (β Hyi) – stars: interiors – stars: oscillations 1 INTRODUCTION Astronomers have been making telescopic observations of sunspots since the time of Galileo, gradually building an historical record showing a periodic rise and fall in the number of sunspots every ∼11 years. We now know that sunspots are regions with an en- hanced local magnetic field, so this 11-year cycle actually traces a variation in surface magnetism. Attempts to understand this be- havior theoretically often invoke a combination of differential rota- tion, convection, and meridional flow to modulate the field through a magnetic dynamo (e.g., see Rempel 2006; Dikpati & Gilman 2006). Although we can rarely observe spots on other solar-type stars directly, these areas of concentrated magnetic field produce strong emission in the Ca II H and K resonance lines in the optical, and the Mg II h and k lines in the ultraviolet. Wilson (1978) was the first to demonstrate that many solar-type stars exhibit long-term cyclic variations in their Ca II H and K emission, analogous to those seen in full-disc solar observations through the magnetic activity cycle. Early analysis of these data revealed an empirical correlation between the mean level of magnetic activity and the rotation pe- riod normalized by the convective timescale (Noyes et al. 1984a), as well as a relation between the rotation rate and the period of the observed activity cycle (Noyes et al. 1984b), which generally supports a dynamo interpretation. Significant progress in dynamo modeling unfolded after he- lioseismology provided detailed constraints on the Sun’s interior structure and dynamics. These observations also established that variations in the mean strength of the solar magnetic field lead to significant shifts (∼0.5 µHz) in the frequencies of even the lowest- degree p-modes (Libbrecht & Woodard 1990; Salabert et al. 2004). These shifts can provide independent constraints on the physical mechanisms that drive the solar dynamo, through their influence on the outer boundary condition for the pulsation modes. They are thought to arise either from changes in the near-surface propagation speed due to a direct magnetic perturbation (Goldreich et al. 1991), or from a slight decrease in the radial component of the turbulent velocity in the outer layers and the associated changes in tempera- ture (Dziembowski & Goode 2004, 2005). Space-based asteroseismology missions, such as MOST (Walker et al. 2003), CoRoT (Baglin et al. 2006), and Kepler (Christensen-Dalsgaard et al. 2007) will soon allow additional tests of dynamo models using other solar-type stars (see Chaplin et al. 2007). High precision time-series photometry from MOST has al- ready revealed latitudinal differential rotation in two solar-type stars (Croll et al. 2006; Walker et al. 2007), and the long-term mon- itoring from future missions is expected to produce asteroseismic measurements of stellar convection zone depths (Monteiro et al. 2000; Verner et al. 2006). By combining such observations with the stellar magnetic activity cycles documented from long-term sur- veys of the Ca II or Mg II lines, we can extend the calibration of dynamo models from the solar case to dozens of independent sets of physical conditions. The G2 subgiant β Hyi is the only solar-type star that presently has both a known magnetic activity cycle (Dravins et al. 1993) and multiple epochs of asteroseismic observations (Bedding et al. 2001, 2007). In this paper we reanalyze archival International Ul- traviolet Explorer (IUE) spectra for an improved characterization of the magnetic cycle in this star, and we use it to predict the activity related shifts in the observed radial p-mode oscillations. We com- pare these predictions with recently published asteroseismic data, c© 0000 RAS http://arxiv.org/abs/0704.1606v1 2 Metcalfe et al. and we suggest the optimal timing of future observations to maxi- mize the amplitude of the expected p-mode frequency shifts. 2 ARCHIVAL IUE SPECTRA The activity cycle of β Hyi was studied by Dravins et al. (1993), who used high resolution IUE data of the Mg II resonance lines over 11 years, from June 1978 to the end of October 1989. They consid- ered these data to be consistent with a cycle period between 15 and 18 years. Since the work of Dravins et al., a significant number of additional IUE spectra were obtained by E. Guinan from early 1992 to the end of 1995. Our analysis of all of these spectra reveals the beginning of a new cycle in 1993-1994. In 1997, the IUE project reprocessed the entire database using improved and uniform reduction procedures (“NEWSIPS”). Using the NEWSIPS merged high resolution extracted spectra, we have reanalyzed the entire IUE dataset containing useful echelle data of the Mg II lines. Data were excluded when the NEWSIPS soft- ware mis-registered the spectral orders, when continuum data near 279.67 nm were saturated, or when continuum data were more than 1σ below the mean (to reject additional poorly registered spectral orders) or more than 1.5σ above the mean. The classic definition of the Mg II index (Heath & Schlesinger 1986) uses wing irradiances at 276 and 283 nm. The wings in their formulation had to be so far away from the cores due to the 1.1 nm spectral bandpass of their instrument. In the IUE spectra, pixels at those wavelengths are saturated, so the photospheric reference levels need to be measured much closer to the emission cores. Snow & McClintock (2005) have shown that at moderate resolu- tion the variability of the inner wings of the Mg II absorption fea- ture is very similar to the variability of the classic wing irradiances. Therefore, we can construct a modified Mg II index using only the unsaturated IUE data that still captures the full chromospheric vari- ability. The chromospheric line cores (0.14 and 0.12 nm wide band- passes centered at 279.65 and 280.365 nm, in vacuo) and two bands in the photospheric wings of the lines (0.4 nm wide bands, edge- smoothed with cosine functions, centered at 279.20 and 280.70 nm) were integrated, and the ratio of total core to total wing fluxes was determined. Figure 1 shows the core to wing indices determined from each usable spectrum from 1978 to the end of 1995. The post- 1992 data permit us to revise the cycle period estimate downwards to ∼12.0 years, with more confidence than was previously possi- ble. This period was derived by fitting a simple sinusoid to the data using the genetic algorithm PIKAIA (Charbonneau 1995). The op- timal fit yields minima at 1980.9 and 1992.8, and a maximum at 1986.9. The fit suggests that the next maximum occurred in 1998.8, a minimum in 2004.8, and a future maximum predicted for 2010.8. The reduced χ2 of the fit was calculated using flux uncertainties for individual IUE observations of 7%, estimated from the varia- tion in the ratios of the two wing fluxes, which vary far less than this in the SOLSTICE solar data. This reduced χ2 has a minimum value of 0.77. The probability of such a value occurring at random is 24%, whereas a χ2 of 1.1 has a random probability of 76%. The 2 = 1.1 hypersurface contours suggests that the uncertainties are roughly ±1 yr for the phase and +3.0−1.7 yr for the period, making our new period estimate marginally consistent with the range quoted by Dravins et al. (1993). The χ2 contours are ovals because these uncertainties are correlated, allowing us to set the following formal limits on the epochs of maximum: 1986.1–1988.0, 1997.5–2002.4, and 2007.8–2017.8. These large uncertainties reinforce the need for Figure 1. Core to wing ratios of the summed Mg II h and k lines determined from IUE high dispersion observations of β Hyi (small points) and 3-month seasonal averages with the uncertainties used in the evaluation of χ2 shown as error bars (large points). The curve is an optimized simple sinusoid fit obtained using a genetic algorithm applied to the seasonally averaged data, intended only to estimate the period and phase of the stellar activity cycle, which is listed in the legend. The IUE index is shown on the left, while the corresponding NOAA index is shown on the right. an activity cycle monitoring program specifically for the southern hemisphere. To compare the IUE Mg II index measurements to the National Oceanic & Atmospheric Administration (NOAA) composite data of solar activity1, we must determine the appropriate scaling fac- tor. The SOLar-STellar Irradiance Comparison Experiment (SOL- STICE) on the SOlar Radiation and Climate Experiment (SORCE; McClintock et al. 2005) measures the solar irradiance every day, and has a resolution of 0.1 nm in this region. We convolved the IUE spectra with the SOLSTICE instrument function and then mea- sured the wings and cores of both solar and stellar data in exactly the same way. In particular, we used 279.15-279.35 nm as the blue wing, and 280.65-280.85 nm for the red wing. The emission cores were defined as 279.47-279.65 nm and 280.21-280.35 nm. We de- termined the relation between the SOLSTICE modified Mg II index and the NOAA long-term record using a standard linear regression method (see Snow et al. 2005; Viereck et al. 2004). Since the mod- ified IUE data has the same bandpass as the SOLSTICE data, the scaling factors derived from SOLSTICE solar data will also ap- ply to the stellar IUE data2. The Mg II index for β Hyi scaled to the NOAA composite data is shown on the right axis of Fig- ure 1. For the analysis in Section 3, we adopt a full amplitude of ∆iMgII = 0.015 in the NOAA index. 3 SCALING P-MODE SHIFTS FROM SOLAR DATA In general, we can evaluate activity related frequency shifts from the variational expression, ∆ν j = d3xK jS 2I jν j , (1) where 1 http://www.sec.noaa.gov/ftpdir/sbuv/NOAAMgII.dat 2 To transform between indices: NOAA = 0.211 + 0.0708 SOLSTICE; SOLSTICE = 0.297 + 1.11 IUE; NOAA = 0.232 + 0.079 IUE c© 0000 RAS, MNRAS 000, 000–000 Asteroseismic Signatures of Activity Cycles 3 I j = d3xρ|ξ|3 ≡ R5ρ̄Ĩ (2) is the mode inertia, j ≡ (n,ℓ,m), and we need to know both the source S(x) and the corresponding kernels K(x). The source must include the direct influence of the growing mean magnetic field, as well as its indirect effect on the convective velocities and tem- perature distribution. Separate kernels for these effects were cal- culated by Dziembowski & Goode (2004), but there is no theory available to calculate the combined source. Moreover, it is unclear whether the model of small-scale magnetic fields, adopted from Goldreich et al. (1991), is adequate. Therefore, we will attempt to formulate an extrapolation of the solar p-mode frequency shifts based on changes in the Mg II activity index measured for the Sun and for β Hyi in Section 2. For p-modes, the dominant terms in all of the kernels are pro- portional to |divξ|2. Thus, we write K j(x) = |divξ j| 2 = q j(D)Y ℓ , (3) where D is the depth beneath the photosphere. A model-dependent coefficient will be absorbed into the source, which we write in the S(x) = Sk(D)P2k(cosθ). (4) Solar data imply that S is strongly concentrated near the photo- sphere. Therefore data from all p-modes, regardless of their ℓ value, may be used to constrain S . We might also expect that the source normalization is correlated with the Mg II index. If we want to calculate ∆ν j according to Eq. (1), we need all terms of S up to k = ℓ. To assess the solar S , we have mea- surements of the centroid shifts and the even-a coefficients (see Dziembowski & Goode 2004). For the ℓ = 0 modes we only need to know the k = 0 term, and for this the centroid data are sufficient. Let us begin with this simple case. 3.1 Radial modes Theoretical arguments and the observed pattern of solar frequency changes suggest that the dominant source must be localized near the photosphere. Therefore, it seems reasonable to try to fit the mea- sured p-mode frequency shifts by adopting S0(D) = 1.5× 10 −11A0δ(D − Dc) µHz , (5) with adjustable parameters A0 and Dc. The numerical coefficient is arbitrary, and was chosen for future convenience. With Eqs. (3-5), we get from Eq. (1) ∆ν j = A0 Q j(Dc), (6) where R and M (as well as L below) are expressed in solar units, frequencies are expressed in µHz, and Q j = 1.5× 10 −11 q j ν j Ĩ j . (7) The solar values of A0 and Dc can be determined by fitting the centroid frequency shifts ∆ν j from SOHO MDI data for p-modes with various spherical degrees, ℓ. Since at n = 1 the approximation inherent in Eq. (3) is questionable, we use data only for the higher orders. For the Sun, we have A0,⊙(Dc) = w j∆ν j , (8) Figure 2. Determinations of A0,⊙ from SOHO MDI data with Dc fixed at 0.3 Mm (top panel), and using the optimal value of Dc for each set (middle panel), with the corresponding changes in the Mg II index from the NOAA composite data (bottom panel). where ∆ν j are the measured shifts and w j are the relative weights. The values of Q j are calculated from a solar model. The best value of Dc is that which minimizes the dispersion, σ(Dc) = A0,⊙ − . (9) It is also reasonable to assume that A0 should be proportional to the change in the Mg II activity index, ∆iMgII, and that Dc is pro- portional to the pressure scale height at the photosphere. Thus, we A0 = A0,⊙ ∆iMgII ∆iMgII,⊙ Dc ∝ Hp = Dc,⊙L 0.25 R . (11) To determine A0,⊙, we used SOHO MDI frequencies for all pn modes with n > 1, ℓ from 0 to ∼181, and ν between 2.5 and 4.2 mHz. The data were combined into 38 sets, typically cover- ing 0.2 years. We averaged the frequencies from the first 5 sets, corresponding to solar minimum (1996.3-1997.3), and subtracted them from the frequencies in subsequent sets to evaluate A0,⊙ using Eq. (8). The results are shown in Figure 2, where the points in the top panel were obtained at fixed Dc = 0.3 Mm, which is represen- tative of the highest activity period. In the middle panel, the value of Dc was determined separately for each set and the error bars represent the dispersion. In the bottom panel, we show the corre- sponding changes in the solar Mg II index calculated from NOAA composite data. A tight correlation between A0,⊙ and ∆iMgII,⊙ is clearly visible. Although the optimum value of Dc is weakly correlated with the activity level, the dispersion changes very little between Dc = 0.2 and 0.4 Mm, so we fixed the value of Dc,⊙ to 0.3 Mm. In Fig- c© 0000 RAS, MNRAS 000, 000–000 4 Metcalfe et al. Figure 3. The observed p-mode frequency shifts averaged from four sets of data obtained near the solar maximum in 2002.0 (top panel), and the same shifts normalized by Q j (bottom panel) showing that most of the frequency and ℓ-dependence is included in our parametrization. Different symbols show the roughly equal number of modes with ℓ ≤ 30 (circles), 31 ≤ ℓ≤ 75 (squares), and 76 ≤ ℓ≤ 181 (triangles). ure 3, we show the quality of the fit to the observed frequency shifts for selected p-modes using Eq. (6) with the adopted value of Dc. The upper panel shows the frequency shifts averaged from four sets of data near the activity maximum in 2002.0, while the lower panel shows ∆ν j/Q j . Note that most of the frequency and ℓ-dependence appears to have been fit by our parametrization. The slight rise at frequencies below 3 mHz could be eliminated by al- lowing a spread of the kernel toward lower depths. However, since the signal is more significant at higher frequencies, we believe that adding a finite radial extent would be an unnecessary complication. There were two activity maxima during solar cycle 23. The first was centered near 2000.6 and the second at 2002.0. The av- erage values of (A0,⊙,∆iMgII,⊙) are (0.3116, 0.0135) for five data sets around the first maximum and (0.3669, 0.0178) for four data sets around the second maximum. For future applications, we adopt A0,⊙/∆iMgII,⊙ = 22. With this specification, we get from Eqs. (6) and (10) ∆ν j = 3.3× 10 ∆iMgII q j(Dc) ν j Ĩ j , (12) where Dc = 0.3L 0.25 R mM (13) and again frequencies are expressed in µHz, while R, M, and L are in solar units. This is our expression for predicting the radial p-mode frequency shifts on the basis of changes in the NOAA composite Mg II index. In Table 1, we list the frequency shifts (∆ν j) calculated from Eq. (12) for the radial modes of β Hyi observed by Bedding et al. (2007), adopting ∆iMgII = 0.015. The mode parameters q j and Ĩ j were calculated from a model of β Hyi generated using the Aarhus STellar Evolution Code (ASTEC; Christensen-Dalsgaard 1982). Table 1. Predicted radial p-mode frequency shifts between activity max- imum and minimum for β Hyi, calculated with Eq. (12) and adopting ∆iMgII = 0.015. Frequencies are from Table 1 of Bedding et al. (2007). n Frequency (µHz) ∆ν j (µHz) 13 833.72 0.061 14 889.87 0.091 15 945.64 0.116 16 1004.21 0.139 17 1062.06 0.168 18 1118.93 0.199 19 1176.48 0.234 3.2 Non-radial modes Now from Eqs. (1) and (4). we have ∆νnlm = Skκk,lm 2Inlνnl , (14) where, κk,lm = dθdφ|Y mℓ | 2P2k(cosθ) sinθ. (15) As in Eq. (5), we can adopt Sk(D) = 1.5× 10 −11Akδ(D − Dc,k) µHz . (16) For k > 0, the solar amplitudes Ak and effective depths, Dc,k can be determined by fitting measurements of shifts in the a2k co- efficients. The relation is ∆a2k,ℓm = AkZk,ℓQk,nℓ(Dc,k), (17) where Zk,ℓ = (−1) k (2k − 1)!! (2ℓ+ 1)!! (2ℓ+ 2k + 1)!! (ℓ− 1)! (ℓ− k)! (cf. Dziembowski & Goode 2004, their Eq. 2), and Qk,nℓ = 1.5× 10 −11 qnℓ(Dc,k) ν j Ĩnℓ (compare to our Eq. 7). The prediction of frequency shifts for non-radial modes re- quires an additional assumption of the same scaling for all required Ak amplitudes, which amounts to assuming the same Butterfly di- agram as observed on the Sun. Moreover, since the shifts depend on |m| and multiplets are not expected to be resolved, we need to adopt the inclination angle (i) to correctly weight the contributions from all of the components. Since we do not know i for β Hyi, we restrict our numerical predictions to the radial modes. 4 ASTEROSEISMIC OBSERVATIONS The detection of solar-like oscillations in β Hyi was first reported by Bedding et al. (2001), and later confirmed by Carrier et al. (2001). These two detections of excess power were based on data obtained during a dual-site campaign organized in June 2000 using the 3.9-m Anglo-Australian Telescope (AAT) at Siding Spring Ob- servatory and the 1.2-m Swiss telescope at the European Southern Observatory (ESO) in Chile. Both sets of observations measured a large frequency separation between 56-58 µHz, but neither was sufficient for unambiguous identification of individual oscillation modes. c© 0000 RAS, MNRAS 000, 000–000 Asteroseismic Signatures of Activity Cycles 5 Nearly 30 individual modes in β Hyi with ℓ=0-2 were detected during a second dual-site campaign organized in September 2005, and reported by Bedding et al. (2007). The authors also reanalyzed the combined 2000 observations using an improved extraction al- gorithm for the AAT data, allowing them to identify some of the same oscillation modes at this earlier epoch. Motivated by the first tentative detection of a systematic frequency offset between two asteroseismic data sets for α Cen A (0.6±0.3 µHz; Fletcher et al. 2006), they compared the two epochs of observation for β Hyi and found the 2005 frequencies to be systematically lower than those in 2000 by 0.1±0.4 µHz, consistent with zero but also with the mean value in Table 1. A comparison of the individual modes from these two data sets (T. Bedding, private communication) allows a further test of our predictions. Of the 14 modes that were detected with S/N> 4 in both 2000 and 2005, only one was known to be a radial (ℓ = 0) mode, while four had ℓ = 1, three had ℓ = 2, two were mixed modes, and four had no certain identification. Without a known inclination, we can only calculate the shifts for radial modes, but the magnitude of the shift is largest at high frequencies (see Table 1). Fortunately, the radial mode that is common to both data sets (ℓ = 0, n = 18) has a frequency above the peak in the envelope of power, improving our chances of measuring a shift. The best estimate of the mode fre- quency from each data set comes from the noise-optimized power spectrum, since this maximizes the S/N of the observed peaks. The noise-optimized frequency for the ℓ = 0, n = 18 mode was 1119.06 and 1118.89 µHz in the 2000 and 2005 data sets, respectively. Considering the quoted uncertainty for this mode from Table 1 of Bedding et al. (2007), the frequency was 0.17±0.62 µHz lower in 2005 than in 2000, again consistent with zero but similar to the predicted shift for this mode in Table 1. 5 DISCUSSION Our reanalysis of archival IUE spectra for β Hyi allows us to test our predictions of the relationship between the stellar activity cy- cle and the systematic frequency shift measured from multi-epoch asteroseismic observations. The optimal period and phase of the ac- tivity cycle from Section 2 suggest that β Hyi was near magnetic minimum (2004.8) during the 2005 observations (2005.7), while it was descending from magnetic maximum (1998.8) during the 2000 campaign (2000.5). The systematic frequency shift of 0.1±0.4 µHz reported by Bedding et al. (2007) between these two epochs, and the observed shift of 0.17±0.62 µHz in the only radial mode (ℓ = 0, n = 18) common to both data sets are not statistically significant. They are both nominally in the direction predicted by our analysis of the activity cycle (lower frequencies during magnetic minimum) and they have approximately the expected magnitude (cf. Table 1), but the formal uncertainties on the period and phase of the activity cycle do not permit a definitive test. Future asteroseismic observations of β Hyi would sample the largest possible frequency shift relative to the 2005 data if timed to coincide with the magnetic maximum predicted for 2010.8+7−3 . Long-term monitoring of the stellar activity cycles of this and other southern asteroseismic targets (e.g. α Cen A/B, µ Ara, ν Ind), which are not included in the Mt. Wilson sample, would allow further tests of our predictions. For asteroseismic targets that have known activity cycles from long-term Ca II H and K measurements (e.g. ǫ Eri, Procyon), it would be straightforward to calibrate our predictions to this index from comparable solar observations. While our current analysis involves a simple scaling from so- lar data, future observations may allow us to refine magnetic dy- namo models by looking for deviations from this scaling relation and attempting to rectify the discrepancies. By requiring the models to reproduce the observed activity cycle periods and amplitudes— along with the resulting p-mode shifts and their frequency depen- dence for a variety of solar-type stars at various stages in their evolution—we can gradually provide a broader context for our un- derstanding of the dynamo operating in our own Sun. ACKNOWLEDGMENTS We would like to thank D. Salabert for inspiring this work with an HAO colloquium on low-degree solar p-mode shifts in May 2005, Keith MacGregor and Margarida Cunha for thoughtful discussions, and the Copernicus Astronomical Centre for fostering this collab- oration during a sponsored visit in September 2006. We also thank the SOHO/MDI team, and especially Jesper Schou for easy access to the solar frequency data, Tim Bedding for providing frequency data for β Hyi, and Jørgen Christensen-Dalsgaard for the use of his stellar evolution code. This work was supported in part by an NSF Astronomy & Astrophysics Fellowship under award AST-0401441, by Polish MNiI grant No. 1 P03D 021 28, and by NASA contract NAS5-97045 at the University of Colorado. The National Center for Atmospheric Research is a federally funded research and devel- opment center sponsored by the U.S. National Science Foundation. REFERENCES Baglin, A., et al. 2006, ESA SP-624: Proceedings of SOHO 18/GONG 2006/HELAS I, Beyond the spherical Sun, 18 Bedding, T. R., et al. 2001, ApJ, 549, L105 Bedding, T. R., et al. 2007, ApJ, in press (astro-ph/0703747) Carrier, F., et al. 2001, A&A, 378, 142 Chaplin, W. 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Phys., 230, 325 Verner, G. A., Chaplin, W. J., & Elsworth, Y. 2006, ApJ, 638, 440 Viereck, R. A., et al. 2004, Space Weather, 2, 5 Walker, G., et al. 2003, PASP, 115, 1023 Walker, G., et al. 2007, ApJ, in press Wilson, O. C. 1978, ApJ, 226, 379 c© 0000 RAS, MNRAS 000, 000–000 http://arxiv.org/abs/astro-ph/0703747 http://arxiv.org/abs/astro-ph/0701323 Introduction Archival IUE spectra Scaling p-mode shifts from solar data Radial modes Non-radial modes Asteroseismic Observations Discussion

Observations of stellar activity cycles provide an opportunity to study magnetic dynamos under many different physical conditions. Space-based asteroseismology missions will soon yield useful constraints on the interior conditions that nurture such magnetic cycles, and will be sensitive enough to detect shifts in the oscillation frequencies due to the magnetic variations. We derive a method for predicting these shifts from changes in the Mg II activity index by scaling from solar data. We demonstrate this technique on the solar-type subgiant beta Hyi, using archival International Ultraviolet Explorer spectra and two epochs of ground-based asteroseismic observations. We find qualitative evidence of the expected frequency shifts and predict the optimal timing for future asteroseismic observations of this star.

Introduction Archival IUE spectra Scaling p-mode shifts from solar data Radial modes Non-radial modes Asteroseismic Observations Discussion

Diffractive parton distributions from the analysis with higher twist Krzysztof Golec-Biernat(a,b)∗ and Agnieszka Luszczak(b)† aInstitute of Nuclear Physics Polish Academy of Sciences, Cracow, Poland bInstitute of Physics, University of Rzeszów, Rzeszów, Poland (October 29, 2018) Abstract Diffractive parton distributions of the proton are determined from fits to diffractive data from HERA. In addition to the twist–2 contribution, the twist–4 contribution from longi- tudinally polarised virtual photons is considered, which is important in the region of small diffractive masses. A new prediction for the longitudinal diffractive structure function is presented which differs significantly from that obtained in the pure twist–2 analyses. e-mail: golec@ifj.edu.pl e-mail: agnieszka.luszczak@ifj.edu.pl http://arxiv.org/abs/0704.1608v3 1 Introduction The diffractive deep inelastic scattering (DDIS) at HERA provide a very interesting example of the interplay between hard and soft aspects of QCD interactions. On one side, the virtuality of the photon probe is large (Q2 ≫ Λ2QCD), while on the other side, the scattered proton remains almost intact, loosing only a small fraction of its initial momentum. Its transverse momentum with respect to the photon-proton collision axis is also small. In addition to the scattered incident particles, a diffractive system forms which is well separated in rapidity from the scattered proton. The most important observation made at HERA is that diffractive processes in DIS are not rare, quite the contrary, they constitute up to 15% of deep inelastic events. What’s more, the ratio of the diffractive and inclusive cross sections is constant as a function of energy of the γ∗p system or as a function of the photon virtuality. The latter fact reflects the logarithmic dependence on Q2 of diffractive structure functions in the Bjorken limit. In the t-channel picture, the diffractive interactions can be viewed as a vacuum quantum number exchange between the diffractive system and the proton. In old days of Regge phe- nomenology such a mechanism of interactions was termed a pomeron. With the advent of quantum chromodynamic we gain a new way of understanding the pomeron by modelling it with the help of gluon exchanges projected onto the color singlet state. In the lowest approxi- mation, the pomeron is a two gluon exchange which is independent of energy. By considering radiative corrections to this process in the high energy limit, the famous BFKL pomeron [1–4] was constructed with a strong, power-like dependence on energy. This dependence ultimately violates unitarity which means that exchanges with more gluons have to be considered. A sys- tematic program to sum exchanges with gluon number changing vertices was formulated in [5,6] and developed in [7–10]. Other, somewhat more intuitive formulation, called Color Glass Con- densate [11–14], is based on the idea of parton saturation [15] in which deep inelastic scattering occurs on a dense gluonic system in the proton. In these approaches unitarization is supposed to change the asymptotic energy behaviour of the cross sections involving the pomeron from power-like to logarithmic. DDIS is particularly sensitive to the pomeron energy behaviour since diffractive scattering amplitudes are squared in diffractive cross sections. Thus, unitarization effects play more im- portant role than for the total cross section which is proportional to the imaginary part of the scattering amplitude. This observation is a basis of a successful description of the first diffractive data from HERA in which the diffractive system was formed by the quark-antiquark (qq) and quark-antiquark-gluon (qqg) systems. They can be viewed in the space of Fourier transformed transverse momenta as color dipoles [16,17]. In the approach we follow in the forthcoming pre- sentation, the pomeron interaction is modelled by a two-gluon exchange which is subsequently substituted by the effective dipole–proton cross section fitted to inclusive DIS data [18, 19]. In this way unitary is achieved. In an alternative approach to DDIS, the diffractive structure functions are defined in terms of diffractive parton distributions (DPD). They are evolved in Q2 with the help of the Dokshitzer- Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations [20–22]. Thus in the Bjorken limit, the diffractive structure functions depend logarithmically on Q2, i.e. they provide the twist–2 de- scription of DDIS. The theoretical justification of this approach is provided by the collinear fac- torisation theorem which is valid for hard diffractive scattering in ep collisions [23–27]. However, collinear factorisation fails in hadron–hadron scattering due to nonfactorizable soft interactions between incident hadrons [28,29]. Thus, unlike inclusive parton distributions, the DPD are not Figure 1: Kinematic variables relevant for diffractive DIS. universal objects and in general can only be used for diffractive processes in the ep deep inelas- tic scattering. Nevertheless, the scale of nonuniversality can be estimated by applying them to hadronic reactions. The relation between the color dipole approach with the qq and qqg diffractive components and the DGLAP based description was studied in detail in [30]. In short, after extracting the twist–2 part, the dipole approach provides Q2-independent quark and gluon DPD. In addition, the qqg component, which was computed assuming strong ordering between transverse momenta of the gluon and the qq pair, gives the first step in the Q2-evolution of the gluon distribution. The twist–2 approach, which is based on the DGLAP evolution equations, extends the two component dipole picture by taking into account more complicated diffractive final state. In the performed up till now twist-2 analyses, the diffractive parton distributions are determined through fits to diffractive data from HERA [31] We will follow this approach with an important modification. The dipole approach teaches us one important lesson concerning the seemingly subleading twist–4 contribution, given by the qq pair from longitudinally polarised virtual photons (Lqq). Formally, it is is suppressed by a power of 1/Q2 with respect to the leading twist–2 transverse contribution. However, the perturbative QCD calculation shows that for small diffractive masses, M2 ≪ Q2, the longitudinal contribution dominates over the twist–2 one which tends to zero in this limit. The effect of the Lqq component is particularly important for the longitudinal diffractive structure function FDL which is supposed to be determined from the high luminosity run data at HERA. Thus, we claim that it is absolutely necessary to consider the twist–4 contribution in the determination of the diffractive parton distributions. The analysis which we present confirms its relevance for the prediction for FDL , which differs significantly from that based on the pure DGLAP analysis. This is the main result of our paper. The paper is organised as follows. In Section 2 we provide basic formulae for the kinematical variables and quantities measured in diffractive deep inelastic scattering. We also describe the three contributions which we include in the description of the diffractive structure functions, i.e. the twist–2, twist–4 and Regge contributions. In Section 3 we describe performed fits while in Section 4 we present their impact on the determination of the diffractive parton distributions and diffractive structure functions. We finish with conclusions and outlook. 2 Basic formulae We consider diffractive deep inelastic scattering: ep → e′p′X, shown schematically in Fig. 1. After averaging over the azimuthal angle of the scattered proton, the four-fold differential cross section is given in terms of the diffractive structure functions FD2 and F dβ dQ2 dxIP dt 2πα2em 1 + (1− y)2 FD2 − 1 + (1− y)2 where y = Q2/(xBs) and s is the ep centre-of-mass energy squared. The expression in the curly bracket is called reduced cross section: σDr = F 1 + (1− y)2 FDL . (2) Both structure functions depend on four kinematic variables (β,Q2, xIP , t), defined as follows xIP = Q2 +M2 − t Q2 +W 2 , β = Q2 +M2 − t , (3) where −Q2 is virtuality of the photon, t = (p − p′)2 < 0 is the square of four-momentum transferred into the diffractive system, M is invariant diffractive mass and W is invariant energy of the γ∗p system. The Bjorken variable xB = xIP β. For most of the diffractive events |t| is much smaller then other scales, thus it can be neglected in eqs. (3). The diffractive structure functions are measured in a limited range of t, thus the integrated structure functions are defined 2,L (β,Q 2, xIP ) = ∫ tmax dt FD2,L(β,Q 2, xIP , t) , (4) The integrated reduced cross section σ r is defined in a similar way. 2.1 Twist–2 contribution In the QCD approach based on collinear factorisation, the diffractive structure functions are decomposed into the leading and higher twist contributions FD2,L = F D(tw2) 2,L + F D(tw4) 2,L + . . . . (5) The twist–2 part is given in terms of the diffractive parton distributions through the standard collinear factorisation formulae [23,32–34]. In the next-to-leading logarithmic approximation we D(tw2) 2 (x,Q 2, xIP , t) = SD + CS2 ⊗ SD +C 2 ⊗GD D(tw2) L (x,Q 2, xIP , t) = CSL ⊗ SD + C L ⊗GD where αs is the strong coupling constant and C 2,L are coefficients functions known from inclusive DIS [35,36]. The integral convolution is performed for the longitudinal momentum fraction (C ⊗ F )(β) = dz C (β/z)F (z) . (8) Notice that in the leading order, when terms proportional to αs are neglected, the longitudinal structure function F D(tw2) L = 0. The functions SD and GD are given by diffractive quark and gluon distributions, q D and gD: e2f β D(β,Q 2, xIP , t) + q D(β,Q 2, xIP , t) GD = βgD(β,Q 2, xIP , t) (10) Note that β = x/xIP plays the role of the Bjorken variable in DDIS. In the infinite momentum frame, the DPD are interpreted as conditional probabilities to find a parton with the momentum fraction x = βxIP in a proton under the condition that the incoming proton stays intact losing a small fraction xIP of its momentum. A formal definition of the diffractive parton distributions based on the quark and gluon twist-2 operators is given in [23,25]. The DPD are evolved in Q2 by the DGLAP evolution equations [37] for which the variables (xIP , t) play the role of external parameters. In this analysis we assume Regge factorisation for these variables: D(β,Q 2, xIP , t) = fIP (xIP , t) q IP (β,Q 2) (11) gD(β,Q 2, xIP , t) = fIP (xIP , t) gIP (β,Q 2) . (12) For convenience, the functions q IP (β,Q 2) and gIP (β,Q 2) are called pomeron parton distributions. The motivation for such a factorisation is a model of diffractive interactions with a pomeron exchange [38]. In this model fIP is the pomeron flux fIP (xIP , t) = F 2IP (t) 1−2αIP (t) IP , (13) where αIP (t) = αIP (0) + α IP t is the pomeron Regge trajectory and the formfactor F 2IP (t) = F IP (0) e −BD |t| (14) describes the pomeron coupling to the proton. We set F 2IP (0) = 54.4 GeV −2 [34], BD = 5.5 GeV−2 and α′IP = 0.06 GeV −2 [31], while the pomeron intercept αIP (0) is fitted to data. The pomeron quark distributions are flavour independent, thus they are given by one function, a singlet quark distribution ΣIP : IP (β,Q 2) = q IP (β,Q ΣIP (β,Q 2) (15) where Nf is a number of active flavours. The question about Regge factorisation is an issue which should be tested experimentally. In our approach, the pomeron is a model of diffractive interactions which only provides energy dependence through the xIP -dependent pomeron flux. Its normalisation is only a useful convention since the normalisations of the pomeron parton distributions in eqs. (11) are fitted to data. 2.2 Twist-2 charm contribution We describe the charm quark diffractive production using twist-2 formulae for the cc pair gener- ation from a gluon. These are formulae analogous to the inclusive case in which the diffractive Figure 2: The qq̄ and qq̄g components of the diffractive system in the dipole approach. gluon distribution gD is substituted for the inclusive one [39]: D(cc) 2,L (β,Q 2, xIP , t) = 2β e gD(z, µ2c , xIP , t) , (16) where a = 1 + 4m2c/Q 2 and the factorisation scale µ2c = 4m c with the charm quark mass mc = 1.4 GeV. The coefficient functions read C2(z, r) = z2 + (1− z)2 + 4z(1− 3z)r − 8z2r2 1 + α α {−1 + 8z(1 − z)− 4z(1 − z)r} (17) CL(z, r) = −4z 2r ln 1 + α + 2αz(1 − z) (18) with α = 1− 4rz/(1 − z). The cc pair can only be produced if invariant mass of the diffractive system M2 fulfils the following condition M2 = Q2 > 4m2c . (19) 2.3 Twist–4 contribution The computation of the twist–4 contribution, proportional to 1/Q2, is a nontrivial task and one could be tempted to assume that this contribution is suppressed at large Q2 as in inclusive DIS. However, by analysing diffractive final states in the dipole approach it was found that for diffractive mass M2 ≪ Q2 (β → 1), the twist–4 contribution dominates over the vanishing twist–2 one [19,40,41]. This observation is made on the basis of the perturbative QCD calculations in which the diffractive state is formed by the qq and qqg systems interacting with a proton through a colorless gluonic exchange which is a model of the pomeron interactions in QCD. In the simplest case, two gluons projected onto the color singlet state are exchanged, see Fig. 2. The computed amplitudes do not depend on energy in such a case which problem can be cured in a more sophisticated approach by modelling the dipole-proton cross section which fulfils unitarity conditions [19]. Independ of the details of the pomeron description, the diffractive mass (or β) dependence is a genuine prediction of pQCD calculations. It appears that the leading in Q2 behaviour components, qq and qqg from transverse virtual photons, vanish for β → 1. This is not the case for the qq production from longitudinal photons (Lqq) which is formally suppressed by 1/Q2 with respect to the leading components. Thus, this particular β-dependence makes the Lqq contribution dominant for β → 1, see Fig. 3. The presence of the Lqq component has important consequence for the longitudinal diffractive structure function which is supposed to be determined from the HERA data. The formula given below is an important element in the description of FDL in the region of large β: FDLqq̄ = 16π4xIP e−BD |t| (1− β)4 2(1−β) k2/Q2 φ20(k, xIP ) (20) where the function φ0(k, xIP ) is given in terms of the dipole cross section σ̂(xIP , r) and the Bessel functions K0 and J0: φ0(k, xIP ) = k dr r K0 J0(kr) σ̂(xIP , r) . (21) Strictly speaking, eq. (20) contains all inverse powers of Q2 but the part proportional to 1/Q2 (called twist–4) dominates. The dipole-proton cross section describes the interaction of a color dipole, formed by the qq or qqg systems, with a proton. Following [18] we choose σ̂(xIP , r) = σ0 {1− exp (−r 2Q2s/4)} (22) where Q2s = (xIP /x0) −λ GeV2 is a saturation scale which provides the energy dependence of the twist–4 contribution. The parameters σ0 = 29 mb, x0 = 4 · 10 −5 and λ = 0.28 are taken from [18] (Fit 2 with charm). This form of the dipole cross section provides successful description of the first HERA data on both inclusive and diffractive structure functions [18,19]. A different parametrisation of σ̂, without the saturation scale, is also given in [42–44]. We checked that a very similar description of FDLqq̄ was found in a recent analysis [45] based on the Color Glass Condensate parameterisation of the dipole scattering amplitude [46]. The relation between the dipole approach with three diffractive components and the DGLAP approach with diffractive parton distributions was analysed at length in [30]. Summarising this relation, the twist–2 part of the qq component gives a diffractive quark distribution. The twist-2 part of the qqg component forms a first step of the DGLAP evolution which starts from a given gluon distribution. Both diffractive parton distributions do not depend on Q2, thus they may serve as initial conditions for the DGLAP equations at the scale which is not determined. From this perspective, the DGLAP approach offers a description of more complicated diffractive state with any number of partons ordered in transverse momenta. However, the pQCD calculations tell us that the twist–2 analysis of diffractive data should include the twist–4 contribution since it cannot be neglected at large β. This is the strategy which we follow in our analysis. We also borrow from the dipole approach a general form in β of the initial quark distribution which vanishes at the endpoints β = 0, 1 (see eq. (31) in which Aq and Cq are positive). A very important aspect of Regge factorisation (11) can also be motivated by the dipole approach. It is a consequence of geometric scaling of the dipole cross section (22) [30,47]. 2.4 Reggeon contribution The diffractive data from the H1 collaboration for higher values of xIP hints towards a contribu- tion which decreases with energy. This effect can be described by reggeon exchanges in addition to the rising with energy pomeron exchange. Following [48, 49], we consider the dominant isoscalar (f2, ω) reggeon exchanges which lead to the following contribution to F fR(xIP , t)FR(β,Q 2) . (23) This contribution breaks Regge factorisation of the diffractive structure function, however, its presence is necessary for xIP > 0.01 [50]. The reggeon flux fR is given by the formula analogous to eq. (13) fR(xIP , t) = F 2R(0) e−|t|/Λ R |ηR(t)| 1−2αR(t) IP , (24) where αR(t) = 0.5475 + 1 · t is the reggeon trajectory. From the Regge phenomenology of hadronic reactions ΛR = 0.65GeV and the reggeon–proton couplings are given by [49]: F (0) = 194GeV−2 and F 2ω(0) = 52GeV −2. The functions |ηR(t)| 2 = 4cos2[παR(t)/2] , |ηR(t)| 2 = 4 sin2[παR(t)/2] (25) are signature factors for even (f2) and odd (ω) reggeons, respectively. We could also consider isovector reggeons (a2, ρ) but their couplings to the proton are much smaller and we neglect them. Finally, the reggeon structure function FR is given by [49] FR(β) = AR β −0.08 (1− β)2 , (26) where the normalisation AR is a fitted parameter. Thus, in the first approximation, we neglect the Q2-dependence of the reggeon contribution. 3 Fit details Collab. No. points Data |t|-range Q2-range β-range H1 [50] 72 LP [0.08, 0.5] [2 , 50] [0.02 , 0.7] ZEUS [51] 80 LP [0.075 , 0.35] [2 , 100] [0.007 , 0.48] H1 [31] 461 MY < 1.6 [|tmin| , 1] [3.5 , 1600] [0.01 , 0.9] ZEUS [52] 198 MY < 2.3 [|tmin| ,∞] [2.2 , 80] [0.003 , 0.975] Table 1: Kinematic regions of diffractive data from HERA. LP means leading proton data and MY is invariant mass of a dissociated proton. Dimensionfull quantities are in units of 1 GeV. In our analysis we use diffractive data from the H1 [31,50] and ZEUS [51,52] collaborations. In Table 1 we show their kinematic limits in which they have been measured. The minimal value value of |t| is given by |tmin| ≃ 1− xIP m2p , (27) where mp is the proton mass. The leading proton data from H1, measured in the range given in Table 1, were corrected by the H1 collaboration to the range |tmin| < |t| < 1 GeV The ZEUS data are given for the diffractive structure function FD2 , thus we use in our analysis the following formulae FD2 = F D(tw2) 2 + F 2 + F Lqq̄ (28) FDL = F D(tw2) L + F Lqq̄ . (29) No Data Fit αIP (0) AR Aq Bq Cq Ag Bg Cg χ 1 H1 (LP) tw-2 1.098 0.29 1.75 1.49 0.5∗ 2.09 0.67 0.80 0.48 2 ZEUS (LP) tw-2 1.145 1.05 2.13 1.51 0.5∗ 10.0* 1.03 2.26 0.40 3 H1 tw-2 1.117 0.49 1.33 1.63 0.34 0.17 -0.16 -1.10 1.04 4 tw-(2+4) 1.119 0.48 1.62 1.98 0.59 0.04 -0.56 -1.68 1.17 5 ZEUS tw-2 1.093 0.0∗ 1.68 1.01 0.5∗ 0.49 -0.03 -0.40 1.35 6 tw-(2+4) 1.092 0.0∗ 1.20 0.85 0.57 0.07 -0.52 -1.48 1.82 Table 2: The fit parameters to H1 nd ZEUS data. The presence of twist–4 in the fits is marked by tw-(2+4). The parameters with an asterisk are fixed in the fits. The longitudinal twist-4 contribution is present on the r.h.s. of eq. (28) since FD2 is the sum of the contributions from the transverse and longitudinal polarised virtual photon. The H1 data, however, are presented for the reduced cross section (2). Thus we substitute relations (28) and (29) in there and use σDr = D(tw2) 2 + F 1 + (1− y)2 D(tw2) 2(1− y) 1 + (1− y)2 FDLqq . (30) The expression in the curly brackets is the twist–2 contribution while the last term is the twist–4 one. Notice that the difference between FD2 and σ r is most important for y → 1. We fit the diffractive parton distributions at the initial scale Q20 = 1.5 GeV 2, assuming the Regge factorised form (11) with the following pomeron parton distributions [31]: βΣIP (β) = Aq β Bq (1− β)Cq (31) βgIP (β) = Ag β Bg (1− β)Cg . (32) The six indicated parameters are fitted to data. We additionally multiplied both distributions by a factor exp{−a/(1 − β)} with a = 0.01 to secure their vanishing for β = 1. This factor is only important when Cq or Cg becomes negative in the fits. For the evolution, we use the next-to-leading order DGLAP equations with ΛQCD = 407 MeV for Nf = 3 flavours [53]. The pomeron flux in eq. (11) is integrated over t in the limits given in Table 1 which leads to the form fIP (xIP ) = F 2IP (0) e−B|tmin| − e−B|tmax| 1−2αIP (0) IP . (33) The shrinkage parameter B equals B = BD + 2α IP ln(1/xIP ) (34) with BD = 5.5GeV −2 and α′IP = 0.06GeV −2 [50]. In summary, we have eight fit parameters altogether: the pomeron intercept αIP (0), reggeon normalisation AR in eq. (26) and six parameters in eqs. (31,32) 4 Fit results The data sets from Table 1 were obtained in different kinematical regions, using different methods of their analysis. Thus, we decided to perform fits to each data set separately. The values of the fit parameter are shown in Table 2. The difference between them can be attributed to the scale of uncertainty of our analysis. In each case we preformed two fits: with and without the twist–4 formula added to the twist–2 contribution. 4.1 Leading proton data We started from fits to the leading proton data. The fit parameters in this case are displayed in the first two rows of Table 2. We only show the twist–2 fit results since they are not changed in fits with the twist–4 term. This happenes because the leading proton data comes from the region of β values where the twist–4 contribution is small (β ≤ 0.7 for H1 and β < 0.5 for ZEUS), see Fig. 3. The data with a dissociated proton (DP) which are measured in the region of large β influence most the value of the parameter Cg which controls the behaviour of the gluon distribution at β → 1. For the LP data Cg is positive and the gluon distribution is suppressed near β ≈ 1, while for the DP data Cg is negative and the gluon distribution is strongly enhanced. This shows that the data with β > 0.7 are crucial for the proper analysis. Without this kinematic region we lose important information about diffractive interactions. Thus, from now on we concentrate on the analysis of the DP data. 4.2 H1 data The fit parameters to the H1 data with a dissociate proton are given in the third and fourth rows of Table 2. We see that the fit quality is practically the same for both fits, with and without the twist–4 contribution. The presence of the reggeon term improves fit quality by 30 units of χ2 for 461 experimental points. A good quality of the fits is illustrated in Fig. 4 which also shows that the reduced cross sections (30) from the twist–2 (solid lines) and twist–(2+4) fits (dashed lines) are very close to each other. In Fig. 5 we show our results on the reduced cross section for the largest measured value of β = 0.9. In this region, the twist–4 contribution, shown by the dotted lines, cannot be neglected. We see that the curves from both the twist–2 (solid) and twist–(2+4) (dashed) fits describe data reasonable well. However, the curves with twist–4 have a steeper dependence on xIP (energy) than in the pure twist–2 analysis. This observation is by far more pronounced in the analysis of the ZEUS data performed for the structure function FD2 . The diffractive parton distributions from our fits are shown in Fig. 6 in terms of the pomeron parton distributions, βΣIP (β,Q 2) and βgIP (β,Q 2). Being independent of the pomeron flux, such a presentation allows for a direct comparison of the results from fits to different data sets. We see that the singlet quark distributions are quite similar while the gluon distributions are different. In the fit with twist–4, the gluon distribution is stronger peaked near β ≈ 1. This somewhat surprising result can be understood by looking at the logarithmic slope of FD2 for fixed values of β. From the LO DGLAP equations we have schematically: ∂ lnQ2 ∂ lnQ2 = Pqq ⊗ ΣIP + PqG ⊗GIP − ΣIP Pqq (35) where the negative term describes virtual corrections. For large β, the measured slope is negative which means that the virtual emission term must dominate over the real emission ones. The addition of the twist–4 contribution to FD2 , proportional to 1/Q 2, contributes a negative value to the slope which has to be compensated by a larger gluon distribution in order to describe the same data. In Fig. 7 we present our most important results. On the left panel, the FD2 structure function is shown from both fits, with and without the twist–4 contribution (shown by the dotted lines). We see no significant difference between these two results. However, the longitudinal structure function FDL differs significantly for the two fits (right panel) due to the twist–4 contribution. Let us emphasise that both sets of curves were found in the fits which well describe the existing data on σDr , including the large β region. Thus, an independent measurement of F L in this region would be an important test of the QCD mechanism of diffraction. 4.3 ZEUS data The results of same fits performed for the ZEUS data are shown in the last two rows of Table 2. This time the Regge contribution (26) is not necessary since fits give the reggeon normalisation AR ≈ 0. In general, the fit quality is worse than for the H1 data. As shown in Fig. 8, the biggest difference between the twist–2 and twist–(2+4) results occurs at large β values. This is analysed in detail in Fig. 9. We see that the presence of the twist–4 term in the fit (dashed lines) improves the agreement with the data in this region. In particular, a steep dependence of FD2 on xIP is better reproduced by the twist–(2+4) fit then by the twist–2 one (solid lines). This dependence is to large extend driven by the twist–4 contribution (dotted lines). The behaviour of the diffractive parton distributions and structure functions, shown in Figs. 10 and 11, respectively, is very similar to that found for the H1 data. The gluon dis- tribution from the fit with twist–4 is stronger peaked near β ≈ 1 and the longitudinal structure functions in the large β region is dominated by the twist–4 contribution. We summarise the effect of the twist–4 contribution in Fig. 12 showing the predictions for the longitudinal diffractive structure function FDL . Ignoring this contribution, we find the two solid curves coming from the pure twist–2 analysis of the H1 (upper) and ZEUS (lower) data. With twist–4, the dashed curves are found, the upper one from the H1 data and the lower one from the ZEUS data. There is a significant difference between these two predictions in the region of large β. We believe that the effect of the twist–4 contribution will be confirmed by the forthcoming analysis of the HERA data. 5 Conclusions We performed fits of the diffractive parton distributions to new diffractive data from the H1 and ZEUS collaborations at HERA. In addition to the standard twist–2 formulae, we also considered the twist–4 contribution which dominates in the region of large β. This contribution comes from the diffractive production of the qq pair by the longitudinally polarised virtual photons. The effect of the twist–4 contribution on the diffractive parton distributions and structure functions was carefully examined. The twist–4 contribution leads to the gluon distribution which is peaked stronger at β ≈ 1 than in the case without twist–4. The main result of our analysis is a new prediction for the longitudinal diffractive struc- ture function FDL . The twist–4 term significantly enhances F L in the region of large β. 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The twist–4 contribution Lqq̄ is indicated by the yellow band. Old ZEUS data points are shown. H1 DATA β=0.01 β=0.01 β=0.04β=0.04 β=0.1β=0.1 β=0.2β=0.2 β=0.4β=0.4 β=0.65β=0.65 β=0.9 Q β=0.9 Q 6.56.5 8.58.5 200200 Figure 4: Reduced cross section σ r for H1 data as a function of xIP . Solid lines: twist–2 fit, dashed lines: twist–(2+4) fit. H1 DATA (β=0.9) Q2= 3.5 GeV2 dashed: tw-(2+4) fit solid: tw-2 fit Q2= 5 GeV2 dotted: tw-4 contribution Q2= 6.5 GeV2 Q2= 8.5 GeV2 Figure 5: Reduced cross section σ r for H1 data at β = 0.9 for four values of Q 2 against fit curves. DPD (H1) solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 6: Pomeron parton distributions: singlet βΣIP (β,Q 2) (left) and gluon βgIP (β,Q 2) (right) from H1 data. DSF (H1) solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit dotted: tw-4 contribution dotted: tw-4 contribution dotted: tw-4 contribution 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 7: Diffractive structure functions F 2 (left) and F L (right) from fits to H1 data for xIP = 10 −3. The band shows the effect of twist–4 on the predictions for F ZEUS DATA β=0.652 β=0.652 β=0.231β=0.231 β=0.07β=0.07 β=0.022β=0.022 β=0.007β=0.007 β=0.003 β=0.003 β=0.735β=0.735 β=0.308β=0.308 β=0.1β=0.1 β=0.032β=0.032 β=0.010β=0.010 β=0.004 β=0.004 β=0.807β=0.807 β=0.4β=0.4 β=0.143β=0.143 β=0.047β=0.047 β=0.015β=0.015 β=0.007 β=0.007 β=0.848β=0.848 β=0.471β=0.471 β=0.182β=0.182 β=0.062β=0.062 β=0.020β=0.020 β=0.009 β=0.009 β=0.907β=0.907 β=0.61β=0.61 β=0.28β=0.28 β=0.104β=0.104 β=0.034β=0.034 β=0.015 β=0.015 β=0.949β=0.949 β=0.75β=0.75 β=0.43β=0.43 β=0.182β=0.182 β=0.063β=0.063 β=0.029 β=0.029 β=0.975β=0.975 β=0.86β=0.86 β=0.604β=0.604 β=0.313β=0.313 β=0.121β=0.121 Figure 8: Diffractive structure function F 2 as a function xIP for ZEUS data. Solid lines: twist–2 fit, dashed lines: twist–(2+4) fit. ZEUS DATA dotted: tw-4 contribution β = 0.86 Q2= 55 GeV2 solid: tw-2 fit dashed: tw-(2+4) fit β = 0.91 Q2= 14 GeV2 β = 0.95 Q2= 27 GeV2 β = 0.975 Q2= 55 GeV2 Figure 9: Diffractive structure function F 2 for ZEUS data at large values of β against fit curves. DPD (ZEUS) solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 10: Pomeron parton distributions βΣIP (β,Q 2) (left) and βgIP (β,Q 2) (right) from fits to ZEUS data. DSF (ZEUS) solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit dotted: tw-4 contribution dotted: tw-4 contribution dotted: tw-4 contribution 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 11: Diffractive structure functions F 2 (left) and F L (right) from fits to ZEUS data for xIP = 10 −3. The band shows the effect of twist–4 on the predictions for F Diffractive FL twist-(2+4) twist-2 Q2=10 GeV2 xP=10 twist-(2+4) twist-2 Q2=10 GeV2 xP=10 Q2=10 GeV2 xP=10 twist-(2+4) twist-2 Q2=10 GeV2 xP=10 twist-(2+4) twist-2 Q2=10 GeV2 xP=10 0.005 0.015 0.025 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 12: Predictions for F L for xIP = 10 −3 and Q2 = 10 GeV2 from the twist–(2+4) fits to the H1 (upper dashed line) and ZEUS (lower dashed line) data. The solid lines show predictions from pure twist–2 fits to the H1 (upper) and ZEUS (lower) data. Introduction Basic formulae Twist–2 contribution Twist-2 charm contribution Twist–4 contribution Reggeon contribution Fit details Fit results Leading proton data H1 data ZEUS data Conclusions

Diffractive parton distributions of the proton are determined from fits to diffractive data from HERA. In addition to the twist--2 contribution, the twist--4 contribution from longitudinally polarised virtual photons is considered, which is important in the region of small diffractive masses. A new prediction for the longitudinal diffractive structure function is presented which differs significantly from that obtained in the pure twist--2 analyses.

Introduction The diffractive deep inelastic scattering (DDIS) at HERA provide a very interesting example of the interplay between hard and soft aspects of QCD interactions. On one side, the virtuality of the photon probe is large (Q2 ≫ Λ2QCD), while on the other side, the scattered proton remains almost intact, loosing only a small fraction of its initial momentum. Its transverse momentum with respect to the photon-proton collision axis is also small. In addition to the scattered incident particles, a diffractive system forms which is well separated in rapidity from the scattered proton. The most important observation made at HERA is that diffractive processes in DIS are not rare, quite the contrary, they constitute up to 15% of deep inelastic events. What’s more, the ratio of the diffractive and inclusive cross sections is constant as a function of energy of the γ∗p system or as a function of the photon virtuality. The latter fact reflects the logarithmic dependence on Q2 of diffractive structure functions in the Bjorken limit. In the t-channel picture, the diffractive interactions can be viewed as a vacuum quantum number exchange between the diffractive system and the proton. In old days of Regge phe- nomenology such a mechanism of interactions was termed a pomeron. With the advent of quantum chromodynamic we gain a new way of understanding the pomeron by modelling it with the help of gluon exchanges projected onto the color singlet state. In the lowest approxi- mation, the pomeron is a two gluon exchange which is independent of energy. By considering radiative corrections to this process in the high energy limit, the famous BFKL pomeron [1–4] was constructed with a strong, power-like dependence on energy. This dependence ultimately violates unitarity which means that exchanges with more gluons have to be considered. A sys- tematic program to sum exchanges with gluon number changing vertices was formulated in [5,6] and developed in [7–10]. Other, somewhat more intuitive formulation, called Color Glass Con- densate [11–14], is based on the idea of parton saturation [15] in which deep inelastic scattering occurs on a dense gluonic system in the proton. In these approaches unitarization is supposed to change the asymptotic energy behaviour of the cross sections involving the pomeron from power-like to logarithmic. DDIS is particularly sensitive to the pomeron energy behaviour since diffractive scattering amplitudes are squared in diffractive cross sections. Thus, unitarization effects play more im- portant role than for the total cross section which is proportional to the imaginary part of the scattering amplitude. This observation is a basis of a successful description of the first diffractive data from HERA in which the diffractive system was formed by the quark-antiquark (qq) and quark-antiquark-gluon (qqg) systems. They can be viewed in the space of Fourier transformed transverse momenta as color dipoles [16,17]. In the approach we follow in the forthcoming pre- sentation, the pomeron interaction is modelled by a two-gluon exchange which is subsequently substituted by the effective dipole–proton cross section fitted to inclusive DIS data [18, 19]. In this way unitary is achieved. In an alternative approach to DDIS, the diffractive structure functions are defined in terms of diffractive parton distributions (DPD). They are evolved in Q2 with the help of the Dokshitzer- Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations [20–22]. Thus in the Bjorken limit, the diffractive structure functions depend logarithmically on Q2, i.e. they provide the twist–2 de- scription of DDIS. The theoretical justification of this approach is provided by the collinear fac- torisation theorem which is valid for hard diffractive scattering in ep collisions [23–27]. However, collinear factorisation fails in hadron–hadron scattering due to nonfactorizable soft interactions between incident hadrons [28,29]. Thus, unlike inclusive parton distributions, the DPD are not Figure 1: Kinematic variables relevant for diffractive DIS. universal objects and in general can only be used for diffractive processes in the ep deep inelas- tic scattering. Nevertheless, the scale of nonuniversality can be estimated by applying them to hadronic reactions. The relation between the color dipole approach with the qq and qqg diffractive components and the DGLAP based description was studied in detail in [30]. In short, after extracting the twist–2 part, the dipole approach provides Q2-independent quark and gluon DPD. In addition, the qqg component, which was computed assuming strong ordering between transverse momenta of the gluon and the qq pair, gives the first step in the Q2-evolution of the gluon distribution. The twist–2 approach, which is based on the DGLAP evolution equations, extends the two component dipole picture by taking into account more complicated diffractive final state. In the performed up till now twist-2 analyses, the diffractive parton distributions are determined through fits to diffractive data from HERA [31] We will follow this approach with an important modification. The dipole approach teaches us one important lesson concerning the seemingly subleading twist–4 contribution, given by the qq pair from longitudinally polarised virtual photons (Lqq). Formally, it is is suppressed by a power of 1/Q2 with respect to the leading twist–2 transverse contribution. However, the perturbative QCD calculation shows that for small diffractive masses, M2 ≪ Q2, the longitudinal contribution dominates over the twist–2 one which tends to zero in this limit. The effect of the Lqq component is particularly important for the longitudinal diffractive structure function FDL which is supposed to be determined from the high luminosity run data at HERA. Thus, we claim that it is absolutely necessary to consider the twist–4 contribution in the determination of the diffractive parton distributions. The analysis which we present confirms its relevance for the prediction for FDL , which differs significantly from that based on the pure DGLAP analysis. This is the main result of our paper. The paper is organised as follows. In Section 2 we provide basic formulae for the kinematical variables and quantities measured in diffractive deep inelastic scattering. We also describe the three contributions which we include in the description of the diffractive structure functions, i.e. the twist–2, twist–4 and Regge contributions. In Section 3 we describe performed fits while in Section 4 we present their impact on the determination of the diffractive parton distributions and diffractive structure functions. We finish with conclusions and outlook. 2 Basic formulae We consider diffractive deep inelastic scattering: ep → e′p′X, shown schematically in Fig. 1. After averaging over the azimuthal angle of the scattered proton, the four-fold differential cross section is given in terms of the diffractive structure functions FD2 and F dβ dQ2 dxIP dt 2πα2em 1 + (1− y)2 FD2 − 1 + (1− y)2 where y = Q2/(xBs) and s is the ep centre-of-mass energy squared. The expression in the curly bracket is called reduced cross section: σDr = F 1 + (1− y)2 FDL . (2) Both structure functions depend on four kinematic variables (β,Q2, xIP , t), defined as follows xIP = Q2 +M2 − t Q2 +W 2 , β = Q2 +M2 − t , (3) where −Q2 is virtuality of the photon, t = (p − p′)2 < 0 is the square of four-momentum transferred into the diffractive system, M is invariant diffractive mass and W is invariant energy of the γ∗p system. The Bjorken variable xB = xIP β. For most of the diffractive events |t| is much smaller then other scales, thus it can be neglected in eqs. (3). The diffractive structure functions are measured in a limited range of t, thus the integrated structure functions are defined 2,L (β,Q 2, xIP ) = ∫ tmax dt FD2,L(β,Q 2, xIP , t) , (4) The integrated reduced cross section σ r is defined in a similar way. 2.1 Twist–2 contribution In the QCD approach based on collinear factorisation, the diffractive structure functions are decomposed into the leading and higher twist contributions FD2,L = F D(tw2) 2,L + F D(tw4) 2,L + . . . . (5) The twist–2 part is given in terms of the diffractive parton distributions through the standard collinear factorisation formulae [23,32–34]. In the next-to-leading logarithmic approximation we D(tw2) 2 (x,Q 2, xIP , t) = SD + CS2 ⊗ SD +C 2 ⊗GD D(tw2) L (x,Q 2, xIP , t) = CSL ⊗ SD + C L ⊗GD where αs is the strong coupling constant and C 2,L are coefficients functions known from inclusive DIS [35,36]. The integral convolution is performed for the longitudinal momentum fraction (C ⊗ F )(β) = dz C (β/z)F (z) . (8) Notice that in the leading order, when terms proportional to αs are neglected, the longitudinal structure function F D(tw2) L = 0. The functions SD and GD are given by diffractive quark and gluon distributions, q D and gD: e2f β D(β,Q 2, xIP , t) + q D(β,Q 2, xIP , t) GD = βgD(β,Q 2, xIP , t) (10) Note that β = x/xIP plays the role of the Bjorken variable in DDIS. In the infinite momentum frame, the DPD are interpreted as conditional probabilities to find a parton with the momentum fraction x = βxIP in a proton under the condition that the incoming proton stays intact losing a small fraction xIP of its momentum. A formal definition of the diffractive parton distributions based on the quark and gluon twist-2 operators is given in [23,25]. The DPD are evolved in Q2 by the DGLAP evolution equations [37] for which the variables (xIP , t) play the role of external parameters. In this analysis we assume Regge factorisation for these variables: D(β,Q 2, xIP , t) = fIP (xIP , t) q IP (β,Q 2) (11) gD(β,Q 2, xIP , t) = fIP (xIP , t) gIP (β,Q 2) . (12) For convenience, the functions q IP (β,Q 2) and gIP (β,Q 2) are called pomeron parton distributions. The motivation for such a factorisation is a model of diffractive interactions with a pomeron exchange [38]. In this model fIP is the pomeron flux fIP (xIP , t) = F 2IP (t) 1−2αIP (t) IP , (13) where αIP (t) = αIP (0) + α IP t is the pomeron Regge trajectory and the formfactor F 2IP (t) = F IP (0) e −BD |t| (14) describes the pomeron coupling to the proton. We set F 2IP (0) = 54.4 GeV −2 [34], BD = 5.5 GeV−2 and α′IP = 0.06 GeV −2 [31], while the pomeron intercept αIP (0) is fitted to data. The pomeron quark distributions are flavour independent, thus they are given by one function, a singlet quark distribution ΣIP : IP (β,Q 2) = q IP (β,Q ΣIP (β,Q 2) (15) where Nf is a number of active flavours. The question about Regge factorisation is an issue which should be tested experimentally. In our approach, the pomeron is a model of diffractive interactions which only provides energy dependence through the xIP -dependent pomeron flux. Its normalisation is only a useful convention since the normalisations of the pomeron parton distributions in eqs. (11) are fitted to data. 2.2 Twist-2 charm contribution We describe the charm quark diffractive production using twist-2 formulae for the cc pair gener- ation from a gluon. These are formulae analogous to the inclusive case in which the diffractive Figure 2: The qq̄ and qq̄g components of the diffractive system in the dipole approach. gluon distribution gD is substituted for the inclusive one [39]: D(cc) 2,L (β,Q 2, xIP , t) = 2β e gD(z, µ2c , xIP , t) , (16) where a = 1 + 4m2c/Q 2 and the factorisation scale µ2c = 4m c with the charm quark mass mc = 1.4 GeV. The coefficient functions read C2(z, r) = z2 + (1− z)2 + 4z(1− 3z)r − 8z2r2 1 + α α {−1 + 8z(1 − z)− 4z(1 − z)r} (17) CL(z, r) = −4z 2r ln 1 + α + 2αz(1 − z) (18) with α = 1− 4rz/(1 − z). The cc pair can only be produced if invariant mass of the diffractive system M2 fulfils the following condition M2 = Q2 > 4m2c . (19) 2.3 Twist–4 contribution The computation of the twist–4 contribution, proportional to 1/Q2, is a nontrivial task and one could be tempted to assume that this contribution is suppressed at large Q2 as in inclusive DIS. However, by analysing diffractive final states in the dipole approach it was found that for diffractive mass M2 ≪ Q2 (β → 1), the twist–4 contribution dominates over the vanishing twist–2 one [19,40,41]. This observation is made on the basis of the perturbative QCD calculations in which the diffractive state is formed by the qq and qqg systems interacting with a proton through a colorless gluonic exchange which is a model of the pomeron interactions in QCD. In the simplest case, two gluons projected onto the color singlet state are exchanged, see Fig. 2. The computed amplitudes do not depend on energy in such a case which problem can be cured in a more sophisticated approach by modelling the dipole-proton cross section which fulfils unitarity conditions [19]. Independ of the details of the pomeron description, the diffractive mass (or β) dependence is a genuine prediction of pQCD calculations. It appears that the leading in Q2 behaviour components, qq and qqg from transverse virtual photons, vanish for β → 1. This is not the case for the qq production from longitudinal photons (Lqq) which is formally suppressed by 1/Q2 with respect to the leading components. Thus, this particular β-dependence makes the Lqq contribution dominant for β → 1, see Fig. 3. The presence of the Lqq component has important consequence for the longitudinal diffractive structure function which is supposed to be determined from the HERA data. The formula given below is an important element in the description of FDL in the region of large β: FDLqq̄ = 16π4xIP e−BD |t| (1− β)4 2(1−β) k2/Q2 φ20(k, xIP ) (20) where the function φ0(k, xIP ) is given in terms of the dipole cross section σ̂(xIP , r) and the Bessel functions K0 and J0: φ0(k, xIP ) = k dr r K0 J0(kr) σ̂(xIP , r) . (21) Strictly speaking, eq. (20) contains all inverse powers of Q2 but the part proportional to 1/Q2 (called twist–4) dominates. The dipole-proton cross section describes the interaction of a color dipole, formed by the qq or qqg systems, with a proton. Following [18] we choose σ̂(xIP , r) = σ0 {1− exp (−r 2Q2s/4)} (22) where Q2s = (xIP /x0) −λ GeV2 is a saturation scale which provides the energy dependence of the twist–4 contribution. The parameters σ0 = 29 mb, x0 = 4 · 10 −5 and λ = 0.28 are taken from [18] (Fit 2 with charm). This form of the dipole cross section provides successful description of the first HERA data on both inclusive and diffractive structure functions [18,19]. A different parametrisation of σ̂, without the saturation scale, is also given in [42–44]. We checked that a very similar description of FDLqq̄ was found in a recent analysis [45] based on the Color Glass Condensate parameterisation of the dipole scattering amplitude [46]. The relation between the dipole approach with three diffractive components and the DGLAP approach with diffractive parton distributions was analysed at length in [30]. Summarising this relation, the twist–2 part of the qq component gives a diffractive quark distribution. The twist-2 part of the qqg component forms a first step of the DGLAP evolution which starts from a given gluon distribution. Both diffractive parton distributions do not depend on Q2, thus they may serve as initial conditions for the DGLAP equations at the scale which is not determined. From this perspective, the DGLAP approach offers a description of more complicated diffractive state with any number of partons ordered in transverse momenta. However, the pQCD calculations tell us that the twist–2 analysis of diffractive data should include the twist–4 contribution since it cannot be neglected at large β. This is the strategy which we follow in our analysis. We also borrow from the dipole approach a general form in β of the initial quark distribution which vanishes at the endpoints β = 0, 1 (see eq. (31) in which Aq and Cq are positive). A very important aspect of Regge factorisation (11) can also be motivated by the dipole approach. It is a consequence of geometric scaling of the dipole cross section (22) [30,47]. 2.4 Reggeon contribution The diffractive data from the H1 collaboration for higher values of xIP hints towards a contribu- tion which decreases with energy. This effect can be described by reggeon exchanges in addition to the rising with energy pomeron exchange. Following [48, 49], we consider the dominant isoscalar (f2, ω) reggeon exchanges which lead to the following contribution to F fR(xIP , t)FR(β,Q 2) . (23) This contribution breaks Regge factorisation of the diffractive structure function, however, its presence is necessary for xIP > 0.01 [50]. The reggeon flux fR is given by the formula analogous to eq. (13) fR(xIP , t) = F 2R(0) e−|t|/Λ R |ηR(t)| 1−2αR(t) IP , (24) where αR(t) = 0.5475 + 1 · t is the reggeon trajectory. From the Regge phenomenology of hadronic reactions ΛR = 0.65GeV and the reggeon–proton couplings are given by [49]: F (0) = 194GeV−2 and F 2ω(0) = 52GeV −2. The functions |ηR(t)| 2 = 4cos2[παR(t)/2] , |ηR(t)| 2 = 4 sin2[παR(t)/2] (25) are signature factors for even (f2) and odd (ω) reggeons, respectively. We could also consider isovector reggeons (a2, ρ) but their couplings to the proton are much smaller and we neglect them. Finally, the reggeon structure function FR is given by [49] FR(β) = AR β −0.08 (1− β)2 , (26) where the normalisation AR is a fitted parameter. Thus, in the first approximation, we neglect the Q2-dependence of the reggeon contribution. 3 Fit details Collab. No. points Data |t|-range Q2-range β-range H1 [50] 72 LP [0.08, 0.5] [2 , 50] [0.02 , 0.7] ZEUS [51] 80 LP [0.075 , 0.35] [2 , 100] [0.007 , 0.48] H1 [31] 461 MY < 1.6 [|tmin| , 1] [3.5 , 1600] [0.01 , 0.9] ZEUS [52] 198 MY < 2.3 [|tmin| ,∞] [2.2 , 80] [0.003 , 0.975] Table 1: Kinematic regions of diffractive data from HERA. LP means leading proton data and MY is invariant mass of a dissociated proton. Dimensionfull quantities are in units of 1 GeV. In our analysis we use diffractive data from the H1 [31,50] and ZEUS [51,52] collaborations. In Table 1 we show their kinematic limits in which they have been measured. The minimal value value of |t| is given by |tmin| ≃ 1− xIP m2p , (27) where mp is the proton mass. The leading proton data from H1, measured in the range given in Table 1, were corrected by the H1 collaboration to the range |tmin| < |t| < 1 GeV The ZEUS data are given for the diffractive structure function FD2 , thus we use in our analysis the following formulae FD2 = F D(tw2) 2 + F 2 + F Lqq̄ (28) FDL = F D(tw2) L + F Lqq̄ . (29) No Data Fit αIP (0) AR Aq Bq Cq Ag Bg Cg χ 1 H1 (LP) tw-2 1.098 0.29 1.75 1.49 0.5∗ 2.09 0.67 0.80 0.48 2 ZEUS (LP) tw-2 1.145 1.05 2.13 1.51 0.5∗ 10.0* 1.03 2.26 0.40 3 H1 tw-2 1.117 0.49 1.33 1.63 0.34 0.17 -0.16 -1.10 1.04 4 tw-(2+4) 1.119 0.48 1.62 1.98 0.59 0.04 -0.56 -1.68 1.17 5 ZEUS tw-2 1.093 0.0∗ 1.68 1.01 0.5∗ 0.49 -0.03 -0.40 1.35 6 tw-(2+4) 1.092 0.0∗ 1.20 0.85 0.57 0.07 -0.52 -1.48 1.82 Table 2: The fit parameters to H1 nd ZEUS data. The presence of twist–4 in the fits is marked by tw-(2+4). The parameters with an asterisk are fixed in the fits. The longitudinal twist-4 contribution is present on the r.h.s. of eq. (28) since FD2 is the sum of the contributions from the transverse and longitudinal polarised virtual photon. The H1 data, however, are presented for the reduced cross section (2). Thus we substitute relations (28) and (29) in there and use σDr = D(tw2) 2 + F 1 + (1− y)2 D(tw2) 2(1− y) 1 + (1− y)2 FDLqq . (30) The expression in the curly brackets is the twist–2 contribution while the last term is the twist–4 one. Notice that the difference between FD2 and σ r is most important for y → 1. We fit the diffractive parton distributions at the initial scale Q20 = 1.5 GeV 2, assuming the Regge factorised form (11) with the following pomeron parton distributions [31]: βΣIP (β) = Aq β Bq (1− β)Cq (31) βgIP (β) = Ag β Bg (1− β)Cg . (32) The six indicated parameters are fitted to data. We additionally multiplied both distributions by a factor exp{−a/(1 − β)} with a = 0.01 to secure their vanishing for β = 1. This factor is only important when Cq or Cg becomes negative in the fits. For the evolution, we use the next-to-leading order DGLAP equations with ΛQCD = 407 MeV for Nf = 3 flavours [53]. The pomeron flux in eq. (11) is integrated over t in the limits given in Table 1 which leads to the form fIP (xIP ) = F 2IP (0) e−B|tmin| − e−B|tmax| 1−2αIP (0) IP . (33) The shrinkage parameter B equals B = BD + 2α IP ln(1/xIP ) (34) with BD = 5.5GeV −2 and α′IP = 0.06GeV −2 [50]. In summary, we have eight fit parameters altogether: the pomeron intercept αIP (0), reggeon normalisation AR in eq. (26) and six parameters in eqs. (31,32) 4 Fit results The data sets from Table 1 were obtained in different kinematical regions, using different methods of their analysis. Thus, we decided to perform fits to each data set separately. The values of the fit parameter are shown in Table 2. The difference between them can be attributed to the scale of uncertainty of our analysis. In each case we preformed two fits: with and without the twist–4 formula added to the twist–2 contribution. 4.1 Leading proton data We started from fits to the leading proton data. The fit parameters in this case are displayed in the first two rows of Table 2. We only show the twist–2 fit results since they are not changed in fits with the twist–4 term. This happenes because the leading proton data comes from the region of β values where the twist–4 contribution is small (β ≤ 0.7 for H1 and β < 0.5 for ZEUS), see Fig. 3. The data with a dissociated proton (DP) which are measured in the region of large β influence most the value of the parameter Cg which controls the behaviour of the gluon distribution at β → 1. For the LP data Cg is positive and the gluon distribution is suppressed near β ≈ 1, while for the DP data Cg is negative and the gluon distribution is strongly enhanced. This shows that the data with β > 0.7 are crucial for the proper analysis. Without this kinematic region we lose important information about diffractive interactions. Thus, from now on we concentrate on the analysis of the DP data. 4.2 H1 data The fit parameters to the H1 data with a dissociate proton are given in the third and fourth rows of Table 2. We see that the fit quality is practically the same for both fits, with and without the twist–4 contribution. The presence of the reggeon term improves fit quality by 30 units of χ2 for 461 experimental points. A good quality of the fits is illustrated in Fig. 4 which also shows that the reduced cross sections (30) from the twist–2 (solid lines) and twist–(2+4) fits (dashed lines) are very close to each other. In Fig. 5 we show our results on the reduced cross section for the largest measured value of β = 0.9. In this region, the twist–4 contribution, shown by the dotted lines, cannot be neglected. We see that the curves from both the twist–2 (solid) and twist–(2+4) (dashed) fits describe data reasonable well. However, the curves with twist–4 have a steeper dependence on xIP (energy) than in the pure twist–2 analysis. This observation is by far more pronounced in the analysis of the ZEUS data performed for the structure function FD2 . The diffractive parton distributions from our fits are shown in Fig. 6 in terms of the pomeron parton distributions, βΣIP (β,Q 2) and βgIP (β,Q 2). Being independent of the pomeron flux, such a presentation allows for a direct comparison of the results from fits to different data sets. We see that the singlet quark distributions are quite similar while the gluon distributions are different. In the fit with twist–4, the gluon distribution is stronger peaked near β ≈ 1. This somewhat surprising result can be understood by looking at the logarithmic slope of FD2 for fixed values of β. From the LO DGLAP equations we have schematically: ∂ lnQ2 ∂ lnQ2 = Pqq ⊗ ΣIP + PqG ⊗GIP − ΣIP Pqq (35) where the negative term describes virtual corrections. For large β, the measured slope is negative which means that the virtual emission term must dominate over the real emission ones. The addition of the twist–4 contribution to FD2 , proportional to 1/Q 2, contributes a negative value to the slope which has to be compensated by a larger gluon distribution in order to describe the same data. In Fig. 7 we present our most important results. On the left panel, the FD2 structure function is shown from both fits, with and without the twist–4 contribution (shown by the dotted lines). We see no significant difference between these two results. However, the longitudinal structure function FDL differs significantly for the two fits (right panel) due to the twist–4 contribution. Let us emphasise that both sets of curves were found in the fits which well describe the existing data on σDr , including the large β region. Thus, an independent measurement of F L in this region would be an important test of the QCD mechanism of diffraction. 4.3 ZEUS data The results of same fits performed for the ZEUS data are shown in the last two rows of Table 2. This time the Regge contribution (26) is not necessary since fits give the reggeon normalisation AR ≈ 0. In general, the fit quality is worse than for the H1 data. As shown in Fig. 8, the biggest difference between the twist–2 and twist–(2+4) results occurs at large β values. This is analysed in detail in Fig. 9. We see that the presence of the twist–4 term in the fit (dashed lines) improves the agreement with the data in this region. In particular, a steep dependence of FD2 on xIP is better reproduced by the twist–(2+4) fit then by the twist–2 one (solid lines). This dependence is to large extend driven by the twist–4 contribution (dotted lines). The behaviour of the diffractive parton distributions and structure functions, shown in Figs. 10 and 11, respectively, is very similar to that found for the H1 data. The gluon dis- tribution from the fit with twist–4 is stronger peaked near β ≈ 1 and the longitudinal structure functions in the large β region is dominated by the twist–4 contribution. We summarise the effect of the twist–4 contribution in Fig. 12 showing the predictions for the longitudinal diffractive structure function FDL . Ignoring this contribution, we find the two solid curves coming from the pure twist–2 analysis of the H1 (upper) and ZEUS (lower) data. With twist–4, the dashed curves are found, the upper one from the H1 data and the lower one from the ZEUS data. There is a significant difference between these two predictions in the region of large β. We believe that the effect of the twist–4 contribution will be confirmed by the forthcoming analysis of the HERA data. 5 Conclusions We performed fits of the diffractive parton distributions to new diffractive data from the H1 and ZEUS collaborations at HERA. In addition to the standard twist–2 formulae, we also considered the twist–4 contribution which dominates in the region of large β. This contribution comes from the diffractive production of the qq pair by the longitudinally polarised virtual photons. The effect of the twist–4 contribution on the diffractive parton distributions and structure functions was carefully examined. The twist–4 contribution leads to the gluon distribution which is peaked stronger at β ≈ 1 than in the case without twist–4. The main result of our analysis is a new prediction for the longitudinal diffractive struc- ture function FDL . The twist–4 term significantly enhances F L in the region of large β. 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The twist–4 contribution Lqq̄ is indicated by the yellow band. Old ZEUS data points are shown. H1 DATA β=0.01 β=0.01 β=0.04β=0.04 β=0.1β=0.1 β=0.2β=0.2 β=0.4β=0.4 β=0.65β=0.65 β=0.9 Q β=0.9 Q 6.56.5 8.58.5 200200 Figure 4: Reduced cross section σ r for H1 data as a function of xIP . Solid lines: twist–2 fit, dashed lines: twist–(2+4) fit. H1 DATA (β=0.9) Q2= 3.5 GeV2 dashed: tw-(2+4) fit solid: tw-2 fit Q2= 5 GeV2 dotted: tw-4 contribution Q2= 6.5 GeV2 Q2= 8.5 GeV2 Figure 5: Reduced cross section σ r for H1 data at β = 0.9 for four values of Q 2 against fit curves. DPD (H1) solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 6: Pomeron parton distributions: singlet βΣIP (β,Q 2) (left) and gluon βgIP (β,Q 2) (right) from H1 data. DSF (H1) solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit dotted: tw-4 contribution dotted: tw-4 contribution dotted: tw-4 contribution 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 7: Diffractive structure functions F 2 (left) and F L (right) from fits to H1 data for xIP = 10 −3. The band shows the effect of twist–4 on the predictions for F ZEUS DATA β=0.652 β=0.652 β=0.231β=0.231 β=0.07β=0.07 β=0.022β=0.022 β=0.007β=0.007 β=0.003 β=0.003 β=0.735β=0.735 β=0.308β=0.308 β=0.1β=0.1 β=0.032β=0.032 β=0.010β=0.010 β=0.004 β=0.004 β=0.807β=0.807 β=0.4β=0.4 β=0.143β=0.143 β=0.047β=0.047 β=0.015β=0.015 β=0.007 β=0.007 β=0.848β=0.848 β=0.471β=0.471 β=0.182β=0.182 β=0.062β=0.062 β=0.020β=0.020 β=0.009 β=0.009 β=0.907β=0.907 β=0.61β=0.61 β=0.28β=0.28 β=0.104β=0.104 β=0.034β=0.034 β=0.015 β=0.015 β=0.949β=0.949 β=0.75β=0.75 β=0.43β=0.43 β=0.182β=0.182 β=0.063β=0.063 β=0.029 β=0.029 β=0.975β=0.975 β=0.86β=0.86 β=0.604β=0.604 β=0.313β=0.313 β=0.121β=0.121 Figure 8: Diffractive structure function F 2 as a function xIP for ZEUS data. Solid lines: twist–2 fit, dashed lines: twist–(2+4) fit. ZEUS DATA dotted: tw-4 contribution β = 0.86 Q2= 55 GeV2 solid: tw-2 fit dashed: tw-(2+4) fit β = 0.91 Q2= 14 GeV2 β = 0.95 Q2= 27 GeV2 β = 0.975 Q2= 55 GeV2 Figure 9: Diffractive structure function F 2 for ZEUS data at large values of β against fit curves. DPD (ZEUS) solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 10: Pomeron parton distributions βΣIP (β,Q 2) (left) and βgIP (β,Q 2) (right) from fits to ZEUS data. DSF (ZEUS) solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit solid: tw-2 fit dashed: tw-(2+4) fit dotted: tw-4 contribution dotted: tw-4 contribution dotted: tw-4 contribution 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure 11: Diffractive structure functions F 2 (left) and F L (right) from fits to ZEUS data for xIP = 10 −3. The band shows the effect of twist–4 on the predictions for F Diffractive FL twist-(2+4) twist-2 Q2=10 GeV2 xP=10 twist-(2+4) twist-2 Q2=10 GeV2 xP=10 Q2=10 GeV2 xP=10 twist-(2+4) twist-2 Q2=10 GeV2 xP=10 twist-(2+4) twist-2 Q2=10 GeV2 xP=10 0.005 0.015 0.025 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 12: Predictions for F L for xIP = 10 −3 and Q2 = 10 GeV2 from the twist–(2+4) fits to the H1 (upper dashed line) and ZEUS (lower dashed line) data. The solid lines show predictions from pure twist–2 fits to the H1 (upper) and ZEUS (lower) data. Introduction Basic formulae Twist–2 contribution Twist-2 charm contribution Twist–4 contribution Reggeon contribution Fit details Fit results Leading proton data H1 data ZEUS data Conclusions

Draft version November 21, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 GRB 061121: BROADBAND SPECTRAL EVOLUTION THROUGH THE PROMPT AND AFTERGLOW PHASES OF A BRIGHT BURST. K.L. Page1, R. Willingale1, J.P. Osborne1, B. Zhang2, O. Godet1, F.E. Marshall3, A. Melandri4, J.P. Norris5,6, P.T. O’Brien1, V. Pal’shin7, E. Rol1, P. Romano8,9, R.L.C. Starling1, P. Schady10, S.A. Yost11, S.D. Barthelmy3, A.P. Beardmore1, G. Cusumano12, D.N. Burrows13, M. De Pasquale10, M. Ehle14, P.A. Evans1, N. Gehrels3, M.R. Goad1, S. Golenetskii7, C. Guidorzi8,9, C. Mundell4, M.J. Page10, G. Ricker15, T. Sakamoto3, B.E. Schaefer16, M. Stamatikos3, E. Troja1,12, M.Ulanov7, F. Yuan11 & H. Ziaeepour9 Draft version November 21, 2018 ABSTRACT Swift triggered on a precursor to the main burst of GRB 061121 (z = 1.314), allowing observations to be made from the optical to gamma-ray bands. Many other telescopes, including Konus-Wind, XMM-Newton, ROTSE and the Faulkes Telescope North, also observed the burst. The gamma-ray, X-ray and UV/optical emission all showed a peak ∼ 75 s after the trigger, although the optical and X-ray afterglow components also appear early on – before, or during, the main peak. Spectral evolution was seen throughout the burst, with the prompt emission showing a clear positive correlation between brightness and hardness. The Spectral Energy Distribution (SED) of the prompt emission, stretching from 1 eV up to 1 MeV, is very flat, with a peak in the flux density at ∼ 1 keV. The optical- to-X-ray spectra at this time are better fitted by a broken, rather than single, power-law, similar to previous results for X-ray flares. The SED shows spectral hardening as the afterglow evolves with time. This behaviour might be a symptom of self-Comptonisation, although circumstellar densities similar to those found in the cores of molecular clouds would be required. The afterglow also decays too slowly to be accounted for by the standard models. Although the precursor and main emission show different spectral lags, both are consistent with the lag-luminosity correlation for long bursts. GRB 061121 is the instantaneously brightest long burst yet detected by Swift. Using a combination of Swift and Konus-Wind data, we estimate an isotropic energy of 2.8 × 1053 erg over 1 keV – 10 MeV in the GRB rest frame. A probable jet break is detected at ∼ 2 × 105 s, leading to an estimate of ∼ 1051 erg for the beaming-corrected gamma-ray energy. Subject headings: gamma-rays: bursts — X-rays: individual (GRB 061121) 1. INTRODUCTION Electronic address: kpa@star.le.ac.uk 1 Department of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH, UK 2 Department of Physics & Astronomy, University of Nevada, Las Vegas, NV 89154-4002, USA 3 NASA/Goddard Space Flight Center, Greenbelt, MD 20771, 4 Astrophysics Research Institute, Liverpool John Moores Uni- versity, Twelve Quays House, Egerton Wharf, Birkenhead, CH41 5 Denver Research Institute, University of Denver, Denver, CO 80208, USA 6 Visiting Scholar, Stanford University 7 Ioffe Physico-Technical Institute, Laboratory for Experimen- tal Astrophysics, 26 Polytekhnicheskaya, Saint Petersburg 194021, Russian Federation 8 INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807, Merate (LC), Italy 9 Dipartimento di Fisica, Universitá di Milano-Bicocca, Piazza delle Scienze 3, I-20126, Milano, Italy 10 Mullard Space Science Laboratory, University College Lon- don, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK 11 University of Michigan, 2477 Randall Laboratory, 450 Church St., Ann Arbor, MI 48104, USA 12 INAF-IASF, Sezione di Palermo, via Ugo La Malfa 153, 90146, Palermo, Italy 13 Department of Astronomy and Astrophysics, Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA 14 XMM-Newton Science Operations Centre, European Space Agency, Villafranca del Castillo, Apartado 50727, E-28080 Madrid, Spain 15 Center for Space Research, Massachusetts Institute of Tech- nology, 70 Vassar Street, Cambridge, MA 02139, USA 16 Department of Physics and Astronomy, Louisiana State Uni- versity, Baton Rouge, LA 70803, USA Gamma-Ray Bursts (GRBs) are intrinsically extremely luminous objects, approaching values of 1054 erg s−1 if the radiation is isotropic (e.g., Frail et al. 2001; Bloom et al. 2003). This energy is emitted over all bands in the electromagnetic spectrum; to understand GRBs as fully as possible, panchromatic observations are required over all time frames of the burst. The Swift multi-wavelength observatory (Gehrels et al. 2004) is designed to detect and follow-up GRBs. With its rapid slewing ability, Swift is able to follow bursts and their afterglows from less than a minute af- ter the initial trigger, and can often still detect them weeks, and sometimes months, later. On rare occa- sions, such as when Swift triggers on a precursor to the main burst, the prompt emission, as well as the after- glow, can be observed at X-ray and UV/optical wave- lengths. GRB 061121, the subject of this paper, is only the third GRB Swift has detected in this manner (af- ter GRB 050117 – Hill et al. 2006 and GRB 060124 – Romano et al. 2006), out of the almost 200 bursts trig- gered on in the first two years of the mission.17 Of these, GRB 061121 is the second well-sampled event (GRB 060124 was the first), and the first for which the UV/Optical Telescope (UVOT) was in event mode. In addition to the small number of precursor triggers, around 10% of Swift bursts show detectable emission over 17 GRB 050820A would possibly have also been in this category, but Swift entered the South Atlantic Anomaly (SAA) just as a dramatic increase in count rate began (Cenko et al. 2006; Page et al. 2005a; Cummings et al. 2005; Page et al. 2005b; Chester et al. 2005); Swift does not actively collect data during SAA passages. http://arxiv.org/abs/0704.1609v1 mailto:kpa@star.le.ac.uk 2 K.L. Page et al. the BAT bandpass by the time the narrow field instru- ments (NFIs) are on target. Besides the Swift observations of prompt emission, there have been a small number of prompt optical mea- surements of GRBs, thanks to the increasing number of robotic telescopes around the world. A variety of behaviours has been found, with some optical (and in- frared) light-curves tracking the gamma-ray emission (e.g., GRB 041219A – Vestrand et al. 2005; Blake et al. 2005), while others appear uncorrelated (e.g., GRB 990123 – Akerlof et al. 1999, Panaitescu & Kumar 2007, though see also Tang & Zhang 2006; GRB 050904 – Boër et al. 2006; GRB 060111B – Klotz et al. 2006; GRB 060124 – Romano et al. 2006). GRB 050820A (Ves- trand et al. 2006) showed a mixture of both correlated and uncorrelated optical flux. Where correlations exist between different energy bands, it is likely that there is a common origin for the components. In the uncorrelated cases, the opti- cal emission may be due to an external reverse shock (e.g., Sari & Piran 1999; Mészáros & Rees 1999), while the prompt gamma-rays are caused by internal shocks. Cenko et al. (2006) suggest that the early optical data for GRB 050820A are produced by the forward shock passing through the band. In the case of GRB 990123, Panaitescu & Kumar (2007) have suggested that the gamma-rays arose from inverse Comptonisation, while the optical emission was due to synchrotron processes; they do not assume a specific mechanism for the energy dissipation, allowing for the possibility of either internal or reverse-external shocks. It is unclear whether precursors are ubiquitous fea- tures of GRBs, often remaining undetected because of a low signal-to-noise ratio or being outside the energy bandpass of the detector, or whether only some bursts exhibit them. A detailed discussion of the precursor phe- nomenon is beyond the scope of this paper and will be addressed in a future publication. In this paper, we report on the multi-wavelength ob- servations of both the prompt and afterglow emission of GRB 061121. §2 details the observations made by Swift, Konus-Wind, XMM-Newton, ROTSE18 and the Faulkes Telescope North (FTN), with multi-band comparisons being made. In §3, we discuss the precursor, prompt and afterglow emission, with a summary given in §4. Throughout the paper, the main burst (∼ 60–200 s after the trigger) will be referred to as the prompt emis- sion, and the emission seen over −5 to +10 s as the precursor, where the BAT trigger time T0 = 0 s. Er- rors are given at 90% confidence (e.g., ∆χ2 = 2.7 for one interesting parameter) unless otherwise stated, and the convention Fν,t ∝ ν −βt−α (with the photon spec- tral index, Γ = β + 1 where dN/dE ∝ E−Γ) has been followed. We have assumed a flat Universe, with Hubble constant, H0 = 70 km s −1 Mpc−1, cosmological constant, ΩΛ = 0.73 and Ωmatter = 1−ΩΛ. 2. OBSERVATIONS AND ANALYSES Two years and one day after launch, the Burst Alert Telescope (BAT; Barthelmy et al. 2005) triggered on a precursor to GRB 061121 at 15:22:29 UT on 21st Novem- ber, 2006. Swift slewed immediately, resulting in the 18 Robotic Optical Transient Search Experiment NFIs being on target and beginning to collect data 55 s (X-ray Telescope: XRT; Burrows et al. 2005a) and 62 s (UVOT; Roming et al. 2005) later. This enabled broad- band observations of the main burst event, which peaked ∼ 75 s after the trigger, leading to spectacular multi- wavelength coverage of the prompt emission. The most accurate Swift position for this burst was that determined by the UVOT: RA = 09h 48m 54.s55, decl = −13◦ 11′ 42.′′4 (J2000.0; 90% confidence radius of 0.′′6; Marshall et al. 2006); the refined XRT position is only 0.′′1 from these coordinates (Page et al. 2006b). GRB 061121 was declared a ‘burst of interest’ by the Swift team (Gehrels et al. 2006a), to encourage an inten- sive ground- and space-based follow-up programme. In addition to the Swift observations, the prompt emission of GRB 061121 was detected by RHESSI19 (Bellm et al. 2006), Konus-Wind and Konus-A (Golenetskii et al. 2006). Later afterglow observations were obtained in the X-ray (XMM-Newton – Schartel 2006) and radio (VLA20 – Chandra & Frail 2006) bands. ATCA21 and WSRT22 also observed in the radio band between ∼5.2 day and ∼6.2 day after the burst, but did not detect the after- glow (van der Horst et al. 2006a,b), implying it had faded since the VLA observation. Likewise, extensive optical follow-up observations were performed: ROTSE-IIIa (Yost et al. 2006), FTN (Me- landri et al. 2006), Kanata 1.5-m telescope (Uemura et al. 2006), the University of Miyazaki 30-cm telescope (Sonoda et al. 2006), MDM23 (Halpern et al. 2006a,b; Halpern & Armstrong 2006a,b), P6024 (Cenko 2006), ART25 (Torii 2006), the CrAO26 2.6-m telescope (Efimov et al. 2006a,b) and SMARTS/ANDICAM27 (at infrared wavelengths, too; Cobb 2006) all detected the optical afterglow. Spectroscopic observations were performed at the Keck telescope about 12 minute after the trig- ger, finding a redshift of z = 1.314 for the optical af- terglow, based on absorption features (Perley & Bloom 2006; Bloom et al. 2006). GRB 061121 has the highest instantaneous peak flux of all the long bursts detected by Swift to date (e.g., Angelini et al. in prep). 2.1. Gamma-ray Data 2.1.1. BAT Temporal Analysis— After the initial precursor, the BAT count rate returned to close to the instrumental back- ground level, until T0+60 s, at which point the much brighter main burst began. This is characterised by a series of overlapping peaks, each brighter than the previ- ous one, after which the gamma-ray flux decayed (from ∼T0+75 s to ∼T0+140 s). Event data were collected un- til almost 1 ks after the trigger, thus covering the entire emission period. 19 Reuven Ramaty High Energy Solar Spectroscopic Imager 20 Very Large Array 21 Australia Telescope Compact Array 22 Westerbork Synthesis Radio Telescope 23 Michigan-Dartmouth-MIT Observatory 24 Palomar 60 inch 25 Automated Response Telescope 26 Crimean Astrophysical Observatory 27 Small and Moderate Aperture Research Telescope System/A Novel Double-Imaging CAMera GRB 061121: Broadband observations 3 T90, over 15-150 keV, and incorporating both the pre- cursor and main emission, is 81 ± 5 s, measured from 8.8–89.8 s after the trigger28. Figure 1 shows the mask- weighted BAT light-curve in the four standard energy bands [15–25, 25–50, 50–100, 100-150 keV; 64 ms bin- ning between 50-80 s after the trigger, with 1 s bins at all other times; units of count s−1 (fully illuminated detector)−1], with light-curves from other instruments: the precursor and the pulses of the main burst are de- tected over all gamma-ray bands, although the precursor is only marginal over the 100-150 keV BAT band. There is also a soft tail (detected below ∼ 50 keV, when suffi- ciently coarse time bins are used) visible until about 140 s after the trigger (see bottom panel of Figure 1), corre- sponding to a similar feature in the X-ray light-curves. Spectral analysis— For the precursor, T90,pre = 7.7 ± 0.5 s (15–150 keV). A spectrum ex- tracted over this interval can be well modelled by a single power-law, with Γ = 1.68 ± 0.09 (χ2/dof = 26.2/23); no significant improvement was found by using the Band function (Band et al. 1993) or a cut-off power-law and a thermal model led to a slightly (χ2 ∼ 8) worse fit. The 15–150 keV fluence for this time interval is 4 × 10−7 erg cm−2. Considering only the main event, T90,main = 18.2 ± 1.1 s (measured from 61.8–80.0 s post-trigger). Fitting a power-law to the mean spectrum during this time also results in a good fit (Γ = 1.40 ± 0.01; fluence = 1.31 × 10−5 erg cm−2 over 15–150 keV; χ2/dof =51.6/56 ); again, neither the Band function nor a cut-off power-law improves upon this. There is significant spectral evolution during the T90 period, as shown in Figure 2: at times when the count rate is higher, the spectrum is harder. This behaviour was also common in earlier bursts, as well as previous Swift detections (e.g. Golenetskii et al. 1983; Ford et al. 1995; Borgonovo & Ryde 2001; Goad et al. 2007). The precursor shows a similar dependence of hardness ratio on count rate, suggesting that the emission processes in the precursor and the main burst are the same or similar. 2.1.2. Konus-Wind Temporal Analysis— Konus-Wind (Aptekar et al. 1995) triggered on the main episode of GRB 061121, while Konus-A triggered on the precursor (Golenetskii et al. 2006). Because of the spatial separation of Swift and Wind, the light travel-time between the spacecraft is 1.562 s: the Konus-Wind trigger time, T0,K−W = T0,BAT + 61.876 s. All Konus light-curves have been plotted with respect to the BAT trigger, corrected for the light travel-time. Figure 1 shows the Konus-Wind data plotted over the standard energy bands, with 64 ms binning; the bottom panel plots the coarser time resolution (2.944 s) ‘waiting mode’ data, showing that Konus-Wind did see slightly enhanced emission at the time of the precursor. The background levels (which have been subtracted in each case) were 1005, 370 and 193.4 count s−1 for bands 21–83, 83–360 and 360–1360 keV, respectively. 28 Errors on the BAT T90 are estimated to be typically 5–10%, depending on the shape of the light-curve. Spectral analysis— Table 1 gives the spectral fits to the Konus-Wind data in three separate time intervals shown by vertical lines in Figure 1 (Konus-Wind spec- tral intervals are automatically selected onboard): up to the end of the ‘bump’ around 70 s (the ‘start’ of the burst), the burst maximum and, finally, until most of the emission has died away (the burst tail). The data were fitted with a cut-off power-law, where dN/dE ∼ E−Γ × e[−(2−Γ)E/Epeak], leading to the photon indices and peak energies given in the table. The Band function was used to estimate upper limits for the pho- ton index above the peak; the values for the peak energy and Γ obtained from the Band function were the same as when fitting the cut-off power-law. Little variation in the spectral slope for energies below the peak is seen over these intervals, though the peak itself may have moved to somewhat higher energies during the burst emission. Ex- tracting BAT spectra over the same time intervals, and fitting with the same model (fixing Epeak at the value determined from the Konus-Wind data) results in con- sistent spectral indices. 2.2. X-ray Data 2.2.1. XRT Temporal Analysis— The XRT identified and centroided on an uncatalogued X-ray source in a 2.5 s Image Mode (IM) frame, as soon as the instrument was on target. This was quickly followed by a pseudo Piled-up Photo Diode (PuPD) mode frame. Following damage from a micrometeoroid impact in May 2005 (Abbey et al. 2005), the Photo Diode mode (Low Rate and Piled-up) has been disabled [see Hill et al. (2004) for details on the different XRT modes]; however, the XRT team are currently work- ing on a method to re-implement these science modes and to update the ground software to process the files. The pseudo PuPD point presented here is the first use of such data. Data were then collected in Windowed Timing (WT) mode starting at a count rate of ∼ 1280 count s−1 (pile- up corrected – see below); the rate rapidly increased to a maximum of ∼ 2500 count s−1 at T0 + 75 s, mak- ing GRB 061121 the brightest burst yet detected by the XRT. Following this peak, the count-rate decreased, with a number of small flares superimposed on the underlying decay (see Figure 1). Photon Counting (PC) mode was automatically selected when the count rate was below about 10 count s−1. Around 1.5 ks, the XRT switched back into WT mode briefly, due to an enhanced back- ground linked to the sunlit Earth and a relatively high CCD temperature. Because of the high count rate, the early WT data were heavily piled-up; see Romano et al. (2006) for informa- tion about pile-up in this mode. To account for this, an extraction region was used which excluded the central 20 pixels (diameter; 1 pixel = 2.′′36) and extended out to a total width of 60 pixels. Likewise, the first three orbits of PC data were piled-up, and the data were thus extracted using annular regions (inner exclusion diame- ter decreasing from 12 to 6 to 4 pixels as the afterglow faded; outer diameter 60 pixels). The count rate was then corrected for the excluded photons by a comparison of the Ancillary Response Files (ARFs) generated with and without a correction for the Point Spread Function 4 K.L. Page et al. White UVOT 0.3−2 keV XRT 2−10 keV 15−25 keV BAT 25−50 keV 50−100 keV 100−150 keV 21−83 keV Konus 83−360 keV 360−1360 keV 0 50 100 150 time since BAT trigger (s) 21−1360 keV (waiting mode) 50 100 150 time since BAT trigger (s) 15−50 keV Fig. 1.— Top panels: Swift UVOT, XRT, BAT and Konus-Wind light-curves of GRB 061121; 1σ error bars are shown for the UVOT and XRT data. Each instrument detected the peak of the main burst, with the precursor being detected over all gamma-ray energies. The vertical lines in the 360–1360 keV panel indicate the start and stop times for the spectra given in Table 1. Bottom panel: The 15-50 keV BAT light-curve, with 10-s bins, showing a tail out to ∼140 s. GRB 061121: Broadband observations 5 start time (s) stop time (s) Γ Epeak (keV) ΓBand χ 2/dof 61.876 70.324 1.40 +0.08 −0.09 <2.1 72/75 70.324 75.188 1.23 +0.05 −0.06 <2.9 88/75 75.188 83.380 1.30 +0.11 −0.13 <2.3 81/75 61.876 83.380 1.32 +0.04 −0.05 <2.7 95/75 TABLE 1 Konus-Wind cut-off power-law spectral fit results. Times are given with respect to the BAT trigger. ΓBand is the upper limit obtained for the spectral index above Epeak when fitting with the Band function. 6 K.L. Page et al. 65 70 75 80 time since trigger (s) 0 1 2 3 BAT count rate (count s−1 detector−1) Fig. 2.— Top panels: Light-curves, hardness ratios (HR) and the variation in Γ using a single power-law fit during the main emission. The BAT light-curve (top panel) is in units of count s−1 (fully il- luminated detector)−1 , and the corresponding hardness ratio plots (50–150 keV)/(15–50 keV) using 1-s binning. The XRT light-curve shows counts over 0.3–10 keV, while the hardness ratio compares (1–10 keV)/(0.3–1 keV) over 1-s bins. Bottom panel: BAT hard- ness ratio versus count rate, showing that the emission is harder when brighter. Data from the precursor are shown as grey circles, with the main burst in black. The grey line shows a fit to the data, of the form HR = 0.14 CR + 0.39. (PSF); the ratio of these files provides an estimate of the correction factor. Nousek et al. (2006) give more details on this method. Occasionally, the afterglow was partially positioned over the CCD columns disabled by microm- eteoroid damage mentioned above. In these cases, the data were corrected using an exposure map. From T0 + 3 × 10 5 s onwards, the afterglow had faded sufficiently for a nearby (41.′′5 away), constant (count rate ∼ 0.003 count s−1) source to contaminate the GRB region; this source is coincident with a faint object in the Digitized Sky Survey and is marginally detected in the UVOT V filter. Thus, beyond this time, the ex- traction region was decreased to a diameter of 30 pix- els, and the count rates corrected for the loss in PSF (a factor of ∼ 1.08). The spectrum of this nearby source can be modelled with a single power-law of Γ = 1.5+0.2 −0.1, 100 1000 104 105 1061 time since trigger (s) Fig. 3.— Swift-XRT light-curve of GRB 061121. The star and triangle show the initial Image Mode and pseudo PuPD point (see text for details), followed by WT mode data (black) during the main burst (and at the end of the first orbit) and PC mode data (in grey). 10050 200 500 time since trigger (s) Fig. 4.— Swift flux light-curve of GRB 061121, showing the early X-ray data (star, triangle and crosses) and the BAT data (grey histogram) extrapolated into the 0.3–10 keV band pass in units of erg cm−2 s−1, together with the UVOT flux density light-curve (light grey circles – V -band; dark grey circles – White filter) in units of erg cm−2 s−1 Å−1, scaled to match the XRT flux observed at the start of the ‘plateau’ phase. with NH = (1.8 −1.2) × 10 21 cm−2, in comparison with the Galactic value in this direction of 5.09 × 1020 cm−2 (Dickey & Lockman 1990). Figure 3 shows the XRT light-curve, starting with the IM point (see Hill et al. 2006 for details on how IM data are converted to a count rate) and followed by the pseudo PuPDmode data. The importance of these early pre-WT data is clear, confirming that the XRT caught the rise of the main burst. After the bright burst, the afterglow began to follow the ‘canonical’ decay, seen in many Swift bursts (Nousek et al. 2006; Zhang et al. 2006a). Such a decay can be parameterised by a series of power-law segments; in this case, fitting the data beyond 200 s after the trig- ger (= 125 s after the main peak), two breaks in the light-curve were identified, with the decay starting off very flat (α = 0.38 ± 0.08) and eventually steepening GRB 061121: Broadband observations 7 α1 0.38 ± 0.08 Plateau phase Tbreak,1 2258 α2 1.07 +0.04 −0.06 Shallow phase Tbreak,2 (3.2 ) × 104 s α3 1.53 +0.09 −0.04 Steep phase TABLE 2 XRT power-law light-curve fits from 200 s after the trigger onwards; times are referenced to the BAT trigger. The names used in the text for the different epochs of the light-curve are listed in the last column. to α = 1.07+0.04 −0.06 at ∼ 2.3 ks and then α =1.53 +0.09 −0.04 at ∼ 32 ks (Table 2). The addition of the second break vastly improved the fit by ∆χ2 = 112.4 for two degrees of freedom. However, we note that O’Brien et al. (2006) and Willingale et al. (2007) advocate a different descrip- tion of the temporal decline; we return to this in §3. Fitting the decay of the main peak (75–200 s, keeping T0 as the trigger time) with a power-law, the slope is very steep, with α0 = 5.1 ± 0.2. However, both Zhang et al. (2006a) and Liang et al. (2006) have shown that the appropriate time origin is the start of the last pulse. Thus, a model of the form f(t) ∝ (t−t0) −α0 was used, finding t0 = 58 ± 1 s and a slope of α0 = 2.2 −0.3; this is a statistically significant improvement on the power- law fit using the precursor T0 (∆χ 2 = 32 for one extra parameter). Figure 4 plots the Swift data in terms of flux (the BAT data have been extrapolated into the 0.3–10 keV band, using the joint fits with the XRT described in §2.4.1) and flux density for UVOT. The BAT and XRT data are fully consistent with each other at all overlapping times. Spectral Analysis— The XRT data also show that strong spectral evolution was present throughout the period of the prompt emission; this is discussed in conjunction with the BAT data in §2.4.1. Considering the X-ray data alone, there is some indication that the spectra may be better modelled with a broken, rather than single, power- law, although the break energies cannot always be well constrained (see Figure 5). For each spectrum [covering periods of 2 s during the main pulse, followed by two spectra of 5 s (80–85 s and 85–90 s) where the emission is fainter], the low-energy slopes were tied together for each spectrum (i.e., the slope measured is that averaged over all of the spectra), as were the high-energy indices, and the rest-frame column density, NH,z, was fixed at (9.2 ± 1.2) × 1021 cm−2 from the best fit to the data from later times (see below); only the break energy and the normalisation were allowed to vary. When simul- taneously fitting all 11 spectra, χ2/dof decreased from 142/134 to 127/132. Individually, the spectral fits were typically improved by χ2 of between 2–5. The X-ray data during the GRB 051117A flares (Goad et al. 2007) were found to be better modelled with bro- ken power-laws, with the break energy moving to harder energies during each flare rise, and then softening again as the flux decayed. Likewise, Guetta et al. (2006) found breaks in the X-ray spectra obtained during the flares in GRB 050713A. The same pattern may be occurring here, and there is certainly an indication of spectral curvature. The observed flux calculated from the spectrum corre- sponding to the peak of the emission (74–76 s) was mea- 70 80 90 time since trigger (s) 70 80 90 time since trigger (s) 70 80 90 time since trigger (s) Fig. 5.— Fitting the X-ray data over 0.3–10 keV with a broken power-law (Γ1 =0.69 +0.13 −0.07 and Γ2 =1.61 +0.14 −0.13 for all spectra), the break energy seems to move through the band, towards higher energies when the emission is brighter. Arrows indicate upper or lower 90% limits. sured to be 1.66 × 10−7 erg cm−2 s−1 (over 0.3–10 keV); the unabsorbed value was 1.77 × 10−7 erg cm−2 s−1. The PC spectra were also extracted for the various phases of the light-curve (‘plateau’, ‘shallow’ and ‘steep’ – defined in Table 2); the results of the fitting are pre- sented in Table 3. In each phase, the spectrum could be well modelled by a single power-law (no break re- quired), with excess absorption in the rest-frame of the GRB (modelled using ztbabs and the ‘Wilms’ abun- dance in xspec; Wilms et al. 2000). Together with the WT spectrum from ∼ 200–590 s after the trigger (in the plateau stage), the first two PC spectra (plateau and shallow) are fully consistent with a constant photon in- dex of Γ = 2.07 ± 0.06 and NH,z = (9.2 ± 1.2) × 10 cm−2. Following the second apparent break in the light-curve, around 3.2 × 104 s, the spectrum hardened slightly, to a photon index of Γ = 1.83 ± 0.11 (or 1.87 ± 0.08 using NH,z = 9.2 × 10 21 cm−2). 2.2.2. XMM-Newton XMM-Newton (Jansen et al. 2001) performed a Target of Opportunity observation of GRB 061121 (Observation ID 0311792101) less than 6.5 hr after the trigger (Schartel 2006) and collected data for ∼ 38 ks (MOS1, MOS2; Turner et al. 2001) and ∼ 35 ks (PN; Strüder et al. 2001). This observation is mainly during the ‘shallow’ phase, though also covers a short timespan after the break at around 32 ks. Figure 6 plots the PN flux light-curve and hardness ratio during the XMM-Newton observation, showing the lack of spectral evolution during this time frame; a hard- ness ratio calculated for the Swift data was in agreement with this finding. The decay slope over this time (MOS1, MOS2, PN and joint) is consistent with the Swift results (α ∼ 1.3; note this crosses the time of the second break in the decay). The XMM-Newton EPIC29 spectra show clear evidence for excess NH, in agreement with the Swift data. In addi- tion, fitting with excess NH in the rest-frame of the GRB 29 European Photon Imaging Camera 8 K.L. Page et al. Epoch time since Γ NH,z χ 2/ν corresponding trigger (s) (1021 cm−2) α Plateau 590–1560 2.14 ± 0.12 10.8 62.5/52 0.38 ± 0.08 Shallow 4900–22245 2.04 ± 0.10 8.9 67.5/70 1.07 +0.04 −0.06 Steep 34550–1152750 1.83 ± 0.11 8.0 48.0/55 1.53 +0.09 −0.04 Plateau 590–1560 2.09 ± 0.08 9.2 ± 1.2 (tied) 63.5/53 0.38 ± 0.08 Shallow 4900–22245 2.05 ± 0.06 9.2 (tied) 67.6/71 1.07 +0.04 −0.06 Steep 34550–1152750 1.87 ± 0.08 9.2 (tied) 48.7/56 1.53 +0.09 −0.04 TABLE 3 XRT PC spectral fits - rest-frame NH free and then tied between all three spectra. The temporal decay slopes, α, corresponding to each stage are also given. The Galactic absorbing column of NH = 5.09 × 10 20 cm−2 was always included in the model. 3×104 3.5×104 4×104 4.5×104 5×104 5.5×104 6×104 time since BAT trigger (s) Fig. 6.— XMM-Newton EPIC-PN light-curve and hardness ratio of GRB 061121. The horizontal line shows the hardness ratio is consistent with a constant value of ∼ 1.46, indicating there is no spectral evolution during this time. gives a significantly better fit than at z = 0, as shown in Figure 7. When fitting in the observer’s frame there is a noticeable bump in the residuals around 0.6 keV; fitting with NH at z = 1.314 removes this feature. The data are of sufficiently high signal-to-noise that the redshift of the absorber can be estimated from the spectrum. Limits can be placed on the redshift and absorbing column, re- spectively, of z > 1.2 and NH,z > 4.6 × 10 21 cm−2 at 99% confidence, in agreement with the spectroscopic redshift from Bloom et al. (2006) within the statistial uncertain- ties. At their value of z = 1.314, the excess NH,z from the EPIC-PN spectrum is (5.3 ± 0.2)× 1021 cm−2, lower than the best fit to the Swift data from the simultaneous ‘shallow’ decay section, but more similar to the values ob- tained from fitting the optical-to-X-ray Spectral Energy Distributions (SEDs) in §2.4.2. In agreement with the simultaneous XRT PC mode data, there is no evidence for a break in the EPIC spectrum over this time period. Spectra from neither the Reflection Grating Spectrom- eter (den Herder et al. 2001) nor EPIC show obvious absorption or emission lines. 2.2.3. Chandra Chandra performed a 33 ks Target of Opportunity ob- servation at ∼ 61 day after the trigger. No source was detected at the position of the X-ray afterglow, with a 3σ upper limit of 2.5 × 10−15 erg cm−2 s−1. 2.3. Optical/UV Data zNH = 5.3x10 21 cm−2 10.5 2 5 channel energy (keV) NH = 1.3x10 21 cm−2 Fig. 7.— EPIC-PN spectrum of the late-time afterglow of GRB 061121, with an excess absorbing column both in the rest- frame of the GRB and the observer’s frame. The spectrum is much better modelled with an excess column at z = 1.314. 2.3.1. UVOT The UVOT detected an optical counterpart in the ini- tial White filter30 observation, starting 62 s after the trigger, and subsequently in all other filters (optical and UV). The UVOT followed the typical sequence for GRB observations, with the early data being collected in event mode, which has a frame time of 8.3 ms during this ob- 30 The White filter covers a broad bandpass of λ ∼ 1600−6500 Å. GRB 061121: Broadband observations 9 servation.31 Photometric measurements were obtained from the UVOT data using a circular source extraction region with a 5− 6′′ radius. uvotmaghist was used to convert count rates to magnitudes and flux; no normali- sation between the different filters was applied. As in the gamma-ray and X-ray bands, the main burst was detected, with an increase in count rate seen between ∼ 50–75 s after the trigger (Figures 1 and 4). However, although an increase in count rate is seen for the UVOT data, it is by a smaller factor than observed for the XRT. After ∼ 110 s, the UVOT emission stops decaying and re- brightens slightly, until 140 s after the trigger, at which time it flattens off and then starts to fade again (Fig- ure 4). The slower decay between ∼ 100–200 s may be indicative of the contribution of an additional (afterglow) component beginning to dominate. A single UV/optical light curve was created from all the UVOT filters in order to get the best measurement of the optical temporal decay. This was done by fitting each filter dataset individually (between 200 and 1 × 105 s) and finding the normalisation, which was then modified to correspond to that of the V -band light-curve. The decay across all the filters beyond 200 s after the trigger can be fitted with a single slope of αUVOT = 0.68 ± 0.02; the individual U , B and V decay rates are consistent with one another. No break in the light-curve is seen out to ∼ 100 ks. 2.3.2. ROTSE ROTSE-IIIa, at the Siding Spring Observatory in Aus- tralia, first imaged GRB 061121 21.6 s after the trigger time under poor (windy) seeing conditions. A variable source was immediately identified, at a position coinci- dent with that determined by the UVOT (Yost et al. 2006). The ROTSE data (unfiltered, but calibrated to the R- band) have been included in Figure 10 (discussed later). It is noticeable that the peak around 75 s seen in the Swift data is not readily apparent in the ROTSE measure- ments. The bandpass of the UVOT White filter is more sensitive to photons with wavelengths of λ < 4500Å32, while the ROTSE bandpass is redder. This, together with poor seeing conditions during the observation, may explain why the ROTSE light-curve does not clearly show the main emission. 2.3.3. Faulkes Telescope North The FTN, at Haleakala on Maui, Hawaii, began ob- servations of GRB 061121 225 s after the burst trigger, performing a BV Ri′ multi-colour sequence (Melandri et al. 2006). R-band photometry was performed rela- tive to the USNO-B 1.0 ‘R2’ magnitudes. Magnitudes were then corrected for Galactic extinction using the dust-extinction maps by Schlegel et al. (1998), and con- verted to fluxes using the absolute flux calibration from Fukugita et al. (1995). The photometric R-band points have been included in Figure 10. 31 The data have been adjusted to take into account an incor- rect onboard setting (between 2006-11-10 and 2006-11-22), which resulted in the wrong frame times being stored in the headers of the UVOT files (Marshall 2006a). 32 See http://swiftsc.gsfc.nasa.gov/docs/heasarc/caldb/swift/ 2.4. Broadband Modelling 2.4.1. Gamma-rays – X-rays Spectral Analysis— Because the BAT was in event mode throughout the observation of the main burst of GRB 061121, detailed spectroscopy could be performed. Unfortunately this was not the case during the prompt observation of GRB 060124 (Romano et al. 2006). Figure 2 demonstrates the spectral evolution seen in both the BAT and XRT during the prompt emission. Spectra were extracted over 2 s intervals, in an attempt to obtain sufficient signal to noise while not binning over too much of the rapid variability. The BAT data are hardest around 68 s and 75 s (the second of these times corresponding to the peak of the main emission); the XRT hardness peaks about 70 s, which could be a further indication of the softer data lagging the harder. The joint spectrum (Γjoint comes from a simple absorbed power- law fit to the simultaneous BAT and XRT data) is at its hardest during the brightest part of the emission. The joint fit also hardens around 68–70 s, between the times when the BAT and XRT data respectively are at their hardest. The onboard spectral time-bin selection pre- vents the Konus-Wind data from being sliced into corre- sponding times, so constraints have not been placed on the high energy cut-off, Epeak. Breaks in the XRT-BAT power-laws can only be poorly constrained. In Figure 4, the BAT and XRT data were converted to 0.3–10 keV fluxes using the time-sliced power-law fits to the simultaneous BAT and XRT spectra. Without the use of such varying conversion factors, the derived BAT and XRT fluxes would have been inconsistent with each other. A broadband spectrum, covering 0.3 keV to 4 MeV in the observer’s frame (XRT, BAT and Konus-Wind) for ∼ 70–75 s post trigger was fitted by the absorbed cut-off power-law model described in §2.1.2. A constant factor of up to 10% was included between the BAT and Konus- Wind data, to allow for calibration uncertainties. The best fit (χ2/dof = 301/167) gives Γ = 1.19 ± 0.01, with Epeak = 670 −47 keV. NH,z was fixed at 9.2 × 10 21 cm−2 (from the X-ray fits in §2.2.1). Allowing Γ to vary be- tween the three spectra hints at further spectral curva- ture, although the differences are marginal, significant at only the 2σ level. The isotropic equivalent energy (calculated using the time-integrated flux over the full T90 period) is 2.8 × 1053 erg in the 1 keV – 10 MeV band (GRB rest frame), meaning that GRB 061121 is consistent with the Amati relationship (Amati et al. 2002). See §3.3.2 for a beaming-corrected gamma-ray energy limit. Lag Analysis— A lag analysis (e.g., Norris et al. 1996) between the BAT bands leads to interesting results. Comparing bands 50–100 keV and 15–25 keV, the precur- sor emission yields a spectral lag of 600 ± 100 ms, while the main emission has a much smaller lag of 1 ± 6 ms. Note that the calculation was performed using 64 ms binning for the precursor and 4 ms binning for the main burst; see Norris (2002) and Norris & Bonnell (2006) for more details on the procedure. This lag for the main emission is rather small for a typical long burst, however both lags are consistent with the long-burst luminosity- lag relationship generally seen (Norris et al. 2000). The http://swiftsc.gsfc.nasa.gov/docs/heasarc/caldb/swift/ 10 K.L. Page et al. 0 2 4 6 time delay (s) 1−4 keV 4−10 keV 15−25 keV 25−50 keV 50−100 keV 100−150 keV Fig. 8.— Autocorrelation function of the BAT and XRT data during the prompt emission of GRB 061121, showing that the main burst peak is broader at softer energies. short spectral lag for the main emission, and the longer value for the precursor are also found when comparing the 100–350 keV and 25–50 keV bands. Similarly, comparison of the hard and soft (2–10 keV and 0.3–2 keV) XRT bands reveals a lag of approximately 2.5 s, as the emission softens through the main burst. The X-ray data also lag behind the gamma-ray data, and the optical behind the X-ray. Link et al. (1993) and Fenimore et al. (1995) used a sample of BATSE33 (Paciesas et al. 1999) bursts to in- vestigate the relationship between the duration of bursts and the energy band considered. They found that the bursts, and smaller structures within the main emission, generally become shorter with increasing energy (see also Cheng et al. 1995; Norris et al. 1996; in’t Zand & Fen- imore 1996; Piro et al. 1998). Figure 8 plots the auto- correlation function over various X-ray and gamma-ray bands, to reinforce the point that the peak is narrower the harder the band – over X-ray as well as gamma-ray energies. Comparison of the light-curves over the differ- ent energy bands in Figure 1 demonstrates this as well. A similar behaviour was also found for GRB 060124, where Romano et al. (2006) compared the T90 values obtained for the main burst over the X-ray and gamma-ray bands. Fenimore et al. (1995) found that the width of the auto- correlation function, W ∝ E−0.4, where E is the energy at which the function was determined; the six measure- ments from GRB 061121 are consistent with this finding. 2.4.2. Optical – X-rays Using the Swift X-ray and UV/optical data, R and i′ band data from the Faulkes Telescope and Rc data from the Kanata telescope (Uemera et al. 2006), SEDs were produced at epochs corresponding to the peak of the emission (72–75 s post BAT trigger), the plateau stage and during the shallow decay. Fitting at the differ- ent epochs gives an estimation of the broadband spectral variation. For each of the UVOT lenticular filters, the tool uvot2pha was used to produce spectral files compatible with xspec, and for the latter two epochs the count rate 33 Burst And Transient Source Experiment in each band was set to that determined from a power-law fit to the individual filter light curves over the time inter- val in question, using α = 0.68. To determine the Faulkes Telescope R and i′ band flux during the plateau stage, a power law was fitted to the complete data set (220–1229 s post BAT trigger for R and 467–1401 s for i′) with the decay index left as a free parameter. The R magnitude at the mid-time of the shallow stage (6058 s) was deter- mined from the Kanata R-band magnitude reported at 6797 s (Uemera et al. 2006), assuming the same decay index as observed in the UVOT data. An uncertainty of 0.2 mag was assumed as the systematic uncertainty for the photometric calibration of the ground based data. At a redshift of z = 1.314, the beginning of the Lyman- α forest is redshifted to an observer-frame wavelength of ∼ 2812 Å which falls within the UVW1 filter bandpass, the reddest of the UV filters. A correction was applied to the three UV filter fluxes to account for this absorption, based on parameters from Madau (1995) and Madau et al. (1996); see also Curran et al. (in prep). The methods used for simultaneous fitting of the SED components are described in detail in Schady et al. (2007a). The SEDs were fitted with a power-law, or a broken power-law, as expected from the synchrotron emission, and two dust and gas components, to model the Galactic and host galaxy photoelectric absorption and dust extinction. The column density and reddening in the first absorption system were fixed at the Galactic values. [The Galactic extinction along this line of sight is E(B − V ) = 0.046 (Schlegel et al. 1998).] The second photoelectric absorption system was set to the redshift of the GRB, and the neutral hydrogen col- umn density in the host galaxy was determined assuming Solar abundances. The dependence of dust extinction on wavelength in the GRB host galaxy was modelled us- ing three extinction laws, taken from observations of the Milky Way (MW), the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC) and parame- terised by Pei (1992) and Cardelli et al. (1989). The greatest differences observed in these extinction laws are the amount of far UV extinction (which is greatest in the SMC and least in the MW) and the strength of the 2175 Å absorption feature (which is most prominent in the MW and negligible in the SMC). Fitting these data together, a measurement of the spec- tral slope and optical and X-ray intrinsic extinctions (for the second two epochs) were obtained (Table 4); the AV values given in the table are in addition to the AV = 0.151 associated with the Milky Way itself. The slope above the break energy (which lies towards the low energy end of the X-ray bandpass for each phase) was assumed to be exactly 0.5 steeper than the spectral slope below the break (the condition required for a cooling break), since allowing all of the parameters to vary leads to uncon- strained fits. Figure 9 shows, as an example, the fit to the data in the plateau stage. A Milky Way dust extinction law provides the best overall fit to the data, using a broken power-law model, although the LMC model is equally acceptable. During the plateau phase, and adopting the bro- ken power-law model parameters given in Table 4, we find gas-to-dust ratios of (1.6 ± 0.7), (2.6 ± 0.7) and (3.0 ± 0.7) ×1022 cm−2 mag−1 for MW, LMC and SMC GRB 061121: Broadband observations 11 1015 1016 1017 1018 Frequency (Hz) Fig. 9.— Broken power-law fit to the UVOT, XRT and ground- based R and i′ spectral energy distribution of GRB 061121 between ∼ 596–1566 s after the trigger (plateau phase) plotted in the ob- server’s frame. The arrows indicate the beginning of the Lyman-α forest (1215Å in the rest-frame) and the absorption feature in the MW dust extinction law (2175Å), which is shown by a dotted line. The solid line corresponds to the LMC extinction, and the dashed one to the SMC extinction. fits respectively. We can compare these estimates to the measured values for the MW of (4.93 ± 0.45) × 1021 cm−2 mag−1 (Diplas & Savage 1994) and the LMC and SMC of (2.0 ± 0.8) and (4.4 ± 1.1) ×1022 cm−2 mag−1 , respectively (Koornneef 1982; Bouchet et al. 1985). The MW fit to GRB 061121, which is found to be marginally the best model, is consistent with the LMC gas-to-dust ratio only, at the 90% confidence level. The ratios derived from the LMC and SMC fits are consistent with both the LMC and SMC gas-to-dust ratios. We note that all fits are inconsistent with the MW ratio at this confidence level, following the trend seen in pre-Swift bursts (e.g., Starling et al. 2007 and references therein), and that if a metallicity below Solar were adopted, the gas-to-dust ratio of GRB 061121 would increase, moving it further towards the SMC value. 3. DISCUSSION Swift triggered on a precursor to GRB 061121 leading to comprehensive broadband observations of the prompt emission, as well as the later afterglow. We discuss these here, together with possible mechanisms involved. 3.1. Precursor Lazzati (2005) found that about 20% of BATSE bursts showed evidence for gamma-ray emission above the back- ground between 10 to ∼200 s before the main burst, typi- cally with non-thermal spectra which tended to be softer than the main burst. GRB 060124 (Romano et al. 2006) and GRB 061121 show the same behaviour. Precursor models have been proposed for emission well- separated from, or just prior to, the main burst. Early emission occurring only a few seconds before the main burst has been explained by the fireball interacting with the massive progenitor star – though the spectrum of such emission is expected to be thermal (Ramirez-Ruiz et al. 2002a). Lazzati et al. (2007) investigated shocks in a cocoon around the main burst; their model predicts a non-thermal precursor as the jet breaks out of the sur- face of the star. A high-pressure cocoon is formed as the sub-relativistic jet head forces its way out of the star. As the head of the jet breaks through the surface, the energy of this cocoon is released through a nozzle and can give rise to a precursor (Ramirez-Ruiz et al. 2002a,b). Within the framework of this model, observers located at view- ing angles of 5◦ < θ < 11◦ are expected to see first a relatively bright precursor, then a dark phase with lit- tle emission, followed, when the jet enters the unshocked phase, by a bright GRB; this is very similar to the light- curve observed for GRB 061121. Waxman & Mészáros (2003) demonstrate that both a series of thermal X-ray precursors (becoming progressively shorter and harder) and nonthermal emission can be produced by an emerg- ing shocked jet, although the nonthermal component is expected to be in the MeV range. There could also be an accompanying inverse Compton component, formed by the thermal X-rays being upscattered by the jet. The same type of smooth, wide-pulse, low intensity emission as seen in some precursors, but occurring af- ter the main emission is also occasionally seen (e.g., Hakkila & Giblin 2004; Nakamura 2000). Hakkila & Gib- lin (2004) discuss two examples where postcursor emis- sion is found to have a longer lag than expected from the lag-luminosity relation, smoother shape and to be softer. In the case of the GRB 061121 precursor, the spectrum is, indeed, softer than the main event, and shows a compar- atively smooth profile. The emission does have a longer lag than the main emission, but it is still consistent with the lag-luminosity relation. There are two expected effects which could lead to such a difference in lags for separate parts of a single burst: the much lower luminosity for the precursor (resulting from a much smaller Lorentz factor; the measured fluence of the precursor is about a factor of 30 smaller than the fluence of the main emission) is a natural explanation, while the precursor being emitted at a greater off-axis angle could also have an effect. In this second case, ejecta are considered to emerge at different angles with respect to the jet axis; not all of the solid angle of the jet will be ‘filled’ uniformly. Such late postcursor emission is unlikely to be linked to the jet breakout from the stellar surface, and it may not be sensible to attribute apparently similar phenomena (in the form of pre- and postcursors) to entirely different processes. Pre/postcursor emission could be due to the decelera- tion of a faster front shell, resulting in slower shells catch- ing up and colliding with it (Fenimore & Ramirez-Ruiz 1999; Umeda et al. 2005; note, however, that a faster shell would be inconsistent with the precursor having a smaller Lorentz factor as suggested to explain the lag discrepancy), or late activity of the central engine. The presence of flares in about 50% of Swift bursts is gener- ally attributed to continuing activity of the central en- gine (Burrows et al. 2005b; Zhang et al. 2006a) and the appearance of broken power-laws in the X-ray spectra of 12 K.L. Page et al. X-ray Model Extinction NH,z Γ1 Ebreak Γ χ2/dof Epoch (1021 cm−2) (keV) Peak PL SMC 1.6 0.99 ± 0.01 · · · · · · 0.64 25/27 LMC 1.9 1.06 ± 0.01 · · · · · · 0.98 23/27 MW 2.4 1.16 ± 0.01 · · · · · · 1.51 22/27 BKN PL SMC 2.7+9.5 0.72+0.08 −0.15 0.17+0.79 −0.15 1.22 0.51 22/26 LMC 3.0+7.7 0.77+0.08 −0.20 0.18+0.53 −0.17 1.27 0.72 22/26 MW 3.0+7.7 0.77+0.10 −0.21 0.09+0.30 −0.09 1.27 1.03 22/26 Plateau PL SMC 1.42± 0.51 1.58± 0.02 · · · · · · 0.62± 0.05 167/59 LMC 1.98± 0.54 1.64± 0.03 · · · · · · 0.94± 0.08 152/59 MW 2.71± 0.69 1.71± 0.03 · · · · · · 1.39± 0.10 136/59 BKN PL SMC 3.89 +0.72 −1.01 +0.03 −0.02 +0.36 −0.12 1.96 0.52± 0.04 84/58 LMC 4.40 +0.77 −1.30 +0.04 −0.02 +0.16 −0.14 2.01 0.74± 0.06 80/58 MW 3.91 +0.77 −0.75 +0.04 −0.03 +0.25 −0.20 2.08 1.03 +0.09 −0.08 79/58 Shallow PL SMC 2.72± 0.49 1.69± 0.02 · · · · · · 0.65± 0.04 162/77 LMC 3.37 +0.53 −0.49 1.75± 0.03 · · · · · · 0.98 +0.07 −0.06 146/77 MW 4.60 +0.65 −0.60 1.87± 0.04 · · · · · · 1.63 +0.12 −0.11 127/77 BKN PL SMC 4.02+0.62 −0.67 1.58+0.02 −0.03 1.30+0.19 −0.11 2.08 0.50± 0.04 101/76 LMC 4.41+0.69 −0.63 1.62± 0.03 1.30+0.16 −0.14 2.12 0.72± 0.06 99/76 MW 4.78+0.75 −0.65 1.67± 0.04 1.35+0.16 −0.17 2.17 1.02+0.11 −0.10 102/76 a Γ2 is set to be equal to Γ1 + 0.5 in each broken power-law fit, as would be expected if the change in index were due to a cooling break. b In the fit to the peak epoch, AV is fixed to the average best-fit value found in the same model fits to plateau and shallow stage data. The AV values are given for the observer’s frame of reference. TABLE 4 Power-law (PL) and broken power-law (BKN PL) fits to the simultaneous UVOT and XRT spectra of GRB 061121, for three different dust extinction models: Small and Large Magellanic Clouds (SMC and LMC) and the Milky Way (MW). Γ1 and Γ2 are the photon indices below and above the spectral break for the BKN PL models. The data points have not been corrected for reddening. both flares and the prompt emission (Guetta et al. 2006; Goad et al. 2007) hints of a common mechanism. 3.2. Prompt Emission The prompt emission mechanism for GRBs is still de- bated and the origin of Epeak is not fully understood (Mészáros et al. 1994; Pilla & Loeb 1998; Lloyd & Pet- rosian 2000; Zhang & Meszaros 2002; Rees & Mészáros 2005; Pe’er et al. 2005). The standard synchrotron model predicts fast cooling (Ghisellini et al. 2000) with a photon index, Γ, of 3/2 and (p/2)+1 below and above the peak energy, respectively (e.g., Zhang & Mészáros 2004). The Konus-Wind spectral index below Epeak is shallower than 3/2, which may suggest a slow cooling spectrum with p < 2 [Epeak being the cooling frequency and Γ =(p+1)/2] or additional heating. A slow-cooling spectrum can be retained by assuming that the magnetic fields behind the shock decay significantly in 104–105 cm, so that synchrotron emission happens in small scale mag- netic fields (Pe’er & Zhang 2006). The SED at the peak time (SED 2 in Figure 11, discussed below) has a peak flux density of around 1 keV, below which the optical to X-ray spectral slope is 0.11 ± 0.09. This slope is harder than expected from the standard synchrotron model (which predicts an in- dex of 1/3). There should, however, be spectral cur- vature around the break, which could flatten the index (Lloyd & Petrosian 2000), so the data could still be con- sistent with the synchrotron model. An alternative to synchrotron emission, in the form of ‘jitter’ radiation is discussed by Medvedev (2000), though that model pre- dicts an even steeper index of 1 below the jitter break frequency. Figures 4 and 10 show that all three instruments on- board Swift saw the prompt emission around 75 s after the BAT trigger. However, it is noticeable that most of the emission is in the gamma-ray and X-ray bands, with the optical showing a relatively small increase in bright- ness in comparison. Assuming the observed process is synchrotron, then the prompt emission which is detected by the UVOT will be the low-frequency extension of this in the internal shock. No reverse shock is apparent. 3.3. Afterglow Emission 3.3.1. Broken Power-law Decline Models The afterglow of GRB 061121 was observed over an even broader energy range (from radio to X-rays) than the prompt emission, with multi-colour data being ob- tained from ∼ 100–105 s after the trigger. The X-ray light-curve shows evidence for substantial curvature at later times (see Figure 3), as has been found for other Swift GRBs (e.g., GRBs 050315 – Vaughan et al. 2006; 060614 – Gehrels et al. 2006b). The standard practice has been to fit such a decay using a series of power- law segments as a function of time. An alternative exponential-to-power-law description of the light-curve is given in §3.3.2. Nousek et al. (2006) and Zhang et al. (2006a) have both discussed the canonical shape that many Swift af- terglows seem to follow: steep to plateau to shallow, with some light-curves showing a further steepening. In these previous works, the extrapolation of the BAT data into the XRT band was incorporated into the derivation of the steep decay at the start of the canonical light- curve shape. In the case of GRB 061121, the full curve GRB 061121: Broadband observations 13 can be seen entirely in X-rays, suggesting that the pre- vious extrapolations are reliable. For the afterglow of GRB 061121, only data after the end of the main burst have been modelled with power-laws. The early steep decline, which might be attributable to the curvature ef- fect (Kumar & Panaitescu 2000; Dermer 2004; Fan & Wei 2005), is not considered here. According to the model proposed in Nousek et al. (2006) and Zhang et al. (2006a), the plateau phase of the light-curve is due to energy injection in the fireball. The plateau phase of GRB 061121 is consistent with an injection of energy since the luminosity index, q, is nega- tive, which is the requirement for injection to modify the afterglow (Zhang et al. 2006a); the later two stages both have q > 1. However, as will be discussed in §3.3.2, the plateau and final transition to the power-law decay are only visible in the X-ray data for GRB 061121; the start of the final decay is much earlier in the V and R-bands (see Figure 10). One might expect that energy injection would affect all the energy bands simultaneously, rather than just the X-rays. From the standard afterglowmodel computations (e.g., Zhang & Mészáros 2004), we find that none of the closure relations fit the entire dataset completely: although the shallow phase (after the end of energy injection, between T + 2.3 ks and T + 32 ks) could be consistent with the evolution of a blast-wave which had already entered the slow cooling regime when deceleration started [i.e., ν > max(νm, νc) where νc is the cooling frequency and νm is the synchrotron injection frequency; Sari et al. 1998; Chevalier & Li 2000], the steeper part of the decay curve (T> 32 ks) is not consistent with any of the models. This lack of consistency suggests that a different approach is required. The change in decay slope between the shallow to steep phases (∼ 32 ks) cannot be easily identified with a jet-break. It certainly seems unlikely that the simplest side-spreading jet model could be applicable, since the post-break decay index (α ∼ 1.5) is not steep enough (a post-jet decay has α = p, where p is the electron in- dex). There is also some indication that the X-ray spec- tral slope hardens after the break, whereas no change in spectral signature is expected over a jet-break. In the case of a non-laterally expanding jet (Panaitescu & Mészáros 1999), α = (3β/2) + 0.25 [for a homoge- neous circumstellar medium (CSM); Panaitescu et al. (2006)], which does, indeed, fit the data after this break: [1.5 × (0.9 ± 0.08)] + 0.25 = 1.6 ± 0.1; the measured α is 1.53. Such a confined jet has been suggested as an explanation for the observed decay in a number of pre- vious bursts (e.g., GRB 990123 – Kulkarni et al. 1999; GRB 050525A – Blustin et al. 2006; GRB 061007 – Schady et al. 2007b). The UVOT data obtained around this time show little evidence for a break, whereas jet breaks should occur across all energy bands simultane- ously. However non-simultaneity could be explained by a multi-component outflow, where the X-ray emission is produced within a narrow jet, while the optical compo- nent comes from a wider jet with lower Lorentz factor (Panaitescu & Kumar 2004; Oates et al. 2007). There remains the issue, however, that α should steepen by 0.75 over a jet break (Mészáros & Rees 1999), whereas the maximum observed change (within the 90% errors) is only ∆α < 0.61, excluding ∆α = 0.75 at almost 3σ; also, again there should be no spectral evolution across the break. There is, however, a probable jet break at later times, which will be covered in the next Section. Other multi-component models [see, e.g., Oates et al. (2007) and references therein] also fail to explain the data, because of the lack of observed energy injection (plateau phase) in the optical data. Panaitescu et al. (2006a) discuss chromatic breaks in Swift light-curves, and postulate that these could be due to a change in microphysical parameters within a wind environment. However, this model requires the cooling frequency to lie between the X-ray and optical bands and, as will be discussed in §3.3.2, this does not seem to be the case here. 3.3.2. Exponential-to-power-law Decline Model As first described by O’Brien et al. (2006), and further expanded by Willingale et al. (2007), GRB light-curves can be well modelled by one or two components com- prised of an early exponential rise followed by a power- law decay phase. Of these components, the first repre- sents the prompt gamma-ray emission and early X-ray decay. The second, when detected, dominates at later times, forming what we see as the afterglow. These re- sults show that fitting an intrinsically curved decay with multiple power-law segments runs the risk of incorrectly identifying temporal breaks (see also Sakamoto et al. in prep). In this Section the models of O’Brien et al. (2006) and Willingale et al. (2007) are applied to the multi-band afterglow data of GRB 061121. Figure 10 brings together the BAT, XRT, UVOT, FTN and ROTSE data, along with further optical and ra- dio points taken from the GCN Circulars (Halpern et al. 2006a,b; Halpern & Armstrong 2006a,b; Chandra & Frail 2006; van der Horst et al. 2006a,b) and the upper limit from Chandra, to form a multi-energy decay plot. The data have been plotted as ‘time since trigger + 4 s’ in order to include the precursor on a log time-scale. The optical points have all been corrected for extinction using AV = 1.2 (a combination of the Galactic value of 0.151 and an estimate of AV ∼ 1 for the GRB host galaxy – see §2.4.2). The contribution from the host galaxy reported by Malesani et al. (2006) and Cobb (2006) has been sub- tracted from the V - and R-band flux values. The magni- tude of the host in the V -band is 22.4, which only changes the last two or three V -band points by a small amount. For the R-band we have no direct measurement, but the last group of MDM exposures gave an R magnitude of 22.7, corresponding to a flux level of 2.8 µJy, and the flux level is still declining at that epoch (∼ 3.3 × 105 s), so an R-band flux level of 2.5 µJy was adopted for the host. The error bars shown on the last few points reflect the large uncertainty in the galaxy contribution subtracted. The curved dotted lines in Figure 10 are the fits to the data using the exponential-to-power-law model, fol- lowed by a break to a steeper decay around 105 s. These models are parameterised by the power-law decay, α, and Ta, the time at which this decay is established. For the X-ray data, Ta,X is found to be 5250 −460 s and αa,X = 1.32 ± 0.03. Fits were also performed to the V - and R-band data, yielding: αa,V = 0.66 ± 0.04 (with Ta,V = 70 −70 s) and αa,R = 0.84 ± 0.03 14 K.L. Page et al. Fig. 10.— Flux density light-curves for the gamma-ray, X-ray, optical and radio data obtained for GRB 061121. The vertical dotted lines indicate the times used for the SED plots shown in Figure 11, while the curved dotted lines show the fit to the X-ray and optical data, including a late-time break, as described in the text. Fig. 11.— SEDs for the four time intervals indicated in Figure 10. SED 2 (the peak of the burst emission) includes the Konus-Wind data, although these have not been included in Figure 10. The solid lines represent the power-law fits to the BAT, XRT and Konus data, while the dashed lines join the radio, optical and 1 keV points. Spectral evolution over time is clearly seen. (Ta,R = 230 −230 s). The non-detection by Chandra almost two months af- ter the burst shows there must have been a further steep- ening in the X-ray regime, and the optical data are not inconsistent with this finding. Constraining the tempo- ral index after the late break to be α = 2 (a typical slope for a post-jet-break decay), break times of ∼ 2.5 × 105, ∼ 2.5 × 104 and ∼ 105 s are estimated for the X-ray, V - and R-band respectively; note that the UVOT V -band value is particularly uncertain, given the small number of data points at late times. Within the uncertainties, these times are likely to be consistent, so the turnover could be achromatic, as required for a jet break. From Willingale et al. (2007), a jet break might be expected at ∼ 100 × Ta,X – i.e., 5.5 × 10 5 s, which is in agreement with these fits. As can be seen from these numbers and the mod- els plotted in Figure 10, the X-ray data clearly show the transition from the plateau to the power-law decay, whereas the start of the final decay is much earlier in the V - and R-bands. The V -band decay is also significantly flatter (by α ∼ 0.2) than that estimated for the R-band. As previously stated, the V , B and U light-curves are all consistent with this slow decay. There have been few multi-colour optical decay curves obtained for GRB af- terglows, and, of these, the different filters [in the case of GRB 061007 (Schady et al. 2007; Mundell et al. 2007) X-ray and gamma-ray data as well as the optical] tend to track each other (e.g., Guidorzi et al. 2005; Blustin et al. 2006; de Ugarte Postigo et al. 2007). In the case of GRB 061121, we find that the R-band data are fading more rapidly than the V . GRB 060218, which was as- sociated with a supernova (e.g., Campana et al. 2006a), shows changes throughout the optical spectra, because of a combination of shock break-out and radioactive heat- ing of the supernova ejecta. There is a large difference between the decays of the blue (V , U , B) and red (R) data for GRB 061121, which cannot be easily explained by a synchrotron spectrum. Although no supernova has been detected in this case, we speculate that some form of pre-supernova thermal emission could possibly be af- fecting the optical data, adding energy into the blue end of the spectrum, thus slowing its decline. After the break in the decays around 105 s, the light- curves across all bands become more consistent with one GRB 061121: Broadband observations 15 another, although there are only limited data at such a late time. The vertical dotted lines in Figure 10 show the times of the SEDs plotted in Figure 11; again, all points were corrected for an extinction of AV = 1.2, so that they represent the true SEDs (with the frequency in the ob- server’s frame). The solid lines represent actual fits to the X-ray and gamma-ray data, while the dashed lines just join the separate radio, optical and 1 keV points. The times of these SEDs, which clearly show spectral evolution, correspond to (1) before the main BAT peak, 56 s after trigger; (2) at the BAT peak, 76 s after trig- ger; (3) just after the start of the plateau, 300 s after the trigger; (4) in the main decay at 65 ks (chosen because radio measurements were taken at this time). SEDs 3 and 4 do not contain any BAT or Konus data, since the gamma-ray flux had decayed by this point; the highest energy point in these corresponds to the maximum en- ergy (10 keV) of the X-ray fits. Table 4 demonstrates that the optical and X-ray spec- tra during the peak emission are best fitted with a broken power-law model, with the break energy at the very low energy end of the X-ray bandpass. SED 2 in Figure 11 shows that this spectral break corresponds to the peak frequency in a flux density plot (β1 is less than zero in this case). Only during SED 2 is the optical flux density lower than that of the higher energy data. Figure 4 also shows that the optical emission is less strong than the X-ray and gamma-ray data during the main burst. Table 5 shows the values of α for the X-ray and opti- cal decays (i.e., before and after the break) in SED 4, at 65 ks, with their corresponding spectral indices. For the initial stages of the power-law decay (Ta < t < 65000 s) the evolution of the afterglow SED and the coupling be- tween the temporal and spectral indices are not com- pletely consistent with the standard model: although the R-band decay, with αa,R = 0.84 ± 0.03, is in good agree- ment with the homogeneous CSM model below the cool- ing break, the X-ray and V -band flux decays are slower than expected from the measured spectral indices; they are in best agreement with the same constant density model below νc, however. The point at which the power-law decay dominates the exponential in the optical bands is noticeably earlier than in the X-ray (< few hundred seconds, rather than ∼ 5000 s) and, as mentioned above, the decay indices are significantly different for all three (X-ray, V and R) bands (see Figure 10). At the time of SED 3, the X-ray data are not decaying (i.e., this is during the plateau), yet both the V and R-band data have already entered the power-law decline phase. The R-band is decaying faster than the V -band, so the spectral index through the optical range is becoming harder. The X-ray spec- tral index shows a similar hardening trend (see Table 3), so the SED measured from optical to 10 keV is gradually getting harder. Such spectral hardening from the plateau to the final decay is a feature of many X-ray afterglows (Willingale et al. 2007). This slow hardening of the broadband spectrum with time could be a signature of synchrotron self-Compton emission (Sari & Esin 2001; Panaitescu & Kumar 2000). The strength of the self-Compton component in the af- terglow depends on the flux of low energy photons (radio- optical) and the electron density in the shock. Using the formulation in Sari & Esin (2001) the density required is given by n1 = 3× 10 f ICmax (E52tday) −1/3cm−3 (1) where f ICmax/f max is the ratio of the peak flux of the seed synchrotron spectrum (i.e., the source of low energy photons) and the peak flux of the self-Compton emis- sion; E52 is the isotropic burst energy in units of 10 52 erg; tday is the time in days after the burst (which determines the distance through the CSM swept up by the external shock). From Figure 11 (SEDs 1, 3 and 4) we see that f ICmax/f max ∼ 0.001 if the X-ray flux has a significant con- tribution from a self-Compton component at tday = 0.75. A value of E52 = 30 gives n1 ≈ 10 5 cm−3. Even as- suming the emission at 0.75 days is not dominated by the self-Comptonisation, and so taking the f ICmax/f ratio to be a factor of ten smaller, the density would be ∼ 5× 103 cm−3, which is still high. It seems unlikely that self-Compton emission is the cause of the spectral hard- ening of the SED unless the CSM density encountered by the external shock is extremely large. However, there have been suggestions that GRBs may form in molecular clouds (Galama & Wijers 2001; Campana et al. 2006b,c), which have densities of 104 or more particles per cubic centimetre in the cores (Miyazaki & Tsuboi 1999; Wil- son et al. 1999). Typically one might expect greater red- dening than is found here (Table 4), though Waxman & Draine (2000) discuss the possibility of dust destruction. The spectrum will be redshifted as the jet slows down, so the optical and X-ray spectral indices should, if any- thing, become softer – the opposite of what is seen here. Although spectral hardening with time is suggested from the data, it is not be easily explained by current models. Whether or not there is a Comptonised component, the later SEDs clearly indicate that there is a break in the spectrum somewhere between the optical and the X- ray; this is also shown by the fits in Table 4, where the UVOT–XRT spectra are better fitted with broken power- laws, with Ebreak towards the low energy end of the X-ray bandpass. Since both the optical and X-ray bands ap- pear to be below the cooling frequency, from the closure relations given in Table 5, this change in slope cannot be identified with a cooling break; its origin remains un- clear. The redshift of z = 1.314 and the isotropic energy of Eiso ∼ 3 × 10 53 erg (§2.4.1) can be used to place con- straints on the jet opening angle. From Sari et al. (1999), and assuming that the jet break occurs at T0+2× 10 we have θj ∼ 4 )1/8 ( n where n and ηγ are the density of the CSM and the efficiency of the fireball in converting the energy in the ejecta into gamma-rays. Taking ηγ = 0.2 and n = 3 cm −3 (following Ghirlanda et al. 2004), this gives Eγ ∼ 1.7 × 10 51 erg for the beaming- corrected gamma-ray energy released, which is within the range previously determined (e.g., Frail et al. 2001) and consistent with the Ghirlanda relationship (Ghirlanda et al. 2004). 4. SUMMARY AND CONCLUSIONS Swift triggered on a precursor to GRB 061121, leading 16 K.L. Page et al. GRB models α(β) α(βa,X) α(βopt) a αbopt V -band R-band CSM SCc (νm < ν < νc) 1.49 ± 0.10 1.32 ± 0.03 0.80 ± 0.09 0.66 ± 0.04 0.84 ± 0.03 Wind SCc (νm < ν < νc) 1.99 ± 0.10 1.30 ± 0.09 CSM or Wind SCc & FCd 0.99 ± 0.10 0.30 ± 0.09 (ν > max(νc, νm)) a Decay calculated from the measured spectral index b Observed power-law decay index. c Slow cooling. d Fast cooling. TABLE 5 Closure relations for exponential-plus-power-law model fits to the X-ray data (βa,X = 0.99± 0.07) and the optical-to-X-ray band (βopt = 0.53 ± 0.06) from the time of SED 4 (65 ks after the burst). to unprecedented coverage of the prompt emission by all three instruments onboard, with the gamma-ray, X-ray and optical/UV bands all tracking the main peak of the burst. GRB 061121 is the instantaneously brightest long Swift burst detected thus far, both in gamma-ray and X- rays. The precursor and main burst show spectral lags of different lengths, though both are consistent with the lag- luminosity relation for long GRBs (Gehrels et al. 2006b). The SED of the prompt emission, stretching from 1 eV to 1 MeV shows a peak flux density at around 1 keV and is harder than the standard model predicts. There is def- inite curvature in the spectra, with the prompt optical- to-X-ray spectrum being better fitted by a broken power- law, similar to results found for fitting X-ray flares (e.g., Guetta et al. 2006; Goad et al. 2007). The afterglow component, in both the optical and X-ray, starts early on – before, or during, the main burst peak (see also O’Brien et al. 2006; Willingale et al. 2007; Zhang et al. 2006b). The broadband SEDs reveal gradual spectral hardening as the afterglow evolves, both within the X-ray regime (Γ flattening from ∼ 2.05 to ∼ 1.87) and between the V - and R-band op- tical data (αV ∼ 0.66 compared with αR ∼ 0.84). Self- Comptonisation could explain the hardening, although a molecular-cloud-core density would be required. A prob- able jet-break occurs around T0 + 2 × 10 5 s, shown by a late-time non-detection by Chandra. Before this break, the X-ray and V -band decays are too slow to be readily explained by the standard models. This extremely well-sampled burst shows clearly that there remains much work to be done in the field of GRB models. A single, unified model for all GRB emission observed should be the ultimate goal. 5. ACKNOWLEDGMENTS The authors gratefully acknowledge support for this work at the University of Leicester by PPARC, at PSU by NASA and in Italy by funding from ASI. This work is partly based on observations with the Konus-Wind ex- periment (supported by the Russian Space Agency con- tract and RFBR grant 06-02-16070) and on data ob- tained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. We thank the Liverpool GRB group at ARI, Liverpool John Moores University, in par- ticular C.J. Mottram, D. Carter, R.J. Smith and A. Gomboc for their assistance with the FTN data acqui- sition and interpretation. The Faulkes Telescopes are operated by the Las Cumbres Observatory Global Tele- scope Network. We also thank J.E. 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Swift triggered on a precursor to the main burst of GRB 061121 (z=1.314), allowing observations to be made from the optical to gamma-ray bands. Many other telescopes, including Konus-Wind, XMM-Newton, ROTSE and the Faulkes Telescope North, also observed the burst. The gamma-ray, X-ray and UV/optical emission all showed a peak ~75s after the trigger, although the optical and X-ray afterglow components also appear early on - before, or during, the main peak. Spectral evolution was seen throughout the burst, with the prompt emission showing a clear positive correlation between brightness and hardness. The Spectral Energy Distribution (SED) of the prompt emission, stretching from 1eV up to 1MeV, is very flat, with a peak in the flux density at ~1keV. The optical-to-X-ray spectra at this time are better fitted by a broken, rather than single, power-law, similar to previous results for X-ray flares. The SED shows spectral hardening as the afterglow evolves with time. This behaviour might be a symptom of self-Comptonisation, although circumstellar densities similar to those found in the cores of molecular clouds would be required. The afterglow also decays too slowly to be accounted for by the standard models. Although the precursor and main emission show different spectral lags, both are consistent with the lag-luminosity correlation for long bursts. GRB 061121 is the instantaneously brightest long burst yet detected by Swift. Using a combination of Swift and Konus-Wind data, we estimate an isotropic energy of 2.8x10^53 erg over 1keV - 10MeV in the GRB rest frame. A probable jet break is detected at ~2x10^5s, leading to an estimate of ~10^51 erg for the beaming-corrected gamma-ray energy.

Draft version November 21, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 GRB 061121: BROADBAND SPECTRAL EVOLUTION THROUGH THE PROMPT AND AFTERGLOW PHASES OF A BRIGHT BURST. K.L. Page1, R. Willingale1, J.P. Osborne1, B. Zhang2, O. Godet1, F.E. Marshall3, A. Melandri4, J.P. Norris5,6, P.T. O’Brien1, V. Pal’shin7, E. Rol1, P. Romano8,9, R.L.C. Starling1, P. Schady10, S.A. Yost11, S.D. Barthelmy3, A.P. Beardmore1, G. Cusumano12, D.N. Burrows13, M. De Pasquale10, M. Ehle14, P.A. Evans1, N. Gehrels3, M.R. Goad1, S. Golenetskii7, C. Guidorzi8,9, C. Mundell4, M.J. Page10, G. Ricker15, T. Sakamoto3, B.E. Schaefer16, M. Stamatikos3, E. Troja1,12, M.Ulanov7, F. Yuan11 & H. Ziaeepour9 Draft version November 21, 2018 ABSTRACT Swift triggered on a precursor to the main burst of GRB 061121 (z = 1.314), allowing observations to be made from the optical to gamma-ray bands. Many other telescopes, including Konus-Wind, XMM-Newton, ROTSE and the Faulkes Telescope North, also observed the burst. The gamma-ray, X-ray and UV/optical emission all showed a peak ∼ 75 s after the trigger, although the optical and X-ray afterglow components also appear early on – before, or during, the main peak. Spectral evolution was seen throughout the burst, with the prompt emission showing a clear positive correlation between brightness and hardness. The Spectral Energy Distribution (SED) of the prompt emission, stretching from 1 eV up to 1 MeV, is very flat, with a peak in the flux density at ∼ 1 keV. The optical- to-X-ray spectra at this time are better fitted by a broken, rather than single, power-law, similar to previous results for X-ray flares. The SED shows spectral hardening as the afterglow evolves with time. This behaviour might be a symptom of self-Comptonisation, although circumstellar densities similar to those found in the cores of molecular clouds would be required. The afterglow also decays too slowly to be accounted for by the standard models. Although the precursor and main emission show different spectral lags, both are consistent with the lag-luminosity correlation for long bursts. GRB 061121 is the instantaneously brightest long burst yet detected by Swift. Using a combination of Swift and Konus-Wind data, we estimate an isotropic energy of 2.8 × 1053 erg over 1 keV – 10 MeV in the GRB rest frame. A probable jet break is detected at ∼ 2 × 105 s, leading to an estimate of ∼ 1051 erg for the beaming-corrected gamma-ray energy. Subject headings: gamma-rays: bursts — X-rays: individual (GRB 061121) 1. INTRODUCTION Electronic address: kpa@star.le.ac.uk 1 Department of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH, UK 2 Department of Physics & Astronomy, University of Nevada, Las Vegas, NV 89154-4002, USA 3 NASA/Goddard Space Flight Center, Greenbelt, MD 20771, 4 Astrophysics Research Institute, Liverpool John Moores Uni- versity, Twelve Quays House, Egerton Wharf, Birkenhead, CH41 5 Denver Research Institute, University of Denver, Denver, CO 80208, USA 6 Visiting Scholar, Stanford University 7 Ioffe Physico-Technical Institute, Laboratory for Experimen- tal Astrophysics, 26 Polytekhnicheskaya, Saint Petersburg 194021, Russian Federation 8 INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807, Merate (LC), Italy 9 Dipartimento di Fisica, Universitá di Milano-Bicocca, Piazza delle Scienze 3, I-20126, Milano, Italy 10 Mullard Space Science Laboratory, University College Lon- don, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK 11 University of Michigan, 2477 Randall Laboratory, 450 Church St., Ann Arbor, MI 48104, USA 12 INAF-IASF, Sezione di Palermo, via Ugo La Malfa 153, 90146, Palermo, Italy 13 Department of Astronomy and Astrophysics, Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA 14 XMM-Newton Science Operations Centre, European Space Agency, Villafranca del Castillo, Apartado 50727, E-28080 Madrid, Spain 15 Center for Space Research, Massachusetts Institute of Tech- nology, 70 Vassar Street, Cambridge, MA 02139, USA 16 Department of Physics and Astronomy, Louisiana State Uni- versity, Baton Rouge, LA 70803, USA Gamma-Ray Bursts (GRBs) are intrinsically extremely luminous objects, approaching values of 1054 erg s−1 if the radiation is isotropic (e.g., Frail et al. 2001; Bloom et al. 2003). This energy is emitted over all bands in the electromagnetic spectrum; to understand GRBs as fully as possible, panchromatic observations are required over all time frames of the burst. The Swift multi-wavelength observatory (Gehrels et al. 2004) is designed to detect and follow-up GRBs. With its rapid slewing ability, Swift is able to follow bursts and their afterglows from less than a minute af- ter the initial trigger, and can often still detect them weeks, and sometimes months, later. On rare occa- sions, such as when Swift triggers on a precursor to the main burst, the prompt emission, as well as the after- glow, can be observed at X-ray and UV/optical wave- lengths. GRB 061121, the subject of this paper, is only the third GRB Swift has detected in this manner (af- ter GRB 050117 – Hill et al. 2006 and GRB 060124 – Romano et al. 2006), out of the almost 200 bursts trig- gered on in the first two years of the mission.17 Of these, GRB 061121 is the second well-sampled event (GRB 060124 was the first), and the first for which the UV/Optical Telescope (UVOT) was in event mode. In addition to the small number of precursor triggers, around 10% of Swift bursts show detectable emission over 17 GRB 050820A would possibly have also been in this category, but Swift entered the South Atlantic Anomaly (SAA) just as a dramatic increase in count rate began (Cenko et al. 2006; Page et al. 2005a; Cummings et al. 2005; Page et al. 2005b; Chester et al. 2005); Swift does not actively collect data during SAA passages. http://arxiv.org/abs/0704.1609v1 mailto:kpa@star.le.ac.uk 2 K.L. Page et al. the BAT bandpass by the time the narrow field instru- ments (NFIs) are on target. Besides the Swift observations of prompt emission, there have been a small number of prompt optical mea- surements of GRBs, thanks to the increasing number of robotic telescopes around the world. A variety of behaviours has been found, with some optical (and in- frared) light-curves tracking the gamma-ray emission (e.g., GRB 041219A – Vestrand et al. 2005; Blake et al. 2005), while others appear uncorrelated (e.g., GRB 990123 – Akerlof et al. 1999, Panaitescu & Kumar 2007, though see also Tang & Zhang 2006; GRB 050904 – Boër et al. 2006; GRB 060111B – Klotz et al. 2006; GRB 060124 – Romano et al. 2006). GRB 050820A (Ves- trand et al. 2006) showed a mixture of both correlated and uncorrelated optical flux. Where correlations exist between different energy bands, it is likely that there is a common origin for the components. In the uncorrelated cases, the opti- cal emission may be due to an external reverse shock (e.g., Sari & Piran 1999; Mészáros & Rees 1999), while the prompt gamma-rays are caused by internal shocks. Cenko et al. (2006) suggest that the early optical data for GRB 050820A are produced by the forward shock passing through the band. In the case of GRB 990123, Panaitescu & Kumar (2007) have suggested that the gamma-rays arose from inverse Comptonisation, while the optical emission was due to synchrotron processes; they do not assume a specific mechanism for the energy dissipation, allowing for the possibility of either internal or reverse-external shocks. It is unclear whether precursors are ubiquitous fea- tures of GRBs, often remaining undetected because of a low signal-to-noise ratio or being outside the energy bandpass of the detector, or whether only some bursts exhibit them. A detailed discussion of the precursor phe- nomenon is beyond the scope of this paper and will be addressed in a future publication. In this paper, we report on the multi-wavelength ob- servations of both the prompt and afterglow emission of GRB 061121. §2 details the observations made by Swift, Konus-Wind, XMM-Newton, ROTSE18 and the Faulkes Telescope North (FTN), with multi-band comparisons being made. In §3, we discuss the precursor, prompt and afterglow emission, with a summary given in §4. Throughout the paper, the main burst (∼ 60–200 s after the trigger) will be referred to as the prompt emis- sion, and the emission seen over −5 to +10 s as the precursor, where the BAT trigger time T0 = 0 s. Er- rors are given at 90% confidence (e.g., ∆χ2 = 2.7 for one interesting parameter) unless otherwise stated, and the convention Fν,t ∝ ν −βt−α (with the photon spec- tral index, Γ = β + 1 where dN/dE ∝ E−Γ) has been followed. We have assumed a flat Universe, with Hubble constant, H0 = 70 km s −1 Mpc−1, cosmological constant, ΩΛ = 0.73 and Ωmatter = 1−ΩΛ. 2. OBSERVATIONS AND ANALYSES Two years and one day after launch, the Burst Alert Telescope (BAT; Barthelmy et al. 2005) triggered on a precursor to GRB 061121 at 15:22:29 UT on 21st Novem- ber, 2006. Swift slewed immediately, resulting in the 18 Robotic Optical Transient Search Experiment NFIs being on target and beginning to collect data 55 s (X-ray Telescope: XRT; Burrows et al. 2005a) and 62 s (UVOT; Roming et al. 2005) later. This enabled broad- band observations of the main burst event, which peaked ∼ 75 s after the trigger, leading to spectacular multi- wavelength coverage of the prompt emission. The most accurate Swift position for this burst was that determined by the UVOT: RA = 09h 48m 54.s55, decl = −13◦ 11′ 42.′′4 (J2000.0; 90% confidence radius of 0.′′6; Marshall et al. 2006); the refined XRT position is only 0.′′1 from these coordinates (Page et al. 2006b). GRB 061121 was declared a ‘burst of interest’ by the Swift team (Gehrels et al. 2006a), to encourage an inten- sive ground- and space-based follow-up programme. In addition to the Swift observations, the prompt emission of GRB 061121 was detected by RHESSI19 (Bellm et al. 2006), Konus-Wind and Konus-A (Golenetskii et al. 2006). Later afterglow observations were obtained in the X-ray (XMM-Newton – Schartel 2006) and radio (VLA20 – Chandra & Frail 2006) bands. ATCA21 and WSRT22 also observed in the radio band between ∼5.2 day and ∼6.2 day after the burst, but did not detect the after- glow (van der Horst et al. 2006a,b), implying it had faded since the VLA observation. Likewise, extensive optical follow-up observations were performed: ROTSE-IIIa (Yost et al. 2006), FTN (Me- landri et al. 2006), Kanata 1.5-m telescope (Uemura et al. 2006), the University of Miyazaki 30-cm telescope (Sonoda et al. 2006), MDM23 (Halpern et al. 2006a,b; Halpern & Armstrong 2006a,b), P6024 (Cenko 2006), ART25 (Torii 2006), the CrAO26 2.6-m telescope (Efimov et al. 2006a,b) and SMARTS/ANDICAM27 (at infrared wavelengths, too; Cobb 2006) all detected the optical afterglow. Spectroscopic observations were performed at the Keck telescope about 12 minute after the trig- ger, finding a redshift of z = 1.314 for the optical af- terglow, based on absorption features (Perley & Bloom 2006; Bloom et al. 2006). GRB 061121 has the highest instantaneous peak flux of all the long bursts detected by Swift to date (e.g., Angelini et al. in prep). 2.1. Gamma-ray Data 2.1.1. BAT Temporal Analysis— After the initial precursor, the BAT count rate returned to close to the instrumental back- ground level, until T0+60 s, at which point the much brighter main burst began. This is characterised by a series of overlapping peaks, each brighter than the previ- ous one, after which the gamma-ray flux decayed (from ∼T0+75 s to ∼T0+140 s). Event data were collected un- til almost 1 ks after the trigger, thus covering the entire emission period. 19 Reuven Ramaty High Energy Solar Spectroscopic Imager 20 Very Large Array 21 Australia Telescope Compact Array 22 Westerbork Synthesis Radio Telescope 23 Michigan-Dartmouth-MIT Observatory 24 Palomar 60 inch 25 Automated Response Telescope 26 Crimean Astrophysical Observatory 27 Small and Moderate Aperture Research Telescope System/A Novel Double-Imaging CAMera GRB 061121: Broadband observations 3 T90, over 15-150 keV, and incorporating both the pre- cursor and main emission, is 81 ± 5 s, measured from 8.8–89.8 s after the trigger28. Figure 1 shows the mask- weighted BAT light-curve in the four standard energy bands [15–25, 25–50, 50–100, 100-150 keV; 64 ms bin- ning between 50-80 s after the trigger, with 1 s bins at all other times; units of count s−1 (fully illuminated detector)−1], with light-curves from other instruments: the precursor and the pulses of the main burst are de- tected over all gamma-ray bands, although the precursor is only marginal over the 100-150 keV BAT band. There is also a soft tail (detected below ∼ 50 keV, when suffi- ciently coarse time bins are used) visible until about 140 s after the trigger (see bottom panel of Figure 1), corre- sponding to a similar feature in the X-ray light-curves. Spectral analysis— For the precursor, T90,pre = 7.7 ± 0.5 s (15–150 keV). A spectrum ex- tracted over this interval can be well modelled by a single power-law, with Γ = 1.68 ± 0.09 (χ2/dof = 26.2/23); no significant improvement was found by using the Band function (Band et al. 1993) or a cut-off power-law and a thermal model led to a slightly (χ2 ∼ 8) worse fit. The 15–150 keV fluence for this time interval is 4 × 10−7 erg cm−2. Considering only the main event, T90,main = 18.2 ± 1.1 s (measured from 61.8–80.0 s post-trigger). Fitting a power-law to the mean spectrum during this time also results in a good fit (Γ = 1.40 ± 0.01; fluence = 1.31 × 10−5 erg cm−2 over 15–150 keV; χ2/dof =51.6/56 ); again, neither the Band function nor a cut-off power-law improves upon this. There is significant spectral evolution during the T90 period, as shown in Figure 2: at times when the count rate is higher, the spectrum is harder. This behaviour was also common in earlier bursts, as well as previous Swift detections (e.g. Golenetskii et al. 1983; Ford et al. 1995; Borgonovo & Ryde 2001; Goad et al. 2007). The precursor shows a similar dependence of hardness ratio on count rate, suggesting that the emission processes in the precursor and the main burst are the same or similar. 2.1.2. Konus-Wind Temporal Analysis— Konus-Wind (Aptekar et al. 1995) triggered on the main episode of GRB 061121, while Konus-A triggered on the precursor (Golenetskii et al. 2006). Because of the spatial separation of Swift and Wind, the light travel-time between the spacecraft is 1.562 s: the Konus-Wind trigger time, T0,K−W = T0,BAT + 61.876 s. All Konus light-curves have been plotted with respect to the BAT trigger, corrected for the light travel-time. Figure 1 shows the Konus-Wind data plotted over the standard energy bands, with 64 ms binning; the bottom panel plots the coarser time resolution (2.944 s) ‘waiting mode’ data, showing that Konus-Wind did see slightly enhanced emission at the time of the precursor. The background levels (which have been subtracted in each case) were 1005, 370 and 193.4 count s−1 for bands 21–83, 83–360 and 360–1360 keV, respectively. 28 Errors on the BAT T90 are estimated to be typically 5–10%, depending on the shape of the light-curve. Spectral analysis— Table 1 gives the spectral fits to the Konus-Wind data in three separate time intervals shown by vertical lines in Figure 1 (Konus-Wind spec- tral intervals are automatically selected onboard): up to the end of the ‘bump’ around 70 s (the ‘start’ of the burst), the burst maximum and, finally, until most of the emission has died away (the burst tail). The data were fitted with a cut-off power-law, where dN/dE ∼ E−Γ × e[−(2−Γ)E/Epeak], leading to the photon indices and peak energies given in the table. The Band function was used to estimate upper limits for the pho- ton index above the peak; the values for the peak energy and Γ obtained from the Band function were the same as when fitting the cut-off power-law. Little variation in the spectral slope for energies below the peak is seen over these intervals, though the peak itself may have moved to somewhat higher energies during the burst emission. Ex- tracting BAT spectra over the same time intervals, and fitting with the same model (fixing Epeak at the value determined from the Konus-Wind data) results in con- sistent spectral indices. 2.2. X-ray Data 2.2.1. XRT Temporal Analysis— The XRT identified and centroided on an uncatalogued X-ray source in a 2.5 s Image Mode (IM) frame, as soon as the instrument was on target. This was quickly followed by a pseudo Piled-up Photo Diode (PuPD) mode frame. Following damage from a micrometeoroid impact in May 2005 (Abbey et al. 2005), the Photo Diode mode (Low Rate and Piled-up) has been disabled [see Hill et al. (2004) for details on the different XRT modes]; however, the XRT team are currently work- ing on a method to re-implement these science modes and to update the ground software to process the files. The pseudo PuPD point presented here is the first use of such data. Data were then collected in Windowed Timing (WT) mode starting at a count rate of ∼ 1280 count s−1 (pile- up corrected – see below); the rate rapidly increased to a maximum of ∼ 2500 count s−1 at T0 + 75 s, mak- ing GRB 061121 the brightest burst yet detected by the XRT. Following this peak, the count-rate decreased, with a number of small flares superimposed on the underlying decay (see Figure 1). Photon Counting (PC) mode was automatically selected when the count rate was below about 10 count s−1. Around 1.5 ks, the XRT switched back into WT mode briefly, due to an enhanced back- ground linked to the sunlit Earth and a relatively high CCD temperature. Because of the high count rate, the early WT data were heavily piled-up; see Romano et al. (2006) for informa- tion about pile-up in this mode. To account for this, an extraction region was used which excluded the central 20 pixels (diameter; 1 pixel = 2.′′36) and extended out to a total width of 60 pixels. Likewise, the first three orbits of PC data were piled-up, and the data were thus extracted using annular regions (inner exclusion diame- ter decreasing from 12 to 6 to 4 pixels as the afterglow faded; outer diameter 60 pixels). The count rate was then corrected for the excluded photons by a comparison of the Ancillary Response Files (ARFs) generated with and without a correction for the Point Spread Function 4 K.L. Page et al. White UVOT 0.3−2 keV XRT 2−10 keV 15−25 keV BAT 25−50 keV 50−100 keV 100−150 keV 21−83 keV Konus 83−360 keV 360−1360 keV 0 50 100 150 time since BAT trigger (s) 21−1360 keV (waiting mode) 50 100 150 time since BAT trigger (s) 15−50 keV Fig. 1.— Top panels: Swift UVOT, XRT, BAT and Konus-Wind light-curves of GRB 061121; 1σ error bars are shown for the UVOT and XRT data. Each instrument detected the peak of the main burst, with the precursor being detected over all gamma-ray energies. The vertical lines in the 360–1360 keV panel indicate the start and stop times for the spectra given in Table 1. Bottom panel: The 15-50 keV BAT light-curve, with 10-s bins, showing a tail out to ∼140 s. GRB 061121: Broadband observations 5 start time (s) stop time (s) Γ Epeak (keV) ΓBand χ 2/dof 61.876 70.324 1.40 +0.08 −0.09 <2.1 72/75 70.324 75.188 1.23 +0.05 −0.06 <2.9 88/75 75.188 83.380 1.30 +0.11 −0.13 <2.3 81/75 61.876 83.380 1.32 +0.04 −0.05 <2.7 95/75 TABLE 1 Konus-Wind cut-off power-law spectral fit results. Times are given with respect to the BAT trigger. ΓBand is the upper limit obtained for the spectral index above Epeak when fitting with the Band function. 6 K.L. Page et al. 65 70 75 80 time since trigger (s) 0 1 2 3 BAT count rate (count s−1 detector−1) Fig. 2.— Top panels: Light-curves, hardness ratios (HR) and the variation in Γ using a single power-law fit during the main emission. The BAT light-curve (top panel) is in units of count s−1 (fully il- luminated detector)−1 , and the corresponding hardness ratio plots (50–150 keV)/(15–50 keV) using 1-s binning. The XRT light-curve shows counts over 0.3–10 keV, while the hardness ratio compares (1–10 keV)/(0.3–1 keV) over 1-s bins. Bottom panel: BAT hard- ness ratio versus count rate, showing that the emission is harder when brighter. Data from the precursor are shown as grey circles, with the main burst in black. The grey line shows a fit to the data, of the form HR = 0.14 CR + 0.39. (PSF); the ratio of these files provides an estimate of the correction factor. Nousek et al. (2006) give more details on this method. Occasionally, the afterglow was partially positioned over the CCD columns disabled by microm- eteoroid damage mentioned above. In these cases, the data were corrected using an exposure map. From T0 + 3 × 10 5 s onwards, the afterglow had faded sufficiently for a nearby (41.′′5 away), constant (count rate ∼ 0.003 count s−1) source to contaminate the GRB region; this source is coincident with a faint object in the Digitized Sky Survey and is marginally detected in the UVOT V filter. Thus, beyond this time, the ex- traction region was decreased to a diameter of 30 pix- els, and the count rates corrected for the loss in PSF (a factor of ∼ 1.08). The spectrum of this nearby source can be modelled with a single power-law of Γ = 1.5+0.2 −0.1, 100 1000 104 105 1061 time since trigger (s) Fig. 3.— Swift-XRT light-curve of GRB 061121. The star and triangle show the initial Image Mode and pseudo PuPD point (see text for details), followed by WT mode data (black) during the main burst (and at the end of the first orbit) and PC mode data (in grey). 10050 200 500 time since trigger (s) Fig. 4.— Swift flux light-curve of GRB 061121, showing the early X-ray data (star, triangle and crosses) and the BAT data (grey histogram) extrapolated into the 0.3–10 keV band pass in units of erg cm−2 s−1, together with the UVOT flux density light-curve (light grey circles – V -band; dark grey circles – White filter) in units of erg cm−2 s−1 Å−1, scaled to match the XRT flux observed at the start of the ‘plateau’ phase. with NH = (1.8 −1.2) × 10 21 cm−2, in comparison with the Galactic value in this direction of 5.09 × 1020 cm−2 (Dickey & Lockman 1990). Figure 3 shows the XRT light-curve, starting with the IM point (see Hill et al. 2006 for details on how IM data are converted to a count rate) and followed by the pseudo PuPDmode data. The importance of these early pre-WT data is clear, confirming that the XRT caught the rise of the main burst. After the bright burst, the afterglow began to follow the ‘canonical’ decay, seen in many Swift bursts (Nousek et al. 2006; Zhang et al. 2006a). Such a decay can be parameterised by a series of power-law segments; in this case, fitting the data beyond 200 s after the trig- ger (= 125 s after the main peak), two breaks in the light-curve were identified, with the decay starting off very flat (α = 0.38 ± 0.08) and eventually steepening GRB 061121: Broadband observations 7 α1 0.38 ± 0.08 Plateau phase Tbreak,1 2258 α2 1.07 +0.04 −0.06 Shallow phase Tbreak,2 (3.2 ) × 104 s α3 1.53 +0.09 −0.04 Steep phase TABLE 2 XRT power-law light-curve fits from 200 s after the trigger onwards; times are referenced to the BAT trigger. The names used in the text for the different epochs of the light-curve are listed in the last column. to α = 1.07+0.04 −0.06 at ∼ 2.3 ks and then α =1.53 +0.09 −0.04 at ∼ 32 ks (Table 2). The addition of the second break vastly improved the fit by ∆χ2 = 112.4 for two degrees of freedom. However, we note that O’Brien et al. (2006) and Willingale et al. (2007) advocate a different descrip- tion of the temporal decline; we return to this in §3. Fitting the decay of the main peak (75–200 s, keeping T0 as the trigger time) with a power-law, the slope is very steep, with α0 = 5.1 ± 0.2. However, both Zhang et al. (2006a) and Liang et al. (2006) have shown that the appropriate time origin is the start of the last pulse. Thus, a model of the form f(t) ∝ (t−t0) −α0 was used, finding t0 = 58 ± 1 s and a slope of α0 = 2.2 −0.3; this is a statistically significant improvement on the power- law fit using the precursor T0 (∆χ 2 = 32 for one extra parameter). Figure 4 plots the Swift data in terms of flux (the BAT data have been extrapolated into the 0.3–10 keV band, using the joint fits with the XRT described in §2.4.1) and flux density for UVOT. The BAT and XRT data are fully consistent with each other at all overlapping times. Spectral Analysis— The XRT data also show that strong spectral evolution was present throughout the period of the prompt emission; this is discussed in conjunction with the BAT data in §2.4.1. Considering the X-ray data alone, there is some indication that the spectra may be better modelled with a broken, rather than single, power- law, although the break energies cannot always be well constrained (see Figure 5). For each spectrum [covering periods of 2 s during the main pulse, followed by two spectra of 5 s (80–85 s and 85–90 s) where the emission is fainter], the low-energy slopes were tied together for each spectrum (i.e., the slope measured is that averaged over all of the spectra), as were the high-energy indices, and the rest-frame column density, NH,z, was fixed at (9.2 ± 1.2) × 1021 cm−2 from the best fit to the data from later times (see below); only the break energy and the normalisation were allowed to vary. When simul- taneously fitting all 11 spectra, χ2/dof decreased from 142/134 to 127/132. Individually, the spectral fits were typically improved by χ2 of between 2–5. The X-ray data during the GRB 051117A flares (Goad et al. 2007) were found to be better modelled with bro- ken power-laws, with the break energy moving to harder energies during each flare rise, and then softening again as the flux decayed. Likewise, Guetta et al. (2006) found breaks in the X-ray spectra obtained during the flares in GRB 050713A. The same pattern may be occurring here, and there is certainly an indication of spectral curvature. The observed flux calculated from the spectrum corre- sponding to the peak of the emission (74–76 s) was mea- 70 80 90 time since trigger (s) 70 80 90 time since trigger (s) 70 80 90 time since trigger (s) Fig. 5.— Fitting the X-ray data over 0.3–10 keV with a broken power-law (Γ1 =0.69 +0.13 −0.07 and Γ2 =1.61 +0.14 −0.13 for all spectra), the break energy seems to move through the band, towards higher energies when the emission is brighter. Arrows indicate upper or lower 90% limits. sured to be 1.66 × 10−7 erg cm−2 s−1 (over 0.3–10 keV); the unabsorbed value was 1.77 × 10−7 erg cm−2 s−1. The PC spectra were also extracted for the various phases of the light-curve (‘plateau’, ‘shallow’ and ‘steep’ – defined in Table 2); the results of the fitting are pre- sented in Table 3. In each phase, the spectrum could be well modelled by a single power-law (no break re- quired), with excess absorption in the rest-frame of the GRB (modelled using ztbabs and the ‘Wilms’ abun- dance in xspec; Wilms et al. 2000). Together with the WT spectrum from ∼ 200–590 s after the trigger (in the plateau stage), the first two PC spectra (plateau and shallow) are fully consistent with a constant photon in- dex of Γ = 2.07 ± 0.06 and NH,z = (9.2 ± 1.2) × 10 cm−2. Following the second apparent break in the light-curve, around 3.2 × 104 s, the spectrum hardened slightly, to a photon index of Γ = 1.83 ± 0.11 (or 1.87 ± 0.08 using NH,z = 9.2 × 10 21 cm−2). 2.2.2. XMM-Newton XMM-Newton (Jansen et al. 2001) performed a Target of Opportunity observation of GRB 061121 (Observation ID 0311792101) less than 6.5 hr after the trigger (Schartel 2006) and collected data for ∼ 38 ks (MOS1, MOS2; Turner et al. 2001) and ∼ 35 ks (PN; Strüder et al. 2001). This observation is mainly during the ‘shallow’ phase, though also covers a short timespan after the break at around 32 ks. Figure 6 plots the PN flux light-curve and hardness ratio during the XMM-Newton observation, showing the lack of spectral evolution during this time frame; a hard- ness ratio calculated for the Swift data was in agreement with this finding. The decay slope over this time (MOS1, MOS2, PN and joint) is consistent with the Swift results (α ∼ 1.3; note this crosses the time of the second break in the decay). The XMM-Newton EPIC29 spectra show clear evidence for excess NH, in agreement with the Swift data. In addi- tion, fitting with excess NH in the rest-frame of the GRB 29 European Photon Imaging Camera 8 K.L. Page et al. Epoch time since Γ NH,z χ 2/ν corresponding trigger (s) (1021 cm−2) α Plateau 590–1560 2.14 ± 0.12 10.8 62.5/52 0.38 ± 0.08 Shallow 4900–22245 2.04 ± 0.10 8.9 67.5/70 1.07 +0.04 −0.06 Steep 34550–1152750 1.83 ± 0.11 8.0 48.0/55 1.53 +0.09 −0.04 Plateau 590–1560 2.09 ± 0.08 9.2 ± 1.2 (tied) 63.5/53 0.38 ± 0.08 Shallow 4900–22245 2.05 ± 0.06 9.2 (tied) 67.6/71 1.07 +0.04 −0.06 Steep 34550–1152750 1.87 ± 0.08 9.2 (tied) 48.7/56 1.53 +0.09 −0.04 TABLE 3 XRT PC spectral fits - rest-frame NH free and then tied between all three spectra. The temporal decay slopes, α, corresponding to each stage are also given. The Galactic absorbing column of NH = 5.09 × 10 20 cm−2 was always included in the model. 3×104 3.5×104 4×104 4.5×104 5×104 5.5×104 6×104 time since BAT trigger (s) Fig. 6.— XMM-Newton EPIC-PN light-curve and hardness ratio of GRB 061121. The horizontal line shows the hardness ratio is consistent with a constant value of ∼ 1.46, indicating there is no spectral evolution during this time. gives a significantly better fit than at z = 0, as shown in Figure 7. When fitting in the observer’s frame there is a noticeable bump in the residuals around 0.6 keV; fitting with NH at z = 1.314 removes this feature. The data are of sufficiently high signal-to-noise that the redshift of the absorber can be estimated from the spectrum. Limits can be placed on the redshift and absorbing column, re- spectively, of z > 1.2 and NH,z > 4.6 × 10 21 cm−2 at 99% confidence, in agreement with the spectroscopic redshift from Bloom et al. (2006) within the statistial uncertain- ties. At their value of z = 1.314, the excess NH,z from the EPIC-PN spectrum is (5.3 ± 0.2)× 1021 cm−2, lower than the best fit to the Swift data from the simultaneous ‘shallow’ decay section, but more similar to the values ob- tained from fitting the optical-to-X-ray Spectral Energy Distributions (SEDs) in §2.4.2. In agreement with the simultaneous XRT PC mode data, there is no evidence for a break in the EPIC spectrum over this time period. Spectra from neither the Reflection Grating Spectrom- eter (den Herder et al. 2001) nor EPIC show obvious absorption or emission lines. 2.2.3. Chandra Chandra performed a 33 ks Target of Opportunity ob- servation at ∼ 61 day after the trigger. No source was detected at the position of the X-ray afterglow, with a 3σ upper limit of 2.5 × 10−15 erg cm−2 s−1. 2.3. Optical/UV Data zNH = 5.3x10 21 cm−2 10.5 2 5 channel energy (keV) NH = 1.3x10 21 cm−2 Fig. 7.— EPIC-PN spectrum of the late-time afterglow of GRB 061121, with an excess absorbing column both in the rest- frame of the GRB and the observer’s frame. The spectrum is much better modelled with an excess column at z = 1.314. 2.3.1. UVOT The UVOT detected an optical counterpart in the ini- tial White filter30 observation, starting 62 s after the trigger, and subsequently in all other filters (optical and UV). The UVOT followed the typical sequence for GRB observations, with the early data being collected in event mode, which has a frame time of 8.3 ms during this ob- 30 The White filter covers a broad bandpass of λ ∼ 1600−6500 Å. GRB 061121: Broadband observations 9 servation.31 Photometric measurements were obtained from the UVOT data using a circular source extraction region with a 5− 6′′ radius. uvotmaghist was used to convert count rates to magnitudes and flux; no normali- sation between the different filters was applied. As in the gamma-ray and X-ray bands, the main burst was detected, with an increase in count rate seen between ∼ 50–75 s after the trigger (Figures 1 and 4). However, although an increase in count rate is seen for the UVOT data, it is by a smaller factor than observed for the XRT. After ∼ 110 s, the UVOT emission stops decaying and re- brightens slightly, until 140 s after the trigger, at which time it flattens off and then starts to fade again (Fig- ure 4). The slower decay between ∼ 100–200 s may be indicative of the contribution of an additional (afterglow) component beginning to dominate. A single UV/optical light curve was created from all the UVOT filters in order to get the best measurement of the optical temporal decay. This was done by fitting each filter dataset individually (between 200 and 1 × 105 s) and finding the normalisation, which was then modified to correspond to that of the V -band light-curve. The decay across all the filters beyond 200 s after the trigger can be fitted with a single slope of αUVOT = 0.68 ± 0.02; the individual U , B and V decay rates are consistent with one another. No break in the light-curve is seen out to ∼ 100 ks. 2.3.2. ROTSE ROTSE-IIIa, at the Siding Spring Observatory in Aus- tralia, first imaged GRB 061121 21.6 s after the trigger time under poor (windy) seeing conditions. A variable source was immediately identified, at a position coinci- dent with that determined by the UVOT (Yost et al. 2006). The ROTSE data (unfiltered, but calibrated to the R- band) have been included in Figure 10 (discussed later). It is noticeable that the peak around 75 s seen in the Swift data is not readily apparent in the ROTSE measure- ments. The bandpass of the UVOT White filter is more sensitive to photons with wavelengths of λ < 4500Å32, while the ROTSE bandpass is redder. This, together with poor seeing conditions during the observation, may explain why the ROTSE light-curve does not clearly show the main emission. 2.3.3. Faulkes Telescope North The FTN, at Haleakala on Maui, Hawaii, began ob- servations of GRB 061121 225 s after the burst trigger, performing a BV Ri′ multi-colour sequence (Melandri et al. 2006). R-band photometry was performed rela- tive to the USNO-B 1.0 ‘R2’ magnitudes. Magnitudes were then corrected for Galactic extinction using the dust-extinction maps by Schlegel et al. (1998), and con- verted to fluxes using the absolute flux calibration from Fukugita et al. (1995). The photometric R-band points have been included in Figure 10. 31 The data have been adjusted to take into account an incor- rect onboard setting (between 2006-11-10 and 2006-11-22), which resulted in the wrong frame times being stored in the headers of the UVOT files (Marshall 2006a). 32 See http://swiftsc.gsfc.nasa.gov/docs/heasarc/caldb/swift/ 2.4. Broadband Modelling 2.4.1. Gamma-rays – X-rays Spectral Analysis— Because the BAT was in event mode throughout the observation of the main burst of GRB 061121, detailed spectroscopy could be performed. Unfortunately this was not the case during the prompt observation of GRB 060124 (Romano et al. 2006). Figure 2 demonstrates the spectral evolution seen in both the BAT and XRT during the prompt emission. Spectra were extracted over 2 s intervals, in an attempt to obtain sufficient signal to noise while not binning over too much of the rapid variability. The BAT data are hardest around 68 s and 75 s (the second of these times corresponding to the peak of the main emission); the XRT hardness peaks about 70 s, which could be a further indication of the softer data lagging the harder. The joint spectrum (Γjoint comes from a simple absorbed power- law fit to the simultaneous BAT and XRT data) is at its hardest during the brightest part of the emission. The joint fit also hardens around 68–70 s, between the times when the BAT and XRT data respectively are at their hardest. The onboard spectral time-bin selection pre- vents the Konus-Wind data from being sliced into corre- sponding times, so constraints have not been placed on the high energy cut-off, Epeak. Breaks in the XRT-BAT power-laws can only be poorly constrained. In Figure 4, the BAT and XRT data were converted to 0.3–10 keV fluxes using the time-sliced power-law fits to the simultaneous BAT and XRT spectra. Without the use of such varying conversion factors, the derived BAT and XRT fluxes would have been inconsistent with each other. A broadband spectrum, covering 0.3 keV to 4 MeV in the observer’s frame (XRT, BAT and Konus-Wind) for ∼ 70–75 s post trigger was fitted by the absorbed cut-off power-law model described in §2.1.2. A constant factor of up to 10% was included between the BAT and Konus- Wind data, to allow for calibration uncertainties. The best fit (χ2/dof = 301/167) gives Γ = 1.19 ± 0.01, with Epeak = 670 −47 keV. NH,z was fixed at 9.2 × 10 21 cm−2 (from the X-ray fits in §2.2.1). Allowing Γ to vary be- tween the three spectra hints at further spectral curva- ture, although the differences are marginal, significant at only the 2σ level. The isotropic equivalent energy (calculated using the time-integrated flux over the full T90 period) is 2.8 × 1053 erg in the 1 keV – 10 MeV band (GRB rest frame), meaning that GRB 061121 is consistent with the Amati relationship (Amati et al. 2002). See §3.3.2 for a beaming-corrected gamma-ray energy limit. Lag Analysis— A lag analysis (e.g., Norris et al. 1996) between the BAT bands leads to interesting results. Comparing bands 50–100 keV and 15–25 keV, the precur- sor emission yields a spectral lag of 600 ± 100 ms, while the main emission has a much smaller lag of 1 ± 6 ms. Note that the calculation was performed using 64 ms binning for the precursor and 4 ms binning for the main burst; see Norris (2002) and Norris & Bonnell (2006) for more details on the procedure. This lag for the main emission is rather small for a typical long burst, however both lags are consistent with the long-burst luminosity- lag relationship generally seen (Norris et al. 2000). The http://swiftsc.gsfc.nasa.gov/docs/heasarc/caldb/swift/ 10 K.L. Page et al. 0 2 4 6 time delay (s) 1−4 keV 4−10 keV 15−25 keV 25−50 keV 50−100 keV 100−150 keV Fig. 8.— Autocorrelation function of the BAT and XRT data during the prompt emission of GRB 061121, showing that the main burst peak is broader at softer energies. short spectral lag for the main emission, and the longer value for the precursor are also found when comparing the 100–350 keV and 25–50 keV bands. Similarly, comparison of the hard and soft (2–10 keV and 0.3–2 keV) XRT bands reveals a lag of approximately 2.5 s, as the emission softens through the main burst. The X-ray data also lag behind the gamma-ray data, and the optical behind the X-ray. Link et al. (1993) and Fenimore et al. (1995) used a sample of BATSE33 (Paciesas et al. 1999) bursts to in- vestigate the relationship between the duration of bursts and the energy band considered. They found that the bursts, and smaller structures within the main emission, generally become shorter with increasing energy (see also Cheng et al. 1995; Norris et al. 1996; in’t Zand & Fen- imore 1996; Piro et al. 1998). Figure 8 plots the auto- correlation function over various X-ray and gamma-ray bands, to reinforce the point that the peak is narrower the harder the band – over X-ray as well as gamma-ray energies. Comparison of the light-curves over the differ- ent energy bands in Figure 1 demonstrates this as well. A similar behaviour was also found for GRB 060124, where Romano et al. (2006) compared the T90 values obtained for the main burst over the X-ray and gamma-ray bands. Fenimore et al. (1995) found that the width of the auto- correlation function, W ∝ E−0.4, where E is the energy at which the function was determined; the six measure- ments from GRB 061121 are consistent with this finding. 2.4.2. Optical – X-rays Using the Swift X-ray and UV/optical data, R and i′ band data from the Faulkes Telescope and Rc data from the Kanata telescope (Uemera et al. 2006), SEDs were produced at epochs corresponding to the peak of the emission (72–75 s post BAT trigger), the plateau stage and during the shallow decay. Fitting at the differ- ent epochs gives an estimation of the broadband spectral variation. For each of the UVOT lenticular filters, the tool uvot2pha was used to produce spectral files compatible with xspec, and for the latter two epochs the count rate 33 Burst And Transient Source Experiment in each band was set to that determined from a power-law fit to the individual filter light curves over the time inter- val in question, using α = 0.68. To determine the Faulkes Telescope R and i′ band flux during the plateau stage, a power law was fitted to the complete data set (220–1229 s post BAT trigger for R and 467–1401 s for i′) with the decay index left as a free parameter. The R magnitude at the mid-time of the shallow stage (6058 s) was deter- mined from the Kanata R-band magnitude reported at 6797 s (Uemera et al. 2006), assuming the same decay index as observed in the UVOT data. An uncertainty of 0.2 mag was assumed as the systematic uncertainty for the photometric calibration of the ground based data. At a redshift of z = 1.314, the beginning of the Lyman- α forest is redshifted to an observer-frame wavelength of ∼ 2812 Å which falls within the UVW1 filter bandpass, the reddest of the UV filters. A correction was applied to the three UV filter fluxes to account for this absorption, based on parameters from Madau (1995) and Madau et al. (1996); see also Curran et al. (in prep). The methods used for simultaneous fitting of the SED components are described in detail in Schady et al. (2007a). The SEDs were fitted with a power-law, or a broken power-law, as expected from the synchrotron emission, and two dust and gas components, to model the Galactic and host galaxy photoelectric absorption and dust extinction. The column density and reddening in the first absorption system were fixed at the Galactic values. [The Galactic extinction along this line of sight is E(B − V ) = 0.046 (Schlegel et al. 1998).] The second photoelectric absorption system was set to the redshift of the GRB, and the neutral hydrogen col- umn density in the host galaxy was determined assuming Solar abundances. The dependence of dust extinction on wavelength in the GRB host galaxy was modelled us- ing three extinction laws, taken from observations of the Milky Way (MW), the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC) and parame- terised by Pei (1992) and Cardelli et al. (1989). The greatest differences observed in these extinction laws are the amount of far UV extinction (which is greatest in the SMC and least in the MW) and the strength of the 2175 Å absorption feature (which is most prominent in the MW and negligible in the SMC). Fitting these data together, a measurement of the spec- tral slope and optical and X-ray intrinsic extinctions (for the second two epochs) were obtained (Table 4); the AV values given in the table are in addition to the AV = 0.151 associated with the Milky Way itself. The slope above the break energy (which lies towards the low energy end of the X-ray bandpass for each phase) was assumed to be exactly 0.5 steeper than the spectral slope below the break (the condition required for a cooling break), since allowing all of the parameters to vary leads to uncon- strained fits. Figure 9 shows, as an example, the fit to the data in the plateau stage. A Milky Way dust extinction law provides the best overall fit to the data, using a broken power-law model, although the LMC model is equally acceptable. During the plateau phase, and adopting the bro- ken power-law model parameters given in Table 4, we find gas-to-dust ratios of (1.6 ± 0.7), (2.6 ± 0.7) and (3.0 ± 0.7) ×1022 cm−2 mag−1 for MW, LMC and SMC GRB 061121: Broadband observations 11 1015 1016 1017 1018 Frequency (Hz) Fig. 9.— Broken power-law fit to the UVOT, XRT and ground- based R and i′ spectral energy distribution of GRB 061121 between ∼ 596–1566 s after the trigger (plateau phase) plotted in the ob- server’s frame. The arrows indicate the beginning of the Lyman-α forest (1215Å in the rest-frame) and the absorption feature in the MW dust extinction law (2175Å), which is shown by a dotted line. The solid line corresponds to the LMC extinction, and the dashed one to the SMC extinction. fits respectively. We can compare these estimates to the measured values for the MW of (4.93 ± 0.45) × 1021 cm−2 mag−1 (Diplas & Savage 1994) and the LMC and SMC of (2.0 ± 0.8) and (4.4 ± 1.1) ×1022 cm−2 mag−1 , respectively (Koornneef 1982; Bouchet et al. 1985). The MW fit to GRB 061121, which is found to be marginally the best model, is consistent with the LMC gas-to-dust ratio only, at the 90% confidence level. The ratios derived from the LMC and SMC fits are consistent with both the LMC and SMC gas-to-dust ratios. We note that all fits are inconsistent with the MW ratio at this confidence level, following the trend seen in pre-Swift bursts (e.g., Starling et al. 2007 and references therein), and that if a metallicity below Solar were adopted, the gas-to-dust ratio of GRB 061121 would increase, moving it further towards the SMC value. 3. DISCUSSION Swift triggered on a precursor to GRB 061121 leading to comprehensive broadband observations of the prompt emission, as well as the later afterglow. We discuss these here, together with possible mechanisms involved. 3.1. Precursor Lazzati (2005) found that about 20% of BATSE bursts showed evidence for gamma-ray emission above the back- ground between 10 to ∼200 s before the main burst, typi- cally with non-thermal spectra which tended to be softer than the main burst. GRB 060124 (Romano et al. 2006) and GRB 061121 show the same behaviour. Precursor models have been proposed for emission well- separated from, or just prior to, the main burst. Early emission occurring only a few seconds before the main burst has been explained by the fireball interacting with the massive progenitor star – though the spectrum of such emission is expected to be thermal (Ramirez-Ruiz et al. 2002a). Lazzati et al. (2007) investigated shocks in a cocoon around the main burst; their model predicts a non-thermal precursor as the jet breaks out of the sur- face of the star. A high-pressure cocoon is formed as the sub-relativistic jet head forces its way out of the star. As the head of the jet breaks through the surface, the energy of this cocoon is released through a nozzle and can give rise to a precursor (Ramirez-Ruiz et al. 2002a,b). Within the framework of this model, observers located at view- ing angles of 5◦ < θ < 11◦ are expected to see first a relatively bright precursor, then a dark phase with lit- tle emission, followed, when the jet enters the unshocked phase, by a bright GRB; this is very similar to the light- curve observed for GRB 061121. Waxman & Mészáros (2003) demonstrate that both a series of thermal X-ray precursors (becoming progressively shorter and harder) and nonthermal emission can be produced by an emerg- ing shocked jet, although the nonthermal component is expected to be in the MeV range. There could also be an accompanying inverse Compton component, formed by the thermal X-rays being upscattered by the jet. The same type of smooth, wide-pulse, low intensity emission as seen in some precursors, but occurring af- ter the main emission is also occasionally seen (e.g., Hakkila & Giblin 2004; Nakamura 2000). Hakkila & Gib- lin (2004) discuss two examples where postcursor emis- sion is found to have a longer lag than expected from the lag-luminosity relation, smoother shape and to be softer. In the case of the GRB 061121 precursor, the spectrum is, indeed, softer than the main event, and shows a compar- atively smooth profile. The emission does have a longer lag than the main emission, but it is still consistent with the lag-luminosity relation. There are two expected effects which could lead to such a difference in lags for separate parts of a single burst: the much lower luminosity for the precursor (resulting from a much smaller Lorentz factor; the measured fluence of the precursor is about a factor of 30 smaller than the fluence of the main emission) is a natural explanation, while the precursor being emitted at a greater off-axis angle could also have an effect. In this second case, ejecta are considered to emerge at different angles with respect to the jet axis; not all of the solid angle of the jet will be ‘filled’ uniformly. Such late postcursor emission is unlikely to be linked to the jet breakout from the stellar surface, and it may not be sensible to attribute apparently similar phenomena (in the form of pre- and postcursors) to entirely different processes. Pre/postcursor emission could be due to the decelera- tion of a faster front shell, resulting in slower shells catch- ing up and colliding with it (Fenimore & Ramirez-Ruiz 1999; Umeda et al. 2005; note, however, that a faster shell would be inconsistent with the precursor having a smaller Lorentz factor as suggested to explain the lag discrepancy), or late activity of the central engine. The presence of flares in about 50% of Swift bursts is gener- ally attributed to continuing activity of the central en- gine (Burrows et al. 2005b; Zhang et al. 2006a) and the appearance of broken power-laws in the X-ray spectra of 12 K.L. Page et al. X-ray Model Extinction NH,z Γ1 Ebreak Γ χ2/dof Epoch (1021 cm−2) (keV) Peak PL SMC 1.6 0.99 ± 0.01 · · · · · · 0.64 25/27 LMC 1.9 1.06 ± 0.01 · · · · · · 0.98 23/27 MW 2.4 1.16 ± 0.01 · · · · · · 1.51 22/27 BKN PL SMC 2.7+9.5 0.72+0.08 −0.15 0.17+0.79 −0.15 1.22 0.51 22/26 LMC 3.0+7.7 0.77+0.08 −0.20 0.18+0.53 −0.17 1.27 0.72 22/26 MW 3.0+7.7 0.77+0.10 −0.21 0.09+0.30 −0.09 1.27 1.03 22/26 Plateau PL SMC 1.42± 0.51 1.58± 0.02 · · · · · · 0.62± 0.05 167/59 LMC 1.98± 0.54 1.64± 0.03 · · · · · · 0.94± 0.08 152/59 MW 2.71± 0.69 1.71± 0.03 · · · · · · 1.39± 0.10 136/59 BKN PL SMC 3.89 +0.72 −1.01 +0.03 −0.02 +0.36 −0.12 1.96 0.52± 0.04 84/58 LMC 4.40 +0.77 −1.30 +0.04 −0.02 +0.16 −0.14 2.01 0.74± 0.06 80/58 MW 3.91 +0.77 −0.75 +0.04 −0.03 +0.25 −0.20 2.08 1.03 +0.09 −0.08 79/58 Shallow PL SMC 2.72± 0.49 1.69± 0.02 · · · · · · 0.65± 0.04 162/77 LMC 3.37 +0.53 −0.49 1.75± 0.03 · · · · · · 0.98 +0.07 −0.06 146/77 MW 4.60 +0.65 −0.60 1.87± 0.04 · · · · · · 1.63 +0.12 −0.11 127/77 BKN PL SMC 4.02+0.62 −0.67 1.58+0.02 −0.03 1.30+0.19 −0.11 2.08 0.50± 0.04 101/76 LMC 4.41+0.69 −0.63 1.62± 0.03 1.30+0.16 −0.14 2.12 0.72± 0.06 99/76 MW 4.78+0.75 −0.65 1.67± 0.04 1.35+0.16 −0.17 2.17 1.02+0.11 −0.10 102/76 a Γ2 is set to be equal to Γ1 + 0.5 in each broken power-law fit, as would be expected if the change in index were due to a cooling break. b In the fit to the peak epoch, AV is fixed to the average best-fit value found in the same model fits to plateau and shallow stage data. The AV values are given for the observer’s frame of reference. TABLE 4 Power-law (PL) and broken power-law (BKN PL) fits to the simultaneous UVOT and XRT spectra of GRB 061121, for three different dust extinction models: Small and Large Magellanic Clouds (SMC and LMC) and the Milky Way (MW). Γ1 and Γ2 are the photon indices below and above the spectral break for the BKN PL models. The data points have not been corrected for reddening. both flares and the prompt emission (Guetta et al. 2006; Goad et al. 2007) hints of a common mechanism. 3.2. Prompt Emission The prompt emission mechanism for GRBs is still de- bated and the origin of Epeak is not fully understood (Mészáros et al. 1994; Pilla & Loeb 1998; Lloyd & Pet- rosian 2000; Zhang & Meszaros 2002; Rees & Mészáros 2005; Pe’er et al. 2005). The standard synchrotron model predicts fast cooling (Ghisellini et al. 2000) with a photon index, Γ, of 3/2 and (p/2)+1 below and above the peak energy, respectively (e.g., Zhang & Mészáros 2004). The Konus-Wind spectral index below Epeak is shallower than 3/2, which may suggest a slow cooling spectrum with p < 2 [Epeak being the cooling frequency and Γ =(p+1)/2] or additional heating. A slow-cooling spectrum can be retained by assuming that the magnetic fields behind the shock decay significantly in 104–105 cm, so that synchrotron emission happens in small scale mag- netic fields (Pe’er & Zhang 2006). The SED at the peak time (SED 2 in Figure 11, discussed below) has a peak flux density of around 1 keV, below which the optical to X-ray spectral slope is 0.11 ± 0.09. This slope is harder than expected from the standard synchrotron model (which predicts an in- dex of 1/3). There should, however, be spectral cur- vature around the break, which could flatten the index (Lloyd & Petrosian 2000), so the data could still be con- sistent with the synchrotron model. An alternative to synchrotron emission, in the form of ‘jitter’ radiation is discussed by Medvedev (2000), though that model pre- dicts an even steeper index of 1 below the jitter break frequency. Figures 4 and 10 show that all three instruments on- board Swift saw the prompt emission around 75 s after the BAT trigger. However, it is noticeable that most of the emission is in the gamma-ray and X-ray bands, with the optical showing a relatively small increase in bright- ness in comparison. Assuming the observed process is synchrotron, then the prompt emission which is detected by the UVOT will be the low-frequency extension of this in the internal shock. No reverse shock is apparent. 3.3. Afterglow Emission 3.3.1. Broken Power-law Decline Models The afterglow of GRB 061121 was observed over an even broader energy range (from radio to X-rays) than the prompt emission, with multi-colour data being ob- tained from ∼ 100–105 s after the trigger. The X-ray light-curve shows evidence for substantial curvature at later times (see Figure 3), as has been found for other Swift GRBs (e.g., GRBs 050315 – Vaughan et al. 2006; 060614 – Gehrels et al. 2006b). The standard practice has been to fit such a decay using a series of power- law segments as a function of time. An alternative exponential-to-power-law description of the light-curve is given in §3.3.2. Nousek et al. (2006) and Zhang et al. (2006a) have both discussed the canonical shape that many Swift af- terglows seem to follow: steep to plateau to shallow, with some light-curves showing a further steepening. In these previous works, the extrapolation of the BAT data into the XRT band was incorporated into the derivation of the steep decay at the start of the canonical light- curve shape. In the case of GRB 061121, the full curve GRB 061121: Broadband observations 13 can be seen entirely in X-rays, suggesting that the pre- vious extrapolations are reliable. For the afterglow of GRB 061121, only data after the end of the main burst have been modelled with power-laws. The early steep decline, which might be attributable to the curvature ef- fect (Kumar & Panaitescu 2000; Dermer 2004; Fan & Wei 2005), is not considered here. According to the model proposed in Nousek et al. (2006) and Zhang et al. (2006a), the plateau phase of the light-curve is due to energy injection in the fireball. The plateau phase of GRB 061121 is consistent with an injection of energy since the luminosity index, q, is nega- tive, which is the requirement for injection to modify the afterglow (Zhang et al. 2006a); the later two stages both have q > 1. However, as will be discussed in §3.3.2, the plateau and final transition to the power-law decay are only visible in the X-ray data for GRB 061121; the start of the final decay is much earlier in the V and R-bands (see Figure 10). One might expect that energy injection would affect all the energy bands simultaneously, rather than just the X-rays. From the standard afterglowmodel computations (e.g., Zhang & Mészáros 2004), we find that none of the closure relations fit the entire dataset completely: although the shallow phase (after the end of energy injection, between T + 2.3 ks and T + 32 ks) could be consistent with the evolution of a blast-wave which had already entered the slow cooling regime when deceleration started [i.e., ν > max(νm, νc) where νc is the cooling frequency and νm is the synchrotron injection frequency; Sari et al. 1998; Chevalier & Li 2000], the steeper part of the decay curve (T> 32 ks) is not consistent with any of the models. This lack of consistency suggests that a different approach is required. The change in decay slope between the shallow to steep phases (∼ 32 ks) cannot be easily identified with a jet-break. It certainly seems unlikely that the simplest side-spreading jet model could be applicable, since the post-break decay index (α ∼ 1.5) is not steep enough (a post-jet decay has α = p, where p is the electron in- dex). There is also some indication that the X-ray spec- tral slope hardens after the break, whereas no change in spectral signature is expected over a jet-break. In the case of a non-laterally expanding jet (Panaitescu & Mészáros 1999), α = (3β/2) + 0.25 [for a homoge- neous circumstellar medium (CSM); Panaitescu et al. (2006)], which does, indeed, fit the data after this break: [1.5 × (0.9 ± 0.08)] + 0.25 = 1.6 ± 0.1; the measured α is 1.53. Such a confined jet has been suggested as an explanation for the observed decay in a number of pre- vious bursts (e.g., GRB 990123 – Kulkarni et al. 1999; GRB 050525A – Blustin et al. 2006; GRB 061007 – Schady et al. 2007b). The UVOT data obtained around this time show little evidence for a break, whereas jet breaks should occur across all energy bands simultane- ously. However non-simultaneity could be explained by a multi-component outflow, where the X-ray emission is produced within a narrow jet, while the optical compo- nent comes from a wider jet with lower Lorentz factor (Panaitescu & Kumar 2004; Oates et al. 2007). There remains the issue, however, that α should steepen by 0.75 over a jet break (Mészáros & Rees 1999), whereas the maximum observed change (within the 90% errors) is only ∆α < 0.61, excluding ∆α = 0.75 at almost 3σ; also, again there should be no spectral evolution across the break. There is, however, a probable jet break at later times, which will be covered in the next Section. Other multi-component models [see, e.g., Oates et al. (2007) and references therein] also fail to explain the data, because of the lack of observed energy injection (plateau phase) in the optical data. Panaitescu et al. (2006a) discuss chromatic breaks in Swift light-curves, and postulate that these could be due to a change in microphysical parameters within a wind environment. However, this model requires the cooling frequency to lie between the X-ray and optical bands and, as will be discussed in §3.3.2, this does not seem to be the case here. 3.3.2. Exponential-to-power-law Decline Model As first described by O’Brien et al. (2006), and further expanded by Willingale et al. (2007), GRB light-curves can be well modelled by one or two components com- prised of an early exponential rise followed by a power- law decay phase. Of these components, the first repre- sents the prompt gamma-ray emission and early X-ray decay. The second, when detected, dominates at later times, forming what we see as the afterglow. These re- sults show that fitting an intrinsically curved decay with multiple power-law segments runs the risk of incorrectly identifying temporal breaks (see also Sakamoto et al. in prep). In this Section the models of O’Brien et al. (2006) and Willingale et al. (2007) are applied to the multi-band afterglow data of GRB 061121. Figure 10 brings together the BAT, XRT, UVOT, FTN and ROTSE data, along with further optical and ra- dio points taken from the GCN Circulars (Halpern et al. 2006a,b; Halpern & Armstrong 2006a,b; Chandra & Frail 2006; van der Horst et al. 2006a,b) and the upper limit from Chandra, to form a multi-energy decay plot. The data have been plotted as ‘time since trigger + 4 s’ in order to include the precursor on a log time-scale. The optical points have all been corrected for extinction using AV = 1.2 (a combination of the Galactic value of 0.151 and an estimate of AV ∼ 1 for the GRB host galaxy – see §2.4.2). The contribution from the host galaxy reported by Malesani et al. (2006) and Cobb (2006) has been sub- tracted from the V - and R-band flux values. The magni- tude of the host in the V -band is 22.4, which only changes the last two or three V -band points by a small amount. For the R-band we have no direct measurement, but the last group of MDM exposures gave an R magnitude of 22.7, corresponding to a flux level of 2.8 µJy, and the flux level is still declining at that epoch (∼ 3.3 × 105 s), so an R-band flux level of 2.5 µJy was adopted for the host. The error bars shown on the last few points reflect the large uncertainty in the galaxy contribution subtracted. The curved dotted lines in Figure 10 are the fits to the data using the exponential-to-power-law model, fol- lowed by a break to a steeper decay around 105 s. These models are parameterised by the power-law decay, α, and Ta, the time at which this decay is established. For the X-ray data, Ta,X is found to be 5250 −460 s and αa,X = 1.32 ± 0.03. Fits were also performed to the V - and R-band data, yielding: αa,V = 0.66 ± 0.04 (with Ta,V = 70 −70 s) and αa,R = 0.84 ± 0.03 14 K.L. Page et al. Fig. 10.— Flux density light-curves for the gamma-ray, X-ray, optical and radio data obtained for GRB 061121. The vertical dotted lines indicate the times used for the SED plots shown in Figure 11, while the curved dotted lines show the fit to the X-ray and optical data, including a late-time break, as described in the text. Fig. 11.— SEDs for the four time intervals indicated in Figure 10. SED 2 (the peak of the burst emission) includes the Konus-Wind data, although these have not been included in Figure 10. The solid lines represent the power-law fits to the BAT, XRT and Konus data, while the dashed lines join the radio, optical and 1 keV points. Spectral evolution over time is clearly seen. (Ta,R = 230 −230 s). The non-detection by Chandra almost two months af- ter the burst shows there must have been a further steep- ening in the X-ray regime, and the optical data are not inconsistent with this finding. Constraining the tempo- ral index after the late break to be α = 2 (a typical slope for a post-jet-break decay), break times of ∼ 2.5 × 105, ∼ 2.5 × 104 and ∼ 105 s are estimated for the X-ray, V - and R-band respectively; note that the UVOT V -band value is particularly uncertain, given the small number of data points at late times. Within the uncertainties, these times are likely to be consistent, so the turnover could be achromatic, as required for a jet break. From Willingale et al. (2007), a jet break might be expected at ∼ 100 × Ta,X – i.e., 5.5 × 10 5 s, which is in agreement with these fits. As can be seen from these numbers and the mod- els plotted in Figure 10, the X-ray data clearly show the transition from the plateau to the power-law decay, whereas the start of the final decay is much earlier in the V - and R-bands. The V -band decay is also significantly flatter (by α ∼ 0.2) than that estimated for the R-band. As previously stated, the V , B and U light-curves are all consistent with this slow decay. There have been few multi-colour optical decay curves obtained for GRB af- terglows, and, of these, the different filters [in the case of GRB 061007 (Schady et al. 2007; Mundell et al. 2007) X-ray and gamma-ray data as well as the optical] tend to track each other (e.g., Guidorzi et al. 2005; Blustin et al. 2006; de Ugarte Postigo et al. 2007). In the case of GRB 061121, we find that the R-band data are fading more rapidly than the V . GRB 060218, which was as- sociated with a supernova (e.g., Campana et al. 2006a), shows changes throughout the optical spectra, because of a combination of shock break-out and radioactive heat- ing of the supernova ejecta. There is a large difference between the decays of the blue (V , U , B) and red (R) data for GRB 061121, which cannot be easily explained by a synchrotron spectrum. Although no supernova has been detected in this case, we speculate that some form of pre-supernova thermal emission could possibly be af- fecting the optical data, adding energy into the blue end of the spectrum, thus slowing its decline. After the break in the decays around 105 s, the light- curves across all bands become more consistent with one GRB 061121: Broadband observations 15 another, although there are only limited data at such a late time. The vertical dotted lines in Figure 10 show the times of the SEDs plotted in Figure 11; again, all points were corrected for an extinction of AV = 1.2, so that they represent the true SEDs (with the frequency in the ob- server’s frame). The solid lines represent actual fits to the X-ray and gamma-ray data, while the dashed lines just join the separate radio, optical and 1 keV points. The times of these SEDs, which clearly show spectral evolution, correspond to (1) before the main BAT peak, 56 s after trigger; (2) at the BAT peak, 76 s after trig- ger; (3) just after the start of the plateau, 300 s after the trigger; (4) in the main decay at 65 ks (chosen because radio measurements were taken at this time). SEDs 3 and 4 do not contain any BAT or Konus data, since the gamma-ray flux had decayed by this point; the highest energy point in these corresponds to the maximum en- ergy (10 keV) of the X-ray fits. Table 4 demonstrates that the optical and X-ray spec- tra during the peak emission are best fitted with a broken power-law model, with the break energy at the very low energy end of the X-ray bandpass. SED 2 in Figure 11 shows that this spectral break corresponds to the peak frequency in a flux density plot (β1 is less than zero in this case). Only during SED 2 is the optical flux density lower than that of the higher energy data. Figure 4 also shows that the optical emission is less strong than the X-ray and gamma-ray data during the main burst. Table 5 shows the values of α for the X-ray and opti- cal decays (i.e., before and after the break) in SED 4, at 65 ks, with their corresponding spectral indices. For the initial stages of the power-law decay (Ta < t < 65000 s) the evolution of the afterglow SED and the coupling be- tween the temporal and spectral indices are not com- pletely consistent with the standard model: although the R-band decay, with αa,R = 0.84 ± 0.03, is in good agree- ment with the homogeneous CSM model below the cool- ing break, the X-ray and V -band flux decays are slower than expected from the measured spectral indices; they are in best agreement with the same constant density model below νc, however. The point at which the power-law decay dominates the exponential in the optical bands is noticeably earlier than in the X-ray (< few hundred seconds, rather than ∼ 5000 s) and, as mentioned above, the decay indices are significantly different for all three (X-ray, V and R) bands (see Figure 10). At the time of SED 3, the X-ray data are not decaying (i.e., this is during the plateau), yet both the V and R-band data have already entered the power-law decline phase. The R-band is decaying faster than the V -band, so the spectral index through the optical range is becoming harder. The X-ray spec- tral index shows a similar hardening trend (see Table 3), so the SED measured from optical to 10 keV is gradually getting harder. Such spectral hardening from the plateau to the final decay is a feature of many X-ray afterglows (Willingale et al. 2007). This slow hardening of the broadband spectrum with time could be a signature of synchrotron self-Compton emission (Sari & Esin 2001; Panaitescu & Kumar 2000). The strength of the self-Compton component in the af- terglow depends on the flux of low energy photons (radio- optical) and the electron density in the shock. Using the formulation in Sari & Esin (2001) the density required is given by n1 = 3× 10 f ICmax (E52tday) −1/3cm−3 (1) where f ICmax/f max is the ratio of the peak flux of the seed synchrotron spectrum (i.e., the source of low energy photons) and the peak flux of the self-Compton emis- sion; E52 is the isotropic burst energy in units of 10 52 erg; tday is the time in days after the burst (which determines the distance through the CSM swept up by the external shock). From Figure 11 (SEDs 1, 3 and 4) we see that f ICmax/f max ∼ 0.001 if the X-ray flux has a significant con- tribution from a self-Compton component at tday = 0.75. A value of E52 = 30 gives n1 ≈ 10 5 cm−3. Even as- suming the emission at 0.75 days is not dominated by the self-Comptonisation, and so taking the f ICmax/f ratio to be a factor of ten smaller, the density would be ∼ 5× 103 cm−3, which is still high. It seems unlikely that self-Compton emission is the cause of the spectral hard- ening of the SED unless the CSM density encountered by the external shock is extremely large. However, there have been suggestions that GRBs may form in molecular clouds (Galama & Wijers 2001; Campana et al. 2006b,c), which have densities of 104 or more particles per cubic centimetre in the cores (Miyazaki & Tsuboi 1999; Wil- son et al. 1999). Typically one might expect greater red- dening than is found here (Table 4), though Waxman & Draine (2000) discuss the possibility of dust destruction. The spectrum will be redshifted as the jet slows down, so the optical and X-ray spectral indices should, if any- thing, become softer – the opposite of what is seen here. Although spectral hardening with time is suggested from the data, it is not be easily explained by current models. Whether or not there is a Comptonised component, the later SEDs clearly indicate that there is a break in the spectrum somewhere between the optical and the X- ray; this is also shown by the fits in Table 4, where the UVOT–XRT spectra are better fitted with broken power- laws, with Ebreak towards the low energy end of the X-ray bandpass. Since both the optical and X-ray bands ap- pear to be below the cooling frequency, from the closure relations given in Table 5, this change in slope cannot be identified with a cooling break; its origin remains un- clear. The redshift of z = 1.314 and the isotropic energy of Eiso ∼ 3 × 10 53 erg (§2.4.1) can be used to place con- straints on the jet opening angle. From Sari et al. (1999), and assuming that the jet break occurs at T0+2× 10 we have θj ∼ 4 )1/8 ( n where n and ηγ are the density of the CSM and the efficiency of the fireball in converting the energy in the ejecta into gamma-rays. Taking ηγ = 0.2 and n = 3 cm −3 (following Ghirlanda et al. 2004), this gives Eγ ∼ 1.7 × 10 51 erg for the beaming- corrected gamma-ray energy released, which is within the range previously determined (e.g., Frail et al. 2001) and consistent with the Ghirlanda relationship (Ghirlanda et al. 2004). 4. SUMMARY AND CONCLUSIONS Swift triggered on a precursor to GRB 061121, leading 16 K.L. Page et al. GRB models α(β) α(βa,X) α(βopt) a αbopt V -band R-band CSM SCc (νm < ν < νc) 1.49 ± 0.10 1.32 ± 0.03 0.80 ± 0.09 0.66 ± 0.04 0.84 ± 0.03 Wind SCc (νm < ν < νc) 1.99 ± 0.10 1.30 ± 0.09 CSM or Wind SCc & FCd 0.99 ± 0.10 0.30 ± 0.09 (ν > max(νc, νm)) a Decay calculated from the measured spectral index b Observed power-law decay index. c Slow cooling. d Fast cooling. TABLE 5 Closure relations for exponential-plus-power-law model fits to the X-ray data (βa,X = 0.99± 0.07) and the optical-to-X-ray band (βopt = 0.53 ± 0.06) from the time of SED 4 (65 ks after the burst). to unprecedented coverage of the prompt emission by all three instruments onboard, with the gamma-ray, X-ray and optical/UV bands all tracking the main peak of the burst. GRB 061121 is the instantaneously brightest long Swift burst detected thus far, both in gamma-ray and X- rays. The precursor and main burst show spectral lags of different lengths, though both are consistent with the lag- luminosity relation for long GRBs (Gehrels et al. 2006b). The SED of the prompt emission, stretching from 1 eV to 1 MeV shows a peak flux density at around 1 keV and is harder than the standard model predicts. There is def- inite curvature in the spectra, with the prompt optical- to-X-ray spectrum being better fitted by a broken power- law, similar to results found for fitting X-ray flares (e.g., Guetta et al. 2006; Goad et al. 2007). The afterglow component, in both the optical and X-ray, starts early on – before, or during, the main burst peak (see also O’Brien et al. 2006; Willingale et al. 2007; Zhang et al. 2006b). The broadband SEDs reveal gradual spectral hardening as the afterglow evolves, both within the X-ray regime (Γ flattening from ∼ 2.05 to ∼ 1.87) and between the V - and R-band op- tical data (αV ∼ 0.66 compared with αR ∼ 0.84). Self- Comptonisation could explain the hardening, although a molecular-cloud-core density would be required. A prob- able jet-break occurs around T0 + 2 × 10 5 s, shown by a late-time non-detection by Chandra. Before this break, the X-ray and V -band decays are too slow to be readily explained by the standard models. This extremely well-sampled burst shows clearly that there remains much work to be done in the field of GRB models. A single, unified model for all GRB emission observed should be the ultimate goal. 5. ACKNOWLEDGMENTS The authors gratefully acknowledge support for this work at the University of Leicester by PPARC, at PSU by NASA and in Italy by funding from ASI. This work is partly based on observations with the Konus-Wind ex- periment (supported by the Russian Space Agency con- tract and RFBR grant 06-02-16070) and on data ob- tained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. We thank the Liverpool GRB group at ARI, Liverpool John Moores University, in par- ticular C.J. Mottram, D. Carter, R.J. Smith and A. Gomboc for their assistance with the FTN data acqui- sition and interpretation. The Faulkes Telescopes are operated by the Las Cumbres Observatory Global Tele- scope Network. We also thank J.E. 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Microsoft Word - Karlsson et al 2007.doc High-altitude signatures of ionospheric density depletions caused by field-aligned currents T. Karlsson1, N. Brenning1, O. Marghitu2,3, G. Marklund1, S. Buchert4 1 Space and Plasma Physics, Royal Institute of Technology (KTH), Stockholm, Sweden. 2 Institute for Space Sciences, Bucharest, Romania. 3 Also at Max-Planck-Institut für extraterrestrische Physik, Garching, Germany. 4 Swedish Institute of Space Physics, Uppsala, Sweden. Abstract. We present Cluster measurements of large electric fields correlated with intense downward field-aligned currents, and show that the data can be reproduced by a simple model of ionospheric plasma depletion caused by the currents. This type of magnetosphere-ionosphere interaction may be important when considering the mapping between these two regions of space. 1. Introduction A system of magnetic field-aligned current sheets closing via Pedersen currents in the ionosphere will set up an ionospheric electric field. For constant conductivity, and for sheets extending to infinity along the field-line and one of the perpendicular directions, we get: 1 1 1P P P P P E j d d Bτν τν νμ ν μ = = = = Σ Σ Σ ∂ Σ∫ ∫ (1) where ν is the direction perpendicular to the sheet, τ the tangential direction, Eν is the normal electric field, JP and ΣP, the height integrated Pedersen current and conductivity, Bτ the tangential magnetic field, j// the field-aligned current (positive for downward currents) and μ0 the magnetic permeability of vacuum. This kind of correlation between Eν and Bτ can be seen rather often in the dayside auroral oval (e.g. Ishii et al., 1992). When the conductivity is not constant, the above correlation breaks down; in this paper we will present data from the Cluster spacecrafts, where this correlation is replaced with a correlation between Eν and j//, i.e. the derivative of Bτ . 2. Cluster data We present electric and magnetic field data from the EFW (Gustafsson et al., 1997) and FGM (Balogh et al., 2001) instruments on the Cluster satellites, which have an apogee of 19.8 RE and a perigee of 4.0 RE, in radial distance. We first present data from a northern hemisphere auroral oval crossing, on Feb 18, 2004, from 08:58:20 to 09:10:00 UT. The Cluster radial distance during this time period was about 4.2 RE, and the satellite separations between approximately 350 and 1100 km. In Figure 1 we show the residual magnetic field vectors along the satellite tracks projected onto a plane perpendicular to the geomagnetic field. The two perpendicular directions in the figure roughly correspond to geomagnetic North, and East. The diamonds at the bottom end of the tracks indicate the satellite positions at 08:58:20 UT. (The data is color coded: black – S/C 1, red – S/C 2, green – S/C 3, blue – S/C 4.) The satellites move relatively close to a pearls-on-a-string configuration. The main feature of the data is the crossing of three sheets of field-aligned current, from bottom to top a relatively smooth sheet of upward current approximately 800 km wide, a thinner sheet of downward current (≈250 km), and finally a wider sheet of predominantly upward currents (~1000 km wide). (The meridional mapping factor to ionospheric altitude is 11.6.) This current system remains essentially stationary in space for the whole 200 s period between the crossings of the central current sheet by S/C 1 and S/C 4, which is the reason we have chosen to present this event. We have applied minimum variance analysis on the magnetic field data from all four S/C, and have used the average resulting angle of 5.8° to establish the sheet-aligned coordinate system. We have then used the infinite current sheet approximation to calculate the field-aligned current j// from the tangential component of the residual magnetic field Bτ . In Figure 2 we present j// and the normal electric field Eν measured by Cluster. All values are mapped to ionospheric altitude. Also presented is the result of a model calculation described in Section 3. The correlation between Eν and j// is clear for all S/C in the downward current region. This type of correlation is rather uncommon, but a manual inspection of around 300 auroral zone crossings resulted in identification of 23 similar events, i.e. in about 8% of the crossings, all for downward currents. 17 of the 23 events where encountered during winter conditions and 15 on the night side. 3. Comparison data – model The close relation between the electric field and the local downward field-aligned current (DFAC) suggests that there is a relation between the DFAC and the conductivity, since an infinitesimally thin current sheet gives a negligible contribution to the ionospheric closure current across the sheet, Jν. However, with a coupling to a local decrease in the conductivity it can produce a local increase in Eν (Figure 3). Such decreases in the conductivity coupled to DFACs have been modeled by Doe at al. (1995), Blixt and Brekke (1996), Karlsson and Marklund (1998, 2005), and Streltsov and Marklund (2006). A few radar observations of ionospheric density cavities which may be related to this mechanism have been reported by Doe et al. (1993), Aikio et al. (2002), and Nilsson et al. (2005). The reason that a cavity is formed in DFAC regions is that the parallel current is mainly carried by electrons, whereas the Pedersen current is carried by ions. In regions where the downward parallel and perpendicular currents couple there will then be a net outflow of current carriers. Here we model this interaction in a heuristic way by prescribing the conductances by , // // , for downward 0, for ward down s P P 0 k j j Σ = Σ − ⎨ Σ = Σ (2) where ΣP,0 and kdown,s (>0) are constants, with s = 1-4, for the four spacecraft crossings. We ignore any effects on the conductance from the upward currents, since we will concentrate on the electric field behavior in the downward current region. We also set a minimum value for the Pedersen conductivity of 0.2 S, which represents the background conductivity due to galactic cosmic rays, which are always present. Current continuity and the assumption of an infinite current sheet yields ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0P HJ j d E E ν ν τν ν ν ν ν ν ν ν′ ′= + Σ − Σ∫ (3) where Jν is the height-integrated ionospheric current normal to the sheet. Eτ is constant if we use the electrostatic approximation ( 0∇× ≡E ). (2) and (3) then give ( ) ( ) ( ) ( ) ( ) 1P H H P P P E E E j d ν ν τ ν ν ν ν ν ν ν ν Σ Σ − Σ ′ ′= + + Σ Σ Σ ∫ (4) Using the observed values for j// along each of the satellite tracks, we can calculate Eν, as a function of ΣP,0 = ΣP(ν0), Eν (ν0), Eτ, and kdown,s. Before t ≈ 320 s the electric field is small and rather constant and we can assume that it can be mapped to the ionosphere and be taken as the background field of our model. However, there is an offset in the electric field component aligned with the direction towards the sun, due to a photo electron sheet. Using data from the electron drift instrument (EDI) on S/C 1 we correct for this and then take the average electric field for 60 s prior to the crossing of the large DFAC, which we use as our background ionospheric electric field: Eν (ν0) = 0, Eτ = -6 mV/m (values mapped to the ionosphere). In principle the conductance could be calculated from the electron data, but this is a very uncertain procedure in the absence of energetic precipitating electrons, and outside the scope of this paper. Instead we assume a reasonable background conductance. The results are rather robust with respect to the chosen value of ΣP,0, but the numerical value of kdown will of course vary within a factor of 2-3 depending on the choice of conductance. By trial and errorr we then find that the following parameters reproduce the electric field behavior in the DFAC region well:, ΣP,0 = 5 S and kdown,1 = 0.33 Sm2/μA, kdown,2 = 0.43 Sm2/μA, kdown,3 = 0.44 Sm2/μA, kdown,4 = 0.68 Sm2/μA, where the subscript on the k’s indicate S/C number. Eν thus calculated is plotted in green in Figure 2. Thus the same set of parameters, except for kdown, reproduces the DFAC electric field quite well. It is interesting that kdown has an increasing trend with time; in Figure 4 we plot the values of kdown as a function of time from the first crossing of the current sheet. The crossing time is defined as the time when the current maximum is encountered, and the error bars in the t-direction indicate when the current is half the maximum value. A linear fit is reasonable which means that we can write 0( )downk t tκ= − , with κ = 1.4·10 -3 Sm2/μAs, and t0 ≈ -200 s, consistent with a gradual deepening of the density cavity, beginning about 200 s before the first satellite crossing. Revisiting the data from the simulations by Karlsson et al. [1998] we can calculate κ. In the simulations, the development of kdown settles down to a reasonably linear dependence on time after the first tens of seconds, from which we can estimate κ. The value of κ depends on various initial conditions of the simulations but for some realistic situations varied from around 1·10-5 to 2·10-3 Sm2/μAs, which is in agreement with the above measurement. For this event the horizontal ionospheric current Jν ,//, resulting from the feeding field-aligned currents was comparable to the current associated with the background electric field: | Jν ,//| ≈ 20 mA/m, |Eτ ΣH(ν0)| ≈ 30 mA/m. Below we show two cases where one of these current contributions dominates over the other one. First (Figure 5a) we show data from a northern hemisphere auroral oval pass on Apr 27, 2002, with MLT ≈ 22, ILat ≈ 66º, and the geocentric distance 4.9 RE. We show data only from S/C 4, but similar signatures can be seen on S/C 2 and 3. Using the same method as above we calculate j//, ΣP, and Eν. In the figure the modeled Eν and ΣP is plotted in red. j// is not shown, but has a maximum (downward) value of 34 μA/m2. For this event upward accelerated electrons are observed from t ≈ 70950 s, which complicates the mapping of the background ionospheric electric field. We instead here consider it as a free parameter. The fact that the constant background current (driven by the background electric field) dominates over Jν ,// (440 mA/m vs. 90 mA/m) means that the electric field traces out the form of the conductivity, which in turn traces out the DFAC. We thus get a very detailed correlation between the electric field and the DFAC, and a unipolar Eν field signature at the density cavity. In Figure 5b we present data from an auroral crossing on Jan 11, 2005. MLT ≈ 22, ILat ≈ 66º, geocentric distance 4.3 RE, max(j//,down) = 32 μA/m2. Here the background ionospheric current is dominated by Jν ,//. This means that the ionospheric current is not constant across the low-conductivity region, and we should not expect such a detailed correlation between Eν and j// as in the above case. In fact, what we see is that the electric field is large inside the low-conductivity region of the DFAC, but since the ionospheric current changes sign inside this region, the electric field also does, and produces a bipolar electric field signature. A small westward background electric field shifts the zero crossing of the total current Jν slightly from that of Jν ,//. 4. Discussion and conclusions The correlation between large electric fields and DFACs presented here is consistent with them being associated with ionospheric low-conductivity regions. A correlation between the electric field and the derivative of the magnetic field could also be the result of a partially reflected Alfvén wave, but this would not explain why we only observe this correlation for downward currents, or the preference for night- /wintertime conditions. The correlation is also not consistent with the signatures of a U-shaped potential structure. There, the largest current is associated with the centre of the structure, where the perpendicular electric field has its minimum. In fact, in order for the electric field correlation with the DFAC to map all the way out to Cluster altitudes, we must assume that there is no field-aligned potential drop along the magnetic field line. In that case the correlation represents the naked high-altitude signature of the ionospheric density depletion. In many cases we would expect large DFACs to be associated with such a parallel potential drop [e.g. Elphic et al, 1998]; this may be one of the reasons why events of the type we have presented here are relatively rare; we will only see them before such a potential drop has developed. Another reason could be that generally rather low background conductivities will be required. Reversing the argument, observations of large perpendicular electric fields at magnetospheric altitudes is generally taken as proof that there is a parallel potential drop above the ionosphere. Our results show that this is not necessarily true, but that at least part of this potential drop may be situated deep in the ionosphere, in the E and lower F regions, where the currents partially close through the developing density cavity [Karlsson and Marklund, 1998]. This should be taken into account when interpreting high-altitude electric field data. For the first event, the current system is stable for around 200 s. The close to linear evolution of kdown, can be seen as a first observational comparison with modeling of the temporal evolution of ionospheric density cavities. The 200 s time scale is, according to the modeling work quoted above, a typical time scale for creating a deep ionospheric plasma depletion. We would expect to see this type of events for conditions of some moderate geomagnetic activity (to create large DFACs), but not during e.g. the substorm expansion phase, where the current systems would probably move around too much on time scales faster than the depletion time. We have checked this by inspecting the Auroral Electrojet index for the 23 events. Only four of the events where encountered during the expansion phase, whereas the rest were observed during periods that had a medium level of activity; growth or recovery phase or steady magnetospheric convection events. This is further support for the model presented above. Acknowledgements The authors are grateful to G. Haerendel for some suggestions and comments. References Aikio, A. T., K. Mursula, S. Buchert, F. Forme, O. Amm, G. Marklund, M. Dunlop, D. Fontaine, A. Vaivads, and A. Fazakerley (2004), Temporal evolution of two auroral arcs as measured by the Cluster satellite and coordinated ground-based instruments, Ann. Geophys., 22, 4089–4101. Balogh, A., Carr, C.M., Acuña, M-H., Dunlop, M.W., Beek, T.J., Brown, P., Fornacon, K.-H., Georgescu, E., Glassmeier, K.-H., Harris, J.P., Musmann, G., Oddy, T.M. and Schwingenschuh, K (2001), The Cluster magnetic field investigation: Overview of in-flight performance and initial results, Ann. Geophys., 19, 1207-1217. Blixt, E. M., A. Brekke (1996), A model of currents and electric fields in a discrete auroral arc, Geophys. Res. Lett., 23, 2553. Doe, R. A., Mendillo, M., Vickrey, J. F., Zanetti, L. J., Eastes, R. W. (1993), Observations of nightside auroral cavities, J. Geophys. Res., 98, 293-310. Doe, R. A., J. F. Vickrey, M. Mendillo (1995), Electrodynamic model for the formation of auroral ionospheric cavities, J. Geophys. Res., 100, 9683. Elphic, R. C. J. W. Bonnell, R. J. Strangeway, L. Kepko, R. E. Ergun, J. P. McFadden, C. W. Carlson, W. Peria, C. A. Cattell, D. Klumpar, E. Shelley, W. Peterson. E. Moebius, L. Kistler, R. Pfaff (1998), The auroral current circuit and field-aligned currents observed by FAST, Geophys. Res. Lett., 25, 2033-2036. Gustafsson, G., R. Bostrom, B. Holback, G. Holmgren, A. Lundgren, K. Stasiewicz, L. Åhlen, F. S. Mozer, D. Pankow, P. Harvey, P. Berg, R. Ulrich., A. Pedersen, R. Schmidt, A. Butler, A. W. C. Fransen, D. Klinge, M. Thomsen, C.-G. Fälthammar, P.-A. Lindqvist, S. Christenson, J. Holtet, B. Lybekk, T. A. Sten , P. Tanskanen, K. Lappalainen, and J. Wygant (1997), The electric field and wave experiment for the Cluster mission, Space Sci. Rev., 79, 137-156. Ishii, M., M. Sugiura, T. Iyemori, J. A. Slavin (1992), Correlation between magnetic and electric field perturbations in the field-aligned current regions deduced from DE-2 observations, J. Geophys. Res., 97, 13877-13887. Karlsson, T. and G. T. Marklund (1998), Simulations of Small-Scale Auroral Current Closure in the Return Current Region, in Physics of Space Plasmas, No. 15, eds. T. Chang and J. R. Jasperse, MIT Center for Theoretical Geo/Cosmo Plasma Physics, Cambridge, Massachusetts, 401-406. Karlsson, T., G. Marklund, N. Brenning, I. Axnäs (2005), On Enhanced Aurora and Low-Altitude Parallel Electric Fields, Phys. Scr., 72, 419-422. Nilsson, H., A. Kozlovsky, T. Sergienko, A. Kotikov (2005), Radar observations in the vicinity of pree-noon auroral arcs, Ann. Geophys., 23, 1785-1796. Streltsov, A. V., and G. T. Marklund (2006), Divergent electric fields in downward current channels, J. Geophys. Res., 111, A07204, doi:10.1029/2005JA011196. Figure 1. Magnetic field perpendicular to the geomagnetic field for the time period 2004-02-18 08:58:20-09:10:00 UT. Figure 2. Measured (blue) and modeled (green) normal electric field, and measured field-aligned current (red) for the same time period as Figure 1. Note that in this figure downward current is plotted as negative current. Figure 3. Schematic of the connection between DFAC, current closure and iono- spheric cavity formation. Figure 4. Temporal evolution of kdown. Figure 5. Measured (black) and modeled (red) quantities. Time is in seconds from 00:00:00 UT.

We present Cluster measurements of large electric fields correlated with intense downward field-aligned currents, and show that the data can be reproduced by a simple model of ionospheric plasma depletion caused by the currents. This type of magnetosphere-ionosphere interaction may be important when considering the mapping between these two regions of space.

Introduction A system of magnetic field-aligned current sheets closing via Pedersen currents in the ionosphere will set up an ionospheric electric field. For constant conductivity, and for sheets extending to infinity along the field-line and one of the perpendicular directions, we get: 1 1 1P P P P P E j d d Bτν τν νμ ν μ = = = = Σ Σ Σ ∂ Σ∫ ∫ (1) where ν is the direction perpendicular to the sheet, τ the tangential direction, Eν is the normal electric field, JP and ΣP, the height integrated Pedersen current and conductivity, Bτ the tangential magnetic field, j// the field-aligned current (positive for downward currents) and μ0 the magnetic permeability of vacuum. This kind of correlation between Eν and Bτ can be seen rather often in the dayside auroral oval (e.g. Ishii et al., 1992). When the conductivity is not constant, the above correlation breaks down; in this paper we will present data from the Cluster spacecrafts, where this correlation is replaced with a correlation between Eν and j//, i.e. the derivative of Bτ . 2. Cluster data We present electric and magnetic field data from the EFW (Gustafsson et al., 1997) and FGM (Balogh et al., 2001) instruments on the Cluster satellites, which have an apogee of 19.8 RE and a perigee of 4.0 RE, in radial distance. We first present data from a northern hemisphere auroral oval crossing, on Feb 18, 2004, from 08:58:20 to 09:10:00 UT. The Cluster radial distance during this time period was about 4.2 RE, and the satellite separations between approximately 350 and 1100 km. In Figure 1 we show the residual magnetic field vectors along the satellite tracks projected onto a plane perpendicular to the geomagnetic field. The two perpendicular directions in the figure roughly correspond to geomagnetic North, and East. The diamonds at the bottom end of the tracks indicate the satellite positions at 08:58:20 UT. (The data is color coded: black – S/C 1, red – S/C 2, green – S/C 3, blue – S/C 4.) The satellites move relatively close to a pearls-on-a-string configuration. The main feature of the data is the crossing of three sheets of field-aligned current, from bottom to top a relatively smooth sheet of upward current approximately 800 km wide, a thinner sheet of downward current (≈250 km), and finally a wider sheet of predominantly upward currents (~1000 km wide). (The meridional mapping factor to ionospheric altitude is 11.6.) This current system remains essentially stationary in space for the whole 200 s period between the crossings of the central current sheet by S/C 1 and S/C 4, which is the reason we have chosen to present this event. We have applied minimum variance analysis on the magnetic field data from all four S/C, and have used the average resulting angle of 5.8° to establish the sheet-aligned coordinate system. We have then used the infinite current sheet approximation to calculate the field-aligned current j// from the tangential component of the residual magnetic field Bτ . In Figure 2 we present j// and the normal electric field Eν measured by Cluster. All values are mapped to ionospheric altitude. Also presented is the result of a model calculation described in Section 3. The correlation between Eν and j// is clear for all S/C in the downward current region. This type of correlation is rather uncommon, but a manual inspection of around 300 auroral zone crossings resulted in identification of 23 similar events, i.e. in about 8% of the crossings, all for downward currents. 17 of the 23 events where encountered during winter conditions and 15 on the night side. 3. Comparison data – model The close relation between the electric field and the local downward field-aligned current (DFAC) suggests that there is a relation between the DFAC and the conductivity, since an infinitesimally thin current sheet gives a negligible contribution to the ionospheric closure current across the sheet, Jν. However, with a coupling to a local decrease in the conductivity it can produce a local increase in Eν (Figure 3). Such decreases in the conductivity coupled to DFACs have been modeled by Doe at al. (1995), Blixt and Brekke (1996), Karlsson and Marklund (1998, 2005), and Streltsov and Marklund (2006). A few radar observations of ionospheric density cavities which may be related to this mechanism have been reported by Doe et al. (1993), Aikio et al. (2002), and Nilsson et al. (2005). The reason that a cavity is formed in DFAC regions is that the parallel current is mainly carried by electrons, whereas the Pedersen current is carried by ions. In regions where the downward parallel and perpendicular currents couple there will then be a net outflow of current carriers. Here we model this interaction in a heuristic way by prescribing the conductances by , // // , for downward 0, for ward down s P P 0 k j j Σ = Σ − ⎨ Σ = Σ (2) where ΣP,0 and kdown,s (>0) are constants, with s = 1-4, for the four spacecraft crossings. We ignore any effects on the conductance from the upward currents, since we will concentrate on the electric field behavior in the downward current region. We also set a minimum value for the Pedersen conductivity of 0.2 S, which represents the background conductivity due to galactic cosmic rays, which are always present. Current continuity and the assumption of an infinite current sheet yields ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0P HJ j d E E ν ν τν ν ν ν ν ν ν ν′ ′= + Σ − Σ∫ (3) where Jν is the height-integrated ionospheric current normal to the sheet. Eτ is constant if we use the electrostatic approximation ( 0∇× ≡E ). (2) and (3) then give ( ) ( ) ( ) ( ) ( ) 1P H H P P P E E E j d ν ν τ ν ν ν ν ν ν ν ν Σ Σ − Σ ′ ′= + + Σ Σ Σ ∫ (4) Using the observed values for j// along each of the satellite tracks, we can calculate Eν, as a function of ΣP,0 = ΣP(ν0), Eν (ν0), Eτ, and kdown,s. Before t ≈ 320 s the electric field is small and rather constant and we can assume that it can be mapped to the ionosphere and be taken as the background field of our model. However, there is an offset in the electric field component aligned with the direction towards the sun, due to a photo electron sheet. Using data from the electron drift instrument (EDI) on S/C 1 we correct for this and then take the average electric field for 60 s prior to the crossing of the large DFAC, which we use as our background ionospheric electric field: Eν (ν0) = 0, Eτ = -6 mV/m (values mapped to the ionosphere). In principle the conductance could be calculated from the electron data, but this is a very uncertain procedure in the absence of energetic precipitating electrons, and outside the scope of this paper. Instead we assume a reasonable background conductance. The results are rather robust with respect to the chosen value of ΣP,0, but the numerical value of kdown will of course vary within a factor of 2-3 depending on the choice of conductance. By trial and errorr we then find that the following parameters reproduce the electric field behavior in the DFAC region well:, ΣP,0 = 5 S and kdown,1 = 0.33 Sm2/μA, kdown,2 = 0.43 Sm2/μA, kdown,3 = 0.44 Sm2/μA, kdown,4 = 0.68 Sm2/μA, where the subscript on the k’s indicate S/C number. Eν thus calculated is plotted in green in Figure 2. Thus the same set of parameters, except for kdown, reproduces the DFAC electric field quite well. It is interesting that kdown has an increasing trend with time; in Figure 4 we plot the values of kdown as a function of time from the first crossing of the current sheet. The crossing time is defined as the time when the current maximum is encountered, and the error bars in the t-direction indicate when the current is half the maximum value. A linear fit is reasonable which means that we can write 0( )downk t tκ= − , with κ = 1.4·10 -3 Sm2/μAs, and t0 ≈ -200 s, consistent with a gradual deepening of the density cavity, beginning about 200 s before the first satellite crossing. Revisiting the data from the simulations by Karlsson et al. [1998] we can calculate κ. In the simulations, the development of kdown settles down to a reasonably linear dependence on time after the first tens of seconds, from which we can estimate κ. The value of κ depends on various initial conditions of the simulations but for some realistic situations varied from around 1·10-5 to 2·10-3 Sm2/μAs, which is in agreement with the above measurement. For this event the horizontal ionospheric current Jν ,//, resulting from the feeding field-aligned currents was comparable to the current associated with the background electric field: | Jν ,//| ≈ 20 mA/m, |Eτ ΣH(ν0)| ≈ 30 mA/m. Below we show two cases where one of these current contributions dominates over the other one. First (Figure 5a) we show data from a northern hemisphere auroral oval pass on Apr 27, 2002, with MLT ≈ 22, ILat ≈ 66º, and the geocentric distance 4.9 RE. We show data only from S/C 4, but similar signatures can be seen on S/C 2 and 3. Using the same method as above we calculate j//, ΣP, and Eν. In the figure the modeled Eν and ΣP is plotted in red. j// is not shown, but has a maximum (downward) value of 34 μA/m2. For this event upward accelerated electrons are observed from t ≈ 70950 s, which complicates the mapping of the background ionospheric electric field. We instead here consider it as a free parameter. The fact that the constant background current (driven by the background electric field) dominates over Jν ,// (440 mA/m vs. 90 mA/m) means that the electric field traces out the form of the conductivity, which in turn traces out the DFAC. We thus get a very detailed correlation between the electric field and the DFAC, and a unipolar Eν field signature at the density cavity. In Figure 5b we present data from an auroral crossing on Jan 11, 2005. MLT ≈ 22, ILat ≈ 66º, geocentric distance 4.3 RE, max(j//,down) = 32 μA/m2. Here the background ionospheric current is dominated by Jν ,//. This means that the ionospheric current is not constant across the low-conductivity region, and we should not expect such a detailed correlation between Eν and j// as in the above case. In fact, what we see is that the electric field is large inside the low-conductivity region of the DFAC, but since the ionospheric current changes sign inside this region, the electric field also does, and produces a bipolar electric field signature. A small westward background electric field shifts the zero crossing of the total current Jν slightly from that of Jν ,//. 4. Discussion and conclusions The correlation between large electric fields and DFACs presented here is consistent with them being associated with ionospheric low-conductivity regions. A correlation between the electric field and the derivative of the magnetic field could also be the result of a partially reflected Alfvén wave, but this would not explain why we only observe this correlation for downward currents, or the preference for night- /wintertime conditions. The correlation is also not consistent with the signatures of a U-shaped potential structure. There, the largest current is associated with the centre of the structure, where the perpendicular electric field has its minimum. In fact, in order for the electric field correlation with the DFAC to map all the way out to Cluster altitudes, we must assume that there is no field-aligned potential drop along the magnetic field line. In that case the correlation represents the naked high-altitude signature of the ionospheric density depletion. In many cases we would expect large DFACs to be associated with such a parallel potential drop [e.g. Elphic et al, 1998]; this may be one of the reasons why events of the type we have presented here are relatively rare; we will only see them before such a potential drop has developed. Another reason could be that generally rather low background conductivities will be required. Reversing the argument, observations of large perpendicular electric fields at magnetospheric altitudes is generally taken as proof that there is a parallel potential drop above the ionosphere. Our results show that this is not necessarily true, but that at least part of this potential drop may be situated deep in the ionosphere, in the E and lower F regions, where the currents partially close through the developing density cavity [Karlsson and Marklund, 1998]. This should be taken into account when interpreting high-altitude electric field data. For the first event, the current system is stable for around 200 s. The close to linear evolution of kdown, can be seen as a first observational comparison with modeling of the temporal evolution of ionospheric density cavities. The 200 s time scale is, according to the modeling work quoted above, a typical time scale for creating a deep ionospheric plasma depletion. We would expect to see this type of events for conditions of some moderate geomagnetic activity (to create large DFACs), but not during e.g. the substorm expansion phase, where the current systems would probably move around too much on time scales faster than the depletion time. We have checked this by inspecting the Auroral Electrojet index for the 23 events. Only four of the events where encountered during the expansion phase, whereas the rest were observed during periods that had a medium level of activity; growth or recovery phase or steady magnetospheric convection events. This is further support for the model presented above. Acknowledgements The authors are grateful to G. Haerendel for some suggestions and comments. References Aikio, A. T., K. Mursula, S. Buchert, F. Forme, O. Amm, G. Marklund, M. Dunlop, D. Fontaine, A. Vaivads, and A. Fazakerley (2004), Temporal evolution of two auroral arcs as measured by the Cluster satellite and coordinated ground-based instruments, Ann. Geophys., 22, 4089–4101. Balogh, A., Carr, C.M., Acuña, M-H., Dunlop, M.W., Beek, T.J., Brown, P., Fornacon, K.-H., Georgescu, E., Glassmeier, K.-H., Harris, J.P., Musmann, G., Oddy, T.M. and Schwingenschuh, K (2001), The Cluster magnetic field investigation: Overview of in-flight performance and initial results, Ann. Geophys., 19, 1207-1217. Blixt, E. M., A. Brekke (1996), A model of currents and electric fields in a discrete auroral arc, Geophys. Res. Lett., 23, 2553. Doe, R. A., Mendillo, M., Vickrey, J. F., Zanetti, L. J., Eastes, R. W. (1993), Observations of nightside auroral cavities, J. Geophys. Res., 98, 293-310. Doe, R. A., J. F. Vickrey, M. Mendillo (1995), Electrodynamic model for the formation of auroral ionospheric cavities, J. Geophys. Res., 100, 9683. Elphic, R. C. J. W. Bonnell, R. J. Strangeway, L. Kepko, R. E. Ergun, J. P. McFadden, C. W. Carlson, W. Peria, C. A. Cattell, D. Klumpar, E. Shelley, W. Peterson. E. Moebius, L. Kistler, R. Pfaff (1998), The auroral current circuit and field-aligned currents observed by FAST, Geophys. Res. Lett., 25, 2033-2036. Gustafsson, G., R. Bostrom, B. Holback, G. Holmgren, A. Lundgren, K. Stasiewicz, L. Åhlen, F. S. Mozer, D. Pankow, P. Harvey, P. Berg, R. Ulrich., A. Pedersen, R. Schmidt, A. Butler, A. W. C. Fransen, D. Klinge, M. Thomsen, C.-G. Fälthammar, P.-A. Lindqvist, S. Christenson, J. Holtet, B. Lybekk, T. A. Sten , P. Tanskanen, K. Lappalainen, and J. Wygant (1997), The electric field and wave experiment for the Cluster mission, Space Sci. Rev., 79, 137-156. Ishii, M., M. Sugiura, T. Iyemori, J. A. Slavin (1992), Correlation between magnetic and electric field perturbations in the field-aligned current regions deduced from DE-2 observations, J. Geophys. Res., 97, 13877-13887. Karlsson, T. and G. T. Marklund (1998), Simulations of Small-Scale Auroral Current Closure in the Return Current Region, in Physics of Space Plasmas, No. 15, eds. T. Chang and J. R. Jasperse, MIT Center for Theoretical Geo/Cosmo Plasma Physics, Cambridge, Massachusetts, 401-406. Karlsson, T., G. Marklund, N. Brenning, I. Axnäs (2005), On Enhanced Aurora and Low-Altitude Parallel Electric Fields, Phys. Scr., 72, 419-422. Nilsson, H., A. Kozlovsky, T. Sergienko, A. Kotikov (2005), Radar observations in the vicinity of pree-noon auroral arcs, Ann. Geophys., 23, 1785-1796. Streltsov, A. V., and G. T. Marklund (2006), Divergent electric fields in downward current channels, J. Geophys. Res., 111, A07204, doi:10.1029/2005JA011196. Figure 1. Magnetic field perpendicular to the geomagnetic field for the time period 2004-02-18 08:58:20-09:10:00 UT. Figure 2. Measured (blue) and modeled (green) normal electric field, and measured field-aligned current (red) for the same time period as Figure 1. Note that in this figure downward current is plotted as negative current. Figure 3. Schematic of the connection between DFAC, current closure and iono- spheric cavity formation. Figure 4. Temporal evolution of kdown. Figure 5. Measured (black) and modeled (red) quantities. Time is in seconds from 00:00:00 UT.

BurgersTurbulence Jérémie Bec Laboratoire Cassiopée UMR6202, CNRS, OCA; BP4229, 06304 Nice Cedex 4, France. Konstantin Khanin Department of Mathematics, University of Toronto, M5S 3G3 Toronto, Ontario, Canada. Abstract The last decades witnessed a renewal of interest in the Burgers equation. Much activities focused on extensions of the original one-dimensional pressureless model introduced in the thirties by the Dutch scientist J.M. Burgers, and more precisely on the problem of Burgers turbulence, that is the study of the solutions to the one- or multi-dimensional Burgers equation with random initial conditions or random forcing. Such work was frequently motivated by new emerging applications of Burgers model to statistical physics, cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the simplest instances of a nonlinear system out of equilibrium. The study of random Lagrangian systems, of stochastic partial differential equations and their invariant measures, the theory of dynamical systems, the applications of field theory to the understanding of dissipative anomalies and of multiscaling in hydrodynamic turbulence have benefited significantly from progress in Burgers turbulence. The aim of this review is to give a unified view of selected work stemming from these rather diverse disciplines. Key words: Burgers equation, turbulence, Lagrangian systems. Contents 1 From interface dynamics to cosmology 2 1.1 The Burgers equation in statistical mechanics 2 1.2 The adhesion model in cosmology 3 1.3 A benchmark for hydrodynamical turbulence 3 2 Basic tools 4 2.1 Inviscid limit and variational principle 4 2.2 Variational principle for the viscous case 5 2.3 Singularities of Burgers turbulence 5 2.4 Remarks on numerical methods 7 3 Decaying Burgers turbulence 8 3.1 Geometrical constructions of the solution 8 Email addresses: jeremie.bec@obs-nice.fr (Jérémie Bec), khanin@math.toronto.edu (Konstantin Khanin). 3.2 Kida’s law for energy decay 10 3.3 Brownian initial velocities 12 4 Transport of mass in the Burgers/adhesion model 14 4.1 Mass density and singularities 15 4.2 Evolution of matter inside shocks 16 4.3 Connections with convex optimization problems 19 5 Forced Burgers turbulence 21 5.1 Stationary régime and global minimizer 21 5.2 Topological shocks 22 5.3 Hyperbolicity of the global minimizer 24 5.4 The case of extended systems 26 6 Time-periodic forcing 29 Preprint submitted to Physics Reports 1 February 2008 http://arXiv.org/abs/0704.1611v1 6.1 Kicked Burgers turbulence 29 6.2 Connections with Aubry–Mather theory 33 7 Velocity statistics in randomly forced Burgers turbulence 37 7.1 Shocks and bifractality – a replica variational approach 38 7.2 Dissipative anomaly and operator product expansion 39 7.3 Tails of the velocity gradient PDF 41 7.4 Self-similar forcing and multiscaling 43 8 Concluding remarks and open questions 45 Acknowledgements 46 References 46 1 From interface dynamics to cosmology At the end of the thirties, the Dutch scientist J.M. Burg- ers [26] introduced a one-dimensional model for pressure- less gas dynamics. He was hoping that the use of a simple model having much in common with the Navier–Stokes equation would significantly contribute to the study of fluid turbulence. This model now known as the Burgers equation ∂tv + v ∂xv = ν ∂ xv, (1.1) has not only the same type of hydrodynamical (or advec- tive) quadratic nonlinearity as the Navier–Stokes equa- tion that is balanced by a diffusive term, but it also has similar invariances and conservation laws (invariance un- der translations in space and time, parity invariance, conservation of energy and momentum in one dimension for ν = 0). Such hopes appeared to be shattered when in the fifties, Hopf [67] and Cole [33] showed that the Burgers equation can be integrated explicitly. This model thus lacks one of the essential properties of Navier–Stokes turbulence: sensitivity to small perturbations in the initial data and thus the spontaneous arise of randomness by chaotic dy- namics. Unable to cope with such a fundamental aspect, the Burgers equation then lost its interest in “explain- ing” fluid turbulence. In spite of this, the Burgers equation reappeared in the eighties as the asymptotic form of various nonlinear dis- sipative systems. Physicists and astrophysicists then de- voted important effort to the understanding of its multi- dimensional form and to the study of its random solu- tions arising from random initial conditions or a random forcing. The goal of this paper is to review selected works that exemplify this strong renewal of interest in Burgers turbulence. The rest of this section is dedicated to the description of several physical situations where the Burgers equation arises. We will then see in section 2 that in any dimen- sion and in the limit of vanishing viscosity, the solutions to the Burgers equation can be expressed in an explicit manner in the decaying case or in an implicit manner in the forced case, in terms of a variational principle that permits a systematic classification of its various singu- larities (shocks and others) and of their local structure (normal form). Section 3 is dedicated to the study of the decay of the solutions to the one-dimensional unforced Burgers equation with random initial data. The multi- dimensional decaying problem is discussed in section 4. The motivation comes from cosmology where large-scale structures can be described in terms of mass transport by solutions to the Burgers equation. The basic prin- ciples of the forced Burgers turbulence are discussed in section 5 where the notions of global minimizer and topological shocks are introduced. Section 6 is dedicated to the study of the solutions to the periodically kicked Burgers equation and their relation with Aubry–Mather theory for commensurate-incommensurate phase transi- tions. Section 7 reviews various studies of the stochas- tically forced Burgers equation in one dimension with a particular emphasize on questions that are arising from the statistical study of turbulent flows. Finally, section 8 encompasses concluding remarks and a non-exhaustive list of open questions on the problem of the Burgers tur- bulence. 1.1 The Burgers equation in statistical mechanics The Burgers equation appears in condensed matter, in statistical physics, and also beyond physics in vehicle traffic models (see [32], for a review on this topic). When a random forcing term is added - usually a white noise in time - it is used to describe various problems of in- terface deposition and growth (see, for instance, [5]). An instance frequently studied is the Kardar–Parisi–Zhang (KPZ) model [74]. This continuous version of ballistic deposition models accounts for the lateral growth of the interface. Let us indeed consider an interface where par- ticles deposit with a random flux F that depends both on time t and on the horizontal position ~x. The growth of the local height h happens in the direction normal to the interface and its time evolution is given by |∇h|2 = ν∇2h+ F, (1.2) where the first term of the right-hand side represents the relaxation due to a surface tension ν. The gradient of (1.2) gives the multidimensional Burgers equation ∂t~v + ~v · ∇~v = ν∇2~v −∇F, ~v = −∇h, (1.3) forced by the random potential F . As we will see later, shocks generically appear in the solution to the Burgers equation in the inviscid limit ν → 0. They correspond to discontinuities of the derivative of the height h. The KPZ model is hence frequently used to understand the appearence of roughness in various interface problems, as for instance front propagation in randomly distributed forests (see, e.g., [101]). The Hopf–Cole transformation Z = exp(h/2ν) allows rewriting (1.2) as a linear problem with random coeffi- cients. ∂tZ = ν∇2Z + F Z, (1.4) This equation appears in many complex systems, as for instance directed polymers in random media [75,22]. Indeed the solution Z(~x, t) is exactly the partition function of an elastic string in the random potential (1/2ν)F (~x, t), subject to the constraint that its bound- ary is fixed at (~x, t). Note that here, the time variable t is actually a space variable in the main direction of the polymer. 1.2 The adhesion model in cosmology The multidimensional Burgers equation has important applications in cosmology where it is closely linked to what is usually referred to as the Zel’dovich approxi- mation [112]. In the limit of vanishing viscosity ν → 0 the Burgers equation is known as the adhesion model [62]. Right after the decoupling between baryons and photons, the primitive Universe is a rarefied medium without pressure composed mainly of non-collisional dust interacting through Newtonian gravity. The initial density of this dark matter fluctuates around a mean value ρ̄. These fluctuations are responsible for the for- mation of the large-scale structures in which both the dark non-baryonic matter and the luminous baryonic matter concentrate. A hydrodynamical formulation of the cosmological problem leads to a description where matter evolves with a velocity ~v, solution of the Euler– Poisson equation (see, e.g., [98], for further details). In the linear theory of the gravitational instability, that is for infinitesimally small initial fluctuations of the den- sity field, an instability is obtained with potential dom- inant modes (i.e. ~v = −∇Ψ) and, in the suitable coordi- nates, the gravitational interactions can be neglected. In 1970, Zel’dovich proposed to extend these two proper- ties to the nonlinear régimes where density fluctuations become important. In this approximation, he also pos- tulates that the acceleration is a Lagrangian invariant, leading to the formation of caustics. N-body simulations however show that the large-scale structures of the Uni- verse are much simpler than caustics: they resemble sort of thin layers in which the particles tend to be trapped (see figure 1(a)). Fig. 1. (a) Projection of the matter distribution in a slice obtained from an N-body simulation by the Virgo consor- tium [71]. (b) Composite picture showing the superposition of the results of an N-body simulation with the skeleton of the results obtained from the adhesion model (from [78]). It was shown by Gurbatov and Saichev [62] that these structures are very well approximated by those obtained when constraining the particles not to cross each other but to stick together. Even if this mechanism is not col- lisional but rather gravitational (probably due to insta- bilities at small spatial scales), its effect can be modeled by a small viscous diffusive term in the Euler–Poisson equation and thus amounts to considering the Burgers equation in the limit of vanishing viscosity. 1.3 A benchmark for hydrodynamical turbulence As a nonlinear conservation law, and since its solution can be easily known explicitly, the one-dimensional Burgers equation frequently serves as a testing ground for numerical schemes, and especially for those dedi- cated to compressible hydrodynamics. For instance, it is a central example for the validation of finite-volumes schemes. The Burgers equation was also used for testing statisti- cal theories of turbulence. For instance, field theoretical methods have frequently been applied to turbulence (see [96,102]). These approaches had very little impact until recently when they led to significant advances in the un- derstanding of intermittency in passive scalar advection (see, e.g., [46] for a review). In the past such attempts were mostly based on a formal expansion of the non- linearity using, for instance, Feynman graphs. Since the Burgers equation has the same type of quadratic nonlin- earity as the Navier–Stokes equation, such methods are applicable in both instances. From this point of view, it is important to know answers for Burgers turbulence to questions that are generally asked for Navier–Stokes turbulence. For instance, Burgers turbulence with a ran- dom forcing is the counterpart of the hydrodynamical turbulence model where a steady state is maintained by an external forcing. The Burgers equation has frequently been used as a model where the dissipation of kinetic energy remains finite in the limit of vanishing viscosity (dissipative anomaly). This allows singling out artifacts arising from manipulation that ignore shock waves (see, for instance, [51,40]). Beyond statistical theory, Burgers turbulence gives a simple hydrodynamical training ground for developing mathematical tools to study not only Navier–Stokes tur- bulence but also various hydrodynamical or Lagrangian problems. The forced Burgers equation has recently been at the center of studies that allowed unifying different branches of mathematics. Mainly used in the past as a simple illustration of the notion of entropy (or viscos- ity) solution for conservation laws [83,95,85], the Burg- ers equation was related in the eighties to the theory of Hamiltonian systems developed by Kolmogorov [80], Arnold [2] and Moser [93] (KAM), through the introduc- tion of the weak KAM theory [43,47,48]. More recently, the study of the solutions to the Burgers equation with a random forcing was at the center of a “random” Aubry– Mather theory related to random Lagrangian systems [38,69]. A particular emphasis on these aspects of Burg- ers turbulence is given throughout the present review. For the application of the Burgers equation to the prop- agation of random nonlinear waves in nondispersive me- dia, we refer the reader to the book written by Gurba- tov, Malakhov, and Saichev [61]. For a complete state of the art on most mathematical apsects of Burgers turbu- lence, we refer the reader to the lecture notes by Woy- czyński [110]. 2 Basic tools In this section we introduce various analytical, geomet- rical and numerical tools that are useful for constructing solutions to the Burgers equation, with and without forc- ing, in the limit of vanishing viscosity. All these tools are derived from a variational principle that allows writing in an implicit way the solution at any time. This varia- tional principle leads to a straightforward classification of the various singularities that are generically present in the solution to the Burgers equation. 2.1 Inviscid limit and variational principle We consider here the multidimensional viscous Burgers equation with forcing ∂t~v + (~v · ∇)~v = ν∇2~v −∇F (~x, t), (2.1) where ~x lives on a prescribed configuration space Ω of dimension d. For a potential initial condition, ~v(~x, t0) = −∇Ψ0(~x), the velocity field remains potential by con- struction at any later time, ~v = −∇Ψ, where the poten- tial Ψ satisfies the equation ∂tΨ − |∇Ψ|2 = ν∇2Ψ + F. (2.2) Note that if one sets abruptly ν = 0 in (2.2), then −Ψ solves the Hamilton–Jacobi equation associated to the Hamiltonian H(~q, ~p) = |~p|2 + F (~q, t). In the unforced case, −Ψ is a solution of the Hamilton–Jacobi equation associated to the dynamics of free particles. The Hopf– Cole transformation [67,33] uses a change of unknown Ψ(~x, t) = 2ν ln Θ(~x, t). The new unknown scalar field Θ is solution of the (imaginary-time) Schrödinger equation ∂tΘ = ν∇2Θ + F Θ, (2.3) with the initial condition Θ(~x, t0) = exp(Ψ0(~x)/(2ν)). The solution can be expressed through the Feynman- Kac formula Θ(~x, t)= Ψ0( ~Wt0)− F ( ~Ws, s) ds , (2.4) where the brackets 〈·〉 denote the ensemble average with respect to the realizations of the d-dimensional Brown- ian motion ~Ws with variance 2ν defined on the configu- ration space Ω and which starts at ~x at time t. The limit ν → 0 is obtained by a classical saddle-point argument. The main contribution will come from the trajectories ~W minimizing the argument of the exponential; the ve- locity potential can then be expressed as a solution of the variational principle Ψ(~x, t) = − inf ~γ(·) [A(~γ, t0, t) − Ψ0(~γ(t0))] , (2.5) where the infimum is taken over all trajectories ~γ that are absolutely continuous (e.g. piece-wise differentiable) with respect to the time variable and that satisfy ~γ(t) = ~x. The action A(~γ, t0, t) associated to the trajectory ~γ is defined by A(~γ, t0, t) = |~̇γ(s)|2 − F (~γ(s), s) ds, (2.6) where the dot stands for time derivative. The kinetic energy term |~̇γ|2/2 comes from the propagator of the d- dimensional Brownian motion ~W . This variational for- mulation of the solution to the Burgers equation was obtained first by Hopf [67], Lax [83] and Oleinik [95] for scalar conservation laws. Its generalization to mul- tidimensional Hamilton–Jacobi equations was done by Kruzhkov [82] (see also [85]). In the case of a random forcing potential F , it was shown by E, Khanin, Mazel and Sinai [38] that this formulation is still valid after re- placing the action by a stochastic integral. It is also im- portant to notice that the variational formulation (2.5) in the limit of vanishing viscosity is valid irrespective of the configuration space Ω on which the solution is de- fined. The minimizing trajectories ~γ(·) necessarily satisfy at times s < t the Newton (or Euler–Lagrange) equation ~̈γ = −∇F (~γ(s), s), (2.7) with the boundary conditions (at the final time t) ~γ(t) = ~x and ~̇γ(t) = ~v(~x, t). (2.8) Note that these equations are only valid backward in time. Extending them to times larger than t requires knowing that the Lagrangian particle will neither cross the trajectory of another particle, nor be absorbed by a mature shock. This requires global knowledge of the solution that satisfies the variational principle (2.5). When the forcing term is absent from (2.1), it is easily checked that the variational principle reduces to Ψ(~x, t) = max Ψ0(~x0) − |~x− ~x0|2 , (2.9) where the maximum is taken over all initial positions ~x0 in the configuration space Ω. The Euler–Lagrange equation takes then the particularly simple form ~̈γ = 0, i.e. ~x = ~x0 + t ~v0(~x0), (2.10) which simply means that the initial velocity is conserved along characteristics. Typically there exist Eulerian locations ~xwhere the min- imum in (2.5) – or the maximum in (2.9) in the unforced case – is reached for several different trajectories ~γ. Such locations correspond to singularities in the solution to the Burgers equation. After their appearance, the veloc- ity potential Ψ contains angular points corresponding to discontinuities of the velocity field ~v = −∇Ψ. 2.2 Variational principle for the viscous case The derivation of the variational principle (2.5) makes use of the Hopf–Cole transformation and of the Feynman–Kac formula. There is in fact another ap- proach which also yields a variational formulation of the solution to the viscous Hamilton–Jacobi equation (2.2). Indeed it turns out that the solution to (2.2) can be obtained in the following way. Consider solutions to the stochastic differential equation d~γ~u = ~u(~γ~u, s) ds+ 2ν d ~Ws , (2.11) where ~u is a stochastic control, that is an arbitrary time- dependent velocity field which depends (progressively measurably) on the noise ~W . Limiting ourselves to so- lutions satisfying the final condition ~γ~u(t) = ~x, we can write Ψ(~x, t) = − inf 〈A~u(~γ~u, t0, t) − Ψ0(~γ~u(t0))〉 , (2.12) where the brackets 〈·〉 now denote average with respect to ~Ws and the action is given by A~u(~γ~u, t0, t) = |~u(s)|2 − F (~γ~u(s), s) ds. (2.13) It is obvious that this variational principle gives (2.5) in the inviscid limit ν → 0. Note that this approach has the advantage to be applicable not only to Burgers dynamics but to any convex Lagrangian (see [50,58]). 2.3 Singularities of Burgers turbulence The singularities appearing in the course of time play an essential role in understanding various aspects of the statistical properties in the inviscid limit. The shocks – discontinuities of the velocity field – and other singular- ities, such as preshocks, generally not associated to dis- continuities, are often responsible for non-trivial univer- sal behaviors. In order to understand the contribution of each kind of singularities, it is first important to know in a detailed manner their genericity and their type. As we have seen in the previous section, the potential solutions to the multidimensional Burgers equation can be expressed in the inviscid limit in terms of the vari- ational principle (2.5) (that reduces to (2.9) in the un- forced case). There typically exist Eulerian locations ~x where the minimum is either degenerate or attained for several trajectories. A co-dimension can be associated to such points by counting the number of relations that are necessary to determine them. The singular locations of co-dimension c form manifolds of the Eulerian space- time with dimension (d− c). The singularities with the lower co-dimension are the shocks corresponding to the Eulerian positions where two different trajectories min- imize (2.5); they form Eulerian manifolds of dimension (d− 1): in one dimension the shocks are isolated points, in two dimensions they are lines, in three dimensions surfaces, etc. There also exist Eulerian manifolds with three different minimizing trajectories. In one dimen- sion, they are isolated space-time events corresponding to the merger of two shocks. In two dimensions, they are triple points where three shock lines meet. In three di- mensions they are filaments corresponding to the inter- section of three shock surfaces. There also exist Eulerian locations where the minimum in (2.5) is reached for four different trajectories, etc. points termination shock lines points triple Fig. 2. Typical aspect of the singularities present at a fixed time in the solution for (a) d = 2 and (b) d = 3. The generic form of such singularities and their typi- cal metamorphoses occurring in the course of time were studied in details and classified for d = 2 and d = 3 by Arnold, Baryshnikov and Bogaevsky in the Appendix of [62] and in a more detailed paper by Bogaevsky [17]. This classification is based on two criteria: (i) the num- ber of trajectories minimizing (2.5) and (ii) the multi- plicity of each of these minima. The shocks correspond- ing to locations with two distinct minimizers are hence denoted by A21. At a fixed time, the A 1 singularities are discrete points in one dimension. In two dimensions (see figure 2(a)) they form curve segments with extremities that can be either triple points A31 or isolated termina- tion points of the type A3 corresponding to a degenerate minimum. In three dimensions (see figure 2(b)) the sin- gular manifold is formed by shock surfaces of A21 points. The boundaries of these surfaces are either made of de- generate A3 points or of triple lines made of A 1 points. The triple lines intersect at isolated A41 points or inter- sect shock boundaries at particular singularities called A1A3 where the minimum is attained in two points, one of which is degenerate. It is important to remark here that degenerate singu- larities (of the type A3 or of higher orders A5, A7, etc.) introduce in the solution points where the velocity gra- dients becomes arbitrarily large. This is not the case of the An1 singularities which correspond to discontinuities of the velocity but are associated to bounded values of its gradients. As we will see in sections 4 and 7, these degenerate singularities are responsible for an algebraic behavior of the probability density function of velocity gradients, velocity increments and of the mass density. Fig. 3. Illustration of the similarities between the singular manifold in space time for d = 1 and at fixed time for d = 2 (b). The two manifolds contain the same type of singularities with the same co-dimensions. The restrictions on the possible metamorphoses in dimension d = 1 are the following: a point of the type A3 can only exist at the bottom extremity of a shock trajectory; the A31 points necessarily correspond to the merger of two shocks; shock trajectories cannot have a horizontal tangent. The singularities with co-dimensions (d+ 1) generically appear in the solution at isolated times. They corre- spond to instantaneous changes in the topological struc- ture of the singular manifold, called metamorphoses and can be also classified (see [17]). In one dimension, there are two generic metamorphoses: shock formations (the preshocks) corresponding to a specific space-time loca- tion where the minimum is degenerate (A3 singularities) and shock mergers associated to space-time positions where the minimum is attained for three different tra- jectories (A31 singularities). We see that some of the sin- gularities generically present in two dimensions appear at isolated times in three dimensions. Actually, all the singularities generically present in dimension (d+1) ap- pear in dimension d on a discrete set of space time, that is at isolated positions and instants of time. However, it has been shown in [17] that the irreversible dynamics of the Burgers equation restricts the set of possible meta- morphoses. The admissible metamorphoses are charac- terized by the following property: after the bifurcation, the singular manifold must remain locally contractible (homotopic to a point in the neighborhood of the Eu- lerian location of the metamorphosis). This topological restriction is illustrated for the one-dimensional case in figure 3. Note that this constraint actually holds for all solutions to the Hamilton–Jacobi equation in the limit of vanishing viscosity, as long as the Hamiltonian is a convex function. In order to determine precisely how all these singularities contribute to the statistical properties of the solution, it is important to know the local structure of the velocity (or potential) field in their vicinity. Variousnormal forms can be obtained from the multiplicity of the minimum in the variational formulation of the solution (2.5). In the case without forcing, they can be obtained from a Taylor expansion of the initial velocity potential. This will be used in next section to determine the tail of the probability distribution of a mass density field advected by a velocity solution to the Burgers equation. 2.4 Remarks on numerical methods All the traditional methods used to solve equations of fluid dynamics, or more generally any partial differen- tial equations, can be used to obtain the solutions to the Burgers equation. However, as we have seen above, the solution typically has singularities (discontinuities of the velocity) in the limit of vanishing viscosity. Hence meth- ods which rely on the smoothness of the solution require a non-vanishing viscosity, which is introduced either in an explicit way to ensure stability (as, e.g., for pseudo- spectral methods) or in an implicit way through the discretization procedure (as for finite-differences meth- ods). In both cases the value of the viscosity is deter- mined from the mesh size and, even in one dimension, their uses might be very disadvantageous. We will now demonstrate various numerical methods that allow ap- proximating the solutions to the Burgers equation di- rectly in the limit of vanishing viscosity ν → 0. 2.4.1 Finite volumes The one-dimensional Burgers equation with no forcing is a scalar conservation law. Its entropic solutions (or vis- cosity solutions) can thus be approximated numerically by finite-volume methods. Instead of constructing a dis- crete approximation of the solution on a grid, such meth- ods consist in considering an approximation of its mean value on a discrete partitioning of the system into finite volumes. One then needs to evaluate for each of these volumes the fluxes exchanged with each of its neighbors. Various approximations of these fluxes were introduced by Godunov, Roe, and Lax and Wendroff (see, e.g., [35], Vol. 3, for a review). These methods require to dicretize both space and time. The time step being then related to the spatial mesh size by a Courant–Friedrichs–Lewy type condition. Thus to integrate the equation during times comparable to one eddy turnover time, they re- quire a computational time O(N2) where N is the res- olution. As we now show there actually exist numerical schemes that allow constructing the solution to the de- caying Burgers equation for arbitrary times without any need to compute the solution at intermediate times. 2.4.2 Fast Legendre transform As we have seen in section 2.1, the solution to the un- forced Burgers equation is given explicitly by the varia- tional principle (2.9). A method based on the idea of us- ing this formulation together with a monotonicity prop- erty of the Lagrangian map ~x0 → ~x = ~X(~x0, t) was given in [94]. It is called the fast Legendre transform whose principles were already sketched in [23]. Both Eu- lerian and Lagrangian positions are discretized on reg- ular grids. Then, for a fixed Eulerian location ~x(i) on the grid, one has to find the corresponding Lagrangian coordinate ~x 0 maximizing (2.9). A naive implementa- tion would require O(NdEN L) operations if the Eulerian and the Lagrangian grids contain NdE and N L points re- spectively. Actually the number of operations can be re- duced to O((NE lnNL) d) by using the monotonicity of the Lagrangian map, that is the fact that for any pair of Lagrangian positions ~x 0 and ~x 0 , one has at any time [ ~X(~x 0 , t) − ~X(~x 0 , t)] · (~x 0 − ~x 0 ) ≥ 0. In the case of orthogonal grids, this property allows perform- ing the maximization by exploring along a binary tree the various possibilities; thus the number of operations is reduced to lnNL for each of the NE positions on the Eulerian grid. Such algorithms give access to the solu- tion not only directly in the limit of vanishing viscosity but also by jumping directly from the initial time to an arbitrary time. This method can also be used for the forced Burgers equation, approximating the forcing by a sum of im- pulses at discrete times and letting the solution decay between two such kicks. This gives an efficient algorithm for the forced Burgers equation directly applicable in the limit of vanishing viscosity. 2.4.3 Particle tracking methods In one dimension, Lagrangian methods can be imple- mented in a straightforward manner after noticing that particles cannot cross each other and that it is advis- able to track not only fluid particles but also shocks (see, e.g., [6]). Lagrangian methods can in principle be used to solve the Burgers equation in any dimension. How- ever the shock dynamics is meaningful only for poten- tial solutions. Outside the potential framework almost nothing is known about the construction of the solution beyond the first crossing of trajectories. In the potential case, a particle method can be formulated by choosing to represent the solution in the position-potential (~x,Ψ) space instead of the position-velocity (~x,~v) space. An idea in two dimensions, which was not yet implemented, consists in considering a meshing of the hyper-surface defined by the velocity potential. If such a mesh contains only triple points, such singularities are preserved by the dynamics and can be tracked using the results discussed below in section 4.2 and by checking at all time steps in an exhaustive manner at all the metamorphoses encoun- tered by triple points. 3 Decaying Burgers turbulence We focus in this section on the solutions to the d- dimensional unforced potential Burgers equation ∂t~v + ~v ·∇~v = ν∇2~v, ~v(~x, 0)=~v0(~x)=−∇Ψ0(~x). (3.1) As showed in section 2.1, the solution can be expressed in the limit of vanishing viscosity ν → 0 in terms of a variational principle that relates the velocity potential at time t to its initial value: Ψ(~x, t) = max Ψ0(~x0) − |~x− ~x0|2 . (3.2) The next subsection describes several geometrical con- structions of the solution that are helpful to determine various statistical properties of the decaying prob- lem (3.1). This is illustrated in subsections 3.2 and 3.3 which are devoted to the study of the decay of smooth homogeneous and of Brownian initial data, respectively. The study of the solutions to the Burgers equation trans- porting a density field is of particular interest in the ap- plication of the Burgers equation in cosmology within the framework of the adhesion model. This question will be discussed in section 4. 3.1 Geometrical constructions of the solution 3.1.1 The potential Lagrangian manifold The variational formulation of the solution (3.2) has a simple geometrical interpretation in the position- potential space (~x,Ψ) of dimension d + 1. Indeed, con- sider the d-dimensional manifold parameterized by the Lagrangian coordinate ~x0 and defined by ~x = ~x0 − t∇Ψ0(~x0) Ψ = Ψ0(~x0) − |∇Ψ0(~x0)|2. (3.3) The first line corresponds to the position where the gra- dient of the argument of the maximum function in (3.2) vanishes while the second line is just its argument eval- uated at the maximum. For a sufficiently regular initial potential Ψ0 (at least twice differentiable) and for suf- ficiently small times, equation (3.3) unambiguously de- fines a single-valued function Ψ(~x, t). However, there ex- ists generically a time t⋆ at which the manifold is folding. Figure 4(a) (upper) shows in one space dimension the typical shape of the Lagrangianmanifold defined by (3.3) after the critical time t⋆. For some Eulerian positions ~x, there is more than one branch and cusps are present at Eulerian locations where the number of branches change. The situation is very similar in higher dimensions as il- lustrated for d = 2 in figure 4(b). Clearly from the varia- tional principle (2.9), the correct solution to the inviscid Burgers equation is obtained by taking the maximum, that is the highest branch. The velocity potential is by construction always continuous but it contains angular points corresponding to discontinuities of the velocity ~v = −∇Ψ. Such singularities are located at Eulerian lo- cations where the maximum in (2.9) is degenerate and attained for different ~x0. As already discussed in sec- tion 2.3 the different singularities appearing in the solu- tions can be classified in any dimension. Below we describe other geometrical constructions of the solutions to the decaying Burgers equation in the limit of vanishing viscosity that are based on the variational principle (2.9). 3.1.2 The velocity Lagrangian manifold In one dimension, when the velocity field is always po- tential, the method based on the study of the poten- tial manifold in the (x,Ψ) space described above can be straightforwardly extended to the position-velocity phase space. Consider the Lagrangian manifold defined x = x0 − t v0(x0) v = v0(x0). (3.4) Maxwell rule Fig. 4. (a) Lagrangian manifold for d = 1 in the (x,Ψ) plane (upper) and in the (x, v) plane (lower); the heavy lines correspond to the correct Eulerian solutions. (b) Lagrangian manifold in the (~x,Ψ) space for d = 2. The regular parts of the graph of the solution are nec- essarily contained in this manifold. However, for times larger than t⋆, folding appears and the naive solution would be multi-valued. To construct the true solution one should find a way to choose among the different branches. In one dimension, there is a simple relation between the potential Lagrangian manifold in the (x,Ψ) plane and those of the (x, v) plane defined by (3.4): the potential manifold is obtained by taking the “multi- valued integral” that can be defined by transforming the spatial integral into an integral with respect to the arc length. The maximum representation (2.9) implies that the velocity potential is continuous. Hence a shock cor- responds to an Eulerian position x where two points be- longing to different branches define equal areas in the (x, v) plane. In the case of a single loop of the manifold, this is equivalent to applying the Maxwell rule to deter- mine the shock position (see figure 4(a) - lower). This construction of the solution can become rather involved as soon as the number of shocks becomes large or that several mergers have taken place. For the moment there xshock interval regular points Φ, Φc Fig. 5. Convex hull construction in terms of the Lagrangian potential (a) for d = 1 and (b) for d = 2. is no generalization to dimension higher than one of this Maxwell rule construction of the solution. For such an extension, one needs to develop a geometrical framework to describe the Lagrangian manifold in the (~x,~v) space. Such approaches would certainly shed some light on the problem of constructing non-potential solutions to the Burgers equation in the limit of vanishing viscosity. 3.1.3 The convex hull of the Lagrangian potential Another geometrical construction of the solution, which is valid in any dimension makes use of the Lagrangian potential Φ(~x0, t) = tΨ0(~x0) − |~x0|2 . (3.5) Clearly, the negative gradient of the Lagrangian poten- tial gives the naive Lagrangian map ~X(~x0, t) = −∇~x0Φ(~x0, t) = ~x0 + t~v0(~x0), (3.6) that is satisfied by Lagrangian trajectories as long as they do not enter shocks. The maximum formulation of the solution (2.9) can be rewritten as tΨ(~x, t) + |~x|2 = max (Φ(~x0) + ~x0 · ~x), (3.7) which represents the potential as, basically, a Legendre transform of the Lagrangian potential. An important property of the Legendre transform is that the right- hand side. of (3.7) is unchanged if the Lagrangian po- tential Φ is replaced by its convex hull, that is the in- tersection of all the half planes containing its graph. In other terms, the convex hull Φc of the Lagrangian po- tential Φ is defined as Φc(~x0, t) = inf g(~x0), where the infimum is taken over all convex functions g satisfying g(·) ≥ Φ(·, t). This is illustrated in one dimension in fig- ure 5(a) which shows both regular points (Lagrangian points which have not fallen into a shock) and one shock interval, situated below the segment which is a part of the convex hull. In two dimensions, as illustrated in fig- ure 5(b), the convex hull is typically formed by regular points, by ruled surfaces, and by triangles which corre- spond, to the regular part of the velocity field, the shock lines, and the shock nodes, respectively. Note that in one dimension, there exists an equivalent construction which is directly based on the Lagrangian map x0 7→ X(x0, t) defined by (3.6). Working with the convex hull is equivalent to the Maxwell rule applied to the non-invertible regions of the Lagrangian map. A shock corresponds to a whole Lagrangian interval having a single point as an Eulerian image. One then talks about a Lagrangian shock interval. 3.1.4 The paraboloid construction Finally, the maximum representation (3.7) leads in a straightforwardway to another geometrical construction of the solution. As illustrated in figure 6 in both one and two dimensions, a paraboloid with apex at ~x and ra- dius of curvature proportional to t is moved down in the (~x0,Ψ0) space until it touches the surface defined by the initial velocity potential Ψ0 at the Lagrangian location associated to ~x. The location ~x0 where the paraboloid touches the graph of the potential is exactly the pre- image of ~x. If it touches simultaneously at several loca- tions, a shock is located at the Eulerian position ~x. One constructs in this way the inverse Lagrangian map. 3.2 Kida’s law for energy decay An important issue in turbulence is that of the law of decay at long times when the viscosity is very small. Before turning to the Burgers equation it is useful to re- call some of the features of decay for the incompressible Navier–Stokes case. It is generally believed that high- Reynolds number turbulence has universal and non- trivial small-scale properties. In contrast, large scales, important for practical applications such as transport of heat or pollutants, are believed to be non-universal. This is however so only for the toy model of turbulence main- tained by prescribed large-scale random forces. Very high-Reynolds number turbulence, decaying away from its production source, and far from boundaries can relax Fig. 6. Paraboloid construction of solution for (a) d = 1 and (b) d = 2. under its internal nonlinear dynamics to a (self-similarly evolving) state with universal and non-trivial statistical properties at all scales. Von Kármán and Howarth [109], investigating the decay for the case of high-Reynolds number homogeneous isotropic three-dimensional tur- bulence, proposed a self-preservation (self-similarity) ansatz for the spatial correlation function of the ve- locity: the functional shape of the correlation function remains fixed, while the integral scale L(t) grows in time and the mean kinetic energy E(t) = V 2(t) decays, both following power laws; there are two exponents which can be related by the condition that the energy dissipation per unit mass |Ė(t)| should be proportional to V 3/L. But an additional relation is needed to actu- ally determine the exponents. The invariance in time of the energy spectrum at low wavenumbers, known as the “permanence of large eddies” [53,84,63] can be used to derive the law of self-similar decay when the initial spec- trum E0(k) ∝ kn at small wavenumbers k with n below a critical value equal to 3 or 4, the actual value being slope 1/t Random position Fig. 7. Snapshot of solution of decaying Burgers turbulence at long times when spatial periodicity is assumed. disputed because of the “Gurbatov phenomenon” (see the end of this section). One then obtains a law of decay E(t) ∝ t−2(n+1)/(3+n). (Kolmogorov [79] proposed a law of energy decay V 2(t) ∝ t−10/7, which corresponds to n = 4 and used in its derivation the so-called “Loitsyan- sky invariant”, a quantity actually not conserved, as shown by Proudman and Reid [100].) When the initial energy spectrum at low wavenumbers goes to zero too quickly, the permanence of large eddies cannot be used, because the energy gets backscattered to low wavenum- bers by nonlinear interactions. For Navier–Stokes tur- bulence the true law of decay is then known only within the framework of closure theories (see, e.g., [84]). For one-dimensional Burgers turbulence, many of the above issues are completely settled. First, we observe that the problem of decay is quite simple if spatial peri- odicity is assumed. Indeed, all the shocks appearing in the solution will eventually merge into a single shock per period, as shown in figure 7. The position of this shock is random and the two ramps have slope 1/t, as is easily shown using the parabola construction of subsection 3.1. Hence, the law of decay is simply E(t) ∝ t−2. Nontrivial laws of decay are obtained if the Burgers turbulence is homogeneous in an unbounded domain and has the “mixing” property (which means, roughly, that correla- tions are vanishing when the separation goes to infinity). The number of shocks is then typically infinite but their density per unit length decreases in time because shocks are constantly merging. The E(t) ∝ t−2(n+1)/(3+n) law mentioned above can be derived for Burgers turbulence from the permanence of large eddies when n ≤ 1 [63]. For n = 0, this t−2/3 law was actually derived by Burg- ers himself [27]. The hardest problem is again when permanence of large eddies does not determine the outcome, namely for n > 1. This problem was solved by Kida [77] (see also [51,61,63]). We now give some key ideas regarding the derivation of Kida’s law of energy decay. We assume Gaussian, homo- geneous smooth initial conditions, such that the poten- tial is homogeneous. Note that a homogeneous function is not, in general, the derivative of another homogeneous function. Here this is guaranteed by assuming that the Fig. 8. An initial potential which is everywhere below the parabola x20/(2t) + Ψ. The probability of such events gives the cumulative probability to have a potential at time t less than Ψ. initial spectrum of the kinetic energy is of the form E0(k) ∝ kn for k → 0 with n > 1 . (3.8) This condition implies that the mean square initial po- tential k−2E0(k) dk has no infrared (small-k) diver- gence (the absence of an ultraviolet divergence is guar- anteed by the assumed smoothness). A very useful property of decaying Burgers turbulence, which has no counterpart for Navier–Stokes turbulence, is the relation E(t) = 〈Ψ〉 , (3.9) which follows by taking the mean of the Hamilton– Jacobi equation for the potential (2.2) in the absence of viscosity and of a driving force. Hence, the law of energy decay can be obtained from the law for the mean po- tential. The latter can be derived from the cumulative probability of the potential which, by homogeneity, does not depend on the position. By (2.9), its expression at x = 0 is Prob{Pot.<Ψ}=Prob ∀x0, Ψ0(x0)< . (3.10) Expressed in words, we want to find the probability that the initial potential does not cross the parabola x20/(2t) + Ψ (see figure 8). Since, at large times t, the relevant Ψ is going to be large, the problem becomes that of not crossing a parabola with small curvature and very high apex. The crossings, more precisely the up- crossings, are spatially quite rare and, for large t, form a Poisson process [92] for which Prob. no crossing ≃ e−〈N(t)〉, (3.11) where 〈N(t)〉 is the mean number of up-crossings. By the Rice formula (a consequence of the identity δ(λx) = (1/|λ|)δ(x)), 〈N(t)〉= 〈∫ +∞ dx0 δ(m(x0)−Ψ) , (3.12) where H is the Heaviside function and m(x0) ≡ Ψ0(x0) − . (3.13) Since Ψ0(x0) is Gaussian, the right-hand side of (3.12) can be easily expressed in terms of integrals over the probability densities of Ψ0(x0) and of dΨ0(x0)/dx0 (as a consequence of homogeneity these variables are uncor- related and, hence, independent). The resulting integral can then be expanded by Laplace’s method for large t, yielding 〈N(t)〉 ∼ t1/2Ψ−1/2e−Ψ , t→ ∞. (3.14) When this expression is used in (3.11) and the result is differentiated with respect to Ψ to obtain the probability density function (PDF) of p(Ψ), the latter is found to be concentrated around Ψ⋆ = (ln t) 1/2. It then follows that, at large times, we obtain Kida’s log-corrected 1/t law for the energy decay 〈Ψ〉 ∼ (ln t)1/2, E(t) ∼ t(ln t)1/2 , L(t) ∼ . (3.15) The Eulerian solution, at large times, has the ramp Fig. 9. The Eulerian solution at large times t. The ramps have slope 1/t. In time-independent scales, the figure would be stretched horizontally and squeezed vertically by a factor proportional to t. structure shown in figure 9 with shocks of typical strength V (t) = E1/2(t), separated by a distance L(t). The fact that Kida’s law is valid for any n > 1, and not just for n ≥ 2 as thought originally, gives rise to an inter- esting phenomenon now known as the “Gurbatov effect”: if 1 < n < 2 the large-time evolution of the energy spec- trum cannot be globally self-similar. Indeed, the perma- nence of large eddies, which is valid for any n < 2 dic- tates that the spectrum should preserve exactly its initial n behavior at small wavenumbers k, with a constant- in-time Cn. Global self-similarity would then imply a t−2(n+1)/(3+n) law for the energy decay, which would contradict Kida’s law. Actually, as shown in [63], there are two characteristic wavenumbers with different time dependences, the integral wavenumber kL(t) ∼ (L(t))−1 and a switching wavenumber ks(t) below which holds the permanence of large eddies. It was shown that the same phenomenon is present also in the decay of a pas- sive scalar [45]. Whether or not a similar phenomenon is present in three-dimensional Navier–Stokes incompress- ible turbulence, or even in closure models, is a contro- versial matter [44,97]. For decaying Burgers turbulence, if we leave aside the Gurbatov phenomenon which does not affect energy- carrying scales, the following may be shown. If we rescale distances by a factor L(t) and velocity amplitudes by a factor V (t) = E1/2(t) and then let t → ∞, the spatial (single-time) statistical properties of the whole random velocity field become time-independent. In other words, there is a self-similar evolution at large times. Hence, dimensionless ratios such as the velocity flatness F (t) ≡ v4(t) [〈v2(t)〉]2 (3.16) have a finite limit as t → ∞. A similar property holds for the decay of passive scalars [28]. We do not know if this property holds also for Navier–Stokes incompress- ible turbulence or if, say, the velocity flatness grows with- out bounds at large times. 3.3 Brownian initial velocities Initial conditions in the Burgers equation that are Gaus- sian with a power-law spectrum ∝ k−α have been fre- quently studied because they belong in cosmology to the class of scale-free initial conditions (see [98,34]). We con- sider here the one-dimensional case with Brownian mo- tion as initial velocity, corresponding to α = 2. Brownian motion is continuous but not differentiable; thus, shocks appear after arbitrarily short times and are actually dense (see figure 10(a)). Numerically supported conjectures made in [104] have led to a proof by Sinai [105] of the following result: in Lagrangian coordinates, the regular points, that is fluid particles which have not yet fallen into shocks, form a fractal set of Hausdorff dimension 1/2. This implies that the Lagrangian map forms a Devil’s staircase of dimension 1/2 (see figure 11). Note that when the initial velocity is Brownian, the La- grangian potential has a second space derivative delta- correlated in space; this can be approximately pictured as a situation where the Lagrangian potential has very wild oscillations in curvature. Hence, it is not surprising that very few points of its graph can belong to its convex hull (see figure 10(b)). We will now highlight some aspects of Sinai’s proof of this result. The idea is to use the construction of the solution in terms of the convex hull of the Lagrangian potential (see section 3.1), so that regular points are ex- actly points where the graph of the Lagrangian poten- tial coincides with its convex hull. For this problem, the Hausdorff dimension of the regular points is also equal to its box-counting dimension, which is easier to deter- mine. One obtains it by finding the probability that a −5 −2.5 0 2.5 5 t = 0 t = 1 Fig. 10. Snapshot of the solution resulting from Brownian initial data in one dimension. (a) Velocity profile at initial time t = 0 and at time t = 1; notice the dense proliferation of shocks. (b) Lagrangian potential together with its convex hull. Fig. 11. The Lagrangian map looks like a Devil’s staircase: it is constant almost everywhere, except on a fractal Can- tor-like set (from [107]). small Lagrangian interval of length ℓ contains at least one regular point which belongs simultaneously to the graph of the Lagrangian potential Φ and to its convex hull. In other words, one looks for points, such asR, with the property that the graph of Φ lies below its tangent at R (see figure 12). Following Sinai, this can be equiva- lently formulated by the box construction with the fol- lowing constraints on the graph: Left: graph of the potential below the half line Γ−, Right: graph of the potential below the half line Γ+, 1 : enter (AF ) with a slope larger than that of Γ− by O(ℓ 2 : exit (CB) with a slope less than that of Γ+ by O(ℓ 3 : cross (FC) and stay below (ED). It is obvious that such conditions ensure the existence of at least one regular point, as seen by moving (ED) down parallel to itself until it touches the graph. Note that A and the slope of (AB) are prescribed. Hence, one is cal- culating conditional probabilities; but it may be shown that the conditioning is not affecting the scaling depen- dence on ℓ. Φ( ) Fig. 12. The box construction used to find a regular point R within a Lagrangian interval of length ℓ (from [105,107]). As the Brownian motion v0(x0) is a Markov process, the constraints Left, Box and Right are independent and hence, P reg. (ℓ) ≡ Prob{regular point in interval of length ℓ} = Prob{Left}×Prob{Box}×Prob{Right} . (3.17) The sizes of the box were chosen so that Prob{Box} is independent of ℓ: Prob {Box} ∼ ℓ0. (3.18) Indeed, Brownian motion and its integral have scaling exponent 1/2 and 3/2, respectively, and the problem with ℓ≪ 1 can be rescaled into that with ℓ = 1 without changing probabilities. It is clear by symmetry that Prob{Left} and Prob{Right} have the same scaling in ℓ. Let us concentrate on Prob{Right}. We can write the equation for the half line Γ+ in the form Γ+: x0 7→Φ(x′′0 )+δℓ3/2 (x′′0 )+γℓ (x0−x′′0), (3.19) where γ and δ are positive O(1) quantities. Hence, intro- ducing α ≡ x0 − x′′0 , the condition Right can be written to the leading order as v0(x0) + γℓ dx0 + δℓ 3/2 + > 0, (3.20) for all α > 0. By the change of variable α = βℓ and use of the fact that the Brownian motion has scaling exponent 1/2, one can write the condition Right as (v0(x0) + γ) dx0 > −δ, for all α ∈ [0, ℓ−1]. (3.21) Without affecting the leading order, one can replace the Brownian motion by a stepwise constant random walk with jumps of ±1 at integer x0’s. The integral in (3.21) has a simple geometric interpretation, as highlighted in figure 13, which shows a random walk starting from the ordinate γ and the arches determined by succes- sive zero-passings. The areas of these arches are denoted S⋆, S1, ...Sn, S⋆⋆. Fig. 13. The arches construction which uses the zero-passings of a random walk to estimate the integral of Brownian motion (from [105,107]). It is easily seen that Prob{Right} ∼ Prob{S1 > 0, S1 + S2 > 0, . . . S1 + · · · + Sn > 0 }, (3.22) where n = O(ℓ−1/2) is the number of zero-passings of the random walk in the interval [0, ℓ−1]. The probability (3.22) can be evaluated by random walk methods (see, e.g.,[49], Chap. 12, section 7), yielding Prob{Right} ∼Prob{n first sums>0} ∝ n−1/2 ∝ ℓ1/4. (3.23) By (3.17), (3.18) and (3.23), the probability to have a regular point in a small interval of length ℓ behaves as ℓ1/2 when ℓ → 0. Thus, the regular points have a box- counting dimension 1/2. This rigorous result on the fractal dimension of regular points served as a basis in [4] for a proof of the bifrac- tality of the inverse Lagrangian map when the initial velocity is Brownian. Namely, the moments Mq(ℓ) = 〈(x0(x+ ℓ) − x0(x))〉 behave as ℓτq at small separation ℓ and the exponents τq experience the phase transition τq = 2q for q ≤ 1/2 (3.24) τq = 1 for q ≥ 1/2 (3.25) At the moment, this is the only rigorous result on the bifractal nature of the solutions to the Burgers equation in the case of non-differentiable initial velocity. In par- ticular, the case of fractional Brownian motion is still opened. 4 Transport of mass in the Burgers/adhesion model In the cosmological application of the Burgers equation, i.e. for the adhesion model, it is of particular interest to analyze the behavior of the density of matter, since the large-scale structures are characterized as regions where mass is concentrated. This is done by associat- ing to the velocity field ~v solution to the d-dimensional decaying Burgers equation (3.1), a continuity equation for the transport of a mass density field ρ. In Eulerian coordinates, the mass density ρ satisfies ∂tρ+ ∇ · (ρ~v) = 0 , ρ(~x, 0) = ρ0(~x) . (4.1) A straightforward consequence of (4.1) and of the for- mulation of Burgers dynamics in terms of characteristics ~X(~x0, t) is that, at the Eulerian locations where the La- grangian map is invertible, the mass density field ρ can be expressed as ρ(~x, t)= ρ0(~x0) J(~x0, t) , where ~X(~x0, t)=~x, and J(~x0, t)=det (∂X i)/(∂x . (4.2) Large but finite values of the density will be reached at locations where the Jacobian J of the Lagrangian map becomes very small. As we will see in section 4.1, they contribute a power-law behavior in the tail of the probability density function of ρ. The expression (4.2) is no more valid when the Jacobian vanishes (inside shocks). Then the density field becomes infinite and mass accumulates on the shock. We will see in section 4.2 that the evolution of the mass inside the singularities of the solution can be obtained as the ν → 0 limit of the well-posed viscous problem. Finally, we will discuss in section 4.3 some of the applications of the Burgers equation to cosmology, and in particular how, assuming the dynamics of the adhesion model, the question of reconstruction of the early Universe from its present state can be interpreted as a convex optimal mass transportation problem. 4.1 Mass density and singularities We give here the proof reported in [54] that in any dimen- sion large densities are localized near “kurtoparabolic” singularities residing on space-time manifolds of co- dimension two. In any dimension, such singularities con- tribute universal power-law tails with exponent −7/2 to the mass density probability density function (PDF) p(ρ), provided that the initial conditions are smooth. In one dimension, the mass density at regular points can be written as ρ(X(x0, t), t) = ρ0(x0) 1 − t[(d2Ψ0)/(dx20)] . (4.3) We suppose here that the initial density ρ0 is strictly positive and that both ρ0 and Ψ0 are sufficiently regu- lar statistically homogeneous random fields. Large val- ues of ρ(x, t) are obtained in the neighborhood of La- grangian positions with a vanishing Jacobian, i.e. where d2Ψ0(x0)/dx 0 = 1/t. Once mature shocks have formed, the Lagrangian points with vanishing Jacobian are in- side shock intervals and thus not regular. The only points with a vanishing Jacobian that are at the boundary of the regular points are obtained at the preshocks, that is when a new shock is just born at some time t⋆. Such points, that we denote by x⋆0, are local minima of the ini- tial velocity gradient which have to be negative, so that the following relations are satisfied: (x⋆0) = (x⋆0) = 0, (x⋆0) < 0 . (4.4) There is of course an extra global regularity condition that the preshock Lagrangian location x⋆0 has not been captured by a mature shock at a time previous to t⋆. This global condition affects only constants but not the scaling behavior of p(ρ) at large ρ. We now Taylor-expand the initial density and the initial velocity potential in the vicinity of x⋆0. By adding a suit- able constant to the initial potential, shifting x⋆0 to the origin and making a Galilean transformation canceling the initial velocity at x⋆0, we obtain the following “nor- mal forms” for the Lagrangian potential (3.5) and for the density Φ(x0, t)≃ τx20+ζx 0, ρ(X(x0, t), t)≃ τ+12ζx20 , (4.5) where t− t⋆ and ζ = < 0 . (4.6) The Lagrangian potential bifurcates from a situation where it has a single maximum at τ < 0 through a de- generate maximum with quartic behavior at τ = 0, to a situation where convexity is lost and where it has two maxima at x±0 = ± −τ/(4ζ) for τ > 0. As a result of our choice of coordinates, the symmetry implies that the convex hull contains a horizontal segment joining these two maxima (see. figure 14(a)). , t) τ < 0 τ = 0 τ > 0 Fig. 14. Normal form of the Lagrangian potential. (a) in one dimension, in the time-neighborhood of a preshock; at the time of the preshock (τ = 0), the Lagrangian potential changes from a single extremum to three extrema and devel- ops a non-trivial convex hull (shown as a dashed line). (b) in two dimension, the space neighborhood of a shock ending point has a structure similar to the spatio-temporal normal form of a preshock in one dimension when replacing the x0,2 variable by the time τ ; the continuous line is the separatrix between the regular part and the ruled surface of the convex hull; the dotted line corresponds to the locations where the Jacobian of the Lagrangian map vanishes. We see from (4.5) that the Eulerian density ρ is propor- tional to x20 in Lagrangian coordinates at t = t⋆. Since X = −∂x0Φ, the relation between Lagrangian and Eu- lerian coordinates is cubic, so that at τ = 0, the den- sity has a singularity ∝ x−2/3 in Eulerian coordinates. At any time t 6= t⋆, the density remains bounded ex- cept at the shock position. Before the preshock (τ < 0), it is clear that ρ < −ρ0/τ , while after (τ > 0), exclu- sion of the Lagrangian shock interval [x−0 , x 0 ] implies that ρ < ρ0/(2τ). Clearly, large densities are obtained only in the immediate space-time neighborhood of the preshock. More precisely, it follows from (4.5) that hav- ing ρ(x, t) > µ requires simultaneously |τ | < ρ0 and |x| < (−12ζ)−1/2 . (4.7) The tail of the cumulative probability of the density can be determined from the fraction of Eulerian space-time where ρ exceeds a given value. This leads to P>(µ; x, t) = Prob{ρ(x, t)>µ} ≃ C(x, t)µ−5/2, (4.8) where the constant C can be expressed as C(x, t) = At |ζ|−1/2p3(x, t, ζ) dζ, (4.9) A is a positive numerical constant and p3 designates the joint probability distribution of the preshock space-time position and of its “strength” coefficient ζ (see [54] for details). This algebraic law for the cumulative probabil- ity implies that the PDF of the mass density has a power- law tail with exponent −7/2 at large values. Actually this law was first proposed in [37] for the large-negative tail of velocity gradients in one-dimensional forced Burg- ers turbulence, a subject to which we shall come back in section 7. In higher dimensions it was shown in [54] that the main contribution to the probability distribution tail of the mass density does not stem from preshocks but from “kurtoparabolic” points. Such singularities (called A3 according to the classification of [62], which is summa- rized in section 2.3) correspond to locations which be- long to the regular part of the convex hull of the La- grangian potential Φ(~x0, t) and where its Hessian van- ishes. The name kurtoparabolic comes from the Greek “kurtos” meaning “convex”. These points are located on the spatial boundaries of shocks and generically form space-time manifolds of co-dimension 2 (persisting iso- lated points for d = 2, lines for d = 3, etc.). As in one dimension, the normal form of such singularities is obtained by Taylor-expanding in a suitable coordinate frame the Lagrangian potential to the relevant order Φ(~x0, t) ≃ ζx40,1+ 2≤j≤d x20,j +βjx 0,1x0,j , (4.10) where the different coefficients satisfy inequalities that ensure that the surface is below its tangent plane at ~x0 = 0. The typical shape of the Lagrangian potential in two dimensions is shown in figure 14(b). The positions where the Jacobian of the Lagrangian map vanishes can be easily determined from this normal form. The convex hull of Φ and the area where the mass density exceeds the value µ can also be constructed explicitly. An important observation is that, in any dimension, the scalar product of the vector ~y0 = (x0,2, . . . , x0,d) with the vector ~β = (β2, . . . , βd) plays locally the same role as time does in the analysis of one-dimensional preshocks. When µ → ∞, the cumulative probability can be esti- mated as P>(µ; x, t) ∝ µ−3/2 ︸ ︷︷ ︸ from x0,1 × µ−1 from ~β·~y0 × 1 × · · · × 1 ︸ ︷︷ ︸ from other components of ~y0 from time . (4.11) The only non-trivial contributions come from x0,1 and from the component of ~y0 along the direction of ~β, all the other components and time contributing order-unity factors. Hence, the cumulative probability P>(µ) is pro- portional to µ−5/2 in any dimension, so that the PDF of mass density has a power-law behavior with the univer- sal exponent −7/2. As we have seen, the theory is not very different in one and higher dimension even if kurtoparabolic points are persistent only in the latter case. This is due to the pres- ence of a time-like direction in the case d ≥ 2. 4.2 Evolution of matter inside shocks As we have seen in the previous subsection, the mass density becomes very large in the neighborhood of kur- toparabolic points (A3 singularities) corresponding to the space-time boundaries of shocks. Such singularities dominate the tail of the mass density probability dis- tribution and contribute a power-law behavior with ex- ponent −7/2. However the mass distribution depends strongly on what happens inside the shocks where the density is infinite. Indeed, after the formation of the first singularity a finite fraction of the initial mass gets con- centrated inside these low-dimensional structures. De- scribing the mass distribution requires understanding how matter evolves once concentrated in the shocks. But before it will be useful to explain briefly the time evolu- tion of the shock manifold. 4.2.1 Dynamics of singularities Suppose that ~X(t) denotes the position of a shock at time t. We suppose this singularity to be of type An1 (see section 2.3), so that at this position, the veloc- ity field is discontinuous; we denote by ~v1, . . . , ~vn the n different limiting values it takes at that point. At any time we generically have n ≤ d + 1 and occasionally n = d+2 at the space-time positions of shock metamor- phoses corresponding to instants when two Ad1 singular- ities merge. We first restrict ourselves to persistent sin- gularities, meaning that n ≤ d+ 1. In the neighborhood of ~X(t), it is easily checked that the velocity potential can be written as Ψ(~x, t) = Ψ( ~X(t), t) + max j=1..n ~vj · ( ~X(t) − ~x) +o(‖~x− ~X(t)‖) . (4.12) This expansion divides locally the physical space in n subdomains Ωj where ~vj · ( ~X(t) − ~x) is maximum, i.e. ~y ∈ Ωj ⇔ (~vi − ~vj) · (~y− ~X(t)) ≥ 0, 1 ≤ i ≤ n . (4.13) Writing the expansion (4.12) amounts to approximating the velocity potential by a continuous function which is piecewise linear on the subdomains Ωj . The boundaries between the Ωj ’s define the local shock manifold. The maximum in (4.12) ensures that we are focusing on en- tropic solutions to the Burgers equation (solutions ob- tained in the limit of vanishing viscosity) and results in the convexity of the local approximation of the poten- tial. Note also that the position ~x = ~X(t) of the reference singular point corresponds by construction to the unique intersection of all subdomains Ωj . Remember that we have assumed that locally, the solution does not have higher-order singularity. The approximation (4.12) fully describes the local struc- ture of the singularity. If n = 2, corresponding to ~X(t) being the position of a simple shock, it is easily checked from (4.12) that there will actually exist a whole shock hyper-plane given by the set of positions ~y satisfying (~v1 − ~v2) · ( ~X(t) − ~y) = 0 . (4.14) If n > 2, meaning that ~X(t) is an intersection be- tween different shocks, all the singular manifolds of co-dimension m ≤ n are present in the expansion and are given by the set of positions ~y satisfying ~vi1 · ( ~X(t) − ~y) = · · · = ~vim · ( ~X(t) − ~y) , (4.15) with 1 ≤ i1 < · · · < im ≤ n. We next apply the variational principle (3.2) in order to solve the decaying problem between times t and t + δt with the initial condition given by (4.12). This yields an approximation of the potential at time t+ δt: Ψ(~x, t+ δt) ≃ Ψ( ~X(t), t) + max j=1..n ~vj · ( ~X(t) − ~y) − ‖~x− ~y‖2 . (4.16) Note that here, δt and ‖~x− ~X(t)‖ are chosen sufficiently small in a suitable way to ensure that (i) any singularity of higher co-dimension does not interfere with the posi- tion of ~X(t) between times t and t+ δt and that (ii) the subleading terms are always dominated by the kinetic energy contribution ‖~x− ~y‖2/(2δt). The two maxima in ~y and in j of (4.16) can be inter- changed, under the condition that the maximum in ~y is restricted to the domain Ωj defined in (4.13). The po- tential at time t+ δt can thus be written as Ψ(~x, t+ δt) ≃ Ψ( ~X(t), t) + max j=1..n ~y∈Ωj ~vj · ( ~X(t) − ~y) − ‖~x− ~y‖2 . (4.17) We next remark that for all ~x, j and ~y, one has ~vj · ( ~X(t) − ~y) − ‖~x− ~y‖2 ≤ ~vj · ( ~X(t) − ~x+ δt~vj) − ‖~vj‖2 ,(4.18) which gives an upper-bound to the maximum over ~y ∈ Ωj in (4.17). Suppose now that the maximum over the index j is achieved for j = j0. This means that for all 1 ≤ i ≤ n and ~y ∈ Ωi ~vi · ( ~X(t) − ~y) − ‖~x− ~y‖2 ≤ max ~z∈Ωj0 ~vj0 · ( ~X(t) − ~z) − ‖~x− ~z‖2 ≤ ~vj0 · ( ~X(t) − ~x+ δt~vj0) − ‖~vj‖2 . (4.19) Let Ωi0 be the domain containing the vector (~x− δt~vj0). Then, (4.19) applied to i = i0 and ~y = ~x− δt~vj0 trivially implies that (~vi0 − ~vj0 ) · (~x− δt~vj0 − ~X(t)) ≥ 0 , (4.20) which together with the definition (4.13) for Ωi0 leads to i0 = j0. Hence, to summarize, if the first maximum is reached for j = j0 then the second maximum is nec- essarily reached for ~y = ~x− δt~vj0 . It is clear that the approximation (4.16) of the velocity potential at time t + δt preserves the local structure of the singular manifold. Indeed, for m ≤ n, the positions ~y satisfying ~v1 · ( ~X(t) − ~y) + ‖~v1‖2 = · · · · · ·= ~vm · ( ~X(t) − ~y) + ‖~vm‖2 (4.21) form a (d−m)-dimensional shock manifold. The trajec- tory ~X(t) of the reference singular point satisfies ~v1 · ‖~v1‖2 = · · · = ~vn · ‖~vn‖2 , (4.22) which can be rewritten as ‖d ~X/dt− ~v1‖ = · · · = ‖d ~X/dt− ~vn‖ . (4.23) This gives n relations for the d components of the vector d ~X/dt. These relations allow determining the normal velocity of the singular manifold. The tangent velocity remains undetermined. The velocity of the singularity located at ~X(t) is completely determined only if n = d, i.e. for point singularities. For instance when d = 1, the dynamics of shocks is given by (u1 + u2) , (4.24) meaning that they move with a velocity equal to the half sum of their right and left velocities. For d = 2, only the positions of triple points (singularities of type A31 corre- sponding to the intersection of three shock lines) are well determined. It is easily checked that the two-dimensional velocity vector d ~X/dt is the circumcenter of the trian- gle formed by the three limiting values (~v1, ~v2, ~v3) that are achieved by the velocity field at this position (see figure 15). v 3v node v mass Fig. 15. Determination of the velocity of a triple point and of that of the mass inside it when the three limiting values of the velocity ~v1, ~v2, and ~v3 form an obtuse triangle. The dash-dotted circle is the circumcircle whose center gives the velocity of the singularity and the dashed circle is the small- est circle containing the triangle whose center gives the ve- locity of mass. 4.2.2 Dynamics of the mass inside the singular mani- One of the central themes of this review article is a con- nection between Lagrangian particle dynamics and the inviscid Burgers equation. In the unforced case the ve- locity is conserved along particle trajectories minimizing the Lagrangian action (see section 2). At a given mo- ment of time, all particles whose trajectories are not min- imizers have been absorbed by the shocks. In the one- dimensional case when shocks are isolated points, par- ticles absorbed by shocks just follow the dynamics of a shock point. However, in the multi-dimensional case the geometry of the singular shock manifold can be rather complicated. This results in a non-trivial particle dy- namics inside the singular manifold. In other words, the particle absorbed by shocks have a rich afterlife and the main problem is to describe their dynamical proper- ties inside the singular manifold. This problem was ad- dressed by I. Bogaevsky in [18]. The basic idea is to consider first particle transport by the velocity field given by smooth solutions to the viscous Burgers equation. Indeed, let ~vν(x, t) be a solution to the viscous Burgers equation ν + (~vν · ∇)~vν = ν∇2~vν −∇F (~x, t). Then the dynamics of a Lagrangian particle labeled by its position ~x0 at time t = 0 is described by the system of ordinary differential equations ~̇Xν(~x0, t) = ~v ν( ~Xν(~x0, t), t), ~X ν(~x0, 0) = ~x0, (4.25) where the dots stand for time derivatives. It is possible to show that in the inviscid limit ν → 0 solutions to (4.25) converge to limiting trajectories { ~X(~x0, t)}. These lim- iting trajectories are not disjoint anymore. In fact, two trajectories corresponding to different initial positions ~x10 and ~x 0 can merge: ~X(~x10, t ∗) = ~X(~x20, t ∗). This corre- sponds to absorption of particles by the shock manifold. Of course, two trajectories coincide after they merge: ~X(~x10, t) = ~X(~x20, t) for t ≥ t∗. Particles which until time t never merged with any other particles correspond to minimizers. Such trajectories obviously satisfy the lim- iting differential equation: ~̇X(~x0, t) = ~v( ~X(~x0, t), t), ~X(~x0, 0) = ~x0, (4.26) where ~v(x, t) is the entropic solution of the inviscid Burg- ers equation which is well defined outside of the shock manifold. However, we are mostly interested in the dy- namics of particles which have merged with other par- ticles and thus were absorbed by shocks. One can prove that for such trajectories one-sided time derivatives exist ~X(t) = lim ∆t→0+ ~X(t+ ∆t) − ~X(t) (4.27) and satisfy a “one-sided” differential equation: ~X(t) = ~v(s)( ~X(t), t). (4.28) Here ~v(s)(·, t) is the velocity field on the shock manifold (index s stands for shocks). It turns out that ~v(s)(~x, t) and the corresponding shock trajectories satisfy a vari- ational principle, described hereafter. Denote by Ψ(~x, t) a potential of the viscous velocity field ~v(~x, t): ~v(~x, t) = −∇Ψ(~x, t). As we have pointed out many times be- fore,−Ψ(~x, t) corresponds to a minimum Lagrangian ac- tion among all the Lagrangian trajectories which pass through point ~x at time t. Shocks correspond to a situa- tion where the minimum is attained for several different trajectories. Correspondingly, one has several smooth branches such that Ψ(~x, t) = Ψi(~x, t), 1 ≤ i ≤ k. Sup- pose a particle moves from a point of shock (~x, t) with a velocity ~v. Then at infinitesimally close time t+ δt its position will be ~x+~vδt. In linear approximation (see pre- vious subsection) the Lagrangian action of this infinites- imal piece of trajectory is equal to [|~v|2/2 − F (~x, t)]δt. Of course, the action minimizing trajectory at the point (~x + ~vδt, t + δt) does not pass through a shock point (~x, t). Hence, the minimum action −Ψ(~x + ~vδt, t + δt) is smaller than −Ψ(~x, t) + [‖~v‖2/2 − F (~x, t)]δt for any velocity ~v. However, we can put a variational condition on ~v which requires the difference between −Ψ(~x, t) + [‖~v‖2/2 − F (~x, t)]δt and −Ψ(~x + ~vδt, t + δt) to be as small as possible. This is exactly the variational principle which determines the velocity ~v = ~v(s)(~x, t) at a shock point. It is easy to see that in linear approximation Ψ(~x+~vδt, t+δt) = max 1≤i≤k [Ψi(~x+~vδt, t+δt)] = Ψ(~x, t) − min 1≤i≤k [−∇Ψi(~x, t) · ~v − ∂tΨi(~x, t)] δt. (4.29) Let us denote by ~vi the limiting velocities −∇Ψi(~x, t) at the shock point (~x, t). Then, using Hamilton–Jacobi equation for the velocity potential ∂tΨi(~x, t) = ‖∇Ψi(~x, t)‖2 + F (~x, t) ‖~vi‖2 + F (~x, t) (4.30) we have Ψ(~x+ ~vδt, t+ δt) = Ψ(~x, t) − − min 1≤i≤k ~vi ·~v − ‖~vi‖2 δt−F (~x, t)δt. (4.31) Hence, the difference of actions can be written as ∆A=−Ψ(~x, t)+ 1 ‖~v‖2δt+ Ψ(~x+~vδt, t+δt) ‖~v‖2δt− min 1≤i≤k ~vi · ~v − ‖~vi‖2 1≤i≤k ‖~v − ~vi‖2δt. (4.32) Obviously minimization of ∆A over ~v corresponds to a center of a minimum ball covering ~vi. It implies that such a center gives the velocity ~v(s)(~x(t), t) of particles concentrated at a shock point (~x, t). It is interesting that this variational principle implies that a particle absorbed by a shock cannot leave the singular shock manifold in the future. Let us now consider the first nontrivial generic exam- ple of a shock point, namely a triple point in two di- mensions d = 2. The point ( ~X(t), t) is thus the intersec- tion of three shock lines. In this case there are exactly three smooth branches Ψi(·, t) with limiting velocities ~vi = −∇Ψi, 1 ≤ i ≤ 3. As we have seen in previous sec- tion the motion of the triple point is determined by con- tinuity of the velocity potential at ( ~X, t). The “geomet- rical velocity” d ~X/dt of the triple point is then the cir- cumcenter of the triangle formed by the three velocities ~v1, ~v2, ~v3. It is easy to see that d ~X/dt = ~v (s) only in the case when the vectors ~v1, ~v2, and ~v3 form an acute trian- gle. If so, a cluster of particles follows the triple point. In the opposite case when the triangle is obtuse, the parti- cles leave the node. Such a situation is presented in fig- ure 15, where the mass leaves the node along the shock line delimiting the values ~v1 and ~v3 of the velocity. The analysis presented above was carried out for the Burgers equation jointly with A. Sobolevskĭı as a part of ongoing work on a similar theory for the case of a general Hamilton–Jacobi equation, with a Hamiltonian that is convex in the momentum variable. The formal extension of this analysis to the latter case is straightforward and can be left to the interested reader; however at present a rigorous justification of it, employing methods of [18], is known only for the case of H(x, ẋ, t) = a(x, t)|ẋ|2, with a(x, t) > 0. 4.3 Connections with convex optimization problems As discussed in section 1.2, Burgers dynamics is known in cosmology as the adhesion model and frequently used to understand the formation of the large-scale structures in the Universe. Recently, this model was used as a basis for developing new techniques for one of the most chal- lenging questions in modern cosmology, namely recon- struction. This problem aims at reconstructing the dy- namical history of the Universe through the mass den- sity initial fluctuations that evolved into the distribution of matter and galaxies which is nowadays observed (see, e.g., [98]). The main difficulty encountered is that the velocities of galaxies (the peculiar velocities) are usually unknown, so that most approaches lead to non-unique solutions to this ill-posed problem. The reconstruction technique we present here, which was proposed in [55,25], is based on the observation that, to the leading order, the mass is initially uniformly distributed in space (see, e.g., [98]). This observation, together with the Zeldovich approximation, leads to a reformulation of the prob- lem as a well-posed instance of an optimal mass trans- portation problem between the initial (uniform) and the present (observed) distributions of mass. More precisely it amounts to a convex optimization problem related to the Monge–Ampère equation and dually, as found by Kantorovich [73], to a linear programming problem. This is the reason why the name MAK (Monge–Ampère– Kantorovich) has been proposed for this method in [55]. Namely, one has to find the transformation from initial to current positions (the Lagrangian map) that maps the initial density ρ(~x0, 0) = ρ0 to the field ρ(~x, t) which is nowadays observed. One then use a well-known fact in cosmology: because of the expansion of the Universe, the initial velocity field of the self-gravitating matter is slaved to the initial gravitational field (see, e.g., [25]). This observation implies that the initial velocity field is potential and allows one to deduce from it the sublead- ing fluctuations of the mass density. The MAK reconstruction technique is based on two cru- cial assumptions. First the Lagrangian map ~x0 7→ ~x = ~X(~x0, t) is assumed to be potential, i.e. ~X = ∇x0Φ(~x0). Second, the Lagrangian potential Φ(~x0) is assumed to be a convex function. As explained in [25] these two hy- potheses are motivated by the adhesion model (and thus inviscid Burgers dynamics) where they are trivially sat- isfied. As we will see later the reverse is actually true: the potentiality of the Lagrangian map and the convexity of the potential is equivalent to assuming that the latent ve- locity field is a solution to the Burgers equation. We will now see how, under these hypotheses, the reconstruction problem relates to Monge–Ampère equation. Conserva- tion of mass trivially implies that ρ(~x, t)d3x = ρ0d which can be rewritten in terms of the Jacobian matrix (∂X i)/(∂x 0) as ρ( ~X(~x0, t), t) . (4.33) Potentiality of the Lagrangian map leads to ∂xi0∂x ρ(∇x0Φ, t) . (4.34) The problem with this formulation is that the unknown potential Φ enters the right-hand side of the equation in a non-trivial way. Convexity of the Lagrangian po- tential Φ is next used to reformulate the problem in term of the inverse Lagrangian map. Indeed, if Φ is con- vex, the inverse Lagrangian map is also potential, i.e. ~x0 = ~X0(x, t) = ∇xΘ(~x) with the potential Θ itself con- vex. The two potentials Φ and Θ are moreover related by Legendre transforms: Θ(~x) = max [~x · ~x0 − Φ(~x0)], (4.35) Φ(~x0) = max [~x · ~x0 − Θ(~x)]. (4.36) In terms of the inverse Lagrangian potential Θ the con- servation of mass (4.34) reads ∂xi∂xj = ρ(~x, t), (4.37) which is exactly the elliptic Monge-Ampère equation. This time, the difficulty expressed above has disappeared since the unknown potential Θ does not enter the right- hand side of the equation. Note that we have implic- itly assumed here that the present distribution of mass has no singularity. The case of a singular distribution could actually be treated using a weak formulation of the Monge-Ampère equation, which amounts to apply- ing conservation of mass on any subdomain but requires allowing the inverse Lagrangian map to be multival- ued. The next step in the design of the MAK method is to reformulate (4.37) as an optimal transport problem with quadratic cost. Indeed, as shown in [24], the map ~X(~x0, t) (and its inverse ~X0(~x, t)) minimizing the cost ‖ ~X(~x0, t) − ~x0‖2ρ0 d3x0 ‖~x− ~X0(~x, t)‖2ρ(~x, t) d3x, (4.38) is a potential map whose potential is convex and is the solution to the Monge–Ampère equation (4.37). This can be understood using a variational approach as proposed in [55]. Suppose we perform a small displacement δ ~X0(~x) of the inverse Lagrangian map ~X0(~x, t) solution of the optimal transport problem. On the one hand the only ad- missible displacement are those satisfying the constraint to map the initial density field ρ0 to the final one ρ(~x, t). It is shown in [25] that this is equivalent to require that ∇x · [ρ(~x, t)δ ~X0(~x)] = 0. On the other hand one easily see that the variation of the cost function corresponding to the variation δx reads δI = −2 [~x− ~X0(~x, t)] · [ρ(~x, t)δ ~X0(~x)] d3x. (4.39) This integral can be interpreted as the scalar product (in the L2 sense) between ~x− ~X0(~x0, t) and ρ(x)δ ~X(~x0). Hence the optimal solution, which should satisfy δI = 0 for all δ ~X0, is such that the displacement ~x − ~X0(~x0, t) (or equivalently ~X(~x0) − ~x0) is orthogonal to all divergence-free vector fields. This means that it is nec- essarily the gradient of a potential, from which it follows that ~X(~x0, t) = ∇x0Φ(~x0). Convexity follows from the observation that the Lagrangian map ~x0 7→ ~X has to satisfy (~x0 − ~x′0) · [ ~X(~x0) − ~X(~x′0)] ≥ 0. (4.40) Indeed, if that was not the case, one can easily check that any map where the Lagrangian pre-image of a neighbor- hood of ~x0 and of one of ~x0 are inverted would lead to a smaller cost. Formulated in terms of potential maps, the relation (4.40) straightforwardly implies convexity of Φ. This finishes the proof of equivalence between Monge– Ampère equation and the optimal transport problem with quadratic cost. The goal of reformulating reconstruction as an optimiza- tion problem is mostly algorithmic. Once discretized, the problem of finding the optimal map between initial and final positions amounts is equivalent to solving a so- called assignment problem. An efficient method to deal numerically with such problems is based on the auction algorithm [15] and was used in [25] with data stemming from N -body cosmological simulations. As summarized 0.2 0.3 0.7 0.80.4 0.5 0.6 simulation coordinate 1 2 3 4 distances Fig. 16. Test of the MAK reconstruction for a sample of N ′ = 17, 178 points from a N-body simulation (from [25]). The scatter diagram plots reconstructed versus true initial positions. The histogram inset gives the distribution (in per- centages) of distances between true and reconstructed initial positions; the horizontal unit is the distance between two sampled points. The width of the first bin is less than unity to ensure that only exactly reconstructed points fall in it. More than sixty percent of the points are exactly reconstructed. in figure 16, the MAK reconstruction method leads to very promising results. More than 60% of the discrete points are assigned to their actual Lagrangianpre-image. Such a number has to be compared with other recon- struction methods for which the success rate barely ex- ceed 40% for the same data set. Even if the mapping from initial to final positions is unique, the peculiar velocities are not well defined ex- cept if we have some extra knowledge of what is happen- ing at intermediate times 0 ≤ t′ ≤ t. Of course the den- sity field ρ(~x′, t′) is unknown. However, there are triv- ial physical requirements. First the two mass transport problems between 0 and t′ and between t′ and t have both to be optimal. This means that one looks for two Lagrangian maps, ~X1 from 0 to t ′ and ~X2 from t ′ to t which are minimizing the respective costs ‖ ~X1(~x0) − ~x0‖2ρ0 d3x0, ‖~x− ~X−12 (~x)‖ 2ρ(~x, t) d3x. (4.41) The second physical requirement is that the composition of these two optimal maps have to give the Lagrangian map between times 0 and t, namely ~X(~x0, t) = ~X2( ~X1). Under these two conditions there is equivalence between the optimal transport with a quadratic cost and the Burgers dynamics supplemented by the transport of a density field (see [13] for details). 5 Forced Burgers turbulence 5.1 Stationary régime and global minimizer We consider in this section solutions to the forced Burg- ers equation. As we have seen in section 2, the solution in the limit of vanishing viscosity can be expressed at any time t in terms of the initial condition at time t0 through a variational principle which consists in minimizing an action along particle trajectories. The statistically sta- tionary régime toward which the solution converges at large time can be studied assuming that the by reject- ing the initial time t0 is at minus infinity. The solution is then given by the variational principle Ψ(~x, t)=−inf ~γ(·) ‖~̇γ(s)‖2−F (~γ(s), s) , (5.1) where the infimum is taken over all (absolutely continu- ous) curves ~γ : (−∞, t] → Ω such that ~γ(t) = ~x. In this setting, the action is computed over the whole half line (−∞, t] and the argument of the infimum does not de- pend anymore on the initial condition. Of course, (5.1) defines Ψ up to an additive constant. This means that only the differences Ψ(~x, t)−Ψ(0, t) can actually be de- fined. A trajectory ~γ minimizing (5.1) is called a one- sided minimizer. It is easily seen from (5.1) that all the minimizers are solutions of the Euler–Lagrange equation ~̈γ(s) = −∇F ( ~γ(s), s) , (5.2) where the dots denote time derivatives. This equation defines a 2d-dimensional (possibly random) dynamical system in the position-velocity phase space (~γ, ~̇γ). The Lagrangian one-sided minimizers ~γ defined over the half- infinite interval (−∞, t] play a crucial role in the con- struction of the global solution and of the stationary régime. Namely, a global solution to the randomly forced inviscid Burgers equation is given by~v(~x, t) = ~̇γ(t) where ~γ(t) = ~x. To prove that such half-infinite minimizers ex- ist, one has to take the limit t0 → −∞ for minimizers defined on the finite time interval [t0, t]. The existence of this limit follows from a uniform bound on the absolute value of the velocity |~̇γ| (see, e.g., [38]). Obtaining such a bound becomes the central problem for the theory, as we shall now see. When the configuration space Ω where the solutions live is compact (bounded), one can expect the velocity of a minimizer to be uniformly bounded. Indeed, in this case the displacement of a minimizer for any time interval is then bounded by the diameter of the domain Ω, so that action minimizing trajectories cannot have large veloc- ities. For forcing potential that are delta-correlated in time, it has been shown by E et al. [38] in one dimension and by Iturriaga and Khanin [68,69] in higher dimen- sions that the minimizing problem (5.1) has a unique solution Ψ with the following properties: • Ψ is the unique statistically stationary solution to the Hamilton–Jacobi equation (2.2) in the inviscid limit ν → 0; • Ψ is almost everywhere differentiable with respect to the space variable ~x; • −∇Ψ uniquely defines a statistically stationary solu- tion to the Burgers equation in the inviscid limit; • there exists a unique one-sided minimizer at those Eu- lerian positions ~x where the potential Ψ is differen- tiable; the locations where Ψ is not differentiable cor- respond to shocks. • There exists a unique minimizer ~γ(g) that minimizes the action calculated from −∞ to any time t. It is called the global minimizer (or two-sided minimizer) and corresponds to the trajectory of a fluid particle that is never absorbed by shocks. Moreover, all one- sided minimizers are asymptotic to it as s→ −∞. All the properties above follow from the variational ap- proach. In fact, the variational principle (2.12) imply similar statements in the viscous case. Of course, when viscosity is positive the unique statistically stationary solution is smooth. However, one can show that the sta- tionary distribution corresponding to such solutions con- verges to inviscid stationary distribution in the limit ν → 0 [58]. Although the variational proofs are concep- tual, general and simple, they are based on the fluctua- tion mechanism and therefore do not give a good control of the rate of convergence to the statistically stationary regime. Exponential convergence would follow from the hyperbolicity of the global minimizer. Although one ex- pects hyperbolicity holds in any dimension, mathemat- ically it is an open problem. At present a rigorous proof of hyperbolicity is only available in dimension one [38]. The assumption of compactness of the configuration space Ω is essential in the construction of the stationary régime. As we will see in subsection 5.4, the situation is much more complex in the non-compact case when for instance the solution is defined on the whole space Ω = Rd. 5.2 Topological shocks To introduce the notion of topological shock we first fo- cus on the one-dimensional case in a periodic domain, i.e. in Ω = T = R/Z. If we “unwrap” at a given time t the configuration space to its universal cover R (see fig- ure 17(a)), we then obtain an infinite number of global minimizer γ k , which at all time s ≤ t satisfy γ k+1(s) = k (s) + 1. All the one-sided minimizers converge back- ward in time to one of these global minimizers. The topo- logical shock (or main shock) is defined as the set of x positions giving rise to several minimizers approaching two successive replicas of the global minimizer. This par- ticular shock is also the only shock that has existed for all times. This construction can easily be extended to higher dimensions (see [10]). For this we unwrap the d- dimensional torus Td to its universal cover, the full space Rd (see figure 17(b) for d = 2). Then, the different replicas of the periodic domain define a lattice of global minimizers ~γ parameterized by integer vectors ~k. The backward-in-time convergence on the torus of the one- sided minimizers to the global minimizer implies that a minimizer associated to a location ~x in Rd at time t will be asymptotic to one of the global minimizer ~γ the lattice. Hence, every position ~x which has a unique one-sided minimizer is associated to an integer vector ~k(~x). This defines a tiling of space at time t. The tiles O~k are the sets of points whose associated one-sided minimizers are asymptotic to the ~k-th global minimizer. The boundaries of the O~k’s are the topological shocks. They are the locations from which at least two one- sided minimizers approach different global minimizers on the lattice. Indeed, a point where two tiles O~k1 and O~k2 meet, has at least two one-sided minimizers, one of which is asymptotic to ~γ and another to ~γ course, there are also points on the boundaries where three or more tiles meet and thus where more than two one-sided minimizers are asymptotic to different global minimizers. For d = 2 such locations are generically iso- lated points corresponding to the intersections of three or more topological shock lines, while for d = 3, they form edges and vertices where shock surfaces meet. Note Fig. 17. Space-time sketch of the unwraping of the periodic domain Td to the whole space Rd for d = 1 (a) and d = 2 (b). that, generically, there exist other points inside O~k with several minimizers. They correspond to shocks of “lo- cal” nature because at these locations, all the one-sided minimizers are asymptotic to the same global minimizer and hence, to each other. In terms of Lagrangian dynamics, the topological shocks play a role dual to that of the global minimizer. Indeed, all the fluid particles are converging backward-in-time to the global mini- mizer and are absorbed forward-in-time by the topo- logical shocks. For the transportation of mass when we assume that the Burgers equation is supplemented by a continuity equation for the mass density, all the mass concentrate at large times in the topological shocks. The global structure of the topological shocks is related to the various singularities generically present in the so- lution to the Burgers equation that were detailed in sec- tion 2.3. Generically there are no locations associated to more than (d+1) minimizers. As one expects to see only generic behavior in a random situation, the probability to have points with more than (d + 1) one-sided mini- mizers is zero. It follows that there are no points where (d+ 2) tiles O~k meet, which is an important restriction on the structure of the tiling. For d = 2 it implies that the tiling is constituted of curvilinear hexagons. Indeed, suppose each tile O~k is a curvilinear polygon with s ver- Fig. 18. (a) Position of the topological shock on the torus; the two triple points are represented as dots. (b) Snapshot of the velocity potential ψ(x, y, t) for d = 2 in the statisti- cal steady state, obtained numerically with 2562 grid points. Shock lines, corresponding to locations where ψ is not dif- ferentiable, are represented as black lines on the bottom of the picture; the four gray areas are different tiles separated by the topological shocks; the other lines are local shocks. tices corresponding to triple points. For a large piece of the tiling that consists of N tiles, the total num- ber of vertices is nv ∼ sN/3 and the total number of edges is ne ∼ sN/2. The Euler formula implies that 1 = nv − ne +N ∼ (6− s)N/6, and we necessarily have s = 6, corresponding to an hexagonal tiling. As shown in figure 18(a), this structure corresponds on the peri- odicity torus T2, to two triple points connected by three shock lines that are the curvilinear edges of the hexagon O~0. The connection between the steady-state potential and the topological shocks is illustrated numerically on figure 18(b). The different tiles covering the periodic do- main were obtained by tracking backward in time fluid particle trajectories and by determining to which peri- odic image of the global minimizer they converge. In dimensions higher than two, the structure of topolog- ical shocks is more complicated. For instance it is not possible to determine in a unique manner the shape of the polyhedra forming the tiling. However, it has been shown by Matveev [87] that for d = 3 the minimum poly- hedra forming such tiling has 24 vertices and 36 edges and is composed of 8 hexagons and 6 rectangles (see fig- ure 19). It is of interest to note that the structure of topological shocks is in direct relation with the notions of complexity and minimum spines of manifolds intro- duced by Matveev from a purely topological viewpoint. Fig. 19. Sketch of the simplest configuration of the topolog- ical shock in dimension d = 3. Algebraic characterization of the topological shock In two dimensions, when periodic boundary conditions are considered, very strong constraints are imposed on the structure of the solution. In particular, the topol- ogy of the torus T2 imply that the topological shocks generically form a periodic tiling of R2 with curvilinear hexagons. However, this tiling can be of various alge- braic types. Consider the tile O~0 surrounded by its six immediate neighbors O~ki , where the integer vectors are labeled in anti-clockwise order, ~k1 having the small- est polar angle (see figure 20). It is easily seen that the periodicity of the tiling implies ~k3 = ~k2 − ~k1, ~k4 = −~k1, ~k5 = −~k2 and ~k6 = ~k1 − ~k2, (5.3) so that the whole information on the algebraic struc- ture of the tiling is contained in the vectors ~k1 and ~k2 which form a matrix S from the group SL(2,Z) of 2× 2 integer matrices with unit determinant. The matrix S gives information on the number of times each shock line turns around the torus before reconnecting to another triple point. Figure 18(a) corresponds to the simplest case when S is the identity matrix. When the forcing is stochastic, the matrix S is random and stationary solu- tions to the two-dimensional Burgers equation define a stationary distribution on SL(2,Z). (i, j) (i−1, j+1) (i+1, j) (i+1, j−1) (i, j+1) (i−1, j) (i, j−1) Fig. 20. The algebraic structure of the topological shock in dimension d = 2 is determined by the indexes corresponding to immediate neighbors of the tiling considered. Certainly, topological shocks evolve in time and may change their algebraic structure. This happens through bifurcations (or metamorphoses) described in section 2.3. In two dimensions, the generic mechanism which transforms the algebraic structure of topological shocks is the merger of two triple points. This metamorphosis is called the flipping bifurcation and corresponds to the appearance at time t⋆ of an A 1 singularity in the solu- tion associated to a position with four minimizers. The mechanism transforming the algebraic structure of the topological shock is illustrated in figure 21. Issues such as the minimum number of flips needed to transform the matrix S1 associated to the algebraic structure of the topological shock to another matrix S2 are discussed in in [1]. Fig. 21. Sketch of the tiling before, at the flipping time t∗ and after it. This example corresponds to a bifurcation from the matrix S1 = [ 1] to S2 = [ 2]. The dashed boxes represent the periodicity domain [0, 1]2. 5.3 Hyperbolicity of the global minimizer The nature of the convergence to a statistical steady state is determined by the local properties of the global minimizer. The hyperbolicity of this action-minimizing trajectory implies an exponential convergence, so that the global picture of the solution is reached very rapidly, after just a few turnover times. Since the trajectory of the global minimizer is unique and can be extended to arbitrary large times, it corre- sponds to an ergodic invariant measure for the stochastic flow defined by the Euler–Lagrange equation (5.2). Con- ditioned by the random force, this measure is simply the delta measure sitting at the location (~γ(g)(0), ~̇γ(g)(0)). By the Oseledets ergodic theorem (see, e.g. [98]), 2d non- random Lyapunov exponents can be associated to the global minimizer trajectory. Since the flow is symplectic these non-random exponents come in pairs with oppo- site signs. That is λ1 ≥ · · · ≥ λd ≥ 0 ≥ −λd ≥ · · · ≥ −λ1 . (5.4) Hyperbolicity is defined as the non-vanishing of all these exponents. Thus, the issue of hyperbolicity can be ad- dressed in terms of the backward-in-time convergence of the one-sided minimizers to the global one or, better, in terms of forward-in-time dynamics. In the latter case, this amounts to looking how fast Lagrangian fluid par- ticles are absorbed by shocks. For this we consider the set Ωreg(T ) of locations ~x such that the fluid particle emanating from ~x at time t = 0 survives, i.e. is not ab- sorbed by any shock, until the time t = T . The long-time shrinking of Ωreg as a function of time is asymptotically governed by the Lyapunov exponents. To ensure the ab- sence of vanishing Lyapunov exponents, it is sufficient to show that the diameter of Ωreg(T ) decays exponentially as T → +∞. In one dimension, it has been shown in [38] that this is indeed the case, and particularly that there exists posi- tive constants α, β, A and B such that diamΩreg(T ) ≥ Ae−αT ≤ Be−βT . (5.5) Unfortunately this proof of hyperbolicity is purely one- dimensional and at present time there is no extension of this result to higher dimensions. In two dimensions, the behavior of diamΩreg(T ) at large times was studied numerically in [10] by using the fast Legendre transform described in section 2.4 and a forc- ing that is a sum of independent random impulses con- centrated at discrete times. The ideas of this numerical method are related to the Lagrangian structure of the flow. This easily permits to track numerically the set Ωreg of regular Lagrangian locations. As seen from figure 22, the diameter of this set decays exponentially fast in time for three different types of forcing, providing good ev- idence of the hyperbolicity of the global minimizer for d = 2. 0 0.25 0.5 0.75 1 time T Algebraic [∝ k−3] Gaussian [∝ exp (−k2)] Step [= Const. if k ≤3] Fig. 22. Time evolution of the diameter of the Lagrangian set Ω(T ) (points corresponding to the regular region) for three different types of forcing spectrum; average over 100 realizations and with 2562 grid points (from [10]). Hyperbolicity of the global minimizer implies existence at any time t of two d-dimensional smooth manifolds u x,t( ) ( )tΓ(u) global minimizer shock a preshock Fig. 23. Sketch of the unstable manifold for d = 1 in the (x, v) plane. Shock locations (A21 singularities) are obtained by applying Maxwell rules to the loops. A preshock (A3 singularity) is represented; it corresponds to the formation of a loop in the manifold. The velocity profile which is the actual solution to the Burgers equation is represented as a bold line. in phase space (~γ, ~̇γ) that are invariant by the Euler– Lagrange dynamics (5.2): a stable (attracting) manifold Γ(s)(t) and an unstable (repelling) manifold Γ(u)(t), de- fined as the instantaneous location of trajectories con- verging to the global minimizer forward in time and backward in time, respectively. Since all the minimizers converge backward in time to the global minimizer, the graph in the position-velocity phase space (~x,~v) of the solution in the statistical steady state is made of pieces of the unstable manifold Γ(u)(t) with discontinuities along the shocks lines or surfaces. In other words, shocks ap- pear as jumps between two different folds of the unstable manifold. The smoothness of the unstable manifold is an important property; for instance, it implies that when d = 2, the topological shock lines are smooth curves. In one dimension, where hyperbolicity is ensured, the main shock corresponds to a jump between the right branch and the left branch of the unstable manifold. Its position can be obtained geometrically after observing that the area b covered by the unstable manifold, once the latter is cut by the main shock, should be equal to the first integral of motion which is conserved, i.e. v(x, t) dx = v0(x) dx . (5.6) The other shocks (or secondary shocks that have existed only for a finite time) cut through the double-fold loops of the unstable manifold (see figure 23). Their locations can be obtained by a Maxwell rule applied to those loops. Indeed, the difference of the two areas defined by cutting such a loop at some position x is equal to the difference of actions of the two trajectories emanating from the up- per and lower locations and, thus, vanishes at the shock location. We will see in section 6 that this construction of the solution is also valid when the forcing is periodic in time, problem which can be related to Aubry–Mather theory relative to commensurate-incommensurate phase transitions. The above geometrical construction of the solution has much in common with that appearing in the unforced problem. Indeed, as we have seen in section 3.1, when F = 0 the solution can be obtained geometrically by considering in the (~x,~v) space, the Lagrangian manifold defined by the position and the velocity of the fluid par- ticles at a given time. This analogy gives good ground predicting that some universal properties associated to the unforced problem will still hold in the forced case, as we will indeed see in section 7. Another instance concerns transport of mass in higher dimension. We have seen in section 4.1 that, for the unforced case, large but finite mass densities are localized near boundaries of shocks (“kurtoparabolic” singularities) contributing power-law tails with the exponent −7/2 to the probability density function of the mass density. When a force is applied the smoothness of the unstable manifold associated to the global minimizer should lead to the same universal law. 5.4 The case of extended systems So far, we have discussed the global structure of the solution to the forced Burgers equation with periodic boundary conditions. Is is however of physical interest to understand instances when the size of the domain is much larger than the typical length scale of the forcing. In this section, we will focus on describing, in the one- dimensional case, the singular structure of the solution in unbounded domains. Based on the formalism of [11], we achieve this goal by considering a spatially periodic forcing with a characteristic scale Lf much smaller than the system size L. More precisely, for a fixed size L we consider the stationary régime corresponding to the limit t → ∞ and then study the limit L → ∞ by keeping constant the energy injection rate (i.e. the L2 norm of the forcing grows like L). In order to get an idea of the behavior of the solution, the limit of infinite aspect ration L/Lf was investigated numerically in [11]. As seen from figure 24(a) numerical observations suggest that at any time in the statistical steady state, the shape of the velocity profile is simi- lar to the order-unity aspect ratio problem, duplicated over independent intervals of sizeLf . In particular, when tracking backward in time the trajectories of fluid par- ticles the minimizers converge to each other in a very non-uniform way. Figure 24(b) shows that the minimiz- ers form different branches, which are converging to each other backward in time; in space time a tree structure is obtained. As shown in figure 25(a) a similar behavior is observed for shocks. The velocity field at a given time t, consists of smooth pieces separated by shocks. Let us denote by {Ωj} the set of intervals in [0, L), on which the solution u(·, t) is smooth. The boundaries of the Ωj ’s are the shocks positions. Each of these shocks is associated to a root- like structure formed by the trajectories of the various Fig. 24. (a) Upper: snapshot of the velocity field for L = 256Lf . Lower: zoom of the field in a interval of length 10Lf . (b) Minimizing trajectories in space time for L = 256Lf and over a time interval of length T = 100 shocks that have merged at times less than t to form the shock under consideration (see figure 25(a)). This root- like structure contains the whole history of the shock and in particular its age (i.e. the length of the deeper branch of the root structure). Indeed, if the root has a finite depth, the shock considered has only existed for a finite time. A T -global shock is defined as a shock whose associated root is deeper than −T . They can alterna- tively be defined geometrically by considering the left- most and the rightmost minimizer associated to it. After tracing them backward for a sufficiently long time, these two minimizers are getting close and eventually converge to each other exponentially fast (see figure 24(b)). For a T -global shock, the time when the two minimizers are getting within a distance smaller than the forcing cor- relation length Lf is larger than T . As we have seen in section 5.2, the existence in one dimension of a main X (t)1 X (t)2 space Fig. 25. (a) Shock trajectories for aspect ratio L/Lf = 32 and with T = 10. The different gray areas correspond to the space-time domains associated to the different smooth pieces Ωj of the velocity field at time t = 0. (b) Sketch of the space-time evolution of a given smooth piece Ωj located between two shock trajectories X1(t) and X2(t) that merge at time Tj . shock in the spatially periodic situation follows from a simple topological argument. The main shock can also be defined as the only shock that has existed forever in the past. It is hence infinitely old, contrary to all other shocks, all of them being created at a finite time and having a finite age. When the periodicity condition is dropped, the main shock disappears and it is useful to consider the T -global shocks that mimic the behavior of a main shock over time scales larger than T . One can dually define T -global minimizers. All the smoothness intervals Ωj defined above, except that which contains the global minimizer, will be entirely absorbed by shocks after a sufficient time (see figure 25(b)). For each of these pieces, one can define a life-time Tj as the time when the last fluid particle contained in this piece at time t enters a shock. It corresponds to the first time for which the shock located on the left of this smooth interval at time t merges with the shock located on the right. When the life-time of such an interval is greater than T , the trajectory of the last surviving fluid particle is here called a T -global minimizer. Note that, when T → ∞, the number of T -main shocks and of T -global minimizers is one, recovering respectively the notions of main-shock and of two-sided minimizer. time T = 64 = 128 = 256 slope = −2/3 Fig. 26. Density of T -main shocks as a function of T for three different system sizes L/Lf = 64, 128 and 256; average over 100 realizations. Lower inset: local scaling exponent. Hence, at a given instant t, and for any timelag T , the spatial domain [0, L) contains a certain number of T -objects. We define their spatial density as being the number of such objects, averaged with respect to the forcing realizations, divided by the size of the domain L. The density ρ(T ) of T -main shocks was investigated nu- merically in [11] for the kicked case by using a two-step method: first, the simulation was run until a large time t for which the statistically stationary régime is reached; secondly, each shock present at time t was tracked backward-in-time down to the instant of its creation, giving an easy way to characterize the density ρ(T ). It is seen in figure 26 that, for three different aspect ratios L/Lf , the density ρ(T ) displays a power-law behavior ρ(T ) ∝ T−2/3 for the intermediate time asymptotics Lf/urms ≪ T ≪ L/urms. We now present a simple phenomenological theory aim- ing to explain the scaling exponent 2/3. We consider the solution at a fixed time (t = 0, for instance). Denote by ℓ(T ) the typical spatial separation scale for two nearest T -global shocks. Obviously, ℓ(T ) ∼ 1/ρ(T ). The mean velocity of the spatial segment of length ℓ is given by [y, y+ℓ] u(x, 0) dx (5.7) Since the expected value 〈u(x, 0)〉 = 0, and that the integral in (5.7) is over an interval of size much larger than the forcing correlation length, it is equivalent to a sum of independent centered random variables and scales as the Brownian motion. Hence, for large ℓ one has the following asymptotics [y, y+ℓ] u(x, 0) dx ∼ ℓ, (5.8) which gives bℓ ∼ ℓ−1/2 for mean velocity fluctuations. Consider now the rightmost minimizer corresponding to the left T -global shock and the leftmost minimizer re- lated to the right one. Since there are no T -global shocks in between, it follows that the two minimizers we se- lected get close to each other backward-in-time around times of the order of −T . This means that the backward- in-time displacement of a spatial segment of length O(ℓ) is itself O(ℓ) for time intervals of the order of T . The corresponding displacement is given as the sum of two competing behaviors: the first, which can be understood as a drift induced by the local mean velocity bℓ, is due to the mean velocity fluctuations and is responsible for a displacement ∝ bℓT ; the second contribution is due to a standard diffusive scale ∝ T 1/2 expressing the diffu- sive behavior of the minimizing trajectories. Taking into account both terms we obtain ℓ ∼ B1T ℓ−1/2 +B2T 1/2, (5.9) where B1 and B2 are numerical constants. It is easy to see that the dominant contribution comes from the first term. Indeed, if the second term were to dominate, then ℓ would be much larger than T , which contradicts (5.9). Hence, one has ℓ ∼ B1T ℓ−1/2, leading to the scaling behavior ℓ(T ) ∝ T 2/3, ρ(T ) ∝ T−2/3. (5.10) As we have already discussed, T -global shocks are shocks older than T . Denote by p(A) the probability density function (PDF) for the age of shocks. More precisely, p(A) is a density in the stationary régime of a probability distribution of the age A(t) of a shock, say the nearest to the origin. It follows from (5.10) that the probability of shocks whose age is larger than A decays like A−2/3; this implies the following asymptotics for the PDF p(A): p(A) ∝ A−5/3. (5.11) Actually, the power-law behavior of the density ρ(T ) of T -global shocks can be interpreted in term of an inverse cascade in the spectrum of the solution (although there is no conserved energy-like quantity). Indeed, the fluctu- ations (5.8) of the mean velocity suggest that, for large- enough separations ℓ, the velocity potential increment scales like |ψ(x+ ℓ, t) − ψ(x, t)| ∝ ℓ1/2. (5.12) This behavior is responsible for the presence of an inter- mediate power-law range with exponent −2 in the spec- trum of the velocity potential at wavenumbers smaller than the forcing scale (see figure 27). In order to observe Fig. 27. Spectrum 〈ψ̂2(k)〉 of the velocity potential in the stationary régime for the aspect ratio L/Lf = 128. This spectrum contains two power-law ranges: at wavenumbers k ≫ L/Lf , the traditional ∝ k −4 inertial range connected to the presence of shocks in the solution and, for k ≪ L/Lf , an “inverse cascade” ∝ k−2 associated to the large-scale fluctuations of ψ the k−2 range at small wavenumbers, the spectrum of the forcing potential must decay faster than k−2; other- wise the leading behavior is non-universal but depends on the functional form of the forcing correlation. The one-dimensional randomly forced Burgers equation in an unbounded domain has been studied in [66] with a different type of forcing: it was assumed that the forc- ing potential has at any time its global maximum and its global minimum in a prescribed compact region of space. It was proven that with these particular settings the statistically stationary régime exists and is very sim- ilar to that arising in compact domains. In particular, there exists a unique global minimizer located in a finite spatial interval for all times and all other minimizers are asymptotic to it in the limit t → −∞. The main idea behind considering such type of forcing potential is to ensure that the potential energy plays a dominant role in comparison with the kinetic (elastic) term in the ac- tion. This leads to effective compactification and allows estimates on the velocities of fluid particles. As we al- ready mentioned in section 5.1, these estimates are very important and pave the way to the construction of the whole theory of the statistically stationary régime. Note finally that it was shown in [76] that for special cases of forcing potentials F (x, t), the velocity of a min- imizers can be arbitrarily large. More specifically, one can construct pathological forcing potentials such that minimizers are accelerated and reach infinite velocities. Randomness is of course expected to prevent such a type of non-generic blow-up. 6 Time-periodic forcing This section is devoted to the study of the solutions to the one-dimensional Burgers equation with time- periodic forcing. In this case many of the objects we have discussed above can be constructed almost explic- itly: the global minimizer, the main shock etc. Also, a mathematical analysis is then much simpler. For in- stance, hyperbolicity of the global minimizer follows immediately from first principles. Finally, the case of time-periodic forcing is directly related to the Aubry- Mather theory as we explain below. 6.1 Kicked Burgers turbulence We shall be concerned here with the initial-value prob- lem for the one-dimensional Burgers equation when the force is concentrated in Dirac delta functions at discrete times: f(x, t) = fj(x) δ(t − tj), (6.1) where both the “impulses” fj(x) and the “kicking times” tj are prescribed (deterministic or random). The kicking times are ordered and form a finite or infinite sequence. The impulses fj(x) are always taken spatially smooth, i.e. acting only at large scales. The general scheme we are presenting below holds for any sequence of impulses fj(x) and kicking time. Later on we shall assume that they define a time-periodic forcing. The precise meaning we ascribe to the Burgers equation with such forcing is that at time tj , the solution u(x, t) changes discontinu- ously by the amount fj(x) u(x, tj+) = u(x, tj−) + fj(x), (6.2) while, between tj+ and t(j+1)− the solution evolves ac- cording to the unforced Burgers equation. We shall also make use of the formulation in terms of the velocity potential ψ(x, t) and the force potentials Fj(x) u(x, t) = −∂xψ(x, t), fj(x) = − Fj(x). (6.3) The velocity potential satisfies ∂tψ = (∂xψ) 2 + ν∂xxψ + Fj(x) δ(t − tj), (6.4) ψ(x, t0) = ψ0(x), (6.5) where ψ0(x) is the initial potential. Using the variational principle we obtain the following “minimum representation” for the potential in the limit of vanishing viscosity which relates the solutions at any two times t > t′ between which no force is applied: ψ(x, t) = −min (x− y)2 2(t− t′) − ψ(y, t′) . (6.6) As before, when t′ is the initial time, the position y which minimizes (6.6) is the Lagrangian coordinate associated to the Eulerian coordinate x. The map y 7→ x is called the Lagrangian map. By expanding the quadratic term it is easily shown that the calculation of ψ(·, t) from ψ(·, t′) is equivalent to a Legendre transformation. For details, see [104,107]. We now turn to the forced case with impulses applied at the kicking times tj . Let tJ(t) be the last such time before t. Using (6.6) iteratively between kicks and changing the potential ψ(y, tj+1) discontinuously by the amount Fj+1(y) at times tj+1, we obtain ψ(x, t) = − min {yj}j0≤j≤J [A({yj};x, t; j0)) − ψ0(yj0)] , (6.7) A({yj};x, t; j0) ≡ (x − yJ)2 2(t− tJ) (yj+1 − yj)2 2(tj+1 − tj) −Fj+1(yj+1) , (6.8) where A(j0;x, t; {yj}) is called the action. We shall as- sume that the force potential and the initial condition are periodic in the space variable and the period is taken to be unity. This assumption is very important for the discussion below. For a given initial condition at tj0 we next define a “min- imizing sequence” associated to (x, t) as a sequence of yj’s (j = j0, j0 + 1, . . . , J(t)) at which the right-hand side of (6.7) achieves its minimum. Differentiating the action (6.8) with respect to the yj ’s one gets necessary conditions for such a sequence, which can be written as a sequence of (Euler–Lagrange) maps vj+1 = vj + fj(yj), (6.9) yj+1 = yj + vj+1(tj+1 − tj) = yj + (vj + fj(yj))(tj+1 − tj), (6.10) where yj − yj−1 tj − tj−1 . (6.11) These equations must be supplemented by the initial and final conditions: vj0 = u0(yj0), (6.12) x= yJ + vJ+1(t− tJ). (6.13) It is easily seen that u(x, t) = vJ+1 = (x− yJ)/(t− tJ ). Observe that the “particle velocity” vj is the velocity of the fluid particle which arrives at yj at time tj and which, of course, has remained unchanged since the last kick (in Lagrangian coordinates). Equation (6.9) just expresses that the particle velocity changes by fj(yj) at the the kicking time tj . Note that (6.9)-(6.10) define an area-preserving and (ex- plicitly) invertible map. As in the case of continuous-in-time forcing we can for- mulate the Burgers equation in the half-infinite time in- terval (−∞, t] without fully specifying the initial con- dition u0(x) but only its (spatial) mean value 〈u〉 ≡ u0(x)dx. The construction of the solution in a half-infinite time interval is done by extending the concept of minimizing sequence to the case of dynamics starting at t0 = −∞. For a half-infinite sequence {yj} (j ≤ J), let us define the action A({yj};x, t;−∞) by (6.8) with j0 = −∞. Such a half-infinite sequence will be called a “minimizer” (or “one-sided minimizer”) if it minimizes this action with respect to any modification of a finite number of yj’s. Specifically, for any other sequence {ŷj} which coincides with {yj} except for finitely many j’s (i.e. ŷj = yj , j ≤ J − k, k ≥ 0), we require A({ŷj};x, t; J − k) ≥ A({yj};x, t; J − k). (6.14) Of course, the Euler–Lagrange relations (6.9)-(6.10) still apply to such minimizers. Hence, if for a given x and t we know u(x, t), we can recursively construct the minimizer {yj} backwards in time by using the inverse of (6.9)- (6.10) for all j < J and the final condition – now an initial condition – (6.13) with vJ+1 = u(x, t). This is well defined except where u(x, t) has a shock and thus more than one value. One way to construct minimizers is to take a sequence of initial conditions at different times t0 → −∞. At each such time some initial condition u0(x) is given with the only constraint that it have the same prescribed value for 〈u〉. Then, (finite) minimizing sequences extending from t0 to t are constructed for these different initial conditions. This sequence of minimizing sequences has limiting points (sequences themselves) which are pre- cisely minimizers (E et al. 1998). The uniqueness of such minimizers, which would then imply the uniqueness of a solution to the Burgers equation in the time interval ]−∞, t], can only be shown by using additional assump- tions, for example for the case of random forcing or when the forcing is time-periodic. If 〈u〉 = 0, the sequence {yj} minimizes the action A({yj};x, t;−∞) in a stronger sense. Consider any sequence {ŷj} such that, for some integer P we have ŷj = yj + P , j ≤ J − k, k ≥ 0 and which differs ar- bitrarily from {yj} for j > J − k. (In other words, in a sufficiently remote past the hatted sequence is just shifted by some integer multiple of the spatial period.) We then have A({ŷj};x, t;−∞) ≥ A({yj}, x, t;−∞). (6.15) Indeed, for 〈u〉 = 0, the velocity potential for any initial condition is itself periodic. In this case a particle can be considered as moving on the circle S1 and its trajectory is a curve on the space-time cylinder. The yj ’s are now defined modulo 1 and can be coded on a representative 0 ≤ yj < 1. The Euler–Lagrange map (6.9)-(6.10) is still valid provided (6.10) is defined modulo 1. The condition of minimality implies now that yj and yj+1 are connected by the shortest possible straight seg- ment. It follows that |vj+1| = ρ(yj , yj+1)/(tj+1 − tj), where ρ is the distance on the circle between the points yj, yj+1, namely ρ(a, b) ≡ min{|a−b|, 1−|a−b|}. Hence, the actionA can be rewritten in terms of cyclic variables: A({yj};x, t;−∞) = ρ2(x, yJ ) 2(t− tJ) ρ2(yj+1, yj) 2(tj+1 − tj) − Fj+1(yj+1) . (6.16) The concept of “global minimizers” can be defined in a usual way. Namely, global minimizers correspond to one-sided minimizers that can be continued to a bilat- eral sequence {yj ,−∞ < j < +∞} while keeping the minimizing property. Such global minimizers correspond to trajectories of fluid particles that, from t = −∞ to t = +∞, have never been absorbed in a shock. As before we define a “main shock” as a shock which has always existed in the past. From now on we shall consider exclusively the case where the kicking is periodic in both space and time. Specifi- cally, we assume that the force in the Burgers equation is given by f(x, t) = g(x) δ(t− jT ), (6.17) g(x) ≡− d G(x), (6.18) where G(x), the kicking potential, is a deterministic function of x which is periodic and sufficiently smooth (e.g. analytic) and where T is the kicking period. The initial potential ψinit(x) is also assumed smooth and pe- riodic. This implies that the initial velocity integrates to zero over the period. The case where this assumption is relaxed will be considered later in connection with the Aubry–Mather theory. The numerical experiments of [9] reported here have Fig. 28. Snapshots of the velocity for the unique time-peri- odic solution corresponding to the kicking force g(x) shown in the upper inset; the various graphs correspond to six out- put times equally spaced during one period. The origin of time is taken at a kick. Notice that during each period, two new shocks are born and two mergers occur. (From [9].) been made with the kicking potential G(x) = sin 3x+ cosx, (6.19) and a kicking period T =1. Other experiments were done with (i) G(x) = − cosx and (ii) G(x) = (1/2) cos(2x) − cosx. The former potential produces a single shock and no preshock. As a consequence it displays no −7/2 law in the PDF of gradients. The latter potential gives es- sentially the same results as reported hereafter but has an additional symmetry. To avoid non-generic behaviors that could result from this symmetry, it was chosen to focus on the forcing potential given by (6.19). The number of collocation points chosen for such simu- lations was mostly Nx = 2 17 ≈ 1.31 × 105, with a few simulations done at Nx = 2 20 (for the study of the re- laxation to the periodic régime presented below). Since the numerical method allows going directly to the de- sired output time (from the nearest kicking time) there is no need to specify a numerical time step. However, in order to perform temporal averages, e.g. when calculat- ing PDF’s or structure functions, without missing the most relevant events (which can be sharply localized in time) sufficiently frequent temporal sampling is needed. The total number of output times Nt ≈ 1000, is thus chosen such that the increment between successive out- put times is roughly the two-thirds power of the mesh (this is related to the cubic structure of preshocks, see section 2.3). Figure 28 shows snapshots of the time-periodic solution at various instants. It is seen that shocks are always present (at least two) and that at each period two new shocks are born at t⋆1 ≈ 0.39 and t⋆2 ≈ 0.67. There is one main shock which remains near x = π and which collides space x Main Shock Fig. 29. Evolution of shock positions during one period. The beginnings of lines correspond to births of shocks (preshocks) at times t⋆1 and t⋆2; shock mergers take place at times tc1 and tc2. The “main shock”, which survives for all time, is shown with a thicker line. 0 2 4 6 8 10 12 14 number of kicks sin(x) sin(2x) sin(3x) slope −0.74 Fig. 30. Exponential relaxation to a time-periodic solution for three different initial velocity data as labeled. The hori- zontal axis gives the time elapsed since t = 0. (From [9].) with the newborn shocks at tc1 ≈ 0.44 and tc2 ≈ 0.86. Figure 29 shows the evolution of the positions of shocks during one period. It was found that, for all initial conditions u0(x) used, the solution u(x, t) relaxes exponentially in time to a unique function u∞(x, t) of period 1 in time. Figure 30 shows the variation of |u(x, n−)−u∞(x, 1−)| dx/(2π) for three different initial conditions as a function of the discrete time n. The phenomenon of exponential convergence to a unique space- and time-periodic solution is something quite gen- eral: whenever the kicking potentialG(x) is periodic and analytic and the initial velocity potential is periodic (so that the mean velocity 〈u〉 =0 at all times), there is ex- ponential convergence to a unique piecewise analytic so- lution. This can be proved rigorously (see Appendix to [9]) in the case when the functions G(x) have a unique point of maximum with a non-vanishing second deriva- tive (Morse generic functions). Here, we just explain the main ideas of the proof and give some additional prop- erties of the unique solution. One very elementary property of solutions is that, for any initial condition of zero mean value, the solution after at least one kick satisfies |u(x, t)| ≤ (1/2) + max |dG(x)/dx|. (6.20) Indeed, at a time t = n− just before any kick we have x = y+u(x, n−) where y is the position just after the pre- vious kick of the fluid particle which goes to x at time n−. It follows from the spatial periodicity of the velocity po- tential that the location y which minimizes the action is within less than half a period from x. Thus, |u(x, n−)| ≤ 1/2. The additional maxx |dG(x)/dx| term comes from the maximum change in velocity from one kick. Hence the solution is bounded. Note that if the spatial and tem- poral periods are L and T , respectively, the bound on the velocity becomes L/(2T ) + maxx |dG(x)/dx|. The convergence at large times to a unique solution can be understood in terms of the two-dimensional conserva- tive (area-preserving) dynamical system defined by the Euler–Lagrange map (6.9)-(6.10). By construction, we have u(x, 1+) = û(x) − dG(x)/dx, where û(x) is the so- lution of the unforced Burgers equation at time t = 1− from the initial condition u(x) at time t = 0+. The map u 7→ û(x) + g(x), where g(x) ≡ −dG(x)/dx, here de- notedBg, solves the kicked Burgers equation over a time interval one. The problem is to show that the iterates Bng u0 converge as n→ ∞ to a unique solution. If it were not for the shocks it would suffice to consider the two-dimensional Euler–Lagrange map. Note that, for the case of periodic kicking, this map has an obvi- ous fixed point P , namely (x = xc, v = 0), where xc is the unique point maximizing the kicking potential. It is easily checked that this fixed point is an unstable (hy- perbolic) saddle point of the Euler–Lagrange map with two eigenvalues λ = 1 + c + c2 + 2c and 1/λ, where c = −∂2xxG(xc)/2. Like for any two-dimensional map with a hyperbolic fixed point, there are two curves globally invariant by the map which intersect at the fixed point. The first is the stable manifold Γ(s), i.e. the set of points which con- verge to the fixed point under indefinite iteration of the map; the second is the unstable manifold Γ(u), i.e. the set of points which converge to the fixed point under in- definite iteration of the inverse map, as illustrated in fig- ure 31(a). Any curve which intersects the stable manifold transversally (at the intersection point, the two curves xx xl rc Fig. 31. (a) Sketch of a hyperbolic fixed point P with stable (Γ(s)) and unstable (Γ(u)) manifolds. The dashed line gives the orbit of successive iterates of a point near the stable manifold. (b) Unstable manifold Γ(u) on the (x, v)-cylinder (the x-coordinate is defined modulo 1) which passes through the fixed point P = (xc, 0). The bold line is the graph of u∞(x, 1−). The main shock is located at xl = xr. Another shock at x1 corresponds to a local zig-zag of Γ (u) between A and B. are not tangent to each other) will, after repeated appli- cations of the map, be pushed exponentially against the unstable manifold at a rate determined by the eigenvalue 1/λ. In the language of Burgers dynamics, the curve in the (x, v) plane defined by an initial condition u0(x) will be mapped after time n into a curve very close to the unstable manifold. In fact, for the case studied numeri- cally, 1/λ ≈ 0.18 is within one percent of the value mea- sured from the exponential part of the graph shown in figure 30. Note that if the initial condition u0(x) contains the fixed point, the convergence rate becomes (1/λ) (even higher powers of 1/λ are possible if the initial con- dition is tangent to the unstable manifold). The fixed point P is actually a very simple global min- imizer: (yj = xc, vj = 0) for all positive and negative j’s. It follows indeed by inspection of (6.16) that any deviation from this minimizer can only increase the ac- tion; actually, this trajectory minimizes both the kinetic and the potential part of the action. Note that the cor- z (0)l rz (0) ���������������������������������������������� ���������������������������������������������� ���������������������������������������������� ���������������������������������������������� ���������������������������������������������� z (0)1z c Fig. 32. Minimizers (trajectories of fluid particles) on the (x, t)-cylinder. Time starts at −∞. Shock locations at t = 0− are characterized by the presence of two minimizers (an in- stance is at x1). The main shock is at xl = xr. The fat line x = xc is the global minimizer. responding fluid particle is at rest forever and will never be captured by a shock (it is actually the only particle with this property). It is easy to see that any minimizer is attracted exponentially to such a global minimizer as t→ −∞. Thus, any point (yj , vj) on a minimizer belongs to the unstable manifold Γ(u) and, hence, any regular part of the graph of the limiting solution u∞(x) belongs to the unstable manifold Γ(u). This unstable manifold is analytic but can be quite complicated. It can have sev- eral branches for a given x (see figure 31(b)) and does not by itself define a single-valued function u∞(x). The solution has shocks and is only piecewise analytic. Con- sideration of the minimizers is required to find the po- sition of the shocks in the limiting solution: two points with the same x corresponding to a shock, such as A and B on figure 31(b) should have the same action. Finally, we give the geometric construction of the main shock, the only shock which exists for an infinite time. Since the eigenvalue λ is positive, locally, minimizers which start to the right of xc approach the global min- imizer from the right, and those which start to the left approach it from the left. Take the rightmost and left- most points xr and xl on the periodicity interval such that the corresponding minimizers approach the global minimizer from the right and left respectively (see fig- ure 32). These points are actually identical since there cannot be any gap between them that would have min- imizers approaching the global minimizer neither from the right nor the left. The solution u∞(x) has then its main shock at xl = xr. 6.2 Connections with Aubry–Mather theory In the previous subsection, the study of the solutions to the periodically kicked Burgers equation was limited to initial conditions with a vanishing spatial average b. With a non-vanishing mean velocity b, which in the forced case cannot be eliminated by a Galilean invari- ance, many of the properties of the solutions described above are still valid. However the action now depends on b. Global minimizers {y(g)j , j ∈ Z} exist in this case as well. However generically they are not unique and do not correspond to fixed points of the Euler–Lagrange map (6.9)-(6.10). A global minimizer now minimizes the ac- A∞({yk}) =A({yk}; +∞;−∞) (yk+1−yk−b)2−G(yk+1) . (6.21) This action is exactly the potential energy associated to an infinite chain of atoms linked by elastic springs and embedded in a periodic potential, problem known as the Frenkel–Kontorova model [52]. The parameter b represents the equilibrium length l of the springs and the spatial period L of the external potential (see fig- ure 33) is equal to 1. A global minimizer of (6.21) rep- Fig. 33. Sketch of the Frenkel–Kontorova model for the equi- librium states of an atom chain in a periodic potential. resents an equilibrium configuration of this system. The properties of this equilibrium, or ground states are de- termined by the competition between two tendencies: on the one hand the atoms tend to stabilize at those loca- tions where the potential is minimum; on the other hand, the springs tend to maintain them at a fixed distance of each other. When b = 0 this competition disappears and the equilibrium is given by yk = xc, where xc is the loca- tion at which G attains its global minimum. For b 6= 0, the situation is more delicate and the structure of the ground states involves, as we shall now see, a problem of commensurate-incommensurate transition. The prop- erties of ground states were studied in great details by Aubry [3] and Mather [86]. The relations between the Burgers equation with a time-periodic forcing and Aubry–Mather theory were discussed for the first time in [70] and in [38]. The the- ory was further developed in [36,106]. For integer values of b, the global minimizer is trivially associated to the fixed point (x, v) = (xc, b) of the Euler–Lagrange map (6.9)-(6.10), which corresponds to a fluid trajectory lo- cated at integer times at x = xc and which moves on distance of b spatial periods during one temporal period. A similar argument implies that it is enough to study values of b in the interval [0, 1).To each global minimizer {y(g)j , j ∈ Z} is associated a rotation number defined as ρ ≡ lim j+1 − y , (6.22) which represents the time-average velocity of the mini- mizer. For a fixed value of the spatial average b of the velocity, all global minimizers associated to the solution of the Burgers equation have the same rotation number ρ. Indeed, as the dynamics is restricted to a compact do- main of the configuration space (in our case T), two min- imizers with different rotation numbers necessarily cross each other; this is an obvious obstruction to the action minimization property. In the case of rational rotation numbers the global minimizers correspond to periodic orbits of the dynamical system defined by the Euler– Lagrange map. An important feature is that for rational ρ, the rotation number does not change when varying b over a certain closed interval [bmin, bmax], called the mode-locking interval. On the contrary, irrational ρ cor- respond to a unique value of the parameter b. Such “ir- rational” values of b form a Cantor set of zero Lebesgue measure. In particular, the graph of ρ as a function of the parameter b is a “Devil staircase” (see figure 34). 0 0.2 0.4 0.6 0.8 1 Fig. 34. Rotation number ρ as a function of the spatial mean of the velocity b for the standard map. When ρ is rational (ρ = p/q in irreducible form), global minimizers correspond to a periodic orbit of period q. It is easy to see that such an orbit generates q different but closely related global minimizers. Of course each of these global minimizer is the image of another one by the Euler–Lagrange map and is mapped back to itself after q iterations. This procedure generates a periodic orbit, which turns out to be hyperbolic one. Hence, each of the q global minimizers has a one-dimensional unstable manifold associated to it. The solution to the Burgers equation is formed by branches of these various mani- folds with jumps between them defining q global shocks. The picture is very different for values of b corresponding to irrational rotation numbers. Consider velocities and positions of all global minimizers at a fixed moment of time, say t = 0. They form a subset G of the phase space T × R. Then two cases have to be distinguished: • The set G forms a closed invariant curve for the Euler– Lagrange map. This invariant curve has a one-to-one projection onto the base T and dynamics on the curve is conjugated to a rigid rotation by angle ρ. The lim- iting solution of the Burgers equation is given by the invariant curve and does not contain any shocks. • The set G forms an invariant Cantor set and the limit- ing solution of the Burgers equation contains an infi- nite number of shocks, none of which is a main shock. The Kolmogorov [80], Arnold [2] and Moser [93] the- ory (frequently referred to as KAM) describes invari- ant curves (or invariant tori) for small analytic per- turbations of integrable Hamiltonian systems, and thus the various types of dynamical trajectories. The KAM theory ensures that for sufficiently small perturbations, most of the invariant curves associated to Diophantine irrational rotation numbers are stable with respect to small analytic perturbations of the system. Diophantine irrational numbers possess fast converging approxima- tions by rational numbers (in a suitable technical sense). However, these invariant curves may disappear from the perturbed system when an interaction corresponding to a non-integrable perturbation gets sufficiently strong. Aubry–Mather theory provides another variational de- scription for the KAM invariant curves. But even more importantly, it describes the invariant Cantor sets that appear instead of invariant curves in the case of strong nonlinear interactions. We have mentioned already that these Cantor sets correspond to global minimizers. Thus Aubry–Mather theory provides information about the global minimizers and, hence, allows one to study in such a situation the properties of limiting entropic solutions and, in particular, the structure of shocks. A numerical study of the Burgers equation in the inviscid limit, with periodic forcing and a non-vanishing spatial average of the velocity, reveals the appearance of shock accumulations. Such events occur for the values of the mean velocity b near the end-points of the mode-locking intervals, corresponding to rational rotation numbers. The shock accumulation phenomenon is due to the fact that the end-points bmin, bmax of the mode-locking in- tervals can be approximated by convergent sequences of “irrational” values of the parameter b. This implies ac- cumulation of shocks, since for irrational rotation num- bers the number of shocks is infinite. The limiting solution u∞(x, t) is completely determined by the function û(x) defined in the previous subsection. The function û(x) corresponds to a stroboscopic section of u∞ right after each impulse. The regular parts of û are made of single-valued functions related to the unsta- ble manifolds. The shocks correspond to jumps, either between different branches of the same manifold (sec- ondary shocks), or between the manifolds associated to different global minimizers (main shocks). When the rotation number is rational (ρ = p/q), there are q global minimizers. The positions of the q main shocks of û are determined by a requirement that the area defined by the graph of the solution is equal to the conserved quantity b. The latter constraint shows that the values of b compatible with the rotation number p/q belong to an interval [bmin, bmax] bounded by the mini- mum and maximum areas defined by the unstable man- ifolds, as illustrated in figure 35. The detailed shape of Fig. 35. Sketch of the unstable manifolds of the two global minimizers associated to the rotation number ρ = 1/2. The values bmin and bmax given by this configurations are repre- sented as grey areas. the manifolds can actually not be sketched on a figure. Generically the unstable manifold of a global minimizer corresponding to a particular point of the basic periodic orbit of period q intersects transversally with the stable manifold of another minimizer corresponding to another point of the periodic orbit. Such an intersection leads to formation of a heteroclinic tangle, a notion which can be traced back to the work of Poincaré. The heteroclinic in- tersection results in the formation of an infinite number of zig-zags of the unstable manifolds. These zig-zags are accumulating along the stable manifold and come arbi- trary close to the corresponding point of the periodic or- bit. The zig-zags contract exponentially in one direction (along the stable manifold) and are stretched exponen- tially in the other direction. It is easy to see that the accumulation of zig-zags generates an infinite number of “potential” shocks of smaller and smaller size which also accumulate near the periodic orbit. When the param- eter b is located well inside the mode-locking interval, the position of the main shock cuts off the accumulated shocks of small size so that the total number of shocks is of the order of unity. However, when b gets closer and closer to bmax or bmin, the main shocks move closer to the periodic points and a larger number of the small ac- cumulating shocks appears in the solution. This mecha- nism leads to an infinite number of shocks in the solution when b is equal to bmin or bmax (see figure 36(a)). Both Fig. 36. (a) Accumulations of shocks occurring for b = bmin or b = bmax, due to the presence of an infinite number of loops of the unstable manifold in the homocline or heterocline tangle. (b) Shock accumulation at the fixed point (0, 0) of the standard map. Here, λ = 0.1 and b = 0.15915. The latter value is close to the upper bound of the interval associated to ρ = 0. The upper inset is a zoom near (0, 0), illustrating the accumulation of shocks. the distances between two consecutive shocks and the sizes of the shocks decrease exponentially fast with the number of shocks; the rate is given by the stable eigen- value associated to the hyperbolic periodic orbit. It is interesting to mention that when b = bmin or b = bmax the main shocks merge with the periodic orbit associ- ated to the global minimizers. Hence, for the end-points of the mode-locking interval the main shocks disappear. To illustrate numerically the change in behavior of the solution to the Burgers equation when the mean velocity b changes, we focus here on the simple periodic kicking potential G(x) = (λ/2π) cos(2πx) where λ is a free pa- rameter. The associated Euler–Lagrangemap then reads T : (y, v) 7→(y+v+λ sin(2πy), v+λ sin(2πy)). (6.23) This transformation is usually called the standard map (or Chirikov–Taylormap). It is one of the simplest model for studying the presence of chaos in Hamiltonian dy- Fig. 37. General aspect in position-velocity phase space of the dynamical system defined by the standard map (6.23) for two different values of the parameter (a) λ = 0.1 and (b) λ = 0.3. The corresponding time-periodic solutions to the kicked Burgers equation are represented as bold lines in both cases. The results are presented for the spatial mean velocities b = 0, b = 0.3 and b = 0.5. namical systems and in particular particularly to study the KAM theory. Figure 36(b) illustrates the accumulation of shocks due to the homoclinic or heteroclinic tangling for the first transition (starting from b = 0). This transition cor- responds to a rotation number of the global minimizer changing value from ρ = 0 to ρ > 0. When b is increased and gets close to the critical value, shocks accumulate on the left-hand side of the global minimizer located at (y, v) = (0, 0). Other numerical experiments were performed in order to observe the destruction of invariant curves and the accumulation of shocks on Cantor sets for irrational ro- tation numbers. It is of course impossible numerically to set the rotation number to an irrational value. Indeed, the values of b for which ρ is irrational are in a Cantor set. It is however possible to be very close to irrational rotation numbers. Figure 37 illustrates the changes in the behavior of the solutions to the periodically kicked Burgers equation when varying the parameter λ. The time-asymptotic solutions associated to various values of the mean velocity b are shown for λ = 0.1 and λ = 0.3. For the latter value, all KAM invariant curves have al- ready disappeared. For b = 0 and for all values of λ the global minimizer trivially corresponds to the fixed point (0, 0) with a vanishing rotation number. For b = 0.5 there are two global minimizers associated to the ratio- nal rotation number ρ = 1/2. For λ = 0.1 and b = 0.3 the rotation number is much closer to an irrational than in previous cases. The solution is then very close to the invariant curve associated to this value. Note that the main shock is actually located close to x ≈ 0.85. It is so small that it can hardly be seen. When λ = 0.3 the value b = 0.3 of the mean velocity no more corresponds to a rotation number close to an irrational value; it is now in the mode-locking interval associated to ρ = 1/3. This change in the rotation number reflects the dependence of the mode-locking intervals [bmin, bmax] on the parameter λ. The interval of values of b associated to ρ = 0 is rep- resented as a function of λ in figure 38. Such a structure is frequently called an Arnold tongue (see, e.g., [72]). −1.5 −1 −0.5 0 0.5 1 1.5 Fig. 38. Evolution as a function of the parameter λ of the mode-locking interval [bmin, bmax] associated to the rotation number ρ = 0. Such a graph is frequently referred to as an Arnold tongue. Finally, we discuss the structure of shocks in the case when the global minimizers form a Cantor set. There are then infinitely many gaps with no global minimizers. It is known in this case that all the gaps can be split into the finite number of images of the main gaps. For the standard map there is only one main gap. Its end-points (x1, v1) and (x2, v2) belong to the Cantor set associated to the global minimizers. All other gaps can be obtained by iterating this main gap with the Euler-Lagrange map (Standard map) for both positive and negative times: (x1i , v i ) = T i(x1, v1), (x2i , v2i ) = T i(x2, v2), i ∈ Z. One can show that the length of the ith gap tends to zero as i→ ±∞. Since global minimizers are hyperbolic trajec- tories one can connect the end-points of the main gap by two smooth curves: the stable manifold Γ(s) and the unstable manifold Γ(u). As i→ ∞ the iterates of the sta- ble manifold T iΓ(s) tend to a straight segment connect- ing the i-th gap with end-points at (x1i , v i ) and (x i , v The same is true for iterates of the unstable manifold T iΓ(u) in the limit i → −∞. On the contrary, negative iterates of the stable manifold and positive of the unsta- ble one form exponentially long curves connecting corre- sponding gaps. As usual we are interested in the iterates of the unstable manifold since they appear in the time- periodic solution of the Burgers equation. Such a solu- tion is formed by the iterates of the unstable manifold connecting all the gaps. Note that in the case of large negative i, the unstable manifold is close to a straight segment; hence there are no shocks located inside the corresponding gaps. Conversely, for large positive i, the unstable manifold is exponentially long and possesses large zig-zags. Hence, the solution to the Burgers equa- tion has one or several shocks inside such gaps. Since there are no shocks for gaps with large enough negative i, it follows that all the shocks have a finite age. In other words, the time-periodic solution has no main shocks. At the moment it was not possible to study numeri- cally the strange behavior of the solutions to the Burgers equation corresponding to global minimizers living on Cantor-like sets. Looking for such cases requires a very high spatial resolution in order to minimize the numeri- cal error in the approximation of the solution. Moreover, a large number of values for the parameters b and λ has to be investigated in order to observe such a phenomenon. This would require heavy computer ressources. How- ever, many other aspects of the Aubry–Mather theory for Hamiltonian systems can be studied numerically us- ing the Burgers equation with periodic kicks. For in- stance it could be very useful for analyzing the higher dimensional versions. 7 Velocity statistics in randomly forced Burgers turbulence The universality of small-scale properties in fully devel- oped Navier–Stokes turbulence has frequently been in- vestigated, assuming that a steady state is maintained by an external large-scale forcing. It is generally conjec- tured that the velocity increments have universal sta- tistical properties with respect to such a force. Under- standing this issue in simpler models of turbulence has motivated much work for over ten years. A toy model which has been extensively studied is the passive trans- port of a scalar field by random flows (see, e.g., [46]). Tools borrowed from statistical physics and field theory were used to describe and explain the anomalous scaling laws observed in the scalar spatial distribution. It was shown that the scale invariance symmetry is broken by geometrical constraints on tracer configurations that are statistically conserved by the dynamics. Universality of the intermittent scaling exponents with respect to the forcing was proven for the case where energy is injected at large scales [31,57,103,14]. Issues of universality for the nonlinear Burgers turbu- lence model has also been very much on the focus. The possibility to solve exactly a hydrodynamical problem displaying the same kind of quadratic nonlinearity as Navier–Stokes turbulence constitutes of course the cen- tral motivation. Three independent approaches were published almost simultaneously in 1995 and were at the origin of the growing interest in Burgers turbulence. First, an analogy was made in [22] between forced Burg- ers turbulence and the problem of a directed polymer in a random medium. This analogy was used to show that the shocks appearing in the solution lead to anomalous scaling laws for the structure functions. The strong intermittency could be related to the replica-symmetry- breaking nature of the disordered system associated to Burgers turbulence. This approach is discussed in subsection 7.1. Second, ideas using operator product expansions borrowed from quantum field theory were proposed in [99]. The goal was to close in the inertial range the equations governing the correlations of the ve- locity field in one dimension. This treatment of the dis- sipative anomaly is described in subsection 7.2. It yields a prediction for the probability density function (PDF) of velocity increments and gradients and in particular to a power-law behavior for the PDF of ∂xv at large negative values [99]. However, the value of the exponent of this algebraic tail has been a matter of controversy. An overview of the various works related to this issue is given in subsection 7.3. Finally, the turbulent model of the one-dimensional Burgers equation with a self-similar forcing was proposed in [30] as one of the simplest non- linear hydrodynamical problem displaying multiscaling of the velocity structure function. As stressed in subsec- tion 7.4 this problem is easily tractable numerically and some of the numerical observations can be confirmed by a one-loop renormalization group expansion. In what follows we consider the solutions to the Burg- ers equation with a homogeneous Gaussian random forc- ing that is delta-correlated in time. Namely, the spatio- temporal correlation of the forcing potential is taken to 〈F (~x, t)F (~x′, t′)〉 = B(~x − ~x′) δ(t− t′) . (7.1) The function B contains information on the spatial structure of the forcing. It can be either smooth (i.e. concentrated at large spatial scales) or asymptoti- cally self-similar (i.e. behaving as a power law at small separations). In the former case the solution reaches exponentially fast a statistically stationary régime in any space dimension. The construction of the solution in this régime in terms of global minimizer and main shock is described in detail in section 5. When B does not decrease sufficiently fast at small separations (e.g. B(r) ∼ r2h with h < 1 as r → 0 in one dimension), there is no rigorous proof of the existence of a statistically sta- tionary régime. However we assume in the sequel that such a stationary régime exists in order to perform a statistical analysis of the solutions to Burgers equation. 7.1 Shocks and bifractality – a replica variational ap- proach The replica solution for Burgers turbulence proposed in [22] is based on its analogy with the problem of a di- rected polymer in a random medium. As already stated in the Introduction, the viscous Burgers equation forced by the potential F is equivalent to finding the partition function Z of an elastic string in the quenched spatio- temporal disorder V (~x, t) = F (~x, t)/2ν (remember that t has to be interpreted as the space direction in which the polymer is oriented). This relation is obtained by ap- plying to the velocity potential Ψ the Hopf–Cole trans- formation Z(~x, t)=exp(Ψ(~x, t)/2ν). The solution of the problem can be written in terms of the path integral Z(~x, t) = ~γ(t)=~x exp(−H(~γ)) d[~γ(·)] , with H(~γ) = 1 ∥~̇γ(s) + F (~γ(s), s) ds. (7.2) In the analogy between Burgers turbulence and directed polymers, the polymer temperature is assumed to be unity and its elastic modulus is 1/(2ν). The strength of the potential fluctuations applied to the polymer de- pends on the viscosity and is ∝ ε1/2Lf/(2ν) (where ε is the energy injection rate and Lf is the spatial scale of forcing). In order to calculate the various moments of the velocity field ~v = −∇Ψ, one needs to average the loga- rithm of the partition function Z, a celebrated problem in disordered systems. Bouchaud, Mézard and Parisi proposed in [22] the use of a replica trick in order to estimate the average free energy 〈lnZ〉. The first step is to write the zero-replica limit lnZ = limn→0 (Zn − 1)/n. Then, the moments 〈Zn〉 are used to generate an effective attraction between replicas: they are written as the partition functions of the disorder-averaged Hamiltonian Hn(~γ1, . . . , ~γn) asso- ciated to n replicas of the same system [90] ∥~̇γi(s) B(~γi(s)−~γj(s)) ,(7.3) where B denotes the spatial part of the forcing potential correlation. The next step is to study this problem by a variational approach. The Hamiltonian Hn is replaced by an effective Gaussian quadratic Hamiltonian that can be written as Heff = ~γi(τ)Gij(τ−τ ′)~γj(τ ′)dτdτ ′. (7.4) The kernel Gij is then chosen in such a way that it minimizes the free energy. It is shown in [22] that the optimal Gaussian Hamiltonian is the solution of a sys- tem of equations that can be solved following the ansatz proposed in [89]. When d > 2 this approach singles out two régimes depending on the Reynolds number Re = ε1/3L f /ν. These régimes are separated by the critical value Rec = [2(1−2/d)1−d/2]1/3. When Re < Rec the optimal solution is of the form Gij = G0 δij +G1 and obeys the replica symmetry. In finite-size systems it cor- responds to a linear velocity profile. When Re > Rec the correct solution is given by the one-step replica- symmetry-breaking scheme (see [89]). The off-diagonal elements of Gij are then parameterized with two func- tions depending on whether the indices i and j belong to the same block or to different blocks. Qualitatively, the one-step replica-symmetry-breaking approach amounts to the assumption that the instantaneous velocity po- tential can be written as a weighted sum of Gaussians, leading to an approximation of the velocity field as ~v(~x, t) ≃ α(~x− ~rα) e −Re (fα+‖~x−~rα‖ 2/2L2 −Re (fα+‖~x−~rα‖2/2L2f ) , (7.5) where the fα’s are independent variables with a Poisson distribution of density exp(−f). The ~rα are uniformly and independently distributed in space. In (7.5) the sum over α is running from 1 to a large-enough integer M . The typical shape of the approximation of the veloc- ity field given by (7.5) is represented in figure 39(a) in the two-dimensional case. In the limit of large Reynolds numbers the random velocity field given by (7.5) typi- cally contains cells of width ∝ Lf . The width of a shock separating two cells is of the order of Lf/Re. The replica approximation (7.5) leads to an estimate of the PDF p(∆v, r) of the longitudinal velocity increment ∆v = (~v(~x+ r ~e, t) − ~v(~x, t)) · ~e, where ~e is an arbitrary unitary vector. When Re ≫ 1 and r ≪ Lf this approxi- mation takes the particularly simple asymptotic form p(∆v, r) ≈ δ ∆v − Uf , (7.6) where Uf = Re ν/Lf is the typical velocity associated to the scale Lf and g is a scaling function that is deter- mined explicitly in [22]. This approximation is in agree- ment with the following qualitative picture. With a prob- ability almost equal to one, the two points ~x and ~x+ r ~e lie in the same cell; the velocity increment is then given by the typical velocity gradient which, according to the Fig. 39. (a) Typical shape of the velocity field given by the replica approximation in dimension d = 2 obtained from (7.5) for Re = 103. The contour lines represent the velocity modulus. Note the cell structure of the domain. (b) Scaling exponents of the pth order structure function. approximation (7.5), is order Uf/Lf . With a probabil- ity r/Lf the two points are sitting on different sides of a shock separating two such cells and the associated ve- locity difference is of the order of Uf . The structure functions of the velocity field given by the various moments of ∆v can be straightforwardly estimated from the approximation (7.6). Their scaling behavior 〈∆vp〉 ∼ rζp at small separations r display a bifractal behavior as sketched in figure 39(b). When p < 1, the first term on the right-hand side of (7.6) domi- nates and 〈∆vp〉 ∝ Upf (r/Lf )p. For p > 1 the shock con- tribution is dominating the small-r behavior and thus 〈∆vp〉 ∝ Upf (r/Lf ). This approach, which makes use of replica tricks, is as we have seen able to catch the leading scaling behavior of velocity structure functions in any dimension. It is based on approximations of the velocity field by the superposi- tion (7.5) of Gaussian velocity potentials. A first advan- tage of this method is that it catches the generic aspect of the solution including the hierarchy of high-order sin- gularities appearing in the solution when Re → ∞ which was examined in section 2.3. This method also gives predictions regarding the dependence on Re of the sta- tistical properties of the solution. However, as stressed in [22], the validity of this approximation is expected to hold only in the limit of infinite space dimension d. In particular, it is known that for d ≤ 2 a full continuous replica-symmetry-breaking scheme is needed [89]. Nev- ertheless, as we have seen, there is enough evidence that this approach describes very well the qualitative aspects of the solution. 7.2 Dissipative anomaly and operator product expan- The replica-trick approach described in the previous subsection cannot reproduce one of the main statis- tical features of the solution, namely the tails of the velocity increments PDF p(∆v, r). Indeed the predic- tion (7.6) based on a variational approximation of the velocity field implies that p identically vanishes when ∆v > Uf (r/Lf ). In order to study the quantitative be- havior of the PDF p(∆v, r) in the inviscid limit ν → 0 (or equivalently Re → ∞), Polyakov [99] proposed to use an operator product expansion. This approach leads to an explicit expression for p(∆v, r) and predicts a super-exponential tail at large positive values and a power-law behavior for negative ones. Such predictions have immediate implications for the asymptotics of the PDF p(ξ) of the velocity gradient ξ = ∂xv. The work of Polyakov was the starting point of a controversy on the value of the exponent of the left tail of p(ξ). Before returning to this issue in the next subsection, we give in the sequel a quick overview of the original work by Polyakov. We henceforth focus on the one-dimensional solutions to the Burgers equation with Gaussian forcing whose autocorrelation is given by (7.1). Following [99] (see also [19,20]) we introduce the characteristic function of the n-point velocity distribution Zn(λj , xj ; t) ≡ e λ1 v(x1,t)+···+λn v(xn,t) . (7.7) For a finite value of the viscosity ν, it is easily seen that this quantity is a solution to a Fokker–Planck (master) equation obtained by differentiating Zn with respect to t and using the Burgers equation and the fact that the forcing is Gaussian and δ-correlated in time. This leads b(xi − xj)λi λj Zn + D(n)ν , (7.8) where b ≡ (d2B)/(dr2) denotes the spatial part of the correlation of the forcing applied to the velocity field. D(n)ν denotes the contribution of the dissipative term and reads D(n)ν ≡ ν v(xj , t) λj v(xj ,t) . (7.9) This term does not vanish in the limit ν → 0 since the solutions to the Burgers equation develop singularities with a finite dissipation. It has been proposed in [99] to use an analogy with the anomalies appearing in quantum field theory in order to tackle this term in the inviscid limit. The important assumption is then made that the singular term in the operator product expansion relates linearly to the characteristic function Zn. Since this ex- pansion should preserve the statistical symmetries of the Burgers equation, it leads to the replacement in all av- erages of the singular limit limν→0 ν λ (∂ xv) e λ v by the asymptotic expression b − 1 eλ v , (7.10) where the coefficients a, b and c are parameters that can be determined only indirectly. However their possible values can be restricted by requiring that Zn is the char- acteristic function of a probability distribution which is non-negative, finite, normalizable, and that the dissipa- tive term D(n)ν acts as a positive operator. Finding these coefficients is similar to an eigenvalue problem in quan- tum mechanics. We now come to a crucial point in Polyakov’s approach. Important restrictions on the form of the different anomalous terms in (7.10) result from the fact that the solutions to the Burgers equation obey a certain form of Galilean invariance. A notion of “strong Galilean prin- ciple” is introduced for invariance of the n-point distri- bution of velocity under the transformation v 7→ v + v0 with v0 an arbitrary constant. As a consequence, the n- point characteristic function Zn has to be proportional to δ(λ1 + · · ·+λn). The operators appearing in the limit ν → 0 have to be consistent with such an invariance. In [99] it is argued that this symmetry is automatically broken by the forcing that introduces a typical velocity 〈v2〉1/2 ∝ b1/3(0)L1/3. However Polyakov assumes this “strong Galilean principle” to be asymptotically recov- ered in the limit L → ∞ of infinite-size systems. In the case of finite-size systems, when L is of the order of the correlation length Lf of the forcing, the strong Galilean symmetry is broken because of the conservation of the spatial average of v which introduces a characteristic velocity v0 = (1/L) v(x, t) dx. However, the Galilean symmetry should be recovered when averaging the cor- relation functions with respect to the mean velocity v0. This symmetry restoration was introduced in [20] where it is referred to as the “weak Galilean principle”. The n-point characteristic function associated to an aver- age velocity v0 relates to that associated to a vanishing mean velocity by Zn(λj , xj ; t; v0) = e Zn(λj , xj ; t; 0) . After averaging with respect to v0, one obtains Zn(λj , xj ; t) = 2π δ  Zn(λj , xj ; t; 0) . (7.11) One can easily check that (7.8), together with the dis- sipative term given by (7.10), are compatible with this expression for the n-point characteristic function Zn. Moreover, any higher-order term in the expansion (7.10) of the dissipative anomaly would violate Galilean invari- ance. To obtain the statistical properties of the solution, one needs to further restrict the values of the three free pa- rameters a, b, and c appearing in the expansion (7.10). Following [99] this can be done by considering the case n = 2 that corresponds to the equation for the PDF of velocity differences. Performing the change of variables λ1,2 = Λ ± µ and x1,2 = X ± y/2, and assuming that λ ≪ µ and y ≪ Lf (so that the spatial part of the forcing correlation is to leading order b(y) ≃ b0 − b1y2), the stationary and space-homogeneous solutions to the master equation (7.8)) satisfy − (2b0Λ2 + b1µ2y2)Z2 = = aZ2 + . (7.12) It is next assumed in [99] (see also [20]) that the velocity difference v(x1, t) − v(x2, t) is statistically independent of the mean velocity (v(x1, t) + v(x2, t))/2. This implies that the two-point characteristic function factorizes as Z2 = Z 2 (Λ)Z 2 (µ, y), where the two functions Z 2 and Z−2 satisfy the closed equations − 2b0Λ2Z+2 = cΛ , (7.13) ∂2Z−2 − b1µ2y2Z−2 = aZ . (7.14) The solution to the first equation corresponds to a Gaus- sian distribution which is normalizable only if c < 0. As shown numerically in [20] this distribution is representa- tive of the bulk of the one-point velocity PDF. Informa- tion on the solutions to the second equation can be ob- tained assuming the scaling property Z−2 (µ, y) = Φ(µy), which amounts to considering only those contributions to the distribution of velocity differences stemming from velocity gradients ξ = ∂xv. This yields a prediction the negative and positive tails of the PDF of velocity gradi- ents: p(ξ) ∝ |ξ|−α when ξ → −∞ , (7.15) p(ξ) ∝ ξβ exp(−C ξ3) when ξ → +∞ , (7.16) where C is a constant, which depends only on the strength of the forcing. The two exponents α and β are related to the coefficient b of the anomaly by α = 2b + 1 and β = 2b − 1 . (7.17) The value of b remains undetermined but is prescribed to belong to a certain range. This approach was first designed in [99] for infinite-size systems where strong Galilean invariance holds. In that case consistency with such an invariance leads to dropping the third term in the operator product expansion (i.e. c = 0). Positivity and normalizability of the two-point velocity PDF and non-positivity of the anomalous dissipation operator im- ply that the two other coefficients form a one-parameter family with 3/4 ≤ b ≤ 1. In particular, this implies that the left tail of the velocity gradient PDF with ex- ponent α should be shallower than ξ−3. As we will see in the next section, strong evidence has been obtained that p(ξ) ∝ ξ−7/2 for ξ → −∞. This seems to contradict the approach based on operator product expansion. How- ever, as argued in [20], the breaking of strong Galilean in- variance occurring in finite-size systems and resulting in the presence of the c anomaly broadens the range of ad- missible values for b. In particular it allows for the value b = 5/4 which corresponds to the exponent α = 7/2. 7.3 Tails of the velocity gradient PDF After the numerical work of Chekhlov and Yakhot [29], the asymptotic behavior at large positive and negative values of the PDF of velocity derivatives ξ = ∂xv for the one-dimensional randomly forced Burgers equation attracted much attention. A broad consensus emerged around the prediction of Polyakov [99] that p(ξ) dis- plays tails of the form (7.16) and (7.15), but the values of the exponents α and β were at the center of a contro- versy. Note that the presence of a super-exponential tail ∝ exp(−C ξ3) at large positive arguments has been con- firmed by the use of instanton techniques [60] and that the only remaining uncertainty concerns the exponent of the algebraic prefactor. A standard approach to de- termine the exponents α and β appearing in (7.15) and (7.16) makes use of the stationary solutions to the in- viscid limit of the Fokker–Planck equation for the PDF, namely −ξp+ν∂ξ ∂3xv | ∂xv=ξ = b̃∂2ξp . (7.18) Here the brackets 〈·|·〉 denote conditional averages and the right-hand side expresses the diffusion of probability due to the delta-correlation in time of the forcing. The main difficulty in studying the solutions of (7.18) stems from the treatment of the dissipative term Dν(ξ) = ∂3xv|∂xv=ξ in the limit ν → 0. The value α = 3 is obtained if a piecewise linear approximation is made for the solutions of the Burgers equation [21]. Gotoh and Kraichnan [59] argued that the dissipative term is to leading order negligible and presented analytical and nu- merical arguments in favor of α = 3 and β = 1. However, the inviscid limit of (7.18) contains anomalies due to the singular behavior of Dν(ξ) in the limit ν → 0. As we have seen in previous section, the approach based on the use of an operator product expansion [99] leads to a rela- tion involving unknown coefficients which must be deter- mined, e.g., from numerical simulations [111,19,20], and restricts the possible values to 5/2 ≤ α ≤ 3 [6]. Anoma- lies cannot be studied without a complete description of the singularities of the solutions, such as shocks, and a thorough understanding of their statistical properties. E, Khanin, Mazel and Sinai made a crucial observa- tion in [37] that large negative gradients stem mainly from preshocks, that is the cubic-root singularities in the velocity preceding the formation of shocks (see sec- tion 2.3). They then used a simple argument for de- termining the fraction of space-time where the veloc- ity gradient is less than some large negative value. This leads to α = 7/2, provided preshocks do not cluster. Later on, this approach has been refined by E and Van- den Eijnden who proposed to determine the dissipative anomaly of (7.18) using formal matched asymptotics [39] or bounded variation calculus [42]. As we shall see be- low, with the assumption that shocks are born with a zero amplitude, that their strengths add up during colli- sions, and that there ar no accumulations of preshocks, the value α = 7/2 was confirmed [42]. Other attempts to derive this value using also isolated preshocks have been made [81,6]. Note that there are simpler instances, in- cluding time-periodic forcing [9] (see section 6) and de- caying Burgers turbulence with smooth random initial conditions [8,42] (see section 4.1), which fall in the uni- versality class α = 7/2, as can be shown by systematic asymptotic expansions using a Lagrangian approach. We give here the flavor of the approach used in [39] in order to estimate the dissipative anomaly D0(ξ) = limν→0D ν(ξ). One first notices that for |ξ| ≫ b̃1/3, the forcing term in the right-hand side of (7.18) becomes negligible, so that stationary solutions to the Fokker– Planck equation satisfy p(ξ) ≈ |ξ|−3 dξ′ ξ′Dν(ξ′). (7.19) A straightforward consequence of this asymptotic ex- pression is that, if the integral in the right-hand side de- creases as ξ → −∞ (i.e. if ξDν(ξ) is integrable), then p(ξ) decreases faster than |ξ|−3, and thus α > 3. To get some insight into the behavior ofDν as ν → 0, one next observes that the solutions to the one-dimensional Burgers equation contain smooth regions where viscos- ity is negligible, which are separated by thin shock layers where dissipation takes place. The basic idea consists in splitting the velocity field v into the sum of an outer so- lution away from shocks and of an inner solution near them for which boundary layer theory applies. Matched asymptotics are then used to construct a uniform ap- proximation of v. To construct the inner solution near a shock centered at x = x⋆, one performs the change of variable x 7→ x̃ = (x−x⋆)/ν and looks for an expression of ṽ(x̃, t) = v(x⋆+νx̃, t) in the form of a Taylor expan- sion in powers of ν: ṽ = ṽ0+νṽ1+o(ν). At leading order, the inner solution satisfies [ṽ0 − v⋆] ∂x̃ṽ0 = ∂2x̃ṽ0, (7.20) where v⋆ = (dx⋆)/(dt). This expression leads to the well- known hyperbolic tangent velocity profile ṽ0 = v⋆ − . (7.21) Here, s = v(x⋆+, t)−v(x⋆−, t) denotes here the velocity jump across the shock and is given by matching condi- tions to the outer solution. The term of order ν is then a solution of ∂tṽ0 + [ṽ0 − v⋆] ∂x̃ṽ1 = ∂2x̃ṽ1 + f(x, t). (7.22) In order to evaluate the dissipative anomaly, it is con- venient to assume spatial ergodicity so that the viscous term in (7.18) can be written as Dν(ξ) = ν∂ξ lim dx ∂3xv δ(∂xv−ξ). (7.23) In the limit ν → 0 the only remaining contribution stems from shocks and is thus given by the inner solution. Using the expansion of the solution up to the first order in ν, this leads to writing the dissipative term in the limit of vanishing viscosity as (see Appendix of [41] for details) D0(ξ) = ds s [p+(s, ξ) + p−(s, ξ)] , (7.24) where ρ is the density of shocks and p+ (respectively p−) is the joint probability of the shock jump and of the value of the velocity gradient at the right (respectively left) of the shock. This expression guarantees the finiteness of the dissipative anomaly, and in particular the fact that the integral in the right-hand side of (7.19) is finite in the limit ν → 0 and converges to 0. As a consequence, this gives a proof that the exponent α of the left tail of the velocity gradient PDF is larger than 3. To proceed further, E and Vanden Eijnden proposed to estimate the probability densities p+ and p− by deriving master equations for the joint probability of the shock strength s, its velocity v⋆ and the values ξ ± of the veloc- ity gradient at its left and at its right. This is done in [42] using a formulation of Burgers dynamics stemming from bounded variation calculus. More precisely, it is shown in [108] that the Burgers equation is equivalent to con- sidering the solutions to the partial differential equation ∂tv + v̄∂xv = f , (7.25) where v̄(x, t) = (v(x+, t) + v(x−, t))/2. Basically this means that Burgers dynamics can be formulated in terms of the transport of the velocity field by its average v̄. This formulation straightforwardly yields a master equation for v(x±, t) and ∂xv(x±, t) which is then used to estimate p± and the dissipative anomaly (7.24). Al- though the treatment of the master equation does not involve any closure hypothesis, it is not fully rigorous: in particular it requires the assumption that shocks are created with zero amplitude and that shock amplitudes add up during collision. However such an approaches strongly suggests that α = 7/2 and β = 1. Obtaining numerically clean scaling for the PDF of gra- dients is not easy with standard schemes. Let us ob- serve that any method involving a small viscosity, either introduced explicitly (e.g. in a spectral calculation) or stemming from discretization (e.g. in a finite difference calculation), may lead to the presence of a power-law range with exponent −1 at very large negative gradi- ents [59]. This behavior makes the inviscid |ξ|−α range appear shallower than it actually is, unless extremely high spatial resolution is used. In contrast, methods that directly capture the inviscid limit with the appropriate shock conditions, such as the fast Legendre transform method [94], lead to delicate interpolation problems. They have been overcome in the case of time-periodic forcing [9] but with white-noise-in-time forcing, it is dif- ficult to prevent spurious accumulations of preshocks leading to α = 3. To avoid such pitfalls, a Lagrangian particle and shock tracking method was developed in [6]. This method is able to separate shocks and smooth parts of the solution and is particularly effective for identifying preshocks. The main idea is to consider the evolution of a set of N massless point particles accelerated by a discrete-in- time approximation of the forcing with a uniform time step. When two of these particles intersect, they merge and create a new type of particle, a shock, characterized by its velocity (half sum of the right and left velocities of merging particles) and its amplitude. The particle- like shocks then evolve as ordinary particles, capture further intersecting particles and may merge with other shocks. In order not to run out of particles too quickly, the initial small region where particles have the least chance of being subsequently captured is determined by localization of the global minimizer of the Lagrangian action (see section 5.1). The calculation is then restarted from t = 0 for the same realization of forcing but with a vastly increased number of particles in that region. This particle and shock tracking method gives complete control over shocks and preshocks. slope = −7/2 −3.75 −3.25 −7/2±1% Fig. 40. PDF of the velocity gradient at negative values in log-log coordinates obtained by averaging over 20 realiza- tions and a time interval of 5 units of time (after relaxation of transients). The simulation involves up to N = 105 parti- cles and the forcing is applied at discrete times separated by δt = 10−4. Upper inset: local scaling exponent (from [6]). Figure 40 shows the PDF of the velocity gradients in log-log coordinates at negative values, for a Gaussian forcing restricted to the first three Fourier modes with equal variances such that the large-scale turnover time is order unity. Quantitative information about the value of the exponent is obtained by measuring the “local scaling exponent”, i.e. the logarithmic derivative of the PDF calculated in this case using least-square fits on half- decades. It is seen that over about five decades, the local exponent is within less than 1% of the value α = 7/2 predicted by E et al. [37]. 7.4 Self-similar forcing and multiscaling As we have seen in section 7.1, the solutions to the Burg- ers equation in a finite domain and with a large-scale forcing have structure functions (moments of the ve- locity increment) displaying a bifractal scaling behav- ior. Such a property can be easily interpreted by the presence of a finite number of shocks with a size order unity in the finite system. Somehow this double scal- ing and its relationship with singularities gives some in- sight on the multiscaling properties that are expected in the case of turbulent incompressible hydrodynamics flows. There is a general consensus that the turbulent solutions to the Navier–Stokes equations display a full multifractal spectrum of singularities which are respon- sible for a nonlinear p-dependence of the scaling expo- nents ζp associated to the scaling behavior of the p-th order structure function [53]. The construction of simple tractable models which are able to reproduce such a be- havior has motivated much work during the last decades. Significant progress, both analytical and numerical, has been made in confirming multiscaling in passive-scalar and passive-vector problems (see, e.g., [46] for a review). However, the linearity of the passive-scalar and passive- vector equations is a crucial ingredient of these studies, so it is not clear how they can be generalized to fluid turbulence and the Navier–Stokes equation. After the work of Chekhlov and Yakhot [30], it appeared that the Burgers equation with self-similar forcing could be the simplest nonlinear partial differential equation which has the potential to display multiscaling of veloc- ity structure functions. We report in this section various works that tried to confirm or to weaken this statement. Let us consider the solutions to the one-dimensional Burgers equation with a forcing term f(x, t) which is ran- dom, space-periodic, Gaussian and whose spatial Fourier transform has correlation 〈f̂(k, t)f̂(k′, t′)〉 = 2D0 |k|β δ(t− t′) δ(k + k′). (7.26) The exponent β determines the scaling properties of the forcing. When β > 0 the force acts at small scales; for instance β = 2 corresponds to thermal noise for the ve- locity potential, and thus to the KPZ model for interface growth [74]. It is well known in this case (see, e.g., [5]) that the solution displays simple scaling (usually known as KPZ scaling), such that ζq = q for all q. More gener- ally, the case β > 0 can be exactly solved using a one- loop renormalization group approach [88]. As stressed in [64], renormalization group techniques fail when β < 0 and the forcing acts mostly at large scales and non-linear terms play a crucial role. When β < −3, the forcing is differentiable in the space variable, the so- lution is piecewise smooth and contains a finite number of shocks with sizes order unity. The scaling exponents are then ζp = min (1, p). In the case of non-differentiable forcing (−3 < β < 0), the presence of order-unity shocks and dimensional arguments suggest that the scaling ex- ponents are ζp = min (1, −pβ/3). However, very little is known regarding the distribution of shocks with inter- mediate sizes. In particular, there is no clear evidence whether or not they form a self-similar structure at small scales. We summarize here some studies which were done on Burgers turbulence with self-similar forcing to show how difficult it might be to measure scaling laws of struc- ture functions and in particular how logarithmic correc- tions can masquerade anomalous scaling. For this we focus on the case β = −1 which has attracted much attention; indeed, dimensional analysis suggests that ζp = p/3 when p ≤ 3, leading to a K41-type −5/3 energy spectrum. Early studies [29,30] seemed to confirm this prediction using pseudo-spectral viscous numerical Fig. 41. Representative snapshots of the velocity v (jagged line) in the statistically stationary régime, and of the inte- gral of the force f over a time step (rescaled for plotting purposes). simulations at rather low resolutions (around ten thou- sands gridpoints). It was moreover argued in [64,65] that a self-similar forcing with −1 < β < 0, could lead to gen- uine multifractality. The lack of accuracy in the determi- nation of the scaling exponents left open the question of a weak anomalous deviation from the dimensional pre- diction. This question was recently revisited in [91] with high-resolution inviscid numerical simulations using the fast Legendre transform algorithm (see section 2.4.2). A typical snapshot of the forcing and of the solution in the stationary régime are represented in figure 41. It is clear that because of shocks the velocity develops small- scale fluctuations much stronger than those present in the force. However one notices that shock dynamics and spatial finiteness of the system lead, as predicted, to the presence of few shocks with order-unity sizes. 0 1 2 3 4 5 ξ p 0 1 2 3 4 5 Fig. 42. Scaling exponents ζp versus order p for N = 216(⋄), 218(∗), and 220(◦) grid points. Error bars (see text) are shown for the case N = 220. The deviation of ζp from the exponents for bifractal scaling (full lines), shown as an inset, naively suggests multiscaling (from [91]) Structure functions were measured with high accuracy. They typically exhibit a power-law behavior over nearly three decades in length scale; this is more than two decades better than in [30]. In principle one expects to be able to measure the scaling exponents with enough accuracy to decide between bifractality and multiscal- ing. Surprisingly the naive analysis summarized in fig- ure 42 does suggest multiscaling: the exponents ζp de- viate significantly from the bifractal-scaling prediction (full lines). Since the goal here is to have a precise han- dle on the scaling properties of velocity increments, it is important to carefully define how the scaling exponents are measured. They are estimated from the average log- arithmic derivative of Sabsp (r) = 〈|v(x+r)−v(x)|p〉 over almost two decades in the separation r. The error bars shown are given by the maximum and minimum devia- tions from this mean value in the fitting range. Note also that the observed multiscaling is supported by the fact that there is no substantial change in the value of the exponents when changing the number N of grid points in the simulation from 216 to 220: any dependence of ζp upon N is much less than the error bars determined through the procedure described above. −5 −4 −3 −2 −1 −6 −4 −2 Fig. 43. Log-log plots of Sabs3 (r) (dashed line), S3(r) (crosses), and 〈(δ+v)3〉 (squares) versus r. The continuous line is a least-square fit to the range of points limited by two vertical dashed lines in the plot. Inset: An explicit check of the von Kármán–Howarth relation (7.27) from the simula- tions with N = 220 reported in [91]. The dashed curve is the integral of the spatial part of the forcing correlation and the circles represent the numerical computation of the left-hand side. As found in [91], the observed deviations of the scaling exponents from bifractality are actually due to the con- tamination by subleading terms in Sabsp (r). To quantify this effect, let us focus on the third-order structure func- tion (p = 3) for which one measures ζ3 ≈ 0.85±0.02 over nearly four decades (see figure 43). To estimate sublead- ing terms we first notice that the third-order structure function S3(r) ≡ 〈(v(x + r) − v(x))3〉, which is defined, this time, without the absolute value, obeys an analog of the von Kármán–Howarth relation in fluid turbulence, namely S3(r) = b(r′)dr′, (7.27) where b(·) denotes the spatial part of the force corre- lation function, i.e. 〈f(x + r, t′)f(x, t)〉 = b(r)δ(t − t′). This relation, together with the correlation (7.26) and β = −1, implies the behavior S3(r) ∼ r ln r at small separations r. As seen in figure 43, the graph of S3(r) in log-log coordinates indeed displays a significant cur- vature which is a signature of logarithmic corrections. The next step consists in decomposing the velocity in- crements δrv = v(x + r, t) − v(x, t) into their positive δ+r v and negative δ r v parts. It is clear that Sabs3 (r) = − (δ−r v) (δ+r v) S3(r) = (δ−r v) (δ+r v) , (7.28) so that (δ+r v) = (Sabs3 (r) + S3(r))/2. As seen in fig- ure 43 the log-log plot of (δ+r v) as a function of r is nearly a straight line with slope ≈ 1.07 very close to unity. This observation is confirmed in [91] by indepen- dently measuring the PDFs of positive and negative ve- locity increments. Assuming that (δ+r v) ∼ B r, one obtains the following prediction for the small-r behav- iors of the third-order structure functions Sabs3 (r) ∼ −Ar ln r +B r, S3(r) ∼ Ar ln r +B r. (7.29) This suggests that the only difference in the small- separation behaviors of Sabs3 (r) and S3(r) is the sign in the balance between the leading term ∝ r ln r and the subleading term ∝ r. In a log-log plot this difference amounts to shifting the graph away from where it is most curved and thus makes it straighter, albeit with a (local) slope which is not unity. This explains why signif- icant deviations from 1 are observed for ζ3. Note that a similar approach can be used for higher-order structure functions. It leads for instance to S4(r) ≈ Cr −Dr4/3, where C and D are two positive constants. The nega- tive sign before the sub-leading term (r4/3) is crucial. It implies that, for any finite r, a naive power-law fit to S4 can yield a scaling exponent less than unity. The presence of sub-leading, power-law terms with oppo- site signs also explains the small apparent “anomalous” scaling behavior observed for other values of p in the simulations. Note that similar artifacts involving two competing power-laws have been described in [16,7]. The work reported in this section indicates that a naive interpretation of numerical measurements might result in predicting artificial anomalous scaling laws. In the case of Burgers turbulence for which high-resolution nu- merics are available and statistical convergence of the averages can be guaranteed, we have seen that it is not too difficult to identify the numerical artifacts which are responsible for such a masquerading. However this is not always the case. For instance, it seems reason- able enough to claim that attacking the problem of mul- tiscaling in spatially extended nonlinear systems, such as Navier–Stokes turbulence, requires considerable the- oretical insight that must supplement sophisticated and heavy numerical simulations and experiments. Note fi- nally that, up to now, the question of the presence or not of anomalous scaling laws in the Burgers equation with a self-similar forcing with exponent −1 < β < 0 remains largely open. 8 Concluding remarks and open questions This review summarizes recent work connected with the Burgers equation. Originally this model was introduced as a simplification of the Navier–Stokes equation with the hope of shedding some light on issues such as turbu- lence. This hope did not materialize. Nevertheless many of the interesting questions that have been addressed for Burgers turbulence are eventually transpositions of similar questions for Navier–Stokes turbulence. One par- ticularly important instance is the issue of universality with respect to the form of the forcing and of the initial condition. For Burgers turbulence most of the universal features, such as scaling exponents or functional forms of PDF tails are dominated by the presence of shocks and other singularities in the solution. This applies both to the case of decaying turbulence driven by random ini- tial conditions and randomly forced turbulence. In the latter case one is mostly interested in analysis of station- ary properties of solutions, for example stationary dis- tribution for velocity increments or gradients. Another set of questions is motivated by more mathematical con- siderations. It mainly concerns the construction of a sta- tionary invariant measure when Burgers dynamics in a finite-size domain is supplemented by an external ran- dom source of energy. Again it has been shown that the presence of shocks, and in particular of global shocks, plays a crucial role in the construction of the statisti- cally stationary solution. Both physical and mathemat- ical questions lead to a similar answer: one first needs to describe and control shocks. The main message to retain for hydrodynamical turbulence is hence a strong confir- mation of the common wisdom that it cannot be fully understood without a detailed description of singulari- ties. Moreover, the behavior depends not only on the lo- cal structure of singularities, but also on their distribu- tion at larger scales. Here a word of caution: for incom- pressible fully developped Navier–Stokes turbulence, we have no evidence that the universal scaling properties observed in experiments and simulations stem from real singularities. Indeed the issue of a finite-time blow-up of the three-dimensional Euler equation is still open (see, e.g. [56]). Another important observation that can be drawn from the study of Burgers turbulence is that both the tools used and the answers obtained strongly de- pend on the kind of setting one considers: decay versus forced turbulence, finite-size versus infinite-size systems, smooth versus self-similar forcing, etc. Besides turbulence, the random Burgers equation has various applications in cosmology, in non-equilibrium statistical physics and in disordered media. Among them, the connection to the problem of directed poly- mers has attracted much attention. As already noted in the Introduction, there is a mathematical equivalence between the zero-viscosity limit of the forced Burgers equation and the zero-temperature limit for directed polymers. We have seen in section 5.4 that the so-called KPZ scaling, which usually is derived for a finite tem- perature, can be established can be established also in the zero-temperatur limit, using the action minimizer representation. Such an observation leads to two related questions: to what extent can the limit of zero temper- ature give an insight into finite-temperature polymer dynamics and how can the global minimizer formalism be extended to tackle the finite-temperature setting? It looks plausible that in polymer dynamics, or more generally in the study of random walks in a random potential, the trajectories carrying most of the Gibbs probability weight are defining corridors in space time. These objects can concentrate near the trajectories of global minimizers but, at the moment, there is no for- malism to describe them, nor attempts to quantify their contribution to the Gibbs statistics. Another important open question concerns the multi- dimensional extensions of the Burgers equation. As we have seen, when the forcing is potential, the potential character of the velocity field is conserved by the dynam- ics. This leads to the construction of stationary solutions which carry many similarities with the one-dimensional case. Up to now there is only limited understanding of what happens when the potentiality assumption of the flow is dropped. This problem has of course concrete applications in gas dynamics and for disperse inelastic granular media (see, e.g., [12]). An interesting question concerns the construction of the limit of vanishing vis- cosity, given that the Hopf–Cole transformation is in- applicable in the non-potential case. Understanding ex- tensions of the viscous limiting procedure to the non- potential case might give new insight into the problem of the large Reynolds number limit in incompressible tur- bulence. Another question related to non-potential flow concerns the interactions between vorticity and shocks. For instance, in two dimensions the vorticity is trans- ported by the flow. This results in its growth in the highly compressible regions of the flow. The various singulari- ties of the velocity field should hence be strongly affected by the flow rotation and, in particular, the shocks are expected to have a spiraling structure. We finish with few remarks on open mathematical prob- lems. As we have seen in the one-dimensional case, one can rigorously prove hyperbolicity of the global mini- mizer. In the multi-dimensional case it is also possible to establish the existence and, in many cases, unique- ness of the global minimizer. However, the very impor- tant question of its hyperbolicity is still an open prob- lem. If proven, hyperbolicity would allow for rigorous analysis of the regularity properties of the stationary so- lutions and of the topological shocks. There are many interesting problems – even basic issues of existence and uniqueness – in the non-compact case where at present a mathematical theory is basically absent. 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From interface dynamics to cosmology The Burgers equation in statistical mechanics The adhesion model in cosmology A benchmark for hydrodynamical turbulence Basic tools Inviscid limit and variational principle Variational principle for the viscous case Singularities of Burgers turbulence Remarks on numerical methods Decaying Burgers turbulence Geometrical constructions of the solution Kida's law for energy decay Brownian initial velocities Transport of mass in the Burgers/adhesion model Mass density and singularities Evolution of matter inside shocks Connections with convex optimization problems Forced Burgers turbulence Stationary régime and global minimizer Topological shocks Hyperbolicity of the global minimizer The case of extended systems Time-periodic forcing Kicked Burgers turbulence Connections with Aubry--Mather theory Velocity statistics in randomly forced Burgers turbulence Shocks and bifractality -- a replica variational approach Dissipative anomaly and operator product expansion Tails of the velocity gradient PDF Self-similar forcing and multiscaling Concluding remarks and open questions Acknowledgements References

The last decades witnessed a renewal of interest in the Burgers equation. Much activities focused on extensions of the original one-dimensional pressureless model introduced in the thirties by the Dutch scientist J.M. Burgers, and more precisely on the problem of Burgers turbulence, that is the study of the solutions to the one- or multi-dimensional Burgers equation with random initial conditions or random forcing. Such work was frequently motivated by new emerging applications of Burgers model to statistical physics, cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the simplest instances of a nonlinear system out of equilibrium. The study of random Lagrangian systems, of stochastic partial differential equations and their invariant measures, the theory of dynamical systems, the applications of field theory to the understanding of dissipative anomalies and of multiscaling in hydrodynamic turbulence have benefited significantly from progress in Burgers turbulence. The aim of this review is to give a unified view of selected work stemming from these rather diverse disciplines.

Introduction, the viscous Burgers equation forced by the potential F is equivalent to finding the partition function Z of an elastic string in the quenched spatio- temporal disorder V (~x, t) = F (~x, t)/2ν (remember that t has to be interpreted as the space direction in which the polymer is oriented). This relation is obtained by ap- plying to the velocity potential Ψ the Hopf–Cole trans- formation Z(~x, t)=exp(Ψ(~x, t)/2ν). The solution of the problem can be written in terms of the path integral Z(~x, t) = ~γ(t)=~x exp(−H(~γ)) d[~γ(·)] , with H(~γ) = 1 ∥~̇γ(s) + F (~γ(s), s) ds. (7.2) In the analogy between Burgers turbulence and directed polymers, the polymer temperature is assumed to be unity and its elastic modulus is 1/(2ν). The strength of the potential fluctuations applied to the polymer de- pends on the viscosity and is ∝ ε1/2Lf/(2ν) (where ε is the energy injection rate and Lf is the spatial scale of forcing). In order to calculate the various moments of the velocity field ~v = −∇Ψ, one needs to average the loga- rithm of the partition function Z, a celebrated problem in disordered systems. Bouchaud, Mézard and Parisi proposed in [22] the use of a replica trick in order to estimate the average free energy 〈lnZ〉. The first step is to write the zero-replica limit lnZ = limn→0 (Zn − 1)/n. Then, the moments 〈Zn〉 are used to generate an effective attraction between replicas: they are written as the partition functions of the disorder-averaged Hamiltonian Hn(~γ1, . . . , ~γn) asso- ciated to n replicas of the same system [90] ∥~̇γi(s) B(~γi(s)−~γj(s)) ,(7.3) where B denotes the spatial part of the forcing potential correlation. The next step is to study this problem by a variational approach. The Hamiltonian Hn is replaced by an effective Gaussian quadratic Hamiltonian that can be written as Heff = ~γi(τ)Gij(τ−τ ′)~γj(τ ′)dτdτ ′. (7.4) The kernel Gij is then chosen in such a way that it minimizes the free energy. It is shown in [22] that the optimal Gaussian Hamiltonian is the solution of a sys- tem of equations that can be solved following the ansatz proposed in [89]. When d > 2 this approach singles out two régimes depending on the Reynolds number Re = ε1/3L f /ν. These régimes are separated by the critical value Rec = [2(1−2/d)1−d/2]1/3. When Re < Rec the optimal solution is of the form Gij = G0 δij +G1 and obeys the replica symmetry. In finite-size systems it cor- responds to a linear velocity profile. When Re > Rec the correct solution is given by the one-step replica- symmetry-breaking scheme (see [89]). The off-diagonal elements of Gij are then parameterized with two func- tions depending on whether the indices i and j belong to the same block or to different blocks. Qualitatively, the one-step replica-symmetry-breaking approach amounts to the assumption that the instantaneous velocity po- tential can be written as a weighted sum of Gaussians, leading to an approximation of the velocity field as ~v(~x, t) ≃ α(~x− ~rα) e −Re (fα+‖~x−~rα‖ 2/2L2 −Re (fα+‖~x−~rα‖2/2L2f ) , (7.5) where the fα’s are independent variables with a Poisson distribution of density exp(−f). The ~rα are uniformly and independently distributed in space. In (7.5) the sum over α is running from 1 to a large-enough integer M . The typical shape of the approximation of the veloc- ity field given by (7.5) is represented in figure 39(a) in the two-dimensional case. In the limit of large Reynolds numbers the random velocity field given by (7.5) typi- cally contains cells of width ∝ Lf . The width of a shock separating two cells is of the order of Lf/Re. The replica approximation (7.5) leads to an estimate of the PDF p(∆v, r) of the longitudinal velocity increment ∆v = (~v(~x+ r ~e, t) − ~v(~x, t)) · ~e, where ~e is an arbitrary unitary vector. When Re ≫ 1 and r ≪ Lf this approxi- mation takes the particularly simple asymptotic form p(∆v, r) ≈ δ ∆v − Uf , (7.6) where Uf = Re ν/Lf is the typical velocity associated to the scale Lf and g is a scaling function that is deter- mined explicitly in [22]. This approximation is in agree- ment with the following qualitative picture. With a prob- ability almost equal to one, the two points ~x and ~x+ r ~e lie in the same cell; the velocity increment is then given by the typical velocity gradient which, according to the Fig. 39. (a) Typical shape of the velocity field given by the replica approximation in dimension d = 2 obtained from (7.5) for Re = 103. The contour lines represent the velocity modulus. Note the cell structure of the domain. (b) Scaling exponents of the pth order structure function. approximation (7.5), is order Uf/Lf . With a probabil- ity r/Lf the two points are sitting on different sides of a shock separating two such cells and the associated ve- locity difference is of the order of Uf . The structure functions of the velocity field given by the various moments of ∆v can be straightforwardly estimated from the approximation (7.6). Their scaling behavior 〈∆vp〉 ∼ rζp at small separations r display a bifractal behavior as sketched in figure 39(b). When p < 1, the first term on the right-hand side of (7.6) domi- nates and 〈∆vp〉 ∝ Upf (r/Lf )p. For p > 1 the shock con- tribution is dominating the small-r behavior and thus 〈∆vp〉 ∝ Upf (r/Lf ). This approach, which makes use of replica tricks, is as we have seen able to catch the leading scaling behavior of velocity structure functions in any dimension. It is based on approximations of the velocity field by the superposi- tion (7.5) of Gaussian velocity potentials. A first advan- tage of this method is that it catches the generic aspect of the solution including the hierarchy of high-order sin- gularities appearing in the solution when Re → ∞ which was examined in section 2.3. This method also gives predictions regarding the dependence on Re of the sta- tistical properties of the solution. However, as stressed in [22], the validity of this approximation is expected to hold only in the limit of infinite space dimension d. In particular, it is known that for d ≤ 2 a full continuous replica-symmetry-breaking scheme is needed [89]. Nev- ertheless, as we have seen, there is enough evidence that this approach describes very well the qualitative aspects of the solution. 7.2 Dissipative anomaly and operator product expan- The replica-trick approach described in the previous subsection cannot reproduce one of the main statis- tical features of the solution, namely the tails of the velocity increments PDF p(∆v, r). Indeed the predic- tion (7.6) based on a variational approximation of the velocity field implies that p identically vanishes when ∆v > Uf (r/Lf ). In order to study the quantitative be- havior of the PDF p(∆v, r) in the inviscid limit ν → 0 (or equivalently Re → ∞), Polyakov [99] proposed to use an operator product expansion. This approach leads to an explicit expression for p(∆v, r) and predicts a super-exponential tail at large positive values and a power-law behavior for negative ones. Such predictions have immediate implications for the asymptotics of the PDF p(ξ) of the velocity gradient ξ = ∂xv. The work of Polyakov was the starting point of a controversy on the value of the exponent of the left tail of p(ξ). Before returning to this issue in the next subsection, we give in the sequel a quick overview of the original work by Polyakov. We henceforth focus on the one-dimensional solutions to the Burgers equation with Gaussian forcing whose autocorrelation is given by (7.1). Following [99] (see also [19,20]) we introduce the characteristic function of the n-point velocity distribution Zn(λj , xj ; t) ≡ e λ1 v(x1,t)+···+λn v(xn,t) . (7.7) For a finite value of the viscosity ν, it is easily seen that this quantity is a solution to a Fokker–Planck (master) equation obtained by differentiating Zn with respect to t and using the Burgers equation and the fact that the forcing is Gaussian and δ-correlated in time. This leads b(xi − xj)λi λj Zn + D(n)ν , (7.8) where b ≡ (d2B)/(dr2) denotes the spatial part of the correlation of the forcing applied to the velocity field. D(n)ν denotes the contribution of the dissipative term and reads D(n)ν ≡ ν v(xj , t) λj v(xj ,t) . (7.9) This term does not vanish in the limit ν → 0 since the solutions to the Burgers equation develop singularities with a finite dissipation. It has been proposed in [99] to use an analogy with the anomalies appearing in quantum field theory in order to tackle this term in the inviscid limit. The important assumption is then made that the singular term in the operator product expansion relates linearly to the characteristic function Zn. Since this ex- pansion should preserve the statistical symmetries of the Burgers equation, it leads to the replacement in all av- erages of the singular limit limν→0 ν λ (∂ xv) e λ v by the asymptotic expression b − 1 eλ v , (7.10) where the coefficients a, b and c are parameters that can be determined only indirectly. However their possible values can be restricted by requiring that Zn is the char- acteristic function of a probability distribution which is non-negative, finite, normalizable, and that the dissipa- tive term D(n)ν acts as a positive operator. Finding these coefficients is similar to an eigenvalue problem in quan- tum mechanics. We now come to a crucial point in Polyakov’s approach. Important restrictions on the form of the different anomalous terms in (7.10) result from the fact that the solutions to the Burgers equation obey a certain form of Galilean invariance. A notion of “strong Galilean prin- ciple” is introduced for invariance of the n-point distri- bution of velocity under the transformation v 7→ v + v0 with v0 an arbitrary constant. As a consequence, the n- point characteristic function Zn has to be proportional to δ(λ1 + · · ·+λn). The operators appearing in the limit ν → 0 have to be consistent with such an invariance. In [99] it is argued that this symmetry is automatically broken by the forcing that introduces a typical velocity 〈v2〉1/2 ∝ b1/3(0)L1/3. However Polyakov assumes this “strong Galilean principle” to be asymptotically recov- ered in the limit L → ∞ of infinite-size systems. In the case of finite-size systems, when L is of the order of the correlation length Lf of the forcing, the strong Galilean symmetry is broken because of the conservation of the spatial average of v which introduces a characteristic velocity v0 = (1/L) v(x, t) dx. However, the Galilean symmetry should be recovered when averaging the cor- relation functions with respect to the mean velocity v0. This symmetry restoration was introduced in [20] where it is referred to as the “weak Galilean principle”. The n-point characteristic function associated to an aver- age velocity v0 relates to that associated to a vanishing mean velocity by Zn(λj , xj ; t; v0) = e Zn(λj , xj ; t; 0) . After averaging with respect to v0, one obtains Zn(λj , xj ; t) = 2π δ  Zn(λj , xj ; t; 0) . (7.11) One can easily check that (7.8), together with the dis- sipative term given by (7.10), are compatible with this expression for the n-point characteristic function Zn. Moreover, any higher-order term in the expansion (7.10) of the dissipative anomaly would violate Galilean invari- ance. To obtain the statistical properties of the solution, one needs to further restrict the values of the three free pa- rameters a, b, and c appearing in the expansion (7.10). Following [99] this can be done by considering the case n = 2 that corresponds to the equation for the PDF of velocity differences. Performing the change of variables λ1,2 = Λ ± µ and x1,2 = X ± y/2, and assuming that λ ≪ µ and y ≪ Lf (so that the spatial part of the forcing correlation is to leading order b(y) ≃ b0 − b1y2), the stationary and space-homogeneous solutions to the master equation (7.8)) satisfy − (2b0Λ2 + b1µ2y2)Z2 = = aZ2 + . (7.12) It is next assumed in [99] (see also [20]) that the velocity difference v(x1, t) − v(x2, t) is statistically independent of the mean velocity (v(x1, t) + v(x2, t))/2. This implies that the two-point characteristic function factorizes as Z2 = Z 2 (Λ)Z 2 (µ, y), where the two functions Z 2 and Z−2 satisfy the closed equations − 2b0Λ2Z+2 = cΛ , (7.13) ∂2Z−2 − b1µ2y2Z−2 = aZ . (7.14) The solution to the first equation corresponds to a Gaus- sian distribution which is normalizable only if c < 0. As shown numerically in [20] this distribution is representa- tive of the bulk of the one-point velocity PDF. Informa- tion on the solutions to the second equation can be ob- tained assuming the scaling property Z−2 (µ, y) = Φ(µy), which amounts to considering only those contributions to the distribution of velocity differences stemming from velocity gradients ξ = ∂xv. This yields a prediction the negative and positive tails of the PDF of velocity gradi- ents: p(ξ) ∝ |ξ|−α when ξ → −∞ , (7.15) p(ξ) ∝ ξβ exp(−C ξ3) when ξ → +∞ , (7.16) where C is a constant, which depends only on the strength of the forcing. The two exponents α and β are related to the coefficient b of the anomaly by α = 2b + 1 and β = 2b − 1 . (7.17) The value of b remains undetermined but is prescribed to belong to a certain range. This approach was first designed in [99] for infinite-size systems where strong Galilean invariance holds. In that case consistency with such an invariance leads to dropping the third term in the operator product expansion (i.e. c = 0). Positivity and normalizability of the two-point velocity PDF and non-positivity of the anomalous dissipation operator im- ply that the two other coefficients form a one-parameter family with 3/4 ≤ b ≤ 1. In particular, this implies that the left tail of the velocity gradient PDF with ex- ponent α should be shallower than ξ−3. As we will see in the next section, strong evidence has been obtained that p(ξ) ∝ ξ−7/2 for ξ → −∞. This seems to contradict the approach based on operator product expansion. How- ever, as argued in [20], the breaking of strong Galilean in- variance occurring in finite-size systems and resulting in the presence of the c anomaly broadens the range of ad- missible values for b. In particular it allows for the value b = 5/4 which corresponds to the exponent α = 7/2. 7.3 Tails of the velocity gradient PDF After the numerical work of Chekhlov and Yakhot [29], the asymptotic behavior at large positive and negative values of the PDF of velocity derivatives ξ = ∂xv for the one-dimensional randomly forced Burgers equation attracted much attention. A broad consensus emerged around the prediction of Polyakov [99] that p(ξ) dis- plays tails of the form (7.16) and (7.15), but the values of the exponents α and β were at the center of a contro- versy. Note that the presence of a super-exponential tail ∝ exp(−C ξ3) at large positive arguments has been con- firmed by the use of instanton techniques [60] and that the only remaining uncertainty concerns the exponent of the algebraic prefactor. A standard approach to de- termine the exponents α and β appearing in (7.15) and (7.16) makes use of the stationary solutions to the in- viscid limit of the Fokker–Planck equation for the PDF, namely −ξp+ν∂ξ ∂3xv | ∂xv=ξ = b̃∂2ξp . (7.18) Here the brackets 〈·|·〉 denote conditional averages and the right-hand side expresses the diffusion of probability due to the delta-correlation in time of the forcing. The main difficulty in studying the solutions of (7.18) stems from the treatment of the dissipative term Dν(ξ) = ∂3xv|∂xv=ξ in the limit ν → 0. The value α = 3 is obtained if a piecewise linear approximation is made for the solutions of the Burgers equation [21]. Gotoh and Kraichnan [59] argued that the dissipative term is to leading order negligible and presented analytical and nu- merical arguments in favor of α = 3 and β = 1. However, the inviscid limit of (7.18) contains anomalies due to the singular behavior of Dν(ξ) in the limit ν → 0. As we have seen in previous section, the approach based on the use of an operator product expansion [99] leads to a rela- tion involving unknown coefficients which must be deter- mined, e.g., from numerical simulations [111,19,20], and restricts the possible values to 5/2 ≤ α ≤ 3 [6]. Anoma- lies cannot be studied without a complete description of the singularities of the solutions, such as shocks, and a thorough understanding of their statistical properties. E, Khanin, Mazel and Sinai made a crucial observa- tion in [37] that large negative gradients stem mainly from preshocks, that is the cubic-root singularities in the velocity preceding the formation of shocks (see sec- tion 2.3). They then used a simple argument for de- termining the fraction of space-time where the veloc- ity gradient is less than some large negative value. This leads to α = 7/2, provided preshocks do not cluster. Later on, this approach has been refined by E and Van- den Eijnden who proposed to determine the dissipative anomaly of (7.18) using formal matched asymptotics [39] or bounded variation calculus [42]. As we shall see be- low, with the assumption that shocks are born with a zero amplitude, that their strengths add up during colli- sions, and that there ar no accumulations of preshocks, the value α = 7/2 was confirmed [42]. Other attempts to derive this value using also isolated preshocks have been made [81,6]. Note that there are simpler instances, in- cluding time-periodic forcing [9] (see section 6) and de- caying Burgers turbulence with smooth random initial conditions [8,42] (see section 4.1), which fall in the uni- versality class α = 7/2, as can be shown by systematic asymptotic expansions using a Lagrangian approach. We give here the flavor of the approach used in [39] in order to estimate the dissipative anomaly D0(ξ) = limν→0D ν(ξ). One first notices that for |ξ| ≫ b̃1/3, the forcing term in the right-hand side of (7.18) becomes negligible, so that stationary solutions to the Fokker– Planck equation satisfy p(ξ) ≈ |ξ|−3 dξ′ ξ′Dν(ξ′). (7.19) A straightforward consequence of this asymptotic ex- pression is that, if the integral in the right-hand side de- creases as ξ → −∞ (i.e. if ξDν(ξ) is integrable), then p(ξ) decreases faster than |ξ|−3, and thus α > 3. To get some insight into the behavior ofDν as ν → 0, one next observes that the solutions to the one-dimensional Burgers equation contain smooth regions where viscos- ity is negligible, which are separated by thin shock layers where dissipation takes place. The basic idea consists in splitting the velocity field v into the sum of an outer so- lution away from shocks and of an inner solution near them for which boundary layer theory applies. Matched asymptotics are then used to construct a uniform ap- proximation of v. To construct the inner solution near a shock centered at x = x⋆, one performs the change of variable x 7→ x̃ = (x−x⋆)/ν and looks for an expression of ṽ(x̃, t) = v(x⋆+νx̃, t) in the form of a Taylor expan- sion in powers of ν: ṽ = ṽ0+νṽ1+o(ν). At leading order, the inner solution satisfies [ṽ0 − v⋆] ∂x̃ṽ0 = ∂2x̃ṽ0, (7.20) where v⋆ = (dx⋆)/(dt). This expression leads to the well- known hyperbolic tangent velocity profile ṽ0 = v⋆ − . (7.21) Here, s = v(x⋆+, t)−v(x⋆−, t) denotes here the velocity jump across the shock and is given by matching condi- tions to the outer solution. The term of order ν is then a solution of ∂tṽ0 + [ṽ0 − v⋆] ∂x̃ṽ1 = ∂2x̃ṽ1 + f(x, t). (7.22) In order to evaluate the dissipative anomaly, it is con- venient to assume spatial ergodicity so that the viscous term in (7.18) can be written as Dν(ξ) = ν∂ξ lim dx ∂3xv δ(∂xv−ξ). (7.23) In the limit ν → 0 the only remaining contribution stems from shocks and is thus given by the inner solution. Using the expansion of the solution up to the first order in ν, this leads to writing the dissipative term in the limit of vanishing viscosity as (see Appendix of [41] for details) D0(ξ) = ds s [p+(s, ξ) + p−(s, ξ)] , (7.24) where ρ is the density of shocks and p+ (respectively p−) is the joint probability of the shock jump and of the value of the velocity gradient at the right (respectively left) of the shock. This expression guarantees the finiteness of the dissipative anomaly, and in particular the fact that the integral in the right-hand side of (7.19) is finite in the limit ν → 0 and converges to 0. As a consequence, this gives a proof that the exponent α of the left tail of the velocity gradient PDF is larger than 3. To proceed further, E and Vanden Eijnden proposed to estimate the probability densities p+ and p− by deriving master equations for the joint probability of the shock strength s, its velocity v⋆ and the values ξ ± of the veloc- ity gradient at its left and at its right. This is done in [42] using a formulation of Burgers dynamics stemming from bounded variation calculus. More precisely, it is shown in [108] that the Burgers equation is equivalent to con- sidering the solutions to the partial differential equation ∂tv + v̄∂xv = f , (7.25) where v̄(x, t) = (v(x+, t) + v(x−, t))/2. Basically this means that Burgers dynamics can be formulated in terms of the transport of the velocity field by its average v̄. This formulation straightforwardly yields a master equation for v(x±, t) and ∂xv(x±, t) which is then used to estimate p± and the dissipative anomaly (7.24). Al- though the treatment of the master equation does not involve any closure hypothesis, it is not fully rigorous: in particular it requires the assumption that shocks are created with zero amplitude and that shock amplitudes add up during collision. However such an approaches strongly suggests that α = 7/2 and β = 1. Obtaining numerically clean scaling for the PDF of gra- dients is not easy with standard schemes. Let us ob- serve that any method involving a small viscosity, either introduced explicitly (e.g. in a spectral calculation) or stemming from discretization (e.g. in a finite difference calculation), may lead to the presence of a power-law range with exponent −1 at very large negative gradi- ents [59]. This behavior makes the inviscid |ξ|−α range appear shallower than it actually is, unless extremely high spatial resolution is used. In contrast, methods that directly capture the inviscid limit with the appropriate shock conditions, such as the fast Legendre transform method [94], lead to delicate interpolation problems. They have been overcome in the case of time-periodic forcing [9] but with white-noise-in-time forcing, it is dif- ficult to prevent spurious accumulations of preshocks leading to α = 3. To avoid such pitfalls, a Lagrangian particle and shock tracking method was developed in [6]. This method is able to separate shocks and smooth parts of the solution and is particularly effective for identifying preshocks. The main idea is to consider the evolution of a set of N massless point particles accelerated by a discrete-in- time approximation of the forcing with a uniform time step. When two of these particles intersect, they merge and create a new type of particle, a shock, characterized by its velocity (half sum of the right and left velocities of merging particles) and its amplitude. The particle- like shocks then evolve as ordinary particles, capture further intersecting particles and may merge with other shocks. In order not to run out of particles too quickly, the initial small region where particles have the least chance of being subsequently captured is determined by localization of the global minimizer of the Lagrangian action (see section 5.1). The calculation is then restarted from t = 0 for the same realization of forcing but with a vastly increased number of particles in that region. This particle and shock tracking method gives complete control over shocks and preshocks. slope = −7/2 −3.75 −3.25 −7/2±1% Fig. 40. PDF of the velocity gradient at negative values in log-log coordinates obtained by averaging over 20 realiza- tions and a time interval of 5 units of time (after relaxation of transients). The simulation involves up to N = 105 parti- cles and the forcing is applied at discrete times separated by δt = 10−4. Upper inset: local scaling exponent (from [6]). Figure 40 shows the PDF of the velocity gradients in log-log coordinates at negative values, for a Gaussian forcing restricted to the first three Fourier modes with equal variances such that the large-scale turnover time is order unity. Quantitative information about the value of the exponent is obtained by measuring the “local scaling exponent”, i.e. the logarithmic derivative of the PDF calculated in this case using least-square fits on half- decades. It is seen that over about five decades, the local exponent is within less than 1% of the value α = 7/2 predicted by E et al. [37]. 7.4 Self-similar forcing and multiscaling As we have seen in section 7.1, the solutions to the Burg- ers equation in a finite domain and with a large-scale forcing have structure functions (moments of the ve- locity increment) displaying a bifractal scaling behav- ior. Such a property can be easily interpreted by the presence of a finite number of shocks with a size order unity in the finite system. Somehow this double scal- ing and its relationship with singularities gives some in- sight on the multiscaling properties that are expected in the case of turbulent incompressible hydrodynamics flows. There is a general consensus that the turbulent solutions to the Navier–Stokes equations display a full multifractal spectrum of singularities which are respon- sible for a nonlinear p-dependence of the scaling expo- nents ζp associated to the scaling behavior of the p-th order structure function [53]. The construction of simple tractable models which are able to reproduce such a be- havior has motivated much work during the last decades. Significant progress, both analytical and numerical, has been made in confirming multiscaling in passive-scalar and passive-vector problems (see, e.g., [46] for a review). However, the linearity of the passive-scalar and passive- vector equations is a crucial ingredient of these studies, so it is not clear how they can be generalized to fluid turbulence and the Navier–Stokes equation. After the work of Chekhlov and Yakhot [30], it appeared that the Burgers equation with self-similar forcing could be the simplest nonlinear partial differential equation which has the potential to display multiscaling of veloc- ity structure functions. We report in this section various works that tried to confirm or to weaken this statement. Let us consider the solutions to the one-dimensional Burgers equation with a forcing term f(x, t) which is ran- dom, space-periodic, Gaussian and whose spatial Fourier transform has correlation 〈f̂(k, t)f̂(k′, t′)〉 = 2D0 |k|β δ(t− t′) δ(k + k′). (7.26) The exponent β determines the scaling properties of the forcing. When β > 0 the force acts at small scales; for instance β = 2 corresponds to thermal noise for the ve- locity potential, and thus to the KPZ model for interface growth [74]. It is well known in this case (see, e.g., [5]) that the solution displays simple scaling (usually known as KPZ scaling), such that ζq = q for all q. More gener- ally, the case β > 0 can be exactly solved using a one- loop renormalization group approach [88]. As stressed in [64], renormalization group techniques fail when β < 0 and the forcing acts mostly at large scales and non-linear terms play a crucial role. When β < −3, the forcing is differentiable in the space variable, the so- lution is piecewise smooth and contains a finite number of shocks with sizes order unity. The scaling exponents are then ζp = min (1, p). In the case of non-differentiable forcing (−3 < β < 0), the presence of order-unity shocks and dimensional arguments suggest that the scaling ex- ponents are ζp = min (1, −pβ/3). However, very little is known regarding the distribution of shocks with inter- mediate sizes. In particular, there is no clear evidence whether or not they form a self-similar structure at small scales. We summarize here some studies which were done on Burgers turbulence with self-similar forcing to show how difficult it might be to measure scaling laws of struc- ture functions and in particular how logarithmic correc- tions can masquerade anomalous scaling. For this we focus on the case β = −1 which has attracted much attention; indeed, dimensional analysis suggests that ζp = p/3 when p ≤ 3, leading to a K41-type −5/3 energy spectrum. Early studies [29,30] seemed to confirm this prediction using pseudo-spectral viscous numerical Fig. 41. Representative snapshots of the velocity v (jagged line) in the statistically stationary régime, and of the inte- gral of the force f over a time step (rescaled for plotting purposes). simulations at rather low resolutions (around ten thou- sands gridpoints). It was moreover argued in [64,65] that a self-similar forcing with −1 < β < 0, could lead to gen- uine multifractality. The lack of accuracy in the determi- nation of the scaling exponents left open the question of a weak anomalous deviation from the dimensional pre- diction. This question was recently revisited in [91] with high-resolution inviscid numerical simulations using the fast Legendre transform algorithm (see section 2.4.2). A typical snapshot of the forcing and of the solution in the stationary régime are represented in figure 41. It is clear that because of shocks the velocity develops small- scale fluctuations much stronger than those present in the force. However one notices that shock dynamics and spatial finiteness of the system lead, as predicted, to the presence of few shocks with order-unity sizes. 0 1 2 3 4 5 ξ p 0 1 2 3 4 5 Fig. 42. Scaling exponents ζp versus order p for N = 216(⋄), 218(∗), and 220(◦) grid points. Error bars (see text) are shown for the case N = 220. The deviation of ζp from the exponents for bifractal scaling (full lines), shown as an inset, naively suggests multiscaling (from [91]) Structure functions were measured with high accuracy. They typically exhibit a power-law behavior over nearly three decades in length scale; this is more than two decades better than in [30]. In principle one expects to be able to measure the scaling exponents with enough accuracy to decide between bifractality and multiscal- ing. Surprisingly the naive analysis summarized in fig- ure 42 does suggest multiscaling: the exponents ζp de- viate significantly from the bifractal-scaling prediction (full lines). Since the goal here is to have a precise han- dle on the scaling properties of velocity increments, it is important to carefully define how the scaling exponents are measured. They are estimated from the average log- arithmic derivative of Sabsp (r) = 〈|v(x+r)−v(x)|p〉 over almost two decades in the separation r. The error bars shown are given by the maximum and minimum devia- tions from this mean value in the fitting range. Note also that the observed multiscaling is supported by the fact that there is no substantial change in the value of the exponents when changing the number N of grid points in the simulation from 216 to 220: any dependence of ζp upon N is much less than the error bars determined through the procedure described above. −5 −4 −3 −2 −1 −6 −4 −2 Fig. 43. Log-log plots of Sabs3 (r) (dashed line), S3(r) (crosses), and 〈(δ+v)3〉 (squares) versus r. The continuous line is a least-square fit to the range of points limited by two vertical dashed lines in the plot. Inset: An explicit check of the von Kármán–Howarth relation (7.27) from the simula- tions with N = 220 reported in [91]. The dashed curve is the integral of the spatial part of the forcing correlation and the circles represent the numerical computation of the left-hand side. As found in [91], the observed deviations of the scaling exponents from bifractality are actually due to the con- tamination by subleading terms in Sabsp (r). To quantify this effect, let us focus on the third-order structure func- tion (p = 3) for which one measures ζ3 ≈ 0.85±0.02 over nearly four decades (see figure 43). To estimate sublead- ing terms we first notice that the third-order structure function S3(r) ≡ 〈(v(x + r) − v(x))3〉, which is defined, this time, without the absolute value, obeys an analog of the von Kármán–Howarth relation in fluid turbulence, namely S3(r) = b(r′)dr′, (7.27) where b(·) denotes the spatial part of the force corre- lation function, i.e. 〈f(x + r, t′)f(x, t)〉 = b(r)δ(t − t′). This relation, together with the correlation (7.26) and β = −1, implies the behavior S3(r) ∼ r ln r at small separations r. As seen in figure 43, the graph of S3(r) in log-log coordinates indeed displays a significant cur- vature which is a signature of logarithmic corrections. The next step consists in decomposing the velocity in- crements δrv = v(x + r, t) − v(x, t) into their positive δ+r v and negative δ r v parts. It is clear that Sabs3 (r) = − (δ−r v) (δ+r v) S3(r) = (δ−r v) (δ+r v) , (7.28) so that (δ+r v) = (Sabs3 (r) + S3(r))/2. As seen in fig- ure 43 the log-log plot of (δ+r v) as a function of r is nearly a straight line with slope ≈ 1.07 very close to unity. This observation is confirmed in [91] by indepen- dently measuring the PDFs of positive and negative ve- locity increments. Assuming that (δ+r v) ∼ B r, one obtains the following prediction for the small-r behav- iors of the third-order structure functions Sabs3 (r) ∼ −Ar ln r +B r, S3(r) ∼ Ar ln r +B r. (7.29) This suggests that the only difference in the small- separation behaviors of Sabs3 (r) and S3(r) is the sign in the balance between the leading term ∝ r ln r and the subleading term ∝ r. In a log-log plot this difference amounts to shifting the graph away from where it is most curved and thus makes it straighter, albeit with a (local) slope which is not unity. This explains why signif- icant deviations from 1 are observed for ζ3. Note that a similar approach can be used for higher-order structure functions. It leads for instance to S4(r) ≈ Cr −Dr4/3, where C and D are two positive constants. The nega- tive sign before the sub-leading term (r4/3) is crucial. It implies that, for any finite r, a naive power-law fit to S4 can yield a scaling exponent less than unity. The presence of sub-leading, power-law terms with oppo- site signs also explains the small apparent “anomalous” scaling behavior observed for other values of p in the simulations. Note that similar artifacts involving two competing power-laws have been described in [16,7]. The work reported in this section indicates that a naive interpretation of numerical measurements might result in predicting artificial anomalous scaling laws. In the case of Burgers turbulence for which high-resolution nu- merics are available and statistical convergence of the averages can be guaranteed, we have seen that it is not too difficult to identify the numerical artifacts which are responsible for such a masquerading. However this is not always the case. For instance, it seems reason- able enough to claim that attacking the problem of mul- tiscaling in spatially extended nonlinear systems, such as Navier–Stokes turbulence, requires considerable the- oretical insight that must supplement sophisticated and heavy numerical simulations and experiments. Note fi- nally that, up to now, the question of the presence or not of anomalous scaling laws in the Burgers equation with a self-similar forcing with exponent −1 < β < 0 remains largely open. 8 Concluding remarks and open questions This review summarizes recent work connected with the Burgers equation. Originally this model was introduced as a simplification of the Navier–Stokes equation with the hope of shedding some light on issues such as turbu- lence. This hope did not materialize. Nevertheless many of the interesting questions that have been addressed for Burgers turbulence are eventually transpositions of similar questions for Navier–Stokes turbulence. One par- ticularly important instance is the issue of universality with respect to the form of the forcing and of the initial condition. For Burgers turbulence most of the universal features, such as scaling exponents or functional forms of PDF tails are dominated by the presence of shocks and other singularities in the solution. This applies both to the case of decaying turbulence driven by random ini- tial conditions and randomly forced turbulence. In the latter case one is mostly interested in analysis of station- ary properties of solutions, for example stationary dis- tribution for velocity increments or gradients. Another set of questions is motivated by more mathematical con- siderations. It mainly concerns the construction of a sta- tionary invariant measure when Burgers dynamics in a finite-size domain is supplemented by an external ran- dom source of energy. Again it has been shown that the presence of shocks, and in particular of global shocks, plays a crucial role in the construction of the statisti- cally stationary solution. Both physical and mathemat- ical questions lead to a similar answer: one first needs to describe and control shocks. The main message to retain for hydrodynamical turbulence is hence a strong confir- mation of the common wisdom that it cannot be fully understood without a detailed description of singulari- ties. Moreover, the behavior depends not only on the lo- cal structure of singularities, but also on their distribu- tion at larger scales. Here a word of caution: for incom- pressible fully developped Navier–Stokes turbulence, we have no evidence that the universal scaling properties observed in experiments and simulations stem from real singularities. Indeed the issue of a finite-time blow-up of the three-dimensional Euler equation is still open (see, e.g. [56]). Another important observation that can be drawn from the study of Burgers turbulence is that both the tools used and the answers obtained strongly de- pend on the kind of setting one considers: decay versus forced turbulence, finite-size versus infinite-size systems, smooth versus self-similar forcing, etc. Besides turbulence, the random Burgers equation has various applications in cosmology, in non-equilibrium statistical physics and in disordered media. Among them, the connection to the problem of directed poly- mers has attracted much attention. As already noted in the Introduction, there is a mathematical equivalence between the zero-viscosity limit of the forced Burgers equation and the zero-temperature limit for directed polymers. We have seen in section 5.4 that the so-called KPZ scaling, which usually is derived for a finite tem- perature, can be established can be established also in the zero-temperatur limit, using the action minimizer representation. Such an observation leads to two related questions: to what extent can the limit of zero temper- ature give an insight into finite-temperature polymer dynamics and how can the global minimizer formalism be extended to tackle the finite-temperature setting? It looks plausible that in polymer dynamics, or more generally in the study of random walks in a random potential, the trajectories carrying most of the Gibbs probability weight are defining corridors in space time. These objects can concentrate near the trajectories of global minimizers but, at the moment, there is no for- malism to describe them, nor attempts to quantify their contribution to the Gibbs statistics. Another important open question concerns the multi- dimensional extensions of the Burgers equation. As we have seen, when the forcing is potential, the potential character of the velocity field is conserved by the dynam- ics. This leads to the construction of stationary solutions which carry many similarities with the one-dimensional case. Up to now there is only limited understanding of what happens when the potentiality assumption of the flow is dropped. This problem has of course concrete applications in gas dynamics and for disperse inelastic granular media (see, e.g., [12]). An interesting question concerns the construction of the limit of vanishing vis- cosity, given that the Hopf–Cole transformation is in- applicable in the non-potential case. Understanding ex- tensions of the viscous limiting procedure to the non- potential case might give new insight into the problem of the large Reynolds number limit in incompressible tur- bulence. Another question related to non-potential flow concerns the interactions between vorticity and shocks. For instance, in two dimensions the vorticity is trans- ported by the flow. This results in its growth in the highly compressible regions of the flow. The various singulari- ties of the velocity field should hence be strongly affected by the flow rotation and, in particular, the shocks are expected to have a spiraling structure. We finish with few remarks on open mathematical prob- lems. As we have seen in the one-dimensional case, one can rigorously prove hyperbolicity of the global mini- mizer. In the multi-dimensional case it is also possible to establish the existence and, in many cases, unique- ness of the global minimizer. However, the very impor- tant question of its hyperbolicity is still an open prob- lem. If proven, hyperbolicity would allow for rigorous analysis of the regularity properties of the stationary so- lutions and of the topological shocks. There are many interesting problems – even basic issues of existence and uniqueness – in the non-compact case where at present a mathematical theory is basically absent. Finally, a very challenging open problem concerns the extension of the results on the evolution of matter inside shocks to the case of general Hamilton-Jacobi equations. Acknowledgements Over the years of our work on Burgers turbulence, we profited a lot from numerous discussions with Uriel Frisch whose influence on our work is warmly acknowl- edged. We also want to express our sincere gratitude to all of our collaborators: W. E, U. Frisch, D. Gomes, V.H. Hoang, R. Iturriaga, D. Khmelev, A. Mazel, D. Mi- tra, P. Padilla, R. Pandit, Ya. Sinai, A. Sobolevskĭı, and B. Villone. While writing this article, we benefited from discus- sions with M. Blank, I. Bogaevsky, K. Domelevo, V. Epstein, and A. Sobolevskĭı. Finally, our thanks go to Itamar Procaccia whose encouragements and patience are greatly appreciated. References [1] S. Anisov. Lowerbounds for the complexity of some 3- dimensional manifolds. preprint, 2001. [2] V. Arnol’d. 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Astronomy & Astrophysics manuscript no. 7228 c© ESO 2022 March 1, 2022 Analytical evaluation of the X-ray scattering contribution to imaging degradation in grazing-incidence X-ray telescopes D. Spiga INAF/Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807, Merate (LC) - Italy Received 02 February 2007 / Accepted 27 March 2007 ABSTRACT Aims. The focusing performance of X-ray optics (conveniently expressed in terms of HEW, Half Energy Width) strongly depend on both mirrors deformations and photon scattering caused by the microroughness of reflecting surfaces. In particular, the contribution of X-ray Scattering (XRS) to the HEW of the optic is usually an increasing function H(E) of the photon energy E. Therefore, in future hard X-ray imaging telescopes of the future (SIMBOL-X, NeXT, Constellation-X, XEUS), the X-ray scattering could be the dominant problem since they will operate also in the hard X-ray band (i.e. beyond 10 keV). In order to ensure the imaging quality at all energies, clear requirements have to be established in terms of reflecting surfaces microroughness. Methods. Several methods were proposed in the past years to estimate the scattering contribution to the HEW, dealing with the surface microroughness expressed in terms of its Power Spectral Density (PSD), on the basis of the well-established theory of X-ray scattering from rough surfaces. We faced that problem on the basis on the same theory, but we tried a new approach: the direct, analytical translation of a given surface roughness PSD into a H(E) trend, and – vice versa – the direct translation of a H(E) requirement into a surface PSD. This PSD represents the maximum tolerable microroughness level in order to meet the H(E) requirement in the energy band of a given X-ray telescope. Results. We have thereby found a new, analytical and widely applicable formalism to compute the XRS contribution to the HEW from the surface PSD, provided that the PSD had been measured in a wide range of spatial frequencies. The inverse problem was also solved, allowing the immediate evaluation of the mirror surface PSD from a measured function H(E). The same formalism allows establishing the maximum allowed PSD of the mirror in order to fulfill a given H(E) requirement. Practical equations are firstly developed for the case of a single-reflection optic with a single-layer reflective coating, and then extended to an optical system with N identical reflections. The results are approximately valid also for multilayer-coated mirrors to be adopted in hard X-rays. These results will be extremely useful in order to establish the surface finishing requirements for the optics of future X-ray telescopes. Key words. Telescopes – Methods: analytical – Instrumentation: high angular resolution 1. Introduction The adoption of grazing-incidence optics in X-ray telescopes in the late 70s allowed a great leap forward in X-ray astronomy because they endowed the X-ray instrumentation with imag- ing capabilities in the soft X-ray band (E < 10 keV). The excellent performances of the soft X-ray telescopes ROSAT (Aschenbach 1988), Chandra (Weisskopf 2003) and Newton- XMM (Gondoin et al. 1998) are well known. To date, the utilized technique to focus soft X-rays consists in systems of double-reflection mirrors with a single layer coat- ing (Au, Ir) in total external reflection at shallow grazing inci- dence angles. In this case, the incidence angle θi (as measured from the mirror surface) cannot exceed the critical angle for total reflection, otherwise the mirror reflectivity would be very low. The critical angle is inversely proportional to E, the energy of the photons to be focused. Using Au coatings, for instance, the incidence angle cannot exceed ∼ 0.4 deg for photon energies E ≈ 10 keV. An extension of this technique to the hard X-ray energy band (E > 10 keV) can be pursued by combining long focal lengths (> 10 m), very small incidence angles (0.1 ÷ 0.25 deg), and wideband multilayer coatings to enhance the reflectance of the mirrors at high energies (Joensen et al. 1995; Tawara et al. 1998). A very long focal length is hardly managed using a single Send offprint requests to: daniele.spiga@brera.inaf.it spacecraft, therefore the optics and the focal plane instruments should be carried by two separate spacecrafts in formation- flight configuration. This is the baseline for the future X-ray telescopes SIMBOL-X (Pareschi & Ferrando 2006) and XEUS (Parmar et al. 2004). Other hard X-ray imaging telescopes of the future are NeXT (Ogasaka et al. 2006) and Constellation-X (Petre et al. 2006). The focusing and reflection efficiency of X-ray optics can be tested and calibrated on ground by means of full-illumination X-ray facilities like PANTER (Bräuninger et al. 2004; Freyberg et al. 2006), successfully utilized in the last years to calibrate the optics of a number of soft X-ray telescopes. The PANTER X-ray facility now allows testing in soft (0.2÷ 10 keV) and hard (15 ÷ 50 keV) X-rays multilayer-coated optics pro- totypes for future X-ray telescopes (Pareschi et al. 2005; Romaine et al. 2005). The source distance finiteness causes some departures of the optic performances, with respect to the case with the source at astronomical distance: effective area loss, different incidence angles on paraboloid and hyper- boloid, focal length displacement, a slight focal spot blurring (Van Speybroeck & Chase 1972). However, there effects can be quantified and subtracted from experimental data. After this treatment, the focusing-concentration performances of the optic can be experimentally characterized as a function of the inci- dent photon energy, in terms of Half-Energy Width (HEW) and Effective Area (EA). http://arxiv.org/abs/0704.1612v2 2 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . The focusing performance, in particular, is altered by mirror deformations that may arise in the manufacturing, handling, in- tegration, positioning processes. The consequent imaging degra- dation can be calculated from the measured departures of the mirrors from the nominal profile, by means of a ray-tracing pro- gram. As long as the geometrical optics approximation can be applied, the effect is independent of the photon energy. The fig- ure errors contribution to the HEW can be also directly measured using a highly collimated beam of visible/UV light in a precision optical bench. In this case, however, the light diffraction has to be carefully estimated and subtracted. Another drawback is the X-ray scattering (XRS) caused by the microroughness of reflecting surfaces (Church et al. 1979; Stearns et al. 1998; Stover 1995; and many others). The XRS spreads a variable fraction of the reflected beam intensity in the surrounding directions: the result is the effective area loss in the specular direction (i.e. in the focus) and a degradation of the imaging quality. The XRS is an increasing function of the pho- ton energy; due to the impact that the XRS can have on astro- nomical X-ray images quality, the height fluctuations rms of the mirror surface should not exceed few angströms. Loss of effec- tive area is also caused by interdiffusion of layers in multilayer coatings, which enhances the X-ray transmission and absorption throughout the stack. On the other hand, an uniform interdiffu- sion does not cause X-ray scattering (Spiller 1994), hence it does not contribute to the focusing degradation. The microroughness of an X-ray mirror can be measured on selected samples using several metrological instruments, each of them sensitive to a definite interval of spatial scales l̂: Long Trace Profilometers (10 cm > l̂ > 0.5 mm: Takács et al. 1999), opti- cal interference profilometers (5 mm > l̂ > 10 µm) and Atomic Force Microscopes (100 µm > l̂ > 5 nm) can be suitable in- struments to provide a detailed profile characterization of X-ray mirrors surface. It is convenient to present the deviation of sur- face from the ideality in terms of Power Spectral Density (PSD), because its values do not depend on the measurement technique in use (see ISO 10110 Standard). In addition, the XRS diagram, and consequently the HEW, can be immediately computed from the PSD at any photon energy (Church et al. 1979). In the past years, several approaches were elaborated to relate a mirror PSF (Point Spread Function) to the PSD of its surface. Among a wealth of works, we can cite (De Korte et al. 1981) the assumption of a Lorentzian model for the PSD to fit the mirror PSFs at some photon energies, allowing the derivation of two parameters (roughness rms and correlation length) of the model PSD. Christensen et al. (1988) perform a fit of experimental high-resolution XRS data dealing with the surface correlation function. Harvey et al. (1988) re- late the PSF of Wolter-I optics to the parameters of an exponen- tial self-correlation function along with a transfer function-based approach. Willingale (1988) derived the surface PSD of a mirror from the wings of a few PSFs, measured at PANTER at some soft X-ray photon energies. O’Dell et al. (1993) interpret the PSF of a focusing mirror on the basis of surface roughness and partic- ulate contamination. Zhao & Van Speybroeck (2003) construct from the PSD of a focusing mirror a model surface and compute the X-ray scattering PSF from the Fraunhofer diffraction theory. In the present work that problem is faced in a new and differ- ent way, looking for a general and simple link between measured roughness and mirror HEW. More precisely, we considered the following question: for an X-ray grazing-incidence optic, what is the maximum acceptable PSD of the surface that fulfills the angular resolution (HEW) requirements of the telescope, in all the energy band of sensitivity? In this work we shall give a definite answer to this question. In the sect. 2 we shall summarize the causes of imaging degra- dation. In the sect. 3 we show how to evaluate H(E), the XRS contribution to the HEW of a focusing mirror at the photon en- ergy E, from any surface microroughness PSD, measured over a very wide range of spatial frequencies. We shall see in the sect. 4 that for the special class of fractal surfaces we can even relate the power-law indexes of PSD and HEW, and in the sect. 5 we see how to treat the other cases. Then we prove in the sect. 6 that the formalism can be reversed, providing thereby an independent evaluation of the surface PSD from an analytical calculation over H(E), and in the sect. 7 we extend the results to focusing mirrors with more than one reflection. Finally, an example of computa- tion is provided in the sect. 8. 2. Contributions to the imaging degradation We shall henceforth indicate with λ the wavelength of photons impinging on the mirror, and we shall consider the HEW as a function of λ instead of the photon energy E. For isotropical re- flecting surfaces in grazing incidence, the X-ray scattering dis- tribution lies essentially in the incidence plane, so we denote the incidence angle on the mirror as θi and the scattering angle as θs, both measured from the surface plane (a schematic of the scatter- ing geometry is drawn in fig. 1). If we do not consider the optic roundness errors, the longitudinal deviations from the nominal profile of a focusing mirror can be classified on the basis of their typical length l̂. According to De Korte et al. (1981), they are: 1. Power errors: errors with l̂ equal to the mirror length L. They consist in a single-concavity deformation of the profile with respect to the nominal one. 2. Regularity errors: errors in the spatial range from 0.1 L < l̂ < 0.5 L. 3. Surface roughness: surface defects with l̂ < 0.1 L. However, other criteria were also formulated to separate fig- ure errors from roughness. Consider a single Fourier component of the surface profile with wavelength l̂ and root mean square σ. That Fourier component is dominated by figure error if it fulfills the condition (Aschenbach 2005) 4π sin θiσ > λ. (1) Otherwise, it is dominated by microroughness. In other words, surface defects within the smooth-surface approximation can be mainly considered as microroughness. To understand the impor- tance of this approximation, we write the optical path difference ∆s of X-rays reflected by two points of the surface with a hori- zontal spacing l̂ and vertical spacing σ̂ = 2 2σ (for optically- polished surfaces, σ is a increasing function of l̂, and usually σ≪ 10−3l̂) as ∆s = l̂(cos θs − cos θi) + σ̂(sin θi + sin θs) (2) that, for small incidence angles, becomes ∆s = l̂ sin θi(θs − θi) + σ̂(θi + θs). (3) If that component is responsible for X-ray scattering, it has to be ∆s ≈ λ, to cause the diffraction from surface features with a l̂ spacing and σ̂ height. Conversely, the ”figure errors”, which are treated with the methods of the geometrical optics, should D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 3 Fig. 1. The geometry of X-ray scattering: the strictly speaking ”reflected” rays (i.e. in the focus direction) are characterized by the equality θs = θi, the others are scattered apart. The rough sur- face is a simulated one, assuming a PSD with power-law index n = 2.4 (see sect. 4). be characterized by the inequality ∆s ≫ λ. Note that this condi- tion becomes similar to the eq. 1 in the limit |θi − θs | → 0. The application of this criterion and of the subsequent X-ray scatter- ing theory requires the incident radiation to be spatially coher- ent over the spatial scale l̂, so that the properties of the reflected wavefront are determined only by the coherence properties of the mirror surface. This in turn requires the angular diameter of the source φS to fulfill the inequality (Holý et al. 1999) l̂ sin θi . (4) This equation sets a maximum to the values of l̂ that can be used in the application of the results presented in this work. The limi- tation can affect X-ray sources at finite distance, like those used for X-ray optics calibrations in full-illumination setup. For very distant astronomical X-ray sources, the condition 4 is met even for larger l̂, up to l̂ ≈ L. It is worth pointing out that, for a given reflecting surface, the separation of figure errors from microroughness is strongly affected by the incidence/scattering angles. In fact, even for large l̂, ∆s can become comparable with λ, if θi and θs are sufficiently small: thus, the spatial wavelength window of interest for X-ray scattering can shift to the large l̂ domain (or, equivalently, to the range of low spatial frequencies f = 1/l̂), provided that the con- dition 4 is fulfilled. Let us now consider how to separate the figure and scattering terms in HEW data. In absence of XRS, the mirror PSF would be independent of the energy and due only to figure errors (i.e. in the approximation of the geometrical optics). The resulting HEW would be also constant. Instead, due to the XRS, the figure PSF is convolved with the X-ray scattering PSF to return the PSF(λ) being measured (Willingale 1988; Stearns et al. 1998; and many others), PS F(λ) = PS Ffig ⊗ PS FXRS(λ). (5) The resulting HEW will depend on the photon wavelength, as it does the PSF. In order to isolate the scattering term from the total PSF a deconvolution should be carried out, provided that the PSFfig is known. However, if we assume that the XRS and the mirror deformations are statistically independent, the total HEW can be approximately calculated as the squared sum of the two contributions: HEW2(λ) ≈ HEW2fig + H 2(λ). (6) An estimation of HEWfig can be obtained: 1. from the application of a ray-tracing code to several mea- surements of the mirror profile, 2. from reliable extrapolation of the HEW(λ) curve to E → 0, in absence of low-energy diffraction effects like dust contam- ination, studied in detail by O’Dell et al. (1993) 3. from a direct measurement of the HEW in visible/UV light, provided that the diffraction at the mirror edges can be reli- ably calculated and subtracted. Once known the measured HEW(λ) experimental trend and the HEWfig term, the eq. 6 can be used to isolate the scattering contribution from the experimental HEW trend: we shall prove in the next section that the H(λ) function is immediately related to the reflecting surface 1D Power Spectral Density (PSD) P( f ) P( f ) = z(x)e−2πi f dx where z(x) is a height profile (of length L) of the mirror, mea- sured in any direction (Stover 1995): the surface is assumed to be isotropic, and the spectral properties of the profile to be rep- resentative of the whole surface. The PSD is often measured in nm3 units, and for optically-polished surfaces it is usually a de- creasing function of the frequency f . PSD measurements have always a finite extent [ fmin, fmax], determined by the length and the spatial resolution of the mea- sured profile. As well known, the surface rms σ is simply com- puted from the PSD by integration over the spatial frequencies f : ∫ fmax P( f ) d f (8) note that the integration range should always be specified. 3. Estimation of H(λ) for single-reflection focusing mirrors 3.1. Single-layer coatings Firstly, we suppose the mirror to be plane and single-layer coated. For a surface with roughness rms σ, the specular beam intensity obeys the well-known Debye-Waller formula R = RF exp 16π2σ2 sin2 θi , (9) here RF is the reflectivity at the grazing incidence angle θi, as calculated from Fresnel’s equations (zero roughness). However, it should be noted in the eq. 9 that neither the spatial frequen- cies range where the PSD should be integrated is specified, nor the separation between reflected and scattered ray is clearly in- dicated: these ambiguities can be solved as follows. Let us derive the total scattered intensity Is from the con- servation of the energy: for smooth surfaces, i.e. fulfilling the inequality 2σ sin θi ≪ λ, we can approximate Is = I0RF 1 − exp 16π2σ2 sin2 θi ≈ I0RF 16π2σ2 sin2 θi .(10) 4 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . In grazing incidence, X-ray scattering lies mainly in the inci- dence plane. Moreover, the normalized scattered intensity per ra- dian at the scattering angle θs (either θs > θi or θs < θi) is related to the PSD along with the well-known formula at first-order ap- proximation (Church et al. 1979; Church & Takács 1986), valid for smooth, isotropic surfaces and for scattering directions close to the specular ray (i. e. |θs − θi| ≪ θi), sin3 θiRFP( f ) (11) where P( f ) is the Power Spectral Density of the surface (eq. 7) and I0 is the flux intensity of the incident X-rays. If the scattered intensity is evaluated at the scattering angle θs, the PSD can be immediately evaluated as a function of the spatial frequency f : f = l̂−1 = cos θi − cos θs sin θi(θs − θi) . (12) In the eq. 12 the approximation was justified by the assumption |θs − θi| ≪ θi and the negative frequencies are conventionally assumed to scatter at θs < θi: the assumed approximations make the XRS diagram symmetric, because the PSD is an even func- tion. For a single-reflection mirror shell, the extension of the for- mulae above-mentioned is straightforward by regarding |θs − θi| as the angular distance at which the PSF is evaluated. The fo- cal image is the superposition of many identical XRS diagrams on the image plane, generated by every meridional section of the mirror shell: since a π angle rotation of every meridional plane of the shell sweeps the whole image plane, the scattered intensity is spread over a π angle. The integration on circular coronae used to compute the mirror PSF (at positive angles) compensates this factor multiplying the XRS diagram by 2π (De Korte et al. 1981). The remaining 2-fold factor accounts for the negative frequen- cies in the surface PSD. We shall henceforth suppose that the fac- tor 2 is embedded in the PSD definition. Therefore, the eqs. 11 and 12 can be used to describe the XRS contribution to the PSF. We are now interested in the scattered power at angles larger than a definite angle αmeasured from the focus. Due to the steep fall of scattering intensity for increasing angles, the integral has a finite value I [|θs − θi| > α] = ∫ π−θi dθs. (13) Combining eqs. 11 and 13, one obtains: I [|θs − θi| > α] = I0RF 16π2 sin3 θi ∫ π−θi P( f ) dθs (14) with respect to the definition used in the eqs. 7 and 11, a factor 2 was included in the PSD. The upper integration limit corre- sponds to a photon back-scattering: at first glance, this seems to violate our small-scattering angle assumption (eqs. 11 and 12), but it should be remembered that only the angles close to θi contribute significantly to the integral in eq. 13: hence its value should not be significantly affected by a particular choice of the upper integration limit. After a variable change from θs to f (eq. 12), the eq. 14 becomes (approximating cos θi ≈ 1 in the upper integration limit): I [|θs − θi| > α] = I0RF 16π2 sin2 θi P( f ) d f (15) where f0 = α sin θi/λ is the spatial frequency corresponding to the scattering at the angle α. As expected, this equation equals the integrated scattering according to the eq. 10, provided that we identify I [|θs − θi| > α] with Is, and the squared roughness rms with P( f ) d f . (16) The eq. 16 is in agreement with the eq. 8, but it states clearly the window of spatial frequencies involved in the XRS. Therefore, for a definite angular limit α the ”reflected beam” intensity can be simply calculated by using the Debye-Waller formula, pro- vided that σ2 is computed from the PSD integration beyond the frequency f0, which corresponds to an X-ray scattering at α. The upper integration limit is a very high frequency (close to 1/Å): hence, the atomic structure of the surface is not important in the integral of the eq. 16. Moreover, considering that the PSD trend for optically-polished surfaces decreases steeply for increasing f , the largest contribution to the integral should be given by the frequencies close to f0. Now we can evaluate H(λ), the scattering term of the HEW. For simplicity, in the following we will suppose that the HEW is obtained from the collection of all the reflected/scattered pho- tons: this allows us to avoid problems related to the finite size of the detector, and to extend the surface roughness PSD up to very large spatial frequencies. By definition, H(λ) is twice the angu- lar distance from focus at which the integrated scattered power halves the total reflected intensity: I [|θs − θi| > α] = I0RF (17) we immediately derive, from the eq. 9, 16π2σ2 sin2 θi , (18) where σ2 has now the meaning as per the eq. 16. Solving the eq. 18 for σ2 and equating to the integral of the PSD, P( f ) d f = λ2 ln 2 16π2 sin2 θi , (19) once known the PSD from topography measurements over a wide range of spatial frequencies, the PSD numerical integra- tion in the eq. 19 allows to recover f0. In turn, f0 is related to H(λ) through the eq. 12, that we write in the following form H(λ) = 2λ f0 sin θi , (20) where H is measured in radians. Note that the condition H(λ) ≪ θi is very important, for the eq. 20 to hold. Small scattering angles and grazing incidence are also very important for the considerations that follow. 3.2. Multilayer coatings The obtained result (eq. 19) can be extended to mirrors with multilayer coatings, used to enhance the grazing incidence re- flectivity of mirrors in hard X-rays (E > 10 keV). In general, the multilayer cannot be characterized by means of a single PSD, due to the evolution of the roughness throughout the stack (Spiller et al. 1993; Stearns et al. 1998). Moreover, due to the interference of scattered waves at each multilayer interface, the final scattering pattern is more structured than eq. 11, with peaks whose height depends on the phase coherence of the interfaces D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 5 Fig. 2. Dependence of the spectral exponents for different in- dexes n of a power-law PSD, for a single-reflection focusing mirror. In the forbidden region (n > 3) γ would be negative. (Kozhevnikov 2003). The HEW term can be computed numeri- cally from the XRS diagram. In order to extend the eq. 19 to mirrors coated with a graded multilayer, we have to assume the additional requirements: 1. the PSD is constant and completely coherent throughout the multilayer stack: i.e., the deposition process does not cause additional roughness and replicates simply the profile of the substrate. Therefore, all the PSDs and all the cross- correlation between interface profiles equal the PSD mea- sured at the multilayer surface. This is often observed in the l̂ > 10 µm regime (Canestrari et al. 2006), where most of frequencies f0 fall when the incidence angle is less than 0.5 deg. Most of microroughness growth, indeed, takes place for 10 µm> l̂ > 0.1 µm. 2. the multilayer reflectivity Rλ(θi) at the photon wavelength λ changes gradually over angular scales of H(λ). Ideally, this condition should be fulfilled by wideband multilayer coat- ings for astronomical X-ray mirrors. Under these hypotheses, a quite tedious calculation reported in appendix A shows that the eq. 19 can be approximately ap- plied also with multilayer coatings. The following developments also apply in that case. 4. H(λ) for a fractal surface We apply now the equations 19 and 20 to the typical (monodi- mensional) PSD model for optically-polished surfaces, a power- law (Church 1988) P( f ) = , (21) where the power-law index n is a real number in the interval 1 < n < 3 and Kn is a normalization factor. A power-law PSD is typical of a fractal surface, and it represents the high-frequency regime of a K-correlation model PSD (Stover 1995). This model exhibits a saturation for f → 0 that avoids the PSD divergence. In practice, the fractal behavior dominates in almost all spatial frequencies of interest for X-ray optics. There are interesting reasons for which n can take val- ues on the interval (1:3). In fact, for a surface in the 3D space, n is related to its Hausdorff-Besicovitch dimension D (Barabási & Stanley 1995) along with the equation n = 7 − 2D (see Church 1988; Gouyet 1996). The restriction 1 < n < 3 for a fractal surface is therefore necessary to have 3 > D > 2. A power-law PSD is particularly interesting because the in- tegral on left-hand side of the eq. 19 can be explicitly calculated: f 1−n0 − n − 1 λ2 ln 2 16π2 sin2 θi . (22) As 1 − n < 0, in grazing incidence the (2/λ)1−n term can be neglected with respect to f 1−n0 . By isolating the frequency f0 and using the eq. 20 to derive H(λ), we obtain after some algebra, for the scattering term of the HEW, H(λ) = 2 16π2Kn (n − 1) ln 2 sin θi . (23) This equation states that: 1. The H(λ) function for a power-law PSD has a power-law dependence on the photon energy E ∝ 1/λ, i.e., H(E) ∝ Eγ. The power-law index γ is related to the PSD power-law index n through the simple equation: 3 − n n − 1 . (24) As 1 < n < 3, γ is positive, i.e. H is an increasing function of the photon energy. For a fixed value of Kn, the HEW di- verges quickly for n ≈ 1 but very slowly for n ≈ 3: a PSD power-law index close to 2-3 would hence be preferable in order to reduce the degradation of focusing performances for increasing energies. 2. H(λ) depends on the sine of the incidence angle at the γth power. In other words, the HEW depends only on the ratio sin θi/λ: this scaling relation shows that for a given power- law PSD (with n < 3) at a given photon wavelength λ we can reduce the HEW by decreasing the incidence angle. 3. H(λ) increases with the PSD normalization Kn, as expected: the dependence is also a power law with spectral index n − 1 . (25) As for γ(n), the closeness of n to the maximum allowed value for fractal surfaces makes less severe the roughness effect on imaging degradation. The functions β and γ are plotted in fig. 2. For instance, if n = 2, γ = β = 1, and H(E) increases linearly with both pho- ton energy and Kn coefficient. The divergence of indexes β, γ for n ≈ 1 makes apparent the importance of obtaining steep PSDs in the optical polishing of X-ray mirrors. Finally, it is worth not- ing that for n > 3 there is the theoretical possibility of a slight decrease of H(E) for increasing energy because γ(n) becomes negative. To clarify the dependence of the HEW on the power-law in- dex n and the incidence angle, we depict in fig. 3 and 4 some ex- amples of H(E) simulations (single reflection) for some power- law PSDs in the photon energy range 0.1-50 keV. The H(E) curves were computed using the eq. 23. In fig. 3 the incidence angle θi is fixed at 0.5 deg and the index n is variable; a constant n = 1.8 and a variable θi is instead assumed in the simulations of fig. 4. Note in fig. 3 the slower H(E) increase for larger n and the common intersection point, determined by the particular choice of the incidence angle and the σ = 4 Å value in the window of spatial wavelengths [100 ÷ 0.01 µm]. 6 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . Fig. 3. H(E) simulations assuming power-law PSDs with con- stant σ = 4 Å in the spatial wavelengths range [100 ÷ 0.01µm], but variable power-law index n. The incidence angle is fixed at θi = 0.5 deg. Fig. 4. H(E) simulations assuming a power-law PSD with power-law index n = 1.8 and with σ = 4 Å in the spatial wave- lengths range [100 ÷ 0.01 µm], but variable incidence angle θi. 5. Numerical integration of the PSD A power-law PSD is a modelization that can be used for optically-polished surfaces. If the polishing process is not op- timized or a reflecting layer is grown onto a optically polished substrate, several deviations from a power-law trend can be ob- served. A typical ”bump”, for instance, can be present in the PSD of multilayer coatings, often in the range of spatial wavelengths [10 ÷ 0.1 µm], as a result of the replication of the substrate to- pography and of fluctuations intrinsically related to the random deposition process (Spiller et al. 1993; Stearns et al. 1998). If the PSD deviates significantly from a power-law, the eq. 23 can- not be used. However, if the surface PSD has been extensively measured over a wide range of spatial frequencies [ fm, fM] (wide enough to have fm < f0(λ) for all λ), the HEW scattering term H(λ) can be computed by numerical integration (eqs. 19 and 20), on condition that the following approximation is valid: P( f ) d f ≈ P( f ) d f . (26) The condition above is usually satisfied when f0 ≪ fM i.e. when the following inequality holds: H(λ)≪ 2λ fM sin θi . (27) As we are also interested in computing H(λ) in hard X-rays (small λ), there is the possibility that the two integrals in the eq. 26 differ by a significant factor. In this case the integral can be corrected by adding the remaining term P( f ) d f = P( f ) d f + P( f ) d f , (28) that can be evaluated, in principle, by measuring the mirror re- flectivity within an angular acceptance corresponding to the spa- tial frequency fM, and using the Debye-Waller formula to derive σ2; then, the importance of measuring the PSD in a very wide frequencies interval becomes apparent. The value of f0 depends strongly on both incidence angle and photon energy: for soft X- rays (< 10 keV) and very small angles (< 0.2 deg) the character- istic spatial wavelength l̂ = 1/ f0 often falls in the millimeter or centimeter range. It should be noted that, if the detector is small, a fraction of the scattered photons can be lost; to account for the finite an- gular radius of the detector d (as seen from the optic principal plane), one should integrate the PSD over the smaller interval [ f0, d sin θi/λ] to recover the measured H(λ) trend. As an alter- native method, one can compare the theoretical predictions of eqs. 19 and 20 with the experimental H(λ) values, as calculated from the Encircled Energy normalized to the photon count fore- seen by the Fresnel equations (i.e. with zero roughness), rather than to the maximum of the measured Encircled Energy func- tion. 6. Computation of the PSD from the H(λ) trend If the approach described above can be used to simulate the HEW trend from a measured surface PSD, the reverse problem, i.e. the derivation of surface PSD from the measured HEW trend is also possible. This requires that the figure error contribution had been reliably measured, in order to isolate the scattering term function H(λ) using the eq. 6. This problem is interesting for three reasons at least: 1. it is a quick, non-destructive surface characterization method in terms of its PSD. 2. The measurement is extended to a large portion of the illu- minated optic, hence local surface features are averaged and ruled out from the PSD. 3. For a given HEW(λ) requirement in the telescope sensitiv- ity energy band, it allows establishing the maximum allowed In order to find an analytical expression for the PSD, we note that the spatial frequency f0 that scatters at an angular distance H/2 from the specular beam is a function only of λ, along with the eq. 20. Solving for f0, we have f0(λ) ≈ H(λ) sin θi . (29) We suppose that all scattered photons are collected, so we can assume the eq. 19 as valid. By deriving both sides of eq. 19 with respect to λ, we have P( f ) d f 8π2 sin2 θi λ, (30) that is, P( f0) = 8π2 sin2 θi λ, (31) D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 7 and, using the eq. 29 to compute the derivative of f0: + f0P( f0) sin θi dH(λ) P( f0) = 8π2 sin2 θi λ. (32) Now remember that, in grazing incidence, f0 ≪ 2λ−1 by several orders of magnitude. Even if P( f ) is not a power-law, it is always a steeply decreasing function of f . Moreover, it should have over [ f0, 2λ −1] an average PSD index ñ > 1, for the reasons explained in the sect. 4. This means that P( f0) P(2λ−1) , (33) therefore, in practical cases the 2λ−1P(2λ−1) term in the eq. 32 is negligible with respect to f0P( f0). Consequently, we can neglect the first term of eq. 31: then we have P( f0) ≈ 8π2 sin2 θi λ. (34) Combining this with the eq. 29 and collecting the constants, we obtain the final result P( f0) 4π2 sin3 θi ≈ 0. (35) The eq. 35 enables the computation of the PSD (at the spatial frequency given by the eq. 29) along with the derivative of the ratio H(λ)/λ with respect to λ. The obtained equation shows that P( f ) is inversely propor- tional to the derivative of H(λ)/λ. This result seems strange at first glance, because by decreasing H(λ) one would obtain a larger P( f ) (a rougher surface). One should remember, indeed, that by reducing H(λ) we increase P( f0), but f0 is shifted to- wards the low frequencies domain, where P( f0) is expected to be higher. In fact, the ”rough” or ”smooth” feature of the surface depends on whether f0 or P( f0) varies more rapidly, i.e. on the overall H(λ) trend. We can also check the correctness of the eq. 35 by computing the PSD for the particular case of the H(λ) derived from the inte- gration of a power-law PSD (the eq. 23, derived under the same approximation, the eq. 33). If the results are correct, the substi- tution of the HEW trend of the eq. 23 in the eq. 35 should return the original PSD (eq. 21). The straightforward, but lengthy cal- culation (carried out in appendix B) shows that the substitution returns P( f0) = as expected. The eq. 35 should be approximately valid also for graded multilayers with a slowly-decreasing reflectivity (see sect. 3.2), however, due to the approximations needed to extend the eq. 19 to the multilayers, the resulting PSD should be considered a ”first guess” in this case. Then, the matching of the PSD to the required HEW trend should be checked by means of a detailed computation of the XRS PSF(λ). 7. Extension to X-ray mirrors with multiple reflections The formalism exposed in the previous sections can be extended to a double-reflection optic (like a Wolter-I one). In this op- tical configuration, photons are firstly reflected by a parabolic surface and subsequently by a hyperbolic one. If the smooth- surface condition is satisfied, multiple scattering is often negli- gible (Willingale 1988) and the scattering diagrams of the two reflecting surfaces can be simply summed (De Korte et al. 1981; Stearns et al. 1998). The source is assumed to be at infinite dis- tance, then X-rays impinge on the two surfaces at the same an- gle θi. If the surface PSDs are the same for both reflections, the scattering diagram will be simply doubled. Thus, the integrated scattered intensity is also doubled: Is = 2I0R 1 − exp 16π2σ2 sin2 θi . (37) The RF factor is squared in the eq. 37 because each ray is re- flected twice: in absence of scattering the reflected power would be I0R F, so the half-power scattering angle condition reads |θs − θi| > F, (38) and, combining the eqs. 37 and 38, we obtain 16π2σ2 sin2 θi . (39) Solving for σ2, and using the eq. 16, P( f ) d f = λ2 ln(4/3) 16π2 sin2 θi , (40) that differs from the eq. 19 only in the factor ln(4/3) instead of ln 2 on right-hand side. Consequently, the corresponding differ- ential equation is P( f0) ln(4/3) 4π2 sin3 θi ≈ 0. (41) Similar equations can be derived for an optical system with an arbitrary number of reflections N: to compute the H(λ) from the PSD, P( f ) d f = 16π2 sin2 θi 2N − 1 . (42) If the PSD is a power-law P( f ) = Kn/ f n we can generalize the eq. 23: H(λ) = 2 2N − 1 16π2Kn (n − 1) sin θi , (43) note the divergence of the logaritmic factor for increasing N, due to the negative exponent 1/(1 − n). This indicates that H(λ) increases rapidly with the number of reflections, as expected. Finally, we can also generalize the differential eq. 35 to an arbitrary number of reflections, P( f0) 4π2 sin3 θi ≈ 0. (44) In the eqs. 42 and 44, f0 is always related to H(λ) by the eq. 29. 8 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . Fig. 5. an hypothetical PSD with reasonable values and a PSD break around a 100 µm spatial wavelength (dashed line). This PSD is adopted to compute the corresponding HEW trends for 1,2,3 reflections at a 0.3 deg grazing incidence angle (fig. 6). The achieved HEW trends were used to re-calculate the respective PSDs (solid line). For clarity, we do not plot the single PSDs, but just their overlap. 8. An example As an application of the equations reported above, we shall make use of a simulated surface PSD with reasonable values, that is not a power-law. The PSD (see fig. 5, dashed line) is ex- tended from 105 µm down to a 0.01 µm spatial wavelength, with a break around 100 µm: at the lowest frequencies the PSD is steep (n ≈ 2.3), whereas at the largest frequencies it is smoother (n ≈ 1.3). From the discussion in sect. 4 concerning the relation between the exponents of the PSD and the HEW (eq. 24), we should expect that the PSD break causes a slope change in the function H(λ): however, as the actual PSD is not a power-law, the H(λ) function should be computed by means of the eqs. 20 and 42. Before carrying out the integration, we can remark qual- itatively that, as we increase the photon energy, the highest fre- quencies in the PSD (where the PSD index becomes smaller) become important; hence, we can expect a steeper increase of the HEW at the highest energies. The analysis is made quantitative in fig. 6, where we show the calculated HEW trends from the PSD in fig. 5 (the dashed line) by means of the eqs. 20 and 42, assuming 1,2,3 reflections at the same grazing incidence angle (0.3 deg). The approxima- tion of eq. 26 was adopted. In addition to the scattering term, 15 arcsec of HEW due to figure errors were added in quadrature. The HEW increases slowly (concave downwards) at low ener- gies, corresponding to a frequency f0 in the steeper part of the PSD. Then it increases more steeply (concave upwards) when the energy becomes large enough to set f0 in the portion of the spectrum with n ≈ 1.3. By increasing the number of reflections, the HEW values also increase, and the ”turning point” where the HEW starts to diverge (arrows in fig. 6) shifts at lower X-ray energies. All the calculation is based on the assumption that the contribution of the PSD over the maximum measured frequency fM = 0.01 µm is negligible. Otherwise, the computed HEW val- ues will be underestimated (see sect. 5). In addition to the general trend of the HEW, there are oscil- lations due to small irregularities in the adopted PSD: the cal- culation is, in fact, very sensitive to small variations of the PSD values. Notice that for a definite energy all the frequencies larger than f0 contribute to the HEW value, even if the largest contri- Fig. 6. the HEW trend computed from the PSD for 1,2,3 reflec- tions, plus 15 arcsec of HEW due to figure errors. The HEW trends were used to compute back the PSD (the solid line in fig. 5) to verify the reversibility of the calculation. The energy at which the concavity change takes place is also indicated (ar- rows). bution comes from frequencies near f0: this is a consequence of the steeply decreasing trend of the PSD. We checked the reversibility of the result by computing the PSD from the HEW trends (after subtracting in quadrature 15 arcsec figure error) by means of the eq. 44 with the respec- tive value of N. The resulting PSDs (the solid line in fig. 5) were overplotted to the initial PSD, with a perfect superposition. Each obtained PSD has, indeed, an extent of spatial frequencies smaller than the initial one: the overall PSD ranges from 104 to 11 µm (vs. the initial 105 ÷ 0.01 µm), and the smaller wave- lengths could be computed from the HEW trend with N = 3. The limitation in spatial frequency ranges occurs for two reasons: 1. small f − large l̂: all the power scattered by the lowest fre- quencies is found at angles less than 1/2 HEW even for the lowest energies being considered: therefore, that part of the spectrum is not necessary to compute the HEW in the energy range of interest; 2. large f − small l̂: the PSD is computed from a derivative, therefore the information concerning the absolute magnitude of the HEW is substantially lost. This information was in- cluded in the integral of the PSD (eq. 42) for the maximum considered energy. Therefore, from the integral in the eq. 42 we cannot re- cover the PSD over the minimum computed spatial wavelength (11 µm, using the HEW trend with N = 3), but we can at least calculate the value of σ at spatial wavelengths smaller than 11 µm. Substituting the incidence angle and the minimum pho- ton wavelength being considered (λ = 0.24 Å) in the eq. 42 with N = 3 and with the approximation of the eq. 26, we ob- tain σ = 1.6 Å, in perfect agreement with the value computed from the original PSD. Summing up, for a given incidence angle the H(λ) function in a definite photon energy range is equivalent to the PSD in a corresponding range of spatial frequencies f (or equivalently, spatial wavelengths l̂), plus the integral of the PSD beyond the maximum frequency being computed. Therefore, requirements of a definite HEW(λ) function in designing an X-ray optical sys- tem can be translated in terms of PSD in a frequencies range [ fmin, fmax] plus the surface rms at frequencies beyond fmax. The usefulness of such a relationship is apparent. D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 9 9. Conclusions In the previous pages we have developed useful equations to compute the contribution of the X-ray scattering to the HEW of a grazing incidence X-ray optic, by means of a simple integra- tion. The formalism has been inverted in order to derive the PSD of the surface from the function H(λ), and it can be extended to an arbitrary number of reflections at the same incidence an- gle. The equations are valid for a single-layer coating mirror, but they can be approximately applied to a multilayer-coated mir- ror. This approach is particularly useful in order to establish the surface finishing level needed to keep the X-ray scattering HEW of X-ray optics within the limits fixed by the X-ray telescope requirements. It should be remarked that the reasoning was developed for the Half-Energy Width, but it can be extended to any angular diameter including a fraction η of the energy spread around the focal point. To do this, it is sufficient to substitute the logarithmic factors in equations 42, 43, 44, 2N − 1 N − 1 + η , (45) and for instance, to compute the 90%-energy diameter for a dou- ble reflection mirror, simply substitute η = 0.9 and N = 2. The proof is straightforward: however, one should always keep in mind that the energy diameters computed with this method can be considered valid only if they are much smaller than the inci- dence angle θi. Notice that in the development of the exposed formalism we have supposed, in addition to the smooth-surface condition, two additional hypotheses: 1. the source is at infinite distance from the mirror 2. the X-ray detector is large enough to collect all the scattered photons. In order to apply the mentioned equations to experimental calibrations of X-ray optics at existing facilities (like MPE- PANTER), where the source is at a finite distance and the de- tector has a finite size, some corrections should be taken into account. We will deal with their quantification in a subsequent paper. Appendix A: Extension to multilayer coatings Here we provide with a plausibility argument to extend the for- malism of sect. 3.1 to mirror shells with multilayer coatings (see sect. 3.2). The intensity of a scattered wave at each interface is proportional to its PSD as per the eq. 11, and the overall scatter- ing diagram will be their coherent interference. To simplify the notation, we neglect the X-ray refraction and we suppose that the incidence angle is beyond the critical angles of the multilayer components. The electric field scattered by the kth interface can be written as Ek = E0TkrkXk( f ) exp(−iφk), (A.1) where rk is the single-boundary amplitude reflectivity, E0 the incident electric field amplitude, the weights Tk are the rel- ative amplitudes of the electric field in the stack (in scalar, single-scattering approximation), and account for the extinction of the incident X-rays due to gradual reflection and absorption. Xk( f ) is the single-boundary scattering power (proportional to the PS D( f ) amplitude), and φk is the phase of the scattered wave at θs by the k th interface φk = 2π sin θi + sin θs zk, (A.2) where zk is the depth of the k th interface with respect to the outer surface of the multilayer. Now, the measured intensity is |Escatt|2 = = |E0|2|Xk( f )|2 rkTk exp(−iφk) . (A.3) Now, |E0|2 = I0, the incident X-ray flux intensity, and |Xk( f )|2 is proportional to the interfacial PSD P( f ), which is independent of k by hypothesis. Assuming the proportionality factor of eq. 11 for |Xk( f )|2, we obtain for the scattering diagram sin3 θiP( f ) rkTk exp(−iφk) (A.4) and if we set Kλ(θi, θs) = rkTk exp(−iφk) , (A.5) the eq. A.4 becomes analogous to the eq. 11, with Kλ(θi, θs) playing the role of RF. Note that Kλ(θi, θi) = Rλ(θi), the mul- tilayer reflectivity in single reflection approximation. As before, we write the scattering diagram for a mirror with axial symmetry as a function of the angular distance from the focus α = |θi − θs| averaging the contributions of negative and positive frequencies sin3 θiP( f )[Kλ(θi, θi − α) + Kλ(θi, θi + α)]. (A.6) For a single reflection optic, we can calculate the scattered power over H/2, where H is the scattering term of optic Half-Energy Width: Is[α > H/2] = I0Rλ(θi). (A.7) Now, the steps 18 and 19 can be repeated: Kλ(θi, θi + α) + Kλ(θi, θi − α) Rλ(θi) P( f ) d f = λ2 ln 2 8π2 sin2 θi (A.8) where f0 is still defined by the eq. 20. For small scattering angles (α ≪ θi), since we assumed a slow variation of Rλ over angular scales of H/2 (and the same occurs for Kλ), we can approximate Kλ(θi, θi ± α) ≈ Rλ(θi) ± α ∂Kλ(θi, θs) θs=θi . (A.9) Substituting in the eq. A.8, we obtain P( f ) d f ≈ λ2 ln 2 16π2 sin2 θi (A.10) because the two derivatives have opposite sign and cancel out. This is the same equation found for the case of a single-layer coating (eq. 19). 10 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . Appendix B: Derivation of the PSD from the HEW for a fractal surface (single reflection) We recall here the H(λ) trend for a power-law PSD (eq. 23): H(λ) = 2 16π2Kn (n − 1) ln 2 sin θi (B.1) we verify that it returns a power-law PSD if substituted in the differential eq. 35: P( f0) 4π2 sin3 θi = 0. (B.2) To simplify the notation, we write simply H instead of H(λ): by carrying out the derivation, n − 1 16π2Kn (n − 1) ln 2 (sin θi) n−1 λ n−1−3. (B.3) Using again the eq. B.1: n − 1 −3 (B.4) hence, the related PSD is P( f0) = − 4π2 sin3 θi λ3 ln 2 4π2H sin3 θi n − 1 . (B.5) Now, we can derive (n − 1)/2 from the eq. B.1, n − 1 4π2HKn )−n ( sin θi (B.6) and combining the eqs. B.5-B.6, one obtains P( f0) = Kn H sin θi , (B.7) that is, by recalling the eq. 29, P( f0) = (B.8) i.e., the expected power-law PSD. Acknowledgements. Many thanks to G. Pareschi, O. Citterio, R. Canestrari, S. Basso, F. Mazzoleni, P. Conconi, V. Cotroneo (INAF/OAB) for support and useful discussions. The author is indebted to MIUR (the Italian Ministry for Universities) for the COFIN grant awarded to the development of multilayer coatings for X-ray telescopes. References Aschenbach, B., 1988, Design, construction, and performance of the ROSAT high-resolution X-ray mirror assembly. Applied Optics, Vol. 27, No. 8, p. 1404-13 Aschenbach, B., 2005, Boundary between geometric and wave optical treatment of x-ray mirrors. In Proc. SPIE, vol. 5900, p. 59000D Barabási, A. L., Stanley, H. E., 1995, Fractal Concepts in Surface Growth, Cambridge University Press Bräuninger, H., Burkert, W., Hartner, G. D., et al., 2004, Calibration of hard X- ray (15-50 keV) optics at the MPE test facility PANTER. In Proc. SPIE, vol. 5168, p. 283-293 Canestrari, R., Spiga, D., Pareschi, G., 2006, Analysis of microroughness evolu- tion in X-ray astronomical multilayer mirrors by surface topography with the MPES program and by X-ray scattering. In Proc. SPIE, vol. 6266, p. 626613 Christensen, F. E., Hornstrup, A., Schnopper H. W., 1988, Surface correlation function analysis of high resolution scattering data from mirrored surfaces obtained using a triple-axis X-ray diffractometer. Applied Optics, vol. 27, No. 8, p. 1548-63 Church, E. L., Jenkinson, H. A., Zavada, J. M., 1979, Relationship between sur- face scattering and microtopographic features. Optical Engineering, vol. 18, p. 125-136 Church, E. L., 1988, Fractal surface finish. 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SPIE, vol. 5900, p. 59000S Stearns, D. G., Gaines, D. P., Sweeney, D. W., et. al., 1998, Non-specular X-ray scattering in a multilayer-coated imaging system. J. Appl. Phys, vol. 82 (2), p. 1003-28 Spiller, E., Stearns, D. G., Krumrey, M., 1993, Multilayer X-ray mirrors: inter- facial roughness, scattering, and image quality. J. Appl. Phys, vol. 74 (1), p. 107-118 Spiller, E., 1994, Soft X-rays Optics, SPIE Optical Engineering Press Stover, J. C., 1995, Optical Scattering: measurement and analysis, SPIE Optical Engineering Press Takács, P. Z., Qian, S., Kester, T., et al., 1999, Large-mirror figure measurement by optical profilometry techniques. In Proc. SPIE, vol. 3782, p. 266-274 Tawara, Y., Yamashita, K., Kunieda, H., et al., 1998, Development of a multilayer supermirror for hard x-ray telescopes. In Proc. SPIE, vol. 3444, p. 569-575 Van Speybroeck, L. P., Chase, R. C., 1972, Design Parameters od Paraboloid- Hyperboloid Telescopes for X-ray astronomy. Applied Optics, vol. 11, No. 2, p. 440-445 Weisskopf, M. C., 2003, Three years of operation of the Chandra X-ray Observatory. In Proc. SPIE, vol. 4851, p. 1-16 Willingale, R., 1988, ROSAT wide field camera mirrors. Applied Optics, vol. 27, No. 8, p. 1423-39 Zhao, P., Van Speybroeck, L. P., 2003, New method to model x-ray scattering from random rough surfaces. In Proc. SPIE, vol. 4851, p. 124-139 Introduction Contributions to the imaging degradation Estimation of H() for single-reflection focusing mirrors Single-layer coatings Multilayer coatings H() for a fractal surface Numerical integration of the PSD Computation of the PSD from the H() trend Extension to X-ray mirrors with multiple reflections An example Conclusions Extension to multilayer coatings Derivation of the PSD from the HEW for a fractal surface (single reflection)

The focusing performance of X-ray optics (conveniently expressed in terms of HEW, Half Energy Width) strongly depend on both mirrors deformations and photon scattering caused by the microroughness of reflecting surfaces. In particular, the contribution of X-ray Scattering (XRS) to the HEW of the optic is usually an increasing function H(E) of the photon energy E. Therefore, in future hard X-ray imaging telescopes of the future (SIMBOL-X, NeXT, Constellation-X, XEUS), the X-ray scattering could be the dominant problem since they will operate also in the hard X-ray band (i.e. beyond 10 keV). [...] Several methods were proposed in the past years to estimate the scattering contribution to the HEW, dealing with the surface microroughness expressed in terms of its Power Spectral Density (PSD), on the basis of the well-established theory of X-ray scattering from rough surfaces. We faced that problem on the basis on the same theory, but we tried a new approach: the direct, analytical translation of a given surface roughness PSD into a H(E) trend, and - vice versa - the direct translation of a H(E) requirement into a surface PSD. This PSD represents the maximum tolerable microroughness level in order to meet the H(E) requirement in the energy band of a given X-ray telescope. We have thereby found a new, analytical and widely applicable formalism to compute the XRS contribution to the HEW from the surface PSD, provided that the PSD had been measured in a wide range of spatial frequencies. The inverse problem was also solved, allowing the immediate evaluation of the mirror surface PSD from a measured function H(E). The same formalism allows establishing the maximum allowed PSD of the mirror in order to fulfill a given H(E) requirement. [...]

Introduction The adoption of grazing-incidence optics in X-ray telescopes in the late 70s allowed a great leap forward in X-ray astronomy because they endowed the X-ray instrumentation with imag- ing capabilities in the soft X-ray band (E < 10 keV). The excellent performances of the soft X-ray telescopes ROSAT (Aschenbach 1988), Chandra (Weisskopf 2003) and Newton- XMM (Gondoin et al. 1998) are well known. To date, the utilized technique to focus soft X-rays consists in systems of double-reflection mirrors with a single layer coat- ing (Au, Ir) in total external reflection at shallow grazing inci- dence angles. In this case, the incidence angle θi (as measured from the mirror surface) cannot exceed the critical angle for total reflection, otherwise the mirror reflectivity would be very low. The critical angle is inversely proportional to E, the energy of the photons to be focused. Using Au coatings, for instance, the incidence angle cannot exceed ∼ 0.4 deg for photon energies E ≈ 10 keV. An extension of this technique to the hard X-ray energy band (E > 10 keV) can be pursued by combining long focal lengths (> 10 m), very small incidence angles (0.1 ÷ 0.25 deg), and wideband multilayer coatings to enhance the reflectance of the mirrors at high energies (Joensen et al. 1995; Tawara et al. 1998). A very long focal length is hardly managed using a single Send offprint requests to: daniele.spiga@brera.inaf.it spacecraft, therefore the optics and the focal plane instruments should be carried by two separate spacecrafts in formation- flight configuration. This is the baseline for the future X-ray telescopes SIMBOL-X (Pareschi & Ferrando 2006) and XEUS (Parmar et al. 2004). Other hard X-ray imaging telescopes of the future are NeXT (Ogasaka et al. 2006) and Constellation-X (Petre et al. 2006). The focusing and reflection efficiency of X-ray optics can be tested and calibrated on ground by means of full-illumination X-ray facilities like PANTER (Bräuninger et al. 2004; Freyberg et al. 2006), successfully utilized in the last years to calibrate the optics of a number of soft X-ray telescopes. The PANTER X-ray facility now allows testing in soft (0.2÷ 10 keV) and hard (15 ÷ 50 keV) X-rays multilayer-coated optics pro- totypes for future X-ray telescopes (Pareschi et al. 2005; Romaine et al. 2005). The source distance finiteness causes some departures of the optic performances, with respect to the case with the source at astronomical distance: effective area loss, different incidence angles on paraboloid and hyper- boloid, focal length displacement, a slight focal spot blurring (Van Speybroeck & Chase 1972). However, there effects can be quantified and subtracted from experimental data. After this treatment, the focusing-concentration performances of the optic can be experimentally characterized as a function of the inci- dent photon energy, in terms of Half-Energy Width (HEW) and Effective Area (EA). http://arxiv.org/abs/0704.1612v2 2 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . The focusing performance, in particular, is altered by mirror deformations that may arise in the manufacturing, handling, in- tegration, positioning processes. The consequent imaging degra- dation can be calculated from the measured departures of the mirrors from the nominal profile, by means of a ray-tracing pro- gram. As long as the geometrical optics approximation can be applied, the effect is independent of the photon energy. The fig- ure errors contribution to the HEW can be also directly measured using a highly collimated beam of visible/UV light in a precision optical bench. In this case, however, the light diffraction has to be carefully estimated and subtracted. Another drawback is the X-ray scattering (XRS) caused by the microroughness of reflecting surfaces (Church et al. 1979; Stearns et al. 1998; Stover 1995; and many others). The XRS spreads a variable fraction of the reflected beam intensity in the surrounding directions: the result is the effective area loss in the specular direction (i.e. in the focus) and a degradation of the imaging quality. The XRS is an increasing function of the pho- ton energy; due to the impact that the XRS can have on astro- nomical X-ray images quality, the height fluctuations rms of the mirror surface should not exceed few angströms. Loss of effec- tive area is also caused by interdiffusion of layers in multilayer coatings, which enhances the X-ray transmission and absorption throughout the stack. On the other hand, an uniform interdiffu- sion does not cause X-ray scattering (Spiller 1994), hence it does not contribute to the focusing degradation. The microroughness of an X-ray mirror can be measured on selected samples using several metrological instruments, each of them sensitive to a definite interval of spatial scales l̂: Long Trace Profilometers (10 cm > l̂ > 0.5 mm: Takács et al. 1999), opti- cal interference profilometers (5 mm > l̂ > 10 µm) and Atomic Force Microscopes (100 µm > l̂ > 5 nm) can be suitable in- struments to provide a detailed profile characterization of X-ray mirrors surface. It is convenient to present the deviation of sur- face from the ideality in terms of Power Spectral Density (PSD), because its values do not depend on the measurement technique in use (see ISO 10110 Standard). In addition, the XRS diagram, and consequently the HEW, can be immediately computed from the PSD at any photon energy (Church et al. 1979). In the past years, several approaches were elaborated to relate a mirror PSF (Point Spread Function) to the PSD of its surface. Among a wealth of works, we can cite (De Korte et al. 1981) the assumption of a Lorentzian model for the PSD to fit the mirror PSFs at some photon energies, allowing the derivation of two parameters (roughness rms and correlation length) of the model PSD. Christensen et al. (1988) perform a fit of experimental high-resolution XRS data dealing with the surface correlation function. Harvey et al. (1988) re- late the PSF of Wolter-I optics to the parameters of an exponen- tial self-correlation function along with a transfer function-based approach. Willingale (1988) derived the surface PSD of a mirror from the wings of a few PSFs, measured at PANTER at some soft X-ray photon energies. O’Dell et al. (1993) interpret the PSF of a focusing mirror on the basis of surface roughness and partic- ulate contamination. Zhao & Van Speybroeck (2003) construct from the PSD of a focusing mirror a model surface and compute the X-ray scattering PSF from the Fraunhofer diffraction theory. In the present work that problem is faced in a new and differ- ent way, looking for a general and simple link between measured roughness and mirror HEW. More precisely, we considered the following question: for an X-ray grazing-incidence optic, what is the maximum acceptable PSD of the surface that fulfills the angular resolution (HEW) requirements of the telescope, in all the energy band of sensitivity? In this work we shall give a definite answer to this question. In the sect. 2 we shall summarize the causes of imaging degra- dation. In the sect. 3 we show how to evaluate H(E), the XRS contribution to the HEW of a focusing mirror at the photon en- ergy E, from any surface microroughness PSD, measured over a very wide range of spatial frequencies. We shall see in the sect. 4 that for the special class of fractal surfaces we can even relate the power-law indexes of PSD and HEW, and in the sect. 5 we see how to treat the other cases. Then we prove in the sect. 6 that the formalism can be reversed, providing thereby an independent evaluation of the surface PSD from an analytical calculation over H(E), and in the sect. 7 we extend the results to focusing mirrors with more than one reflection. Finally, an example of computa- tion is provided in the sect. 8. 2. Contributions to the imaging degradation We shall henceforth indicate with λ the wavelength of photons impinging on the mirror, and we shall consider the HEW as a function of λ instead of the photon energy E. For isotropical re- flecting surfaces in grazing incidence, the X-ray scattering dis- tribution lies essentially in the incidence plane, so we denote the incidence angle on the mirror as θi and the scattering angle as θs, both measured from the surface plane (a schematic of the scatter- ing geometry is drawn in fig. 1). If we do not consider the optic roundness errors, the longitudinal deviations from the nominal profile of a focusing mirror can be classified on the basis of their typical length l̂. According to De Korte et al. (1981), they are: 1. Power errors: errors with l̂ equal to the mirror length L. They consist in a single-concavity deformation of the profile with respect to the nominal one. 2. Regularity errors: errors in the spatial range from 0.1 L < l̂ < 0.5 L. 3. Surface roughness: surface defects with l̂ < 0.1 L. However, other criteria were also formulated to separate fig- ure errors from roughness. Consider a single Fourier component of the surface profile with wavelength l̂ and root mean square σ. That Fourier component is dominated by figure error if it fulfills the condition (Aschenbach 2005) 4π sin θiσ > λ. (1) Otherwise, it is dominated by microroughness. In other words, surface defects within the smooth-surface approximation can be mainly considered as microroughness. To understand the impor- tance of this approximation, we write the optical path difference ∆s of X-rays reflected by two points of the surface with a hori- zontal spacing l̂ and vertical spacing σ̂ = 2 2σ (for optically- polished surfaces, σ is a increasing function of l̂, and usually σ≪ 10−3l̂) as ∆s = l̂(cos θs − cos θi) + σ̂(sin θi + sin θs) (2) that, for small incidence angles, becomes ∆s = l̂ sin θi(θs − θi) + σ̂(θi + θs). (3) If that component is responsible for X-ray scattering, it has to be ∆s ≈ λ, to cause the diffraction from surface features with a l̂ spacing and σ̂ height. Conversely, the ”figure errors”, which are treated with the methods of the geometrical optics, should D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 3 Fig. 1. The geometry of X-ray scattering: the strictly speaking ”reflected” rays (i.e. in the focus direction) are characterized by the equality θs = θi, the others are scattered apart. The rough sur- face is a simulated one, assuming a PSD with power-law index n = 2.4 (see sect. 4). be characterized by the inequality ∆s ≫ λ. Note that this condi- tion becomes similar to the eq. 1 in the limit |θi − θs | → 0. The application of this criterion and of the subsequent X-ray scatter- ing theory requires the incident radiation to be spatially coher- ent over the spatial scale l̂, so that the properties of the reflected wavefront are determined only by the coherence properties of the mirror surface. This in turn requires the angular diameter of the source φS to fulfill the inequality (Holý et al. 1999) l̂ sin θi . (4) This equation sets a maximum to the values of l̂ that can be used in the application of the results presented in this work. The limi- tation can affect X-ray sources at finite distance, like those used for X-ray optics calibrations in full-illumination setup. For very distant astronomical X-ray sources, the condition 4 is met even for larger l̂, up to l̂ ≈ L. It is worth pointing out that, for a given reflecting surface, the separation of figure errors from microroughness is strongly affected by the incidence/scattering angles. In fact, even for large l̂, ∆s can become comparable with λ, if θi and θs are sufficiently small: thus, the spatial wavelength window of interest for X-ray scattering can shift to the large l̂ domain (or, equivalently, to the range of low spatial frequencies f = 1/l̂), provided that the con- dition 4 is fulfilled. Let us now consider how to separate the figure and scattering terms in HEW data. In absence of XRS, the mirror PSF would be independent of the energy and due only to figure errors (i.e. in the approximation of the geometrical optics). The resulting HEW would be also constant. Instead, due to the XRS, the figure PSF is convolved with the X-ray scattering PSF to return the PSF(λ) being measured (Willingale 1988; Stearns et al. 1998; and many others), PS F(λ) = PS Ffig ⊗ PS FXRS(λ). (5) The resulting HEW will depend on the photon wavelength, as it does the PSF. In order to isolate the scattering term from the total PSF a deconvolution should be carried out, provided that the PSFfig is known. However, if we assume that the XRS and the mirror deformations are statistically independent, the total HEW can be approximately calculated as the squared sum of the two contributions: HEW2(λ) ≈ HEW2fig + H 2(λ). (6) An estimation of HEWfig can be obtained: 1. from the application of a ray-tracing code to several mea- surements of the mirror profile, 2. from reliable extrapolation of the HEW(λ) curve to E → 0, in absence of low-energy diffraction effects like dust contam- ination, studied in detail by O’Dell et al. (1993) 3. from a direct measurement of the HEW in visible/UV light, provided that the diffraction at the mirror edges can be reli- ably calculated and subtracted. Once known the measured HEW(λ) experimental trend and the HEWfig term, the eq. 6 can be used to isolate the scattering contribution from the experimental HEW trend: we shall prove in the next section that the H(λ) function is immediately related to the reflecting surface 1D Power Spectral Density (PSD) P( f ) P( f ) = z(x)e−2πi f dx where z(x) is a height profile (of length L) of the mirror, mea- sured in any direction (Stover 1995): the surface is assumed to be isotropic, and the spectral properties of the profile to be rep- resentative of the whole surface. The PSD is often measured in nm3 units, and for optically-polished surfaces it is usually a de- creasing function of the frequency f . PSD measurements have always a finite extent [ fmin, fmax], determined by the length and the spatial resolution of the mea- sured profile. As well known, the surface rms σ is simply com- puted from the PSD by integration over the spatial frequencies f : ∫ fmax P( f ) d f (8) note that the integration range should always be specified. 3. Estimation of H(λ) for single-reflection focusing mirrors 3.1. Single-layer coatings Firstly, we suppose the mirror to be plane and single-layer coated. For a surface with roughness rms σ, the specular beam intensity obeys the well-known Debye-Waller formula R = RF exp 16π2σ2 sin2 θi , (9) here RF is the reflectivity at the grazing incidence angle θi, as calculated from Fresnel’s equations (zero roughness). However, it should be noted in the eq. 9 that neither the spatial frequen- cies range where the PSD should be integrated is specified, nor the separation between reflected and scattered ray is clearly in- dicated: these ambiguities can be solved as follows. Let us derive the total scattered intensity Is from the con- servation of the energy: for smooth surfaces, i.e. fulfilling the inequality 2σ sin θi ≪ λ, we can approximate Is = I0RF 1 − exp 16π2σ2 sin2 θi ≈ I0RF 16π2σ2 sin2 θi .(10) 4 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . In grazing incidence, X-ray scattering lies mainly in the inci- dence plane. Moreover, the normalized scattered intensity per ra- dian at the scattering angle θs (either θs > θi or θs < θi) is related to the PSD along with the well-known formula at first-order ap- proximation (Church et al. 1979; Church & Takács 1986), valid for smooth, isotropic surfaces and for scattering directions close to the specular ray (i. e. |θs − θi| ≪ θi), sin3 θiRFP( f ) (11) where P( f ) is the Power Spectral Density of the surface (eq. 7) and I0 is the flux intensity of the incident X-rays. If the scattered intensity is evaluated at the scattering angle θs, the PSD can be immediately evaluated as a function of the spatial frequency f : f = l̂−1 = cos θi − cos θs sin θi(θs − θi) . (12) In the eq. 12 the approximation was justified by the assumption |θs − θi| ≪ θi and the negative frequencies are conventionally assumed to scatter at θs < θi: the assumed approximations make the XRS diagram symmetric, because the PSD is an even func- tion. For a single-reflection mirror shell, the extension of the for- mulae above-mentioned is straightforward by regarding |θs − θi| as the angular distance at which the PSF is evaluated. The fo- cal image is the superposition of many identical XRS diagrams on the image plane, generated by every meridional section of the mirror shell: since a π angle rotation of every meridional plane of the shell sweeps the whole image plane, the scattered intensity is spread over a π angle. The integration on circular coronae used to compute the mirror PSF (at positive angles) compensates this factor multiplying the XRS diagram by 2π (De Korte et al. 1981). The remaining 2-fold factor accounts for the negative frequen- cies in the surface PSD. We shall henceforth suppose that the fac- tor 2 is embedded in the PSD definition. Therefore, the eqs. 11 and 12 can be used to describe the XRS contribution to the PSF. We are now interested in the scattered power at angles larger than a definite angle αmeasured from the focus. Due to the steep fall of scattering intensity for increasing angles, the integral has a finite value I [|θs − θi| > α] = ∫ π−θi dθs. (13) Combining eqs. 11 and 13, one obtains: I [|θs − θi| > α] = I0RF 16π2 sin3 θi ∫ π−θi P( f ) dθs (14) with respect to the definition used in the eqs. 7 and 11, a factor 2 was included in the PSD. The upper integration limit corre- sponds to a photon back-scattering: at first glance, this seems to violate our small-scattering angle assumption (eqs. 11 and 12), but it should be remembered that only the angles close to θi contribute significantly to the integral in eq. 13: hence its value should not be significantly affected by a particular choice of the upper integration limit. After a variable change from θs to f (eq. 12), the eq. 14 becomes (approximating cos θi ≈ 1 in the upper integration limit): I [|θs − θi| > α] = I0RF 16π2 sin2 θi P( f ) d f (15) where f0 = α sin θi/λ is the spatial frequency corresponding to the scattering at the angle α. As expected, this equation equals the integrated scattering according to the eq. 10, provided that we identify I [|θs − θi| > α] with Is, and the squared roughness rms with P( f ) d f . (16) The eq. 16 is in agreement with the eq. 8, but it states clearly the window of spatial frequencies involved in the XRS. Therefore, for a definite angular limit α the ”reflected beam” intensity can be simply calculated by using the Debye-Waller formula, pro- vided that σ2 is computed from the PSD integration beyond the frequency f0, which corresponds to an X-ray scattering at α. The upper integration limit is a very high frequency (close to 1/Å): hence, the atomic structure of the surface is not important in the integral of the eq. 16. Moreover, considering that the PSD trend for optically-polished surfaces decreases steeply for increasing f , the largest contribution to the integral should be given by the frequencies close to f0. Now we can evaluate H(λ), the scattering term of the HEW. For simplicity, in the following we will suppose that the HEW is obtained from the collection of all the reflected/scattered pho- tons: this allows us to avoid problems related to the finite size of the detector, and to extend the surface roughness PSD up to very large spatial frequencies. By definition, H(λ) is twice the angu- lar distance from focus at which the integrated scattered power halves the total reflected intensity: I [|θs − θi| > α] = I0RF (17) we immediately derive, from the eq. 9, 16π2σ2 sin2 θi , (18) where σ2 has now the meaning as per the eq. 16. Solving the eq. 18 for σ2 and equating to the integral of the PSD, P( f ) d f = λ2 ln 2 16π2 sin2 θi , (19) once known the PSD from topography measurements over a wide range of spatial frequencies, the PSD numerical integra- tion in the eq. 19 allows to recover f0. In turn, f0 is related to H(λ) through the eq. 12, that we write in the following form H(λ) = 2λ f0 sin θi , (20) where H is measured in radians. Note that the condition H(λ) ≪ θi is very important, for the eq. 20 to hold. Small scattering angles and grazing incidence are also very important for the considerations that follow. 3.2. Multilayer coatings The obtained result (eq. 19) can be extended to mirrors with multilayer coatings, used to enhance the grazing incidence re- flectivity of mirrors in hard X-rays (E > 10 keV). In general, the multilayer cannot be characterized by means of a single PSD, due to the evolution of the roughness throughout the stack (Spiller et al. 1993; Stearns et al. 1998). Moreover, due to the interference of scattered waves at each multilayer interface, the final scattering pattern is more structured than eq. 11, with peaks whose height depends on the phase coherence of the interfaces D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 5 Fig. 2. Dependence of the spectral exponents for different in- dexes n of a power-law PSD, for a single-reflection focusing mirror. In the forbidden region (n > 3) γ would be negative. (Kozhevnikov 2003). The HEW term can be computed numeri- cally from the XRS diagram. In order to extend the eq. 19 to mirrors coated with a graded multilayer, we have to assume the additional requirements: 1. the PSD is constant and completely coherent throughout the multilayer stack: i.e., the deposition process does not cause additional roughness and replicates simply the profile of the substrate. Therefore, all the PSDs and all the cross- correlation between interface profiles equal the PSD mea- sured at the multilayer surface. This is often observed in the l̂ > 10 µm regime (Canestrari et al. 2006), where most of frequencies f0 fall when the incidence angle is less than 0.5 deg. Most of microroughness growth, indeed, takes place for 10 µm> l̂ > 0.1 µm. 2. the multilayer reflectivity Rλ(θi) at the photon wavelength λ changes gradually over angular scales of H(λ). Ideally, this condition should be fulfilled by wideband multilayer coat- ings for astronomical X-ray mirrors. Under these hypotheses, a quite tedious calculation reported in appendix A shows that the eq. 19 can be approximately ap- plied also with multilayer coatings. The following developments also apply in that case. 4. H(λ) for a fractal surface We apply now the equations 19 and 20 to the typical (monodi- mensional) PSD model for optically-polished surfaces, a power- law (Church 1988) P( f ) = , (21) where the power-law index n is a real number in the interval 1 < n < 3 and Kn is a normalization factor. A power-law PSD is typical of a fractal surface, and it represents the high-frequency regime of a K-correlation model PSD (Stover 1995). This model exhibits a saturation for f → 0 that avoids the PSD divergence. In practice, the fractal behavior dominates in almost all spatial frequencies of interest for X-ray optics. There are interesting reasons for which n can take val- ues on the interval (1:3). In fact, for a surface in the 3D space, n is related to its Hausdorff-Besicovitch dimension D (Barabási & Stanley 1995) along with the equation n = 7 − 2D (see Church 1988; Gouyet 1996). The restriction 1 < n < 3 for a fractal surface is therefore necessary to have 3 > D > 2. A power-law PSD is particularly interesting because the in- tegral on left-hand side of the eq. 19 can be explicitly calculated: f 1−n0 − n − 1 λ2 ln 2 16π2 sin2 θi . (22) As 1 − n < 0, in grazing incidence the (2/λ)1−n term can be neglected with respect to f 1−n0 . By isolating the frequency f0 and using the eq. 20 to derive H(λ), we obtain after some algebra, for the scattering term of the HEW, H(λ) = 2 16π2Kn (n − 1) ln 2 sin θi . (23) This equation states that: 1. The H(λ) function for a power-law PSD has a power-law dependence on the photon energy E ∝ 1/λ, i.e., H(E) ∝ Eγ. The power-law index γ is related to the PSD power-law index n through the simple equation: 3 − n n − 1 . (24) As 1 < n < 3, γ is positive, i.e. H is an increasing function of the photon energy. For a fixed value of Kn, the HEW di- verges quickly for n ≈ 1 but very slowly for n ≈ 3: a PSD power-law index close to 2-3 would hence be preferable in order to reduce the degradation of focusing performances for increasing energies. 2. H(λ) depends on the sine of the incidence angle at the γth power. In other words, the HEW depends only on the ratio sin θi/λ: this scaling relation shows that for a given power- law PSD (with n < 3) at a given photon wavelength λ we can reduce the HEW by decreasing the incidence angle. 3. H(λ) increases with the PSD normalization Kn, as expected: the dependence is also a power law with spectral index n − 1 . (25) As for γ(n), the closeness of n to the maximum allowed value for fractal surfaces makes less severe the roughness effect on imaging degradation. The functions β and γ are plotted in fig. 2. For instance, if n = 2, γ = β = 1, and H(E) increases linearly with both pho- ton energy and Kn coefficient. The divergence of indexes β, γ for n ≈ 1 makes apparent the importance of obtaining steep PSDs in the optical polishing of X-ray mirrors. Finally, it is worth not- ing that for n > 3 there is the theoretical possibility of a slight decrease of H(E) for increasing energy because γ(n) becomes negative. To clarify the dependence of the HEW on the power-law in- dex n and the incidence angle, we depict in fig. 3 and 4 some ex- amples of H(E) simulations (single reflection) for some power- law PSDs in the photon energy range 0.1-50 keV. The H(E) curves were computed using the eq. 23. In fig. 3 the incidence angle θi is fixed at 0.5 deg and the index n is variable; a constant n = 1.8 and a variable θi is instead assumed in the simulations of fig. 4. Note in fig. 3 the slower H(E) increase for larger n and the common intersection point, determined by the particular choice of the incidence angle and the σ = 4 Å value in the window of spatial wavelengths [100 ÷ 0.01 µm]. 6 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . Fig. 3. H(E) simulations assuming power-law PSDs with con- stant σ = 4 Å in the spatial wavelengths range [100 ÷ 0.01µm], but variable power-law index n. The incidence angle is fixed at θi = 0.5 deg. Fig. 4. H(E) simulations assuming a power-law PSD with power-law index n = 1.8 and with σ = 4 Å in the spatial wave- lengths range [100 ÷ 0.01 µm], but variable incidence angle θi. 5. Numerical integration of the PSD A power-law PSD is a modelization that can be used for optically-polished surfaces. If the polishing process is not op- timized or a reflecting layer is grown onto a optically polished substrate, several deviations from a power-law trend can be ob- served. A typical ”bump”, for instance, can be present in the PSD of multilayer coatings, often in the range of spatial wavelengths [10 ÷ 0.1 µm], as a result of the replication of the substrate to- pography and of fluctuations intrinsically related to the random deposition process (Spiller et al. 1993; Stearns et al. 1998). If the PSD deviates significantly from a power-law, the eq. 23 can- not be used. However, if the surface PSD has been extensively measured over a wide range of spatial frequencies [ fm, fM] (wide enough to have fm < f0(λ) for all λ), the HEW scattering term H(λ) can be computed by numerical integration (eqs. 19 and 20), on condition that the following approximation is valid: P( f ) d f ≈ P( f ) d f . (26) The condition above is usually satisfied when f0 ≪ fM i.e. when the following inequality holds: H(λ)≪ 2λ fM sin θi . (27) As we are also interested in computing H(λ) in hard X-rays (small λ), there is the possibility that the two integrals in the eq. 26 differ by a significant factor. In this case the integral can be corrected by adding the remaining term P( f ) d f = P( f ) d f + P( f ) d f , (28) that can be evaluated, in principle, by measuring the mirror re- flectivity within an angular acceptance corresponding to the spa- tial frequency fM, and using the Debye-Waller formula to derive σ2; then, the importance of measuring the PSD in a very wide frequencies interval becomes apparent. The value of f0 depends strongly on both incidence angle and photon energy: for soft X- rays (< 10 keV) and very small angles (< 0.2 deg) the character- istic spatial wavelength l̂ = 1/ f0 often falls in the millimeter or centimeter range. It should be noted that, if the detector is small, a fraction of the scattered photons can be lost; to account for the finite an- gular radius of the detector d (as seen from the optic principal plane), one should integrate the PSD over the smaller interval [ f0, d sin θi/λ] to recover the measured H(λ) trend. As an alter- native method, one can compare the theoretical predictions of eqs. 19 and 20 with the experimental H(λ) values, as calculated from the Encircled Energy normalized to the photon count fore- seen by the Fresnel equations (i.e. with zero roughness), rather than to the maximum of the measured Encircled Energy func- tion. 6. Computation of the PSD from the H(λ) trend If the approach described above can be used to simulate the HEW trend from a measured surface PSD, the reverse problem, i.e. the derivation of surface PSD from the measured HEW trend is also possible. This requires that the figure error contribution had been reliably measured, in order to isolate the scattering term function H(λ) using the eq. 6. This problem is interesting for three reasons at least: 1. it is a quick, non-destructive surface characterization method in terms of its PSD. 2. The measurement is extended to a large portion of the illu- minated optic, hence local surface features are averaged and ruled out from the PSD. 3. For a given HEW(λ) requirement in the telescope sensitiv- ity energy band, it allows establishing the maximum allowed In order to find an analytical expression for the PSD, we note that the spatial frequency f0 that scatters at an angular distance H/2 from the specular beam is a function only of λ, along with the eq. 20. Solving for f0, we have f0(λ) ≈ H(λ) sin θi . (29) We suppose that all scattered photons are collected, so we can assume the eq. 19 as valid. By deriving both sides of eq. 19 with respect to λ, we have P( f ) d f 8π2 sin2 θi λ, (30) that is, P( f0) = 8π2 sin2 θi λ, (31) D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 7 and, using the eq. 29 to compute the derivative of f0: + f0P( f0) sin θi dH(λ) P( f0) = 8π2 sin2 θi λ. (32) Now remember that, in grazing incidence, f0 ≪ 2λ−1 by several orders of magnitude. Even if P( f ) is not a power-law, it is always a steeply decreasing function of f . Moreover, it should have over [ f0, 2λ −1] an average PSD index ñ > 1, for the reasons explained in the sect. 4. This means that P( f0) P(2λ−1) , (33) therefore, in practical cases the 2λ−1P(2λ−1) term in the eq. 32 is negligible with respect to f0P( f0). Consequently, we can neglect the first term of eq. 31: then we have P( f0) ≈ 8π2 sin2 θi λ. (34) Combining this with the eq. 29 and collecting the constants, we obtain the final result P( f0) 4π2 sin3 θi ≈ 0. (35) The eq. 35 enables the computation of the PSD (at the spatial frequency given by the eq. 29) along with the derivative of the ratio H(λ)/λ with respect to λ. The obtained equation shows that P( f ) is inversely propor- tional to the derivative of H(λ)/λ. This result seems strange at first glance, because by decreasing H(λ) one would obtain a larger P( f ) (a rougher surface). One should remember, indeed, that by reducing H(λ) we increase P( f0), but f0 is shifted to- wards the low frequencies domain, where P( f0) is expected to be higher. In fact, the ”rough” or ”smooth” feature of the surface depends on whether f0 or P( f0) varies more rapidly, i.e. on the overall H(λ) trend. We can also check the correctness of the eq. 35 by computing the PSD for the particular case of the H(λ) derived from the inte- gration of a power-law PSD (the eq. 23, derived under the same approximation, the eq. 33). If the results are correct, the substi- tution of the HEW trend of the eq. 23 in the eq. 35 should return the original PSD (eq. 21). The straightforward, but lengthy cal- culation (carried out in appendix B) shows that the substitution returns P( f0) = as expected. The eq. 35 should be approximately valid also for graded multilayers with a slowly-decreasing reflectivity (see sect. 3.2), however, due to the approximations needed to extend the eq. 19 to the multilayers, the resulting PSD should be considered a ”first guess” in this case. Then, the matching of the PSD to the required HEW trend should be checked by means of a detailed computation of the XRS PSF(λ). 7. Extension to X-ray mirrors with multiple reflections The formalism exposed in the previous sections can be extended to a double-reflection optic (like a Wolter-I one). In this op- tical configuration, photons are firstly reflected by a parabolic surface and subsequently by a hyperbolic one. If the smooth- surface condition is satisfied, multiple scattering is often negli- gible (Willingale 1988) and the scattering diagrams of the two reflecting surfaces can be simply summed (De Korte et al. 1981; Stearns et al. 1998). The source is assumed to be at infinite dis- tance, then X-rays impinge on the two surfaces at the same an- gle θi. If the surface PSDs are the same for both reflections, the scattering diagram will be simply doubled. Thus, the integrated scattered intensity is also doubled: Is = 2I0R 1 − exp 16π2σ2 sin2 θi . (37) The RF factor is squared in the eq. 37 because each ray is re- flected twice: in absence of scattering the reflected power would be I0R F, so the half-power scattering angle condition reads |θs − θi| > F, (38) and, combining the eqs. 37 and 38, we obtain 16π2σ2 sin2 θi . (39) Solving for σ2, and using the eq. 16, P( f ) d f = λ2 ln(4/3) 16π2 sin2 θi , (40) that differs from the eq. 19 only in the factor ln(4/3) instead of ln 2 on right-hand side. Consequently, the corresponding differ- ential equation is P( f0) ln(4/3) 4π2 sin3 θi ≈ 0. (41) Similar equations can be derived for an optical system with an arbitrary number of reflections N: to compute the H(λ) from the PSD, P( f ) d f = 16π2 sin2 θi 2N − 1 . (42) If the PSD is a power-law P( f ) = Kn/ f n we can generalize the eq. 23: H(λ) = 2 2N − 1 16π2Kn (n − 1) sin θi , (43) note the divergence of the logaritmic factor for increasing N, due to the negative exponent 1/(1 − n). This indicates that H(λ) increases rapidly with the number of reflections, as expected. Finally, we can also generalize the differential eq. 35 to an arbitrary number of reflections, P( f0) 4π2 sin3 θi ≈ 0. (44) In the eqs. 42 and 44, f0 is always related to H(λ) by the eq. 29. 8 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . Fig. 5. an hypothetical PSD with reasonable values and a PSD break around a 100 µm spatial wavelength (dashed line). This PSD is adopted to compute the corresponding HEW trends for 1,2,3 reflections at a 0.3 deg grazing incidence angle (fig. 6). The achieved HEW trends were used to re-calculate the respective PSDs (solid line). For clarity, we do not plot the single PSDs, but just their overlap. 8. An example As an application of the equations reported above, we shall make use of a simulated surface PSD with reasonable values, that is not a power-law. The PSD (see fig. 5, dashed line) is ex- tended from 105 µm down to a 0.01 µm spatial wavelength, with a break around 100 µm: at the lowest frequencies the PSD is steep (n ≈ 2.3), whereas at the largest frequencies it is smoother (n ≈ 1.3). From the discussion in sect. 4 concerning the relation between the exponents of the PSD and the HEW (eq. 24), we should expect that the PSD break causes a slope change in the function H(λ): however, as the actual PSD is not a power-law, the H(λ) function should be computed by means of the eqs. 20 and 42. Before carrying out the integration, we can remark qual- itatively that, as we increase the photon energy, the highest fre- quencies in the PSD (where the PSD index becomes smaller) become important; hence, we can expect a steeper increase of the HEW at the highest energies. The analysis is made quantitative in fig. 6, where we show the calculated HEW trends from the PSD in fig. 5 (the dashed line) by means of the eqs. 20 and 42, assuming 1,2,3 reflections at the same grazing incidence angle (0.3 deg). The approxima- tion of eq. 26 was adopted. In addition to the scattering term, 15 arcsec of HEW due to figure errors were added in quadrature. The HEW increases slowly (concave downwards) at low ener- gies, corresponding to a frequency f0 in the steeper part of the PSD. Then it increases more steeply (concave upwards) when the energy becomes large enough to set f0 in the portion of the spectrum with n ≈ 1.3. By increasing the number of reflections, the HEW values also increase, and the ”turning point” where the HEW starts to diverge (arrows in fig. 6) shifts at lower X-ray energies. All the calculation is based on the assumption that the contribution of the PSD over the maximum measured frequency fM = 0.01 µm is negligible. Otherwise, the computed HEW val- ues will be underestimated (see sect. 5). In addition to the general trend of the HEW, there are oscil- lations due to small irregularities in the adopted PSD: the cal- culation is, in fact, very sensitive to small variations of the PSD values. Notice that for a definite energy all the frequencies larger than f0 contribute to the HEW value, even if the largest contri- Fig. 6. the HEW trend computed from the PSD for 1,2,3 reflec- tions, plus 15 arcsec of HEW due to figure errors. The HEW trends were used to compute back the PSD (the solid line in fig. 5) to verify the reversibility of the calculation. The energy at which the concavity change takes place is also indicated (ar- rows). bution comes from frequencies near f0: this is a consequence of the steeply decreasing trend of the PSD. We checked the reversibility of the result by computing the PSD from the HEW trends (after subtracting in quadrature 15 arcsec figure error) by means of the eq. 44 with the respec- tive value of N. The resulting PSDs (the solid line in fig. 5) were overplotted to the initial PSD, with a perfect superposition. Each obtained PSD has, indeed, an extent of spatial frequencies smaller than the initial one: the overall PSD ranges from 104 to 11 µm (vs. the initial 105 ÷ 0.01 µm), and the smaller wave- lengths could be computed from the HEW trend with N = 3. The limitation in spatial frequency ranges occurs for two reasons: 1. small f − large l̂: all the power scattered by the lowest fre- quencies is found at angles less than 1/2 HEW even for the lowest energies being considered: therefore, that part of the spectrum is not necessary to compute the HEW in the energy range of interest; 2. large f − small l̂: the PSD is computed from a derivative, therefore the information concerning the absolute magnitude of the HEW is substantially lost. This information was in- cluded in the integral of the PSD (eq. 42) for the maximum considered energy. Therefore, from the integral in the eq. 42 we cannot re- cover the PSD over the minimum computed spatial wavelength (11 µm, using the HEW trend with N = 3), but we can at least calculate the value of σ at spatial wavelengths smaller than 11 µm. Substituting the incidence angle and the minimum pho- ton wavelength being considered (λ = 0.24 Å) in the eq. 42 with N = 3 and with the approximation of the eq. 26, we ob- tain σ = 1.6 Å, in perfect agreement with the value computed from the original PSD. Summing up, for a given incidence angle the H(λ) function in a definite photon energy range is equivalent to the PSD in a corresponding range of spatial frequencies f (or equivalently, spatial wavelengths l̂), plus the integral of the PSD beyond the maximum frequency being computed. Therefore, requirements of a definite HEW(λ) function in designing an X-ray optical sys- tem can be translated in terms of PSD in a frequencies range [ fmin, fmax] plus the surface rms at frequencies beyond fmax. The usefulness of such a relationship is apparent. D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 9 9. Conclusions In the previous pages we have developed useful equations to compute the contribution of the X-ray scattering to the HEW of a grazing incidence X-ray optic, by means of a simple integra- tion. The formalism has been inverted in order to derive the PSD of the surface from the function H(λ), and it can be extended to an arbitrary number of reflections at the same incidence an- gle. The equations are valid for a single-layer coating mirror, but they can be approximately applied to a multilayer-coated mir- ror. This approach is particularly useful in order to establish the surface finishing level needed to keep the X-ray scattering HEW of X-ray optics within the limits fixed by the X-ray telescope requirements. It should be remarked that the reasoning was developed for the Half-Energy Width, but it can be extended to any angular diameter including a fraction η of the energy spread around the focal point. To do this, it is sufficient to substitute the logarithmic factors in equations 42, 43, 44, 2N − 1 N − 1 + η , (45) and for instance, to compute the 90%-energy diameter for a dou- ble reflection mirror, simply substitute η = 0.9 and N = 2. The proof is straightforward: however, one should always keep in mind that the energy diameters computed with this method can be considered valid only if they are much smaller than the inci- dence angle θi. Notice that in the development of the exposed formalism we have supposed, in addition to the smooth-surface condition, two additional hypotheses: 1. the source is at infinite distance from the mirror 2. the X-ray detector is large enough to collect all the scattered photons. In order to apply the mentioned equations to experimental calibrations of X-ray optics at existing facilities (like MPE- PANTER), where the source is at a finite distance and the de- tector has a finite size, some corrections should be taken into account. We will deal with their quantification in a subsequent paper. Appendix A: Extension to multilayer coatings Here we provide with a plausibility argument to extend the for- malism of sect. 3.1 to mirror shells with multilayer coatings (see sect. 3.2). The intensity of a scattered wave at each interface is proportional to its PSD as per the eq. 11, and the overall scatter- ing diagram will be their coherent interference. To simplify the notation, we neglect the X-ray refraction and we suppose that the incidence angle is beyond the critical angles of the multilayer components. The electric field scattered by the kth interface can be written as Ek = E0TkrkXk( f ) exp(−iφk), (A.1) where rk is the single-boundary amplitude reflectivity, E0 the incident electric field amplitude, the weights Tk are the rel- ative amplitudes of the electric field in the stack (in scalar, single-scattering approximation), and account for the extinction of the incident X-rays due to gradual reflection and absorption. Xk( f ) is the single-boundary scattering power (proportional to the PS D( f ) amplitude), and φk is the phase of the scattered wave at θs by the k th interface φk = 2π sin θi + sin θs zk, (A.2) where zk is the depth of the k th interface with respect to the outer surface of the multilayer. Now, the measured intensity is |Escatt|2 = = |E0|2|Xk( f )|2 rkTk exp(−iφk) . (A.3) Now, |E0|2 = I0, the incident X-ray flux intensity, and |Xk( f )|2 is proportional to the interfacial PSD P( f ), which is independent of k by hypothesis. Assuming the proportionality factor of eq. 11 for |Xk( f )|2, we obtain for the scattering diagram sin3 θiP( f ) rkTk exp(−iφk) (A.4) and if we set Kλ(θi, θs) = rkTk exp(−iφk) , (A.5) the eq. A.4 becomes analogous to the eq. 11, with Kλ(θi, θs) playing the role of RF. Note that Kλ(θi, θi) = Rλ(θi), the mul- tilayer reflectivity in single reflection approximation. As before, we write the scattering diagram for a mirror with axial symmetry as a function of the angular distance from the focus α = |θi − θs| averaging the contributions of negative and positive frequencies sin3 θiP( f )[Kλ(θi, θi − α) + Kλ(θi, θi + α)]. (A.6) For a single reflection optic, we can calculate the scattered power over H/2, where H is the scattering term of optic Half-Energy Width: Is[α > H/2] = I0Rλ(θi). (A.7) Now, the steps 18 and 19 can be repeated: Kλ(θi, θi + α) + Kλ(θi, θi − α) Rλ(θi) P( f ) d f = λ2 ln 2 8π2 sin2 θi (A.8) where f0 is still defined by the eq. 20. For small scattering angles (α ≪ θi), since we assumed a slow variation of Rλ over angular scales of H/2 (and the same occurs for Kλ), we can approximate Kλ(θi, θi ± α) ≈ Rλ(θi) ± α ∂Kλ(θi, θs) θs=θi . (A.9) Substituting in the eq. A.8, we obtain P( f ) d f ≈ λ2 ln 2 16π2 sin2 θi (A.10) because the two derivatives have opposite sign and cancel out. This is the same equation found for the case of a single-layer coating (eq. 19). 10 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . Appendix B: Derivation of the PSD from the HEW for a fractal surface (single reflection) We recall here the H(λ) trend for a power-law PSD (eq. 23): H(λ) = 2 16π2Kn (n − 1) ln 2 sin θi (B.1) we verify that it returns a power-law PSD if substituted in the differential eq. 35: P( f0) 4π2 sin3 θi = 0. (B.2) To simplify the notation, we write simply H instead of H(λ): by carrying out the derivation, n − 1 16π2Kn (n − 1) ln 2 (sin θi) n−1 λ n−1−3. (B.3) Using again the eq. B.1: n − 1 −3 (B.4) hence, the related PSD is P( f0) = − 4π2 sin3 θi λ3 ln 2 4π2H sin3 θi n − 1 . (B.5) Now, we can derive (n − 1)/2 from the eq. B.1, n − 1 4π2HKn )−n ( sin θi (B.6) and combining the eqs. B.5-B.6, one obtains P( f0) = Kn H sin θi , (B.7) that is, by recalling the eq. 29, P( f0) = (B.8) i.e., the expected power-law PSD. Acknowledgements. Many thanks to G. Pareschi, O. Citterio, R. Canestrari, S. Basso, F. Mazzoleni, P. Conconi, V. Cotroneo (INAF/OAB) for support and useful discussions. The author is indebted to MIUR (the Italian Ministry for Universities) for the COFIN grant awarded to the development of multilayer coatings for X-ray telescopes. References Aschenbach, B., 1988, Design, construction, and performance of the ROSAT high-resolution X-ray mirror assembly. Applied Optics, Vol. 27, No. 8, p. 1404-13 Aschenbach, B., 2005, Boundary between geometric and wave optical treatment of x-ray mirrors. In Proc. 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SPIE, vol. 5900, p. 59000S Stearns, D. G., Gaines, D. P., Sweeney, D. W., et. al., 1998, Non-specular X-ray scattering in a multilayer-coated imaging system. J. Appl. Phys, vol. 82 (2), p. 1003-28 Spiller, E., Stearns, D. G., Krumrey, M., 1993, Multilayer X-ray mirrors: inter- facial roughness, scattering, and image quality. J. Appl. Phys, vol. 74 (1), p. 107-118 Spiller, E., 1994, Soft X-rays Optics, SPIE Optical Engineering Press Stover, J. C., 1995, Optical Scattering: measurement and analysis, SPIE Optical Engineering Press Takács, P. Z., Qian, S., Kester, T., et al., 1999, Large-mirror figure measurement by optical profilometry techniques. In Proc. SPIE, vol. 3782, p. 266-274 Tawara, Y., Yamashita, K., Kunieda, H., et al., 1998, Development of a multilayer supermirror for hard x-ray telescopes. In Proc. SPIE, vol. 3444, p. 569-575 Van Speybroeck, L. P., Chase, R. C., 1972, Design Parameters od Paraboloid- Hyperboloid Telescopes for X-ray astronomy. Applied Optics, vol. 11, No. 2, p. 440-445 Weisskopf, M. C., 2003, Three years of operation of the Chandra X-ray Observatory. In Proc. SPIE, vol. 4851, p. 1-16 Willingale, R., 1988, ROSAT wide field camera mirrors. Applied Optics, vol. 27, No. 8, p. 1423-39 Zhao, P., Van Speybroeck, L. P., 2003, New method to model x-ray scattering from random rough surfaces. In Proc. SPIE, vol. 4851, p. 124-139 Introduction Contributions to the imaging degradation Estimation of H() for single-reflection focusing mirrors Single-layer coatings Multilayer coatings H() for a fractal surface Numerical integration of the PSD Computation of the PSD from the H() trend Extension to X-ray mirrors with multiple reflections An example Conclusions Extension to multilayer coatings Derivation of the PSD from the HEW for a fractal surface (single reflection)

Reply to “Comment on ‘On the inconsistency of the Bohm-Gadella theory with quantum mechanics’ ” Rafael de la Madrid Department of Physics, University of California at San Diego, La Jolla, CA 92093 E-mail: rafa@physics.ucsd.edu Abstract. In this reply, we show that when we apply standard distribution theory to the Lippmann-Schwinger equation, the resulting spaces of test functions would comply with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov, and of the theory of ultradistributions were wrong. As well, we point out several differences between the “standard method” of constructing rigged Hilbert spaces in quantum mechanics and the method used in Time Asymmetric Quantum Theory. PACS numbers: 03.65.-w, 02.30.Hq 1. Introduction The authors of [1] allege to have shown that the conclusions of [2] regarding the inconsistency of Time Asymmetric Quantum Theory (TAQT) with quantum mechanics are false. In this reply, we will show that the arguments of [1] are missing essential aspects of [2], and that therefore the conclusions of [2] still stand. The most important claims of [1] are the following: 1. There are many examples of TAQT, and the present author has inadvertently constructed another one. 2. The flaws of the Quantum Arrow of Time (QAT) pointed out in [2] are actually not flaws, because the original derivation of the QAT was misquoted from its source [3]. 3. The crucial argument of [2] regarding the exponential blowup of the test functions ϕ̂±(z) does not prevent ϕ̂±(z) from being of Hardy class. As we shall see, all these claims do not stand close scrutiny. In order to show why, in Sec. 2 we will outline the method to construct rigged Hilbert spaces in quantum mechanics based on the theory of distributions [4]. We shall refer to this method as the “standard method” and show that the resulting rigged Hilbert spaces are not of Hardy class. We shall also explain the meaning of the exponential blowup of ϕ̂±(z) and why it implies that the spaces of test functions are not of Hardy class. In Sec. 3, we briefly outline the method to introduce rigged Hilbert spaces of Hardy class in TAQT and compare such method with the “standard method.” It will then be apparent that using the method of TAQT, one can introduce any arbitrary rigged Hilbert space for the http://arxiv.org/abs/0704.1613v1 Gamow states. In order to address claim 2, we show (again) in Sec. 4 that no matter how one introduces it, the Quantum Arrow of Time has little to do with the actual time evolution of a quantum system. To address claim 3, in Sec. 5 we use classic results of Paley and Wiener and of Gelfand and Shilov to show that the “standard method” of dealing with the Lippmann-Schwinger equation leads to rigged Hilbert spaces that are not of Hardy class. Section 7 concludes that the arguments of [2] still stand. 2. The “standard method” In this section, we illustrate the main features of the “standard method” to construct rigged Hilbert spaces in quantum mechanics [5]. Such “standard method” is based on the theory of distributions [4]. For the sake of clarity, we shall use the spherical shell potential of height V0, V (~x) = V (r) = 0 0 < r < a V0 a < r < b 0 b < r < ∞ . (2.1) For l = 0, the Hamiltonian acts as (we take ~2/2m = 1) H = − d + V (r) . (2.2) The regular solution is χ(r;E) = E r) 0 < r < a J1(E)ei E−V0 r + J2(E)e−i E−V0 r a < r < b J3(E)ei E r + J4(E)e−i E r b < r < ∞ . (2.3) The Jost functions and the S matrix are given by J+(E) = −2iJ4(E) , J−(E) = 2iJ3(E) , (2.4) S(E) = J−(E) J+(E) . (2.5) The solutions of the Lippmann-Schwinger equation can be written as 〈r|E±〉 ≡ χ±(r;E) = χ(r;E) J±(E) . (2.6) When V tends to zero, these eigensolutions tend to the “free” eigensolution: 〈r|E〉 ≡ χ0(r;E) = E r) . (2.7) These eigenfunctions are delta-normalized and therefore their associated unitary operators, (U±f)(E) = dr χ±(r;E) f(r) ≡ f̂±(E) , E ≥ 0 , (2.8) (U0f)(E) = dr χ0(r;E) f(r) ≡ f̂0(E) , E ≥ 0 , (2.9) transform from L2([0,∞), dr) onto L2([0,∞), dE). The Lippmann-Schwinger and the “free” eigenfunctions can be analytically continued from the scattering spectrum into the whole complex plane. We shall denote such analytically continued eigenfunctions by χ±(r; z) and χ0(r; z). Whenever they exist, the analytic continuations of (2.8) and (2.9) are denoted by f̂±(z) = dr χ±(r; z) f(r) , (2.10) f̂0(z) = dr χ0(r; z) f(r) , (2.11) where here and in the following z belongs to a two-sheeted Riemann surface. The resonant energies are given by the poles zn of the S matrix, and their associated Gamow states are u(r; zn) = Nn J3(zn) sin( zn r) 0 < r < a J1(zn) J3(zn)e zn−V0 r + J2(zn) J3(zn)e zn−V0 r a < r < b zn r b < r < ∞ , (2.12) where Nn is a normalization factor. The theory of distributions [4] says that a test function ϕ(r) on which a distribution d(r) acts is such that the following integral is finite:‡ 〈ϕ|d〉 ≡ dr ϕ(r)d(r) < ∞ , (2.13) where 〈ϕ|d〉 represents the action of the functional |d〉 on the test function ϕ. With some variations, this is the “standard method” followed by [7–15] to introduce spaces of test functions in quantum mechanics. Thus, contrary to what the authors of [1] assert, the method followed by the present author runs (somewhat) parallel to [8], not to TAQT. In order to use (2.13) to construct the rigged Hilbert spaces for the analytically continued Lippmann-Schwinger eigenfunctions and for the Gamow states, we need to obtain the growth of χ±(r; z), χ0(r; z) and u(r; zn). Because the regular solution blows up exponentially [16], |χ(r; z)| ≤ C |z|1/2 r 1 + |z|1/2 r z |r , (2.14) the growth of the eigenfunctions (2.6), (2.7) and (2.12) blows up exponentially: |χ±(r; z)| ≤ C 1 J±(z) |z|1/4 r 1 + |z|1/2 r z |r , (2.15) ‡ In quantum mechanics, we need to impose a few more requirements, but we will not need to go into such details here. |χ0(r; z)| ≤ C |z|1/4 r 1 + |z|1/2 r z |r , (2.16) |u(r; zn)| ≤ Cn |zn|1/2 r 1 + |zn|1/2 r zn |r . (2.17) When we plug this exponential blowup into the basic requirement (2.13) of the “standard method,” we see that the test functions on which those distributions act must fall off at least exponentially. By using the Gelfand-Shilov theory of M an Ω functions [4], it was shown in [15] that when a and b are positive real numbers satisfying = 1 , (2.18) and when ϕ+(r) is an infinitely differentiable function whose tails fall off like e−r a/a, then ϕ+(z) grows like e|Im( z)|b/b in the infinite arc of the lower half-plane of the Riemann surface: If |ϕ+(r)| < Ce− a as r → ∞, then |ϕ̂+(z)| ≤ Ce b as |z| → ∞ . (2.19) It was shown in [2] that when ϕ+(r) ∈ C∞0 , ϕ̂+(z) blows up exponentially in the infinite arc of the lower half-plane of the Riemann surface: If |ϕ+(r)| = 0 when r > A, then |ϕ̂+(z)| ≤ CeA|Im z | as |z| → ∞ . (2.20) From the above estimates, we concluded in [2] that the ϕ+’s obtained from the “standard method” cannot be Hardy functions, since ϕ̂+(z) does not tend to zero as |z| tends to infinity. The authors of [1] argue that one cannot draw any conclusion on the limit |z| → ∞ from estimates such as (2.19) or (2.20), and therefore they conclude that nothing prevents ϕ̂+(z) from tending to zero and therefore from being Hardy functions. Their conclusion is not true, because their argument does not take the nature of (2.19) and (2.20) into account. After we explain the meaning of those estimates, it will be clear why they prevent ϕ̂±(z) from tending to zero in any infinite arc of the Riemann surface. In order to understand what (2.19) and (2.20) mean, we start with the simple sine function sin( E r). When E ≥ 0, the sine function oscillates between +1 and −1: | sin( E r)| ≤ 1 , E ≥ 0 . (2.21) As E tends to infinity, such oscillatory behavior remains, and in such limit the sine function does not tend to zero. When we analytically continue the sine function, z r) , (2.22) the oscillations are bounded by | sin( z r)| ≤ C |z|1/2 r 1 + |z|1/2 r z |r . (2.23) Thus, as |z| tends to infinity, sin( z r) oscillates wildly, and the magnitude of its oscillation is tightly bounded by the exponential function. It is certain that as |z| tends to infinity, sin( z r) does not tend to zero, even though the function vanishes z r = ±nπ, n = 0, 1, . . . It just happens that the solutions of the Lippmann-Schwinger equation follow the same pattern. When E is positive, the eigensolutions are oscillatory and bounded by |χ±(r;E)| ≤ C 1 J±(E) |E|1/4 r 1 + |E|1/2 r . (2.24) When the energy is complex, their oscillations get wild and are bounded by Eq. (2.15).§ Thus, the analytic continuations of the Lippmann-Schwinger eigenfunctions oscillate wildly, and the magnitude of their oscillation is tightly bounded by an exponential function (multiplied by factors that don’t cancel the exponential blowup when |z| → ∞). Because in Eqs. (2.10) and (2.11) we are integrating over r, the exponentially- bounded oscillations of χ±(r; z) get transmitted into ϕ̂±(z). The estimates (2.19) and (2.20) bound the oscillation of the test functions of the “standard method,” except for factors that don’t cancel the exponential blowup. It is the exponentially-bounded oscillations of ϕ̂±(z) what prevent ϕ̂±(z) from tending to zero in any infinite arc of the Riemann surface and therefore from being of Hardy class. A somewhat simpler way to understand the above estimates is by looking at the “free” incoming and outgoing wave functions ϕin and ϕout. Because in the energy representation such wave functions are the same as the “in” and “out” wave functions, ϕ̂in(E) = 〈E|ϕin〉 = 〈+E|ϕ+〉 = ϕ̂+(E) , (2.25) ϕ̂out(E) = 〈E|ϕout〉 = 〈−E|ϕ−〉 = ϕ̂−(E) , (2.26) in TAQT the analytic continuation of ϕ̂in(E) and ϕ̂out(E) are also of Hardy class. Since ϕ̂in,out(z) = z r)ϕin,out(r) , (2.27) it is evident that the exponential blowup (2.23) of sin( z r) will prevent ϕ̂in,out(z) from tending to zero as |z| → ∞ in any half-plane of the Riemann surface. Thus, ϕ̂in,out(z) are not of Hardy class, contrary to TAQT. Strictly speaking, the bounds (2.19) and (2.20) are not the tightest ones. We should include polynomial corrections, see Eq. (B.15) in [15], and the effect of |z|1/4r 1+|z|1/2r and 1J±(z) to obtain the tightest bounds. We shall not obtain those corrections here, because they do not cancel the exponential blowup at infinity, and because in this reply we shall use instead other classic bounds, see Sec. 5. Let us summarize this section. In standard quantum mechanics, once the Lippmann-Schwinger equation is solved, the properties of ϕ̂±(z) are already determined by Eqs. (2.10) and (2.11), and there is no room for any extra assumption on their properties. This means, in particular, that the Hardy axiom cannot be simply assumed. Rather, the Hardy axiom must be proved using Eqs. (2.10) and (2.11).‖ It simply § The points at which J±(z) = 0 do not affect the essence of the argument. ‖ This is what in [2] it was meant by the assertion that the Hardy axiom is not a matter of assumption but a matter of proof. happens that the “standard method” yields ϕ̂±(z) and ϕ̂in,out(z) that oscillate wildly. Because these oscillations are bounded by exponential functions, ϕ̂±(z) and ϕ̂in,out(z) do not tend to zero as |z| tends to infinity in any half-plane of the Riemann surface—hence they are not of Hardy class. 3. TAQT vs. the “standard method” In TAQT, one doesn’t solve the Lippmann-Schwinger equation in order to afterward obtain the properties of ϕ̂±(z) using Eq. (2.10). Instead, one transforms into the energy representation (using U± in our example) and then imposes the Hardy axiom. If H2± denotes the spaces of Hardy functions from above (+) and below (−), S denotes the Schwartz space, and Φ̃± denote their intersection restricted to the positive real line, Φ̃± = H2± ∩ S|R+ , (3.1) then the Hardy axiom states that the functions ϕ̂±(z) belong to Φ̃∓: ϕ̂±(z) ∈ Φ̃∓ . (3.2) This means that in the position representation, the Gamow states and the analytic continuation of the Lippmann-Schwinger eigenfunctions act on the following spaces: ΦBG∓ = U ± Φ̃∓ . (3.3) It is obvious that the choices (3.2)-(3.3) are arbitrary. One may as well choose another dense subset of L2([0,∞), dE) with different properties and obtain a different space of test functions for the Gamow states. What is more, ΦBG± are different from the spaces of test functions obtained through the “standard method,” because the functions ϕ̂±(z) of the “standard method” are not of Hardy class. The authors of [1] claim that the present author has inadvertently constructed an example of TAQT. That such is not the case can be seen not only from the differences between the “standard method” and the method used in TAQT to introduce rigged Hilbert spaces, but also from the outcomes. For example, whereas in the position representation the “standard method” calls for just one rigged Hilbert space for the Gamow states and for the analytically continued Lippmann-Schwinger eigenfunctions [15], TAQT uses two rigged Hilbert spaces ΦBG± ⊂ L2([0,∞), dr) ⊂ Φ×BG± . (3.4) One of the rigged Hilbert spaces is used for the “in” solutions and for the anti-resonant states, whereas the other one is used for the “out” solutions and for the resonant states. Another difference is that in TAQT, the solutions of the Lippmann-Schwinger equation for scattering energies have a time asymmetric evolution [17], whereas the “standard method” yields that such time evolution runs from t = −∞ to t = +∞, see [14]. Incidentally, this is an instance where TAQT differs not only mathematically but also physically from standard quantum mechanics, because in standard scattering theory, the time evolution of a scattering process goes from the asymptotically remote past (t → −∞) to the asymptotically far future (t → +∞). This is not so in TAQT [17]. It seems hardly necessary to clarify what the present author means by “standard quantum mechanics.” Standard quantum mechanics means the Schrödinger equation, and standard scattering theory means the Lippmann-Schwinger equation. In standard quantum mechanics, one assumes that these equations describe the physics and then solves them. Because of the scattering and resonant spectra, their solutions lie within rigged Hilbert spaces. The construction of such rigged Hilbert spaces follows by application of the “standard method.” By contrast, TAQT simply assumes that the solutions of the Schrödinger and the Lippmann-Schwinger equations comply with the Hardy axiom, without ever showing that the actual solutions of those equations comply with such axiom. It was claimed in [2] that there is no example of TAQT. The authors of [1] dispute such claim and assert that there are many examples. The present author disagrees with their assertion, because assuming that for a large class of potentials the solutions of the Lippmann-Schwinger equation comply with the Hardy axiom is not the same as having an example where it is shown that the actual solutions of the Lippmann- Schwinger equation comply with the Hardy axiom. In fact, to the best of the present author’s knowledge, no advocate of TAQT has ever used Eq. (2.10) to discuss the analytic properties of ϕ̂±(E) = 〈±E|ϕ±〉 in terms of the actual solutions χ±(r;E) of the Lippmann-Schwinger equation. The authors of [1] inadvertently acknowledge that there is no example of TAQT when they say that they still need “to identify the form and properties” of the functions of (3.3), see the last paragraph in section 2 of [1]. By saying so, they are acknowledging that they don’t know whether the standard Gamow states defined in the position representation are well defined as functionals acting on ΦBG±. If TAQT had an example, it would be known. 4. The Quantum Arrow of Time (QAT) Advocates of TAQT argue that their choice (3.3) is not arbitrary but rather is rooted on a causality principle. Such causality principle is the “preparation-registration arrow of time,” sometimes referred to as the “Quantum Arrow of Time” (QAT). For the “in” states ϕ+, the causal statement of the QAT is written as ϕ̃+(t) ≡ dE e−iEtϕ̂+(E) = 0 , for t > 0 . (4.1) By one of the Paley-Wiener theorems, Eq. (4.1) is equivalent to assuming that ϕ̂+(E) is of Hardy class from below. The corresponding causal statement for the “out” wave functions ϕ− implies that ϕ− is of Hardy class from above. Hence, in TAQT, the choice (3.3) is not arbitrary but a consequence of causality. It was pointed out in [2] that the QAT is flawed. The argument was twofold. First, it was pointed out that the original derivation [3] of Eq. (4.1) made use of the following flawed assumption: 0 = 〈E|ϕin(t)〉 = 〈+E|ϕ+(t)〉 = e−iEtϕ̂+(E) , for all energies, (4.2) which can happen only when ϕ+ and ϕin are identically 0. It was then pointed out that even though one may simply assume the causal statement (4.1) and forget about how it was derived, such causal statement says little about the actual time evolution of a quantum system, because the quantum mechanical time evolution of ϕ+ is not given by Eq. (4.1): ϕ+(t) = e−iHtϕ+ 6= ϕ̃+(t) . (4.3) To counter this argument, the authors of [1] claim that the derivation of the QAT was misquoted from the original source [3], and that the flawed assumption (4.2) was never used to derive the QAT (4.1). It seems therefore necessary to quote the original derivation (see [3], page 2597):¶ “We are now in the position to give a mathematical formulation of the QAT: we choose t = 0 to be the time before which all preparations of φin(t) are completed and after which the registration of ψout(t) begins. This means that for t > 0 the energy distribution of the preparation apparatus must vanish: 〈E, η|φin(t)〉 = 0 for all values of the quantum numbers E and η (η are the additional quantum numbers which we usually suppress). As the mathematical statement for ‘no preparations for t > 0’ we therefore write (the slightly weaker condition) dE 〈E|φin(t)〉 = dE 〈+E|φ+(t)〉 = dE 〈+E|e−iHt|φ+〉 (4.4) dE 〈+E|φ+〉e−iEt ≡ F(t) for t > 0 . (4.5) The readers can decide whether or not the flawed hypothesis (4.2) was used to derive the QAT (4.5). Nevertheless, it is actually not very relevant whether the authors of [3] used (4.2) to derive (4.1). As pointed out in [2], and as mentioned above, even though one can forget (4.2) and simply assume (4.1) as the causal condition to be satisfied by ϕ+, such causal condition has little to do with the time evolution of a quantum system, see again Eq. (4.3). In particular, as even the author of [6] has asserted, the t that appears in Eq. (4.1) is not the same as the parametric time t that labels the evolution of a quantum system.+ Thus, as far as standard quantum mechanics is concerned, the causal content of the QAT is physically vacuous, and therefore, regardless of how one motivates it, there is no physical justification for the choice (3.3). 5. TAQT vs. the “classic results” In this section, we are going to compare the Hardy axiom of TAQT with some classic results of Paley and Wiener, of Gelfand and Shilov and of the theory of ultradistributions, which we shall collectively refer to as the “classic results.” More ¶ In this quote, φin, φ+, F(t) and Eq. (4.5) correspond, respectively, to ϕin, ϕ+, ϕ̃+(t) and Eq. (4.1). + All this shows that the new term TAQT is a misnomer. A better name is Bohm-Gadella theory, because it was these two authors who proposed the theory and summarized it in [18]. precisely, we will see that the spaces of test functions ϕ̂± obtained by the “standard method” would be of Hardy class only if the “classic results” were wrong. The direct comparison with the “classic results” is more easily done in one dimension, and therefore we shall use the example of the one-dimensional rectangular barrier potential: V (x) = 0 −∞ < x < a V0 a < x < b 0 b < x < ∞ . (5.1) For this potential, the “in” and “out” eigensolutions are well known and can be found for example in [12]. We shall denote them by χ±l,r(x;E), where the labels l,r denote left and right incidence. When we analytically continue these eigenfunctions, or when we consider the Gamow states for this potential, the “standard method” calls for test functions ϕ±l,r(x) for which the following integrals are finite: ϕ̂±l,r(z) = dxχ±l,r(x; z)ϕ(x) . (5.2) Just as in the example discussed in Sec. 2, the test functions ϕ(x) must at least fall off faster than exponentials. To further simplify the discussion, we need to recall that, because of Eqs. (2.25) and (2.26), the Hardy axiom assumes that the “free” wave functions ϕ̂inl,r(E) and ϕ̂ l,r (E) are also of Hardy class. These “free” functions are given by (hereafter, we just consider ϕinl,r, since the analysis for ϕ l,r is the same) ϕ̂inl (E) = dx e−ikx ϕin(x) , (5.3) ϕ̂inr (E) = dx eikx ϕin(x) , (5.4) where k = E is the wave number. The total wave function is given by the sum of left and right components: ϕ̂in(E) = ϕ̂inl (E) + ϕ̂ r (E) . (5.5) It is simpler to work with k rather than with E and define ϕ̂inl,r(k) ≡ 2k ϕ̂inl,r(E) ; (5.6) that is, ϕ̂inl (k) = dx e−ikx ϕin(x) , k ≥ 0 , (5.7) ϕ̂inr (k) = dx eikx ϕin(x) , k ≥ 0 . (5.8) The “total” wave function in the wave-number representation, ϕ̂in(k) = ϕ̂inl (k)+ ϕ̂ r (k), is thus the Fourier transform of ϕ(x), ϕ̂in(k) = dx e−ikx ϕin(x) , k ∈ R . (5.9) Its analytic continuation will be denoted as ϕ̂in(q) = dx e−iqx ϕin(x) , q ∈ C . (5.10) At this point, we are ready to introduce two classic theorems. The first one is due to Paley and Wiener (see Theorem IX.11 in [19]): Theorem 1 (Paley-Wiener). An entire analytic function ϕ̂(q) is the Fourier transform of a C∞0 (R) function ϕ(x) with support in the segment {x | |x| < A} if, and only if, for each N there is a CN so that |ϕ̂(q)| ≤ CN e A|Im(q)| (1 + |q|)N (5.11) for all q ∈ C. This theorem says that the Fourier transform of a C∞0 function is an analytic function that grows exponentially, and that such exponential growth is mildly corrected (but not canceled) by a polynomial falloff. The second theorem we shall use is due to Gelfand and Shilov [4]. Before stating it, we need some definitions. Let a and b denote two positive real numbers satisfying (2.18). Let us define Φa,b as the set of all differentiable functions ϕ(x) (−∞ < x < ∞) satisfying the inequalities dnϕ(x) ∣∣∣∣ ≤ Cne −α |x| a (5.12) with constants Cn and α > 0 which may depend on the function ϕ. Let us define the space Φ̂a,b as the set of entire analytic functions ϕ̂(q), q = Re(q) + i Im(q), which satisfy the inequalities |qnϕ̂(q)| ≤ Cne+β |Im(q)|b b , (5.13) where the constants Cn and β > 0 depend on the function ϕ. It is obvious that the elements of Φa,b are functions that, together with their derivatives, decrease at infinity faster than e− a , whereas the elements of Φ̂a,b are analytic functions that grow exponentially at infinity as e+ |Im(q)|b b , except for a polynomial correction that doesn’t cancel the exponential blowup. Theorem 2 (Gelfand-Shilov). The space Φ̂a,b is the Fourier transform of Φa,b. This theorem means that the smooth functions that fall off at infinity faster than e−|x| a/a are, in Fourier space, analytic functions that grow exponentially like e+|Im(q)| The bounds (5.11) and (5.13) are to be understood in the same way as the bounds (2.19) and (2.20). That is, the bounds (5.11) and (5.13) mean that ϕ̂(q) is an oscillatory function that grows exponentially in the infinite arc of the q-plane, the oscillation being tightly bounded by Eqs. (5.11) and (5.13) when ϕ(x) belongs to C∞0 and Φa,b, respectively. Note that after the addition of the corresponding polynomial corrections, the bounds (2.19) and (2.20) are entirely analogous to the bounds (5.11) and (5.13)—the operators U± are after all Fourier-like transforms [12]. Let us now apply the above theorems to the functions ϕin(x) obtained by the “standard method.” In order for Eq. (5.10) to make sense, ϕin(x) must fall off faster than exponentials. If we choose ϕin(x) to fall off like e−|x| a/a, then the Gelfand-Shilov theorem tells us that ϕ̂in(q) grows like e+|Im(q)| b/b. Even when we impose that ϕin(x) is C∞0 , which is already a very strict requirement, the Paley-Wiener theorem says that ϕ̂in(q) grows exponentially. This means, in particular, that the ϕ̂in(q) do in general not tend to zero in the infinite arc of the q-plane, because if they did, the Paley-Wiener and the Gelfand-Shilov theorems would be wrong. Because of Eq. (5.6), ϕ̂in(z) does in general not tend to zero as |z| tends to infinity in the lower half-plane of the second sheet. Hence the space of ϕ̂in’s is not of Hardy class from below. The space of ϕ̂+’s cannot be of Hardy class from below either, because if it were, |z|→∞ ϕ̂+(z) = 0 , (5.14) where the limit is taken in the lower half plane of the second sheet. By Eq. (2.25), this implies that also the space of ϕ̂in’s would be of Hardy class and comply with this limit, which we know is not possible due to the “classic results.” Thus, the “standard method” yields spaces of test functions that do not comply with the Hardy axiom. This is precisely what it was meant in [2] by the assertion that TAQT is inconsistent with standard quantum mechanics. To finish this section, we note that if we chose the test functions as in [8], then we would be dealing with ultradistributions. In Fourier space, the test functions for ultradistributions grow faster than any exponential as we follow the imaginary axis, see [8] and references therein. Thus, if the “standard method” yielded spaces of Hardy functions, that property of ultradistributions would be false. 6. Further remarks The authors of [1] claim that it is inaccurate to state that the proponents of TAQT dispense with asymptotic completeness. This statement should be compared with the first quote in section 6 of [2]. The authors of [1] also claim that TAQT obtains the resonant states by solving the Schrödinger equation subject to purely outgoing boundary conditions. This claim should be compared with the second quote in section 6 of [2]. The authors of [1] also dispute the assertion of [2] that TAQT sometimes uses the whole real line as though it coincided with the scattering spectrum of the Hamiltonian. A glance at, for example, the QAT (4.1) seems to support such assertion. 7. Conclusions In standard scattering theory, one assumes that the physics is described by the Lippmann-Schwinger equation. When one solves such equation, one finds that its solutions must be accommodated by a rigged Hilbert space, and that its time evolution runs from t = −∞ till t = +∞ [14]. When one analytically continues the solutions of the Lippmann-Schwinger equation, one finds that they must be accommodated by one rigged Hilbert space, which also accommodates the resonant (Gamow) states. The construction of such rigged Hilbert space is determined by standard distribution theory. By contrast, TAQT assumes that the solutions of the Lippmann-Schwinger equations belong to two rigged Hilbert spaces of Hardy class. In TAQT, one never explicitly solves the Lippmann-Schwinger equation for specific potentials in the position representation. Instead, one assumes that its solutions satisfy the Hardy axiom. Unlike in standard scattering theory, in TAQT the time evolution of the solutions of the Lippmann-Schwinger equation does not run from t = −∞ till t = +∞. By comparing the properties of the actual solutions of the Lippmann-Schwinger equation with the Hardy axiom, we have seen that such actual solutions would comply with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov, and of the theory of ultradistributions were wrong. We have (again) stressed the fact that the Quantum Arrow of Time, which is the justification for using the rigged Hilbert spaces of Hardy class, has little to do with the time evolution of a quantum system. We have stressed that using the method of TAQT to introduce rigged Hilbert spaces, we could accommodate the Gamow states in a landscape of arbitrary rigged Hilbert spaces, see also [5]. Our claim of inconsistency should not be taken as a claim that TAQT is mathematically inconsistent or that TAQT doesn’t have a beautiful mathematical structure. What the present author claims is that TAQT is not applicable in quantum mechanics and is in fact a different theory. To finish, we would like to mention that the “classic theorems” are not in conflict with using Hardy functions in quantum mechanics. They are in conflict only with the Hardy axiom. Thus, our results do not apply to other works that use Hardy functions in a different way [20]. Acknowledgment This research was supported by MEC and DOE. References [1] M. Gadella, S. Wickramasekara, “A Comment on ‘On the inconsistency of the Bohm-Gadella theory with quantum mechanics,’ ” J. Phys. A (to be published). [2] R. de la Madrid, J. Phys. A: Math. Gen. 39, 9255 (2006); quant-ph/0606186. [3] A. Bohm, I. Antoniou, P. Kielanowski, J. Math. Phys. 36, 2593 (1995). http://arxiv.org/abs/quant-ph/0606186 [4] I.M. Gelfand, G.E. Shilov Generalized Functions, Vol. 1-3, Academic Press, New York (1964). I.M. Gelfand, N.Ya. Vilenkin Generalized Functions Vol. 4, Academic Press, New York (1964). [5] R. de la Madrid, “Reply to [6],” submitted to J. Phys. A. [6] H. Baumgartel, “Comment on [2],” submitted to J. Phys. A. [7] G. Parravicini, V. Gorini, E.C.G. Sudarshan, J. Math. Phys. 21, 2208 (1980). [8] C.G. Bollini, O. Civitarese, A.L. De Paoli, M.C. Rocca, Phys. Lett. B382, 205 (1996); J. Math. Phys. 37, 4235 (1996). [9] R. de la Madrid, A. Bohm, M. Gadella, Fortschr. Phys. 50, 185 (2002); quant-ph/0109154. [10] R. de la Madrid, J. Phys. A: Math. Gen. 35, 319 (2002); quant-ph/0110165. [11] R. de la Madrid, Int. J. Theo. Phys. 42, 2441 (2003); quant-ph/0210167. [12] R. de la Madrid, J. Phys. A: Math. Gen. 37, 8129 (2004); quant-ph/0407195. [13] R. de la Madrid, Eur. J. Phys. 26, 287 (2005); quant-ph/0502053. [14] R. de la Madrid, J. Phys. A: Math. Gen. 39, 3949 (2006); quant-ph/0603176. [15] R. de la Madrid, J. Phys. A: Math. Gen. 39, 3981 (2006); quant-ph/0603177. [16] J.R. Taylor, Scattering theory, John Wiley & Sons, Inc., New York (1972). [17] A. Bohm, P. Kielanowski, S. Wickramasekara, “Complex energies and beginnings of time suggest a theory of scattering and decay,” quant-ph/0510060. [18] A. Bohm, M. Gadella, Dirac Kets, Gamow Vectors and Gelfand Triplets, Springer Lecture Notes in Physics 348, Berlin (1989). [19] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. II, Academic Press, New York (1975). [20] Y. Strauss, L.P. Hortwitz, A. Volovick, “Approximate resonance states in the semigroup decomposition of resonance evolution,” quant-ph/0612027. http://arxiv.org/abs/quant-ph/0109154 http://arxiv.org/abs/quant-ph/0110165 http://arxiv.org/abs/quant-ph/0210167 http://arxiv.org/abs/quant-ph/0407195 http://arxiv.org/abs/quant-ph/0502053 http://arxiv.org/abs/quant-ph/0603176 http://arxiv.org/abs/quant-ph/0603177 http://arxiv.org/abs/quant-ph/0510060 http://arxiv.org/abs/quant-ph/0612027 Introduction The ``standard method'' TAQT vs. the ``standard method'' The Quantum Arrow of Time (QAT) TAQT vs. the ``classic results'' Further remarks Conclusions

In this reply, we show that when we apply standard distribution theory to the Lippmann-Schwinger equation, the resulting spaces of test functions would comply with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov, and of the theory of ultradistributions were wrong. As well, we point out several differences between the ``standard method'' of constructing rigged Hilbert spaces in quantum mechanics and the method used in Time Asymmetric Quantum Theory.

Introduction The authors of [1] allege to have shown that the conclusions of [2] regarding the inconsistency of Time Asymmetric Quantum Theory (TAQT) with quantum mechanics are false. In this reply, we will show that the arguments of [1] are missing essential aspects of [2], and that therefore the conclusions of [2] still stand. The most important claims of [1] are the following: 1. There are many examples of TAQT, and the present author has inadvertently constructed another one. 2. The flaws of the Quantum Arrow of Time (QAT) pointed out in [2] are actually not flaws, because the original derivation of the QAT was misquoted from its source [3]. 3. The crucial argument of [2] regarding the exponential blowup of the test functions ϕ̂±(z) does not prevent ϕ̂±(z) from being of Hardy class. As we shall see, all these claims do not stand close scrutiny. In order to show why, in Sec. 2 we will outline the method to construct rigged Hilbert spaces in quantum mechanics based on the theory of distributions [4]. We shall refer to this method as the “standard method” and show that the resulting rigged Hilbert spaces are not of Hardy class. We shall also explain the meaning of the exponential blowup of ϕ̂±(z) and why it implies that the spaces of test functions are not of Hardy class. In Sec. 3, we briefly outline the method to introduce rigged Hilbert spaces of Hardy class in TAQT and compare such method with the “standard method.” It will then be apparent that using the method of TAQT, one can introduce any arbitrary rigged Hilbert space for the http://arxiv.org/abs/0704.1613v1 Gamow states. In order to address claim 2, we show (again) in Sec. 4 that no matter how one introduces it, the Quantum Arrow of Time has little to do with the actual time evolution of a quantum system. To address claim 3, in Sec. 5 we use classic results of Paley and Wiener and of Gelfand and Shilov to show that the “standard method” of dealing with the Lippmann-Schwinger equation leads to rigged Hilbert spaces that are not of Hardy class. Section 7 concludes that the arguments of [2] still stand. 2. The “standard method” In this section, we illustrate the main features of the “standard method” to construct rigged Hilbert spaces in quantum mechanics [5]. Such “standard method” is based on the theory of distributions [4]. For the sake of clarity, we shall use the spherical shell potential of height V0, V (~x) = V (r) = 0 0 < r < a V0 a < r < b 0 b < r < ∞ . (2.1) For l = 0, the Hamiltonian acts as (we take ~2/2m = 1) H = − d + V (r) . (2.2) The regular solution is χ(r;E) = E r) 0 < r < a J1(E)ei E−V0 r + J2(E)e−i E−V0 r a < r < b J3(E)ei E r + J4(E)e−i E r b < r < ∞ . (2.3) The Jost functions and the S matrix are given by J+(E) = −2iJ4(E) , J−(E) = 2iJ3(E) , (2.4) S(E) = J−(E) J+(E) . (2.5) The solutions of the Lippmann-Schwinger equation can be written as 〈r|E±〉 ≡ χ±(r;E) = χ(r;E) J±(E) . (2.6) When V tends to zero, these eigensolutions tend to the “free” eigensolution: 〈r|E〉 ≡ χ0(r;E) = E r) . (2.7) These eigenfunctions are delta-normalized and therefore their associated unitary operators, (U±f)(E) = dr χ±(r;E) f(r) ≡ f̂±(E) , E ≥ 0 , (2.8) (U0f)(E) = dr χ0(r;E) f(r) ≡ f̂0(E) , E ≥ 0 , (2.9) transform from L2([0,∞), dr) onto L2([0,∞), dE). The Lippmann-Schwinger and the “free” eigenfunctions can be analytically continued from the scattering spectrum into the whole complex plane. We shall denote such analytically continued eigenfunctions by χ±(r; z) and χ0(r; z). Whenever they exist, the analytic continuations of (2.8) and (2.9) are denoted by f̂±(z) = dr χ±(r; z) f(r) , (2.10) f̂0(z) = dr χ0(r; z) f(r) , (2.11) where here and in the following z belongs to a two-sheeted Riemann surface. The resonant energies are given by the poles zn of the S matrix, and their associated Gamow states are u(r; zn) = Nn J3(zn) sin( zn r) 0 < r < a J1(zn) J3(zn)e zn−V0 r + J2(zn) J3(zn)e zn−V0 r a < r < b zn r b < r < ∞ , (2.12) where Nn is a normalization factor. The theory of distributions [4] says that a test function ϕ(r) on which a distribution d(r) acts is such that the following integral is finite:‡ 〈ϕ|d〉 ≡ dr ϕ(r)d(r) < ∞ , (2.13) where 〈ϕ|d〉 represents the action of the functional |d〉 on the test function ϕ. With some variations, this is the “standard method” followed by [7–15] to introduce spaces of test functions in quantum mechanics. Thus, contrary to what the authors of [1] assert, the method followed by the present author runs (somewhat) parallel to [8], not to TAQT. In order to use (2.13) to construct the rigged Hilbert spaces for the analytically continued Lippmann-Schwinger eigenfunctions and for the Gamow states, we need to obtain the growth of χ±(r; z), χ0(r; z) and u(r; zn). Because the regular solution blows up exponentially [16], |χ(r; z)| ≤ C |z|1/2 r 1 + |z|1/2 r z |r , (2.14) the growth of the eigenfunctions (2.6), (2.7) and (2.12) blows up exponentially: |χ±(r; z)| ≤ C 1 J±(z) |z|1/4 r 1 + |z|1/2 r z |r , (2.15) ‡ In quantum mechanics, we need to impose a few more requirements, but we will not need to go into such details here. |χ0(r; z)| ≤ C |z|1/4 r 1 + |z|1/2 r z |r , (2.16) |u(r; zn)| ≤ Cn |zn|1/2 r 1 + |zn|1/2 r zn |r . (2.17) When we plug this exponential blowup into the basic requirement (2.13) of the “standard method,” we see that the test functions on which those distributions act must fall off at least exponentially. By using the Gelfand-Shilov theory of M an Ω functions [4], it was shown in [15] that when a and b are positive real numbers satisfying = 1 , (2.18) and when ϕ+(r) is an infinitely differentiable function whose tails fall off like e−r a/a, then ϕ+(z) grows like e|Im( z)|b/b in the infinite arc of the lower half-plane of the Riemann surface: If |ϕ+(r)| < Ce− a as r → ∞, then |ϕ̂+(z)| ≤ Ce b as |z| → ∞ . (2.19) It was shown in [2] that when ϕ+(r) ∈ C∞0 , ϕ̂+(z) blows up exponentially in the infinite arc of the lower half-plane of the Riemann surface: If |ϕ+(r)| = 0 when r > A, then |ϕ̂+(z)| ≤ CeA|Im z | as |z| → ∞ . (2.20) From the above estimates, we concluded in [2] that the ϕ+’s obtained from the “standard method” cannot be Hardy functions, since ϕ̂+(z) does not tend to zero as |z| tends to infinity. The authors of [1] argue that one cannot draw any conclusion on the limit |z| → ∞ from estimates such as (2.19) or (2.20), and therefore they conclude that nothing prevents ϕ̂+(z) from tending to zero and therefore from being Hardy functions. Their conclusion is not true, because their argument does not take the nature of (2.19) and (2.20) into account. After we explain the meaning of those estimates, it will be clear why they prevent ϕ̂±(z) from tending to zero in any infinite arc of the Riemann surface. In order to understand what (2.19) and (2.20) mean, we start with the simple sine function sin( E r). When E ≥ 0, the sine function oscillates between +1 and −1: | sin( E r)| ≤ 1 , E ≥ 0 . (2.21) As E tends to infinity, such oscillatory behavior remains, and in such limit the sine function does not tend to zero. When we analytically continue the sine function, z r) , (2.22) the oscillations are bounded by | sin( z r)| ≤ C |z|1/2 r 1 + |z|1/2 r z |r . (2.23) Thus, as |z| tends to infinity, sin( z r) oscillates wildly, and the magnitude of its oscillation is tightly bounded by the exponential function. It is certain that as |z| tends to infinity, sin( z r) does not tend to zero, even though the function vanishes z r = ±nπ, n = 0, 1, . . . It just happens that the solutions of the Lippmann-Schwinger equation follow the same pattern. When E is positive, the eigensolutions are oscillatory and bounded by |χ±(r;E)| ≤ C 1 J±(E) |E|1/4 r 1 + |E|1/2 r . (2.24) When the energy is complex, their oscillations get wild and are bounded by Eq. (2.15).§ Thus, the analytic continuations of the Lippmann-Schwinger eigenfunctions oscillate wildly, and the magnitude of their oscillation is tightly bounded by an exponential function (multiplied by factors that don’t cancel the exponential blowup when |z| → ∞). Because in Eqs. (2.10) and (2.11) we are integrating over r, the exponentially- bounded oscillations of χ±(r; z) get transmitted into ϕ̂±(z). The estimates (2.19) and (2.20) bound the oscillation of the test functions of the “standard method,” except for factors that don’t cancel the exponential blowup. It is the exponentially-bounded oscillations of ϕ̂±(z) what prevent ϕ̂±(z) from tending to zero in any infinite arc of the Riemann surface and therefore from being of Hardy class. A somewhat simpler way to understand the above estimates is by looking at the “free” incoming and outgoing wave functions ϕin and ϕout. Because in the energy representation such wave functions are the same as the “in” and “out” wave functions, ϕ̂in(E) = 〈E|ϕin〉 = 〈+E|ϕ+〉 = ϕ̂+(E) , (2.25) ϕ̂out(E) = 〈E|ϕout〉 = 〈−E|ϕ−〉 = ϕ̂−(E) , (2.26) in TAQT the analytic continuation of ϕ̂in(E) and ϕ̂out(E) are also of Hardy class. Since ϕ̂in,out(z) = z r)ϕin,out(r) , (2.27) it is evident that the exponential blowup (2.23) of sin( z r) will prevent ϕ̂in,out(z) from tending to zero as |z| → ∞ in any half-plane of the Riemann surface. Thus, ϕ̂in,out(z) are not of Hardy class, contrary to TAQT. Strictly speaking, the bounds (2.19) and (2.20) are not the tightest ones. We should include polynomial corrections, see Eq. (B.15) in [15], and the effect of |z|1/4r 1+|z|1/2r and 1J±(z) to obtain the tightest bounds. We shall not obtain those corrections here, because they do not cancel the exponential blowup at infinity, and because in this reply we shall use instead other classic bounds, see Sec. 5. Let us summarize this section. In standard quantum mechanics, once the Lippmann-Schwinger equation is solved, the properties of ϕ̂±(z) are already determined by Eqs. (2.10) and (2.11), and there is no room for any extra assumption on their properties. This means, in particular, that the Hardy axiom cannot be simply assumed. Rather, the Hardy axiom must be proved using Eqs. (2.10) and (2.11).‖ It simply § The points at which J±(z) = 0 do not affect the essence of the argument. ‖ This is what in [2] it was meant by the assertion that the Hardy axiom is not a matter of assumption but a matter of proof. happens that the “standard method” yields ϕ̂±(z) and ϕ̂in,out(z) that oscillate wildly. Because these oscillations are bounded by exponential functions, ϕ̂±(z) and ϕ̂in,out(z) do not tend to zero as |z| tends to infinity in any half-plane of the Riemann surface—hence they are not of Hardy class. 3. TAQT vs. the “standard method” In TAQT, one doesn’t solve the Lippmann-Schwinger equation in order to afterward obtain the properties of ϕ̂±(z) using Eq. (2.10). Instead, one transforms into the energy representation (using U± in our example) and then imposes the Hardy axiom. If H2± denotes the spaces of Hardy functions from above (+) and below (−), S denotes the Schwartz space, and Φ̃± denote their intersection restricted to the positive real line, Φ̃± = H2± ∩ S|R+ , (3.1) then the Hardy axiom states that the functions ϕ̂±(z) belong to Φ̃∓: ϕ̂±(z) ∈ Φ̃∓ . (3.2) This means that in the position representation, the Gamow states and the analytic continuation of the Lippmann-Schwinger eigenfunctions act on the following spaces: ΦBG∓ = U ± Φ̃∓ . (3.3) It is obvious that the choices (3.2)-(3.3) are arbitrary. One may as well choose another dense subset of L2([0,∞), dE) with different properties and obtain a different space of test functions for the Gamow states. What is more, ΦBG± are different from the spaces of test functions obtained through the “standard method,” because the functions ϕ̂±(z) of the “standard method” are not of Hardy class. The authors of [1] claim that the present author has inadvertently constructed an example of TAQT. That such is not the case can be seen not only from the differences between the “standard method” and the method used in TAQT to introduce rigged Hilbert spaces, but also from the outcomes. For example, whereas in the position representation the “standard method” calls for just one rigged Hilbert space for the Gamow states and for the analytically continued Lippmann-Schwinger eigenfunctions [15], TAQT uses two rigged Hilbert spaces ΦBG± ⊂ L2([0,∞), dr) ⊂ Φ×BG± . (3.4) One of the rigged Hilbert spaces is used for the “in” solutions and for the anti-resonant states, whereas the other one is used for the “out” solutions and for the resonant states. Another difference is that in TAQT, the solutions of the Lippmann-Schwinger equation for scattering energies have a time asymmetric evolution [17], whereas the “standard method” yields that such time evolution runs from t = −∞ to t = +∞, see [14]. Incidentally, this is an instance where TAQT differs not only mathematically but also physically from standard quantum mechanics, because in standard scattering theory, the time evolution of a scattering process goes from the asymptotically remote past (t → −∞) to the asymptotically far future (t → +∞). This is not so in TAQT [17]. It seems hardly necessary to clarify what the present author means by “standard quantum mechanics.” Standard quantum mechanics means the Schrödinger equation, and standard scattering theory means the Lippmann-Schwinger equation. In standard quantum mechanics, one assumes that these equations describe the physics and then solves them. Because of the scattering and resonant spectra, their solutions lie within rigged Hilbert spaces. The construction of such rigged Hilbert spaces follows by application of the “standard method.” By contrast, TAQT simply assumes that the solutions of the Schrödinger and the Lippmann-Schwinger equations comply with the Hardy axiom, without ever showing that the actual solutions of those equations comply with such axiom. It was claimed in [2] that there is no example of TAQT. The authors of [1] dispute such claim and assert that there are many examples. The present author disagrees with their assertion, because assuming that for a large class of potentials the solutions of the Lippmann-Schwinger equation comply with the Hardy axiom is not the same as having an example where it is shown that the actual solutions of the Lippmann- Schwinger equation comply with the Hardy axiom. In fact, to the best of the present author’s knowledge, no advocate of TAQT has ever used Eq. (2.10) to discuss the analytic properties of ϕ̂±(E) = 〈±E|ϕ±〉 in terms of the actual solutions χ±(r;E) of the Lippmann-Schwinger equation. The authors of [1] inadvertently acknowledge that there is no example of TAQT when they say that they still need “to identify the form and properties” of the functions of (3.3), see the last paragraph in section 2 of [1]. By saying so, they are acknowledging that they don’t know whether the standard Gamow states defined in the position representation are well defined as functionals acting on ΦBG±. If TAQT had an example, it would be known. 4. The Quantum Arrow of Time (QAT) Advocates of TAQT argue that their choice (3.3) is not arbitrary but rather is rooted on a causality principle. Such causality principle is the “preparation-registration arrow of time,” sometimes referred to as the “Quantum Arrow of Time” (QAT). For the “in” states ϕ+, the causal statement of the QAT is written as ϕ̃+(t) ≡ dE e−iEtϕ̂+(E) = 0 , for t > 0 . (4.1) By one of the Paley-Wiener theorems, Eq. (4.1) is equivalent to assuming that ϕ̂+(E) is of Hardy class from below. The corresponding causal statement for the “out” wave functions ϕ− implies that ϕ− is of Hardy class from above. Hence, in TAQT, the choice (3.3) is not arbitrary but a consequence of causality. It was pointed out in [2] that the QAT is flawed. The argument was twofold. First, it was pointed out that the original derivation [3] of Eq. (4.1) made use of the following flawed assumption: 0 = 〈E|ϕin(t)〉 = 〈+E|ϕ+(t)〉 = e−iEtϕ̂+(E) , for all energies, (4.2) which can happen only when ϕ+ and ϕin are identically 0. It was then pointed out that even though one may simply assume the causal statement (4.1) and forget about how it was derived, such causal statement says little about the actual time evolution of a quantum system, because the quantum mechanical time evolution of ϕ+ is not given by Eq. (4.1): ϕ+(t) = e−iHtϕ+ 6= ϕ̃+(t) . (4.3) To counter this argument, the authors of [1] claim that the derivation of the QAT was misquoted from the original source [3], and that the flawed assumption (4.2) was never used to derive the QAT (4.1). It seems therefore necessary to quote the original derivation (see [3], page 2597):¶ “We are now in the position to give a mathematical formulation of the QAT: we choose t = 0 to be the time before which all preparations of φin(t) are completed and after which the registration of ψout(t) begins. This means that for t > 0 the energy distribution of the preparation apparatus must vanish: 〈E, η|φin(t)〉 = 0 for all values of the quantum numbers E and η (η are the additional quantum numbers which we usually suppress). As the mathematical statement for ‘no preparations for t > 0’ we therefore write (the slightly weaker condition) dE 〈E|φin(t)〉 = dE 〈+E|φ+(t)〉 = dE 〈+E|e−iHt|φ+〉 (4.4) dE 〈+E|φ+〉e−iEt ≡ F(t) for t > 0 . (4.5) The readers can decide whether or not the flawed hypothesis (4.2) was used to derive the QAT (4.5). Nevertheless, it is actually not very relevant whether the authors of [3] used (4.2) to derive (4.1). As pointed out in [2], and as mentioned above, even though one can forget (4.2) and simply assume (4.1) as the causal condition to be satisfied by ϕ+, such causal condition has little to do with the time evolution of a quantum system, see again Eq. (4.3). In particular, as even the author of [6] has asserted, the t that appears in Eq. (4.1) is not the same as the parametric time t that labels the evolution of a quantum system.+ Thus, as far as standard quantum mechanics is concerned, the causal content of the QAT is physically vacuous, and therefore, regardless of how one motivates it, there is no physical justification for the choice (3.3). 5. TAQT vs. the “classic results” In this section, we are going to compare the Hardy axiom of TAQT with some classic results of Paley and Wiener, of Gelfand and Shilov and of the theory of ultradistributions, which we shall collectively refer to as the “classic results.” More ¶ In this quote, φin, φ+, F(t) and Eq. (4.5) correspond, respectively, to ϕin, ϕ+, ϕ̃+(t) and Eq. (4.1). + All this shows that the new term TAQT is a misnomer. A better name is Bohm-Gadella theory, because it was these two authors who proposed the theory and summarized it in [18]. precisely, we will see that the spaces of test functions ϕ̂± obtained by the “standard method” would be of Hardy class only if the “classic results” were wrong. The direct comparison with the “classic results” is more easily done in one dimension, and therefore we shall use the example of the one-dimensional rectangular barrier potential: V (x) = 0 −∞ < x < a V0 a < x < b 0 b < x < ∞ . (5.1) For this potential, the “in” and “out” eigensolutions are well known and can be found for example in [12]. We shall denote them by χ±l,r(x;E), where the labels l,r denote left and right incidence. When we analytically continue these eigenfunctions, or when we consider the Gamow states for this potential, the “standard method” calls for test functions ϕ±l,r(x) for which the following integrals are finite: ϕ̂±l,r(z) = dxχ±l,r(x; z)ϕ(x) . (5.2) Just as in the example discussed in Sec. 2, the test functions ϕ(x) must at least fall off faster than exponentials. To further simplify the discussion, we need to recall that, because of Eqs. (2.25) and (2.26), the Hardy axiom assumes that the “free” wave functions ϕ̂inl,r(E) and ϕ̂ l,r (E) are also of Hardy class. These “free” functions are given by (hereafter, we just consider ϕinl,r, since the analysis for ϕ l,r is the same) ϕ̂inl (E) = dx e−ikx ϕin(x) , (5.3) ϕ̂inr (E) = dx eikx ϕin(x) , (5.4) where k = E is the wave number. The total wave function is given by the sum of left and right components: ϕ̂in(E) = ϕ̂inl (E) + ϕ̂ r (E) . (5.5) It is simpler to work with k rather than with E and define ϕ̂inl,r(k) ≡ 2k ϕ̂inl,r(E) ; (5.6) that is, ϕ̂inl (k) = dx e−ikx ϕin(x) , k ≥ 0 , (5.7) ϕ̂inr (k) = dx eikx ϕin(x) , k ≥ 0 . (5.8) The “total” wave function in the wave-number representation, ϕ̂in(k) = ϕ̂inl (k)+ ϕ̂ r (k), is thus the Fourier transform of ϕ(x), ϕ̂in(k) = dx e−ikx ϕin(x) , k ∈ R . (5.9) Its analytic continuation will be denoted as ϕ̂in(q) = dx e−iqx ϕin(x) , q ∈ C . (5.10) At this point, we are ready to introduce two classic theorems. The first one is due to Paley and Wiener (see Theorem IX.11 in [19]): Theorem 1 (Paley-Wiener). An entire analytic function ϕ̂(q) is the Fourier transform of a C∞0 (R) function ϕ(x) with support in the segment {x | |x| < A} if, and only if, for each N there is a CN so that |ϕ̂(q)| ≤ CN e A|Im(q)| (1 + |q|)N (5.11) for all q ∈ C. This theorem says that the Fourier transform of a C∞0 function is an analytic function that grows exponentially, and that such exponential growth is mildly corrected (but not canceled) by a polynomial falloff. The second theorem we shall use is due to Gelfand and Shilov [4]. Before stating it, we need some definitions. Let a and b denote two positive real numbers satisfying (2.18). Let us define Φa,b as the set of all differentiable functions ϕ(x) (−∞ < x < ∞) satisfying the inequalities dnϕ(x) ∣∣∣∣ ≤ Cne −α |x| a (5.12) with constants Cn and α > 0 which may depend on the function ϕ. Let us define the space Φ̂a,b as the set of entire analytic functions ϕ̂(q), q = Re(q) + i Im(q), which satisfy the inequalities |qnϕ̂(q)| ≤ Cne+β |Im(q)|b b , (5.13) where the constants Cn and β > 0 depend on the function ϕ. It is obvious that the elements of Φa,b are functions that, together with their derivatives, decrease at infinity faster than e− a , whereas the elements of Φ̂a,b are analytic functions that grow exponentially at infinity as e+ |Im(q)|b b , except for a polynomial correction that doesn’t cancel the exponential blowup. Theorem 2 (Gelfand-Shilov). The space Φ̂a,b is the Fourier transform of Φa,b. This theorem means that the smooth functions that fall off at infinity faster than e−|x| a/a are, in Fourier space, analytic functions that grow exponentially like e+|Im(q)| The bounds (5.11) and (5.13) are to be understood in the same way as the bounds (2.19) and (2.20). That is, the bounds (5.11) and (5.13) mean that ϕ̂(q) is an oscillatory function that grows exponentially in the infinite arc of the q-plane, the oscillation being tightly bounded by Eqs. (5.11) and (5.13) when ϕ(x) belongs to C∞0 and Φa,b, respectively. Note that after the addition of the corresponding polynomial corrections, the bounds (2.19) and (2.20) are entirely analogous to the bounds (5.11) and (5.13)—the operators U± are after all Fourier-like transforms [12]. Let us now apply the above theorems to the functions ϕin(x) obtained by the “standard method.” In order for Eq. (5.10) to make sense, ϕin(x) must fall off faster than exponentials. If we choose ϕin(x) to fall off like e−|x| a/a, then the Gelfand-Shilov theorem tells us that ϕ̂in(q) grows like e+|Im(q)| b/b. Even when we impose that ϕin(x) is C∞0 , which is already a very strict requirement, the Paley-Wiener theorem says that ϕ̂in(q) grows exponentially. This means, in particular, that the ϕ̂in(q) do in general not tend to zero in the infinite arc of the q-plane, because if they did, the Paley-Wiener and the Gelfand-Shilov theorems would be wrong. Because of Eq. (5.6), ϕ̂in(z) does in general not tend to zero as |z| tends to infinity in the lower half-plane of the second sheet. Hence the space of ϕ̂in’s is not of Hardy class from below. The space of ϕ̂+’s cannot be of Hardy class from below either, because if it were, |z|→∞ ϕ̂+(z) = 0 , (5.14) where the limit is taken in the lower half plane of the second sheet. By Eq. (2.25), this implies that also the space of ϕ̂in’s would be of Hardy class and comply with this limit, which we know is not possible due to the “classic results.” Thus, the “standard method” yields spaces of test functions that do not comply with the Hardy axiom. This is precisely what it was meant in [2] by the assertion that TAQT is inconsistent with standard quantum mechanics. To finish this section, we note that if we chose the test functions as in [8], then we would be dealing with ultradistributions. In Fourier space, the test functions for ultradistributions grow faster than any exponential as we follow the imaginary axis, see [8] and references therein. Thus, if the “standard method” yielded spaces of Hardy functions, that property of ultradistributions would be false. 6. Further remarks The authors of [1] claim that it is inaccurate to state that the proponents of TAQT dispense with asymptotic completeness. This statement should be compared with the first quote in section 6 of [2]. The authors of [1] also claim that TAQT obtains the resonant states by solving the Schrödinger equation subject to purely outgoing boundary conditions. This claim should be compared with the second quote in section 6 of [2]. The authors of [1] also dispute the assertion of [2] that TAQT sometimes uses the whole real line as though it coincided with the scattering spectrum of the Hamiltonian. A glance at, for example, the QAT (4.1) seems to support such assertion. 7. Conclusions In standard scattering theory, one assumes that the physics is described by the Lippmann-Schwinger equation. When one solves such equation, one finds that its solutions must be accommodated by a rigged Hilbert space, and that its time evolution runs from t = −∞ till t = +∞ [14]. When one analytically continues the solutions of the Lippmann-Schwinger equation, one finds that they must be accommodated by one rigged Hilbert space, which also accommodates the resonant (Gamow) states. The construction of such rigged Hilbert space is determined by standard distribution theory. By contrast, TAQT assumes that the solutions of the Lippmann-Schwinger equations belong to two rigged Hilbert spaces of Hardy class. In TAQT, one never explicitly solves the Lippmann-Schwinger equation for specific potentials in the position representation. Instead, one assumes that its solutions satisfy the Hardy axiom. Unlike in standard scattering theory, in TAQT the time evolution of the solutions of the Lippmann-Schwinger equation does not run from t = −∞ till t = +∞. By comparing the properties of the actual solutions of the Lippmann-Schwinger equation with the Hardy axiom, we have seen that such actual solutions would comply with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov, and of the theory of ultradistributions were wrong. We have (again) stressed the fact that the Quantum Arrow of Time, which is the justification for using the rigged Hilbert spaces of Hardy class, has little to do with the time evolution of a quantum system. We have stressed that using the method of TAQT to introduce rigged Hilbert spaces, we could accommodate the Gamow states in a landscape of arbitrary rigged Hilbert spaces, see also [5]. Our claim of inconsistency should not be taken as a claim that TAQT is mathematically inconsistent or that TAQT doesn’t have a beautiful mathematical structure. What the present author claims is that TAQT is not applicable in quantum mechanics and is in fact a different theory. To finish, we would like to mention that the “classic theorems” are not in conflict with using Hardy functions in quantum mechanics. They are in conflict only with the Hardy axiom. Thus, our results do not apply to other works that use Hardy functions in a different way [20]. Acknowledgment This research was supported by MEC and DOE. References [1] M. Gadella, S. Wickramasekara, “A Comment on ‘On the inconsistency of the Bohm-Gadella theory with quantum mechanics,’ ” J. Phys. A (to be published). [2] R. de la Madrid, J. Phys. A: Math. Gen. 39, 9255 (2006); quant-ph/0606186. [3] A. Bohm, I. Antoniou, P. Kielanowski, J. Math. Phys. 36, 2593 (1995). http://arxiv.org/abs/quant-ph/0606186 [4] I.M. Gelfand, G.E. Shilov Generalized Functions, Vol. 1-3, Academic Press, New York (1964). I.M. Gelfand, N.Ya. Vilenkin Generalized Functions Vol. 4, Academic Press, New York (1964). [5] R. de la Madrid, “Reply to [6],” submitted to J. Phys. A. [6] H. Baumgartel, “Comment on [2],” submitted to J. Phys. A. [7] G. Parravicini, V. Gorini, E.C.G. Sudarshan, J. Math. Phys. 21, 2208 (1980). [8] C.G. Bollini, O. Civitarese, A.L. De Paoli, M.C. Rocca, Phys. Lett. B382, 205 (1996); J. Math. Phys. 37, 4235 (1996). [9] R. de la Madrid, A. Bohm, M. Gadella, Fortschr. Phys. 50, 185 (2002); quant-ph/0109154. [10] R. de la Madrid, J. Phys. A: Math. Gen. 35, 319 (2002); quant-ph/0110165. [11] R. de la Madrid, Int. J. Theo. Phys. 42, 2441 (2003); quant-ph/0210167. [12] R. de la Madrid, J. Phys. A: Math. Gen. 37, 8129 (2004); quant-ph/0407195. [13] R. de la Madrid, Eur. J. Phys. 26, 287 (2005); quant-ph/0502053. [14] R. de la Madrid, J. Phys. A: Math. Gen. 39, 3949 (2006); quant-ph/0603176. [15] R. de la Madrid, J. Phys. A: Math. Gen. 39, 3981 (2006); quant-ph/0603177. [16] J.R. Taylor, Scattering theory, John Wiley & Sons, Inc., New York (1972). [17] A. Bohm, P. Kielanowski, S. Wickramasekara, “Complex energies and beginnings of time suggest a theory of scattering and decay,” quant-ph/0510060. [18] A. Bohm, M. Gadella, Dirac Kets, Gamow Vectors and Gelfand Triplets, Springer Lecture Notes in Physics 348, Berlin (1989). [19] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. II, Academic Press, New York (1975). [20] Y. Strauss, L.P. Hortwitz, A. Volovick, “Approximate resonance states in the semigroup decomposition of resonance evolution,” quant-ph/0612027. http://arxiv.org/abs/quant-ph/0109154 http://arxiv.org/abs/quant-ph/0110165 http://arxiv.org/abs/quant-ph/0210167 http://arxiv.org/abs/quant-ph/0407195 http://arxiv.org/abs/quant-ph/0502053 http://arxiv.org/abs/quant-ph/0603176 http://arxiv.org/abs/quant-ph/0603177 http://arxiv.org/abs/quant-ph/0510060 http://arxiv.org/abs/quant-ph/0612027 Introduction The ``standard method'' TAQT vs. the ``standard method'' The Quantum Arrow of Time (QAT) TAQT vs. the ``classic results'' Further remarks Conclusions

Mon. Not. R. Astron. Soc. 000, 000–000 (2005) Printed 6 September 2021 (MN LATEX style file v2.2) Modelling the Galactic bar using OGLE-II Red Clump Giant Stars Nicholas J. Rattenbury1, Shude Mao1, Takahiro Sumi2, Martin C. Smith3 ⋆ 1 University of Manchester, Jodrell Bank Observatory, Macclesfield, SK11 9DL, UK 2 Solar-Terrestrial Environment Laboratory, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan 3 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands Accepted ........ Received .......; in original form ...... ABSTRACT Red clump giant stars can be used as distance indicators to trace the mass distribution of the Galactic bar. We use RCG stars from 44 bulge fields from the OGLE-II microlensing collaboration database to constrain analytic tri-axial models for the Galactic bar. We find the bar major axis is oriented at an angle of 24◦ – 27◦ to the Sun-Galactic centre line-of-sight. The ratio of semi-major and semi-minor bar axis scale lengths in the Galactic plane x0, y0, and vertical bar scale length z0, is x0 : y0 : z0 = 10 : 3.5 : 2.6, suggesting a slightly more prolate bar structure than the working model of Gerhard (2002) which gives the scale length ratios as x0 : y0 : z0 = 10 : 4 : 3. Key words: Galaxy: bulge - Galaxy: centre - Galaxy: structure 1 INTRODUCTION It is now generally accepted that the Galactic bulge is a tri-axial, bar-like structure. Observational evidence for a bar has arisen from several sources, such as the study of gas kinematics (e.g. Binney et al. 1991), surface brightness (e.g. Blitz & Spergel 1991), star counts (e.g. Nakada et al. 1991; Stanek et al. 1994) and mi- crolensing (e.g. Udalski et al. 1994); see Gerhard (2002) for a re- view. Observational data have been used to constrain dynami- cal models of the Galaxy. Dwek et al. (1995) used the COBE- DIRBE multi-wavelength observations of the Galactic centre (Weiland et al. 1994) to constrain several analytic bar models. Stanek et al. (1997) used optical observations of red clump gi- ant (RCG) stars to constrain theoretical bar models. Similarly, Babusiaux & Gilmore (2005) and Nishiyama et al. (2005) traced the bulge RCG population in the infrared. This work uses a sam- ple of stars 30 times larger than that of Stanek et al. (1997), with a greater number of fields distributed across a larger area of the Galactic bulge, thus allowing finer constraints to be placed on the bar parameters than those determined by Stanek et al. (1997). Our current understanding of the Galactic bar is that it is ori- entated at about 15− 40◦ to the Sun–Galactic centre line-of-sight, with the near end in the first Galactic longitude quadrant. The bar length is around 3.1 – 3.5 kpc with axis ratio approximately 10 : 4 : 3 (Gerhard 2002). The above bar parameters are gener- ally accepted as a working model, however they are not well de- termined. Our understanding of the complete structure of the inner Galactic regions is similarly incomplete. For example, recent infra- red star counts collected by the Spitzer Space Telescope for Galac- ⋆ e-mail: (njr, smao)@jb.man.ac.uk; sumi@stelab.nagoya-u.ac.jp; msmith@astro.rug.nl tic longitudes l = 10◦ – 30◦ are best explained assuming a long thin bar oriented at an angle of ∼ 44◦ to the Sun–Galactic centre line (Benjamin et al. 2005) while most previous studies (performed at |l| . 12◦) prefer a short bar with an opening angle of ∼ 20◦. Re- cently, Cabrera-Lavers et al. (2007) report that NIR observations of RCGs support the hypothesis that a long thin bar oriented at ∼ 45◦ co-exists with a distinct short tri-axial bulge structure oriented at ∼ 13◦. In addition, there may be some fine features, such as a ring in the Galactic bulge (Babusiaux & Gilmore 2005), or a secondary bar (Nishiyama et al. 2005), that are not yet firmly established. It is therefore crucial to obtain as many constraints as possible in order to better understand the structure of the inner Galaxy. In this paper we present an analysis of RCG stars observed in the Galactic bulge fields during the second phase of the OGLE microlensing project (Udalski et al. 2000). These stars are bright and they are approximately standard candles, hence their magni- tudes can be taken as an approximate measure of their distances. Number counts in 34 central bulge fields with −4◦ 6 l 6 6◦ and −6◦ 6 b 6 3◦ are used to constrain analytic tri-axial bar models, and thereby obtain estimates on bar parameters. We repeat the analysis with 44 fields with −6.8◦ 6 l 6 10.6◦. We find the fitted bar parameters support the general orientation and shape of the bar reported by other groups. This paper is organised as fol- lows: in Section 2 we describe the OGLE microlensing experiment and photometry catalogue and we illustrate how RCG stars can be used as approximate distance indicators; in Section 3 we detail how RCGs in the OGLE-II proper motion catalogue are selected; in Sec- tion 4 we compute the distance modulus to the red clump in 45 OGLE-II fields and thereby trace the central mass density of the Galaxy; in Section 5 we describe how RCG star count histograms for each field can be used to constrain analytic bar models of the inner Galaxy; our results and their comparison to previous works is c© 2005 RAS http://arxiv.org/abs/0704.1614v1 2 Rattenbury et al. given in Section 6 and in Section 7 we discuss the implications and limitations of these results. 2 DATA The OGLE (Udalski et al. 2000) and MOA (Bond et al. 2001; Sumi et al. 2003) microlensing collaborations currently make rou- tine observations of crowded stellar fields towards the Galactic bulge, and issue alerts when a microlensing event is detected. A result of this intense monitoring is the creation of massive photom- etry databases for stars in the Galactic bulge fields. Such databases are extremely useful for kinematic and population studies of the central regions of the Galaxy. Sumi et al. (2004) obtained the proper motions for millions of stars in the OGLE-II database for a large area of the sky. Fig. 1 shows the OGLE-II fields towards the Galactic bulge. In this paper we focus on the population of red clump giant stars at the Galac- tic centre. Red clump giants are metal-rich horizontal branch stars (Stanek et al. 2000 and references therein). Theoretically, one ex- pects their magnitudes to have (small) variations with metallicity, age and initial stellar mass (Girardi & Salaris 2001). Empirically they appear to be reasonable standard candles in the I-band with little dependence on metallicities (Udalski 2000; Zhao et al. 2001). 3 METHODS Stanek et al. (1997) used RCG stars in 12 fields (see Fig. 1) observed during the first phase of the OGLE microlensing ex- periment, OGLE-I, to constrain several analytic models of the Galactic bar density distribution. Babusiaux & Gilmore (2005), Nishiyama et al. (2005) and Cabrera-Lavers et al. (2007) similarly used IR observations of RCGs to trace the bulge stellar density. We follow similar procedures to extract RCG stars from the OGLE-II Galactic bulge fields and to constrain analytic models. 3.1 Sample selection We compute the reddening-independent magnitude IV−I for all stars in each of the 45 OGLE-II fields: IV−I = I − AI/(AV − AI) (V − I) where AI and AV are the extinctions in the I and V bands deter- mined by Sumi (2004). We select stars which have I < 4(V − I)+k, where k is a constant chosen for each field that excludes the main-sequence dwarf stars, and IV−I < 14.66, which corresponds to the magnitude of RCG stars closer than 15 kpc1. Fig. 2 shows the sample of stars selected from the IV−I, (V −I) CMD for OGLE-II field 1. The selected stars are then collected in IV−I magnitude bins, see Fig. 3. A function comprised of quadratic and Gaussian com- ponents is used to model this number count histogram in each of the OGLE-II fields: N(x ≡ IV−I) = a+bx+cx2+ 2πσRC − (IV−I,RC − x) 2σRC2 1 We assume the fiducial RCG star at 15 kpc has an absolute magnitude and colour of I0 = −0.26 and (V − I)0 = 1.0 respectively, with AI/(AV − AI) = 0.96. 0.5 1 1.5 2 2.5 3 17PSfrag replacements V − I Figure 2. Reddening-independent magnitude vs colour diagram for OGLE- II field 1. The red clump is clearly visible. Stars are selected (black dots) using the criteria I < 4(V − I) + k, where k is a constant chosen for each field, and IV−I < 14.66 (solid lines, see text). 12 12.5 13 13.5 14 14.5 15 PSfrag replacements Figure 3. Number count histogram for selected stars in OGLE-II field 1. The heavy solid line is the best-fitting model of the form given by equa- tion (1). The last histogram bin is generally not included in the fitting due to incompleteness effects near the limiting magnitude of IV−I < 14.66. where σRC is the spread of the red clump giant magnitudes, IV −I,RC is the mean apparent magnitude of the red clump, NRC and a, b and c are coefficients for the Gaussian and quadratic com- ponents respectively. These six parameters are allowed to vary for each of the OGLE-II fields and the best-fitting values obtained by minimising χ2 = [(N − Nobs,i)/σi]2 where the sum is taken over all i = 1 . . . 26 histogram bins which cover the range 12 6 IV−I 6 14.6. The error on histogram number counts is as- sumed to be σi = Nobs,i , i.e. Poissonian. The errors on the mean magnitude and distribution width of the red clump stars, ξIV−I,RC and ξσRC respectively, are deter- mined using a maximum-likelihood analysis (see e.g. Lupton et al. 1987). We determine the distance modulus to the red clump in each field as: c© 2005 RAS, MNRAS 000, 000–000 Modelling the Galactic bar 3 −8−6−4−20246810 replacem Galactic longitude (◦) Figure 1. The position of the 45 OGLE-II fields used in this analysis. The solid black regions indicate the location of fields used in Stanek et al. (1997). µ = 5 log(d)− 5 = I − I0,RC = IV −I,RC +R (V − I)0,RC − I0,RC (2) where d is the distance to the red clump measured in parsecs, IV −I,RC is the fitted peak reddening-independent magnitude of the red clump, R is the mean value of AI/(AV − AI) for each field. I0,RC = −0.26 ± 0.03 (Alves et al. 2002) and (V − I)0,RC = 1.0± 0.08 (Paczynski & Stanek 1998) are the mean absolute mag- nitude and colour of the population of local red clump giant stars. We assume that the properties of the local population of red clump giant stars are the same as the Galactic bulge population, however it is likely that population effects are significant. We discuss the effects of red clump population effects in Section 4.1. 4 DISTANCE MODULI OF RCGS Table 1 lists the fitted parameters IV −I,RC ± ξIV−I,RC and σRC ± ξσRC , along with the mean value of R for each field. The dis- tance modulus, µ, computed from equation (2) showed that there was a significant offset: the fitted mean red clump giant magni- tudes were uniformly too faint compared to that expected of typ- ical local RCG stars at 8 kpc, resulting in overly large distance moduli. Sumi (2004) found that the OGLE-II RCGs are approxi- mately 0.3 mag fainter than expected when assuming that the pop- ulation of RCGs in the bulge is the same as local. For this reason, we apply an offset of 0.3 mag to the distance moduli computed above. The shifted distance moduli µ′ = µ − 0.3 are given in Ta- ble 1. The true line-of-sight dispersion, σlos, can be approximated by σlos = (σ RC − σ2RC,0 − σ2e )1/2 where σRC is the Gaussian dis- persion fitted using equation (1), σRC,0 is the intrinsic dispersion of the RCG luminosity function and σe is an estimate of the pho- tometric errors (Babusiaux & Gilmore 2005). We use σRC,0 = 0.2 (see Section 2) and σe = 0.02, along with the tabulated values of σRC to determine σlos. Fig. 4 shows the mean distance to the red clump stars in each of the 45 OGLE-II fields listed in Table 1. The fields with Galactic longitude −4◦ 6 l 6 6◦ show clear evidence of a bar, with a major axis oriented at ≃ 25◦ to the Sun-Galactic centre line-of-sight. For fields with l > 6◦ and l < −4◦ the mean position of the red clump stars do not continue to trace the major axis of a linear structure. Babusiaux & Gilmore (2005) find similar evidence that the position of the red clump stars is not predicted by a bar for l = −9.7◦, suggesting that this is a detection of the end of the bar, or the beginning of a ring-like structure. We investigate these possibilities in Section 6.1. The uncertainties on the mean position of the red clump are large; the largest term in the error expression for ξIV−I,RC arises from the relatively large uncertainty in the intrinsic colour of the RCGs. The true line-of-sight dispersions, σlos, are consistent with a wide range of bar position angles, but the mean position of the red clump in each direction strongly suggest a bar oriented along a direction consistent with that determined by the previous work referred to in Section 1. c© 2005 RAS, MNRAS 000, 000–000 4 Rattenbury et al. Table 1. Fitted values of the red clump mean magnitude, IV −I,RC, and Gaussian dispersion, σRC for RCG stars selected from 45 OGLE-II fields. The mean selective extinction R is also given. The shifted distance modulus µ′ = µ − 0.3 for each field is computed via equation (2), see text. σlos is the true line-of-sight distance dispersion of the red clump giant stars. N is the total number of stars selected from each CMD, see Section 3.1 Field l b IV −I,RC σRC R µ ′ σlos N 1 1.08 -3.62 13.616 ±0.005 0.2936 ±0.0043 0.964 ±0.02 14.55 ±0.09 0.21 31002 2 2.23 -3.46 13.536 ±0.005 0.3130 ±0.0040 0.964 ±0.02 14.47 ±0.09 0.24 33813 3 0.11 -1.93 13.664 ±0.003 0.2461 ±0.0026 0.964 ±0.04 14.60 ±0.09 0.14 66123 4 0.43 -2.01 13.655 ±0.003 0.2517 ±0.0026 0.964 ±0.04 14.59 ±0.09 0.15 65748 5 -0.23 -1.33 13.630 ±0.003 0.2809 ±0.0030 0.964 ±0.06 14.56 ±0.10 0.20 43493 6 -0.25 -5.70 13.606 ±0.011 0.4197 ±0.0088 0.964 ±0.03 14.54 ±0.09 0.37 12085 7 -0.14 -5.91 13.589 ±0.012 0.4255 ±0.0100 0.964 ±0.03 14.52 ±0.09 0.38 11328 8 10.48 -3.78 13.366 ±0.002 1.4277 ±0.0013 0.964 ±0.03 14.30 ±0.09 1.41 10248 9 10.59 -3.98 13.383 ±0.012 0.5855 ±0.0110 0.964 ±0.03 14.32 ±0.09 0.55 9971 10 9.64 -3.44 13.407 ±0.002 1.4178 ±0.0013 0.964 ±0.03 14.34 ±0.09 1.40 12068 11 9.74 -3.64 13.438 ±0.015 0.3902 ±0.0125 0.964 ±0.04 14.37 ±0.09 0.33 11345 12 7.80 -3.37 13.381 ±0.008 0.4692 ±0.0072 0.964 ±0.04 14.31 ±0.09 0.42 15936 13 7.91 -3.58 13.389 ±0.009 0.4234 ±0.0081 0.964 ±0.03 14.32 ±0.09 0.37 15698 14 5.23 2.81 13.550 ±0.006 0.3131 ±0.0053 0.964 ±0.04 14.48 ±0.09 0.24 27822 15 5.38 2.63 13.564 ±0.007 0.3027 ±0.0059 0.964 ±0.04 14.50 ±0.09 0.23 24473 16 5.10 -3.29 13.487 ±0.007 0.3200 ±0.0057 0.964 ±0.03 14.42 ±0.09 0.25 22055 17 5.28 -3.45 13.474 ±0.007 0.3270 ±0.0058 0.964 ±0.03 14.41 ±0.09 0.26 23132 18 3.97 -3.14 13.471 ±0.005 0.2983 ±0.0044 0.964 ±0.02 14.40 ±0.09 0.22 32457 19 4.08 -3.35 13.491 ±0.006 0.3026 ±0.0048 0.964 ±0.03 14.42 ±0.09 0.23 30410 20 1.68 -2.47 13.583 ±0.004 0.2726 ±0.0031 0.964 ±0.03 14.52 ±0.09 0.18 49900 21 1.80 -2.66 13.596 ±0.004 0.2886 ±0.0034 0.964 ±0.02 14.53 ±0.09 0.21 45578 22 -0.26 -2.95 13.741 ±0.004 0.2669 ±0.0034 0.964 ±0.04 14.67 ±0.09 0.18 42914 23 -0.50 -3.36 13.724 ±0.004 0.2778 ±0.0037 0.964 ±0.04 14.66 ±0.09 0.19 36030 24 -2.44 -3.36 13.817 ±0.004 0.2638 ±0.0037 0.964 ±0.04 14.75 ±0.09 0.17 35351 25 -2.32 -3.56 13.810 ±0.004 0.2695 ±0.0039 0.964 ±0.03 14.74 ±0.09 0.18 31801 26 -4.90 -3.37 13.815 ±0.005 0.2897 ±0.0046 0.964 ±0.02 14.75 ±0.09 0.21 26940 27 -4.92 -3.65 13.794 ±0.006 0.2782 ±0.0050 0.964 ±0.02 14.73 ±0.09 0.19 24603 28 -6.76 -4.42 13.785 ±0.010 0.3081 ±0.0082 0.964 ±0.02 14.72 ±0.09 0.23 13702 29 -6.64 -4.62 13.762 ±0.009 0.2792 ±0.0083 0.964 ±0.02 14.70 ±0.09 0.19 12893 30 1.94 -2.84 13.570 ±0.004 0.2746 ±0.0034 0.964 ±0.03 14.50 ±0.09 0.19 41748 31 2.23 -2.94 13.535 ±0.004 0.2892 ±0.0036 0.964 ±0.02 14.47 ±0.09 0.21 40623 32 2.34 -3.14 13.528 ±0.004 0.3001 ±0.0038 0.964 ±0.02 14.46 ±0.09 0.22 35954 33 2.35 -3.66 13.559 ±0.005 0.3206 ±0.0045 0.964 ±0.02 14.49 ±0.09 0.25 30882 34 1.35 -2.40 13.608 ±0.003 0.2711 ±0.0031 0.964 ±0.03 14.54 ±0.09 0.18 52216 35 3.05 -3.00 13.533 ±0.005 0.3049 ±0.0040 0.964 ±0.02 14.47 ±0.09 0.23 36796 36 3.16 -3.20 13.508 ±0.005 0.3104 ±0.0042 0.964 ±0.02 14.44 ±0.09 0.24 34437 37 0.00 -1.74 13.636 ±0.003 0.2488 ±0.0025 0.964 ±0.05 14.57 ±0.10 0.15 72098 38 0.97 -3.42 13.637 ±0.005 0.2911 ±0.0038 0.964 ±0.02 14.57 ±0.09 0.21 34675 39 0.53 -2.21 13.687 ±0.003 0.2524 ±0.0027 0.964 ±0.04 14.62 ±0.09 0.15 60217 40 -2.99 -3.14 13.854 ±0.004 0.2459 ±0.0036 0.964 ±0.04 14.79 ±0.09 0.14 35426 41 -2.78 -3.27 13.857 ±0.004 0.2543 ±0.0035 0.964 ±0.04 14.79 ±0.09 0.16 34118 42 4.48 -3.38 13.494 ±0.006 0.3354 ±0.0051 0.964 ±0.03 14.43 ±0.09 0.27 27377 43 0.37 2.95 13.839 ±0.004 0.2654 ±0.0033 0.964 ±0.05 14.77 ±0.10 0.17 40730 45 0.98 -3.94 13.595 ±0.006 0.3302 ±0.0047 0.964 ±0.03 14.53 ±0.09 0.26 29009 46 1.09 -4.14 13.616 ±0.006 0.3189 ±0.0047 0.964 ±0.03 14.55 ±0.09 0.25 26027 4.1 RCG population effects Sumi (2004) found that the extinction-corrected magnitudes of RCG stars in the OGLE-II bulge fields were 0.3 mag higher than that of a RCG with an intrinsic magnitude equal to those of lo- cal RCGs, placed at a distance of 8kpc. Sumi (2004) notes that the cause of this offset is uncertain but may be resolved when de- tailed V-band OGLE-II photometry of RR Lyrae stars will allow improved extinction zero-point estimations for all fields. There is evidence that the properties of bulge RCGs are different to local RCG stars, in particular age and metallicity, resulting in a differ- ent average absolute magnitude (see e.g. Percival & Salaris (2003); Salaris et al. (2003)). This could in part explain the observed off- set of 0.3 mag between local and bulge RCG stars. The distance modulus plotted in Fig. 4 is as for equation (2): µ = IV −I,RC +R (V − I)0,RC − I0,RC + κ where κ = −0.3 mag is the offset between the observed bulge RCG population and a fiducial local RCG placed at 8 kpc. Includ- ing the difference in intrinsic magnitude between local and bulge RCG populations we have: µ = IV −I,RC +R (V − I)0,RC − I0,RC +∆IRC + ν where ∆IRC is the intrinsic RCG magnitude difference between lo- cal and bulge populations and ν is a magnitude offset arising from effects other than population differences. The theoretical popula- tion models of Girardi & Salaris (2001) estimate ∆IRC ≃ −0.1. c© 2005 RAS, MNRAS 000, 000–000 Modelling the Galactic bar 5 −2−1012 PSfrag replacements x (kpc) l = −10◦l = −5◦l = +5◦l = +10◦ Figure 4. Structure of the inner region of the Milky Way as traced by red clump giant stars extracted from the OGLE-II microlensing survey data. The mean position of the red clump for 45 OGLE-II fields is indicated by black dots with errors shown by black lines. The grey lines along each line of sight indicate the 1-σ spread in distances obtained from fitting equa- tion (1) to the red clump data in each field and correcting for intrinsic red clump luminosity dispersion and photometric errors; see text. A position angle of 25◦ is shown by the thin solid line. Lines-of-sight at l = ±5◦ and l = ±10◦ are indicated along the top axis. The remaining magnitude offset ν = −0.2 can be accounted for by decreasing the assumed value of the Galactocentric distance R0 from 8 kpc to 7.3 kpc. This assumption of R0 is in agreement with the value of 7.52±0.10 (stat) ±0.32 (sys) kpc determined by Nishiyama et al. (2006), and that of 7.63±0.32 kpc determined by Eisenhauer et al. (2005). We note that as long as the stellar popula- tion is uniform in the Galactic bulge (an assumption largely consis- tent with the data), then this unknown offset only affects the zero point (and hence the distance to the Galactic centre), and the abso- lute bar scale lengths. However the ratios between bar scale lengths are very robust, which we demonstrate in Section 6.2 where we determine the bar parameters for several values of R0. A metallicity gradient across the Galactic bulge would have an effect on the intrinsic colour and absolute magnitude of bulge RCG stars as a function of Galactic longitude. Pérez et al. (2006) find that there is a connection between metallicity gradient and struc- tural features in a sample of 6 barred galaxies. Minniti et al. (1995) measured metallicities for K giants in two fields at 1.5 kpc and 1.7 kpc from the Galactic centre. The average metallicity was [Fe/H] = −0.6, lower than that of K giants in Baade’s window. However, no such metallicity gradient was reported for Galactocentric dis- tance range 500 pc – 3.5 kpc by Ibata & Gilmore (1995). Similarly, Santiago et al. (2006) determine that there is no metallicity gradient within angles 2.2◦ to 6.0◦ of the Galactic centre (corresponding to 300 – 800 pc from the GC assuming a Sun-GC distance of 8 kpc). 5 MODELLING THE BAR We continue to investigate the structure of the Galactic bar by fit- ting analytic models of the stellar density to the observed RCG data in the OGLE-II fields. Following Stanek et al. (1997) we use the an- alytic models of Dwek et al. (1995) to fit the observed data. Three model families (Gaussian, exponential and power-law) are used: ρG1 = ρ0 exp(−r2/2) (3) ρG2 = ρ0 exp(−r2s /2) (4) ρG3 = ρ0r exp(−r3) (5) ρE1 = ρ0 exp(−re) (6) ρE2 = ρ0 exp(−r) (7) ρE3 = ρ0K0(rs) (8) ρP1 = ρ0 1 + r ρP2 = ρ0 r(1 + r)3 ρP3 = ρ0 1 + r2 where K0 is the modified Bessel function of the second kind and The co-ordinate system has the origin at the Galactic centre, with the xy plane defining the mid-plane of the Galaxy and the z direction parallel to the direction of the Galactic poles. The x direction defines the semi-major axis of the bar. The functions are rotated by an angle α around the z-axis. An angle of α = ±π corresponds to the major axis of the bar pointing towards the Sun. The functions can also be rotated by an angle β around the y axis, corresponding to the Sun’s position away from the mid-plane of the Galaxy. We aim to fit the observed number count histograms for each field as a function of magnitude. Given a magnitude IV−I, the num- ber of stars with this magnitude is (Stanek et al. 1997): N(IV−I) = c1 Z smax ρ(s)s Φ(L)Lds (12) where the integration is taken over distance smin = 3kpc < s < smax = 13 kpc. We perform the integration over this range of R0± 5 kpc as we do not expect the tri-axial bulge density structure to exceed these limits. The constant c1 is dependent on the solid angle subtended around each line of sight. The luminosity L is given by L = c2s −0.4IV−I c© 2005 RAS, MNRAS 000, 000–000 6 Rattenbury et al. and c2 is a constant. The luminosity function Φ(L) is Φ(L) = N0 2π σRC − (L− LRC) 2σ2RC where LRC is the luminosity of the red clump and σRC is the in- trinsic spread in red clump giant luminosity and is held constant. There are ten parameters to be determined in the above equa- tions: the three bar scale lengths x0, y0, z0; the bar orientation and tilt angles α and β; the luminosity function parameters N0, NRC , γ and LRC and the density function parameter ρ0. There is evidence that the centroid of the bar is offset from the centre of mass of the Galaxy, a feature commonly observed in external galaxies (Stanek et al. 1997; Nishiyama et al. 2006 and references therein). We include an additional parameter in the mod- elling process, δl, which allows for this possible centroid offset. Theoretical number counts are computed for the i = 1 . . .M fields at longitudes li as Ni(li + δl) where the offset parameter δl is de- termined over all fields for a given density model. We apply an exponential cut-off to the density functions simi- lar to that of Dwek et al. (1995): f(r) = 1.0 r = x2 + y2 6 rmax − (r−rmax) r > rmax where r is in kpc, r0 = 0.5 kpc and rmax is a cut-off radius. The theoretical constraint on the maximum radius of stable stellar or- bits is the co-rotation radius, rmax. We adopt the co-rotation value of rmax = 2.4 kpc determined by Binney et al. (1991) in equa- tion (13). We later repeat the modelling process without this theo- retical cut-off, see Section 6.2. Stanek et al. (1997) included a further fitting parameter to ac- count for a possible metallicity gradient across the bulge, but found that this did not significantly affect the bar parameters. The discus- sion in Section 4.1 suggests that there is no appreciable metallic- ity gradient over the bulge region investigated here. We therefore do not include an additional model parameter corresponding to a metallicity gradient in the model fitting analysis. The model fitting was performed using a standard non-linear Neadler-Mead minimisation algorithm. For each of the nine density profile models we minimise χ2: (Nik(IV−I)− bNik(IV−I))2 where the summations are taken over each of the 26 IV−I histogram bins in M OGLE-II fields, Nk(IV−I) is the observed histogram data for field k and bNk(IV−I) is the model number count histogram from equation (12). The error in Nik(IV−I) was taken to be σik = Nik(IV−I). 6 RESULTS The 11 parameters x0, y0, z0, α, β, A, N0, NRC, γ, LRC and δl were fitted for each of the nine density profiles given in equa- tions. (3 – 11) for the 34 OGLE-II fields2 which have −4◦ 6 l 6 6◦. A naive interpretation of Fig. 4 is that fitting all fields including those with l > 6◦ and l < −4◦ with a single bar-like structure 2 Field 5 was excluded due to poor understanding of the dust extinction in this field. would not be successful and indeed initial modelling using data from all fields resulted in bar angles of ≃ 45◦. As seen in Fig. 4 this result is consistent with the mean position angle of all fields, but clearly does not describe the bar correctly. We therefore begin modelling the bar using data from the central 34 fields which have −4◦ 6 l 6 6◦. In Section 6.1 we proceed by including the data from fields with l > 6◦ and l < −4◦ in the modelling process and in Section 6.2 we consider the effect of changing R0. The range of field latitudes is small and therefore the infor- mation available in the latitude direction from the total dataset un- likely to be able to constrain β strongly. Initial modelling runs held β constant at 0◦. Once a χ2 minimum was determined for each model holding β = 0◦, this parameter was allowed to vary with the ten other parameters. Figure 5 shows the number count histograms of observed red clump giants in the 34 OGLE-II fields with −4◦ 6 l 6 6◦, along with the best-fitting model (equation (12)) using the E2 density profile given in equation (7) as an example3. Each set of axes is arranged roughly in order of decreasing Galactic longitude. The magnitude of the red clump peak increases with decreasing longi- tude, consistent with a bar potential. The observed number count histograms are reasonably well- fitted by all models in most fields. The most obvious exceptions are fields 6, 7, 14, 15 and 43. These fields are at the most extreme latitudes represented in the data. In the case of fields 14 and 15 the algorithm fails to fit satisfactorily both observed histograms for a given model. Upon closer inspection of the observed histogram data for these two fields, it is noted that the total number of stars in the field 15 histogram is significantly less than that for field 14. The models preferred by all fields clearly cannot reproduce the decline in total star count between fields 14 and 15. Upon inspection of the CCD pixel position of stars for field 15 it was found that there are no stars recorded in a region along one edge of the field. The lack of stars in this strip is due to missing V -band data for these stars. Similarly, there is a lack of stars in field 15 in the lower right corner, due to heavy dust extinction in this area. All models predict the peak of the red clump to be at a lower magnitude than that observed for field 43. A similar offset is seen in the other high latitude fields 14 and 15. These fields are all at latitude b ≃ 3◦ and the observed offset in all three fields is likely to be related to this common position in latitude. The cause of this effect is currently unknown. The observed offset between the maxi- mum density position predicted by the tri-axial bar models and that observed for some fields may imply some asymmetry of the bar in the latitude direction, as suggested by Nishiyama et al. (2006) and references therein. Fields 6 and 7, both at b ≃ −6◦, are also poorly fitted by all models. The histograms of observed red clump stars in these fields show a curious double peak. Clearly the density functions equations (3 – 11) are inadequate for describing such a feature. This double-peaked structure may be the result of another popu- lation of stars lying along the line of sight, at a distance differ- ent to the main bulge population. The best-fitting model curves typically have a peak at magnitudes corresponding to the highest peak in the observed histogram, thereby suggesting that the sec- ond population, if real, and composed of a significant fraction of RCGs, exists between us and the bulge population. A population of asymptotic giant branch (AGB) stars (Alves & Sarajedini 1999; 3 Similar figures showing the best-fitting models to the data using all den- sity profiles equations (3 – 11) appear in the on-line supplementary material. c© 2005 RAS, MNRAS 000, 000–000 Modelling the Galactic bar 7 15 17 14 16 42 19 18 36 35 33 32 31 2 30 21 4000 20 L 34 L 46 R 1 R 200045 R 38 L 39 R 4 R 3 R 43 L 22 L 23 L 7 R 13 14 15 10006 R 12 13 14 13 14 13 14 13 14 15 PSfrag replacements Figure 5. Red clump number count histograms and best-fitting profiles using density model E2 for 34 OGLE-II fields with −4◦ 6 l 6 6◦ . Fields are arranged roughly in order of descending Galactic longitude. Field numbers are given in each set of axes. A ‘L’ or ‘R’ in the axes of rows 4 – 6 indicates whether the vertical scale corresponds to left or right vertical axis of the row. c© 2005 RAS, MNRAS 000, 000–000 8 Rattenbury et al. Faria et al. 2007) may also be the cause of the secondary peak in the number count histograms for these and similar fields (e.g. field 10, Fig. 9). Closer investigation of the stellar characteristics and kinematics of stars in these fields may help in describing the origin of the second number count peak. The best-fitting parameter values for all the nine density mod- els are presented in Table 2. Two sets of parameter values are given, one where the tilt angle, β, was held at 0.0◦, and the other where β was allowed to vary. The values of χ2 are well in excess of the num- ber of degrees of freedom: Ndof = 26 × 34 − 11 = 873, and can therefore not be used to set reliable errors on the parameters values in the usual way. It is likely that the errors on the histogram data are underestimated, resulting in extreme values of χ2. We assumed that the errors on the number count data would be Poissonian and we have not added any error in quadrature that might arise from any systematic effects. Furthermore, as mentioned above, the extreme latitude fields 6, 7, 14, 15 and 43 are poorly fit by every model. The cause for this is unknown, however the effect is to add a relatively high contribution to the total value of χ2 compared to other fields. The values of χ2 can only be used to differentiate between the various models in a qualitative manner. In terms of relative per- formance, model E3 best reproduces the observed number count histograms for the fields tested here, and model G2 provides the worst fit to the data. The position angle of the bar semi-major axis with respect to the Sun-Galactic centre line-of-sight is α′ and is related to the ro- tation angle α by α′ = α− sgn(α)π/2. We find α′ = 20◦ – 26◦. The addition of β as another variable parameter does not result in a significant improvement in χ2. The lack of information in the lati- tude direction means that the data have little power in constraining parameters such as β. The absolute values of the scale lengths in Table 2 cannot be directly compared between models. The axis ratios x0/y0 and x0/z0 can be compared however. Fig. 6 shows the axis ratios x0/y0 and x0/z0 for the nine models tested. The ratio of the major bar axis scale length to the minor bar axis in the plane of the Galaxy, x0/y0, has values 3.2 – 3.6, with the exception of that for model E1, for which x0/y0 = 4.0. Stanek et al. (1997) found x0/y0 = 2.0 – 2.4, with the exception of model E1 for which x0/y0 = 2.9. It is interesting to note that our values of x0/y0 have a similar range to that of Stanek et al. (1997), when we exclude the outlying result from the same model (E1). Our results do suggest a slimmer bar, i.e. higher values of x0/y0 compared to Stanek et al. (1997). Both these results and the results of Stanek et al. (1997) are consistent with the value of x0/y0 ≃ 3± 1 reported by Dwek et al. (1995). The ratio of the major bar axis scale length to the verti- cal axis scale length x0/z0 was found to be 3.4 – 4.2, with the same exception of model E1, which has an outlying value of x0/z0 = 6.6. Again, the range of ratio values is comparable to that of Stanek et al. (1997) who found x0/z0 = 2.8 – 3.8, with the exception of model E1 again, which had x0/z0 = 5.6. The mean axis ratios are x0 : y0 : z0 are 10 : 2.9 : 2.5. Excluding the outlying results from the E1 model, the ratios are 10 : 3.0 : 2.6. These results suggest a bar more prolate than the general working model with 10 : 4 : 3 (Gerhard 2002). 6.1 Evidence of non-bar structure? Including wide longitude fields Fig. 4 shows the mean position of the bulge red clump stars. At longitudes −4◦ 6 l 6 6◦ the bulge red clump stars follow the main axis of the bar. At greater angular distances, the mean positions of G1 G2 G3 E1 E2 E3 P1 P2 P3 PSfrag replacements Model x0/y0, 34 fields x0/z0, 34 fields x0/y0, 44 fields x0/z0, 44 fields Figure 6. Scale length ratios x0/y0 (squares) and x0/z0 (triangles) for all models. Solid and open symbols show the ratios for the best-fitting models using the 34 fields with −4◦ 6 l 6 6◦, and all 44 fields (see Section 6.1) respectively. the red clump stars are clearly separated from the main bar axis. Babusiaux & Gilmore (2005) find similar evidence and suggest that this could indicate either the end of the bar or the existence of a ring-like structure. We used the OGLE-II data to investigate these possibilities. The OGLE-II data from fields with longitude l < −4◦ and l > 6◦ were excluded from the original bar modelling based on the findings illustrated by Fig. 4, and because initial modelling ef- forts using all data from all fields simultaneously failed to produce satisfactory results. Using the best-fitting model (E3) determined using the central OGLE-II fields, Fig. 7 shows the predicted num- ber count densities for fields with l < −4◦ and l > 6◦, overlaid on the observed number count histograms. It is clear from the left hand column of Fig. 7 that the mean position of the red clump in fields with l > 6◦ are significantly removed from that predicted by the E3 linear bar model. However, the right hand column of Fig. 7 shows that the observed position of the red clump is in rough agreement with that predicted from the model for fields with l < −4◦. We test this further, by taking lines of sight through the best-fitting E3 model and computing the num- ber count profile using equation (12). The mean magnitude of the red clump is converted to a distance for each line of sight. Fig. 8 shows the density contours of the analytic E3 model overlaid with the mean position of the red clump determined in this way. The fea- tures of Fig. 4 at longitudes |l| & 5◦ are qualitatively reproduced in Fig. 8. This suggests that the observed data are likely to be consis- tent with a single bar-like structure, rather than requiring additional structure elements. We note however, that the abrupt departure of the location of the density peak in Fig. 4 at longitudes l . 5◦ is best reproduced in Fig. 8 by the analytical density model when the exponential cut-off of equation (13) is imposed. Fields with l < −4◦ and l > 6◦ should be included in the modelling of the bar, as they are likely to provide further constraints on the final model. Initial attempts to model the bar using all fields failed because the fitting algorithm found a model with a bar angle of 45◦, consistent with the observed data, but not consistent with the current understanding of the bar. Fig. 8 shows that the mor- c© 2005 RAS, MNRAS 000, 000–000 Modelling the Galactic bar 9 13 14 15 13 14 15 PSfrag replacements Figure 7. Predicted number count profiles using the best-fitting E3 model (equation (12)) with the observed number count histograms for OGLE-II fields with l > 6◦ (left column) and l < −4◦ (right column). The predicted number count profiles are significantly different for fields with l > 6◦, however the predicted magnitude peak of the red clump roughly coincides with those observed for fields with l < −4◦ . c© 2005 RAS, MNRAS 000, 000–000 10 Rattenbury et al. Table 2. Best fitting parameter values for all density models (equations 3 – 11) fitted to the number count histograms from the 34 OGLE-II fields with −4◦ 6 l 6 6◦. Two sets of parameter values are given, one where the tilt angle, β, was held at 0.0◦, and the other where β was allowed to vary. α′ is the position angle of the bar semi-major axis with respect to the Sun-Galactic centre line-of-sight. Bar orientation (◦) Scale lengths (pc) Axis ratios Model β α′ x0 y0 z0 χ2 x0 : y0 : z0 G1 0.00 21.82 1469.00 449.28 391.80 15302.31 -0.99 21.82 1469.41 448.78 391.62 15282.57 10.0 : 3.1 : 2.7 G2 0.00 19.76 1206.92 375.94 353.81 16558.28 -0.28 19.79 1203.37 377.22 354.24 16547.35 10.0 : 3.1 : 2.9 G3 0.00 25.57 5289.91 1512.32 1277.52 14306.20 -0.30 25.55 5288.64 1511.72 1277.98 14306.06 10.0 : 2.9 : 2.4 E1 0.00 21.63 2143.51 540.08 325.49 15766.77 0.06 21.62 2135.11 539.86 325.48 15753.98 10.0 : 2.5 : 1.5 E2 0.00 23.68 1034.30 306.39 261.43 11930.61 -0.01 23.68 1034.24 306.38 261.47 11930.58 10.0 : 3.0 : 2.5 E3 0.00 21.92 1039.86 323.80 299.08 10722.69 0.47 21.97 1039.62 323.75 298.75 10711.13 10.0 : 3.1 : 2.9 P1 0.00 24.66 1988.33 562.49 478.93 13952.47 0.08 24.66 1988.10 562.39 478.84 13952.09 10.0 : 2.8 : 2.4 P2 0.00 25.07 3906.98 1088.90 927.00 15183.47 0.76 25.03 3907.42 1089.36 927.61 15180.89 10.0 : 2.8 : 2.4 P3 0.00 23.81 1992.91 580.32 491.30 12123.07 0.10 23.82 1993.85 580.33 491.09 12122.82 10.0 : 2.9 : 2.5 −3 −2 −1 0 1 2 3 PSfrag replacements x (kpc) Figure 8. Mean position of the red clump determined using the best-fitting analytic bar model (E3). Gray lines indicate 11 lines-of-sight through the E3 model, with l = −10◦,−9.8◦, . . . , 10◦ and b = 0.0◦ and contours in- dicating the density profile of the model. The dashed line shows the orienta- tion of the bar major axis, with the Sun positioned at (0,−8) kpc. Number count profiles are generated using equation (12). The mean red clump mag- nitude for each line-of-sight was converted to a distance, and plotted as solid circles. Black circles indicate the mean position of the red clump when the exponential cut-off, equation (13) is applied. Grey circles indicate the mean RCG position when this cut-off is not imposed. The observed features in Fig. 4 at longitudes |l| & 5◦ are more clearly reproduced when the cut-off is applied to the analytic model. phology indicated by the observed data in Fig. 4 can be explained using a bar structure oriented at ∼ 20◦ to the Sun-GC line-of-sight. Cabrera-Lavers et al. (2007) also note that the line-of-sight density for tri-axial bulges reaches a maximum where the line-of-sight is tangential to the ellipsoidal density contours. These authors also quantify the difference between the positions of maximum density and the intersection of the line-of-sight with the major axis of the The fitting procedure for all nine tri-axial models was re- peated, including now the 10 OGLE-II fields with l < −4◦ or l > 6◦. Figure 9 shows the best-fitting number count profiles for the ten non-central fields for the ‘G’ type of analytic tri-axial model4. The best-fitting number count profiles for fields with−4◦ 6 l 6 6◦ are not significantly different to those shown in Figure 5. The main features of the observed number count profiles are reproduced by the best-fitting analytic models. There are however instances where details of the observed number counts are not reproduced. Magni- tude bins with IV−I & 13 are poorly fitted by the G-type models in positive longitude fields. Two of the E-type models (E2, E3) sim- ilarly fail to trace these data. The number count profile for model E1 shows a flattened peak, resulting from the pronounced box-like nature of the density profile. All the P-type models appear to repro- duce the data at these magnitudes with similar profiles. All models predict a RCG peak at magnitudes greater than that observed in fields 28 and 29. This offset between predicted and observed RCG peak location is also seen in fields 26 and 27 but to a lesser degree. The best-fitting parameters are shown in Table 3. The model which results in the lowest value of χ2 is still E3. Model G2 also remains the worst-fitting model to the data. We now consider the change to the best-fitting model param- eters when data from all fields are used in the analysis. The po- sition angle of the bar semi-major axis with respect to the Sun- Galactic centre line-of-sight is now α′ = 24◦ – 27◦. The bar 4 Similar figures showing the best-fitting models using the ‘E’ and ‘P’ type density profiles appear in the on-line supplementary material. c© 2005 RAS, MNRAS 000, 000–000 Modelling the Galactic bar 11 111111 101010 131313 13 14 15 1000 121212 2000262626 272727 292929 13 14 15 282828 PSfrag replacements Figure 9. Best-fitting number count profiles using the Gaussian ‘G’ type tri-axial models with the observed number count histograms for OGLE-II fields with l > 6◦ (left column) and l < −4◦ (right column). Solid, dashed and dot-dashed lines correspond to model subtype 1, 2 and 3 respectively. c© 2005 RAS, MNRAS 000, 000–000 12 Rattenbury et al. Table 3. Best fitting parameter values for all density models (equations 3-11) fitted to the number count histograms from all 44 OGLE-II fields. Bar orientation (◦) Scale lengths (pc) Axis ratios Model β α′ x0 y0 z0 χ2 x0 : y0 : z0 G1 0.75 27.06 1505.20 568.49 392.19 23808.97 10.0 : 3.8 : 2.6 G2 -7.68 24.49 1276.56 473.28 359.91 26571.42 10.0 : 3.7 : 2.8 G3 -0.72 26.56 4787.17 1680.64 1279.94 17397.76 10.0 : 3.5 : 2.7 E1 1.95 23.82 1901.43 626.89 324.93 19170.60 10.0 : 3.3 : 1.7 E2 0.97 26.43 986.60 356.65 260.88 15674.39 10.0 : 3.6 : 2.6 E3 2.50 25.54 1045.67 378.20 294.89 15110.75 10.0 : 3.6 : 2.8 P1 2.19 25.30 1810.98 609.40 478.93 16720.85 10.0 : 3.4 : 2.6 P2 3.67 25.16 3513.34 1160.72 927.54 18069.53 10.0 : 3.3 : 2.6 P3 -0.84 26.06 1876.75 658.13 487.76 15228.11 10.0 : 3.5 : 2.6 G1 G2 G3 E1 E2 E3 P1 P2 P3 PSfrag replacements Model x0,M=44/x0,M=34 y0,M=44/y0,M=34 z0,M=44/z0,M=34 Figure 10. Ratio of bar scale lengths from best-fitting models using data from all 44 OGLE-II fields to those using data from the central 34 OGLE-II fields. scale lengths x0, y0 and z0 also changed; Fig. 10 shows the ra- tio of the best-fitting scale lengths determined using all 44 fields to those found using only the central 34 fields. On average, upon including the data from wide longitude fields, the semi-major axis scale length, x0, decreased by ∼ 5%; the semi-minor axis scale length, y0, increased by ∼ 16% and the vertical scale length, z0, remained essentially unchanged. The relatively large change in the semi-minor axis scale length is intuitively understandable, as a bar position angle of ∼ 25◦ with respect to the Sun-GC line-of-sight mean the semi-minor axis direction has a large vector component in the Galactic longitude direction. The additional constraints of the data from fields at extended Galactic longitudes can therefore have a pronounced effect on the fitted values of y0. Similarly, we expect some weak correlation between the semi-major and semi- minor axis scale lengths, evidenced by the slight decrease on av- erage for the x0 values upon adding the data from wide longitude fields. It is unsurprising that the values of z0 are unchanged, as adding data from the wide longitude fields has little power in fur- ther constraining model elements in the direction perpendicular to the Galactic midplane. As above, we consider the ratio of the axis scale lengths for each model (see Fig. 6). The ratio x0/y0 for all nine models using all field data has values 2.7 – 3.0. This range is now completely consistent with the range 2.5 – 3.3 reported by Bissantz & Gerhard (2002). The ratio x0/z0 now lies in the range 3.6 – 3.9, with the exception of that for model G1 which has x0/z0 = 5.9. The range x0/z0 = 3.6 – 3.9 is narrower than that found previously, 3.4 – 4.2, yet still higher on average than 2.8 – 3.8 reported by Stanek et al. (1997). The scale length ratios are now x0 : y0 : z0 = 10 : 3.5 : 2.6, which compared to those determined using only the central OGLE-II fields (x0 : y0 : z0 = 10 : 3.0 : 2.6) are closer to the working model proposed by Gerhard (2002) which has x0 : y0 : z0 = 10 : 4 : 3. There are features in the number count histograms that are not faithfully reproduced by the analytic tri-axial bar models for wide longitude fields. Specifically, the predicted number count disper- sions around the RCG peak magnitudes for fields with l > 6◦ are too small compared to that observed, see Fig. 9. While the location of the observed maximum line-of-sight density can be reproduced using just a bar, it is possible that the reason why the analytic bar models underestimate the observed number count dispersions for fields with l > 6◦ is because these models do not include elements which correspond to extended aggregations of stars at or near the ends of the bar. Clumps of stars at the ends of the bar would in- crease the line-of-sight density dispersion at wide longitudes. Such extra aggregations of stars have been observed in galaxies with boxy or peanut-shaped bulges (Bureau et al. 2006). Their origin may be due to the superposition of members of the x1 family of orbits (Patsis et al. 2002; Binney & Tremaine 1987) many of which have loops near the end of the bar. Alternatively, the aggregations of stars near the ends of the bar may be the edge-on projection of an inner ring (Bureau et al. 2006 and references therein). The presence of another density structure such as a long thin bar ori- ented at ∼ 45◦ (Cabrera-Lavers et al. 2007) might also contribute to the relatively large line-of-sight density dispersion. The effects of such a structure would be most pronounced at wide longitudes. The presence of a spiral arm might similarly contribute to the large line-of-sight density dispersion for these fields. 6.2 Varying R0 We consider in this section the effect on the fitted model parameters of changing the Galactocentric distance R0. It was noted in Sec- tion 4.1 that there is an offset between the peak magnitude of RCGs observed in the OGLE-II data and that expected for a RCG, with ab- solute magnitude similar to that of local RCGs, placed at a distance of 8.0 kpc. The observed RCGs are systematically 0.3 mag fainter than the fiducial local RCG at 8.0 kpc. There are two possible rea- sons for this magnitude offset. Firstly, the adopted Galactocentric distance of R0 = 8.0 kpc may be incorrect; secondly, there may be population variations between local and bulge RCGs, resulting in different intrinsic magnitudes in the two populations. In Section 4.1 c© 2005 RAS, MNRAS 000, 000–000 Modelling the Galactic bar 13 we noted that the offset predicted from the theoretical population models of Girardi & Salaris (2001) estimate ∆IRC ≃ −0.1. We postulated that if the remaining magnitude offset is completely ac- counted for by a change in the Galactocentric distance, this would mean that R0 = 7.3 kpc. Conversely, if we adopted the value of R0 = 7.6 ± 0.32 kpc determined by Eisenhauer et al. (2005), the magnitude offset would become −0.18 mag, where including the population effect predicted by Girardi & Salaris (2001) would leave an unaccounted-for offset of −0.08 mag. In this section we present the results of further modelling where we apply magnitude offsets which correspond to different values of R0. In Section 6, we note that for the OGLE-II fields 6, 7, 14, 15 and 43, which have latitudes removed from the majority of OGLE- II fields, there are systematic offsets between the RCG peak in ob- served number count histograms and the predictions of all nine tri- axial models tested. For this reason, we limited the fields used to those which have Galactic latitude −5◦ 6 b 6 −2◦. We also noted in Section 6 that due to the lack of field coverage in the latitude direction the data are not effective at constraining the bar tilt an- gle β. For this reason, β was held constant at 0.0◦ in the following modelling analysis. During the previous modelling efforts of Sec- tion 6 it was also found that applying the exponential cut-off of equation (13) had little effect on determining the best-fitting bar parameters. This cut-off was not applied in the following analysis. The best-fitting bar parameters were determined for the nine tri-axial bar models for the OGLE-II fields occupying the latitude strip−5◦ 6 b 6 −2◦ using magnitude offsets corresponding to R0 values of 8.0 kpc, 7.6 kpc and 7.3 kpc. The resulting best-fitting parameters are listed in Table 4. From the results in Table 4 we see that the fitted bar opening angle α increases for all models as R0 decreases. The fitted scale lengths x0, y0 and z0 all decrease linearly as R0 decreases. The val- ues of χ2 in Table 4 indicate that a smaller value of R0 is favoured by all models. The ratio between the scale heights remains remark- ably constant with varying R0, indicating that while the orientation of the bar changes slightly with varying R0, the overall shape of the bar does not change. The mean values of the scale length ratios for all models, and all values of R0 are x0 : y0 : z0 = 10 : 3.6 : 2.7. This result is very close to that determined using data from all 44 OGLE-II fields, see Section 6 above. 7 DISCUSSION Red clump giant stars in the OGLE-II microlensing survey cata- logue can be used as tracers of the bulge density over a large re- gion towards the Galactic centre. Nine analytic tri-axial bar mod- els were initially fitted to the number count histograms of red clump stars observed in 34 OGLE-II fields which have −4◦ 6 l 6 6◦. The models all have the major axis of the bar oriented at 20◦ – 26◦ to the Sun-Galactic centre line-of-sight. This orien- tation is in agreement with the results of Stanek et al. (1997) and Nikolaev & Weinberg (1997) which give a bar angle of 20◦ – 30◦, and is marginally in agreement with the value of 12 ± 6◦ from López-Corredoira et al. (2000). Bissantz & Gerhard (2002) obtain best-fitting non-parametric models to the COBE/DIRBE L band map of the inner Galaxy with bar angles of 20◦ – 25◦. We find the ratio of the bar major axis scale length to minor axis scale length in the plane of the Galaxy to be x0/y0 = 3.2 – 3.6, higher than the value of 2.0 – 2.4 reported by Stanek et al. (1997), but consistent with the upper end of the range 2.5 – 3.3 found by Bissantz & Gerhard (2002). The ratio of bar major axis scale length to vertical axis scale length was found to be x0/z0 = 3.4 – 4.2, again higher on average than that reported by Stanek et al. (1997) who found x0/z0 = 2.8 – 3.8, and higher than the value of ≃ 3.3 generally adopted (Gerhard 2002). The working model proposed by Gerhard (2002) gives the scale length ratios as x0 : y0 : z0 = 10 : 4 : 3. Our results suggest a more prolate model with x0 : y0 : z0 = 10 : 3.0 : 2.6. The observed separation of the mean position of red clump giants from the bar major axis at |l| & 5◦ was shown to be a ge- ometric effect, rather than evidence of a more complicated struc- ture such as a ring. The observed data from these fields were used to further constrain the bar models. The resulting bar position an- gles was found to be 24◦ – 27◦. This narrower range is consis- tent with the several values of the bar position angle found by pre- vious studies. The bar scale length ratios were determined to be x0 : y0 : z0 = 10 : 3.5 : 2.6, which are closer to the working model proposed by Gerhard (2002) than those values found using data from only the central 34 OGLE-II fields. Reasons for the difference between the bar axis ratios deter- mined here and the general working model of Gerhard (2002) may include RCG population effects noted above in Section 4.1. The in- trinsic luminosity of bulge RCG stars was assumed to be the same as the local population, however it was found that an offset of −0.3 mag had to be applied to the computed distance moduli in order to obtain results consistent with standard bar models. Sumi (2004) found the mean magnitude of observed bulge clump stars is found to be 0.3 mag fainter to that expected assuming (i) the properties of bulge RCGs are the same as the local population and (ii) the distance to the Galactic centre is 8 kpc. A possible implication is that the adopted distance to the Galactic centre of 8 kpc may be in- correct. The bar modelling procedure was repeated for three values of the Sun-Galactic centre distance R0: 7.3 kpc, 7.6 kpc and 8.0 kpc, using data from OGLE-II fields which have −5◦ 6 b 6 −2◦. Some fields with latitudes outside this range were found to have number count histograms which were not reproducible by any lin- ear tri-axial model of the bar tested in this work, and were there- fore excluded. In addition, most of the OGLE-II fields latitudes −4◦ < b < −2◦ were excluded. The low amount of informa- tion in the latitude direction means that the data have little leverage in determining some of the model parameters. Without more data in these regions, model fitting algorithms may not be able to refine some model parameters such as the tilt angle β. The three values of R0 used are the ‘default’ value of 8.0 kpc; 7.6 kpc, as deter- mined by Eisenhauer et al. (2005); and 7.3 kpc, which corresponds to the value of R0 consistent with the observed mean RCG magni- tudes in the OGLE-II bulge fields assuming the population effects of Girardi & Salaris (2001). The fitted scale lengths x0, y0 and z0 were found to increase linearly with increasing values of R0, while the bar opening angle α decreased slightly with increasing R0. The shape of the bar, as quantified by the ratio of axis scale lengths, was found to be insensitive to different values of R0. The mean ra- tio of scale lengths over all models and values of R0 was found to be x0 : y0 : z0 = 10 : 3.6 : 2.7, which is slightly closer to the working model of Gerhard (2002) for the Galactic bar than that de- termined using all 44 OGLE-II fields. The goodness-of-fit measure χ2 decreased for all models as the value of R0 was lowered. Improved modelling for the Galactic bar may be possible through the addition of further elements to the methods presented here. The possibility of a metallicity gradient across the bulge was not accounted for in this work. Including spiral terms (see e.g. Evans & Belokurov 2002) in the analytic density profiles may sim- c© 2005 RAS, MNRAS 000, 000–000 14 Rattenbury et al. Table 4. Best-fitting parameters for all density models (equations 3-11) fitted to the number count histograms for OGLE-II fields with Galactic latitude −5◦ 6 b 6−2◦ . Three values of Galactocentric distance R0 were used. The bar tilt angle, β, was held at 0.0◦. Scale lengths (pc) Axis ratios Model R0 (kpc) α (◦) x0 y0 z0 χ2 x0 : y0 : z0 8.0 26.74 1525.19 569.35 382.73 14622.23 10.0 : 3.7 : 2.5 G1 7.6 27.67 1430.76 528.21 363.45 14384.62 10.0 : 3.7 : 2.5 7.3 28.63 1357.09 494.16 349.04 14165.15 10.0 : 3.6 : 2.6 8.0 25.91 1313.82 467.27 337.77 15676.58 10.0 : 3.6 : 2.6 G2 7.6 26.65 1235.00 435.18 320.48 15303.12 10.0 : 3.5 : 2.6 7.3 27.42 1173.14 408.84 307.58 14993.34 10.0 : 3.5 : 2.6 8.0 24.16 4587.68 1658.73 1330.48 9712.92 10.0 : 3.6 : 2.9 G3 7.6 25.32 4266.68 1528.53 1272.76 9502.48 10.0 : 3.6 : 3.0 7.3 26.52 4011.04 1418.58 1226.61 9353.45 10.0 : 3.5 : 3.1 8.0 21.41 1710.52 626.99 343.73 11530.12 10.0 : 3.7 : 2.0 E1 7.6 22.48 1595.52 575.62 327.43 11028.20 10.0 : 3.6 : 2.1 7.3 23.70 1509.95 532.02 316.25 10707.93 10.0 : 3.5 : 2.1 8.0 24.56 974.73 351.07 264.40 9135.87 10.0 : 3.6 : 2.7 E2 7.6 25.63 907.63 323.75 250.86 9085.08 10.0 : 3.6 : 2.8 7.3 26.75 855.05 301.02 240.66 9043.39 10.0 : 3.5 : 2.8 8.0 23.87 1023.09 365.74 297.10 9341.10 10.0 : 3.6 : 2.9 E3 7.6 24.75 954.56 338.68 281.02 9254.02 10.0 : 3.5 : 2.9 7.3 25.69 900.61 316.34 268.93 9180.48 10.0 : 3.5 : 3.0 8.0 22.05 1698.22 609.30 485.75 9418.38 10.0 : 3.6 : 2.9 P1 7.6 23.34 1550.61 554.41 456.64 9288.41 10.0 : 3.6 : 2.9 7.3 24.68 1435.21 509.34 434.79 9191.40 10.0 : 3.5 : 3.0 8.0 21.69 3192.12 1144.07 919.70 10148.09 10.0 : 3.6 : 2.9 P2 7.6 23.03 2894.57 1035.28 861.54 9957.52 10.0 : 3.6 : 3.0 7.3 24.43 2663.48 946.72 817.88 9813.78 10.0 : 3.6 : 3.1 8.0 23.61 1827.34 651.79 487.70 8606.41 10.0 : 3.6 : 2.7 P3 7.6 24.77 1686.69 599.37 460.45 8566.67 10.0 : 3.6 : 2.7 7.3 25.88 1584.40 555.15 444.66 8479.57 10.0 : 3.5 : 2.8 ilarly result in a closer reproduction of the observed number count profiles. The observed maximum line-of-sight density can be repro- duced using just a bar, without requiring additional structure ele- ments. However, the finer details of the number count histograms, especially at wide longitudes, were not reproduced by the tri-axial bar models used here. The predicted number count dispersions around the peak RCG magnitude were underestimated by the ana- lytical models, particularly for fields with l > 6◦. It is possible that these finer features can be reproduced using models which include extra stellar agglomerations at the ends of the bar. The additional stellar densities could arise from specific stellar orbits aligned with the bar, or due to the projection effect of an inner ring. A long thin bar as postulated by Cabrera-Lavers et al. (2007) might similarly increase the line-of-sight density dispersion, particularly for wide longitude fields, as might the presence of a spiral arm. Modelling the bar using non-parametric methods (see e.g. Efstathiou et al. 1988) may provide valuable insight into the existence and nature of such additional features. Some preliminary work applying these methods has begun. Data from current and future surveys of the Galactic bulge region will be useful for refining the constraints on the bar param- eters. The third evolution of the OGLE microlensing experiment, OGLE-III, is currently in progress, covering a larger region of the central Galactic region than OGLE-II. Infra-red observations of the bulge have the advantage of lower extinction effects due to dust, compared to optical observa- tions. Current catalogues which would be suitable for investigat- ing the structure of the Galactic bar include the point source cata- logue from the 2MASS All Sky data release (Skrutskie et al. 2006), the Galactic Plane Survey from the UKIRT Infra-Red Deep Sky Survey (Lawrence et al. 2006; Dye et al. 2006) and data from the ISOGAL (Omont et al. 2003), Spitzer/GLIMPSE (Benjamin et al. 2005) and AKARI (Ishihara & Onaka 2006) space telescope sur- veys. Proposed infra-red surveys towards the Galactic centre in- clude the Galactic Bar Infra-red Time-domain (GABARIT) survey which aims to monitor the Galactic bar region in the K band for mi- crolensing events (Kerins 2006, private communication). The anal- ysis of red clump giant star counts in these surveys may result in tighter constraints on the properties of the bar, and can be combined with constrains on stellar kinematics from proper motion surveys c© 2005 RAS, MNRAS 000, 000–000 Modelling the Galactic bar 15 (Rattenbury et al. 2006) in order to develop dynamical models of the inner Galaxy. ACKNOWLEDGEMENTS We thank D. Faria, Ł. Wyrzykowski, R. James and W. Evans for helpful discussions. NJR acknowledges financial support by a PPARC PDRA fellowship and the LKBF. NJR thanks the Kapteyn Astronomical Institute, RuG, for visitor support. This work was partially supported by the European Community’s Sixth Frame- work Marie Curie Research Training Network Programme, Con- tract No. MRTN-CT-2004-505183 ‘ANGLES’. 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Udalski, A., Zebrun, K., Szymanski, M., et al., 2000, Acta Astro- nomica, 50, 1 Weiland, J.L., Arendt, R.G., Berriman, G.B., et al., 1994, ApJ, 425, L81 Zhao, G., Qiu, H.M., Mao, S., 2001, ApJ, 551, L85 c© 2005 RAS, MNRAS 000, 000–000 http://arxiv.org/abs/astro-ph/0604426 http://arxiv.org/abs/astro-ph/0612159 15 17 14 16 42 19 18 36 35 33 32 31 2 30 21 4000 20 L 34 L 46 R 1 R 200045 R 38 L 39 R 4 R 3 R 43 L 22 L 23 L 7 R 13 14 15 10006 R 12 13 14 13 14 13 14 13 14 15 PSfrag replacements Supplementary Figure 1: As for Fig. 5 using density model G1. c© 0000 RAS, MNRAS 000, 000–000 15 17 14 16 42 19 18 36 35 33 32 31 2 30 21 4000 20 L 34 L 46 R 1 R 200045 R 38 L 39 R 4 R 3 R 43 L 22 L 23 L 7 R 13 14 15 10006 R 12 13 14 13 14 13 14 13 14 15 PSfrag replacements Supplementary Figure 2: As for Fig. 5 using density model G2. c© 0000 RAS, MNRAS 000, 000–000 15 17 14 16 42 19 18 36 35 33 32 31 2 30 21 4000 20 L 34 L 46 R 1 R 200045 R 38 L 39 R 4 R 3 R 43 L 22 L 23 L 7 R 13 14 15 10006 R 12 13 14 13 14 13 14 13 14 15 PSfrag replacements Supplementary Figure 3: As for Fig. 5 using density model G3. c© 0000 RAS, MNRAS 000, 000–000 15 17 14 16 42 19 18 36 35 33 32 31 2 30 21 4000 20 L 34 L 46 R 1 R 200045 R 38 L 39 R 4 R 3 R 43 L 22 L 23 L 7 R 13 14 15 10006 R 12 13 14 13 14 13 14 13 14 15 PSfrag replacements Supplementary Figure 4: As for Fig. 5 using density model E1. c© 0000 RAS, MNRAS 000, 000–000 15 17 14 16 42 19 18 36 35 33 32 31 2 30 21 4000 20 L 34 L 46 R 1 R 200045 R 38 L 39 R 4 R 3 R 43 L 22 L 23 L 7 R 13 14 15 10006 R 12 13 14 13 14 13 14 13 14 15 PSfrag replacements Supplementary Figure 5: As for Fig. 5 using density model E3. c© 0000 RAS, MNRAS 000, 000–000 15 17 14 16 42 19 18 36 35 33 32 31 2 30 21 4000 20 L 34 L 46 R 1 R 200045 R 38 L 39 R 4 R 3 R 43 L 22 L 23 L 7 R 13 14 15 10006 R 12 13 14 13 14 13 14 13 14 15 PSfrag replacements Supplementary Figure 6: As for Fig. 5 using density model P1. c© 0000 RAS, MNRAS 000, 000–000 15 17 14 16 42 19 18 36 35 33 32 31 2 30 21 4000 20 L 34 L 46 R 1 R 200045 R 38 L 39 R 4 R 3 R 43 L 22 L 23 L 7 R 13 14 15 10006 R 12 13 14 13 14 13 14 13 14 15 PSfrag replacements Supplementary Figure 7: As for Fig. 5 using density model P2. c© 0000 RAS, MNRAS 000, 000–000 15 17 14 16 42 19 18 36 35 33 32 31 2 30 21 4000 20 L 34 L 46 R 1 R 200045 R 38 L 39 R 4 R 3 R 43 L 22 L 23 L 7 R 13 14 15 10006 R 12 13 14 13 14 13 14 13 14 15 PSfrag replacements Supplementary Figure 8: As for Fig. 5 using density model P3. c© 0000 RAS, MNRAS 000, 000–000 111111 101010 131313 13 14 15 1000 121212 2000262626 272727 292929 13 14 15 282828 PSfrag replacements Supplementary Figure 9: As for Fig. 9 using the exponential ‘E’ type tri-axial models. c© 0000 RAS, MNRAS 000, 000–000 111111 101010 131313 13 14 15 1000 121212 2000262626 272727 292929 13 14 15 282828 PSfrag replacements Supplementary Figure 10: As for Fig. 9 using the power ‘P’ type tri-axial models. c© 0000 RAS, MNRAS 000, 000–000 Introduction Data Methods Sample selection Distance Moduli of RCGs RCG population effects Modelling the bar Results Evidence of non-bar structure? Including wide longitude fields Varying R0 Discussion

Red clump giant stars can be used as distance indicators to trace the mass distribution of the Galactic bar. We use RCG stars from 44 bulge fields from the OGLE-II microlensing collaboration database to constrain analytic tri-axial models for the Galactic bar. We find the bar major axis is oriented at an angle of 24 - 27 degrees to the Sun-Galactic centre line-of-sight. The ratio of semi-major and semi-minor bar axis scale lengths in the Galactic plane x_0, y_0, and vertical bar scale length z_0, is x_0 : y_0 : z_0 = 10 : 3.5 : 2.6, suggesting a slightly more prolate bar structure than the working model of Gerhard (2002) which gives the scale length ratios as x_0 : y_0 : z_0 = 10 : 4 : 3 .

Introduction Data Methods Sample selection Distance Moduli of RCGs RCG population effects Modelling the bar Results Evidence of non-bar structure? Including wide longitude fields Varying R0 Discussion

Dynamical Coupled-Channel Model of πN Scattering in the W ≤ 2 GeV Nucleon Resonance Region∗ (From EBAC, Thomas Jefferson National Accelerator Facility) B. Juliá-Dı́az,1, 2 T.-S. H. Lee,1, 3 A. Matsuyama,1, 4 and T. Sato1, 5 1 Excited Baryon Analysis Center (EBAC), Thomas Jefferson National Accelerator Facility, Newport News, VA 22901, USA 2Departament d’Estructura i Constituents de la Matèria, Universitat de Barcelona, E–08028 Barcelona, Spain 3Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA 4Department of Physics, Shizuoka University, Shizuoka 422-8529, Japan 5Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Abstract As a first step to analyze the electromagnetic meson production reactions in the nucleon reso- nance region, the parameters of the hadronic interactions of a dynamical coupled-channel model, developed in Physics Reports 439, 193 (2007), are determined by fitting the πN scattering data. The channels included in the calculations are πN , ηN and ππN which has π∆, ρN , and σN reso- nant components. The non-resonant meson-baryon interactions of the model are derived from a set of Lagrangians by using a unitary transformation method. One or two bare excited nucleon states in each of S, P , D, and F partial waves are included to generate the resonant amplitudes in the fits. The parameters of the model are first determined by fitting as much as possible the empirical πN elastic scattering amplitudes of SAID up to 2 GeV. We then refine and confirm the resulting parameters by directly comparing the predicted differential cross section and target polarization asymmetry with the original data of the elastic π±p → π±p and charge-exchange π−p → π0n pro- cesses. The predicted total cross sections of πN reactions and πN → ηN reactions are also in good agreement with the data. Applications of the constructed model in analyzing the electromagnetic meson production data as well as the future developments are discussed. PACS numbers: 13.75.Gx, 13.60.Le, 14.20.Gk ∗ Notice: Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05- 06OR23177. The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce this manuscript for U.S. Government purposes. http://arxiv.org/abs/0704.1615v2 I. INTRODUCTION It is now well recognized that a coupled-channel approach is needed to extract the nucleon resonance (N∗) parameters from the data of πN and electromagnetic meson production re- actions. With the recent experimental developments [1, 2], such a theoretical effort is needed to analyze the very extensive data from Jefferson Laboratory (JLab), Mainz, Bonn, GRAAL, and Spring-8. To cope with this challenge, a dynamical coupled-channel model (MSL) for meson-baryon reactions in the nucleon resonance region has been developed recently [3]. In this paper we report a first-stage determination of the parameters of this model by fitting the πN scattering data up to invariant mass W = 2 GeV. The details of the MSL model are given in Ref. [3]. Here we will only briefly recall its essential features. Similar to the earlier works on meson-exchange models [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] of pion-nucleon scattering, the starting point of the MSL model is a set of Lagrangians describing the interactions between mesons (M =γ, π, η , ρ, ω, σ, . . .) and baryons (B = N,∆, N∗, . . .). By applying a unitary transformation method [13, 27], an effective Hamiltonian is then derived from the considered Lagrangian. It can be cast into the following more transparent form Heff = H0 + ΓV + v22 + hππN , (1) where H0 = m2α + ~p α with mα denoting the mass of particle α, and ΓV = { ΓN∗→MB) + hM∗→ππ}+ {c.c.} , (2) v22 = MB,M ′B′ vMB,M ′B′ + vππ , (3) hππN = ΓN∗→ππN + [(vMB,ππN) + (c.c.)] + vππN,ππN . (4) Here c.c. denotes the complex conjugate of the terms on its left-hand-side. In the above equations, MB = γN, πN , ηN, π∆, ρN, σN , represent the considered meson-baryon states. The resonance associated with the bare baryon state N∗ is induced by the vertex interactions ΓN∗→MB and ΓN∗→ππN . Similarly, the bare meson states M ∗ = ρ, σ can develop into reso- nances through the vertex interaction hM∗→ππ. Note that the masses M N∗ and m M∗ of the bare states N∗ and M∗ are the parameters of the model which must be determined by fit- ting the πN and ππ scattering data. They differ from the empirically determined resonance positions by mass shifts which are due to the coupling of the bare states to the scattering states. The term v22 contains the non-resonant meson-baryon interaction vMB,M ′B′ and ππ interaction vππ. The non-resonant interactions involving ππN states are in hππN . All of these interactions are energy independent, an important feature of the MSL formulation. We note here that the Hamiltonian defined above does not have a πN ↔ N vertex. By applying the unitary transformation method, this un-physical process as well as any vertex interaction A ↔ B + C with a mass relation mA < mB + mC are eliminated from the considered Hilbert space and their effects are absorbed in the effective interactions v22 and hππN . This procedure defines the Hamiltonian in terms of physical nucleons and greatly simplifies the formulation of a unitary reaction model. In particular, the complications due to the nucleon mass and wavefunction renormalizations do not appear in the resulting scattering equations. This makes the numerical calculations involving the ππN channel much more tractable in practice. The details of this approach are discussed in Refs. [13, 27] as well as in the earlier works on πNN interactions [28]. Starting from the above Hamiltonian, the coupled-channel equations for πN and γN reactions are then derived by using the standard projection operator technique [29], as given explicitly in Ref. [3]. The obtained scattering equations satisfy the two-body (πN, ηN , γN) and three-body (ππN) unitarity conditions. The π∆, ρN and σN resonant components of the ππN continuum are generated dynamically by the vertex interaction ΓV of Eq. (2). Accordingly, the ππN cuts are treated more rigorously than the commonly used quasi- particle formulation within which these resonant channels are treated as simple two-particle states with a phenomenological parametrization of their widths. The importance of such a dynamical treatment of unstable particle channels was well known in earlier studies of πN scattering [4, 30] and πNN reactions [31]. A complete determination of the parameters of the model Hamiltonian defined by Eqs.(1)- (4) requires good fits to all of the data of πN and γN reactions up to invariant mass W ≤ about 2 GeV. Obviously, this is a very complex task and can only be accomplished step by step. Our strategy is as follows. We need to first determine the parameters associated with the hadronic interaction parts of the Hamiltonian. With the fits to ππ phase shifts in Ref. [32], the ππ interactions hρ,ππ and hσ,ππ and the corresponding bare masses for ρ and σ have been determined in an isobar model with vππ = 0. We next proceed in two stages. The first-stage is to determine the ranges of the parameters of the interactions ΓN∗→MB and vMB,M ′B′ . This will be achieved by fitting the πN scattering data from performing coupled- channel calculations which neglect the more complex three-body interaction term hππN . This simplification greatly reduces the numerical complexity and the number of parameters to be determined in the fits. This first-stage fit will provide the starting parameters to fit both the data of πN scattering and πN → ππN reactions. In this second-stage, the parameters associated with ΓN∗→MB and vMB,M ′B′ will be refined and the parameters of hππN are then determined. The dynamical coupled-channel calculations for such more extensive fits are numerically more complex, as explained in Ref. [3]. In this work we report on the results from our first-stage determination of the parameters of ΓN∗→MB and vMB,M ′B′ of Eqs.(2)-(3) withMB,M ′B′ = πN, ηN, π∆, ρN, σN . We proceed in two steps. We first locate the range of the model parameters by fitting as much as possible the empirical πN elastic scattering amplitudes up to W = 2 GeV of SAID [33]. We then refine and confirm the resulting parameters by directly comparing our predictions with the original πN scattering data. Our procedures are similar to what have been used in determining the nucleon-nucleon (NN) potentials [34] from fitting NN scattering data. The constructed model can describe well almost all of the empirical πN amplitudes in S, P , D, and F partial waves of SAID [33]. We then show that the predicted differential cross sections and target polarization asymmetry are in good agreement with the original data of elastic π±p → π±p and charge-exchange π−p → π0n processes. Furthermore the predicted total cross sections of the πN reactions and πN → ηN reactions agree well with the data. Thus the constructed model is at least comparable to, if not better than, all of the recent πN models [11, 12, 13, 19, 20, 22, 23, 24, 26]. It can be used to perform a first-stage extraction of the γN → N∗ parameters by analyzing the photo- and electro-production of single π meson. It has also provided us with a starting point for performing the second-stage determination of the model parameters by also fitting the data of πN → ππN reactions. Our efforts in these directions are in progress and will be reported elsewhere. In Section II, we recall the coupled-channel equations presented in Ref. [3]. The calcula- MB,M’B’ t MB,M’B’ t MB,M’B’ t MB,M’B’ vMB,M’B’ FIG. 1: Graphical representation of Eqs.(5)-(21). tions performed in this work are described in Section III. The fitting procedure is described in Section IV and the results are presented in Section V. In Section VI we give a summary and discuss future developments. II. DYNAMICAL COUPLED-CHANNEL EQUATIONS With the simplification that ππN interaction hππN of Eq. (4) is set to zero, the meson- baryon (MB) scattering equations derived in Ref. [3] are illustrated in Fig. 1. Explicitly, they are defined by the following equations TMB,M ′B′(E) = tMB,M ′B′(E) + t MB,M ′B′(E) , (5) where MB = πN, ηN, π∆, ρN, σN . The full amplitudes TπN,πN(E) can be directly used to calculate πN scattering observables. The non-resonant amplitude tMB,M ′B′(E) in Eq. (5) is defined by the coupled-channel equations, tMB,M ′B′(E) = VMB,M ′B′(E) + M ′′B′′ VMB,M ′′B′′(E) GM ′′B′′(E) tM ′′B′′,M ′B′(E) (6) VMB,M ′B′(E) = vMB,M ′B′ + Z MB,M ′B′(E) . (7) Here the interactions vMB,M ′B′ are derived from the tree-diagrams illustrated in Fig. 2 by using a unitary transformation method [13, 27]. It is energy independent and free of singu- larity. On the other hand, Z MB,M ′B′(E) is induced by the decays of the unstable particles v v v vs u t c FIG. 2: Mechanisms for vMB,M ′B′ of Eq. (7): v s direct s-channel, vu crossed u-channel, vt one- particle-exchange t-channel, vc contact interactions. MB,M’B’ FIG. 3: One-particle-exchange interactions Z π∆,π∆(E), Z ρN,π∆ and Z σN,π∆ of Eq. (7). (∆, ρ, σ) and thus contains moving singularities due to the ππN cuts, as illustrated in Fig.3. Here we note that if the ππN interaction term hππN of Eq.(4) is included, the driving term Eq. (7) will have an additional term Z MB,M ′B′(E) which involves a 3-3 ππN amplitude tππN,ππN , as given in Ref. [3], and hence is much more difficult to calculate. As explained in Section I, we neglect this term in this first-stage fit to the πN scattering data. The second term in the right-hand-side of Eq. (5) is the resonant term defined by tRMB,M ′B′(E) = Γ̄MB→N∗ (E)[D(E)]i,jΓ̄N∗ →M ′B′(E) , (8) [D−1(E)]i,j = (E −M0N∗ )δi,j − Σ̄i,j(E) , (9) where M0N∗ is the bare mass of the resonant state N ∗, and the self-energies are Σ̄i,j(E) = →MBGMB(E)Γ̄MB→N∗ (E) . (10) The dressed vertex interactions in Eq. (8) and Eq. (10) are (defining ΓMB→N∗ = Γ N∗→MB) Γ̄MB→N∗(E) = ΓMB→N∗ + M ′B′ tMB,M ′B′(E)GM ′B′(E)ΓM ′B′→N∗ , (11) Γ̄N∗→MB(E) = ΓN∗→MB + M ′B′ ΓN∗→M ′B′GM ′B′(E)tM ′B′,MB(E) . (12) It is useful to mention here that if there is only one N∗ in the considered partial wave, the resonant amplitude (Eq. (8)) can be written as tRMB,M ′B′(E) = Γ̄MB→N∗ (E)Γ̄N∗ →M ′B′(E) E − ER(E) + iΓR(E)2 ER(E) = M N∗ + Re[Σ̄(E)] , (14) ΓR(E) = −2 Im[Σ̄(E)] , (15) where, Σ̄(E) = ΓN∗→MBGMB(E){ M ′B′ [δMB,M ′B′ + tMB,M ′B′(E)GM ′B′(E)]}ΓM ′B′→N∗(E) . The form Eq. (13) is similar to the commonly used Breit-Wigner form, but the resonance position ER(E) and width ΓR(E) are determined by the N ∗ → MB vertex and the non- resonant amplitude tMB,M ′B′ . This is the consequence of the unitarity condition and is an important and well known feature of a dynamical approach. Namely, the resonance amplitude necessarily includes the non-resonant mechanisms. This feature is consistent with the well developed formal reaction theory [29]. Eq. (16) indicates that it is essential to understand the non-resonant mechanisms in extracting the bare vertex functions ΓN∗,MB which contain the information for exploring the N∗ structure. The parameterization used for ΓN∗,MB will be explained in Section III. We also note here that the energy dependence of ER(E) and ΓR(E), defined by Eqs (14)-(15), is essential in determining the resonance poles in the complex E-plane. The meson-baryon propagators GMB in the above equations are GMB(k, E) = E −EM(k)−EB(k) + iǫ for the stable particle channels MB = πN, ηN , and GMB(k, E) = E − EM(k)− EB(k)− ΣMB(k, E) for the unstable particle channels MB = π∆, ρN, σN . The self-energies [36] in Eq. (18) are Σπ∆(k, E) = E∆(k) MπN(q) [M2πN(q) + k 2]1/2 |f∆,πN(q)|2 E − Eπ(k)− [(EN(q) + Eπ(q))2 + k2]1/2 + iǫ ΣρN (k, E) = Eρ(k) Mππ(q) [M2ππ(q) + k 2]1/2 |fρ,ππ(q)|2 E − EN(k)− [(2Eπ(q))2 + k2]1/2 + iǫ , (20) ΣσN (k, E) = Eσ(k) Mππ(q) [M2ππ(q) + k 2]1/2 |fσ,ππ(q)|2 E −EN (k)− [(2Eπ(q))2 + k2]1/2 + iǫ , (21) where MπN(q) = Eπ(q) + EN (q) and Mππ(q) = 2Eπ(q). The vertex function f∆,πN(q) is taken from Ref. [13], fρ,ππ(q) and fσ,ππ(q) are from the isobar fits [32] to the ππ phase shifts. They are also given explicitly in [3]. Here we note that the driving term Z MB,M ′B′ of Eq. (7) is also determined by the same vertex functions f∆,πN(q), fρ,ππ(q) and fρ,ππ(q) of Eqs. (19)-(21). This consistency is essential for the solutions of Eq. (6) to satisfy the unitarity condition. III. CALCULATIONS We solve the coupled-channel equations defined by Eqs.(5)-(21) in the partial-wave repre- sentation. The input of these equations are the partial-wave matrix elements of ΓN∗→MB and vMB,M ′B′ of Eqs.(2)-(3), with MB,M ′B′ = πN, ηN , π∆, ρN, σN , and Z MB,M ′B′ of Eq. (7) with MB,M ′B′ = π∆, ρN, σN . The calculations of these matrix elements have been given explicitly in the appendices of Ref. [3]. Here we only mention a few points which are needed for later discussions. In deriving the non-resonant interactions vMB,M ′B′ of Eq. (7) we consider the tree- diagrams (Fig. 2) generated from a set of Lagrangians with π, η, σ, ρ, ω, N , and ∆ fields. The higher mass mesons, such as a0, a1 included in other meson-exchange πN models, such as the Jülich model [19], are not considered. The employed Lagrangians are ( in the convention of Bjorken and Drell [37]) LπNN = − ψ̄Nγµγ5~τψN · ∂µ ~φπ , (22) LπN∆ = − ~TψN · ∂µ ~φπ , (23) Lπ∆∆ = ψ̄∆µγ νγ5 ~T∆ψ ∆ · ∂ν~φπ , (24) LηNN = − ψ̄Nγµγ5ψN∂ µφη . (25) LρNN = gρNN ψ̄N [γµ − ν ] ~ρµ · ψN , (26) LρN∆ = −i νγ5 ~T · [∂µ ~ρν − ∂ν ~ρµ]ψN + [h.c.] , (27) Lρ∆∆ = gρ∆∆ψ̄∆α[γ µ − κρ∆∆ σµν∂ν ] ~ρµ · ~T∆ψα∆ , (28) Lρππ = gρππ[ ~φπ × ∂µ ~φπ] · ~ρµ , (29) LNNρπ = gρNN ψ̄Nγµγ5~τψN · ~ρµ × ~φπ , (30) LNNρρ = − µν~τψN · ~ρµ × ~ρν . (31) LωNN = gωNN ψ̄N [γµ − ν ]ωµψN , (32) Lωπρ = − ǫµαλν∂ α ~ρµ∂λ ~φπω ν , (33) LσNN = gσNN ψ̄NψNφσ (34) Lσππ = − ∂µ~φπ∂µ~φπφσ . (35) To solve the coupled-channel equations, Eq. (6), we need to regularize the matrix elements of vMB,M ′B′ , illustrated in Fig. 2. Here we follow Ref. [13] in order to use the parameters determined in the ∆ (1232) region as the starting parameters in our fits. For the vs and vu terms of Fig. 2, we include at each meson-baryon-baryon vertex a form factor of the following form F (~k,Λ) = [~k2/[(~k2 + Λ2)]2 (36) with ~k being the meson momentum. For the meson-meson-meson vertex of vt of Fig. 2, the form Eq. (36) is also used with ~k being the momentum of the exchanged meson. For the contact term vc, we regularize it by F (~k,Λ)F (~k′,Λ′). With the non-resonant amplitudes generated from solving Eq. (6), the resonant ampli- tude tRMB,M ′B′ Eq. (8) then depends on the bare mass M N∗ and the bare N ∗ → MB vertex functions. As discussed in Ref. [3], these bare N∗ parameters can perhaps be taken from a hadron structure calculation which does not include coupling with meson-baryon contin- uum states or meson-exchange quark interactions. Unfortunately, such information is not available to us. We thus use the following parameterization ΓN∗,MB(LS)(k) = (2π)3/2 CN∗,MB(LS) Λ2N∗,MB(LS) Λ2N∗,MB(LS) + (k − kR)2 (2+L/2) [ .(37) where L and S are the orbital angular momentum and the total spin of the MB system, respectively. The above parameterization accounts for the threshold kL dependence and the right power (2 + L/2) such that the integration for calculating the dressed vertex Eq. (11)- (12) is finite. Nevertheless as we will discuss in Section V this parameterization could be too naive. The partial-wave quantum numbers for the considered channels are listed in Table I. The numerical methods for handling the moving singularities due to the ππN cuts in Z MB,M ′B′ (Fig. 3) in solving Eq. (6) are explained in detail in Ref [3]. To get the πN elastic scattering amplitudes, we can use either the method of contour rotation by solving the equations on the complex momentum axis k = ke−iθ with θ > 0 or the Spline-function method developed in Refs. [38, 39] and explained in detail in Ref. [3]. We perform the calculations using these two very different methods and they agree within less than 1%. When Z MB,M ′B′ is neglected, Eq. (6) can be solved by the standard subtraction method since the resonant propagators, Eqs. (18), for unstable particle channels π∆, ρN , and σN are free of singularity on the real momentum axis. A code for this simplified case has also been developed to confirm the results from using the other two methods. The method of contour rotation becomes difficult at high W since the required rotation angle θ is very small. The Spline function method has no such limitation and we can perform calculations at W > 1.9 GeV without any difficulty. Typically, 24 and 32 mesh points are needed to get convergent solutions of the coupled-channel integral equation (6). Such mesh points are also needed to get stable integrations in evaluating the dressed resonance quantities Eqs. (10)-(12). (LS) of the considered partial waves πN ηN π∆ σN ρN S11 (0, ) (0, 1 ) (2, 3 ) (1, 1 ) (0, 1 ), (2, 3 S31 (0, ) − (2, 3 ) − (0, 1 ), (2, 3 P11 (1, ) (1, 1 ) (1, 3 ) (0, 1 ) (1, 1 ), (1, 3 P13 (1, ) (1, 1 ) (1, 3 ),(3, 3 ) (2, 1 ) (1, 1 ),(1, 3 ), (3, 3 P31 (1, ) − (1, 3 ) − (1, 1 ), (1, 3 P33 (1, ) − (1, 3 ),(3, 3 ) − (1, 1 ),(1, 3 ), (3, 3 D13 (2, ) (2, 1 ) (0, 3 ),(2, 3 ) (1, 1 ) (2, 1 ), (0, 3 ), (4, 3 D15 (2, ) (2, 1 ) (2, 3 ) , (4, 3 ) (3, 1 ) (2, 1 ), (2, 3 ), (4, 3 D33 (2, ) − (0, 3 ),(2, 3 ) − (2, 1 ), (0, 3 ), (2, 3 D35 (2, ) − (2, 3 ), (4, 3 ) − (2, 1 ), (2, 3 ), (4, 3 F15 (3, ) (3, 1 ) (1, 3 ),(3, 3 ) (2, 1 ) (3, 1 ), (1, 3 ), (3, 3 F17 (3, ) (3, 1 ) (3, 3 ),(5, 3 ) (4, 1 ) (3, 1 ), (3, 3 ), (5, 1 F35 (3, ) − (1, 3 ),(3, 3 ) − (3, 1 ), (1, 3 ), (3, 3 F37 (3, ) − (3, 3 ),(5, 3 ) - (3, 1 ), (3, 3 ), (5, 3 TABLE I: The orbital angular momentum (L) and total spin (S)of the partial waves included in solving the coupled channel Equation (6). IV. FITTING PROCEDURE With the specifications given in Section III, the parameters associated with Z MB,M ′B′ of Eq. (7) are completely determined from fitting the ππ phase shifts in Refs. [13] and [32]. Thus the considered model has the following parameters: (a) the coupling constants associated with the Lagrangians listed in Eqs. (22)-(35), (b) the cutoff Λ for each vertex of vMB,M ′B′ (Fig. 2), (c) the coupling strength CN∗,MB(LS) and range kR and ΛN∗,MB(LS) of the bare N∗ → MB vertex Eq. (37), and (d) the bare mass M0N∗ of each N∗ state. We determine these by fitting the πN scattering data. Our fitting procedure is as follows. We first perform fits to the πN scattering data up to about 1.4 GeV and including only one bare state, the ∆ (1232) resonance. In these fits, the starting coupling constant parameters of vMB,M ′B′ are taken from the previous studies of πN and NN scattering, which are also given in Ref. [3]. Except the πNN coupling constant fπNN all coupling constants and the cutoff parameters are allowed to vary in the χ 2-fit to the πN data. The coupled-channel effects can shift the coupling constants greatly from their starting values. We try to minimize these shifts by allowing the cutoff parameters to vary in a very wide range 500 MeV < Λ < 2000 MeV. Some signs of coupling constants, which could not be fixed by the previous works [40], are also allowed to change. We then use the parameters from these fits at low energies as the starting ones to fit the amplitudes up to 2 GeV by also adjusting the resonance parameters, M0N∗ , CN∗,MB(LS), kR and ΛN∗,MB(LS). Here we need to specify the number of bare N∗ states in each partial wave. The simplest approach is to assume that each of 3-star and 4-star resonances listed by the Particle Data Group [35] is generated from a bare N∗ state of the model Hamiltonian Eq. (1). However, this choice is perhaps not well justified since the situation of the higher mass N∗’s is not so clear. We thus start the fits including only the bare states which generate the lowest and well- established N∗ resonance in each partial wave. The second higher mass bare state is then included when a good fit can not be achieved. We also impose the condition that if the resulting M0N∗ is too high > 2.5 GeV, we remove such a bare state in the fit. This is due to the consideration that the interactions due to such a heavy bare N∗ state could be just the separable representation of some non-resonant mechanisms which should be included in vMB,M ′B′ . In some partial waves the quality of the fits is not very sensitive to the N couplings to π∆, ρN , and σN . But the freedom of varying these coupling parameters is needed to achieve good fits. It is rather difficult to fit all partial waves simultaneously because the number of resonance parameters to be determined is very large. We proceed as follows. We first fit only 3 or 4 partial waves which have well established resonant states, and whose amplitudes have an involved energy dependence. These are the S11, P11, S31 and P33 partial waves. These fits are aimed at identifying the possible ranges of the parameters associated with vMB,M ′B′ . This step is most difficult and time consuming. We then gradually extend the fits to include more partial waves. For some cases, the fits can be reached easily by simply adjusting the bare N∗ parameters. But it often requires some adjustments of the non-resonance parameters to obtain new fits. This procedure has to be repeated many times to explore the parameter space as much as we can. We carry out this very involved numerical task by using the fitting code MINUIT and the parallel computation facilities at NERSC in US and the Barcelona Supercomputing Center in Spain. The most uncertain part of the fitting is to handle the large number of parameters asso- ciated with the bare N∗ states. Here the use of the empirical partial-wave amplitudes from SAID is an essential step in the fit. It allows us to locate the ranges of the N∗ parameters partial-wave by partial-wave for a given set of the parameters for the non-resonant vMB,M ′B,. Even with this, the information is far from complete for pinning down the N∗ parameters. Perhaps the N∗ parameters associated with the πN state are reasonably well determined in this fit to the πN scattering data. The parameters associated with ηN , π∆, ρN and σN can only be better determined by also fitting to the data of πN → ηN and πN → ππN reactions. This will be pursued in our second-stage calculations, as discussed in section I. It is useful to note here that the leading-order effect due to Z(E) of the meson-baryon interaction Eq. (7) on πN elastic scattering is δvπN,πN = MB,M ′B′=π∆,ρN,σN vπN,MBGMB(E)Z MB,M ′B′GM ′B′(E)vM ′B′,πN . (38) We have found by explicit numerical calculations that δvπN,πN is much weaker than vπN,πN and hence the coupled channel effects due to Z MB,M ′B′ on πN elastic scattering amplitude are weak. One example obtained from our model is shown in Table II. Thus we first perform the fits without including Z(E) term to speed up the computation. We then refine the parameters by including this term in the fits. V. RESULTS As mentioned in section I, we first locate the range of the parameters by fitting the empirical πN scattering amplitude of SAID [33]. We then check and refine the resulting parameters by directly comparing our predictions with the original πN scattering data. Re[tπN,πN ] Re[tπN,πN (Z (E) = 0)] Im[tπN,πN ] Im[tπN,πN(Z (E) = 0)] S11 −0.00481 −0.00557 0.0841 0.0827 P11 0.0937 0.103 0.636 0.640 P13 0.169 0.181 0.275 0.275 D13 0.202 0.194 0.299 0.309 D15 0.117 0.116 0.0179 0.0179 F15 0.290 0.291 0.157 0.155 F17 0.0360 0.0359 0.00293 0.00289 S31 −0.433 −0.437 0.496 0.504 P31 −0.253 −0.230 0.434 0.448 P33 0.0506 0.0306 0.510 0.457 D33 −0.00504 −0.0135 0.106 0.104 D35 0.0551 0.0551 0.0540 0.0537 F35 −0.0214 −0.0229 0.0259 0.0283 F37 0.0625 0.0626 0.00502 0.00512 TABLE II: The effect of Z MB,M ′B′ on the πN scattering amplitudes tπN,πN from solving Eq. (6) at W = 1.7 GeV. The normalization is tπN,πN = (e 2iδπN − 1)/(2i), where δπN is the πN scattering phase shift which could be complex at energies above the π production threshold. Our fits to the empirical amplitudes of SAID [33] are given in Figs. 4-5 and Figs. 6-7 for the T = 1/2 and T = 3/2 partial waves, respectively. The resulting parameters are presented in Appendix I. The parameters associated with the non-resonant interactions, vMB,M ′B′ with MB,M ′B′ = πN, ηN , π∆, ρN, σN , are given in Table III for the coupling constants of the starting Lagrangian Eqs.(22)-(35) and Table IV for the cutoffs of the form factors defined by Eq. (36). The resulting bare N∗ parameters are listed in Tables V-VII From Figs. 4-7, one can see that the empirical πN amplitudes can be fitted very well. The most significant discrepancies are in the imaginary part of S31 in Fig.7. The agreement is also poor for the F17 in Fig.4-5 and D35 in Figs.6-7, but there are rather large errors in the data. Our parameters are therefore checked by directly comparing our predictions with the data of differential cross sections dσ/dΩ and target polarization asymmetry P of elastic π±p → π±p and charge-exchange π−p → π0n processes. Our results (solid red curves) are shown in Figs.8-12. Clearly, our model is rather consistent with the available data, and are close to the results (dashed blue curves) calculated from the SAID’s amplitudes. Thus our model is justified despite the differences with the SAID’s amplitudes seen in Fig.4-7. It will be important to further refine our parameters by fitting the data of other πN scattering observables, such as the recoil polarization and double polarization. Hopefully, such data can be obtained from the new hadron facilities at JPARC in Japan. Our model is further checked by examining our predictions of the total cross sections σtot which can be calculated from the forward elastic scattering amplitudes by using the optical theorem. The total elastic scattering cross sections σel can be calculated from the predicted partial wave amplitudes. With the normalization < ~k|~k′ >= δ(~k − ~k′) used in Ref. [3], we σel(W ) = T=1/2,3/2 σelT (W ) (39) 1200 1600 2000 1200 1600 2000 1200 1600 2000 -0.15 -0.05 1200 1600 2000 1200 1600 2000 W (MeV) 1200 1600 2000 W (MeV) 1200 1600 2000 W (MeV) -0.04 -0.02 11 P13 FIG. 4: Real parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 1/2 are compared with the energy independent solutions of Ref. [33]. 1200 1600 2000 1200 1600 2000 1200 1600 2000 1200 1600 2000 1200 1600 2000 W (MeV) 1200 1600 2000 W (MeV) 1200 1600 2000 W (MeV) 11 P13 FIG. 5: Imaginary parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 1/2 are compared with the energy independent solutions of Ref. [33]. 1200 1600 2000 1200 1600 2000 1200 1600 2000 1200 1600 2000 -0.15 -0.05 1200 1600 2000 W (MeV) -0.08 -0.06 -0.04 -0.02 1200 1600 2000 W (MeV) -0.15 -0.05 1200 1600 2000 W (MeV) 31 P33 FIG. 6: Real parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 3/2 are compared with the energy independent solutions of Ref. [33]. σelT (W ) = (4π)2 ρπN (W ) (2J + 1) |T TJπN(LS),πN(LS)(k, k,W )|2 , (40) where ρπN(W ) = πkEπ(k)EN (k)/W with k determined by W = Eπ(k) + EN(k) and the amplitude T TJL′S′(πN),LS(πN)(k, k;W ) is the partial-wave solution of Eq. (5). Similarly, the total πN → ηN cross sections can be calculated from σtotπN→ηN = (4π)2 πN(W )ρ ηN (W ) (2J + 1) |T T=1/2,JηN(LS),πN(LS)(k ′, k,W )|2 (41) where ρηN (W ) = πk ′Eη(k ′)EN(k ′)/W with k′ determined by W = Eη(k ′) + EN (k ′). We can also calculate the contribution from each of the unstable channels, π∆, ρN , and σN , to the total πN → ππN cross sections. For example, we have for the πN → π∆ → ππN contribution in the center of mass frame σrecπ∆(W ) = ∫ W−mπ mN+mπ E∆(k) Γπ∆(k, E)/(2π) |W − Eπ(k)− E∆(k)− Σπ∆(k, E)|2 σπN→π∆(k,W ) (42) where k is defined by MπN = Eπ(k) +EN(k), EπN(k) = [M πN + k 2]1/2, Σπ∆(k, E) is defined in Eq.(19), Γπ∆(k, E) = −2Im(Σπ∆(k, E)), and σπN→π∆(k,W ) = 4πρπN (k0)ρπ∆(k) L′S′,LS,J 2J + 1 (2SN + 1)(2Sπ + 1) |T Jπ∆(L′S′),πN(LS)(k, k0;W )|2 where k0 is defined by W = Eπ(k0) + EN (k0) and ρab(k) = πkEa(k)Eb(k)/W . The am- plitude T JL′S′(π∆),LS(πN)(k, k0;W ) is the partial-wave solution of Eq.(5). The corresponding 1200 1600 2000 1200 1600 2000 1200 1600 2000 1200 1600 2000 1200 1600 2000 W (MeV) 1200 1600 2000 W (MeV) 1200 1600 2000 W (MeV) 31 P33 FIG. 7: Imaginary parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 3/2 are compared with the energy independent solutions of Ref. [33]. 0 40 80 120 160 θ (cm) 0 40 80 120 160 θ (cm) p π p π W=1440 MeV W=1440 MeV W=1650 MeV W=1650 MeV W=1800 MeV W=1800 MeV FIG. 8: Differential cross section for several different center of mass energies. Solid red curve corresponds to our model while blue dashed lines correspond to the SP06 solution of SAID [33]. All data have been obtained through the SAID online applications. Ref. [33]. 0 40 80 120 160 θ (cm) 0 40 80 120 160 θ (cm) p π p π W=1235 MeV W=1235 MeV W=1535 MeV W=1535 MeV W=1680 MeV W=1680 MeV FIG. 9: Differential cross section for several different center of mass energies. Similar description as Fig. 8. All data have been obtained through the SAID online applications. Ref. [33]. 0 40 80 120 160 θ (cm) 0 40 80 120 160 θ (cm) p π p π W=1230 MeV W=1640 MeV W=1440 MeV W=1680 MeV W=1540 MeV W=1800 MeV FIG. 10: Target polarization asymmetry, P , for several different center of mass energies. Similar description as Fig. 8. All data have been obtained through the SAID online applications. Ref. [33]. 0 40 80 120 160 θ (cm) 0 40 80 120 160 θ (cm) p π p π W=1230 MeV W=1640 MeV W=1440 MeV W=1680 MeV W=1540 MeV W=1800 MeV FIG. 11: Target polarization asymmetry, P , for several different center of mass energies. Descrip- tion as in Fig. 8. All data have been obtained through the SAID online applications. Ref. [33]. expressions for the unstable channels ρN and σN can be obtained from Eqs. (42)-(43) by changing the channel labels. The predicted σtot (solid curves) along with the resulting total elastic scattering cross sections σel compared with the data of π+p reaction are shown in Fig. 13. Clearly, the model can account for the data very well within the experimental errors. Here only the T = 3/2 partial waves are relevant. Equally good agreement with the data for π−p reaction are shown in the left side of Fig. 14. In the right side, we show how the contributions from each channel add up to get the total cross sections. The comparison of the contribution from ηN channel with the data is shown in Fig. 15. It is possible to improve the fit to this data by adjusting N∗ → ηN parameters. But this can be done correctly only when the differential cross section data of πN → ηN are included in the fit. This is beyond the scope of this work and will be pursed in our second-stage calculations. The contributions from π∆, ρN and σN intermediate states to the π−p → ππN total cross sections calculated from our model can be seen in the right side of Fig. 14. These predictions remain to be verified by the future experiments. The existing πN → ππN data are not sufficient for extracting model independently the contributions from each unstable channel. The results shown in Figs. 13-15 indicate that our parameters are consistent with the total cross section data. We now discuss the parameters presented in Appendix A. It is rather difficult to compare the resulting non-resonant coupling constants listed in Table III with the values from other works, since the coupling strengths are also determined by the cutoff parameters listed in 0 40 80 120 160 θ (cm) 0 40 80 120 160 θ (cm) p π p π W=1230 MeV W=1640 MeV W=1440 MeV W=1680 MeV W=1540 MeV W=1800 MeV FIG. 12: Target polarization asymmetry, P , for several different center of mass energies. Descrip- tion as in Fig. 8. All data have been obtained through the SAID online applications. Ref. [33]. Table IV. Perhaps it is possible to narrow their differences by using a different parameter- ization of the form factors. However, the fit is a rather time consuming process and hence no attempt is made in this work to try other forms of form factors. In Table V, we see that all of the bare masses are higher than the PDG’s resonance positions. This can be understood from the expression Eq. (14) for the partial waves with only one N∗ since one finds in general that Re[Σ̄(E)] < 0. For the S11, P11, P33 and D13 partial waves, two bare N∗ states are mixed by their interactions, as can be seen in Eq. (10). Thus the relation between their bare masses and the resonance positions identified by PDG is much more complex. As we mentioned above, the fit to πN elastic scattering is can not determine well the bare N∗ → π∆, ρN, σN parameters. Thus the results for these unstable particle channels listed in Tables III-VII must be refined by fitting the πN → ππN data. VI. SUMMARY AND FUTURE DEVELOPMENTS Within the formulation developed in Ref. [3], we have constructed a dynamical coupled- channel model of πN scattering by fitting the πN scattering data. The parameters of the model are first determined by fitting as much as possible the empirical πN elastic scattering amplitudes of SAID up to 2 GeV. We then refine and confirm the resulting parameters by directly comparing the predicted differential cross section and target polarization asymmetry with the original data of the elastic π±p→ π±p and charge-exchange π−p→ π0n processes. 1200 1400 1600 1800 2000 W (MeV) FIG. 13: The predicted total cross sections of the π+p → X (solid curve)and π+p → π+p (dashed curve) reactions are compared with the data. Squares and triangles are the corresponding data from Ref. [35]. The predicted total cross sections of πN reactions and are also in good agreement with the data. The model thus can be used as a starting point for analyzing the very extensive data of electromagnetic π production reactions. The predicted total cross sections of πN → ηN reactions are also in fair agreement with the data. However, the parameters associated with the ηN channel need to be refined to also fit the differential cross section data of πN → ηN before the model can be used to analyze the data of electromagnetic η production reactions. The main shortcoming of this work is that the ππN interaction term hππN of Eq.(4) is not included in the calculations. As derived in Ref. [3], the effects due to this interaction can be included by adding a term Z MB,M ′B′(E), which contains the ππN → ππN scattering amplitude, to the driving term VMB,M ′B′(E) of Eq.(6). Our effort in this direction is in progress along with the development of a more complete determination of the parameters of the model by fitting both the data of πN elastic scattering and πN → ππN reactions. This is also essential to pin down the parameters of the interactions associated with the π∆, ρN and σN states. Only when this second-stage is completed, we then can perform dynamical coupled-channel analysis of the very extensive and complex data of photo- and electro-production of two pions. This is an essential step to probe the W > about 1.7 GeV resonance region where the information on N∗ is very limited and uncertain. Finally, a necessary next step is to extract the resonance poles and the associated residues from the predicted πN amplitudes. This is being pursued and will be published else- where [46]. Acknowledgments We would like to thank M. Paris for his assistance in using the parallel processors at NERSC and A. Parreño for her help and encouragement to use the BSC. This work is sup- ported by the U.S. Department of Energy, Office of Nuclear Physics Division, under contract 1200 1400 1600 1800 W (MeV) 1200 1400 1600 1800 2000 W (MeV) πN, ηN πN, ηN, π∆ πN, ηN, pD, σN πN, ηN, π∆, σN, ρN FIG. 14: Left: The predicted total cross sections of the π−p → X (solid curve) and π−p → π−p + π0n (dashed curve) reactions are compared with the data. Open squares are the data on π−p → X from Ref. [35], open triangles are obtained by adding the π−p → π−p and π−p → π0n data obtained from Ref. [35] and SAID database [41] respectively. Right: Show how the predicted contributions from each channel are added up to the predicted total cross sections of the π−p → X. 1400 1600 1800 2000 W (MeV) FIG. 15: The predicted total cross sections of πp → ηp reaction are compared with the data [42, 43]. No. DE-AC02-06CH11357, and Contract No. DE-AC05-060R23177 under which Jefferson Science Associates operates Jefferson Lab, and by the Japan Society for the Promotion of Sci- ence, Grant-in-Aid for Scientific Research(c) 15540275. This work is also partially supported by Grant No. FIS2005-03142 from MEC (Spain) and FEDER and European Hadron Physics Project RII3-CT-2004-506078. The computations were performed at NERSC (LBNL) and Barcelona Supercomputing Center (BSC/CNS) (Spain). The authors thankfully acknowl- edges the computer resources, technical expertise and assistance provided by the Barcelona Supercomputing Center - Centro Nacional de Supercomputacion (Spain). APPENDIX A: PARAMETERS FROM THE FITS Parameter SL Model f2πNN/(4π) 0.08 0.08 mσ (MeV) 500.1 − fπN∆ 2.2061 2.0490 fηNN 3.8892 − gρNN 8.7214 6.1994 κρ 2.654 1.8250 gωNN 8.0997 10.5 κω 1.0200 0.0 gσNN 6.8147 − gρππ 4. 6.1994 fπ∆∆ 1.0000 − fρN∆ 7.516 − gσππ 2.353 − gωπρ 6.955 − gρ∆∆ 3.3016 − kρ∆∆ 2.0000 − TABLE III: The parameters associated with the Lagrangians Eqs.(22)-(35). The results are from fitting the empirical πN partial-wave amplitudes [33] of a given total isospin T = 1/2 or 3/2. The parameters from the SL model of Ref. [13] are also listed. Parameter (MeV) SL model (MeV) ΛπNN 809.05 642.18 ΛπN∆ 829.17 648.18 ΛρNN 1086.7 1229.1 Λρππ 1093.2 1229.1 ΛωNN 1523.18 − ΛηNN 623.56 − ΛσNN 781.16 − ΛρN∆ 1200.0 − Λπ∆∆ 600.00 − Λσππ 1200.0 − Λωπρ 600.00 − Λρ∆∆ 600.00 − TABLE IV: Cut-offs of the form factors, Eq. (36), of the non-resonant interaction vMB,M ′B′ . The results are from fitting the empirical πN partial-wave amplitudes [33] of a given total isospin T = 1/2 or 3/2. The parameters from the SL model of Ref. [13] are also listed. LTJ PDG’s Mass( MeV) M1 (MeV) M2 (MeV) S11 1535; 1655 1800. 1880. S31 1630 1850. P11 1440; 1710 1763 2037 P13 1720 1711 P31 1910 1900.3 P33 1232; 1600 1391 1602. D13 1520; 1700 1899.1 1988. D15 1675 1898 D33 1700 1976 D35 1960 − F15 1685 2187 F35 1890 2162 F37 1930 2137.8 TABLE V: The masses of the nucleon excited states included in the fits. (second and third columns). The first column contains the masses of the nucleon resonances given by PDG [35]. πN ηN π∆ σN ρN S11 (1) 7.0488 9.1000 −1.8526 −2.7945 2.0280 .02736 S11 (2) 9.8244 .60000 .04470 1.1394 −9.5179 −3.0144 S31 5.275002 − −6.17463 − −4.2989 5.63817 P11 (1) 3.91172 2.62103 −9.90545 −7.1617 −5.1570 3.45590 P11 (2) 9.9978 3.6611 −6.9517 8.62949 −2.9550 −0.9448 P13 3.2702 −.99924 −9.9888 −5.0384 1.0147 −.00343 1.9999 −.08142 P31 6.80277 − 2.11764 − 9.91459 0.15340 P33 (1) 1.31883 − 2.03713 9.53769 − −.3175 1.0358 0.76619 P33 (2) 1.3125 − 1.0783 1.52438 − 2.0118 −1.2490 0.37930 D13 (1) .44527 −.0174 −1.9505 .97755 −.481855 1.1325 −.31396 .17900 D13 (2) .46477 .35700 9.9191 3.8752 −5.4994 .28916 9.6284 −.14089 D15 .31191 −.09594 4.7920 .01988 −.45517 −.17888 1.248 −.10105 D33 .9446 − 3.9993 3.9965 − .16237 3.948 −.85580 F15 .06223 0.0000 1.0395 .00454 1.5269 −1.0353 1.6065 −.0258 F35 .173934 − −2.96090 −1.09339 − −.07581 8.0339 −.06114 F37 0.25378 − −0.3156 −0.0226 − .100 .100 .100 TABLE VI: The coupling constants CN∗,JTLS;MB of Eq. (37) with MB = πN, ηN, π∆, σN, ρN for each of the resonances. When there are more than one value for π∆ and ρN channels, they correspond to the possible quantum numbers (LS) listed in Table 2. πN ηN π∆ σN ρN S11 (1) 1676.4 598.97 554.04 801.03 1999.8 1893.6 S11 (2) 533.48 500.02 1999.1 1849.5 796.83 500.00 S31 2000.00 − 500.00 − 500.031 500.00 P11 (1) 1203.62 1654.85 729.0 1793.0 621.998 1698.90 P11 (2) 646.86 897.84 501.26 1161.20 500.06 922.280 P13 1374.0 500.23 500.00 500.770 640.50 500.00 500.10 1645.2 P31 828.765 − 1999.9 − 1998.8 2000.6 P33 (1) 880.715 − 507.29 501.73 − 606.78 1043.4 528.37 P33 (2) 746.205 − 846.37 780.96 − 584.98 500.240 1369.7 D13 (1) 1658. 1918.2 976.36 1034.5 1315.8 599.79 1615.1 1499.50 D13 (2) 1094.0 678.41 1960.0 660.02 1317.0 550.14 597.57 1408.7 D15 1584.7 1554.0 500.77 820.17 507.07 735.40 749.41 937.53 D33 806.005 − 1359.38 608.090 − 1514.98 1998.99 956.61 F15 1641.6 655.87 1899.5 522.68 500.93 500.76 500.0 1060.9 F35 1035.28 − 1227.999 586.79 − 1514.3 593.84 1506.0 F37 1049.04 − 1180.2 1031.81 − 600.02 600.00 600.02 TABLE VII: The range parameter ΛN∗,JTLS;MB (in unit of (MeV/c)) of Eq. 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[36] The Lorentz boost factors for transforming the matrix elements of ∆ → πN (ρ, σ → ππ) vertex interaction from the πN (ππ) to the ππN center of mass frames, which were not given in Ref. [3], are included here. [37] J. D. Bjorken and S. D. Drell, Relativistic Quantum Field Theory (McGraw-Hill, New York, 1964). [38] A. Matsuyama, Phys. Lett. B152, 42 (1984). [39] A. Matsuyama and T.-S. H. Lee, Phys. Rev. C 34, 1900 (1986). [40] K. Nakayama, Y. Oh, J. Haidenbauer, and T.-S. H. Lee, e-Print: nucl-th/0611101 and to appear in Phys. Lett (2007). [41] CNS Data Analysis Center, GWU, http://gwdac.phys.gwu.edu. [42] S. Prakhov et al., Phys. Rev C 72,015203 (2005). [43] Robert M. Brown et al., Nucl. Phys. B153, 89 (1979). [44] B. Julia-Diaz, D.O. Riska, and F. Coester, Phys. Rev. C69, 035212 (2004). [45] See the review by P. Maris and C.D. Roberts, Int.J.Mod.Phys. E12 297(2003). [46] N. Suzuki, T. Sato and T.-S.H. Lee, in preparation. http://arxiv.org/abs/nucl-th/0611101 http://gwdac.phys.gwu.edu Introduction Dynamical Coupled-channel equations Calculations Fitting Procedure Results Summary and Future Developments Acknowledgments Parameters from the fits References

As a first step to analyze the electromagnetic meson production reactions in the nucleon resonance region, the parameters of the hadronic interactions of a dynamical coupled-channel model, developed in {\it Physics Reports 439, 193 (2007)}, are determined by fitting the $\pi N$ scattering data. The channels included in the calculations are $\pi N$, $\eta N$ and $\pi\pi N$ which has $\pi\Delta$, $\rho N$, and $\sigma N$ resonant components. The non-resonant meson-baryon interactions of the model are derived from a set of Lagrangians by using a unitary transformation method. One or two bare excited nucleon states in each of $S$, $P$, $D$, and $F$ partial waves are included to generate the resonant amplitudes in the fits. The parameters of the model are first determined by fitting as much as possible the empirical $\pi N$ elastic scattering amplitudes of SAID up to 2 GeV. We then refine and confirm the resulting parameters by directly comparing the predicted differential cross section and target polarization asymmetry with the original data of the elastic $\pi^{\pm} p \to \pi^{\pm} p$ and charge-exchange $\pi^- p \to \pi^0 n$ processes. The predicted total cross sections of $\pi N$ reactions and $\pi N\to \eta N$ reactions are also in good agreement with the data. Applications of the constructed model in analyzing the electromagnetic meson production data as well as the future developments are discussed.

Introduction Dynamical Coupled-channel equations Calculations Fitting Procedure Results Summary and Future Developments Acknowledgments Parameters from the fits References

arXiv:0704.1616v1 [hep-ph] 12 Apr 2007 Published in Phys. Rev. Lett. 98, 149104 (2007) Reply to “Comment on ‘Chiral suppression of scalar glueball decay’ ” Michael Chanowitz1 Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720 In [1] I observed that the amplitude for spin zero glue- ball decay is proportional to the quark mass, M(G0 → qq) ∝ mq, to all orders in perturbation theory, so that the ratio Γ(G0 → uu+ dd)/Γ(G0 → ss) is calculable and small, even though the individual rates are not pertur- batively calculable because of soft t and u channel quark exchanges. I noted that if hadronization of G0 → qq is an important mechanism for G0 → ππ and G0 → KK, then Γ(G0 → ππ) is much smaller than Γ(G0 → KK), ex- plaining a previous LQCD result[2] and supporting iden- tification of f0(1710) with G0. A more robust conse- quence, emphasized in [3], is that mixing of G0 with uu + dd (and perhaps also ss) mesons is suppressed, so that the scalar (and pseudoscalar) may be the purest glueballs. In both [1] and [3] I emphasized the neces- sity to verify the existence and consequences of chiral suppression by a reliable nonperturbative method, which today can only be LQCD. Chao et al. agree that G0 → qq is chirally suppressed but propose that G0 → qqqq, which is not chirally sup- pressed, is the dominant mechanism for G0 → ππ. In the preceding Comment[4] and in a previous paper[5] they exhibit an O(αS) amplitude for the exclusive process G0 → ππ using light cone wave functions. Since pQCD for exclusive processes converges much more slowly than inclusive pQCD[6], the estimate is not quantitatively re- liable at the experimentally interesting scale, mG = 1.7 GeV, where even the applicability of ordinary inclusive pQCD is marginal. While the qqqq mechanism might in- deed dilute or remove chiral suppression of G0 → ππ, it is not possible to decide, since the magnitude of neither the qq nor qqqq contributions are reliably calculable. Comparing the amplitudes for M(G0 → qq) and M(G0 → qqqq → ππ) in [1] and [4, 5] it appears that both begin at first order in αS , but this impression is misleading. It is easy to see that M(G0 → qqqq → ππ) vanishes in the chiral limit at O(αS) for on-shell con- stituent gluons. The qqqq mechanism requires the quark from one gluon to combine with the antiquark from the other gluon to form a color singlet pion. But G0 cms (center of mass) kinematics then requires both quarks to have the same energy fraction, x = 2Eq/mG and both antiquarks to have fraction 1−x, with m2π = x(1−x)m One of the q or q constituents of each pion is then mov- ing in the opposite direction to the pion in the G0 cms. Boosting to an infinite momentum frame, one constituent is then at x = 1 and the other at x = 0, where the wave function vanishes. In the chiral limit, mπ = 0, this is al- ready apparent in the G0 cms. Since confining dynamics may put the gluons off-shell of order ΛQCD, the ampli- tude does not actually vanish but is suppressed of order O(ΛQCD/mG). In the revised Comment the authors have responded to this observation with the added stipulation that the G0 constituent gluons are maximally off-shell, of order mG. Although this requirement was not imposed in [5], the result is apparently unchanged. Certainly one conse- quence is that fg, the effective G0gg coupling, cannot be identified with the corresponding coupling f0 in [1] as is claimed in [4, 5], but reflects the off-shell tail of the G0 wave function or implicitly contains a factor αS at the hard scale mG reflecting hard gg → g ∗g∗ scattering to push the gluons maximally off-shell. Alternatively, hard scattering of qqqq can align the quarks suitably with the final state pions, with the amplitude then explicitly of order O(α2S). The relative magnitude of the qq and qqqq mechanisms for G0 → ππ is not obvious. For the qq mechanism we do not know the magnitude of M(G0 → qq) because both αS(Q) and the running mass mq(Q) are evaluated at a soft scale, O(ΛQCD), and thus are not under perturbative control. In addition we do not know the hadronization rate from qq to ππ and KK compared to multi-meson fi- nal states. On the other hand, Γ(G0 → ππ) via the qqqq mechanism cannot be reliably estimated and is addition- ally suppressed by the square of the coupling, αS(Q) evaluated at the largest scale in the problem, Q = mG. It is then important to stress the agreement, expressed in both [1, 3] and [4], on the most important point: re- liable nonperturbative methods are needed to determine whether G0 → ππ is chirally suppressed. We eagerly await LQCD “data” and data from BES II to clarify the issue. Acknowledgments: This work was supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under contract DE- AC02-05CH11231. [1] M. Chanowitz, Phys. Rev. Lett. 95 (2005), hep- ph/0506125 [2] J.Sexton, A.Vaccarino, D.Weingarten, Phys.Rev.Lett.75: 4563,1995, hep-lat/9510022. [3] M. Chanowitz, Talk given at Charm 2006, Beijing, China, 5-7 Jun 2006. Published in Int.J.Mod.Phys.A21, 5535 (2006), hep-ph/0609217. http://arxiv.org/abs/0704.1616v1 [4] K.T. Chao, X-G He, J.P. Ma, preceding Comment. [5] K.T. Chao, X-G He, J.P. Ma, hep-ph/0512327. [6] N. Isgur and C.H. Llewellyn-Smith, Phys. Rev. Lett. 52: 1080 (1984).

Reply to the comment of Chao, He, and Ma.

arXiv:0704.1616v1 [hep-ph] 12 Apr 2007 Published in Phys. Rev. Lett. 98, 149104 (2007) Reply to “Comment on ‘Chiral suppression of scalar glueball decay’ ” Michael Chanowitz1 Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720 In [1] I observed that the amplitude for spin zero glue- ball decay is proportional to the quark mass, M(G0 → qq) ∝ mq, to all orders in perturbation theory, so that the ratio Γ(G0 → uu+ dd)/Γ(G0 → ss) is calculable and small, even though the individual rates are not pertur- batively calculable because of soft t and u channel quark exchanges. I noted that if hadronization of G0 → qq is an important mechanism for G0 → ππ and G0 → KK, then Γ(G0 → ππ) is much smaller than Γ(G0 → KK), ex- plaining a previous LQCD result[2] and supporting iden- tification of f0(1710) with G0. A more robust conse- quence, emphasized in [3], is that mixing of G0 with uu + dd (and perhaps also ss) mesons is suppressed, so that the scalar (and pseudoscalar) may be the purest glueballs. In both [1] and [3] I emphasized the neces- sity to verify the existence and consequences of chiral suppression by a reliable nonperturbative method, which today can only be LQCD. Chao et al. agree that G0 → qq is chirally suppressed but propose that G0 → qqqq, which is not chirally sup- pressed, is the dominant mechanism for G0 → ππ. In the preceding Comment[4] and in a previous paper[5] they exhibit an O(αS) amplitude for the exclusive process G0 → ππ using light cone wave functions. Since pQCD for exclusive processes converges much more slowly than inclusive pQCD[6], the estimate is not quantitatively re- liable at the experimentally interesting scale, mG = 1.7 GeV, where even the applicability of ordinary inclusive pQCD is marginal. While the qqqq mechanism might in- deed dilute or remove chiral suppression of G0 → ππ, it is not possible to decide, since the magnitude of neither the qq nor qqqq contributions are reliably calculable. Comparing the amplitudes for M(G0 → qq) and M(G0 → qqqq → ππ) in [1] and [4, 5] it appears that both begin at first order in αS , but this impression is misleading. It is easy to see that M(G0 → qqqq → ππ) vanishes in the chiral limit at O(αS) for on-shell con- stituent gluons. The qqqq mechanism requires the quark from one gluon to combine with the antiquark from the other gluon to form a color singlet pion. But G0 cms (center of mass) kinematics then requires both quarks to have the same energy fraction, x = 2Eq/mG and both antiquarks to have fraction 1−x, with m2π = x(1−x)m One of the q or q constituents of each pion is then mov- ing in the opposite direction to the pion in the G0 cms. Boosting to an infinite momentum frame, one constituent is then at x = 1 and the other at x = 0, where the wave function vanishes. In the chiral limit, mπ = 0, this is al- ready apparent in the G0 cms. Since confining dynamics may put the gluons off-shell of order ΛQCD, the ampli- tude does not actually vanish but is suppressed of order O(ΛQCD/mG). In the revised Comment the authors have responded to this observation with the added stipulation that the G0 constituent gluons are maximally off-shell, of order mG. Although this requirement was not imposed in [5], the result is apparently unchanged. Certainly one conse- quence is that fg, the effective G0gg coupling, cannot be identified with the corresponding coupling f0 in [1] as is claimed in [4, 5], but reflects the off-shell tail of the G0 wave function or implicitly contains a factor αS at the hard scale mG reflecting hard gg → g ∗g∗ scattering to push the gluons maximally off-shell. Alternatively, hard scattering of qqqq can align the quarks suitably with the final state pions, with the amplitude then explicitly of order O(α2S). The relative magnitude of the qq and qqqq mechanisms for G0 → ππ is not obvious. For the qq mechanism we do not know the magnitude of M(G0 → qq) because both αS(Q) and the running mass mq(Q) are evaluated at a soft scale, O(ΛQCD), and thus are not under perturbative control. In addition we do not know the hadronization rate from qq to ππ and KK compared to multi-meson fi- nal states. On the other hand, Γ(G0 → ππ) via the qqqq mechanism cannot be reliably estimated and is addition- ally suppressed by the square of the coupling, αS(Q) evaluated at the largest scale in the problem, Q = mG. It is then important to stress the agreement, expressed in both [1, 3] and [4], on the most important point: re- liable nonperturbative methods are needed to determine whether G0 → ππ is chirally suppressed. We eagerly await LQCD “data” and data from BES II to clarify the issue. Acknowledgments: This work was supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under contract DE- AC02-05CH11231. [1] M. Chanowitz, Phys. Rev. Lett. 95 (2005), hep- ph/0506125 [2] J.Sexton, A.Vaccarino, D.Weingarten, Phys.Rev.Lett.75: 4563,1995, hep-lat/9510022. [3] M. Chanowitz, Talk given at Charm 2006, Beijing, China, 5-7 Jun 2006. Published in Int.J.Mod.Phys.A21, 5535 (2006), hep-ph/0609217. http://arxiv.org/abs/0704.1616v1 [4] K.T. Chao, X-G He, J.P. Ma, preceding Comment. [5] K.T. Chao, X-G He, J.P. Ma, hep-ph/0512327. [6] N. Isgur and C.H. Llewellyn-Smith, Phys. Rev. Lett. 52: 1080 (1984).

Astronomy & Astrophysics manuscript no. cepa2007˙paper c© ESO 2021 September 15, 2021 High-resolution study of a star-forming cluster in the Cep-A HW2 region C. Comito1, P. Schilke1, U. Endesfelder1 , I. Jiménez-Serra2, and J. Martı́n-Pintado2 1 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany e-mail: ccomito@mpifr-bonn.mpg.de 2 Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientı́ficas, Departamento de Astrofı́sica Molecular e Infrarroja, C/Serrano 121, E-28006 Madrid, Spain Received; accepted ABSTRACT Context. Due to its relatively small distance (725 pc), the Cepheus A East star-forming region is an ideal laboratory to study massive star formation processes. Aims. Based on its morphology, it has been suggested that the flattened molecular gas distribution around the YSO HW2 may be a 350-AU-radius massive protostellar disk. Goal of our work is to ascertain the nature of this structure. Methods. We have employed the Plateau de Bure Interferometer⋆ to acquire (sub-)arcsecond-resolution imaging of high-density and shock tracers, such as methyl cyanide (CH3CN) and silicon monoxide (SiO), towards the HW2 position. Results. On the 1′′ (∼ 725 AU) scale, the flattened distribution of molecular gas around HW2 appears to be due to the projected superposition, on the plane of the sky, of at least three protostellar objects, of which at least one is powering a molecular outflow at a small angle with respect to the line of sight. The presence of a protostellar disk around HW2 is not ruled out, but such structure is likely to be detected on a smaller spatial scale, or using different molecular tracers. 1. Introduction Several theories are being considered to explain the forma- tion of massive (M ≥ 8 M⊙) stars, which can be roughly grouped into accretion-driven and coalescence-driven models (cf. Stahler et al. 2000). In the latter case, high-mass stars would form by merging of two or more lower-mass objects, making the presence of stable massive accretion disks around the protostar very unlikely. However, only models based on disk-protostar interactions are capable of explaining the exis- tence of jets and outflows: hence, the high incidence, in large samples of massive YSOs, of highly collimated outflows (cf. Beuther et al. 2002) has been interpreted as indirect evidence for the existence of high-mass disks. It is undoubted that the direct detection of accretion onto massive protostars through rotating disks constitutes an im- portant tile in the massive-star-formation-theory mosaic. From an observational point of view, this task is made very difficult by two factors: i) massive star-forming regions typically are far away, a few kpc on average, making the direct observa- tion of small-scale structure such as disks virtually impossible with current instruments; and ii), massive stars form in clus- Send offprint requests to: C. Comito ters, making the surrounding region extremely complex, both spatially and kinematically. Located only ∼ 725 pc from the Sun (Johnson 1957), Cepheus A is considered a very promising candidate for the detection of a massive disk. Its well-studied bipolar outflow (cf. Gómez et al. 1999, hereafter G99, and references therein) is thought to be powered by the radio-continuum source HW2 (∼ 104 L⊙, Rodrı́guez et al. 1994). Curiel et al. (2006) report the presence of very large tangential velocities in the HW2 ra- dio jet, consistent with HW2 being a massive Young Stellar Object (YSO). The distribution of H2O masers (Torrelles et al. 1996) and of the SiO emission (G99) around HW2, both oriented perpendicularly with respect to the direction of the flow, have been interpreted as strongly supporting the existence of accretion shocks onto a rotating and contracting molecu- lar disk of ∼ 700-AU diameter, centered on HW2, with the northeast-southwest outflow being triggered by the interaction between such disk and HW2 itself. Similar conclusions have been reached by Patel et al. (2005), based on SMA observa- tions of CH3CN and dust emission. However, the fact that the HW2 vicinities are crowded with YSOs (at least three within an area of 0.′′6 × 0.′′6, Curiel et al. 2002), together with the re- cent detection of an internally heated hot core within 0.′′4 from the center of the outflow (Martı́n-Pintado et al. 2005, hereafter http://arxiv.org/abs/0704.1617v1 2 C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region MP05) cast some doubts on this interpretation. Based on our PdBI observations, we conclude that, on the 1′′ scale, the elon- gated molecular structure around HW2 can be explained with the superposition, on the plane of the sky, of at least three dif- ferent hot-core-type sources, at least one of them being the ex- citing source for a second molecular outflow. 2. Observations In 2003 and 2004, with the Plateau de Bure Interferometer, we have carried out observations of several high-density and shock tracers (also cf. Schilke et al., in prep.), among which silicon monoxide (SiO) and methyl cyanide (CH3CN), towards the HW2 position (αJ2000 = 22 h56m17.9s, δJ2000 = +62 ◦01′49.6′′). A combination of high-spectral-resolution correlator units were employed to achieve a channel width ∆v of up to ∼ 0.3 km s−1. The five antennas in AB (extended) configuration provided a HPBW of 2′′ × 1.′′6 for SiO(2-1) at 86 GHz, and of 0.′′9 × 0.′′7 for CH3CN(12 − 11) at 220 GHz. The data cubes were pro- duced with natural weighting. All maps have been CLEANed. Analysis of all molecular spectra has been performed after sub- traction of the continuum emission. 3. Results Fig. 1 (left panel) shows the Cep-A star-forming region within a 1100-AU radius from HW2. The peak of the 241-GHz dust emission (grey scale) roughly coincides with the HW2 posi- tion and with the center of the large-scale outflow. The inte- grated CH3CN emission is also centered on HW2 (contours), and somewhat elongated almost perpendicularly to the direc- tion of the large-scale outflow. Like other molecular tracers (cf. Brogan et al. 2007), CH3CN displays two different velocity components, centered around −5 and −10 km s−1 respectively. The solid contours in Fig. 1, center panel, show the emission of the CH3CN(123−113) transition, integrated between −7 and −3 km s−1, whereas the emission in the range between −11.5 and −7.5 km s−1 is represented by the dashed contours (see § 3.2). The center of SiO emission, instead, is at ∼ −10 km s−1. Silicon monoxide peaks about 0.′′4 eastwards of HW2, at a po- sition that coincides with the HC source of MP05 (triangle in Fig. 1, see § 3.1), close to the −10-km s−1 CH3CN component. In what follows, we will discuss in more detail the SiO and CH3CN data. 3.1. SiO Our dataset confirms that the spatial distribution of this shock tracer is mainly concentrated in the HW2 region (its presence in the large-scale outflow is limited to a few bullets at large dis- tances from the center), although not centered on the HW2 po- sition. This does indeed suggest that shock processes are taking place in the (projected) immediate vicinities of HW2. However, if the SiO emission were arising from accretion shocks onto a rotating disk (as proposed by G99), we would expect to ob- serve a similar velocity structure to that observed for the other molecular tracers peaking around HW2. Instead, SiO seems to be tracing a completely different kinematic component: unlike any other line in our dataset, the SiO(2-1) line has a velocity spread of at least 35 km s−1at the zero-flux level (∼ 15 km s−1 FWHM). A mass of about 90 M⊙ would be required to pro- duce such large line width in a gravitationally bound enviro- ment (assuming virial equilibrium, and that the emission arises in a region of ∼ 350-AU radius). This value is about one order of magnitude larger than the estimated mass of HW2, which is expected to become a B0.5 star once in ZAMS (Rodriguez et al. 1994). Fig. 2 shows a comparison between SiO(2-1) and CH3CN(124 − 114). We carried out a two-dimensional Gaussian fit of the SiO(2- 1) spatial distribution, for every spectral channel in the velocity range −25 < vlsr −3.5 km s −1. This corresponds to the spec- tral interval in which the signal-to-noise (S/N) ratio of the SiO transition is ≥ 9σ. For smaller S/N ratios, in fact, the error on the fitted centroid position easily exceeds 50%. The result is a distribution of the centroids of SiO emission as a function of velocity. Fig. 1, right panel, shows that the centroid positions are located in a well-defined two-lobed area, centered about 0.′′4 eastwards of HW2 and of the dust continuum emission peak. Although the error on every single centroid position is still relatively large (up to 30%), as a whole their distribution describes a very clear velocity trend, with all the emission at vlsr< −10 km s −1 clustering in the left lobe, and all the emis- sion at vlsr> −10 km s −1 clustering in the right lobe. This re- sult suggests that a second molecular outflow is being ejected in the HW2 region. Our interpretation is supported by the re- cent discovery of an intermediate-mass protostar, surrounded by the hot molecular core HC (MP05) located in the region be- tween the blue and red lobes of SiO emission (white triangle in Fig. 1, right panel), hence a very likely candidate to be its powering engine. With the current dataset it is not possible to establish the exact inclination angle of the flow, but the large velocity spread observed in the SiO(2-1) line, together with the relatively concentrated spatial distribution of the SiO emission, suggests that the inclination angle must be high, i.e., that the SiO flow is being ejected at a small angle with respect to the line of sight. 3.2. CH3CN The dense molecular gas, as traced by CH3CN(123-113) (Fig. 1, left panel) appears to be distributed around the HW2 position, and elongated in a direction roughly perpendicular to the projected direction of the large-scale outflow on the plane of the sky. From a morphological point of view, therefore, the data are very suggestive of the presence of a ∼ 350-AU-radius disk-like structure around HW2. On the other hand, as already pointed out by Torrelles et al. (1999) based on VLA observations of NH3, the kinemati- cal picture describing the dense gas distribution in the region is quite complex. A position-velocity cut along the major axis of the elongated structure (indicated in Fig. 1, left and center panels, with a dashed line) reveals a velocity spread of about 6 km s−1 (see Fig. 3), also observed by Patel et al. (2005). However, the two intensity peaks along the axis share roughly C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region 3 Fig. 1. All panels: the grey levels represent the continuum emission at 241 GHz. Lowest level is 3.3 mJy/beam or 2σ, highest is 22σ. The HW2 position is indicated with a white star. The solid, crossing lines show the opening angle of the large-scale northeast-southwest outflow, inferred from HCN and 13CO observations obtained with PdBI together with the core-tracing data. The contours trace the continuum-subtracted CH3CN emission at 220 GHz, and in the top left corner, the HPBW for the con- tinuum (grey foreground) and CH3CN (black background ellipse) are shown. The position of the intermediate-mass protostar HC (MP05) is indicated with a white triangle. Left panel: the integrated emission of the CH3CN(123-113) is shown in solid contours. Center panel: integrated intensity of CH3CN(123-113) between −7 and −3 km s −1 (solid contours) and between −11.5 and −7.5 km s−1 (dashed contours). The dotted square box indicates the area enlarged in the right panel. Right panel: The circles show the centroid positions of the SiO(2-1) emission for every channel in the range −25 < vlsr −3.5 km s −1, with a channel width of 0.5 km s−1. The black circles stand for the blue-shifted (vlsr < −10 km s −1), white circles for the red-shifted (vlsr > −10 km s emission. Fig. 2. Comparison between the line profiles of SiO(2-1) (grey spectrum) and CH3CN(124 − 114) (transparent spectrum). At vlsr=∼ −10 km s −1, the center of the SiO emission coincides with the position of source HC of MP05 (cf. § 3.2 and Fig. 1). The SiO(2-1) spectrum arises from a region of about 1′′ ra- dius around the HC position (50%-of-peak-emission level). The CH3CN emission arises from the HC3 position (cf. § 3.2 and Fig. 4), and has been scaled by a factor 0.5 for a better visual comparison. No recognizable counterpart to the ∼ −5- km s−1 CH3CN component is observed for SiO. the same systemic velocity (∼ −5 km s−1). The weaker, blue- shifted component of emission (∼ −10 km s−1), appears to trace rather the outskirt of a physically separated component than a rotation-induced velocity gradient along the axis of the alleged “disk”. The peak of the -10 km s−1 CH3CN emission is spa- tially and kinematically close to the center of the small-scale SiO outflow (see Fig. 1, right panel), it is likely associated to it and/or to its exciting source. The CH3CN integrated intensity is dominated by the two −5-km s−1 peaks, which lie respectively about 0.′′6 to the north- west, and 0.′′5 to the southeast of the HW2 position. The fact that the two CH3CN peaks share roughly the same systemic velocity, is not compatible with the “rotating disk” hypothesis. In what follows, we will treat them as independent condensa- tions, and to be consistent with the nomenclature introduced by MP05, we will refer to them respectively as HC2 and HC3. Fig. 4 compares the spectra observed towards the two positions. It is clear that, in both cases, both the −5- and −10-km s−1 com- ponents are present along the line of sight, although the contri- bution from the latter is more substantial towards HC3, i.e., close to the peak of the −10-km s−1 SiO emission. We have assumed the LTE approximation to fit the physical parameters associated with the two different velocity compo- nents. All transitions in the spectrum are fitted simultaneously, in order to take line blending and optical depth effects properly into account (a detailed description of the method can be found in Comito et al. 2005). For the ∼ −5 km s−1component, our fit reveals that the k= 0 through k= 4 transitions are optically thick towards both positions. The data at this velocity can only be reproduced by including a very compact, hot, dense object in the model. The emission centered at ∼ −10-km s−1 can be modeled with a cooler, more extended component. The results of the fit, for the two positions, are summarized in Tab. 1. Although the presence of secondary minima in the χ2 space is unavoidable when so many parameters are varied to achieve 4 C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region Fig. 3. Position-velocity plot for CH3CN(123 − 113), along the major axis of the elongated structure (dashed line in Fig. 1, left and center panels). Levels range from 3σ to 27σ in 3-σ steps. The bracketed velocity ranges show the intervals making up the −5 (solid) and −10 (dashed) km s−1 components, whose spatial distribution is shown in Fig. 1, center panel. minimization, in this case the simultaneous fitting of intensity ratios between optically thick and optically thin lines, between ortho- and para-CH3CN transitions, and between 12C and 13C isotopologues of methyl cyanide (see Fig. 4), places very strin- Fig. 4. In grey, high-resolution (∆v = 0.3 km s−1) spectra of the CH3CN(12-11) emission at 200 GHz, towards the HC2 and HC3 positions (see Fig. 1, center panel). Overlayed in black are the model spectra, resulting from the parameters listed in Tab. 1, assuming LTE approximation and that [12CO]/[13CO]= gent constraints on the viable parameter space, at least as far as the compact (∼ −5 km s−1) component is concerned. Note that the rotational temperatures derived in this fashion are sig- nificantly higher than those derived by Patel et al. (2005). This discrepancy can be explained with the optical depth correction in our fit. 4. Discussion The observed elongation of the molecular gas distribution around HW2, over a radius of ∼ 0.′′5 (∼ 360 AU), appears to be due to the projected superposition, on the plane of the sky, of at least three protostellar objects, of which at least one is triggering a molecular outflow at a small angle with respect to the line of sight (§ 3.1, § 3.2). All lines in our dataset are con- sistent with this interpretation. The distribution of molecular gas around HW2 can, on a 1′′ scale, be interpreted as a clus- ter of high- and intermediate-mass protostars in the Cepheus A HW2 region. The analysis of the CH3CN spectra (§ 3.2) sug- gests the presence of internally heated compact hot-core-type objects like HC, likely hosting protostellar objects, although the 1-mm continuum emission peaks on the HW2 position and does not show any secondary clumps. This may be due to in- sufficient dynamic range in our data, if the contribution, to the 241-GHz continuum, of free-free emission from the HW2 ther- mal jet is large. Fig. 5 shows the variation of the measured HW2 continuum flux density as a function of frequency, S ν, between 1.5 and 327 GHz (data points from: Rodrı́guez et al. 1994; this work; Patel et al. 2005). A two-component least- squares fit of the data yields S jet ∝ ν (0.51±0.12) (consistent with the value inferred by Rodrı́guez et al. 1994, and with the theo- retical predictions for the radio continuum spectrum of a con- fined thermal jet, Reynolds 1986) and S (sub)mm ∝ ν (1.92±0.12) (dashed and dashed-dotted lines respectively in Fig. 5), where S jet + S (sub)mm = S ν (solid curve in Fig. 5). Based on this esti- mate, the thermal jet (free-free) contribution at 241 GHz should be ∼ 10% of the total flux. However, the flux density variation in the (sub)mm- wavelength portion of the spectrum increases basically on a Rayleigh-Jeans slope, suggesting the presence of optically thick emission from an unresolved (with our best spatial res- olution, at 241 GHz with PdBI) continuum source, whose size thus cannot be larger than ∼ 0.′′6. Although we cannot determine the nature of this compact source, it makes sense to hypothesize that it can be described by i), dust emission, and/or ii), free-free emission from a Hii region (for example associated to a photoevaporating disk). For case i), we adopt Beckwith et al.’s (1990) values for the mass absorption coeffi- cient, κν = 0.1(ν/10 12Hz)β, with β = 1, to estimate the lower mass limit for an object with size θdust = 0. ′′5 (∼ 363 AU) to produce optically thick emission at 87 GHz: Mτdust=1 ≥ 1 M⊙. Based on the peak flux at 87, 241 (this work) and 327 GHz (Patel et al. 2005), the above value of θdust yields a brightness temperature, TB ≃ 80 K. Smaller source sizes would lead to higher intrinsic temperatures and lower mass limits. In case ii), we assume TB = 10 4 K (typical for Hii regions), which would translate into a source size of θfree−free ≃ 0. ′′04 (∼ 30 AU). The continuum source VLA-mm (Curiel et al. 2002), which is lo- C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region 5 vlsr Source size N(CH3CN) Trot ∆v (km s−1) (cm−2) (K) (km s−1) HC2 -4.5 0.′′3 ∼ 3 × 1016 250 2.9 -8.9 1′′ ∼ 8 × 1014 150 4.0 HC3 -4.2 0.′′25 ∼ 3 × 1016 250 3.2 -10.0 0.′′45 ∼ 5 × 1015 150 4.5 Table 1. LTE model results for the CH3CN emission towards the HC2 and HC3 cores (Fig. 4). For a discussion on the error estimate, cf. Comito et al. 2005. cated ∼ 0.′′15 south of HW2 and whose size cannot be larger than 30 AU, displays a much too weak emission at cm and mm wavelengths, and we estimate its contribution to cover for at most 5% of the observed (sub)mm flux. In other words, we cannot discard any of the two hypothe- ses for the observed optically thick continuum emission. At higher frequencies, the spectral index would get flatter if the optically thick emission were only due to free-free emission, while it would remain the same for dust. Observations with a resolution of < 0.5′′, which are within the reach of present day interferometers, would be able to shed more light on the nature of this object. Fig. 5. Flux density as a function of frequency for Cep-A HW2. For the Rodrı́guez et al. (1994, triangles) and Patel et al. (2005, square) data points, the errorbars fall within the sym- bols and are therefore not visible. The solid curve shows the 2-component least-squares fit of the data, as descrived in the text. The single components resulting from the fit are also plot- ted separately (dashed and dashed-dotted lines). As these estimates show, our conclusions do not rule out at all the existence, on a smaller scale, of an accretion disk around HW2, which in fact is to be expected, based on the very pres- ence of the HW2 jet. In fact, recent 7-mm VLA observations of SO2 have led Jiménez-Serra et al. (2007) to claim the de- tection of a disk-like structure with a size of 600 × 100 AU, roughly centered on the HW2 position, part of which may be photoevaporating. Although spatially almost coexistent on the plane of the sky, this structure is characterized by a different vlsr (−7.3 km s −1, as opposed to ∼ −5 km s−1) and apparently a different chemistry from the molecular gas traced by CH3CN. Our above estimates on the nature of the black-body emission in the (sub)mm regime are all consistent with Jiménez-Serra et al.’s conclusions. Overall, the Cepheus A HW2 allows, due to its proxim- ity, a view into the heart of a massive star forming region. The emerging picture is anything but simple: including the sources detected by Curiel et al. (2002), and the hot cores HC, HC2 and HC3, at least 6 probable young stellar or protostellar ob- jects are located within a radius of 1′′ or 725 AU. It remains an open issue whether, under such circumstances we can expect to observe a classical accretion disk feeding a single central star, or rather some kind of circum-cluster disk or ring-like structure (analogous perhaps to circumbinary rings like the one around GG Tau, Guilloteau et al. 1999), and what such a structure may look like, both from a morphological and from a kinematical point of view. Higher spatial resolution is needed, but the chal- lenge is to identify the right chemical tracer to investigate the structures one is interested in. CH3CN, otherwise considered a reasonably good disk tracer (e.g. for the disk in IRAS 20126, Cesaroni et al. 1997), does not seem to trace the disk-like struc- ture seen by Jiménez-Serra et al. (2007) at all. Another issue is the physical location of the −10 km s−1 molecular component. Though it seems likely that the peak of −10-km s−1 CH3CN emission is associated with the power- ing source of the small-scale SiO outflow, its connection to the somewhat more extended molecular emission at this systemic velocity (cf. Brogan et al. 2007) remains to be confirmed. Acknowledgements. The authors are grateful to the IRAM staff in Grenoble, particularly to H. Wiesemeyer, J. M. Winters and R. Neri, for their support in the data calibration process. An anonymous referee has given a significant contribution to the improvement of this paper. CC and PS have enjoyed many fruitful discussions with Malcolm Walmsley. JMP and IJS acknowledge the support pro- vided through projects number ESP2004-00665 and S-0505/ESP- 0277 (ASTROCAM). 6 C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region References Beckwith, S. V. W., Sargent, A. I., Chini, R. S., & Guesten, R. 1990, AJ, 99, 924 Beuther, H., Schilke, P., Sridharan, T. K., Menten, K. M., Walmsley, C. M., & Wyrowski, F. 2002, A&A, 383, 892 Brogan, C., Hunter, T. R., Indebetouw, R., Shirley, Y. L., Chandler, C. J., Rao, R., Sarma, A. P. 2007, in Proceedings of Science with ALMA: A new era for Astrophysics, Ed. R. Bachiller (Springer) Cesaroni, R., Felli, M., Testi, L., Walmsley, C. M., & Olmi, L. 1997, A&A, 325, 725 Comito, C., Schilke, P., Phillips, T. G., Lis, D. C., Motte, F., & Mehringer, D. 2005, ApJS, 156, 127 Curiel, S., et al. 2002, ApJ, 564, L35 Curiel, S., et al. 2006, Astrophys. J., 638, 878 Gómez, J. F., Sargent, A. I., Torrelles, J. M., Ho, P. T. P., Rodrı́guez, L. F., Cantó, J., & Garay, G. 1999, Astrophys. J., 514, 287 (G99) Guilloteau, S., Dutrey, A., & Simon, M. 1999, A&A, 348, 570 Jiménez-Serra, I., et al. 2007, Astrophys. J., submitted Johnson, H. L. 1957, Astrophys. J., 126, 121 Martı́n-Pintado, J., Jiménez-Serra, I., Rodrı́guez-Franco, A., Martı́n, S., & Thum, C. 2005, ApJ, 628, L61 (MP05) Patel, N. A., et al. 2005, Nature, 437, 109 Reynolds, S. P. 1986, Astrophys. J., 304, 713 Rodriguez, L. F., Garay, G., Curiel, S., Ramirez, S., Torrelles, J. M., Gomez, Y., & Velazquez, A. 1994, ApJ, 430, L65 Sridharan, T. K., Williams, S. J., & Fuller, G. A. 2005, ApJL, 631, L73 Stahler, S. W., Palla, F., & Ho, P. T. P. 2000, Protostars and Planets IV, Torrelles, J. M., Gomez, J. F., Rodriguez, L. F., Curiel, S., Ho, P. T. P., & Garay, G. 1996, ApJ, 457, L107 Torrelles, J. M., Gómez, J. F., Garay, G., Rodrı́guez, L. F., Miranda, L. F., Curiel, S., & Ho, P. T. P. 1999, Monthly Notices Roy. Astron. Soc., 307, 58 Introduction Observations Results CH3CN Discussion

Due to its relatively small distance (725 pc), the Cepheus A East star-forming region is an ideal laboratory to study massive star formation processes. Based on its morphology, it has been suggested that the flattened molecular gas distribution around the YSO HW2 may be a 350-AU-radius massive protostellar disk. Goal of our work is to ascertain the nature of this structure. We have employed the Plateau de Bure Interferometer to acquire (sub-)arcsecond-resolution imaging of high-density and shock tracers, such as methyl cyanide (CH3CN) and silicon monoxide (SiO), towards the HW2 position. On the 1-arcsecond (about 725 AU) scale, the flattened distribution of molecular gas around HW2 appears to be due to the projected superposition, on the plane of the sky, of at least three protostellar objects, of which at least one is powering a molecular outflow at a small angle with respect to the line of sight. The presence of a protostellar disk around HW2 is not ruled out, but such structure is likely to be detected on a smaller spatial scale, or using different molecular tracers.

Introduction Several theories are being considered to explain the forma- tion of massive (M ≥ 8 M⊙) stars, which can be roughly grouped into accretion-driven and coalescence-driven models (cf. Stahler et al. 2000). In the latter case, high-mass stars would form by merging of two or more lower-mass objects, making the presence of stable massive accretion disks around the protostar very unlikely. However, only models based on disk-protostar interactions are capable of explaining the exis- tence of jets and outflows: hence, the high incidence, in large samples of massive YSOs, of highly collimated outflows (cf. Beuther et al. 2002) has been interpreted as indirect evidence for the existence of high-mass disks. It is undoubted that the direct detection of accretion onto massive protostars through rotating disks constitutes an im- portant tile in the massive-star-formation-theory mosaic. From an observational point of view, this task is made very difficult by two factors: i) massive star-forming regions typically are far away, a few kpc on average, making the direct observa- tion of small-scale structure such as disks virtually impossible with current instruments; and ii), massive stars form in clus- Send offprint requests to: C. Comito ters, making the surrounding region extremely complex, both spatially and kinematically. Located only ∼ 725 pc from the Sun (Johnson 1957), Cepheus A is considered a very promising candidate for the detection of a massive disk. Its well-studied bipolar outflow (cf. Gómez et al. 1999, hereafter G99, and references therein) is thought to be powered by the radio-continuum source HW2 (∼ 104 L⊙, Rodrı́guez et al. 1994). Curiel et al. (2006) report the presence of very large tangential velocities in the HW2 ra- dio jet, consistent with HW2 being a massive Young Stellar Object (YSO). The distribution of H2O masers (Torrelles et al. 1996) and of the SiO emission (G99) around HW2, both oriented perpendicularly with respect to the direction of the flow, have been interpreted as strongly supporting the existence of accretion shocks onto a rotating and contracting molecu- lar disk of ∼ 700-AU diameter, centered on HW2, with the northeast-southwest outflow being triggered by the interaction between such disk and HW2 itself. Similar conclusions have been reached by Patel et al. (2005), based on SMA observa- tions of CH3CN and dust emission. However, the fact that the HW2 vicinities are crowded with YSOs (at least three within an area of 0.′′6 × 0.′′6, Curiel et al. 2002), together with the re- cent detection of an internally heated hot core within 0.′′4 from the center of the outflow (Martı́n-Pintado et al. 2005, hereafter http://arxiv.org/abs/0704.1617v1 2 C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region MP05) cast some doubts on this interpretation. Based on our PdBI observations, we conclude that, on the 1′′ scale, the elon- gated molecular structure around HW2 can be explained with the superposition, on the plane of the sky, of at least three dif- ferent hot-core-type sources, at least one of them being the ex- citing source for a second molecular outflow. 2. Observations In 2003 and 2004, with the Plateau de Bure Interferometer, we have carried out observations of several high-density and shock tracers (also cf. Schilke et al., in prep.), among which silicon monoxide (SiO) and methyl cyanide (CH3CN), towards the HW2 position (αJ2000 = 22 h56m17.9s, δJ2000 = +62 ◦01′49.6′′). A combination of high-spectral-resolution correlator units were employed to achieve a channel width ∆v of up to ∼ 0.3 km s−1. The five antennas in AB (extended) configuration provided a HPBW of 2′′ × 1.′′6 for SiO(2-1) at 86 GHz, and of 0.′′9 × 0.′′7 for CH3CN(12 − 11) at 220 GHz. The data cubes were pro- duced with natural weighting. All maps have been CLEANed. Analysis of all molecular spectra has been performed after sub- traction of the continuum emission. 3. Results Fig. 1 (left panel) shows the Cep-A star-forming region within a 1100-AU radius from HW2. The peak of the 241-GHz dust emission (grey scale) roughly coincides with the HW2 posi- tion and with the center of the large-scale outflow. The inte- grated CH3CN emission is also centered on HW2 (contours), and somewhat elongated almost perpendicularly to the direc- tion of the large-scale outflow. Like other molecular tracers (cf. Brogan et al. 2007), CH3CN displays two different velocity components, centered around −5 and −10 km s−1 respectively. The solid contours in Fig. 1, center panel, show the emission of the CH3CN(123−113) transition, integrated between −7 and −3 km s−1, whereas the emission in the range between −11.5 and −7.5 km s−1 is represented by the dashed contours (see § 3.2). The center of SiO emission, instead, is at ∼ −10 km s−1. Silicon monoxide peaks about 0.′′4 eastwards of HW2, at a po- sition that coincides with the HC source of MP05 (triangle in Fig. 1, see § 3.1), close to the −10-km s−1 CH3CN component. In what follows, we will discuss in more detail the SiO and CH3CN data. 3.1. SiO Our dataset confirms that the spatial distribution of this shock tracer is mainly concentrated in the HW2 region (its presence in the large-scale outflow is limited to a few bullets at large dis- tances from the center), although not centered on the HW2 po- sition. This does indeed suggest that shock processes are taking place in the (projected) immediate vicinities of HW2. However, if the SiO emission were arising from accretion shocks onto a rotating disk (as proposed by G99), we would expect to ob- serve a similar velocity structure to that observed for the other molecular tracers peaking around HW2. Instead, SiO seems to be tracing a completely different kinematic component: unlike any other line in our dataset, the SiO(2-1) line has a velocity spread of at least 35 km s−1at the zero-flux level (∼ 15 km s−1 FWHM). A mass of about 90 M⊙ would be required to pro- duce such large line width in a gravitationally bound enviro- ment (assuming virial equilibrium, and that the emission arises in a region of ∼ 350-AU radius). This value is about one order of magnitude larger than the estimated mass of HW2, which is expected to become a B0.5 star once in ZAMS (Rodriguez et al. 1994). Fig. 2 shows a comparison between SiO(2-1) and CH3CN(124 − 114). We carried out a two-dimensional Gaussian fit of the SiO(2- 1) spatial distribution, for every spectral channel in the velocity range −25 < vlsr −3.5 km s −1. This corresponds to the spec- tral interval in which the signal-to-noise (S/N) ratio of the SiO transition is ≥ 9σ. For smaller S/N ratios, in fact, the error on the fitted centroid position easily exceeds 50%. The result is a distribution of the centroids of SiO emission as a function of velocity. Fig. 1, right panel, shows that the centroid positions are located in a well-defined two-lobed area, centered about 0.′′4 eastwards of HW2 and of the dust continuum emission peak. Although the error on every single centroid position is still relatively large (up to 30%), as a whole their distribution describes a very clear velocity trend, with all the emission at vlsr< −10 km s −1 clustering in the left lobe, and all the emis- sion at vlsr> −10 km s −1 clustering in the right lobe. This re- sult suggests that a second molecular outflow is being ejected in the HW2 region. Our interpretation is supported by the re- cent discovery of an intermediate-mass protostar, surrounded by the hot molecular core HC (MP05) located in the region be- tween the blue and red lobes of SiO emission (white triangle in Fig. 1, right panel), hence a very likely candidate to be its powering engine. With the current dataset it is not possible to establish the exact inclination angle of the flow, but the large velocity spread observed in the SiO(2-1) line, together with the relatively concentrated spatial distribution of the SiO emission, suggests that the inclination angle must be high, i.e., that the SiO flow is being ejected at a small angle with respect to the line of sight. 3.2. CH3CN The dense molecular gas, as traced by CH3CN(123-113) (Fig. 1, left panel) appears to be distributed around the HW2 position, and elongated in a direction roughly perpendicular to the projected direction of the large-scale outflow on the plane of the sky. From a morphological point of view, therefore, the data are very suggestive of the presence of a ∼ 350-AU-radius disk-like structure around HW2. On the other hand, as already pointed out by Torrelles et al. (1999) based on VLA observations of NH3, the kinemati- cal picture describing the dense gas distribution in the region is quite complex. A position-velocity cut along the major axis of the elongated structure (indicated in Fig. 1, left and center panels, with a dashed line) reveals a velocity spread of about 6 km s−1 (see Fig. 3), also observed by Patel et al. (2005). However, the two intensity peaks along the axis share roughly C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region 3 Fig. 1. All panels: the grey levels represent the continuum emission at 241 GHz. Lowest level is 3.3 mJy/beam or 2σ, highest is 22σ. The HW2 position is indicated with a white star. The solid, crossing lines show the opening angle of the large-scale northeast-southwest outflow, inferred from HCN and 13CO observations obtained with PdBI together with the core-tracing data. The contours trace the continuum-subtracted CH3CN emission at 220 GHz, and in the top left corner, the HPBW for the con- tinuum (grey foreground) and CH3CN (black background ellipse) are shown. The position of the intermediate-mass protostar HC (MP05) is indicated with a white triangle. Left panel: the integrated emission of the CH3CN(123-113) is shown in solid contours. Center panel: integrated intensity of CH3CN(123-113) between −7 and −3 km s −1 (solid contours) and between −11.5 and −7.5 km s−1 (dashed contours). The dotted square box indicates the area enlarged in the right panel. Right panel: The circles show the centroid positions of the SiO(2-1) emission for every channel in the range −25 < vlsr −3.5 km s −1, with a channel width of 0.5 km s−1. The black circles stand for the blue-shifted (vlsr < −10 km s −1), white circles for the red-shifted (vlsr > −10 km s emission. Fig. 2. Comparison between the line profiles of SiO(2-1) (grey spectrum) and CH3CN(124 − 114) (transparent spectrum). At vlsr=∼ −10 km s −1, the center of the SiO emission coincides with the position of source HC of MP05 (cf. § 3.2 and Fig. 1). The SiO(2-1) spectrum arises from a region of about 1′′ ra- dius around the HC position (50%-of-peak-emission level). The CH3CN emission arises from the HC3 position (cf. § 3.2 and Fig. 4), and has been scaled by a factor 0.5 for a better visual comparison. No recognizable counterpart to the ∼ −5- km s−1 CH3CN component is observed for SiO. the same systemic velocity (∼ −5 km s−1). The weaker, blue- shifted component of emission (∼ −10 km s−1), appears to trace rather the outskirt of a physically separated component than a rotation-induced velocity gradient along the axis of the alleged “disk”. The peak of the -10 km s−1 CH3CN emission is spa- tially and kinematically close to the center of the small-scale SiO outflow (see Fig. 1, right panel), it is likely associated to it and/or to its exciting source. The CH3CN integrated intensity is dominated by the two −5-km s−1 peaks, which lie respectively about 0.′′6 to the north- west, and 0.′′5 to the southeast of the HW2 position. The fact that the two CH3CN peaks share roughly the same systemic velocity, is not compatible with the “rotating disk” hypothesis. In what follows, we will treat them as independent condensa- tions, and to be consistent with the nomenclature introduced by MP05, we will refer to them respectively as HC2 and HC3. Fig. 4 compares the spectra observed towards the two positions. It is clear that, in both cases, both the −5- and −10-km s−1 com- ponents are present along the line of sight, although the contri- bution from the latter is more substantial towards HC3, i.e., close to the peak of the −10-km s−1 SiO emission. We have assumed the LTE approximation to fit the physical parameters associated with the two different velocity compo- nents. All transitions in the spectrum are fitted simultaneously, in order to take line blending and optical depth effects properly into account (a detailed description of the method can be found in Comito et al. 2005). For the ∼ −5 km s−1component, our fit reveals that the k= 0 through k= 4 transitions are optically thick towards both positions. The data at this velocity can only be reproduced by including a very compact, hot, dense object in the model. The emission centered at ∼ −10-km s−1 can be modeled with a cooler, more extended component. The results of the fit, for the two positions, are summarized in Tab. 1. Although the presence of secondary minima in the χ2 space is unavoidable when so many parameters are varied to achieve 4 C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region Fig. 3. Position-velocity plot for CH3CN(123 − 113), along the major axis of the elongated structure (dashed line in Fig. 1, left and center panels). Levels range from 3σ to 27σ in 3-σ steps. The bracketed velocity ranges show the intervals making up the −5 (solid) and −10 (dashed) km s−1 components, whose spatial distribution is shown in Fig. 1, center panel. minimization, in this case the simultaneous fitting of intensity ratios between optically thick and optically thin lines, between ortho- and para-CH3CN transitions, and between 12C and 13C isotopologues of methyl cyanide (see Fig. 4), places very strin- Fig. 4. In grey, high-resolution (∆v = 0.3 km s−1) spectra of the CH3CN(12-11) emission at 200 GHz, towards the HC2 and HC3 positions (see Fig. 1, center panel). Overlayed in black are the model spectra, resulting from the parameters listed in Tab. 1, assuming LTE approximation and that [12CO]/[13CO]= gent constraints on the viable parameter space, at least as far as the compact (∼ −5 km s−1) component is concerned. Note that the rotational temperatures derived in this fashion are sig- nificantly higher than those derived by Patel et al. (2005). This discrepancy can be explained with the optical depth correction in our fit. 4. Discussion The observed elongation of the molecular gas distribution around HW2, over a radius of ∼ 0.′′5 (∼ 360 AU), appears to be due to the projected superposition, on the plane of the sky, of at least three protostellar objects, of which at least one is triggering a molecular outflow at a small angle with respect to the line of sight (§ 3.1, § 3.2). All lines in our dataset are con- sistent with this interpretation. The distribution of molecular gas around HW2 can, on a 1′′ scale, be interpreted as a clus- ter of high- and intermediate-mass protostars in the Cepheus A HW2 region. The analysis of the CH3CN spectra (§ 3.2) sug- gests the presence of internally heated compact hot-core-type objects like HC, likely hosting protostellar objects, although the 1-mm continuum emission peaks on the HW2 position and does not show any secondary clumps. This may be due to in- sufficient dynamic range in our data, if the contribution, to the 241-GHz continuum, of free-free emission from the HW2 ther- mal jet is large. Fig. 5 shows the variation of the measured HW2 continuum flux density as a function of frequency, S ν, between 1.5 and 327 GHz (data points from: Rodrı́guez et al. 1994; this work; Patel et al. 2005). A two-component least- squares fit of the data yields S jet ∝ ν (0.51±0.12) (consistent with the value inferred by Rodrı́guez et al. 1994, and with the theo- retical predictions for the radio continuum spectrum of a con- fined thermal jet, Reynolds 1986) and S (sub)mm ∝ ν (1.92±0.12) (dashed and dashed-dotted lines respectively in Fig. 5), where S jet + S (sub)mm = S ν (solid curve in Fig. 5). Based on this esti- mate, the thermal jet (free-free) contribution at 241 GHz should be ∼ 10% of the total flux. However, the flux density variation in the (sub)mm- wavelength portion of the spectrum increases basically on a Rayleigh-Jeans slope, suggesting the presence of optically thick emission from an unresolved (with our best spatial res- olution, at 241 GHz with PdBI) continuum source, whose size thus cannot be larger than ∼ 0.′′6. Although we cannot determine the nature of this compact source, it makes sense to hypothesize that it can be described by i), dust emission, and/or ii), free-free emission from a Hii region (for example associated to a photoevaporating disk). For case i), we adopt Beckwith et al.’s (1990) values for the mass absorption coeffi- cient, κν = 0.1(ν/10 12Hz)β, with β = 1, to estimate the lower mass limit for an object with size θdust = 0. ′′5 (∼ 363 AU) to produce optically thick emission at 87 GHz: Mτdust=1 ≥ 1 M⊙. Based on the peak flux at 87, 241 (this work) and 327 GHz (Patel et al. 2005), the above value of θdust yields a brightness temperature, TB ≃ 80 K. Smaller source sizes would lead to higher intrinsic temperatures and lower mass limits. In case ii), we assume TB = 10 4 K (typical for Hii regions), which would translate into a source size of θfree−free ≃ 0. ′′04 (∼ 30 AU). The continuum source VLA-mm (Curiel et al. 2002), which is lo- C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region 5 vlsr Source size N(CH3CN) Trot ∆v (km s−1) (cm−2) (K) (km s−1) HC2 -4.5 0.′′3 ∼ 3 × 1016 250 2.9 -8.9 1′′ ∼ 8 × 1014 150 4.0 HC3 -4.2 0.′′25 ∼ 3 × 1016 250 3.2 -10.0 0.′′45 ∼ 5 × 1015 150 4.5 Table 1. LTE model results for the CH3CN emission towards the HC2 and HC3 cores (Fig. 4). For a discussion on the error estimate, cf. Comito et al. 2005. cated ∼ 0.′′15 south of HW2 and whose size cannot be larger than 30 AU, displays a much too weak emission at cm and mm wavelengths, and we estimate its contribution to cover for at most 5% of the observed (sub)mm flux. In other words, we cannot discard any of the two hypothe- ses for the observed optically thick continuum emission. At higher frequencies, the spectral index would get flatter if the optically thick emission were only due to free-free emission, while it would remain the same for dust. Observations with a resolution of < 0.5′′, which are within the reach of present day interferometers, would be able to shed more light on the nature of this object. Fig. 5. Flux density as a function of frequency for Cep-A HW2. For the Rodrı́guez et al. (1994, triangles) and Patel et al. (2005, square) data points, the errorbars fall within the sym- bols and are therefore not visible. The solid curve shows the 2-component least-squares fit of the data, as descrived in the text. The single components resulting from the fit are also plot- ted separately (dashed and dashed-dotted lines). As these estimates show, our conclusions do not rule out at all the existence, on a smaller scale, of an accretion disk around HW2, which in fact is to be expected, based on the very pres- ence of the HW2 jet. In fact, recent 7-mm VLA observations of SO2 have led Jiménez-Serra et al. (2007) to claim the de- tection of a disk-like structure with a size of 600 × 100 AU, roughly centered on the HW2 position, part of which may be photoevaporating. Although spatially almost coexistent on the plane of the sky, this structure is characterized by a different vlsr (−7.3 km s −1, as opposed to ∼ −5 km s−1) and apparently a different chemistry from the molecular gas traced by CH3CN. Our above estimates on the nature of the black-body emission in the (sub)mm regime are all consistent with Jiménez-Serra et al.’s conclusions. Overall, the Cepheus A HW2 allows, due to its proxim- ity, a view into the heart of a massive star forming region. The emerging picture is anything but simple: including the sources detected by Curiel et al. (2002), and the hot cores HC, HC2 and HC3, at least 6 probable young stellar or protostellar ob- jects are located within a radius of 1′′ or 725 AU. It remains an open issue whether, under such circumstances we can expect to observe a classical accretion disk feeding a single central star, or rather some kind of circum-cluster disk or ring-like structure (analogous perhaps to circumbinary rings like the one around GG Tau, Guilloteau et al. 1999), and what such a structure may look like, both from a morphological and from a kinematical point of view. Higher spatial resolution is needed, but the chal- lenge is to identify the right chemical tracer to investigate the structures one is interested in. CH3CN, otherwise considered a reasonably good disk tracer (e.g. for the disk in IRAS 20126, Cesaroni et al. 1997), does not seem to trace the disk-like struc- ture seen by Jiménez-Serra et al. (2007) at all. Another issue is the physical location of the −10 km s−1 molecular component. Though it seems likely that the peak of −10-km s−1 CH3CN emission is associated with the power- ing source of the small-scale SiO outflow, its connection to the somewhat more extended molecular emission at this systemic velocity (cf. Brogan et al. 2007) remains to be confirmed. Acknowledgements. The authors are grateful to the IRAM staff in Grenoble, particularly to H. Wiesemeyer, J. M. Winters and R. Neri, for their support in the data calibration process. An anonymous referee has given a significant contribution to the improvement of this paper. CC and PS have enjoyed many fruitful discussions with Malcolm Walmsley. JMP and IJS acknowledge the support pro- vided through projects number ESP2004-00665 and S-0505/ESP- 0277 (ASTROCAM). 6 C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region References Beckwith, S. V. W., Sargent, A. I., Chini, R. S., & Guesten, R. 1990, AJ, 99, 924 Beuther, H., Schilke, P., Sridharan, T. K., Menten, K. M., Walmsley, C. M., & Wyrowski, F. 2002, A&A, 383, 892 Brogan, C., Hunter, T. R., Indebetouw, R., Shirley, Y. L., Chandler, C. J., Rao, R., Sarma, A. P. 2007, in Proceedings of Science with ALMA: A new era for Astrophysics, Ed. R. Bachiller (Springer) Cesaroni, R., Felli, M., Testi, L., Walmsley, C. M., & Olmi, L. 1997, A&A, 325, 725 Comito, C., Schilke, P., Phillips, T. G., Lis, D. C., Motte, F., & Mehringer, D. 2005, ApJS, 156, 127 Curiel, S., et al. 2002, ApJ, 564, L35 Curiel, S., et al. 2006, Astrophys. J., 638, 878 Gómez, J. F., Sargent, A. I., Torrelles, J. M., Ho, P. T. P., Rodrı́guez, L. F., Cantó, J., & Garay, G. 1999, Astrophys. J., 514, 287 (G99) Guilloteau, S., Dutrey, A., & Simon, M. 1999, A&A, 348, 570 Jiménez-Serra, I., et al. 2007, Astrophys. J., submitted Johnson, H. L. 1957, Astrophys. J., 126, 121 Martı́n-Pintado, J., Jiménez-Serra, I., Rodrı́guez-Franco, A., Martı́n, S., & Thum, C. 2005, ApJ, 628, L61 (MP05) Patel, N. A., et al. 2005, Nature, 437, 109 Reynolds, S. P. 1986, Astrophys. J., 304, 713 Rodriguez, L. F., Garay, G., Curiel, S., Ramirez, S., Torrelles, J. M., Gomez, Y., & Velazquez, A. 1994, ApJ, 430, L65 Sridharan, T. K., Williams, S. J., & Fuller, G. A. 2005, ApJL, 631, L73 Stahler, S. W., Palla, F., & Ho, P. T. P. 2000, Protostars and Planets IV, Torrelles, J. M., Gomez, J. F., Rodriguez, L. F., Curiel, S., Ho, P. T. P., & Garay, G. 1996, ApJ, 457, L107 Torrelles, J. M., Gómez, J. F., Garay, G., Rodrı́guez, L. F., Miranda, L. F., Curiel, S., & Ho, P. T. P. 1999, Monthly Notices Roy. Astron. Soc., 307, 58 Introduction Observations Results CH3CN Discussion

A Renormalization group approach for highly anisotropic 2D Fermion systems: application to coupled Hubbard chains S. Moukouri Department of Physics and Michigan Center for Theoretical Physics University of Michigan, 2477 Randall Laboratory, Ann Arbor MI 48109 I apply a two-step density-matrix renormalization group method to the anisotropic two- dimensional Hubbard model. As a prelude to this study, I compare the numerical results to the exact one for the tight-binding model. I find a ground-state energy which agrees with the exact value up to four digits for systems as large as 24 × 25. I then apply the method to the interact- ing case. I find that for strong Hubbard interaction, the ground-state is dominated by magnetic correlations. These correlations are robust even in the presence of strong frustration. Interchain pair tunneling is negligible in the singlet and triplet channels and it is not enhanced by frustration. For weak Hubbard couplings, interchain non-local singlet pair tunneling is enhanced and magnetic correlations are strongly reduced. This suggests a possible superconductive ground state. I. INTRODUCTION Quasi-one dimensional organic1 and inorganic2 mate- rials have been the object of an important theoretical in- terest for the last three decades. The essential features of their phase diagram may be captured by the anisotropic Hubbard model (AHM), H = −t‖ i,l,σ i,l,σci+1,l,σ + h.c.) + U ni,l,↑ni,l,↓ i,l,σ ni,l,σni+1,l,σ − µ i,l,σ ni,l,σ i,l,σ i,l,σci,l+1,σ + h.c.). (1) or a more general Hubbard-like model including longer range Coulomb interactions. The indices i and l label the sites and the chains respectively. For these highly anisotropic materials, t⊥ ≪ t‖. Over the years, the AHM has remained a formidable challenge to condensed- matter theorists. Some important insights on this model or its low energy version, the g-ology model, have been obtained through the work of Bourbonnais and Caron3,4 and others. They used a perturbative renormalization group approach to analyze the crossover from 1D to 2D at low temperatures. More recently, Biermann et al.5 ap- plied the chain dynamical mean-field approach to study the crossover from Luttinger liquid to Fermi liquid in this model. Despite this important progress, crucial informa- tion such as the ground-state phase diagram, or most no- tably, whether the AHM displays superconductivity, are still unknown. So far it has remained beyond the reach of numerical methods such as the exact diagonalization (ED) or the quantum Monte Carlo (QMC) methods. ED cannot exceed lattices of about 4 × 5. It is likely to re- main so for many years unless there is a breakthrough in quantum computations. The QMC method is plagued by the minus sign problem and will not be helpful at low temperatures. The small value of t⊥ implies that, in or- der to see the 2D behavior, it will be necessary to reach lower temperatures than those usually studied for the isotropic 2D Hubbard model. Hence, even in the absence of the minus sign problem, in order to work in this low temperature regime, the QMC algorithm requires spe- cial stabilization schemes which lead to prohibitive cpu time.6 II. TWO-STEP DMRG I have shown in Ref. 7 that this class of anisotropic models may be studied using a two-step density-matrix renormalization group (TSDMRG) method. The TS- DMRG method is a perturbative approach in which the standard 1D DMRG is applied twice. In the first step, the usual 1D DMRG method9 is applied to find a set of low lying eigenvalues ǫn and eigenfunctions |φn〉 of a single chain. In the second step, the 2D Hamiltonian is then projected onto the basis constructed from the tensor product of the |φn〉’s. This projection yields an effective one-dimensional Hamiltonian for the 2D lattice, E‖[n]|Φ‖[n]〉〈Φ‖[n]| − t⊥ i,l,σ i,l,σ c̃i,l+1,σ + h.c.)(2) where E‖[n] is the sum of eigenvalues of the different chains, E‖[n] = l ǫnl ; |Φ‖[n]〉 are the corresponding eigenstates, |Φ‖[n]〉 = |φn1〉|φn2〉...|φnL〉; c̃ i,l,σ, c̃i,l,σ, and ñi,l,σ are the renormalized matrix elements in the single chain basis. They are given by i,l,σ) nl,ml = (−1)ni〈φnl |c i,l,σ|φml〉, (3) (c̃i,l,σ) nl,ml = (−1)ni〈φnl |ci,l,σ|φml〉, (4) (ñi,l,σ) nl,ml = 〈φnl |ni,l,σ|φml〉, (5) where ni represents the total number of fermions from sites 1 to i− 1. For each chain, operators for all the sites are stored in a single matrix http://arxiv.org/abs/0704.1618v1 l,σ = (c̃ 1,l,σ, ..., c̃ L,l,σ), (6) c̃l,σ = (c̃1,l,σ, ..., c̃L,l,σ), (7) ñl,σ = (ñ1,l,σ, ..., ñL,l,σ). (8) Since the in-chain degrees of freedom have been inte- grated out, the interchain couplings are between the block matrix operators in Eq.( 6, 7) which depend only on the chain index l. In this matrix notation, the effec- tive Hamiltonian is one-dimensional and it is also studied by the DMRG method. The only difference compared to a normal 1D situation is that the local operators are now ms2 ×ms2 matrices, where ms2 is the number of states kept during the second step. The two-step method has previously been applied to anisotropic two-dimensional Heisenberg models.7 In Ref. 8, it was applied to the t− J model but due to the absence of an exact result in certain limits, it was tested against ED results on small ladders only. A systematic analysis of its performance on a fermionic model on 2D lattices of various size has not been done. In this paper, as a prelude to the study of the AHM, I will apply the TSDMRG to the anisotropic tight-binding model on a 2D lattice, i.e., model (1) with U = V = 0. I perform a comparison with the exact result of the tight-binding model. I was able to obtain agreement for the ground- state energies on the order of 10−4 for lattices of up to 24 × 25. I then discuss how these calculations may be extended to the interacting case, before presenting the U 6= 0 results. III. WARM UP: THE TIGHT-BINDING MODEL The tight-binding Hamiltonian is diagonal in the mo- mentum space, the single particle energies are, ǫk = −2t‖coskx − 2t⊥cosky − µ, (9) with k = (kx, ky), kx = nxπ/(Lx+1) and ky = nyπ/(Ly+ 1) for open boundary conditions (OBC); Lx, Ly are re- spectively the linear dimensions of the lattice in the par- allel and transverse directions. The ground-state energy of an N electron system is obtained by filling the low- est states up to the Fermi level, E[0](N) = However in real space, this problem is not trivial and it constitutes, for any real space method such as the TS- DMRG, a test having the same level of difficulty as the case with U 6= 0. This is because the term involving U is diagonal in real space and the challenge of diagonalizing the AHM arises from the hopping term. I will study the tight-binding model at quarter filling, N/LxLy = 1/2, the nominal density of the organic con- ductors known as the Bechgaard salts. Systems of up to Lx×Ly = L×(L+1) = 24×25 will be studied. During the first step, I keep enough states (ms1 is a few hundred) so 0 20 40 60 80 100 FIG. 1: Low-lying states of the 1D tight-binding model (full line) and of the 1D Heisenberg spin chain (dotted line) for L = 16 and ms2 = 96. that the truncation error ρ1 is less than 10 −6. I target the lowest state in each charge-spin sectorsNx±2, Nx±1, Nx and Sz ± 1, Sz ± 2, Nx is the number of electrons within the chain. It is fixed such that Nx/Lx = 1/2. There is a total of 22 charge-spin states targeted at each iteration. For the tight-binding model, the chains remain discon- nected if t⊥ < ǫ0(Nx + 1) − ǫ0(Nx) or t⊥ < ǫ0(Nx) − ǫ0(Nx − 1), where Nx is the number of electons on sin- gle chain. In order to observe transverse motion, it is necessary that at least t⊥ >∼ ǫ0(Nx + 1) − ǫ0(Nx) and t⊥ >∼ ǫ0(Nx)− ǫ0(Nx − 1). These two conditions are sat- isfied only if µ is appropietly chosen. The values listed in Table (I) corresponds to µ = (ǫ0(Nx+1)−ǫ0(Nx−1))/2. This treshold varies with L. I give in Table (I) the val- ues of t⊥ chosen for different lattice sizes. In principle, for the TSDMRG to be accurate, it is necessary that ∆ǫ = ǫnc − ǫ0, where ǫnc is the cut-off, be such that ∆ǫ/t⊥ ≫ 1. But in practice, I find that I can achieve accuracy up to the fourth digit even if ∆ǫ/t⊥ ≈ 5 using the finite system method. Five sweeps were necessary to reach convergence. Note that this conclusion is some- what different from my earlier estimate of ∆ǫ/t⊥ ≈ 10 for spin systems.8 This is because in Ref. 8, I used the infinite system method during the second step. The ultimate success of the TSDMRG depends on the density of the low-lying states in the 1D model. For fixed ms2 and L, it is, for instance, easier to reach larger ∆ǫ/J⊥ in the anisotropic spin one-half Heisen- berg model, studied in Ref. 7, than ∆ǫ/t⊥ for the tight- binding model as shown in Fig.1. For L = 16, ms2 = 96, and J⊥ = t⊥ = 0.15, I find that ∆ǫ/J⊥ ≈ 10, while ∆ǫ/t⊥ ≈ 5. Hence, the TSDMRG method will be more accurate for a spin model than for the tight-binding model. Using the infinite system method during the second step on the anisotropic Heisenberg model with J⊥ = 0.1, I can now reach an agreement of about 10 with the stochastic QMC method. Two possible sources of error can contribute to reduce the accuracy in the TSDMRG with respect to the conven- 8× 9 16× 17 24× 25 t⊥ 0.28 0.15 0.1 µ -1.2660 -1.3411 -1.3657 ∆ǫ/t⊥ 6.42 5.40 5.78 TABLE I: Transverse hopping and chemical potential used in the simulations for different lattice sizes ms2 8× 3 16× 3 24× 3 64 -0.241524 -0.211929 0.204040 Exact -0.241524 -0.211931 0.204049 TABLE II: Ground-state energies of three-leg ladders. tional DMRG. They are the truncation of the superblock from 4×ms1 states to onlyms2 states and the use of three blocks instead of four during the second step. In Table (II) I analyze the impact of the reduction of the number of states to ms2 for three-leg ladders. The choice of three- leg ladders is motivated by the fact that at this point, the TSDMRG is equivalent to the exact diagonalization of three reduced superblocks. It can be seen that as far as t⊥ >∼ ǫ0(Nx+1)−ǫ0(Nx) and t⊥ >∼ ǫ0(Nx)−ǫ0(Nx−1), the TSDMRG at this point is as accurate as the 1D DMRG. Note that the accuracy remains nearly the same irrespec- tive of L as far as the ratio ∆ǫ/t⊥ remains nearly con- stant. Since ∆ǫ decreases when L increases, t⊥ must be decreased in order to keep the same level of accuracy for fixed ms2. In principle, following this prescription, much larger systems may be studied. ∆ǫ/t⊥ does not have to be very large, in this case it is about 5, to obtain very good agreement with the exact result. The second source of error is related to the fact that the effective single site during the second step is now a chain having ms2 states, I am thus forced to use three blocks instead of four to reduce the computational bur- den. In Table (III), it can be seen that this results in a reduction in accuracy of about two orders of magnitude with respect to those of three leg-ladders. These results are nevertheless very good given the relatively modest computer power involved. All calculations were done on a workstation. The DMRG is less accurate when three blocks are used instead of four. This can be understood by applying the following view on the formation of the reduced density matrix. The construction of the reduced density matrix ms2 8× 9 16× 17 24× 25 64 -0.24761 -0.21401 0.20504 100 -0.24819 -0.21414 0.20509 120 -0.24832 -0.21419 Exact -0.24857 -0.21432 0.20519 TABLE III: Ground-state energies for different lattice sizes; a single state was targeted in the second step. may be regarded as a linear mapping uΨ : F ∗ → E, where E is the system, F is the environment and, F∗ is the dual space of F. Using the decomposition of the superblock wave function Ψ[0] = i ⊗ φRi , with φLi ∈ E and ΦRi ∈ F, for any φ∗ ∈ F ∗, 〈φ∗|φRi 〉φLi . (10) Let |k〉, k = 1, ...dimE and |l〉, l = 1, ...dimF be the basis of E and F respectively. Then, |l〉 has a dual basis 〈l∗| such that 〈l∗|l〉 = δl,l∗ . The matrix elements of uΨ in this basis are just the coordinates of the superblock wave function Φ[0]k,l . The rank r of this mapping, which is also the rank of the reduced density matrix is always smaller or equal to the smallest dimension of E or F, r < Min(dimE, dimF). Hence, if ms2 states are kept in the two external blocks, the number of non-zero eigenval- ues of ρ cannot be larger than ms2. Consequently, some states which have non-zero eigenvalues in the normal four block configuration will be missing. A possible cure to this problem is to target additional low-lying states above Ψ[0](N). The weight of these states in ρ must be small so that their role is simply to add the missing states not to be described accurately themselves. A larger weight on these additional states would lead to the reduction of the accuracy for a fixed ms2. In table (IV), I show the improved energies when, besides the ground state, I target the lowest states of the spin sectors Sz = −1 and Sz = +1 with N electrons. The weights were re- spectively 0.995, 0.0025, and 0.0025 for the three states. This lowers E[0](N) in all cases, but the gain does not appear to be spectacular. But I do not know whether this is due to my choice of perturbation of ρ or whether even the algorithm with four blocks would not yield bet- ter E[0](N). If the lowest sectors with N + 1 and N − 1 electrons which have Sz = ±0.5 are projected instead, I find that the results are similar to those with Sz = ±1 sectors, there are possibly many ways to add the missing states. A more systematic approach to this problem has recently been suggested.10 It is based on using a local perturbation to build a correction to the density matrix from the site at the edge of the system. Here, such a perturbation would be ∆ρ = αc l ρcl, where α is a con- stant, α ≈ 10−3 − 10−2, and c†l , cl are the creation and annihilation operators of the chain at the edge of the sys- tem. This type of perturbation resulted in an accuracy gain of more than an order of magnitude in the case of a spin chain.10 The three block method was found to be on par with the four block method. It will be interesting to see in a future study how this type of local perturbation performs within the TSDMRG. To conclude this section, as a first step to the investi- gation of interacting electron models, I have shown that the TSDMRG can successfuly be applied to the tight- binding model. The agreement with the exact result is very good and can be improved since the computational power involved in this study was modest. The extension ms2 8× 9 16× 17 64 -0.24803 -0.21401 100 -0.24828 -0.21417 Exact -0.24857 -0.21432 TABLE IV: Ground-state energies for different lattice sizes; three states were targeted in the second step: the ground state itself and the lowest states of Sz = 0 and Sz = 1 sectors. 0 1 2 3 4 FIG. 2: Width ∆ǫ for the low-lying states of the 1D Hubbard chain as function of U for L = 16 and ms2 = 128. to the AHM with U 6= 0 is straightforward. There is no additional change in the algorithm since the term involv- ing U is local and thus treated during the 1D part of the TSDMRG. The role of U is to reduce ∆ǫ as shown in Fig.2. For fixed L and ms2, ∆ǫ decreases linearly with increasing U . For L = 16 and ms2 = 128, I anticipate that for U <∼ 3 the interacting system results will be on the same level or better than those of the non-interacting case with ms2 = 100 for the same value of L. IV. GROUND-STATE PROPERTIES OF COUPLED HUBBARD CHAINS I now proceed to the study of U 6= 0. One of the main motivations for such a study is the possibility to gain insight into the mechanism of superconductivity in quasi 1D systems. The mechanism of superconductivity in the quasi 1D organic materials Bechgaard and Fabre salts, is still an open issue.13 Since these materials are 1D above a crossover temperature Tx ≈ t⊥/π, it is broadly accepted that the starting point for the the understand- ing of their low T behavior should be pure 1D physics. The occurence of the low T ordered phases is driven by the interchain hopping t⊥. Two main hypotheses have been suggested concerning superconductivity. The first hypothesis (see a recent review in Ref. 13) relies on a more conventional physics: t⊥ drives the system to a 2D electron gas which is an anisotropic Fermi liquid which becomes superconductive through a conventional BCS 0 2 4 6 8 -0.004 -0.003 -0.002 -0.001 FIG. 3: Transverse Green’s function G(y) for td = 0 (circles), td = 0.1 (squares). mechanism. However, it has been argued11 that given the smallness of t⊥, the resulting electron-phonon cou- pling would not be enough to account for the observed Tc. The second hypothesis, which has gained strength over the years given the absence of a clear phonon sig- nature, is that the pairing mechanism originates from an exchange of spin fluctuation.11 Interest in this issue was recently revived by the NMR Knight shift experimental finding that the symmetry of the Cooper pairs is triplet12 in (TMTSF )2(PF )6. No shift was found in the magnetic susceptibility at the tran- sition for measurement made under a magnetic field of about 1.4 Tesla. A triplet pairing scenario was subse- quently supported by the persistence of superconduc- tivity under fields far exceeding the Pauli breaking-pair limit19. However there is no simple explanation of this scenario. Triplet pairing would be unfavorable in a BCS like scenario for which a singlet s-wave is most likely. Triplet pairing is also less likely in the spin fluctuation mechanism for which a singlet d-wave is predicted by an- lytical RG13 or by perturbative approaches17. It has be argued that these difficulties in both mechanisms can be circumvented. In the BCS case, the association of AFM fluctuations with an open Fermi surface to the electron- phonon mechanism may lead to a triplet pairing18. In the spin fluctuation case, the addition of interchain Coulomb interactions may favor a triplet f-wave in lieu of the sin- glet d-wave13,17. The more exotic Fulde-Ferrel-Larkin- Ovchinnikov phase can also been invoked to account for the large paramagnetic limit. However, the Knight shift result which was thought to bring a conclusion to this long standing issue has only revived the old controversy. The conclusion of this experiment itself has been recently challenged. In Ref. 15, it was pointed out that the ob- servation of triplet superconductivity claimed in Ref. 12 could be a spurious effect due to the lack of thermaliza- tion of the samples. A recent Knight shift experimenent performed at lower fields reveals a decrease in the spin susceptibility. This is consistent with singlet pairing.16 0 2 4 6 8 -0.002 -0.001 0.001 FIG. 4: Transverse spin-spin correlation C(y) for td = 0 (cir- cles), td = 0.1 (squares). The 1D interacting electron gas is now fairly well understood.3 There is no phase with long range order. There are essentially four regions in the phase diagram, characterized by the dominant correlations i.e., SDW, charge density wave (CDW), singlet superconductivity (SS) and triplet superconductivity (TS). The essential question is whether the interchain hopping will simply freeze the dominant 1D fluctuation into long-range order (LRO) or create new 2D physics. The estimated values of U and V for the Bechgaard salts suggest that they are in the SDW region in their 1D regime. This suggests that superconductivity in these materials is a 2D phe- nomenon. Interchain pair tunneling was suggested soon after the discovery of superconductivity in an organic compound.14 Emery argued instead that a mechanism similar to the Kohn-Luttinger mechanism might be re- sponsible for superconductivity in the organic materials. When t⊥ is turned on, pairing can arise from exchange of short-range SDW fluctuations. The reason is that the os- cillating SDW susceptibility atQ = (2kF , k⊥) would have an attractive region if k⊥ 6= 0. In particular if k⊥ = π as I found, then the interaction would be attractive be- tween particles in neighboring chains. In this study, I will restrain myself to the study of interchain pair tun- neling. I was unable to compute correlation functions of pairs in which each electron belongs to a different chain. The reason is that in the DMRG method, for the correla- tion functions to be accurate, at least two different blocks should be involved. This means that for pair correlation for which each electron of the pair is on a different chain, at least four blocks are needed. However, the introduc- tion of four blocks in the second step of the TSDMRG leads to a prohibitive CPU time. With the hope of frustrating an SDW ordering which is usually expected, I will add an extra terms to model (1) . These are the diagonal interchain hopping, Hd = −td i,l,σ i,l,σci+1,l+1,σ + 0 2 4 6 8 -0.002 -0.001 0.000 0.001 0 1 2 3 4 5 6 7 8 -0.0001 0.0000 0.0001 FIG. 5: Transverse local singlet correlation SS(y) for td = 0 (circles), td = 0.1 (squares). h.c) + (c i+1,l,σci,l−1,σ + h.c), (11) and the next-nearrest neighbor interchain hopping, H ′⊥ = −t′⊥ i,l,σ i,l,σci,l+2,σ + h.c). I will also add the interchain Coulomb interaction, HV = V⊥ i,l,σ ni,l,σ.ni,l+1,σ (12) I set t⊥ = 0.2, ms1 = 256, ms2 = 128, and (L × (L + 1) = 16×17. A second set of calculations with t⊥ = 0.15, same values of ms1 and ms2, and (L× (L+1) = 24× 25 lead to the same conclusions. Therefore, they will not be shown here. In order to analyze the physics induced by the transverse couplings, I compute the following inter- chain correlations: the transverse single-particle Green’s function, shown in Fig.3, G(y) = 〈cL/2,L/2+yc†L/2,L/2+1〉, (13) the transverse spin-spin correlation function, shown in Fig.4, C(y) = 〈SL/2,L/2+ySL/2,L/2+1〉, (14) the transverse local pairs singlet superconductive corre- lation, shown in Fig.5, SS(y) = 〈ΣL/2,L/2+yΣ†L/2,L/2+1〉, (15) where Σi,l = ci,l↑ci,l↓, (16) the transverse triplet superconductive correlation, shown in Fig.6, ST (y) = 2〈ΘL/2,L/2+yΘ†L/2,L/2+1〉, (17) where Θi,l = (ci,l↑ci+1,l↓ + ci,l↓ci+1,l↑), (18) and the transverse non-local singlet pair superconductive correlation function, shown in Fig.7, SD(y) = 2〈∆L/2,L/2+y∆†L/2,L/2+1〉, (19) where ∆i,l = (ci,l↑ci+1,l↓ − ci,l↓ci+1,l↑). (20) A. Strong-coupling regime Let us first consider, the regime U >∼ 4, I choose for instance U = 4, V = 0.85, µ = 0, and td = t ⊥ = V⊥ = 0; besides single-particle hop- ping, t⊥ also generates two-particle hopping both in the particle-hole and particle-particle channels. These two-particle correlation functions are roughly given by the average values t2⊥〈c i,lσci,l−σc i,l+j−σci,l+jσ〉 and t2⊥〈c i,lσc i,l−σci,l+jσci,l+j−σ〉 for an on-site pair created at (i, l) and then destroyed at (i, l + j). It is expected that the dominant two-particle correlation are SDW with k⊥ = π. This is seen in Fig.(4-7). The transverse pair- ing correlations are all found to be small with respect to C(y). Among the pairing correlations, SS(y) decays faster then ST (y) and SD(y). These results are consis- tent with the view that the role of t⊥ is to freeze the dominant 1D correlations into LRO. When td 6= 0, it is expected that for a strong enough td, the magnetic order will vanish because of the frustration induced by td. A simple argument is that td induces an AFM exchange between next-nearest neigbhors on chains l and l + 1 which compete with the AFM exchange be- tween nearest neigbhors. The hope is that there could be a region of the phase diagram where superconductivity could ultimately win either by pair tunneling between the chains or by the Emery’s mechanism. However, in Fig.(4-7) it can be seen that, while td slightly reduces C(y), the dominant correlations are still SDW even for a strong td/t⊥ = 0.5. SS(y), ST (y) and SD(y) are barely affected by td. The fact that td does not strongly af- fect the SDW order can be understood in the light of recent study of coupled t− J chains8. It was shown that the frustration strongly suppresses magnetic LRO only 0 1 2 3 4 5 6 7 -0.002 -0.001 0.000 0.001 0 1 2 3 4 5 6 7 8 -0.0002 -0.0001 0.0000 0.0001 FIG. 6: Transverse triplet superconductive correlation ST (y) for td = 0 (circles), td = 0.1 (squares). 0 1 2 3 4 5 6 7 -0.002 -0.001 0.000 0.001 0 1 2 3 4 5 6 7 8 -0.0001 0.0000 0.0001 0.0002 FIG. 7: Transverse singlet non-local superconductive correla- tion SD(y) for td = 0 (circles), td = 0.1 (squares). close to half-filling. For large dopings, two neighboring spins in a chain do not always points to opposite direc- tion as the consequence, td does not necessarily frustrate the magnetic order. This is illustrated in a simple sketch in Fig.(8). td could even enhance it as seen in the study of t− J chains. In Fig.3, it can be seen that td enhances G(y). This enhancement, together with the decrease of C(y), suggests a possible widening of an eventual Fermi liquid region at finite T above the ordered phase. When t⊥ 6= 0, I also found (not shown) that magnetic correla- tion are not effectively suppressed even when t′⊥ = t⊥/2. For this value, it would be expected that the ratio of the effective exchange term generated by t′⊥ to that generated by t⊥ is about one quarter. In the frustrated J1−J2 spin chain, a spin gap opens around this ratio. This simple picture does not seem to work here. FIG. 8: sketch of the spin texture (arrows) in two consecutive chains in an SDW. The bold horizontal lines represent the chains. The full diagonal lines show bonds for which td tends to increase the SDW order. The diagonal dotted lines show bonds for which td frustrates the magnetic order. B. Weak-coupling regime I now turn in to the regime where U <∼ 4. I set U = 2, V = 0, µ = −0.9271, t⊥ = 0.2, td = 0, and V⊥ = 0.4, where V⊥ is the interchain Coulomb interaction between nearest neighbors. It can be seen in Fig.9 that C(y) is now strongly reduced with respect to its strong coupling values. It is already within our numerical error for the next-nearest neighbor in the transverse direction. This is an indication that the ground state is probably not an SDW. It is to be noted that this occurs even in the ab- sence of td or t ⊥. This seems to be at variance with the RG analysis which requires t′⊥ to destroy the magnetic order. A possible explanation of this is that at half-filling the perfect nesting occurs at the wave vector Q = (π, π) for the spectrum of equation (9). Away from half-filling the nesting is no longer perfect this leads to the reduc- tion of magnetic correlations. The first correction to the nesting is an effective frustration term which is roughly t2⊥cos2k⊥. This expression is identical to a term that could be generated by an explicit frustration t′⊥ = t The discrepancy between the TSDMRG and the RG re- sults could be that this nesting deviation is undereval- uated in the RG analysis. This mechanism cannot be invoked in the strong coupling regime where band effects are small. The suppression of magnetism is concommitant to a strong enhancement of the singlet pairing correlations as seen in Fig. 11. Triplet correlations, shown in Fig. 10, remain very small. However, while it is clear from the behavior of C(y) that the ground state is non magnetic. This result strongly suggests that the ground state is a superconductor in this regime. A finite size analysis is, however, necessary to conclude whether this persists to the thermodynamic limit. I cannot rule out the possibil- ity of a Fermi liquid ground state, which is implied by 0 2 4 6 8 -0.002 -0.001 0.001 FIG. 9: Transverse spin-spin correlation C(y) for U = 4 (cir- cles), U = 2 and V⊥ = 0.4 (squares). 0 1 2 3 4 5 6 7 -0.001 0.001 0.002 FIG. 10: Transverse triplet superconductive correlation ST (y) for U = 4 (circles), U = 2 and V⊥ = 0.4 (squares). strong single particle correlations. V. CONCLUSION In this paper, I have presented a TSDMRG study of the competition between magnetism and superconductiv- ity in an anisotropic Hubbard model. I have analyzed the effect of the interchain hopping in the strong and weak U regimes. In the strong-coupling regime, the results are consistent with earlier predictions that the role of t⊥ is to freeze the dominant 1D SDW correlations into a 2D ordered state. But at variance with analytical predic- tions, this is only true in the strong U regime. In this regime, I find that even the introduction of frustration does not disrupt the SDW order which remain robust up to large values of the frustration. In the weak cou- pling regime singlet pair correlations are dominant. The ground state seems to be a superconductor. This behav- ior is somewhat in agreement with experiments in the Bechgaard or Fabre salts. The phase diagram is domi- 0 1 2 3 4 5 6 7 -0.001 0.001 0.002 FIG. 11: Transverse singlet non-local superconductive cor- relation SD(y) for U = 4 (circles), U = 2 and V⊥ = 0.4 (squares). nated by magnetism at low pressure (strong U) and by superconductivity at high pressure (weak U). Because of experimental relevance, I restricted myself to the com- petion between magnetism and superconductivity. I did not analyze CDW correlations. These are likely to be important given that I applied open boundary conditions which are known to generate Friedel oscillations20 that very decay slowly from the boundaries. They may also genuinely generated by V⊥, leading to a CDW ground state instead of a superconductor. Acknowledgments I am very grateful to C. Bourbonnais for very helpful exchanges. I wish to thank A.M.-S. Tremblay for helpful discussions. This work was supported by the NSF Grant No. DMR-0426775. 1 C. Bourbonnais and D. Jérome in ”Advance in Synthetic Metals” Eds. P. Bernier, S. Lefrant and G. Bidan (Elsevier, New York), 206 (1999). 2 J.W. Allen, Sol. St. Comm. 123, 469 (2002). 3 C. Bourbonnais and L.G. Caron, Int. J. Mod. Phys. B 5, 1033 (1991). 4 T. Giamarchi in ”Quantum Physics in One Dimension”, Clarendon Press Eds, P. 254-269 (2004). 5 S. Biermann, A. Georges, A. Lichtenstein, and T. Gia- marchi, Phys. Rev. Lett. 87, 276405 (2001). 6 S.R. White, D.J. Scalapino, R.L. Sugar, E.Y. Loh, J.E. Gu- bernatis, and R.T. Scalettar, Phys. Rev. B 40, 506 (1989). 7 S. Moukouri, Phys. Rev. B 70, 014403 (2004). 8 S. Moukouri, J. Stat. Mech. P02002 (2006) 9 S.R. White, Phys. Rev. Lett. 69, 2863 (1992). Phys. Rev. B 48, 10 345 (1993). 10 S.R. White, Phys. Rev. B 72, 180403 (2005). 11 V.J. Emery, Synthetic Metals, 13, 21 (1986). 12 I.J. Lee et al., Phys. Rev. Lett. 88, 017004 (2002). 13 N. Dupuis, C. Bourbonnais and J.C. Nickel, cond-mat/0510544. 14 D. Jerome and H.J. Schulz, Adv. Phys. 31, 299 (1982). 15 D. Jerome, Chem. Rev. 104, 5565 (2004). D. Jerome and C.R. Pasquier, in Superconductors, edited by A.V. Narlikar (Springer Verlag, Berlin, 2005). 16 Y. Shinagawa, et al., cond-mat/0701566 (2007). 17 Y. Tanaka and K. Kuroki, Phys. Rev.B 70, 060502 (2004). 18 Mahito Kohmoto and Masatoshi Sato, cond-mat/0001331 (2000). 19 I.J. Lee, M.J. Naughton, P.M. Chaikin, Physica B 294- 295, 413 (2001). 20 S. R. White, Ian Affleck, and D. J. Scalapino, Phys. Rev. B 65, 165122 (2002). http://arxiv.org/abs/cond-mat/0510544 http://arxiv.org/abs/cond-mat/0701566 http://arxiv.org/abs/cond-mat/0001331

I apply a two-step density-matrix renormalization group method to the anisotropic two-dimensional Hubbard model. As a prelude to this study, I compare the numerical results to the exact one for the tight-binding model. I find a ground-state energy which agrees with the exact value up to four digits for systems as large as $24 \times 25$. I then apply the method to the interacting case. I find that for strong Hubbard interaction, the ground-state is dominated by magnetic correlations. These correlations are robust even in the presence of strong frustration. Interchain pair tunneling is negligible in the singlet and triplet channels and it is not enhanced by frustration. For weak Hubbard couplings, interchain non-local singlet pair tunneling is enhanced and magnetic correlations are strongly reduced. This suggests a possible superconductive ground state.

A Renormalization group approach for highly anisotropic 2D Fermion systems: application to coupled Hubbard chains S. Moukouri Department of Physics and Michigan Center for Theoretical Physics University of Michigan, 2477 Randall Laboratory, Ann Arbor MI 48109 I apply a two-step density-matrix renormalization group method to the anisotropic two- dimensional Hubbard model. As a prelude to this study, I compare the numerical results to the exact one for the tight-binding model. I find a ground-state energy which agrees with the exact value up to four digits for systems as large as 24 × 25. I then apply the method to the interact- ing case. I find that for strong Hubbard interaction, the ground-state is dominated by magnetic correlations. These correlations are robust even in the presence of strong frustration. Interchain pair tunneling is negligible in the singlet and triplet channels and it is not enhanced by frustration. For weak Hubbard couplings, interchain non-local singlet pair tunneling is enhanced and magnetic correlations are strongly reduced. This suggests a possible superconductive ground state. I. INTRODUCTION Quasi-one dimensional organic1 and inorganic2 mate- rials have been the object of an important theoretical in- terest for the last three decades. The essential features of their phase diagram may be captured by the anisotropic Hubbard model (AHM), H = −t‖ i,l,σ i,l,σci+1,l,σ + h.c.) + U ni,l,↑ni,l,↓ i,l,σ ni,l,σni+1,l,σ − µ i,l,σ ni,l,σ i,l,σ i,l,σci,l+1,σ + h.c.). (1) or a more general Hubbard-like model including longer range Coulomb interactions. The indices i and l label the sites and the chains respectively. For these highly anisotropic materials, t⊥ ≪ t‖. Over the years, the AHM has remained a formidable challenge to condensed- matter theorists. Some important insights on this model or its low energy version, the g-ology model, have been obtained through the work of Bourbonnais and Caron3,4 and others. They used a perturbative renormalization group approach to analyze the crossover from 1D to 2D at low temperatures. More recently, Biermann et al.5 ap- plied the chain dynamical mean-field approach to study the crossover from Luttinger liquid to Fermi liquid in this model. Despite this important progress, crucial informa- tion such as the ground-state phase diagram, or most no- tably, whether the AHM displays superconductivity, are still unknown. So far it has remained beyond the reach of numerical methods such as the exact diagonalization (ED) or the quantum Monte Carlo (QMC) methods. ED cannot exceed lattices of about 4 × 5. It is likely to re- main so for many years unless there is a breakthrough in quantum computations. The QMC method is plagued by the minus sign problem and will not be helpful at low temperatures. The small value of t⊥ implies that, in or- der to see the 2D behavior, it will be necessary to reach lower temperatures than those usually studied for the isotropic 2D Hubbard model. Hence, even in the absence of the minus sign problem, in order to work in this low temperature regime, the QMC algorithm requires spe- cial stabilization schemes which lead to prohibitive cpu time.6 II. TWO-STEP DMRG I have shown in Ref. 7 that this class of anisotropic models may be studied using a two-step density-matrix renormalization group (TSDMRG) method. The TS- DMRG method is a perturbative approach in which the standard 1D DMRG is applied twice. In the first step, the usual 1D DMRG method9 is applied to find a set of low lying eigenvalues ǫn and eigenfunctions |φn〉 of a single chain. In the second step, the 2D Hamiltonian is then projected onto the basis constructed from the tensor product of the |φn〉’s. This projection yields an effective one-dimensional Hamiltonian for the 2D lattice, E‖[n]|Φ‖[n]〉〈Φ‖[n]| − t⊥ i,l,σ i,l,σ c̃i,l+1,σ + h.c.)(2) where E‖[n] is the sum of eigenvalues of the different chains, E‖[n] = l ǫnl ; |Φ‖[n]〉 are the corresponding eigenstates, |Φ‖[n]〉 = |φn1〉|φn2〉...|φnL〉; c̃ i,l,σ, c̃i,l,σ, and ñi,l,σ are the renormalized matrix elements in the single chain basis. They are given by i,l,σ) nl,ml = (−1)ni〈φnl |c i,l,σ|φml〉, (3) (c̃i,l,σ) nl,ml = (−1)ni〈φnl |ci,l,σ|φml〉, (4) (ñi,l,σ) nl,ml = 〈φnl |ni,l,σ|φml〉, (5) where ni represents the total number of fermions from sites 1 to i− 1. For each chain, operators for all the sites are stored in a single matrix http://arxiv.org/abs/0704.1618v1 l,σ = (c̃ 1,l,σ, ..., c̃ L,l,σ), (6) c̃l,σ = (c̃1,l,σ, ..., c̃L,l,σ), (7) ñl,σ = (ñ1,l,σ, ..., ñL,l,σ). (8) Since the in-chain degrees of freedom have been inte- grated out, the interchain couplings are between the block matrix operators in Eq.( 6, 7) which depend only on the chain index l. In this matrix notation, the effec- tive Hamiltonian is one-dimensional and it is also studied by the DMRG method. The only difference compared to a normal 1D situation is that the local operators are now ms2 ×ms2 matrices, where ms2 is the number of states kept during the second step. The two-step method has previously been applied to anisotropic two-dimensional Heisenberg models.7 In Ref. 8, it was applied to the t− J model but due to the absence of an exact result in certain limits, it was tested against ED results on small ladders only. A systematic analysis of its performance on a fermionic model on 2D lattices of various size has not been done. In this paper, as a prelude to the study of the AHM, I will apply the TSDMRG to the anisotropic tight-binding model on a 2D lattice, i.e., model (1) with U = V = 0. I perform a comparison with the exact result of the tight-binding model. I was able to obtain agreement for the ground- state energies on the order of 10−4 for lattices of up to 24 × 25. I then discuss how these calculations may be extended to the interacting case, before presenting the U 6= 0 results. III. WARM UP: THE TIGHT-BINDING MODEL The tight-binding Hamiltonian is diagonal in the mo- mentum space, the single particle energies are, ǫk = −2t‖coskx − 2t⊥cosky − µ, (9) with k = (kx, ky), kx = nxπ/(Lx+1) and ky = nyπ/(Ly+ 1) for open boundary conditions (OBC); Lx, Ly are re- spectively the linear dimensions of the lattice in the par- allel and transverse directions. The ground-state energy of an N electron system is obtained by filling the low- est states up to the Fermi level, E[0](N) = However in real space, this problem is not trivial and it constitutes, for any real space method such as the TS- DMRG, a test having the same level of difficulty as the case with U 6= 0. This is because the term involving U is diagonal in real space and the challenge of diagonalizing the AHM arises from the hopping term. I will study the tight-binding model at quarter filling, N/LxLy = 1/2, the nominal density of the organic con- ductors known as the Bechgaard salts. Systems of up to Lx×Ly = L×(L+1) = 24×25 will be studied. During the first step, I keep enough states (ms1 is a few hundred) so 0 20 40 60 80 100 FIG. 1: Low-lying states of the 1D tight-binding model (full line) and of the 1D Heisenberg spin chain (dotted line) for L = 16 and ms2 = 96. that the truncation error ρ1 is less than 10 −6. I target the lowest state in each charge-spin sectorsNx±2, Nx±1, Nx and Sz ± 1, Sz ± 2, Nx is the number of electrons within the chain. It is fixed such that Nx/Lx = 1/2. There is a total of 22 charge-spin states targeted at each iteration. For the tight-binding model, the chains remain discon- nected if t⊥ < ǫ0(Nx + 1) − ǫ0(Nx) or t⊥ < ǫ0(Nx) − ǫ0(Nx − 1), where Nx is the number of electons on sin- gle chain. In order to observe transverse motion, it is necessary that at least t⊥ >∼ ǫ0(Nx + 1) − ǫ0(Nx) and t⊥ >∼ ǫ0(Nx)− ǫ0(Nx − 1). These two conditions are sat- isfied only if µ is appropietly chosen. The values listed in Table (I) corresponds to µ = (ǫ0(Nx+1)−ǫ0(Nx−1))/2. This treshold varies with L. I give in Table (I) the val- ues of t⊥ chosen for different lattice sizes. In principle, for the TSDMRG to be accurate, it is necessary that ∆ǫ = ǫnc − ǫ0, where ǫnc is the cut-off, be such that ∆ǫ/t⊥ ≫ 1. But in practice, I find that I can achieve accuracy up to the fourth digit even if ∆ǫ/t⊥ ≈ 5 using the finite system method. Five sweeps were necessary to reach convergence. Note that this conclusion is some- what different from my earlier estimate of ∆ǫ/t⊥ ≈ 10 for spin systems.8 This is because in Ref. 8, I used the infinite system method during the second step. The ultimate success of the TSDMRG depends on the density of the low-lying states in the 1D model. For fixed ms2 and L, it is, for instance, easier to reach larger ∆ǫ/J⊥ in the anisotropic spin one-half Heisen- berg model, studied in Ref. 7, than ∆ǫ/t⊥ for the tight- binding model as shown in Fig.1. For L = 16, ms2 = 96, and J⊥ = t⊥ = 0.15, I find that ∆ǫ/J⊥ ≈ 10, while ∆ǫ/t⊥ ≈ 5. Hence, the TSDMRG method will be more accurate for a spin model than for the tight-binding model. Using the infinite system method during the second step on the anisotropic Heisenberg model with J⊥ = 0.1, I can now reach an agreement of about 10 with the stochastic QMC method. Two possible sources of error can contribute to reduce the accuracy in the TSDMRG with respect to the conven- 8× 9 16× 17 24× 25 t⊥ 0.28 0.15 0.1 µ -1.2660 -1.3411 -1.3657 ∆ǫ/t⊥ 6.42 5.40 5.78 TABLE I: Transverse hopping and chemical potential used in the simulations for different lattice sizes ms2 8× 3 16× 3 24× 3 64 -0.241524 -0.211929 0.204040 Exact -0.241524 -0.211931 0.204049 TABLE II: Ground-state energies of three-leg ladders. tional DMRG. They are the truncation of the superblock from 4×ms1 states to onlyms2 states and the use of three blocks instead of four during the second step. In Table (II) I analyze the impact of the reduction of the number of states to ms2 for three-leg ladders. The choice of three- leg ladders is motivated by the fact that at this point, the TSDMRG is equivalent to the exact diagonalization of three reduced superblocks. It can be seen that as far as t⊥ >∼ ǫ0(Nx+1)−ǫ0(Nx) and t⊥ >∼ ǫ0(Nx)−ǫ0(Nx−1), the TSDMRG at this point is as accurate as the 1D DMRG. Note that the accuracy remains nearly the same irrespec- tive of L as far as the ratio ∆ǫ/t⊥ remains nearly con- stant. Since ∆ǫ decreases when L increases, t⊥ must be decreased in order to keep the same level of accuracy for fixed ms2. In principle, following this prescription, much larger systems may be studied. ∆ǫ/t⊥ does not have to be very large, in this case it is about 5, to obtain very good agreement with the exact result. The second source of error is related to the fact that the effective single site during the second step is now a chain having ms2 states, I am thus forced to use three blocks instead of four to reduce the computational bur- den. In Table (III), it can be seen that this results in a reduction in accuracy of about two orders of magnitude with respect to those of three leg-ladders. These results are nevertheless very good given the relatively modest computer power involved. All calculations were done on a workstation. The DMRG is less accurate when three blocks are used instead of four. This can be understood by applying the following view on the formation of the reduced density matrix. The construction of the reduced density matrix ms2 8× 9 16× 17 24× 25 64 -0.24761 -0.21401 0.20504 100 -0.24819 -0.21414 0.20509 120 -0.24832 -0.21419 Exact -0.24857 -0.21432 0.20519 TABLE III: Ground-state energies for different lattice sizes; a single state was targeted in the second step. may be regarded as a linear mapping uΨ : F ∗ → E, where E is the system, F is the environment and, F∗ is the dual space of F. Using the decomposition of the superblock wave function Ψ[0] = i ⊗ φRi , with φLi ∈ E and ΦRi ∈ F, for any φ∗ ∈ F ∗, 〈φ∗|φRi 〉φLi . (10) Let |k〉, k = 1, ...dimE and |l〉, l = 1, ...dimF be the basis of E and F respectively. Then, |l〉 has a dual basis 〈l∗| such that 〈l∗|l〉 = δl,l∗ . The matrix elements of uΨ in this basis are just the coordinates of the superblock wave function Φ[0]k,l . The rank r of this mapping, which is also the rank of the reduced density matrix is always smaller or equal to the smallest dimension of E or F, r < Min(dimE, dimF). Hence, if ms2 states are kept in the two external blocks, the number of non-zero eigenval- ues of ρ cannot be larger than ms2. Consequently, some states which have non-zero eigenvalues in the normal four block configuration will be missing. A possible cure to this problem is to target additional low-lying states above Ψ[0](N). The weight of these states in ρ must be small so that their role is simply to add the missing states not to be described accurately themselves. A larger weight on these additional states would lead to the reduction of the accuracy for a fixed ms2. In table (IV), I show the improved energies when, besides the ground state, I target the lowest states of the spin sectors Sz = −1 and Sz = +1 with N electrons. The weights were re- spectively 0.995, 0.0025, and 0.0025 for the three states. This lowers E[0](N) in all cases, but the gain does not appear to be spectacular. But I do not know whether this is due to my choice of perturbation of ρ or whether even the algorithm with four blocks would not yield bet- ter E[0](N). If the lowest sectors with N + 1 and N − 1 electrons which have Sz = ±0.5 are projected instead, I find that the results are similar to those with Sz = ±1 sectors, there are possibly many ways to add the missing states. A more systematic approach to this problem has recently been suggested.10 It is based on using a local perturbation to build a correction to the density matrix from the site at the edge of the system. Here, such a perturbation would be ∆ρ = αc l ρcl, where α is a con- stant, α ≈ 10−3 − 10−2, and c†l , cl are the creation and annihilation operators of the chain at the edge of the sys- tem. This type of perturbation resulted in an accuracy gain of more than an order of magnitude in the case of a spin chain.10 The three block method was found to be on par with the four block method. It will be interesting to see in a future study how this type of local perturbation performs within the TSDMRG. To conclude this section, as a first step to the investi- gation of interacting electron models, I have shown that the TSDMRG can successfuly be applied to the tight- binding model. The agreement with the exact result is very good and can be improved since the computational power involved in this study was modest. The extension ms2 8× 9 16× 17 64 -0.24803 -0.21401 100 -0.24828 -0.21417 Exact -0.24857 -0.21432 TABLE IV: Ground-state energies for different lattice sizes; three states were targeted in the second step: the ground state itself and the lowest states of Sz = 0 and Sz = 1 sectors. 0 1 2 3 4 FIG. 2: Width ∆ǫ for the low-lying states of the 1D Hubbard chain as function of U for L = 16 and ms2 = 128. to the AHM with U 6= 0 is straightforward. There is no additional change in the algorithm since the term involv- ing U is local and thus treated during the 1D part of the TSDMRG. The role of U is to reduce ∆ǫ as shown in Fig.2. For fixed L and ms2, ∆ǫ decreases linearly with increasing U . For L = 16 and ms2 = 128, I anticipate that for U <∼ 3 the interacting system results will be on the same level or better than those of the non-interacting case with ms2 = 100 for the same value of L. IV. GROUND-STATE PROPERTIES OF COUPLED HUBBARD CHAINS I now proceed to the study of U 6= 0. One of the main motivations for such a study is the possibility to gain insight into the mechanism of superconductivity in quasi 1D systems. The mechanism of superconductivity in the quasi 1D organic materials Bechgaard and Fabre salts, is still an open issue.13 Since these materials are 1D above a crossover temperature Tx ≈ t⊥/π, it is broadly accepted that the starting point for the the understand- ing of their low T behavior should be pure 1D physics. The occurence of the low T ordered phases is driven by the interchain hopping t⊥. Two main hypotheses have been suggested concerning superconductivity. The first hypothesis (see a recent review in Ref. 13) relies on a more conventional physics: t⊥ drives the system to a 2D electron gas which is an anisotropic Fermi liquid which becomes superconductive through a conventional BCS 0 2 4 6 8 -0.004 -0.003 -0.002 -0.001 FIG. 3: Transverse Green’s function G(y) for td = 0 (circles), td = 0.1 (squares). mechanism. However, it has been argued11 that given the smallness of t⊥, the resulting electron-phonon cou- pling would not be enough to account for the observed Tc. The second hypothesis, which has gained strength over the years given the absence of a clear phonon sig- nature, is that the pairing mechanism originates from an exchange of spin fluctuation.11 Interest in this issue was recently revived by the NMR Knight shift experimental finding that the symmetry of the Cooper pairs is triplet12 in (TMTSF )2(PF )6. No shift was found in the magnetic susceptibility at the tran- sition for measurement made under a magnetic field of about 1.4 Tesla. A triplet pairing scenario was subse- quently supported by the persistence of superconduc- tivity under fields far exceeding the Pauli breaking-pair limit19. However there is no simple explanation of this scenario. Triplet pairing would be unfavorable in a BCS like scenario for which a singlet s-wave is most likely. Triplet pairing is also less likely in the spin fluctuation mechanism for which a singlet d-wave is predicted by an- lytical RG13 or by perturbative approaches17. It has be argued that these difficulties in both mechanisms can be circumvented. In the BCS case, the association of AFM fluctuations with an open Fermi surface to the electron- phonon mechanism may lead to a triplet pairing18. In the spin fluctuation case, the addition of interchain Coulomb interactions may favor a triplet f-wave in lieu of the sin- glet d-wave13,17. The more exotic Fulde-Ferrel-Larkin- Ovchinnikov phase can also been invoked to account for the large paramagnetic limit. However, the Knight shift result which was thought to bring a conclusion to this long standing issue has only revived the old controversy. The conclusion of this experiment itself has been recently challenged. In Ref. 15, it was pointed out that the ob- servation of triplet superconductivity claimed in Ref. 12 could be a spurious effect due to the lack of thermaliza- tion of the samples. A recent Knight shift experimenent performed at lower fields reveals a decrease in the spin susceptibility. This is consistent with singlet pairing.16 0 2 4 6 8 -0.002 -0.001 0.001 FIG. 4: Transverse spin-spin correlation C(y) for td = 0 (cir- cles), td = 0.1 (squares). The 1D interacting electron gas is now fairly well understood.3 There is no phase with long range order. There are essentially four regions in the phase diagram, characterized by the dominant correlations i.e., SDW, charge density wave (CDW), singlet superconductivity (SS) and triplet superconductivity (TS). The essential question is whether the interchain hopping will simply freeze the dominant 1D fluctuation into long-range order (LRO) or create new 2D physics. The estimated values of U and V for the Bechgaard salts suggest that they are in the SDW region in their 1D regime. This suggests that superconductivity in these materials is a 2D phe- nomenon. Interchain pair tunneling was suggested soon after the discovery of superconductivity in an organic compound.14 Emery argued instead that a mechanism similar to the Kohn-Luttinger mechanism might be re- sponsible for superconductivity in the organic materials. When t⊥ is turned on, pairing can arise from exchange of short-range SDW fluctuations. The reason is that the os- cillating SDW susceptibility atQ = (2kF , k⊥) would have an attractive region if k⊥ 6= 0. In particular if k⊥ = π as I found, then the interaction would be attractive be- tween particles in neighboring chains. In this study, I will restrain myself to the study of interchain pair tun- neling. I was unable to compute correlation functions of pairs in which each electron belongs to a different chain. The reason is that in the DMRG method, for the correla- tion functions to be accurate, at least two different blocks should be involved. This means that for pair correlation for which each electron of the pair is on a different chain, at least four blocks are needed. However, the introduc- tion of four blocks in the second step of the TSDMRG leads to a prohibitive CPU time. With the hope of frustrating an SDW ordering which is usually expected, I will add an extra terms to model (1) . These are the diagonal interchain hopping, Hd = −td i,l,σ i,l,σci+1,l+1,σ + 0 2 4 6 8 -0.002 -0.001 0.000 0.001 0 1 2 3 4 5 6 7 8 -0.0001 0.0000 0.0001 FIG. 5: Transverse local singlet correlation SS(y) for td = 0 (circles), td = 0.1 (squares). h.c) + (c i+1,l,σci,l−1,σ + h.c), (11) and the next-nearrest neighbor interchain hopping, H ′⊥ = −t′⊥ i,l,σ i,l,σci,l+2,σ + h.c). I will also add the interchain Coulomb interaction, HV = V⊥ i,l,σ ni,l,σ.ni,l+1,σ (12) I set t⊥ = 0.2, ms1 = 256, ms2 = 128, and (L × (L + 1) = 16×17. A second set of calculations with t⊥ = 0.15, same values of ms1 and ms2, and (L× (L+1) = 24× 25 lead to the same conclusions. Therefore, they will not be shown here. In order to analyze the physics induced by the transverse couplings, I compute the following inter- chain correlations: the transverse single-particle Green’s function, shown in Fig.3, G(y) = 〈cL/2,L/2+yc†L/2,L/2+1〉, (13) the transverse spin-spin correlation function, shown in Fig.4, C(y) = 〈SL/2,L/2+ySL/2,L/2+1〉, (14) the transverse local pairs singlet superconductive corre- lation, shown in Fig.5, SS(y) = 〈ΣL/2,L/2+yΣ†L/2,L/2+1〉, (15) where Σi,l = ci,l↑ci,l↓, (16) the transverse triplet superconductive correlation, shown in Fig.6, ST (y) = 2〈ΘL/2,L/2+yΘ†L/2,L/2+1〉, (17) where Θi,l = (ci,l↑ci+1,l↓ + ci,l↓ci+1,l↑), (18) and the transverse non-local singlet pair superconductive correlation function, shown in Fig.7, SD(y) = 2〈∆L/2,L/2+y∆†L/2,L/2+1〉, (19) where ∆i,l = (ci,l↑ci+1,l↓ − ci,l↓ci+1,l↑). (20) A. Strong-coupling regime Let us first consider, the regime U >∼ 4, I choose for instance U = 4, V = 0.85, µ = 0, and td = t ⊥ = V⊥ = 0; besides single-particle hop- ping, t⊥ also generates two-particle hopping both in the particle-hole and particle-particle channels. These two-particle correlation functions are roughly given by the average values t2⊥〈c i,lσci,l−σc i,l+j−σci,l+jσ〉 and t2⊥〈c i,lσc i,l−σci,l+jσci,l+j−σ〉 for an on-site pair created at (i, l) and then destroyed at (i, l + j). It is expected that the dominant two-particle correlation are SDW with k⊥ = π. This is seen in Fig.(4-7). The transverse pair- ing correlations are all found to be small with respect to C(y). Among the pairing correlations, SS(y) decays faster then ST (y) and SD(y). These results are consis- tent with the view that the role of t⊥ is to freeze the dominant 1D correlations into LRO. When td 6= 0, it is expected that for a strong enough td, the magnetic order will vanish because of the frustration induced by td. A simple argument is that td induces an AFM exchange between next-nearest neigbhors on chains l and l + 1 which compete with the AFM exchange be- tween nearest neigbhors. The hope is that there could be a region of the phase diagram where superconductivity could ultimately win either by pair tunneling between the chains or by the Emery’s mechanism. However, in Fig.(4-7) it can be seen that, while td slightly reduces C(y), the dominant correlations are still SDW even for a strong td/t⊥ = 0.5. SS(y), ST (y) and SD(y) are barely affected by td. The fact that td does not strongly af- fect the SDW order can be understood in the light of recent study of coupled t− J chains8. It was shown that the frustration strongly suppresses magnetic LRO only 0 1 2 3 4 5 6 7 -0.002 -0.001 0.000 0.001 0 1 2 3 4 5 6 7 8 -0.0002 -0.0001 0.0000 0.0001 FIG. 6: Transverse triplet superconductive correlation ST (y) for td = 0 (circles), td = 0.1 (squares). 0 1 2 3 4 5 6 7 -0.002 -0.001 0.000 0.001 0 1 2 3 4 5 6 7 8 -0.0001 0.0000 0.0001 0.0002 FIG. 7: Transverse singlet non-local superconductive correla- tion SD(y) for td = 0 (circles), td = 0.1 (squares). close to half-filling. For large dopings, two neighboring spins in a chain do not always points to opposite direc- tion as the consequence, td does not necessarily frustrate the magnetic order. This is illustrated in a simple sketch in Fig.(8). td could even enhance it as seen in the study of t− J chains. In Fig.3, it can be seen that td enhances G(y). This enhancement, together with the decrease of C(y), suggests a possible widening of an eventual Fermi liquid region at finite T above the ordered phase. When t⊥ 6= 0, I also found (not shown) that magnetic correla- tion are not effectively suppressed even when t′⊥ = t⊥/2. For this value, it would be expected that the ratio of the effective exchange term generated by t′⊥ to that generated by t⊥ is about one quarter. In the frustrated J1−J2 spin chain, a spin gap opens around this ratio. This simple picture does not seem to work here. FIG. 8: sketch of the spin texture (arrows) in two consecutive chains in an SDW. The bold horizontal lines represent the chains. The full diagonal lines show bonds for which td tends to increase the SDW order. The diagonal dotted lines show bonds for which td frustrates the magnetic order. B. Weak-coupling regime I now turn in to the regime where U <∼ 4. I set U = 2, V = 0, µ = −0.9271, t⊥ = 0.2, td = 0, and V⊥ = 0.4, where V⊥ is the interchain Coulomb interaction between nearest neighbors. It can be seen in Fig.9 that C(y) is now strongly reduced with respect to its strong coupling values. It is already within our numerical error for the next-nearest neighbor in the transverse direction. This is an indication that the ground state is probably not an SDW. It is to be noted that this occurs even in the ab- sence of td or t ⊥. This seems to be at variance with the RG analysis which requires t′⊥ to destroy the magnetic order. A possible explanation of this is that at half-filling the perfect nesting occurs at the wave vector Q = (π, π) for the spectrum of equation (9). Away from half-filling the nesting is no longer perfect this leads to the reduc- tion of magnetic correlations. The first correction to the nesting is an effective frustration term which is roughly t2⊥cos2k⊥. This expression is identical to a term that could be generated by an explicit frustration t′⊥ = t The discrepancy between the TSDMRG and the RG re- sults could be that this nesting deviation is undereval- uated in the RG analysis. This mechanism cannot be invoked in the strong coupling regime where band effects are small. The suppression of magnetism is concommitant to a strong enhancement of the singlet pairing correlations as seen in Fig. 11. Triplet correlations, shown in Fig. 10, remain very small. However, while it is clear from the behavior of C(y) that the ground state is non magnetic. This result strongly suggests that the ground state is a superconductor in this regime. A finite size analysis is, however, necessary to conclude whether this persists to the thermodynamic limit. I cannot rule out the possibil- ity of a Fermi liquid ground state, which is implied by 0 2 4 6 8 -0.002 -0.001 0.001 FIG. 9: Transverse spin-spin correlation C(y) for U = 4 (cir- cles), U = 2 and V⊥ = 0.4 (squares). 0 1 2 3 4 5 6 7 -0.001 0.001 0.002 FIG. 10: Transverse triplet superconductive correlation ST (y) for U = 4 (circles), U = 2 and V⊥ = 0.4 (squares). strong single particle correlations. V. CONCLUSION In this paper, I have presented a TSDMRG study of the competition between magnetism and superconductiv- ity in an anisotropic Hubbard model. I have analyzed the effect of the interchain hopping in the strong and weak U regimes. In the strong-coupling regime, the results are consistent with earlier predictions that the role of t⊥ is to freeze the dominant 1D SDW correlations into a 2D ordered state. But at variance with analytical predic- tions, this is only true in the strong U regime. In this regime, I find that even the introduction of frustration does not disrupt the SDW order which remain robust up to large values of the frustration. In the weak cou- pling regime singlet pair correlations are dominant. The ground state seems to be a superconductor. This behav- ior is somewhat in agreement with experiments in the Bechgaard or Fabre salts. The phase diagram is domi- 0 1 2 3 4 5 6 7 -0.001 0.001 0.002 FIG. 11: Transverse singlet non-local superconductive cor- relation SD(y) for U = 4 (circles), U = 2 and V⊥ = 0.4 (squares). nated by magnetism at low pressure (strong U) and by superconductivity at high pressure (weak U). Because of experimental relevance, I restricted myself to the com- petion between magnetism and superconductivity. I did not analyze CDW correlations. These are likely to be important given that I applied open boundary conditions which are known to generate Friedel oscillations20 that very decay slowly from the boundaries. They may also genuinely generated by V⊥, leading to a CDW ground state instead of a superconductor. Acknowledgments I am very grateful to C. Bourbonnais for very helpful exchanges. I wish to thank A.M.-S. Tremblay for helpful discussions. This work was supported by the NSF Grant No. DMR-0426775. 1 C. Bourbonnais and D. Jérome in ”Advance in Synthetic Metals” Eds. P. Bernier, S. 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Pasquier, in Superconductors, edited by A.V. Narlikar (Springer Verlag, Berlin, 2005). 16 Y. Shinagawa, et al., cond-mat/0701566 (2007). 17 Y. Tanaka and K. Kuroki, Phys. Rev.B 70, 060502 (2004). 18 Mahito Kohmoto and Masatoshi Sato, cond-mat/0001331 (2000). 19 I.J. Lee, M.J. Naughton, P.M. Chaikin, Physica B 294- 295, 413 (2001). 20 S. R. White, Ian Affleck, and D. J. Scalapino, Phys. Rev. B 65, 165122 (2002). http://arxiv.org/abs/cond-mat/0510544 http://arxiv.org/abs/cond-mat/0701566 http://arxiv.org/abs/cond-mat/0001331

Mon. Not. R. Astron. Soc. 000, 000–000 (2005) Printed 2 November 2018 (MN LATEX style file v2.2) Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge: Observations and Model Comparisons Nicholas J. Rattenbury1, Shude Mao1, Victor P. Debattista2, Takahiro Sumi3 Ortwin Gerhard4, Flavio De Lorenzi4 ⋆ 1 University of Manchester, Jodrell Bank Observatory, Macclesfield, Cheshire, SK11 9DL, UK 2 Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195-1580, USA 3 Solar-Terrestrial Environment Laboratory, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan 4 Max-Planck-Institut fuer extraterrestrische Physik, P.O. Box 1312,D-85741 Garching, Germany Accepted ........ Received .......; in original form ...... ABSTRACT Red clump giants in the Galactic bulge are approximate standard candles and hence they can be used as distance indicators. We compute the proper motion dispersions of RCG stars in the Galactic bulge using the proper motion catalogue from the second phase of the Optical Gravitational Microlensing Experiment (OGLE-II, Sumi et al. 2004) for 45 fields. The proper motion dispersions are measured to a few per cent accuracy due to the large number of stars in the fields. The observational sample is comprised of 577736 stars. These observed data are compared to a state-of-the-art particle simulation of the Galactic bulge region. The predictions are in rough agreement with observations, but appear to be too anisotropic in the velocity ellipsoid. We note that there is significant field-to-field variation in the observed proper motion dispersions. This could either be a real feature, or due to some unknown systematic effect. Key words: gravitational lensing - Galaxy: bulge - Galaxy: centre - Galaxy: kinematics and dynamics - Galaxy: structure 1 INTRODUCTION Many lines of evidence suggest the presence of a bar at the Galac- tic centre, such as infrared maps (Dwek et al. 1995; Binney et al. 1997) and star counts (Stanek et al. 1997; Nikolaev & Weinberg 1997; Unavane & Gilmore 1998), see Gerhard (2002) for a review. However, the bar parameters are not well determined. For example, recent infra-red star counts collected by the Spitzer Space Tele- scope are best explained assuming a bar at a ∼ 44◦ angle to the Sun–Galactic centre line (Benjamin et al. 2005) while most previ- ous studies prefer a bar at ∼ 20◦. In addition, there may be some fine features, such as a ring in the Galactic bulge, that are not yet firmly established (Babusiaux & Gilmore 2005). It is therefore cru- cial to obtain as many constraints as possible in order to better un- derstand the structure of the inner Galaxy. Many microlensing groups monitor the Galactic bulge, in- cluding the EROS (Aubourg et al. 1993), MACHO (Alcock et al. 2000), MOA (Bond et al. 2001; Sumi et al. 2003a) and OGLE (Udalski et al. 2000) collaborations. In addition to discovering mi- crolensing events, these groups have also accumulated a huge amount of data about the stars in the Galactic bulge spanning sev- eral years to a decade and a half. ⋆ e-mail: (njr, smao)@jb.man.ac.uk; debattis@astro.washington.edu; gerhard@exgal.mpe.mpg.de; sumi@stelab.nagoya-u.ac.jp; lorenzi@exgal.mpe.mpg.de Eyer & Woźniak (2001) first demonstrated that the data can be used to infer the proper motions of stars, down to ∼ mas yr−1. Sumi et al. (2004) obtained the proper motions for millions of stars in the OGLE-II database for a large area of the sky. In this pa- per, we focus on the red clump giants. These stars are bright and they are approximately standard candles, hence their magnitudes can be taken as a crude measure of their distances. As the OGLE-II proper motions are relative, in this paper we compute the proper motion dispersions of bulge stars for all field data presented by Sumi et al. (2004), as they are independent of the unknown proper motion zero-points. These results could aid theoretical modelling efforts for the central regions of the Galaxy. The structure of the paper is as follows. In section 2, we describe the OGLE-II proper motion catalogue and compute the proper motion dispersions for bulge stars in 45 OGLE-II fields. In section 3 we describe the stellar-dynamical model of the Galaxy used in this work and detail how the model was used to generate proper motion dispersions. These model predictions are compared to the observational results in section 4 and in section 5 we discuss the implications of the results. 2 OBSERVED PROPER MOTION DISPERSIONS The second phase of the OGLE experiment observed the Galactic Centre in 49 fields using the 1.3m Warsaw telescope at the Las c© 2005 RAS http://arxiv.org/abs/0704.1619v1 2 Rattenbury et al. −0.5 0 0.5 1 1.5 2 PSfrag replacements (V − I)0 Figure 2. Extinction-corrected colour-magnitude diagram for stars in the OGLE-II field 1. The ellipse defines the selection criteria for RCG stars based on colour and magnitude, see text. Sample stars are also required to have proper motion errors sl,b < 1 mas yr −1 and total proper motion µ < 10 mas yr−1. Campanas Observatory, Chile. Data were collected over an interval of almost four years, between 1997 and 2000. Each field is 0.24◦× 0.95◦ in size. Fig. 1 shows the position of the OGLE-II Galactic Bulge fields which returned data used in this paper. 2.1 Red Clump Giants The red clump giants are metal-rich horizontal branch stars (Stanek et al. 2000 and references therein). Theoretically, one ex- pects their magnitudes to have (small) variations with metallicity, age and initial stellar mass (Girardi & Salaris 2001). Empirically they appear to be reasonable standard candles in the I-band with little dependence on metallicities (Udalski 2000; Zhao et al. 2001). Below we describe the selection of RCG stars in more detail. 2.2 OGLE-II proper motion data Bulge RCG stars are selected from the OGLE-II proper motion cat- alogue by applying a cut in magnitude and colour to all stars in each of the OGLE-II fields. We corrected for extinction and reddening using the maps presented by Sumi (2004) for each field. Stars were selected which are located in an ellipse with centre (V −I)0 = 1.0 , I0 = 14.6; and semi-major (magnitude) and semi-minor (colour) axes of 0.9 and 0.4 respectively, see Fig. 2; a similar selection cri- terion was used by Sumi (2004). Stars with errors in proper mo- tion greater than 1 mas yr−1 in either the l or b directions were excluded. Stars with total proper motion greater than 10 mas yr−1 where similarly excluded, as these are likely to be nearby disk stars, see also section 3.2. Fields 44, 47-49 were not analysed due to the low number of RCG stars appearing in these fields. The proper motion dispersions for the longitude and latitude directions (σl and σb) were computed for each field via a maxi- mum likelihood analysis following Lupton et al. (1987). Assuming a Gaussian distribution of proper motions with mean µ̄ and intrin- sic proper motion dispersion σ, the probability of a single observed proper motion µi with measurement error ξi is: 2π(σ2 + ξ2i ) (µi − µ̄) 2(σ2 + ξ2i ) Maximising the likelihood ln(L) = ln( pi) for µ̄ and σ over all observations we find: ∂ lnL (µi − µ̄) σ2 + ξ2i = 0 (2) ⇒ µ̄ = σ2 + ξ2i −1 (3) ∂ lnL σ2 + ξ2i (µi − µ̄) (σ2 + ξ2i ) = 0 (4) which can be solved numerically to find σ2. The values of µ̄ and σ obtained using the above maximum- likelihood analysis are virtually identical to those obtained via the equations in Spaenhauer et al. (1992). The errors on the observed proper motion dispersion values were determined from a bootstrap analysis using 500 samplings of the observed dataset. 2.3 Extinction In order to ensure the correction for extinction and reddening above does not affect the kinematic measurements, σl and σb were recom- puted for each OGLE-II field using reddening-independent mag- nitudes. Following Stanek et al. (1997) we define the reddening- independent magnitude IV−I: IV−I = I − AI/(AV − AI) (V − I) (5) where AI and AV are the extinctions in the I and V bands deter- mined by Sumi (2004). The position of the red clump in the IV−I, (V − I) CMD varies from field to field. The red clump stars were extracted by iteratively applying a selection ellipse computed from the moments of the data (Rocha et al. 2002) rather than centred on a fixed colour and magnitude. The selection ellipse was recomputed iteratively for each sample until convergence. The proper motion dispersions σl and σb computed using RCG stars selected in this way are consistent with those determined using the original selec- tion criteria on corrected magnitudes and colours. 2.4 Results Table 1 lists the observed proper motion dispersions along with errors for each of the 45 OGLE-II fields considered in this paper. Figures 3 and 4 show the proper motion dispersions σl and σb as a function of Galactic longitude and latitude. A typical value of σl or σb of 3.0 mas yr −1 corresponds to ∼ 110 kms−1, assum- ing a distance to the Galactic centre of 8 kpc. The proper motion dispersion profiles as a function of Galactic longitude shows some slight asymmetry about the Galactic centre. This asymmetry may be related to the tri-axial Galactic bar structure (Stanek et al. 1997; Nishiyama et al. 2005; Babusiaux & Gilmore 2005). The most dis- crepant points in Fig. 3 correspond to the low-latitude fields num- bers 6 and 7 (see Fig. 1). The varying field latitude accounts for some of the scatter in Fig. 3, however we note below in section 4.1 that there are significant variations in the observed proper motion dispersion between some pairs of adjacent fields. Owing to the the c© 2005 RAS, MNRAS 000, 000–000 Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 3 −8−6−4−20246810 replacem Galactic longitude (◦) Figure 1. The position of the 45 OGLE-II fields used in this analysis. The field used in Spaenhauer et al. (1992) is shown, located within OGLE-II field 45 with (l, b) = (1.0245◦,−3.9253◦). lack of fields at positive Galactic latitude, any asymmetry about the Galactic centre in the proper motion dispersions as a function of Galactic latitude is not obvious, see Fig. 4. Field-to-field variations in the proper motion dispersions similarly contribute to the scat- ter seen in Fig. 4, along with the wide range of field longitudes, especially for fields with −4◦ < b < −3◦. Table 2 lists the proper motion dispersions and cross- correlation term Clb in the OGLE-II Baade’s Window fields 45 and 46 along with those found by Kozłowski et al. (2006) using HST data in four BW fields. The two sets of proper motion dis- persions results are consistent at the ∼ 2σ level. It is important to note that the errors on the proper motion dispersions in Table 1 do not include systematic errors. We also note that the selection cri- teria applied to stars in the HST data are very different to those for the ground-based data, in particular the magnitude limits ap- plied in each case. The bulge kinematics from the HST data of Kozłowski et al. (2006) were determined for stars with magnitudes 18.0 < IF814W < 21.5. The approximate reddening-independent magnitude range for the OGLE-II data was 12.5 . IV−I . 14.6. The effects of blending are also very different in the two datasets. It is therefore very reassuring that our results are in general agreement with those obtained by Kozłowski et al. (2006) using higher reso- lution data from the HST. For more comparisons between ground and HST RCG proper motion dispersions, see section 4. Figure 5 shows the cross-correlation term Clb as a function of Galactic co-ordinate. There is a clear sinusoidal structure in the Clb data as a function of Galactic longitude, with the degree of corre- lation between σl and σb changing most rapidly near l ≃ 0 ◦. The Clb data as a function of Galactic latitude may also show some ev- idence of structure. It is possible however, that this apparent struc- ture is due to the different number of fields at each latitude, rather than some real physical cause. 3 GALACTIC MODEL The stellar-dynamical model used in this work was produced us- ing the made-to-measure method (Syer & Tremaine 1996). The model is constrained to reproduce the density distribution con- structed from the dust-corrected L-band COBE/DIRBE map of Spergel et al. (1996). An earlier dynamical model was built to match the total column density of the disk (Bissantz & Gerhard 2002). This dynamical model matched the radial velocity and proper motion data in two fields (including Baade’s window) quite well. No kinematic constraints were imposed during the construc- tion of the model. We refer the readers to Bissantz et al. (2004) for more detailed descriptions. The model used here is constructed as in that case with the further refinement that the vertical den- sity distribution is also included. This is necessary as the vertical kinematics (σb) will also be compared with observations in this paper. However the density distribution near the mid-plane is con- siderably more uncertain, in part because of the dust extinction cor- rection. Thus the model used in this paper can only be considered illustrative, not final. Further efforts to model the vertical density distribution are currently under way and will be reported elsewhere (Debattista et al. 2007, in preparation). In Fig. 6, we present the mean motion of stars in the mid-plane of the Galaxy from this model. A bar position angle of θ = 20◦ is shown here, as this is the orientation favoured both by optical depth measurements (Evans & Belokurov 2002) and by the red clump gi- ant brightness distribution (Stanek et al. 1997) and was the angle c© 2005 RAS, MNRAS 000, 000–000 4 Rattenbury et al. Table 1. Observed proper motion dispersions in the longitude and latitude directions, σl, σb , and cross-correlation term Clb for bulge stars in 45 OGLE-II fields. High precision proper motion data for bulge stars were extracted from the OGLE-II proper motion catalogue (Sumi et al. 2004). N is the number of stars selected from each field. Fields 44, 47-49 were not analysed due to the low number of RCG stars appearing in these fields. Field Field centre PM Dispersions (mas yr−1) Clb N l (◦) b (◦) Longitude σl Latitude σb 1 1.08 -3.62 3.10 ±0.02 2.83 ±0.02 -0.13 ±0.01 15434 2 2.23 -3.46 3.21 ±0.02 2.80 ±0.02 -0.14 ±0.01 16770 3 0.11 -1.93 3.40 ±0.01 3.30 ±0.02 -0.08 ±0.01 26763 4 0.43 -2.01 3.43 ±0.02 3.26 ±0.01 -0.11 ±0.01 26382 5 -0.23 -1.33 3.23 ±0.03 3.00 ±0.04 -0.04 ±0.02 3145 6 -0.25 -5.70 2.61 ±0.02 2.36 ±0.03 -0.06 ±0.01 7027 7 -0.14 -5.91 2.70 ±0.03 2.43 ±0.02 -0.05 ±0.01 6236 8 10.48 -3.78 2.80 ±0.03 2.29 ±0.02 -0.08 ±0.01 5136 9 10.59 -3.98 2.73 ±0.02 2.16 ±0.03 -0.06 ±0.01 5114 10 9.64 -3.44 2.77 ±0.02 2.27 ±0.02 -0.07 ±0.01 5568 11 9.74 -3.64 2.84 ±0.02 2.32 ±0.02 -0.10 ±0.01 5369 12 7.80 -3.37 2.66 ±0.03 2.31 ±0.03 -0.08 ±0.01 6035 13 7.91 -3.58 2.66 ±0.03 2.24 ±0.02 -0.07 ±0.01 5601 14 5.23 2.81 2.97 ±0.02 2.60 ±0.02 0.04 ±0.01 10427 15 5.38 2.63 3.02 ±0.02 2.64 ±0.03 -0.00 ±0.01 8989 16 5.10 -3.29 2.87 ±0.02 2.53 ±0.02 -0.12 ±0.01 9799 17 5.28 -3.45 2.81 ±0.02 2.42 ±0.01 -0.12 ±0.01 10268 18 3.97 -3.14 2.92 ±0.02 2.62 ±0.02 -0.13 ±0.01 14019 19 4.08 -3.35 2.90 ±0.02 2.60 ±0.02 -0.17 ±0.01 13256 20 1.68 -2.47 3.27 ±0.01 2.82 ±0.01 -0.12 ±0.01 17678 21 1.80 -2.66 3.31 ±0.02 2.90 ±0.02 -0.13 ±0.01 17577 22 -0.26 -2.95 3.17 ±0.02 2.84 ±0.02 -0.01 ±0.01 19787 23 -0.50 -3.36 3.15 ±0.01 2.84 ±0.02 -0.04 ±0.01 17996 24 -2.44 -3.36 2.96 ±0.01 2.48 ±0.01 0.02 ±0.01 16397 25 -2.32 -3.56 2.91 ±0.01 2.50 ±0.01 0.02 ±0.01 16386 26 -4.90 -3.37 2.68 ±0.02 2.17 ±0.01 0.02 ±0.01 13099 27 -4.92 -3.65 2.63 ±0.02 2.15 ±0.01 0.03 ±0.01 12728 28 -6.76 -4.42 2.63 ±0.03 2.12 ±0.02 -0.01 ±0.01 8367 29 -6.64 -4.62 2.66 ±0.03 2.09 ±0.02 -0.02 ±0.01 8108 30 1.94 -2.84 3.04 ±0.02 2.70 ±0.02 -0.12 ±0.01 17774 31 2.23 -2.94 3.11 ±0.02 2.74 ±0.01 -0.12 ±0.01 17273 32 2.34 -3.14 3.10 ±0.02 2.78 ±0.01 -0.13 ±0.01 15966 33 2.35 -3.66 3.08 ±0.02 2.77 ±0.02 -0.14 ±0.01 15450 34 1.35 -2.40 3.36 ±0.02 2.92 ±0.01 -0.11 ±0.01 16889 35 3.05 -3.00 3.09 ±0.02 2.72 ±0.02 -0.14 ±0.01 15973 36 3.16 -3.20 3.19 ±0.02 2.77 ±0.02 -0.16 ±0.01 14955 37 0.00 -1.74 3.29 ±0.02 3.04 ±0.01 -0.05 ±0.01 20233 38 0.97 -3.42 3.15 ±0.01 2.84 ±0.02 -0.12 ±0.01 15542 39 0.53 -2.21 3.21 ±0.01 3.00 ±0.01 -0.07 ±0.01 24820 40 -2.99 -3.14 2.84 ±0.01 2.47 ±0.02 0.05 ±0.01 13581 41 -2.78 -3.27 2.78 ±0.01 2.41 ±0.02 0.04 ±0.01 14070 42 4.48 -3.38 2.89 ±0.02 2.63 ±0.02 -0.15 ±0.01 10099 43 0.37 2.95 3.17 ±0.02 2.87 ±0.01 0.02 ±0.01 11467 45 0.98 -3.94 2.97 ±0.04 2.61 ±0.04 -0.13 ±0.02 2380 46 1.09 -4.14 2.90 ±0.04 2.67 ±0.04 -0.16 ±0.03 1803 Table 2. Comparison between proper motion dispersions and cross-correlation term Clb in two of the OGLE-II fields (45 and 46) with proper motion dispersions computed from four nearby HST fields (Kozłowski et al. 2006). Field l (◦) b (◦) σl (mas yr −1) σb (mas yr −1) Clb Ref 119-A 1.32 -3.77 2.89 ±0.10 2.44 ±0.08 -0.14 ±0.04 1 119-C 0.85 -3.89 2.79 ±0.10 2.65 ±0.08 -0.14 ±0.04 1 OGLE-II 45 0.98 -3.94 2.97 ±0.04 2.61 ±0.04 -0.13 ±0.02 2 119-D 1.06 -4.12 2.75 ±0.10 2.56 ±0.09 -0.05 ±0.06 1 95-BLG-11 0.99 -4.21 2.82 ±0.09 2.62 ±0.09 -0.14 ±0.04 1 OGLE-II 46 1.09 -4.14 2.90 ±0.04 2.67 ±0.04 -0.16 ±0.03 2 1Kozłowski et al. (2006) 2This work. c© 2005 RAS, MNRAS 000, 000–000 Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 5 −10−5051015 PSfrag replacements Galactic longitude (◦) Figure 3. Proper motion dispersion in the Galactic longitude (σl) and lati- tude (σb) directions for 45 OGLE-II Galactic bulge fields as a function of field Galactic longitude. Open circles correspond to fields 6, 7, 14, 15 and 43 which have relatively extreme galactic latitudes, see Fig. 1. used in deriving the model. Clearly one can see that the mean mo- tion follows elliptical paths around the Galactic bar. The analysis of OGLE-II proper motions by Sumi et al. (2003b) is consistent with this streaming motion. 3.1 Model stellar magnitudes The model has a four-fold symmetry, obtained by a rotation of π radians around the vertical axis and by positioning the Sun above or below the mid-plane. The kinematics of model particles falling within the solid angle of each OGLE-II field were combined to those from the three other equivalent lines-of-sight. This procedure allows an increase in the number of model particles used for the predictions of stellar kinematics. We assign magnitudes to stars in the Galactic model described above which appear in the same fields as that observed by the OGLE collaboration. Number counts as a function of I-band ap- parent magnitude, I , were used to compute the fraction of RCG stars in each of the OGLE-II fields. Figure 7 shows an example of the fitted number count function Nk(I) for one of the k = 1 . . . 49 OGLE-II fields, where Nk(I) is of the form of a power-law and a Gaussian (Sumi 2004): Nk(I) = ak10 (bkI) + ck exp −(I − Ip,k) −6 −4 −2 0 2 4 PSfrag replacements Galactic latitude (◦) Figure 4. Proper motion dispersion in the Galactic longitude (σl) and lati- tude (σb) directions for 45 OGLE-II Galactic bulge fields as a function of field Galactic latitude. Open circles correspond to fields 6, 7, 14, 15 and 43 which have relatively extreme galactic latitudes, see Fig. 1. where the constants ak, bk, ck, Ip,k, σk are determined for each of the k OGLE-II fields, see Table 3. The fraction Rk of RCG stars is evaluated as the ratio of the area under the Gaussian component of equation (6) to the area under the full expression. The integrals are taken over ±3σk around the RCG peak in Nk(I) for each of the k OGLE-II fields. Fields 44 and 47-49 are not included as there are insufficient RCGs in the OGLE-II fields to fit equation (6). Fig- ure 7 shows that the model number count function fails to fit the observed number counts well for magnitudes I ≃ 15.4. In order to convert stellar density to a distribution of apparent magnitude, the relevant quantity is ρr3 (Bissantz & Gerhard 2002). Depending on the line-of-sight, this quantity can give asymmetric magnitude dis- tributions through the bulge. Using the best-fitting analytic tri-axial density models for the bulge (Rattenbury et al. 2007, in prepara- tion), this asymmetry is observed and may explain the excess of stars in the number count histograms, compared to the best-fitting two-component fit of equation (6). The inability of equation (6) to model completely all features in the observed number counts in some cases leads to an additional uncertainty in the magnitude lo- cation of the fitted Gaussian peak. Computing the apparent magni- tude distribution as ∝ ρr3 also produces a small shift in the peak of the magnitude distribution. This shift is ∼ +0.04 mag for l = 0◦, b = 0◦. The proper motion dispersions computed here are unlikely to be sensitive to these small offsets. c© 2005 RAS, MNRAS 000, 000–000 6 Rattenbury et al. Table 3. Values of fitted parameters in equation (6) for all 45 OGLE-II fields used in this analysis. R is the ratio of observed RCG stars to the total number of stars in each field, evaluated over ±3σ around the RCG peak magnitude, Ip, where σ is the fitted Gaussian spread in equation (6). The magnitudes of the model RCG stars are shifted by ∆m to correspond with the observed mean RCG magnitude in each field. The total number of model stars in each field assigned RCG magnitudes and colours is nrcg and the total number of model stars in each field is nall. The corresponding total model weight values for each field are given by wrcg and wall respectively. The large values of σ for fields 8-11 might be related to their position at large positive longitudes, and could indicate a structure such as the end of the bar, a ring or spiral arm. An analysis of the bar morphology based on these results is underway (Rattenbury et al. 2007, in preparation). Field a b c Ip σ R ∆m nrcg nall wrcg wall 1 0.11 0.27 1735.70 14.62 0.29 0.40 0.43 585 1773 277.2 842.4 2 0.15 0.26 1876.47 14.54 -0.29 0.43 0.41 621 1802 298.1 853.9 3 0.16 0.28 4692.78 14.66 0.25 0.44 0.54 1264 3626 668.5 1911.1 4 0.17 0.28 4438.63 14.65 0.24 0.44 0.52 1298 3653 670.8 1922.2 5 0.05 0.33 4581.59 14.70 0.28 0.33 0.55 1342 4668 755.7 2685.7 6 0.04 0.27 519.71 14.57 0.37 0.34 0.36 152 583 69.5 270.8 7 0.03 0.28 457.42 14.55 0.39 0.32 0.36 143 527 71.9 243.8 8 0.04 0.27 259.65 14.37 -0.51 0.22 0.35 96 561 41.7 236.2 9 0.04 0.27 270.90 14.34 0.51 0.25 -0.05 96 497 46.1 230.9 10 0.08 0.26 321.32 14.44 0.52 0.22 0.40 131 654 49.1 260.1 11 0.04 0.28 316.25 14.45 0.50 0.23 0.28 128 695 57.5 339.4 12 0.12 0.25 546.85 14.43 0.38 0.28 0.41 238 908 100.7 393.1 13 0.10 0.25 520.45 14.45 0.37 0.29 0.15 190 863 83.9 392.4 14 0.09 0.28 1309.28 14.55 0.32 0.35 0.34 458 1587 216.0 767.4 15 0.05 0.29 1154.52 14.57 0.33 0.31 0.55 421 1661 185.2 761.8 16 0.12 0.27 1042.72 14.50 0.35 0.33 0.50 397 1383 172.8 601.1 17 0.12 0.26 1069.07 14.48 0.34 0.35 0.25 406 1443 212.4 753.4 18 0.17 0.26 1569.83 14.49 0.31 0.40 0.35 527 1564 234.7 702.4 19 0.17 0.26 1429.23 14.51 0.32 0.40 0.44 434 1365 184.4 608.5 20 0.20 0.27 3012.09 14.58 0.26 0.42 0.53 939 2728 480.3 1398.3 21 0.15 0.27 2793.36 14.58 0.26 0.43 0.45 900 2554 443.5 1260.0 22 0.12 0.28 2574.77 14.74 0.28 0.42 0.51 830 2419 382.5 1113.3 23 0.09 0.28 2147.71 14.73 0.29 0.42 0.47 767 2126 384.2 1060.6 24 0.12 0.27 2130.41 14.82 0.28 0.42 0.50 595 1864 269.6 905.4 25 0.07 0.28 2002.91 14.82 0.28 0.42 0.51 581 1782 289.5 885.1 26 0.09 0.27 1452.89 14.83 0.31 0.38 0.55 375 1325 159.7 570.5 27 0.07 0.27 1319.67 14.81 0.32 0.39 0.40 387 1238 172.5 578.9 28 0.04 0.28 563.00 14.79 0.31 0.31 0.62 162 649 72.3 293.5 29 0.05 0.27 559.86 14.78 0.31 0.32 0.44 156 607 70.7 267.5 30 0.18 0.27 2533.75 14.57 0.27 0.42 0.41 754 2195 362.4 1026.7 31 0.17 0.27 2354.64 14.53 0.28 0.43 0.32 763 2229 361.9 1122.1 32 0.17 0.26 2062.96 14.53 0.28 0.42 0.41 638 1962 291.8 938.5 33 0.13 0.27 1614.83 14.56 0.31 0.41 0.34 559 1586 265.5 760.7 34 0.18 0.27 3210.56 14.60 0.27 0.43 0.42 990 2936 503.0 1473.9 35 0.16 0.26 1963.53 14.53 0.29 0.41 0.45 663 1925 307.7 913.7 36 0.16 0.26 1773.62 14.51 0.30 0.41 0.47 574 1902 301.1 943.5 37 0.18 0.28 4901.22 14.64 0.25 0.42 0.43 1439 4077 794.9 2218.5 38 0.12 0.27 2091.19 14.64 0.28 0.43 0.46 662 1945 319.2 948.1 39 0.18 0.28 3919.30 14.69 0.26 0.44 0.65 1217 3456 631.8 1804.2 40 0.09 0.28 2181.18 14.87 0.29 0.41 0.62 668 1936 315.1 933.3 41 0.10 0.28 2180.49 14.87 0.28 0.42 0.55 626 1905 318.2 965.4 42 0.13 0.26 1215.38 14.52 0.35 0.37 0.40 425 1389 190.2 637.7 43 0.10 0.28 2659.91 14.84 0.27 0.41 0.79 777 2290 345.8 1074.6 45 0.11 0.27 1541.36 14.59 0.31 0.40 0.38 485 1568 228.3 767.7 46 0.09 0.27 1428.63 14.60 0.30 0.41 0.38 454 1400 221.6 669.5 Each star in the galactic model is assigned a RCG magnitude with probability Rk for each field. The apparent magnitude is com- puted using the model distance. Stars which are not assigned a RCG magnitude are assigned a magnitude using the power-law compo- nent of equation (6), defined over the same limits used to compute Rk. Here we implicitly assume that the RCG stars trace the overall Galactic disk and bulge populations. The RCG luminosity function is approximated by a Gaussian distribution with mean magnitude −0.26 and σ = 0.2. These as- sumptions are mostly consistent with observations (Stanek et al. 1997) and the fitted distribution from Udalski (2000), but there may be small offsets between local and bulge red clump giants. It was noted in Sumi (2004) that there is some as-yet unexplained off- set (0.3 mag) in the extinction-corrected mean RCG magnitudes in the OGLE fields. A possible explanation for this offset is that the RCG population effects are large: so that the absolute magnitude of RCG stars is significantly different for RCGs in the bulge com- pared to local RCGs, as claimed by Percival & Salaris (2003) and c© 2005 RAS, MNRAS 000, 000–000 Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 7 −10−5051015 −0.15 −0.05 −6 −4 −2 0 2 4 −0.15 −0.05 PSfrag replacements Galactic longitude (◦) Galactic latitude (◦) Figure 5. Cross-correlation term Clb for 45 OGLE-II Galactic bulge fields as a function of field Galactic longitude (top) and latitude (bottom). Open circles in the top plot of Clb vs. l correspond to fields 6, 7, 14, 15 and 43 which have relatively extreme galactic latitudes, see Fig. 1. Salaris et al. (2003). A different value of the distance to the Galac- tic centre to that assumed here (8 kpc) would in part account for the discrepancy, however would not remove it completely. Using a value of 7.6 kpc (Eisenhauer et al. 2005; Nishiyama et al. 2006) as the distance to the Galactic centre would change the zero-point by 0.12 mag, resulting in an offset value of 0.18 mag. It is also possi- ble that reddening toward the Galactic centre is more complicated than assumed in Sumi (2004). In order to compare the model proper motion results with the observed data, it was necessary to shift the mean model RCG magnitudes to correspond with that observed in each of the OGLE fields. The model RCG magnitudes were fitted with a Gaussian curve. The mean of the model RCG magnitudes was then shifted by a value ∆m, see Table 3, to correspond with the observed mean RCG magnitude in each of the OGLE fields. No- tice that we concentrate on second-order moments (proper motion dispersions) of the proper motion, so a small shift in the zero-point has little effect on our results. Every model particle has an associated weight, wi. The par- ticle weight can take values 0 < wi . 20. In order to account for this weighting, ⌈wi⌉ stars are generated for each particle with the same kinematics but magnitudes determined as above. ⌈wi⌉ is the nearest integer toward +∞. Each model star is then assigned a weight, γi = wi/⌈wi⌉. Notice this procedure allows us to in- crease the effective number of particles to better sample the lumi- nosity function. The total number of stars and the number of stars assigned RCG magnitudes in each field are listed in Table 3 as nall Figure 6. Galactic kinematics from the model of Debattista et al (2007, in preparation). Bulk stellar motion in the mid-plane of the Galaxy is shown super-imposed on the stellar density. The Sun is located at the origin (not shown). An example line of sight is shown. The model can be rotated to four equivalent positions for each line of sight due to symmetry (see section 3.1). 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 PSfrag replacements stars non-RCG stars Figure 7. Number count as function of apparent magnitude, I , for OGLE-II field 1. The number count histogram is shown along with the fitted function equation (6). The fraction of RCG stars, Rk , is evaluated over the magni- tude range Ip ±3σ for each of the (k = 1 . . . 49) OGLE-II fields. The ratio Rk is assumed to be the same at all stellar distances for each field. and nrcg respectively. 81806 stars from the model were used to compare model kinematics to observed values. 3.2 Model kinematics Stars with apparent magnitudes within the limits mmin = 13.7 and mmax = 15.5, were selected from the model data. This magnitude range corresponds to the selection criteria imposed on the observed data sample, see section 2.2. Model stars with total proper motions greater than 10 mas yr−1 (corresponding to > 380 kms−1 at a c© 2005 RAS, MNRAS 000, 000–000 8 Rattenbury et al. distance of the Galactic centre) were excluded on the basis that such stars would be similarly excluded from any observed sample. The fraction of weight removed and number of stars removed in this way only amounted to a few per cent of the total weight and number of stars in each field. Bulge model stars were selected by requiring a distance d > 6 kpc. The mean proper motion and proper motion dispersions in the latitude and longitude directions were computed along with their errors for all model stars in each field which obey the above selec- tion criteria. The weights on model stars, γi, were used to compute these values. We then tested whether the finite and discrete nature of the model data gives rise to uncertainties in the measured proper mo- tion dispersion values. We measured the intrinsic noise in the model by comparing the proper motion dispersions computed for four equivalent lines-of-sight through the model for each field. The spread of the proper motion dispersions for each field was then used as the estimate of the intrinsic noise in the model. The mean (me- dian) value of these errors in the longitude and latitude directions are 0.08 (0.06) and 0.12 (0.097) mas yr−1 respectively. The statistical error for the proper motion dispersions in the longitude and latitude directions for each field were combined in quadrature with the error arising from the finite discrete nature of the model data to give the total error on the proper motion disper- sions computed from the model. 4 COMPARISON BETWEEN THEORETICAL MODEL AND OBSERVED DATA The observed and predicted proper motion dispersions for each of the OGLE-II fields are shown in Table 4. Fig. 8 shows the observed proper motion dispersions for each of the analysed OGLE-II fields plotted against the predicted model proper motion dispersions. Fig. 8 shows that the model predictions are in general agree- ment with observed proper motion dispersions for the OGLE- II fields. The model has been used previously to predict the proper motion dispersions of 427 stars1 entries observed by Spaenhauer et al. (1992) in a single 6′× 6′ field toward the bulge (Bissantz et al. 2004). The model value of σl in this previous anal- ysis was in agreement with the observed value, yet the model and observed values of σb were significantly different. The 6 ′× 6′ field used by Spaenhauer et al. (1992) falls within the OGLE-II field number 45. The model prediction of σl for stars in OGLE field 45 is completely consistent with the measured value. The model pre- diction of σb shows a similar discrepancy to the previous analysis of Bissantz et al. (2004). Fig. 9 shows the ratio R = σl/σb and cross-correlation term Clb = σlb/(σlσb) computed using the model and observed data. Typically the model predicts more anisotropic motion with R > 1 than what is observed. The model predictions for stellar kinematics in the latitude di- rection may be problematic. This is not surprising as the model is not well constrained toward the plane due to a lack of observational data because of the heavy dust extinction. The problem is currently under investigation. Similarly, the model predictions for σl degrade as l increases. This is because the model performance has been op- timised for regions close to the Galactic centre. 1 There are two repeated entries in Table 2 of Spaenhauer et al. (1992). The significant difference between the observed proper mo- tion dispersions of adjacent fields (e.g. fields 1 and 45) might hint at some fine-scale population effect, whereby a group of stars surviv- ing the selection criteria have a significant and discrepant kinematic signature. Higher-accuracy observations using the HST support this evidence of such population effects (Kozłowski et al. 2006). No attempt has been made to account for the blending of flux inherent in the OGLE-II crowded-field photometry. It is certain that a fraction of stars in each OGLE-II field suffers from some degree of blending (Kozłowski et al. 2006). To investigate this ef- fect, we checked one field covering the lens MACHO-95-BLG- 37 (l = 2.54◦, b = 3.33◦, Thomas et al. 2005) from the HST proper motion survey of Kozłowski et al. (2006), which falls inside OGLE-II field number 2. HST images suffer much less blending, but the field of view is small, and so it has only a dozen or so clump giants. We derive a proper motion of σl = 3.13 ± 0.57 mas yr and σb = 2.17±0.40 mas yr −1. These values agree with our kine- matics in field 2 within 0.2σ for σl and 1.6σ for σb. The errors in our proper motion dispersions are very small (∼ km s−1 at a dis- tance of the Galactic centre), but it is likely that we underestimate the error bars on the observed data due to systematic effects such as blending. 4.1 Understanding the differences We now seek to understand the cause of the differences between the model and the Milky Way, at least at a qualitative level. We first consider the possibility that the difference can be explained by some systematic effect. We compute the differences between ob- served proper motion dispersions of nearest fields for fields with separations less than 0.25 degrees. No pair of fields is used twice, and the difference ∆ = σi − σj is always plotted such that |bi| ≥ |bj |. ∆l,obs and ∆b,obs denote the difference in observed proper motion dispersions between adjacent fields in the longitude and latitude directions respectively. The equivalent quantities pre- dicted from the model are denoted ∆l,mod and ∆b,mod. In Fig. 10 we see that the deviations ∆l,obs and ∆b,obs scatter about 0, but have a quite broad distribution in both the l and b directions, with several fields inconsistent with zero difference at 1σ (defined as the sum in quadrature of the uncertainties of the corresponding quanti- ties of the two fields under comparison). Several deviations are as large as 0.2 mas yr−1 (corresponding to ≃ 8 kms−1 at the Galac- tic centre) and many σ’s away from zero. In view of the fact that these differences have mean close to zero, it is possible that these deviations are due to some systematic effect rather than to intrinsic substructure in the Milky Way. We return to this point briefly in the discussion. In the case of the model uncertainties, however, Fig. 11 shows that in most cases the differences ∆l,mod and ∆b,mod are consistent with zero at the 1σ level, indicating that these error estimates are robust. We now seek to explore the correlations of the residuals with properties of the model. We plot residuals δl,b = (σmod − σobs), where σmod and σobs are the model and observed proper motion dispersions in the corresponding Galactic co-ordinate. The errorbar length is (u2mod+u 1/2 where umod and uobs are the uncertain- ties in the model and observed proper motion dispersions, respec- tively. Plotting these quantities as a function of l, we note that there is no significant correlation, but that the largest deviations in the latitude proper motion dispersion occur close to l = 0, see Fig. 12. In plotting δl,b as a function of b, the reason which becomes evi- dent is that the fields closest to the mid-plane have the largest δb, c© 2005 RAS, MNRAS 000, 000–000 Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 9 2.6 2.8 3 3.2 3.4 27 28 PSfrag replacements Model σl (mas yr Model σb (mas yr Observed σb (mas yr BW fields |l|>5◦ |l|<5◦ BW fields |l|>5◦ |l|<5◦ 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 11 12 24 25 35 36 46PSfrag replacements Model σl (mas yr Observed σl (mas yr Model σb (mas yr BW fields |l|>5◦ |l|<5◦ BW fields |l|>5◦ |l|<5◦ Figure 8. Comparison between observed and predicted proper motion dispersions for stars in the OGLE-II proper motion catalogue of Sumi et al. (2004). Left: Proper motion dispersions in the galactic longitude direction, σl. The OGLE-II field number is indicated adjacent to each point, see also Fig. 1. Fields with galactic longitude |l| > 5◦ are shown in magenta; fields within Baade’s window are shown in red; all other fields in blue. Right: Proper motion dispersions in the galactic latitude direction, σb, shown with the same colour scheme. 0.8 1 1.2 1.4 1.6 1.8 PSfrag replacements Model σl/σb Model Clb Observed Clb BW fields |l|>5◦ |l|<5◦ BW fields |l|>5◦ |l|<5◦ −0.1 0 0.1 PSfrag replacements Model σl/σb Observed σl/σb Model Clb BW fields |l|>5◦ |l|<5◦ BW fields |l|>5◦ |l|<5◦ Figure 9. Left: Ratio of proper motion dispersions R = σl/σb for the observed OGLE-II proper motion data and model predictions. The model generally predicts more anisotropic motion, i.e. R > 1 than is observed in the data. Right: The cross-correlation term Clb = σlb/σlσb. see Fig. 13. The density distribution in this region is uncertain due to presence of dust and the large extinction corrections required. This may explain why the residuals of σb seem to correlate more with b than those of σl. We note that the σl residuals also seem to have some dependence on b. A possible explanation is that there is some additional effect due to dust which has not been accounted 5 DISCUSSION Red clump giant stars in the dense fields observed by the OGLE-II microlensing survey can be used as tracers of the bulge density and motion over a large region toward the Galactic centre. The proper motion dispersions of bulge RCG stars in the OGLE-II proper mo- tion catalogue of Sumi et al. (2004) were calculated for 45 OGLE- II fields. The kinematics derived from the ground-based OGLE-II data were found to be in agreement with HST observations in two fields from Kozłowski et al. (2006). It is reassuring that the results presented here are consistent with those derived from the higher resolution HST data, despite possible selection effects and blend- The observed values of σl and σb were compared to predic- tions from the made-to-measure stellar-dynamical model of De- battista et al. (2007, in preparation). In general, the model gives predictions qualitatively similar to observed values of σl and σb for fields close to the Galactic centre. The model is in agree- ment with observed OGLE-II data in the direction previously tested by Bissantz et al. (2004). Using the definition of De Lorenzi et al. (2007), the effective number of particles in the model used here c© 2005 RAS, MNRAS 000, 000–000 10 Rattenbury et al. Table 4. Proper motion dispersions in the longitude and latitude directions, σl, σb , and cross-correlation term Clb for bulge stars in 45 OGLE-II fields. High precision proper motion data for bulge stars were extracted from the OGLE-II proper motion catalogue (Sumi et al. 2004). N is the number of stars selected from each field. Field 44 was not used due to the low number of RCGs in this field. PM Dispersions (mas yr−1) Clb Field Field centre Longitude σl Latitude σb l (◦) b (◦) Model Observed Model Observed Model Observed N 1 1.08 -3.62 3.02 ±0.08 3.10 ±0.02 2.46 ±0.08 2.83 ±0.02 0.01 ±0.08 -0.13 ±0.01 15434 2 2.23 -3.46 3.02 ±0.06 3.21 ±0.02 2.68 ±0.15 2.80 ±0.02 0.12 ±0.03 -0.14 ±0.01 16770 3 0.11 -1.93 3.19 ±0.08 3.40 ±0.01 2.64 ±0.02 3.30 ±0.02 0.02 ±0.01 -0.08 ±0.01 26763 4 0.43 -2.01 3.26 ±0.05 3.43 ±0.02 2.80 ±0.06 3.26 ±0.01 0.01 ±0.03 -0.11 ±0.01 26382 5 -0.23 -1.33 3.22 ±0.15 3.23 ±0.03 2.30 ±0.07 3.00 ±0.04 -0.01 ±0.03 -0.04 ±0.02 3145 6 -0.25 -5.70 3.26 ±0.16 2.61 ±0.02 2.42 ±0.23 2.36 ±0.03 -0.02 ±0.13 -0.06 ±0.01 7027 7 -0.14 -5.91 2.95 ±0.15 2.70 ±0.03 2.49 ±0.12 2.43 ±0.02 -0.15 ±0.16 -0.05 ±0.01 6236 8 10.48 -3.78 3.07 ±0.09 2.80 ±0.03 1.98 ±0.14 2.29 ±0.02 0.00 ±0.09 -0.08 ±0.01 5136 9 10.59 -3.98 3.28 ±0.21 2.73 ±0.02 2.03 ±0.07 2.16 ±0.03 -0.03 ±0.07 -0.06 ±0.01 5114 10 9.64 -3.44 3.30 ±0.32 2.77 ±0.02 2.89 ±0.62 2.27 ±0.02 0.09 ±0.08 -0.07 ±0.01 5568 11 9.74 -3.64 3.01 ±0.20 2.84 ±0.02 2.22 ±0.29 2.32 ±0.02 -0.08 ±0.09 -0.10 ±0.01 5369 12 7.80 -3.37 3.31 ±0.10 2.66 ±0.03 2.29 ±0.06 2.31 ±0.03 -0.09 ±0.06 -0.08 ±0.01 6035 13 7.91 -3.58 3.26 ±0.18 2.66 ±0.03 2.29 ±0.12 2.24 ±0.02 0.05 ±0.02 -0.07 ±0.01 5601 14 5.23 2.81 3.21 ±0.05 2.97 ±0.02 2.62 ±0.13 2.60 ±0.02 0.06 ±0.04 0.04 ±0.01 10427 15 5.38 2.63 3.31 ±0.12 3.02 ±0.02 2.46 ±0.07 2.64 ±0.03 0.04 ±0.04 -0.00 ±0.01 8989 16 5.10 -3.29 3.19 ±0.07 2.87 ±0.02 2.23 ±0.08 2.53 ±0.02 0.03 ±0.08 -0.12 ±0.01 9799 17 5.28 -3.45 3.09 ±0.09 2.81 ±0.02 2.50 ±0.07 2.42 ±0.01 -0.01 ±0.11 -0.12 ±0.01 10268 18 3.97 -3.14 3.20 ±0.09 2.92 ±0.02 2.48 ±0.08 2.62 ±0.02 0.02 ±0.03 -0.13 ±0.01 14019 19 4.08 -3.35 3.06 ±0.13 2.90 ±0.02 2.49 ±0.26 2.60 ±0.02 0.01 ±0.03 -0.17 ±0.01 13256 20 1.68 -2.47 3.12 ±0.06 3.27 ±0.01 2.66 ±0.05 2.82 ±0.01 0.07 ±0.03 -0.12 ±0.01 17678 21 1.80 -2.66 3.12 ±0.06 3.31 ±0.02 2.57 ±0.08 2.90 ±0.02 -0.02 ±0.03 -0.13 ±0.01 17577 22 -0.26 -2.95 3.17 ±0.04 3.17 ±0.02 2.46 ±0.12 2.84 ±0.02 0.01 ±0.03 -0.01 ±0.01 19787 23 -0.50 -3.36 3.13 ±0.17 3.15 ±0.01 2.62 ±0.10 2.84 ±0.02 -0.02 ±0.14 -0.04 ±0.01 17996 24 -2.44 -3.36 2.77 ±0.04 2.96 ±0.01 2.32 ±0.10 2.48 ±0.01 -0.04 ±0.04 0.02 ±0.01 16397 25 -2.32 -3.56 2.76 ±0.07 2.91 ±0.01 2.47 ±0.15 2.50 ±0.01 -0.04 ±0.03 0.02 ±0.01 16386 26 -4.90 -3.37 2.80 ±0.17 2.68 ±0.02 2.22 ±0.04 2.17 ±0.01 -0.00 ±0.03 0.02 ±0.01 13099 27 -4.92 -3.65 2.78 ±0.07 2.63 ±0.02 2.19 ±0.04 2.15 ±0.01 -0.06 ±0.02 0.03 ±0.01 12728 28 -6.76 -4.42 3.02 ±0.11 2.63 ±0.03 2.44 ±0.36 2.12 ±0.02 0.05 ±0.06 -0.01 ±0.01 8367 29 -6.64 -4.62 3.02 ±0.21 2.66 ±0.03 1.79 ±0.14 2.09 ±0.02 -0.00 ±0.11 -0.02 ±0.01 8108 30 1.94 -2.84 3.13 ±0.07 3.04 ±0.02 2.59 ±0.11 2.70 ±0.02 -0.04 ±0.08 -0.12 ±0.01 17774 31 2.23 -2.94 3.08 ±0.05 3.11 ±0.02 2.68 ±0.11 2.74 ±0.01 0.08 ±0.05 -0.12 ±0.01 17273 32 2.34 -3.14 3.10 ±0.09 3.10 ±0.02 2.56 ±0.04 2.78 ±0.01 0.11 ±0.02 -0.13 ±0.01 15966 33 2.35 -3.66 2.82 ±0.11 3.08 ±0.02 2.57 ±0.12 2.77 ±0.02 0.07 ±0.06 -0.14 ±0.01 15450 34 1.35 -2.40 3.18 ±0.06 3.36 ±0.02 2.62 ±0.03 2.92 ±0.01 0.04 ±0.02 -0.11 ±0.01 16889 35 3.05 -3.00 3.05 ±0.05 3.09 ±0.02 2.59 ±0.07 2.72 ±0.02 0.08 ±0.03 -0.14 ±0.01 15973 36 3.16 -3.20 3.00 ±0.06 3.19 ±0.02 2.95 ±0.40 2.77 ±0.02 -0.05 ±0.08 -0.16 ±0.01 14955 37 0.00 -1.74 3.29 ±0.04 3.29 ±0.02 2.70 ±0.04 3.04 ±0.01 -0.01 ±0.01 -0.05 ±0.01 20233 38 0.97 -3.42 3.01 ±0.07 3.15 ±0.01 2.60 ±0.14 2.84 ±0.02 0.07 ±0.07 -0.12 ±0.01 15542 39 0.53 -2.21 3.22 ±0.03 3.21 ±0.01 2.69 ±0.06 3.00 ±0.01 0.01 ±0.04 -0.07 ±0.01 24820 40 -2.99 -3.14 2.84 ±0.04 2.84 ±0.01 2.28 ±0.07 2.47 ±0.02 -0.09 ±0.07 0.05 ±0.01 13581 41 -2.78 -3.27 2.86 ±0.06 2.78 ±0.01 2.60 ±0.19 2.41 ±0.02 -0.16 ±0.07 0.04 ±0.01 14070 42 4.48 -3.38 3.07 ±0.05 2.89 ±0.02 2.44 ±0.15 2.63 ±0.02 0.02 ±0.02 -0.15 ±0.01 10099 43 0.37 2.95 3.13 ±0.06 3.17 ±0.02 2.72 ±0.10 2.87 ±0.01 0.04 ±0.07 0.02 ±0.01 11467 45 0.98 -3.94 3.02 ±0.05 2.97 ±0.04 2.42 ±0.14 2.61 ±0.04 0.06 ±0.11 -0.13 ±0.02 2380 46 1.09 -4.14 2.87 ±0.08 2.90 ±0.04 2.53 ±0.21 2.67 ±0.04 -0.03 ±0.06 -0.16 ±0.03 1803 is 3986. This relatively low number results in large errors on the model proper motion dispersions and we therefore recommend re- garding interpretations based on this model with some caution. An improved model with a larger number of particles (the recent study by De Lorenzi et al. (2007) has an effective particle number ∼ 106) will decrease the errors on the model predictions and allow a more useful comparison between model and observed proper motion dis- persions. The OGLE-II fields mostly extend over ∼ 17◦ in longi- tude and about 5◦ in latitude across the Galactic bulge region and can therefore provide a more powerful set of constraints on stellar motions predicted by galactic models. The high-accuracy proper motion data for the 45 fields and those obtained with HST (Kozłowski et al. 2006) can be used as direct input in the made-to- measure method to construct a better constrained dynamical model of the Milky Way. The statistical errors of our proper motion dispersions are small (∼ km s−1), but systematic uncertainties (for example due to incorrect dust extinction treatment) which were not included in the analysis may be significant. Nevertheless, it is interesting to note that there appears to be significant difference between the ob- served proper motion dispersions of adjacent fields (e.g. fields 1 c© 2005 RAS, MNRAS 000, 000–000 Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 11 0.22 0.23 0.24 0.25 PSfrag replacements Separation (◦) Figure 10. Difference between observed proper motion dispersions for pairs of fields with separations less than 0.25 degrees (corresponding to ≃ 40 pc at the Galactic centre). and 45). This might hint at some fine-scale population effect,where the kinematics of the bulge may be not in total equilibrium (e.g. due to a small accretion event). Higher-accuracy observations using the HST may provide further evidence of such population effects. We note that Rich et al. (2006) suggest the possible existence of cold structures using data from a radial velocity survey of Galactic bulge M giant stars although their conclusion could be strengthened by a larger sample of stars. The OGLE-II proper motion catalogue (Sumi et al. 2004) for millions of bulge stars is still somewhat under-explored. For exam- ple, it will be interesting to explore the nature of the high proper motion stars (µ > 10 mas yr−1) and search for wide binaries in the catalogue. Some exploration along these lines is under way. ACKNOWLEDGEMENTS We thank Drs. Vasily Belokurov, Wyn Evans and Martin Smith for helpful discussions, and the anonymous referee for their helpful suggestions. NJR acknowledges financial support by a PPARC PDRA fellowship. This work was partially supported by the European Community’s Sixth Framework Marie Curie Research Training Network Programme, Contract No. MRTN-CT-2004-505183 ‘AN- GLES’. VPD is supported by a Brooks Prize Fellowship at the Uni- 0.22 0.23 0.24 0.25 PSfrag replacements Separation (◦) Figure 11. Difference between model proper motion dispersions for pairs of fields with separations less than 0.25 degrees. versity of Washington and receives partial support from NSF ITR grant PHY-0205413. 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Red clump giants in the Galactic bulge are approximate standard candles and hence they can be used as distance indicators. We compute the proper motion dispersions of RCG stars in the Galactic bulge using the proper motion catalogue from the second phase of the Optical Gravitational Microlensing Experiment (OGLE-II, Sumi et al. 2004) for 45 fields. The proper motion dispersions are measured to a few per cent accuracy due to the large number of stars in the fields. The observational sample is comprised of 577736 stars. These observed data are compared to a state-of-the-art particle simulation of the Galactic bulge region. The predictions are in rough agreement with observations, but appear to be too anisotropic in the velocity ellipsoid. We note that there is significant field-to-field variation in the observed proper motion dispersions. This could either be a real feature, or due to some unknown systematic effect.

Introduction Observed Proper Motion Dispersions Red Clump Giants OGLE-II proper motion data Extinction Results Galactic Model Model stellar magnitudes Model kinematics Comparison between theoretical model and observed data Understanding the differences Discussion

Intramolecular long-range correlations in polymer melts: The segmental size distribution and its moments J.P. Wittmer,∗ P. Beckrich, H. Meyer, A. Cavallo, A. Johner, and J. Baschnagel† Institut Charles Sadron, CNRS, 23 rue du Loess, 67037 Strasbourg Cédex, France (Dated: November 1, 2018) Abstract Presenting theoretical arguments and numerical results we demonstrate long-range intrachain correlations in concentrated solutions and melts of long flexible polymers which cause a systematic swelling of short chain segments. They can be traced back to the incompressibility of the melt leading to an effective repulsion u(s) ≈ s/ρR3(s) ≈ ce/ s when connecting two segments together where s denotes the curvilinear length of a segment, R(s) its typical size, ce ≈ 1/ρb3e the “swelling coefficient”, be the effective bond length and ρ the monomer density. The relative deviation of the segmental size distribution from the ideal Gaussian chain behavior is found to be proportional to u(s). The analysis of different moments of this distribution allows for a precise determination of the effective bond length be and the swelling coefficient ce of asymptotically long chains. At striking variance to the short-range decay suggested by Flory’s ideality hypothesis the bond-bond correlation function of two bonds separated by s monomers along the chain is found to decay algebraically as 1/s3/2. Effects of finite chain length are considered briefly. PACS numbers: 61.25.Hq,64.60.Ak,05.40.Fb ∗Electronic address: jwittmer@ics.u-strasbg.fr †URL: http://www-ics.u-strasbg.fr/~etsp/welcome.php http://arxiv.org/abs/0704.1620v2 mailto:jwittmer@ics.u-strasbg.fr http://www-ics.u-strasbg.fr/~etsp/welcome.php I. FLORY’S IDEALITY HYPOTHESIS REVISITED A cornerstone of polymer physics. Polymer melts are dense disordered systems con- sisting of macromolecular chains [1]. Theories that predict properties of chains in a melt or concentrated solutions generally start from the “Flory ideality hypothesis” formulated already in the 1940s by Flory [2, 3, 4]. This cornerstone of polymer physics states that chain conformations correspond to “ideal” random walks on length scales much larger than the monomer diameter [1, 4, 5, 6]. The commonly accepted justification of this mean-field result is that intrachain and interchain excluded volume forces compensate each other if many chains strongly overlap which is the case for three-dimensional melts [5]. Since these systems are essentially incompressible, density fluctuations are known to be small. Hence, all correlations are supposed to be short-ranged as has been systematically discussed first by Edwards who developed the essential statistical mechanical tools [6, 7, 8, 9, 10] also used in this paper. One immediate consequence of Flory’s hypothesis is that the mean-squared size of chain segments of curvilinear length s = m − n (with 1 ≤ n < m < N) should scale as R2e(s) ≡ 〈r2〉 = b2es if the two monomers n and m on the same chain are sufficiently separated along the chain backbone, and local correlations may be neglected (1 ≪ s). For the total chain (s = N−1 ≫ 1) this implies obviously that R2e(N−1) = b2e(N−1) ≈ b2eN . Here, N denotes the number of monomers per chain, r the end-to-end vector of the segment, r = ||r|| its length and be the “effective bond length” of asymptotically long chains [6]. (See Fig. 1 for an illustration of some notations used in this paper.) For the 2p-th moment (p = 0, 1, 2, . . .) of the segmental size distribution G(r, s) in three dimensions one may write more generally Kp(s) ≡ 1− (2p+ 1)! 〈r2p〉 (b2es) = 0 (1) which is, obviously, consistent with a Gaussian segmental size distribution G0(r, s) = 2πsb2e . (2) Both equations are expected to hold as long as the moment is not too high for a given segment length and the finite-extensibility of the polymer strand remains irrelevant [6]. Deviations caused by the segmental correlation hole effect. Recently, Flory’s hypothe- sis has been challenged both theoretically [11, 12, 13, 14, 15] and numerically for three- dimensional solutions [16, 17, 18, 19, 20] and ultrathin films [21, 22]. These studies suggest that intra- and interchain excluded volume forces do not fully compensate each other on intermediate length scales, leading to long-range intrachain correlations. The general phys- ical idea behind these correlations is related to the “segmental correlation hole” of a typical chain segment [19]. As sketched in Fig. 2, this induces an effective repulsive interaction when bringing two segments together, and swells (to some extent) the chains causing, hence, a systematic violation of Eq. (1). Elaborating and clarifying various points already presented briefly elsewhere [18, 19, 20], we focus here on melts of long and flexible polymers. Us- ing two well-studied coarse-grained polymer models [23] various intrachain properties are investigated numerically as functions of s and compared with predictions from first-order perturbation theory. (For a discussion of intrachain correlations in reciprocal space see Refs. [14, 15, 19].) Central results tested in this study. The key claim verified here concerns the deviation δG(r, s) = G(r, s) − G0(r, s) of the segmental size distribution G(r, s) from Gaussianity, Eq. (2), for asymptotically long chains (N → ∞) in the “scale-free regime” (1 ≪ s ≪ N). We show that the relative deviation divided by ce/ s scales as a function f(n) of n = r/be δG(r, s)/G0(r, s) = f(n) = + 9n− . (3) As we shall see, this scaling holds indeed for sufficiently large segment size r and curvilinear length s. The indicated “swelling coefficient” ce has been predicted analytically, 24/π3 (ρ being the monomer number density), where we shall argue that the bond length of the Gaussian reference chain of the perturbation calculation must be renormalized to the effective bond length be. Accepting Eq. (3) the swelling of the segment size is readily obtained by computing 〈r2p〉 = 4π dr r2+2pG(r, s). For the 2p-th moment this yields Kp(s) = 3(2pp!p)2 2(2p+ 1)! . (5) For instance, for the second moment (p = 1) this reduces to K1(s) = 1 − Re(s)2/b2es = s ≈ ce/ s. We have replaced in Eq. (5) the theoretically expected swelling coefficient ce by empirically determined coefficients cp. It will be shown, however, that cp/ce is close to unity for all moments. Effectively, this reduces Eq. (5) to an efficient one-parameter extrapolation formula for the effective bond length be of asymptotically long chains albeit empirical and theoretical swelling coefficients may slightly differ. While we show how be may be fitted, no attempt is made to predict it from the operational model parameters and other measured properties such as the microscopic structure or the bulk compression modulus [6, 9, 10]. Outline. We begin our discussion by sketching the central theoretical ideas in Sec. II. There we will give a simple scaling argument and outline very briefly some elements of the standard perturbation calculations we have performed to derive them (Sec. II B). Details of the analytical treatment are relegated to Appendix A. The numerical models and algorithms allowing the computation of dense melts containing the large chain lengths needed for a clear- cut test are presented in Sec. III. Our computational results are given in Sec. IV. While focusing on long chains in dense melts, we explain also briefly effects of finite chain size. The general background of this work and possible consequences for other problems of polymer science are discussed in the final Sec. V. II. PHYSICAL IDEA AND SKETCH OF THE PERTURBATION CALCULATION A. Scaling arguments Incompressibility and correlation of composition fluctuations. Polymer melts are essen- tially incompressible on length scales large compared to the monomer diameter, and the density ρ of all monomers does not fluctuate. On the other hand, composition fluctuations of labeled chains or subchains may certainly occur, however, subject to the total density constraint. Composition fluctuations are therefore coupled and segments feel an entropic penalty when their distance becomes comparable to their size [12, 19]. As sketched in Fig. 2(a), we consider two independent test chains of length s in a melt of very long chains (N → ∞). If s is sufficiently large, their typical size, R(s) ≈ be s, is set by the effective bond length be of the surrounding melt (taking apart finite chain-size effects). The test chains interact with each other directly and through the density fluctuations of the sur- rounding melt. The scaling of their effective interaction may be obtained from the potential of mean force U(r, s) ≡ − ln(g(r, s)/g(∞, s)) where g(r) is the probability to find the second chain at a distance r assuming the first segment at the origin (r = 0). Since the correlation hole is shallow for large s, expansion leads to U(r, s) ≈ 1− g(r, s)/g(∞, s) ≈ c(r, s)/ρ with c(r, s) being the density distribution of a test chain around its center of mass. This distri- bution scales as c(r ≈ 0, s) ≈ s/R(s)d close to the center of mass (d being the dimension of space) and decays rapidly at distances of order R(s) [5]. Hence, the interaction strength at r/R(s) ≪ 1 is set by u(s) ≡ U(0, s) ≡ c(0, s)/ρ ≈ s/ρR(s)d ∼ s1−d/2 [12, 19]. Interestingly, u(s) does not depend explicitly on the bulk compression modulus v. It is dimensionless and independent of the definition of the monomer unit, i.e. it does not change if λ monomers are regrouped to form an effective monomer (ρ → ρ/λ, s → s/λ) while keeping the segment size R fixed. Connectivity and swelling. To connect both test chains to form a chain of length 2s the effective energy u(s) has to be paid and this repulsion will push the half-segments apart. We consider next a segment of length s in the middle of a very long chain. All interactions between the test segment and the rest of the chain are first switched off but we keep all other interactions, especially within the segment and between the segment monomers and monomers of surrounding chains. The typical size R(s) of the test segment remains essentially unchanged from the size of an independent chain of same strand length. If we now switch on the interactions between the segment and monomers on adjacent segments of same length s, this corresponds to an effective interaction of order u(s) as before. (The effect of switching on the interaction to all other monomers of the chain is inessential at scaling level, since these other monomers are more distant.) Since this repels the respective segments from each other, the corresponding subchain is swollen compared to a Gaussian chain of non-interacting segments. It is this effect we want to characterize. Perturbation approach in three dimensions. In the following we will exclusively consider chain segments s which are much larger than the number of monomers g ≡ 1/vρ contained in a blob [5], i.e. we will look on a scale where incompressibility matters. (The number g is also sometimes called “dimensionless compressibility” [14].) Interestingly, when taken at s = g the interaction strength takes the value u(s = g) ≈ ρbdeg (vρ)d/2−1 ≈ Gz (6) with Gz being the standard Ginzburg parameter used for the perturbation calculation of strongly interacting polymers [6]. Hence, the segmental correlation hole potential u(s) ≈ Gz(g/s)d/2−1 ≪ Gz for d > 2 and s ≫ g. Although for real polymer melts as for computational systems large values of Gz ≈ 1 may sometimes be found, u(s) ∼ 1/ decreases rapidly with s in three dimensions, as illustrated in Fig. 2(b), and standard per- turbation calculations can be successfully performed. As sketched in the next paragraph these calculations yield quantities K[u] which are defined such that they vanish (K[u = 0] = 0) if the perturbation potential u(s) is switched off and are then shown to scale, to leading order, linearly with u. For instance, for the quantity Kp(s), defined in Eq. (1), characterizing the deviation of the chain segment size from Flory’s hypothesis one thus expects the scaling Kp[u(s)] ≈ +u(s) ≈ + ρRd(s) . (7) The +-sign indicated marks the fact that the prefactor has to be positive to be consis- tent with the expected swelling of the chains. Consequently, the typical segment size, R(s)/be s ≈ 1 − u(s), must approach the asymptotic limit for large s from below. For three dimensional solutions Eq. (7) implies that Kp(s) should vanish rapidly as 1/(ρb (This is different in thin films where u(s) ≈ Gz decays only logarithmically [12] as may be seen from Eq. (33) given below.) Taking apart the prefactors — which require a full calcu- lation — this corresponds exactly to Eq. (5) with a swelling coefficient ce ≈ cp ≈ 1/ρb3e in agreement with Eq. (4). Note also that the predicted deviations are inversely proportional to b3e , i.e. the more flexible the chains, the more pronounced the effect. Similar relations K[u] ∼ u may also be formulated for other quantities and will be tested numerically in Sec. IV. There, we will also check that the linear order is sufficient. B. Perturbation calculation Generalities. Before delving more into our computational results we summarize here how Eqs. (3-5) and related relations have been obtained using standard one-loop perturbation calculation. The general task is to determine 〈A〉 ≈ 〈A〉0 (1+ 〈U〉0)−〈AU〉0 for measurable quantities A such as the squared distance between two monomers n andm on the same chain, A = r2nm. Here, 〈...〉0 denotes the average over the distribution function of the unperturbed ideal chain of bond length b and U = dl ṽ(rkl) the effective perturbation potential. We discuss first the general results in the scale free regime (1 ≪ s ≪ N), argue then that b should be renormalized to the effective bond length be and sketch finally the calculation of finite chain-size effects. The scale free regime. Following Edwards [6, 7, 8], the Gaussian (or “Random Phase” [5]) approximation of the pair interaction potential in real space is ṽ(r) = v δ(r)− exp(−r/ξ) 4πrξ2 where v is a parameter which tunes the monomer interaction. (It is commonly associated with the bare excluded volume of the monomers [6], but should more correctly be identified with the bulk modulus effectively measured for the system. See the discussion of Eq. (15) of Ref. ([13]).) The effective potential consists of a strongly repulsive part vδ(r) of very short range, and a weak attractive part of range ξ where the correlation length of the density fluctuations is given by ξ2 = b2g/12 with g = 1/ρv. In Fourier space Eq. (8) is equivalent to ṽ(q) = v q2 + ξ−2 with q being the wave vector. This is sufficient for calculating the scale free regime corre- sponding to asymptotically long chains where chain end effects may be ignored. The different graphs one has to compute are indicated in Fig. 1. For A = r2 (with 1 ≪ n < m ≪ N) this yields, e.g., (m− n)− 24/π3 = b2es . (10) In the second line we have used the definition of the swelling coefficient ce indicated in Eq. (4) and have set b2e ≡ b2 1 + p 1 + p with Gz ≡ vρ/b3ρ and p = 1. (The prefactor p has been added for convenience.) The coefficient be of the leading Gaussian term in Eq. (10) — entirely due to the graph Ii describing the interactions of monomers inside the segment between n and m — has been predicted long ago by Edwards [6]. It describes how the effective bond length is increased from b to be under the influence of a small excluded volume interaction. The second term in Eq. (10) entails the 1/ s-swelling which is investigated numerically in this paper. It does only depend on b and ρ but, more importantly, not on v — in agreement with the scaling of u(s) discussed in Sec. IIA. The relative weights contributing to this term are indicated in Fig. 1 in units of − 6/π3vξ2/b3 s. The diagrams I− and I+ are obviously identical in the scale free limit. Note that the interactions described by the strongest graph Ii align the bonds ln and lm while the others tend to reduce the effect. For higher moments of the segment size distribution G(r, s) it is convenient to calculate first the deviation of the Fourier-Laplace transformation of δG(r, s) and to obtain the mo- ments from the coefficients of the expansion of this “generating function” in terms of the squared wave vector q2. As explained in detail in the Appendix A this yields more generally (2p+ 1)! (b2es) 3(2pp!p)2 2(2p+ 1)! )2p−3 where we have used Eq. (11) with general p. Obviously, Eq. (12) is consistent with our previous finding Eq. (10) for p = 1. The corresponding segmental size distribution is G(r, s) = 2πb2es 2πb2s f(n) (13) with n = r/b s and f(n) being the same function as indicated in Eq. (3). The leading Gaussian terms in Eqs. (12) and (13) depend on the effective bond length be, the second only on the Kuhn length b of the reference chain. When comparing these result with Eqs. (3) and (5) proposed in the Introduction, one sees that both equations are essentially identical — taken apart, however, that they depend on b and be. Note the conspicuous factor (b/be) in Eq. (12) which would strongly reduce the empirical swelling coefficients cp = ce(b/be) for large p if b and be were different. Interpretation of first-loop results in different contexts. The above perturbation results may be used directly to describe the effect of a weak excluded volume v on a reference system of perfectly ideal polymer melts with Kuhn segment length b where all interactions have been switched off (v = 0). It is expected to give a good estimation for the effective bond length be only for a small Ginzburg parameter: Gz ≪ 1. For the dense melts we want to describe this does not hold (Sec. III) and one cannot hope to find a good quantitative agreement with Eq. (11). Note also that large wave vectors contribute strongly to the leading Gaussian term. The effective bond length be is, hence, strongly influenced by local and non-universal effects and is very difficult to predict in general. Our much more modest goal is to predict the coefficient of the 1/ s-perturbation and to express it in terms of a suitable variational reference Hamiltonian characterized by a conveniently chosen Kuhn segment b and the measured effective bond length be (instead of Eq. (11)). Following Muthukumar and Edwards [10], we argue that for dense melts b should be renormalized to be to take into account higher order graphs. No strict mathematical proof can be given at present that the infinite number of possible graphs must add up in this manner. Our hypothesis relies on three observations: • The general scaling argument discussed in Sec. IIA states that we have only one relevant length scale in this problem, the typical segment size R(s) ≈ be s itself. The incompressibility constraint cannot generate an additional scale. It is this size R(s) which sets the strength of the effective interaction which then in turn feeds back to the deviations of R(s) from Gaussianity. Having a bond length b in addition to the effective bond length be associated with R(s) would imply incorrectly a second length scale b s varying independently with the bulk modulus v. (We will check explicitly below in Fig. 13 that there is only one length scale.) This implies b/be = const v • Thus, since by construction b/be = 1 for v → 0, it follows that both lengths should be equal for all v. • We know from Eq. (12) that the empirical coefficients cp = ce(b/be)2p−3 should depend strongly on the moment considered if the ratio b/be is not close to unity. It will be shown below (Fig. 6) that cp/ce ≈ 1 for all p. This implies b ≈ be. Finite chain size effects. To describe properly finite chain size corrections Eq. (9) must be replaced by the general linear response formula ṽ(q) + ρF (q) (14) with F (q) = NfD(x) being the form factor of the Gaussian reference chain given by Debye’s function fD(x) = 2(e −x − 1 + x)/x2 with x = (qb)2N/6 [6]. This approximation allows in principle to compute, for instance, the (mean-squared) total chain end-to-end distance, A = (rN −r1)2. One verifies readily (see [6], Eq. (5.III.9)) that the effect of the perturbation may be expressed as 〈A〉0 〈U〉0 − 〈AU〉0 = (2π)3 ṽ(q) ds s2(N − s) exp q2b2s . (15) We take now first the integral over s. In the remaining integral over q small q wave vectors contribute to the N -swelling while large q renormalize the effective bond length of the dominant Gaussian behaviour linear in N (as discussed above). Since we wish to determine the non-Gaussian corrections, we may focus on small wave vectors q ≪ 1/ξ. Since in this limit 1/v = ρg ≪ ρF (q), one can neglect in Eq. (14) the 1/v contribution to the inverse effective interaction potential. We thus continue the calculation using the much simpler ṽ(N, x) = 1/(NρfD(x)). This allows us to express the swelling as 〈(rN − r1)2〉 I(xu). (16) To simplify the notation we have set here finally b = be in agreement with the hypothesis discussed above. The numerical integral I(xu) = dx . . . over x is slowly convergent at infinity. As a consequence the estimate I(∞) = 1.59 may be too large for moderate chain lengths. In practice, convergence is not achieved for values xu(N) ≈ (b/ξ)2N corresponding to the screening length ξ. We remark finally that numerical integration can be avoided for various properties if the Padé approximation of the form factor, F (q) = N/(1 + (qb)2N/12), is used. This allows analytical calculations by means of the simplified effective interaction potential ṽ(q) = 12ρb3 . (17) This has been used for instance for the calculation of finite chain size effects for the bond- bond correlation function discussed in Sec. IVC below [61]. III. COMPUTATIONAL MODELS AND TECHNICAL DETAILS A. Bond fluctuation model A widely-used lattice Monte Carlo scheme for coarse-grained polymers. The body of our numerical data comes from the three dimensional bond fluctuation model (BFM) — a lattice Monte Carlo (MC) algorithm where each monomer occupies eight sites of a unit cell of a simple cubic lattice [24, 25, 26]. Our version of the BFM with 108 bond vectors corresponds to flexible athermal chain configurations [23]. All length scales are given in units of the lattice constant and time in units of Monte Carlo Steps (MCS). We use cubic periodic simulation boxes of linear size L = 256 containing nmon = ρL 3 = 220 ≈ 106 monomers. This monomer number corresponds to a monomer number density ρ = 0.5/8 where half of the lattice sites are occupied (volume fraction 0.5). The large system sizes used allow us to suppress finite box-size effects for systems with large chains. Using a mix of local, slithering snake [27, 28, 29], and double-bridging [17, 23, 30, 31] MC moves we were able to equilibrate dense systems with chain lengths up to N = 8192. Equilibration and sampling of high-molecular BFM melts. Standard BFM implementa- tions [26, 32, 33] use local MC jumps to the 6 closest lattice sites to prevent the crossing of chains and conserve therefore the chain topology. These “L06” moves lead to very large relaxation times, scaling at least as τe ∼ N3, as may be seen from Fig. 3 (stars). The relax- ation time τe = R e/6Ds indicated in this figure has been estimated from the self-diffusion coefficient Ds obtained from the mean-square displacements of all monomers in the free diffusion limit. (For the largest chain indicated for L06 dynamics only a lower bound for τe is given.) Instead of this more realistic but very slow dynamical scheme we make jump attempts to the 26 sites of the cube surrounding the current monomer position (called “L26” moves). This allows the chains to cross each other which dramatically speeds up the dynam- ics, especially for long chains (N > 512). If only local moves are considered, the dynamics is perfectly consistent with the Rouse model [6]. As shown in Fig. 3, we find τe ≈ 530N2 for L26 dynamics. This is, however, still prohibitive by large for sampling configurations with the longest chain length N we aim to characterize [62]. Slithering snake moves. In addition to the local moves one slithering snake move per chain is attempted on average per MCS corresponding to the displacement of N monomer along the chain backbone. Note that in our units two spatial displacement attempts per MCS are performed on average per monomer, one for a local move and one for a snake move. (In practice, it is computationally more efficient for large N to take off a monomer at one chain end and to paste it at the other leaving all other monomers unaltered. Before dynamical measurements are performed the original order of beads must then be restored.) Interestingly, a significantly larger slithering snake attempt frequency would not be useful since the relaxation time of slithering snakes without or only few local moves increases exponentially with mass [29, 34] due to the correlated motion of snakes [35]. In order to obtain an efficient free snake diffusion (with a chain length independent curvilinear diffusion coefficient Dc(N) ∼ N0 and τe ≈ N2/Dc(N) ∼ N2 [28, 29]) it is important to relax density fluctuations rapidly by local dynamical pathways. As shown in Fig. 3 (squares), we find a much reduced relaxation time τe ≈ 40N2 which is, however, still unconveniently large for our longest chains. Note that most of the CPU time is still used by local moves. The computational load per MCS remains therefore essentially chain length independent. Advantages and pitfalls of double-bridging moves. Double-bridging (DB) moves are very useful for high densities and help us to extend the accessible molecular masses close to 104. As for slithering snake moves we use all 108 bond vectors to switch chain segments between two different chains. Only chain segments of equal length are swapped to conserve monodispersity. Topolocial constraints are again systematically and deliberately violated. Since more than one swap partner is possible for a selected first monomer, delicate detailed balance questions arise. This is particularly important for short chains and is discussed in detail in Ref. [23]. Technically, the simplest solution to this problem is to refuse all moves with more than one swap partner (to be checked both for forward and back move). The configurations are screened with a frequency fDB for possible DB moves where we scan in random order over the monomers. The frequency should not be too large to avoid (more or less) immediate back swaps and monomers should move at least out of the local monomer cage and over a couple of lattice sites. We use fDB = 0.1 between DB updates for the configurations reported here. (The influence of fDB on the performance has not been explored systematically, but preliminary results suggest a slightly smaller DB frequency for future studies.) The diffusion times over the end-to-end distance for this case are indicated in Tab. I. As shown in Fig. 3, we find empirically τe(N) ≈ 13N1.62. For N = 8192 this corresponds to 3 · 107 MCS. This allows us even for the largest chain lengths to observe monomer diffusion over several Re within the 10 8 MCS which are feasible on our XEON-PC processor cluster. The efficiency of DB moves is commonly characterized in terms of the relaxation time τee of the end-to-end vector correlation function [30, 31]. For normal chain dynamics this would indeed characterize the longest relaxation time of the system, i.e. τe ≈ τee. For the double-bridging this is, however, not sufficient since density fluctuations do not couple to the bridging moves and can not be relaxed. We find therefore that configurations equilibrate on time scales given by τe rather than by τee ≪ τe. This may be verified, for instance, from the time needed for the distribution Re(s) (and especially its spatial components) to equilibrate. The criterion given in the literature [30] is clearly not satisfactory and may lead to insufficiently equilibrated configurations. In summary, equilibration with DB moves still requires monomer diffusion over the typical chain size, however at a much reduced price. Some properties of our configurations. The Tables I and II summarize some system properties obtained for our reference density ρ = 0.5/8. Averages are performed over all chains and 1000 configurations. These configurations may be considered to be independent for N < 4096. Only a few independent configurations exist for the largest chain length N = 8192 which has to be considered with some care. Taking apart this system, chains are always much smaller than the box size. For asymptotically long chains, we obtain an average bond length 〈|l|〉 ≈ 2.604, a root-mean-squared bond length l ≡ 〈l2〉1/2 ≈ 2.635 and an effective bond length be ≈ 3.244 — as we will determine below in Sec. IVA. This corresponds to a ratio C∞ ≡ b2e/l2 ≈ 1.52 and, hence, to a persistence length lp = l(C∞ + 1)/2 ≈ 3.32 [23]. Especially, we find from the zero wave vector limit of the total structure factor S(q) a low (dimensionless) compressibility g = S(q → 0)/ρ ≈ 0.246 which compares well with real experimental melts. From the measured bulk compression modulus v ≡ 1/g(ρ)ρ and the effective bond length be one may estimate a Ginzburg parameter Gz = vρ/b3eρ ≈ 0.96. Following Ref. [13] the interaction parameter v is supposed here to be given by the full inverse compressibility and not just by the second virial coefficient. B. Bead spring model Hamiltonian. Additionally, molecular dynamics simulations of a bead-spring model (BSM) [36] were performed to dispel concerns that our results are influenced by the un- derlying lattice structure of the BFM. The model is derived from a coarse-grained model for polyvinylalcohol which has been employed to study polymer crystallization [37]. It is characterized by two potentials: a non-bonded potential of Lennard-Jones (LJ) type and a harmonic bond potential. While the often employed Kremer-Grest model [38] uses a 12− 6 LJ potential to describe the non-bonded interactions Unb(r), our non-bonded potential has a softer repulsive part. It is given by Unb(r) = 1.511 , (18) which is truncated and shifted at the minimum at rmin ≈ 1.15. Note that all length scales are given in units of σ0 and we use LJ units [39] for all BSM data (mass m = 1, Boltzmann constant kB = 1). The parameters of the bond potential, Ub(r) = 1120(r− lb)2, are adjusted so that the average bond length l(ρ = 0.84) ≈ lb = 0.97 is approximately the same as in the standard Kremer-Grest model [38]. The average bond length and the root-mean-squared bond length are almost identical for the BSM due to the very stiff bond potential. Since rmin/l ≈ 1.16 bonded monomers penetrate each other significantly. Equilibration and sampling. We perform standard molecular dynamics simulations in the canonical ensemble with a Langevin thermostat (friction constant Γ = 0.5) at temperature T = 1. The equations of motion are integrated by the velocity-Verlet algorithm [39]. To improve the statistics for large chain length, we have implemented additional double-bridging moves. Since only few of these MC moves are accepted per unit time, this does affect neither the stability nor the accuracy of the molecular dynamics sweeps. Some properties obtained. For clarity, we show only data for chain length N = 1024 and number density ρ = 0.84, the typical melt density of the Kremer-Grest model [38]. For the reported data we use periodic simulation boxes of linear size L ≈ 62 containing nmon = 196608 monomers, but we have also sampled different boxes sizes (up to L = 77.5) to check for finite box-size effects. For the reference density a dimensionless compressibility g ≈ 0.08 is found which is about three times smaller than for our BFM melt. For the effective bond length we obtain be ≈ 1.34, i.e. BSM chains (C∞ ≈ 1.91, lp ≈ 1.41) are slightly stiffer than the corresponding BFM polymers. Fortunately, the product ρb3e ≈ 2 is roughly similar in both models and one expects from Eq. (4) a similar swelling for large s. Note finally that the Ginzburg parameter Gz ≈ 1.8 is much larger than for the BFM systems. As we have emphasized in Sec. II, this should, however, not influence the validity of the perturbation prediction of the expected 1/ s-swelling of the chains when expressed in terms of the measured effective bond length. IV. NUMERICAL RESULTS As illustrated in Fig. 1, a chain segment of curvilinear length s > 0 is identified by two monomers n and m = n + s on the same chain. We compute here various moments of chain segment properties where we ensemble-average over all chains and all start points n. The statistical accuracy must therefore always decrease for large s. We concentrate first on the second moment (p = 1) of the segmental size distribution. Higher moments and the segmental size distribution are discussed in Sec. IVE. A. The swelling of chain segments Scale free regime for 1 ≪ s ≪ N . The mean-squared segment size R2e(s) = 〈r2〉 is presented in the Figs. 4, 5 and 6. The first plot shows clearly that chain segments are swollen, i.e. R2e(s)/s increases systematically and this up to very large curvilinear distances s. Only BFM data are shown for clarity. A similar plots exists for the BSM data. In agreement with Eq. (5) for p = 1, the asymptotic Gaussian behavior (dashed line) is approached from below and the deviation decays as u(s) ∝ 1/ s (bold line). The bold line indicated corresponds to be = 3.244 and c1 ≈ ce ≈ 0.41 which fits nicely the data over several decades in s — provided that chain end effects can be neglected (s ≪ N). Note that a systematic underestimation of the true effective bond length would be obtained by taking simply the largest R2e(s)/s ≈ 3.232 value available, say, for monodisperse chains of length N = 2048. Finite chain-size effects. Interestingly, R2e(s)/s does not approach the asymptotic limit monotonicly. Especially for short chains one finds a non-monotonic behavior for s → N . This means that the total chain end-to-end distance Re(s = N − 1) must show even more pronounced deviations from the asymptotic limit. This is confirmed by the dashed line representing the b2e(N) ≡ R2e(N − 1)/(N − 1) data points given in Tab. I. We emphasize that the non-monotonicity of R2e(s)/s becomes weaker with increasing N and that, as one expects, the inner distances, as well as the total chain size, are characterized by the same effective bond length be for large s or N . The non-monotonic behavior may be qualitatively understood by the reduced self-interactions at chain ends which lessens the swelling on these scales. These finite-N corrections have been calculated analytically using the full Debye function for the effective interaction potential ṽ(q), Eq. (14). The prediction for the total chain end-to-end vector given in Eq. (16) is indicated in Fig. 4 (dash-dotted line) where we have replaced the weakly N -dependent integral I(xu) by its upper bound value for infinite chains R2e(N − 1) b2e (N − 1) 1.59ce√ N − 1 . (19) We have changed here the chain length N in the analytical formula (obtained for large chains where N ≈ N − 1) to the curvilinear length N − 1. This is physically reasonable and allows to take better into account the behavior of small chains. Note that Eq. (19) is similar to Eq. (5) — apart from a slightly larger prefactor explaining the observed stronger deviations. Theory compares well with the measured data for large N . It does less so for smaller N , as expected, where the chain length dependence of the numerical integral I(xu(N)) ≤ 1.59 must become visible. This explains why the data points are above the dash-dotted line. Note also that additional non-universal finite-N effects not accounted for by the theory are likely for small N . In contrast to this, Re(s) is well described by the theory even for rather small s provided that N is large and chain end effects can be neglected. In summary, it is clear that one should use the segment size Re(s) rather than the total chain size to obtain in a computational study a reliable fit of the effective bond length be. Extrapolation of the effective bond length of asymptotically long chains. The represen- tation chosen in Fig. 4 is not the most convenient one for an accurate determination of be and c1. How precise coefficients may be obtained according to Eq. (5) is addressed in the Figs. 5 and 6. The fitting of the effective bond length be and its accuracy is illustrated in Fig. 5 for BFM chains of length N = 2048. This may be first done approximately in linear coordinates by plotting R2e(s)/s as a function of 1/ s (not shown). Since data for large s are less visible in this representation, we recommend for the fine-tuning of be to switch then to logarithmic coordinates with a vertical axis y = 1 − R2e(s)/b2es for different trial values of be. The correct value of be is found by adjusting the vertical axis y such that the data extrapolates linearly as a function of 1/ s to zero for large s. We assume for the fine-tuning that higher order perturbation corrections may be neglected, i.e. we take Eq. (5) literally. (We show below that higher order corrections must indeed be very small.) The plot shows that this method is very sensitive, yielding a best value that agrees with the theory over more than one order of magnitude without curvature. As expected, it is not possible to rationalize the numerically obtained values be ≈ 3.244 for the BFM and be ≈ 1.34 for the BSM using Eq. (11). According to Eq. (4) these fit values imply the theoretical swelling coefficients ce = 0.41 for the BFM and ce = 0.44 for the BSM. Empirical swelling coefficients. As a next step the horizontal axis is rescaled such that all data sets collapse on the bisection line, i.e. using Eq. (5) we fit for the empirical swelling coefficient c1 and compare it to the predicted value ce. This rescaling of the axes allows to compare both models in Fig. 6. For clarity the BSM data have been shifted upwards. For the BFM we find c1/ce ≈ 1.0, as expected, while our BSM simulations yield a slightly more pronounced swelling with c1/ce ≈ 1.2. Segmental radius of gyration. Also indicated in Fig. 6 is the segmental radius of gyration Rg(s) (filled circles) computed as usual [6] as the variance of the positions of the segment monomers around their center of mass. Being the sum over all s + 1 monomers, it has a much better statistics compared to Re(s). The scaling used can be understood by expressing the radius of gyration R2g(s) = (s+1)2 l=n r in terms of displacement vectors rkl [6]. Using Eq. (5) and integrating twice this yields 6R2g(s) b2e(s+ 1) . (20) Plotting the l.h.s. of this relation against the r.h.s. we obtain a perfect data collapse on the bisection line where we have used the same parameters be and c1 as for the mean-squared segment size. This is an important cross-check which we strongly recommend. Different values indicate insufficient sample equilibration. B. Chain connectivity and recursion relation As was emphasized in Sec. IIA the observed swelling is due to an entropic repulsion between chain segments induced by the incompressibility of the melt. To stress the role of chain connectivity we repeat the general scaling argument given above in a form originally proposed by Semenov and Johner for ultrathin films [12]. As shown in Fig. 7 we test the relation Kλ(s) ≡ R2e(λs)− λR2e(s) λ)R2e(s) ≈ u(s) ≡ s ρRe(s)d with Kλ(s) being a direct measure of the non-Gaussianity (λ being a positive number) comparing the size of a segment of length λs with the size of λ segments of length s joined together. (The prefactor 1/(λ − λ) in the definition of Kλ(s) has been introduced for convenience.) Equivalently, this can be read as a measure for the swelling of a chain where initially the interaction energy u between the segments has been switched off. Kλ is a functional of u(s) with Kλ[u = 0] = 0. The analytic expansion of the functional must be dominated by the linear term (as indicated by ≈ in the above relation) simply because u is very small. Altogether, Eq. (21) yields a recursion relation relating Re(λs) with Re(s) for any λ provided 1 ≪ s < λs ≪ N . It can be solved, leading (in lowest order) to Eq. (5) with p = 1. This may be seen from the ansatz R2e(s) = b es(1− ce/sω−1+ . . .) which readily yields ω = 3/2 and ce ≈ 1/ρb3e . Eq. (21) has been validated directly in Fig. 7 for λ = 2 (corresponding to two segments of length s joined together) for the BFM and the BSM as indicated. In addition, for BFM chains of length N = 2048 several values of λ have been given. As suggested by Eq. (4), we have plotted Kλ(s) as a function of (c1/ce) u(s) with u(s) ≡ 24/π3s/ρR3e(s) ≈ ce/ The prefactor of u(s) allows a convenient comparison with Fig. 5. Note the perfect data collapse for all data sets. More importantly, the predicted linearity is well confirmed for large segments (1 ≪ s) and this without any tunable parameter for the vertical axis, as was needed in the previous Figs. 5 and 6. C. Intrachain bond-bond correlations Expectation from Flory’s hypothesis. An even more striking violation of Flory’s ideality hypothesis may be obtained by computing the bond-bond correlation function, defined by the first Legendre polynomial P (s) = 〈lm=n+s · ln〉 /l2 where the average is performed, as before, over all possible pairs of monomers (n,m = n+ s) [63]. Here, li = ri+1 − ri denotes the bond vector between two adjacent monomers i and i + 1 and l2 = 〈l2n〉n the mean- squared bond length. The bond-bond correlation function is generally believed to decrease exponentially [4]. This belief is based on the few simple single chain models which have been solved rigorously [4, 40] and on the assumption that all long range interactions are negligible on distances larger than the screening length ξ. Hence, only correlations along the backbone of the chains are expected to matter and it is then straightforward to work out that an exponential cut-off is inevitable due to the multiplicative loss of any information transferred recursively along the chain [4]. Asymptotic behavior in the melt. That this reasoning must be incorrect follows imme- diately from the relation P (s) = R2e(s) (22) expressing the bond-bond correlation function as the curvature of the second moment of the segment size distribution. It is obtained from the identity 〈ln · lm〉 = 〈∂nrn · ∂mrm〉 = −∂n∂m 〈r2nm〉 /2. (Note that the velocity correlation function is similarly related to the second derivative of the mean-square displacement with respect to time [41].) Hence, P (s) allows us to probe directly the non-Gaussian corrections without any ideal contribution. This relation together with Eq. (5) suggests an algebraical decay P (s) = cP/s ω with ω = 3/2 , cP = c1 (be/l) 2/8 ≈ ρl2be of the bond-bond correlation function for dense solutions and melts, rather than the ex- ponential cut-off expected from Flory’s hypothesis. This prediction (bold line) is perfectly confirmed by the larger chains (N > 256) indicated in Fig. 8. In principle, the swelling coefficient, c1 ∼ cP, may also be obtained from the power law amplitude of the bond-bond correlation function, however, to lesser accuracy than by the previous method (Fig. 5). One reason is that P (s) decays very rapidly and does not allow a precise fit beyond s ≈ 102. The values of cP obtained from c1 are indicated in Tab. II. Data from the BSM have also been included in the figure to demonstrate the universality of the result. The vertical axis has been rescaled with cP which allows to collapse the data of both models. Finite chain-size corrections. As can be seen for N = 16, exponentials are compatible with the data of short chains. This might explain how the power law scaling has been overlooked in previous numerical studies, since good statistics for large chains (N > 1000) has only become available recently. However, it is clearly shown that P (s) approaches systematically the scale free asymptote with increasing N . The departure from this limit is fully accounted for by the theory if chain end effects are carefully considered (dashed lines). Generalizing Eq. (23) and using the Padé approximation, Eq. (17), perturbation theory yields P (s) = 1 + 3x+ 5x2 1 + x (1− x)2 (24) where we have set x = s/N . For x ≪ 1 this is consistent with Eq. (23). In the limit of large s → N , the correlation functions vanish rigorously as P (s) ∝ (1 − x)2. Considering that non-universal features cannot be neglected for short chain properties and that the theory does not allow for any free fitting parameter, the agreement found in Fig. 8 is rather satisfactory. D. Higher moments and associated coefficients Effective bond length and empirical swelling coefficients. The preceding discussion fo- cused on the second moment of the segmental size distribution G(r, s). We have also com- puted for both models higher moments 〈r2p〉 with p ≤ 5. If traced in log-linear coordinates as y = (6pp! 〈r2p〉 /(2p + 1)!sp)1/p vs. x = s higher moments approach b2e from below — just as the second moment presented in Fig. 4. The deviations from ideality are now more pronounced and increase with p (not shown). The moments are compared in Fig. 6 with Eq. (5) where they are rescaled as y = Kp(s) as defined in Eq. (1) and plotted as functions of x = 3(2pp!p)2 2(2p+1)! . The prediction is indicated by bold lines. It is important that the same effective bond length be is obtained from the analysis of all functions Kp(s) as illustrated in Fig. 6. Otherwise we would regard equilibration and statistics as insufficient. The empirical swelling coefficients cp are obtained, as above in Sec. IVA, by shifting the data horizontally. A good agreement with the expected cp/ce ≈ 1 is found for both models and all moments as may be seen from Tab. II. This confirms the renormalization of the Kuhn segment b → be of the Gaussian reference chain in agreement with our discussion in Sec. II B. Otherwise we would have measured empirical coefficients decreasing strongly as cp/ce ≈ (b/be)2p−3 with p. Since the effective bond length of non-interacting chains are known for the BFM (b ≈ 2.688) and the BSM (b ≈ 0.97), one can simply check, say for p = 5, that the non-renormalized values would correspond to the ratios c5/ce ≈ (2.688/3.244)7 ≈ 0.3 for the BFM and c5/ce ≈ (0.97/1.34)7 ≈ 0.1 for the BSM. This is clearly not consistent with our data. It should be emphasized that both coefficients be and cp are more difficult to determine for large p, since the linear regime for x ≪ 1 in the representation chosen in Fig. 6 becomes reduced. For large x ≫ 1 one finds that y(x) → 1, i.e. 〈r2p〉 /b2pe sp → 0. This trivial departure from both Gaussianity and the 1/ s-deviations we try to describe, is due to the finite extensibility of chain segments of length s which becomes more marked for larger moments probing larger segment sizes. The data collapse for both x-regimes is remarkable, however. Incidentally, it should be noted that for the BSM the empirical swelling coefficients are slightly larger than expected. At present we do not have a satisfactory explanation for this altogether minor effect, but it might be attributed to the fact that neighbouring BSM beads along the chain strongly interpenetrate — an effect not considered by the theory. Non-Gaussian parameter αp. The failure of Flory’s hypothesis can also be demonstrated by means of the standard non-Gaussian parameter αp(s) ≡ 1− (2p+ 1)! 〈r2pnm〉 〈r2nm〉 p (25) comparing the 2p-th moment with the second moment (p = 1). In contrast to the closely related parameter Kp(s) this has the advantage that here two measured properties are com- pared without any tuneable parameter, such as be, which has to be fitted first. Fig. 9 presents αp(s) vs. ce/ s for the three moments with p = 2, 3, 4. For each p we find perfect data col- lapse for all chain lengths and both models and confirm the linear relationship αp(s) ≈ u(s) expected. The lines indicate the theoretical prediction αp(s) = 3 (2pp!p)2 2 (2p+ 1)! which can be derived from Eq. (5) by expanding the second moment in the denominator. An alternative derivation based on the coefficients of the expansion of the generating function G(q, s) in q2 is indicated by Eq. (A2) in the Appendix. Having confirmed above that cp/ce ≈ 1, we assume in Eq. (26) that cp = ce to simplify the notation. The prefactors 6/5, 111/35 and 604/105 for p = 2, 3 and 4 respectively are nicely confirmed. They increase strongly with p, i.e. the non-Gaussianity becomes more pronounced for larger moments as already mentioned. Note also the curvature of the data at small s due to the finite extensibility of the segments which becomes more marked for higher moments. If one plots αp(s) as a function of the r.h.s. of Eq. (26) all data points for all moments and even for too small s collapse on one master curve (not shown) — just as we have seen before in Fig. (6). Correlations of different directions. A similar correlation function is presented in Fig. 10 which measures the non-Gaussian correlations of different spatial directions. It is defined by Kxy(s) ≡ 1− 〈x2 y2〉 〈x2〉 〈y2〉 for the two spatial components x and y of the vector r as illustrated by the sketch given at the bottom of Fig. 10. Symmetry allows to average over the three pairs of directions (x, y), (x, z) and (x, z). Following the general scaling argument given in Sec. II we expect Kxy(s) ≈ u(s) ≈ ce/ s which is confirmed by the perturbation result Kxy(s) = = K2(s). (28) This is nicely confirmed by the linear relationship found (bold line) on which all data from both simulation models collapse perfectly. The different directions of chain segments are therefore coupled. As explained in the Appendix (Eq. (A3)), Kxy(s) and α2(s) must be identical if the Fourier transformed segmental size distribution G(q, s) can be expanded in terms of q2 and this irrespective of the values the expansion coefficients take. Fig. 10 confirms, hence, that our computational systems are perfectly isotropic and tests the validity of the general analytical expansion. The correlation function Kxy is of particular interest since the zero-shear viscosity should be proportional to ∼ 〈x2y2〉 = 〈x2〉 〈y2〉 (1 − Kxy(s)). We assume here following Edwards [6] that only intrachain stresses contribute to the shear stress σxy. Hence, our results suggest that the classical calculations [6] — assuming incorrectly Kxy = 0 — should be revisited. E. The segmental size distribution We turn finally to the segmental size distribution G(r, s) itself which is presented in Figs. 11, 12 and 13. From the theoretical point of view G(r, s) is the most fundamental property from which all others can be derived. It is presented last since it is computationally more demanding — at least if high accuracy is needed — and coefficients such as be may be best determined directly from the moments. The normalized histograms G(r, s) are computed by counting the number of segment vectors between r − dr/2 and r + dr/2 with dr being the width of the bin and one divides then by the spherical bin volume. Since the BFM model is a lattice model, this volume is not 4πr2dr but given by the number of lattice sites the segment vector can actually point to for being allocated to the bin. Incorrect histograms are obtained for small r if this is not taken into account. (Averages are taken over all segments and chains, just as before.) Clearly, non-universal physics must show up for small vector length r and small curvilinear distance s and we concentrate therefore on values r ≫ σ and s ≥ 31. When plotted in linear coordinates as in Fig. 11, G(r, s) compares roughly with the Gaussian prediction G0(r, s) given by Eq. (2), but presents a distinct depletion for small segment sizes with n ≡ r/be s ≪ 1 and an enhanced regime for n ≈ 1. A second depletion region for large n ≫ 1 — expected from the finite extensibility of the segments — can be best seen in the log-log representation of the data (not shown). To analyse the data it is better to consider instead of G(r, s) the relative deviation δG(r, s)/G0(r, s) = G(r, s)/G0(r, s)− 1 which should further be divided by the strength of the segmental correlation hole, ce/ As presented in Fig. 12 this yields a direct test of the key relation Eq. (3) announced in the Introduction. The figure demonstrates nicely the scaling of the data for all s and for both models. It shows further a good collapse of the data close to the universal function f(n) predicted by theory (bold line). Note that the depletion scales as 1/n for small segment sizes (dashed line). The agreement of simulation and theory is by all standards remarkable. (Obviously, error bars increase strongly for n ≫ 1 where G0(r, s) decreases strongly. The regime for very large n where the finite extensibility of segments matters has been omitted for clarity.) We emphasize that this scaling plot depends very strongly on the value be which is used to calculate the Gaussian reference distribution. If a precise value is not available we recommend to use instead the scaling variable m = r/Re(s) for the horizontal axis, i.e. to replace the scale be s estimated from the behaviour of asymptotically long chains by the measured (mean-squared) segment size for the given s. The Gaussian reference distribution is then accordingly G0(m,Re(s)) = (3/2πRe(s) 2) exp(−3 m2). The corresponding scaling plot is given in Fig. 13. It is simi- lar and of comparable quality as the previous plot. Changing the scaling variable from n = r/be s to m = r/Re(s) ≈ (r/be s)(1 + ce/2 s) changes somewhat the universal func- tion. Expanding the previous result, Eq. (3), this adds even powers of m to the function f(n) given in Eq. (3) f(n) ⇒ f(m) = + 9m+ m2 − 9 . (29) That the two additional terms in the function are correct can be seen by computing the second moment 4π drr4δG(r, s) which must vanish by construction. The rescaled relative deviation is somewhat broader than in the previous plot due to the additional term scaling as m2. As already stressed this scaling does not rely on the effective bond length be and is therefore more robust. It has the nice feature that it underlines that there is only one characteristic length scale relevant for the swelling induced by the segmental correlation hole, the typical size of the chain segment itself. V. CONCLUSION Issues covered and central theoretical claims. We have revisited Flory’s famous ideality hypothesis for long polymers in the melt by analyzing both analytically and numerically the segmental size distribution G(r, s) and its moments for chain segments of curvilinear length s. We have first identified the general mechanism that gives rise to deviations from ideal chain behavior in dense polymer solutions and melts (Sec. II). This mechanism rests upon the interplay of chain connectivity and the incompressibility of the system which generates an effective repulsion between chain segments (Fig. 2). This repulsion scales like u(s) ≈ ce/ s where the “swelling coefficient” ce ≈ 1/b3eρ sets the strength of the interaction. It is strong for small segment length s, but becomes weak for s → N in the large-N limit. The overall size of a long chain thus remains almost ‘ideal’, whereas subchains are swollen as described by Eq. (5). Most notably, the relative deviation δG(r, s)/G0(r, s) of the segmental size distribution from Gaussianity should be proportional to u(s). As a function of segment size r, the repulsion manifests itself by a strong 1/r-depletion at short distances r ≪ be and a subsequent shift of the histogram to larger distances (Eq. (3)). Summary of computational results. Using Monte Carlo and molecular dynamics simu- lation of two coarse-grained polymer models we have verified numerically the theoretical predictions for long and flexible polymers in the bulk. We have explicitly checked (e.g., Figs. 7, 9, 13) that the relative deviations from Flory’s hypothesis scale indeed as 1/ Especially, the measurement of the bond-bond correlation function P (s), being the second derivative of the second moment of G(r, s) with respect of s, allows a very precise verifi- cation (Fig. 8) and shows that higher order corrections beyond the first-order perturbation approximation must be small. The most central and highly non-trivial numerical verification concerns the data collapse presented in Figs. 12 and 13 for the segmental size distribution of both computational models. All other statements made in this paper can be derived and understood from this key finding. It shows especially that the swelling coefficient ce must be close to the predicted value, Eq. (4). It is well known [10] that the effective bond length is difficult to predict at low com- pressibility and no attempt has been done to do so in this paper. We show instead how the systematic swelling of chain segments – once understood – may be used to extrapolate for the effective bond length of asymptotically long chains. Figs. 5 and 6 indicate how this may be done using Eq. (5). The high precision of our data is demonstrated in Fig. 12 by the successful scaling of the segmental size distribution. For several moments 〈r2p〉 we have also fitted empirical swelling coefficients cp using Eq. (5). In contrast to the effective bond length be these coefficients are rather well pre- dicted by one-loop perturbation theory if the bond length b of the reference Hamiltonian is renormalized to the effective bond length be, as we have conjectured in Sec. II B. Since the empirical swelling coefficients, cp ≈ ce(b/be)2p−3, would otherwise strongly depend on the moment taken, as shown in Eq. (12), our numerical data (Tab. II) clearly imply b/be ≈ 1. Minor deviations found for the BSM samples may be attributed to the fact that monomers along the BSM chains do strongly overlap — an effect not taken into account by the theory. To clarify ultimately this issue we are currently performing a numerical study where we systematically vary both the compressibility and the bond length of the BSM. General background and outlook. The most striking result presented in this work con- cerns the power law decay found for the bond-bond correlation function, P (s) ∝ 1/s3/2 (Fig. 8). This result suggests an analogy with the well-known long-range velocity correla- tions found in dense fluids by Alder and Wainwright nearly fourty years ago [41, 42]. In both cases, the ideal uncorrelated object is a random walker which is weakly perturbed (for d > 2) by the self-interactions generated by global constraints. Although these constraints are different (momentum conservation for the fluid, incompressibility for polymer melts) the weight with which these constraints increase the stiffness of the random walker is always proportional to the return probability. It can be shown that the correspondence of both problems is mathematically rigorous if the fluid dynamics is described on the level of the linearized Navier-Stokes equations [43]. We point out that the physical mechanism which has been sketched above is rather general and should not be altered by details such as a finite persistence length — at least not as long as nematic ordering remains negligible and the polymer chains are sufficiently long. (Similarly, velocity correlations in dense liquids must show an analytical decay for sufficiently large times irrespective of the particle mass and the local static structure of the solution.) While this paper focused exclusively on scales beyond the correlation length of the density fluctuations, i.e. qξ ≪ 1 or s/g ≫ 1, where the polymer solution appears incompressible, effects of finite density and compressibility can be readily described within the same theoretical framework and will be presented elsewhere [43]. To test our predictions, flexible chains should be studied preferentially, since the chain length required for a clear- cut description increases strongly with persistence length. This is in fact confirmed by preliminary and on-going simulations using the BSM algorithm. In this work we have only discussed properties in real space as a function of the curvilin- ear distance s. These quantities are straightforward to compute in a computer simulation but are barely experimentally relevant. The non-Gaussian deviations induced by the seg- mental correlation hole arise, however, also for an experimentally accessible property, the intramolecular form factor (single chain scattering function) F (q). As explained at the end of the Appendix, the form factor can be readily obtained by integrating the Fourier transformed segmental size distribution given in Eq. (3). This yields q2F (q) ≈ 12 in agreement with the result obtained in Refs. [14, 19] by direct calculation of the form factor for very long equilibrium polymers. As a consequence of this, the Kratky plot (q2F (q) vs. wave vector q) should not exhibit the plateau expected for Gaussian chains in the scale-free regime, but rather noticeable non-monotonic deviations. See Fig. 3 of [19]. This result suggests to revisit experimentally this old pivotal problem of polymer science. Our work is part of a broader attempt to describe systematically the effects of correlated density fluctuations in dense polymer systems, both for static [12, 13, 44, 45] and dynam- ical [29, 35, 46] properties. An important unresolved question is for instance whether the predicted long-range repulsive forces of van der Waals type (“Anti-Casimir effect”) [13, 45] are observable, for instance in the oscillatory decay of the standard density pair-correlation function of dense polymer solutions. Since the results presented here challenge an important concept of polymer physics, they should hopefully be useful for a broad range of theoreti- cal approaches which commonly assume the validity of the Gaussian chain model down to molecular scales [47, 48, 49]. This study shows that a polymer in dense solutions should not be viewed as one soft sphere (or ellipsoid) [50, 51, 52], but as a hierarchy of nested segmental correlation holes of all sizes aligned and correlated along the chain backbone (Fig. 2 (b)). We note that similar deviations from Flory’s hypothesis have been reported recently for linear polymers [16, 17, 47] and polymer gels and networks [53, 54]. The repulsive interactions should also influence the polymer dynamics, since strong deviations from Gaussianity are expected on the scale where entanglements become important, hence, quantitative predic- tions for the entanglement length Ne have to be regarded with more care. The demonstrated swelling of chains should be included in the popular primitive path analysis for obtaining Ne [55], especially if ‘short’ chains (N < 500) are considered. The effect could be responsible for observed deviations from Rouse behavior [26, 56] as may be seen by considering the cor- relation function Cpq ≡ 〈Xp ·Xq〉 of the Rouse modes Xp = 1N dnrn cos(npπ/N) where p, q = 0, . . . , N − 1 [6, 57]. Using (rn − rm)2 = r2n + r2m − 2rn · rm for the segment size, this correlation function can be readily expressed as an integral over the second moment of the segmental size distribution Cpq = − (rn − rm)2 cos(npπ/N) cos(mpπ/N) (31) which can be solved using our result Eq. (5). This implies for instance for p = q that Cpp = 2(πp)2 1− π√ . (32) The bracket entails an important correction with respect to the classical description given by the prefactor [6]. We are currently working out how static corrections, such as those for Cpp, may influence the dynamics for polymer chains without topological constraints. (This may be realized, e.g., within the BFM algorithm by using the L26 moves described in Sec. IIIA.) Moreover, for thin polymer films of width H the repulsive interactions are known to be stronger than in the bulk [12]. This provides a mechanism to rationalize the trend towards swelling observed experimentally [58] and confirmed computationally [21]: = log(s)/H. (33) (Prefactors omitted for clarity.) Here Rx(s) and bx denote the components of the segment size and the effective bond length parallel to the film. It also explains the (at first sight surprising) systematic increase of the polymer dynamics with decreasing film thickness [22]. Specifically, the parallel component of the monomer mean-squared displacement gx(t) is expected to scale as gx(t) ≈ R2x(s(t)) ∝ t1/4(1 + log(t)/H) for long reptating chains where s(t) ∝ t1/4 [6]. (The corresponding effect for the three-dimensional bulk should be small, however.) For the same reason (flexible) polymer chains close to container walls must be more swollen and, hence, faster on intermediate time scales than their peers in the bulk. Acknowledgments We thank T. Kreer, S. Peter and A.N. Semenov (all ICS, Strasbourg, France), S.P. Obukhov (Gainesville, Florida) and M. Müller (Göttingen, Germany) for helpful discus- sions. A generous grant of computer time by the IDRIS (Orsay) is also gratefully acknowl- edged. J.B. acknowledges financial support by the IUF and from the European Community’s “Marie-Curie Actions” under contract MRTN-CT-2004-504052. APPENDIX A: MOMENTS OF THE SEGMENTAL SIZE DISTRIBUTION AND THEIR GENERATING FUNCTION Higher moments of the segmental size distribution G(r, s) can be systematically obtained from its Fourier transformation G(q, s) = d3r G(r, s) exp(iq · r), which is in this context sometimes called the “generating function” [59]. For an ideal Gaussian chain, the generating function is then G0(q, s) = exp(−sq2a2) where we have used a2 = b2/6 instead of the bond length b2 to simplify the notation. Moments of the size distribution are given by proper derivatives of G(q, s) taken at q = 0. For example, 〈r2p〉 = (−1)p∆pG(q, s)q=0 (with ∆ being the Laplace operator with respect to the wave vec- tor q). A moment of order 2p is, hence, linked to only one coefficient A2p in the systematic expansion, G(q, s) = p=0A2pq 2p, of G(q, s) around q = 0. For our example this implies = (−1)p(2p+ 1)! A2p (A1) in general and more specifically for a Gaussian distribution 〈r2p〉0 = (2p+1)! spa2p. The non- Gaussian parameters read, hence, αp(s) ≡ 1− (2p+ 1)! 〈r2p〉 〈r2〉p = 1− p! A2p , (A2) which implies (by construction) αp = 0 for a Gaussian distribution. As various moments of the same global order 2p are linked to the same A2p they differ by a multiplicative constant independent of the details of the (isotropic) distribution G(q, s). For example, 〈r2〉 = 6|A2|, 〈r4〉 = 120A4, 〈x2〉 = 〈y2〉 = 2|A2|, 〈x2y2〉 = 8A4 with x and y denoting the spatial components of the segment vector r. Using Eq. (A2) for p = 2 it follows that Kxy(s) ≡ 1− 〈x2y2〉 〈x2〉〈y2〉 = 1− 2A4 = α2(s), (A3) i.e. the properties α2(s) and Kxy(s) discussed in Figs. 9 and 10 must be identical in general provided that G(q, s) is isotropic and can be expanded in q2. We turn now to specific properties ofG(q, s) computed for formally infinite polymer chains in the melt. In practice, these results are also relevant for small segments in large chains, N ≫ s ≫ 1, and, especially, for segments located far from the chain ends. These chains are nearly Gaussian and the generating function can be written as G(q, s) = G0(q, s) + δG(q, s) where δG(q, s) = −〈UG〉0 + 〈U〉0〈G〉0 is a small perturbation under the effective interaction potential ṽ(q) given by Eq. (9). To compute the different integrals it is more convenient to work in Fourier-Laplace space (q, t) with t being the Laplace variable conjugate to s: δG(q, t) = ds δG(q, s)e−st. As illustrated in Fig. 14, there a three contributions to this perturbation: one due to in- teractions between two monomers inside the segment (left panel), one due to interactions between an internal monomer and an external one (middle panel) and one due to interac- tions between two external monomers located on opposite sides (right panel). In analogy to the derivation of the form factor described in Ref. [14] this yields: δG(q, t) = − (q2a2 + t)2 q2a2 + t− (q2a2 + t)2 4πqa2ξ2 Arctan a/ξ + a/ξ + q2a2 + t q2a2 + t 4πqa4 Arctan a/ξ + a/ξ + q2a2 + t 4πqa6 Arctan q2a2 + t . (A4) The graph given in the left panel of Fig. 14 corresponds to the first two lines, the middle panel to the third line and the right panel to the last one. Seeking for the moments we expand δG(q, t) around q = 0. Having in mind chain strands counting many monomers (s ≫ 1), we need only to retain the most singular terms for t → 0. Defining the two dimensionless constants d = vξ/3πa4 = 12vξ/πb4 and c = (3π3/2a3ρ)−1 = 24/π3/b3ρ this expansion can be written as δG(q, t) = − d a2q2 + Γ(3/2) c a2q2 + . . . (A5) d a4q4 − 1 Γ(5/2) c a4q4 + . . . d a6q6 + Γ(7/2) c a6q6 + . . . + . . . where we have used Euler’s Gamma function Γ(α) [60]. The first leading term at each order in q2 — being proportional to the coefficient d — ensures the renormalization of the effective bond length. The next term scaling with the coefficient c corresponds to the leading finite strand size correction. Performing the inverse Laplace transformation Γ(α)/tα → sα−1 and adding the Gaussian reference distribution G0(q, s) this yields the A2p-coefficients for the expansion of G(q, s) around q = 0: A0 = 1 A2 = −a2s 1 + d− c√ 1 + 2d− A6 = − 1 + 3d− 216 A8 = . . . (A6) More generally, one finds A2p = (−1)p (sa2)p 1 + pd− 3(2 pp!p)2 2(2p+ 1)! From this result and using Eq. (A1) one immediately verifies that the moments of the distribution are given by the Eqs. (11) and (12). Using Eq. (A2) one justifies similarly Eq. (26) for the non-Gaussian parameter αp. These moments completely determine the segmental distribution G(r, s) which is indi- cated in Eq. (13). While at least in principle this may be done directly by inverse Fourier- Laplace transformation of the correction δG(q, t) to the generating function it is helpful to simplify further Eq. (A4). We observe first that δG(q, t) does diverge for strictly incom- pressible systems (v → ∞) and one must keep v finite in the effective potential whenever necessary to ensure convergence (actually everywhere but in the diagram corresponding to the interaction between two external monomers). Since we are not interested in the wave vectors larger than 1/ξ we expand δG(q, t) for ξ → 0 which leads to the much simpler expression δG(q, t) ≈ − vξq 3πa2(a2q2 + t)2 t(3a2q2 + t) (a2q2 + t)2 Arctan[ aq√ + o(vξ3). (A8) The first term diverges as v for diverging v. It renormalizes the effective bond length in the zero order term which is indicated in the first line of Eq. (13). The next two terms scale both as v0. Subsequent terms must all vanish for diverging v and can be discarded. It is then easy to perform an inverse Fourier-Laplace transformation of the two relevant v0 terms. This yields δG(x, s) = G0(x, s) with x = r/a 6n. This is consistent with the expression given in the second line of Eq. (13). We note finally that the intramolecular form factor F (q) = 1 n,m=1 〈exp(iq · (rn − rm)〉 of asymptotically long chains can be readily obtained from Eq. (A8). Observing that 〈exp(iq · (rn − rm)〉 = d3r exp(iq · r)G(r, s) = G(q, s) one finds δF (q) = 2 ds δG(q, s) = 2 δG(q, t = 0) = −2 vξ , (A10) where we used the third term of Eq. (A8) in the last step. The first term in Eq. (A8) is discarded as before, since it renormalizes the effective bond length in the reference form factor: F0(q) = 12/b 2q2 ⇒ 12/b2eq2. It follows, hence, that within first-order perturbation theory F (q) = F0(q) + δF (q) ≈ F0(q) (A11) as indicated by Eq. (30) in the Conclusion. This is equivalent to the result 1/F (q)−1/F0(q) ≈ q3/32ρ discussed in Refs. [14, 19] for polymer melts and anticipated by Schäfer [11] by renormalization group calculations of semidilute solutions. N nch τe Re Rg be(N) 6bg(N) 16 216 1214 11.7 4.8 2.998 2.939 32 215 3485 17.1 7.0 3.066 3.030 64 214 1.1 · 104 24.8 10.1 3.116 3.094 128 8192 3.3 · 104 35.6 14.5 3.153 3.139 256 4096 1.0 · 105 50.8 20.7 3.179 3.171 512 2048 3.2 · 105 72.2 29.5 3.200 3.193 1024 1024 1.0 · 106 103 42.0 3.216 3.212 2048 512 3.2 · 106 146 59.5 3.227 3.223 4096 256 9.7 · 106 207 85.0 3.235 3.253 8192 128 2.9 · 107 294 120 3.249 3.248 TABLE I: Various static properties of dense BFM melts of number density ρ = 0.5/8: the chain length N , the number of chains nch per box, the relaxation time τe characterized by the diffusion of the monomers over the end-to-end distance and corresponding to the circles indicated in Fig. 3, the root-mean-squared chain end-to-end distance Re and the radius of gyration Rg of the total chain (s = N − 1). The last two columns give estimates for the effective bond length from the end-to-end distance, be(N) ≡ Re/(N − 1)1/2, and the radius of gyration, bg(N) ≡ Rg/ N . The dashed line in Fig. 4 indicates be(N) 2. Apparently, both estimates increase monotonicly with N reaching be(N) ≈ 6bg(N) ≈ 3.2 for the largest chains available. Note that 6bg(N) < be(N) for smaller N . Property BFM BSM Length unit lattice constant bead diameter Temperature kBT 1 1 Number density ρ 0.5/8 0.84 Linear box size L 256 ≤ 62 Number of monomers nmon 1048576 ≤ 196608 Largest chain length N 8192 1024 Mean bond length 〈|ln|〉 2.604 0.97 l = 〈l2n〉1/2 2.636 0.97 Effective bond length be 3.244 1.34 ρb3e 2.13 2.02 C∞ = (be/l) 2 1.52 1.91 lp = l(C∞ + 1)/2 3.32 1.41 24/π3/ρb3e 0.41 0.44 c1/ce 1.0 1.2 c2/ce 1.0 1.1 c3/ce 1.0 1.0 c4/ce 1.1 1.2 c5/ce 1.1 0.9 cP = c1(be/l) 2/8 0.078 0.124 Dimensionless compressibility g 0.245 0.08 Compression modulus v ≡ 1/gρ 66.7 14.9 vρ/b3eρ 0.96 1.8 TABLE II: Comparison of some static properties of dense BFM and BSM melts. The first six rows indicate conventions and operational parameters. The effective bond length be and the swelling coefficients cp (defined in Eq. (5)) are determined from the first five even moments of the segmental size distribution. The dimensionless compressibility g = S(q → 0)/ρ has been obtained from the total static structure factor S(q) = 1 ∑nmon k,l=1 〈exp(iq · (rk − rl))〉 in the zero wave vector limit as shown at the end of Ref. [14]. The values indicated correspond to the asymptotic long chain behavior. Properties of very small chains deviate slightly. I ~ −3 I ~ −9 I ~ 45 I ~ −9− r n m FIG. 1: (Color online) Sketch of a polymer chain of length N in a dense melt in d = 3 dimensions. As notations we use ri for the position vector of a monomer i, li = ri+1 − ri for its bond vector, r = rm−rn for the end-to-end vector of the chain segment between the monomers n and m = n+s and r = ||r|| for its length. Segment properties, such as the 2p-th moments , are averaged over all possible pairs of monomers (n,m) of a chain and over all chains. The second moment (p = 1) is denoted Re(s) = , the total chain end-to-end distance is Re(s = N − 1). The dashed lines show the relevant graphs of the analytical perturbation calculation outlined in Sec. II B. The numerical factors indicate for infinite chains (without chain end effects) the relative weights contributing to the 1/ s-swelling of Re(s) indicated in Eq. (10). a bdensity ρ c(r,s) correlation hole Segmental = const R(s) Repulsion FIG. 2: (Color online) Role of incompressibility and chain connectivity in dense polymer solutions and melts. (a) Sketch of the segmental correlation hole of a marked chain segment of curvilin- ear length s. Density fluctuations of chain segments must be correlated, since the total density fluctuations (dashed line) are small. Consequently, a second chain segment feels an entropic re- pulsion when both correlation holes start to overlap. (b) Self-similar pattern of nested segmental correlation holes of decreasing strength u(s) ≈ s/ρR(s)3 ≈ ce/ s aligned along the backbone of a reference chain. The large dashed circle represents the classical correlation hole of the total chain (s ≈ N) [5]. This is the input of recent approaches to model polymer chains as soft spheres [50, 52]. We argue that incompressibility on all scales and chain connectivity leads to a short distance repulsion of the segmental correlation holes, which increases with decreasing s. L06 (conserved topology) L26,SS L26,SS,DB FIG. 3: (Color online) Diffusion time τe over the (root-mean-squared) chain end-to-end distance Re(N − 1) as a function of chain length N for different versions of the Bond Fluctuation Model (BFM). All data indicated are for the high number density (ρ = 0.5/8) corresponding to a polymer melt with half the lattice sites being occupied. We have obtained τe = R e(N−1)/6Ds from the self- diffusion coefficient Ds measured from the free diffusion limit of the mean-squared displacement of all monomers δr(t)2 = 6Dst. Data from the classical BFM with topology conserving local Monte Carlo (MC) moves in 6 spatial directions (L06) [26] are represented as stars. All other data sets use topology violating local MC moves in 26 lattice directions (L26). If only local moves are used, L26-dynamics is even at relatively short times perfectly Rouse like which allows the accurate determination of Ds although the monomers possibly have not yet moved over Re(N − 1) for the largest chain lengths considered. Additional slithering snake (SS) moves increase the efficiency of the algorithm by approximately an order of magnitude (squares,bold line). The power law exponent is changed from 2 to an empirical 1.62 (dashed line) if in addition we perform double-bridging (DB) moves. N=128 N=256 N=512 N=1024 N=2048 N=4096 N=8192 =3.244 Eq.(5) FIG. 4: (Color online) Mean-squared segment size Re(s) 2/s vs. curvilinear distance s. We present BFM data for different chain length N at number density ρ = 0.5/8. The averages are taken over all possible monomer pairs (n,m = n+s). The statistics deteriorates, hence, for large s. Log-linear coordinates are used to emphasize the power law swelling over several orders of magnitude of s. The data approach the asymptotic limit (horizontal line) from below, i.e. the chains are swollen. This behavior is well fitted by Eq. (5) for 1 ≪ s ≪ N (bold line). Non-monotonic behavior is found for s → N , especially for small N . The dashed line indicates the measured total chain end-to-end distances, be(N) 2 ≡ Re(N − 1)2/(N − 1) from Tab. I, showing even more pronounced deviations from the asymptotic limit. The dash-dotted line compares this data with Eq. (19). =3.235 =3.240 =3.244 =3.250 =3.255 BFM ρ=0.5/8, N=2048: =3.244, c =0.41=c FIG. 5: (Color online) Replot of the mean-squared segment size as y = K1(s) = 1 − Re(s)2/b2es vs. x = c1/ s, as suggested by Eq. (5), for different trial effective bond lengths be as indicated. Only BFM chains of length N = 2048 are considered for clarity. This procedure is very sensitive to the value chosen and allows for a precise determination. It assumes, however, that higher order terms in the expansion of K1(s) may be neglected. The value be is confirmed from a similar test for higher moments (Fig. 6). x ~1/s BFM b =3.244 BSM b =1.34 too small s ! FIG. 6: (Color online) Critial test of Eq. (5) where the rescaled moments y = Kp(s) of the segment size distribution (defined in Eq. (1)) are plotted vs. x = 3(2pp!p)2 2(2p+1)! . We consider the first five even moments (p = 1, . . . , 5) for the BFM with N = 2048 and the BSM with N = 1024. Also indicated is the rescaled radius of gyration, y = 5/8 (1 − 6R2g(s)/b2e(s + 1)), as a function of x = c1/ (filled circles). The BSM data has been shifted upwards for clarity. Without this shift a perfect data collapse is found for both models and all moments. Keeping the same effective bond length be for all moments of each model we fit for the swelling coefficients cp by rescaling the horizontal axis. We find be ≈ 3.244 for the BFM and 1.34 for the BSM. If be is chosen correctly, all data sets extrapolate linearly to zero for large s (x → 0). The swelling coefficients found are close the theoretical prediction ce, as indicated in Tab. II. The plot demonstrates that the non-Gaussian deviations scale as the segmental correlation hole, u(s) ∼ ce/ s and this for all moments as long as x ≪ 1. The saturation at large x is due to the finite extensibility of short chain segments. Since this effect becomes more marked for larger moments, the fit of be is best performed for p = 1. u(s) c Kλ(s) λ=2: N=1024 λ=2: N=2048 λ=2: N=4096 λ=4: N=2048 λ=8: N=2048 λ=16: N=2048 Insufficient statistics & Finite chain length effects ⇒ ω = 3/2 FIG. 7: (Color online) Plot of Kλ(s) as a function of u(s)c1/ce ∼ 1/ s using the measured u(s) ≡ 24/π3s/ρRe(s) 3. For λ = 2 (corresponding to two segments being connected) BFM and BSM data are compared. Several λ values are given for N = 2048 BFM chains. For chain segments with 1 ≪ s ≪ N all data sets collapse on the bisection line confirming the so-called “recursion relation” Kλ ≈ u proposed by Semenov and Johner [12]. The statistics becomes insufficient for large s (left bottom corner). Systematic deviations arise for s → N due to additional finite-N effects. N=128 N=512 N=2048 N=4096 N=8192 Slope ω=3/2 Chain end effects FIG. 8: (Color online) The bond-bond correlation function P (s)/cP as a function of the curvilinear distance s. Various chain lengths are given for BFM. Provided that 1 ≪ s ≪ N , all data sets collapse on the power law slope with exponent ω = 3/2 (bold line) as predicted by Eq. (23). The dash-dotted curve P (s) ≈ exp(−s/1.5) shows that exponential behavior is only compatible with very small chain lengths. The dashed lines correspond to the theoretical prediction, Eq. (24), for short chains with N = 16, 32, 64 and 128 (from left to right). BFM N=1024 BFM N=2048 BFM N=4096 BFM N=8192 BSM N=1024 y = 6x/5 y = 111x/35 y = 605x/105 small s ! Noise ! FIG. 9: (Color online) Non-Gaussian parameter αp(s) computed for the end-to-end distance of chain segments as a function of ce/ s. Perfect data collapse for all chain lengths and both sim- ulation models is obtained for each p. A linear relationship over nearly two orders of magnitude is found as theoretically expected. Data for three moments (p = 2, 3, 4) are indicated showing a systematic increase of non-Gaussianity with p. The data curvature for small s becomes more pronounced for larger p. N=256 N=512 N=1024 N=2048 N=4096 N=8192 m=n+s FIG. 10: (Color online) Plot of Kxy(s) = 1− averaged over all pairs of monomers (n,m = n+s) and three different direction pairs as a function of ce/ s. As indicated by the sketch at the bottom of the figure, Kxy(s) measures the correlation of the components of the segment vector r. All data points collapse and show again a linear relationship Kxy ≈ u(s). Different directions are therefore coupled! No curvature is observed over two orders of magnitude confirming that higher order perturbation corrections are negligible. Noise cannot be neglected for large s > 100 and finite segment-size effects are visible for s ≈ 1. 0.0 0.5 1.0 1.5 2.0 n=r/b BFM s=32 BFM s=64 BFM s=128 BSM s=32 BSM s=64 BSM s=128 GaussEnhancem entDepletion FIG. 11: (Color online) Segment size distribution y = G(r, s)(bes 1/2)3 vs. n = r/bes 1/2 for several s as indicated in the figure. Only data for BFM with N = 2048 and BSM with N = 1024 are presented. (A similar plot can be achieved by renormalizing the axes using Re(s) instead of bes 1/2). The bold line denotes the Gaussian behaviour y = (3/2π)3/2 exp(−3n2/2). One sees that compared to this reference the measured distributions are depleted for small n ≪ 1 (where the data does not scale) and enhanced for n ≈ 1. 0.0 0.5 1.0 1.5 2.0 2.5 n=r/b BFM s=31 BFM s=63 BFM s=127 BSM s=32 BSM s=64 BSM s=128 f(n) from Eq. (3) Enhancement epletion FIG. 12: (Color online) Deviation δG(r, s) = G(r, s) − G0(r, s) of the measured segmental size distribution from the Gaussian behavior G0(r, s) expected from Flory’s hypothesis for sev- eral s and both models as indicated in the figure. As suggested by Eq. (3), we have plotted y = (δG(r, s)/G0(r, s))/(ce/ s) as a function of n = r/be s. The Gaussian reference distribution has been computed according to Eq. (2) for the measured effective bond length be. A close to perfect data collapse is found for both models. This shows that the deviation scales linearly with u(s) ≈ ce/ s, as expected. The bold line indicates the universal function of f(n) predicted by Eq. (3). 0.0 0.5 1.0 1.5 2.0 2.5 m=r/R BFM s=31 BFM s=63 BFM s=127 BSM s=32 BSM s=64 BSM s=128 f(m) from Eq. (29) Enhancement letio FIG. 13: (Color online) Replot of the relative deviation of the measured segment size distribution, y = (δG(r, s)/G0(r, s)))/(ce/ s), as a function of m = r/Re(s). The figure highlights that the measured segment size is the only length scale relevant for describing the deviation from Flory’s hypothesis. The same data sets and symbols are used as in the previous Fig. 12. v(k) v(k) v(k)~ ~ ~ q q q q q q−k q−k FIG. 14: Interaction diagrams used in reciprocal space for the calculation of δG(q, t) in the scale free limit. 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(16) with the coefficients one would obtain by computing Eq. (15) either with the effective potential ṽ(q) for infinite chains given by Eq. (9) or with the Padé approximation, Eq. (17). Within these approximations of the full linear response formula, Eq. (14), the coefficients can be obtained directly without numerical integration yielding overall similar values. In the first case we obtain 15/8 ≈ 1.87 and in the second 11/8 ≈ 1.37. While the first value is clearly not compatible with the measured end-to-end distances, the second yields a reasonable fit, especially for small N < 1000, when the data is plotted as in Fig. 5. Ultimately, for very long chains the correct coefficient should be 1.59 as is indicated by the dash-dotted line in Fig. 4. [62] These topology non-conserving moves yield configurations which are not accessible with the classical scheme with jumps in 6 directions only. Concerning the static properties we are interested in this paper both system classes are practically equivalent. This has been confirmed by comparing various static properties and by counting the number of monomers which become “blocked” (in absolute space or with respect to an initial group of neighbor monomers) once one returns to the original local scheme. Typically we find about 10 blocked monomers for a system of 220 monomers. The relative difference of microstates is therefore tiny and irrelevant for static properties. Care is needed, however, if the equilibrated configurations are used to investigate the dynamics of the topology conserving BFM version. The same caveats arise for the slithering snake and double-bridging moves. [63] We have verified that the alternative definition 〈em=n+s · en〉 with en being the normalized bond vector yields very similar results. This is due to the weak bond length fluctuations, specifically at high densities, in both coarse-grained models under consideration. It is possi- ble that other models show slightly different power law amplitudes cP depending on which definition is taken. Flory's ideality hypothesis revisited Physical idea and sketch of the perturbation calculation Scaling arguments Perturbation calculation Computational models and technical details Bond fluctuation model Bead spring model Numerical results The swelling of chain segments Chain connectivity and recursion relation Intrachain bond-bond correlations Higher moments and associated coefficients The segmental size distribution Conclusion Acknowledgments Moments of the segmental size distribution and their generating function References

Presenting theoretical arguments and numerical results we demonstrate long-range intrachain correlations in concentrated solutions and melts of long flexible polymers which cause a systematic swelling of short chain segments. They can be traced back to the incompressibility of the melt leading to an effective repulsion $u(s) \approx s/\rho R^3(s) \approx ce/\sqrt{s}$ when connecting two segments together where $s$ denotes the curvilinear length of a segment, $R(s)$ its typical size, $ce \approx 1/\rho be^3$ the ``swelling coefficient", $be$ the effective bond length and $\rho$ the monomer density. The relative deviation of the segmental size distribution from the ideal Gaussian chain behavior is found to be proportional to $u(s)$. The analysis of different moments of this distribution allows for a precise determination of the effective bond length $be$ and the swelling coefficient $ce$ of asymptotically long chains. At striking variance to the short-range decay suggested by Flory's ideality hypothesis the bond-bond correlation function of two bonds separated by $s$ monomers along the chain is found to decay algebraically as $1/s^{3/2}$. Effects of finite chain length are considered briefly.

Introduction, one sees that both equations are essentially identical — taken apart, however, that they depend on b and be. Note the conspicuous factor (b/be) in Eq. (12) which would strongly reduce the empirical swelling coefficients cp = ce(b/be) for large p if b and be were different. Interpretation of first-loop results in different contexts. The above perturbation results may be used directly to describe the effect of a weak excluded volume v on a reference system of perfectly ideal polymer melts with Kuhn segment length b where all interactions have been switched off (v = 0). It is expected to give a good estimation for the effective bond length be only for a small Ginzburg parameter: Gz ≪ 1. For the dense melts we want to describe this does not hold (Sec. III) and one cannot hope to find a good quantitative agreement with Eq. (11). Note also that large wave vectors contribute strongly to the leading Gaussian term. The effective bond length be is, hence, strongly influenced by local and non-universal effects and is very difficult to predict in general. Our much more modest goal is to predict the coefficient of the 1/ s-perturbation and to express it in terms of a suitable variational reference Hamiltonian characterized by a conveniently chosen Kuhn segment b and the measured effective bond length be (instead of Eq. (11)). Following Muthukumar and Edwards [10], we argue that for dense melts b should be renormalized to be to take into account higher order graphs. No strict mathematical proof can be given at present that the infinite number of possible graphs must add up in this manner. Our hypothesis relies on three observations: • The general scaling argument discussed in Sec. IIA states that we have only one relevant length scale in this problem, the typical segment size R(s) ≈ be s itself. The incompressibility constraint cannot generate an additional scale. It is this size R(s) which sets the strength of the effective interaction which then in turn feeds back to the deviations of R(s) from Gaussianity. Having a bond length b in addition to the effective bond length be associated with R(s) would imply incorrectly a second length scale b s varying independently with the bulk modulus v. (We will check explicitly below in Fig. 13 that there is only one length scale.) This implies b/be = const v • Thus, since by construction b/be = 1 for v → 0, it follows that both lengths should be equal for all v. • We know from Eq. (12) that the empirical coefficients cp = ce(b/be)2p−3 should depend strongly on the moment considered if the ratio b/be is not close to unity. It will be shown below (Fig. 6) that cp/ce ≈ 1 for all p. This implies b ≈ be. Finite chain size effects. To describe properly finite chain size corrections Eq. (9) must be replaced by the general linear response formula ṽ(q) + ρF (q) (14) with F (q) = NfD(x) being the form factor of the Gaussian reference chain given by Debye’s function fD(x) = 2(e −x − 1 + x)/x2 with x = (qb)2N/6 [6]. This approximation allows in principle to compute, for instance, the (mean-squared) total chain end-to-end distance, A = (rN −r1)2. One verifies readily (see [6], Eq. (5.III.9)) that the effect of the perturbation may be expressed as 〈A〉0 〈U〉0 − 〈AU〉0 = (2π)3 ṽ(q) ds s2(N − s) exp q2b2s . (15) We take now first the integral over s. In the remaining integral over q small q wave vectors contribute to the N -swelling while large q renormalize the effective bond length of the dominant Gaussian behaviour linear in N (as discussed above). Since we wish to determine the non-Gaussian corrections, we may focus on small wave vectors q ≪ 1/ξ. Since in this limit 1/v = ρg ≪ ρF (q), one can neglect in Eq. (14) the 1/v contribution to the inverse effective interaction potential. We thus continue the calculation using the much simpler ṽ(N, x) = 1/(NρfD(x)). This allows us to express the swelling as 〈(rN − r1)2〉 I(xu). (16) To simplify the notation we have set here finally b = be in agreement with the hypothesis discussed above. The numerical integral I(xu) = dx . . . over x is slowly convergent at infinity. As a consequence the estimate I(∞) = 1.59 may be too large for moderate chain lengths. In practice, convergence is not achieved for values xu(N) ≈ (b/ξ)2N corresponding to the screening length ξ. We remark finally that numerical integration can be avoided for various properties if the Padé approximation of the form factor, F (q) = N/(1 + (qb)2N/12), is used. This allows analytical calculations by means of the simplified effective interaction potential ṽ(q) = 12ρb3 . (17) This has been used for instance for the calculation of finite chain size effects for the bond- bond correlation function discussed in Sec. IVC below [61]. III. COMPUTATIONAL MODELS AND TECHNICAL DETAILS A. Bond fluctuation model A widely-used lattice Monte Carlo scheme for coarse-grained polymers. The body of our numerical data comes from the three dimensional bond fluctuation model (BFM) — a lattice Monte Carlo (MC) algorithm where each monomer occupies eight sites of a unit cell of a simple cubic lattice [24, 25, 26]. Our version of the BFM with 108 bond vectors corresponds to flexible athermal chain configurations [23]. All length scales are given in units of the lattice constant and time in units of Monte Carlo Steps (MCS). We use cubic periodic simulation boxes of linear size L = 256 containing nmon = ρL 3 = 220 ≈ 106 monomers. This monomer number corresponds to a monomer number density ρ = 0.5/8 where half of the lattice sites are occupied (volume fraction 0.5). The large system sizes used allow us to suppress finite box-size effects for systems with large chains. Using a mix of local, slithering snake [27, 28, 29], and double-bridging [17, 23, 30, 31] MC moves we were able to equilibrate dense systems with chain lengths up to N = 8192. Equilibration and sampling of high-molecular BFM melts. Standard BFM implementa- tions [26, 32, 33] use local MC jumps to the 6 closest lattice sites to prevent the crossing of chains and conserve therefore the chain topology. These “L06” moves lead to very large relaxation times, scaling at least as τe ∼ N3, as may be seen from Fig. 3 (stars). The relax- ation time τe = R e/6Ds indicated in this figure has been estimated from the self-diffusion coefficient Ds obtained from the mean-square displacements of all monomers in the free diffusion limit. (For the largest chain indicated for L06 dynamics only a lower bound for τe is given.) Instead of this more realistic but very slow dynamical scheme we make jump attempts to the 26 sites of the cube surrounding the current monomer position (called “L26” moves). This allows the chains to cross each other which dramatically speeds up the dynam- ics, especially for long chains (N > 512). If only local moves are considered, the dynamics is perfectly consistent with the Rouse model [6]. As shown in Fig. 3, we find τe ≈ 530N2 for L26 dynamics. This is, however, still prohibitive by large for sampling configurations with the longest chain length N we aim to characterize [62]. Slithering snake moves. In addition to the local moves one slithering snake move per chain is attempted on average per MCS corresponding to the displacement of N monomer along the chain backbone. Note that in our units two spatial displacement attempts per MCS are performed on average per monomer, one for a local move and one for a snake move. (In practice, it is computationally more efficient for large N to take off a monomer at one chain end and to paste it at the other leaving all other monomers unaltered. Before dynamical measurements are performed the original order of beads must then be restored.) Interestingly, a significantly larger slithering snake attempt frequency would not be useful since the relaxation time of slithering snakes without or only few local moves increases exponentially with mass [29, 34] due to the correlated motion of snakes [35]. In order to obtain an efficient free snake diffusion (with a chain length independent curvilinear diffusion coefficient Dc(N) ∼ N0 and τe ≈ N2/Dc(N) ∼ N2 [28, 29]) it is important to relax density fluctuations rapidly by local dynamical pathways. As shown in Fig. 3 (squares), we find a much reduced relaxation time τe ≈ 40N2 which is, however, still unconveniently large for our longest chains. Note that most of the CPU time is still used by local moves. The computational load per MCS remains therefore essentially chain length independent. Advantages and pitfalls of double-bridging moves. Double-bridging (DB) moves are very useful for high densities and help us to extend the accessible molecular masses close to 104. As for slithering snake moves we use all 108 bond vectors to switch chain segments between two different chains. Only chain segments of equal length are swapped to conserve monodispersity. Topolocial constraints are again systematically and deliberately violated. Since more than one swap partner is possible for a selected first monomer, delicate detailed balance questions arise. This is particularly important for short chains and is discussed in detail in Ref. [23]. Technically, the simplest solution to this problem is to refuse all moves with more than one swap partner (to be checked both for forward and back move). The configurations are screened with a frequency fDB for possible DB moves where we scan in random order over the monomers. The frequency should not be too large to avoid (more or less) immediate back swaps and monomers should move at least out of the local monomer cage and over a couple of lattice sites. We use fDB = 0.1 between DB updates for the configurations reported here. (The influence of fDB on the performance has not been explored systematically, but preliminary results suggest a slightly smaller DB frequency for future studies.) The diffusion times over the end-to-end distance for this case are indicated in Tab. I. As shown in Fig. 3, we find empirically τe(N) ≈ 13N1.62. For N = 8192 this corresponds to 3 · 107 MCS. This allows us even for the largest chain lengths to observe monomer diffusion over several Re within the 10 8 MCS which are feasible on our XEON-PC processor cluster. The efficiency of DB moves is commonly characterized in terms of the relaxation time τee of the end-to-end vector correlation function [30, 31]. For normal chain dynamics this would indeed characterize the longest relaxation time of the system, i.e. τe ≈ τee. For the double-bridging this is, however, not sufficient since density fluctuations do not couple to the bridging moves and can not be relaxed. We find therefore that configurations equilibrate on time scales given by τe rather than by τee ≪ τe. This may be verified, for instance, from the time needed for the distribution Re(s) (and especially its spatial components) to equilibrate. The criterion given in the literature [30] is clearly not satisfactory and may lead to insufficiently equilibrated configurations. In summary, equilibration with DB moves still requires monomer diffusion over the typical chain size, however at a much reduced price. Some properties of our configurations. The Tables I and II summarize some system properties obtained for our reference density ρ = 0.5/8. Averages are performed over all chains and 1000 configurations. These configurations may be considered to be independent for N < 4096. Only a few independent configurations exist for the largest chain length N = 8192 which has to be considered with some care. Taking apart this system, chains are always much smaller than the box size. For asymptotically long chains, we obtain an average bond length 〈|l|〉 ≈ 2.604, a root-mean-squared bond length l ≡ 〈l2〉1/2 ≈ 2.635 and an effective bond length be ≈ 3.244 — as we will determine below in Sec. IVA. This corresponds to a ratio C∞ ≡ b2e/l2 ≈ 1.52 and, hence, to a persistence length lp = l(C∞ + 1)/2 ≈ 3.32 [23]. Especially, we find from the zero wave vector limit of the total structure factor S(q) a low (dimensionless) compressibility g = S(q → 0)/ρ ≈ 0.246 which compares well with real experimental melts. From the measured bulk compression modulus v ≡ 1/g(ρ)ρ and the effective bond length be one may estimate a Ginzburg parameter Gz = vρ/b3eρ ≈ 0.96. Following Ref. [13] the interaction parameter v is supposed here to be given by the full inverse compressibility and not just by the second virial coefficient. B. Bead spring model Hamiltonian. Additionally, molecular dynamics simulations of a bead-spring model (BSM) [36] were performed to dispel concerns that our results are influenced by the un- derlying lattice structure of the BFM. The model is derived from a coarse-grained model for polyvinylalcohol which has been employed to study polymer crystallization [37]. It is characterized by two potentials: a non-bonded potential of Lennard-Jones (LJ) type and a harmonic bond potential. While the often employed Kremer-Grest model [38] uses a 12− 6 LJ potential to describe the non-bonded interactions Unb(r), our non-bonded potential has a softer repulsive part. It is given by Unb(r) = 1.511 , (18) which is truncated and shifted at the minimum at rmin ≈ 1.15. Note that all length scales are given in units of σ0 and we use LJ units [39] for all BSM data (mass m = 1, Boltzmann constant kB = 1). The parameters of the bond potential, Ub(r) = 1120(r− lb)2, are adjusted so that the average bond length l(ρ = 0.84) ≈ lb = 0.97 is approximately the same as in the standard Kremer-Grest model [38]. The average bond length and the root-mean-squared bond length are almost identical for the BSM due to the very stiff bond potential. Since rmin/l ≈ 1.16 bonded monomers penetrate each other significantly. Equilibration and sampling. We perform standard molecular dynamics simulations in the canonical ensemble with a Langevin thermostat (friction constant Γ = 0.5) at temperature T = 1. The equations of motion are integrated by the velocity-Verlet algorithm [39]. To improve the statistics for large chain length, we have implemented additional double-bridging moves. Since only few of these MC moves are accepted per unit time, this does affect neither the stability nor the accuracy of the molecular dynamics sweeps. Some properties obtained. For clarity, we show only data for chain length N = 1024 and number density ρ = 0.84, the typical melt density of the Kremer-Grest model [38]. For the reported data we use periodic simulation boxes of linear size L ≈ 62 containing nmon = 196608 monomers, but we have also sampled different boxes sizes (up to L = 77.5) to check for finite box-size effects. For the reference density a dimensionless compressibility g ≈ 0.08 is found which is about three times smaller than for our BFM melt. For the effective bond length we obtain be ≈ 1.34, i.e. BSM chains (C∞ ≈ 1.91, lp ≈ 1.41) are slightly stiffer than the corresponding BFM polymers. Fortunately, the product ρb3e ≈ 2 is roughly similar in both models and one expects from Eq. (4) a similar swelling for large s. Note finally that the Ginzburg parameter Gz ≈ 1.8 is much larger than for the BFM systems. As we have emphasized in Sec. II, this should, however, not influence the validity of the perturbation prediction of the expected 1/ s-swelling of the chains when expressed in terms of the measured effective bond length. IV. NUMERICAL RESULTS As illustrated in Fig. 1, a chain segment of curvilinear length s > 0 is identified by two monomers n and m = n + s on the same chain. We compute here various moments of chain segment properties where we ensemble-average over all chains and all start points n. The statistical accuracy must therefore always decrease for large s. We concentrate first on the second moment (p = 1) of the segmental size distribution. Higher moments and the segmental size distribution are discussed in Sec. IVE. A. The swelling of chain segments Scale free regime for 1 ≪ s ≪ N . The mean-squared segment size R2e(s) = 〈r2〉 is presented in the Figs. 4, 5 and 6. The first plot shows clearly that chain segments are swollen, i.e. R2e(s)/s increases systematically and this up to very large curvilinear distances s. Only BFM data are shown for clarity. A similar plots exists for the BSM data. In agreement with Eq. (5) for p = 1, the asymptotic Gaussian behavior (dashed line) is approached from below and the deviation decays as u(s) ∝ 1/ s (bold line). The bold line indicated corresponds to be = 3.244 and c1 ≈ ce ≈ 0.41 which fits nicely the data over several decades in s — provided that chain end effects can be neglected (s ≪ N). Note that a systematic underestimation of the true effective bond length would be obtained by taking simply the largest R2e(s)/s ≈ 3.232 value available, say, for monodisperse chains of length N = 2048. Finite chain-size effects. Interestingly, R2e(s)/s does not approach the asymptotic limit monotonicly. Especially for short chains one finds a non-monotonic behavior for s → N . This means that the total chain end-to-end distance Re(s = N − 1) must show even more pronounced deviations from the asymptotic limit. This is confirmed by the dashed line representing the b2e(N) ≡ R2e(N − 1)/(N − 1) data points given in Tab. I. We emphasize that the non-monotonicity of R2e(s)/s becomes weaker with increasing N and that, as one expects, the inner distances, as well as the total chain size, are characterized by the same effective bond length be for large s or N . The non-monotonic behavior may be qualitatively understood by the reduced self-interactions at chain ends which lessens the swelling on these scales. These finite-N corrections have been calculated analytically using the full Debye function for the effective interaction potential ṽ(q), Eq. (14). The prediction for the total chain end-to-end vector given in Eq. (16) is indicated in Fig. 4 (dash-dotted line) where we have replaced the weakly N -dependent integral I(xu) by its upper bound value for infinite chains R2e(N − 1) b2e (N − 1) 1.59ce√ N − 1 . (19) We have changed here the chain length N in the analytical formula (obtained for large chains where N ≈ N − 1) to the curvilinear length N − 1. This is physically reasonable and allows to take better into account the behavior of small chains. Note that Eq. (19) is similar to Eq. (5) — apart from a slightly larger prefactor explaining the observed stronger deviations. Theory compares well with the measured data for large N . It does less so for smaller N , as expected, where the chain length dependence of the numerical integral I(xu(N)) ≤ 1.59 must become visible. This explains why the data points are above the dash-dotted line. Note also that additional non-universal finite-N effects not accounted for by the theory are likely for small N . In contrast to this, Re(s) is well described by the theory even for rather small s provided that N is large and chain end effects can be neglected. In summary, it is clear that one should use the segment size Re(s) rather than the total chain size to obtain in a computational study a reliable fit of the effective bond length be. Extrapolation of the effective bond length of asymptotically long chains. The represen- tation chosen in Fig. 4 is not the most convenient one for an accurate determination of be and c1. How precise coefficients may be obtained according to Eq. (5) is addressed in the Figs. 5 and 6. The fitting of the effective bond length be and its accuracy is illustrated in Fig. 5 for BFM chains of length N = 2048. This may be first done approximately in linear coordinates by plotting R2e(s)/s as a function of 1/ s (not shown). Since data for large s are less visible in this representation, we recommend for the fine-tuning of be to switch then to logarithmic coordinates with a vertical axis y = 1 − R2e(s)/b2es for different trial values of be. The correct value of be is found by adjusting the vertical axis y such that the data extrapolates linearly as a function of 1/ s to zero for large s. We assume for the fine-tuning that higher order perturbation corrections may be neglected, i.e. we take Eq. (5) literally. (We show below that higher order corrections must indeed be very small.) The plot shows that this method is very sensitive, yielding a best value that agrees with the theory over more than one order of magnitude without curvature. As expected, it is not possible to rationalize the numerically obtained values be ≈ 3.244 for the BFM and be ≈ 1.34 for the BSM using Eq. (11). According to Eq. (4) these fit values imply the theoretical swelling coefficients ce = 0.41 for the BFM and ce = 0.44 for the BSM. Empirical swelling coefficients. As a next step the horizontal axis is rescaled such that all data sets collapse on the bisection line, i.e. using Eq. (5) we fit for the empirical swelling coefficient c1 and compare it to the predicted value ce. This rescaling of the axes allows to compare both models in Fig. 6. For clarity the BSM data have been shifted upwards. For the BFM we find c1/ce ≈ 1.0, as expected, while our BSM simulations yield a slightly more pronounced swelling with c1/ce ≈ 1.2. Segmental radius of gyration. Also indicated in Fig. 6 is the segmental radius of gyration Rg(s) (filled circles) computed as usual [6] as the variance of the positions of the segment monomers around their center of mass. Being the sum over all s + 1 monomers, it has a much better statistics compared to Re(s). The scaling used can be understood by expressing the radius of gyration R2g(s) = (s+1)2 l=n r in terms of displacement vectors rkl [6]. Using Eq. (5) and integrating twice this yields 6R2g(s) b2e(s+ 1) . (20) Plotting the l.h.s. of this relation against the r.h.s. we obtain a perfect data collapse on the bisection line where we have used the same parameters be and c1 as for the mean-squared segment size. This is an important cross-check which we strongly recommend. Different values indicate insufficient sample equilibration. B. Chain connectivity and recursion relation As was emphasized in Sec. IIA the observed swelling is due to an entropic repulsion between chain segments induced by the incompressibility of the melt. To stress the role of chain connectivity we repeat the general scaling argument given above in a form originally proposed by Semenov and Johner for ultrathin films [12]. As shown in Fig. 7 we test the relation Kλ(s) ≡ R2e(λs)− λR2e(s) λ)R2e(s) ≈ u(s) ≡ s ρRe(s)d with Kλ(s) being a direct measure of the non-Gaussianity (λ being a positive number) comparing the size of a segment of length λs with the size of λ segments of length s joined together. (The prefactor 1/(λ − λ) in the definition of Kλ(s) has been introduced for convenience.) Equivalently, this can be read as a measure for the swelling of a chain where initially the interaction energy u between the segments has been switched off. Kλ is a functional of u(s) with Kλ[u = 0] = 0. The analytic expansion of the functional must be dominated by the linear term (as indicated by ≈ in the above relation) simply because u is very small. Altogether, Eq. (21) yields a recursion relation relating Re(λs) with Re(s) for any λ provided 1 ≪ s < λs ≪ N . It can be solved, leading (in lowest order) to Eq. (5) with p = 1. This may be seen from the ansatz R2e(s) = b es(1− ce/sω−1+ . . .) which readily yields ω = 3/2 and ce ≈ 1/ρb3e . Eq. (21) has been validated directly in Fig. 7 for λ = 2 (corresponding to two segments of length s joined together) for the BFM and the BSM as indicated. In addition, for BFM chains of length N = 2048 several values of λ have been given. As suggested by Eq. (4), we have plotted Kλ(s) as a function of (c1/ce) u(s) with u(s) ≡ 24/π3s/ρR3e(s) ≈ ce/ The prefactor of u(s) allows a convenient comparison with Fig. 5. Note the perfect data collapse for all data sets. More importantly, the predicted linearity is well confirmed for large segments (1 ≪ s) and this without any tunable parameter for the vertical axis, as was needed in the previous Figs. 5 and 6. C. Intrachain bond-bond correlations Expectation from Flory’s hypothesis. An even more striking violation of Flory’s ideality hypothesis may be obtained by computing the bond-bond correlation function, defined by the first Legendre polynomial P (s) = 〈lm=n+s · ln〉 /l2 where the average is performed, as before, over all possible pairs of monomers (n,m = n+ s) [63]. Here, li = ri+1 − ri denotes the bond vector between two adjacent monomers i and i + 1 and l2 = 〈l2n〉n the mean- squared bond length. The bond-bond correlation function is generally believed to decrease exponentially [4]. This belief is based on the few simple single chain models which have been solved rigorously [4, 40] and on the assumption that all long range interactions are negligible on distances larger than the screening length ξ. Hence, only correlations along the backbone of the chains are expected to matter and it is then straightforward to work out that an exponential cut-off is inevitable due to the multiplicative loss of any information transferred recursively along the chain [4]. Asymptotic behavior in the melt. That this reasoning must be incorrect follows imme- diately from the relation P (s) = R2e(s) (22) expressing the bond-bond correlation function as the curvature of the second moment of the segment size distribution. It is obtained from the identity 〈ln · lm〉 = 〈∂nrn · ∂mrm〉 = −∂n∂m 〈r2nm〉 /2. (Note that the velocity correlation function is similarly related to the second derivative of the mean-square displacement with respect to time [41].) Hence, P (s) allows us to probe directly the non-Gaussian corrections without any ideal contribution. This relation together with Eq. (5) suggests an algebraical decay P (s) = cP/s ω with ω = 3/2 , cP = c1 (be/l) 2/8 ≈ ρl2be of the bond-bond correlation function for dense solutions and melts, rather than the ex- ponential cut-off expected from Flory’s hypothesis. This prediction (bold line) is perfectly confirmed by the larger chains (N > 256) indicated in Fig. 8. In principle, the swelling coefficient, c1 ∼ cP, may also be obtained from the power law amplitude of the bond-bond correlation function, however, to lesser accuracy than by the previous method (Fig. 5). One reason is that P (s) decays very rapidly and does not allow a precise fit beyond s ≈ 102. The values of cP obtained from c1 are indicated in Tab. II. Data from the BSM have also been included in the figure to demonstrate the universality of the result. The vertical axis has been rescaled with cP which allows to collapse the data of both models. Finite chain-size corrections. As can be seen for N = 16, exponentials are compatible with the data of short chains. This might explain how the power law scaling has been overlooked in previous numerical studies, since good statistics for large chains (N > 1000) has only become available recently. However, it is clearly shown that P (s) approaches systematically the scale free asymptote with increasing N . The departure from this limit is fully accounted for by the theory if chain end effects are carefully considered (dashed lines). Generalizing Eq. (23) and using the Padé approximation, Eq. (17), perturbation theory yields P (s) = 1 + 3x+ 5x2 1 + x (1− x)2 (24) where we have set x = s/N . For x ≪ 1 this is consistent with Eq. (23). In the limit of large s → N , the correlation functions vanish rigorously as P (s) ∝ (1 − x)2. Considering that non-universal features cannot be neglected for short chain properties and that the theory does not allow for any free fitting parameter, the agreement found in Fig. 8 is rather satisfactory. D. Higher moments and associated coefficients Effective bond length and empirical swelling coefficients. The preceding discussion fo- cused on the second moment of the segmental size distribution G(r, s). We have also com- puted for both models higher moments 〈r2p〉 with p ≤ 5. If traced in log-linear coordinates as y = (6pp! 〈r2p〉 /(2p + 1)!sp)1/p vs. x = s higher moments approach b2e from below — just as the second moment presented in Fig. 4. The deviations from ideality are now more pronounced and increase with p (not shown). The moments are compared in Fig. 6 with Eq. (5) where they are rescaled as y = Kp(s) as defined in Eq. (1) and plotted as functions of x = 3(2pp!p)2 2(2p+1)! . The prediction is indicated by bold lines. It is important that the same effective bond length be is obtained from the analysis of all functions Kp(s) as illustrated in Fig. 6. Otherwise we would regard equilibration and statistics as insufficient. The empirical swelling coefficients cp are obtained, as above in Sec. IVA, by shifting the data horizontally. A good agreement with the expected cp/ce ≈ 1 is found for both models and all moments as may be seen from Tab. II. This confirms the renormalization of the Kuhn segment b → be of the Gaussian reference chain in agreement with our discussion in Sec. II B. Otherwise we would have measured empirical coefficients decreasing strongly as cp/ce ≈ (b/be)2p−3 with p. Since the effective bond length of non-interacting chains are known for the BFM (b ≈ 2.688) and the BSM (b ≈ 0.97), one can simply check, say for p = 5, that the non-renormalized values would correspond to the ratios c5/ce ≈ (2.688/3.244)7 ≈ 0.3 for the BFM and c5/ce ≈ (0.97/1.34)7 ≈ 0.1 for the BSM. This is clearly not consistent with our data. It should be emphasized that both coefficients be and cp are more difficult to determine for large p, since the linear regime for x ≪ 1 in the representation chosen in Fig. 6 becomes reduced. For large x ≫ 1 one finds that y(x) → 1, i.e. 〈r2p〉 /b2pe sp → 0. This trivial departure from both Gaussianity and the 1/ s-deviations we try to describe, is due to the finite extensibility of chain segments of length s which becomes more marked for larger moments probing larger segment sizes. The data collapse for both x-regimes is remarkable, however. Incidentally, it should be noted that for the BSM the empirical swelling coefficients are slightly larger than expected. At present we do not have a satisfactory explanation for this altogether minor effect, but it might be attributed to the fact that neighbouring BSM beads along the chain strongly interpenetrate — an effect not considered by the theory. Non-Gaussian parameter αp. The failure of Flory’s hypothesis can also be demonstrated by means of the standard non-Gaussian parameter αp(s) ≡ 1− (2p+ 1)! 〈r2pnm〉 〈r2nm〉 p (25) comparing the 2p-th moment with the second moment (p = 1). In contrast to the closely related parameter Kp(s) this has the advantage that here two measured properties are com- pared without any tuneable parameter, such as be, which has to be fitted first. Fig. 9 presents αp(s) vs. ce/ s for the three moments with p = 2, 3, 4. For each p we find perfect data col- lapse for all chain lengths and both models and confirm the linear relationship αp(s) ≈ u(s) expected. The lines indicate the theoretical prediction αp(s) = 3 (2pp!p)2 2 (2p+ 1)! which can be derived from Eq. (5) by expanding the second moment in the denominator. An alternative derivation based on the coefficients of the expansion of the generating function G(q, s) in q2 is indicated by Eq. (A2) in the Appendix. Having confirmed above that cp/ce ≈ 1, we assume in Eq. (26) that cp = ce to simplify the notation. The prefactors 6/5, 111/35 and 604/105 for p = 2, 3 and 4 respectively are nicely confirmed. They increase strongly with p, i.e. the non-Gaussianity becomes more pronounced for larger moments as already mentioned. Note also the curvature of the data at small s due to the finite extensibility of the segments which becomes more marked for higher moments. If one plots αp(s) as a function of the r.h.s. of Eq. (26) all data points for all moments and even for too small s collapse on one master curve (not shown) — just as we have seen before in Fig. (6). Correlations of different directions. A similar correlation function is presented in Fig. 10 which measures the non-Gaussian correlations of different spatial directions. It is defined by Kxy(s) ≡ 1− 〈x2 y2〉 〈x2〉 〈y2〉 for the two spatial components x and y of the vector r as illustrated by the sketch given at the bottom of Fig. 10. Symmetry allows to average over the three pairs of directions (x, y), (x, z) and (x, z). Following the general scaling argument given in Sec. II we expect Kxy(s) ≈ u(s) ≈ ce/ s which is confirmed by the perturbation result Kxy(s) = = K2(s). (28) This is nicely confirmed by the linear relationship found (bold line) on which all data from both simulation models collapse perfectly. The different directions of chain segments are therefore coupled. As explained in the Appendix (Eq. (A3)), Kxy(s) and α2(s) must be identical if the Fourier transformed segmental size distribution G(q, s) can be expanded in terms of q2 and this irrespective of the values the expansion coefficients take. Fig. 10 confirms, hence, that our computational systems are perfectly isotropic and tests the validity of the general analytical expansion. The correlation function Kxy is of particular interest since the zero-shear viscosity should be proportional to ∼ 〈x2y2〉 = 〈x2〉 〈y2〉 (1 − Kxy(s)). We assume here following Edwards [6] that only intrachain stresses contribute to the shear stress σxy. Hence, our results suggest that the classical calculations [6] — assuming incorrectly Kxy = 0 — should be revisited. E. The segmental size distribution We turn finally to the segmental size distribution G(r, s) itself which is presented in Figs. 11, 12 and 13. From the theoretical point of view G(r, s) is the most fundamental property from which all others can be derived. It is presented last since it is computationally more demanding — at least if high accuracy is needed — and coefficients such as be may be best determined directly from the moments. The normalized histograms G(r, s) are computed by counting the number of segment vectors between r − dr/2 and r + dr/2 with dr being the width of the bin and one divides then by the spherical bin volume. Since the BFM model is a lattice model, this volume is not 4πr2dr but given by the number of lattice sites the segment vector can actually point to for being allocated to the bin. Incorrect histograms are obtained for small r if this is not taken into account. (Averages are taken over all segments and chains, just as before.) Clearly, non-universal physics must show up for small vector length r and small curvilinear distance s and we concentrate therefore on values r ≫ σ and s ≥ 31. When plotted in linear coordinates as in Fig. 11, G(r, s) compares roughly with the Gaussian prediction G0(r, s) given by Eq. (2), but presents a distinct depletion for small segment sizes with n ≡ r/be s ≪ 1 and an enhanced regime for n ≈ 1. A second depletion region for large n ≫ 1 — expected from the finite extensibility of the segments — can be best seen in the log-log representation of the data (not shown). To analyse the data it is better to consider instead of G(r, s) the relative deviation δG(r, s)/G0(r, s) = G(r, s)/G0(r, s)− 1 which should further be divided by the strength of the segmental correlation hole, ce/ As presented in Fig. 12 this yields a direct test of the key relation Eq. (3) announced in the Introduction. The figure demonstrates nicely the scaling of the data for all s and for both models. It shows further a good collapse of the data close to the universal function f(n) predicted by theory (bold line). Note that the depletion scales as 1/n for small segment sizes (dashed line). The agreement of simulation and theory is by all standards remarkable. (Obviously, error bars increase strongly for n ≫ 1 where G0(r, s) decreases strongly. The regime for very large n where the finite extensibility of segments matters has been omitted for clarity.) We emphasize that this scaling plot depends very strongly on the value be which is used to calculate the Gaussian reference distribution. If a precise value is not available we recommend to use instead the scaling variable m = r/Re(s) for the horizontal axis, i.e. to replace the scale be s estimated from the behaviour of asymptotically long chains by the measured (mean-squared) segment size for the given s. The Gaussian reference distribution is then accordingly G0(m,Re(s)) = (3/2πRe(s) 2) exp(−3 m2). The corresponding scaling plot is given in Fig. 13. It is simi- lar and of comparable quality as the previous plot. Changing the scaling variable from n = r/be s to m = r/Re(s) ≈ (r/be s)(1 + ce/2 s) changes somewhat the universal func- tion. Expanding the previous result, Eq. (3), this adds even powers of m to the function f(n) given in Eq. (3) f(n) ⇒ f(m) = + 9m+ m2 − 9 . (29) That the two additional terms in the function are correct can be seen by computing the second moment 4π drr4δG(r, s) which must vanish by construction. The rescaled relative deviation is somewhat broader than in the previous plot due to the additional term scaling as m2. As already stressed this scaling does not rely on the effective bond length be and is therefore more robust. It has the nice feature that it underlines that there is only one characteristic length scale relevant for the swelling induced by the segmental correlation hole, the typical size of the chain segment itself. V. CONCLUSION Issues covered and central theoretical claims. We have revisited Flory’s famous ideality hypothesis for long polymers in the melt by analyzing both analytically and numerically the segmental size distribution G(r, s) and its moments for chain segments of curvilinear length s. We have first identified the general mechanism that gives rise to deviations from ideal chain behavior in dense polymer solutions and melts (Sec. II). This mechanism rests upon the interplay of chain connectivity and the incompressibility of the system which generates an effective repulsion between chain segments (Fig. 2). This repulsion scales like u(s) ≈ ce/ s where the “swelling coefficient” ce ≈ 1/b3eρ sets the strength of the interaction. It is strong for small segment length s, but becomes weak for s → N in the large-N limit. The overall size of a long chain thus remains almost ‘ideal’, whereas subchains are swollen as described by Eq. (5). Most notably, the relative deviation δG(r, s)/G0(r, s) of the segmental size distribution from Gaussianity should be proportional to u(s). As a function of segment size r, the repulsion manifests itself by a strong 1/r-depletion at short distances r ≪ be and a subsequent shift of the histogram to larger distances (Eq. (3)). Summary of computational results. Using Monte Carlo and molecular dynamics simu- lation of two coarse-grained polymer models we have verified numerically the theoretical predictions for long and flexible polymers in the bulk. We have explicitly checked (e.g., Figs. 7, 9, 13) that the relative deviations from Flory’s hypothesis scale indeed as 1/ Especially, the measurement of the bond-bond correlation function P (s), being the second derivative of the second moment of G(r, s) with respect of s, allows a very precise verifi- cation (Fig. 8) and shows that higher order corrections beyond the first-order perturbation approximation must be small. The most central and highly non-trivial numerical verification concerns the data collapse presented in Figs. 12 and 13 for the segmental size distribution of both computational models. All other statements made in this paper can be derived and understood from this key finding. It shows especially that the swelling coefficient ce must be close to the predicted value, Eq. (4). It is well known [10] that the effective bond length is difficult to predict at low com- pressibility and no attempt has been done to do so in this paper. We show instead how the systematic swelling of chain segments – once understood – may be used to extrapolate for the effective bond length of asymptotically long chains. Figs. 5 and 6 indicate how this may be done using Eq. (5). The high precision of our data is demonstrated in Fig. 12 by the successful scaling of the segmental size distribution. For several moments 〈r2p〉 we have also fitted empirical swelling coefficients cp using Eq. (5). In contrast to the effective bond length be these coefficients are rather well pre- dicted by one-loop perturbation theory if the bond length b of the reference Hamiltonian is renormalized to the effective bond length be, as we have conjectured in Sec. II B. Since the empirical swelling coefficients, cp ≈ ce(b/be)2p−3, would otherwise strongly depend on the moment taken, as shown in Eq. (12), our numerical data (Tab. II) clearly imply b/be ≈ 1. Minor deviations found for the BSM samples may be attributed to the fact that monomers along the BSM chains do strongly overlap — an effect not taken into account by the theory. To clarify ultimately this issue we are currently performing a numerical study where we systematically vary both the compressibility and the bond length of the BSM. General background and outlook. The most striking result presented in this work con- cerns the power law decay found for the bond-bond correlation function, P (s) ∝ 1/s3/2 (Fig. 8). This result suggests an analogy with the well-known long-range velocity correla- tions found in dense fluids by Alder and Wainwright nearly fourty years ago [41, 42]. In both cases, the ideal uncorrelated object is a random walker which is weakly perturbed (for d > 2) by the self-interactions generated by global constraints. Although these constraints are different (momentum conservation for the fluid, incompressibility for polymer melts) the weight with which these constraints increase the stiffness of the random walker is always proportional to the return probability. It can be shown that the correspondence of both problems is mathematically rigorous if the fluid dynamics is described on the level of the linearized Navier-Stokes equations [43]. We point out that the physical mechanism which has been sketched above is rather general and should not be altered by details such as a finite persistence length — at least not as long as nematic ordering remains negligible and the polymer chains are sufficiently long. (Similarly, velocity correlations in dense liquids must show an analytical decay for sufficiently large times irrespective of the particle mass and the local static structure of the solution.) While this paper focused exclusively on scales beyond the correlation length of the density fluctuations, i.e. qξ ≪ 1 or s/g ≫ 1, where the polymer solution appears incompressible, effects of finite density and compressibility can be readily described within the same theoretical framework and will be presented elsewhere [43]. To test our predictions, flexible chains should be studied preferentially, since the chain length required for a clear- cut description increases strongly with persistence length. This is in fact confirmed by preliminary and on-going simulations using the BSM algorithm. In this work we have only discussed properties in real space as a function of the curvilin- ear distance s. These quantities are straightforward to compute in a computer simulation but are barely experimentally relevant. The non-Gaussian deviations induced by the seg- mental correlation hole arise, however, also for an experimentally accessible property, the intramolecular form factor (single chain scattering function) F (q). As explained at the end of the Appendix, the form factor can be readily obtained by integrating the Fourier transformed segmental size distribution given in Eq. (3). This yields q2F (q) ≈ 12 in agreement with the result obtained in Refs. [14, 19] by direct calculation of the form factor for very long equilibrium polymers. As a consequence of this, the Kratky plot (q2F (q) vs. wave vector q) should not exhibit the plateau expected for Gaussian chains in the scale-free regime, but rather noticeable non-monotonic deviations. See Fig. 3 of [19]. This result suggests to revisit experimentally this old pivotal problem of polymer science. Our work is part of a broader attempt to describe systematically the effects of correlated density fluctuations in dense polymer systems, both for static [12, 13, 44, 45] and dynam- ical [29, 35, 46] properties. An important unresolved question is for instance whether the predicted long-range repulsive forces of van der Waals type (“Anti-Casimir effect”) [13, 45] are observable, for instance in the oscillatory decay of the standard density pair-correlation function of dense polymer solutions. Since the results presented here challenge an important concept of polymer physics, they should hopefully be useful for a broad range of theoreti- cal approaches which commonly assume the validity of the Gaussian chain model down to molecular scales [47, 48, 49]. This study shows that a polymer in dense solutions should not be viewed as one soft sphere (or ellipsoid) [50, 51, 52], but as a hierarchy of nested segmental correlation holes of all sizes aligned and correlated along the chain backbone (Fig. 2 (b)). We note that similar deviations from Flory’s hypothesis have been reported recently for linear polymers [16, 17, 47] and polymer gels and networks [53, 54]. The repulsive interactions should also influence the polymer dynamics, since strong deviations from Gaussianity are expected on the scale where entanglements become important, hence, quantitative predic- tions for the entanglement length Ne have to be regarded with more care. The demonstrated swelling of chains should be included in the popular primitive path analysis for obtaining Ne [55], especially if ‘short’ chains (N < 500) are considered. The effect could be responsible for observed deviations from Rouse behavior [26, 56] as may be seen by considering the cor- relation function Cpq ≡ 〈Xp ·Xq〉 of the Rouse modes Xp = 1N dnrn cos(npπ/N) where p, q = 0, . . . , N − 1 [6, 57]. Using (rn − rm)2 = r2n + r2m − 2rn · rm for the segment size, this correlation function can be readily expressed as an integral over the second moment of the segmental size distribution Cpq = − (rn − rm)2 cos(npπ/N) cos(mpπ/N) (31) which can be solved using our result Eq. (5). This implies for instance for p = q that Cpp = 2(πp)2 1− π√ . (32) The bracket entails an important correction with respect to the classical description given by the prefactor [6]. We are currently working out how static corrections, such as those for Cpp, may influence the dynamics for polymer chains without topological constraints. (This may be realized, e.g., within the BFM algorithm by using the L26 moves described in Sec. IIIA.) Moreover, for thin polymer films of width H the repulsive interactions are known to be stronger than in the bulk [12]. This provides a mechanism to rationalize the trend towards swelling observed experimentally [58] and confirmed computationally [21]: = log(s)/H. (33) (Prefactors omitted for clarity.) Here Rx(s) and bx denote the components of the segment size and the effective bond length parallel to the film. It also explains the (at first sight surprising) systematic increase of the polymer dynamics with decreasing film thickness [22]. Specifically, the parallel component of the monomer mean-squared displacement gx(t) is expected to scale as gx(t) ≈ R2x(s(t)) ∝ t1/4(1 + log(t)/H) for long reptating chains where s(t) ∝ t1/4 [6]. (The corresponding effect for the three-dimensional bulk should be small, however.) For the same reason (flexible) polymer chains close to container walls must be more swollen and, hence, faster on intermediate time scales than their peers in the bulk. Acknowledgments We thank T. Kreer, S. Peter and A.N. Semenov (all ICS, Strasbourg, France), S.P. Obukhov (Gainesville, Florida) and M. Müller (Göttingen, Germany) for helpful discus- sions. A generous grant of computer time by the IDRIS (Orsay) is also gratefully acknowl- edged. J.B. acknowledges financial support by the IUF and from the European Community’s “Marie-Curie Actions” under contract MRTN-CT-2004-504052. APPENDIX A: MOMENTS OF THE SEGMENTAL SIZE DISTRIBUTION AND THEIR GENERATING FUNCTION Higher moments of the segmental size distribution G(r, s) can be systematically obtained from its Fourier transformation G(q, s) = d3r G(r, s) exp(iq · r), which is in this context sometimes called the “generating function” [59]. For an ideal Gaussian chain, the generating function is then G0(q, s) = exp(−sq2a2) where we have used a2 = b2/6 instead of the bond length b2 to simplify the notation. Moments of the size distribution are given by proper derivatives of G(q, s) taken at q = 0. For example, 〈r2p〉 = (−1)p∆pG(q, s)q=0 (with ∆ being the Laplace operator with respect to the wave vec- tor q). A moment of order 2p is, hence, linked to only one coefficient A2p in the systematic expansion, G(q, s) = p=0A2pq 2p, of G(q, s) around q = 0. For our example this implies = (−1)p(2p+ 1)! A2p (A1) in general and more specifically for a Gaussian distribution 〈r2p〉0 = (2p+1)! spa2p. The non- Gaussian parameters read, hence, αp(s) ≡ 1− (2p+ 1)! 〈r2p〉 〈r2〉p = 1− p! A2p , (A2) which implies (by construction) αp = 0 for a Gaussian distribution. As various moments of the same global order 2p are linked to the same A2p they differ by a multiplicative constant independent of the details of the (isotropic) distribution G(q, s). For example, 〈r2〉 = 6|A2|, 〈r4〉 = 120A4, 〈x2〉 = 〈y2〉 = 2|A2|, 〈x2y2〉 = 8A4 with x and y denoting the spatial components of the segment vector r. Using Eq. (A2) for p = 2 it follows that Kxy(s) ≡ 1− 〈x2y2〉 〈x2〉〈y2〉 = 1− 2A4 = α2(s), (A3) i.e. the properties α2(s) and Kxy(s) discussed in Figs. 9 and 10 must be identical in general provided that G(q, s) is isotropic and can be expanded in q2. We turn now to specific properties ofG(q, s) computed for formally infinite polymer chains in the melt. In practice, these results are also relevant for small segments in large chains, N ≫ s ≫ 1, and, especially, for segments located far from the chain ends. These chains are nearly Gaussian and the generating function can be written as G(q, s) = G0(q, s) + δG(q, s) where δG(q, s) = −〈UG〉0 + 〈U〉0〈G〉0 is a small perturbation under the effective interaction potential ṽ(q) given by Eq. (9). To compute the different integrals it is more convenient to work in Fourier-Laplace space (q, t) with t being the Laplace variable conjugate to s: δG(q, t) = ds δG(q, s)e−st. As illustrated in Fig. 14, there a three contributions to this perturbation: one due to in- teractions between two monomers inside the segment (left panel), one due to interactions between an internal monomer and an external one (middle panel) and one due to interac- tions between two external monomers located on opposite sides (right panel). In analogy to the derivation of the form factor described in Ref. [14] this yields: δG(q, t) = − (q2a2 + t)2 q2a2 + t− (q2a2 + t)2 4πqa2ξ2 Arctan a/ξ + a/ξ + q2a2 + t q2a2 + t 4πqa4 Arctan a/ξ + a/ξ + q2a2 + t 4πqa6 Arctan q2a2 + t . (A4) The graph given in the left panel of Fig. 14 corresponds to the first two lines, the middle panel to the third line and the right panel to the last one. Seeking for the moments we expand δG(q, t) around q = 0. Having in mind chain strands counting many monomers (s ≫ 1), we need only to retain the most singular terms for t → 0. Defining the two dimensionless constants d = vξ/3πa4 = 12vξ/πb4 and c = (3π3/2a3ρ)−1 = 24/π3/b3ρ this expansion can be written as δG(q, t) = − d a2q2 + Γ(3/2) c a2q2 + . . . (A5) d a4q4 − 1 Γ(5/2) c a4q4 + . . . d a6q6 + Γ(7/2) c a6q6 + . . . + . . . where we have used Euler’s Gamma function Γ(α) [60]. The first leading term at each order in q2 — being proportional to the coefficient d — ensures the renormalization of the effective bond length. The next term scaling with the coefficient c corresponds to the leading finite strand size correction. Performing the inverse Laplace transformation Γ(α)/tα → sα−1 and adding the Gaussian reference distribution G0(q, s) this yields the A2p-coefficients for the expansion of G(q, s) around q = 0: A0 = 1 A2 = −a2s 1 + d− c√ 1 + 2d− A6 = − 1 + 3d− 216 A8 = . . . (A6) More generally, one finds A2p = (−1)p (sa2)p 1 + pd− 3(2 pp!p)2 2(2p+ 1)! From this result and using Eq. (A1) one immediately verifies that the moments of the distribution are given by the Eqs. (11) and (12). Using Eq. (A2) one justifies similarly Eq. (26) for the non-Gaussian parameter αp. These moments completely determine the segmental distribution G(r, s) which is indi- cated in Eq. (13). While at least in principle this may be done directly by inverse Fourier- Laplace transformation of the correction δG(q, t) to the generating function it is helpful to simplify further Eq. (A4). We observe first that δG(q, t) does diverge for strictly incom- pressible systems (v → ∞) and one must keep v finite in the effective potential whenever necessary to ensure convergence (actually everywhere but in the diagram corresponding to the interaction between two external monomers). Since we are not interested in the wave vectors larger than 1/ξ we expand δG(q, t) for ξ → 0 which leads to the much simpler expression δG(q, t) ≈ − vξq 3πa2(a2q2 + t)2 t(3a2q2 + t) (a2q2 + t)2 Arctan[ aq√ + o(vξ3). (A8) The first term diverges as v for diverging v. It renormalizes the effective bond length in the zero order term which is indicated in the first line of Eq. (13). The next two terms scale both as v0. Subsequent terms must all vanish for diverging v and can be discarded. It is then easy to perform an inverse Fourier-Laplace transformation of the two relevant v0 terms. This yields δG(x, s) = G0(x, s) with x = r/a 6n. This is consistent with the expression given in the second line of Eq. (13). We note finally that the intramolecular form factor F (q) = 1 n,m=1 〈exp(iq · (rn − rm)〉 of asymptotically long chains can be readily obtained from Eq. (A8). Observing that 〈exp(iq · (rn − rm)〉 = d3r exp(iq · r)G(r, s) = G(q, s) one finds δF (q) = 2 ds δG(q, s) = 2 δG(q, t = 0) = −2 vξ , (A10) where we used the third term of Eq. (A8) in the last step. The first term in Eq. (A8) is discarded as before, since it renormalizes the effective bond length in the reference form factor: F0(q) = 12/b 2q2 ⇒ 12/b2eq2. It follows, hence, that within first-order perturbation theory F (q) = F0(q) + δF (q) ≈ F0(q) (A11) as indicated by Eq. (30) in the Conclusion. This is equivalent to the result 1/F (q)−1/F0(q) ≈ q3/32ρ discussed in Refs. [14, 19] for polymer melts and anticipated by Schäfer [11] by renormalization group calculations of semidilute solutions. N nch τe Re Rg be(N) 6bg(N) 16 216 1214 11.7 4.8 2.998 2.939 32 215 3485 17.1 7.0 3.066 3.030 64 214 1.1 · 104 24.8 10.1 3.116 3.094 128 8192 3.3 · 104 35.6 14.5 3.153 3.139 256 4096 1.0 · 105 50.8 20.7 3.179 3.171 512 2048 3.2 · 105 72.2 29.5 3.200 3.193 1024 1024 1.0 · 106 103 42.0 3.216 3.212 2048 512 3.2 · 106 146 59.5 3.227 3.223 4096 256 9.7 · 106 207 85.0 3.235 3.253 8192 128 2.9 · 107 294 120 3.249 3.248 TABLE I: Various static properties of dense BFM melts of number density ρ = 0.5/8: the chain length N , the number of chains nch per box, the relaxation time τe characterized by the diffusion of the monomers over the end-to-end distance and corresponding to the circles indicated in Fig. 3, the root-mean-squared chain end-to-end distance Re and the radius of gyration Rg of the total chain (s = N − 1). The last two columns give estimates for the effective bond length from the end-to-end distance, be(N) ≡ Re/(N − 1)1/2, and the radius of gyration, bg(N) ≡ Rg/ N . The dashed line in Fig. 4 indicates be(N) 2. Apparently, both estimates increase monotonicly with N reaching be(N) ≈ 6bg(N) ≈ 3.2 for the largest chains available. Note that 6bg(N) < be(N) for smaller N . Property BFM BSM Length unit lattice constant bead diameter Temperature kBT 1 1 Number density ρ 0.5/8 0.84 Linear box size L 256 ≤ 62 Number of monomers nmon 1048576 ≤ 196608 Largest chain length N 8192 1024 Mean bond length 〈|ln|〉 2.604 0.97 l = 〈l2n〉1/2 2.636 0.97 Effective bond length be 3.244 1.34 ρb3e 2.13 2.02 C∞ = (be/l) 2 1.52 1.91 lp = l(C∞ + 1)/2 3.32 1.41 24/π3/ρb3e 0.41 0.44 c1/ce 1.0 1.2 c2/ce 1.0 1.1 c3/ce 1.0 1.0 c4/ce 1.1 1.2 c5/ce 1.1 0.9 cP = c1(be/l) 2/8 0.078 0.124 Dimensionless compressibility g 0.245 0.08 Compression modulus v ≡ 1/gρ 66.7 14.9 vρ/b3eρ 0.96 1.8 TABLE II: Comparison of some static properties of dense BFM and BSM melts. The first six rows indicate conventions and operational parameters. The effective bond length be and the swelling coefficients cp (defined in Eq. (5)) are determined from the first five even moments of the segmental size distribution. The dimensionless compressibility g = S(q → 0)/ρ has been obtained from the total static structure factor S(q) = 1 ∑nmon k,l=1 〈exp(iq · (rk − rl))〉 in the zero wave vector limit as shown at the end of Ref. [14]. The values indicated correspond to the asymptotic long chain behavior. Properties of very small chains deviate slightly. I ~ −3 I ~ −9 I ~ 45 I ~ −9− r n m FIG. 1: (Color online) Sketch of a polymer chain of length N in a dense melt in d = 3 dimensions. As notations we use ri for the position vector of a monomer i, li = ri+1 − ri for its bond vector, r = rm−rn for the end-to-end vector of the chain segment between the monomers n and m = n+s and r = ||r|| for its length. Segment properties, such as the 2p-th moments , are averaged over all possible pairs of monomers (n,m) of a chain and over all chains. The second moment (p = 1) is denoted Re(s) = , the total chain end-to-end distance is Re(s = N − 1). The dashed lines show the relevant graphs of the analytical perturbation calculation outlined in Sec. II B. The numerical factors indicate for infinite chains (without chain end effects) the relative weights contributing to the 1/ s-swelling of Re(s) indicated in Eq. (10). a bdensity ρ c(r,s) correlation hole Segmental = const R(s) Repulsion FIG. 2: (Color online) Role of incompressibility and chain connectivity in dense polymer solutions and melts. (a) Sketch of the segmental correlation hole of a marked chain segment of curvilin- ear length s. Density fluctuations of chain segments must be correlated, since the total density fluctuations (dashed line) are small. Consequently, a second chain segment feels an entropic re- pulsion when both correlation holes start to overlap. (b) Self-similar pattern of nested segmental correlation holes of decreasing strength u(s) ≈ s/ρR(s)3 ≈ ce/ s aligned along the backbone of a reference chain. The large dashed circle represents the classical correlation hole of the total chain (s ≈ N) [5]. This is the input of recent approaches to model polymer chains as soft spheres [50, 52]. We argue that incompressibility on all scales and chain connectivity leads to a short distance repulsion of the segmental correlation holes, which increases with decreasing s. L06 (conserved topology) L26,SS L26,SS,DB FIG. 3: (Color online) Diffusion time τe over the (root-mean-squared) chain end-to-end distance Re(N − 1) as a function of chain length N for different versions of the Bond Fluctuation Model (BFM). All data indicated are for the high number density (ρ = 0.5/8) corresponding to a polymer melt with half the lattice sites being occupied. We have obtained τe = R e(N−1)/6Ds from the self- diffusion coefficient Ds measured from the free diffusion limit of the mean-squared displacement of all monomers δr(t)2 = 6Dst. Data from the classical BFM with topology conserving local Monte Carlo (MC) moves in 6 spatial directions (L06) [26] are represented as stars. All other data sets use topology violating local MC moves in 26 lattice directions (L26). If only local moves are used, L26-dynamics is even at relatively short times perfectly Rouse like which allows the accurate determination of Ds although the monomers possibly have not yet moved over Re(N − 1) for the largest chain lengths considered. Additional slithering snake (SS) moves increase the efficiency of the algorithm by approximately an order of magnitude (squares,bold line). The power law exponent is changed from 2 to an empirical 1.62 (dashed line) if in addition we perform double-bridging (DB) moves. N=128 N=256 N=512 N=1024 N=2048 N=4096 N=8192 =3.244 Eq.(5) FIG. 4: (Color online) Mean-squared segment size Re(s) 2/s vs. curvilinear distance s. We present BFM data for different chain length N at number density ρ = 0.5/8. The averages are taken over all possible monomer pairs (n,m = n+s). The statistics deteriorates, hence, for large s. Log-linear coordinates are used to emphasize the power law swelling over several orders of magnitude of s. The data approach the asymptotic limit (horizontal line) from below, i.e. the chains are swollen. This behavior is well fitted by Eq. (5) for 1 ≪ s ≪ N (bold line). Non-monotonic behavior is found for s → N , especially for small N . The dashed line indicates the measured total chain end-to-end distances, be(N) 2 ≡ Re(N − 1)2/(N − 1) from Tab. I, showing even more pronounced deviations from the asymptotic limit. The dash-dotted line compares this data with Eq. (19). =3.235 =3.240 =3.244 =3.250 =3.255 BFM ρ=0.5/8, N=2048: =3.244, c =0.41=c FIG. 5: (Color online) Replot of the mean-squared segment size as y = K1(s) = 1 − Re(s)2/b2es vs. x = c1/ s, as suggested by Eq. (5), for different trial effective bond lengths be as indicated. Only BFM chains of length N = 2048 are considered for clarity. This procedure is very sensitive to the value chosen and allows for a precise determination. It assumes, however, that higher order terms in the expansion of K1(s) may be neglected. The value be is confirmed from a similar test for higher moments (Fig. 6). x ~1/s BFM b =3.244 BSM b =1.34 too small s ! FIG. 6: (Color online) Critial test of Eq. (5) where the rescaled moments y = Kp(s) of the segment size distribution (defined in Eq. (1)) are plotted vs. x = 3(2pp!p)2 2(2p+1)! . We consider the first five even moments (p = 1, . . . , 5) for the BFM with N = 2048 and the BSM with N = 1024. Also indicated is the rescaled radius of gyration, y = 5/8 (1 − 6R2g(s)/b2e(s + 1)), as a function of x = c1/ (filled circles). The BSM data has been shifted upwards for clarity. Without this shift a perfect data collapse is found for both models and all moments. Keeping the same effective bond length be for all moments of each model we fit for the swelling coefficients cp by rescaling the horizontal axis. We find be ≈ 3.244 for the BFM and 1.34 for the BSM. If be is chosen correctly, all data sets extrapolate linearly to zero for large s (x → 0). The swelling coefficients found are close the theoretical prediction ce, as indicated in Tab. II. The plot demonstrates that the non-Gaussian deviations scale as the segmental correlation hole, u(s) ∼ ce/ s and this for all moments as long as x ≪ 1. The saturation at large x is due to the finite extensibility of short chain segments. Since this effect becomes more marked for larger moments, the fit of be is best performed for p = 1. u(s) c Kλ(s) λ=2: N=1024 λ=2: N=2048 λ=2: N=4096 λ=4: N=2048 λ=8: N=2048 λ=16: N=2048 Insufficient statistics & Finite chain length effects ⇒ ω = 3/2 FIG. 7: (Color online) Plot of Kλ(s) as a function of u(s)c1/ce ∼ 1/ s using the measured u(s) ≡ 24/π3s/ρRe(s) 3. For λ = 2 (corresponding to two segments being connected) BFM and BSM data are compared. Several λ values are given for N = 2048 BFM chains. For chain segments with 1 ≪ s ≪ N all data sets collapse on the bisection line confirming the so-called “recursion relation” Kλ ≈ u proposed by Semenov and Johner [12]. The statistics becomes insufficient for large s (left bottom corner). Systematic deviations arise for s → N due to additional finite-N effects. N=128 N=512 N=2048 N=4096 N=8192 Slope ω=3/2 Chain end effects FIG. 8: (Color online) The bond-bond correlation function P (s)/cP as a function of the curvilinear distance s. Various chain lengths are given for BFM. Provided that 1 ≪ s ≪ N , all data sets collapse on the power law slope with exponent ω = 3/2 (bold line) as predicted by Eq. (23). The dash-dotted curve P (s) ≈ exp(−s/1.5) shows that exponential behavior is only compatible with very small chain lengths. The dashed lines correspond to the theoretical prediction, Eq. (24), for short chains with N = 16, 32, 64 and 128 (from left to right). BFM N=1024 BFM N=2048 BFM N=4096 BFM N=8192 BSM N=1024 y = 6x/5 y = 111x/35 y = 605x/105 small s ! Noise ! FIG. 9: (Color online) Non-Gaussian parameter αp(s) computed for the end-to-end distance of chain segments as a function of ce/ s. Perfect data collapse for all chain lengths and both sim- ulation models is obtained for each p. A linear relationship over nearly two orders of magnitude is found as theoretically expected. Data for three moments (p = 2, 3, 4) are indicated showing a systematic increase of non-Gaussianity with p. The data curvature for small s becomes more pronounced for larger p. N=256 N=512 N=1024 N=2048 N=4096 N=8192 m=n+s FIG. 10: (Color online) Plot of Kxy(s) = 1− averaged over all pairs of monomers (n,m = n+s) and three different direction pairs as a function of ce/ s. As indicated by the sketch at the bottom of the figure, Kxy(s) measures the correlation of the components of the segment vector r. All data points collapse and show again a linear relationship Kxy ≈ u(s). Different directions are therefore coupled! No curvature is observed over two orders of magnitude confirming that higher order perturbation corrections are negligible. Noise cannot be neglected for large s > 100 and finite segment-size effects are visible for s ≈ 1. 0.0 0.5 1.0 1.5 2.0 n=r/b BFM s=32 BFM s=64 BFM s=128 BSM s=32 BSM s=64 BSM s=128 GaussEnhancem entDepletion FIG. 11: (Color online) Segment size distribution y = G(r, s)(bes 1/2)3 vs. n = r/bes 1/2 for several s as indicated in the figure. Only data for BFM with N = 2048 and BSM with N = 1024 are presented. (A similar plot can be achieved by renormalizing the axes using Re(s) instead of bes 1/2). The bold line denotes the Gaussian behaviour y = (3/2π)3/2 exp(−3n2/2). One sees that compared to this reference the measured distributions are depleted for small n ≪ 1 (where the data does not scale) and enhanced for n ≈ 1. 0.0 0.5 1.0 1.5 2.0 2.5 n=r/b BFM s=31 BFM s=63 BFM s=127 BSM s=32 BSM s=64 BSM s=128 f(n) from Eq. (3) Enhancement epletion FIG. 12: (Color online) Deviation δG(r, s) = G(r, s) − G0(r, s) of the measured segmental size distribution from the Gaussian behavior G0(r, s) expected from Flory’s hypothesis for sev- eral s and both models as indicated in the figure. As suggested by Eq. (3), we have plotted y = (δG(r, s)/G0(r, s))/(ce/ s) as a function of n = r/be s. The Gaussian reference distribution has been computed according to Eq. (2) for the measured effective bond length be. A close to perfect data collapse is found for both models. This shows that the deviation scales linearly with u(s) ≈ ce/ s, as expected. The bold line indicates the universal function of f(n) predicted by Eq. (3). 0.0 0.5 1.0 1.5 2.0 2.5 m=r/R BFM s=31 BFM s=63 BFM s=127 BSM s=32 BSM s=64 BSM s=128 f(m) from Eq. (29) Enhancement letio FIG. 13: (Color online) Replot of the relative deviation of the measured segment size distribution, y = (δG(r, s)/G0(r, s)))/(ce/ s), as a function of m = r/Re(s). The figure highlights that the measured segment size is the only length scale relevant for describing the deviation from Flory’s hypothesis. The same data sets and symbols are used as in the previous Fig. 12. v(k) v(k) v(k)~ ~ ~ q q q q q q−k q−k FIG. 14: Interaction diagrams used in reciprocal space for the calculation of δG(q, t) in the scale free limit. 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(16) with the coefficients one would obtain by computing Eq. (15) either with the effective potential ṽ(q) for infinite chains given by Eq. (9) or with the Padé approximation, Eq. (17). Within these approximations of the full linear response formula, Eq. (14), the coefficients can be obtained directly without numerical integration yielding overall similar values. In the first case we obtain 15/8 ≈ 1.87 and in the second 11/8 ≈ 1.37. While the first value is clearly not compatible with the measured end-to-end distances, the second yields a reasonable fit, especially for small N < 1000, when the data is plotted as in Fig. 5. Ultimately, for very long chains the correct coefficient should be 1.59 as is indicated by the dash-dotted line in Fig. 4. [62] These topology non-conserving moves yield configurations which are not accessible with the classical scheme with jumps in 6 directions only. Concerning the static properties we are interested in this paper both system classes are practically equivalent. This has been confirmed by comparing various static properties and by counting the number of monomers which become “blocked” (in absolute space or with respect to an initial group of neighbor monomers) once one returns to the original local scheme. Typically we find about 10 blocked monomers for a system of 220 monomers. The relative difference of microstates is therefore tiny and irrelevant for static properties. Care is needed, however, if the equilibrated configurations are used to investigate the dynamics of the topology conserving BFM version. The same caveats arise for the slithering snake and double-bridging moves. [63] We have verified that the alternative definition 〈em=n+s · en〉 with en being the normalized bond vector yields very similar results. This is due to the weak bond length fluctuations, specifically at high densities, in both coarse-grained models under consideration. It is possi- ble that other models show slightly different power law amplitudes cP depending on which definition is taken. Flory's ideality hypothesis revisited Physical idea and sketch of the perturbation calculation Scaling arguments Perturbation calculation Computational models and technical details Bond fluctuation model Bead spring model Numerical results The swelling of chain segments Chain connectivity and recursion relation Intrachain bond-bond correlations Higher moments and associated coefficients The segmental size distribution Conclusion Acknowledgments Moments of the segmental size distribution and their generating function References

to appear with shortened appendicies in The Astrophysical Journal Stability Properties of Strongly Magnetized Spine Sheath Relativistic Jets Philip E. Hardee Department of Physics & Astronomy, The University of Alabama, Tuscaloosa, AL 35487, USA phardee@bama.ua.edu ABSTRACT The linearized relativistic magnetohydrodynamic (RMHD) equations describing a uniform axially magnetized cylindrical relativistic jet spine embedded in a uniform axi- ally magnetized relativistically moving sheath are derived. The displacement current is retained in the equations so that effects associated with Alfvén wave propagation near light speed can be studied. A dispersion relation for the normal modes is obtained. An- alytical solutions for the normal modes in the low and high frequency limits are found and a general stability condition is determined. A trans-Alfvénic and even a super- Alfénic relativistic jet spine can be stable to velocity shear driven Kelvin-Helmholtz modes. The resonance condition for maximum growth of the normal modes is obtained in the kinetically and magnetically dominated regimes. Numerical solution of the dis- persion relation verifies the analytical solutions and is used to study the regime of high sound and Alfvén speeds. Subject headings: galaxies: jets — gamma rays: bursts — ISM: jets and outflows — methods: analytical — MHD — relativity — instabilities 1. Introduction Relativistic jets are associated with active galactic nuclei and quasars (AGN), with black hole binary star systems (microquasars), and are thought responsible for the gamma-ray bursts (GRBs). In microquasar and AGN jets proper motions of intensity enhancements show mildly superluminal for the microquasar jets ∼ 1.2 c (Mirabel & Rodriquez 1999), range from subluminal (≪ c) to superluminal (. 6 c) along the M87 jet (Biretta et al. 1995, 1999), are up to ∼ 25 c along the 3C 345 jet (Zensus et al. 1995; Steffen et al. 1995), and have inferred Lorentz factors γ > 100 in the GRBs (e.g., Piran 2005). The observed proper motions along microquasar and AGN jets imply speeds from ∼ 0.9 c up to ∼ 0.999 c, and the speeds inferred for the GRBs are ∼ 0.99999 c. Jets at the larger scales may be kinetically dominated and contain relatively weak magnetic fields, e.g., equipartition between magnetic and gas pressure or less, but the possibility of much stronger magnetic fields exists close to the acceleration and collimation region. Here general rel- ativistic magnetohydrodynamic (GRMHD) simulations of jet formation (e.g., Koide et al. 2000; Nishikawa et al. 2005; De Villiers, Hawley & Krolik 2003; De Villiers et al. 2005; Hawley & Krolik 2006; McKinney 2006; Mizuno et al. 2006) and earlier theoretical work (e.g., Lovelace 1976; Bland- ford 1976; Blandford & Znajek 1977; Blandford & Payne 1982) invoke strong magnetic fields. In addition to strong magnetic fields, GRMHD simulation studies of jet formation indicate that highly collimated high speed jets driven by the magnetic fields threading the ergosphere may themselves reside within a broader wind or sheath outflow driven by the magnetic fields anchored in the ac- cretion disk (e.g., McKinney 2006; Hawley & Krolik 2006; Mizuno et al. 2006). This configuration might additionally be surrounded by a less collimated accretion disk wind from the hot corona (e.g., Nishikawa et al. 2005). http://arxiv.org/abs/0704.1621v1 – 2 – That relativistic jets may have jet-wind structure is indicated by recent observations of high speed winds in several QSO’s with speeds, ∼ 0.1 − 0.4c, (Chartas, Brandt & Gallagher 2002, Chartas et al. 2003; Pounds et al. 2003a; Pounds et al. 2003b; Reeves, O’Brien &Ward 2003). Other observational evidence such as limb brightening has been interpreted as evidence for a slower external sheath flow surrounding a faster jet spine, e.g., Mkn 501 (Giroletti et al. 2004), M 87 (Perlman et al. 2001), and a few other radio galaxy jets (e.g., Swain, Bridle & Baum 1998; Giovannini et al. 2001). Additional circumstantial evidence such as the requirement for large Lorentz factors suggested by the TeV BL Lacs when contrasted with much slower observed motions suggests the presence of a spine-sheath morphology (Ghisellini, Tavecchio & Chiaberge 2005). At hundreds of kiloparsec scales Siemignowska et al. (2007) have proposed a two component (spine-sheath) model to explain the broad-band emission from the PKS 1127-145 jet. A spine-sheath jet structure has been proposed based on theoretical arguments (e.g., Sol et al. 1989; Henri & Pelletier 1991; Laing 1996; Meier 2003). Similar type structure has been investigated in the context of GRB jets (e.g., Rossi, Lazzati & Rees 2002; Lazzatti & Begelman 2005; Zhang, Wooseley & MacFadyen 2003; Zhang, Woosley & Heger 2004; Morsony, Lazzati & Begelman 2006). In order to study the effect of strong magnetic fields and the effect of a moving wind or sheath around a jet or jet spine, I begin by adopting a simple system with no radial dependence of quantities inside the jet spine and no radial dependence of quantities outside the jet in the sheath. This “top hat” configuration with magnetic fields parallel to the flow can be described exactly by the linearized relativistic magnetohydrodynamic (RMHD) equations. This system with no magnetic and flow helicity is stable to current driven (CD) modes of instability (Istomin & Pariev 1994, 1996; Lyubarskii 1999). However, this system can be unstable to Kelvin-Helmholtz (KH) modes of instability (Hardee 2004). This approach allows us to look at the potential KH modes without complications arising from coexisting CD modes (see Baty, Keppens & Compte 2004) and predictions can be verified by numerical simulations (Mizuno, Hardee & Nishikawa 2006). This paper is organized as follows. In §2, I present the dispersion relation arising from a normal mode analysis of the linearized RMHD equations. Analytical approximate solutions to the dispersion relation for various limiting cases are given in §3. I verify the analytical solution through numerical solution of the dispersion relation in §4. I summarize the stability results in §5 and discuss the applicability of the present results in §6. Derivation of the linearized RMHD equations is shown in Appendix A, derivation of the normal mode dispersion relation is presented in Appendix B, and derivation of the analytical solutions is shown in Appendix C. 2. The RMHD Normal Mode Dispersion Relation Let us analyze the stability of a spine-sheath system by modeling the jet spine as a cylinder of radius R, having a uniform proper density, ρj, a uniform axial magnetic field, Bj = Bj,z, and a uniform velocity, uj = uj,z. The external sheath is assumed to have a uniform proper density, ρe, a uniform axial magnetic field, Be = Be,z, a uniform velocity ue = ue,z, and extends to infinity. The sheath velocity corresponds to an outflow around the central spine if ue,z > 0 or represents backflow when ue,z < 0. The jet spine is established in static total pressure balance with the external sheath where the total static uniform pressure is P ∗e ≡ Pe + B2e/8π = P ∗j ≡ Pj + B2j /8π, and the initial equilibrium satisfies the zeroth order equations. Formally, the assumption of an infinite sheath means that a dispersion relation could be derived in the reference frame of the sheath with results transformed to the source/observer reference frame. However, it is not much more difficult to derive a dispersion relation in the source/observer frame in which analytical solutions to the dispersion relation take on simple revealing forms. Additionally, this approach lends itself to modeling the propagation and appearance of jet structures viewed in the source/observer frame, e.g., helical structures in the 3C 120 jet (Hardee, Walker & Gómez 2005). The general approach to analyzing the time dependent properties of this system is to linearize – 3 – the ideal RMHD and Maxwell equations, where the density, velocity, pressure and magnetic field are written as ρ = ρ0 + ρ1, v = u + v1 (we use v0 ≡ u for notational reasons), P = P0 + P1, and B = B0+B1, where subscript 1 refers to a perturbation to the equilibrium quantity with subscript 0. Additionally, the Lorentz factor γ2 = (γ0 + γ1) 2 ≃ γ20 + 2γ40u · v1/c2 where γ1 = γ30u · v1/c2. The linearization is shown in Appendix A. In cylindrical geometry a random perturbation ρ1, v1 B1 and P1 can be considered to consist of Fourier components of the form f1(r, φ, z, t) = f1(r) exp[i(kz ± nφ− ωt)] (1) where flow is along the z-axis, and r is in the radial direction with the flow bounded by r = R. In cylindrical geometry n is an integer azimuthal wavenumber, for n > 0 waves propagate at an angle to the flow direction, and +n and −n give wave propagation in the clockwise and counter-clockwise sense, respectively, when viewed in the flow direction. In equation (1) n = 0, 1, 2, 3, 4, etc. correspond to pinching, helical, elliptical, triangular, rectangular, etc. normal mode distortions of the jet, respectively. Propagation and growth or damping of the Fourier components can be described by a dispersion relation of the form n(βjR) Jn(βjR) n (βeR) n (βeR) . (2) Derivation of this dispersion relation is given in Appendix B. In the dispersion relation Jn and H are Bessel and Hankel functions, the primes denote derivatives of the Bessel and Hankel functions with respect to their arguments. In equation (2) χj ≡ γ2j γ2AjWj ̟2j − κ2jv2Aj , (3a) χe ≡ γ2eγ2AeWe ̟2e − κ2ev2Ae , (3b) β2j ≡ ̟2j − κ2ja2j ̟2j − κ2jv2Aj v2msj̟ j − κ2jv2Aja2j  , (4a) β2e ≡ ̟2ex − κ2ea2e ̟2e − κ2ev2Ae v2mse̟ e − κ2ev2Aea2e . (4b) In equations (3a & 3b) and equations (4a & 4b) ̟2j,e ≡ (ω − kuj,e) and κ2j,e ≡ k − ωuj,e/c2 γj,e ≡ (1 − u2j,e/c2)−1/2 is the flow Lorentz factor, γAj,e ≡ (1 − v2Aj,e/c2)−1/2 is the Alfvén Lorentz factor, W ≡ ρ+[Γ/ (Γ− 1)]P/c2 is the enthalpy, a is the sound speed, vA is the Alfvén wave speed, and vms is a magnetosonic speed. The sound speed is defined by where 4/3 ≤ Γ ≤ 5/3 is the adiabatic index. The Alfvén wave speed is defined by 1 + V 2A/c where V 2A ≡ B20/(4πW0). A magnetosonic speed corresponding to the fast magnetosonic speed for propagation perpendicular to the magnetic field (e.g., Vlahakis & Königl 2003) is defined by vms ≡ a2 + v2A − a2v2A/c2 a2/γ2A + v – 4 – 3. Analytical Solutions to the Dispersion Relation In this section analytical solutions to the dispersion relation in the low frequency limit, in the fluid and magnetic limits at resonance (maximum growth), and in the high frequency limit are summarized. The analytical solutions are derived in Appendix C. 3.1. Low Frequency Limit Analytically each normal mode n contains a single fundamental/surface wave (ω −→ 0, k −→ 0, ω/k > 0) solution and multiple body wave (ω −→ 0, k > 0, ω/k −→ 0) solutions that satisfy the dispersion relation. In the low frequency limit the fundamental pinch mode (n = 0) solution is given by uj ± vw 1± vwuj/c2 where the pinch fundamental mode wave speed v2w ≈ a2j v2msj v2msj , (6) δ ≡ − ̟2e − κ2ev2Ae v2msj with |δ| ∝ ∣k2R2 ∣ << 1. In this limit δ is complex and this mode consists of a growing and damped wave pair. The imaginary part of the solution is vanishingly small in the low frequency limit. The above form indicates that growth, which arises from the complex value of δ, will be reduced as (v2Aj/v msj) −→ 1. The unstable growing solution is associated with the backwards moving (in the jet fluid reference frame) wave. In the low frequency limit the surface helical, elliptical, and higher order normal modes (n > 0) have a solution given by [ηuj + ue]± iη1/2 (uj − ue)2 − V 2As/γ2j γ2e (1 + V 2Ae/γ 2) + η(1 + V 2Aj/γ where η ≡ γ2jWj γ2eWe and a “surface” Alfvén speed is defined by V 2As ≡ γ2AjWj + γ ) B2j +B 4πWjWe . (9) In equation (9) note that the Alfvén Lorentz factor γ2Aj,e = 1 + V Aj,e/c 2. Thus, the jet is stable to n > 0 surface wave mode perturbations when γ2j γ e (uj − ue) < γ2Ajγ Ae +We/γ ) B2j +B 4πWjWe . (10) For example, with uj ≈ c >> ue, γ2e ≈ 1, γ2Aj >> γ2Ae ≈ 1, B2j >> B2e , and using γ2Aj = 1 +B2j /4πWjc 2 the jet is stable when γ2j < 4πWec2 γ2Aj (11a) – 5 – or with Be = Bj , We = Wj , so that vA,j = vA,e, and with γA ≡ γA,e = γA,j the jet is stable when γ2j γ e (uj − ue)2 < 4γ2A(γ2A − 1)c2 . (11b) Thus, the jet can remain stable to the surface wave modes even when the jet Lorentz factor exceeds the Alfvén Lorentz factor. In the low frequency limit the real part of the body wave solutions is given by kR ≈ kminnmR ≡ v2msju j − v2Aja2j γ2j (u j − a2j)(u2j − v2Aj) × [(n+ 2m− 1/2)π/2 + (−1)mǫn] (12) where n specifies the normal mode, m = 1, 2, 3, ... specify the first, second, third, etc. body wave solutions, and n(βjR) n (βeR) n (βeR) In the absence of a significant external magnetic field and a significant external flow ǫn = 0 as χe = γ u2e − v2Ae k2 = 0. In this low frequency limit the body wave solutions are either purely real or damped, exist only when kminnmR has a positive real part, and with |ǫn| << 1 require v2msju j − v2Aja2j γ2j (u j − a2j )(u2j − v2Aj) > 0 . (13) Thus, the body modes can exist when the jet is supersonic and super-Alfvénic, i.e., u2j − a2j > 0 and u2j − v2Aj > 0, or in a limited velocity range given approximately by a2j > u2j > [γ2sj/(1+ γ2sj)]a2j when v2Aj ≈ a2j , where γsj ≡ (1− a2j/c2)−1/2 is a sonic Lorentz factor. 3.2. Resonance With the exception of the pinch fundamental mode which can have a relatively broad plateau in the growth rate, all body modes, and all surface modes can have a distinct maximum in the growth rate at some resonant frequency. The resonance condition can be evaluated analytically in either the fluid limit where a >> VA or in the magnetic limit where VA >> a. Note that in the magnetic limit, magnetic pressure balance implies that Bj = Be. In these cases a necesary condition for resonance is that uj − ue 1− ujue/c2 vwj + vwe 1 + vwjvwe/c2 , (14) where vwj ≡ (aj , vAj) and vwe ≡ (ae, vAe) in the fluid or magnetic limits, respectively. When this condition is satisfied it can be shown that the wave speed at resonance is vw ≈ v∗w ≡ γj(γwevwe)uj + γe(γwjvwj)ue γj(γwevwe) + γe(γwjvwj) where γw ≡ (1 − v2w/c2)−1/2 is the sonic or Alfvénic Lorentz factor accompanying vwj ≡ (aj , vAj) and vwe ≡ (ae, vAe) in the fluid or magnetic limits, respectively. – 6 – The resonant wave speed and maximum growth rate occur at a frequency given by ωR/vwe ≈ ω∗nmR/vwe ≡ (2n + 1)π/4 +mπ (1− ue/v∗w) 2 − (vwe/v∗w − uevwe/c2) . (16) In equation (16) n specifies the normal mode, m = 0 specifies the surface wave, and m ≥ 1 specifies the body waves. In the limit of insignificant sheath flow, ue = 0, and using eq. (15) for v w in eq. (16) allows the resonant frequency to be written as ω∗nmRj/vwe = (2n + 1)π/4 +mπ v2we/u j + 2 vwevwj/γju v2wj/γ )]1/2 and this predicts a resonant frequency that is primarily a function of the sound and Alfvén wave speeds in the sheath. The effect of sheath flow is best illustrated by assuming comparable conditions in the spine and sheath, γwjvwj ∼ γwevwe, and assuming that γjuj >> γeue in which case ω∗nmRj/vwe ∼ (2n + 1)π/4 +mπ 1− 2 (ue/uj) (1− v2we/c2)− (v2we − u2e) /u2j The term ue/uj in the denominator indicates that the resonant frequency increases as the shear speed, uj − ue, declines. In the limit uj − ue 1− ujue/c2 vwj + vwe 1 + vwjvwe/c2 the resonant frequency ω∗nmR/vwe −→ ∞. The resonant wavelength is given by λ ≈ λ∗nm ≡ 2πv∗w/ω∗nm and can be calculated from λ∗nm ≡ (2n+ 1)π/4 +mπ (v∗w − ue) vwe − (vweue/c2)v∗w R . (17) Equations (15 - 17) provide the proper functional dependence of the resonant wave speed, frequency and wavelength provided (ue/uj) 2 << 1 and (vwe/uj) 2 << 1. With the exception of the n = 0, m = 0, fundamental pinch mode, a maximum spatial growth rate, kmaxI , is approximated by kmaxI R ≈ k∗IR ≡ − ln |R| , (18) where |R| ≈ 4 (ω∗nmR/vwe) (1− 2ue/uj) + (ln |R| /2)2 (ln |R| /2)2 . (19) Equations (18) and (19) show that the maximum growth rate is primarily a function of the jet sound, Alfvén and flow speed through vwj/γjuj, and secondarily a function of the sheath sound, Alfvén and flow speed through (ω∗nmR/vwe) (1− 2ue/uj). – 7 – I can illustrate the dependencies of the maximum growth rate on sound, Alfvén and flow speeds by using ω∗nmR (1− 2ue/uj) ≈ (1− 2ue/uj) 1− 2 (ue/uj) (1− v2we/c2)− (v2we − u2e)/u2j ] × [(2n + 1)π/4 +mπ]2 and if say ue = 0, then |R|2 − 1 ln |R| ≈ 4 1− v2we/u2j × [(2n + 1)π/4 +mπ] . (20) Thus, |R| increases as ω∗nm increases for higher order modes with larger n and larger m and this result indicates an increase in the growth rate for larger n and larger m. When the sound or Alfvén wave speed, vwe, increases |R| increases. This result indicates an increase in the growth rate at the higher resonant frequency accompanying an increase in the sound or Alfvén wave speed in the sheath. The behavior of the maximum growth rate as the shear speed, uj−ue, declines is best illustrated by considering the effect of an increasing wind speed where (v2we − u2e)/u2j << 1 is ignored. In this |R|2 − 1 ln |R| ≈ 4 [(2n + 1)π/4 +mπ] (21) and |R| will remain relatively independent of ω∗nm even as ω∗nm −→ ∞ as the shear speed decreases. This result indicates a relatively constant resonant growth rate as the shear speed decreases. In the fluid limit decline in the shear speed ultimately results in a decrease in the growth rate and increase in the spatial growth length. This decline in the growth rate is also indicated by equation (8) which, in the fluid limit, becomes ηuj + ue 1 + η ± i η 1 + η (uj − ue) . (22) Equation (22) applies to frequencies below the resonant frequency ω∗nm and directly reveals the decline in growth rates as uj − ue −→ 0. In the magnetic limit the resonant frequency ω∗nmR/vAe −→ ∞ as uj − ue 1− ujue/c2 vAj + vAe 1 + vAjvAe/c2 . (23) Here equation (8) indicates that the jet is stable when γ2j γ e (uj − ue) < V 2As , and the jet will be stable as ω∗nm −→ ∞ when γ2j γ 1− ujue/c2 < 2γ2Ajγ v2Ae + v (vAj + vAe) 1 + vAjvAe/c , (24) where I have used an equality in equation (23) in equation (8) to obtain equation (24). Equation (24) indicates that a high jet speed relative to the Alfvén wave speed is necessary for instability. For example, if vA ≡ vAj = vAe and ue = 0, the jet is stable at high frequencies provided γ2j < 1 + v2A/c γ4A . (25a) – 8 – This high frequency condition is slightly different from the low frequency stabilization condition found when vA ≡ vAj = vAe and ue = 0 from equation (11b) γ2j (uj/c) 2 < 4γ2A(γ A − 1) . (25b) Note that eqs. (25a & 25b) are identical in the large Lorentz factor limit. Equations (25) predict that stabilization at high frequencies occurs at somewhat higher jet speeds than stabilization at lower frequencies. Determination of stabilization at intermediate frequencies requires numerical solution of the dispersion relation. A non-negligable postive external flow requires even higher jet speeds for the jet to be unstable. Thus, a strongly magnetized relativistic trans-Alfvénic jet is predicted to be KH stable and a super-Alfvénic jet can be KH stable. 3.3. High Frequency Limit Provided the condition, eq. (14), for resonance is met, the real part of the solutions to the dispersion relation in the high frequency limit for fundamental, surface, and body modes is given uj ± vwj 1± vwjuj/c2 . (26) and describes sound waves vwj = aj or Alfvén waves vwj = vAj propagating with and against the jet flow inside the jet. Unstable growing solutions are associated with the backwards moving (in the jet fluid reference frame) wave but the growth rate is vanishingly small in this limit. 4. Numerical Solution of the Dispersion Relation The detailed behavior of solutions within an order of magnitude of the resonant frequency and for comparable sound and Alfvén wave speeds must be investigated by numerical solution of the dispersion relation. Analytical solutions found in the previous section can be used for initial estimates and to provide the functional behavior of solutions. Numerical solution of the dispersion relation also allows a determination of the accuracy and applicability of the analytical expressions in §3. In this section pinch fundamental, helical surface and elliptical surface, and the associated first body modes are investigated in the fluid, magnetic and magnetosonic regimes. These modes are chosen as they have been identified with structure seen in relativistic hydrodynamic (RHD) numerical simulations or tentatively identified with structures in resolved AGN jets. For example, trailing shocks in a numerical simulation (Agudo et al. 2002) and in the 3C 120 jet (Gómez et al. 2001) have been identified with the first pinch body mode. The development of large scale helical twisting of jets has been attributed to or may be associated with growth of the helical surface mode, e.g., 3C 449 (Hardee 1981) and Cygnus A (Hardee 1996) Additionally, the development of twisted filamentary structures has been attributed to helical and elliptical surface and first body modes, e.g., 3C 273 (Lobanov & Zensus 2001), M87 (Lobanov, Hardee & Eilek 2003), 3C 120 (Hardee, Walker & Gómez 2005), and have been studied in RHD numerical simulations, e.g., Hardee & Hughes (2003); Perucho et al. (2006). 4.1. Fluid Limit In this section the basic behavior of the pinch (F) fundamental, helical (S) surface and elliptical (S) surface modes is investigated: (1) as a function of varying sound speed in the external sheath or jet spine for a fixed sound speed in the jet spine or external sheath and no sheath flow, (2) as a function of equal sound speeds in the jet spine and external sheath for no sheath flow, and (3) as – 9 – a function of sheath flow for a relatively high sound speed equal in jet spine and external sheath. In general only growing solutions are shown and complexities associated with multiple crossing solutions are not shown. For all solutions shown the jet spine Lorentz factor and speed are set to γ = 2.5 and uj = 0.9165 c. Sound speeds are input directly with the only constant being the sheath number density. Total pressure and spine density are quantities computed for the specified sound speeds. The adiabatic index is chosen to be Γ = 13/9 when 0.1 ≤ aj,e/c ≤ 0.5 consistent with relativistically hot electrons and cold protons (Synge 1957). For sound speeds aj,e ∼ c/ 3 the adiabatic index is set to Γ = 4/3. Solutions shown assume zero magnetic field. Test calculations with magnetic fields giving magnetic pressures a few percent of the gas pressure and Alfvén wave speeds an order of magnitude less than the sound speeds give almost identical results. In Figure 1 solutions in the left column are for a fixed jet spine sound speed aj = 0.3 c and in the right column are for a fixed external sheath sound speed ae = 0.3 c. The solutions shown in Figure 1 confirm the accuracy of the low frequency solutions to the pinch fundamental mode, eqs. (5 & 6), and the helical and elliptical surface modes, eq. (8). Note that fast or slow wave speeds are possible at low frequencies depending on whether η ≃ (γjae/γeaj)2 in eq. (8) is much greater or much less than one, respectively. The numerical solutions to the dispersion relation show that the maximum growth rate is primarily a function of the jet spine sound speed and only secondarily a function of the external sheath sound speed as indicated by eqs. (18 - 20). Where a distinct supersonic resonance exists, the resonant frequency is primarily a function of the external sheath sound speed as predicted from eq. (16). The analytical expression for the resonant frequency for the helical and elliptical surface modes provides the correct functional variation to within a constant multiplier provided ae ≤ c/ 3 and aj < c/3. A dramatic increase in the resonant frequency and modest increase in the growth rate for larger jet spine sound speeds indicates the transition to transonic behavior. Equation (15) for the resonant wave speed and equation (17) for the resonant wavelength also provide a reasonable approximation to the functional variations provided ae ≤ c/ and aj < c/3. These results confirm the resonant solutions found in §3.2. At frequencies more than an order of magnitude above resonance the growth rate is greatly reduced and solutions approach the high frequency limiting form given by eq. (26). Note that eq. (26) allows only relatively high wave speeds at high frequencies because aj ≤ c/ In Figure 2 the behavior of solutions to the fundamental/surface (left column) and associated first body mode (right column) shows how solutions change as the sound speed increases in both the jet spine and external sheath. Here I illustrate the transition from supersonic to transonic behavior for no flow in the sheath. At low frequencies the modes behave as predicted by the analytic solutions given in §3.1. The solutions show the expected shift to a higher resonant frequency that is primarily a function of the increased external sheath sound speed and an accompanying increase in the resonant growth rate that is primarily a function of the increased jet spine sound speed. The resonance disappears as sound speeds approach c/ 3 as the jet becomes transonic as predicted by the resonance condition in §3.2. In the transonic regime high frequency fundamental/surface mode growth rates and wave speeds are identical with wave speeds given by eq. (26). Provided the jet is sufficiently supersonic, i.e., aj,e < 0.5 c, the maximum growth rate of the first body mode is greater than that of the pinch fundamental mode, is comparable to that of the helical surface mode, and is less than that of the elliptical surface mode. A narrow damping peak shown for the helical first body (B1) solution when aj,e = 0.4 c is indicative of complexities in the body mode solution structure. In the transonic regime growth of the first body mode is less than that of the pinch fundamental, helical surface and elliptical surface modes. Figure 3 illustrates the behavior of fundamental/surface and first body modes as a function of the sheath speed for equal sound speeds in spine and sheath of aj,e = 0.4 c. For this value of the sound speeds a sheath speed ue = 0 provides a supersonic solution structure baseline. At low frequencies the surface modes behave as predicted by eq. (8), and the wave speed rises as ue – 10 – Fig. 1.— Solutions to the dispersion relation for pinch fundamental, helical surface and elliptical surface modes for different sound speeds in the sheath (left column) and in the spine (right column) are shown for no sheath flow. The real part of the wavenumber, krRj , is shown by the dashed lines and the imaginary part , kiRj , is shown by the dash-dot lines as a function of the dimensionless angular frequency, ωRj/uj . For the pinch mode the vertical lines indicate the maximum growth rate range. Otherwise, the vertical lines indicate the location of maximum growth. Immediately under the dispersion relation solution panel is a panel that shows the relativistic wave speed, γwvw/c. Line colors indicate the sound speed in units of c: (black) 0.10, (blue) 0.20, (cyan) 0.30, (green) 0.40, & (red) 0.577. – 11 – Fig. 2.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column) and the first body (right column) modes are shown for equal sound speeds in spine and sheath and no sheath flow. Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 1. Locations of the maximum growth rate are indicated by the vertical solid lines. A vertical arrow (helical B1) indicates a narrow damping feature. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sound speed in units of c: (black) 0.10, (blue) 0.20, (green) 0.40, & (red) 0.577. – 12 – Fig. 3.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column) and first body (right column) modes as a function of the sheath speed for equal sound speeds in spine and sheath. Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 1. Locations of the maximum growth rate are indicated by the vertical solid lines. Vertical arrows (helical B1) indicate damping features. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sheath speed in units of c: (black) 0.0, (blue) 0.20, (cyan) 0.35, (green) 0.40, & (red) 0.60. – 13 – increases. As ue increases the resonant frequency increases in accordance with eq. (16). On the other hand, the growth rate at resonance does not vary significantly in accordance with eqs. (18 & 19). When the sheath speed exceeds the sound speed, solutions make a transition from supersonic to transonic structure. Note that the transition point between supersonic and transonic behavior is similar but not identical for the helical and elliptical surface modes, i.e., ocurs at a slightly lower sheath speed for the elliptical mode. The first body modes also show an increase in resonant frequency with little change in the maximum growth rate provided the sheath speed remains below the sound speed. A significant damping feature in the helical first body (B1) panel, is found. While a similar damping feature was not found for the pinch and elliptical first body mode, this does not indicate a significant difference as the root finding technique does not find all structure associated with the body modes. The body mode solution structure is complex with multiple solutions not shown here and modest damping or growth can occur where solutions cross, e.g., Mizuno, Hardee & Nishikawa (2006). When the sheath speed exceeds the sound speed the maximum body mode growth rate delines significantly. This result is quite different from the transonic solution behavior illustrated in Figure 2 when aj,e = 0.577 c for no sheath flow. Thus, sheath flow effects stability of the relativistic jet beyond that accompanying an increase to the maximum sound speed in the absence of sheath flow. The reduction in growth of the body modes in the presence of sheath flow provides the relativistic jet equivalent of non-relativistic transonic/subsonic jet solution behavior. At the higher frequencies wave speeds are identical, with wave speeds given by eq. (26). Note that the high frequency wave speeds are nearly independent of ue. 4.2. Magnetic Limit In this subsection the basic behavior of pinch, helical and elliptical modes is investigated: (1) as a function of varying Alfvén speed in the external sheath or jet spine for a fixed Alfvén speed in the jet spine or external sheath and no sheath flow, (2) as a function of equal Alfvén speeds in the jet spine and external sheath for no sheath flow, and (3) as a function of sheath speed for a relatively high Alfvén speed equal in jet spine and external sheath. In general only growing solutions are shown and complexities associated with multiple crossing solutions are not shown. For all solutions shown the jet spine Lorentz factor and speed are set to γ = 2.5 and uj = 0.9165 c. Alfvén speeds are on the order of two magnitudes larger than the sound speed and are determined by varying the sound speeds but with a gas pressure fraction on the order of 0.01% of the total pressure. Only the sheath number density is held constant. The adiabatic index is set to Γ = 5/3 when aj,e/c << 0.1 consistent with low gas pressures and temperatures. The solutions shown in Figure 4 confirm the theoretical predictions in the magnetic limit with behavior depending on the Alfvén speed like the behavior found for the sound speed (see Figure 1). The pinch fundamental mode (not shown) has a growth rate almost entirely dependent on sound speeds and is negligable in the magnetic limit as predicted by eq. (6). In Figure 4 solutions in the left column are for a fixed jet spine Alfvén speed vAj = 0.3 c and in the right column are for a fixed external sheath Alfvén speed vAe = 0.3 c. The solutions shown confirm the accuracy of the low frequency solutions for helical and elliptical surface modes given by eq. (8). Note that low frequency wave speeds can be high or low depending on the values of η = γ2jWj/γ eWe, VAe/γe and VAj/γj . The numerical solutions to the dispersion relation show that the maximum growth rate is primarily a function of the jet spine Alfvén speed and only secondarily a function of the external sheath Alfvén speed as predicted by eqs. (18 - 20). The resonant frequency is primarily a function of the external sheath Alfvén speed as predicted by eq. (16). The analytical expression for the resonant frequency of the helical and elliptical surface modes provides the correct functional variation to within a constant multiplier provided vAj,e < 0.5 c. Decrease in the growth rate for jet sheath Alfvén speeds vAe > 0.5 c indicates the transition towards trans-Alfvénic behavior. Equation (15) for the resonant wave speed and equation (17) for the resonant wavelength also – 14 – Fig. 4.— Solutions to the dispersion relation for pinch fundamental, helical surface and elliptical surface modes for different Alfvén speeds in the sheath (left column) and in the spine (right column) are shown for no sheath flow. Sound speeds are aj,e ∼ 0.01vAj,e. As in Figures 1 - 3, the real part of the wavenumber, krRj , is shown by the dashed lines and the imaginary part , kiRj , is shown by the dash-dot lines as a function of the dimensionless angular frequency, ωRj/uj . The vertical lines indicate the location of maximum growth. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the Alfvén speed in units of c: (black) 0.10, (cyan) 0.30, (green) 0.50, & (red) 0.80. provide a reasonable approximation to the functional variations for vAj,e < 0.5 c. At frequencies more than an order of magnitude above resonance the growth rate is greatly reduced and solutions approach the high frequency limiting form given by eq. (26). The surface modes have relatively slow wave speeds, γwvw/c < 1 at high frequencies when the Alfvén wave speed vAj > 0.5 c. Unlike the fluid case, the helical and elliptical surface modes are stabilized for Alfvén speeds somewhat in excess of vAj,e ∼ 0.8 c in accordance with eqs. (8 & 24). In Figure 5 the behavior of solutions to the pinch fundamental mode is shown in addition to the helical and elliptical surface (left column) and associated first body modes (right column) and the figure shows how solutions change as the Alfvén speed increases in both the jet spine and external sheath. The sound speed is aj,e = 0.2 c for the pinch fundamental mode panel in order to – 15 – Fig. 5.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column) and the first body (right column) modes are shown for equal sound speeds in jet and sheath and no sheath flow. Pinch fundamental mode sound speed is aj,e = 0.2 c. Sound speeds for all other cases are aj,e ∼ 0.01vAj,e. Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 4. Locations of the maximum growth rate are indicated by the vertical solid lines. A vertical arrow (elliptical B1) indicates a narrow damping feature. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the Alfvén speed in units of c: (black) 0.10, (blue) 0.20, (green) 0.40, & (red) 0.60. – 16 – Fig. 6.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column) and the first body (right column) modes are shown for equal sound speeds in jet and sheath for different sheath flow speeds. The pinch fundamental mode sound speed is aj,e = 0.2 c. Sound speeds for all other cases are aj,e ∼ 0.01vAj,e = 0.005 c. Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 4. Locations of the maximum growth rate are indicated by the vertical solid lines. A vertical arrow indicates low frequency damping of the pinch B1 solutions. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sheath speed in units of c: (black) 0.0, (blue) 0.20, (cyan) 0.30, (green) 0.40, & (red) 0.60. – 17 – illustrate the mode behavior with increasing Alfvén speed. Sound speeds for all body modes and for helical and elliptical surface modes are aj,e ∼ 0.01vAj,e. Here the transition from super-Alfvénic towards trans-Alfvénic behavior for no flow in the sheath is illustrated. At low frequencies the modes behave as predicted by the analytic solutions given in §3.1. The growth rate of the pinch fundamental mode is reduced as the Alfvén speed increases as predicted by eq. (6). The surface and body mode solutions show the expected shift to a higher resonant frequency that is primarily a function of the increased sheath Alfvén speed and an accompanying increase in the resonant growth rate that is primarily a function of the increased spine Alfvén speed. The resonance moves to higher frequency but the maximum growth rate is reduced for Alfvén speeds vAj,e > 0.60 c and all modes become stable at higher Alfvén speeds in accordance with eqs. (8 & 24). At high frequencies wave speeds are given by eq. (26). Provided the jet is sufficiently super-Alfvénic, i.e., vAj,e < 0.6 c, the maximum growth rate of the first body mode is much greater than that of the pinch fundamental mode, is comparable to that of the helical surface mode, and is less than that of the elliptical surface mode. A narrow damping peak shown for the elliptical body mode (B1) solution when vAj,e = 0.6 c indicated by the arrow is indicative of complexities in the body mode solution structure. Figure 6 illustrates the behavior of fundamental/surface and first body modes as a function of the sheath speed for an equal Alfvén speed in spine and sheath of vAj,e = 0.5 c. For this value of the Alfvén speeds a sheath speed ue = 0 provides a super-Alfvénic solution structure baseline. The sound speed is aj,e = 0.2 c for the pinch fundamental mode panel in order to illustrate the mode behavior with increasing sheath speed. Sound speeds for all other cases are aj,e ∼ 0.01vAj,e. Solutions in the first pinch body mode panel show the damping solution as opposed to the purely real solution at the lower frequencies (indicated by the arrow). At higher frequencies the body mode is growing. At low frequencies the surface modes behave as predicted by the analytic solutions given in §3.1 and the growth rate of the surface modes decreases as ue increases. Additionally, the growth rate at resonance decreases as expected for this relatively high Alfvén speed as the sheath speed increases. At the higher frequencies wave speeds are identical, with wave speeds given by eq. (26). Note that the high frequency wave speeds are relatively independent of ue. When the velocity shear speed drops to less than the “surface” Alfvén speed, see eq. (11b), the helical and elliptical surface modes and the first body modes are stabilized. This surface and body mode mode stabilization occurs when sheath speeds exceed ue ∼ 0.5 c. However, note that the maximum pinch fundmental mode growth rate is insensitive to the sheath speed and remains unstable at ue = 0.6 c even when all other modes are stabilized. 4.3. A High Sound and Alfvén Speed Magnetosonic Case In this subsection the basic behavior of the pinch fundamental, helical surface, elliptical surface and associated first body modes is illustrated for different sheath speeds. The sheath speeds span a solution structure from supersonic to transonic but still super-Alfvénic flow. Here the sound speed in jet spine and external sheath are set equal with aj,e = 0.577 c and Alfvén speeds are set equal with vAj,e = 0.5 c. The solutions for this case are shown in Figure 7. With no sheath flow the fundamental/surface and first body modes show a typical supersonic and super-Alfvénic structure albeit the pinch fundamental mode now has a maximum growth rate comparable to the helical and elliptical surface modes as a consequence of the high sound speed. The associated first body modes also have maximum growth rates comparable to the fundamental/surface modes. Increase in the sheath speed results in a decrease in the growth rate of the helical and elliptical surface modes at low frequencies as predicted by eq. (8). The low frequency growth rate of the pinch fundamental also declines with increasing sheath speed. The resonant frequency increases with increasing sheath speed as expected from the analytical and numerical studies performed in the fluid and magnetic limits and the fundamental/surface modes take on a transonic structure for – 18 – Fig. 7.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column), and the associated first body (right column) modes are shown for a maximal spine and sheath sound speed, aj,e = 0.577 c, and a slightly smaller spine and sheath Alfvén speed, vAj,e = 0.5 c, for different sheath flow speeds. As in previous figures the real part of the wavenumber, krRj , is shown by the dashed lines, the imaginary part , kiRj , is shown by the dash-dot lines, and the vertical lines indicate the location of maximum growth. Arrows indicate damping features. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sheath speed in units of c: (black) 0.0, (blue) 0.20, (green) 0.40, & (red) 0.50. Fast and slow refer to the faster and slower moving solutions and the yellow extension indicates a damped solution. – 19 – sheath speeds 0.4 c ≤ ue ≤ 0.1 c. At high frequencies the fundamental/surface modes exhibit very high growth rates provided sheath flow remains below the Alfvén speed. On the other hand, the maximum growth rate of the first body modes declines as the sheath speed increases and is reduced severely when ue > 0.1 c. This behavior is similar to what is found for non-relativistic jets as flow enters the transonic and super-Alfvénic regime (Hardee & Rosen 1999). Additional increase in the sheath flow speed to ue > 0.4 c results in a decrease in the growth rate of the fundamental/surface modes. Solutions for the helical and elliptical surface modes shown in Figure 7 for a sheath speed ue = 0.5 c equal to the Alfvén speed illustrate some of the complexity associated with barely super-Alfvénic flow. Here limited growth is associated with both the slow and fast helical and elliptical surface solution pair. At slower sheath speeds in the super-Alfvénic regime growth is associated with the slow surface solution, i.e., backwards moving in the jet fluid reference frame. The yellow dash-dot line extension at higher frequencies in the helical and elliptical surface panels indicates a damped solution. Solutions were very difficult to follow in this parameter regime and it is possible that some solutions were not found. When the sheath speed ue > 0.5 c all modes are stabilized. A choice of Alfvén speeds greater than sound speeds results in a more magnetic like solution structure like that shown in §4.2. A choice of Alfvén speeds more than a factor of two less than sound speeds produces a more fluid like solution structure like that shown in §4.1. The more complicated solution structure illustrated in Figure 7 only occurs for a relatively narrow range of high sound speeds with similar or slightly lesser Alfvén speeds. In general, the detailed solution structure for situations in which sound and Alfvén speeds are comparable must by examined individually, e.g., Mizuno, Hardee & Nishikawa (2006), and further investigation of these cases is beyond the scope of the present paper. 5. Summary The analytical and numerical work performed here provides for the first time a detailed analysis of the KH stability properities of a RMHD jet spine-sheath configuration that allows for relativistic motions of the sheath, sound speeds up to c/ 3, and, by keeping the displacement current in the analysis, Alfvén wave speeds approaching lightspeed and large Alfvén Lorentz factors. In the fluid limit, the present results confirm an earlier more restricted low frequency analytical and numerical simulation study performed by Hardee & Hughes (2003). Provided the jet spine is super-sonic and super-Alfvénic internally and also relative to the sheath, the helical, elliptical and higher order surface modes and the pinch, helical, elliptical and higher order first body modes have a maximum growth rate at a resonant frequency. The pinch fundamental growth rate is significant only when the sound speeds, aj,e ∼ c/ 3. In general, the first body mode maximum growth rate is: greater than the pinch fundamental mode, slightly greater than the helical surface mode, slightly less than the elliptical surface mode, and occurs at a higher frequency than the maximum growth rate for the fundamental/surface mode. The basic KH stability behavior as a function of spine-sheath parameters is indicated by the analytic low frequency surface mode solution and by the behavior of the resonant frequency. The analytic surface mode solution valid at frequencies below resonance is given by ± iωi [ηuj + ue]± iη1/2 (uj − ue)2 − V 2As/γ2j γ2e (1 + V 2Ae/γ 2) + η(1 + V 2Aj/γ where V 2As ≡ γ2AjWj + γ ) B2j +B 4πWjWe , (28) – 20 – and η ≡ γ2jWj γ2eWe , V A ≡ B2/4πW , W ≡ ρ + [Γ/ (Γ− 1)]P/c2 and γA ≡ (1 − v2A/c2)−1/2. Equation (27) provides a temporal growth rate, ωi(k), and a wave speed, vw = ωr/k. The reciprocal provides a spatial growth rate ki(ω), and growth length ℓ = k i . Increase or decrease of the growth rate, dependence on physical parameters and stabilization at frequencies/wavenumbers below resonance is directly revealed by ωi in eq. (27). Note that higher jet Lorentz factors reduce ωi through the dependence on η. The resonant frequency is (1− ue/v∗w) 2 − (vwe/v∗w − uevwe/c2) , (29) where v∗w is the wave speed at resonance, eq. (15). The resonant frequency increases as the sheath sound or Alfvén wave speed, vwe ≡ (ae, vAe) increases and ω∗ −→ ∞ when the denominator decreases to zero as uj − ue 1− ujue/c2 vwj + vwe 1 + vwjvwe/c2 where vwj,e ≡ (aj,e, vAj,e) in the fluid and magnetic limits, respectively. Since eq. (27) applies below resonance the overall behavior of the growth rate is indicated by ωi. Thus, growth rates decline to zero as (uj − ue)2 − V 2As/γ2j γ2e −→ 0. The numerical analysis of the dispersion relation shows that the pinch fundamental and all first body modes are comparably or more readily stabilized and thus the jet is KH stable when (uj − ue)2 − V 2As/γ2j γ2e < 0 . (30) This stability condition takes on a particularly simple form when conditions in spine and sheath are equal, i.e., Be = Bj, We = Wj, so that vA,j = vA,e, and with γA ≡ γA,e = γA,j γ2j γ e (uj − ue) < 4γ2A γ2A − 1 c2 (31) indicates stability. This result implies that a trans-Alfvénic relativistic jet with γjuj & γAvA will be KH stable, and that even a super-Alfvénic jet with γj >> γA can be KH stable. 6. Discussion Formally, the present results and expressions apply only to magnetic fields parallel to an axial spine-sheath flow in which conditions within the spine and within the sheath are independent of radius and the sheath extends to infinity. A rapid decline in perturbation amplitudes in the sheath as a function of radius, governed by the Hankel function in the dispersion relation, suggests that the present results will apply to sheaths more than about three times the spine radius in thickness. The relativistic jet is transonic in the absence of sheath flow only for spine and sheath sound speeds ∼ c/ 3. Only in this regime does the pinch fundamental have a significant growth rate and, in general, we do not expect the pinch fundamental to grow significantly on relativistic jets. On the other hand, the pinch first body mode can have a significant maximum growth rate and would dominate any axisymmetric structure. The elliptical and higher order surface modes have increas- ingly larger maximum growth rates at resonant frequencies higher than the helical surface mode, and the maximum first body mode growth rates for helical and elliptical modes are comparable to that of the surface modes. Nevertheless, we expect the helical surface mode to achieve the largest amplitudes in the non-linear limit as a result of the reduced saturation amplitudes that accompany the higher resonant frequency and shorter resonant wavelengths associated with the higher order surface modes and all body modes. – 21 – In astrophysical jets we expect a toroidal magnetic field component, and possibly an ordered helical structure and accompanying flow helicity. Jet rotation (e.g., Bodo et al. 1996), or a radial velocity profile (e.g., Birkinshaw 1991) will modify the present results but will not stabilize the helical mode. Two dimensional non-relativistic slab jet theoretical results, indicate that KH sta- bilization occurs when the velocity shear projected on the wavevector is less than the projected Alfvén speed (Hardee et al. 1992). In the work presented here magnetic and flow field are parallel and project equally on the wavevector which for the helical (n = 1) and elliptical (n = 2) mode lies at an angle θ = tan−1(n/kR) relative to the jet axis. Provided magnetic and flow helicity and radial gradients in jet spine/sheath properties are not too large we expect the present results to remain valid where uj,e and Bj,e refer to the poloidal velocity and field components. KH driven normal mode structures move at less than the jet speed. The fundamental pinch mode moves backwards in the jet frame at about the sound speed nearly independent of the sheath properites and thus moves at nearly the jet speed in the source/observer frame. Low frequency and long wavelength helical and higer order surface modes are advected with wave speed indicated by eq. (27) and move slowly in the source/observer frame for light, i.e. η ≡ γ2jWj γ2eWe < 1, and/or for magnetically dominated flows. Higher frequency (above resonance) and shorter wavelength normal mode structures move backwards in the jet frame at the sound/Alfvén wave speed, have a wave speed nearly independent of the sheath properties, and can move slowly in the source/observer frame only for magnetically dominated flows. Where flow and magentic fields are parallel, current driven (CD) modes are stable (Isotomin & Pariev 1994, 1996). Where magnetic and flow fields are helical CD modes can be unstable (Lyubarskii 1999) in addition to the KH modes. CD and KH instability are expected to produce helically twisted structure. However, the conditions for instability, the radial structure, the growth rate and the pattern motions are different. For example, KH modes grow more rapidly when the magnetic field is force-free (e.g., Appl 1996), and non-relativistic simulation work (e.g., Lery et al. 2000; Baty & Keppens 2003; Nakamura & Meier 2004) indicates that CD driven structure is internal to any spine-sheath interface and moves at the jet speed. The differences between KH and CD instability can serve to identify the source of helical structure on relativistic jets and allow determination of jet properties near to the central engine. Perhaps the observation of relatively low proper motions in the TeV BL Lacs when intensity mod- eling requires high flow Lorentz factors (Ghisellini et al. 2005) is an indication of a magnetically dominated KH unstable spine-sheath configuration. The author acknowledges partial support through National Space Science and Technology Center (NSSTC/NASA) cooperative agreement NCC8-256 and by National Science Foundation (NSF) award AST-0506666 to the University of Alabama. A. Linearization of the RMHD Equations In vector notation the relativistic MHD continuity equation, energy equation, and momentum equation can be written as: [γρ] +∇ · [γρv] = 0 , (A1) γ2W − P )− (v/c ·B) γ2Wv + v−(v ·B) B = 0 , (A2) v + v · ∇v = −∇P − v P + ρqE+ [j×B] . (A3) – 22 – These equations along with Maxwell’s equations ∇ ·B = 0 ∇ ·E = 4πρq ∇×B = 1 E+ 4π j ∇×E = −1 and assuming ideal MHD with comoving electric field equal to zero E = −v ×B provide the complete set of ideal RMHD equations. In the above W is the enthalpy, the Lorentz factor γ = (1 − v · v/c2)−1/2, and ρ is the proper density. In what follows I will assume that the effects of radiation can be ignored, the enthalpy is given by W = ρ+ and the condition for isentropic flow is given by + v · ∇ = 0 . The general approach to analyzing the time dependent properties of this system is to linearize the ideal RMHD equations, where the density, velocity, pressure and magnetic field are written as ρ = ρ0 + ρ1, v = u + v1 (we use v0 ≡ u for notational reasons), P = P0 + P1 E = E0 + E1, and B = B0+B1, where subscript 1 refers to a perturbation to the equilibrium quantity with subscript 0. Additionally, W = W0 + W1, γ 2 = (γ0 + γ1) 2 ≃ γ20 + 2γ40u · v1/c2 and γ1 ≃ γ30u · v1/c2. It is assumed that the initial equilibrium system satisfies the zero order equations. The linearized continuity, energy and momentum equation become [γ0ρ1 + γ1ρ0] +∇ · [γ0ρ1u+ γ0ρ0v1 + γ1ρ0u] = 0 , (A4) γ20W1 − P1/c2 + 2γ40 u · v1/c2 γ20W1u+ 2γ u · v1/c2 W0u+ γ 0W0v1 u · v1/c2 + (1 + u2/c2)B0·B1 − (u ·B1/c+ v1·B0/c)u ·B0/c 2(B0·B1)u+B20v1 − (u ·B0)B1 − (u ·B1)B0 − (v1·B0)B0 = 0 , γ20W0 + u · ∇v1 = −∇P1 − (j0×B1) + (j1×B0) . (A6) The linearized Maxwell equations become: ∇ ·B1 = 0 ∇ ·E1 = 4πρq1 ∇×B1 = 1c j1 ∇×E1 = −1c – 23 – where I keep the displacement current in order to allow for strong magnetic fields and Alfvén wave speeds comparable to lightspeed. Under the assumption of ideal MHD, the comoving electric field is zero, the equilibrium charge density ρq,0 = 0, and the electric field E1= − u×B1 + v1×B0 is first order, the charge density ρq1 = (▽ ·E1) /4π is also first order, and the electrostatic force term, ρq1E1, is second order and dropped from the linearized momentum equation. The condition for isentropic perturbations becomes P1 = ã 2ρ1 = This basic set of linearized RMHD equations is similar to those found in Begelman (1998) but allows a relativistic zeroth order velocity, i.e., v = u + v1 and u . c whereas Begelman allowed only for relativistic first order motions, v1. In what follows let us model a jet as a cylinder of radius R, having a uniform proper density, ρj , a uniform axial magnetic field, Bj = Bz,j, and a uniform velocity, uj = uz,j. The external medium is assumed to have a uniform proper density, ρe, a uniform axial magnetic field, Be = Bz,e, and a uniform velocity, ue = uz,e. An external velocity could be the result of a wind or sheath outflow around a central jet, ue > 0, or could represent backflow, ue < 0, in a cocoon surrounding the jet. The jet is established in static total pressure balance with the external medium where the total static uniform pressure is P ∗e ≡ Pe + B2e/8π = P ∗j ≡ Pj + B2j /8π. Under these assumptions the linearized continuity equation becomes [γ0ρ1 + γ1ρ0] + u [γ0ρ1 + γ1ρ0] + γ0ρ0∇ · v1 = 0 . (A7) The linearized energy equation becomes γ20W1 − + 2γ40 γ20W1 + 2γ + γ20W0∇ · v1 = 0 . (A8) This result for the linearized energy equation is found by noting that the magnetic terms in the energy equation linearize to Bz1 + u − (uB0)∇ ·B1 +B20∇ · v1 −B20 ∂∂zvz1 = ∇ · v1 − ∂∂zvz1 ∇ · v1 − ∂∂zvz1 where I have used Bz1 + u Bz1 = − (rvr1) + = −B0 ∇ · v1 − from ∂B1/∂t = ∇× (u×B1) +∇× (v1×B0). The linearized momentum equation becomes γ20W0 v1 + u · ∇v1 − 14πc2γ2 v1×B0 −▽P1 − uc2 [(▽×B0)×B1 + (▽×B1)×B0] + 14πc2 – 24 – where I have used j0×B1 (∇×B0)×B1 j1×B0 (∇×B1)×B0 E1×B0 = (∇×B1)×B0 v1×B0 ×B0 , which includes the displacement current. The components of the linearized momentum equation can be written as γ20W0 vr1 + u Br1 + Br1 − , (A10a) γ20W0 vφ1 + u Bφ1 + Bφ1 − (A10b) γ20W0 vz1 + u = − ∂ P1 (A10c) where V 2A ≡ B20/(4πW0). B. Normal Mode Dispersion Relation In cylindrical geometry perturbations ρ1, v1, P1, andB1 can be considered to consist of Fourier components of the form f1(r, φ, z, t) = f1(r)e i(kz±nφ−ωt) where the flow is in the z direction, and r is in the radial direction with the jet bounded by r = R. In cylindrical geometry n, an integer, is the azimuthal wavenumber, for n > 0 waves propagate at an angle to the flow direction, where +n and −n refer to wave propagation in the clockwise and counterclockwise sense, respectively, when viewed outwards along the flow direction. In general the goal is to write a differential equation for the radial dependence of the total pressure perturbation P ∗1 ≡ P1 + (B1 · B0)/4π = P ∗1 (r) exp[i(kz ± nφ − ωt)]. The differential equation can be obtained from the energy equation by using the momentum equation and writing the velocity components vr1, vφ1, vz1 in terms of P 1 , u,B0. The components of the linearized momentum equation (eqs. A10a, b, & c) written in the form γ20W0 vr1 + u = − ∂ P ∗1 + Br1 + , (B1a) γ20W0 vφ1 + u P ∗1 + Bφ1 + , (B1b) γ20W0 vz1 + u P ∗1 + Bz1 + (B1c) along with B1 = −c(∇×E1) = ∇×(u×B1) +∇×(v1×B0) – 25 – are used to provide relations between Br1 and vr1, and Bφ1 and vφ1 Br1 + u Br1 = B0 vr1 , (B2a) Bφ1 + u Bφ1 = B0 vφ1 , (B2b) and to provide a relation between Bz1, vz1, and P Bz1 + u Bz1 = vz1 + 2B0γ vz1 + u P ∗1 + u P ∗1 . (B2c) To obtain equation (B2c) I have used Bz1 + u Bz1 = − (rvr1) + = −B0 ▽ · v1 − where −γ20W0▽ · v1 = γ20 P1 + u P1 + 2γ vz1 + u from the energy equation (eq. A8), and W1 = ρ1 + Using equations (B1a, b, & c) combined with f1(r, φ, z, t) = −iωf1(r)ei(kz±nφ−ωt) f1(r, φ, z, t) = f1(r)e i(kz±nφ−ωt) f1(r, φ, z, t) = ±inf1(r)ei(kz±nφ−ωt) f1(r, φ, z, t) = +ikf1(r)e i(kz±nφ−ωt) allows the velocity components to be written as iγ20W0 ku− ω vr1 = − P ∗1 + i k − ω u Br1 , (B3a) iγ20W0 ku− ω vφ1 = − P ∗1 + i k − ω Bφ1 , (B3b) γ20W0 [ku− ω] vz1 = − k − ω P ∗1 + k − ω Bz1 . (B3c) – 26 – The perturbed magnetic field components from equations (B2a, b,& c) become Br1 = ku− ω B0 , (B4a) Bφ1 = ku− ω B0 , (B4b) Bz1 = (ku−ω) k − ωu/c2 P ∗1 + γ k − ωu/c2 + (ku− ω) u (ku− ω) + V 2A (ku−ω) − (k − ωu/c2) u ] B0 (B4c) where I have used k + 2γ20 (ku− ω)u/c2 = γ20 k − ωu/c2 + (ku− ω)u/c2 γ20 (ku− ω) ã−2 + Γ (Γ− 1)−1 c−2 + ω/c2 = γ20 (ku− ω) /a2 − k − ωu/c2 to obtain the expression for Bz1. Using equations (B4a, b, & c) for the perturbed magnetic field components, I obtain the following relations between the perturbed velocity components v1 and the total pressure perturbation P ∗1 : vr1 ≡ Cr P ∗1 = i P ∗1 = i (ku− ω) γ20W0γ (ku− ω)2 − (k − ωu/c2)2 v2A P ∗1 , (B5a) vφ1 ≡ CφP ∗1 = ∓ P ∗1 = ∓ (ku− ω) γ20W0γ (ku− ω)2 − (k − ωu/c2)2 v2A ]P ∗1 , (B5b) vz1 ≡ CzP ∗1 = − (ku− ω) k − ωu/c2 γ20W0 (ku− ω)2 + γ2Av2A (ku−ω)2 − (k − ωu/c2)2 ]}P ∗1 . (B5c) To obtain the above relationships I have used (ku− ω)− k − ωu/c2 ku− ω = γ2A (ku− ω)− k − ωu/c2 (ku− ω) in addition to γ20 (ku− ω) ã−2 + Γ (Γ− 1)−1 c−2 + ω/c2 = γ20 (ku− ω) /a2 − k − ωu/c2 where v2A ≡ 1 + V 2A/c – 27 – is the Alfvén wave speed and γ2A = 1− v2A/c2 is an Alfvénic Lorentz factor. Note that V 2A = A and γ A = 1 + V 2. Thus we have that ∇ · v1 = Cr ∂ P ∗1 + P ∗1 + P ∗1 + Cz P ∗1 + P ∗1 − n P ∗1 + ikCzP Using the energy equation (eq. A8) written in the form −γ20W0▽ · v1 = γ20 P ∗1 + u Bz1 + u Bz1 + 2γ vz1 + u inserting Bz1 + u Bz1 = −B0 ▽ · v1 − and using vz1 = CzP 1 gives ∇ · v1 = −i Yγ2 (ku−ω) (ku−ω) ]−1 [ 2γ20 (ku− ω) uc2 − (ku−ω) where Y = γ20 (ku− ω) /a2 − k − ωu/c2 . (B8) Setting equations (B6) and (B7) equal gives us a differential equation for P ∗1 in the form of Bessel’s equation P ∗1 + r P ∗1 + β2r2 − n2 P ∗1 = 0 (B9) where β2 = Y X (ku−ω) +kXCz + (ku−ω) ]−1 [ 2γ20 (ku− ω) uc2 − (ku−ω) XCz . (B10) I can simplify the expression for β2 by writing (ku− ω) + V (ku− ω) γ20W0 (ku− ω) + kCz + 2γ20 (ku− ω) kY Cz from which it follows that (ku− ω) + V (ku− ω) γ20W0 Y + γ20 (ku− ω) k − ωu/c2 + (ku− ω) u/c2 – 28 – where I have used 2γ20 (ku− ω)u/c2 = γ20 k − ωu/c2 + (ku− ω)u/c2 − k. Substituting the ex- pressions for X and Cz from equations (B5a, b, & c), and Y from equation (B8) and modest algebraic manipulation yields (ku−ω)2−(k−ωu/c2) (a2+γ2Av A)(ku−ω) a2(ku−ω)(k−ωu/c2)u/c2 (ku− ω)2 + (ku− ω) k − ωu/c2 a2u/c2 − (ku−ω)2 (k−ωu/c2) a2+(ku−ω)(k−ωu/c2)ua2/c2 (a2+γ2Av A)(ku−ω) a2(k−ωu/c2) (B11) Additional regrouping provides the following form (ku−ω)2−(k−ωu/c2) (a2+γ2Av A)(ku−ω) a2(ku−ω)(k−ωu/c2)u/c2 [(a2+γ2Av A)(ku−ω) a2(ku−ω)(k−ωu/c2)u/c2] (ku−ω)2−(k−ωu/c2) (a2+γ2Av A)(ku−ω) a2(k−ωu/c2) } (B12) from which I find that β2 can be written in the compact form: ̟2 − κ2a2 ̟2 − κ2v2A v2ms̟ 2 − κ2v2Aa2 . (B13) where ̟2 ≡ (ω − ku)2, κ2 ≡ k − ωu/c2 , and where the fast magnetosonic speed perpendicular to the magentic field is given by (e.g., Vlahakis & Königl 2003) vms ≡ a2 + v2A − a2v2A/c2 a2/γ2A + v It is easily seen that this expression for β2 reduces to the relativistic pure fluid form β2 −→ ̟2 − κ2a2 = γ20 (ku− ω)2 k − ωu/c2 given in Hardee (2000) and that this expression for β2 reduces to the non-relativistic MHD form β2 −→ ̟2 − κ2a2 ̟2 − κ2v2A a2 + v2A ̟2 − κ2v2Aa2 (ku− ω)4 a2 + V 2A (ku− ω)2 − k2V 2Aa2 where κ2 −→ k2 and v2A −→ V 2A given in Hardee, Clarke & Rosen (1997). The solutions that are well behaved at jet center and at infinity are P ∗j1(r ≤ R) = CjJ±n(βjr), and P ∗e1(r ≥ R) = CeH ±n(βer), respectively, where J±n and H ±n are the Bessel and Hankel functions with arguments defined as β2j ≡ ̟2j − κ2ja2j ̟2j − κ2jv2Aj v2msj̟ j − κ2jv2Aja2j  , (B14a) – 29 – β2e ≡ ̟2e − κ2ea2e ̟2e − κ2ev2Ae v2mse̟ e − κ2ev2Aea2e , (B14b) where̟2j,e ≡ (ω − kuj,e) , κ2j,e ≡ k − ωuj,e/c2 , γ2j,e ≡ 1− u2j,e/c2 and γ2Aj,e ≡ 1− v2Aj,e/c2 The jet flow speed and external flow speed are positive if flow is in the +z direction. The condition that the total pressure be continuous across the jet boundary requires that CjJ±n(βjR) = CeH ±n(βeR) . (B15) The first derivative of the total pressure is given by P ∗1 = −iXvr1 . and with vr1 ≡ + u · ∇ ξr = −i (ω − ku) ξr where ξr is the fluid displacement in the radial direction it follows that ∂P ∗1 = − (ω − ku)Xξr . (B16) The radial displacement of the jet and external medium must be equal at the jet boundary, i.e., r(R) = ξ r(R), from which it follows that − (ω − kuj)Xj ∂Jn(βjr) ∂ (βjr) − (ω − kue)Xe n (βer) ∂ (βer) . (B17) Inserting Cj and Ce in terms of the Bessel and Hankel functions leads to a dispersion relation describing the propagation of Fourier components which can be written in the following form: n(βjR) Jn(βjR) n (βeR) n (βeR) . (B18) where the primes denote derivatives of the Bessel and Hankel functions with respect to their argu- ments. The expressions χj ≡ γ2j γ2AjWj ̟2j − κ2jv2Aj (B19a) χe ≡ γ2eγ2AeWe ̟2e − κ2ev2Ae (B19b) readily reduce to the non-relativistic form χ = ρ0[(ω − ku)2 − k2V 2A] where W0 −→ ρ0 given in Hardee, Clarke & Rosen (1997). This dispersion relation describes the normal modes of a cylindrical jet where n = 0, 1, 2, 3, 4, etc. involve pinching, helical, elliptical, triangular, rectangular, etc. normal mode distortions of the jet, respectively. – 30 – C. Analytic Solutions and Approximations Each normal mode n contains a fundamental/surface wave and multiple body wave solutions to the dispersion relation. The low-frequency limiting form for the fundamental/surface modes are obtained in the limit where ω −→ 0 and k −→ 0 but with ω/k 6= 0. In this limit the dispersion relation for the fundamental (n = 0) and surface (n > 0) modes is given by χj ≈ −12χe (βjR) ) + π ǫ− iπ n = 0 (C1) χj ≈ −χe n > 0 (C2) where in this limit βeR −→ 0 and βjR −→ 0, and I have used the small argument forms for the Bessel and Hankel functions to write n(βjR) Jn(βjR) n (βeR) n (βeR) (βeR) (βjR) ) + π ǫ− iπ n = 0 −βe/βj n > 0 where ǫ is Euler’s constant. C.1. Fundamental Pinch Mode (n = 0 ; m = 0) in the low frequency limit In the low frequency limit, dispersion relation solutions for the fundamental axisymmetric pinch mode are obtained from equation (C1) γ2j γ ̟2j − κ2jv2Aj ̟2e − κ2ev2Ae ) + π ǫ− iπ Here we have the trivial solution ̟2j − κ2jv2Aj = 0 with v2w = v2Aj and the more interesting zeroth order solution ̟2j ≈ κ2j v2Aja v2msj with wave speed in the proper frame given by v2w = ̟ v2Aja v2msj . (C5) To first order this magnetosonic wave solution (eq. C3) can be written as ̟2j [1− δ] ≃ κ2ja2j v2msj , (C6) – 31 – where δ ≡ − ̟2e − κ2ev2Ae v2msj and δ is complex. Thus, in the low frequency limit this fundamental pinch mode (n = 0) solution consists of a growing and damped wave pair with wave speed in the observer frame uj ± vw 1± vwuj/c2 where v2w ≃ a2j v2msj v2msj . (C9) Previous work has shown the unstable growing solution associated with the backwards moving (in the jet fluid reference frame) wave. C.2. Surface Modes (n > 0 ; m = 0) in the low frequency limit In the low frequency limit the fundamental dispersion relation solution for all higher order modes (n > 0) is most easily obtained from equation (C2) written in the form γ2jWj (ω − kuj)2 − V 2Aj k2 − ω2/c2 = −γ2eWe (ω − kue)2 − V 2Ae k2 − ω2/c2 (C10) where I have used χ ≡ γ20γ2AW0 ̟2 − κ2v2A = γ20W0 (ω − ku)2 − k2 − ω2/c2 V 2A/γ . The solu- tion can be put in the form [ηuj + ue]± iη1/2 (uj − ue)2 − V 2As/γ2j γ2e (1 + V 2Ae/γ 2) + η(1 + V 2Aj/γ (C11) where γ2jWj γ2eWe and a “surface” Alfvén speed is defined by V 2As ≡ γ2AjWj + γ ) B2j +B 4πWjWe The jet is stable to higher order n > 0 fundamental mode perturbations when γ2j γ e (uj − ue) < γ2Ajγ Ae +We/γ ) B2j +B 4πWjWe . (C12) Equation (C11) reduces to the relativistic fluid expression ηuj + ue 1 + η 1 + η (uj − ue) (C13a) – 32 – given in Hardee & Hughes (2003) equation (6a) where for pressure balance and equal adiabatic index in jet and external medium η −→ γ2j ae/γ2eaj . Similarly equation (C11) reduces to the non-relativistic MHD form ηuj + ux 1 + η 1 + η (uj − ue)2 − V 2As (C13b) given by Hardee & Rosen (2002) eq. (4) where V 2As −→ (ρj + ρe) B2j +B / (4πρjρe) and η −→ ρj/ρe. C.3. Body Modes (n ≥ 0 ; m ≥ 1) in the low frequency limit In the low frequency limit the real part of the body wave solutions can be obtained in the limit ω = 0, k 6= 0 where the dispersion relation can be written in the form cos [βjR− (2n + 1)π/4] ≈ ǫn ≡ n(βjR) n (βeR) n (βeR) . (C14) Here I have assumed that the large argument form Jn(βjR) ≈ (2/πβjR)1/2 cos [βjR− (2n + 1)π/4] applies. In the absence of a magnetic field and a flow surrounding the jet, χe = 0, ǫn = 0, and solutions are found from βjR − (2n + 1)π/4 = ±mπ ± π/2, where m is an integer. Provided ǫn << π/2 and θ ≈ cos−1 ǫn ≈ ± (π/2− ǫn), solutions can be found from βjR − (2n + 1)π/4 = ± [mπ + (π/2 ± ǫn)], where for ±ǫn the plus or minus sign is for m odd or even, respectively. In the limit ω = 0 βjR ≈ γ2j (u j − a2j)(u2j − v2Aj) v2msju j − v2Aja2j kR , (C15) and the solutions are given by kR ≈ kminnmR ≡ v2msju j − v2Aja2j γ2j (u j − a2j )(u2j − v2Aj) × [(n+ 2m− 1/2)π/2 + (−1)mǫn] (C16) where I have set m −→ m+1 to be consistent with previous notation so m = 1 corresponds to the first body mode. In the limit a2j >> v Aj equation (C16) reduces to the relativistic purely fluid form found in Hardee & Hughes (2003) kR ≈ kminnmR ≡ [(n+ 2m− 1/2)π/2 + (−1)mǫn] M2j − 1 (C17a) where M2j = u j . In the limit v Aj >> a j equation (C16) becomes kR ≈ kminnmR ≡ [(n+ 2m− 1/2)π/2 + (−1)mǫn] M2Aj − 1 (C17b) where M2Aj = u – 33 – Equation (C16) reduces to the non-relativistic MHD form found in Hardee & Rosen (2002) kR ≈ kminnmR ≡ 1−M2ms/M2AjM2j × [(n+ 2m− 1/2)π/2 + (−1)mǫn] (C17c) where M2ms = u j + v Aj) and I have used γ2j (u j − a2j)(u2j − v2Aj) v2msju j − v2Aja2j = γ2j a2+v2 1−M2ms/M2AjM2j a2+v2 We note here that there is an error in equations (5) in previous articles in the treatment of the sign on ǫn for even values of m. C.4. The Resonance Condition The resonance conditions are found by evaluating the transmittance, T , and reflectance, R, of waves at the jet boundary where T = 1 +R. With the dispersion relation written as n(βjR) Jn(βjR) n (βeR) n (βeR) (C18) where Z = χ/β with Z = γ2γ2AW ̟2 − κ2v2A a2 + γ2Av ̟2 − γ2Aκ2v2Aa2 γ2γ2A (̟ 2 − κ2a2) ̟2 − κ2v2A (C19) the reflectance R = (Ze − Zj)/(Ze + Zj) . (C20) For a fluid containing no magnetic field Z is a quantity related to the acoustic normal impedance (Gill 1965). When Ze + Zj ≈ 0, R and T are large, and the reflected and transmitted waves have an amplitude much larger than the incident wave. C.4.1. The Fluid Limit (Alfvén speed ≪ sound speed) For the case of a pure fluid ζ2e + γ , (C21a) ζ2j + γ , (C21b) where χ/k2 = W ζ2 + γ2κ2/k2 a2 and ζ ≡ β/k. For non-relativistic flows where (u2/c2)(ω/ku) << 1, γ ≈ 1, and with adiabatic indicies Γj = Γe the reflectance (ζe − ζj)(ζeζj − 1) (ζe + ζj)(ζeζj + 1) (C22) – 34 – and a supersonic resonance (Miles 1957) occurs when βe + βj = k(ζe + ζj) = 0. This supersonic resonance corresponds to the maximum growth rate of the normal mode solutions to the dispersion relation. I now generalize the results in Hardee (2000) to include flow in the external medium relative to the source/observer frame. Here Ze + Zj = 0 becomes Γeζjχe + Γjζeχj = Γeζj + Γjζe = 0 . (C23) A necessary condition for resonance is ζj < 0 and ζe > 0, and on the real axis uj − aj 1− ujaj/c2 ue + ae 1 + ueae/c2 . (C24) It follows that the resonance only exists when uj − aj 1− ujaj/c2 ue + ae 1 + ueae/c2 (C25a) or equivalently uj − ue 1− ujue/c2 aj + ae 1 + ajae/c2 . (C25b) To find the resonant solution for the real part of the phase velocity I solve ζ2j = ε 2ζ2e where here I set ε ≡ (Γjγ2j̟2j/k2a2j )/ = 1 so that ζ2j = γ (ω/k − uj)2 = ζ2e = γ (ω/k − ue)2 . (C26) The resulting quadratic equation can be written in the form a2e/(γ sj)− a2j/(γ2j γ2se) )2 − 2 γ2j a euj/γ sj − γ2ea2jue/γ2se γ2j a u2j − a2j − γ2ea2j u2e − a2e = 0 , (C27) where I have used γ2j a − γ2ea2j a2e/(γ sj)− a2j/(γ2j γ2se) and γ2s ≡ 1− a2/c2 . The solutions to equation (C27) are given by γ2j a euj/γ sj − γ2ea2jue/γ2se a2e/(γ sj)− a2j/(γ2j γ2se) γseγsj u2j − 2ujue + u2e a2e/(γ sj)− a2j/(γ2j γ2se) ] (C28) with the resonant solution given by v∗w = (γseae)γjuj + (γsjaj)γeue γj(γseae) + γe(γsjaj) . (C29) – 35 – Inserting the resonant solution (eq. C29) into the expression for ε gives 0.695 ≤ ε2 = ≤ 1.44 where 2.78 ≤ Γ2γ4s ≤ 4. When aj = ae and Γj = Γe, ε2 = 1, and the resonant solution is exact. The small range on ε (0.83 ≤ ε ≤ 1.2) suggests that this solution remains relatively robust for unequal values of the sound speed and adiabatic index in the jet and external medium. In the absence of an external flow the resonant solution v∗w = (γseae)γjuj γj(γseae) + (γsjaj) M2j − β2 M2j − β2 + (M2e − β2) is equivalent to the form given in Hardee (2000). The resonant frequencies can be estimated using the large argument forms for the Bessel and Hankel functions. In this limit the dispersion relation becomes n(βjR)H n (βeR) Jn(βjR)H n (βeR) ≈ i tan(βjR− 2n+ 1 . (C30) From the dispersion relation with Ze + Zj ≈ 0, and (χj/βj)(βe/χe) = Zj/Ze ≈ βe/βj ≈ −1, tan[βjR− (2n + 1)π/4]Re ≈ 0 on the real axis. It follows that |βjR| ≈ |βeR| ≈ (2n+ 1)π/4 +mπ can be used to obtain an estimate for the resonant frequencies from |βeR| ≈ (2n + 1)π/4 + mπ, with result that the resonant frequencies are given by ω∗nmR (2n + 1)π/4 +mπ (1− ue/v∗w) 2 − (ae/v∗w − ueae/c2) . (C31) In the absence of external flow, ue = 0, and for uj >> ae and 1 >> (kae/ω) 2 this expression reduces to the form given in Hardee (2000). When γj(γseae) >> γe(γsjaj), the resonant wave speed becomes v w ≈ uj, ue/v∗w ≈ ue/uj and provided ue << uj and ae << uj , the resonant frequency increases with increasing ue/uj and ae/uj ω∗nmR (2n + 1)π/4 +mπ 1− 2ue/uj(1− a2e/c2)− (a2e − u2e)/u2j . (C32) In general, the resonant frequency ω∗nm −→ ∞ as (1− ue/v∗w) w − ueae/c2 )2 −→ 0. An equivalent condition for (1− ue/v∗w) w − ueae/c2 = 0 is uj − ue 1− ujue/c2 aj + ae 1 + ajae/c2 , (C33) and the resonance moves to higher frequencies with ω∗nm −→ ∞ when the“shear” speed drops below a “surface” sound speed. – 36 – The behavior of the growth rate at resonance also can be found using the large argument forms for the Bessel and Hankel functions. In this limit the reflectance can be written as (Ze − Zj) (Ze + Zj) Jn(βjR)H n (βeR)− J n(βjR)H n (βeR) Jn(βjR)H n (βeR) + J n(βjR)H n (βeR) ≈ exp[−2i(βjR− 2n+ 1 π)] , (C34) 2n+ 1 π ≈ i ln |R| − φ (C35) where R ≡ |R| eiφ. It follows that (βjR)I ≈ ln |R| (C36) and since typically at resonance, |ω − kRuj | /aj > ∣kR − ωuj/c2 ∣ I can approximate βj by βj ≡ βRj + iβIj ≈ γj (ω − kRuj) − ikI . (C37) It follows that (βjR)I ≈ −γj kIR , (C38) kIR ≈ − ln |R| . (C39) At resonance (Ze − Zj) (Ze + Zj) βj − βe(χj/χe) βj + βe(χj/χe) βj − βe βj + βe ≈ −2βe βj + βe (C40) |R| ≈ βj − βe βj + βe −2βRe βIj − βIe βIj + β (C41) where I have used βRj − βRe ≈ −2βRe from the resonance condition on the real axis. It follows |R| ≈ (ω−kRue) + k2I − γe ueae − γeae ω−kRue ω−kRue (C42) where I have used βe ≡ βRe + iβIe ≈ γe (ω − kRue) − ikI ω − kRue If I assume that γj(γseae) >> γe(γsjaj), with resonant wave speed ω/kR ≈ uj and ue/uj << 1 |R| ≈ ω∗nmR (1− 2ue/uj) + k2IR2 (1 + ue/uj) (1 + ue/uj) , (C43) – 37 – and since kIR ≈ − (aj/2γjuj) ln |R| |R| ≈ ω∗nmR (1− 2ue/uj) + [ln |R| /2]2 [ln |R| /2]2 . (C44) From equation (C32) ω∗nmR (1− 2ue/uj) ≈ (1− 2ue/uj) 1− 2 (ue/uj) (1− a2e/c2)− a2e/u2j ] [(2n+ 1)π/4 +mπ] and if say ue = 0, then |R|2 − 1 ln |R| ≈ 4 1− a2e/u2j [(2n + 1)π/4 +mπ] . (C45) Formally |R| −→ ∞ as ω∗nm −→ ∞ when the jet speed drops below the “surface” sound speed given by equation (C33). This result applies only to the surface modes and not to the body modes as, in the fluid limit, the body modes do not exist when the jet speed drops below the jet sound speed, see equation (C17). On the other hand, if say, a2e/u j << 1, then |R|2 − 1 ln |R| ≈ 4 [(2n + 1)π/4 +mπ] . (C46) Formally |R| ≈ constant as ω∗nm −→ ∞ when the wind speed becomes comparable to the jet speed, ue . uj , as must be the case for the velocity shear driven Kelvin-Helmholtz instability. C.4.2. The Magnetic Limit (Alfvén speed ≫ sound speed) For the magnetic limit in which magnetic pressure dominates gas pressure Ze = γeγ AeWevAe ̟2e − κ2ev2Ae , (C47a) Zj = γjγ AjWjvAj ̟2j − κ2jv2Aj , (C47b) A necessary condition for resonance is ̟2e − κ2ev2Ae > 0 and ̟2j − κ2jv2Aj < 0 on the real axis with result that Ze + Zj = 0 when uj − vAj 1− ujvAj/c2 ue + vAe 1 + uevAe/c2 . (C48) It follows that the resonance only exists when uj − ue 1− ujue/c2 vAj + vAe 1 + vAjvAe/c2 . (C49) This result is identical in form to the sonic case with sound speeds replaced by Alfvén wave speeds. – 38 – The resonant solution for the real part of the phase velocity is obtained from Z2j = γ ̟2j − k2 − ω2/c2 V 2Aj/γ = Z2e = γ ̟2e − k2 − ω2/c2 V 2Ae/γ (C50) where I have used γ2γ2A ̟2 − κ2v2A k2 − ω2/c2 V 2A/γ , and recall that v2A = V The resulting quadratic equation can be written in the form Aj − γ2eW 2e V 2Ae )2 − 2 Ajuj − γ2eW 2e V 2Aeue j − γ2eW 2e V 2Aeu2e (C51) where I have used k2 − ω2/c2 V 2Aj/γ j = γ k2 − ω2/c2 V 2Ae/γ because pressure balance in the magnetically dominated case requires WjV Aj = WeV Ae. The solutions are given by Ajuj − γ2eW 2e V 2Aeue ± γjγeWjWeVAjVAe (uj − ue) Aj − γ2eW 2e V 2Ae , (C52) and the resonant solution becomes v∗w = γjWjVAjuj + γeWeVAeue γjWjVAj + γeWeVAe (γAevAe) γjuj + (γAjvAj) γeue γj (γAevAe) + γe (γAjvAj) (C53) where I have used WVA = WV A/ (γAvA), and WjV Aj = WeV Ae. This resonant solution has the same form as the sonic case with sound speeds and sonic Lorentz factors replaced by Alfvén wave speeds and Alfvénic Lorentz factors. As in the sonic case the resonant frequencies are found from |βeR| ≈ (2n + 1)π/4 +mπ with result that the resonant frequencies are given by ω∗nmR (2n + 1)π/4 +mπ (1− ue/v∗w) 2 − (vAe/v∗w − uevAe/c2) . (C54) When γj (γAevAe) >> γe (γAjvAj) the resonant wave speed becomes v w ≈ uj and ue/v∗w ≈ ue/uj , and provided ue << uj and vAe << uj the resonant frequency increases with increasing ue/uj and vAe/uj as ω∗nmR ≈ (2n + 1)π/4 +mπ 1− 2 (ue/uj) (1− v2Ae/c2)− (v2Ae − u2e)/u2j . (C55) Here the resonant frequency ω∗nm −→ ∞ as (1− ue/v∗w) vAe/v w − uevAe/c2 )2 −→ 0. An equivalent condition for (1− ue/v∗w) vAe/v w − uevAe/c2 = 0 is uj − ue 1− ujue/c2 vAj + vAe 1 + vAjvAe/c2 , (C56) – 39 – and the resonance moves to higher frequencies as the “shear” speed becomes trans-Alfvénic. The behavior of the growth rate at resonance proceeds in the same manner as for the fluid limit but with sound speeds replaced by Alfvén wave speeds. The resonant growth rate is now given by kIR ≈ − ln |R| . (C57) If I assume that γj(γAevAe) >> γe(γAjvAj), with resonant wave speed ω/kR ≈ uj and ue/uj << 1 |R| ≈ ω∗nmR (1− 2ue/uj) + k2IR2 − vAe (1 + ue/uj) + vAe (1 + ue/uj) , (C58) From equation (C54) ω∗nmR (1− 2ue/uj) ≈ (1− 2ue/uj) 1− 2 (ue/uj) (1− v2Ae/c2)− (v2Ae − u2e)/u2j ] [(2n+ 1)π/4 +mπ] and if say ue = 0, then |R|2 − 1 ln |R| ≈ 4 1− v2Ae/u2j [(2n + 1)π/4 +mπ] (C59) and |R| increases as ω∗nm increases when the jet speed decreases. However, when the shear speed drops below the “surface” Alfvén speed, see equations (C11 & C12) the jet is stable. This result is quite different from the fluid limit where the jet remains unstable when the shear speed drops below the “surface” sound speed. If I insert uj − ue = 1− ujue/c2 1 + vAjvAe/c2 (vAj + vAe) . from equation (C56) into equation (C12), it follows that the jet will be unstable when resonance disappears only when γ2j γ 1− ujue/c2 > 2γ2Ajγ v2Ae + v (vAj + vAe) 1 + vAjvAe/c . (C60) where I have used v2Ae + v Ae +We/γ B2j +B / (4πWjWe) as Be = Bj from magnetic pressure balance. Formally |R| −→ ∞ as ω∗nm −→ ∞ only for jet Lorentz factors greatly in excess of the Alfvénic Lorentz factor. C.5. Wave modes at high frequency To obtain the behavior of wave modes at high frequency I begin with the dispersion relation written in the form Jn(βjR) n(βjR) n (βeR) n (βeR) Jn(βjR) ∓Jn±1(βjR)± nβjRJn(βjR) n−1(βeR)− n (βeR) n (βeR) (C61) – 40 – and assume a large argument in the Hankel function with H n (βeR) ≈ exp i [βeR− (2n+ 1) π/4] and a small argument βjR << 1 in the Bessel function to write J0(βjR) −J1(βjR) e−iπ/2 n = 0 e−iπ/2 n > 0 The small arguement form for the Bessel function gives J0(βjR)/J1(βjR) ≈ 2/βjR with result that the dispersion relation becomes βjR ≈ e−iπ/2 n = 0 e−iπ/2 n > 0 . (C62) At high frequency and large wavenumber χj and χe, are proportional to k 2, βj and βe are pro- portional to k, and βjR = ζjkR ∝ (kR)1/2 for n = 0. 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The linearized relativistic magnetohydrodynamic (RMHD) equations describing a uniform axially magnetized cylindrical relativistic jet spine embedded in a uniform axially magnetized relativistically moving sheath are derived. The displacement current is retained in the equations so that effects associated with Alfven wave propagation near light speed can be studied. A dispersion relation for the normal modes is obtained. Analytical solutions for the normal modes in the low and high frequency limits are found and a general stability condition is determined. A trans-Alfvenic and even a super-Alfvenic relativistic jet spine can be stable to velocity shear driven Kelvin-Helmholtz modes. The resonance condition for maximum growth of the normal modes is obtained in the kinetically and magnetically dominated regimes. Numerical solution of the dispersion relation verifies the analytical solutions and is used to study the regime of high sound and Alfven speeds.

Introduction Relativistic jets are associated with active galactic nuclei and quasars (AGN), with black hole binary star systems (microquasars), and are thought responsible for the gamma-ray bursts (GRBs). In microquasar and AGN jets proper motions of intensity enhancements show mildly superluminal for the microquasar jets ∼ 1.2 c (Mirabel & Rodriquez 1999), range from subluminal (≪ c) to superluminal (. 6 c) along the M87 jet (Biretta et al. 1995, 1999), are up to ∼ 25 c along the 3C 345 jet (Zensus et al. 1995; Steffen et al. 1995), and have inferred Lorentz factors γ > 100 in the GRBs (e.g., Piran 2005). The observed proper motions along microquasar and AGN jets imply speeds from ∼ 0.9 c up to ∼ 0.999 c, and the speeds inferred for the GRBs are ∼ 0.99999 c. Jets at the larger scales may be kinetically dominated and contain relatively weak magnetic fields, e.g., equipartition between magnetic and gas pressure or less, but the possibility of much stronger magnetic fields exists close to the acceleration and collimation region. Here general rel- ativistic magnetohydrodynamic (GRMHD) simulations of jet formation (e.g., Koide et al. 2000; Nishikawa et al. 2005; De Villiers, Hawley & Krolik 2003; De Villiers et al. 2005; Hawley & Krolik 2006; McKinney 2006; Mizuno et al. 2006) and earlier theoretical work (e.g., Lovelace 1976; Bland- ford 1976; Blandford & Znajek 1977; Blandford & Payne 1982) invoke strong magnetic fields. In addition to strong magnetic fields, GRMHD simulation studies of jet formation indicate that highly collimated high speed jets driven by the magnetic fields threading the ergosphere may themselves reside within a broader wind or sheath outflow driven by the magnetic fields anchored in the ac- cretion disk (e.g., McKinney 2006; Hawley & Krolik 2006; Mizuno et al. 2006). This configuration might additionally be surrounded by a less collimated accretion disk wind from the hot corona (e.g., Nishikawa et al. 2005). http://arxiv.org/abs/0704.1621v1 – 2 – That relativistic jets may have jet-wind structure is indicated by recent observations of high speed winds in several QSO’s with speeds, ∼ 0.1 − 0.4c, (Chartas, Brandt & Gallagher 2002, Chartas et al. 2003; Pounds et al. 2003a; Pounds et al. 2003b; Reeves, O’Brien &Ward 2003). Other observational evidence such as limb brightening has been interpreted as evidence for a slower external sheath flow surrounding a faster jet spine, e.g., Mkn 501 (Giroletti et al. 2004), M 87 (Perlman et al. 2001), and a few other radio galaxy jets (e.g., Swain, Bridle & Baum 1998; Giovannini et al. 2001). Additional circumstantial evidence such as the requirement for large Lorentz factors suggested by the TeV BL Lacs when contrasted with much slower observed motions suggests the presence of a spine-sheath morphology (Ghisellini, Tavecchio & Chiaberge 2005). At hundreds of kiloparsec scales Siemignowska et al. (2007) have proposed a two component (spine-sheath) model to explain the broad-band emission from the PKS 1127-145 jet. A spine-sheath jet structure has been proposed based on theoretical arguments (e.g., Sol et al. 1989; Henri & Pelletier 1991; Laing 1996; Meier 2003). Similar type structure has been investigated in the context of GRB jets (e.g., Rossi, Lazzati & Rees 2002; Lazzatti & Begelman 2005; Zhang, Wooseley & MacFadyen 2003; Zhang, Woosley & Heger 2004; Morsony, Lazzati & Begelman 2006). In order to study the effect of strong magnetic fields and the effect of a moving wind or sheath around a jet or jet spine, I begin by adopting a simple system with no radial dependence of quantities inside the jet spine and no radial dependence of quantities outside the jet in the sheath. This “top hat” configuration with magnetic fields parallel to the flow can be described exactly by the linearized relativistic magnetohydrodynamic (RMHD) equations. This system with no magnetic and flow helicity is stable to current driven (CD) modes of instability (Istomin & Pariev 1994, 1996; Lyubarskii 1999). However, this system can be unstable to Kelvin-Helmholtz (KH) modes of instability (Hardee 2004). This approach allows us to look at the potential KH modes without complications arising from coexisting CD modes (see Baty, Keppens & Compte 2004) and predictions can be verified by numerical simulations (Mizuno, Hardee & Nishikawa 2006). This paper is organized as follows. In §2, I present the dispersion relation arising from a normal mode analysis of the linearized RMHD equations. Analytical approximate solutions to the dispersion relation for various limiting cases are given in §3. I verify the analytical solution through numerical solution of the dispersion relation in §4. I summarize the stability results in §5 and discuss the applicability of the present results in §6. Derivation of the linearized RMHD equations is shown in Appendix A, derivation of the normal mode dispersion relation is presented in Appendix B, and derivation of the analytical solutions is shown in Appendix C. 2. The RMHD Normal Mode Dispersion Relation Let us analyze the stability of a spine-sheath system by modeling the jet spine as a cylinder of radius R, having a uniform proper density, ρj, a uniform axial magnetic field, Bj = Bj,z, and a uniform velocity, uj = uj,z. The external sheath is assumed to have a uniform proper density, ρe, a uniform axial magnetic field, Be = Be,z, a uniform velocity ue = ue,z, and extends to infinity. The sheath velocity corresponds to an outflow around the central spine if ue,z > 0 or represents backflow when ue,z < 0. The jet spine is established in static total pressure balance with the external sheath where the total static uniform pressure is P ∗e ≡ Pe + B2e/8π = P ∗j ≡ Pj + B2j /8π, and the initial equilibrium satisfies the zeroth order equations. Formally, the assumption of an infinite sheath means that a dispersion relation could be derived in the reference frame of the sheath with results transformed to the source/observer reference frame. However, it is not much more difficult to derive a dispersion relation in the source/observer frame in which analytical solutions to the dispersion relation take on simple revealing forms. Additionally, this approach lends itself to modeling the propagation and appearance of jet structures viewed in the source/observer frame, e.g., helical structures in the 3C 120 jet (Hardee, Walker & Gómez 2005). The general approach to analyzing the time dependent properties of this system is to linearize – 3 – the ideal RMHD and Maxwell equations, where the density, velocity, pressure and magnetic field are written as ρ = ρ0 + ρ1, v = u + v1 (we use v0 ≡ u for notational reasons), P = P0 + P1, and B = B0+B1, where subscript 1 refers to a perturbation to the equilibrium quantity with subscript 0. Additionally, the Lorentz factor γ2 = (γ0 + γ1) 2 ≃ γ20 + 2γ40u · v1/c2 where γ1 = γ30u · v1/c2. The linearization is shown in Appendix A. In cylindrical geometry a random perturbation ρ1, v1 B1 and P1 can be considered to consist of Fourier components of the form f1(r, φ, z, t) = f1(r) exp[i(kz ± nφ− ωt)] (1) where flow is along the z-axis, and r is in the radial direction with the flow bounded by r = R. In cylindrical geometry n is an integer azimuthal wavenumber, for n > 0 waves propagate at an angle to the flow direction, and +n and −n give wave propagation in the clockwise and counter-clockwise sense, respectively, when viewed in the flow direction. In equation (1) n = 0, 1, 2, 3, 4, etc. correspond to pinching, helical, elliptical, triangular, rectangular, etc. normal mode distortions of the jet, respectively. Propagation and growth or damping of the Fourier components can be described by a dispersion relation of the form n(βjR) Jn(βjR) n (βeR) n (βeR) . (2) Derivation of this dispersion relation is given in Appendix B. In the dispersion relation Jn and H are Bessel and Hankel functions, the primes denote derivatives of the Bessel and Hankel functions with respect to their arguments. In equation (2) χj ≡ γ2j γ2AjWj ̟2j − κ2jv2Aj , (3a) χe ≡ γ2eγ2AeWe ̟2e − κ2ev2Ae , (3b) β2j ≡ ̟2j − κ2ja2j ̟2j − κ2jv2Aj v2msj̟ j − κ2jv2Aja2j  , (4a) β2e ≡ ̟2ex − κ2ea2e ̟2e − κ2ev2Ae v2mse̟ e − κ2ev2Aea2e . (4b) In equations (3a & 3b) and equations (4a & 4b) ̟2j,e ≡ (ω − kuj,e) and κ2j,e ≡ k − ωuj,e/c2 γj,e ≡ (1 − u2j,e/c2)−1/2 is the flow Lorentz factor, γAj,e ≡ (1 − v2Aj,e/c2)−1/2 is the Alfvén Lorentz factor, W ≡ ρ+[Γ/ (Γ− 1)]P/c2 is the enthalpy, a is the sound speed, vA is the Alfvén wave speed, and vms is a magnetosonic speed. The sound speed is defined by where 4/3 ≤ Γ ≤ 5/3 is the adiabatic index. The Alfvén wave speed is defined by 1 + V 2A/c where V 2A ≡ B20/(4πW0). A magnetosonic speed corresponding to the fast magnetosonic speed for propagation perpendicular to the magnetic field (e.g., Vlahakis & Königl 2003) is defined by vms ≡ a2 + v2A − a2v2A/c2 a2/γ2A + v – 4 – 3. Analytical Solutions to the Dispersion Relation In this section analytical solutions to the dispersion relation in the low frequency limit, in the fluid and magnetic limits at resonance (maximum growth), and in the high frequency limit are summarized. The analytical solutions are derived in Appendix C. 3.1. Low Frequency Limit Analytically each normal mode n contains a single fundamental/surface wave (ω −→ 0, k −→ 0, ω/k > 0) solution and multiple body wave (ω −→ 0, k > 0, ω/k −→ 0) solutions that satisfy the dispersion relation. In the low frequency limit the fundamental pinch mode (n = 0) solution is given by uj ± vw 1± vwuj/c2 where the pinch fundamental mode wave speed v2w ≈ a2j v2msj v2msj , (6) δ ≡ − ̟2e − κ2ev2Ae v2msj with |δ| ∝ ∣k2R2 ∣ << 1. In this limit δ is complex and this mode consists of a growing and damped wave pair. The imaginary part of the solution is vanishingly small in the low frequency limit. The above form indicates that growth, which arises from the complex value of δ, will be reduced as (v2Aj/v msj) −→ 1. The unstable growing solution is associated with the backwards moving (in the jet fluid reference frame) wave. In the low frequency limit the surface helical, elliptical, and higher order normal modes (n > 0) have a solution given by [ηuj + ue]± iη1/2 (uj − ue)2 − V 2As/γ2j γ2e (1 + V 2Ae/γ 2) + η(1 + V 2Aj/γ where η ≡ γ2jWj γ2eWe and a “surface” Alfvén speed is defined by V 2As ≡ γ2AjWj + γ ) B2j +B 4πWjWe . (9) In equation (9) note that the Alfvén Lorentz factor γ2Aj,e = 1 + V Aj,e/c 2. Thus, the jet is stable to n > 0 surface wave mode perturbations when γ2j γ e (uj − ue) < γ2Ajγ Ae +We/γ ) B2j +B 4πWjWe . (10) For example, with uj ≈ c >> ue, γ2e ≈ 1, γ2Aj >> γ2Ae ≈ 1, B2j >> B2e , and using γ2Aj = 1 +B2j /4πWjc 2 the jet is stable when γ2j < 4πWec2 γ2Aj (11a) – 5 – or with Be = Bj , We = Wj , so that vA,j = vA,e, and with γA ≡ γA,e = γA,j the jet is stable when γ2j γ e (uj − ue)2 < 4γ2A(γ2A − 1)c2 . (11b) Thus, the jet can remain stable to the surface wave modes even when the jet Lorentz factor exceeds the Alfvén Lorentz factor. In the low frequency limit the real part of the body wave solutions is given by kR ≈ kminnmR ≡ v2msju j − v2Aja2j γ2j (u j − a2j)(u2j − v2Aj) × [(n+ 2m− 1/2)π/2 + (−1)mǫn] (12) where n specifies the normal mode, m = 1, 2, 3, ... specify the first, second, third, etc. body wave solutions, and n(βjR) n (βeR) n (βeR) In the absence of a significant external magnetic field and a significant external flow ǫn = 0 as χe = γ u2e − v2Ae k2 = 0. In this low frequency limit the body wave solutions are either purely real or damped, exist only when kminnmR has a positive real part, and with |ǫn| << 1 require v2msju j − v2Aja2j γ2j (u j − a2j )(u2j − v2Aj) > 0 . (13) Thus, the body modes can exist when the jet is supersonic and super-Alfvénic, i.e., u2j − a2j > 0 and u2j − v2Aj > 0, or in a limited velocity range given approximately by a2j > u2j > [γ2sj/(1+ γ2sj)]a2j when v2Aj ≈ a2j , where γsj ≡ (1− a2j/c2)−1/2 is a sonic Lorentz factor. 3.2. Resonance With the exception of the pinch fundamental mode which can have a relatively broad plateau in the growth rate, all body modes, and all surface modes can have a distinct maximum in the growth rate at some resonant frequency. The resonance condition can be evaluated analytically in either the fluid limit where a >> VA or in the magnetic limit where VA >> a. Note that in the magnetic limit, magnetic pressure balance implies that Bj = Be. In these cases a necesary condition for resonance is that uj − ue 1− ujue/c2 vwj + vwe 1 + vwjvwe/c2 , (14) where vwj ≡ (aj , vAj) and vwe ≡ (ae, vAe) in the fluid or magnetic limits, respectively. When this condition is satisfied it can be shown that the wave speed at resonance is vw ≈ v∗w ≡ γj(γwevwe)uj + γe(γwjvwj)ue γj(γwevwe) + γe(γwjvwj) where γw ≡ (1 − v2w/c2)−1/2 is the sonic or Alfvénic Lorentz factor accompanying vwj ≡ (aj , vAj) and vwe ≡ (ae, vAe) in the fluid or magnetic limits, respectively. – 6 – The resonant wave speed and maximum growth rate occur at a frequency given by ωR/vwe ≈ ω∗nmR/vwe ≡ (2n + 1)π/4 +mπ (1− ue/v∗w) 2 − (vwe/v∗w − uevwe/c2) . (16) In equation (16) n specifies the normal mode, m = 0 specifies the surface wave, and m ≥ 1 specifies the body waves. In the limit of insignificant sheath flow, ue = 0, and using eq. (15) for v w in eq. (16) allows the resonant frequency to be written as ω∗nmRj/vwe = (2n + 1)π/4 +mπ v2we/u j + 2 vwevwj/γju v2wj/γ )]1/2 and this predicts a resonant frequency that is primarily a function of the sound and Alfvén wave speeds in the sheath. The effect of sheath flow is best illustrated by assuming comparable conditions in the spine and sheath, γwjvwj ∼ γwevwe, and assuming that γjuj >> γeue in which case ω∗nmRj/vwe ∼ (2n + 1)π/4 +mπ 1− 2 (ue/uj) (1− v2we/c2)− (v2we − u2e) /u2j The term ue/uj in the denominator indicates that the resonant frequency increases as the shear speed, uj − ue, declines. In the limit uj − ue 1− ujue/c2 vwj + vwe 1 + vwjvwe/c2 the resonant frequency ω∗nmR/vwe −→ ∞. The resonant wavelength is given by λ ≈ λ∗nm ≡ 2πv∗w/ω∗nm and can be calculated from λ∗nm ≡ (2n+ 1)π/4 +mπ (v∗w − ue) vwe − (vweue/c2)v∗w R . (17) Equations (15 - 17) provide the proper functional dependence of the resonant wave speed, frequency and wavelength provided (ue/uj) 2 << 1 and (vwe/uj) 2 << 1. With the exception of the n = 0, m = 0, fundamental pinch mode, a maximum spatial growth rate, kmaxI , is approximated by kmaxI R ≈ k∗IR ≡ − ln |R| , (18) where |R| ≈ 4 (ω∗nmR/vwe) (1− 2ue/uj) + (ln |R| /2)2 (ln |R| /2)2 . (19) Equations (18) and (19) show that the maximum growth rate is primarily a function of the jet sound, Alfvén and flow speed through vwj/γjuj, and secondarily a function of the sheath sound, Alfvén and flow speed through (ω∗nmR/vwe) (1− 2ue/uj). – 7 – I can illustrate the dependencies of the maximum growth rate on sound, Alfvén and flow speeds by using ω∗nmR (1− 2ue/uj) ≈ (1− 2ue/uj) 1− 2 (ue/uj) (1− v2we/c2)− (v2we − u2e)/u2j ] × [(2n + 1)π/4 +mπ]2 and if say ue = 0, then |R|2 − 1 ln |R| ≈ 4 1− v2we/u2j × [(2n + 1)π/4 +mπ] . (20) Thus, |R| increases as ω∗nm increases for higher order modes with larger n and larger m and this result indicates an increase in the growth rate for larger n and larger m. When the sound or Alfvén wave speed, vwe, increases |R| increases. This result indicates an increase in the growth rate at the higher resonant frequency accompanying an increase in the sound or Alfvén wave speed in the sheath. The behavior of the maximum growth rate as the shear speed, uj−ue, declines is best illustrated by considering the effect of an increasing wind speed where (v2we − u2e)/u2j << 1 is ignored. In this |R|2 − 1 ln |R| ≈ 4 [(2n + 1)π/4 +mπ] (21) and |R| will remain relatively independent of ω∗nm even as ω∗nm −→ ∞ as the shear speed decreases. This result indicates a relatively constant resonant growth rate as the shear speed decreases. In the fluid limit decline in the shear speed ultimately results in a decrease in the growth rate and increase in the spatial growth length. This decline in the growth rate is also indicated by equation (8) which, in the fluid limit, becomes ηuj + ue 1 + η ± i η 1 + η (uj − ue) . (22) Equation (22) applies to frequencies below the resonant frequency ω∗nm and directly reveals the decline in growth rates as uj − ue −→ 0. In the magnetic limit the resonant frequency ω∗nmR/vAe −→ ∞ as uj − ue 1− ujue/c2 vAj + vAe 1 + vAjvAe/c2 . (23) Here equation (8) indicates that the jet is stable when γ2j γ e (uj − ue) < V 2As , and the jet will be stable as ω∗nm −→ ∞ when γ2j γ 1− ujue/c2 < 2γ2Ajγ v2Ae + v (vAj + vAe) 1 + vAjvAe/c , (24) where I have used an equality in equation (23) in equation (8) to obtain equation (24). Equation (24) indicates that a high jet speed relative to the Alfvén wave speed is necessary for instability. For example, if vA ≡ vAj = vAe and ue = 0, the jet is stable at high frequencies provided γ2j < 1 + v2A/c γ4A . (25a) – 8 – This high frequency condition is slightly different from the low frequency stabilization condition found when vA ≡ vAj = vAe and ue = 0 from equation (11b) γ2j (uj/c) 2 < 4γ2A(γ A − 1) . (25b) Note that eqs. (25a & 25b) are identical in the large Lorentz factor limit. Equations (25) predict that stabilization at high frequencies occurs at somewhat higher jet speeds than stabilization at lower frequencies. Determination of stabilization at intermediate frequencies requires numerical solution of the dispersion relation. A non-negligable postive external flow requires even higher jet speeds for the jet to be unstable. Thus, a strongly magnetized relativistic trans-Alfvénic jet is predicted to be KH stable and a super-Alfvénic jet can be KH stable. 3.3. High Frequency Limit Provided the condition, eq. (14), for resonance is met, the real part of the solutions to the dispersion relation in the high frequency limit for fundamental, surface, and body modes is given uj ± vwj 1± vwjuj/c2 . (26) and describes sound waves vwj = aj or Alfvén waves vwj = vAj propagating with and against the jet flow inside the jet. Unstable growing solutions are associated with the backwards moving (in the jet fluid reference frame) wave but the growth rate is vanishingly small in this limit. 4. Numerical Solution of the Dispersion Relation The detailed behavior of solutions within an order of magnitude of the resonant frequency and for comparable sound and Alfvén wave speeds must be investigated by numerical solution of the dispersion relation. Analytical solutions found in the previous section can be used for initial estimates and to provide the functional behavior of solutions. Numerical solution of the dispersion relation also allows a determination of the accuracy and applicability of the analytical expressions in §3. In this section pinch fundamental, helical surface and elliptical surface, and the associated first body modes are investigated in the fluid, magnetic and magnetosonic regimes. These modes are chosen as they have been identified with structure seen in relativistic hydrodynamic (RHD) numerical simulations or tentatively identified with structures in resolved AGN jets. For example, trailing shocks in a numerical simulation (Agudo et al. 2002) and in the 3C 120 jet (Gómez et al. 2001) have been identified with the first pinch body mode. The development of large scale helical twisting of jets has been attributed to or may be associated with growth of the helical surface mode, e.g., 3C 449 (Hardee 1981) and Cygnus A (Hardee 1996) Additionally, the development of twisted filamentary structures has been attributed to helical and elliptical surface and first body modes, e.g., 3C 273 (Lobanov & Zensus 2001), M87 (Lobanov, Hardee & Eilek 2003), 3C 120 (Hardee, Walker & Gómez 2005), and have been studied in RHD numerical simulations, e.g., Hardee & Hughes (2003); Perucho et al. (2006). 4.1. Fluid Limit In this section the basic behavior of the pinch (F) fundamental, helical (S) surface and elliptical (S) surface modes is investigated: (1) as a function of varying sound speed in the external sheath or jet spine for a fixed sound speed in the jet spine or external sheath and no sheath flow, (2) as a function of equal sound speeds in the jet spine and external sheath for no sheath flow, and (3) as – 9 – a function of sheath flow for a relatively high sound speed equal in jet spine and external sheath. In general only growing solutions are shown and complexities associated with multiple crossing solutions are not shown. For all solutions shown the jet spine Lorentz factor and speed are set to γ = 2.5 and uj = 0.9165 c. Sound speeds are input directly with the only constant being the sheath number density. Total pressure and spine density are quantities computed for the specified sound speeds. The adiabatic index is chosen to be Γ = 13/9 when 0.1 ≤ aj,e/c ≤ 0.5 consistent with relativistically hot electrons and cold protons (Synge 1957). For sound speeds aj,e ∼ c/ 3 the adiabatic index is set to Γ = 4/3. Solutions shown assume zero magnetic field. Test calculations with magnetic fields giving magnetic pressures a few percent of the gas pressure and Alfvén wave speeds an order of magnitude less than the sound speeds give almost identical results. In Figure 1 solutions in the left column are for a fixed jet spine sound speed aj = 0.3 c and in the right column are for a fixed external sheath sound speed ae = 0.3 c. The solutions shown in Figure 1 confirm the accuracy of the low frequency solutions to the pinch fundamental mode, eqs. (5 & 6), and the helical and elliptical surface modes, eq. (8). Note that fast or slow wave speeds are possible at low frequencies depending on whether η ≃ (γjae/γeaj)2 in eq. (8) is much greater or much less than one, respectively. The numerical solutions to the dispersion relation show that the maximum growth rate is primarily a function of the jet spine sound speed and only secondarily a function of the external sheath sound speed as indicated by eqs. (18 - 20). Where a distinct supersonic resonance exists, the resonant frequency is primarily a function of the external sheath sound speed as predicted from eq. (16). The analytical expression for the resonant frequency for the helical and elliptical surface modes provides the correct functional variation to within a constant multiplier provided ae ≤ c/ 3 and aj < c/3. A dramatic increase in the resonant frequency and modest increase in the growth rate for larger jet spine sound speeds indicates the transition to transonic behavior. Equation (15) for the resonant wave speed and equation (17) for the resonant wavelength also provide a reasonable approximation to the functional variations provided ae ≤ c/ and aj < c/3. These results confirm the resonant solutions found in §3.2. At frequencies more than an order of magnitude above resonance the growth rate is greatly reduced and solutions approach the high frequency limiting form given by eq. (26). Note that eq. (26) allows only relatively high wave speeds at high frequencies because aj ≤ c/ In Figure 2 the behavior of solutions to the fundamental/surface (left column) and associated first body mode (right column) shows how solutions change as the sound speed increases in both the jet spine and external sheath. Here I illustrate the transition from supersonic to transonic behavior for no flow in the sheath. At low frequencies the modes behave as predicted by the analytic solutions given in §3.1. The solutions show the expected shift to a higher resonant frequency that is primarily a function of the increased external sheath sound speed and an accompanying increase in the resonant growth rate that is primarily a function of the increased jet spine sound speed. The resonance disappears as sound speeds approach c/ 3 as the jet becomes transonic as predicted by the resonance condition in §3.2. In the transonic regime high frequency fundamental/surface mode growth rates and wave speeds are identical with wave speeds given by eq. (26). Provided the jet is sufficiently supersonic, i.e., aj,e < 0.5 c, the maximum growth rate of the first body mode is greater than that of the pinch fundamental mode, is comparable to that of the helical surface mode, and is less than that of the elliptical surface mode. A narrow damping peak shown for the helical first body (B1) solution when aj,e = 0.4 c is indicative of complexities in the body mode solution structure. In the transonic regime growth of the first body mode is less than that of the pinch fundamental, helical surface and elliptical surface modes. Figure 3 illustrates the behavior of fundamental/surface and first body modes as a function of the sheath speed for equal sound speeds in spine and sheath of aj,e = 0.4 c. For this value of the sound speeds a sheath speed ue = 0 provides a supersonic solution structure baseline. At low frequencies the surface modes behave as predicted by eq. (8), and the wave speed rises as ue – 10 – Fig. 1.— Solutions to the dispersion relation for pinch fundamental, helical surface and elliptical surface modes for different sound speeds in the sheath (left column) and in the spine (right column) are shown for no sheath flow. The real part of the wavenumber, krRj , is shown by the dashed lines and the imaginary part , kiRj , is shown by the dash-dot lines as a function of the dimensionless angular frequency, ωRj/uj . For the pinch mode the vertical lines indicate the maximum growth rate range. Otherwise, the vertical lines indicate the location of maximum growth. Immediately under the dispersion relation solution panel is a panel that shows the relativistic wave speed, γwvw/c. Line colors indicate the sound speed in units of c: (black) 0.10, (blue) 0.20, (cyan) 0.30, (green) 0.40, & (red) 0.577. – 11 – Fig. 2.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column) and the first body (right column) modes are shown for equal sound speeds in spine and sheath and no sheath flow. Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 1. Locations of the maximum growth rate are indicated by the vertical solid lines. A vertical arrow (helical B1) indicates a narrow damping feature. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sound speed in units of c: (black) 0.10, (blue) 0.20, (green) 0.40, & (red) 0.577. – 12 – Fig. 3.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column) and first body (right column) modes as a function of the sheath speed for equal sound speeds in spine and sheath. Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 1. Locations of the maximum growth rate are indicated by the vertical solid lines. Vertical arrows (helical B1) indicate damping features. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sheath speed in units of c: (black) 0.0, (blue) 0.20, (cyan) 0.35, (green) 0.40, & (red) 0.60. – 13 – increases. As ue increases the resonant frequency increases in accordance with eq. (16). On the other hand, the growth rate at resonance does not vary significantly in accordance with eqs. (18 & 19). When the sheath speed exceeds the sound speed, solutions make a transition from supersonic to transonic structure. Note that the transition point between supersonic and transonic behavior is similar but not identical for the helical and elliptical surface modes, i.e., ocurs at a slightly lower sheath speed for the elliptical mode. The first body modes also show an increase in resonant frequency with little change in the maximum growth rate provided the sheath speed remains below the sound speed. A significant damping feature in the helical first body (B1) panel, is found. While a similar damping feature was not found for the pinch and elliptical first body mode, this does not indicate a significant difference as the root finding technique does not find all structure associated with the body modes. The body mode solution structure is complex with multiple solutions not shown here and modest damping or growth can occur where solutions cross, e.g., Mizuno, Hardee & Nishikawa (2006). When the sheath speed exceeds the sound speed the maximum body mode growth rate delines significantly. This result is quite different from the transonic solution behavior illustrated in Figure 2 when aj,e = 0.577 c for no sheath flow. Thus, sheath flow effects stability of the relativistic jet beyond that accompanying an increase to the maximum sound speed in the absence of sheath flow. The reduction in growth of the body modes in the presence of sheath flow provides the relativistic jet equivalent of non-relativistic transonic/subsonic jet solution behavior. At the higher frequencies wave speeds are identical, with wave speeds given by eq. (26). Note that the high frequency wave speeds are nearly independent of ue. 4.2. Magnetic Limit In this subsection the basic behavior of pinch, helical and elliptical modes is investigated: (1) as a function of varying Alfvén speed in the external sheath or jet spine for a fixed Alfvén speed in the jet spine or external sheath and no sheath flow, (2) as a function of equal Alfvén speeds in the jet spine and external sheath for no sheath flow, and (3) as a function of sheath speed for a relatively high Alfvén speed equal in jet spine and external sheath. In general only growing solutions are shown and complexities associated with multiple crossing solutions are not shown. For all solutions shown the jet spine Lorentz factor and speed are set to γ = 2.5 and uj = 0.9165 c. Alfvén speeds are on the order of two magnitudes larger than the sound speed and are determined by varying the sound speeds but with a gas pressure fraction on the order of 0.01% of the total pressure. Only the sheath number density is held constant. The adiabatic index is set to Γ = 5/3 when aj,e/c << 0.1 consistent with low gas pressures and temperatures. The solutions shown in Figure 4 confirm the theoretical predictions in the magnetic limit with behavior depending on the Alfvén speed like the behavior found for the sound speed (see Figure 1). The pinch fundamental mode (not shown) has a growth rate almost entirely dependent on sound speeds and is negligable in the magnetic limit as predicted by eq. (6). In Figure 4 solutions in the left column are for a fixed jet spine Alfvén speed vAj = 0.3 c and in the right column are for a fixed external sheath Alfvén speed vAe = 0.3 c. The solutions shown confirm the accuracy of the low frequency solutions for helical and elliptical surface modes given by eq. (8). Note that low frequency wave speeds can be high or low depending on the values of η = γ2jWj/γ eWe, VAe/γe and VAj/γj . The numerical solutions to the dispersion relation show that the maximum growth rate is primarily a function of the jet spine Alfvén speed and only secondarily a function of the external sheath Alfvén speed as predicted by eqs. (18 - 20). The resonant frequency is primarily a function of the external sheath Alfvén speed as predicted by eq. (16). The analytical expression for the resonant frequency of the helical and elliptical surface modes provides the correct functional variation to within a constant multiplier provided vAj,e < 0.5 c. Decrease in the growth rate for jet sheath Alfvén speeds vAe > 0.5 c indicates the transition towards trans-Alfvénic behavior. Equation (15) for the resonant wave speed and equation (17) for the resonant wavelength also – 14 – Fig. 4.— Solutions to the dispersion relation for pinch fundamental, helical surface and elliptical surface modes for different Alfvén speeds in the sheath (left column) and in the spine (right column) are shown for no sheath flow. Sound speeds are aj,e ∼ 0.01vAj,e. As in Figures 1 - 3, the real part of the wavenumber, krRj , is shown by the dashed lines and the imaginary part , kiRj , is shown by the dash-dot lines as a function of the dimensionless angular frequency, ωRj/uj . The vertical lines indicate the location of maximum growth. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the Alfvén speed in units of c: (black) 0.10, (cyan) 0.30, (green) 0.50, & (red) 0.80. provide a reasonable approximation to the functional variations for vAj,e < 0.5 c. At frequencies more than an order of magnitude above resonance the growth rate is greatly reduced and solutions approach the high frequency limiting form given by eq. (26). The surface modes have relatively slow wave speeds, γwvw/c < 1 at high frequencies when the Alfvén wave speed vAj > 0.5 c. Unlike the fluid case, the helical and elliptical surface modes are stabilized for Alfvén speeds somewhat in excess of vAj,e ∼ 0.8 c in accordance with eqs. (8 & 24). In Figure 5 the behavior of solutions to the pinch fundamental mode is shown in addition to the helical and elliptical surface (left column) and associated first body modes (right column) and the figure shows how solutions change as the Alfvén speed increases in both the jet spine and external sheath. The sound speed is aj,e = 0.2 c for the pinch fundamental mode panel in order to – 15 – Fig. 5.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column) and the first body (right column) modes are shown for equal sound speeds in jet and sheath and no sheath flow. Pinch fundamental mode sound speed is aj,e = 0.2 c. Sound speeds for all other cases are aj,e ∼ 0.01vAj,e. Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 4. Locations of the maximum growth rate are indicated by the vertical solid lines. A vertical arrow (elliptical B1) indicates a narrow damping feature. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the Alfvén speed in units of c: (black) 0.10, (blue) 0.20, (green) 0.40, & (red) 0.60. – 16 – Fig. 6.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column) and the first body (right column) modes are shown for equal sound speeds in jet and sheath for different sheath flow speeds. The pinch fundamental mode sound speed is aj,e = 0.2 c. Sound speeds for all other cases are aj,e ∼ 0.01vAj,e = 0.005 c. Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 4. Locations of the maximum growth rate are indicated by the vertical solid lines. A vertical arrow indicates low frequency damping of the pinch B1 solutions. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sheath speed in units of c: (black) 0.0, (blue) 0.20, (cyan) 0.30, (green) 0.40, & (red) 0.60. – 17 – illustrate the mode behavior with increasing Alfvén speed. Sound speeds for all body modes and for helical and elliptical surface modes are aj,e ∼ 0.01vAj,e. Here the transition from super-Alfvénic towards trans-Alfvénic behavior for no flow in the sheath is illustrated. At low frequencies the modes behave as predicted by the analytic solutions given in §3.1. The growth rate of the pinch fundamental mode is reduced as the Alfvén speed increases as predicted by eq. (6). The surface and body mode solutions show the expected shift to a higher resonant frequency that is primarily a function of the increased sheath Alfvén speed and an accompanying increase in the resonant growth rate that is primarily a function of the increased spine Alfvén speed. The resonance moves to higher frequency but the maximum growth rate is reduced for Alfvén speeds vAj,e > 0.60 c and all modes become stable at higher Alfvén speeds in accordance with eqs. (8 & 24). At high frequencies wave speeds are given by eq. (26). Provided the jet is sufficiently super-Alfvénic, i.e., vAj,e < 0.6 c, the maximum growth rate of the first body mode is much greater than that of the pinch fundamental mode, is comparable to that of the helical surface mode, and is less than that of the elliptical surface mode. A narrow damping peak shown for the elliptical body mode (B1) solution when vAj,e = 0.6 c indicated by the arrow is indicative of complexities in the body mode solution structure. Figure 6 illustrates the behavior of fundamental/surface and first body modes as a function of the sheath speed for an equal Alfvén speed in spine and sheath of vAj,e = 0.5 c. For this value of the Alfvén speeds a sheath speed ue = 0 provides a super-Alfvénic solution structure baseline. The sound speed is aj,e = 0.2 c for the pinch fundamental mode panel in order to illustrate the mode behavior with increasing sheath speed. Sound speeds for all other cases are aj,e ∼ 0.01vAj,e. Solutions in the first pinch body mode panel show the damping solution as opposed to the purely real solution at the lower frequencies (indicated by the arrow). At higher frequencies the body mode is growing. At low frequencies the surface modes behave as predicted by the analytic solutions given in §3.1 and the growth rate of the surface modes decreases as ue increases. Additionally, the growth rate at resonance decreases as expected for this relatively high Alfvén speed as the sheath speed increases. At the higher frequencies wave speeds are identical, with wave speeds given by eq. (26). Note that the high frequency wave speeds are relatively independent of ue. When the velocity shear speed drops to less than the “surface” Alfvén speed, see eq. (11b), the helical and elliptical surface modes and the first body modes are stabilized. This surface and body mode mode stabilization occurs when sheath speeds exceed ue ∼ 0.5 c. However, note that the maximum pinch fundmental mode growth rate is insensitive to the sheath speed and remains unstable at ue = 0.6 c even when all other modes are stabilized. 4.3. A High Sound and Alfvén Speed Magnetosonic Case In this subsection the basic behavior of the pinch fundamental, helical surface, elliptical surface and associated first body modes is illustrated for different sheath speeds. The sheath speeds span a solution structure from supersonic to transonic but still super-Alfvénic flow. Here the sound speed in jet spine and external sheath are set equal with aj,e = 0.577 c and Alfvén speeds are set equal with vAj,e = 0.5 c. The solutions for this case are shown in Figure 7. With no sheath flow the fundamental/surface and first body modes show a typical supersonic and super-Alfvénic structure albeit the pinch fundamental mode now has a maximum growth rate comparable to the helical and elliptical surface modes as a consequence of the high sound speed. The associated first body modes also have maximum growth rates comparable to the fundamental/surface modes. Increase in the sheath speed results in a decrease in the growth rate of the helical and elliptical surface modes at low frequencies as predicted by eq. (8). The low frequency growth rate of the pinch fundamental also declines with increasing sheath speed. The resonant frequency increases with increasing sheath speed as expected from the analytical and numerical studies performed in the fluid and magnetic limits and the fundamental/surface modes take on a transonic structure for – 18 – Fig. 7.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column), and the associated first body (right column) modes are shown for a maximal spine and sheath sound speed, aj,e = 0.577 c, and a slightly smaller spine and sheath Alfvén speed, vAj,e = 0.5 c, for different sheath flow speeds. As in previous figures the real part of the wavenumber, krRj , is shown by the dashed lines, the imaginary part , kiRj , is shown by the dash-dot lines, and the vertical lines indicate the location of maximum growth. Arrows indicate damping features. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sheath speed in units of c: (black) 0.0, (blue) 0.20, (green) 0.40, & (red) 0.50. Fast and slow refer to the faster and slower moving solutions and the yellow extension indicates a damped solution. – 19 – sheath speeds 0.4 c ≤ ue ≤ 0.1 c. At high frequencies the fundamental/surface modes exhibit very high growth rates provided sheath flow remains below the Alfvén speed. On the other hand, the maximum growth rate of the first body modes declines as the sheath speed increases and is reduced severely when ue > 0.1 c. This behavior is similar to what is found for non-relativistic jets as flow enters the transonic and super-Alfvénic regime (Hardee & Rosen 1999). Additional increase in the sheath flow speed to ue > 0.4 c results in a decrease in the growth rate of the fundamental/surface modes. Solutions for the helical and elliptical surface modes shown in Figure 7 for a sheath speed ue = 0.5 c equal to the Alfvén speed illustrate some of the complexity associated with barely super-Alfvénic flow. Here limited growth is associated with both the slow and fast helical and elliptical surface solution pair. At slower sheath speeds in the super-Alfvénic regime growth is associated with the slow surface solution, i.e., backwards moving in the jet fluid reference frame. The yellow dash-dot line extension at higher frequencies in the helical and elliptical surface panels indicates a damped solution. Solutions were very difficult to follow in this parameter regime and it is possible that some solutions were not found. When the sheath speed ue > 0.5 c all modes are stabilized. A choice of Alfvén speeds greater than sound speeds results in a more magnetic like solution structure like that shown in §4.2. A choice of Alfvén speeds more than a factor of two less than sound speeds produces a more fluid like solution structure like that shown in §4.1. The more complicated solution structure illustrated in Figure 7 only occurs for a relatively narrow range of high sound speeds with similar or slightly lesser Alfvén speeds. In general, the detailed solution structure for situations in which sound and Alfvén speeds are comparable must by examined individually, e.g., Mizuno, Hardee & Nishikawa (2006), and further investigation of these cases is beyond the scope of the present paper. 5. Summary The analytical and numerical work performed here provides for the first time a detailed analysis of the KH stability properities of a RMHD jet spine-sheath configuration that allows for relativistic motions of the sheath, sound speeds up to c/ 3, and, by keeping the displacement current in the analysis, Alfvén wave speeds approaching lightspeed and large Alfvén Lorentz factors. In the fluid limit, the present results confirm an earlier more restricted low frequency analytical and numerical simulation study performed by Hardee & Hughes (2003). Provided the jet spine is super-sonic and super-Alfvénic internally and also relative to the sheath, the helical, elliptical and higher order surface modes and the pinch, helical, elliptical and higher order first body modes have a maximum growth rate at a resonant frequency. The pinch fundamental growth rate is significant only when the sound speeds, aj,e ∼ c/ 3. In general, the first body mode maximum growth rate is: greater than the pinch fundamental mode, slightly greater than the helical surface mode, slightly less than the elliptical surface mode, and occurs at a higher frequency than the maximum growth rate for the fundamental/surface mode. The basic KH stability behavior as a function of spine-sheath parameters is indicated by the analytic low frequency surface mode solution and by the behavior of the resonant frequency. The analytic surface mode solution valid at frequencies below resonance is given by ± iωi [ηuj + ue]± iη1/2 (uj − ue)2 − V 2As/γ2j γ2e (1 + V 2Ae/γ 2) + η(1 + V 2Aj/γ where V 2As ≡ γ2AjWj + γ ) B2j +B 4πWjWe , (28) – 20 – and η ≡ γ2jWj γ2eWe , V A ≡ B2/4πW , W ≡ ρ + [Γ/ (Γ− 1)]P/c2 and γA ≡ (1 − v2A/c2)−1/2. Equation (27) provides a temporal growth rate, ωi(k), and a wave speed, vw = ωr/k. The reciprocal provides a spatial growth rate ki(ω), and growth length ℓ = k i . Increase or decrease of the growth rate, dependence on physical parameters and stabilization at frequencies/wavenumbers below resonance is directly revealed by ωi in eq. (27). Note that higher jet Lorentz factors reduce ωi through the dependence on η. The resonant frequency is (1− ue/v∗w) 2 − (vwe/v∗w − uevwe/c2) , (29) where v∗w is the wave speed at resonance, eq. (15). The resonant frequency increases as the sheath sound or Alfvén wave speed, vwe ≡ (ae, vAe) increases and ω∗ −→ ∞ when the denominator decreases to zero as uj − ue 1− ujue/c2 vwj + vwe 1 + vwjvwe/c2 where vwj,e ≡ (aj,e, vAj,e) in the fluid and magnetic limits, respectively. Since eq. (27) applies below resonance the overall behavior of the growth rate is indicated by ωi. Thus, growth rates decline to zero as (uj − ue)2 − V 2As/γ2j γ2e −→ 0. The numerical analysis of the dispersion relation shows that the pinch fundamental and all first body modes are comparably or more readily stabilized and thus the jet is KH stable when (uj − ue)2 − V 2As/γ2j γ2e < 0 . (30) This stability condition takes on a particularly simple form when conditions in spine and sheath are equal, i.e., Be = Bj, We = Wj, so that vA,j = vA,e, and with γA ≡ γA,e = γA,j γ2j γ e (uj − ue) < 4γ2A γ2A − 1 c2 (31) indicates stability. This result implies that a trans-Alfvénic relativistic jet with γjuj & γAvA will be KH stable, and that even a super-Alfvénic jet with γj >> γA can be KH stable. 6. Discussion Formally, the present results and expressions apply only to magnetic fields parallel to an axial spine-sheath flow in which conditions within the spine and within the sheath are independent of radius and the sheath extends to infinity. A rapid decline in perturbation amplitudes in the sheath as a function of radius, governed by the Hankel function in the dispersion relation, suggests that the present results will apply to sheaths more than about three times the spine radius in thickness. The relativistic jet is transonic in the absence of sheath flow only for spine and sheath sound speeds ∼ c/ 3. Only in this regime does the pinch fundamental have a significant growth rate and, in general, we do not expect the pinch fundamental to grow significantly on relativistic jets. On the other hand, the pinch first body mode can have a significant maximum growth rate and would dominate any axisymmetric structure. The elliptical and higher order surface modes have increas- ingly larger maximum growth rates at resonant frequencies higher than the helical surface mode, and the maximum first body mode growth rates for helical and elliptical modes are comparable to that of the surface modes. Nevertheless, we expect the helical surface mode to achieve the largest amplitudes in the non-linear limit as a result of the reduced saturation amplitudes that accompany the higher resonant frequency and shorter resonant wavelengths associated with the higher order surface modes and all body modes. – 21 – In astrophysical jets we expect a toroidal magnetic field component, and possibly an ordered helical structure and accompanying flow helicity. Jet rotation (e.g., Bodo et al. 1996), or a radial velocity profile (e.g., Birkinshaw 1991) will modify the present results but will not stabilize the helical mode. Two dimensional non-relativistic slab jet theoretical results, indicate that KH sta- bilization occurs when the velocity shear projected on the wavevector is less than the projected Alfvén speed (Hardee et al. 1992). In the work presented here magnetic and flow field are parallel and project equally on the wavevector which for the helical (n = 1) and elliptical (n = 2) mode lies at an angle θ = tan−1(n/kR) relative to the jet axis. Provided magnetic and flow helicity and radial gradients in jet spine/sheath properties are not too large we expect the present results to remain valid where uj,e and Bj,e refer to the poloidal velocity and field components. KH driven normal mode structures move at less than the jet speed. The fundamental pinch mode moves backwards in the jet frame at about the sound speed nearly independent of the sheath properites and thus moves at nearly the jet speed in the source/observer frame. Low frequency and long wavelength helical and higer order surface modes are advected with wave speed indicated by eq. (27) and move slowly in the source/observer frame for light, i.e. η ≡ γ2jWj γ2eWe < 1, and/or for magnetically dominated flows. Higher frequency (above resonance) and shorter wavelength normal mode structures move backwards in the jet frame at the sound/Alfvén wave speed, have a wave speed nearly independent of the sheath properties, and can move slowly in the source/observer frame only for magnetically dominated flows. Where flow and magentic fields are parallel, current driven (CD) modes are stable (Isotomin & Pariev 1994, 1996). Where magnetic and flow fields are helical CD modes can be unstable (Lyubarskii 1999) in addition to the KH modes. CD and KH instability are expected to produce helically twisted structure. However, the conditions for instability, the radial structure, the growth rate and the pattern motions are different. For example, KH modes grow more rapidly when the magnetic field is force-free (e.g., Appl 1996), and non-relativistic simulation work (e.g., Lery et al. 2000; Baty & Keppens 2003; Nakamura & Meier 2004) indicates that CD driven structure is internal to any spine-sheath interface and moves at the jet speed. The differences between KH and CD instability can serve to identify the source of helical structure on relativistic jets and allow determination of jet properties near to the central engine. Perhaps the observation of relatively low proper motions in the TeV BL Lacs when intensity mod- eling requires high flow Lorentz factors (Ghisellini et al. 2005) is an indication of a magnetically dominated KH unstable spine-sheath configuration. The author acknowledges partial support through National Space Science and Technology Center (NSSTC/NASA) cooperative agreement NCC8-256 and by National Science Foundation (NSF) award AST-0506666 to the University of Alabama. A. Linearization of the RMHD Equations In vector notation the relativistic MHD continuity equation, energy equation, and momentum equation can be written as: [γρ] +∇ · [γρv] = 0 , (A1) γ2W − P )− (v/c ·B) γ2Wv + v−(v ·B) B = 0 , (A2) v + v · ∇v = −∇P − v P + ρqE+ [j×B] . (A3) – 22 – These equations along with Maxwell’s equations ∇ ·B = 0 ∇ ·E = 4πρq ∇×B = 1 E+ 4π j ∇×E = −1 and assuming ideal MHD with comoving electric field equal to zero E = −v ×B provide the complete set of ideal RMHD equations. In the above W is the enthalpy, the Lorentz factor γ = (1 − v · v/c2)−1/2, and ρ is the proper density. In what follows I will assume that the effects of radiation can be ignored, the enthalpy is given by W = ρ+ and the condition for isentropic flow is given by + v · ∇ = 0 . The general approach to analyzing the time dependent properties of this system is to linearize the ideal RMHD equations, where the density, velocity, pressure and magnetic field are written as ρ = ρ0 + ρ1, v = u + v1 (we use v0 ≡ u for notational reasons), P = P0 + P1 E = E0 + E1, and B = B0+B1, where subscript 1 refers to a perturbation to the equilibrium quantity with subscript 0. Additionally, W = W0 + W1, γ 2 = (γ0 + γ1) 2 ≃ γ20 + 2γ40u · v1/c2 and γ1 ≃ γ30u · v1/c2. It is assumed that the initial equilibrium system satisfies the zero order equations. The linearized continuity, energy and momentum equation become [γ0ρ1 + γ1ρ0] +∇ · [γ0ρ1u+ γ0ρ0v1 + γ1ρ0u] = 0 , (A4) γ20W1 − P1/c2 + 2γ40 u · v1/c2 γ20W1u+ 2γ u · v1/c2 W0u+ γ 0W0v1 u · v1/c2 + (1 + u2/c2)B0·B1 − (u ·B1/c+ v1·B0/c)u ·B0/c 2(B0·B1)u+B20v1 − (u ·B0)B1 − (u ·B1)B0 − (v1·B0)B0 = 0 , γ20W0 + u · ∇v1 = −∇P1 − (j0×B1) + (j1×B0) . (A6) The linearized Maxwell equations become: ∇ ·B1 = 0 ∇ ·E1 = 4πρq1 ∇×B1 = 1c j1 ∇×E1 = −1c – 23 – where I keep the displacement current in order to allow for strong magnetic fields and Alfvén wave speeds comparable to lightspeed. Under the assumption of ideal MHD, the comoving electric field is zero, the equilibrium charge density ρq,0 = 0, and the electric field E1= − u×B1 + v1×B0 is first order, the charge density ρq1 = (▽ ·E1) /4π is also first order, and the electrostatic force term, ρq1E1, is second order and dropped from the linearized momentum equation. The condition for isentropic perturbations becomes P1 = ã 2ρ1 = This basic set of linearized RMHD equations is similar to those found in Begelman (1998) but allows a relativistic zeroth order velocity, i.e., v = u + v1 and u . c whereas Begelman allowed only for relativistic first order motions, v1. In what follows let us model a jet as a cylinder of radius R, having a uniform proper density, ρj , a uniform axial magnetic field, Bj = Bz,j, and a uniform velocity, uj = uz,j. The external medium is assumed to have a uniform proper density, ρe, a uniform axial magnetic field, Be = Bz,e, and a uniform velocity, ue = uz,e. An external velocity could be the result of a wind or sheath outflow around a central jet, ue > 0, or could represent backflow, ue < 0, in a cocoon surrounding the jet. The jet is established in static total pressure balance with the external medium where the total static uniform pressure is P ∗e ≡ Pe + B2e/8π = P ∗j ≡ Pj + B2j /8π. Under these assumptions the linearized continuity equation becomes [γ0ρ1 + γ1ρ0] + u [γ0ρ1 + γ1ρ0] + γ0ρ0∇ · v1 = 0 . (A7) The linearized energy equation becomes γ20W1 − + 2γ40 γ20W1 + 2γ + γ20W0∇ · v1 = 0 . (A8) This result for the linearized energy equation is found by noting that the magnetic terms in the energy equation linearize to Bz1 + u − (uB0)∇ ·B1 +B20∇ · v1 −B20 ∂∂zvz1 = ∇ · v1 − ∂∂zvz1 ∇ · v1 − ∂∂zvz1 where I have used Bz1 + u Bz1 = − (rvr1) + = −B0 ∇ · v1 − from ∂B1/∂t = ∇× (u×B1) +∇× (v1×B0). The linearized momentum equation becomes γ20W0 v1 + u · ∇v1 − 14πc2γ2 v1×B0 −▽P1 − uc2 [(▽×B0)×B1 + (▽×B1)×B0] + 14πc2 – 24 – where I have used j0×B1 (∇×B0)×B1 j1×B0 (∇×B1)×B0 E1×B0 = (∇×B1)×B0 v1×B0 ×B0 , which includes the displacement current. The components of the linearized momentum equation can be written as γ20W0 vr1 + u Br1 + Br1 − , (A10a) γ20W0 vφ1 + u Bφ1 + Bφ1 − (A10b) γ20W0 vz1 + u = − ∂ P1 (A10c) where V 2A ≡ B20/(4πW0). B. Normal Mode Dispersion Relation In cylindrical geometry perturbations ρ1, v1, P1, andB1 can be considered to consist of Fourier components of the form f1(r, φ, z, t) = f1(r)e i(kz±nφ−ωt) where the flow is in the z direction, and r is in the radial direction with the jet bounded by r = R. In cylindrical geometry n, an integer, is the azimuthal wavenumber, for n > 0 waves propagate at an angle to the flow direction, where +n and −n refer to wave propagation in the clockwise and counterclockwise sense, respectively, when viewed outwards along the flow direction. In general the goal is to write a differential equation for the radial dependence of the total pressure perturbation P ∗1 ≡ P1 + (B1 · B0)/4π = P ∗1 (r) exp[i(kz ± nφ − ωt)]. The differential equation can be obtained from the energy equation by using the momentum equation and writing the velocity components vr1, vφ1, vz1 in terms of P 1 , u,B0. The components of the linearized momentum equation (eqs. A10a, b, & c) written in the form γ20W0 vr1 + u = − ∂ P ∗1 + Br1 + , (B1a) γ20W0 vφ1 + u P ∗1 + Bφ1 + , (B1b) γ20W0 vz1 + u P ∗1 + Bz1 + (B1c) along with B1 = −c(∇×E1) = ∇×(u×B1) +∇×(v1×B0) – 25 – are used to provide relations between Br1 and vr1, and Bφ1 and vφ1 Br1 + u Br1 = B0 vr1 , (B2a) Bφ1 + u Bφ1 = B0 vφ1 , (B2b) and to provide a relation between Bz1, vz1, and P Bz1 + u Bz1 = vz1 + 2B0γ vz1 + u P ∗1 + u P ∗1 . (B2c) To obtain equation (B2c) I have used Bz1 + u Bz1 = − (rvr1) + = −B0 ▽ · v1 − where −γ20W0▽ · v1 = γ20 P1 + u P1 + 2γ vz1 + u from the energy equation (eq. A8), and W1 = ρ1 + Using equations (B1a, b, & c) combined with f1(r, φ, z, t) = −iωf1(r)ei(kz±nφ−ωt) f1(r, φ, z, t) = f1(r)e i(kz±nφ−ωt) f1(r, φ, z, t) = ±inf1(r)ei(kz±nφ−ωt) f1(r, φ, z, t) = +ikf1(r)e i(kz±nφ−ωt) allows the velocity components to be written as iγ20W0 ku− ω vr1 = − P ∗1 + i k − ω u Br1 , (B3a) iγ20W0 ku− ω vφ1 = − P ∗1 + i k − ω Bφ1 , (B3b) γ20W0 [ku− ω] vz1 = − k − ω P ∗1 + k − ω Bz1 . (B3c) – 26 – The perturbed magnetic field components from equations (B2a, b,& c) become Br1 = ku− ω B0 , (B4a) Bφ1 = ku− ω B0 , (B4b) Bz1 = (ku−ω) k − ωu/c2 P ∗1 + γ k − ωu/c2 + (ku− ω) u (ku− ω) + V 2A (ku−ω) − (k − ωu/c2) u ] B0 (B4c) where I have used k + 2γ20 (ku− ω)u/c2 = γ20 k − ωu/c2 + (ku− ω)u/c2 γ20 (ku− ω) ã−2 + Γ (Γ− 1)−1 c−2 + ω/c2 = γ20 (ku− ω) /a2 − k − ωu/c2 to obtain the expression for Bz1. Using equations (B4a, b, & c) for the perturbed magnetic field components, I obtain the following relations between the perturbed velocity components v1 and the total pressure perturbation P ∗1 : vr1 ≡ Cr P ∗1 = i P ∗1 = i (ku− ω) γ20W0γ (ku− ω)2 − (k − ωu/c2)2 v2A P ∗1 , (B5a) vφ1 ≡ CφP ∗1 = ∓ P ∗1 = ∓ (ku− ω) γ20W0γ (ku− ω)2 − (k − ωu/c2)2 v2A ]P ∗1 , (B5b) vz1 ≡ CzP ∗1 = − (ku− ω) k − ωu/c2 γ20W0 (ku− ω)2 + γ2Av2A (ku−ω)2 − (k − ωu/c2)2 ]}P ∗1 . (B5c) To obtain the above relationships I have used (ku− ω)− k − ωu/c2 ku− ω = γ2A (ku− ω)− k − ωu/c2 (ku− ω) in addition to γ20 (ku− ω) ã−2 + Γ (Γ− 1)−1 c−2 + ω/c2 = γ20 (ku− ω) /a2 − k − ωu/c2 where v2A ≡ 1 + V 2A/c – 27 – is the Alfvén wave speed and γ2A = 1− v2A/c2 is an Alfvénic Lorentz factor. Note that V 2A = A and γ A = 1 + V 2. Thus we have that ∇ · v1 = Cr ∂ P ∗1 + P ∗1 + P ∗1 + Cz P ∗1 + P ∗1 − n P ∗1 + ikCzP Using the energy equation (eq. A8) written in the form −γ20W0▽ · v1 = γ20 P ∗1 + u Bz1 + u Bz1 + 2γ vz1 + u inserting Bz1 + u Bz1 = −B0 ▽ · v1 − and using vz1 = CzP 1 gives ∇ · v1 = −i Yγ2 (ku−ω) (ku−ω) ]−1 [ 2γ20 (ku− ω) uc2 − (ku−ω) where Y = γ20 (ku− ω) /a2 − k − ωu/c2 . (B8) Setting equations (B6) and (B7) equal gives us a differential equation for P ∗1 in the form of Bessel’s equation P ∗1 + r P ∗1 + β2r2 − n2 P ∗1 = 0 (B9) where β2 = Y X (ku−ω) +kXCz + (ku−ω) ]−1 [ 2γ20 (ku− ω) uc2 − (ku−ω) XCz . (B10) I can simplify the expression for β2 by writing (ku− ω) + V (ku− ω) γ20W0 (ku− ω) + kCz + 2γ20 (ku− ω) kY Cz from which it follows that (ku− ω) + V (ku− ω) γ20W0 Y + γ20 (ku− ω) k − ωu/c2 + (ku− ω) u/c2 – 28 – where I have used 2γ20 (ku− ω)u/c2 = γ20 k − ωu/c2 + (ku− ω)u/c2 − k. Substituting the ex- pressions for X and Cz from equations (B5a, b, & c), and Y from equation (B8) and modest algebraic manipulation yields (ku−ω)2−(k−ωu/c2) (a2+γ2Av A)(ku−ω) a2(ku−ω)(k−ωu/c2)u/c2 (ku− ω)2 + (ku− ω) k − ωu/c2 a2u/c2 − (ku−ω)2 (k−ωu/c2) a2+(ku−ω)(k−ωu/c2)ua2/c2 (a2+γ2Av A)(ku−ω) a2(k−ωu/c2) (B11) Additional regrouping provides the following form (ku−ω)2−(k−ωu/c2) (a2+γ2Av A)(ku−ω) a2(ku−ω)(k−ωu/c2)u/c2 [(a2+γ2Av A)(ku−ω) a2(ku−ω)(k−ωu/c2)u/c2] (ku−ω)2−(k−ωu/c2) (a2+γ2Av A)(ku−ω) a2(k−ωu/c2) } (B12) from which I find that β2 can be written in the compact form: ̟2 − κ2a2 ̟2 − κ2v2A v2ms̟ 2 − κ2v2Aa2 . (B13) where ̟2 ≡ (ω − ku)2, κ2 ≡ k − ωu/c2 , and where the fast magnetosonic speed perpendicular to the magentic field is given by (e.g., Vlahakis & Königl 2003) vms ≡ a2 + v2A − a2v2A/c2 a2/γ2A + v It is easily seen that this expression for β2 reduces to the relativistic pure fluid form β2 −→ ̟2 − κ2a2 = γ20 (ku− ω)2 k − ωu/c2 given in Hardee (2000) and that this expression for β2 reduces to the non-relativistic MHD form β2 −→ ̟2 − κ2a2 ̟2 − κ2v2A a2 + v2A ̟2 − κ2v2Aa2 (ku− ω)4 a2 + V 2A (ku− ω)2 − k2V 2Aa2 where κ2 −→ k2 and v2A −→ V 2A given in Hardee, Clarke & Rosen (1997). The solutions that are well behaved at jet center and at infinity are P ∗j1(r ≤ R) = CjJ±n(βjr), and P ∗e1(r ≥ R) = CeH ±n(βer), respectively, where J±n and H ±n are the Bessel and Hankel functions with arguments defined as β2j ≡ ̟2j − κ2ja2j ̟2j − κ2jv2Aj v2msj̟ j − κ2jv2Aja2j  , (B14a) – 29 – β2e ≡ ̟2e − κ2ea2e ̟2e − κ2ev2Ae v2mse̟ e − κ2ev2Aea2e , (B14b) where̟2j,e ≡ (ω − kuj,e) , κ2j,e ≡ k − ωuj,e/c2 , γ2j,e ≡ 1− u2j,e/c2 and γ2Aj,e ≡ 1− v2Aj,e/c2 The jet flow speed and external flow speed are positive if flow is in the +z direction. The condition that the total pressure be continuous across the jet boundary requires that CjJ±n(βjR) = CeH ±n(βeR) . (B15) The first derivative of the total pressure is given by P ∗1 = −iXvr1 . and with vr1 ≡ + u · ∇ ξr = −i (ω − ku) ξr where ξr is the fluid displacement in the radial direction it follows that ∂P ∗1 = − (ω − ku)Xξr . (B16) The radial displacement of the jet and external medium must be equal at the jet boundary, i.e., r(R) = ξ r(R), from which it follows that − (ω − kuj)Xj ∂Jn(βjr) ∂ (βjr) − (ω − kue)Xe n (βer) ∂ (βer) . (B17) Inserting Cj and Ce in terms of the Bessel and Hankel functions leads to a dispersion relation describing the propagation of Fourier components which can be written in the following form: n(βjR) Jn(βjR) n (βeR) n (βeR) . (B18) where the primes denote derivatives of the Bessel and Hankel functions with respect to their argu- ments. The expressions χj ≡ γ2j γ2AjWj ̟2j − κ2jv2Aj (B19a) χe ≡ γ2eγ2AeWe ̟2e − κ2ev2Ae (B19b) readily reduce to the non-relativistic form χ = ρ0[(ω − ku)2 − k2V 2A] where W0 −→ ρ0 given in Hardee, Clarke & Rosen (1997). This dispersion relation describes the normal modes of a cylindrical jet where n = 0, 1, 2, 3, 4, etc. involve pinching, helical, elliptical, triangular, rectangular, etc. normal mode distortions of the jet, respectively. – 30 – C. Analytic Solutions and Approximations Each normal mode n contains a fundamental/surface wave and multiple body wave solutions to the dispersion relation. The low-frequency limiting form for the fundamental/surface modes are obtained in the limit where ω −→ 0 and k −→ 0 but with ω/k 6= 0. In this limit the dispersion relation for the fundamental (n = 0) and surface (n > 0) modes is given by χj ≈ −12χe (βjR) ) + π ǫ− iπ n = 0 (C1) χj ≈ −χe n > 0 (C2) where in this limit βeR −→ 0 and βjR −→ 0, and I have used the small argument forms for the Bessel and Hankel functions to write n(βjR) Jn(βjR) n (βeR) n (βeR) (βeR) (βjR) ) + π ǫ− iπ n = 0 −βe/βj n > 0 where ǫ is Euler’s constant. C.1. Fundamental Pinch Mode (n = 0 ; m = 0) in the low frequency limit In the low frequency limit, dispersion relation solutions for the fundamental axisymmetric pinch mode are obtained from equation (C1) γ2j γ ̟2j − κ2jv2Aj ̟2e − κ2ev2Ae ) + π ǫ− iπ Here we have the trivial solution ̟2j − κ2jv2Aj = 0 with v2w = v2Aj and the more interesting zeroth order solution ̟2j ≈ κ2j v2Aja v2msj with wave speed in the proper frame given by v2w = ̟ v2Aja v2msj . (C5) To first order this magnetosonic wave solution (eq. C3) can be written as ̟2j [1− δ] ≃ κ2ja2j v2msj , (C6) – 31 – where δ ≡ − ̟2e − κ2ev2Ae v2msj and δ is complex. Thus, in the low frequency limit this fundamental pinch mode (n = 0) solution consists of a growing and damped wave pair with wave speed in the observer frame uj ± vw 1± vwuj/c2 where v2w ≃ a2j v2msj v2msj . (C9) Previous work has shown the unstable growing solution associated with the backwards moving (in the jet fluid reference frame) wave. C.2. Surface Modes (n > 0 ; m = 0) in the low frequency limit In the low frequency limit the fundamental dispersion relation solution for all higher order modes (n > 0) is most easily obtained from equation (C2) written in the form γ2jWj (ω − kuj)2 − V 2Aj k2 − ω2/c2 = −γ2eWe (ω − kue)2 − V 2Ae k2 − ω2/c2 (C10) where I have used χ ≡ γ20γ2AW0 ̟2 − κ2v2A = γ20W0 (ω − ku)2 − k2 − ω2/c2 V 2A/γ . The solu- tion can be put in the form [ηuj + ue]± iη1/2 (uj − ue)2 − V 2As/γ2j γ2e (1 + V 2Ae/γ 2) + η(1 + V 2Aj/γ (C11) where γ2jWj γ2eWe and a “surface” Alfvén speed is defined by V 2As ≡ γ2AjWj + γ ) B2j +B 4πWjWe The jet is stable to higher order n > 0 fundamental mode perturbations when γ2j γ e (uj − ue) < γ2Ajγ Ae +We/γ ) B2j +B 4πWjWe . (C12) Equation (C11) reduces to the relativistic fluid expression ηuj + ue 1 + η 1 + η (uj − ue) (C13a) – 32 – given in Hardee & Hughes (2003) equation (6a) where for pressure balance and equal adiabatic index in jet and external medium η −→ γ2j ae/γ2eaj . Similarly equation (C11) reduces to the non-relativistic MHD form ηuj + ux 1 + η 1 + η (uj − ue)2 − V 2As (C13b) given by Hardee & Rosen (2002) eq. (4) where V 2As −→ (ρj + ρe) B2j +B / (4πρjρe) and η −→ ρj/ρe. C.3. Body Modes (n ≥ 0 ; m ≥ 1) in the low frequency limit In the low frequency limit the real part of the body wave solutions can be obtained in the limit ω = 0, k 6= 0 where the dispersion relation can be written in the form cos [βjR− (2n + 1)π/4] ≈ ǫn ≡ n(βjR) n (βeR) n (βeR) . (C14) Here I have assumed that the large argument form Jn(βjR) ≈ (2/πβjR)1/2 cos [βjR− (2n + 1)π/4] applies. In the absence of a magnetic field and a flow surrounding the jet, χe = 0, ǫn = 0, and solutions are found from βjR − (2n + 1)π/4 = ±mπ ± π/2, where m is an integer. Provided ǫn << π/2 and θ ≈ cos−1 ǫn ≈ ± (π/2− ǫn), solutions can be found from βjR − (2n + 1)π/4 = ± [mπ + (π/2 ± ǫn)], where for ±ǫn the plus or minus sign is for m odd or even, respectively. In the limit ω = 0 βjR ≈ γ2j (u j − a2j)(u2j − v2Aj) v2msju j − v2Aja2j kR , (C15) and the solutions are given by kR ≈ kminnmR ≡ v2msju j − v2Aja2j γ2j (u j − a2j )(u2j − v2Aj) × [(n+ 2m− 1/2)π/2 + (−1)mǫn] (C16) where I have set m −→ m+1 to be consistent with previous notation so m = 1 corresponds to the first body mode. In the limit a2j >> v Aj equation (C16) reduces to the relativistic purely fluid form found in Hardee & Hughes (2003) kR ≈ kminnmR ≡ [(n+ 2m− 1/2)π/2 + (−1)mǫn] M2j − 1 (C17a) where M2j = u j . In the limit v Aj >> a j equation (C16) becomes kR ≈ kminnmR ≡ [(n+ 2m− 1/2)π/2 + (−1)mǫn] M2Aj − 1 (C17b) where M2Aj = u – 33 – Equation (C16) reduces to the non-relativistic MHD form found in Hardee & Rosen (2002) kR ≈ kminnmR ≡ 1−M2ms/M2AjM2j × [(n+ 2m− 1/2)π/2 + (−1)mǫn] (C17c) where M2ms = u j + v Aj) and I have used γ2j (u j − a2j)(u2j − v2Aj) v2msju j − v2Aja2j = γ2j a2+v2 1−M2ms/M2AjM2j a2+v2 We note here that there is an error in equations (5) in previous articles in the treatment of the sign on ǫn for even values of m. C.4. The Resonance Condition The resonance conditions are found by evaluating the transmittance, T , and reflectance, R, of waves at the jet boundary where T = 1 +R. With the dispersion relation written as n(βjR) Jn(βjR) n (βeR) n (βeR) (C18) where Z = χ/β with Z = γ2γ2AW ̟2 − κ2v2A a2 + γ2Av ̟2 − γ2Aκ2v2Aa2 γ2γ2A (̟ 2 − κ2a2) ̟2 − κ2v2A (C19) the reflectance R = (Ze − Zj)/(Ze + Zj) . (C20) For a fluid containing no magnetic field Z is a quantity related to the acoustic normal impedance (Gill 1965). When Ze + Zj ≈ 0, R and T are large, and the reflected and transmitted waves have an amplitude much larger than the incident wave. C.4.1. The Fluid Limit (Alfvén speed ≪ sound speed) For the case of a pure fluid ζ2e + γ , (C21a) ζ2j + γ , (C21b) where χ/k2 = W ζ2 + γ2κ2/k2 a2 and ζ ≡ β/k. For non-relativistic flows where (u2/c2)(ω/ku) << 1, γ ≈ 1, and with adiabatic indicies Γj = Γe the reflectance (ζe − ζj)(ζeζj − 1) (ζe + ζj)(ζeζj + 1) (C22) – 34 – and a supersonic resonance (Miles 1957) occurs when βe + βj = k(ζe + ζj) = 0. This supersonic resonance corresponds to the maximum growth rate of the normal mode solutions to the dispersion relation. I now generalize the results in Hardee (2000) to include flow in the external medium relative to the source/observer frame. Here Ze + Zj = 0 becomes Γeζjχe + Γjζeχj = Γeζj + Γjζe = 0 . (C23) A necessary condition for resonance is ζj < 0 and ζe > 0, and on the real axis uj − aj 1− ujaj/c2 ue + ae 1 + ueae/c2 . (C24) It follows that the resonance only exists when uj − aj 1− ujaj/c2 ue + ae 1 + ueae/c2 (C25a) or equivalently uj − ue 1− ujue/c2 aj + ae 1 + ajae/c2 . (C25b) To find the resonant solution for the real part of the phase velocity I solve ζ2j = ε 2ζ2e where here I set ε ≡ (Γjγ2j̟2j/k2a2j )/ = 1 so that ζ2j = γ (ω/k − uj)2 = ζ2e = γ (ω/k − ue)2 . (C26) The resulting quadratic equation can be written in the form a2e/(γ sj)− a2j/(γ2j γ2se) )2 − 2 γ2j a euj/γ sj − γ2ea2jue/γ2se γ2j a u2j − a2j − γ2ea2j u2e − a2e = 0 , (C27) where I have used γ2j a − γ2ea2j a2e/(γ sj)− a2j/(γ2j γ2se) and γ2s ≡ 1− a2/c2 . The solutions to equation (C27) are given by γ2j a euj/γ sj − γ2ea2jue/γ2se a2e/(γ sj)− a2j/(γ2j γ2se) γseγsj u2j − 2ujue + u2e a2e/(γ sj)− a2j/(γ2j γ2se) ] (C28) with the resonant solution given by v∗w = (γseae)γjuj + (γsjaj)γeue γj(γseae) + γe(γsjaj) . (C29) – 35 – Inserting the resonant solution (eq. C29) into the expression for ε gives 0.695 ≤ ε2 = ≤ 1.44 where 2.78 ≤ Γ2γ4s ≤ 4. When aj = ae and Γj = Γe, ε2 = 1, and the resonant solution is exact. The small range on ε (0.83 ≤ ε ≤ 1.2) suggests that this solution remains relatively robust for unequal values of the sound speed and adiabatic index in the jet and external medium. In the absence of an external flow the resonant solution v∗w = (γseae)γjuj γj(γseae) + (γsjaj) M2j − β2 M2j − β2 + (M2e − β2) is equivalent to the form given in Hardee (2000). The resonant frequencies can be estimated using the large argument forms for the Bessel and Hankel functions. In this limit the dispersion relation becomes n(βjR)H n (βeR) Jn(βjR)H n (βeR) ≈ i tan(βjR− 2n+ 1 . (C30) From the dispersion relation with Ze + Zj ≈ 0, and (χj/βj)(βe/χe) = Zj/Ze ≈ βe/βj ≈ −1, tan[βjR− (2n + 1)π/4]Re ≈ 0 on the real axis. It follows that |βjR| ≈ |βeR| ≈ (2n+ 1)π/4 +mπ can be used to obtain an estimate for the resonant frequencies from |βeR| ≈ (2n + 1)π/4 + mπ, with result that the resonant frequencies are given by ω∗nmR (2n + 1)π/4 +mπ (1− ue/v∗w) 2 − (ae/v∗w − ueae/c2) . (C31) In the absence of external flow, ue = 0, and for uj >> ae and 1 >> (kae/ω) 2 this expression reduces to the form given in Hardee (2000). When γj(γseae) >> γe(γsjaj), the resonant wave speed becomes v w ≈ uj, ue/v∗w ≈ ue/uj and provided ue << uj and ae << uj , the resonant frequency increases with increasing ue/uj and ae/uj ω∗nmR (2n + 1)π/4 +mπ 1− 2ue/uj(1− a2e/c2)− (a2e − u2e)/u2j . (C32) In general, the resonant frequency ω∗nm −→ ∞ as (1− ue/v∗w) w − ueae/c2 )2 −→ 0. An equivalent condition for (1− ue/v∗w) w − ueae/c2 = 0 is uj − ue 1− ujue/c2 aj + ae 1 + ajae/c2 , (C33) and the resonance moves to higher frequencies with ω∗nm −→ ∞ when the“shear” speed drops below a “surface” sound speed. – 36 – The behavior of the growth rate at resonance also can be found using the large argument forms for the Bessel and Hankel functions. In this limit the reflectance can be written as (Ze − Zj) (Ze + Zj) Jn(βjR)H n (βeR)− J n(βjR)H n (βeR) Jn(βjR)H n (βeR) + J n(βjR)H n (βeR) ≈ exp[−2i(βjR− 2n+ 1 π)] , (C34) 2n+ 1 π ≈ i ln |R| − φ (C35) where R ≡ |R| eiφ. It follows that (βjR)I ≈ ln |R| (C36) and since typically at resonance, |ω − kRuj | /aj > ∣kR − ωuj/c2 ∣ I can approximate βj by βj ≡ βRj + iβIj ≈ γj (ω − kRuj) − ikI . (C37) It follows that (βjR)I ≈ −γj kIR , (C38) kIR ≈ − ln |R| . (C39) At resonance (Ze − Zj) (Ze + Zj) βj − βe(χj/χe) βj + βe(χj/χe) βj − βe βj + βe ≈ −2βe βj + βe (C40) |R| ≈ βj − βe βj + βe −2βRe βIj − βIe βIj + β (C41) where I have used βRj − βRe ≈ −2βRe from the resonance condition on the real axis. It follows |R| ≈ (ω−kRue) + k2I − γe ueae − γeae ω−kRue ω−kRue (C42) where I have used βe ≡ βRe + iβIe ≈ γe (ω − kRue) − ikI ω − kRue If I assume that γj(γseae) >> γe(γsjaj), with resonant wave speed ω/kR ≈ uj and ue/uj << 1 |R| ≈ ω∗nmR (1− 2ue/uj) + k2IR2 (1 + ue/uj) (1 + ue/uj) , (C43) – 37 – and since kIR ≈ − (aj/2γjuj) ln |R| |R| ≈ ω∗nmR (1− 2ue/uj) + [ln |R| /2]2 [ln |R| /2]2 . (C44) From equation (C32) ω∗nmR (1− 2ue/uj) ≈ (1− 2ue/uj) 1− 2 (ue/uj) (1− a2e/c2)− a2e/u2j ] [(2n+ 1)π/4 +mπ] and if say ue = 0, then |R|2 − 1 ln |R| ≈ 4 1− a2e/u2j [(2n + 1)π/4 +mπ] . (C45) Formally |R| −→ ∞ as ω∗nm −→ ∞ when the jet speed drops below the “surface” sound speed given by equation (C33). This result applies only to the surface modes and not to the body modes as, in the fluid limit, the body modes do not exist when the jet speed drops below the jet sound speed, see equation (C17). On the other hand, if say, a2e/u j << 1, then |R|2 − 1 ln |R| ≈ 4 [(2n + 1)π/4 +mπ] . (C46) Formally |R| ≈ constant as ω∗nm −→ ∞ when the wind speed becomes comparable to the jet speed, ue . uj , as must be the case for the velocity shear driven Kelvin-Helmholtz instability. C.4.2. The Magnetic Limit (Alfvén speed ≫ sound speed) For the magnetic limit in which magnetic pressure dominates gas pressure Ze = γeγ AeWevAe ̟2e − κ2ev2Ae , (C47a) Zj = γjγ AjWjvAj ̟2j − κ2jv2Aj , (C47b) A necessary condition for resonance is ̟2e − κ2ev2Ae > 0 and ̟2j − κ2jv2Aj < 0 on the real axis with result that Ze + Zj = 0 when uj − vAj 1− ujvAj/c2 ue + vAe 1 + uevAe/c2 . (C48) It follows that the resonance only exists when uj − ue 1− ujue/c2 vAj + vAe 1 + vAjvAe/c2 . (C49) This result is identical in form to the sonic case with sound speeds replaced by Alfvén wave speeds. – 38 – The resonant solution for the real part of the phase velocity is obtained from Z2j = γ ̟2j − k2 − ω2/c2 V 2Aj/γ = Z2e = γ ̟2e − k2 − ω2/c2 V 2Ae/γ (C50) where I have used γ2γ2A ̟2 − κ2v2A k2 − ω2/c2 V 2A/γ , and recall that v2A = V The resulting quadratic equation can be written in the form Aj − γ2eW 2e V 2Ae )2 − 2 Ajuj − γ2eW 2e V 2Aeue j − γ2eW 2e V 2Aeu2e (C51) where I have used k2 − ω2/c2 V 2Aj/γ j = γ k2 − ω2/c2 V 2Ae/γ because pressure balance in the magnetically dominated case requires WjV Aj = WeV Ae. The solutions are given by Ajuj − γ2eW 2e V 2Aeue ± γjγeWjWeVAjVAe (uj − ue) Aj − γ2eW 2e V 2Ae , (C52) and the resonant solution becomes v∗w = γjWjVAjuj + γeWeVAeue γjWjVAj + γeWeVAe (γAevAe) γjuj + (γAjvAj) γeue γj (γAevAe) + γe (γAjvAj) (C53) where I have used WVA = WV A/ (γAvA), and WjV Aj = WeV Ae. This resonant solution has the same form as the sonic case with sound speeds and sonic Lorentz factors replaced by Alfvén wave speeds and Alfvénic Lorentz factors. As in the sonic case the resonant frequencies are found from |βeR| ≈ (2n + 1)π/4 +mπ with result that the resonant frequencies are given by ω∗nmR (2n + 1)π/4 +mπ (1− ue/v∗w) 2 − (vAe/v∗w − uevAe/c2) . (C54) When γj (γAevAe) >> γe (γAjvAj) the resonant wave speed becomes v w ≈ uj and ue/v∗w ≈ ue/uj , and provided ue << uj and vAe << uj the resonant frequency increases with increasing ue/uj and vAe/uj as ω∗nmR ≈ (2n + 1)π/4 +mπ 1− 2 (ue/uj) (1− v2Ae/c2)− (v2Ae − u2e)/u2j . (C55) Here the resonant frequency ω∗nm −→ ∞ as (1− ue/v∗w) vAe/v w − uevAe/c2 )2 −→ 0. An equivalent condition for (1− ue/v∗w) vAe/v w − uevAe/c2 = 0 is uj − ue 1− ujue/c2 vAj + vAe 1 + vAjvAe/c2 , (C56) – 39 – and the resonance moves to higher frequencies as the “shear” speed becomes trans-Alfvénic. The behavior of the growth rate at resonance proceeds in the same manner as for the fluid limit but with sound speeds replaced by Alfvén wave speeds. The resonant growth rate is now given by kIR ≈ − ln |R| . (C57) If I assume that γj(γAevAe) >> γe(γAjvAj), with resonant wave speed ω/kR ≈ uj and ue/uj << 1 |R| ≈ ω∗nmR (1− 2ue/uj) + k2IR2 − vAe (1 + ue/uj) + vAe (1 + ue/uj) , (C58) From equation (C54) ω∗nmR (1− 2ue/uj) ≈ (1− 2ue/uj) 1− 2 (ue/uj) (1− v2Ae/c2)− (v2Ae − u2e)/u2j ] [(2n+ 1)π/4 +mπ] and if say ue = 0, then |R|2 − 1 ln |R| ≈ 4 1− v2Ae/u2j [(2n + 1)π/4 +mπ] (C59) and |R| increases as ω∗nm increases when the jet speed decreases. However, when the shear speed drops below the “surface” Alfvén speed, see equations (C11 & C12) the jet is stable. This result is quite different from the fluid limit where the jet remains unstable when the shear speed drops below the “surface” sound speed. If I insert uj − ue = 1− ujue/c2 1 + vAjvAe/c2 (vAj + vAe) . from equation (C56) into equation (C12), it follows that the jet will be unstable when resonance disappears only when γ2j γ 1− ujue/c2 > 2γ2Ajγ v2Ae + v (vAj + vAe) 1 + vAjvAe/c . (C60) where I have used v2Ae + v Ae +We/γ B2j +B / (4πWjWe) as Be = Bj from magnetic pressure balance. Formally |R| −→ ∞ as ω∗nm −→ ∞ only for jet Lorentz factors greatly in excess of the Alfvénic Lorentz factor. C.5. Wave modes at high frequency To obtain the behavior of wave modes at high frequency I begin with the dispersion relation written in the form Jn(βjR) n(βjR) n (βeR) n (βeR) Jn(βjR) ∓Jn±1(βjR)± nβjRJn(βjR) n−1(βeR)− n (βeR) n (βeR) (C61) – 40 – and assume a large argument in the Hankel function with H n (βeR) ≈ exp i [βeR− (2n+ 1) π/4] and a small argument βjR << 1 in the Bessel function to write J0(βjR) −J1(βjR) e−iπ/2 n = 0 e−iπ/2 n > 0 The small arguement form for the Bessel function gives J0(βjR)/J1(βjR) ≈ 2/βjR with result that the dispersion relation becomes βjR ≈ e−iπ/2 n = 0 e−iπ/2 n > 0 . (C62) At high frequency and large wavenumber χj and χe, are proportional to k 2, βj and βe are pro- portional to k, and βjR = ζjkR ∝ (kR)1/2 for n = 0. 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MATLAB codes for teaching quantum physics: Part 1 R. Garcia,∗ A. Zozulya, and J. Stickney Department of Physics, Worcester Polytechnic Institute, Worcester, MA 01609 (Dated: February 1, 2008) Among the ideas to be conveyed to students in an introductory quantum course, we have the pivotal idea championed by Dirac that functions correspond to column vectors (kets) and that differential operators correspond to matrices (ket-bras) acting on those vectors. The MATLAB (matrix-laboratory) programming environment is especially useful in conveying these concepts to students because it is geared towards the type of matrix manipulations useful in solving introductory quantum physics problems. In this article, we share MATLAB codes which have been developed at WPI, focusing on 1D problems, to be used in conjunction with Griffiths’ introductory text. Two key concepts underpinning quantum physics are the Schrodinger equation and the Born probability equa- tion. In 1930 Dirac introduced bra-ket notation for state vectors and operators.1 This notation emphasized and clarified the role of inner products and linear function spaces in these two equations and is fundamental to our modern understanding of quantum mechanics. The Schrodinger equation tells us how the state Ψ of a particle evolves in time. In bra-ket notation, it reads |Ψ〉 = H |Ψ〉 (1) where H is the Hamiltonian operator and |Ψ〉 is a ket or column vector representing the quantum state of the par- ticle. When a measurement of a physical quantity A is made on a particle initially in the state Ψ, the Born equa- tion provides a way to calculate the probability P (Ao) that a particular result Ao is obtained from the measure- ment. In bra-ket notation, it reads2 P (Ao) ∼ |〈Ao|Ψ〉|2 (2) where if |Ao〉 is the state vector corresponding to the particular result Ao having been measured, 〈Ao| = |Ao〉† is the corresponding bra or row vector and 〈Ao|Ψ〉 is thus the inner product between |Ao〉 and |Ψ〉. In the Dirac formalism, the correspondence between the wavefunction Ψ(~x) and the ket |Ψ〉 is set by the relation Ψ(~x) = 〈~x|Ψ〉, where |~x〉 is the state vector corresponding to the particle being located at ~x. Thus we regard Ψ(~x) as a component of a state vector |Ψ〉, just as we usually3 regard ai = ı̂·~a as a component of ~a along the direction ı̂. Similarly, we think of the Hamiltonian operator as a matrix d3~x |~x〉 Ψ(~x, t) + U(~x) 〈~x| (3) acting on the space of kets. While an expert will necessarily regard Eqs.(1-3) as a great simplification when thinking of the content of quan- tum physics, the novice often understandably reels under the weight of the immense abstraction. We learn much about student thinking from from the answers given by our best students. For example, we find a common error when studying 1D quantum mechanics is a student treat- ing Ψ(x) and |Ψ〉 interchangeably, ignoring the fact that the first is a scalar but the ket corresponds to a column vector. For example, they may write incorrectly 〈p||Ψ〉|x〉 = |Ψ〉〈p|x〉 (incorrect!) (4) or some similar abberation. To avoid these types of mis- conceptions, a number of educators and textbook authors have stressed incorporating a numerical calculation as- pect to quantum courses.4,5,6,7,8,9 The motive is simple. Anyone who has done numerical calculations can’t help but regard a ket |Ψ〉 as a column vector, a bra 〈Ψ| as a row vector and an operatorH as a matrix because that is how they concretely represented in the computer. Intro- ducing a computational aspect to the course provides one further benefit: it gives the beginning quantum student the sense that he or she is being empowered to solve real problems that may not have simple, analytic solutions. With these motivations in mind, we have developed MATLAB codes10 for solving typical 1 D problems found in the first part of a junior level quantum course based on Griffith’s book.11 We chose MATLAB for our pro- gramming environment because the MATLAB syntax is especially simple for the typical matrix operations used in 1D quantum mechanics problems and because of the ease of plotting functions. While some MATLAB numeri- cal recipes have previously been published by others,12,13 the exercises we share here are special because they em- phasize simplicity and quantum pedagogy, not numerical efficiency. Our point has been to provide exercises which show students how to numerically solve 1 D problems in such a way that emphasizes the column vector aspect of kets, the row vector aspect of bras and the matrix aspect of operators. Exercises using more efficient MATLAB ODE solvers or finite-element techniques are omitted be- cause they do not serve this immediate purpose. In part II of this article, we hope to share MATLAB codes which can be used in conjunction with teaching topics pertain- ing to angular momentum and non-commuting observ- ables. http://arXiv.org/abs/0704.1622v1 I. FUNCTIONS AS VECTORS To start students thinking of functions as column vector-like objects, it is very useful to introduce them to plotting and integrating functions in the MATLAB environment. Interestingly enough, the plot command in MATLAB takes vectors as its basic input element. As shown in Program 1 below, to plot a function f(x) in MATLAB, we first generate two vectors: a vector of x values and a vector of y values where y = f(x). The com- mand plot(x, y,′ r′) then generates a plot window con- taining the points (xi, yi) displayed as red points ( ′r′). Having specified both x and y, to evaluate the definite integral ydx, we need only sum all the y values and multiply by dx. %**************************************************************** % Program 1: Numerical Integration and Plotting using MATLAB %**************************************************************** N=1000000; % No. of points L=500; % Range of x: from -L to L x=linspace(-L,L,N)’; % Generate column vector with N % x values ranging from -L to L dx=x(2)-x(1); % Distance between adjacent points % Alternative Trial functions: % To select one, take out the comment command % at the beginning. %y=exp(-x.^2/16); % Gaussian centered at x=0 %y=((2/pi)^0.5)*exp(-2*(x-1).^2); % Normed Gaussian at x=1 %y=(1-x.^2).^-1; % Symmetric fcn which blows up at x=1 %y=(cos(pi*x)).^2; % Cosine fcn %y=exp(i*pi*x); % Complex exponential %y=sin(pi*x/L).*cos(pi*x/L);% Product of sinx times cosx %y=sin(pi*x/L).*sin(pi*x/L);% Product of sin(nx) times sin(mx) %A=100; y=sin(x*A)./(pi*x); % Rep. of delta fcn A=20; y=(sin(x*A).^2)./(pi*(A*x.^2));% Another rep. of delta fcn % Observe: numerically a function y(x) is represented by a vector! % Plot a vector/function plot(x,y); % Plots vector y vs. x %plot(x,real(y),’r’, x, imag(y), ’b’); % Plots real&imag y vs. x axis([-2 2 0 7]); % Optimized axis parameters for sinx^2/pix^2 %axis([-2 2 -8 40]); % Optimized axis parameters for sinx/pix % Numerical Integration sum(y)*dx % Simple numerical integral of y trapz(y)*dx % Integration using trapezoidal rule II. DIFFERENTIAL OPERATORS AS MATRICES Just as f(x) is represented by a column vector |f〉 in the computer, for numerical purposes a differential opera- tor D̂ acting on f(x) is reresented by a matrixD that acts on |f〉. As illustrated in Program 2, MATLAB provides many useful, intuitive, well-documented commands for generating matrices D that correspond to a given D̂.10 Two examples are the commands ones and diag. The command ones(a, b) generates an a × b matrix of ones. The command diag(A, n) generates a matrix with the el- ements of the vector A placed along the nth diagonal and zeros everywhere else. The central diagonal corresponds to n = 0, the diagonal above the center one corresponds to n = 1, etc...). An exercise we suggest is for students to verify that the derivative matrix is not Hermitian while the deriva- tive matrix times the imaginary number i is. This can be very valuable for promoting student understanding if done in conjunction with the proof usually given for the differential operator. %**************************************************************** % Program 2: Calculate first and second derivative numerically % showing how to write differential operator as a matrix %**************************************************************** % Parameters for solving problem in the interval 0 < x < L L = 2*pi; % Interval Length N = 100; % No. of coordinate points x = linspace(0,L,N)’; % Coordinate vector dx = x(2) - x(1); % Coordinate step % Two-point finite-difference representation of Derivative D=(diag(ones((N-1),1),1)-diag(ones((N-1),1),-1))/(2*dx); % Next modify D so that it is consistent with f(0) = f(L) = 0 D(1,1) = 0; D(1,2) = 0; D(2,1) = 0; % So that f(0) = 0 D(N,N-1) = 0; D(N-1,N) = 0; D(N,N) = 0; % So that f(L) = 0 % Three-point finite-difference representation of Laplacian Lap = (-2*diag(ones(N,1),0) + diag(ones((N-1),1),1) ... + diag(ones((N-1),1),-1))/(dx^2); % Next modify Lap so that it is consistent with f(0) = f(L) = 0 Lap(1,1) = 0; Lap(1,2) = 0; Lap(2,1) = 0; % So that f(0) = 0 Lap(N,N-1) = 0; Lap(N-1,N) = 0; Lap(N,N) = 0;% So that f(L) = 0 % To verify that D*f corresponds to taking the derivative of f % and Lap*f corresponds to taking a second derviative of f, % define f = sin(x) or choose your own f f = sin(x); % And try the following: Df = D*f; Lapf = Lap*f; plot(x,f,’b’,x,Df,’r’, x,Lapf,’g’); axis([0 5 -1.1 1.1]); % Optimized axis parameters % To display the matrix D on screen, simply type D and return ... D % Displays the matrix D in the workspace Lap % Displays the matrix Lap III. INFINITE SQUARE WELL When solving Eq. (1), the method of separation of vari- ables entails that as an intermediate step we look for the separable solutions |ΨE(t)〉 = |ΨE(0)〉exp(−iEt/~) (5) where |ΨE(0)〉 satisfies the time-independent Schrodinger equation H |ΨE(0)〉 = E |ΨE(0)〉. (6) In solving Eq. (6) we are solving for the eigenvalues E and eigenvectors |ΨE(0)〉 of H . In MATLAB, the com- mand [V,E] = eig(H) does precisely this: it generates two matrices. The first matrix V has as its columns the eigenvectors |ΨE(0)〉. The second matrix E is a diago- nal matrix with the eigenvalues Ei corresponding to the eigenvectors |ΨEi(0)〉 placed along the central diagonal. We can use the command E = diag(E) to convert this matrix into a column vector. In Program 3, we solve for the eigenfunctions and eigenvalues for the infinite square well Hamiltonian. For brevity, we omit the commands setting the parameters L,N, x, and dx. %**************************************************************** % Program 3: Matrix representation of differential operators, % Solving for Eigenvectors & Eigenvalues of Infinite Square Well %**************************************************************** % For brevity we omit the commands setting the parameters L, N, % x and dx; We also omit the commands defining the matrix Lap. % These would be the same as in Program 2 above. % Total Hamiltonian where hbar=1 and m=1 hbar = 1; m = 1; H = -(1/2)*(hbar^2/m)*Lap; % Solve for eigenvector matrix V and eigenvalue matrix E of H [V,E] = eig(H); % Plot lowest 3 eigenfunctions plot(x,V(:,3),’r’,x,V(:,4),’b’,x,V(:,5),’k’); shg; E % display eigenvalue matrix diag(E) % display a vector containing the eigenvalues Note that in the MATLAB syntax the object V (:, 3) specifies the column vector consisting of all the elements in column 3 of matrix V . Similarly V (2, :) is the row vector consisting of all elements in row 2 of V ; V (3, 1) is the element at row 3, column 1 of V ; and V (2, 1 : 3) is a row vector consisting of elements V (2, 1), V (2, 2) and V (2, 3). IV. ARBITRARY POTENTIALS Numerical solution of Eq. (1) is not limited to any par- ticular potential. Program 4 gives example MATLAB codes solving the time independent Schrodinger equa- tion for finite square well potentials, the harmonic os- cillator potential and even for potentials that can only solved numerically such as the quartic potential U = x4. In order to minimize the amount of RAM required, the codes shown make use of sparse matrices, where only the non-zero elements of the matrices are stored. The commands for sparse matrices are very similar to those for non-sparse matrices. For example, the command [V,E] = eigs(H,nmodes..) provides the nmodes lowest energy eigenvectors V of of the sparse matrix H . Fig. 1 shows the plot obtained from Program 4 for the potential U = 1 ·100 ·x2. Note that the 3 lowest energies displayed in the figure are just as expected due to the analytic formula E = ~ω with n = integer and ω = = 10 rad/s. V. A NOTE ON UNITS IN OUR PROGRAMS When doing numerical calculations, it is important to minimize the effect of rounding errors by choosing units such that the variables used in the simulation are of the %**************************************************************** % Program 4: Find several lowest eigenmodes V(x) and % eigenenergies E of 1D Schrodinger equation % -1/2*hbar^2/m(d2/dx2)V(x) + U(x)V(x) = EV(x) % for arbitrary potentials U(x) %**************************************************************** % Parameters for solving problem in the interval -L < x < L % PARAMETERS: L = 5; % Interval Length N = 1000; % No of points x = linspace(-L,L,N)’; % Coordinate vector dx = x(2) - x(1); % Coordinate step % POTENTIAL, choose one or make your own U = 1/2*100*x.^(2); % quadratic harmonic oscillator potential %U = 1/2*x.^(4); % quartic potential % Finite square well of width 2w and depth given %w = L/50; %U = -500*(heaviside(x+w)-heaviside(x-w)); % Two finite square wells of width 2w and distance 2a apart %w = L/50; a=3*w; %U = -200*(heaviside(x+w-a) - heaviside(x-w-a) ... % + heaviside(x+w+a) - heaviside(x-w+a)); % Three-point finite-difference representation of Laplacian % using sparse matrices, where you save memory by only % storing non-zero matrix elements e = ones(N,1); Lap = spdiags([e -2*e e],[-1 0 1],N,N)/dx^2; % Total Hamiltonian hbar = 1; m = 1; % constants for Hamiltonian H = -1/2*(hbar^2/m)*Lap + spdiags(U,0,N,N); % Find lowest nmodes eigenvectors and eigenvalues of sparse matrix nmodes = 3; options.disp = 0; [V,E] = eigs(H,nmodes,’sa’,options); % find eigs [E,ind] = sort(diag(E));% convert E to vector and sort low to high V = V(:,ind); % rearrange corresponding eigenvectors % Generate plot of lowest energy eigenvectors V(x) and U(x) Usc = U*max(abs(V(:)))/max(abs(U)); % rescale U for plotting plot(x,V,x,Usc,’--k’); % plot V(x) and rescaled U(x) % Add legend showing Energy of plotted V(x) lgnd_str = [repmat(’E = ’,nmodes,1),num2str(E)]; legend(lgnd_str) % place lengend string on plot order of unity. In the programs presented here, our fo- cus being undergraduate physics students, we wanted to avoid unnecessarily complicating matters. To make the equations more familiar to the students, we explicitly left constants such as ~ in the formulas and chose units such that ~ = 1 and m = 1. We recognize that others may have other opinions on how to address this issue. An al- ternative approach used in research is to recast the equa- tions in terms of dimensionless variables, for example by rescaling the energy to make it dimensionless by express- ing it in terms of some characteristic energy in the prob- lem. In a more advanced course for graduate students or in a course in numerical methods, such is an approach which would be preferable. VI. TIME DEPENDENT PHENOMENA The separable solutions |ΨE(t)〉 are only a subset of all possible solutions of Eq. (1). Fortunately, they are complete set so that we can construct the general solution −1.5 −1 −0.5 0 0.5 1 1.5 −0.05 x [m] E = 4.99969 E = 14.9984 E = 24.9959 FIG. 1: Output of Program 4, which plots the energy eigen- functions V (x) and a scaled version of the potential U(x) = 1/2 · 100 · x2. The corresponding energies displayed within the figure legend, 4.99969, 14.9984 and 24.9959, are, within rounding error, precisely those expected from Eq. (7) for the three lowest-energy modes. via the linear superposition |Ψ(t)〉 = aE |ΨE(0)〉exp(−iEt/~) (8) where aE are constants and the sum is over all possible values of E. The important difference between the sep- arable solutions (5) and the general solution (8) is that the probability densities derived from the general solu- tions are time-dependent whereas those derived from the separable solutions are not. A very apt demonstration of this is provided in the Program 5 which calculates the time-dependent probability density ρ(x, t) for a particle trapped in a a pair of finite-square wells whose initial state |Ψ(0)〉 is set equal to the the equally-weighted su- perposition |Ψ(0)〉 = 1√ (|ΨE0〉 + |ΨE1〉) (9) of the ground state |ΨE0〉 and first excited state |ΨE1〉 of the double well system. As snapshots of the program output show in Fig. 2, the particle is initially completely localized in the rightmost well. However, due to E0 6= E1, the probability density ρ(x, t) = [ |ψE0(x)|2 + |ψE1(x)|2 + 2|ψE0(x)||ψE1 (x)|cos2 ((E1 − E2)t/~) ] (10) is time-dependent, oscillating between the ρ(x) that cor- responds to the particle being entirely in the right well ρ(x) = | |ψE0(x)| + |ψE1(x)| | and ρ(x) for the particle being entirely in the left well ρ(x) = | |ψE0(x)| − |ψE1(x)| | . (12) By observing the period with which ρ(x, t) oscillates in the simulation output shown in Fig. 2 students can verify that it is the same as the period of oscillation 2π~/(E1 − E2) expected from Eq. (10). %**************************************************************** % Program 5: Calculate Probability Density as a function of time % for a particle trapped in a double-well potential %**************************************************************** % Potential due to two square wells of width 2w % and a distance 2a apart w = L/50; a = 3*w; U = -100*( heaviside(x+w-a) - heaviside(x-w-a) ... + heaviside(x+w+a) - heaviside(x-w+a)); % Finite-difference representation of Laplacian and Hamiltonian, % where hbar = m = 1. e = ones(N,1); Lap = spdiags([e -2*e e],[-1 0 1],N,N)/dx^2; H = -(1/2)*Lap + spdiags(U,0,N,N); % Find and sort lowest nmodes eigenvectors and eigenvalues of H nmodes = 2; options.disp = 0; [V,E] = eigs(H,nmodes,’sa’,options); [E,ind] = sort(diag(E));% convert E to vector and sort low to high V = V(:,ind); % rearrange coresponding eigenvectors % Rescale eigenvectors so that they are always % positive at the center of the right well for c = 1:nmodes V(:,c) = V(:,c)/sign(V((3*N/4),c)); %**************************************************************** % Compute and display normalized prob. density rho(x,T) %**************************************************************** % Parameters for solving the problem in the interval 0 < T < TF TF = 10; % Length of time interval NT = 100; % No. of time points T = linspace(0,TF,NT); % Time vector % Compute probability normalisation constant (at T=0) psi_o = 0.5*V(:,1)+0.5*V(:,2); % wavefunction at T=0 sq_norm = psi_o’*psi_o*dx; % square norm = |<ff|ff>|^2 Usc = U*max(abs(V(:)))/max(abs(U)); % rescale U for plotting % Compute and display rho(x,T) for each time T for t=1:NT; % time index parameter for stepping through loop % Compute wavefunction psi(x,T) and rho(x,T) at T=T(t) psi = 0.5*V(:,1)*exp(-i*E(1)*T(t)) ... + 0.5*V(:,2)*exp(-i*E(2)*T(t)); rho = conj(psi).*psi/sq_norm; % normalized probability density % Plot rho(x,T) and rescaled potential energy Usc plot(x,rho,’o-k’, x, Usc,’.-b’); axis([-L/8 L/8 -1 6]); lgnd_str = [repmat(’T = ’,1,1),num2str(T)]; text(-0.12,5.5,lgnd_str, ’FontSize’, 18); xlabel(’x [m]’, ’FontSize’, 24); ylabel(’probability density [1/m]’,’FontSize’, 24); pause(0.05); % wait 0.05 seconds −0.6 −0.4 −0.2 0 0.2 0.4 0.6 T = 0s x [m] −0.6 −0.4 −0.2 0 0.2 0.4 0.6 T = 2.8189s x [m] −0.6 −0.4 −0.2 0 0.2 0.4 0.6 T = 5.6378s x [m] −0.6 −0.4 −0.2 0 0.2 0.4 0.6 T = 8.4566s x [m] −0.6 −0.4 −0.2 0 0.2 0.4 0.6 T = 11.2755s x [m] FIG. 2: The probability density ρ(x, t) output from Program 5 for a particle trapped in a pair of finite-height square well potentials that are closely adjacent. The initial state of the particle is chosen to be |Ψ(0)〉 ∼ |E0〉+ |E1〉. Shown is ρ(x, t) plotted for times T = {0, 0.25, 0.5, 0.75, 1.0}×2π~/(E1 − E2). VII. WAVEPACKETS AND STEP POTENTIALS Wavepackets are another time-dependent phenomenon encountered in undergraduate quantum mechanics for which numerical solution techniques have been typically advocated in the hopes of promoting intuitive acceptance and understanding of approximations necessarily invoked in more formal, analytic treatments. Program 6 calcu- lates and displays the time evolution of a wavepacket for one of two possible potentials, either U = 0 or a step potential U = UoΘ(x − L). The initial wavepacket is generated as the Fast Fourier Transform of a Gaussian momentum distribution centered on a particular value of the wavevector ko. Because the wavepacket is com- posed of a distribution of different ks, different parts of the wavepacket move with different speeds, which leads to the wave packet spreading out in space as it moves. While there is a distribution of velocities within the wavepacket, two velocities in particular are useful in char- acterizing it. The phase velocity vw = ω/k = E/p = ~ko/2m is the velocity of the plane wave component which has wavevector ko. The group velocity vg = ~ko/m is the velocity with which the expectation value < x > moves and is the same as the classical particle velocity as- sociated with the momentum p = ~k. Choosing U = 0, students can modify this program to plot < x > vs t. They can extract the group velocity from their numerical simulation and observe that indeed vg = 2vw for a typ- ical wave packet. Students can also observe that, while vg matches the particle speed from classical mechanics, the wavepacket spreads out as time elapses. In Program 6, we propagate the wave function forward via the formal solution |Ψ(t)〉 = exp(−iHt/~)|Ψ(0)〉, (13) where the Hamiltonian matrix H is in the exponential. This solution is equivalent to Eq. (6), as as can be shown by simple substitution. Moreover, MATLAB has no trou- ble exponentiating the matrix that numerically repre- senting the Hamiltonian operator as long as the matrix is small enough to fit in the available computer memory. Even more interestingly, students can use this method to investigate scattering of wavepackets from various po- tentials, including the step potential U = UoΘ(x−L/2). In Fig. 3, we show the results of what happens as the wavepacket impinges on the potential barrier. The pa- rameters characterizing the initial wavepacket have been deliberately chosen so that the wings do not fall outside the simulation area and initially also do not overlap the barrier on the right. If 〈E〉 ≪ Uo, the wavepacket is completely reflected from the barrier. If 〈E〉 ≈ Uo, a portion of the wave is is reflected and a portion is trans- mitted through. If 〈E〉 ≫ Uo, almost all of the wave is transmitted. In Fig. 4 we compare the reflection probability R cal- culated numerically using Program 6 with R calculated by averaging the single-mode11 expression R(E) = E − Uo√ E − Uo over the distribution of energies in the initial wavepacket. While the numerically and analytically estimated R are found to agree for large and small 〈E〉/Uo, there is a noticeable discrepancy due to the shortcomings of the %**************************************************************** % Program 6: Wavepacket propagation using exponential of H %**************************************************************** % Parameters for solving the problem in the interval 0 < x < L L = 100; % Interval Length N = 400; % No of points x = linspace(0,L,N)’; % Coordinate vector dx = x(2) - x(1); % Coordinate step % Parameters for making intial momentum space wavefunction phi(k) ko = 2; % Peak momentum a = 20; % Momentum width parameter dk = 2*pi/L; % Momentum step km=N*dk; % Momentum limit k=linspace(0,+km,N)’; % Momentum vector % Make psi(x,0) from Gaussian kspace wavefunction phi(k) using % fast fourier transform : phi = exp(-a*(k-ko).^2).*exp(-i*6*k.^2); % unnormalized phi(k) psi = ifft(phi); % multiplies phi by expikx and integrates vs. x psi = psi/sqrt(psi’*psi*dx); % normalize the psi(x,0) % Expectation value of energy; e.g. for the parameters % chosen above <E> = 2.062. avgE = phi’*0.5*diag(k.^2,0)*phi*dk/(phi’*phi*dk); % CHOOSE POTENTIAL U(X): Either U = 0 OR % U = step potential of height avgE that is located at x=L/2 %U = 0*heaviside(x-(L/2)); % free particle wave packet evolution U = avgE*heaviside(x-(L/2)); % scattering off step potential % Finite-difference representation of Laplacian and Hamiltonian e = ones(N,1); Lap = spdiags([e -2*e e],[-1 0 1],N,N)/dx^2; H = -(1/2)*Lap + spdiags(U,0,N,N); % Parameters for computing psi(x,T) at different times 0 < T < TF NT = 200; % No. of time steps TF = 29; T = linspace(0,TF,NT); % Time vector dT = T(2)-T(1); % Time step hbar = 1; % Time displacement operator E=exp(-iHdT/hbar) E = expm(-i*full(H)*dT/hbar); % time desplacement operator %*************************************************************** % Simulate rho(x,T) and plot for each T %*************************************************************** for t = 1:NT; % time index for loop % calculate probability density rho(x,T) psi = E*psi; % calculate new psi from old psi rho = conj(psi).*psi; % rho(x,T) plot(x,rho,’k’); % plot rho(x,T) vs. x axis([0 L 0 0.15]); % set x,y axis parameters for plotting xlabel(’x [m]’, ’FontSize’, 24); ylabel(’probability density [1/m]’,’FontSize’, 24); pause(0.05); % pause between each frame displayed % Calculate Reflection probability for a=1:N/2; R=R+rho(a); R=R*dx numerical simulation for 〈E〉/Uo ≈ 1. This discrepancy can be reduced significantly by increasing the number of points in the simulation to 1600 but only at the cost of significantly slowing down the speed of the computation. For our purposes, the importance comparing the analyt- ical and numerical calculations is that it gives student a baseline from which to form an opinion or intuition regarding the accuracy of Eq. (14). 0 10 20 30 40 50 60 70 80 90 100 x [m] T = 0s 0 10 20 30 40 50 60 70 80 90 100 x [m] T = 14.4271s 0 10 20 30 40 50 60 70 80 90 100 x [m] T = 28.8543s FIG. 3: Output of Program 6 showing a wavepacket encoun- tering step potential of height ∼ 〈E〉 located at x/L = 0.5 at different times. VIII. CONCLUSIONS One benefit of incorporating numerical simulation into the teaching of quantum mechanics, as we have men- tioned, is the development of student intuition. Another is showing students that non-ideal, real-world problems can be solved using the concepts they learn in the class- room. However, our experimentation incorporating these simulations in quantum physics at WPI during the past year has shown us that the most important benefit is a type of side-effect to doing numerical simulation: the ac- ceptance on an intuitive level by the student that func- tions are like vectors and differential operators are like matrices. While in the present paper, we have only had sufficient space to present the most illustrative MATLAB ��� ��� ��� ��� ��� ��� �� ���� �� ��� ��� ��� ������������ ������������� �!��"�# ����������� FIG. 4: The reflection probability R vs. 〈E〉/Uo. The dashed line is simply Eq. (14) where we subsitute E = 〈E〉, the solid line is Eq. (14) averaged over the energy distribution in the incident wavepacket, and the points are numerical results ob- tained using Program 6, where the horizontal distance be- tween points is σE/Uo where σE is the standard deviation of the energy distribution in the wavepacket. codes, our goal is to eventualy make available a more complete set of polished codes is available for download- ing either from the authors or directly from the file ex- change at MATLAB Central.14 ∗ Electronic address: garcia@wpi.edu 1 P. A. M. Dirac, The Principles of Quantum Mechanics, 1st ed., (Oxford University Press, 1930). 2 Born’s law is stated as a proportionality because an addi- tional factor is necessary depending on the units of |Ψ〉 3 C. C. Silva and R. de Andrade Martins, “Polar and ax- ial vectors versus quaternions,” Am. J. Phys. 70, 958-963 (2002). 4 R. W. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples, (Oxford Uni- versity Press, 1997). 5 H. Gould, “Computational physics and the undergraduate curriculum,” Comput. Phys. Commun. 127, 610 (2000); J. Tobochnik and H. Gould, “Teaching computational physics to undergraduates,” in Ann. Rev. Compu. Phys. IX, edited by D. Stauffer (World Scientific, Singapore, 2001), p. 275; H. Gould, J. Tobochnik, W. Christian, An Introduction to Computer Simulation Methods: Applica- tions to Physical Systems, (Benjamin Cummings, Upper Saddle River, NJ, 2006) 3rd ed.. 6 R. Spenser, “Teaching computational physics as a labora- tory sequence,” Am. J. Phys. 73, 151-153 (2005). 7 D. Styer, “Common misconceptions regarding quantum mechanics,” Am. J. Phys. 64, 31-34 (1996). 8 A. Goldberg, H. M. Schey, J. L. Schwartz, “Computer- Generated Motion Pictures of One-Dimensional Quantum- Mechanical Transmission and Reflection Phenomena,” Am. J. Phys. 35, 177-186 (1967). 9 C. Singh, M. Belloni, and W. Christian, “Improving stu- dents’ understanding of quantum mechanics,” Physics To- day, August 2006, p. 43. 10 These MATLAB commands are explained in an exten- sive on-line, tutorial within MATLAB and which is also independently available on the MATHWORKS website, http://www.mathworks.com/. 11 D. J. Griffiths, Introduction to Quantum Mechanics, 2nd Ed., Prentice Hall 2003. 12 A. Garcia, Numerical Methods for Physics, 2nd Ed., (Pren- tice Hall, 1994). 13 G. Lindblad, “Quantum Mechanics with MATLAB,” available on internet, http://mathphys.physics.kth.se/schrodinger.html . 14 See the user file exchange at http://www.mathworks.com/matlabcentral/. mailto:garcia@wpi.edu http://www.mathworks.com/ http://mathphys.physics.kth.se/schrodinger.html http://www.mathworks.com/matlabcentral/

Among the ideas to be conveyed to students in an introductory quantum course, we have the pivotal idea championed by Dirac that functions correspond to column vectors (kets) and that differential operators correspond to matrices (ket-bras) acting on those vectors. The MATLAB (matrix-laboratory) programming environment is especially useful in conveying these concepts to students because it is geared towards the type of matrix manipulations useful in solving introductory quantum physics problems. In this article, we share MATLAB codes which have been developed at WPI, focusing on 1D problems, to be used in conjunction with Griffiths' introductory text.

Introduction to Computer Simulation Methods: Applica- tions to Physical Systems, (Benjamin Cummings, Upper Saddle River, NJ, 2006) 3rd ed.. 6 R. Spenser, “Teaching computational physics as a labora- tory sequence,” Am. J. Phys. 73, 151-153 (2005). 7 D. Styer, “Common misconceptions regarding quantum mechanics,” Am. J. Phys. 64, 31-34 (1996). 8 A. Goldberg, H. M. Schey, J. L. Schwartz, “Computer- Generated Motion Pictures of One-Dimensional Quantum- Mechanical Transmission and Reflection Phenomena,” Am. J. Phys. 35, 177-186 (1967). 9 C. Singh, M. Belloni, and W. Christian, “Improving stu- dents’ understanding of quantum mechanics,” Physics To- day, August 2006, p. 43. 10 These MATLAB commands are explained in an exten- sive on-line, tutorial within MATLAB and which is also independently available on the MATHWORKS website, http://www.mathworks.com/. 11 D. J. Griffiths, Introduction to Quantum Mechanics, 2nd Ed., Prentice Hall 2003. 12 A. Garcia, Numerical Methods for Physics, 2nd Ed., (Pren- tice Hall, 1994). 13 G. Lindblad, “Quantum Mechanics with MATLAB,” available on internet, http://mathphys.physics.kth.se/schrodinger.html . 14 See the user file exchange at http://www.mathworks.com/matlabcentral/. mailto:garcia@wpi.edu http://www.mathworks.com/ http://mathphys.physics.kth.se/schrodinger.html http://www.mathworks.com/matlabcentral/

07041623rev_mod arXiv:cond-mat/0704.1623 Lecture Notes in Nanoscale Science and Technology, Vol. 2, Nanoscale Phenomena: Basic Science to Device Applications, Eds. Z.K. Tang and P.Sheng, Springer (2008) Nanodevices and Maxwell’s Demon Supriyo Datta School of Electrical and Computer Engineering, NSF Network for Computational Nanotechnology, Purdue Universtiy, West Lafayette, IN-47906, USA. (Date: April 12, 2007) In the last twenty years there has been significant progress in our understanding of quantum transport far from equilibrium and a conceptual framework has emerged through a combination of the Landauer approach with the non-equilibrium Green function (NEGF) method, which is now being widely used in the analysis and design of nanoscale devices. It provides a unified description for all kinds of devices from molecular conductors to carbon nanotubes to silicon transistors covering different transport regimes from the ballistic to the diffusive limit. In this talk I use a simple version of this model to analyze a specially designed device that could be called an electronic Maxwell’s demon, one that lets electrons go preferentially in one direction over another. My objective is to illustrate the fundamental role of “contacts” and “demons” in transport and energy conversion. The discussion is kept at an academic level steering clear of real world details, but the illustrative devices we use are very much within the capabilities of present-day technology. For example, recent experiments on thermoelectric effects in molecular conductors agree well with the predictions from our model. The Maxwell’s demon device itself is very similar to the pentalayer spin-torque device which has been studied by a number of groups though we are not aware of any discussion of the possibility of using the device as a nanoscale heat engine or as a refrigerator as proposed here. However, my objective is not to evaluate possible practical applications. Rather it is to introduce a simple transparent model showing how out-of-equibrium demons suitably incorporated into nanodevices can achieve energy conversion. 1. INTRODUCTION Maxwell invented his famous demon to illustrate the subtleties of the second law of thermodynamics and his conjecture has inspired much discussion ever since [see for example, Leff and Rex,1990,2003, Nikulov and Sheehan, 2004]. When the subject of thermodynamics was relatively new, it was not clear that heat was a form of energy since heat could never be converted entirely into useful work. Indeed the second law asserts that none of it can be converted to work if it is all available at a single temperature. Heat engines can only function by operating between two reservoirs at two different temperatures. Maxwell’s demon is supposed to get around this fundamental principle by creating a temperature differential between two sides of a reservoir that is initially at a uniform temperature. This is achieved by opening and closing a little door separating the two sides at just the right times to allow fast molecules (white) to cross to the left but not the slow molecules (black, Fig.1a). As a result, faster molecules crowd onto the left side making its temperature higher than that of the right side. Technology has now reached a point where one can think of building an electronic Maxwell’s demon that can be interposed between the two contacts (labeled source and drain) of a nanoscale conductor (Fig.1b) to allow electrons to flow preferentially in one direction so that a current will flow in the external circuit even without any external source of power. Such a device can be built (indeed one could argue has already been built) though not surprisingly it is expected to operate in conformity with the second law. Here I would like to use this device simply to illustrate the fundamental role of “contacts” and “demons” in transport and energy conversion. I will try to keep the discussion at an academic level steering clear of real world details. But it should be noted that the illustrative devices we will discuss are very much within the capabilities of present-day technology. For example, recent pioneering experiments on thermoelectric effects in molecular conductors [Reddy et.al.2007] seem to agree well with the predictions from our model [Paulsson and Datta,2003]. The device discussed in Section 3 (Figs.5,6) incorporating an elastic Maxwell’s demon (Datta 2005b,c) is being investigated experimentally by introducing manganese impurities into a GaAs spin-valve device with MnAs contacts [Saha et.al.2006]. The device incorporating an inelastic demon described in Section 4 (Figs.9,10) is very similar to the pentalayer spin-torque device which has been studied by a number of groups both experimentally [Fuchs et.al.2006, Nanodevices and Maxwell’s demon Supriyo Datta (a) Maxwell’s demon (b) Electronic demon Fig.1. (a) Maxwell’s demon opens and closes a trapdoor to separate fast (white) molecules from slow (black) molecules making the left warmer than the right, thus creating a temperature differential without expending energy . (b) Electronic Maxwell’s demon discussed in this talk lets electrons go preferentially from right to left thus creating a current in the external circuit without any external source of energy. Fig.2. Schematic representing the general approach used to model nanoscale devices: (a) Simple version with numbers γ , D used in this talk and (b) Complete version with matrices Σ ,H (Adapted from Datta2005a). Channel Source Drain <---- L ----> Σ1 Σ2 µ1 µ2 µ1 µ2 1γ 2γ D(E) 3 arXiv:cond-mat/0704.1623 datta@purdue.edu Huai et.al.2004] and theoretically [Salahuddin and Datta,2006a,b] though we are not aware of any discussion of the possibility of using the device as a nanoscale heat engine or as a refrigerator as proposed here. We leave it to future work to assess whether these possibilities are of any practical importance. Here my objective is to lay out a simple transparent model showing how out-of-equilibrium demons suitably incorporated into nanodevices can achieve energy conversion. In the last twenty years there has been significant progress in our ability to tackle the problem of quantum transport far from equilibrium and a conceptual framework has emerged through a combination of the Landauer approach with the non-equilibrium Green function (NEGF) method [Datta,1989,1990, Meir and Wingreen,1992],which is being widely used in the analysis and design of nanoscale devices [see Datta,2005a and references therein]. It provides a unified description for all kinds of devices from molecular conductors to carbon nanotubes to silicon transistors in terms of the Hamiltonian [H] describing the channel, the self-energies [ Σ1,2] describing the connection to the contacts and the [ Σ s] describing interactions inside the channel (Fig.2b). In each case the details are very different, but once these matrices (whose size depends on the number of basis functions needed to describe the channel) have been written down, the procedure for obtaining quantities of interest such as current flow and energy dissipation are the same regardless of the specifics of the problem at hand. The model covers different transport regimes from the ballistic to the diffusive limit depending on the relative magnitudes of Σ1,2 and Σ s . In this paper I will use a particularly simple version of this approach (Fig.2a) where matrices like Σ1,2 and Σs are replaced with numbers like γ1,2 and γ s having simple physical interpretations: γ1,2 /ℏ represents the rate of transfer of channel electrons in and out of the contacts while γ s /ℏ represents the rate at which they interact with any “demons” that inhabit the channel. In the past it was common to have γ1,2 << γ s so that transport was dominated by the interactions within the channel, with contacts playing a minor enough role that theorists seldom drew them prior to 1990! By contrast, today’s nanodevices have reached a point where γ1,2 >> γ s , placing them in what we could call the ballistic or “Landauer limit”. An appealing feature of this limit is that it physically separates the dynamics from the dissipation: reversible dynamics dominates the channel while dissipation dominates the contact. Usually these two aspects of transport are conceptually so intertwined that it is difficult to see how irreversibility is introduced into a problem described by reversible dynamic equations (Newton or Schrodinger) and this issue continues to spark debate and discussion ever since the path-breaking work of Boltzmann many years ago [see for example, McQuarrie,1976]. Let me start with a brief summary of the basic framework shown in Fig.2a that we call the bottom-up viewpoint (Section 2). We will then use this approach to discuss a specially designed device which is in the Landauer limit except for a particularly simple version of Maxwell’s demon that interacts with the channel electrons but does not exchange any energy with them (Section 3).We then consider a more sophisticated demon that exchanges energy as well and show how it can be used to build nanoscale heat engines and refrigerators (Section 4). We conclude with a few words about entangled demons and related conceptual issues that I believe need to be clarified in order to take transport physics to its next level (Section 5). Readers may find the related video lectures posted on the nanoHUB useful [Datta,2006] and I will be happy to share the MATLAB codes used to generate the figures in this paper. 2. BOTTOM-UP VIEWPOINT Consider the device shown in Fig.1b without the “demon” but with a voltage V applied across two contacts (labeled “source” and “drain”) made to a conductor (“channel”). How do we calculate the current I, as the length of the channel L is made shorter and shorter, down to a few atoms? This is not just an academic question since experimentalists are actually making current measurements through “channels” that are only a few atoms long. Indeed, this is also a question of great interest from an applied point of view, since every laptop computer contains about one billion nanotransistors, each of which is basically a conductor like the one in Figure 1b with L ~ 50 nm, but with the demon replaced by a third terminal that can be used to control the resistance of the channel. As the channel length L is reduced from macroscopic dimensions (~ millimeters) to atomic dimensions (~ nanometers), the nature of electron transport that is, current flow, changes significantly (Fig.3). At one end, it is described by a diffusion equation in which electrons are viewed as particles that are repeatedly scattered by various obstacles causing them to perform a “random walk” from the source to the drain. The resistance obeys Ohm’s law: a sample twice as long has twice the resistance. At the other end, there is the regime of ballistic transport where the resistance of a sample can be independent of length. Indeed due to wavelike interference effects it is even possible for a longer sample to have a lower resistance! The subject of current flow is commonly approached using a “top-down” viewpoint. Students start in high school from the macroscopic limit (large L) and seldom reach the atomic limit, except late in graduate school if at all. I believe that this is primarily for historical reasons. After all, twenty years ago, no one knew what the resistance was for an atomic scale conductor, or if it even made sense to Nanodevices and Maxwell’s demon Supriyo Datta Fig.3. Evolution of devices from the regime of diffusive transport to ballistic transport as the channel length L is scaled down from millimeters to nanometers. Fig.4. Plot of energy current in a 1-D ballistic conductor with G = e2 / h ≈ 40µS and an applied voltage of V = 0.05 volts. Energy dissipated is given by the drop in the energy current, showing that the Joule heating V 2G = 0.1 µW is divided equally between the two contact-channel interfaces. Diffusive Transport 1 mm 100 µm 10 µm 1 µm 100 nm 10 nm 1 nm 1 Ballistic Transport Macroscopic dimensions Atomic dimensions <----- L -----> 0 0.2 0.4 0.6 0.8 1 Energy Current Normalized Distance Along Device Source Channel Drain Dissipationless channel Newton’s law/ Schrodinger equation Dissipation DissipationContacts assumed to remain in equilibrium 5 arXiv:cond-mat/0704.1623 datta@purdue.edu ask about its resistance But now that the bottom-line is known, I believe that a “bottom-up” approach is needed if only because nanoscale devices like the ones I want to talk about look too complicated from the “top-down” viewpoint. In the top-down view we start by learning that the conductance G=I/V is related to a material property called conductivity σ through a relation that depends on the sample geometry and for a rectangular conductor of cross- section A and length L is given by G = σA /L . We then learn that the conductivity is given by σ = e2nτ /m where e is the electronic charge, n is the electron density, τ is the mean free time and m is the electron mass. Unfortunately from this point of view it is very difficult to understand the ballistic limit. Since electrons get from one contact to the other without scattering it is not clear what the mean free time τ is. Neither is it clear what one should use for ‘n’ since it stands for the density of free electrons and with molecular scale conductors it may not be clear which electrons are free. Even the mass is not very clear since the effective mass is deduced from the bandstructure of an infinite periodic solid and cannot be defined for really small conductors. 2.1. Conductance formula: the “bottom-up” version A more transparent approach at least for small conductors is provided by the bottom-up viewpoint [Datta, 2005a, Chapter 1] which leads to an expression for conductance in terms of two basic factors, namely the density of states D around the equilibrium electrochemical potential and the effective escape rate γ /ℏ from the channel into the contacts ( ℏ = h / 2π , h being Planck’s constant): (1a) The escape rate appearing above is the series combination of the escape rates associated with each contact: (1b) This is an expression that we can apply to the smallest of conductors, even a hydrogen molecule. Although it looks very different from the expression for conductivity mentioned earlier, it is closer in spirit to another well- known expression for the conductivity (2) in terms of the density of states per unit volume ˜ N and the diffusion coefficient ˜ D . Indeed we could obtain Eq.(1a) from Eq.(2) if we make the replacements (3) which look reasonable since the density of states for a large conductor is expected to be proportional to the volume AL and the time taken to escape from a diffusive channel is ~ ˜ D /L2. 2.2 Current-voltage relation: without demons The result cited above (Eq.(1a) is a linear response version of a more general set of equations that can be used to calculate the full current-voltage characteristics, which in turn follow from the NEGF-Landauer formulation (Fig.2b). For our purpose in this talk the simpler version (Fig.2a) will be adequate and in this version the basic equations are fairly intuitive: (4) These equations relate the currents per unit energy at contacts 1 and 2 to the density of states D(E) , the Fermi functions at the two contacts related to their electrochemical potentials (5) and the distribution function f(E) inside the channel. The total currents at the source and drain contacts are obtained by integrating the corresponding energy resolved currents (6) If the electrons in the channel do not interact with any demons, we can simply set i1(E) = i2 (E) , calculate f(E) and substitute back into Eq.(4) to obtain (7) The conductance expression stated earlier (Eq.(1a)) follows from Eqs.(6) and (7) by noting that (8) G = (e2 / h) 2πD γ γ = γ D → ˜ N . AL and γ /ℏ → ˜ D / L2 i1(E) = (e /ℏ) γ1 D(E) [ f1 (E) − f (E)] i2 (E) = (e /ℏ) γ 2 D(E) [ f (E) − f2 (E)] f1,2 (E) = 1+ exp((E − µ1,2) / kBT1,2) i1(E) = i2 (E) = (e /ℏ) γ1γ 2 γ1 + γ 2 D(E) [ f1(E) − f2 (E)] dE [ f1(E) − f2 (E)] = µ1 − µ2∫ I1 = dE∫ i1(E) , I2 = dE∫ i2 (E) σ = e2 ˜ N ˜ D Nanodevices and Maxwell’s demon Supriyo Datta and assuming the density of states D(E) to be nearly constant over the energy range of transport where f1(E) − f2 (E) is significantly different from zero. One-level conductor: Eq.(7) can be used more generally even when the density of states has sharp structures in energy. For example, for a very small conductor with just one energy level in the energy range of interest, the density of states is a “delta function” that is infinitely large at a particular energy. Eqs.(6,7) then yield a current-voltage characteristic as sketched below. The maximum current is equal to eγ1 / 2ℏ , assuming γ 2= γ1. It might appear that the maximum conductance can be infinitely large, since the voltage scale over which the current rises is ~ kBT , so that dI/dV can increase without limit as the temperature tends to zero. However, the uncertainty principle requires that the escape rate of γ /ℏ into the contacts from an energy level also broadens the level by γ as shown below. This means that the voltage scale over which the current rises is at least ~ (γ1 +γ2) /e = 2γ1 /e , even at zero temperature. This means that a small device has a maximum conductance of This rough estimate is not too far from the correct result (9) which is one of the seminal results of mesoscopic physics that was not known before 1988. One could view this as a consequence of the energy broadening required by the uncertainty principle which comes out automatically in the full Landauer-NEGF approach (Fig.2b), but has to be inserted by hand into the simpler version we are using where the density of states D(E) is an input parameter (Fig.2a). For our examples in this paper we will use a density of states that is constant in the energy range of interest, for which the current-voltage characteristic is basically linear. 2.3 Current-voltage relation: with demons Defining is (E) as the “scattering current” induced by interaction of the electrons with the “demon”, we can write (10) This current can be modeled in general as a difference between two processes one involving a loss of energy ε from the demon and the other involving a gain of energy by the demon. (11)) The basic principle of equilibrium statistical mechanics requires that if the demon is in equilibrium at some temperature TD then the strength F (ε) of the energy loss processes is related to the strength F (−ε) of energy gain processes by the ratio: (12) With a little algebra one can show that this relation ensures that if the electron distribution f(E) is given by an equilibrium Fermi function with the same temperature T is (E) = (e /ℏ) γ s D(E) dε∫ D(E +ε) [F (ε) f (E) (1− f (E +ε) − F (−ε) f (E +ε) (1− f (E)] i1(E) = i2 (E) + is (E) F (ε) = F (−ε) exp (−ε / kBTD ) -1 -0.5 0 0.5 1 Normalized voltage Normalized current -1 -0.5 0 0.5 1 Normalized voltage Normalized current eγ1 /2ℏ Gmax = e 2 /h ≈ 25.8 KΩ eγ1 / 2ℏ eγ1 / 2ℏ µ1 Normalized voltage Normalized current D(E) -1 -0.5 0 0.5 1 7 arXiv:cond-mat/0704.1623 datta@purdue.edu the two terms in Eq.(11) will cancel out. This result is independent of the detailed shape of the function F (ε) describing the spectrum of the demon, as long as Eq.(12) is true. This means that if the demon is in equilibrium with the electrons with the same temperature, there can be no net flow of energy either to or from the demon. Indeed one could view this as the basic principle of equilibrium statistical mechanics and work backwards to obtain Eq.(12) as the condition needed to ensure compliance with this principle. To summarize, if the electrons in the channel do not interact with any “demons”, the current voltage characteristics are obtained from Eq.(7) using the Fermi functions from Eq.(5). For the more interesting case with interactions, we solve for the distribution f(E) inside the channel from Eqs.(4),(10) and (11) and then calculate the currents. Usually the current flow is driven by an external voltage that separates the electrochemical potentials µ1 and µ2 in Eq.(7). But thermoelectric currents driven by a difference in temperatures T1 and T2 can also be calculated from this model [Paulsson and Datta,2003] as mentioned in the introduction. Our focus here is on a different possibility for energy conversion, namely through out-of-equilibrium demons. 2.4. Where is the heat? Before we move on, let me say a few words about an important conceptual issue that caused much argument in the early days: Where is the heat dissipated in a ballistic conductor? After all, if there is no demon to take up the energy, there cannot be any dissipation inside the channel. The answer is that the transiting electron appears as a hot electron in the drain (right) contact and leaves behind a hot hole in the source (left) contact (see Fig.4). The contacts are immediately restored to their equilibrium states by unspecified dissipative processes operative within each contact. These processes can be quite complicated but are usually incorporated surreptitiously into the model through what appears to be an innocent boundary condition, namely that the electrons in the contacts are always maintained in thermal equilibrium described by Fermi distributions (Eq.(5)) with electrochemical potentials µ1 and µ2 and temperatures T1 and T2 . To understand the spatial distribution of the dissipated energy it is useful to look at the energy current which is obtained by replacing the charge ‘e’ with the energy E of the electron: (13) The energy currents at the source and drain contacts are written simply as (14) assuming that the entire current flows around the corresponding electrochemical potentials, Fig.4 shows a spatial plot of the energy current from the source end to the drain end for a uniform 1-D ballistic conductor with a voltage of 50 mV applied across it. For a conductor with no demon for electrons to exchange energy with, i1(E) = i2 (E) making the energy current uniform across the entire channel implying that no energy is dissipated inside the channel. But the energy current entering the source contact is larger than this value while that leaving the drain contact is lower. Wherever the energy current drops, it means that the rate at which energy flows in is greater than the rate at which it flows out, indicating a net energy dissipation. Clearly in this example, 0.05 µW is dissipated in each of the two contacts thus accounting for the expected Joule heating given by V 2G = 0.1 µW. Real conductors have distributed demons throughout the channel so that dissipation occurs not just in the contacts but in the channel as well. Indeed we commonly assume the Joule heating to occur uniformly across a conductor. But there are now experimental examples of nanoscale conductors that would have been destroyed if all the heat were dissipated internally and it is believed that the conductors survive only because most of the heat is dissipated in the contacts which are large enough to get rid of it efficiently. The idealized model depicted in Fig.4 thus may not be too far from real nanodevices of today. As I mentioned in the introduction, what distinguishes the Landauer model (Fig.4) is the physical separation of dynamics and dissipation clearly showing that what makes transport an irreversible process is the continual restoration of the contacts back to equilibrium. Without this repeated restoration, all flow would cease once a sufficient number of electrons have transferred from the source to the drain. Over a hundred years ago Boltzmann showed how pure Newtonian dynamics could be supplemented to describe transport processes, and his celebrated equation stands today as the centerpiece for the flow of all kinds of particles. Boltzmann’s approach too involved “repeated restoration” through an assumption referred to as the “Stohsslansatz” [see for example, McQuarrie 1976] Today’s devices often involve Schrodinger dynamics in place of Newtonian dynamics and the non-equilibrium Green function (NEGF) method that we use (Fig.2b) supplements the Schrodinger equation with similar assumptions about the repeated restoration of the I E s (E) = (µ1 /e) I1 , I E d (E) = (µ 2 /e) I2 IE1 = (1 /e) dE∫ E i1(E) , IE 2 = (1 /e) dE∫ E i2 (E) Nanodevices and Maxwell’s demon Supriyo Datta surroundings that enter the evaluation of the various self- energy functions Σ or the corresponding quantities γ in the simpler model that we are using. “Contacts” and “demons” are an integral part of all devices, the most common demon being the phonon bath for which the relation in Eq.(12) is satisfied by requiring that energy loss ~ N and energy gain ~ (N+1), N being the number of phonons given by the Bose Einstein factor if the bath is maintained in equlibrium. Typically such demons add channels for dissipation, but our purpose here is to show how suitably engineered out-of-equilibrium demons can act as sources of energy. For this purpose, it is convenient to study a device specially designed to accentuate the impact of the demon. Usually the interactions with the demon do not have any clear distinctive effect on the terminal current that can be easily detected. But in this special device, ideally no current flows unless the channel electrons interact with the demon. Let me now describe how such a device can be engineered. 3. ELASTIC DEMON Let us start with a simple 1-D ballistic device but having two rather special kinds of contacts. The source contact allows one type of spin, say “black” (upspins, drawn pointing to the right), to go in and out of the channel much more easily than the other type, say “white” (downspins, drawn pointing to the left, Fig.5a). Devices like this are called spin-valves and are widely used to detect magnetic fields in magnetic memory devices [see for example the articles in Heinreich and Bland, 2004]. Although today’s spin-valves operate with contacts that are far from perfect, since we are only trying to make a conceptual point, let us simplify things by assuming contacts that are perfect in their discrimination between the two spins. One only lets black spins while the other only lets white spins to go in and out of the channel. We then have the situation shown in Fig.5b and no current can flow since neither black nor white spins communicate with both contacts. But if we introduce impurities into the channel (the demon) with which electrons can interact and flip their spin, then current flow should be possible as indicated: black spins come in from the left, interact with impurities to flip into white spins and go out the right contact (Fig.5c). Consider now what happens if the impurities are say all white (Fig.6a). Electrons can now flow only when the bias is such that the source injects and drain collects (positive drain voltage), but not if the drain injects and the source collects (negative drain voltage). This is because the source injects black spins which interact with the white impurities, flip into whites and exit through the drain. But the drain only injects white spins which cannot interact with the white impurities and cannot cross over into the source. Similarly if the impurities are all black, current flows only for negative drain voltage (Fig.6b). At non-zero temperatures the cusps in the current-voltage characteristics get smoothed out and we get the smoother curves shown in Figs.6a,b. Note the surprising feature of the plots at T=300K: there is a non-zero current even at zero voltage! This I believe is correct. Devices like those shown in Figs.6a,b with polarized impurities could indeed be used to generate power and one could view the system of impurities as a Maxwell’s demon that lets electrons go preferentially from source to drain or from drain to source. The second law is in no danger, since the energy is extracted at the expense of the entropy of the system of impurities that collectively constitute the demon. Assuming the spins are all non- interacting and it takes no energy to flip one, the polarized state of the demon represents an unnatural low entropy state. Every time an electron goes through, a spin gets flipped raising the entropy and the flow will eventually stop when demon has been restored to its natural unpolarized state with the highest entropy of N I kB ln 2 , where N I is the number of impurities. To perpetuate the flow an external source will have to spend the energy needed to maintain the demon in its low entropy state. 3.1. Current versus voltage: Model summary Let me now summarize the equations that I am using to analyze structures like the one in Fig.6 quantitatively. Basically it is an extension of the equations described earlier (see Eq.(4), Section 2) to include the two spin channels denoted by the subscripts ‘u’ (up or black) and ‘d’ (down or white): (15) (16) For the scattering current caused by the demon we write (see Eq.(11)) (17) i1,u (E) = (e /ℏ) γ1,u Du (E) [ f1(E) − fu (E)] i2,u (E) = (e /ℏ) γ 2,u Du (E) [ fu (E) − f2 (E)] i1,d (E) = (e /ℏ) γ1,d Dd (E) [ f1(E) − fd (E)] i2,d (E) = (e /ℏ) γ 2,d Dd (E) [ fd (E) − f2 (E)] is,u (E) = − is,d (E) = (e /ℏ) (2π J 2N I ) Du (E) Dd (E) [Fd fu (E) (1− fd (E) − Fu fd (E) (1− fu (E)] 9 arXiv:cond-mat/0704.1623 datta@purdue.edu Fig.5. Anti-parallel (AP) Spin-valve: (a) Physical structure, (b) no current flows if the contacts can discriminate between the two spins perfectly, (c) current flow is possible if impurities are present to induce spin- flip processes, Adapted from Datta, 2005b,c. Source Drain Spin-flip impurities Source Drain (b (c) Current Voltage (b) w/o spin-flip (c) with spin-flip DrainSource Channel Nanodevices and Maxwell’s demon Supriyo Datta Fig.6. Perfect AP Spin-valve with (a) white (down spin, drawn as pointing to the left) impurities and (b) black (upspin, drawn as pointing to the right) impurities. Note the non-zero current at zero voltage for non-zero temperatures. Adapted from Datta, 2005b,c. -0.1 -0.05 0 0.05 0.1 5x 10 Voltage -0.1 -0.05 0 0.05 0.1 5x 10 Voltage Current (A) 300K 300K 0 K 0 K Source Drain (a) “White” impurities Source Drain (b) “Black” impurities 11 arXiv:cond-mat/0704.1623 datta@purdue.edu Fig.7. An anti-parallel (AP) spin-valve with spin-flip impurities and a simple equivalent circuit to help visualize the equations we use to describe it, namely Eqs.(15) through (17), Adapted from Datta, 2005b,c. DrainSource Magnetic Impurities down spin up spin g2,dg1,d ⇑ idem Nanodevices and Maxwell’s demon Supriyo Datta noting that we are considering an elastic demon that can neither absorb nor give up energy (ε =0). The first term within parenthesis in Eq.(17) represents an up electron flipping down by interacting with a down-impurity while the second term represents a down electron flipping up by interacting with an up-impurity. The strengths of the two processes are proportional to the fractions Fd and Fu of down and up-spin impurities.The overall strength of the interaction is governed by the number of impurities N I and the square of the matrix element J governing the electron- impurity interaction. Eqs.(15),(16) can be visualized in terms of an equivalent circuit (Fig.7) if we think of the various f’s as “voltages” since the currents are proportional to differences in ‘f’ just as we expect for conductances etc. The scattering current (Eq.(17)) too could be represented with a conductance gs if we set Fd = Fu = 0.5 (18a) However, this is true only if the impurities are in equilibrium, while the interesting current-voltage characteristics shown in Fig.6 require an out-of- equilibrium demon with Fd ≠ Fu . So we write the total scattering current from Eq.(17) as a sum of two components, one given by Eq.(18a) and another proportional to ( Fd - Fu ) which we denote with a subscript ‘dem’: (18b) Eqs.(16) and (17) can be solved to obtain the distribution functions fu (E) and fd (E) by imposing the requirement of overall current conservation (cf. Eq.(11)): (19) The currents are then calculated and integrated over energy to obtain the terminal currents shown in Fig.6. We can also find the energy currents using equations like Eqs.(14), (15) and the results are shown in Fig.8 for Fd - Fu = -1 and for Fd - Fu = 0 each with a voltage difference of 2 kBT =50 mV between the two terminals. With Fd - Fu = 0 the direction of current is in keeping with an external battery driving the device. But with Fd - Fu = -1, the external current flows against the terminal voltage indicating that the device is acting as a source of energy driving a load as shown in the inset. This is also borne out by the energy current flow which shows a step up at each interface indicating that energy is being absorbed from the contacts (~ 10 nW from each) and delivered to the external load. But isn’t this exactly what the second law forbids? After all if we could just absorb energy from our surroundings (the contacts) and do useful things, there would be no energy problem. The reason this device is able to perform this impossible feat is that the impurities are assumed to be held in a non-equilibrium state with very low entropy. A collection of N I impurities can be unpolarized in 2 different ways having an entropy of S = N I kB ln 2 . But it can be completely polarized ( Fd - Fu = ± 1) in only one way with an entropy of S = 0. What this device does is to exchange entropy for energy. Many believe that the universe as a whole is evolving the same way, with constant total energy, from a particularly low entropy state continually towards a higher entropy one. But that is a different matter. To have our device continue delivering energy indefinitely we will need an external source to maintain the impurities in their low entropy state which will cost energy. The details will depend on the actual mechanism used for the purpose but we will not go into this. Note that if we do not have such a mechanism, the current will die out as the spins get unpolarized. This depolarization process can be described by an equation of the form: (20) where ˜ τ 0 and si are related to the scattering current as defined in Eqs.(18), while the additional time constant represents processes unrelated to the channel electrons by which impurities can relax their spins. 4. INELASTIC DEMON We have argued above that although one can extract energy from polarized impurities, energy is needed to keep them in that state since their natural high entropy state is the unpolarized one. It would seem that one way to keep the spins naturally in a polarized state is to use a nanomagnet, a collection of spins driven by a ferromagnetic interaction that keeps them all pointed in the is = (e /ℏ) (πJ 2NI ) Du (E)Dd (E) ( fd − fu ) idem = (Fd − Fu ) eNI / ˜ τ 0(E) ˜ τ 0(E ) = (1/ℏ) (πJ2) Du (E)Dd (E) [ fd (1− fu ) + fu (1− fd )] i1,u (E) = i2,u (E) + is,u (E) i1,d (E) = i2,d (E) + is,d (E) g1,u = (e 2 /ℏ) γ1,u Du dP ∫= 13 arXiv:cond-mat/0704.1623 datta@purdue.edu same direction. Could such a magnet remain polarized naturally and enable us to extract energy from the contacts Fig.8. Energy currents with Fd - Fu = -1 and with Fd - Fu = 0 each with a voltage difference of 2 kBT =50 mV between the two terminals. With Fd - Fu = 0 the direction of current is in keeping with an external battery driving the device. But with Fd - Fu = -1, the external current flows against the terminal voltage indicating that the device is acting as a source of energy driving a load as shown in the inset. This is also borne out by the energy current flow which shows a step up at each interface indicating that energy is being absorbed from the contacts (~ 10 nW from each) and delivered to the external load. 0 0.2 0.4 0.6 0.8 1 4x 10 Energy Current Normalized Distance Along Device Source Channel Drain 0.05 V Fd - Fu = 0 ++++ 0.05 V ---- Drain Source Fd - Fu = -1 Load Resistor Nanodevices and Maxwell’s demon Supriyo Datta forever? The answer can be “yes” if the magnet is at a different temperature from the electrons. What we then have is a heat engine working between two reservoirs (the electrons and the magnet) at different temperatures and we will show that its operation is in keeping with Carnot’s principle as required by the second Law. To model the interaction of the electrons with the nanomagnet we need to modify the expression for the scattering current (Eq.(17)) for it now takes energy to flip a spin. We can write (21) where F (ε) denotes the magnon spectrum obeying the general law stated earlier (see Eq.(12)) if the magnet has a temperature TD . Eq.(21) can be solved along with Eqs.(15,(16),(19) and (21) as before to obtain currents, energy currents etc. But let us first try to get some insight using simple approximations. If we assume that the electron distribution functions fu (E) and fd (E) are described by Fermi functions with electrochemical potentials µu and µd respectively and temperature T, and make use of Eq.(12) we can rewrite the scattering current from Eq.(21) in the form (22) If we further assume the exponent to be small, so that 1 – exp(-x) ≈ x , we can write (23) The first term represents a dissipative current proportional to the potential difference and can be represented by a conductance like the gs in Fig.7, while the second term is the demon induced source term that can be harnessed to do external work. It vanishes when the demon temperature TD equals the electron temperature T. To be specific, let us assume that the demon is cooler than the rest of the device (TD < T) so that giving rise to a flow of electrons out of the upspin node and back into the downspin node. If we use this to drive an external load then µu − µd > 0, which will tend to reduce the net current given by Eq.(23) and the maximum output voltage one can get corresponds to “open circuit” conditions with zero current: (24) This expression has a simple interpretation in terms of the Carnot principle. Every time an electron flows around the circuit giving up energy ε to the demon, it delivers energy µu − µd to the load. All this energy µu − µd + ε is absorbed from the contacts and the Carnot principle requires that Energy from contacts / T ≤ Energy to demon / TD that is, which is just a restatement of Eq.(24). Note that usual derivations of the Carnot principle do not put the system simultaneously in contact with two reservoirs at different temperatures as we have done. Our treatment is closer in spirit to the classic discussion of the ratchet and pawl in the Feynman Lectures [Feynman et.al. 1963]. Fig.9 shows numerical results obtained by solving Eqs.(21),(19),(15) and (16). The extracted power is a maximum (Fig.9d) when the output voltage is somewhere halfway between zero and the maximum output voltage of 80 mV (a few kBT ).Fig.9b shows the energy current profile at an output voltage of 50 mV: energy is absorbed from the source and drain contacts and given up partly to is,u (E) = − is,d (E) = (e /ℏ) (2π J 2N I ) dε∫ Du (E) Dd (E +ε) [F (ε) fu (E) (1− fd (E +ε)) − F (−ε) fd (E +ε) (1− fu (E)] is,u (E) = − is,d (E) = (e /ℏ) (2π J 2N I ) dε∫ Du (E) Dd (E +ε) F (ε) fu (E) (1− fd (E +ε) F (−ε) F (+ε) fd (E +ε) 1− fd (E +ε) 1− fu (E) fu (E) = (e /ℏ) (2π J 2N I ) dε∫ Du (E) Dd (E +ε)F (ε) fu (E) (1− fd (E +ε) 1 − exp µd −ε − µu is,u = − is,d µu − µd is,u < 0 is,u µu − µd +ε µu − µd 15 arXiv:cond-mat/0704.1623 datta@purdue.edu Fig.9. Heat engine: (a) AP Spin-valve with a cooled magnet as Maxwell’s demon controlling the flow of electrons. (b) Output current and output power versus output voltage as the load is varied from short circuit (V=0) to open circuit conditions (I=0). The spectrum of the magnet is assumed to consist of a single energy ε = 2kBT = 50 meV. (c) Energy current profile assuming a load such that the output voltage is 50 mV. Output Voltage (V) Output Voltage (V) Current (A) Power (W) 0 0.02 0.04 0.06 0.08 14x 10 0 0.02 0.04 0.06 0.08 2x 10 Normalized Distance Along Device 0 0.2 0.4 0.6 0.8 1 2x 10 Source Drain “Demon” Energy Current Drain at 600 K Source at 600 K “Magnet” at 300 K Nanodevices and Maxwell’s demon Supriyo Datta Fig.10. (a) Heat engine from Fig.9 operated as a refrigerator by applying an external battery to inject downspin (white) electrons from the drain that flip down the thermally created upspins in the magnet, thus cooling it. (b) Energy current profile showing that energy is absorbed from the external battery and the demon and dissipated in the source and drain contacts. Fig.11. We have described spin-flip processes in terms of a direct conversion from state A to state B. Quantum mechanics, however, requires an intermediate state consisting of a superposition of A and B before wavefunction collapse reduces it to a B. This state may have a significant role in devices with weak contacts and strong interactions in the channel, requiring a model that can deal with entangled states. 0 0.2 0.4 0.6 0.8 1 1.5 x 10 Normalized Distance Along Device Energy Current Source Drain “Demon” Source Drain Source Drain Drain at 300 K Source at 300 K “Magnet” at 250 K 17 arXiv:cond-mat/0704.1623 datta@purdue.edu the demon and the rest is delivered to the external load. The efficiency (energy given to load / energy absorbed from contact) is a maximum close to open circuit conditions, but the energy delivered is very small at that point as evident from Fig.9d. As one might expect, one can also operate the same device as a refrigerator by using an external source that seeks to inject downspins (white) from the drain contact that flip back thermally created upspins in the magnet thus cooling it. It is evident from the energy current profile shown in Fig.10 that in this case, energy is absorbed from the demon and from the battery and given up to the source and drain contacts. 5. ENTANGLED DEMON In this talk I have tried to introduce a simple transparent model showing how out-of-equibrium demons suitably incorporated into nanodevices can achieve energy conversion. At the same time this model illustrates the fundamental role played by “contacts” and “demons” in these processes. I would like to end by pointing out another aspect of contacts that I believe is important in taking us to our next level of understanding. The basic point can be appreciated by considering a simple version of the spin capacitor we started with (see Fig.5) but having just one impurity (Fig.11). No current can flow in this structure without spin-flip processes since the source injects black (up) electrons while the drain only collects white (down) electrons. But if the black electron interacts with the white impurity(A) to produce a black impurity then the white electron can be collected resulting in a flow of current. The process of conversion From A: Black electron ⊗ White impurity To B: White electron ⊗ Black impurity is incorporated into our model through the scattering current (see Eq.(17)). A more complete quantum transport model involving matrices (Fig.3a) rather than numbers (Fig.3b) could be used to describe this effect, but the essential underlying assumption in either case is that the state of the electron-impurity system changes from A to B. Quantum mechanics, however, paints a different picture of the process involving an intermediate entangled state. It says that the system goes from A into a state consisting of a superposition of A and B and it is only when the electron is collected by the drain that the wavefunction collapses to a B.If the collection rates γ1,2 /ℏ are much larger than the interaction rate per impurity γ s /ℏN I , we expect the entangled state to play a minor role. But this may not be true of devices with weak contacts and strong interactions in the channel, requiring a model that can deal with entangled states. Entangled states are difficult to describe within the conceptual framework we have been using where both the electrons and the impurity are assumed to exist in independent states. It is hard to describe a “conditional state” where the electron is black if the impurity is white or vice versa, let alone a superposition of the two. To account for this entangled state we need to treat the electron and impurity as one big system and write rate equations for it, in the spirit of the many-electron rate equations widely used to treat Coulomb blockade but the standard approach [Beenakker 1991, Likharev 1999] needs to be extended to include coherences and broadening. This is an area of active research [see for example Braun et.al. 2004, Braig and Brouwer 2005] where an adequate general approach does not yet exist. Actually correlated states (classical version of entanglement) were an issue even before the advent of quantum mechanics. Boltzmann ignored them through his assumption of “molecular chaos” or “Stohsslansatz”,and it is believed that it is precisely this assumption that leads to irreversibility [see for example, McQuarrie 1976]. An intervening entangled or correlated state is characteristic of all “channel”-“contact" interactions, classical or quantum, and the increase in entropy characteristic of irreversible processes can be associated with the destruction or neglect of the correlation and/or entanglement generated by the interaction (see for example, Zhang 2007, Datta 2007). This aspect is largely ignored in today’s transport theory, just as even the presence of a contact was barely acknowledged before the advent of mesoscopic physics. But new experiments showing the effect of entanglements on current flow are on the horizon and will hopefully lead us to the next level of understanding. This work was supported by the Office of Naval Research under Grant No. N00014-06-1-0025 and the Network for Computational Nanotechnology. REFERENCES Beenakker C.W.J., 1991, Theory of Coulomb blockade oscillations in the conductance of a quantum dot. In Physical Review, B44, 1646. Braig S. and Brouwer P.W., 2005, Rate equations for Coulomb blockade with ferromagnetic leads. In Physical Review, B71, 195324. Braun M., Koenig J. and Martinek J., 2004, Theory of transport through quantum-dot spin valves in the weak coupling regime. In Physical Review, B70, 195345. Datta S., 1989, Steady-state quantum kinetic equation. In Physical Review, B40, 5830. Datta S., 1990, A simple kinetic equation for steady-state quantum transport. In Journal of Physics: Condensed Matter, 2, 8023. Nanodevices and Maxwell’s demon Supriyo Datta Datta S., 2005a, Quantum Transport: Atom to Transistor, Cambridge University Press. Datta S., 2005b, Proposal for a spin capacitor. In Applied Physics Letters, 87, 012115. Datta S., 2005c, Spin dephasing and hot spins, In Proceedings of the International School of Physics "Enrico Fermi" Course CLX edited by A. D'Amico, G. Balestrino and A. Paoletti, IOS Press Amsterdam and SIF Bologna. Datta S., 2006, Concepts of Quantum Transport, a series of video lectures, http://www.nanohub.org/courses/cqt Datta S., 2007, Quantum Transport as a Process of Repeated Disentanglement, preprint. Feynman R.P., Leighton R.B. and Sands M., 1963, Lectures on Physics, vol.1, Chapter 46, Addison-Wesley. Fuchs G.D., Krivotorov I.N., Braganca P.M.,Emley N.C., Garcia A.G.F., Ralph D.C., Buhrman R.A., 2005, Adjustable spin torque in magnetic tunnel junctions with two fixed layers, In Applied Physics Letters, 86, 152509. Huai Y., Pakala M., Diao Z., Ding Y., 2005. Spin transfer switching current reduction in magnetic tunnel junction based dual spin filter structures. In Applied Physics Letters, 87 222510. Heinreich B. and Bland J.A.C., 2004, eds. Introduction to micromagnetics in ultrathin magnetic structures, vol.IV, Springer-Verlag, Berlin. Leff H. S. and Rex A.F. (eds) 1990, Maxwell’s demon, entropy, information and computing, Princeton Series in Physics. Leff H. S. and Rex A.F. (eds) 2003, Maxwell’s demon 2, entropy, classical and quantum information and computing, IOP Publishing, Bristol. Likharev K., 1999, Single electron devices and their applications. In Proceedings IEEE, 87, 606. McQuarrie D.A., 1976. Statistical Mechanics, Harper and Row. Meir Y. and Wingreen N.S. 1992. Landauer formula for the current through an interacting electron region. In Physical Review Letters 68, 2512. Nikulov A. and Sheehan D. (eds.) 2004, Entropy, 6. Paulsson M. and Datta S., 2003, Thermoelectric effect in molecular electronics. In Physical Review, B67, 241403(R). Reddy P., Jang S.Y., Segalman R. and Majumdar A., 2007, Thermoelectricity in molecular junctions. In Science, 315,1568. Saha D., Holub M., Bhattacharya P. and Liao Y.C., 2006, Epitaxially grown MnAs/GaAs lateral spin valves. In Applied Physics Letters, 89, 142504. Salahuddin S. and Datta S., 2006a, Electrical Detection of Spin Excitations. In Physical Review, B73, 081201(R). Salahuddin S. and Datta S., 2006b, Self-consistent simulation of quantum transport and magnetization dynamics in spin-torque based devices, In Applied.Physics Letters, 89, 153504. Zhang Qi-Ren, 2007, A General Information Theoretical Proof for the Second Law of Thermodynamics, arXiv:quantph/0610005v3.

In the last twenty years there has been significant progress in our understanding of quantum transport far from equilibrium and a conceptual framework has emerged through a combination of the Landauer approach with the non-equilibrium Green function (NEGF) method, which is now being widely used in the analysis and design of nanoscale devices. It provides a unified description for all kinds of devices from molecular conductors to carbon nanotubes to silicon transistors covering different transport regimes from the ballistic to the diffusive limit. In this talk I use a simple version of this model to analyze a specially designed device that could be called an electronic Maxwell's demon, one that lets electrons go preferentially in one direction over another. My objective is to illustrate the fundamental role of contacts and demons in transport and energy conversion. The discussion is kept at an academic level steering clear of real world details, but the illustrative devices we use are very much within the capabilities of present-day technology. For example, recent experiments on thermoelectric effects in molecular conductors agree well with the predictions from our model. The Maxwell's demon device itself is very similar to the pentalayer spin-torque device which has been studied by a number of groups though we are not aware of any discussion of the possibility of using the device as a nanoscale heat engine or as a refrigerator as proposed here. However, my objective is not to evaluate possible practical applications. Rather it is to introduce a simple transparent model showing how out-of-equibrium demons suitably incorporated into nanodevices can achieve energy conversion.

Introduction to micromagnetics in ultrathin magnetic structures, vol.IV, Springer-Verlag, Berlin. Leff H. S. and Rex A.F. (eds) 1990, Maxwell’s demon, entropy, information and computing, Princeton Series in Physics. Leff H. S. and Rex A.F. (eds) 2003, Maxwell’s demon 2, entropy, classical and quantum information and computing, IOP Publishing, Bristol. Likharev K., 1999, Single electron devices and their applications. In Proceedings IEEE, 87, 606. McQuarrie D.A., 1976. Statistical Mechanics, Harper and Row. Meir Y. and Wingreen N.S. 1992. Landauer formula for the current through an interacting electron region. In Physical Review Letters 68, 2512. Nikulov A. and Sheehan D. (eds.) 2004, Entropy, 6. Paulsson M. and Datta S., 2003, Thermoelectric effect in molecular electronics. In Physical Review, B67, 241403(R). Reddy P., Jang S.Y., Segalman R. and Majumdar A., 2007, Thermoelectricity in molecular junctions. In Science, 315,1568. Saha D., Holub M., Bhattacharya P. and Liao Y.C., 2006, Epitaxially grown MnAs/GaAs lateral spin valves. In Applied Physics Letters, 89, 142504. Salahuddin S. and Datta S., 2006a, Electrical Detection of Spin Excitations. In Physical Review, B73, 081201(R). Salahuddin S. and Datta S., 2006b, Self-consistent simulation of quantum transport and magnetization dynamics in spin-torque based devices, In Applied.Physics Letters, 89, 153504. Zhang Qi-Ren, 2007, A General Information Theoretical Proof for the Second Law of Thermodynamics, arXiv:quantph/0610005v3.

COMPLETE SEGAL SPACES ARISING FROM SIMPLICIAL CATEGORIES JULIA E. BERGNER Abstract. In this paper, we compare several functors which take simplicial categories or model categories to complete Segal spaces, which are particularly nice simplicial spaces which, like simplicial categories, can be considered to be models for homotopy theories. We then give a characterization, up to weak equivalence, of complete Segal spaces arising from these functors. 1. Introduction and Overview The idea that simplicial categories, or categories enriched over simplicial sets, model homotopy theories goes back to a series of several papers by Dwyer and Kan [8], [10], [11], [12]. Taking the viewpoint that a model category, or more generally a category with weak equivalences, can be considered to be a model for a homotopy theory, they develop two methods to obtain from a model category a simplicial category. This “simplicial localization” encodes higher-order structure which is lost when we pass to the homotopy category associated to the model category. Furthermore, they prove that, up to a natural notion of weak equivalence of simplicial categories (see Definition 3.1), every simplicial category arises as the simplicial localization of some category with weak equivalences [10, 2.1]. Thus, the category of all (small) simplicial categories with these weak equivalences can be regarded as the “homotopy theory of homotopy theories.” This notion was mentioned briefly at the end of Dwyer and Spalinski’s introduc- tion to model categories [13, 11.6], and was made precise by the author in [2, 1.1], in which the category SC of small simplicial categories with these weak equivalences is shown to have the structure of a model category. However, this category is not practically useful for many purposes. Simplicial categories are not particularly easy objects to work with, and the weak equivalences, while natural generalizations of equivalences of categories, are difficult to identify. Motivated by this problem, Rezk defines a model structure, which we denote CSS, on the category of simplicial spaces, in which the fibrant-cofibrant objects are called complete Segal spaces (see Definition 4.3). This model structure is especially nice, in that the objects are just diagrams of simplicial sets and the weak equivalences, at least between complete Segal spaces, are just levelwise weak equivalences of simplicial sets. Furthermore, this model structure has the additional structures of a simplicial model category and a monoidal model category. Because it is given by a Date: October 25, 2018. 2000 Mathematics Subject Classification. Primary: 55U40; Secondary: 55U35, 18G55, 18G30, 18D20. Key words and phrases. simplicial categories, model categories, complete Segal spaces, homo- topy theories. http://arxiv.org/abs/0704.1624v2 2 J.E. BERGNER localization of a model structure on the category of simplicial spaces with levelwise weak equivalences, it is an example of a presentation for a homotopy theory as described by Dugger [7]. In his paper, Rezk defines two different functors, one from the category of sim- plicial categories, and one from the category of model categories, to the category of complete Segal spaces. Thus, he describes a relationship between simplicial cat- egories and complete Segal spaces, but he does not give an inverse construction. However, the author was able to show in [5] that the model categories SC and CSS are Quillen equivalent to one another, and therefore are models for the same homotopy theory. Thus, the hope is that we can answer questions about simplicial categories by working with complete Segal spaces. This paper is the beginning of that project. However, the functor that we use to show that the two model categories are Quillen equivalent is not the same as Rezk’s functor. Furthermore, the question arises whether Rezk’s functor on model categories agrees with the composite of the simplicial localization functor with his functor on simplicial categories. Rezk gives the beginning of a proof that the two functors agree when the model category in question has the additional structure of a simplicial model category. Our goal in this paper is prove that, up to weak equivalence in CSS, the two functors from SC to CSS are the same, that Rezk’s result holds for model categories which are not necessarily simplicial, and that this result does imply that the functor on model categories agrees with the one on their corresponding simplicial categories up to weak equivalence. We then go on to characterize, up to weak equivalence, the complete Segal space arising from any simplicial category. It can be described at each level as the nerve of the monoid of self weak equivalences of representing objects of the category. We should mention that the model categories SC and CSS are only two of several known models for the homotopy theory of homotopy theories. Our proof that the two are Quillen equivalent actually uses two intermediate model structures SeCatc and SeCatf on the category of Segal precategories, or simplicial spaces with a discrete Space at level zero [5]. The fibrant-cofibrant objects in these structures are known as Segal categories. Furthermore, Joyal and Tierney have shown that there is a Quillen equivalence between each of these model structures and a model structure QCat on the category of simplicial sets [17], [19]. The fibrant-cofibrant objects of this model structure are known as quasi-categories and are generalizations of Kan complexes [18]. One can compare our composite functor SC → CSS to one they define which factors through QCat rather than through SeCatc and SeCatf . It is a consequence of Joyal and Tierney’s work that these two different functors give rise to weakly equivalent complete Segal spaces, as we will describe further in the section on complete Segal spaces. An introduction to each of these model structures and the Quillen equivalences can be found in the survey paper [4]. In [1], we use the results of this paper to consider the complete Segal spaces arising from the homotopy fiber product construction for model categories, as de- scribed by Toën in his work on derived Hall algebras [24]. It seems that results relating diagrams of model categories to diagrams of complete Segal spaces will prove to be useful. COMPLETE SEGAL SPACES 3 Acknowledgments. I would like to thank Bill Dwyer, Charles Rezk, André Joyal, and Myles Tierney for conversations about the work in this paper, as well as the referee for helpful comments on the exposition. 2. Background on Model Categories and Simplicial Objects In this section, we summarize some facts about model categories, simplicial sets, and other simplicial objects that we need in the course of this paper. A model category M is a category with three distinguished classes of mor- phisms, fibrations, cofibrations, and weak equivalences. A morphism which is both a (co)fibration and a weak equivalence is called an acyclic (co)fibration. The cate- goryM with these choices of classes is required to satisfy five axioms [13, 3.3]. Axiom MC1 guarantees that M has small limits and colimits, so in particular M has an initial object and a terminal object. An object X in a model category M is fibrant if the unique map X → ∗ to the terminal object is a fibration. Dually, X is cofibrant if the unique map from the initial object φ→ X is a cofibration. The factorization axiom (MC5) can be applied in such a way that, given any object X ofM, we can factor the map X → ∗ as a composite ∼ //X ′ //∗ of an acyclic cofibration followed by a fibration. In this case, X ′ is called a fibrant replacement of X , since it is weakly equivalent to X and fibrant. This replacement is not necessarily unique, but in all the model categories we consider here it can be assumed to be functorial [16, 1.1.1]. Cofibrant replacements can be defined dually. The structure of a model category enables us to invert the weak equivalences formally in such a way that we still have a set, rather than a proper class, of mor- phisms between any two objects. If we were merely to take the localizationW−1M, there would be no guarantee that we would not have a proper class. However, with the structure of a model category, we can define the homotopy category Ho(M) to have the same objects asM, and as morphisms HomHo(M)(X,Y ) = [X cf , Y cf ]M, where the right-hand side denotes homotopy classes of maps between fibrant-cofibrant replacements of X and Y , respectively. The standard notion of equivalence of model categories is given by the following definitions. First, recall that an adjoint pair of functors F : C ⇆ D : G satisfies the property that, for any objects X of C and Y of D, there is a natural isomorphism ϕ : HomD(FX, Y )→ HomC(X,GY ). The functor F is called the left adjoint and G the right adjoint [20, IV.1]. Definition 2.1. [16, 1.3.1] An adjoint pair of functors F : M ⇆ N : G between model categories is a Quillen pair if F preserves cofibrations and G preserves fibra- tions. Definition 2.2. [16, 1.3.12] A Quillen pair of model categories is a Quillen equiv- alence if if for all cofibrant X in M and fibrant Y in N , a map f : FX → Y is a weak equivalence in D if and only if the map ϕf : X → GY is a weak equivalence 4 J.E. BERGNER An important example of a model category is that of the standard model struc- ture on the category of simplicial sets SSets. Recall that a simplicial set is a functor X : ∆op → Sets, where ∆op is the opposite of the category ∆ of finite ordered sets [n] = {0→ 1→ · · · → n} and order-preserving maps between them. We denote the set X([n]) by Xn. In particular in X we have face maps di : Xn → Xn−1 and de- generacy maps si : Xn → Xn+1 for each 0 ≤ i ≤ n, satisfying several compatibility conditions. Three particularly useful examples of simplicial sets are the n-simplex ∆[n] and its boundary ∆̇[n] for each n ≥ 0, and the boundary with the kth face removed, V [n, k], for each n ≥ 1 and 0 ≤ k ≤ n. Given a simplicial set X , we can take its geometric realization |X |, which is a topological space [14, I.1]. In the standard model category structure on SSets, the weak equivalences are the maps f : X → Y for which the geometric realization |f | : |X | → |Y | is a weak homotopy equivalence of topological spaces [14, I.11.3]. In fact, this model structure on simplicial sets is Quillen equivalent to the standard model structure on the category of topological spaces [16, 3.6.7]. More generally, a simplicial object in a category C is a functor ∆op → C. The two main examples which we consider in this paper are those of simplicial spaces (also called bisimplicial sets), or functors ∆op → SSets, and simplicial groups. We denote the category of simplicial spaces by SSets∆ A simplicial set X can be regarded as a simplicial space in two ways. It can be considered a constant simplicial space with the simplicial set X at each level, and in this case we will denote the constant simplicial set by cX or just X if no confusion will arise. Alternatively, we can take the simplicial space, which we denote Xt, for which (Xt)n is the discrete simplicial set Xn. The superscript t is meant to suggest that this simplicial space is the “transpose” of the constant simplicial space. A natural choice for the weak equivalences in the category SSets∆ is the class of levelwise weak equivalences of simplicial sets. If we define the cofibrations to be levelwise also, we obtain a model structure which is usually referred to as the Reedy model structure on SSets∆ [22]. The Reedy model structure has the additional structure of a simplicial model category. In particular, given any two objects X and Y of SSets∆ , there is a mapping space, or simplicial set Map(X,Y ) satisfying compatibility conditions [15, 9.1.6]. In the case where X is cofibrant (as is true of all objects in the Reedy model structure) and Y is fibrant, this choice of mapping space is homotopy invariant. One way to obtain other model structures on the category of simplicial spaces is to localize the Reedy structure with respect to a set of maps. While this process works for much more general model categories [15, 3.3.1], we will focus here on this particular case. Let S = {f : A → B} be a set of maps of simplicial spaces. A Reedy fibrant simplicial space W is S-local if for each map f ∈ S, the induced map Map(f,W ) : Map(B,W )→ Map(A,W ) is a weak equivalence of simplicial sets. A map g : X → Y is then an S-local equivalence if for any S-local object W , the induced map Map(g,W ) : Map(Y,W )→ Map(X,W ) is a weak equivalence of simplicial sets. Theorem 2.3. [15, 4.1.1] There is a model structure LSSSets on the category of simplicial spaces in which COMPLETE SEGAL SPACES 5 • the weak equivalences are the S-local equivalences, • the cofibrations are levelwise cofibrations of simplicial sets, and • the fibrant objects are the S-local objects. Furthermore, this model category has the additional structure of a simplicial model category. We now turn to a few facts about simplicial groups, or functors from ∆op to the category of groups. Given a simplicial group G, we can take its nerve, a simplicial space with nerve(G)n,m = Hom([m], Gn). Taking the diagonal of this simplicial space, we obtain a simplicial set, also often called the nerve of G. From another perspective, for G a simplicial group (or, more generally, a sim- plicial monoid), we can find a classifying complex of G, a simplicial set whose geometric realization is the classifying space BG. A precise construction can be made for this classifying space by the WG construction [14, V.4.4], [21]. However, we are not so concerned here with the precise construction as with the fact that such a classifying space exists, so for simplicity we will simply write BG for the classifying complex of G. 3. Simplicial Categories and Simplicial Localizations In this section, we consider simplicial categories and show how they arise from Dwyer and Kan’s simplicial localization techniques. We then discuss model category structures, first on the category of simplicial categories with a fixed object set, and then on the category of all small simplicial categories. First of all, we clarify some terminology. In this paper, by (small) “simplicial category” we will mean a category with a set of objects and a simplicial set of morphisms Map(x, y) between any two objects x and y, also known as a category enriched over simplicial sets. This notion does not coincide with the more general one of a simplicial object in the category of small categories, in which we would also have a simplicial set of objects. Using this more general definition, if we impose the additional condition that all face and degeneracy maps are the identity on the objects, then we get our more restricted notion. A simplicial category can be seen as a generalization of a category, since any ordinary category can be regarded as a simplicial category with a discrete mapping space. Given any simplicial category C, we can consider its category of components π0C, which has the same objects as C and whose morphisms are given by Homπ0C(x, y) = π0MapC(x, y). The following definition of weak equivalence of simplicial categories is a natural generalization of the notion of equivalence of categories. Definition 3.1. A simplicial functor f : C → D is a Dwyer-Kan equivalence or DK-equivalence if the following two conditions hold: (1) For any objects x and y of C, the induced map Map(x, y) → Map(fx, fy) is a weak equivalence of simplicial sets. (2) The induced map on the categories of components π0f : π0C → π0D is an equivalence of categories. 6 J.E. BERGNER The idea of obtaining a simplicial category from a model categoryM goes back to several papers of Dwyer and Kan [8], [11], [12]. In fact, they define two different methods of doing so, the simplicial localization LM [12] and the hammock local- ization LHM [11]. The first has the advantage of being easier to describe, while the second is more convenient for making calculations. It should be noted that these constructions can be made for more general cate- gories with weak equivalences, and do not depend on the model structure if we are willing to ignore the potential set-theoretic difficulties. However, as with the homo- topy category construction, the hammock localization in particular can be defined much more nicely when we have the additional structure of a model category. We begin with the construction of the simplicial localization LM. Recall that, given a category M with some choice of weak equivalences W , we denote the localization ofM with respect toW byW−1M. This localization is obtained from M by formally inverting the maps of W . Further, recall that, given a categoryM, we denote by FM the free category on M, or category with the same objects as M and morphisms freely generated by the non-identity morphisms of M. Note in particular that there are natural functors FM → M and FM → F 2M [12, 2.4]. These functors can be used to define a simplicial resolution F∗M, which is a simplicial category with the category F k+1M at level k [12, 2.5]. We can apply this same construction to the subcategoryW to obtain a simplicial resolution F∗W . Using these two resolutions, we have the following definition. Definition 3.2. [12, 4.1] The simplicial localization ofM with respect toW is the localization (F∗W) −1(F∗M). This simplicial localization is denoted L(M,W) or simply LM. The following result gives interesting information about the mapping spaces in LM in the case where W is all ofM. Proposition 3.3. [12, 5.5] Suppose that W =M and nerve(M) is connected. (i) The simplicial localization LM is a simplicial groupoid, so for all objects x and y, the simplicial sets MapLM(x, y) are all isomorphic. In particular, the simplicial sets MapLM(x, x) are all isomorphic simplicial groups. (ii) The classifying complex BMapLM(x, x) has the homotopy type of nerve(M), and thus each simplicial set MapLM(x, y) has the homotopy type of the loop space Ω(nerve(M)). We now turn to the other construction, that of the hammock localization. Again, letM be a category with a specified subcategory W of weak equivalences. Definition 3.4. [11, 3.1] The hammock localization of M with respect to W , denoted LH(M,W), or simply LHM, is the simplicial category defined as follows: (1) The simplicial category LHM has the same objects asM. (2) Given objects X and Y of M, the simplicial set MapLHM(X,Y ) has as k-simplices the reduced hammocks of width k and any length between X COMPLETE SEGAL SPACES 7 and Y , or commutative diagrams of the form · · · C0,n−1 · · · C1,n−1 ~~~~~~~~~ CCCCCCCCC Ck,1 Ck,2 · · · Ck,n−1 in which (i) the length of the hammock is any integer n ≥ 0, (ii) the vertical maps are all in W , (iii) in each column all the horizontal maps go the same direction, and if they go to the left, then they are in W , (iv) the maps in adjacent columns go in opposite directions, and (v) no column contains only identity maps. Proposition 3.5. [8, 2.2] For a given model categoryM, the simplicial categories LM and LHM are DK-equivalent. We should add that the description of the hammock localization can be greatly simplified if we make use of the model category structure on M. In this case, Dwyer and Kan prove that it suffices to consider hammocks of length 3 such as the following [8, §8]: X C0,1 ≃oo //C0,2 Y ≃oo . Restricting to the category of simplicial categories with a fixed set O of ob- jects, Dwyer and Kan prove the existence of a model structure on this category, which we denote SCO [12, 7.2]. In this situation, the weak equivalences are the DK-equivalences, but with the objects fixed the second condition follows imme- diately from the first. The fibrations in this model structure are given by the functors f : C → D inducing, for any objects x and y, fibrations of simplicial sets MapC(x, y)→ MapD(x, y). The cofibrations are then defined to be the maps with the left lifting property with respect to the acyclic fibrations. However, they can be more precisely charac- terized. To do so, we recall the definition of a free map of simplicial categories. Definition 3.6. [12, 7.4] A map f : C → D in SCO is free if (1) f is a monomorphism, (2) if ∗ denotes the free product, then in each simplicial dimension k, the category Dk admits a unique free factorization Dk = f(Ck) ∗ Fk, where Fk is a free category, and (3) for each k ≥ 0, all degeneracies of generators of Fk are generators of Fk+1. 8 J.E. BERGNER Definition 3.7. [12, 7.5] A map f : C → D of simplicial categories is a strong retract of a map f ′ : C → D′ if there exists a commutative diagram >>}}}}}}} id // D Using these definitions, Dwyer and Kan prove the following result. Proposition 3.8. [12, 7.6] The cofibrations of SCO are precisely the strong retracts of free maps. In particular, a cofibrant simplicial category is a retract of a free category. This result can then be generalized to the category of all simplicial categories, in which the DK-equivalences are the weak equivalences. If C is a simplicial category, a morphism e ∈ HomC(a, b)0 is a homotopy equivalence if it becomes an isomorphism in π0C. Theorem 3.9. [2, 1.1] There is a model category structure SC on the category of small simplicial categories in which • the weak equivalences are the Dwyer-Kan equivalences, and • the fibrations are the maps f : C → D satisfying the following two condi- tions: (i) For any objects a1 and a2 in C, the map HomC(a1, a2)→ HomD(fa1, fa2) is a fibration of simplicial sets. (ii) For any object a1 in C, b in D, and homotopy equivalence e : fa1 → b in D, there is an object a2 in C and homotopy equivalence d : a1 → a2 in C such that fd = e. 4. Complete Segal Spaces Here we define complete Segal spaces and describe Rezk’s model structure on the category of simplicial spaces, in which the complete Segal spaces are the fibrant- cofibrant objects. Recall that by a simplicial space we mean a simplicial object in the category of simplicial sets, or functor ∆op → SSets. In section 2, we described the Reedy model category structure on this category, in which both the weak equivalences and cofibrations are defined levelwise. The model structure CSS is given by a localization of this structure with respect to a set of maps. We begin with the definition of a Segal space. In [23, 4.1], Rezk defines for each 0 ≤ i ≤ n − 1 a map αi : [1] → [n] in ∆ such that 0 7→ i and 1 7→ i + 1. There is a corresponding map αi : ∆[1] → ∆[n]. Then for each n he defines the simplicial space G(n)t = αi∆[1] t ⊂ ∆[n]t. COMPLETE SEGAL SPACES 9 Let X be a Reedy fibrant simplicial space. There is a weak equivalence of simplicial sets MapSSets∆opc (G(n) t, X)→ X1 ×X0 · · · ×X0 X1︸ ︷︷ ︸ where the right hand side is the limit of the diagram d0 // X0 X1 d1oo d0 // . . . d0 // X0 X1 with n copies of X1. Now, given any n, define the map ϕn : G(n)t → ∆[n]t to be the inclusion map. Then for any Reedy fibrant simplicial space W there is a map ϕn = MapSSets∆opc (ϕ n,W ) : MapSSets∆opc (∆[n] t,W )→ MapSSets∆opc (G(n) t,W ). More simply written, this map is ϕn : Wn → W1 ×W0 · · · ×W0 W1︸ ︷︷ ︸ and is often called a Segal map. The Segal map is actually defined for any simplicial space W , but here we assume Reedy fibrancy so that the mapping spaces involved are homotopy invariant. Definition 4.1. [23, 4.1] A Reedy fibrant simplicial space W is a Segal space if for each n ≥ 2 the Segal map ϕn : Wn →W1 ×W0 · · · ×W0 W1 is a weak equivalence of simplicial sets. In fact, there is a model category structure SeSp on the category of simplicial spaces in which the fibrant objects are precisely the Segal spaces [23, 7.1]. This model structure is obtained from the Reedy structure via localization. The idea is that in a Segal space there is a notion of “composition,” at least up to homotopy. In fact, given a Segal space, we can sensibly use many categorical notions. We summarize some of these ideas here; a detailed description is given by Rezk [23]. The objects of a Segal space W are given by the set W0,0. Given the (d1, d0) : W1 →W0 ×W0, themapping space mapW (x, y) is given by the fiber of this map over (x, y). (The fact that W is Reedy fibrant guarantees that this mapping space is homotopy invariant.) Two maps f, g ∈ mapW (x, y)0 are homotopic if they lie in the same component of the simplicial set mapW (x, y). Thus, we define the space of homotopy equivalences Whoequiv ⊆W1 to consist of all the components containing homotopy equivalences. Given any (x0, . . . , xn) ∈ W 0,0 , let mapW (x0, . . . , xn) denote the fiber of the (α0, . . . , αn) : Wn →W over (x0, . . . , xn). Consider the commutative diagram Wn = Map(∆[n] t,W ) ϕk // Map(G(n)t,W ) wwppp Wn+10 10 J.E. BERGNER and notice that, since W is a Segal space, the horizontal arrow is a weak equivalence and a fibration. In particular, this map induces an acyclic fibration on the fibers of the two vertical arrows, mapW (x0, . . . , xn)→ mapW (xn−1, xn)× · · · ×mapW (x0, x1). Given f ∈ map(x, y)0 and g ∈ map(y, z)0, their composite is a lift of (g, f) ∈ map(y, z) × map(x, y) along ϕ2 to some k ∈ map(x, y, z)0. The result of this composition is defined to be d1(k) ∈ map(x, z)0. It can be shown that any two results are homotopic, so we can use g ◦ f unambiguously. Then, the homotopy category of W , denoted Ho(W ), has as objects the elements of the set W0,0, and HomHo(W )(x, y) = π0mapW (x, y). A homotopy equivalence in W is a 0-simplex of W1 whose image in Ho(W ) is an isomorphism. Definition 4.2. A map f : W → Z of Segal spaces is a Dwyer-Kan equivalence if (1) for any objects x and y ofW , the induced map mapW (x, y)→ mapZ(fx, fy) is a weak equivalence of simplicial sets, and (2) the induced map Ho(W )→ Ho(Z) is an equivalence of categories. Notice that the definition of these maps bears a striking resemblance to that of the Dwyer-Kan equivalences between simplicial categories, hence the use of the same name. For a Segal space W , note that the degeneracy map s0 : W0 → W1 factors through the space of homotopy equivalences Whoequiv, since the image of s0 consists of “identity maps.” Given this fact, we are now able to give a definition of complete Segal space. Definition 4.3. [23, §6] A Segal space W is a complete Segal space if the map W0 →Whoequiv given above is a weak equivalence of simplicial sets. The idea behind this notion is that, although W0 is not required to be discrete, as the objects are for a simplicial category, it is not heuristically too different from a simplicial space with discrete 0-space. (This viewpoint is further confirmed by the comparison of complete Segal spaces with Segal categories, which are essentially the analogues of Segal spaces with discrete 0-space [5, 6.3].) Now, we give a description of the model structure CSS. We do not give all the details here, such as a description of an arbitrary weak equivalence, but refer the interested reader to Rezk’s paper [23, §7]. Theorem 4.4. [23, 7.2] There is a model structure CSS on the category of simpli- cial spaces, obtained as localization of the Reedy model structure, such that (1) the fibrant objects are precisely the complete Segal spaces, (2) the cofibrations are the monomorphisms; in particular, every object is cofi- brant, (3) the weak equivalences between Segal spaces are Dwyer-Kan equivalences, (4) the weak equivalences between complete Segal spaces are levelwise weak equivalences of simplicial sets. COMPLETE SEGAL SPACES 11 Furthermore, CSS has the additional structure of a simplicial model category and is cartesian closed. The fact that CSS is cartesian closed allows us to consider, for any complete Segal spaceW and simplicial space X , the complete Segal space WX . In particular, using the simplicial structure, the simplicial set at level n is given by (WX)n = Map(X ×∆[n] t,W ). If W is a (not necessarily complete) Segal space, then WX is again a Segal space; in other words, the model category SeSp is also cartesian closed. We denote the functorial fibrant replacement functor in CSS by LCSS . Thus, given any simplicial space X , there is a weakly equivalent complete Segal space LCSSX . This model structure is connected to the model structure SC by a chain of Quillen equivalences as follows. Each of these model categories is Quillen equivalent to a model structure SeCatf on the category of Segal precategories. A Segal precategory is a simplicial space X with X0 a discrete space. A Segal category is then a Segal precategory with the Segal maps weak equivalences. In the model structure SeCatf , the fibrant objects are Segal categories, and so it is considered a Segal category model structure on the category of Segal precategories. We have the following chain of Quillen equivalences, with the left adjoint functors topmost: SC ⇆ SeCatf ⇄ CSS. The right adjoint SC → SeCatf is given by the nerve functor, and the left adjoint SeCatf → CSS is given by the inclusion functor. There is actually another chain of Quillen equivalences connecting the two model structures; in this case, both SC and CSS are Quillen equivalent to Joyal’s model structure QCat on the category of simplicial sets [18]. The fibrant objects in QCat are quasi-categories, or simplicial sets K such that a dotted arrow lift exists making the diagram V [m, k] // commute for any 0 < k < m. The chain of Quillen equivalences in this case is given SC ⇄ QCat ⇆ CSS. The right adjoint SC → QCat is given by Cordier and Porter’s coherent nerve functor [6], [17, 2.10], and the right adjoint CSS → QCat is given given by sending a simplicial space W to the simplicial set W∗,0 [19, 4.11]. It is a consequence of work of Joyal [17, §1-2] and of Joyal and Tierney [19, §4-5] that the simplicial space obtained from a simplicial category via these functors is weakly equivalent to the one obtained from the composite functor described in the previous paragraph. 5. Obtaining Complete Segal Spaces from Simplicial Categories and Model Categories In this section, we describe several different ways of obtaining a complete Segal space. First, we look at a particularly nice functor which Rezk uses to modify 12 J.E. BERGNER the notion of a nerve of a category. Then we look at how this functor can be generalized to one on any simplicial category, and how a similar idea can be used to get a complete Segal space from any model category. We then consider the functors used in the Quillen equivalences connecting SC and CSS. Let us begin with Rezk’s classifying diagram construction, which associates to any small category C a complete Segal space NC. First, we denote by nerve(C) the ordinary nerve, which is the simplicial set given by (nerve(C))n = Hom([n], C). Further, we denote by iso(C) the maximal subgroupoid of C, or subcategory of C with all objects of C and whose only morphisms are the isomorphisms of C. By C[n] we denote the category of functors [n]→ C, or the category whose objects are n-chains of composable morphisms in C. Definition 5.1. [23, 3.5] The classifying diagram NC is the simplicial space given by (NC)n = nerve(iso(C [n])). Thus, (NC)0 is simply the nerve of iso(C), and (NC)1 is the nerve of the maxi- mal subgroupoid of the morphism category of C. In particular, information about invertible morphisms of C is encoded at level 0, while information about the other morphisms of C does not appear until level 1. Thus, the classifying diagram of a category can be regarded as a more refined version of the nerve, since, unlike the ordinary nerve construction, it enables one to recover information about whether morphisms are invertible or not. This con- struction is also particularly useful for our purposes due to the following result. Proposition 5.2. [23, 6.1] If C is a small category, then its classifying diagram NC is a complete Segal space. However, this construction, as defined above, cannot be used to assign a complete Segal space to any simplicial category, since, beginning with level 1, we would have homotopy invariance problems with a simplicial set of objects in C[1]. Rezk defines an analogous functor, though, from the category of small simplicial categories which is similar in spirit to the classifying diagram but avoids these difficulties. Let I[m] denote the category with m + 1 objects and a single isomorphism between any two objects, and let E(m) = nerve(I[m])t. If W is a Segal space and X is any simplicial space, recall that WX denotes the internal hom object, which is a Segal space. With these notations in place, we can give the definition of Rezk’s completion functor. Let W be a Segal space. Then its completion Ŵ is defined as a fibrant replacement in CSS of the simplicial space W̃ defined by W̃n = diag([m] 7→ MapSSets∆op (E(m),W ∆[n]t)) = diag([m] 7→ (WE(m))n). From a simplicial category C, then, we can takes its nerve to obtain a simplicial space, followed by a fibrant replacement functor in the Segal space model structure, to obtain a Segal space W . From W we can then pass to a complete Segal space via this completion functor. We will denote this complete Segal space LC(W ), or LC(C) where W comes from a simplicial category as just described. The first important fact about this completion functor is that the completion map iW : W → Ŵ = LC(W ) is not only a weak equivalence in the model category CSS, but is also a Dwyer-Kan equivalence of Segal spaces [23, §14]. Furthermore, this completion functor restricts nicely to the classifying diagram in the case where C is a discrete category. COMPLETE SEGAL SPACES 13 Proposition 5.3. [23, 14.2] If C is a discrete category, then LC(C) is isomorphic to NC. If we begin with a model category M with subcategory of weak equivalences W , a functor analogous to the classifying diagram functor can be used to obtain a complete Segal space. In this case, rather than taking the subcategory iso(M) of isomorphisms ofM, we take the subcategory of weak equivalences, denoted we(M). Thus, Rezk defines the classification diagram of (M,W), denoted N(M,W), by N(M,W)n = nerve(we(M [n])). Unlike the classifying diagram, the classification diagram of a model category is not necessarily a complete Segal space as stated, but taking a Reedy fibrant replacement of it results in a complete Segal space, as we show in the next section. Lastly, we have the two functors given by the two different chains of Quillen equivalences between the model categories SC and CSS. As mentioned in the previous section, these two functors are equivalent. In each case, the resulting simplicial space is not Reedy fibrant in general, and so not a complete Segal space, but applying the fibrant replacement functor LCSS results in a complete Segal space. The first of these composite functors, in particular, is simple to describe ab- stractly, as in the previous section, but it has a disadvantage over Rezk’s functor in that it gives very little insight into what the resulting complete Segal space looks like. In the next section, we prove that the two functors from SC to CSS result in weakly equivalent complete Segal spaces, and that if we use Rezk’s classification diagram construction we get a weakly equivalent complete Segal space to the one we would obtain by taking the simplicial localization followed by his completion functor. We then use Rezk’s functor to describe what the complete Segal space corresponding to a simplicial category looks like. 6. Comparison of Functors from SC to CSS Here we prove that each of the functors we have described all give rise to com- plete Segal spaces weakly equivalent to those given in the previous section. We begin by stating the result that establishes the equivalence between Rezk’s com- pletion functor LC : SC → CSS and the functor arising from the chain of Quillen equivalences factoring through SeCatf . Let LCSS denote the functorial fibrant re- placement functor in CSS. Theorem 6.1. If C is a simplicial category, then the complete Segal spaces LC(C) and LCSS(nerve(C)) are weakly equivalent in CSS. Proof. Let LS denote a fibrant replacement functor in the Segal space model struc- ture SeSp on the category of simplicial spaces. The fact that the two functors in question result in weakly equivalent complete Segal spaces can be shown by considering the following chain of weak equivalences: LCSS(nerve(C))← nerve(C)→ LSnerve(C)→ LC(C). The map on the left is the localization functor in CSS and so is a weak equivalence in CSS. The middle map is a weak equivalence in SeSp and therefore also a weak equivalence in CSS, since the latter model category is a localization of the former. The map on the right is Rezk’s completion, and it is a weak equivalence in CSS, as given in the previous section. Therefore, the objects at the far left and right of this 14 J.E. BERGNER zigzag, both of which are complete Segal spaces, are weakly equivalent as objects of CSS. � Now, we would like to compare either of these functors to the classifying diagram construction for a model category M. In other words, we want to show that N(M,W) is equivalent to LC(L HM), where we first take the hammock localization ofM to obtain a simplicial category, and then apply Rezk’s functor LC . An initial problem here is that N(M,W) is not necessarily Reedy fibrant, and so it is not necessarily a complete Segal space. We prove that a Reedy fibrant replacement of it, denoted N(M,W)f , is in fact a complete Segal space in the process of comparing the “mapping spaces” in this Reedy fibrant replacement to the mapping spaces of the hammock localization LHM. Theorem 6.2. Let M be a model category, and let W denote its subcategory of weak equivalences. Then N(M,W)f , is a complete Segal space. Furthermore, for any objects x, y of M, there is a weak equivalence of spaces mapN(M,W)f (x, y) ≃ MapLHM(x, y), and there is an equivalence of categories Ho(N(M,W) f ) ≈ Ho(M). This result was proved by Rezk in the case where M is a simplicial model category [23, 8.3], namely, in the case where we do not need to pass to the simplicial localization ofM to consider its function complexes. However, here we prove that, as he conjectured [23, 8.4], the result continues to hold in this more general case. We prove this theorem very similarly to the way Rezk proves it in the more restricted case, using a proposition of Dwyer and Kan. To begin, we introduce some terminology. LetM be a model category. A classification complex ofM, as defined in [9, 1.2], is the nerve of any subcategory C ofM such that (1) every map in C is a weak equivalence, (2) if f : X → Y inM is a weak equivalence and either X or Y is in C, then f is in C, and (3) nerve(C) is homotopically small; i.e., each homotopy group of |nerve(C)| is small [11, 2.2]. The special classification complex sc(X) of an object X in M is a connected classification complex containing X . LetM be a model category and X a fibrant-cofibrant object ofM. Denote by Auth(X) the simplicial monoid of weak equivalences given by Auth (X) in the hammock localization LHM, and by BAuth(X) its classifying complex. The following proposition was proved by Dwyer and Kan in [9, 2.3] in the case thatM is a simplicial model category. However, the proof does not actually require the simplicial structure; in fact, their proof is essentially the one given below, with the extra step showing that the mapping spaces in the hammock localization are equivalent to those given by the simplicial structure ofM [11, 4.8]. Proposition 6.3. Let X be an object of a model category M. The classifying complex BAuth(X) is weakly equivalent to the special classification complex of X, sc(X), and the two can be connected by a finite zig-zag of weak equivalences. Proof. Let W be the subcategory of weak equivalences of M. Consider the con- nected component of nerve(W) containing X . For the rest of this proof, we assume that W is such that its nerve is connected. We further assume that nerve(W) is homotopically small, taking an appropriate subcategory, as described in [11, 2.3], if necessary. COMPLETE SEGAL SPACES 15 In this case, by Proposition 3.3, the function complexes MapLW(X,X) are all iso- morphic. Furthermore, by the same result, the classifying complex BMapLW(X,X) has the homotopy type of nerve(W). Thus, we can take nerve(W) as sc(X). Now, as in the statement of the proposition, we take Auth(X) to consist of the components of MapLHM(X,X) which are invertible in π0mapL HM(X,X). But, by [11, 4.6(ii)], the map BMapLHW(X,X) → BAut h(X) is a weak equivalence of simplicial sets. Since LHW can be connected to LW by a finite string of weak equivalences, it follows that so can MapLHW(X,X) and MapLW(X,X). Thus, BMapLW(X,X) and BAut h(X) can also be connected by such a string. It follows that sc(X) has the same homotopy type as BAuth(X). � Proof of Theorem 6.2. Consider the category M[n] of functors [n] → M. If M is a model category, then M[n] can be given the structure of a model category with the weak equivalences and fibrations given by levelwise weak equivalences and fibrations inM. Given any map [m]→ [n], we obtain a functorM[n] →M[m]. Let Y = (y0 → y1 · · · → yn) be a fibrant-cofibrant object ofM [n]. It restricts to an object Y ′ = (y0 → y1 · · · → yn−1) inM [n−1]. From this map, we obtain a map of simplicial sets BAuth LHM[n](Y )→ BAut LHM(yn)×BAut LHM[n−1](Y The homotopy fiber of this map is weakly equivalent to the union of those compo- nents of MapLHM(yn−1, yn) containing the conjugates of the map fn−1 : yn−1 → yn, or maps j ◦ fn−1 ◦ i, where i and j are self-homotopy equivalences. Iterating this process, we can take the homotopy fiber of the map BAuth LHM[n](Y )→ BAut LHM(yn)× · · · ×BAut LHM(y0), which is weakly equivalent to the union of the components of MapLHM(yn−1, yn)× · · · ×MapLHM(y0, y1) containing conjugates of the sequence of maps fi : yi → yi+1, 0 ≤ i ≤ n − 1. However, applying Proposition 6.3 to the map in question shows that this simplicial set is also the homotopy fiber of the map sc(Y )→ sc(yn)× · · · × sc(y0). Let U denote the simplicial space N(M,W) so that Un = nerve(we(M [n])). Then, let V be a Reedy fibrant replacement of U , from which we get weak equiva- lences Un → Vn for all n ≥ 0. For each n ≥ 0, there exists a map pn : Un → U 0 given by iterated face maps to the “objects.” Then, for every (n+1)-tuple of objects (x0, x1, . . . , xn), the homotopy fiber of pn over (x0, . . . , xn), given by mapV (xn−1, xn)× · · · ×mapV (x0, x1), is weakly equivalent to MapLHM(x n−1, x n )× · · · ×MapLHM(x 0 , x where xcf denotes a fibrant-cofibrant replacement of X inM. It follows that once we take the Reedy fibrant replacement V of U , it is a Segal space. Now, consider the set π0U0, which consists of the weak equivalence classes of objects inM; it follows that π0V0 is an isomorphic set. Further, note that HomHo(M)(x, y) = π0MapLHM(x cf , ycf). 16 J.E. BERGNER Thus, we have shown that Ho(M) is equivalent to Ho(V ). It remains to show that V is a complete Segal space. Consider the space Vhoequiv ⊆ V1, and define Uhoequiv to be the preimage of Vhoequiv under the natural map U → V . Since V is a Reedy fibrant replacement for U , it suffices to show that the complete Segal space condition holds, i.e., that U0 → Uhoequiv is a weak equivalence of simplicial sets. Notice that Uhoequiv must consist precisely of the components of U1 whose 0-simplices come from weak equivalences inM. In other words, Uhoequiv = nerve(we(we(M)) [1]). There is an adjoint pair of functors F :M[1] ⇄M : G for which F (x→ y) = x and G(x) = idx. This adjoint pair can be restricted to an adjoint pair F : nerve(we(we(M))[1]) ⇄ we(M) : G which in turn induces a weak equivalence of simplicial sets on the nerves, Uhoequiv ≃ U0, which completes the proof. � Now that we have proved that the mapping spaces and homotopy categories agree for V and for LHM, it remains to show that they agree for LHM and LC(L Theorem 6.4. Let M be a model category. For any x and y objects of LHM, there is a weak equivalence of simplicial sets MapLHM(x, y) ≃ mapLC(LHM)(x, y), and there is an equivalence of categories π0L HM≈ Ho(LC(L HM)). Note in particular that x and y are just objects of M, and that π0L HM is equivalent to the homotopy category Ho(M). Proof. Given the hammock localization LHM of the model categoryM, we have the following composite map of simplicial spaces: X = nerve(LHM)→ Xf → LC(L Here, Xf denotes a Reedy fibrant replacement of X . This composite is just Rezk’s method for assigning the complete Segal space LC(L HM) to the simplicial category On the left-hand side, the mapping spaces of X = nerve(LHM) are precisely those of LHM, by the definition of the nerve functor. In the nerve, one of these mapping spaces, say mapX(x, y) for some objects x and y ofM, is given by the fiber over (x, y) of the map (d1, d0) : X1 → X0×X0. Although these mapping spaces can be defined for X , there is no reason that they are homotopy invariant. When we take a Reedy fibrant replacement Xf of X , however, this map becomes a fibration, and hence this fiber is actually a homotopy fiber and so homotopy invariant. For a general simplicial space, we cannot assume that the mapping spaces of the Reedy fibrant replacement are equivalent to the original ones. However, if the 0-space of the simplicial space in question is discrete in degree zero, then the map above is a fibration. Using an argument similar to the one in [5, §5], we can find a Reedy fibrant replacement functor which leaves the 0-space discrete. While the space in degree one might be changed in this process of passing to X 1 , it will still be weakly equivalent X1. In particular, the mapping spaces in X f will be weakly equivalent to those in X . COMPLETE SEGAL SPACES 17 Since the objects of Xf are just the objects of LHM, or the objects ofM, this equivalence of mapping spaces gives us also an equivalence of homotopy categories. The right-most map is the one defined by Rezk, iXf : X f → X̂f , which takes a Segal space to a complete Segal space. But, he defines this map in such a way that it is in fact a Dwyer-Kan equivalence. In other words, it induces weak equivalences on mapping spaces and an equivalence of homotopy categories. Thus, the composite map induces equivalences on mapping spaces and an equivalence on homotopy categories. � 7. A Characterization of Complete Segal Spaces Arising from Simplicial Categories In this section, we give a thorough description of the weak equivalence type of complete Segal spaces which occur as images of Rezk’s functor from the category of simplicial categories. We consider several different cases, beginning with ones for which we can use the classifying diagram construction, i.e., discrete categories, and then proceed to more general simplicial categories. It should be noted that we are characterizing these complete Segal spaces up to weak equivalence, and so the resulting descriptions are of the homotopy type of the spaces in each simplicial degree. For example, in the case of a discrete category, we describe the corresponding complete Segal space in terms of the isomorphism classes of objects, rather than in terms of individual objects, in order to simplify the description. One could just take all objects, and generally get much larger spaces, if the more precise description were needed for the complete Segal space corresponding to a given category. Furthermore, notice that determining the homotopy type of the spaces in degrees zero and one are sufficient to determine the homotopy type of all the spaces, since we are considering Segal spaces. Thus, we focus our attention on these spaces, adding in a few comments about how to continue the process with the higher-degree spaces. 7.1. Case 1: C is a discrete groupoid. If C = G is a group, then applying Rezk’s classifying diagram construction results in a complete Segal space equivalent to BG, i.e., the constant simplicial space which is the simplicial set BG at each level. In particular, since all morphisms are invertible, we obtain essentially no new information at level 1 that we didn’t have already at level 0. Example 7.1. Let G = Z/2. Then (NG)0 is just the nerve, or BZ/2. Then (NG)1 has two 0-simplices, given by the two morphisms (elements) of G. However, these two objects of G[1] are isomorphic, and the automorphism group of either one of them is Z/2. Thus, (NG)1 is also equivalent to BZ/2. If C has more than one object but only one isomorphism class of objects, we get instead a simplicial space weakly equivalent to the constant simplicial space which is BAut(x) at each level, for a representative object x. If C has more than one isomorphism class 〈x〉, then the result will be weakly equivalent to the constant simplicial space 〈x〉 BAut(x). 7.2. Case 2: C is a discrete category. Since in the classifying diagram NC, (NC)0 picks out the isomorphisms of C only, we still essentially have 〈x〉 BAut(x) 18 J.E. BERGNER at level 0. However, if C is not a groupoid, then there is new information at level 1. It instead looks like ∐ 〈x〉,〈y〉 BAut( Hom(x, y)α) where the α index the isomorphism classes of elements of Hom(x, y). The subspace of (NC)1 corresponding to 〈x〉, 〈y〉, denoted (NC)1(x, y), fits into a fibration Hom(x, y)→ (NC)1(x, y)→ BAut(x)×BAut(y). The space in dimension 2 is determined, then, by the spaces at levels 0 and 1. The subspace corresponding to isomorphism classes of objects 〈x〉, 〈y〉, 〈z〉, denoted (NC)2(x, y, z), fits into a fibration Hom(x, y)×Hom(y, z)→ (NC)2(x, y, z)→ BAut(x) ×BAut(y)×BAut(z). The whole space (NC)2, up to homotopy, looks like 〈x〉,〈y〉,〈z〉 〈α〉,〈β〉 Hom(x, y)α ×Hom(y, z)β We could describe each (NC)n analogously. Example 7.2. Let C denote the category with two objects and one nontrivial morphism between them (· → ·). If {e} denotes the trivial group, then (NC)0 ≃ B{e} ∐ B{e} and (NC)1 ≃ B{e} ∐ B{e} ∐ B{e}. In particular, NC is not equiv- alent to the classifying diagram of the trivial category with one object and one morphism, which would be the constant simplicial space B{e}. However, note that the nerves of these two categories are homotopy equivalent. Thus, we can see that the classifying diagram is more refined than the nerve in distinguishing between these two categories. 7.3. Case 3: C is a simplicial groupoid. First, consider the case where we have a simplicial group G. Let Gn denote the group of n-simplices of G. Then hocolim∆op(nerve(Gn) t) = nerve(G). Let LC denote Rezk’s completion functor which makes the nerve into a complete Segal space. We claim that LC(hocolim∆op(nerve(Gn) t)) ≃ LC(hocolim∆opLC(nerve(Gn) We actually prove the more general statement that, for any X = hocolim∆opXn, LC(hocolim∆opXn) ≃ LC(hocolim∆opLCXn). To prove this claim, first note that we have Rezk’s completion map i : hocolim∆opXn → LC(hocolim∆opXn) which is a weak equivalence. Furthermore, since in CSS any complete Segal space Y is a local object and every object is cofibrant, we have a weak equivalence of spaces Map(LC(hocolim∆opXn), Y ) ≃Map(hocolim∆opXn, Y ). COMPLETE SEGAL SPACES 19 So, for any complete Segal space Y , we have that Map(LChocolim∆op(LCXn), Y ) ≃Map(hocolim∆op(LCXn), Y ) ≃ holim∆Map(LCXn, Y ) ≃ holim∆Map(Xn, Y ) ≃Map(hocolim∆opXn, Y ) ≃Map(LChocolim∆opXn, Y ). Note that the above calculation depends on the fact that, Map(hocolim∆opXn, Y ) ≃ holim∆Map(Xn, Y ), which follows from working levelwise on simplicial sets. Then, since Gn is a discrete group, completing its nerve is the same as taking the classifying diagram NGn which, by case 1, is weakly equivalent to the constant simplicial space BGn, denoted here cBGn. Thus we have: LC(nerve(Gn)) ≃ LC [hocolim∆op(nerve(Gn))] ≃ LC [hocolim∆op(LC(nerve(Gn)))] ≃ LC [hocolim∆op(cBGn)] ≃ LC(BG) ≃ BG. So, we obtain a simplicial space weakly equivalent to the constant simplicial space with BG at each level. (Recall, however, that BG here is obtained by taking the diagonal of the simplicial nerve, so it is not quite the identical case.) If we have a simplicial groupoid, rather than a simplicial group, we obtain the analogous result, replacing BG with ∐ BAut(x). 7.4. Case 4: C is a simplicial category with every morphism invertible up to homotopy. Alternatively stated, this case covers the situation in which π0(C) is a groupoid. Recall that we have a model structure SCO on the category of categories with a fixed object set O, in which the cofibrant objects are retracts of free objects. So, taking a cofibrant replacement of C in this model category structure SCO essen- tially gives a free replacement of C, denoted F (C), which is weakly equivalent to C. (This cofibrant category can be obtained by taking a simplicial resolution F∗C and then taking a diagonal [12, 6.1].) Now, taking the localization with respect to all morphisms results in a simplicial groupoid. So, we have Dwyer-Kan equivalences F (C)−1F (C) F (C) ≃oo ≃ //C But, now F (C)−1F (C) is a simplicial groupoid weakly equivalent to C, so we have now reduced this situation to case 3. Note that, to write down a description of this complete Segal space in terms of the original category C, we need to take isomorphism classes of objects in π0(C), or weak equivalence classes, as well as self-maps which are invertible up to homotopy rather than strict automorphisms. While we will still use 〈x〉 to denote the equivalence class of a given object, we will use Auth(x) to signify homotopy automorphisms of 20 J.E. BERGNER x. Thus, the complete Segal space corresponding to C in this case essentially looks like ∐ BAuth(x) at each level. 7.5. Case 5: C is any simplicial category. First consider the subcategory of C containing all the objects of C and only the morphisms of C which are invertible up to homotopy. Apply case 5 to get a complete Segal space, but take only the 0-space of it, to be the 0-space of the desired complete Segal space. To find the 1-space, first recall the definition of the completion functor as applied to a Segal space W : LC(W ) = LCSS(diag([m] 7→ (W E(m))n)). Recall further that (WE(m))n = Map(E(m)×∆[n] t,W ). Thus, the Segal space we obtain (before applying the functor LCSS) looks like Map(E(0)×∆[0]t,W )⇐ Map(E(1)×∆[1]t,W ) ⇚ Map(E(2)×∆[2]t,W ) · · · . If the Segal space W is a fibrant replacement of nerve(C), then the space at level 1 consists of diagrams x′ // y with the maps in the appropriate simplicial level. For simplicity, we restrict to a given pair of objects x and y, representing given equivalence classes. Consider the homotopy automorphisms of x and y. If they are not all invertible, we take a cofibrant replacement and group completion as in case 4. So, without loss of generality, assume that Aut(x) and Aut(y) are simplicial groups. Note that we have Aut(x) = hocolim∆opAut(x)n Aut(y) = hocolim∆opAut(y)n. Now look at Map(x, y) = hocolim∆opMap(x, y)n. Consider for each n ≥ 0 the discrete category C(x, y)n which has as objects Map(x, y)n and as morphisms pairs (α, β) of automorphisms in Aut(x)n × Aut(y)n making a commutative square with f, f ′ ∈Map(x, y)n. Thus, the 1-space that we are interested in is also the 1-space of the complete Segal space given by LCSS(hocolim∆op(nerve(C(x, y)n))). COMPLETE SEGAL SPACES 21 Using a straightforward argument about localization functors similar to the one in case 3 (which can be found in [3, 4.1]), we can also apply the functor LCSS on the inside to get an equivalent simplicial space LCSS(hocolim∆op(LCSSnerve(C(x, y)n))). But, since C(x, y)n is a discrete category, this space is just LCSShocolim∆op(NC(x, y)n) ≃ hocolim∆op(NC(x, y)n). Now, we restrict to the 1-space here, which is hocolim∆op 〈x〉,〈y〉 Mapn(x, y)α 〈x〉,〈y〉 Map(x, y)α As with the previous case, we can then go back and weaken to homotopy auto- morphisms and equivalence classes of objects to consider categories before taking a group completion, so our space looks like 〈x〉,〈y〉 BAuth Map(x, y)α We could then obtain the 2-space of our complete Segal space by considering categories C(x, y, z)n defined similarly, and the description of the 2-space of the classifying diagram of a discrete category as given in case 2. We can summarize these results in the following theorem. For an object x of a simplicial category C, let 〈x〉 denote the weak equivalence class of x in C, and for a morphism α : x → y, let 〈α〉 denote the weak equivalence class of α in the morphism category C[1]. Let Auth(x) denote the space of self-maps of x which are invertible in π0C. Theorem 7.3. Let C be a simplicial category. The complete Segal space corre- sponding to C has the form BAuth(x)⇐ 〈x〉,〈y〉 BAuth Map(x, y)α  ⇚ · · · . References [1] J.E. Bergner, Homotopy fiber products of homotopy theories, in preparation. [2] J.E. Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), 2043-2058. [3] J.E. Bergner, Simplicial monoids and Segal categories, Contemp. Math. 431 (2007) 59-83. [4] J.E. Bergner, A survey of (∞, 1)-categories, preprint available at math.AT/0610239. [5] J.E. Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007), 397-436. [6] J.M. Cordier and T. Porter, Vogt’s theorem on categories of homotopy coherent diagrams, Math. Proc. Camb. Phil. Soc. (1986), 100, 65-90. [7] Daniel Dugger, Universal homotopy theories, Adv. Math. 164 (2001), no. 1, 144–176. [8] W.G. Dwyer and D.M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980), 17-35. [9] W.G. Dwyer and D.M. Kan, A classification theorem for diagrams of simplicial sets, Topology 23(1984), 139-155. [10] W.G. Dwyer and D.M. Kan, Equivalences between homotopy theories of diagrams, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), 180–205, Ann. of Math. Stud., 113, Princeton Univ. Press, Princeton, NJ, 1987. http://arxiv.org/abs/math/0610239 22 J.E. BERGNER [11] W.G. Dwyer and D.M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427-440. [12] W.G. Dwyer and D.M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980), no. 3, 267–284. [13] W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, in Handbook of Algebraic Topology, Elsevier, 1995. [14] P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Math, vol. 174, Birkhauser, 1999. [15] Philip S. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and Monographs 99, AMS, 2003. [16] Mark Hovey, Model Categories, Mathematical Surveys and Monographs, 63. American Math- ematical Society 1999. [17] A. Joyal, Simplicial categories vs quasi-categories, in preparation. [18] A. Joyal, The theory of quasi-categories I, in preparation. [19] André Joyal and Myles Tierney, Quasi-categories vs Segal spaces, Contemp. Math. 431 (2007) 277-326. [20] Saunders Mac Lane, Categories for the Working Mathematician, Second Edition, Graduate Texts in Mathematics 5, Springer-Verlag, 1997. [21] J.P. May, Simplicial Objects in Algebraic Topology, University of Chicago Press, 1967. [22] C.L. Reedy, Homotopy theory of model categories, unpublished manuscript, available at http://www-math.mit.edu/∼psh. [23] Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353(3) (2001), 973-1007. [24] Bertrand Toën, Derived Hall algebras, Duke Math. J. 135, no. 3 (2006), 587-615. Kansas State University, 138 Cardwell Hall Manhattan, KS 66506 E-mail address: bergnerj@member.ams.org http://www-math.mit.edu/~psh 1. Introduction and Overview 2. Background on Model Categories and Simplicial Objects 3. Simplicial Categories and Simplicial Localizations 4. Complete Segal Spaces 5. Obtaining Complete Segal Spaces from Simplicial Categories and Model Categories 6. Comparison of Functors from SC to CSS 7. A Characterization of Complete Segal Spaces Arising from Simplicial Categories 7.1. Case 1: C is a discrete groupoid 7.2. Case 2: C is a discrete category 7.3. Case 3: C is a simplicial groupoid 7.4. Case 4: C is a simplicial category with every morphism invertible up to homotopy 7.5. Case 5: C is any simplicial category References

In this paper, we compare several functors which take simplicial categories or model categories to complete Segal spaces, which are particularly nice simplicial spaces which, like simplicial categories, can be considered to be models for homotopy theories. We then give a characterization, up to weak equivalence, of complete Segal spaces arising from these functors.

Introduction and Overview The idea that simplicial categories, or categories enriched over simplicial sets, model homotopy theories goes back to a series of several papers by Dwyer and Kan [8], [10], [11], [12]. Taking the viewpoint that a model category, or more generally a category with weak equivalences, can be considered to be a model for a homotopy theory, they develop two methods to obtain from a model category a simplicial category. This “simplicial localization” encodes higher-order structure which is lost when we pass to the homotopy category associated to the model category. Furthermore, they prove that, up to a natural notion of weak equivalence of simplicial categories (see Definition 3.1), every simplicial category arises as the simplicial localization of some category with weak equivalences [10, 2.1]. Thus, the category of all (small) simplicial categories with these weak equivalences can be regarded as the “homotopy theory of homotopy theories.” This notion was mentioned briefly at the end of Dwyer and Spalinski’s introduc- tion to model categories [13, 11.6], and was made precise by the author in [2, 1.1], in which the category SC of small simplicial categories with these weak equivalences is shown to have the structure of a model category. However, this category is not practically useful for many purposes. Simplicial categories are not particularly easy objects to work with, and the weak equivalences, while natural generalizations of equivalences of categories, are difficult to identify. Motivated by this problem, Rezk defines a model structure, which we denote CSS, on the category of simplicial spaces, in which the fibrant-cofibrant objects are called complete Segal spaces (see Definition 4.3). This model structure is especially nice, in that the objects are just diagrams of simplicial sets and the weak equivalences, at least between complete Segal spaces, are just levelwise weak equivalences of simplicial sets. Furthermore, this model structure has the additional structures of a simplicial model category and a monoidal model category. Because it is given by a Date: October 25, 2018. 2000 Mathematics Subject Classification. Primary: 55U40; Secondary: 55U35, 18G55, 18G30, 18D20. Key words and phrases. simplicial categories, model categories, complete Segal spaces, homo- topy theories. http://arxiv.org/abs/0704.1624v2 2 J.E. BERGNER localization of a model structure on the category of simplicial spaces with levelwise weak equivalences, it is an example of a presentation for a homotopy theory as described by Dugger [7]. In his paper, Rezk defines two different functors, one from the category of sim- plicial categories, and one from the category of model categories, to the category of complete Segal spaces. Thus, he describes a relationship between simplicial cat- egories and complete Segal spaces, but he does not give an inverse construction. However, the author was able to show in [5] that the model categories SC and CSS are Quillen equivalent to one another, and therefore are models for the same homotopy theory. Thus, the hope is that we can answer questions about simplicial categories by working with complete Segal spaces. This paper is the beginning of that project. However, the functor that we use to show that the two model categories are Quillen equivalent is not the same as Rezk’s functor. Furthermore, the question arises whether Rezk’s functor on model categories agrees with the composite of the simplicial localization functor with his functor on simplicial categories. Rezk gives the beginning of a proof that the two functors agree when the model category in question has the additional structure of a simplicial model category. Our goal in this paper is prove that, up to weak equivalence in CSS, the two functors from SC to CSS are the same, that Rezk’s result holds for model categories which are not necessarily simplicial, and that this result does imply that the functor on model categories agrees with the one on their corresponding simplicial categories up to weak equivalence. We then go on to characterize, up to weak equivalence, the complete Segal space arising from any simplicial category. It can be described at each level as the nerve of the monoid of self weak equivalences of representing objects of the category. We should mention that the model categories SC and CSS are only two of several known models for the homotopy theory of homotopy theories. Our proof that the two are Quillen equivalent actually uses two intermediate model structures SeCatc and SeCatf on the category of Segal precategories, or simplicial spaces with a discrete Space at level zero [5]. The fibrant-cofibrant objects in these structures are known as Segal categories. Furthermore, Joyal and Tierney have shown that there is a Quillen equivalence between each of these model structures and a model structure QCat on the category of simplicial sets [17], [19]. The fibrant-cofibrant objects of this model structure are known as quasi-categories and are generalizations of Kan complexes [18]. One can compare our composite functor SC → CSS to one they define which factors through QCat rather than through SeCatc and SeCatf . It is a consequence of Joyal and Tierney’s work that these two different functors give rise to weakly equivalent complete Segal spaces, as we will describe further in the section on complete Segal spaces. An introduction to each of these model structures and the Quillen equivalences can be found in the survey paper [4]. In [1], we use the results of this paper to consider the complete Segal spaces arising from the homotopy fiber product construction for model categories, as de- scribed by Toën in his work on derived Hall algebras [24]. It seems that results relating diagrams of model categories to diagrams of complete Segal spaces will prove to be useful. COMPLETE SEGAL SPACES 3 Acknowledgments. I would like to thank Bill Dwyer, Charles Rezk, André Joyal, and Myles Tierney for conversations about the work in this paper, as well as the referee for helpful comments on the exposition. 2. Background on Model Categories and Simplicial Objects In this section, we summarize some facts about model categories, simplicial sets, and other simplicial objects that we need in the course of this paper. A model category M is a category with three distinguished classes of mor- phisms, fibrations, cofibrations, and weak equivalences. A morphism which is both a (co)fibration and a weak equivalence is called an acyclic (co)fibration. The cate- goryM with these choices of classes is required to satisfy five axioms [13, 3.3]. Axiom MC1 guarantees that M has small limits and colimits, so in particular M has an initial object and a terminal object. An object X in a model category M is fibrant if the unique map X → ∗ to the terminal object is a fibration. Dually, X is cofibrant if the unique map from the initial object φ→ X is a cofibration. The factorization axiom (MC5) can be applied in such a way that, given any object X ofM, we can factor the map X → ∗ as a composite ∼ //X ′ //∗ of an acyclic cofibration followed by a fibration. In this case, X ′ is called a fibrant replacement of X , since it is weakly equivalent to X and fibrant. This replacement is not necessarily unique, but in all the model categories we consider here it can be assumed to be functorial [16, 1.1.1]. Cofibrant replacements can be defined dually. The structure of a model category enables us to invert the weak equivalences formally in such a way that we still have a set, rather than a proper class, of mor- phisms between any two objects. If we were merely to take the localizationW−1M, there would be no guarantee that we would not have a proper class. However, with the structure of a model category, we can define the homotopy category Ho(M) to have the same objects asM, and as morphisms HomHo(M)(X,Y ) = [X cf , Y cf ]M, where the right-hand side denotes homotopy classes of maps between fibrant-cofibrant replacements of X and Y , respectively. The standard notion of equivalence of model categories is given by the following definitions. First, recall that an adjoint pair of functors F : C ⇆ D : G satisfies the property that, for any objects X of C and Y of D, there is a natural isomorphism ϕ : HomD(FX, Y )→ HomC(X,GY ). The functor F is called the left adjoint and G the right adjoint [20, IV.1]. Definition 2.1. [16, 1.3.1] An adjoint pair of functors F : M ⇆ N : G between model categories is a Quillen pair if F preserves cofibrations and G preserves fibra- tions. Definition 2.2. [16, 1.3.12] A Quillen pair of model categories is a Quillen equiv- alence if if for all cofibrant X in M and fibrant Y in N , a map f : FX → Y is a weak equivalence in D if and only if the map ϕf : X → GY is a weak equivalence 4 J.E. BERGNER An important example of a model category is that of the standard model struc- ture on the category of simplicial sets SSets. Recall that a simplicial set is a functor X : ∆op → Sets, where ∆op is the opposite of the category ∆ of finite ordered sets [n] = {0→ 1→ · · · → n} and order-preserving maps between them. We denote the set X([n]) by Xn. In particular in X we have face maps di : Xn → Xn−1 and de- generacy maps si : Xn → Xn+1 for each 0 ≤ i ≤ n, satisfying several compatibility conditions. Three particularly useful examples of simplicial sets are the n-simplex ∆[n] and its boundary ∆̇[n] for each n ≥ 0, and the boundary with the kth face removed, V [n, k], for each n ≥ 1 and 0 ≤ k ≤ n. Given a simplicial set X , we can take its geometric realization |X |, which is a topological space [14, I.1]. In the standard model category structure on SSets, the weak equivalences are the maps f : X → Y for which the geometric realization |f | : |X | → |Y | is a weak homotopy equivalence of topological spaces [14, I.11.3]. In fact, this model structure on simplicial sets is Quillen equivalent to the standard model structure on the category of topological spaces [16, 3.6.7]. More generally, a simplicial object in a category C is a functor ∆op → C. The two main examples which we consider in this paper are those of simplicial spaces (also called bisimplicial sets), or functors ∆op → SSets, and simplicial groups. We denote the category of simplicial spaces by SSets∆ A simplicial set X can be regarded as a simplicial space in two ways. It can be considered a constant simplicial space with the simplicial set X at each level, and in this case we will denote the constant simplicial set by cX or just X if no confusion will arise. Alternatively, we can take the simplicial space, which we denote Xt, for which (Xt)n is the discrete simplicial set Xn. The superscript t is meant to suggest that this simplicial space is the “transpose” of the constant simplicial space. A natural choice for the weak equivalences in the category SSets∆ is the class of levelwise weak equivalences of simplicial sets. If we define the cofibrations to be levelwise also, we obtain a model structure which is usually referred to as the Reedy model structure on SSets∆ [22]. The Reedy model structure has the additional structure of a simplicial model category. In particular, given any two objects X and Y of SSets∆ , there is a mapping space, or simplicial set Map(X,Y ) satisfying compatibility conditions [15, 9.1.6]. In the case where X is cofibrant (as is true of all objects in the Reedy model structure) and Y is fibrant, this choice of mapping space is homotopy invariant. One way to obtain other model structures on the category of simplicial spaces is to localize the Reedy structure with respect to a set of maps. While this process works for much more general model categories [15, 3.3.1], we will focus here on this particular case. Let S = {f : A → B} be a set of maps of simplicial spaces. A Reedy fibrant simplicial space W is S-local if for each map f ∈ S, the induced map Map(f,W ) : Map(B,W )→ Map(A,W ) is a weak equivalence of simplicial sets. A map g : X → Y is then an S-local equivalence if for any S-local object W , the induced map Map(g,W ) : Map(Y,W )→ Map(X,W ) is a weak equivalence of simplicial sets. Theorem 2.3. [15, 4.1.1] There is a model structure LSSSets on the category of simplicial spaces in which COMPLETE SEGAL SPACES 5 • the weak equivalences are the S-local equivalences, • the cofibrations are levelwise cofibrations of simplicial sets, and • the fibrant objects are the S-local objects. Furthermore, this model category has the additional structure of a simplicial model category. We now turn to a few facts about simplicial groups, or functors from ∆op to the category of groups. Given a simplicial group G, we can take its nerve, a simplicial space with nerve(G)n,m = Hom([m], Gn). Taking the diagonal of this simplicial space, we obtain a simplicial set, also often called the nerve of G. From another perspective, for G a simplicial group (or, more generally, a sim- plicial monoid), we can find a classifying complex of G, a simplicial set whose geometric realization is the classifying space BG. A precise construction can be made for this classifying space by the WG construction [14, V.4.4], [21]. However, we are not so concerned here with the precise construction as with the fact that such a classifying space exists, so for simplicity we will simply write BG for the classifying complex of G. 3. Simplicial Categories and Simplicial Localizations In this section, we consider simplicial categories and show how they arise from Dwyer and Kan’s simplicial localization techniques. We then discuss model category structures, first on the category of simplicial categories with a fixed object set, and then on the category of all small simplicial categories. First of all, we clarify some terminology. In this paper, by (small) “simplicial category” we will mean a category with a set of objects and a simplicial set of morphisms Map(x, y) between any two objects x and y, also known as a category enriched over simplicial sets. This notion does not coincide with the more general one of a simplicial object in the category of small categories, in which we would also have a simplicial set of objects. Using this more general definition, if we impose the additional condition that all face and degeneracy maps are the identity on the objects, then we get our more restricted notion. A simplicial category can be seen as a generalization of a category, since any ordinary category can be regarded as a simplicial category with a discrete mapping space. Given any simplicial category C, we can consider its category of components π0C, which has the same objects as C and whose morphisms are given by Homπ0C(x, y) = π0MapC(x, y). The following definition of weak equivalence of simplicial categories is a natural generalization of the notion of equivalence of categories. Definition 3.1. A simplicial functor f : C → D is a Dwyer-Kan equivalence or DK-equivalence if the following two conditions hold: (1) For any objects x and y of C, the induced map Map(x, y) → Map(fx, fy) is a weak equivalence of simplicial sets. (2) The induced map on the categories of components π0f : π0C → π0D is an equivalence of categories. 6 J.E. BERGNER The idea of obtaining a simplicial category from a model categoryM goes back to several papers of Dwyer and Kan [8], [11], [12]. In fact, they define two different methods of doing so, the simplicial localization LM [12] and the hammock local- ization LHM [11]. The first has the advantage of being easier to describe, while the second is more convenient for making calculations. It should be noted that these constructions can be made for more general cate- gories with weak equivalences, and do not depend on the model structure if we are willing to ignore the potential set-theoretic difficulties. However, as with the homo- topy category construction, the hammock localization in particular can be defined much more nicely when we have the additional structure of a model category. We begin with the construction of the simplicial localization LM. Recall that, given a category M with some choice of weak equivalences W , we denote the localization ofM with respect toW byW−1M. This localization is obtained from M by formally inverting the maps of W . Further, recall that, given a categoryM, we denote by FM the free category on M, or category with the same objects as M and morphisms freely generated by the non-identity morphisms of M. Note in particular that there are natural functors FM → M and FM → F 2M [12, 2.4]. These functors can be used to define a simplicial resolution F∗M, which is a simplicial category with the category F k+1M at level k [12, 2.5]. We can apply this same construction to the subcategoryW to obtain a simplicial resolution F∗W . Using these two resolutions, we have the following definition. Definition 3.2. [12, 4.1] The simplicial localization ofM with respect toW is the localization (F∗W) −1(F∗M). This simplicial localization is denoted L(M,W) or simply LM. The following result gives interesting information about the mapping spaces in LM in the case where W is all ofM. Proposition 3.3. [12, 5.5] Suppose that W =M and nerve(M) is connected. (i) The simplicial localization LM is a simplicial groupoid, so for all objects x and y, the simplicial sets MapLM(x, y) are all isomorphic. In particular, the simplicial sets MapLM(x, x) are all isomorphic simplicial groups. (ii) The classifying complex BMapLM(x, x) has the homotopy type of nerve(M), and thus each simplicial set MapLM(x, y) has the homotopy type of the loop space Ω(nerve(M)). We now turn to the other construction, that of the hammock localization. Again, letM be a category with a specified subcategory W of weak equivalences. Definition 3.4. [11, 3.1] The hammock localization of M with respect to W , denoted LH(M,W), or simply LHM, is the simplicial category defined as follows: (1) The simplicial category LHM has the same objects asM. (2) Given objects X and Y of M, the simplicial set MapLHM(X,Y ) has as k-simplices the reduced hammocks of width k and any length between X COMPLETE SEGAL SPACES 7 and Y , or commutative diagrams of the form · · · C0,n−1 · · · C1,n−1 ~~~~~~~~~ CCCCCCCCC Ck,1 Ck,2 · · · Ck,n−1 in which (i) the length of the hammock is any integer n ≥ 0, (ii) the vertical maps are all in W , (iii) in each column all the horizontal maps go the same direction, and if they go to the left, then they are in W , (iv) the maps in adjacent columns go in opposite directions, and (v) no column contains only identity maps. Proposition 3.5. [8, 2.2] For a given model categoryM, the simplicial categories LM and LHM are DK-equivalent. We should add that the description of the hammock localization can be greatly simplified if we make use of the model category structure on M. In this case, Dwyer and Kan prove that it suffices to consider hammocks of length 3 such as the following [8, §8]: X C0,1 ≃oo //C0,2 Y ≃oo . Restricting to the category of simplicial categories with a fixed set O of ob- jects, Dwyer and Kan prove the existence of a model structure on this category, which we denote SCO [12, 7.2]. In this situation, the weak equivalences are the DK-equivalences, but with the objects fixed the second condition follows imme- diately from the first. The fibrations in this model structure are given by the functors f : C → D inducing, for any objects x and y, fibrations of simplicial sets MapC(x, y)→ MapD(x, y). The cofibrations are then defined to be the maps with the left lifting property with respect to the acyclic fibrations. However, they can be more precisely charac- terized. To do so, we recall the definition of a free map of simplicial categories. Definition 3.6. [12, 7.4] A map f : C → D in SCO is free if (1) f is a monomorphism, (2) if ∗ denotes the free product, then in each simplicial dimension k, the category Dk admits a unique free factorization Dk = f(Ck) ∗ Fk, where Fk is a free category, and (3) for each k ≥ 0, all degeneracies of generators of Fk are generators of Fk+1. 8 J.E. BERGNER Definition 3.7. [12, 7.5] A map f : C → D of simplicial categories is a strong retract of a map f ′ : C → D′ if there exists a commutative diagram >>}}}}}}} id // D Using these definitions, Dwyer and Kan prove the following result. Proposition 3.8. [12, 7.6] The cofibrations of SCO are precisely the strong retracts of free maps. In particular, a cofibrant simplicial category is a retract of a free category. This result can then be generalized to the category of all simplicial categories, in which the DK-equivalences are the weak equivalences. If C is a simplicial category, a morphism e ∈ HomC(a, b)0 is a homotopy equivalence if it becomes an isomorphism in π0C. Theorem 3.9. [2, 1.1] There is a model category structure SC on the category of small simplicial categories in which • the weak equivalences are the Dwyer-Kan equivalences, and • the fibrations are the maps f : C → D satisfying the following two condi- tions: (i) For any objects a1 and a2 in C, the map HomC(a1, a2)→ HomD(fa1, fa2) is a fibration of simplicial sets. (ii) For any object a1 in C, b in D, and homotopy equivalence e : fa1 → b in D, there is an object a2 in C and homotopy equivalence d : a1 → a2 in C such that fd = e. 4. Complete Segal Spaces Here we define complete Segal spaces and describe Rezk’s model structure on the category of simplicial spaces, in which the complete Segal spaces are the fibrant- cofibrant objects. Recall that by a simplicial space we mean a simplicial object in the category of simplicial sets, or functor ∆op → SSets. In section 2, we described the Reedy model category structure on this category, in which both the weak equivalences and cofibrations are defined levelwise. The model structure CSS is given by a localization of this structure with respect to a set of maps. We begin with the definition of a Segal space. In [23, 4.1], Rezk defines for each 0 ≤ i ≤ n − 1 a map αi : [1] → [n] in ∆ such that 0 7→ i and 1 7→ i + 1. There is a corresponding map αi : ∆[1] → ∆[n]. Then for each n he defines the simplicial space G(n)t = αi∆[1] t ⊂ ∆[n]t. COMPLETE SEGAL SPACES 9 Let X be a Reedy fibrant simplicial space. There is a weak equivalence of simplicial sets MapSSets∆opc (G(n) t, X)→ X1 ×X0 · · · ×X0 X1︸ ︷︷ ︸ where the right hand side is the limit of the diagram d0 // X0 X1 d1oo d0 // . . . d0 // X0 X1 with n copies of X1. Now, given any n, define the map ϕn : G(n)t → ∆[n]t to be the inclusion map. Then for any Reedy fibrant simplicial space W there is a map ϕn = MapSSets∆opc (ϕ n,W ) : MapSSets∆opc (∆[n] t,W )→ MapSSets∆opc (G(n) t,W ). More simply written, this map is ϕn : Wn → W1 ×W0 · · · ×W0 W1︸ ︷︷ ︸ and is often called a Segal map. The Segal map is actually defined for any simplicial space W , but here we assume Reedy fibrancy so that the mapping spaces involved are homotopy invariant. Definition 4.1. [23, 4.1] A Reedy fibrant simplicial space W is a Segal space if for each n ≥ 2 the Segal map ϕn : Wn →W1 ×W0 · · · ×W0 W1 is a weak equivalence of simplicial sets. In fact, there is a model category structure SeSp on the category of simplicial spaces in which the fibrant objects are precisely the Segal spaces [23, 7.1]. This model structure is obtained from the Reedy structure via localization. The idea is that in a Segal space there is a notion of “composition,” at least up to homotopy. In fact, given a Segal space, we can sensibly use many categorical notions. We summarize some of these ideas here; a detailed description is given by Rezk [23]. The objects of a Segal space W are given by the set W0,0. Given the (d1, d0) : W1 →W0 ×W0, themapping space mapW (x, y) is given by the fiber of this map over (x, y). (The fact that W is Reedy fibrant guarantees that this mapping space is homotopy invariant.) Two maps f, g ∈ mapW (x, y)0 are homotopic if they lie in the same component of the simplicial set mapW (x, y). Thus, we define the space of homotopy equivalences Whoequiv ⊆W1 to consist of all the components containing homotopy equivalences. Given any (x0, . . . , xn) ∈ W 0,0 , let mapW (x0, . . . , xn) denote the fiber of the (α0, . . . , αn) : Wn →W over (x0, . . . , xn). Consider the commutative diagram Wn = Map(∆[n] t,W ) ϕk // Map(G(n)t,W ) wwppp Wn+10 10 J.E. BERGNER and notice that, since W is a Segal space, the horizontal arrow is a weak equivalence and a fibration. In particular, this map induces an acyclic fibration on the fibers of the two vertical arrows, mapW (x0, . . . , xn)→ mapW (xn−1, xn)× · · · ×mapW (x0, x1). Given f ∈ map(x, y)0 and g ∈ map(y, z)0, their composite is a lift of (g, f) ∈ map(y, z) × map(x, y) along ϕ2 to some k ∈ map(x, y, z)0. The result of this composition is defined to be d1(k) ∈ map(x, z)0. It can be shown that any two results are homotopic, so we can use g ◦ f unambiguously. Then, the homotopy category of W , denoted Ho(W ), has as objects the elements of the set W0,0, and HomHo(W )(x, y) = π0mapW (x, y). A homotopy equivalence in W is a 0-simplex of W1 whose image in Ho(W ) is an isomorphism. Definition 4.2. A map f : W → Z of Segal spaces is a Dwyer-Kan equivalence if (1) for any objects x and y ofW , the induced map mapW (x, y)→ mapZ(fx, fy) is a weak equivalence of simplicial sets, and (2) the induced map Ho(W )→ Ho(Z) is an equivalence of categories. Notice that the definition of these maps bears a striking resemblance to that of the Dwyer-Kan equivalences between simplicial categories, hence the use of the same name. For a Segal space W , note that the degeneracy map s0 : W0 → W1 factors through the space of homotopy equivalences Whoequiv, since the image of s0 consists of “identity maps.” Given this fact, we are now able to give a definition of complete Segal space. Definition 4.3. [23, §6] A Segal space W is a complete Segal space if the map W0 →Whoequiv given above is a weak equivalence of simplicial sets. The idea behind this notion is that, although W0 is not required to be discrete, as the objects are for a simplicial category, it is not heuristically too different from a simplicial space with discrete 0-space. (This viewpoint is further confirmed by the comparison of complete Segal spaces with Segal categories, which are essentially the analogues of Segal spaces with discrete 0-space [5, 6.3].) Now, we give a description of the model structure CSS. We do not give all the details here, such as a description of an arbitrary weak equivalence, but refer the interested reader to Rezk’s paper [23, §7]. Theorem 4.4. [23, 7.2] There is a model structure CSS on the category of simpli- cial spaces, obtained as localization of the Reedy model structure, such that (1) the fibrant objects are precisely the complete Segal spaces, (2) the cofibrations are the monomorphisms; in particular, every object is cofi- brant, (3) the weak equivalences between Segal spaces are Dwyer-Kan equivalences, (4) the weak equivalences between complete Segal spaces are levelwise weak equivalences of simplicial sets. COMPLETE SEGAL SPACES 11 Furthermore, CSS has the additional structure of a simplicial model category and is cartesian closed. The fact that CSS is cartesian closed allows us to consider, for any complete Segal spaceW and simplicial space X , the complete Segal space WX . In particular, using the simplicial structure, the simplicial set at level n is given by (WX)n = Map(X ×∆[n] t,W ). If W is a (not necessarily complete) Segal space, then WX is again a Segal space; in other words, the model category SeSp is also cartesian closed. We denote the functorial fibrant replacement functor in CSS by LCSS . Thus, given any simplicial space X , there is a weakly equivalent complete Segal space LCSSX . This model structure is connected to the model structure SC by a chain of Quillen equivalences as follows. Each of these model categories is Quillen equivalent to a model structure SeCatf on the category of Segal precategories. A Segal precategory is a simplicial space X with X0 a discrete space. A Segal category is then a Segal precategory with the Segal maps weak equivalences. In the model structure SeCatf , the fibrant objects are Segal categories, and so it is considered a Segal category model structure on the category of Segal precategories. We have the following chain of Quillen equivalences, with the left adjoint functors topmost: SC ⇆ SeCatf ⇄ CSS. The right adjoint SC → SeCatf is given by the nerve functor, and the left adjoint SeCatf → CSS is given by the inclusion functor. There is actually another chain of Quillen equivalences connecting the two model structures; in this case, both SC and CSS are Quillen equivalent to Joyal’s model structure QCat on the category of simplicial sets [18]. The fibrant objects in QCat are quasi-categories, or simplicial sets K such that a dotted arrow lift exists making the diagram V [m, k] // commute for any 0 < k < m. The chain of Quillen equivalences in this case is given SC ⇄ QCat ⇆ CSS. The right adjoint SC → QCat is given by Cordier and Porter’s coherent nerve functor [6], [17, 2.10], and the right adjoint CSS → QCat is given given by sending a simplicial space W to the simplicial set W∗,0 [19, 4.11]. It is a consequence of work of Joyal [17, §1-2] and of Joyal and Tierney [19, §4-5] that the simplicial space obtained from a simplicial category via these functors is weakly equivalent to the one obtained from the composite functor described in the previous paragraph. 5. Obtaining Complete Segal Spaces from Simplicial Categories and Model Categories In this section, we describe several different ways of obtaining a complete Segal space. First, we look at a particularly nice functor which Rezk uses to modify 12 J.E. BERGNER the notion of a nerve of a category. Then we look at how this functor can be generalized to one on any simplicial category, and how a similar idea can be used to get a complete Segal space from any model category. We then consider the functors used in the Quillen equivalences connecting SC and CSS. Let us begin with Rezk’s classifying diagram construction, which associates to any small category C a complete Segal space NC. First, we denote by nerve(C) the ordinary nerve, which is the simplicial set given by (nerve(C))n = Hom([n], C). Further, we denote by iso(C) the maximal subgroupoid of C, or subcategory of C with all objects of C and whose only morphisms are the isomorphisms of C. By C[n] we denote the category of functors [n]→ C, or the category whose objects are n-chains of composable morphisms in C. Definition 5.1. [23, 3.5] The classifying diagram NC is the simplicial space given by (NC)n = nerve(iso(C [n])). Thus, (NC)0 is simply the nerve of iso(C), and (NC)1 is the nerve of the maxi- mal subgroupoid of the morphism category of C. In particular, information about invertible morphisms of C is encoded at level 0, while information about the other morphisms of C does not appear until level 1. Thus, the classifying diagram of a category can be regarded as a more refined version of the nerve, since, unlike the ordinary nerve construction, it enables one to recover information about whether morphisms are invertible or not. This con- struction is also particularly useful for our purposes due to the following result. Proposition 5.2. [23, 6.1] If C is a small category, then its classifying diagram NC is a complete Segal space. However, this construction, as defined above, cannot be used to assign a complete Segal space to any simplicial category, since, beginning with level 1, we would have homotopy invariance problems with a simplicial set of objects in C[1]. Rezk defines an analogous functor, though, from the category of small simplicial categories which is similar in spirit to the classifying diagram but avoids these difficulties. Let I[m] denote the category with m + 1 objects and a single isomorphism between any two objects, and let E(m) = nerve(I[m])t. If W is a Segal space and X is any simplicial space, recall that WX denotes the internal hom object, which is a Segal space. With these notations in place, we can give the definition of Rezk’s completion functor. Let W be a Segal space. Then its completion Ŵ is defined as a fibrant replacement in CSS of the simplicial space W̃ defined by W̃n = diag([m] 7→ MapSSets∆op (E(m),W ∆[n]t)) = diag([m] 7→ (WE(m))n). From a simplicial category C, then, we can takes its nerve to obtain a simplicial space, followed by a fibrant replacement functor in the Segal space model structure, to obtain a Segal space W . From W we can then pass to a complete Segal space via this completion functor. We will denote this complete Segal space LC(W ), or LC(C) where W comes from a simplicial category as just described. The first important fact about this completion functor is that the completion map iW : W → Ŵ = LC(W ) is not only a weak equivalence in the model category CSS, but is also a Dwyer-Kan equivalence of Segal spaces [23, §14]. Furthermore, this completion functor restricts nicely to the classifying diagram in the case where C is a discrete category. COMPLETE SEGAL SPACES 13 Proposition 5.3. [23, 14.2] If C is a discrete category, then LC(C) is isomorphic to NC. If we begin with a model category M with subcategory of weak equivalences W , a functor analogous to the classifying diagram functor can be used to obtain a complete Segal space. In this case, rather than taking the subcategory iso(M) of isomorphisms ofM, we take the subcategory of weak equivalences, denoted we(M). Thus, Rezk defines the classification diagram of (M,W), denoted N(M,W), by N(M,W)n = nerve(we(M [n])). Unlike the classifying diagram, the classification diagram of a model category is not necessarily a complete Segal space as stated, but taking a Reedy fibrant replacement of it results in a complete Segal space, as we show in the next section. Lastly, we have the two functors given by the two different chains of Quillen equivalences between the model categories SC and CSS. As mentioned in the previous section, these two functors are equivalent. In each case, the resulting simplicial space is not Reedy fibrant in general, and so not a complete Segal space, but applying the fibrant replacement functor LCSS results in a complete Segal space. The first of these composite functors, in particular, is simple to describe ab- stractly, as in the previous section, but it has a disadvantage over Rezk’s functor in that it gives very little insight into what the resulting complete Segal space looks like. In the next section, we prove that the two functors from SC to CSS result in weakly equivalent complete Segal spaces, and that if we use Rezk’s classification diagram construction we get a weakly equivalent complete Segal space to the one we would obtain by taking the simplicial localization followed by his completion functor. We then use Rezk’s functor to describe what the complete Segal space corresponding to a simplicial category looks like. 6. Comparison of Functors from SC to CSS Here we prove that each of the functors we have described all give rise to com- plete Segal spaces weakly equivalent to those given in the previous section. We begin by stating the result that establishes the equivalence between Rezk’s com- pletion functor LC : SC → CSS and the functor arising from the chain of Quillen equivalences factoring through SeCatf . Let LCSS denote the functorial fibrant re- placement functor in CSS. Theorem 6.1. If C is a simplicial category, then the complete Segal spaces LC(C) and LCSS(nerve(C)) are weakly equivalent in CSS. Proof. Let LS denote a fibrant replacement functor in the Segal space model struc- ture SeSp on the category of simplicial spaces. The fact that the two functors in question result in weakly equivalent complete Segal spaces can be shown by considering the following chain of weak equivalences: LCSS(nerve(C))← nerve(C)→ LSnerve(C)→ LC(C). The map on the left is the localization functor in CSS and so is a weak equivalence in CSS. The middle map is a weak equivalence in SeSp and therefore also a weak equivalence in CSS, since the latter model category is a localization of the former. The map on the right is Rezk’s completion, and it is a weak equivalence in CSS, as given in the previous section. Therefore, the objects at the far left and right of this 14 J.E. BERGNER zigzag, both of which are complete Segal spaces, are weakly equivalent as objects of CSS. � Now, we would like to compare either of these functors to the classifying diagram construction for a model category M. In other words, we want to show that N(M,W) is equivalent to LC(L HM), where we first take the hammock localization ofM to obtain a simplicial category, and then apply Rezk’s functor LC . An initial problem here is that N(M,W) is not necessarily Reedy fibrant, and so it is not necessarily a complete Segal space. We prove that a Reedy fibrant replacement of it, denoted N(M,W)f , is in fact a complete Segal space in the process of comparing the “mapping spaces” in this Reedy fibrant replacement to the mapping spaces of the hammock localization LHM. Theorem 6.2. Let M be a model category, and let W denote its subcategory of weak equivalences. Then N(M,W)f , is a complete Segal space. Furthermore, for any objects x, y of M, there is a weak equivalence of spaces mapN(M,W)f (x, y) ≃ MapLHM(x, y), and there is an equivalence of categories Ho(N(M,W) f ) ≈ Ho(M). This result was proved by Rezk in the case where M is a simplicial model category [23, 8.3], namely, in the case where we do not need to pass to the simplicial localization ofM to consider its function complexes. However, here we prove that, as he conjectured [23, 8.4], the result continues to hold in this more general case. We prove this theorem very similarly to the way Rezk proves it in the more restricted case, using a proposition of Dwyer and Kan. To begin, we introduce some terminology. LetM be a model category. A classification complex ofM, as defined in [9, 1.2], is the nerve of any subcategory C ofM such that (1) every map in C is a weak equivalence, (2) if f : X → Y inM is a weak equivalence and either X or Y is in C, then f is in C, and (3) nerve(C) is homotopically small; i.e., each homotopy group of |nerve(C)| is small [11, 2.2]. The special classification complex sc(X) of an object X in M is a connected classification complex containing X . LetM be a model category and X a fibrant-cofibrant object ofM. Denote by Auth(X) the simplicial monoid of weak equivalences given by Auth (X) in the hammock localization LHM, and by BAuth(X) its classifying complex. The following proposition was proved by Dwyer and Kan in [9, 2.3] in the case thatM is a simplicial model category. However, the proof does not actually require the simplicial structure; in fact, their proof is essentially the one given below, with the extra step showing that the mapping spaces in the hammock localization are equivalent to those given by the simplicial structure ofM [11, 4.8]. Proposition 6.3. Let X be an object of a model category M. The classifying complex BAuth(X) is weakly equivalent to the special classification complex of X, sc(X), and the two can be connected by a finite zig-zag of weak equivalences. Proof. Let W be the subcategory of weak equivalences of M. Consider the con- nected component of nerve(W) containing X . For the rest of this proof, we assume that W is such that its nerve is connected. We further assume that nerve(W) is homotopically small, taking an appropriate subcategory, as described in [11, 2.3], if necessary. COMPLETE SEGAL SPACES 15 In this case, by Proposition 3.3, the function complexes MapLW(X,X) are all iso- morphic. Furthermore, by the same result, the classifying complex BMapLW(X,X) has the homotopy type of nerve(W). Thus, we can take nerve(W) as sc(X). Now, as in the statement of the proposition, we take Auth(X) to consist of the components of MapLHM(X,X) which are invertible in π0mapL HM(X,X). But, by [11, 4.6(ii)], the map BMapLHW(X,X) → BAut h(X) is a weak equivalence of simplicial sets. Since LHW can be connected to LW by a finite string of weak equivalences, it follows that so can MapLHW(X,X) and MapLW(X,X). Thus, BMapLW(X,X) and BAut h(X) can also be connected by such a string. It follows that sc(X) has the same homotopy type as BAuth(X). � Proof of Theorem 6.2. Consider the category M[n] of functors [n] → M. If M is a model category, then M[n] can be given the structure of a model category with the weak equivalences and fibrations given by levelwise weak equivalences and fibrations inM. Given any map [m]→ [n], we obtain a functorM[n] →M[m]. Let Y = (y0 → y1 · · · → yn) be a fibrant-cofibrant object ofM [n]. It restricts to an object Y ′ = (y0 → y1 · · · → yn−1) inM [n−1]. From this map, we obtain a map of simplicial sets BAuth LHM[n](Y )→ BAut LHM(yn)×BAut LHM[n−1](Y The homotopy fiber of this map is weakly equivalent to the union of those compo- nents of MapLHM(yn−1, yn) containing the conjugates of the map fn−1 : yn−1 → yn, or maps j ◦ fn−1 ◦ i, where i and j are self-homotopy equivalences. Iterating this process, we can take the homotopy fiber of the map BAuth LHM[n](Y )→ BAut LHM(yn)× · · · ×BAut LHM(y0), which is weakly equivalent to the union of the components of MapLHM(yn−1, yn)× · · · ×MapLHM(y0, y1) containing conjugates of the sequence of maps fi : yi → yi+1, 0 ≤ i ≤ n − 1. However, applying Proposition 6.3 to the map in question shows that this simplicial set is also the homotopy fiber of the map sc(Y )→ sc(yn)× · · · × sc(y0). Let U denote the simplicial space N(M,W) so that Un = nerve(we(M [n])). Then, let V be a Reedy fibrant replacement of U , from which we get weak equiva- lences Un → Vn for all n ≥ 0. For each n ≥ 0, there exists a map pn : Un → U 0 given by iterated face maps to the “objects.” Then, for every (n+1)-tuple of objects (x0, x1, . . . , xn), the homotopy fiber of pn over (x0, . . . , xn), given by mapV (xn−1, xn)× · · · ×mapV (x0, x1), is weakly equivalent to MapLHM(x n−1, x n )× · · · ×MapLHM(x 0 , x where xcf denotes a fibrant-cofibrant replacement of X inM. It follows that once we take the Reedy fibrant replacement V of U , it is a Segal space. Now, consider the set π0U0, which consists of the weak equivalence classes of objects inM; it follows that π0V0 is an isomorphic set. Further, note that HomHo(M)(x, y) = π0MapLHM(x cf , ycf). 16 J.E. BERGNER Thus, we have shown that Ho(M) is equivalent to Ho(V ). It remains to show that V is a complete Segal space. Consider the space Vhoequiv ⊆ V1, and define Uhoequiv to be the preimage of Vhoequiv under the natural map U → V . Since V is a Reedy fibrant replacement for U , it suffices to show that the complete Segal space condition holds, i.e., that U0 → Uhoequiv is a weak equivalence of simplicial sets. Notice that Uhoequiv must consist precisely of the components of U1 whose 0-simplices come from weak equivalences inM. In other words, Uhoequiv = nerve(we(we(M)) [1]). There is an adjoint pair of functors F :M[1] ⇄M : G for which F (x→ y) = x and G(x) = idx. This adjoint pair can be restricted to an adjoint pair F : nerve(we(we(M))[1]) ⇄ we(M) : G which in turn induces a weak equivalence of simplicial sets on the nerves, Uhoequiv ≃ U0, which completes the proof. � Now that we have proved that the mapping spaces and homotopy categories agree for V and for LHM, it remains to show that they agree for LHM and LC(L Theorem 6.4. Let M be a model category. For any x and y objects of LHM, there is a weak equivalence of simplicial sets MapLHM(x, y) ≃ mapLC(LHM)(x, y), and there is an equivalence of categories π0L HM≈ Ho(LC(L HM)). Note in particular that x and y are just objects of M, and that π0L HM is equivalent to the homotopy category Ho(M). Proof. Given the hammock localization LHM of the model categoryM, we have the following composite map of simplicial spaces: X = nerve(LHM)→ Xf → LC(L Here, Xf denotes a Reedy fibrant replacement of X . This composite is just Rezk’s method for assigning the complete Segal space LC(L HM) to the simplicial category On the left-hand side, the mapping spaces of X = nerve(LHM) are precisely those of LHM, by the definition of the nerve functor. In the nerve, one of these mapping spaces, say mapX(x, y) for some objects x and y ofM, is given by the fiber over (x, y) of the map (d1, d0) : X1 → X0×X0. Although these mapping spaces can be defined for X , there is no reason that they are homotopy invariant. When we take a Reedy fibrant replacement Xf of X , however, this map becomes a fibration, and hence this fiber is actually a homotopy fiber and so homotopy invariant. For a general simplicial space, we cannot assume that the mapping spaces of the Reedy fibrant replacement are equivalent to the original ones. However, if the 0-space of the simplicial space in question is discrete in degree zero, then the map above is a fibration. Using an argument similar to the one in [5, §5], we can find a Reedy fibrant replacement functor which leaves the 0-space discrete. While the space in degree one might be changed in this process of passing to X 1 , it will still be weakly equivalent X1. In particular, the mapping spaces in X f will be weakly equivalent to those in X . COMPLETE SEGAL SPACES 17 Since the objects of Xf are just the objects of LHM, or the objects ofM, this equivalence of mapping spaces gives us also an equivalence of homotopy categories. The right-most map is the one defined by Rezk, iXf : X f → X̂f , which takes a Segal space to a complete Segal space. But, he defines this map in such a way that it is in fact a Dwyer-Kan equivalence. In other words, it induces weak equivalences on mapping spaces and an equivalence of homotopy categories. Thus, the composite map induces equivalences on mapping spaces and an equivalence on homotopy categories. � 7. A Characterization of Complete Segal Spaces Arising from Simplicial Categories In this section, we give a thorough description of the weak equivalence type of complete Segal spaces which occur as images of Rezk’s functor from the category of simplicial categories. We consider several different cases, beginning with ones for which we can use the classifying diagram construction, i.e., discrete categories, and then proceed to more general simplicial categories. It should be noted that we are characterizing these complete Segal spaces up to weak equivalence, and so the resulting descriptions are of the homotopy type of the spaces in each simplicial degree. For example, in the case of a discrete category, we describe the corresponding complete Segal space in terms of the isomorphism classes of objects, rather than in terms of individual objects, in order to simplify the description. One could just take all objects, and generally get much larger spaces, if the more precise description were needed for the complete Segal space corresponding to a given category. Furthermore, notice that determining the homotopy type of the spaces in degrees zero and one are sufficient to determine the homotopy type of all the spaces, since we are considering Segal spaces. Thus, we focus our attention on these spaces, adding in a few comments about how to continue the process with the higher-degree spaces. 7.1. Case 1: C is a discrete groupoid. If C = G is a group, then applying Rezk’s classifying diagram construction results in a complete Segal space equivalent to BG, i.e., the constant simplicial space which is the simplicial set BG at each level. In particular, since all morphisms are invertible, we obtain essentially no new information at level 1 that we didn’t have already at level 0. Example 7.1. Let G = Z/2. Then (NG)0 is just the nerve, or BZ/2. Then (NG)1 has two 0-simplices, given by the two morphisms (elements) of G. However, these two objects of G[1] are isomorphic, and the automorphism group of either one of them is Z/2. Thus, (NG)1 is also equivalent to BZ/2. If C has more than one object but only one isomorphism class of objects, we get instead a simplicial space weakly equivalent to the constant simplicial space which is BAut(x) at each level, for a representative object x. If C has more than one isomorphism class 〈x〉, then the result will be weakly equivalent to the constant simplicial space 〈x〉 BAut(x). 7.2. Case 2: C is a discrete category. Since in the classifying diagram NC, (NC)0 picks out the isomorphisms of C only, we still essentially have 〈x〉 BAut(x) 18 J.E. BERGNER at level 0. However, if C is not a groupoid, then there is new information at level 1. It instead looks like ∐ 〈x〉,〈y〉 BAut( Hom(x, y)α) where the α index the isomorphism classes of elements of Hom(x, y). The subspace of (NC)1 corresponding to 〈x〉, 〈y〉, denoted (NC)1(x, y), fits into a fibration Hom(x, y)→ (NC)1(x, y)→ BAut(x)×BAut(y). The space in dimension 2 is determined, then, by the spaces at levels 0 and 1. The subspace corresponding to isomorphism classes of objects 〈x〉, 〈y〉, 〈z〉, denoted (NC)2(x, y, z), fits into a fibration Hom(x, y)×Hom(y, z)→ (NC)2(x, y, z)→ BAut(x) ×BAut(y)×BAut(z). The whole space (NC)2, up to homotopy, looks like 〈x〉,〈y〉,〈z〉 〈α〉,〈β〉 Hom(x, y)α ×Hom(y, z)β We could describe each (NC)n analogously. Example 7.2. Let C denote the category with two objects and one nontrivial morphism between them (· → ·). If {e} denotes the trivial group, then (NC)0 ≃ B{e} ∐ B{e} and (NC)1 ≃ B{e} ∐ B{e} ∐ B{e}. In particular, NC is not equiv- alent to the classifying diagram of the trivial category with one object and one morphism, which would be the constant simplicial space B{e}. However, note that the nerves of these two categories are homotopy equivalent. Thus, we can see that the classifying diagram is more refined than the nerve in distinguishing between these two categories. 7.3. Case 3: C is a simplicial groupoid. First, consider the case where we have a simplicial group G. Let Gn denote the group of n-simplices of G. Then hocolim∆op(nerve(Gn) t) = nerve(G). Let LC denote Rezk’s completion functor which makes the nerve into a complete Segal space. We claim that LC(hocolim∆op(nerve(Gn) t)) ≃ LC(hocolim∆opLC(nerve(Gn) We actually prove the more general statement that, for any X = hocolim∆opXn, LC(hocolim∆opXn) ≃ LC(hocolim∆opLCXn). To prove this claim, first note that we have Rezk’s completion map i : hocolim∆opXn → LC(hocolim∆opXn) which is a weak equivalence. Furthermore, since in CSS any complete Segal space Y is a local object and every object is cofibrant, we have a weak equivalence of spaces Map(LC(hocolim∆opXn), Y ) ≃Map(hocolim∆opXn, Y ). COMPLETE SEGAL SPACES 19 So, for any complete Segal space Y , we have that Map(LChocolim∆op(LCXn), Y ) ≃Map(hocolim∆op(LCXn), Y ) ≃ holim∆Map(LCXn, Y ) ≃ holim∆Map(Xn, Y ) ≃Map(hocolim∆opXn, Y ) ≃Map(LChocolim∆opXn, Y ). Note that the above calculation depends on the fact that, Map(hocolim∆opXn, Y ) ≃ holim∆Map(Xn, Y ), which follows from working levelwise on simplicial sets. Then, since Gn is a discrete group, completing its nerve is the same as taking the classifying diagram NGn which, by case 1, is weakly equivalent to the constant simplicial space BGn, denoted here cBGn. Thus we have: LC(nerve(Gn)) ≃ LC [hocolim∆op(nerve(Gn))] ≃ LC [hocolim∆op(LC(nerve(Gn)))] ≃ LC [hocolim∆op(cBGn)] ≃ LC(BG) ≃ BG. So, we obtain a simplicial space weakly equivalent to the constant simplicial space with BG at each level. (Recall, however, that BG here is obtained by taking the diagonal of the simplicial nerve, so it is not quite the identical case.) If we have a simplicial groupoid, rather than a simplicial group, we obtain the analogous result, replacing BG with ∐ BAut(x). 7.4. Case 4: C is a simplicial category with every morphism invertible up to homotopy. Alternatively stated, this case covers the situation in which π0(C) is a groupoid. Recall that we have a model structure SCO on the category of categories with a fixed object set O, in which the cofibrant objects are retracts of free objects. So, taking a cofibrant replacement of C in this model category structure SCO essen- tially gives a free replacement of C, denoted F (C), which is weakly equivalent to C. (This cofibrant category can be obtained by taking a simplicial resolution F∗C and then taking a diagonal [12, 6.1].) Now, taking the localization with respect to all morphisms results in a simplicial groupoid. So, we have Dwyer-Kan equivalences F (C)−1F (C) F (C) ≃oo ≃ //C But, now F (C)−1F (C) is a simplicial groupoid weakly equivalent to C, so we have now reduced this situation to case 3. Note that, to write down a description of this complete Segal space in terms of the original category C, we need to take isomorphism classes of objects in π0(C), or weak equivalence classes, as well as self-maps which are invertible up to homotopy rather than strict automorphisms. While we will still use 〈x〉 to denote the equivalence class of a given object, we will use Auth(x) to signify homotopy automorphisms of 20 J.E. BERGNER x. Thus, the complete Segal space corresponding to C in this case essentially looks like ∐ BAuth(x) at each level. 7.5. Case 5: C is any simplicial category. First consider the subcategory of C containing all the objects of C and only the morphisms of C which are invertible up to homotopy. Apply case 5 to get a complete Segal space, but take only the 0-space of it, to be the 0-space of the desired complete Segal space. To find the 1-space, first recall the definition of the completion functor as applied to a Segal space W : LC(W ) = LCSS(diag([m] 7→ (W E(m))n)). Recall further that (WE(m))n = Map(E(m)×∆[n] t,W ). Thus, the Segal space we obtain (before applying the functor LCSS) looks like Map(E(0)×∆[0]t,W )⇐ Map(E(1)×∆[1]t,W ) ⇚ Map(E(2)×∆[2]t,W ) · · · . If the Segal space W is a fibrant replacement of nerve(C), then the space at level 1 consists of diagrams x′ // y with the maps in the appropriate simplicial level. For simplicity, we restrict to a given pair of objects x and y, representing given equivalence classes. Consider the homotopy automorphisms of x and y. If they are not all invertible, we take a cofibrant replacement and group completion as in case 4. So, without loss of generality, assume that Aut(x) and Aut(y) are simplicial groups. Note that we have Aut(x) = hocolim∆opAut(x)n Aut(y) = hocolim∆opAut(y)n. Now look at Map(x, y) = hocolim∆opMap(x, y)n. Consider for each n ≥ 0 the discrete category C(x, y)n which has as objects Map(x, y)n and as morphisms pairs (α, β) of automorphisms in Aut(x)n × Aut(y)n making a commutative square with f, f ′ ∈Map(x, y)n. Thus, the 1-space that we are interested in is also the 1-space of the complete Segal space given by LCSS(hocolim∆op(nerve(C(x, y)n))). COMPLETE SEGAL SPACES 21 Using a straightforward argument about localization functors similar to the one in case 3 (which can be found in [3, 4.1]), we can also apply the functor LCSS on the inside to get an equivalent simplicial space LCSS(hocolim∆op(LCSSnerve(C(x, y)n))). But, since C(x, y)n is a discrete category, this space is just LCSShocolim∆op(NC(x, y)n) ≃ hocolim∆op(NC(x, y)n). Now, we restrict to the 1-space here, which is hocolim∆op 〈x〉,〈y〉 Mapn(x, y)α 〈x〉,〈y〉 Map(x, y)α As with the previous case, we can then go back and weaken to homotopy auto- morphisms and equivalence classes of objects to consider categories before taking a group completion, so our space looks like 〈x〉,〈y〉 BAuth Map(x, y)α We could then obtain the 2-space of our complete Segal space by considering categories C(x, y, z)n defined similarly, and the description of the 2-space of the classifying diagram of a discrete category as given in case 2. We can summarize these results in the following theorem. For an object x of a simplicial category C, let 〈x〉 denote the weak equivalence class of x in C, and for a morphism α : x → y, let 〈α〉 denote the weak equivalence class of α in the morphism category C[1]. Let Auth(x) denote the space of self-maps of x which are invertible in π0C. Theorem 7.3. Let C be a simplicial category. The complete Segal space corre- sponding to C has the form BAuth(x)⇐ 〈x〉,〈y〉 BAuth Map(x, y)α  ⇚ · · · . References [1] J.E. Bergner, Homotopy fiber products of homotopy theories, in preparation. [2] J.E. Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), 2043-2058. [3] J.E. Bergner, Simplicial monoids and Segal categories, Contemp. Math. 431 (2007) 59-83. [4] J.E. Bergner, A survey of (∞, 1)-categories, preprint available at math.AT/0610239. [5] J.E. Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007), 397-436. [6] J.M. Cordier and T. Porter, Vogt’s theorem on categories of homotopy coherent diagrams, Math. Proc. Camb. Phil. Soc. (1986), 100, 65-90. [7] Daniel Dugger, Universal homotopy theories, Adv. Math. 164 (2001), no. 1, 144–176. [8] W.G. Dwyer and D.M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980), 17-35. [9] W.G. Dwyer and D.M. Kan, A classification theorem for diagrams of simplicial sets, Topology 23(1984), 139-155. [10] W.G. Dwyer and D.M. Kan, Equivalences between homotopy theories of diagrams, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), 180–205, Ann. of Math. Stud., 113, Princeton Univ. Press, Princeton, NJ, 1987. http://arxiv.org/abs/math/0610239 22 J.E. BERGNER [11] W.G. Dwyer and D.M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427-440. [12] W.G. Dwyer and D.M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980), no. 3, 267–284. [13] W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, in Handbook of Algebraic Topology, Elsevier, 1995. [14] P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Math, vol. 174, Birkhauser, 1999. [15] Philip S. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and Monographs 99, AMS, 2003. 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Kansas State University, 138 Cardwell Hall Manhattan, KS 66506 E-mail address: bergnerj@member.ams.org http://www-math.mit.edu/~psh 1. Introduction and Overview 2. Background on Model Categories and Simplicial Objects 3. Simplicial Categories and Simplicial Localizations 4. Complete Segal Spaces 5. Obtaining Complete Segal Spaces from Simplicial Categories and Model Categories 6. Comparison of Functors from SC to CSS 7. A Characterization of Complete Segal Spaces Arising from Simplicial Categories 7.1. Case 1: C is a discrete groupoid 7.2. Case 2: C is a discrete category 7.3. Case 3: C is a simplicial groupoid 7.4. Case 4: C is a simplicial category with every morphism invertible up to homotopy 7.5. Case 5: C is any simplicial category References

A Systematic Scan for 7-colourings of the Grid Markus Jalsenius Department of Computer Science, University of Liverpool Ashton Street, Liverpool, L69 3BX, United Kingdom Kasper Pedersen Department of Computer Science, University of Liverpool Ashton Street, Liverpool, L69 3BX, United Kingdom Abstract We study the mixing time of a systematic scan Markov chain for sampling from the uniform distribution on proper 7-colourings of a finite rectangular sub-grid of the infinite square lattice, the grid. A systematic scan Markov chain cycles through finite-size subsets of vertices in a deterministic order and updates the colours assigned to the vertices of each subset. The systematic scan Markov chain that we present cycles through subsets consisting of 2×2 sub-grids and updates the colours assigned to the vertices using a procedure known as heat-bath. We give a computer-assisted proof that this systematic scan Markov chain mixes in O(log n) scans, where n is the size of the rectangular sub-grid. We make use of a heuristic to compute required couplings of colourings of 2×2 sub-grids. This is the first time the mixing time of a systematic scan Markov chain on the grid has been shown to mix for less than 8 colours. We also give partial results that underline the challenges of proving rapid mixing of a systematic scan Markov chain for sampling 6-colourings of the grid by considering 2×3 and 3×3 sub-grids. 1 Introduction This paper is concerned with sampling from the uniform distribution, π, on the set of proper q-colourings of a finite-size rectangular grid. A q-colouring of a graph is an assignment of a colour from a finite set of q distinct colours to each vertex and we say that a colouring is a proper colouring if no two adjacent vertices are assigned the same colour. Proper q-colourings of the grid correspond to the zero-temperature anti-ferromagnetic q-state Potts model on the square lattice, a model of significant importance in statistical physics (see for example Salas and Sokal [14]). Sampling from π is computationally challenging, however it remains an important task and it is frequently carried out in experimental work by physicists by simulating some suitable random dynamics that converges to π. Ensuring that a dynamics converges to π is generally straight forward, but obtaining good upper bounds on the number of steps http://arxiv.org/abs/0704.1625v3 required for the dynamics to become sufficiently close to π is a much more difficult prob- lem. Physicists are at times forced to “guess” (using some heuristic methods) the number of steps required for their dynamics to be sufficiently close to the uniform distribution in order to carry out their experiments. By establishing rigorous bounds on the convergence rates (mixing time) of these dynamics computer scientists can provide underpinnings for this type of experimental work and also allow a more structured approach to be taken. Providing bounds on the mixing time of Markov chains is a well-studied problem in theoretical computer science. However, the types of Markov chains frequently considered by computer scientists do not always correspond to the dynamics usually used in the experimental work by physicists. In computer science, the mixing time of various types of random update Markov chains have been frequently analysed; notably on the grid by Achlioptas, Molloy, Moore and van Bussel [1] and Goldberg, Martin and Paterson [9]. We say that a Markov chain on the set of colourings is a random update Markov chain when one step of the the process consists of randomly selecting a set of vertices (often a single vertex) and updating the colours assigned to those vertices according to some well-defined distribution induced by π. Experimental work is, however, often carried out by cycling through and updating the vertices (or subsets of vertices) in a deterministic order. This type of dynamics has recently been studied by computer scientists in the form of systematic scan Markov chains (systematic scan for short). For results regarding systematic scan see for instance Dyer, Goldberg and Jerrum [5, 4] and Pedersen [12] although these papers are not considering the grid specifically. It is important to note that systematic scan remains a random process since the method used to update the colour assigned to the selected set of vertices is a randomised procedure drawing from some well-defined distribution induced by π. In Section 3 we present a computer assisted proof that systematic scan mixes rapidly when considering 7-colourings of the grid. Previously eight was the least number of colours for which systematic scan on the grid was known to be rapidly mixing, due to Pedersen [12], a result which we hence improve on in this paper. We will make use of a recent result by Pedersen [12] to prove rapid mixing of systematic scan by bounding the influence on a vertex (note that the literature traditionally talks about sites rather than vertices). We will provide bounds on this influence parameter by using a heuristic to mechanically construct sufficiently good couplings of proper colourings of a 2×2 sub- grid. We will hence use a heuristic based computation in order to establish a rigorous result about the mixing time of a systematic scan Markov chain. Finally, in Section 4, we consider the possibility of proving rapid mixing of systematic scan for 6-colourings of the grid by increasing the size of the sub-grids. We give lower bounds on the appropriate influence parameter that imply that the proof technique we employ does not imply rapid mixing of systematic scan for 6-colourings of the grid when using 2×2, 2×3 and 3×3 sub-grids. 1.1 Preliminaries and statement of results Let Q = {1, . . . , 7} be the set of colours and V = {1, . . . , n} the set of vertices of a finite rectangular grid G with toroidal boundary conditions. Working on the torus is common practice as it avoids treating several technicalities regarding the vertices on the boundary of a finite grid as special cases and hence lets us present the proof in a more “clean” way. We point out however that these technicalities are straightforward to deal with (more on this in Section 2). We formally say that a colouring σ of G is a function from V to Q. Let Ω+ be the set of all colourings of G and Ω be the set of all proper q-colourings. Then the distribution π, described earlier, is the uniform distribution on Ω. If σ ∈ Ω+ is a colouring and j ∈ V is a vertex then σj denotes the colour assigned to vertex j in colouring σ. Furthermore, for a subset of vertices Λ ⊆ V and a colouring σ ∈ Ω+ we let σΛ denote the colouring of the vertices in Λ under σ. For each vertex j ∈ V , let Sj denote the set of pairs (σ, τ) ∈ Ω+ × Ω+ of colourings that only differ on the colour assigned to vertex j, that is σi = τi for all i 6= j. Let M be a Markov chain with state space Ω+ and stationary distribution π. Suppose that the transition matrix of M is P . Then the mixing time from an initial colouring σ ∈ Ω+ is the number of steps, that is applications of P , required for M to become sufficiently close to π. Formally the mixing time of M from an initial colouring σ ∈ Ω+ is defined, as a function of the deviation ε from stationarity, by Mixσ(M, ε) = min{t > 0 : dTV(P t(σ, ·), π) ≤ ε}, (1) where dTV(θ1, θ2) = |θ1(i)− θ2(i)| = max |θ1(A)− θ2(A)| (2) is the total variation distance between two distributions θ1 and θ2 on Ω +. The mixing time Mix(M, ε) of M is then obtained my maximising over all possible initial colourings Mix(M, ε) = max Mixσ(M, ε). (3) We say that M is rapidly mixing if the mixing time of M is polynomial in n and log(ε−1). We will make use of a recent result by Pedersen [12] to study the mixing time of a systematic scan Markov chain for 7-colourings of the grid using block updates. We need the following notation in order to define our systematic scan Markov chain. Define the following set Θ = {Θ1, . . . ,Θm} of m blocks. Each block Θk ⊆ V is a 2×2 sub-grid and m is the smallest integer such that k=1Θk = V . For any block Θk and a pair of colourings σ, τ ∈ Ω+ we write “σ = τ on Θk” if σi = τi for each i ∈ Θk and similarly “σ = τ off Θk” if σi = τi for each i ∈ V \ Θk. We also let ∂Θk denote the set of vertices in V \ Θk that are adjacent to some vertex in Θk, and we will refer to ∂Θk as the boundary of Θk. Note from our previous definitions that σ∂Θk denotes the colouring of the boundary of Θk under a colouring σ ∈ Ω +. We will refer to σ∂Θk as a boundary colouring. Finally we say that a 7-colouring of the 2×2 sub-grid Θk agrees with a boundary colouring σ∂Θk if (1) no adjacent sites in Θk are assigned the same colour and (2) each vertex j ∈ Θk is assigned a colour that is different to the colours of all boundary vertices adjacent to j. For each block Θk and colouring σ ∈ Ω + let Ωk(σ) be the subset of Ω + such that if σ′ ∈ Ωk(σ) then σ ′ = σ off Θk and σ agrees with σ∂Θk . Let πk(σ) be the uniform distribution on Ωk(σ). We then define P [k] to be the transition matrix on the state space Ω+ for performing a so-called heat-bath move on Θk. A heat-bath move on a block Θk, given a colouring σ ∈ Ω+, is performed by drawing a new colouring from the distribution πk(σ). Note in particular that applying P [k] to a colouring σ ∈ Ω+ results in a colouring σ′ ∈ Ω+ such that σ′ = σ off Θk and the colouring σ of Θk is proper and agrees with the colouring σ′∂Θk of the boundary of Θk (which is identical to σ∂Θk). We formally define the following systematic scan Markov chain for 7-colourings of G, which systematically performs heat-bath moves on 2×2 sub-grids, as follows. It is worth pointing out that this holds for any ordering of the set of blocks. Definition 1. The systematic scan dynamics for 7-colourings of G is a Markov chain Mgrid with state space Ω + and transition matrix Pgrid = Π It can be shown that the stationary distribution of Mgrid is π by considering the construction of Pgrid. It is customary to refer to one application of Pgrid (that is updating each block once) as one scan. One scan takes |Θk| vertex updates and by construction of Θ this sum is clearly of order O(n). We will prove the following theorem and point out that this is the first proof of rapid mixing of systematic scan for 7-colourings on the grid. Theorem 2. Let Mgrid be the Markov chain from Definition 1 on 7-colourings of G. Then the mixing time of Mgrid is Mix(Mgrid, ε) ≤ 63 log(nε −1). (4) 1.2 Context and related work We now provide an overview of previous achievements for colourings of the grid. Previ- ously it was known that systematic scan for q-colourings on general graphs with maximum vertex degree ∆ mixes in O(logn) scans when q ≥ 2∆ due to Pedersen [12]. That result is a hand-proof and uses block updates that updates the colour at each endpoint of an edge during each step. Earlier Dyer et al. [4] had shown that a single-site systematic scan Markov chain (where one vertex is updated at a time) mixes in O(logn) scans when q > 2∆ and in O(n2 logn) scans when q = 2∆. It is hence well-established that system- atic scan is rapidly mixing for q-colourings of the grid when q ≥ 8 but nothing has been known about the mixing time for smaller q. The results of both Pedersen [12] and Dyer et al. [4] bound the mixing time by studying the influence on a vertex. We will use that technique in this paper as well, however we will construct the required couplings using a heuristic. We defer the required definitions to Section 2 which also contains the proof of Theorem 2. Recent results have revealed that, in a single-site setting, one is not restricted use the total influence on a vertex when analysing the mixing time of systematic scan by bounding influence parameters. In a single-site setting one can define an n×n-matrix whose entries are the influences that all vertices have on each other. Hayes [10] has shown that providing a sufficiently small upper bound on the spectral gap of this matrix implies rapid mixing of both systematic scan and random update. Dyer, Goldberg and Jerrum [6] furthermore showed that an upper bound on any matrix norm also implies rapid mixing of both types of Markov chains. These techniques are however not known to apply to Markov chains using block moves. See the PhD thesis by Pedersen [13] for more comprehensive review of the above results and for the difficulties in extending them to cover block dynamics. As random update Markov chains have received more attention than systematic scan we also summarise some mixing results of interest regarding q-colourings of the grid (recall that a random update Markov chain selects randomly a subset of sites to be updated at each step). Achlioptas et al. [1] give a computer-assisted proof of mixing in O(n logn) updates when q = 6 by considering blocks consisting of 2×3 sub-grids. Our computations are similar in nature to the ones of Achlioptas et al. however their computations are not sufficient to imply mixing of systematic scan as we will discuss in due course. More recently Goldberg, Martin and Paterson [9] gave a hand-proof of mixing in O(n logn) updates when q ≥ 7 using the technique of strong spatial mixing. Previously Salas and Sokal [14] gave a computer-assisted proof of the q = 7 case, a result which was also implied by another computer-assisted result due to Bubley, Dyer and Greenhill [3] that applies to 4-regular triangle-free graphs. Finally it is worth pointing out that, in the special case when q = 3, two complementary results of Luby, Randall and Sinclair [11] and Goldberg, Martin and Paterson [8] give rapid mixing of random update. 2 Bounding the mixing time of systematic scan This section will contain a proof of Theorem 2 although the proof of a crucial lemma, which requires computer-assistance, is deferred to Section 3. We will bound the mixing time of Mgrid by bounding the influence on a vertex, a parameter which we denote by α and will define formally in due course. If α is sufficiently small then Theorem 2 from Pedersen [12] implies that any systematic scan Markov chain, whose transition matrices for updating each block satisfy two simple properties, mixes in O(logn) scans. For completeness we restate this theorem (Theorem 3 below) and in the statement we let M→ denote a systematic scan Markov chain whose transition matrices for each block update satisfy the required properties. Theorem 3. If α < 1 then the mixing time of M→ is Mix(M→, ε) ≤ log(nε−1) . (5) For each block Θk the transition matrix P [k] needs to satisfy the following two prop- erties in order for Theorem 3 to apply. 1. If P [k](σ, τ) > 0 then σ = τ off Θk, and 2. π is invariant with respect to P [k]. It is pointed out in Pedersen [12] that if P [k] is a transition matrix performing a heat-bath move then both of these properties are easily satisfied. Furthermore, it is pointed out that when Ω is the set of proper colourings of a graph, then π is the uniform distribution on Ω as we require. Since the transition matrices P [k] used in the definition of Mgrid perform heat-bath updates we are hence able to use Theorem 3 to bound the mixing time of Mgrid. We are now ready to formally define the parameter α denoting the influence on a vertex. For any pair of colourings (σ, τ) ∈ Si let Ψk(σ, τ) be a coupling of the distributions induced by P [k](σ, ·) and P [k](τ, ·), namely πk(σ) and πk(τ) respectively. We remind the reader that a coupling of two distributions π1 and π2 on state space Ω + is a joint distribution Ω+ × Ω+ such that the marginal distributions are π1 and π2. For ease of reference we also let pj(Ψk(σ, τ)) denote the probability that a vertex j ∈ Θk is assigned a different colour in a pair of colourings drawn from some coupling Ψk(σ, τ). We then let ρki,j = max (σ,τ)∈Si pj(Ψk(σ, τ)) (6) be the influence of i on j under Θk. Finally the parameter α denoting the influence on any vertex is defined as α = max ρki,j. (7) Pedersen [12] actually defines α with a weight associated with each vertex, however as we will not use weights in our proof we have omitted them from the above account. So, in order to upper bound α we are required to upper bound the probability of a discrepancy at each vertex j ∈ Θk under a coupling Ψk(σ, τ) of the distributions πk(σ) and πk(τ) for any pair of colourings (σ, τ) ∈ Si that only differ at the colour of vertex i. Our main task is hence to specify a coupling Ψk(σ, τ) of πk(σ) and πk(τ) for each pair of colourings (σ, τ) ∈ Si and upper bound the probability of assigning a different colour to each vertex in a pair of colourings drawn from that coupling. Consider any block Θk and any pair of colourings (σ, τ) ∈ Si that differ only on the colour assigned to some vertex i. Clearly the distribution on colourings of Θk, induced by πk(σ) only depends on the boundary colouring σ∂Θk . Similarly, the distribution on colourings of Θk, induced by πk(τ) depends only on τ∂Θk . If i 6∈ ∂Θk then the distributions on the colourings of Θk, induced by πk(σ) and πk(τ), respectively, are the same and we let Ψk(σ, τ) be the coupling in which any pair of colourings drawn from Ψk(σ, τ) agree on Θk. That is, if the pair (σ ′, τ ′) of colourings are drawn from Ψk(σ, τ) then σ ′ = σ off Θk, τ ′ = τ off Θk and σ ′ = τ ′ on Θk. This gives ρ i,j = 0 for any i 6∈ ∂Θk and j ∈ Θk. We now need to construct Ψk(σ, τ) for the case when i ∈ ∂Θk. For each j ∈ Θk we need pj(Ψk(σ, τ)) to be sufficiently small in order to avoid ρ i,j being too big. If the ρki,j-values are too big the parameter α will be too big (that is greater than one) and we cannot make use of Theorem 3 to show rapid mixing. Constructing Ψk(σ, τ) by hand such that pj(Ψk(σ, τ)) is sufficiently small is a difficult task. It is, however, straight forward to mechanically determine which colourings have positive measure in the distributions πk(σ) and πk(τ) for a given pair of boundary colourings σ∂Θk and τ∂Θk . From these distributions we can then use some suitable heuristic to construct a coupling that is good enough for our purposes. We hence need to construct a specific coupling for each individual pair of colourings differing only at a single vertex. In order to do this we will make use of the following lemma, which is proved in Section 3. Lemma 4. Let v1, . . . , v4 be the four vertices in a 2×2-block and z1, . . . , z8 be the boundary vertices of the block and let the labeling be as in Figure 1. Let Z and Z ′ be any two 7- colourings of the boundary vertices such that Z and Z ′ agree on each vertex except on z1. Let πZ and πZ′ be the uniform distributions on proper 7-colourings of the block that agree with Z and Z ′, respectively. For i = 1, . . . , 4 let pvi(Ψ) denote the probability that the colour of vertex vi differ in a pair of colourings drawn from a coupling Ψ of πZ and πZ′. Then there exists a coupling Ψ such that pv1(Ψ) < 0.283, pv2(Ψ) < 0.079, pv3(Ψ) < 0.051 and pv4(Ψ) < 0.079. z4 z5 Figure 1: General labeling of the vertices in a 2×2-block Θk and the vertices ∂Θk on the boundary of the block. i (b) Figure 2: A 2×2-block Θk showing all eight positions of a vertex i ∈ ∂Θk on the boundary of the block in relation to a vertex j ∈ Θk in the block. Thus if i ∈ ∂Θk we let Ψk(σ, τ) be the coupling of πk(σ) and πk(τ) that draws the colouring of Θk from the coupling Ψ in Lemma 4, where Z is the boundary colouring obtained from σ∂Θk and Z ′ is obtained from τ∂Θk , and leaves the colour of the remaining vertices, V \Θk, unchanged. That is, if the pair (σ ′, τ ′) of colourings are drawn from Ψk(σ, τ) then σ ′ = σ off Θk, τ ′ = τ off Θk and the colourings of Θk in σ ′ and τ ′ are drawn from the coupling Ψ in Lemma 4 (see the proof for details on how to construct Ψ). It is straightforward to verify that this is indeed a coupling of πk(σ) and πk(τ). Note that due to the symmetry of the 2×2-block, with respect to rotation and mirroring, we can always label the vertices of Θk and ∂Θk such that label z1 in Figure 1 represents the discrepancy vertex i on the boundary. Hence we can make use of Lemma 4 to compute upper bounds on the parameters ρki,j. We summarise the ρ i,j-values in the following Corollary of Lemma 4. Note that due to the symmetry of the block we can assume that vertex j ∈ Θk in the corollary is located in the bottom left corner, as Figure 2 shows. Corollary 5. Let Θk be any 2×2-block, let j ∈ Θk be any vertex in the block and let i ∈ ∂Θk be a vertex on the boundary of the block. Then ρki,j = max (σ,τ)∈Si pj(Ψk(σ, τ)) < 0.283, if i and j as in Figure 2(a) or (b), 0.079, if i and j as in Figure 2(c) or (h), 0.051, if i and j as in Figure 2(e) or (f), 0.079, if i and j as in Figure 2(d) or (g). If i /∈ ∂Θk is not on the boundary of the block then ρ i,j = 0. We can then use Corollary 5 to prove Theorem 2. The proof of Theorem 2 is given here: Proof of Theorem 2. Let αk,j = ρki,j be the influence on j under Θk. We need αk,j to be upper bounded by one for each block Θk and vertex j ∈ Θk in order to ensure that α = maxk maxj∈Θk αk,j is less than one. Fix any block Θk and any vertex j ∈ Θk. A vertex i ∈ ∂Θk on the boundary of the block can occupy eight different positions on the boundary in relation to j as showed in Figure 2(a)–(h). Recall that we are working on the torus, and hence every vertex on the boundary of the block will belong to G. Thus, using the bounds from Corollary 5 we have αk,j = ρki,j < 2(0.283 + 0.079 + 0.051 + 0.079) = 0.984. (9) Then α = maxk maxj∈Θk αk,j < maxk 0.984 = 0.984 < 1 and we obtain the stated bound on the mixing time of Mgrid by Theorem 3. We make the following remark. In the proof of Theorem 2 above, we assume that G is a finite rectangular grid with toroidal boundary conditions. Hence, every block is a 2×2-sub-grid and each vertex on the block boundary belongs to V . We note that if G is a finite rectangular grid without toroidal boundary conditions then some vertices on the boundary ∂Θk of a block Θk might fall outside G. The sum in Equation (9) is over boundary vertices i that do belong to V , and hence the number of terms in this sum is reduced if some boundary vertices do not belong to V , making α smaller. Furthermore, if G is a non-rectangular region of the grid then a block next to the boundary might be smaller than 2×2 vertices. Suppose Θk is a block that is smaller than 2×2 vertices. Then the vertices that are missing in order to make Θk a full 2×2-block are boundary vertices. Suppose i ∈ ∂Θk belongs to V and i ′ ∈ ∂Θk does not belong to V . When constructing couplings Ψk(σ, τ), where (σ, τ) ∈ Si, we must consider the vertex i ′ as “colourless”, which would decrease the value of pki,j . A more rigorous analysis yields that our mixing result with seven colours and 2×2-blocks holds for arbitrary finite regions G of the grid. Of course we have yet to establish a proof of Lemma 4, and the rest of this paper will be concerned with this. Our method of proof uses some ideas of Goldberg, Jalsenius, Martin and Paterson [7] in so far as it is computer assisted and we will be focusing on minimising the probability of assigning different colours to vertex v1 in the constructed couplings. We will however be required to construct a coupling on the 2×2 sub-grid, rather than establishing bounds on the disagreement probability of a vertex adjacent to the initial discrepancy and then extending this to a coupling on the whole block recursively. Our approach is similar to the one Achlioptas et al. [1] take, however we do not have the option of constructing an “optimal” coupling using a suitable linear program (even when feasible) since our probabilities will be maximised over all boundary colourings. The crucial difference between the approaches is that Achlioptas et al. [1] are using path coupling (see Bubley and Dyer [2]) as a proof technique which requires them to bound the expected Hamming distance between a pair of colourings drawn from a coupling. This in turn enables them to, for a given boundary colouring, specify an “optimal” coupling which minimises Hamming distance. We are, however, required to bound the influence of i on j for each boundary colouring and sum over the maximum of these influences. The reason for this is the inherit maximisation over boundary colourings in the definition of ρki,j as described above. Finally it is worth mentioning that providing bounds on the expected Hamming dis- tance is similar to showing that the influence of a vertex is small and it is known that this condition implies rapid mixing of a random update Markov chain, see for example Weitz [15]. In a single-site setting the condition “the influence of a vertex is small” also implies rapid mixing of systematic scan (Dyer et al. [4]), however, in a block setting this condition is not sufficient to give rapid mixing of systematic scan (Pedersen [13]), which is why we need to bound the influence on a vertex. 3 Constructing the coupling by machine In order to prove Lemma 4 we will construct a coupling Ψ of πZ and πZ′ for all pairs of boundary colourings Z and Z ′ that are identical on all boundary vertices but vertex z1, on which Z and Z ′ differ. For each coupling constructed we verify that the probabilities pvi(Ψ), i = 1, . . . , 4, are within the bounds of the lemma. The method is well suited to be carried out with the help of a computer and we have implemented a program in C to do so. Before stating the proof of Lemma 4 we will discuss how a coupling can be represented by an edge-weighted complete bipartite graph. We make use of this representation of Ψ in the proof of the lemma. 3.1 Representing a coupling as a bipartite graph Let S be a set of objects and let W be a set of |S| pairs (s, ws) such that s ∈ S and ws ≥ 0 is a non-negative value representing the weight of s. Each element s ∈ S is contained in exactly one of the pairs in W . If the value ws is an integer (which it is in our case) it can be regarded as the multiplicity of s in a multiset. The set W is referred to as a weighted set of S. Let πS,W be the distribution on S such that the probability of s is proportional to ws, where (s, ws) is a pair in W . More precisely, the probability of s in πS,W is PrπS,W (s) = ws/ (t,wt)∈W wt. For example, let W be a weighted set of S and let S ′ ⊆ S be a subset of S. Assume the weight ws = 0 if s ∈ S\S ′ and ws = k if s ∈ S where k > 0 is a positive constant. Then πS,W is the uniform distribution on S The reason for introducing the notion of a weighted set is that it can be used when specifying a coupling of two distributions. Let S be a set and let W and W ′ be two weighted sets of S such that the sum of the weights in W equals the sum of the weights in W ′. Let wtot denote this sum. That is, wtot = (s,ws)∈W (s′,w′ )∈W ′ w s′. The two weighted sets W and W ′ define two distributions πS,W and πS,W ′ on S. We want to specify a coupling Ψ of πS,W and πS,W ′. Let K|S|,|S| be an edge-weighted complete bipartite graph with vertex sets W and W ′. That is, for each pair (s, ws) ∈ W there is an edge to every pair in W ′. Every edge e of K|S|,|S| has a weight we ≥ 0 such that the following condition holds. Let (s, ws) be any pair in W ∪ W ′ and let E be the set of all |S| edges incident to (s, ws). Then we = ws. It follows that the sum of the edge weights of all |S|2 edges in K|S|,|S| equals wtot, the sum of the weights in W (and W ′). The idea is that K|S|,|S| represents a coupling Ψ of πS,W and πS,W ′. In order to draw a pair of elements from Ψ we randomly select an edge e in K|S|,|S| proportional to its weight. The endpoints of e represent the elements in S drawn from πS,W and πS,W ′. More precisely, the probability of choosing edge e in K|S|,|S| with weight we is we/wtot. If edge e = ((s, ws), (s ′, w′s′)) is chosen it means that we have drawn s from πS,W and s from πS,W ′, the marginal distributions of Ψ. The bipartite graph representation of a coupling will be used when we construct couplings of colourings of 2×2-blocks in the proof of Lemma 4. 3.2 The proof of Lemma 4 Here is the proof of Lemma 4: Proof of Lemma 4. Fix two colourings Z and Z ′ of the boundary that differ on vertex z1. Let c be the colour of vertex z1 in Z and let c ′ 6= c be the colour of z1 in Z ′. Let CZ and CZ′ be the two sets of proper 7-colourings of the block that agree with Z and Z respectively. Let C+ be the set of all 7-colourings of the block. Let WZ and WZ′ be two weighted sets of C+. The weights are assigned as follows. • For the pair (σ, wσ) ∈ WZ let the weight wσ = |CZ′| if σ ∈ CZ , otherwise let wσ = 0. • For the pair (σ, wσ) ∈ WZ′ let the weight wσ = |CZ | if σ ∈ CZ′, otherwise let wσ = 0. It follows from the assignment of the weights that the distribution πC+,WZ is the uniform distribution on CZ . That is, πC+,WZ = πZ . Similarly, πC+,WZ′ is the uniform distribution πZ′ on CZ′. Note that the sum of the weights is |CZ||CZ′| in both WZ and WZ′. Then a coupling Ψ of πC+,WZ and πC+,WZ′ can be specified with an edge-weighted complete bipartite graph K = K|C+|,|C+|. For a given valid assignment of the weights of the edges of K, making K represent a coupling Ψ, we can compute the probabilities of having a mismatch on a vertex vi of the block when two colourings are drawn from Ψ. Let E be the set of all edges e = ((σ, wσ), (σ ′, w′σ′)) in K such that σ and σ ′ differ on vertex vi. Then pvi(Ψ) = e∈E we/|CZ||CZ′|. In order to obtain sufficiently small upper bounds on pvi(Ψ) for the four vertices v1, . . . , v4 in the block we would like to assign weights to the edges of K such that much weight is assigned to edges between colourings that agree on many vertices in the block. In general it is not clear exactly how to assign weights to the edges. For instance, if we assign too much weight to edges between colourings that are identical on vertex v2 we might not be able to assign as much weight as we would like to on edges between colourings that are identical on vertex v4. Thus, the probability of having a mismatch on v4 would increase. Intuitively a good strategy would be to assign as much weight as possible to edges between colourings that are identical on the whole block. This implies that we try to assign as much weight as possible to edges between colourings that are identical on vertex v1, the vertex adjacent to the discrepancy vertex z1 on the boundary. If there is a mismatch on vertex v1 it should be a good idea to assign as much weight as possible to edges between colourings that are identical on the whole block apart from vertex v1. This idea leads to a heuristic in which the assignment of the edge weights is divided into three phases. The exact procedure is described as follows. In phase one we match identical colourings. For all colourings σ ∈ C+ of the block the edge e = ((σ, wσ), (σ, w σ)) in K will be given weight we = min(wσ, w σ). That is, we maximise the probability of drawing the same colouring σ from both πC+,WZ and πC+,WZ′ . For the following two phases we define an ordering of the colourings in C+. We order the colourings lexicographically with respect to the vertex order v3, v2, v4, v1. That is, if the seven colours are 1, . . . , 7 the colouring of v3, v2, v4, v1 will start with 1, 1, 1, 1, respectively. The next colouring will be 1, 1, 1, 2, and so on. This ordering of colourings in C+ carries over to an ordering of the pairs in WZ and WZ′. That is, we order the pairs (σ, wσ) in WZ with respect to the lexicographical ordering of σ. Similarly we order the pairs in WZ′ . This ordering of the pairs will be important in the next two phases. It provides some control of how colourings are being paired up in terms of the assignment of the weights on edges between pairs. Edges will be considered with respect to this ordering because choosing an arbitrary ordering of the edges would not necessarily result in probabilities pvi(Ψ) that would be within the bounds of the lemma. In the second phase we ignore the colour of vertex v1 and match colourings that are identical on all of the remaining three vertices v2, v3 and v4. More precisely, for each pair (σ, wσ) ∈ WZ , considered in the ordering explained above, we consider the edges e = ((σ, wσ), (σ ′, w′σ′)) where σ and σ ′ are identical on all vertices but v1. The edges are considered in the ordering of the second component (σ′, w′σ′) ∈ WZ′. We assign as much weight as possible to e such that the total weight on edges incident to (σ, wσ) ∈ WZ does not exceed wσ and such that the total weight on edges incident to (σ ′, w′σ′) ∈ WZ′ does not exceed w′σ′ . Note that in the lexicographical ordering of the colourings, vertex v1 is the least significant vertex and therefore the ordering provides some level of control of pairing up colourings that are similar on the remaining three vertices. It turns out that the resulting coupling is sufficiently good for proving the lemma. In the third and last phase we assign the remaining weights on the edges. As in phase two, for each pair (σ, wσ) ∈ WZ we consider the edges e = ((σ, wσ), (σ ′, w′σ′)). The pairs and edges are considered in accordance with the ordering explained above. The difference between the second and third phase is that now we do not have any restrictions on the colourings σ and σ′. We assign as much weight as possible to e such that the total weight on edges incident to (σ, wσ) ∈ WZ does not exceed wσ and such that the total weight on edges incident to (σ′, w′σ′) ∈ WZ′ does not exceed w σ′. After phase three we have assigned all weights to the edges of K and hence K represents a coupling Ψ of πZ and πZ′. From K we compute the probabilities pv1(Ψ), pv2(Ψ), pv3(Ψ) and pv4(Ψ) as described above. We have written a C-program which loops through all colourings Z and Z ′ of the boundary of the block and constructs the bipartite graph K as described above. For each boundary the probabilities pv1(Ψ), pv2(Ψ), pv3(Ψ) and pv4(Ψ) are successfully verified to be within the bounds of the lemma. For details on the C-program, see http://www.csc.liv.ac.uk/∼markus/systematicscan/. 4 Partial results for 6-colourings of the grid In previous sections we have seen that systematic scan on the grid using 2×2-blocks and seven colours mixes rapidly. An immediate question is whether we can do better and show rapid mixing with six colours. This matter will be discussed in this section and we will show that, even with bigger block sizes (up to 3×3), it is not possible to show rapid mixing using the technique of this paper. More precisely, we will establish lower bounds on the parameter α for 2×2-blocks, 2×3-blocks and 3×3-blocks. All three lower bounds are greater than one and hence we cannot make use of Theorem 3 to show rapid mixing. 4.1 Establishing lower bounds for 2×2 blocks We start by examining the 2×2-block again but this time with six colours. Lemma 4 provides upper bounds (under any colourings of the boundary) on the probabilities of having discrepancies at each of the four vertices of the block when two 7-colourings are drawn from the specified coupling. For six colours we will show lower bounds on these probabilities under any coupling and a specified pair of boundary colourings. Once again, let v1, . . . , v4 be the four vertices in a 2×2-block and let z1, . . . , z8 be the boundary vertices of the block and let the labeling be as in Figure 1. Let Z and Z ′ be any two 6-colourings of the boundary vertices that assign the same colour to each vertex except for z1. Let πZ and πZ′ be the uniform distributions on the sets of proper 6-colourings of the block that agree with Z and Z ′, respectively. Let Ψminvk (Z,Z ′) be a coupling of πZ and πZ′ that minimises pvk(Ψ). That is, pvk(Ψ) ≥ pvk(Ψ (Z,Z ′)) for all couplings Ψ of πZ and πZ′. Also let p = maxZ,Z′ pvk(Ψ (Z,Z ′)). We can hence say that there exist two 6-colourings Z and Z ′ of the boundary of a 2×2 block, that assign the same colour to each vertex except for z1, such that pvk(Ψ) ≥ p for any coupling Ψ of πZ and πZ′ . We have the following lemma, which is proved by computation. Lemma 6. Consider 6-colourings of the 2×2-block in Figure 1. Then plowv1 ≥ 0.379, plowv2 ≥ 0.107, p ≥ 0.050 and plowv4 ≥ 0.107. Proof. Fix one vertex vk in the block and fix two colourings Z and Z ′ of the boundary of the block that differ only on the colour of vertex z1. Let CZ and CZ′ be the two sets of proper 6-colourings of the block that agree with Z and Z ′, respectively. For c = 1, . . . , 6 let nc be the number of colourings in CZ in which vertex vk is assigned colour c. Similarly let n′c be the number of colourings in CZ′ in which vertex vk is assigned colour c. It is clear that the probability that vk is assigned colour c in a colouring σ ′ drawn from πZ is PrπZ(σ = c) = nc/|CZ|. For c = 1, . . . , 6 define mc = nc|CZ′|, m c = n c|CZ | and M = |CZ||CZ′|. It follows that PrπZ(σ = c) = mc/M and PrπZ′ (τ = c) = m′c/M , where σ′ and τ ′ are colourings drawn from πZ and πZ′, respectively. Observe that the quantities mc, m c and M can be easily computed for a given pair of boundary colourings. Now let Ψ be any coupling of πZ and πZ′ . It is easy to see that the probability that vertex vk is coloured c in both colourings drawn from Ψ can be at most min(mc, m c)/M . Therefore, the probability of drawing two colourings from Ψ such that the colour of vertex vk is the same in both colourings is at most c=1,...,6 min(mc, m c)/M , and the probability of assigning different colours to vertex vk is at least pvk(Ψ) ≥ 1− c=1,...,6min(mc, m c)/M . We have successfully verified the bounds in the statement of the lemma by maximising the lower bound on pvk(Ψ) over all boundary colourings Z and Z ′ for each vertex vk in the block. The computations are carried out with the help of a computer program written in C. For details on the program, see http://www.csc.liv.ac.uk/∼markus/ systematicscan/. For seven colours, Corollary 5 makes use of Lemma 4 to establish upper bounds on the influence parameters ρki,j . These parameters are used in the proof of Theorem 2 to obtain an upper bound on the parameter α. The upper bound on α is shown to be less than one which implies rapid mixing for seven colours when applying Theorem 3. We can use Lemma 6 to obtain lower bounds on the influence parameters ρki,j by completing the coupling in a way analogous to the coupling in Corollary 5. This in turn will result in a v1 v3v2z2 z4 z5 z6 z1 z9z10 v4 v5 v6 Figure 3: (a) General labeling of the vertices in a 2×3-block Θk and the vertices ∂Θk on the boundary of the block. (b)–(c) All ten positions of a vertex i ∈ ∂Θk on the boundary of the block in relation to a vertex j ∈ Θk in the corner of the block. lower bound on the parameter α that is greater than one. That is, following the proof of Theorem 2 and making use of Lemma 6, a lower bound on α will be α ≥ 2(0.379 + 0.107 + 0.050 + 0.107) = 1.286 > 1. (10) Hence we fail to show rapid mixing of systematic scan with six colours using 2×2-blocks. 4.2 Bigger blocks We failed to show rapid mixing of systematic scan with six colours and 2×2-blocks and we will now show that increasing the block size to both 2×3 and 3×3 will not be suf- ficient either. Lemma 7 below considers 2×3-blocks and is analogous to Lemma 6. We make use of the same notation as for Lemma 6, only the block is bigger and the label- ing of the vertices is different (see Figure 3(a)). Lemma 7 is proved by computation in the same way as Lemma 6. For details on the C-program used in the proof, see http://www.csc.liv.ac.uk/∼markus/systematicscan/. Lemma 7. Consider 6-colourings of the 2×3-block in Figure 3(a). Then plowv1 ≥ 0.3671, plowv3 ≥ 0.0298, p ≥ 0.0997 and plowv6 ≥ 0.0174. We will now use Lemma 7 to show that α > 1 for 2×3 blocks. Let Θk be any 2×3- block and let j ∈ Θk be a vertex in a corner of the block. A vertex i ∈ ∂Θk on the boundary of the block can occupy ten different positions on the boundary in relation to j. See Figure 3(b) and (c). We can again determine lower bounds on the influences ρki,j of i on j under Θk from Lemma 7. However, Lemma 7 provides lower bounds on ρ i,j only when i ∈ ∂Θk is adjacent to a corner vertex of the block, as in Figure 3(b). If i is located as in Figure 3(c) we do not know more than that ρki,j is bounded from below by zero. Nevertheless, the lower bound on α exceeds one. Let αk,j = ρki,j be the influence on j under Θk. Following the proof of Theorem 2 and using the lower bounds in Lemma 7 we αk,j = i in Fig. 3(b) ρki,j + i in Fig. 3(c) ρki,j ≥ 2(0.3671 + 0.0298 + 0.0997 + 0.0174) = 1.028, (11) where we set the lower bound on the second sum to zero. Now, α = max αk,j ≥ 1.028 > 1. (12) v1 v3v2z2 z5 z6 z7 v4 v5 v6 z12 z11 v7 v8 v9 v1 v3v2 z6 z7 z8 z10v4 v5 v6 v7 v8 v9 z1 (c) Figure 4: (a)–(b) General labeling of the vertices in a 3×3-block Θk and two different labellings of the vertices ∂Θk on the boundary of the block. The discrepancy vertex on the boundary has label z1. (b)–(c) All twelve positions of a vertex i ∈ ∂Θk on the boundary of the block in relation to a vertex j ∈ Θk in the corner of the block. Hence we cannot use Theorem 3 to show rapid mixing of systematic scan with six colours and 2×3-blocks. It is interesting to note that considering 2×3-blocks was sufficient for Achlioptas et al. [1] to prove mixing of a random update Markov chain for sampling 6-colourings of the grid. Lastly, we increase the block size to 3×3 and show that a lower bound on α is still greater than one. We have the following lemma which is proved by computation in the same way as Lemmas 6 and 7. For details on the C-program used in the proof see http://www.csc.liv.ac.uk/∼markus/systematicscan/. Lemma 8. For 6-colourings of the 3×3-block with vertices labeled as in Figure 4(a) we have plowv1 ≥ 0.3537, p ≥ 0.0245, plowv7 ≥ 0.0245 and p ≥ 0.0071. Furthermore, for 6-colourings of the 3×3-block in Figure 4(b) we have plowv1 ≥ 0.0838, p ≥ 0.0838, plowv7 ≥ 0.0138 and p ≥ 0.0138. Note that Lemma 8 provides lower bounds on the probabilities of having a mismatch on a corner vertex of the block when the discrepancy vertex on the boundary (labeled z1) is adjacent to a corner vertex (Figure 4(a)) and adjacent to a middle vertex (Figure 4(b)). Let Θk be any 3×3-block and let j ∈ Θk be a vertex in a corner of the block. A vertex i ∈ ∂Θk on the boundary of the block can occupy twelve different positions on the boundary in relation to j. See Figure 4(c) and (d). Analogous to Corollary 5 lower bounds on the influences ρki,j of i on j under Θk can be determined from Lemma 8. Let αk,j = ρki,j be the influence on j under Θk. Following the proof of Theorem 2 and using the lower bounds in Lemma 8 we have αk,j = i in Fig. 4(c) ρki,j + i in Fig. 4(d) ρki,j ≥ 2(0.3537 + 0.0245 + 0.0245 + 0.0071) + (0.0838 + 0.0838 + 0.0138 + 0.0138) = 1.0148. (13) Thus, α = maxk maxj∈Θk αk,j ≥ 1.0148 > 1. Hence, we cannot use Theorem 3 to show rapid mixing of systematic scan with six colours and 3×3-blocks. A natural question is whether we can show rapid mixing using even bigger blocks. It seems possible to do this although the computations rapidly become intractable as the block size increases. Already with a 3×3-block the number of boundary colourings we need to consider (after removing isomorphisms) is in excess of 106 and for each boundary colouring there are more than 107 colourings of the block to consider. In addition to simply generating the distributions on colourings of the block, the time it would take to actually construct the required couplings, as we did in the proof of Lemma 4, would also increase. Finally when using a larger block size, different positions of vertex j in the block need to be considered whereas we could make use of to the symmetry of the 2×2-block to only consider one position of vertex j in the block. If different positions of j have to be considered this has to be captured in the construction of the coupling and would likely require more computations. The conclusion is that in order to show rapid mixing for six colours of systematic scan on the grid we would most likely have to rely on a different approach than the one presented in this paper. References [1] D. Achlioptas, M. Molloy, C. Moore, and F. Van Bussel. Sampling grid colorings with fewer colours. In LATIN 2004: Theoretical Informatics, volume 2976 of Lecture Notes in Computer Science, pages 80–89. Springer, 2004. [2] R. Bubley and M. Dyer. Path coupling: a technique for proving rapid mixing in Markov chains. In FOCS ’97: Proceedings of the 38th Symposium on Foundations of Computer Science, pages 223–231. IEEE Computer Society Press, 1997. [3] R. Bubley, M. Dyer, and C. Greenhill. Beating the 2∆ bound for approximately counting colourings: A computer-assisted proof of rapid mixing. In SODA ’98: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 355–363. Society for Industrial and Applied Mathematics, 1998. [4] M. Dyer, L. A. Goldberg, and M. Jerrum. Dobrushin conditions and systematic scan. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 4110 of Lecture Notes in Computer Science, pages 327–338. Springer, 2006. [5] M. Dyer, L. A. Goldberg, and M. Jerrum. Systematic scan and sampling colourings. Annals of Applied Probability, 16(1):185–230, 2006. [6] M. Dyer, L. A. Goldberg, and M. Jerrum. Matrix norms and rapid mixing for spin systems. To appear in the Annals of Applied Probability, 2008. [7] L. A. Goldberg, M. Jalsenius, R. Martin, and M. Paterson. Improved mixing bounds for the anti-ferromagnetic potts model on Z2. LMS Journal of Computation and Mathematics, 9:1–20, 2006. [8] L. A. Goldberg, R. Martin, and M. Paterson. Random sampling of 3-colourings in 2. Random Structures and Algorithms, 24(3):279–302, 2004. [9] L. A. Goldberg, R. Martin, and M. Paterson. Strong spatial mixing with fewer colours for lattice graphs. SIAM Journal on Computing, 35(2):486–517, 2005. [10] T. P. Hayes. A simple condition implying rapid mixing of single-site dynamics on spin systems. In FOCS ’06: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 39–46. IEEE Computer Society Press, 2006. [11] M. Luby, D. Randall, and A. Sinclair. Markov chain algorithms for planar lattice structures. SIAM Journal on Computing, 31(1):167–192, 2001. [12] K. Pedersen. Dobrushin conditions for systematic scan with block dynamics. In Mathematical Foundations of Computer Science 2007, volume 4708 of Lecture Notes in Computer Science, pages 264–275. Springer, 2007. [13] K. Pedersen. On Systematic Scan. PhD thesis, University of Liverpool, 2008. [14] J. Salas and A. D. Sokal. Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. Journal of Statistical Physics, 86(3– 4):551, 1997. [15] D. Weitz. Combinatorial criteria for uniqueness of Gibbs measures. Random Struc- tures and Algorithms, 27(4):445–475, 2005. Introduction Preliminaries and statement of results Context and related work Bounding the mixing time of systematic scan Constructing the coupling by machine Representing a coupling as a bipartite graph The proof of Lemma ?? Partial results for 6-colourings of the grid Establishing lower bounds for 22 blocks Bigger blocks

We study the mixing time of a systematic scan Markov chain for sampling from the uniform distribution on proper 7-colourings of a finite rectangular sub-grid of the infinite square lattice, the grid. A systematic scan Markov chain cycles through finite-size subsets of vertices in a deterministic order and updates the colours assigned to the vertices of each subset. The systematic scan Markov chain that we present cycles through subsets consisting of 2x2 sub-grids and updates the colours assigned to the vertices using a procedure known as heat-bath. We give a computer-assisted proof that this systematic scan Markov chain mixes in O(log n) scans, where n is the size of the rectangular sub-grid. We make use of a heuristic to compute required couplings of colourings of 2x2 sub-grids. This is the first time the mixing time of a systematic scan Markov chain on the grid has been shown to mix for less than 8 colours. We also give partial results that underline the challenges of proving rapid mixing of a systematic scan Markov chain for sampling 6-colourings of the grid by considering 2x3 and 3x3 sub-grids.

Introduction This paper is concerned with sampling from the uniform distribution, π, on the set of proper q-colourings of a finite-size rectangular grid. A q-colouring of a graph is an assignment of a colour from a finite set of q distinct colours to each vertex and we say that a colouring is a proper colouring if no two adjacent vertices are assigned the same colour. Proper q-colourings of the grid correspond to the zero-temperature anti-ferromagnetic q-state Potts model on the square lattice, a model of significant importance in statistical physics (see for example Salas and Sokal [14]). Sampling from π is computationally challenging, however it remains an important task and it is frequently carried out in experimental work by physicists by simulating some suitable random dynamics that converges to π. Ensuring that a dynamics converges to π is generally straight forward, but obtaining good upper bounds on the number of steps http://arxiv.org/abs/0704.1625v3 required for the dynamics to become sufficiently close to π is a much more difficult prob- lem. Physicists are at times forced to “guess” (using some heuristic methods) the number of steps required for their dynamics to be sufficiently close to the uniform distribution in order to carry out their experiments. By establishing rigorous bounds on the convergence rates (mixing time) of these dynamics computer scientists can provide underpinnings for this type of experimental work and also allow a more structured approach to be taken. Providing bounds on the mixing time of Markov chains is a well-studied problem in theoretical computer science. However, the types of Markov chains frequently considered by computer scientists do not always correspond to the dynamics usually used in the experimental work by physicists. In computer science, the mixing time of various types of random update Markov chains have been frequently analysed; notably on the grid by Achlioptas, Molloy, Moore and van Bussel [1] and Goldberg, Martin and Paterson [9]. We say that a Markov chain on the set of colourings is a random update Markov chain when one step of the the process consists of randomly selecting a set of vertices (often a single vertex) and updating the colours assigned to those vertices according to some well-defined distribution induced by π. Experimental work is, however, often carried out by cycling through and updating the vertices (or subsets of vertices) in a deterministic order. This type of dynamics has recently been studied by computer scientists in the form of systematic scan Markov chains (systematic scan for short). For results regarding systematic scan see for instance Dyer, Goldberg and Jerrum [5, 4] and Pedersen [12] although these papers are not considering the grid specifically. It is important to note that systematic scan remains a random process since the method used to update the colour assigned to the selected set of vertices is a randomised procedure drawing from some well-defined distribution induced by π. In Section 3 we present a computer assisted proof that systematic scan mixes rapidly when considering 7-colourings of the grid. Previously eight was the least number of colours for which systematic scan on the grid was known to be rapidly mixing, due to Pedersen [12], a result which we hence improve on in this paper. We will make use of a recent result by Pedersen [12] to prove rapid mixing of systematic scan by bounding the influence on a vertex (note that the literature traditionally talks about sites rather than vertices). We will provide bounds on this influence parameter by using a heuristic to mechanically construct sufficiently good couplings of proper colourings of a 2×2 sub- grid. We will hence use a heuristic based computation in order to establish a rigorous result about the mixing time of a systematic scan Markov chain. Finally, in Section 4, we consider the possibility of proving rapid mixing of systematic scan for 6-colourings of the grid by increasing the size of the sub-grids. We give lower bounds on the appropriate influence parameter that imply that the proof technique we employ does not imply rapid mixing of systematic scan for 6-colourings of the grid when using 2×2, 2×3 and 3×3 sub-grids. 1.1 Preliminaries and statement of results Let Q = {1, . . . , 7} be the set of colours and V = {1, . . . , n} the set of vertices of a finite rectangular grid G with toroidal boundary conditions. Working on the torus is common practice as it avoids treating several technicalities regarding the vertices on the boundary of a finite grid as special cases and hence lets us present the proof in a more “clean” way. We point out however that these technicalities are straightforward to deal with (more on this in Section 2). We formally say that a colouring σ of G is a function from V to Q. Let Ω+ be the set of all colourings of G and Ω be the set of all proper q-colourings. Then the distribution π, described earlier, is the uniform distribution on Ω. If σ ∈ Ω+ is a colouring and j ∈ V is a vertex then σj denotes the colour assigned to vertex j in colouring σ. Furthermore, for a subset of vertices Λ ⊆ V and a colouring σ ∈ Ω+ we let σΛ denote the colouring of the vertices in Λ under σ. For each vertex j ∈ V , let Sj denote the set of pairs (σ, τ) ∈ Ω+ × Ω+ of colourings that only differ on the colour assigned to vertex j, that is σi = τi for all i 6= j. Let M be a Markov chain with state space Ω+ and stationary distribution π. Suppose that the transition matrix of M is P . Then the mixing time from an initial colouring σ ∈ Ω+ is the number of steps, that is applications of P , required for M to become sufficiently close to π. Formally the mixing time of M from an initial colouring σ ∈ Ω+ is defined, as a function of the deviation ε from stationarity, by Mixσ(M, ε) = min{t > 0 : dTV(P t(σ, ·), π) ≤ ε}, (1) where dTV(θ1, θ2) = |θ1(i)− θ2(i)| = max |θ1(A)− θ2(A)| (2) is the total variation distance between two distributions θ1 and θ2 on Ω +. The mixing time Mix(M, ε) of M is then obtained my maximising over all possible initial colourings Mix(M, ε) = max Mixσ(M, ε). (3) We say that M is rapidly mixing if the mixing time of M is polynomial in n and log(ε−1). We will make use of a recent result by Pedersen [12] to study the mixing time of a systematic scan Markov chain for 7-colourings of the grid using block updates. We need the following notation in order to define our systematic scan Markov chain. Define the following set Θ = {Θ1, . . . ,Θm} of m blocks. Each block Θk ⊆ V is a 2×2 sub-grid and m is the smallest integer such that k=1Θk = V . For any block Θk and a pair of colourings σ, τ ∈ Ω+ we write “σ = τ on Θk” if σi = τi for each i ∈ Θk and similarly “σ = τ off Θk” if σi = τi for each i ∈ V \ Θk. We also let ∂Θk denote the set of vertices in V \ Θk that are adjacent to some vertex in Θk, and we will refer to ∂Θk as the boundary of Θk. Note from our previous definitions that σ∂Θk denotes the colouring of the boundary of Θk under a colouring σ ∈ Ω +. We will refer to σ∂Θk as a boundary colouring. Finally we say that a 7-colouring of the 2×2 sub-grid Θk agrees with a boundary colouring σ∂Θk if (1) no adjacent sites in Θk are assigned the same colour and (2) each vertex j ∈ Θk is assigned a colour that is different to the colours of all boundary vertices adjacent to j. For each block Θk and colouring σ ∈ Ω + let Ωk(σ) be the subset of Ω + such that if σ′ ∈ Ωk(σ) then σ ′ = σ off Θk and σ agrees with σ∂Θk . Let πk(σ) be the uniform distribution on Ωk(σ). We then define P [k] to be the transition matrix on the state space Ω+ for performing a so-called heat-bath move on Θk. A heat-bath move on a block Θk, given a colouring σ ∈ Ω+, is performed by drawing a new colouring from the distribution πk(σ). Note in particular that applying P [k] to a colouring σ ∈ Ω+ results in a colouring σ′ ∈ Ω+ such that σ′ = σ off Θk and the colouring σ of Θk is proper and agrees with the colouring σ′∂Θk of the boundary of Θk (which is identical to σ∂Θk). We formally define the following systematic scan Markov chain for 7-colourings of G, which systematically performs heat-bath moves on 2×2 sub-grids, as follows. It is worth pointing out that this holds for any ordering of the set of blocks. Definition 1. The systematic scan dynamics for 7-colourings of G is a Markov chain Mgrid with state space Ω + and transition matrix Pgrid = Π It can be shown that the stationary distribution of Mgrid is π by considering the construction of Pgrid. It is customary to refer to one application of Pgrid (that is updating each block once) as one scan. One scan takes |Θk| vertex updates and by construction of Θ this sum is clearly of order O(n). We will prove the following theorem and point out that this is the first proof of rapid mixing of systematic scan for 7-colourings on the grid. Theorem 2. Let Mgrid be the Markov chain from Definition 1 on 7-colourings of G. Then the mixing time of Mgrid is Mix(Mgrid, ε) ≤ 63 log(nε −1). (4) 1.2 Context and related work We now provide an overview of previous achievements for colourings of the grid. Previ- ously it was known that systematic scan for q-colourings on general graphs with maximum vertex degree ∆ mixes in O(logn) scans when q ≥ 2∆ due to Pedersen [12]. That result is a hand-proof and uses block updates that updates the colour at each endpoint of an edge during each step. Earlier Dyer et al. [4] had shown that a single-site systematic scan Markov chain (where one vertex is updated at a time) mixes in O(logn) scans when q > 2∆ and in O(n2 logn) scans when q = 2∆. It is hence well-established that system- atic scan is rapidly mixing for q-colourings of the grid when q ≥ 8 but nothing has been known about the mixing time for smaller q. The results of both Pedersen [12] and Dyer et al. [4] bound the mixing time by studying the influence on a vertex. We will use that technique in this paper as well, however we will construct the required couplings using a heuristic. We defer the required definitions to Section 2 which also contains the proof of Theorem 2. Recent results have revealed that, in a single-site setting, one is not restricted use the total influence on a vertex when analysing the mixing time of systematic scan by bounding influence parameters. In a single-site setting one can define an n×n-matrix whose entries are the influences that all vertices have on each other. Hayes [10] has shown that providing a sufficiently small upper bound on the spectral gap of this matrix implies rapid mixing of both systematic scan and random update. Dyer, Goldberg and Jerrum [6] furthermore showed that an upper bound on any matrix norm also implies rapid mixing of both types of Markov chains. These techniques are however not known to apply to Markov chains using block moves. See the PhD thesis by Pedersen [13] for more comprehensive review of the above results and for the difficulties in extending them to cover block dynamics. As random update Markov chains have received more attention than systematic scan we also summarise some mixing results of interest regarding q-colourings of the grid (recall that a random update Markov chain selects randomly a subset of sites to be updated at each step). Achlioptas et al. [1] give a computer-assisted proof of mixing in O(n logn) updates when q = 6 by considering blocks consisting of 2×3 sub-grids. Our computations are similar in nature to the ones of Achlioptas et al. however their computations are not sufficient to imply mixing of systematic scan as we will discuss in due course. More recently Goldberg, Martin and Paterson [9] gave a hand-proof of mixing in O(n logn) updates when q ≥ 7 using the technique of strong spatial mixing. Previously Salas and Sokal [14] gave a computer-assisted proof of the q = 7 case, a result which was also implied by another computer-assisted result due to Bubley, Dyer and Greenhill [3] that applies to 4-regular triangle-free graphs. Finally it is worth pointing out that, in the special case when q = 3, two complementary results of Luby, Randall and Sinclair [11] and Goldberg, Martin and Paterson [8] give rapid mixing of random update. 2 Bounding the mixing time of systematic scan This section will contain a proof of Theorem 2 although the proof of a crucial lemma, which requires computer-assistance, is deferred to Section 3. We will bound the mixing time of Mgrid by bounding the influence on a vertex, a parameter which we denote by α and will define formally in due course. If α is sufficiently small then Theorem 2 from Pedersen [12] implies that any systematic scan Markov chain, whose transition matrices for updating each block satisfy two simple properties, mixes in O(logn) scans. For completeness we restate this theorem (Theorem 3 below) and in the statement we let M→ denote a systematic scan Markov chain whose transition matrices for each block update satisfy the required properties. Theorem 3. If α < 1 then the mixing time of M→ is Mix(M→, ε) ≤ log(nε−1) . (5) For each block Θk the transition matrix P [k] needs to satisfy the following two prop- erties in order for Theorem 3 to apply. 1. If P [k](σ, τ) > 0 then σ = τ off Θk, and 2. π is invariant with respect to P [k]. It is pointed out in Pedersen [12] that if P [k] is a transition matrix performing a heat-bath move then both of these properties are easily satisfied. Furthermore, it is pointed out that when Ω is the set of proper colourings of a graph, then π is the uniform distribution on Ω as we require. Since the transition matrices P [k] used in the definition of Mgrid perform heat-bath updates we are hence able to use Theorem 3 to bound the mixing time of Mgrid. We are now ready to formally define the parameter α denoting the influence on a vertex. For any pair of colourings (σ, τ) ∈ Si let Ψk(σ, τ) be a coupling of the distributions induced by P [k](σ, ·) and P [k](τ, ·), namely πk(σ) and πk(τ) respectively. We remind the reader that a coupling of two distributions π1 and π2 on state space Ω + is a joint distribution Ω+ × Ω+ such that the marginal distributions are π1 and π2. For ease of reference we also let pj(Ψk(σ, τ)) denote the probability that a vertex j ∈ Θk is assigned a different colour in a pair of colourings drawn from some coupling Ψk(σ, τ). We then let ρki,j = max (σ,τ)∈Si pj(Ψk(σ, τ)) (6) be the influence of i on j under Θk. Finally the parameter α denoting the influence on any vertex is defined as α = max ρki,j. (7) Pedersen [12] actually defines α with a weight associated with each vertex, however as we will not use weights in our proof we have omitted them from the above account. So, in order to upper bound α we are required to upper bound the probability of a discrepancy at each vertex j ∈ Θk under a coupling Ψk(σ, τ) of the distributions πk(σ) and πk(τ) for any pair of colourings (σ, τ) ∈ Si that only differ at the colour of vertex i. Our main task is hence to specify a coupling Ψk(σ, τ) of πk(σ) and πk(τ) for each pair of colourings (σ, τ) ∈ Si and upper bound the probability of assigning a different colour to each vertex in a pair of colourings drawn from that coupling. Consider any block Θk and any pair of colourings (σ, τ) ∈ Si that differ only on the colour assigned to some vertex i. Clearly the distribution on colourings of Θk, induced by πk(σ) only depends on the boundary colouring σ∂Θk . Similarly, the distribution on colourings of Θk, induced by πk(τ) depends only on τ∂Θk . If i 6∈ ∂Θk then the distributions on the colourings of Θk, induced by πk(σ) and πk(τ), respectively, are the same and we let Ψk(σ, τ) be the coupling in which any pair of colourings drawn from Ψk(σ, τ) agree on Θk. That is, if the pair (σ ′, τ ′) of colourings are drawn from Ψk(σ, τ) then σ ′ = σ off Θk, τ ′ = τ off Θk and σ ′ = τ ′ on Θk. This gives ρ i,j = 0 for any i 6∈ ∂Θk and j ∈ Θk. We now need to construct Ψk(σ, τ) for the case when i ∈ ∂Θk. For each j ∈ Θk we need pj(Ψk(σ, τ)) to be sufficiently small in order to avoid ρ i,j being too big. If the ρki,j-values are too big the parameter α will be too big (that is greater than one) and we cannot make use of Theorem 3 to show rapid mixing. Constructing Ψk(σ, τ) by hand such that pj(Ψk(σ, τ)) is sufficiently small is a difficult task. It is, however, straight forward to mechanically determine which colourings have positive measure in the distributions πk(σ) and πk(τ) for a given pair of boundary colourings σ∂Θk and τ∂Θk . From these distributions we can then use some suitable heuristic to construct a coupling that is good enough for our purposes. We hence need to construct a specific coupling for each individual pair of colourings differing only at a single vertex. In order to do this we will make use of the following lemma, which is proved in Section 3. Lemma 4. Let v1, . . . , v4 be the four vertices in a 2×2-block and z1, . . . , z8 be the boundary vertices of the block and let the labeling be as in Figure 1. Let Z and Z ′ be any two 7- colourings of the boundary vertices such that Z and Z ′ agree on each vertex except on z1. Let πZ and πZ′ be the uniform distributions on proper 7-colourings of the block that agree with Z and Z ′, respectively. For i = 1, . . . , 4 let pvi(Ψ) denote the probability that the colour of vertex vi differ in a pair of colourings drawn from a coupling Ψ of πZ and πZ′. Then there exists a coupling Ψ such that pv1(Ψ) < 0.283, pv2(Ψ) < 0.079, pv3(Ψ) < 0.051 and pv4(Ψ) < 0.079. z4 z5 Figure 1: General labeling of the vertices in a 2×2-block Θk and the vertices ∂Θk on the boundary of the block. i (b) Figure 2: A 2×2-block Θk showing all eight positions of a vertex i ∈ ∂Θk on the boundary of the block in relation to a vertex j ∈ Θk in the block. Thus if i ∈ ∂Θk we let Ψk(σ, τ) be the coupling of πk(σ) and πk(τ) that draws the colouring of Θk from the coupling Ψ in Lemma 4, where Z is the boundary colouring obtained from σ∂Θk and Z ′ is obtained from τ∂Θk , and leaves the colour of the remaining vertices, V \Θk, unchanged. That is, if the pair (σ ′, τ ′) of colourings are drawn from Ψk(σ, τ) then σ ′ = σ off Θk, τ ′ = τ off Θk and the colourings of Θk in σ ′ and τ ′ are drawn from the coupling Ψ in Lemma 4 (see the proof for details on how to construct Ψ). It is straightforward to verify that this is indeed a coupling of πk(σ) and πk(τ). Note that due to the symmetry of the 2×2-block, with respect to rotation and mirroring, we can always label the vertices of Θk and ∂Θk such that label z1 in Figure 1 represents the discrepancy vertex i on the boundary. Hence we can make use of Lemma 4 to compute upper bounds on the parameters ρki,j. We summarise the ρ i,j-values in the following Corollary of Lemma 4. Note that due to the symmetry of the block we can assume that vertex j ∈ Θk in the corollary is located in the bottom left corner, as Figure 2 shows. Corollary 5. Let Θk be any 2×2-block, let j ∈ Θk be any vertex in the block and let i ∈ ∂Θk be a vertex on the boundary of the block. Then ρki,j = max (σ,τ)∈Si pj(Ψk(σ, τ)) < 0.283, if i and j as in Figure 2(a) or (b), 0.079, if i and j as in Figure 2(c) or (h), 0.051, if i and j as in Figure 2(e) or (f), 0.079, if i and j as in Figure 2(d) or (g). If i /∈ ∂Θk is not on the boundary of the block then ρ i,j = 0. We can then use Corollary 5 to prove Theorem 2. The proof of Theorem 2 is given here: Proof of Theorem 2. Let αk,j = ρki,j be the influence on j under Θk. We need αk,j to be upper bounded by one for each block Θk and vertex j ∈ Θk in order to ensure that α = maxk maxj∈Θk αk,j is less than one. Fix any block Θk and any vertex j ∈ Θk. A vertex i ∈ ∂Θk on the boundary of the block can occupy eight different positions on the boundary in relation to j as showed in Figure 2(a)–(h). Recall that we are working on the torus, and hence every vertex on the boundary of the block will belong to G. Thus, using the bounds from Corollary 5 we have αk,j = ρki,j < 2(0.283 + 0.079 + 0.051 + 0.079) = 0.984. (9) Then α = maxk maxj∈Θk αk,j < maxk 0.984 = 0.984 < 1 and we obtain the stated bound on the mixing time of Mgrid by Theorem 3. We make the following remark. In the proof of Theorem 2 above, we assume that G is a finite rectangular grid with toroidal boundary conditions. Hence, every block is a 2×2-sub-grid and each vertex on the block boundary belongs to V . We note that if G is a finite rectangular grid without toroidal boundary conditions then some vertices on the boundary ∂Θk of a block Θk might fall outside G. The sum in Equation (9) is over boundary vertices i that do belong to V , and hence the number of terms in this sum is reduced if some boundary vertices do not belong to V , making α smaller. Furthermore, if G is a non-rectangular region of the grid then a block next to the boundary might be smaller than 2×2 vertices. Suppose Θk is a block that is smaller than 2×2 vertices. Then the vertices that are missing in order to make Θk a full 2×2-block are boundary vertices. Suppose i ∈ ∂Θk belongs to V and i ′ ∈ ∂Θk does not belong to V . When constructing couplings Ψk(σ, τ), where (σ, τ) ∈ Si, we must consider the vertex i ′ as “colourless”, which would decrease the value of pki,j . A more rigorous analysis yields that our mixing result with seven colours and 2×2-blocks holds for arbitrary finite regions G of the grid. Of course we have yet to establish a proof of Lemma 4, and the rest of this paper will be concerned with this. Our method of proof uses some ideas of Goldberg, Jalsenius, Martin and Paterson [7] in so far as it is computer assisted and we will be focusing on minimising the probability of assigning different colours to vertex v1 in the constructed couplings. We will however be required to construct a coupling on the 2×2 sub-grid, rather than establishing bounds on the disagreement probability of a vertex adjacent to the initial discrepancy and then extending this to a coupling on the whole block recursively. Our approach is similar to the one Achlioptas et al. [1] take, however we do not have the option of constructing an “optimal” coupling using a suitable linear program (even when feasible) since our probabilities will be maximised over all boundary colourings. The crucial difference between the approaches is that Achlioptas et al. [1] are using path coupling (see Bubley and Dyer [2]) as a proof technique which requires them to bound the expected Hamming distance between a pair of colourings drawn from a coupling. This in turn enables them to, for a given boundary colouring, specify an “optimal” coupling which minimises Hamming distance. We are, however, required to bound the influence of i on j for each boundary colouring and sum over the maximum of these influences. The reason for this is the inherit maximisation over boundary colourings in the definition of ρki,j as described above. Finally it is worth mentioning that providing bounds on the expected Hamming dis- tance is similar to showing that the influence of a vertex is small and it is known that this condition implies rapid mixing of a random update Markov chain, see for example Weitz [15]. In a single-site setting the condition “the influence of a vertex is small” also implies rapid mixing of systematic scan (Dyer et al. [4]), however, in a block setting this condition is not sufficient to give rapid mixing of systematic scan (Pedersen [13]), which is why we need to bound the influence on a vertex. 3 Constructing the coupling by machine In order to prove Lemma 4 we will construct a coupling Ψ of πZ and πZ′ for all pairs of boundary colourings Z and Z ′ that are identical on all boundary vertices but vertex z1, on which Z and Z ′ differ. For each coupling constructed we verify that the probabilities pvi(Ψ), i = 1, . . . , 4, are within the bounds of the lemma. The method is well suited to be carried out with the help of a computer and we have implemented a program in C to do so. Before stating the proof of Lemma 4 we will discuss how a coupling can be represented by an edge-weighted complete bipartite graph. We make use of this representation of Ψ in the proof of the lemma. 3.1 Representing a coupling as a bipartite graph Let S be a set of objects and let W be a set of |S| pairs (s, ws) such that s ∈ S and ws ≥ 0 is a non-negative value representing the weight of s. Each element s ∈ S is contained in exactly one of the pairs in W . If the value ws is an integer (which it is in our case) it can be regarded as the multiplicity of s in a multiset. The set W is referred to as a weighted set of S. Let πS,W be the distribution on S such that the probability of s is proportional to ws, where (s, ws) is a pair in W . More precisely, the probability of s in πS,W is PrπS,W (s) = ws/ (t,wt)∈W wt. For example, let W be a weighted set of S and let S ′ ⊆ S be a subset of S. Assume the weight ws = 0 if s ∈ S\S ′ and ws = k if s ∈ S where k > 0 is a positive constant. Then πS,W is the uniform distribution on S The reason for introducing the notion of a weighted set is that it can be used when specifying a coupling of two distributions. Let S be a set and let W and W ′ be two weighted sets of S such that the sum of the weights in W equals the sum of the weights in W ′. Let wtot denote this sum. That is, wtot = (s,ws)∈W (s′,w′ )∈W ′ w s′. The two weighted sets W and W ′ define two distributions πS,W and πS,W ′ on S. We want to specify a coupling Ψ of πS,W and πS,W ′. Let K|S|,|S| be an edge-weighted complete bipartite graph with vertex sets W and W ′. That is, for each pair (s, ws) ∈ W there is an edge to every pair in W ′. Every edge e of K|S|,|S| has a weight we ≥ 0 such that the following condition holds. Let (s, ws) be any pair in W ∪ W ′ and let E be the set of all |S| edges incident to (s, ws). Then we = ws. It follows that the sum of the edge weights of all |S|2 edges in K|S|,|S| equals wtot, the sum of the weights in W (and W ′). The idea is that K|S|,|S| represents a coupling Ψ of πS,W and πS,W ′. In order to draw a pair of elements from Ψ we randomly select an edge e in K|S|,|S| proportional to its weight. The endpoints of e represent the elements in S drawn from πS,W and πS,W ′. More precisely, the probability of choosing edge e in K|S|,|S| with weight we is we/wtot. If edge e = ((s, ws), (s ′, w′s′)) is chosen it means that we have drawn s from πS,W and s from πS,W ′, the marginal distributions of Ψ. The bipartite graph representation of a coupling will be used when we construct couplings of colourings of 2×2-blocks in the proof of Lemma 4. 3.2 The proof of Lemma 4 Here is the proof of Lemma 4: Proof of Lemma 4. Fix two colourings Z and Z ′ of the boundary that differ on vertex z1. Let c be the colour of vertex z1 in Z and let c ′ 6= c be the colour of z1 in Z ′. Let CZ and CZ′ be the two sets of proper 7-colourings of the block that agree with Z and Z respectively. Let C+ be the set of all 7-colourings of the block. Let WZ and WZ′ be two weighted sets of C+. The weights are assigned as follows. • For the pair (σ, wσ) ∈ WZ let the weight wσ = |CZ′| if σ ∈ CZ , otherwise let wσ = 0. • For the pair (σ, wσ) ∈ WZ′ let the weight wσ = |CZ | if σ ∈ CZ′, otherwise let wσ = 0. It follows from the assignment of the weights that the distribution πC+,WZ is the uniform distribution on CZ . That is, πC+,WZ = πZ . Similarly, πC+,WZ′ is the uniform distribution πZ′ on CZ′. Note that the sum of the weights is |CZ||CZ′| in both WZ and WZ′. Then a coupling Ψ of πC+,WZ and πC+,WZ′ can be specified with an edge-weighted complete bipartite graph K = K|C+|,|C+|. For a given valid assignment of the weights of the edges of K, making K represent a coupling Ψ, we can compute the probabilities of having a mismatch on a vertex vi of the block when two colourings are drawn from Ψ. Let E be the set of all edges e = ((σ, wσ), (σ ′, w′σ′)) in K such that σ and σ ′ differ on vertex vi. Then pvi(Ψ) = e∈E we/|CZ||CZ′|. In order to obtain sufficiently small upper bounds on pvi(Ψ) for the four vertices v1, . . . , v4 in the block we would like to assign weights to the edges of K such that much weight is assigned to edges between colourings that agree on many vertices in the block. In general it is not clear exactly how to assign weights to the edges. For instance, if we assign too much weight to edges between colourings that are identical on vertex v2 we might not be able to assign as much weight as we would like to on edges between colourings that are identical on vertex v4. Thus, the probability of having a mismatch on v4 would increase. Intuitively a good strategy would be to assign as much weight as possible to edges between colourings that are identical on the whole block. This implies that we try to assign as much weight as possible to edges between colourings that are identical on vertex v1, the vertex adjacent to the discrepancy vertex z1 on the boundary. If there is a mismatch on vertex v1 it should be a good idea to assign as much weight as possible to edges between colourings that are identical on the whole block apart from vertex v1. This idea leads to a heuristic in which the assignment of the edge weights is divided into three phases. The exact procedure is described as follows. In phase one we match identical colourings. For all colourings σ ∈ C+ of the block the edge e = ((σ, wσ), (σ, w σ)) in K will be given weight we = min(wσ, w σ). That is, we maximise the probability of drawing the same colouring σ from both πC+,WZ and πC+,WZ′ . For the following two phases we define an ordering of the colourings in C+. We order the colourings lexicographically with respect to the vertex order v3, v2, v4, v1. That is, if the seven colours are 1, . . . , 7 the colouring of v3, v2, v4, v1 will start with 1, 1, 1, 1, respectively. The next colouring will be 1, 1, 1, 2, and so on. This ordering of colourings in C+ carries over to an ordering of the pairs in WZ and WZ′. That is, we order the pairs (σ, wσ) in WZ with respect to the lexicographical ordering of σ. Similarly we order the pairs in WZ′ . This ordering of the pairs will be important in the next two phases. It provides some control of how colourings are being paired up in terms of the assignment of the weights on edges between pairs. Edges will be considered with respect to this ordering because choosing an arbitrary ordering of the edges would not necessarily result in probabilities pvi(Ψ) that would be within the bounds of the lemma. In the second phase we ignore the colour of vertex v1 and match colourings that are identical on all of the remaining three vertices v2, v3 and v4. More precisely, for each pair (σ, wσ) ∈ WZ , considered in the ordering explained above, we consider the edges e = ((σ, wσ), (σ ′, w′σ′)) where σ and σ ′ are identical on all vertices but v1. The edges are considered in the ordering of the second component (σ′, w′σ′) ∈ WZ′. We assign as much weight as possible to e such that the total weight on edges incident to (σ, wσ) ∈ WZ does not exceed wσ and such that the total weight on edges incident to (σ ′, w′σ′) ∈ WZ′ does not exceed w′σ′ . Note that in the lexicographical ordering of the colourings, vertex v1 is the least significant vertex and therefore the ordering provides some level of control of pairing up colourings that are similar on the remaining three vertices. It turns out that the resulting coupling is sufficiently good for proving the lemma. In the third and last phase we assign the remaining weights on the edges. As in phase two, for each pair (σ, wσ) ∈ WZ we consider the edges e = ((σ, wσ), (σ ′, w′σ′)). The pairs and edges are considered in accordance with the ordering explained above. The difference between the second and third phase is that now we do not have any restrictions on the colourings σ and σ′. We assign as much weight as possible to e such that the total weight on edges incident to (σ, wσ) ∈ WZ does not exceed wσ and such that the total weight on edges incident to (σ′, w′σ′) ∈ WZ′ does not exceed w σ′. After phase three we have assigned all weights to the edges of K and hence K represents a coupling Ψ of πZ and πZ′. From K we compute the probabilities pv1(Ψ), pv2(Ψ), pv3(Ψ) and pv4(Ψ) as described above. We have written a C-program which loops through all colourings Z and Z ′ of the boundary of the block and constructs the bipartite graph K as described above. For each boundary the probabilities pv1(Ψ), pv2(Ψ), pv3(Ψ) and pv4(Ψ) are successfully verified to be within the bounds of the lemma. For details on the C-program, see http://www.csc.liv.ac.uk/∼markus/systematicscan/. 4 Partial results for 6-colourings of the grid In previous sections we have seen that systematic scan on the grid using 2×2-blocks and seven colours mixes rapidly. An immediate question is whether we can do better and show rapid mixing with six colours. This matter will be discussed in this section and we will show that, even with bigger block sizes (up to 3×3), it is not possible to show rapid mixing using the technique of this paper. More precisely, we will establish lower bounds on the parameter α for 2×2-blocks, 2×3-blocks and 3×3-blocks. All three lower bounds are greater than one and hence we cannot make use of Theorem 3 to show rapid mixing. 4.1 Establishing lower bounds for 2×2 blocks We start by examining the 2×2-block again but this time with six colours. Lemma 4 provides upper bounds (under any colourings of the boundary) on the probabilities of having discrepancies at each of the four vertices of the block when two 7-colourings are drawn from the specified coupling. For six colours we will show lower bounds on these probabilities under any coupling and a specified pair of boundary colourings. Once again, let v1, . . . , v4 be the four vertices in a 2×2-block and let z1, . . . , z8 be the boundary vertices of the block and let the labeling be as in Figure 1. Let Z and Z ′ be any two 6-colourings of the boundary vertices that assign the same colour to each vertex except for z1. Let πZ and πZ′ be the uniform distributions on the sets of proper 6-colourings of the block that agree with Z and Z ′, respectively. Let Ψminvk (Z,Z ′) be a coupling of πZ and πZ′ that minimises pvk(Ψ). That is, pvk(Ψ) ≥ pvk(Ψ (Z,Z ′)) for all couplings Ψ of πZ and πZ′. Also let p = maxZ,Z′ pvk(Ψ (Z,Z ′)). We can hence say that there exist two 6-colourings Z and Z ′ of the boundary of a 2×2 block, that assign the same colour to each vertex except for z1, such that pvk(Ψ) ≥ p for any coupling Ψ of πZ and πZ′ . We have the following lemma, which is proved by computation. Lemma 6. Consider 6-colourings of the 2×2-block in Figure 1. Then plowv1 ≥ 0.379, plowv2 ≥ 0.107, p ≥ 0.050 and plowv4 ≥ 0.107. Proof. Fix one vertex vk in the block and fix two colourings Z and Z ′ of the boundary of the block that differ only on the colour of vertex z1. Let CZ and CZ′ be the two sets of proper 6-colourings of the block that agree with Z and Z ′, respectively. For c = 1, . . . , 6 let nc be the number of colourings in CZ in which vertex vk is assigned colour c. Similarly let n′c be the number of colourings in CZ′ in which vertex vk is assigned colour c. It is clear that the probability that vk is assigned colour c in a colouring σ ′ drawn from πZ is PrπZ(σ = c) = nc/|CZ|. For c = 1, . . . , 6 define mc = nc|CZ′|, m c = n c|CZ | and M = |CZ||CZ′|. It follows that PrπZ(σ = c) = mc/M and PrπZ′ (τ = c) = m′c/M , where σ′ and τ ′ are colourings drawn from πZ and πZ′, respectively. Observe that the quantities mc, m c and M can be easily computed for a given pair of boundary colourings. Now let Ψ be any coupling of πZ and πZ′ . It is easy to see that the probability that vertex vk is coloured c in both colourings drawn from Ψ can be at most min(mc, m c)/M . Therefore, the probability of drawing two colourings from Ψ such that the colour of vertex vk is the same in both colourings is at most c=1,...,6 min(mc, m c)/M , and the probability of assigning different colours to vertex vk is at least pvk(Ψ) ≥ 1− c=1,...,6min(mc, m c)/M . We have successfully verified the bounds in the statement of the lemma by maximising the lower bound on pvk(Ψ) over all boundary colourings Z and Z ′ for each vertex vk in the block. The computations are carried out with the help of a computer program written in C. For details on the program, see http://www.csc.liv.ac.uk/∼markus/ systematicscan/. For seven colours, Corollary 5 makes use of Lemma 4 to establish upper bounds on the influence parameters ρki,j . These parameters are used in the proof of Theorem 2 to obtain an upper bound on the parameter α. The upper bound on α is shown to be less than one which implies rapid mixing for seven colours when applying Theorem 3. We can use Lemma 6 to obtain lower bounds on the influence parameters ρki,j by completing the coupling in a way analogous to the coupling in Corollary 5. This in turn will result in a v1 v3v2z2 z4 z5 z6 z1 z9z10 v4 v5 v6 Figure 3: (a) General labeling of the vertices in a 2×3-block Θk and the vertices ∂Θk on the boundary of the block. (b)–(c) All ten positions of a vertex i ∈ ∂Θk on the boundary of the block in relation to a vertex j ∈ Θk in the corner of the block. lower bound on the parameter α that is greater than one. That is, following the proof of Theorem 2 and making use of Lemma 6, a lower bound on α will be α ≥ 2(0.379 + 0.107 + 0.050 + 0.107) = 1.286 > 1. (10) Hence we fail to show rapid mixing of systematic scan with six colours using 2×2-blocks. 4.2 Bigger blocks We failed to show rapid mixing of systematic scan with six colours and 2×2-blocks and we will now show that increasing the block size to both 2×3 and 3×3 will not be suf- ficient either. Lemma 7 below considers 2×3-blocks and is analogous to Lemma 6. We make use of the same notation as for Lemma 6, only the block is bigger and the label- ing of the vertices is different (see Figure 3(a)). Lemma 7 is proved by computation in the same way as Lemma 6. For details on the C-program used in the proof, see http://www.csc.liv.ac.uk/∼markus/systematicscan/. Lemma 7. Consider 6-colourings of the 2×3-block in Figure 3(a). Then plowv1 ≥ 0.3671, plowv3 ≥ 0.0298, p ≥ 0.0997 and plowv6 ≥ 0.0174. We will now use Lemma 7 to show that α > 1 for 2×3 blocks. Let Θk be any 2×3- block and let j ∈ Θk be a vertex in a corner of the block. A vertex i ∈ ∂Θk on the boundary of the block can occupy ten different positions on the boundary in relation to j. See Figure 3(b) and (c). We can again determine lower bounds on the influences ρki,j of i on j under Θk from Lemma 7. However, Lemma 7 provides lower bounds on ρ i,j only when i ∈ ∂Θk is adjacent to a corner vertex of the block, as in Figure 3(b). If i is located as in Figure 3(c) we do not know more than that ρki,j is bounded from below by zero. Nevertheless, the lower bound on α exceeds one. Let αk,j = ρki,j be the influence on j under Θk. Following the proof of Theorem 2 and using the lower bounds in Lemma 7 we αk,j = i in Fig. 3(b) ρki,j + i in Fig. 3(c) ρki,j ≥ 2(0.3671 + 0.0298 + 0.0997 + 0.0174) = 1.028, (11) where we set the lower bound on the second sum to zero. Now, α = max αk,j ≥ 1.028 > 1. (12) v1 v3v2z2 z5 z6 z7 v4 v5 v6 z12 z11 v7 v8 v9 v1 v3v2 z6 z7 z8 z10v4 v5 v6 v7 v8 v9 z1 (c) Figure 4: (a)–(b) General labeling of the vertices in a 3×3-block Θk and two different labellings of the vertices ∂Θk on the boundary of the block. The discrepancy vertex on the boundary has label z1. (b)–(c) All twelve positions of a vertex i ∈ ∂Θk on the boundary of the block in relation to a vertex j ∈ Θk in the corner of the block. Hence we cannot use Theorem 3 to show rapid mixing of systematic scan with six colours and 2×3-blocks. It is interesting to note that considering 2×3-blocks was sufficient for Achlioptas et al. [1] to prove mixing of a random update Markov chain for sampling 6-colourings of the grid. Lastly, we increase the block size to 3×3 and show that a lower bound on α is still greater than one. We have the following lemma which is proved by computation in the same way as Lemmas 6 and 7. For details on the C-program used in the proof see http://www.csc.liv.ac.uk/∼markus/systematicscan/. Lemma 8. For 6-colourings of the 3×3-block with vertices labeled as in Figure 4(a) we have plowv1 ≥ 0.3537, p ≥ 0.0245, plowv7 ≥ 0.0245 and p ≥ 0.0071. Furthermore, for 6-colourings of the 3×3-block in Figure 4(b) we have plowv1 ≥ 0.0838, p ≥ 0.0838, plowv7 ≥ 0.0138 and p ≥ 0.0138. Note that Lemma 8 provides lower bounds on the probabilities of having a mismatch on a corner vertex of the block when the discrepancy vertex on the boundary (labeled z1) is adjacent to a corner vertex (Figure 4(a)) and adjacent to a middle vertex (Figure 4(b)). Let Θk be any 3×3-block and let j ∈ Θk be a vertex in a corner of the block. A vertex i ∈ ∂Θk on the boundary of the block can occupy twelve different positions on the boundary in relation to j. See Figure 4(c) and (d). Analogous to Corollary 5 lower bounds on the influences ρki,j of i on j under Θk can be determined from Lemma 8. Let αk,j = ρki,j be the influence on j under Θk. Following the proof of Theorem 2 and using the lower bounds in Lemma 8 we have αk,j = i in Fig. 4(c) ρki,j + i in Fig. 4(d) ρki,j ≥ 2(0.3537 + 0.0245 + 0.0245 + 0.0071) + (0.0838 + 0.0838 + 0.0138 + 0.0138) = 1.0148. (13) Thus, α = maxk maxj∈Θk αk,j ≥ 1.0148 > 1. Hence, we cannot use Theorem 3 to show rapid mixing of systematic scan with six colours and 3×3-blocks. A natural question is whether we can show rapid mixing using even bigger blocks. It seems possible to do this although the computations rapidly become intractable as the block size increases. Already with a 3×3-block the number of boundary colourings we need to consider (after removing isomorphisms) is in excess of 106 and for each boundary colouring there are more than 107 colourings of the block to consider. In addition to simply generating the distributions on colourings of the block, the time it would take to actually construct the required couplings, as we did in the proof of Lemma 4, would also increase. Finally when using a larger block size, different positions of vertex j in the block need to be considered whereas we could make use of to the symmetry of the 2×2-block to only consider one position of vertex j in the block. If different positions of j have to be considered this has to be captured in the construction of the coupling and would likely require more computations. The conclusion is that in order to show rapid mixing for six colours of systematic scan on the grid we would most likely have to rely on a different approach than the one presented in this paper. References [1] D. Achlioptas, M. Molloy, C. Moore, and F. Van Bussel. Sampling grid colorings with fewer colours. In LATIN 2004: Theoretical Informatics, volume 2976 of Lecture Notes in Computer Science, pages 80–89. Springer, 2004. [2] R. Bubley and M. Dyer. Path coupling: a technique for proving rapid mixing in Markov chains. In FOCS ’97: Proceedings of the 38th Symposium on Foundations of Computer Science, pages 223–231. IEEE Computer Society Press, 1997. [3] R. Bubley, M. Dyer, and C. Greenhill. Beating the 2∆ bound for approximately counting colourings: A computer-assisted proof of rapid mixing. In SODA ’98: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 355–363. Society for Industrial and Applied Mathematics, 1998. [4] M. Dyer, L. A. Goldberg, and M. Jerrum. Dobrushin conditions and systematic scan. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 4110 of Lecture Notes in Computer Science, pages 327–338. Springer, 2006. [5] M. Dyer, L. A. Goldberg, and M. Jerrum. Systematic scan and sampling colourings. Annals of Applied Probability, 16(1):185–230, 2006. [6] M. Dyer, L. A. Goldberg, and M. Jerrum. Matrix norms and rapid mixing for spin systems. To appear in the Annals of Applied Probability, 2008. [7] L. A. Goldberg, M. Jalsenius, R. Martin, and M. Paterson. Improved mixing bounds for the anti-ferromagnetic potts model on Z2. LMS Journal of Computation and Mathematics, 9:1–20, 2006. [8] L. A. Goldberg, R. Martin, and M. Paterson. Random sampling of 3-colourings in 2. Random Structures and Algorithms, 24(3):279–302, 2004. [9] L. A. Goldberg, R. Martin, and M. Paterson. Strong spatial mixing with fewer colours for lattice graphs. SIAM Journal on Computing, 35(2):486–517, 2005. [10] T. P. Hayes. A simple condition implying rapid mixing of single-site dynamics on spin systems. In FOCS ’06: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 39–46. IEEE Computer Society Press, 2006. [11] M. Luby, D. Randall, and A. Sinclair. Markov chain algorithms for planar lattice structures. SIAM Journal on Computing, 31(1):167–192, 2001. [12] K. Pedersen. Dobrushin conditions for systematic scan with block dynamics. In Mathematical Foundations of Computer Science 2007, volume 4708 of Lecture Notes in Computer Science, pages 264–275. Springer, 2007. [13] K. Pedersen. On Systematic Scan. PhD thesis, University of Liverpool, 2008. [14] J. Salas and A. D. Sokal. Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. Journal of Statistical Physics, 86(3– 4):551, 1997. [15] D. Weitz. Combinatorial criteria for uniqueness of Gibbs measures. Random Struc- tures and Algorithms, 27(4):445–475, 2005. Introduction Preliminaries and statement of results Context and related work Bounding the mixing time of systematic scan Constructing the coupling by machine Representing a coupling as a bipartite graph The proof of Lemma ?? Partial results for 6-colourings of the grid Establishing lower bounds for 22 blocks Bigger blocks

Magnetic exponents of two-dimensional Ising spin glasses F. Liers1 and O. C. Martin2 Institut für Informatik, Universität zu Köln, Pohligstraße 1, D-50969 Köln,Germany. Univ Paris-Sud, UMR8626, LPTMS, Orsay, F-91405; CNRS, Orsay, F-91405, France. (Dated: October 30, 2018) The magnetic critical properties of two-dimensional Ising spin glasses are controversial. Using exact ground state determination, we extract the properties of clusters flipped when increasing con- tinuously a uniform field. We show that these clusters have many holes but otherwise have statistical properties similar to those of zero-field droplets. A detailed analysis gives for the magnetization ex- ponent δ ≈ 1.30 ± 0.02 using lattice sizes up to 80 × 80; this is compatible with the droplet model prediction δ = 1.282. The reason for previous disagreements stems from the need to analyze both singular and analytic contributions in the low-field regime. PACS numbers: 75.10.Nr, 75.40.-s, 75.40.Mg Spin glasses [1, 2] have been the focus of much interest because of their many remarkable features: they undergo a subtle freezing transition as temperature is lowered, their relaxational dynamics is slow (non-Arrhenius), they exhibit ageing, memory effects, etc. Although there are still some heated disputes concerning three-dimensional spin glasses, the case of two dimensions is relatively consensual, at least in the absence of a magnetic field. Indeed, two recent studies [3, 4] found that the ther- mal properties of two-dimensional Ising spin glasses with Gaussian couplings agreed very well with the predictions of the scaling/droplet pictures [5, 6]. Interestingly, the situation in the presence of a magnetic field remains un- clear; in particular, some Monte Carlo simulations [7] and basically all ground state studies [8, 9, 10] seem to go against the scaling/droplet pictures. Nevertheless, since spin glasses often have large corrections to scaling, the apparent disagreement with the droplet picture resulting from these studies may be misleading and tests in one dimension give credence to this claim [11]. In this study we use state of the art algorithms for de- termining exact ground states in the presence of a mag- netic field and treat significantly larger lattice sizes than in previous work. By finding the precise points where the ground states change as a function of the field, we extract the excitations relevant in the presence of a field which can then be compared to the zero-field droplets. Although for small size lattices we agree with previous studies, at our larger ones a careful analysis, taking into account both the analytic and the singular terms, gives excellent agreement with the droplet picture. The model and its properties — We work on an L×L square lattice having Ising spins on its sites and couplings Jij on its bonds. The Hamiltonian is H({σi}) ≡ − Jijσiσj −B σi (1) The first sum runs over all nearest neighbor sites using periodic boundary conditions to minimize finite size ef- fects. The Jij are independent random variables of either Gaussian or exponential distribution. It is generally agreed that two-dimensional spin glasses have a unique critical point at T = B = 0. There, the free energy is non-analytic and in fact, standard argu- ments [12] suggest that as T → 0 and B → 0 the free energy goes as βF (L, β) ∼ βE0+Gs(TLyT , BLyB ) where E0 is the ground-state energy, β the inverse temperature, while yT and yB are the thermal and magnetic exponents. Previous work when B = 0 is compatible with this form and in fact also agrees with the scaling/droplet picture of Ising spin glasses in which one has yT = −θ ≈ 0.282. The stumbling block concerns the behavior when B 6= 0; there, the droplet prediction in general dimension d is yB = yT + d/2 (2) but the numerical evidence for this is muddled at best. It is thus worth reviewing the hypotheses assumed by the droplet model so that they can be tested directly. We begin with the fact that in any dimension d, a mag- netic field destabilizes the ground state beyond a charac- teristic length scale ξB. To see this, consider an infinites- imal field and zero-field droplets of scale ℓ. These are expected to be compact. The interfacial energy of such droplets is O(ℓθ) while their total magnetization goes as ℓd/2. The magnetic and interfacial energy are then bal- anced when B reaches a value O(1/ℓd/2−θ): at that value of the field, some of the droplets will flip and the ground state will be destabilized. We then see that for each field strength there is an associated magnetic length scale ξB ξB ≈ B −θ (3) This leads to the identification yB = d/2−θ in agreement with Eq. 2, giving yb ≈ 1.282 at d = 2. The droplet model also predicts the scaling of the mag- netization in the B → 0 limit via the exponent δ: m(B) ∼ B1/δ (4) If this form also holds for infinitesimal fields at finite L, we can consider the field B∗ for which system-size http://arxiv.org/abs/0704.1626v2 droplets flip; this happens when B = O(1/LyB) and then the magnetization is O(L−d/2), the droplets having ran- dom magnetizations. This leads to m(B∗) ∼ L−d/2 and m(B∗) ∼ [1/LyB ]1/δ so that dδ = 2yB (5) Although the droplet model arguments are not proofs, they seem quite convincing. Nevertheless, the numerical studies measuring δ do not give good agreement with the prediction δ = 1.282. For instance, using Monte Carlo at “low enough” temperatures, Kinzel and Binder [7] find δ ≈ 1.39. Since thermalization is difficult at low tem- peratures, it is preferable to work directly with ground states, at least when that is possible. This was done by three independent groups [8, 9, 10] with increasing power, leading to δ ≈ 1.48, δ ≈ 1.54 and δ ≈ 1.48. Taken together, these studies show a real discrepancy with the droplet prediction. To save the droplet model from this thorny situation, one can appeal to large corrections to scaling. Such potential effects have been considered [11] in dimension one where it was shown that ξB was poorly fitted by a pure power law unless the fields were very small. Here we revisit the two-dimensional case to reveal either the size of the corrections to scaling or a cause for the break down of the droplet reasoning. Computation of ground states — We determine the exact ground state of the Hamiltonian (1) by computing a maximum cut in the graph of interactions [13], a promi- nent problem in combinatorial optimization. Whereas it is polynomially solvable on two-dimensional grids with- out a field and couplings bounded by a polynomial in the size of the input, it is NP-hard with an external field. In practice, we rely on a branch-and-cut algorithm [14, 15]. Let the ground state at a field B be denoted as {σ(G)(B)}. To study the magnetization, we computed the ground states at increasing values of B, in steps of size 0.02. When focusing instead on the flipping clusters, we had to determine the intervals in which the ground state was constant and in what manner it changed when going from one interval to the next. In Fig. 1 we show the associated piecewise constant magnetization curves for three samples of the disorder variables Jij at L = 10. To get the sequence of intervals or break points associ- ated with such a function exactly, we start by computing the ground state in zero field. By applying postoptimal- ity analysis from linear programming theory, we deter- mine [10, 15] a range ∆B such that the ground state at a field B remains the optimum in the interval [B,B+∆B]. We reoptimize at B+∆B+ ǫ, with ǫ being a sufficiently small number. By repeatedly applying this procedure, we get a new ground state configuration, and increase B until all spins are aligned with the field. This procedure works for system sizes in which the branch-and-cut pro- gram can prove optimality without branching, i.e., with- out dividing the problem into smaller sub-problems. If the algorithm branches (this occurs only for the largest 0 0.5 1 1.5 2 sample 1 sample 2 sample 3 FIG. 1: Magnetization as a function of B for three typical L = 10 samples. system sizes studied here), we apply a divide-and-conquer strategy for determining {σ(G)(B)} in an interval, say [B1, B2]. For a fixed configuration the Hamiltonian (1) is linear in the field, the slope being the system’s magneti- zation. Let f1, f2 be the two linear functions associated with {σ(G)(B1)} and {σ(G)(B2)}. If f1 and f2 are equal, we are done. Otherwise, we determine the field B3 at which the functions intersect and recursively solve the problem in the intervals [B1, B3] and [B3, B2]. A typical sample at L = 80 requires about 2 hours of cpu on a work station for determining the ground states when B goes through the multiples of 0.02. The more time consuming computation of the exact break points takes about 4 hours on typical samples with L = 60, but less than a minute if L ≤ 30 because the ground- state determinations are fast and branching almost never arises. For our work, we considered mainly the case of Gaussian Jij , analyzing 2500 samples at L = 80, 5000 at L = 70, and from 2000 to 11000 instances for sizes L = 60, 50, 40, 30, 24, 20, 14. We also analyzed a smaller number of samples for Jij taken from an exponential distribution; exponents showed no significant differences when comparing to the Gaussian case. The exponent δ — Given the Hamiltonian, it is easy to see that for each sample the magnetization (density) mJ(B) = must be an increasing function of B. (The index J on the magnetization is to recall that it depends on the disorder realisation, but in the large L limit mJ is self averag- ing; also, without loss of generality, we shall work with B > 0.) At large fields mJ saturates to 1, while at low fields, its growth law must be above a linear function of B. Indeed, for continuous Jij , the distribution of local fields has a finite density at zero and so small clusters of spins will flip and will lead to a linear contribution to the magnetization. A more singular behavior is in fact predicted by the droplet model since δ > 1, indicating that the system is anomalously sensitive to the magnetic field perturbation. If B is not too small, the convergence to the thermo- dynamic limit (L → ∞) is rapid, and in fact one expects exponential convergence in L/ξB. We should thus see an envelope curve m(B) appear as L increases; to make a power dependence on B manifest, we show in Fig. 2 a log-log plot of the ratio m(B)/B1/δ where δ is set to its droplet scaling value of 1.282. For that value of δ there 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 FIG. 2: Magnetization divided by B1/δ as a function of B; the B1/1.45 line is to guide the eye. From top to bottom, L = 14, 20, 24, 30, 40, 50, 60, 70, and 80. Inset: m − χ1B divided by B1/δ as a function of B. (Same L and symbols as in main part of the figure.) In both cases, δ is set to its droplet model value, δ = 1.282. is not much indication that a flat region is developping when L increases, while at L = 50 a direct fit to a power gives δ = 1.45 (cf. line displayed in the figure to guide the eye), as found in previous work [8, 9, 10]. The prob- lem with this simple analysis is that m has both analytic and non-analytic contributions; to lowest order we have m = χ1B + χSB 1/δ + . . . (7) Although χ1B is sub-dominant, it is far from negligible in practice; for instance for it to contribute to less than 10% of m, one would need B < (0.1χS/χ1) 1/0.282. This could easily mean B < 10−3 for which there would be huge finite size effects since L would then be much smaller than the magnetic length ξB . We thus must take into account the term χ1B; we have done this, adjusting χ1 so that (m−χ1B)/B1/δ has an envelope as flat as possible. The result is displayed in the inset of Fig. 2, showing that the droplet scaling fits very well the data as long as the χ1B term is included. In fact, direct fits to the form of Eq. 7 give δ in the range 1.28 to 1.32 depending on the sets of L’s included in the fits. The clusters that flip are like zero-field droplets — The fundamental hypothesis in the droplet argument re- lating δ or yB to θ is the fact that in an infinitesimal field one flips droplets defined in zero field, droplets which are compact and have random (except for the sign) magneti- zations. We therefore now focus on the properties of the actual clusters that are flipped at low fields. At zero field, the droplet of lowest energy almost al- ways is a single spin (this follows from the large number of such droplets, in spite of their typically higher energy). Thus as the field is turned on, the ground state changes first mainly via single spin flips, and when large clusters do flip (they finally do so but at larger fields), they nec- essarily have many “holes” and thus do not correspond exactly to zero-field droplets. This is not a problem for the droplet argument as long as these clusters are com- pact and have random magnetizations. To test this, we 1.14 1.16 1.18 1.22 1.24 10 20 30 40 50 60 FIG. 3: The cluster magnetization divided by the square root of cluster volume — for the largest cluster flipped in each sample — is insensitive to L. Inset: The clusters’s mean surface scales as LdS with dS ≈ 1.32. consider for each realization of the Jij disorder the largest cluster that flips during the whole passage from B = 0 to B = ∞. According to the droplet picture, this clus- ter should contain a number of spins V that scales as L2 (compactness) and have a total magnetization M that scales as V (randomness). This is confirmed by our data where we find M/V ∼ 2/L; in Fig. 3 we plot the disorder mean ofM/ V for increasing L; manifestly, this mean is remarkably insensitive to L. Similar conclusions apply to V/L2. For completeness, we show in the inset of the figure that the surface of these clusters, defined as the number of lattice bonds connecting them to their com- plement, grows as LdS with dS ≈ 1.32; this is to be com- pared to the value dS = 1.27 for zero-field droplets [16], in spite of the fact that our clusters have holes. All in all, we find that the clusters considered have statistical properties that are completely compatible with those as- sumed in the droplet scaling argument, thereby directly validating the associated hypotheses. The magnetic exponent yB and finite size scaling of the magnetization — One can also measure the exponent yB directly via the magnetic length which scales as ξB ∼ B−1/yB . For each sample, define B∗J as that field where the ground state changes by the largest cluster of spins as described in the previous paragraph. Since these clusters involve a number of spins growing as L2, we can identify J ) with L. Let B ∗ be the disorder average of B∗J ; then B∗ ∼ L−yB from which we can estimate yB. We find that a pure power with yB set to its value in the droplet picture describes the data quite well; in the inset of Fig. 4 we display the product L1.282B∗ as a function of 0.01 0.04 0.07 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 FIG. 4: Inset: Field B∗ times L1/δ as a function of 1/L shows a limit at large L as expected in the droplet model (δ = 1.282). Main figure: Data collapse plot exhibiting finite size scaling of the singular part of the magnetization (L = 10, 14, 20, 24, 30, 40, 50, and 60). 1/L and see that the behavior is compatible with a large L limit with O(1/L) finite size effects. Direct fits to the form B∗(L) = uL−yB(1+v/L) give yBs in the range 1.28 to 1.30 depending on the points included in the fit. Given the magnetic length, one can perform finite size scaling (FSS) on the magnetization data m(B,L). Since FSS applies to the singular part of an observable, we should have a data collapse according to m(B,L)− χ1B m(B∗, L)− χ1B∗ = W (B/B∗) (8) W being a universal function, W (0) = O(1) and W (x) ∼ x1/δ at large x. Using the value of χ1 previously deter- mined, we display in Fig. 4 the associated data. The collapse is excellent and we have checked that this also holds when the Jij are drawn from an exponential distri- bution. Added to the figure is the function x1/δ to guide the eye (δ = 1.282 as predicted by the droplet model). Conclusions — We have investigated the 2d Ising spin glass with Gaussian and exponential couplings at zero temperature as a function of the magnetic field. The magnetization exponent δ can be measured; previ- ous studies did not find good agreement with the droplet model prediction δ = 1.282 because the analytic con- tributions to the magnetization curve were mishandled, while in this work we found instead 1.28 ≤ δ ≤ 1.32. We also performed a direct measurement of the magnetic length, obtaining for the associated exponent 1.28 ≤ yB ≤ 1.30, again in excellent agreement with the droplet prediction. With this length we showed that finite size scaling is realized without going to infinitesimal fields or huge lattices. Finally, we validated the hypotheses un- derlying the arguments of the droplet model inherent to the in-field case; we find in particular that in the low-field limit the spin clusters that are relevant are compact and have random magnetizations. In summary, by combin- ing improved computational techniques and greater care in the analysis, we have lifted the discrepancy on the magnetic exponents that has existed for over a decade between numerics and droplet scaling. We thank T. Jorg for helpful comments. The computa- tions were performed on the cliot cluster of the Regional Computing Center and on the scale cluster of E. Speck- enmeyer’s group, both in Cologne. FL has been sup- ported by the German Science Foundation in the projects Ju 204/9 and Li 1675/1 and by the Marie Curie RTN ADONET 504438 funded by the EU. This work was sup- ported also by the EEC’s HPP under contract HPRN- CT-2002-00307 (DYGLAGEMEM). [1] M. Mézard, G. Parisi, and M. A. Virasoro, Spin-Glass Theory and Beyond, vol. 9 of Lecture Notes in Physics (World Scientific, Singapore, 1987). [2] A. Young, ed., Spin Glasses and Random Fields (World Scientific, Singapore, 1998). [3] H. G. Katzgraber, L. Lee, and A. Young, Phys. Rev. B 70, 014417 (2004). [4] A. Hartmann and J. Houdayer, Phys. Rev. B 70, 014418 (2004), cond-mat/0402036. [5] A. J. Bray and M. A. Moore, in Heidelberg Colloquium on Glassy Dynamics, edited by J. L. van Hemmen and I. Morgenstern (Springer, Berlin, 1986), vol. 275 of Lec- ture Notes in Physics, pp. 121–153. [6] D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601 (1986). [7] W. Kinzel and K. Binder, Phys. Rev. Lett. 50, 1509 (1983). [8] N. Kawashima and M. Suzuki, J. Phys. A 25, 1055 (1992). [9] F. Barahona, Phys. Rev. B 49, 12864 (1994). [10] H. Rieger, L. Santen, U. Blasum, M. Diehl, M. Jünger, and G. Rinaldi, J. Phys. A 29, 3939 (1996). [11] A. Carter, A. Bray, and M. Moore, J. Phys. A 36, 5699 (2003). [12] J. Cardy, Scaling and renormalization in statistical physics (Cambridge University Press, Cambridge, 1996). [13] F. Barahona, J. Phys. A 15, 3241 (1982). [14] F. Barahona, M. Grötschel, M. Jünger, and G. Reinelt, Oper. Res. 36, 493 (1988). [15] F. Liers, M. Jünger, G. Reinelt, and G. Rinaldi, in New Optimization Algorithms in Physics, edited by A. Hart- mann and H. Rieger (Wiley-VCH, Berlin, 2004). [16] A. Hartmann and A. Young, Phys. Rev. B 66, 094419 (2002), cond-mat/0205659.

The magnetic critical properties of two-dimensional Ising spin glasses are controversial. Using exact ground state determination, we extract the properties of clusters flipped when increasing continuously a uniform field. We show that these clusters have many holes but otherwise have statistical properties similar to those of zero-field droplets. A detailed analysis gives for the magnetization exponent delta = 1.30 +/- 0.02 using lattice sizes up to 80x80; this is compatible with the droplet model prediction delta = 1.282. The reason for previous disagreements stems from the need to analyze both singular and analytic contributions in the low-field regime.

Magnetic exponents of two-dimensional Ising spin glasses F. Liers1 and O. C. Martin2 Institut für Informatik, Universität zu Köln, Pohligstraße 1, D-50969 Köln,Germany. Univ Paris-Sud, UMR8626, LPTMS, Orsay, F-91405; CNRS, Orsay, F-91405, France. (Dated: October 30, 2018) The magnetic critical properties of two-dimensional Ising spin glasses are controversial. Using exact ground state determination, we extract the properties of clusters flipped when increasing con- tinuously a uniform field. We show that these clusters have many holes but otherwise have statistical properties similar to those of zero-field droplets. A detailed analysis gives for the magnetization ex- ponent δ ≈ 1.30 ± 0.02 using lattice sizes up to 80 × 80; this is compatible with the droplet model prediction δ = 1.282. The reason for previous disagreements stems from the need to analyze both singular and analytic contributions in the low-field regime. PACS numbers: 75.10.Nr, 75.40.-s, 75.40.Mg Spin glasses [1, 2] have been the focus of much interest because of their many remarkable features: they undergo a subtle freezing transition as temperature is lowered, their relaxational dynamics is slow (non-Arrhenius), they exhibit ageing, memory effects, etc. Although there are still some heated disputes concerning three-dimensional spin glasses, the case of two dimensions is relatively consensual, at least in the absence of a magnetic field. Indeed, two recent studies [3, 4] found that the ther- mal properties of two-dimensional Ising spin glasses with Gaussian couplings agreed very well with the predictions of the scaling/droplet pictures [5, 6]. Interestingly, the situation in the presence of a magnetic field remains un- clear; in particular, some Monte Carlo simulations [7] and basically all ground state studies [8, 9, 10] seem to go against the scaling/droplet pictures. Nevertheless, since spin glasses often have large corrections to scaling, the apparent disagreement with the droplet picture resulting from these studies may be misleading and tests in one dimension give credence to this claim [11]. In this study we use state of the art algorithms for de- termining exact ground states in the presence of a mag- netic field and treat significantly larger lattice sizes than in previous work. By finding the precise points where the ground states change as a function of the field, we extract the excitations relevant in the presence of a field which can then be compared to the zero-field droplets. Although for small size lattices we agree with previous studies, at our larger ones a careful analysis, taking into account both the analytic and the singular terms, gives excellent agreement with the droplet picture. The model and its properties — We work on an L×L square lattice having Ising spins on its sites and couplings Jij on its bonds. The Hamiltonian is H({σi}) ≡ − Jijσiσj −B σi (1) The first sum runs over all nearest neighbor sites using periodic boundary conditions to minimize finite size ef- fects. The Jij are independent random variables of either Gaussian or exponential distribution. It is generally agreed that two-dimensional spin glasses have a unique critical point at T = B = 0. There, the free energy is non-analytic and in fact, standard argu- ments [12] suggest that as T → 0 and B → 0 the free energy goes as βF (L, β) ∼ βE0+Gs(TLyT , BLyB ) where E0 is the ground-state energy, β the inverse temperature, while yT and yB are the thermal and magnetic exponents. Previous work when B = 0 is compatible with this form and in fact also agrees with the scaling/droplet picture of Ising spin glasses in which one has yT = −θ ≈ 0.282. The stumbling block concerns the behavior when B 6= 0; there, the droplet prediction in general dimension d is yB = yT + d/2 (2) but the numerical evidence for this is muddled at best. It is thus worth reviewing the hypotheses assumed by the droplet model so that they can be tested directly. We begin with the fact that in any dimension d, a mag- netic field destabilizes the ground state beyond a charac- teristic length scale ξB. To see this, consider an infinites- imal field and zero-field droplets of scale ℓ. These are expected to be compact. The interfacial energy of such droplets is O(ℓθ) while their total magnetization goes as ℓd/2. The magnetic and interfacial energy are then bal- anced when B reaches a value O(1/ℓd/2−θ): at that value of the field, some of the droplets will flip and the ground state will be destabilized. We then see that for each field strength there is an associated magnetic length scale ξB ξB ≈ B −θ (3) This leads to the identification yB = d/2−θ in agreement with Eq. 2, giving yb ≈ 1.282 at d = 2. The droplet model also predicts the scaling of the mag- netization in the B → 0 limit via the exponent δ: m(B) ∼ B1/δ (4) If this form also holds for infinitesimal fields at finite L, we can consider the field B∗ for which system-size http://arxiv.org/abs/0704.1626v2 droplets flip; this happens when B = O(1/LyB) and then the magnetization is O(L−d/2), the droplets having ran- dom magnetizations. This leads to m(B∗) ∼ L−d/2 and m(B∗) ∼ [1/LyB ]1/δ so that dδ = 2yB (5) Although the droplet model arguments are not proofs, they seem quite convincing. Nevertheless, the numerical studies measuring δ do not give good agreement with the prediction δ = 1.282. For instance, using Monte Carlo at “low enough” temperatures, Kinzel and Binder [7] find δ ≈ 1.39. Since thermalization is difficult at low tem- peratures, it is preferable to work directly with ground states, at least when that is possible. This was done by three independent groups [8, 9, 10] with increasing power, leading to δ ≈ 1.48, δ ≈ 1.54 and δ ≈ 1.48. Taken together, these studies show a real discrepancy with the droplet prediction. To save the droplet model from this thorny situation, one can appeal to large corrections to scaling. Such potential effects have been considered [11] in dimension one where it was shown that ξB was poorly fitted by a pure power law unless the fields were very small. Here we revisit the two-dimensional case to reveal either the size of the corrections to scaling or a cause for the break down of the droplet reasoning. Computation of ground states — We determine the exact ground state of the Hamiltonian (1) by computing a maximum cut in the graph of interactions [13], a promi- nent problem in combinatorial optimization. Whereas it is polynomially solvable on two-dimensional grids with- out a field and couplings bounded by a polynomial in the size of the input, it is NP-hard with an external field. In practice, we rely on a branch-and-cut algorithm [14, 15]. Let the ground state at a field B be denoted as {σ(G)(B)}. To study the magnetization, we computed the ground states at increasing values of B, in steps of size 0.02. When focusing instead on the flipping clusters, we had to determine the intervals in which the ground state was constant and in what manner it changed when going from one interval to the next. In Fig. 1 we show the associated piecewise constant magnetization curves for three samples of the disorder variables Jij at L = 10. To get the sequence of intervals or break points associ- ated with such a function exactly, we start by computing the ground state in zero field. By applying postoptimal- ity analysis from linear programming theory, we deter- mine [10, 15] a range ∆B such that the ground state at a field B remains the optimum in the interval [B,B+∆B]. We reoptimize at B+∆B+ ǫ, with ǫ being a sufficiently small number. By repeatedly applying this procedure, we get a new ground state configuration, and increase B until all spins are aligned with the field. This procedure works for system sizes in which the branch-and-cut pro- gram can prove optimality without branching, i.e., with- out dividing the problem into smaller sub-problems. If the algorithm branches (this occurs only for the largest 0 0.5 1 1.5 2 sample 1 sample 2 sample 3 FIG. 1: Magnetization as a function of B for three typical L = 10 samples. system sizes studied here), we apply a divide-and-conquer strategy for determining {σ(G)(B)} in an interval, say [B1, B2]. For a fixed configuration the Hamiltonian (1) is linear in the field, the slope being the system’s magneti- zation. Let f1, f2 be the two linear functions associated with {σ(G)(B1)} and {σ(G)(B2)}. If f1 and f2 are equal, we are done. Otherwise, we determine the field B3 at which the functions intersect and recursively solve the problem in the intervals [B1, B3] and [B3, B2]. A typical sample at L = 80 requires about 2 hours of cpu on a work station for determining the ground states when B goes through the multiples of 0.02. The more time consuming computation of the exact break points takes about 4 hours on typical samples with L = 60, but less than a minute if L ≤ 30 because the ground- state determinations are fast and branching almost never arises. For our work, we considered mainly the case of Gaussian Jij , analyzing 2500 samples at L = 80, 5000 at L = 70, and from 2000 to 11000 instances for sizes L = 60, 50, 40, 30, 24, 20, 14. We also analyzed a smaller number of samples for Jij taken from an exponential distribution; exponents showed no significant differences when comparing to the Gaussian case. The exponent δ — Given the Hamiltonian, it is easy to see that for each sample the magnetization (density) mJ(B) = must be an increasing function of B. (The index J on the magnetization is to recall that it depends on the disorder realisation, but in the large L limit mJ is self averag- ing; also, without loss of generality, we shall work with B > 0.) At large fields mJ saturates to 1, while at low fields, its growth law must be above a linear function of B. Indeed, for continuous Jij , the distribution of local fields has a finite density at zero and so small clusters of spins will flip and will lead to a linear contribution to the magnetization. A more singular behavior is in fact predicted by the droplet model since δ > 1, indicating that the system is anomalously sensitive to the magnetic field perturbation. If B is not too small, the convergence to the thermo- dynamic limit (L → ∞) is rapid, and in fact one expects exponential convergence in L/ξB. We should thus see an envelope curve m(B) appear as L increases; to make a power dependence on B manifest, we show in Fig. 2 a log-log plot of the ratio m(B)/B1/δ where δ is set to its droplet scaling value of 1.282. For that value of δ there 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 FIG. 2: Magnetization divided by B1/δ as a function of B; the B1/1.45 line is to guide the eye. From top to bottom, L = 14, 20, 24, 30, 40, 50, 60, 70, and 80. Inset: m − χ1B divided by B1/δ as a function of B. (Same L and symbols as in main part of the figure.) In both cases, δ is set to its droplet model value, δ = 1.282. is not much indication that a flat region is developping when L increases, while at L = 50 a direct fit to a power gives δ = 1.45 (cf. line displayed in the figure to guide the eye), as found in previous work [8, 9, 10]. The prob- lem with this simple analysis is that m has both analytic and non-analytic contributions; to lowest order we have m = χ1B + χSB 1/δ + . . . (7) Although χ1B is sub-dominant, it is far from negligible in practice; for instance for it to contribute to less than 10% of m, one would need B < (0.1χS/χ1) 1/0.282. This could easily mean B < 10−3 for which there would be huge finite size effects since L would then be much smaller than the magnetic length ξB . We thus must take into account the term χ1B; we have done this, adjusting χ1 so that (m−χ1B)/B1/δ has an envelope as flat as possible. The result is displayed in the inset of Fig. 2, showing that the droplet scaling fits very well the data as long as the χ1B term is included. In fact, direct fits to the form of Eq. 7 give δ in the range 1.28 to 1.32 depending on the sets of L’s included in the fits. The clusters that flip are like zero-field droplets — The fundamental hypothesis in the droplet argument re- lating δ or yB to θ is the fact that in an infinitesimal field one flips droplets defined in zero field, droplets which are compact and have random (except for the sign) magneti- zations. We therefore now focus on the properties of the actual clusters that are flipped at low fields. At zero field, the droplet of lowest energy almost al- ways is a single spin (this follows from the large number of such droplets, in spite of their typically higher energy). Thus as the field is turned on, the ground state changes first mainly via single spin flips, and when large clusters do flip (they finally do so but at larger fields), they nec- essarily have many “holes” and thus do not correspond exactly to zero-field droplets. This is not a problem for the droplet argument as long as these clusters are com- pact and have random magnetizations. To test this, we 1.14 1.16 1.18 1.22 1.24 10 20 30 40 50 60 FIG. 3: The cluster magnetization divided by the square root of cluster volume — for the largest cluster flipped in each sample — is insensitive to L. Inset: The clusters’s mean surface scales as LdS with dS ≈ 1.32. consider for each realization of the Jij disorder the largest cluster that flips during the whole passage from B = 0 to B = ∞. According to the droplet picture, this clus- ter should contain a number of spins V that scales as L2 (compactness) and have a total magnetization M that scales as V (randomness). This is confirmed by our data where we find M/V ∼ 2/L; in Fig. 3 we plot the disorder mean ofM/ V for increasing L; manifestly, this mean is remarkably insensitive to L. Similar conclusions apply to V/L2. For completeness, we show in the inset of the figure that the surface of these clusters, defined as the number of lattice bonds connecting them to their com- plement, grows as LdS with dS ≈ 1.32; this is to be com- pared to the value dS = 1.27 for zero-field droplets [16], in spite of the fact that our clusters have holes. All in all, we find that the clusters considered have statistical properties that are completely compatible with those as- sumed in the droplet scaling argument, thereby directly validating the associated hypotheses. The magnetic exponent yB and finite size scaling of the magnetization — One can also measure the exponent yB directly via the magnetic length which scales as ξB ∼ B−1/yB . For each sample, define B∗J as that field where the ground state changes by the largest cluster of spins as described in the previous paragraph. Since these clusters involve a number of spins growing as L2, we can identify J ) with L. Let B ∗ be the disorder average of B∗J ; then B∗ ∼ L−yB from which we can estimate yB. We find that a pure power with yB set to its value in the droplet picture describes the data quite well; in the inset of Fig. 4 we display the product L1.282B∗ as a function of 0.01 0.04 0.07 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 FIG. 4: Inset: Field B∗ times L1/δ as a function of 1/L shows a limit at large L as expected in the droplet model (δ = 1.282). Main figure: Data collapse plot exhibiting finite size scaling of the singular part of the magnetization (L = 10, 14, 20, 24, 30, 40, 50, and 60). 1/L and see that the behavior is compatible with a large L limit with O(1/L) finite size effects. Direct fits to the form B∗(L) = uL−yB(1+v/L) give yBs in the range 1.28 to 1.30 depending on the points included in the fit. Given the magnetic length, one can perform finite size scaling (FSS) on the magnetization data m(B,L). Since FSS applies to the singular part of an observable, we should have a data collapse according to m(B,L)− χ1B m(B∗, L)− χ1B∗ = W (B/B∗) (8) W being a universal function, W (0) = O(1) and W (x) ∼ x1/δ at large x. Using the value of χ1 previously deter- mined, we display in Fig. 4 the associated data. The collapse is excellent and we have checked that this also holds when the Jij are drawn from an exponential distri- bution. Added to the figure is the function x1/δ to guide the eye (δ = 1.282 as predicted by the droplet model). Conclusions — We have investigated the 2d Ising spin glass with Gaussian and exponential couplings at zero temperature as a function of the magnetic field. The magnetization exponent δ can be measured; previ- ous studies did not find good agreement with the droplet model prediction δ = 1.282 because the analytic con- tributions to the magnetization curve were mishandled, while in this work we found instead 1.28 ≤ δ ≤ 1.32. We also performed a direct measurement of the magnetic length, obtaining for the associated exponent 1.28 ≤ yB ≤ 1.30, again in excellent agreement with the droplet prediction. With this length we showed that finite size scaling is realized without going to infinitesimal fields or huge lattices. Finally, we validated the hypotheses un- derlying the arguments of the droplet model inherent to the in-field case; we find in particular that in the low-field limit the spin clusters that are relevant are compact and have random magnetizations. In summary, by combin- ing improved computational techniques and greater care in the analysis, we have lifted the discrepancy on the magnetic exponents that has existed for over a decade between numerics and droplet scaling. We thank T. Jorg for helpful comments. The computa- tions were performed on the cliot cluster of the Regional Computing Center and on the scale cluster of E. Speck- enmeyer’s group, both in Cologne. FL has been sup- ported by the German Science Foundation in the projects Ju 204/9 and Li 1675/1 and by the Marie Curie RTN ADONET 504438 funded by the EU. This work was sup- ported also by the EEC’s HPP under contract HPRN- CT-2002-00307 (DYGLAGEMEM). [1] M. Mézard, G. Parisi, and M. A. Virasoro, Spin-Glass Theory and Beyond, vol. 9 of Lecture Notes in Physics (World Scientific, Singapore, 1987). [2] A. Young, ed., Spin Glasses and Random Fields (World Scientific, Singapore, 1998). [3] H. G. Katzgraber, L. Lee, and A. Young, Phys. Rev. B 70, 014417 (2004). [4] A. Hartmann and J. Houdayer, Phys. Rev. B 70, 014418 (2004), cond-mat/0402036. [5] A. J. Bray and M. A. Moore, in Heidelberg Colloquium on Glassy Dynamics, edited by J. L. van Hemmen and I. Morgenstern (Springer, Berlin, 1986), vol. 275 of Lec- ture Notes in Physics, pp. 121–153. [6] D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601 (1986). [7] W. Kinzel and K. Binder, Phys. Rev. Lett. 50, 1509 (1983). [8] N. Kawashima and M. Suzuki, J. Phys. A 25, 1055 (1992). [9] F. Barahona, Phys. Rev. B 49, 12864 (1994). [10] H. Rieger, L. Santen, U. Blasum, M. Diehl, M. Jünger, and G. Rinaldi, J. Phys. A 29, 3939 (1996). [11] A. Carter, A. Bray, and M. Moore, J. Phys. A 36, 5699 (2003). [12] J. Cardy, Scaling and renormalization in statistical physics (Cambridge University Press, Cambridge, 1996). [13] F. Barahona, J. Phys. A 15, 3241 (1982). [14] F. Barahona, M. Grötschel, M. Jünger, and G. Reinelt, Oper. Res. 36, 493 (1988). [15] F. Liers, M. Jünger, G. Reinelt, and G. 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Thin elastic shells with variable thickness for lithospheric flexure of one-plate planets Mikael Beuthe Royal Observatory of Belgium, Brussels, Belgium. E-mail: mbeuthe@oma.be Abstract Planetary topography can either be modeled as a load supported by the lithosphere, or as a dynamical effect due to lithospheric flexure caused by mantle convection. In both cases the response of the lithosphere to external forces can be calculated with the theory of thin elastic plates or shells. On one-plate planets the spherical geometry of the lithospheric shell plays an important role in the flexure mechanism. So far the equations governing the deformations and stresses of a spherical shell have only been derived under the assumption of a shell of constant thickness. However local studies of gravity and topography data suggest large variations in the thickness of the lithosphere. In this article we obtain the scalar flexure equations governing the deformations of a thin spherical shell with variable thickness or variable Young’s modulus. The resulting equations can be solved in succession, except for a system of two simultaneous equations, the solutions of which are the transverse deflection and an associated stress function. In order to include bottom loading generated by mantle convection, we extend the method of stress functions to include loads with a toroidal tangential component. We further show that toroidal tangential displacement always occurs if the shell thickness varies, even in the absence of toroidal loads. We finally prove that the degree-one harmonic components of the transverse deflection and of the toroidal tangential displacement are independent of the elastic properties of the shell and are associated with translational and rotational freedom. While being constrained by the static assumption, degree-one loads can deform the shell and generate stresses. The flexure equations for a shell of variable thickness are useful not only for the prediction of the gravity signal in local admittance studies, but also for the construction of stress maps in tectonic analysis. 1 Introduction Terrestrial planets are flattened spheres only at first sight: close-up views reveal a rich topography with unique characteristics for each planet. Unity in diversity is found by studying the support mechanism for topographic deviations from the hydrostatic planetary shape. A simple mechanism, called isostasy, postulates mountains floating with iceberg-like roots in the higher density mantle. Another simple mech- anism assumes that mountains stand on a rigid membrane called the mechanical lithosphere. The two simple models predict very different gravity signals, from very weak in the first to very strong in the second. The truth lies in-between: an encompassing model views topographic structures as loads on an elastic shell with finite rigidity, called the elastic lithosphere (the elastic lithosphere is a subset of the mechanical lithosphere). The rigidity depends on the elastic properties of the rocks and on the apparent elastic thickness of the lithosphere. The latter parameter is the main objective of many studies, since its value can be related to the lithospheric composition and temperature. Another important application of the model of lithospheric flexure is the determination of stress maps, which can then be compared with the observed distribution of tectonic features. http://arxiv.org/abs/0704.1627v2 Besides the important assumption of elasticity, the model of lithospheric flexure is often simplified by two approximations. The first one states that the area to be analyzed is sufficiently small so that the curved lithospheric shell can be modeled as a flat plate. The second approximation states that the shell (or plate) is thin, which means that deformations are small and that the elastic thickness is small with respect to the wavelength of the load. On Earth, the model of lithospheric flexure has been very successful for the understanding of the topography of the oceanic plates, whereas the analysis of continental plates is fraught with difficulties due to the very old and complex structure of the continents. As far as we know, plate tectonics do not occur at the present time on other terrestrial planets, which are deemed one-plate planets [Solomon , 1978], although the term single-shell would fit better because of the curvature. A single shell can support loads of much larger extent and greater weight, the best example of which is the huge Tharsis volcanic formation covering a large portion of Mars. Indeed such a load cannot be supported by the bending moments present in a thin flat plate, whereas it can be supported by stresses tangent to the shell: the shell acts as a membrane. The first application of thin shell theory to a planet was done by Brotchie and Silvester [1969] for the lithosphere of the Earth (see also Brotchie [1971]). However the approximation of flat plate theory was seen to be sufficient when it was understood that the lithosphere of the Earth is broken into several plates [Tanimoto, 1998]. Brotchie and Silvester [1969] do not consider tangential loads and their flexure equation only includes dominant terms in derivatives; their equation is thus a special case of the equations of Kraus [1967] and Vlasov [1964] discussed below. The articles reviewed hereafter use the thin shell theory of Kraus or Vlasov unless mentioned otherwise. On the Moon, Solomon and Head [1979] used Brotchie’s equation to study displacement and stress in mare basins. Turcotte et al. [1981] estimated gravity-topography ratios for the mascons and discussed the type of stress supporting topographic loads. Arkani-Hamed [1998] modeled the support of mascons with Brotchie’s equation. Sugano and Heki [2004] and Crosby and McKenzie [2005] estimated the elastic thickness of the lithosphere from Lunar Prospector data. On Mars, the dominance of the Tharsis rise in the topography led to numerous applications of the theory of thin elastic shells to lithospheric flexure. Thurber and Toksöz [1978], Comer et al. [1985], Hall et al. [1986] and Janle and Jannsen [1986] used Brotchie’s equation to estimate the lithospheric thickness under Martian volcanoes. Turcotte et al. [1981] studied the transition between bending and membrane regimes. Willemann and Turcotte [1982] analyzed the lithospheric support of Tharsis. Sleep and Phillips [1985] analyzed the membrane stress distribution on the whole surface. Banerdt et al. [1992], Banerdt and Golombek [2000] and Phillips et al. [2001] used a model of lithospheric flexure including membrane support, bending stresses and tangent loads [Banerdt , 1986] in order to study the global stress distribution. Arkani-Hamed [2000] determined the elastic thickness beneath large volcanoes with Brotchie’s equation whereas Johnson et al. [2000] estimated the elastic thickness beneath the North Polar Cap. Using local admittance analysis with spatiospectral methods, McGovern et al. [2002] determined the elastic thickness at various locations [see also McGovern et al., 2004]. McKenzie et al. [2002] made local admittance analyses of line-of-sight grav- ity data both with flat plates and with spherical shell models. Turcotte et al. [2002] used the spherical shell formula for a one-dimensional wavelet analysis of the admittance in order to determine the average elastic thickness of the lithosphere. Zhong and Roberts [2003] and Lowry and Zhong [2003] studied the support of the Tharsis rise with an hybrid model including the flexure of a thin elastic shell as well as the internal loading of a thermal plume in the mantle. Belleguic et al. [2005] determined the elastic thickness and the density beneath large volcanoes. Searls et al. [2006] investigated the elastic thickness and the density beneath the Utopia and Hellas basins. Venus is considered as a one-plate planet but does not have giant volcanic or tectonic structures comparable to Tharsis. The spherical shell model has thus not been used as often for Venus as for Mars. Banerdt [1986] studied the global stress distribution. Janle and Jannsen [1988] and Johnson and Sandwell [1994] used Brotchie’s equation to estimate the lithospheric thickness in various locations. Sandwell et al. [1997] computed global strain trajectories for comparison with observed tectonics. Lawrence and Phillips [2003] inverted the admittance in order to estimate the elastic thickness and mantle density anomalies over two lowland regions and one volcanic rise. Anderson and Smrekar [2006] established a global map of the elastic thickness based on local admittance analysis. Mercury’s topography and gravity fields are not yet known well enough to warrant the application of a thin elastic shell model. We refer to Wieczorek [2007] for a review of recent results regarding the lithosphere of terrestrial planets. The elastic thickness of the lithosphere is not at all homogeneous over the surface of a planet. For example, McGovern et al. [2004] (for Mars) and Anderson and Smrekar [2006] (for Venus) find litho- spheric thickness variations of more than 100 km. The former study explains the variation in lithospheric thickness in terms of different epochs of loading, as the lithosphere is thickening with time. Other studies however inferred that spatial variations in lithospheric thickness on Mars are as important as temporal variations [Solomon and Head , 1982, 1990; Comer et al., 1985]. Thin spherical shell models have always been applied with a constant elastic thickness for the whole lithosphere. Local studies are done by win- dowing both the data (gravity and topography) and the model predictions for gravity [Simons et al., 1997; Wieczorek and Simons , 2005]. The assumptions behind these methods are that the elastic thick- ness is constant within the window and that the area outside the window can be neglected. The first assumption is of course true for a small enough window, but the size of the window is limited from below by the resolution of the data [Wieczorek and Simons , 2005]. Even if the first assumption were true, the second assumption is violated in two ways (unless the elastic thickness is spatially constant). First, the deformation of the shell within the window as well as the associated stress field are both modified if the elastic thickness is changed in the area outside the window. Second, the value of the predicted gravity field within the window depends on the shell deflection outside the window. These reasons make it interesting to develop a model of the lithospheric flexure for a spherical shell of variable thickness. Although a full inversion of the gravity and topography data is impractical with such a model because of the huge size of the parameter space, other applications are of high interest. For example a two-stage inversion can be considered: in the first stage a constant elastic thickness is assumed, and the resulting values are used in the second stage as a starting point for an inversion with variable elastic thickness (the parameter space can also be constrained with an a priori). Moreover this model can be used to produce synthetic data and thus allows us to check the validity of inversions assuming a constant elastic thickness. Finally, stress and strain fields can be computed for given variations of the elastic thickness, with the aim of comparing stress and strain maps with tectonic features. General equations governing the deformations of a thin elastic shell have been given by various authors [e.g. Love, 1944; Vlasov , 1964; Kraus, 1967]. However the possibility of variable shell thickness is only considered at the early stage where the strain-displacement relationships, Hooke’s law and the equilibrium equations are separately derived for the thin shell. The combination of these three sets of equations into a unique equation for the transverse deflection is made under the restriction of constant elastic thickness, ‘owing to the analytical complications which would otherwise arise’ [see Kraus, 1967, p. 199]. This article is dedicated to the derivation of the minimum set of equations governing the deformations of a thin elastic spherical shell with variable thickness. Using Kraus’ method of stress functions, we find that the transverse deflection is the solution of a simultaneous system of two differential equations of the fourth order. Contrary to the case of constant thickness, these equations cannot be combined due to the presence of products of derivatives of the thickness and derivatives of the deflection or of the stress function. In order to include bottom loading generated by mantle convection, we extend the method of stress functions to include toroidal tangential loads, which were not considered by Kraus [1967]. Non- toroidal loading is for example generated by tangential lithostatic forces whereas toroidal loading could be due to mantle flow generating drag at the base of the lithosphere (mantle flow also produces non- toroidal loading). With applications to tectonics in mind, we derive the equations relating the tangential displacements and the stresses to the transverse deflection and the stress functions. We further show that toroidal tangential displacement occurs even if there is no toroidal loading (unless the shell thickness is constant). Finally we prove three properties specific to the degree-one harmonic components: (1) the degree-one transverse deflection and the degree-one toroidal tangential displacement drop from the elasticity equations because they represent rigid displacements of the whole shell, (2) the transverse and tangential components of degree-one loads are related so that the shell does not accelerate, (3) degree- one loads can deform the shell and generate stresses. Though our aim is to introduce a variable shell thickness, the final equations are also valid for a variable Young’s modulus (Poisson’s ratio must be kept constant). Another way to take into account variations of the lithospheric thickness consists in treating the lithosphere as a three-dimensional spherical solid that is elastic [Métivier et al., 2006] or viscoelastic [e.g. Zhong et al., 2003; Latychev et al., 2005; Wang and Wu, 2006]. The resulting equations are exact (no thin shell approximation) and can be solved with finite element methods. The thin shell assumption is probably satisfied for known planetary lithospheres; in case of doubt, it is advisable to compare the results of thin shell theory with three-dimensional models assuming constant elastic thickness: static thick shell models [e.g. Banerdt et al., 1982; Janes and Melosh , 1990; Arkani-Hamed and Riendler , 2002] or time-dependent viscoelastic models [e.g. Zhong and Zuber , 2000; Zhong , 2002]. The advantage of thin shell equations is their two-dimensional character, making them much easier to program and quicker to solve on a computer. Solving faster either gives access to finer two-dimensional grids or allows to examine a larger parameter space. We choose to work within the formalism of differential calculus on curved surfaces, without which the final equations would be cumbersome. Actually the only tool used in this formalism is the covariant derivative, which can be seen by geophysicists as just a way of combining several terms into one ‘deriva- tive’. All necessary formulas are given in the Appendix. The possibility of writing a differential equation in terms of covariant derivatives (in a tensorial form) also provides a consistency check. The presence of derivatives that cannot be included into covariant derivatives is simply forbidden. This is not without meaning for differential equations of the fourth order including products of derivatives. In section 2, we show how to obtain the strain-displacement relationships, Hooke’s law and the equilibrium equations for a thin spherical shell. These equations are available in the literature for the general case of a thin shell [e.g. Kraus, 1967], but we derive them for the spherical case in a simpler way, starting directly with the metric for the spherical shell. We examine in detail the various approximations made to obtain the thin shell theory of flexure, refraining until the end from taking the ‘thin shell’ quantitative limit in order to ascertain its influence on the final equations. In section 3, we use the method of stress functions to obtain the flexure equations governing the displacements and the stresses. In section 4, we give the final form of the flexure equations in the thin shell approximation. We also study the covariance and the degree-one projection of the flexure equations. In section 5, we examine various limit cases in which the flexure equations take a simpler form: the membrane limit, the Euclidean limit and the limit of constant thickness. 2 Fundamental equations of elasticity 2.1 Three-dimensional elasticity theory Linear elasticity theory is based on three sets of equations. We directly state them in tensorial form for an isotropic material, since they are derived in Cartesian coordinates in many books [e.g. Ranalli , 1987; Synge and Schild , 1978]. Recall that, in tensorial notation, there is an implicit summation on indices that are repeated on the same side of an equation. The first set of equations includes strain-displacement relationships: ǫij = (ui,j + uj,i) , (1) where ǫij is the infinitesimal strain tensor and ui are the finite displacements. The ‘comma’ notation denotes the spatial derivative (see Appendix 7.2). The second set includes the constitutive equations of elasticity, or Hooke’s law, relating the strain tensor and the stress tensor σij : σij = λ ǫ δij + 2Gǫij , (2) where ǫ = ǫ11 + ǫ22 + ǫ33 and δij = 1 if i= j, otherwise it equals zero. The parameter λ is known as the first Lamé constant. The parameter G is known as the second Lamé constant, or the shear modulus, or the modulus of rigidity. Boundary conditions are given by σij nj = Ti , (3) with nj being the normal unit vector of the surface element and Ti being the surface force per unit area. The third set includes equations of motion which reduce to equilibrium equations for stresses if the problem is static: σij,j = 0 . (4) Body forces, such as gravity, are assumed to be absent. Both strain and stress tensors are symmetric: ǫij = ǫji and σij = σji. These three-dimensional equations do not yet have the right form for the description of the defor- mations of a two-dimensional spherical shell. Various methods have been used to generate appropriate equations. Love [1944] and Timoshenko and Woinowsky-Krieger [1964] derive strain-displacements and equilibrium equations directly on the surface of the sphere (the latter only for the special cases of no bending or axisymmetrical loading). Sokolnikoff [1956] derives strain-displacement equations in three- dimensional curvilinear coordinates using an arbitrary diagonal metric but states without proof the equi- librium equations in curvilinear coordinates. Kraus [1967] uses Sokolnikoff’s form of strain-displacement equations and derives equilibrium equations for an arbitrary two-dimensional surface using Hamilton’s principle (i.e. virtual displacements). Instead of directly deriving equations on the two-dimensional surface of the sphere, we will first obtain their form in three-dimensional curvilinear coordinates and then restrict them to the surface of the sphere. The first step can elegantly be done through the use of tensors [Synge and Schild , 1978]. Equations (1)-(4) are tensorial with respect to orthogonal transformations, but not with respect to other coordinate transforms (one reason being the presence of usual derivatives). In other words, they are only valid in Cartesian coordinates. In a three-dimensional Euclidean space, tensorial equations have a simplified form in Cartesian coordinates because supplementary terms that make them tensorial with respect to arbitrary coordinate transformations are zero. The missing terms can be reconstructed by using a set of rules, such as the replacement of usual derivatives by covariant derivatives and the substitution of tensorial contraction to sum on components. Correspondence rules lead to the following three sets of equations: ǫij = ui|j + uj|i , (5) σij = λ ǫ gij + 2Gǫij , (6) gjk σij|k = 0 , (7) where ǫ = gkl ǫkl. The notation ui|j denotes the covariant derivative of ui (see Appendix 7.2). The metric and its inverse are noted gij and g ij , respectively. Tensorial components cannot be expressed in a normalized basis (except for Cartesian coordinates) which is more common for physical interpretation (see Appendix 7.1). Covariant components in equations (5)-(7) are related to components defined in a normalized basis (written with a hat) by: gii ûi , ǫij = giigjj ǫ̂ij , σij = giigjj σ̂ij , where there is no implicit summation on repeated indices. In the next section we will introduce additional assumptions in order to restrict the equations to the two-dimensional surface of a spherical shell. 2.2 Spherical shell 2.2.1 Assumptions of the thin shell theory Suppose that the two first coordinates are the colatitude θ and longitude ϕ on the surface of the sphere, whereas the third coordinate ζ is radial. R is the shell radius. Assumptions of the thin shell theory are [see Kraus, 1967, chap. 2.2]: 1. The shell is thin (say less than one tenth of the radius of the sphere). 2. The deflections of the shell are small. 3. The transverse normal stress is negligible: σζζ = 0. 4. Normals to the reference surface of the shell remain normal to it and undergo no change of length during deformation: ǫθζ = ǫϕζ = ǫζζ = 0. The second assumption allows us to use linear equations to describe the deflections. The third and fourth assumptions are not fully consistent: we refer to Kraus [1967] for more details. We will relax them in the derivation of the equations for the deflection of a spherical shell. The crucial assumption is σζζ = 0 which is essential for the restriction of Hooke’s law to the two-dimensional shell. As we will see later, σζζ cannot be zero since it is related to the non-zero transverse load (besides the fact that it is incompatible with a vanishing transverse strain). What is absolutely necessary is that σζζ ≪ σii for i = (θ, ϕ). In section 5.3.3, we will show that this condition is satisfied if the wavelength of the load is much larger than the thickness of the shell. The reference surface is the middle surface of the shell. With the aim of integrating out the third coordinate, a coordinate system is chosen so that the radial coordinate ζ is zero on the reference surface. The metric is given by ds2 = (R+ ζ) dθ2 + sin2 θ dϕ2 + dζ2 . (8) Christoffel symbols necessary for the computation of the covariant derivatives are given in Appendix 7.3. 2.2.2 Strain-displacement equations With the metric (8), the strain-displacement equations (5) become ǫ̂θθ = (ûθ,θ + ûζ) , ǫ̂ϕϕ = (csc θ ûϕ,ϕ + cot θ ûθ + ûζ) , ǫ̂θϕ = (csc θ ûθ,ϕ − cot θ ûϕ + ûϕ,θ) , (9) ǫ̂ζζ = ûζ,ζ , ǫ̂θζ = ((R + ζ) ûθ,ζ − ûθ + ûζ,θ) , ǫ̂ϕζ = ((R + ζ) ûϕ,ζ − ûϕ + csc θ ûζ,ϕ) , where all quantities are given in a normalized basis. The fourth assumption of the thin shell theory implies that the displacements are linearly distributed across the thickness of the shell, with the transverse displacement being constant. Displacements can thus be expanded to first order in ζ: (ûθ, ûϕ, ûζ) = (vθ + ζβθ, vϕ + ζβϕ, w) . (10) The coefficients (vθ , vϕ, w, βθ, βϕ) are independent of ζ: (vθ, vϕ, w) represent the components of the displacement vector of a point on the reference surface, whereas (βθ, βϕ) represent the rotations of tangents to the reference surface oriented along the tangent axes. We determine βθ and βϕ by applying the fourth assumption of the thin shell theory and the expansion (10) to the last two strain-displacement equations (vθ − w,θ) , (vϕ − csc θ w,ϕ) . (11) The substitution of the expansion (10) into the fourth strain-displacement equation (ǫ̂ζζ = ûζ,ζ) leads to the condition ǫ̂ζζ = 0, as postulated by the thin shell theory. After the insertion of the expansion formulas (10) and (11), the first three strain-displacement equa- tions (9) become ǫ̂θθ = ǫ 1 + ζ/R κ0θ , ǫ̂ϕϕ = ǫ 1 + ζ/R κ0ϕ , (12) 2 ǫ̂θϕ = γ 1 + ζ/R The extensional strains ǫ0θ, ǫ ϕ and γ θϕ are defined by ǫ0θ = (vθ,θ + w) , ǫ0ϕ = (csc θ vϕ,ϕ + cot θ vθ + w) , (13) γ0θϕ = (vϕ,θ − cot θ vϕ + csc θ vθ,ϕ) . They represent the normal and shearing strains of the reference surface. The flexural strains κ0θ, κ ϕ and τ0 are given by κ0θ = − O1 w , κ0ϕ = − O2 w , (14) τ0 = − O3 w . They represent the changes in curvature and the torsion of the reference surface during deformation [Kraus, 1967]. The differential operators O1,2,3 are defined by + 1 , O2 = csc2 θ + cot θ + 1 , (15) O3 = csc θ − cot θ The zero upper index in (ǫ0θ, ǫ θϕ, κ 0) refers to the reference surface and appears in order to follow Kraus’ notation. In Love’s theory, one makes the approximation 1 ∼= 1R in equations (12). However this does not simplify the calculations when the shell is a sphere because it has only one radius of curvature (for a shell with two radii of curvature, this approximation is a great simplification). Our choice to keep the factor 1 leads to the same results as the theory of Flügge-Lur’e-Byrne, explained in Kraus [1967, chap. 3.3a] or Novozhilov [1964, p. 53], in which this factor is expanded up to second order. In any case the choice between the approximation or the expansion of this factor does not affect the equations (derived in the next section) relating the stress and moment resultants to the strains: they are the same is approximated to zeroth order, expanded to second order or fully kept. 2.2.3 Hooke’s law When the metric is diagonal and the basis is normalized, Hooke’s law (6) becomes σ̂ii = λ ǫ̂kk + 2G ǫ̂ii (i = j) , σ̂ij = 2G ǫ̂ij (i 6= j) . There is no implicit summation on repeated indices. The third assumption of the thin shell theory, σ̂ζζ = 0, can be used to eliminate ǫ̂ζζ from Hooke’s σ̂θθ = 1− ν2 (ǫ̂θθ + ν ǫ̂ϕϕ) , σ̂ϕϕ = 1− ν2 (ǫ̂ϕϕ + ν ǫ̂θθ) . Young’s modulus E and Poisson’s ratio ν are related to Lamé parameters by (1 + ν)(1 − 2ν) 2(1 + ν) In principle, the fourth assumption of the thin shell theory, ǫ̂θζ = ǫ̂ϕζ = 0, leads to σ̂θζ = σ̂ϕζ = 0 but non-vanishing values must be retained for purposes of equilibrium. Figure 1: Stress resultants and stress couples acting on a small element of the shell. The directions of the stress resultants (simple arrows) and the rotation sense of the stress couples (double arrows) correspond to positive components (tensile stress is positive). Loads (qθ, qϕ, q) act on the reference surface. The substitution of the expansion (12) into the thin shell approximation of Hooke’s law gives σ̂θθ = 1− ν2 ǫ0θ + νǫ 1 + ζ/R κ0θ + ν κ σ̂ϕϕ = 1− ν2 ǫ0ϕ + νǫ 1 + ζ/R κ0ϕ + ν κ , (16) σ̂θϕ = 2(1 + ν) γ0θϕ + 1 + ζ/R with ǫ0θ, ǫ θϕ, κ ϕ and τ 0 defined by equations (13) and (14). The expressions for σ̂θζ and σ̂ϕζ will not be needed. We now integrate the stress distributions across the thickness h of the shell (see Figure 1). The stress resultants and couples obtained in this way are defined per unit of arc length on the reference surface: ∫ h/2 σ̂ii (1 + ζ/R) dζ (i = θ, ϕ) , Nθϕ = Nϕθ = ∫ h/2 σ̂θϕ (1 + ζ/R) dζ , ∫ h/2 σ̂iζ (1 + ζ/R) dζ (i = θ, ϕ) , (17) ∫ h/2 σ̂ii (1 + ζ/R) ζ dζ (i = θ, ϕ) , Mθϕ = Mϕθ = ∫ h/2 σ̂θϕ (1 + ζ/R) ζ dζ . We evaluate these integrals with the expansion (16). The tangential stress resultants are Nθ = K ǫ0θ + ν ǫ Nϕ = K ǫ0ϕ + ν ǫ , (18) Nθϕ = K γ0θϕ . Explicit expressions for the transverse shearing stress resultants Qi are not needed since these quantities will be determined from the equilibrium equations. The moment resultants are Mθ = D κ0θ + ν κ ǫ0θ + ν ǫ Mϕ = D κ0ϕ + ν κ ǫ0ϕ + ν ǫ , (19) Mθϕ = D The extensional rigidity K and the bending rigidity D are defined by 1− ν2 , (20) 12(1− ν2) . (21) Their dimensionless ratio ξ is a large number, ξ = R2 , (22) the inverse of which will serve as an expansion parameter for thin shell theory. 2.2.4 Equilibrium equations With the metric (8), the components θ, ϕ and ζ of the equilibrium equations (7) respectively become (R+ ζ) (sin θ σ̂θθ),θ + σ̂θϕ,ϕ − cos θ σ̂ϕϕ + sin θ σ̂ζθ + sin θ (R+ ζ) = 0 , (R + ζ) (sin θ σ̂θϕ),θ + σ̂ϕϕ,ϕ + cos θ σ̂θϕ + sin θ σ̂ζϕ + sin θ (R+ ζ) = 0 , (R + ζ) (sin θ σ̂θζ),θ + σ̂ϕζ,ϕ − sin θ (σ̂θθ + σ̂ϕϕ) + sin θ (R+ ζ) = 0 , where the equations have been multiplied by sin θ (R + ζ), (R + ζ) and sin θ (R + ζ)2, respectively. The stress components are given in a normalized basis. The integration on ζ of these three equations in the range [−h/2, h/2] yields the equilibrium equations for the forces: (sin θ Nθ),θ +Nθϕ,ϕ − cos θ Nϕ + sin θ Qθ +R sin θ qθ = 0 , (23) (sin θNθϕ),θ +Nϕ,ϕ + cos θ Nθϕ + sin θ Qϕ +R sin θ qϕ = 0 , (24) (sin θ Qθ),θ +Qϕ,ϕ − sin θ (Nθ +Nϕ)−R sin θ q = 0 , (25) where qθ and qϕ are the components of the tangential load vector per unit area of the reference surface: (R+ ζ) = R2 qi (i = θ, ϕ) . We choose the convention that tensile stresses are positive (see Figure 1). The transverse load per unit area of the reference surface is noted q and is taken to be positive toward the center of the sphere: (R+ ζ) = −R2 q . (26) The first two equilibrium equations for the stresses can also be multiplied by ζ before the integration to yield the equilibrium equations for the moments: (sin θMθ),θ +Mθϕ,ϕ − cos θMϕ −R sin θ Qθ = 0 , (27) (sin θMθϕ),θ +Mϕ,ϕ + cos θMθϕ −R sin θ Qϕ = 0 . (28) We have neglected small terms in ζ (R+ ζ) where i = (θ, ϕ). A third equilibrium equation for the moments exists but has the form of an identity: Mθϕ =Mϕθ. 3 Resolution 3.1 Available methods At this stage the elastic theory for a thin spherical shell involves 17 equations: six strain-displacement relationships (13)-14), six stress-strain relations (18)-(19) making Hooke’s law, and five equilibrium equa- tions (23)-(23) and (27)-(28). There are 17 unknowns: six strain components (ǫ0θ, ǫ θϕ, κ three displacements (w, vθ , vϕ), three tangential stress resultants (Nθ, Nϕ, Nθϕ), two transverse shearing stress resultants (Qθ, Qϕ), and three moment resultants (Mθ,Mϕ,Mθϕ). The three equations (16) are also needed if the tangential stresses (σ̂θθ, σ̂θϕ, σ̂ϕϕ) are required. The quantities of primary interest to us are the transverse deflection and the tangential stresses (sometimes tangential strain is preferred, as in Sandwell et al. [1997] or Banerdt and Golombek [2000]). We thus want to find the minimum set of equations that must be solved to determine these quantities. We are aware of two methods of resolution [Novozhilov , 1964, p. 66]. In the first one, we insert the strain-displacement relationships into Hooke’s law, and substitute in turn Hooke’s law into the equilibrium equations. This method yields three simultaneous differential equations for the displacements. Once the displacements are known, it is possible to compute the strains and the stresses. The second method supplements the equilibrium equations with the equations of compatibility [Novozhilov , 1964, p. 27] that relate the partial derivatives of the strain components. It is then convenient to introduce the so-called stress functions [Kraus , 1967, p. 243], without direct physical interpretation, which serve to define the stress resultants without introducing the tangential displacements. Equations relating the transverse displacement and the stress functions are then found by applying the third equation of equilibrium and the third equation of compatibility (it is also possible to use all three equations of compatibility in order to directly solve for the stress and moment resultants). Once the transverse displacement and the stress functions are known, stresses can be computed. If the shell thickness is constant, the deformations of a thin spherical shell can be completely calcu- lated with both methods. If the shell thickness is variable, the three equations governing displacements, obtained with the first method, cannot be decoupled and are not easy to solve. Kraus’ method with stress functions leads to a system of three equations (relating the transverse displacement and the two stress functions), in which the first equation is decoupled and solved before the other two. This method thus provides a system of equations much easier to solve and will be chosen in this article. When solving the equations, one usually assumes from the beginning the large ξ limit, i.e. 1+ ξ ∼= ξ where ξ is defined by equation (22). We will only take this limit at the end of the resolution. This procedure will not complicate the computations, since we have to compute anyway many new terms because of the variable shell thickness. 3.2 Differential operators We will repeatedly encounter the operators Oi which intervene in the expressions (14) for the flexural strains (κ0θ, κ 0). Since we are looking for scalar equations, we need to find out how the operators Oi can be combined in order to yield scalar expressions, i.e. expressions that are invariant with respect to changes of coordinates on the sphere. The first thing is to relate the Oi to tensorial operators. Starting from the covariant derivatives on the sphere ∇i, we construct the following tensorial differential operators of the second degree in derivatives: Dij = ∇i∇j + gij , (29) where ∇i denotes the covariant derivative (see Appendix 7.2). These operators give zero when applied on spherical harmonics of degree one (considered as scalars): Dij Y1m = 0 (m = −1, 0, 1) . (30) This property can be explicitly checked on the spherical harmonics (109) with the metric and the formulas for the double covariant derivatives given in Appendix 7.4. In two-dimensional spherical coordinates (θ, ϕ), the three operators Oi defined by equations (15) are related to the operators Dij acting on a scalar function f through O1 f = Dθθ f , O2 f = csc2 θDϕϕ f , (31) O3 f = csc θDθϕ f . The operators O1,2,3f actually correspond to normalized Dijf , i.e. Dijf/( giigjj). The usual derivatives of the operators Oi satisfy the useful identities (112)-(113) which are proven in Appendix 7.7. These identities are the consequence of the path dependence of the parallel transport of vectors on the curved surface of the sphere. Invariant expressions are built by contracting all indices of the differential operators in their tensorial form. The indices can be contracted with the inverse metric gij or with the antisymmetric tensor εij (see Appendix 7.4), which should not be confused with the strain tensor ǫij . In the following, a and b are scalar functions on the sphere. At degree 2, the only non-zero contraction of the Dij is related to the Laplacian (104): ∆′a ≡ gij Dij a = (∆+ 2) a (32) = (O1 +O2) a = a,θ,θ + cot θ a,θ + csc 2 θ a,ϕ,ϕ + 2 a . At degree 4, a scalar expression symmetric in (a, b) is given by A(a ; b) ≡ [∆′a][∆′b]− [Dij a][Dij b] , = [∆ a][∆ b]− [∇i∇j a][∇i∇j b] + [∆ a] b+ a [∆ b] + 2 a b (33) = [O1 a][O2 b] + [O2 a][O1 b]− 2 [O3 a][O3 b] = (a,θ,θ + a) csc2 θ b,ϕ,ϕ + cot θ b,θ + b csc2 θ a,ϕ,ϕ + cot θ a,θ + a (b,θ,θ + b) −2 csc2 θ (a,θ,ϕ − cot θ a,ϕ) (b,θ,ϕ − cot θ b,ϕ) . where upper indices are raised with the inverse metric: Dij = gikgjlDkl. The action of an operator does not extend beyond the brackets enclosing it. If a is constant, A(a ; b) = a∆′b. It is useful to define an associated operator A0 that gives zero if its first argument is constant: A0(a ; b) = A(a ; b)− a [∆′b] . (34) A scalar expression of degree 4 antisymmetric in (a, b) is given by B1(a ; b) ≡ gij εkl [Dik a] [Djl b] = gij εkl [∇i∇k a] [∇j∇l b] (35) = [(O1 −O2) a] [O3b]− [O3a] [(O1 −O2) b] = csc θ a,θ,θ − csc2 θ a,ϕ,ϕ − cot θ a,θ (b,θ,ϕ − cot θ b,ϕ) − csc θ (a,θ,ϕ − cot θ a,ϕ) b,θ,θ − csc2 θ b,ϕ,ϕ − cot θ b,θ We will also need another operator of degree 4: B2(a ; b) ≡ εij [∇ia ] [∇j ∆′b] = csc θ a,θ [∆ ′b],ϕ − a,ϕ [∆ ′b],θ . (36) The sum of the operators B1 and B2 is noted B: B(a ; b) = B1(a ; b) + B2(a ; b) . (37) If either a or b is constant, B(a ; b) = 0. The operators A and B have an interesting property proven in Appendix 7.8: for arbitrary scalar functions a and b, A(a ; b) and B(a ; b) do not have a degree-one term in their spherical harmonic expansion. This is not true of A0, B1 and B2. 3.3 Transverse displacement 3.3.1 Resolution of the equations of equilibrium The first step consists in finding expressions for the moment resultants (Mθ,Mϕ,Mθϕ) in terms of the transverse displacement and the stress resultants. The extensional strains (ǫ0θ, ǫ θϕ) can be eliminated from the equations for stress and moment resultants (18)-(19). The flexural strains (κ0θ, κ 0) depend on the transverse displacement w through equations (14). We thus obtain Mθ = − (O1 + νO2)w + Mϕ = − (O2 + νO1)w + Nϕ , (38) Mθϕ = − (1− ν)O3 w + Nθϕ , where the parameter ξ is defined by equation (22). The second step consists in solving the equilibrium equations for moments in order to find the transverse shearing stress resultants (Qθ, Qϕ). We substitute expressions (38) into equations (27)-(28). Knowing that the stress resultants satisfy the equilibrium equations (23)-(24), we obtain new expressions for Qθ and Qϕ (identities (112)-(113) are helpful): Qθ = − (D∆′w),θ − (1− η)Rqθ (1− ν) (D,θ O2 − csc θD,ϕO3) w − (η,θNθ + csc θ η,ϕNθϕ) , (39) Qϕ = − csc θ (D∆′w),ϕ − (1− η)Rqϕ (1− ν) (−D,θ O3 + csc θD,ϕ O1) w − (η,θ Nθϕ + csc θ η,ϕNϕ) . (40) The operator ∆′ is defined by equation (32) and η is a parameter close to 1: 1 + ξ . (41) The third step consists in finding expressions for the stress resultants (Nθ, Nϕ, Nθϕ) in terms of stress functions by solving the first two equilibrium equations (23)-(24). Let us define the following linear combinations of the stress and moment resultants: (Pθ, Pϕ, Pθϕ) = Mθ, Nϕ + Mϕ, Nθϕ + . (42) We observe that these linear combinations satisfy simplified equations of equilibrium: (sin θ Pθ),θ + Pθϕ,ϕ − cos θ Pϕ +R sin θ qθ = 0 , (sin θ Pθϕ),θ + Pϕ,ϕ + cos θ Pθϕ +R sin θ qϕ = 0 . (43) Comparing these equilibrium equations with the identities (112)-(113), we see that the homogeneous equations (i.e. equations (43) with a zero tangential load qθ = qϕ = 0) are always satisfied if (Pθ, Pϕ, Pθϕ) = (O2,O1,−O3)F , (44) where F is an auxiliary function called stress function. For the moment, this function is completely arbitrary apart from being scalar and differentiable. Particular solutions of the full equations (43) can be found if we express the tangential load qT = qθθ̂ + qϕϕ̂ in terms of the surface gradient of a scalar potential Ω (consoidal or poloidal component) and the surface curl of a vector potential V r̂ (toroidal component): qT = − ∇̄Ω + 1 ∇̄ × (V r̂) . (45) Surface operators are defined in Appendix 7.5, where the terms consoidal/poloidal are also discussed. The covariant components of qT are (qθ, sin θqϕ) and can be expressed as − 1RΩ,i + gjkεikV,j , which gives qθ = − Ω,θ + R sin θ V,ϕ , sin θ qϕ = − Ω,ϕ − sin θ V,θ . (46) If the tangential load is consoidal (V = 0), a particular solution of equations (43) is given by (Pθ, Pϕ, Pθϕ) = (1, 1, 0)Ω . (47) If the tangential load is toroidal (Ω = 0), a particular solution of equations (43) is given by (Pθ, Pϕ, Pθϕ) = (2O3,−2O3,O2 −O1)H , (48) where we have introduced a second stress function H which satisfies the constraint ∆′H = −V + V0 , (49) where V0 is a constant (identities (112)-(113) are useful). This equation allows us to determine the stress function H if the toroidal source V is known. The general solution of the equations (43) is given by the sum of the general solution (44) of the homogeneous equations and the two particular solutions (47)-(48) of the full equations: (Pθ, Pϕ, Pθϕ) = (O2F +Ω + 2O3H,O1F +Ω− 2O3H,−O3F + (O2 −O1)H) . (50) The stress resultants (Nθ, Nϕ, Nθϕ) can now be obtained from (Pθ, Pϕ, Pθϕ) by using equations (42) and (38): (Nθ, Nϕ, Nθϕ) = η (Pθ, Pϕ, Pθϕ) + η (O1 + νO2,O2 + νO1, (1− ν)O3)w , (51) which finally give Nθ = η O2 F + (∆′ − (1− ν)O2)w +Ω+ 2O3H Nϕ = η O1 F + (∆′ − (1− ν)O1)w +Ω− 2O3H , (52) Nθϕ = η −O3 F + (1 − ν) O3 w + (O2 −O1)H The fourth step consists in expressing the third equation of equilibrium (25) in terms of the transverse displacement w and the stress functions F and H . For this purpose, it is handy to express the transverse shearing stress resultants (Qθ, Qϕ) in terms of (w,F,H). We thus substitute Nϕ, Nθ, Nθϕ, given by equations (52), into the expressions for Qθ and Qϕ, given by equations (39)-(40): Qθ = − (ηD∆′w),θ + (ηD),θ O2 − csc θ (ηD),ϕ O3 − (η,θ O2 − csc θ η,ϕO3)F + Ω,θ − (ηΩ),θ − csc θ V,ϕ − 2 (η,θ O3 + csc θ η,ϕ O2)H + csc θ (η,ϕ ∆′H − η(∆′H),ϕ) , (53) Qϕ = − csc θ (ηD∆′w),ϕ + − (ηD),θ O3 + csc θ (ηD),ϕ O1 − (−η,θ O3 + csc θ η,ϕ O1)F + csc θ (Ω,ϕ − (ηΩ),ϕ) + V,θ +2 (η,θ O1 + csc θ η,ϕ O3)H − (η,θ ∆′H − η(∆′H),θ) . (54) The various terms present in Qθ and Qϕ can be classified into generic types according to their differential structure. Each generic type will contribute in a characteristic way to the third equation of equilibrium. It is worthwhile to compute the generic contribution of each type before using the full expressions of the stress resultants with their multiple terms. Identities (112)-(113) are helpful in this calculation. We thus write the third equation of equilibrium (25) as follows: I (Qθ ;Qϕ ; 0)− (Nθ +Nϕ)−Rq = 0 , (55) where the differential operator I is defined for arbitrary expressions (X,Y, Z) by I(X ;Y ;Z) = csc θ (sin θX),θ + csc θ Y,ϕ − cot θ Z . (56) This operator must be evaluated for the following generic types present in (Qθ, Qϕ): I (a,θ ; csc θ a,ϕ ; 0) = ∆a , I (− csc θ a,ϕ ; a,θ ; 0) = 0 , I (a,θ O2b− csc θ a,ϕ O3b ;−a,θ O3b+ csc θ a,ϕO1b ; 0) = A0(a ; b) , (57) I (a,θ O3b+ csc θ a,ϕ O2b ;−a,θ O1b− csc θ a,ϕO3b ; 0) = B1(a ; b) , I (− csc θ (a,ϕ∆′b − a[∆′b],ϕ) ; a,θ ∆′b− a[∆′b],θ ; 0) = 2B2(a ; b) , where a and b are scalar functions. The operators A0, B1 and B2 are defined in section 3.2. We now substitute (Qθ, Qϕ) and (Nϕ, Nθ) into the third equation of equilibrium (55) and use formulas (57). We thus obtain the first of the differential equations that relate w and the stress functions F and ∆′ (ηD∆′w)− (1 − ν)A(ηD ;w) +R3 A(η ;F ) + 2R3 B(η ;H) = −R4 q +R3 (∆Ω−∆′(ηΩ)) . (58) The operators ∆′, A and B are defined in section 3.2. 3.3.2 Compatibility relation A second equation relating the transverse displacement and the stress functions comes from the compat- ibility relation which is derived by eliminating (vθ, vϕ) from the strain-displacement equations (13): sin θ γ0θϕ,ϕ sin2 θ ǫ0ϕ,θ + ǫ0θ,ϕ,ϕ − sin θ cos θ ǫ0θ,θ + 2 sin2 θ ǫ0θ − sin2 θ ∆′w . (59) The strain components are related to the stress resultants through Hooke’s law (18). The substitution of equations (18) into the compatibility equation (59) gives ∆′ (α (Nθ +Nϕ))− ∆′w − (1 + ν)J (αNθ ;αNϕ ;αNθϕ) = 0 . (60) where α is the reciprocal of the extensional rigidity: α ≡ 1 K(1− ν2) . (61) For arbitrary expressions (X,Y, Z), J (X ;Y ;Z) is defined by J (X ;Y ;Z) = csc2 θ sin2 θX,θ + Y,ϕ,ϕ + 2 (sin θ Z,ϕ),θ − cot θ Y,θ + 2 Y . (62) As in the case of the third equation of equilibrium, it is practical to classify the terms present in (Nθ, Nϕ, Nθϕ) into generic types and evaluate separately their contribution to the equation of compati- bility. There are three types of terms for which we must evaluate the operator J (identities (112)-(115) are helpful in this calculation): J (a ; a ; 0) = ∆′a , J (aO2b ; aO1b ;−aO3b) = A(a ; b) , (63) J (2aO3b ;−2aO3b ; a (O2 −O1)b) = 2B(a ; b) , where a and b are scalar functions. The operators ∆′, A and B are defined in section 3.2. We now evaluate J in equation (60) with expressions (52) and formulas (63): J (αNθ ;αNϕ ;αNθϕ) = A(ηα ;F ) + ∆′ (ηαD∆′w)− 1 (1− ν)A(ηαD ;w) + ∆′ (ηαΩ) + 2B(ηα ;H) . (64) The term A(ηαD ;w) can be rewritten with the help of the following equality: 1− ν2 A(ηαD ;w) = ∆′w −A(η ;w) , (65) since (1 − ν2)ηαD/R2 = η/ξ and (η/ξ),i = −η,i. We finally substitute Nθ + Nϕ, given by equations (52), into the compatibility equation (60) and use expressions (64)-(65). We thus obtain the second of the differential equations that relate w and the stress functions F and H : ∆′ (ηα∆′F )− (1 + ν)A(ηα ;F )− A(η ;w) − 2(1 + ν)B(ηα ;H) = −(1− ν)∆′(ηαΩ) . (66) 3.4 Tangential displacements Assuming that the flexure equations (58) and (66) for the transverse displacement w and the stress functions (F,H) have been solved, we now show how to calculate the tangential displacements. In analogy with the decomposition of the tangential load in equations (45)-(46), the tangential displacement can be separated into consoidal and toroidal components: v = ∇̄S + ∇̄ × (T r̂) , (67) where S and T are the consoidal and toroidal scalars, respectively. The covariant components of v are (vθ, sin θ vϕ) and can be expressed as S,i + g jkεikT,j (see Appendix 7.5), which gives vθ = S,θ + csc θ T,ϕ , sin θ vϕ = S,ϕ − sin θ T,θ . (68) The strain-displacement equations (13) become ǫ0θ = ((O1 − 1)S +O3 T + w) , ǫ0ϕ = ((O2 − 1)S −O3 T + w) , (69) γ0θϕ = (2O3S + (O2 −O1)T ) . The stress resultants (Nθ, Nϕ, Nθϕ) given by equations (18) become (∆S + (1 + ν)w − (1− ν) ((O2 − 1)S −O3 T )) , (∆S + (1 + ν)w − (1− ν) ((O1 − 1)S +O3 T )) , (70) Nθϕ = (2O3 S + (O2 −O1)T ) . The toroidal potential T cancels in the sum Nθ +Nϕ: Nθ +Nϕ = (1 + ν) (∆S + 2w) . The consoidal displacement potential S can thus be related to (w,F,Ω) by using expressions (52) for the stress resultants: ∆S = Rηα (1− ν) (∆′F + 2Ω) + η ∆′w − 2w , (71) where α is defined by equation (61). It is more difficult to extract the toroidal displacement potential T . When the shell thickness is constant, decoupled equations for the displacements can be found by suitable differentiation and combi- nation of the three equilibrium equations (23)-(25) for the stress resultants. This method does not work if the shell thickness is variable because the resulting equations are coupled. The trick consists in relating the tangential displacements to (w,F,H) by the way of equations similar to the homogeneous part of the first two equilibrium equations, but with (Nθ, Nϕ, Nθϕ) replaced by (Nθ, Nϕ, Nθϕ), so that derivatives do not mix with derivatives of T . We will thus calculate the following expression in two different ways (from equations (52) and (70)): = csc2 θ (−X,ϕ + sin θ Y,θ) , where X and Y are defined by X = sin θ I Nθϕ ; Y = sin θ I Nθϕ ; Nϕ ;− The operator I is defined by equation (56). As before, it is easier to begin with the evaluation of the operator Z for generic contributions: Z(a ; a ; 0) = 0 , Z(aO2b ; aO1b ;−aO3b) = −B(a ; b) , Z(2aO3b ;−2aO3b ; a(O2 −O1)b) = 2A(a ; b)−∆′ (a∆′b) , On the one hand, the evaluation Z for (Nθ, Nϕ, Nθϕ) given by equations (70) gives −1− ν ∆∆′ T . On the other hand, the evaluation of Z for (Nθ, Nϕ, Nθϕ) given by equations (52) gives + (1− ν)B + 2RA The equality of the two previous formulas yields the sought equation for T : ∆∆′ T = 2R (1 + ν) (B (ηα ;F )− 2A (ηα ;H) + ∆′ (ηα∆′H)) + 2B (η ;w) . (72) We have used the relation (η/ξ),i = −η,i. The equation for T shows that toroidal displacement always occurs when the shell thickness varies. The right-hand side of equation (72) only vanishes when two conditions are met: (1) there is no toroidal source (so that the stress function H vanishes) and (2) the shell thickness is constant (so that the terms in B vanish). 3.5 Stresses Stresses can be computed from (w,F,H) and Ω by substituting equations (14), (18) and (52) into equa- tions (16): σ̂θθ = (O2 F +Ω+ 2O3H) + R(1− ν2) (∆′ − (1− ν)O2)w , σ̂ϕϕ = (O1 F +Ω− 2O3H) + R(1− ν2) (∆′ − (1− ν)O1)w , (73) σ̂θϕ = (−O3 F + (O2 −O1)H) + R(1 + ν) O3w . Stresses at the surface are obtained by setting ζ = h/2. 4 Flexure equations and their properties 4.1 Thin shell approximation The flexure equations derived in section 3 already include several assumptions of the thin shell theory, but not yet the first one stating that the shell is thin. Of course, the three other assumptions can be seen to be consequences of the first one [see Kraus, 1967, chap. 2.2], but we have not imposed in a quantitative way the thinness condition on the equations. We thus impose the limit of small h/R or, equivalently, the limit of large ξ (defined by equation (22)) on the flexure equations for (H,F,w, S, T ). This procedure amounts to expand η ≈ 1 − 1/ξ (neglecting terms in 1/ξ wherever appropriate) and to neglect the derivatives of η in equations (49), (58), (66), (71) and (72). We thus obtain the final flexure equations for the displacements (w, S, T ) and for the stress functions (F,H): ∆′H = −V + V0 , (74) ∆′ (D∆′w)− (1 − ν)A(D ;w) +R3∆′F = −R4 q − 2R3Ω+ R ∆Ω , (75) ∆′ (α∆′F )− (1 + ν)A(α ;F )− 1 ∆′w = −(1− ν)∆′(αΩ) + 2(1 + ν)B(α ;H) , (76) ∆S = Rα (1 − ν) (∆′F + 2Ω) + 1 ∆′w − 2w , (77) ∆∆′ T = 2R (1 + ν) (B (α ;F )− 2A (α ;H) + ∆′ (α∆′H)) . (78) Recall that the differential operators ∆′, A and B are defined by equations (32), (33) and (37). The potential pairs (Ω, V ) and (S, T ) are related to the tangential load and displacement by equations (45) and (67), respectively. In the second equation, the term R3∆Ω/ξ has been kept since it could be large if Ω has a short wavelength. For the same reason, the term ∆′w/ξ has been kept in the fourth equation. In the third and fifth equations, the terms depending on H belong to the right-hand sides since they can be considered as a source once H has been calculated from the first equation. The same can be said of the terms depending on w and F in the fourth and fifth equations: they are supposed to be known from the simultaneous resolution of the second and third equations. The difficulty in solving the equations thus lies with the two flexure equations for w and F which are linear with non-constant coefficients (all other equations are linear - in their unknowns - with constant coefficients). Once these two core equations have been solved, all other quantities are easily derived from them. The bending rigidity D, defined by equation (21), characterizes the bending regime: the shell locally bends in a similar way as a flat plate undergoing small deflections with negligible stretching. The pa- rameter α, defined by equation (61), is the reciprocal of the extensional rigidity K and characterizes the membrane regime of the shell in which bending moments are negligible and the load is mainly supported by internal stresses tangent to the shell. Since D and K are respectively proportional to the third and first power of the shell thickness, the membrane regime (in which the D-depending terms are neglected) is obtained in the limit of an extremely thin shell. This observation and the fact that such a shell, lacking rigidity, cannot support bending moments justify the use of the term ‘membrane’. The weight of the various terms in the equations depends on two competing factors: the magnitude of the coefficient multiplying the derivative and the number of derivatives. On the one hand, a coefficient containing D will be smaller than a coefficient containing α−1 since αD/R2 ∼ ξ−1 is a small number (see equation (22)). On the other hand, a large number of derivatives will increase the weight of the term if the derived function has a small wavelength. The transition between the membrane and the bending regimes thus depends on the wavelength of the load: if the load has a large wavelength (with respect to the shell radius), the flexure of the shell will be well described by equations without the terms depending on D (see section 5.1), whereas the flexure under loads of small wavelength is well described by equations keeping only the D-depending terms with the largest number of derivatives (see section 5.2). Stress functions are associated with membrane stretching and give negligible contributions in the bending regime. Formulas (52) show that F (respectively H) plays the role of potential for the stress resultants in the membrane regime when the load is transversal (respectively tangential toroidal). There is no stress function associated with the tangential consoidal component of the load because it is identical to Ω, the consoidal potential of the load. The variation of the shell thickness has two effects. First, it couples the spherical harmonic modes that are solutions to the equations for a shell of constant thickness. Second, the toroidal part remains intertwined with the transversal and consoidal parts, whereas it decouples if the thickness is constant. For example, the toroidal load is a source for the transverse deflection through the term B in the third flexure equation. Furthermore, the stress function F is a source for the toroidal displacement in the fifth flexure equation. For numerical computations, the flexure equations (74)-(78) obtained in the thin shell approximation (to which we can add the equations (73) for the stresses) are adequate. For this purpose, it is not useful to keep small terms in 1/ξ since the theory rests on assumptions only true for a thin shell. In the rest of the article, we will continue to work with the equations (49), (58), (66), (71) and (72) for more generality. 4.2 Covariance Because of their tensorial form, the flexure equations (74)-(78) are covariant; this is also true of the more general equations (49), (58), (66), (71) and (72). This property means that their form is valid in all systems of coordinates on the sphere, though the tensor components (the covariant derivatives of scalar functions, the metric and the antisymmetric tensor) will have a different expression in each system. The scalar functions will also have a different dependence on the coordinates in each system. The covariance of the final equations was expected. We indeed started with tensorial equations in section 2.1; their restriction to the sphere in principle respected the tensoriality with respect to changes of coordinates on the sphere. However the covariance of the two-dimensional theory was not made explicit until we obtained the final equations. This property is thus a strong constraint on the form of the solution and a check of its validity, though only necessary and not sufficient (other covariant terms may have been ignored). Another advantage of the covariant form is the facility to express the final equations in different systems of coordinates (even non-orthogonal ones) with the aim of solving them. For example, the fi- nite difference method in spherical coordinates (θ, ϕ) suffers from a very irregular grid and from pole singularities. These problems can be avoided with the ‘cubed sphere’ coordinate system [Ronchi et al., 1996]. Operators including covariant derivatives (here the Laplacian and the operators A and B) can ex- pressed in any system of coordinates whose metric is known. Christoffel symbols and tensorial differential operators can be computed with symbolic mathematical software. 4.3 Degree one Displacements of degree one require special consideration. All differential operators acting on (w,F,H) (that is ∆′, A and B) in the flexure equations (58) and (66) can be expressed in terms of the operators Dij = ∇i∇j+gij (see section 3.2). The Dij have the interesting property that they give zero when acting on spherical harmonics of degree one (see equation (30)). Therefore the degree-one terms in the spherical harmonic expansion of w vanish from the flexure equations. The magnitude of the transverse deflection of degree one neither depends on the load nor on the elastic properties of the spherical shell. More generally, the homogeneous (q = Ω = V = 0) flexure equations (49), (58) and (66) are satisfied by w and F being both of degree one. According to equations (71)-(72), the corresponding tangential displacement is constrained by ∆S = −2w and ∆∆′T = 0, so that S and T are also of degree one, with S = w. These conditions lead to vanishing strains (see equations (69)) which indicate a rigid displacement. The total displacement is then given by u = w r̂+ ∇̄w + ∇̄ × (T r̂) , where w and T are of degree one. In Appendix 7.6, we show that the first two terms represent a rigid translation whereas the last term represents a rigid rotation. As expected, stresses vanish for such displacements (see equation (73)). This freedom in translating or rotating the solution reflects the freedom in the choice of the reference frame (in practice the reference frame is centered at the center of the undeformed shell). The same freedom of translation is also found in the theory of deformations of a spherical, radially stratified, gravitating solid [e.g. Farrell , 1972; Greff-Lefftz and Legros , 1997; Blewitt , 2003]. What can we say about degree-one loading? Let us first examine what happens when the flexure equations are projected on the spherical harmonics of degree one. Since the operator ∆′ annihilates the degree one in any spherical harmonic expansion, terms of the form ∆′f vanish when they are projected on the spherical harmonics of degree one. Moreover, the operators A and B also vanish in this projection since they do not contain a degree-one term (see equations (116)-(117) of Appendix 7.8). Therefore the degree-one component of the flexure equation (66) is identically zero, whereas the degree-one component of the flexure equation (58) is q + ∇̄ · qT Yi = 0 (i = x, y, z) , (79) where dω = sin θ dθ dϕ and Yi are the real spherical harmonics of degree one. The integral is taken over the whole spherical surface. We have used the relation ∆Ω = −R ∇̄ ·qT derived from equations (45) and (106). The first term in the integrand of equation (79) is the projection on the Cartesian axes (x̂, ŷ, ẑ) of the vector field q r̂: (q Yx, q Yy, q Yz) = (q r̂ · x̂, q r̂ · ŷ, q r̂ · ẑ) , where we have used formulas (109) for the spherical harmonics. The second term in equation (79) can be rewritten with the identity (102) and Gauss’ theorem (107): ∇̄ · qT Yi = − dω qT · ∇̄Yi (i = x, y, z) , where the Yi are considered as scalars. Since qT is orthogonal to r̂, the integrand is the projection of qT on the Cartesian axes (x̂, ŷ, ẑ): qT · ∇̄Yx,qT · ∇̄Yy,qT · ∇̄Yz = (qT · x̂,qT · ŷ,qT · ẑ) , where we have used formulas (110) for the gradients of the spherical harmonics. Recalling that the transverse load q was defined positive towards the center of the sphere (see equation (26)), we define a total load vector q = −q r̂+qT . With the above results, we can rewrite the degree-one projection (79) as dω (q · x̂ ,q · ŷ ,q · ẑ) = (0, 0, 0) , (80) which means that the integral (over the whole spherical surface) of the projection on the coordinate axes of the total load vector vanishes. This result is the consequence of the static assumption in the equations of motion (7), since a non-zero sum of the external forces would accelerate the sphere. In practice, degree-one loads on planetary surfaces are essentially due to mass redistribution [Greff-Lefftz and Legros , 1997] and have a tangential consoidal component (for example the gravitational force is not directed to- ward the center of figure of the shell). If the shell thickness is variable, a non-zero Ω of degree one will induce degrees higher than one in w and in S. If the shell thickness is constant, the degree-one load drops from the flexure equations (58)-(66) and w is not affected. However, the degree-one Ω generates (assuming a constant shell thickness) a degree-one tangential displacement through equation (71), so that S 6= w. Whether the shell thickness is variable or not, a degree-one Ω thus generates a total displacement which is not only a translation but also a tangential deformation [Blewitt , 2003], in which case stresses do not vanish as shown by equations (73). 5 Limit cases 5.1 Membrane limit A shell is in a membrane state of stress if bending moments (Mθ,Mϕ,Mθϕ) can be neglected, in analogy with a membrane which cannot support bending moments. Equations (19) show that this is true if the bending rigidity vanishes: D = 0. Consistency with equation (22) imposes the limit of infinite ξ (or η = 1). With these approximations, the first flexure equation (58) for (w,F,H) becomes ∆′F = −Rq − 2Ω . (81) The second flexure equation (66) for (w,F,H) becomes ∆′w = ∆′ (α∆′F )− (1 + ν)A(α ;F )− 2(1 + ν)B(α ;H) + (1− ν)∆′(αΩ) . (82) If q is independent of w, F and w can be successively determined with spherical harmonic transforms from equations (81) and (82) (though the right-hand side of the latter equation must be computed with another method). If q has a linear dependence in w (such as when the sphere is filled with a fluid), w can be eliminated between equations (81) and (82), so that F and w can also be computed in succession (however the equation for F cannot be solved by a spherical harmonic transform). H is supposed to be known since equation (49) is not modified and can be solved with spherical harmonics. The stresses are obtained from equations (73) with the additional approximation of neglecting the term in ζ: (σ̂θθ, σ̂ϕϕ, σ̂θϕ) = (O2F +Ω+ 2O3H,O1F +Ω− 2O3H,−O3F + (O2 −O1)H) . Bending moments play a small role if the load has a large wavelength. In practice, the threshold at which bending moments become significant can be evaluated from the constant thickness equation for w (see section 5.3.1). One must compare the magnitudes of the terms in D and 1/α in the left-hand side of equation (88). Bending moments are negligible (i.e. the term in D) if the spherical harmonic degree ℓ of the transverse displacement w is such that ℓ . k , (83) where k = (12(1− ν2))1/4 ≈ 1.8 if we take ν = 1/4. This threshold is about 10 for a planet with a radius of 3400 km and a lithospheric thickness equal to 100 km. Flexure equations in the membrane limit of a shell with constant thickness have been used by Sleep and Phillips [1985] to study the lithospheric stress in the Tharsis region of the planet Mars. 5.2 Euclidean limit The equation for the deflection of a rectangular plate with variable thickness has been derived by Timoshenko and Woinowsky-Krieger [1964, p. 173]. We will check here that the Euclidean limit of our equations gives the same answer. Let us define the coordinates (x, y) = (Rϕ,R θ′), where θ′ = θ/2 − θ is the latitude. We work in a small latitude band around the equator (so that θ′ ≪ 1) and in the limit of large spherical radius R and large ξ (η = 1). Under this change of coordinate, each derivative introduces a factor R so that terms with the largest number of derivatives dominate. In particular, covariant derivatives can be approximated by usual derivatives. The surface Laplacian (104) can be approximated as follows: ∆ ≈ R2 ≡ R2∆e . We assume that there are no tangential loads (Ω = V = H = 0). The flexure equations (75)-(76) for the transverse displacement become: R4∆e (D∆ew)− (1 − ν)R4 Ae(D ;w) +R5∆eF = −R4 q , (84) R4 ∆e (α∆eF )− (1 + ν)R4 Ae(α ;F )−R ∆ew = 0 , (85) where the operator Ae is defined by Ae(a ; b) = (∆e a) (∆e b)− a,i,j b,i,j Equation (85) gives a relation between the magnitudes of F and w: O(R3 ∆eF ) ∼ O(w/α) . In the large R limit, equation (84) thus becomes ∆e (D∆ew) − (1− ν)Ae(D ;w) = −q , which is the equation derived by Timoshenko and Woinowsky-Krieger [1964]. This equation has been used by Stark et al. [2003], Kirby and Swain [2004] and Pérez-Gussinyé et al. [2004] for the local analysis of the lithosphere of the Earth. Its one-dimensional version, in which Ae vanishes, has been used by Sandwell [1984] to describe the flexure of the oceanic lithosphere on Earth and by Stewart and Watts [1997] to model the flexure at mountain ranges. 5.3 Shell with constant thickness 5.3.1 Displacements If the thickness of the shell is constant, the toroidal part of the tangential displacement decouples. The terms in B indeed drop from the flexure equations (58) and (66) so that the equations for (w,F ) depend only on (q,Ω): ηD∆′∆′w − (1− ν) ηD∆′w + ηR3 ∆′F = −R4 q +R3 ((1 − η)∆Ω− 2ηΩ) , (86) ∆′∆′F − (1 + ν)∆′F − 1 ∆′w = −(1− ν)∆′Ω , (87) where we used the property A(a ; b) = a∆′b valid for constant a. We eliminate F from these equations and obtain a sixth order equation relating w to (q,Ω): ηD∆∆′∆′w + ∆′w = −R4 (∆′ − 1− ν) q +R3 1 + ξ ∆′ − 1− ν ∆Ω . (88) The elimination of ∆′F between equations (77) and (86) gives an equation relating the consoidal displacement potential S to (w, q,Ω): ∆S = − 1 1 + ν 1 + ξ ∆∆′w − 2w − (1 − ν)R2α q + 1− ν 1 + ξ Rα∆Ω . Terms not including a Laplacian can be eliminated with equation (88), so that we obtain an explicit solution for S in terms of (w, q,Ω): 1 + ξ 1− ν2 (∆ + 1 + ν)∆′w + w +R2α q − Rα 1 + ξ (∆− ξ(1 + ν))Ω , (89) where the integration constant has been set to zero. Equations (49) and (72) give an equation for the toroidal displacement potential: ∆′ T = −2ηRα (1 + ν)V , (90) where the integration constant has been set to zero. We have assumed that ∆V 6= 0, otherwise we get ∆′T = 0. The differential equations given in this section can be solved with spherical harmonics so that the co- efficients of the spherical harmonic expansions of (w, S, T ) can be expressed in terms of the corresponding coefficients of the loads (q,Ω, V ) (see Kraus [1967], Turcotte et al. [1981], Banerdt [1986]). 5.3.2 Comparison with the literature We now compare our equations for a shell of constant thickness with those found in the literature. The formulas of Banerdt [1986] (taken from the work of Vlasov [1964]) are the most general: ∆3 + 4∆2 (∆ + 2)w = −R4 (∆ + 1− ν) q +R3 ∆− 1− ν ∆Ω , (91) (∆ + 2)χ = Dξ(1− ν) ∆V . (92) Banerdt’s notation is slightly different: his formulas are obtained with the substitutions ξ → ψ, Ω → RΩ and V → RV . The normal rotation χ is proportional to the radial component (in a normalized basis) of the curl of the tangential displacement: (∇× v)r̂ . The curl ∇× v is related to our surface curl (103) by ∇× v = ∇̄ × v + csc θ (sin θv̂ϕ),θ − v̂θ,ϕ With the formulas (68) and (104), we get ∇× v = −∆T r̂ so that equation (92) becomes ∆′ T = −2Rα(1 + ν)V . (93) We see that Banerdt’s equations (91) and (93) coincide with our equations (88) and (90) in the limit of large ξ (η = 1), with one exception: the bending term for w is written D(∆3 + 4∆2)w instead of D∆∆′∆′w = (∆3 + 4∆2 + 4∆)w. This error has propagated in many articles and is of consequence for the degree-one harmonic component, since it violates the static assumption and spoils the translation invariance discussed in section 4.3. The impact on higher degrees is negligible. Because of this mistake, many authors give a separate treatment to the first harmonic degree. Banerdt also gives formulas for the tangential displacements in terms of consoidal and toroidal scalars (A,B) corresponding to our scalars (S, T ): his formula (A10) is equivalent to our equation (89) in the limit of large ξ. If we ignore temperature effects, Kraus’ first equation for (w,F ) is equivalent to our equation (86) in the limit of large ξ, whereas his second equation for (w,F ) is equivalent to the combination eq.(87)+ 1+ν eq.(86) in the limit of large ξ [see Kraus , 1967, eq. 6.54h and 6.55d]. Note that the definition of Kraus’ stress function F [Kraus, 1967, p. 243] differs from ours: FKraus = F − k(1 − ν) with k = 1. This freedom of redefining F for arbitrary k remains as long as D is constant. The flexure equation for w, equation (88), is unaffected so that the solution for w is unchanged. In the final step, Kraus makes a mistake when combining the two equations for (w,F ) and thus obtains a flexure equation for w with the same error as in equation (91). Kraus does not include toroidal loading. The flexure equation of Turcotte et al. [1981] is taken from Kraus [1967] without the tangential loading and is the same as equation (91) with Ω = 0. The flexure equation of Brotchie and Silvester [1969] is given in our notation by D∆2w + w = −R4 q , (94) where q includes their term γw describing the response of the enclosed liquid. This equation can be obtained from our equation (88) as follows: keep only the derivatives of the highest order in each term, set Ω = 0, take the limit of large ξ (η = 1) and integrate. Brotchie and Silvester choose to work in the approximation of a shallow shell and with axisymmetrical loading, solving their equation in polar coordinates with Bessel-Kelvin functions. The reduction to fourth order in equation (94), the shallow shell approximation and the axisymmetrical assumption are not justified nowadays since the full equation (88) can be quickly solved with computer-generated spherical harmonics. The contraction due to a transverse load of degree 0, w = −R2α(1 − ν)q/2, is equivalent to the radial displacement computed by Love in the limit of a thin shell [Love, 1944, p.142]. However ad- ditional assumptions about the initial state of stress and the internal density changes are necessary [Willemann and Turcotte, 1982] so that the degree 0 is usually excluded from the analysis. 5.3.3 Breakdown of the third assumption of thin shell theory With the spherical harmonic solutions of the equations for a shell of constant thickness, it is possible to check the thin shell assumption stating that the transverse normal stress is negligible with respect to the tangential normal stress. The magnitude of the former can be estimated by the load q (see definition (26)) whereas the magnitude of the latter can be approximated with formulas (73) evaluated on the outer surface: (σ̂θθ + σ̂ϕϕ) |h ∆′F − Eh 4R2(1− ν) ∆′w . where we have assumed the absence of tangential loads (Ω = 0) and the limit of large ξ. We can relate σT to q by using the solution in spherical harmonics of equations (87) and (88). Since the thin shell assumption is expected to fail for a load of sufficiently small wavelength, we assume that the spherical harmonic degree ℓ is large. Assuming ℓ≫ 1, we obtain (∆′F )ℓm ≈ wℓm , wℓm ≈ − 1 + ℓ ξ(1−ν2) qℓm , where the spherical harmonic coefficients are indexed by their degree ℓ and their order m. If the shell is not in a membrane state of stress (see equation (83)), ℓ2 > 2R/h so that σT can be approximated by (σT )lm ≈ ξ(1 + ν) qlm . The thin shell assumption holds if q < σT , that is if 3(1 + ν) or λ > 3(1 + ν) h , (95) where λ is the load wavelength (λ ≈ 2πR/ℓ). We have 3(1 + ν) ≈ 1.9 and 2π/ 3(1 + ν) ≈ 3.2 if we take ν = 1/4. This condition on λ is consistent with the transition zone between the thin and thick shell responses analyzed in Janes and Melosh [1990] and Zhong and Zuber [2000], but does not coincide with the constraint given in Willemann and Turcotte [1982], which is ℓ < 2π R/h (this last condition looks more like the threshold (83) for the membrane regime). Though the stress distribution is affected, the limit (95) on the degree ℓ is not important for the displacements, since they tend to zero at small wavelengths. Therefore the theory does not break down at short wavelength if one is interested in the computation of the gravity field associated to the transverse deflection of the lithosphere. 6 Conclusion The principal results of this article are the five flexure equations (74)-(78) governing the three displace- ments of the thin spherical shell and the two auxiliary stress functions. Stresses are derived quantities which can be obtained from equations (73). The shell thickness and Young’s modulus can vary, but Poisson’s ratio must be constant. The loads acting on the shell can be of any type since we extend the method of stress functions to include not only transverse and consoidal tangential loads, but also toroidal tangential loads. The flexure equations can be solved one after the other, except the two equations (75)-(76) for the transverse deflection w and the stress function F , which must be simultaneously solved. Tangential loading is usually neglected when solving for the deflection because of its small effect. In that case, it is sufficient to solve the two equations (75)-(76) with Ω = H = 0: ∆′ (D∆′w)− (1− ν)A(D ;w) +R3∆′F = −R4 q , ∆′ (α∆′F )− (1 + ν)A(α ;F )− 1 ∆′w = 0 . However tangential loading must be taken into account when computing stress fields [Banerdt , 1986]. In the long-wavelength limit (i.e. membrane regime), all equations can be solved one after the other because it is possible to solve for F before solving for the transverse deflection. If a small part of the shell is considered, the flexure equations reduce to the equations governing the deflection of a flat plate with variable thickness. If the shell thickness is constant, the flexure equations reduce to equations available in the literature which can be completely solved with spherical harmonics. Our rigorous treatment of the thin shell approximation has clarified the effect of the shell thickness on the flexure equations. We emphasize the need to use the correct form for the equations (without the common mistake in the differential operator acting on w) in order to have the correct properties for the degree-one deflection and degree-one load. We have also obtained two general properties of the flexure equations. First we have shown that there is always a toroidal component in the tangential displacement if the shell thickness is variable. Second we have proven that the degree-one harmonic components of the transverse deflection and of the toroidal component of the tangential displacement do not depend on the elastic properties of the shell. This property reflects the freedom under translations and rotations of the reference frame. Besides we have shown that degree-one loads are constrained by the static assumption but can deform the shell and generate stresses. This article was dedicated to the theoretical treatment of the flexure of a thin elastic shell with variable thickness. While the special case of constant thickness admits an analytical solution in terms of spherical harmonics, the general flexure equations must be solved with numerical methods such as finite differences, finite elements or pseudospectral methods. In a forthcoming paper, we will give a practical method of solution and discuss applications to real cases. Acknowledgments M. Beuthe is supported by a PRODEX grant of the Belgian Science Federal Policy. The author thanks Tim Van Hoolst for his help and Jeanne De Jaegher for useful comments. Special thanks are due to Patrick Wu for his constructive criticisms which helped to improve the manuscript. 7 Appendix 7.1 Covariant, contravariant and normalized components Tensors can be defined by their transformation law under changes of coordinates. The two types of tensor components, namely covariant and contravariant components, transform in a reciprocal way under changes of coordinates. Tensor components cannot be expressed in a normalized basis: the space must have a coordinate vector basis (for contravariant components) and a dual basis (for covariant components) which are not normalized. The only exception is a flat space with Cartesian coordinates, where covariant, contravariant and normalized components are identical. Since the metric is the scalar product of the elements of the coordinate vector basis, the covariant components are related to components defined in a normalized basis (written with a hat) by gii ûi , whereas the relation for contravariant components is ûi . The normalized Cartesian basis (x̂, ŷ, ẑ) is related to the normalized basis for spherical coordinates (r̂, θ̂, ϕ̂) by x̂ = cos θ cosϕ θ̂ − sinϕ ϕ̂+ sin θ cosϕ r̂ , ŷ = cos θ sinϕ θ̂ + cosϕ ϕ̂+ sin θ sinϕ r̂ , (96) ẑ = − sin θ θ̂ + cos θ r̂ . 7.2 Covariant derivatives Usual derivatives are indicated by a ‘comma’: vi,j = Covariant derivatives (defined below) are indicated by a ‘bar’ or by the operator ∇i: vi|j = ∇j vi . The former notation emphasizes the tensorial character of the covariant derivative since the covariant derivative adds a covariant index to the vector. The latter notation is more adapted when we are interested by the properties of the operator. The covariant derivative of a scalar function f is equal to the usual derivative, f|i = f,i, and is itself a covariant vector: f|i = vi. Covariant derivatives on covariant and contravariant vector components are defined by vi|j = vi,j − Γkij vk , (97) vi |j = v ,j + Γ k , (98) where the summation on repeated indices is implicit. The symbols Γkij are the Christoffel symbols of the second kind [Synge and Schild , 1978]. Their expressions for the metrics used in this article are given in sections 7.3 and 7.4. Covariant differentiation of higher order tensors is explained in Synge and Schild [1978] but we only need the rule for a covariant tensor of second order: σij|k = σij,k − Γlik σlj − Γljk σil . If some of the indices of the tensor are contravariant, the rule is changed according to equation (98). The covariant derivatives of the metric and of the inverse metric are zero: gij|k = 0 and g 7.3 Three-dimensional spherical geometry The geometry of a thin spherical shell of average radius R can be described with coordinates θ, ϕ and ζ, respectively representing the colatitude, longitude and radial coordinates. The radial coordinate ζ is zero on the reference surface (i.e. the sphere of radius R) of the shell. The non-zero components of the metric are given by gθθ = (R + ζ) gϕϕ = (R + ζ) 2 sin2 θ , gζζ = 1 . The non-zero Christoffel symbols are given by θθ = −(R+ ζ) , Γζϕϕ = −(R+ ζ) sin2 θ , Γθζθ = Γ θζ = Γ ζϕ = Γ Γθϕϕ = − sin θ cos θ , ϕθ = Γ θϕ = cot θ . 7.4 Two-dimensional spherical geometry If θ and ϕ respectively represent the colatitude and longitude coordinates, the non-zero components of the metric on the surface of the sphere are given by gθθ = 1 , gϕϕ = sin 2 θ . (99) The non-zero Christoffel symbols are given by Γθϕϕ = − sin θ cos θ , ϕθ = Γ θϕ = cot θ . The double covariant derivatives of a scalar function f are thus given by f|θ|θ = f,θ,θ , f|θ|ϕ = f|ϕ|θ = f,θ,ϕ − cot θ f,ϕ , f|ϕ|ϕ = f,ϕ,ϕ + sin θ cos θ f,θ . An antisymmetric tensor εij is defined by εij ≡ det gij ε̄ij , where ε̄ij is the antisymmetric symbol invariant under coordinate transformations: ε̄θϕ = −ε̄ϕθ = 1, ε̄θθ = ε̄ϕϕ = 0 (ε̄ij is usually called a tensor density; Synge and Schild [1978] call it a relative tensor of weight -1). The non-zero covariant components of εij are given for the metric of the spherical surface by εθϕ = −εϕθ = sin θ . The non-zero contravariant components, εij = gikgjlεkl, are given by εθϕ = −εϕθ = csc θ . The covariant derivative of the tensor εij is zero: εij|k = 0. 7.5 Gradient, divergence, curl and Laplacian Various differential operators on the surface of the sphere can be constructed with covariant derivatives. In this section, f and t are scalar functions defined on the sphere and v is a vector tangent to the sphere. Backus [1986] gives more details on surface operators and on Helmholtz’s theorem. As mentioned in Appendix 7.2, the covariant derivative of a scalar function f defined on the sphere is a covariant vector tangent to the sphere whose components are f,θ and f,ϕ. The surface gradient of f is the same vector with its components expressed in the normalized basis (θ̂, ϕ̂): ∇̄f = f,θ θ̂ + csc θ f,ϕ ϕ̂ . (100) The contraction of the covariant derivative with the components of a vector v yields a scalar: vi |i = v ,θ + cot θ v θ + vϕ,ϕ . The surface divergence is the corresponding operation on the vector with its components expressed in the normalized basis (θ̂, ϕ̂): ∇̄ · v = csc θ (sin θ v̂θ),θ + v̂ϕ,ϕ . (101) Since the result is a scalar, vi = ∇̄ · v. A useful identity is ∇̄ · (f v) = ∇̄f · v + f ∇̄ · v . (102) The contraction of the antisymmetric tensor εij with the covariant derivative of a scalar t yields the covariant components of a vector v: vi = g jk εik t,j . The components are given for the metric (99) by vθ = csc θ t,ϕ and vϕ = − sin θ t,θ. If t is considered as the radial component of the radial vector t = t r̂ (the covariant radial component is equal to the normalized one), vi are the non-zero covariant components of the three-dimensional curl of t, which is tangent to the sphere. This fact justifies the definition of the surface curl of t, which is equal to the vector v but with components given in the normalized basis (θ̂, ϕ̂): ∇̄ × t = csc θ t,ϕ θ̂ − t,θ ϕ̂ . (103) The contraction of the double covariant derivative acting on a scalar f defines the surface Laplacian: ∆f = gij f|i|j = f,θ,θ + cot θ f,θ + csc 2 θ f,ϕ,ϕ . (104) The surface Laplacian can also be seen as the composition of the surface divergence with the surface gradient: ∆f = ∇̄ · ∇̄f . According to Helmholtz’s theorem, a vector tangent to the sphere can be written as the sum of the surface gradient of a scalar f and the surface curl of a radial vector t r̂: v = ∇̄f + ∇̄ × (t r̂) . (105) While t is always called the toroidal scalar (or potential) for v, there is no standard terminology for f . Backus [1986] calls f the consoidal scalar for v. Some authors [e.g. Banerdt , 1986] call f the poloidal potential for v. The origin of this use lies in the theory of mantle convection, in which plate tectonics are assumed to be driven by mantle flow. Under the assumption of an incompressible mantle fluid, the velocity field of the fluid is solenoidal, i.e. its 3-dimensional divergence vanishes. In such a case, the velocity field can be decomposed into a poloidal part (∇×∇×(P r̂)) and a toroidal part (∇×(Q r̂)), where differential operators are 3-dimensional [Backus , 1986]. If the velocity field is tangent to the spherical surface, the poloidal component at the surface is also the consoidal component [Forte and Peltier , 1987]. However the fields for which we use Helmholtz’s theorem, i.e. the tangential surface load and the tangential surface displacement, do not belong to 3-dimensional solenoidal vector fields. We thus prefer to use the term ‘consoidal’. The surface divergence of v depends only on the consoidal scalar f : ∇̄ · v = ∆f . (106) The two-dimensional version of Gauss theorem is dω ∇̄ · v = 0 . (107) where dω = sin θ dθ dϕ and the integral is taken over the whole spherical surface. It can be proven with formula (101). 7.6 Rigid displacements At the surface of a sphere subjected to deformation, the displacement u of a point can be expressed with the help of Helmholtz’s theorem (105) in terms of three scalar functions (w, S, T ) depending on θ and ϕ: u = w r̂+ ∇̄S + ∇̄ × (T r̂) . (108) Strains (and stresses) vanish for rigid displacements. Equations (69) show that strains vanish when (w, S, T ) are of degree one, with S = w (recall that the operatorsOi annihilate the degree one). Assuming these conditions, we now show that utransl = w r̂ + ∇̄w represents a rigid translation whereas urot = ∇̄ × (t r̂) represents a rigid rotation of the sphere. We choose as basis the real spherical harmonics of degree one which form the components of the radial unit vector in Cartesian coordinates: (Yx, Yy, Yz) = (sin θ cosϕ, sin θ sinϕ, cos θ) = (x̂, ŷ, ẑ) · r̂ . (109) We need the surface gradient of the real spherical harmonics which can be computed with formulas (96) and (100): ∇̄Yx, ∇̄Yy, ∇̄Yz = (x̂− sin θ cosϕ r̂, ŷ − sin θ sinϕ r̂, ẑ− cos θ r̂) . (110) If the expansion of w in the degree-one basis is w = a Yx + b Yy + c Yz, then w r̂+ ∇̄w = a x̂+ b ŷ + c ẑ , so that utransl is indeed a rigid translation of the sphere. If the expansion of T in the degree-one basis is T = a′Yx + b ′Yy + c ′Yz, then ∇̄ × (T r̂) = a′ − sinϕ θ̂ − cos θ cosϕ ϕ̂ cosϕ θ̂ − cos θ sinϕ ϕ̂ + c′ sin θ ϕ̂ , so that urot includes a rigid rotation of the sphere, with (a ′, b′, c′) being the angles of rotation around the axes (x̂, ŷ, ẑ), respectively. Though urot seems to include a uniform radial expansion, one should recall that linearized strain-displacement equations are not valid for large displacements. Since strains vanish, the radial expansion is not physical and urot represents a pure rotation. Finite deformations are for example discussed in Love [1944][pp. 66-73] and Sokolnikoff [1956][pp. 29-33]. 7.7 Differential identities for the operators Oi The differential operators Oi defined by equations (15) satisfy differential identities useful when obtaining the flexure equations. They are special cases of differential identities valid in curved spaces. The presence of curvature makes the parallel transport of vectors path-dependent; this property quantifies the curvature of space and can be expressed as the lack of commutativity of the covariant derivatives of a vector v: vi|j|k − vi|k|j = Riljk vl , (111) where Riljk are the covariant components of the Riemann tensor. On the sphere, the Riemann tensor has only one independent component that is non-zero, Rθϕθϕ = − sin2 θ. Other components are related by the symmetries Rαβγδ = −Rβαγδ = −Rαβδγ = Rγδαβ. The substitution of f,i to vi in the commutation relation (111) provides two differential identities satisfied by double covariant derivatives acting on scalar functions: csc θ f|ϕ|ϕ − csc θ f|ϕ|θ,ϕ − cos θ f|θ|θ + sin θ f,θ = 0 , f|θ|θ,ϕ − f|ϕ|θ,θ − cot θ f|ϕ|θ + f,ϕ = 0 . The replacement in the above equations of the double covariant derivatives by the normalized dif- ferential operators (31) yields the following identities: (sin θO2f),θ − (O3f),ϕ − cos θO1f = 0 (I1) , (112) (sin θO3f),θ − (O1f),ϕ + cos θO3f = 0 (I2) . (113) These identities can also be directly checked with the definitions (15) of the operators Oi. The identities (I1)-(I2) can be differentiated to generate identities of higher order. A first useful identity is obtained from sin θ(I1),θ − (I2),ϕ = 0: csc2 θ sin2 θ (O2f),θ + (O1f),ϕ,ϕ − 2 sin θ (O3f),ϕ − cot θ (O1f),θ + 2O1f = ∆ ′f . (114) A second useful identity is obtained from (I1),ϕ + sin θ(I2),θ = 0: csc2 θ sin2 θ (O3f),θ − (O3f),ϕ,ϕ + sin θ ((O2 −O1) f),ϕ + cot θ (O3f),θ − 2O3f = 0 . (115) 7.8 No degree one in operators A and B We want to prove that the operators A and B defined by equations (33) and (37) do not have any degree-one term in their spherical harmonic expansion: dω A(a ; b)Y ∗1p = 0 (p = −1, 0, 1) , (116) dω B(a ; b)Y ∗1p = 0 (p = −1, 0, 1) , (117) where (a, b) are arbitrary scalar functions on the sphere, dω = sin θ dθ dϕ and the integral is taken over the whole spherical surface. This property is not a straightforward consequence of constructing A and B with Dij as a building block. Although A and B can be factored into terms without degree one (such as Dija or ∆′a), the product of the factors may contain degree-one terms in its spherical harmonic expansion. Without loss of generality, we can prove the above identities with the arguments (a, b) being spherical harmonics of given degree and order. The general result is then obtained by superposition. Let a and b be spherical harmonics of order m and n: a ∼ eimϕ and b ∼ einϕ (we will not use their harmonic degree in the proof). All derivatives with respect to ϕ in the operators A and B can then be replaced with the rules a,ϕ → ima and b,ϕ → inb. The integral over ϕ in equations (116)-(117) gives dϕ ei(m+n−p)ϕ = 2π δm+n−p,0 , so that the integral is zero unless n = p−m. First consider the case p = 0 (n = −m), that is the projection on the zonal spherical harmonic of degree one. We thus have to calculate dθ A0 and dθ B0 with A0 ≡ sin θ cos θA(a ; b) , B0 ≡ sin θ cos θ B(a ; b) . The trick consists in rewriting the integrands as total derivatives: cos2 θ a,θ b,θ + cot θ sin2 θ −m2 (ab),θ + sin2 θ +m2 csc2 θ cos 2θ B0 = −im cos θ a,θ b,θ + sin θ a b,θ − csc θ cos2 θ a,θ b+ cos θ a b,θ,θ The sought integrals are thus given by dθ A0 = a,θ b,θ −m2 cot θ (ab),θ +m2 csc2 θ ab dθ B0 = −im cos θ a,θ b,θ − csc θ a,θ b+ cos θ a b,θ,θ where we have dropped the terms containing at least one power of sin θ which vanish at the limits; we have also replaced cos2 θ and cos 2θ by their value at the limits. The remaining terms can be evaluated by recalling the dependence in sin θ of the spherical harmonics: a = (sin θ)|m| a0 and b = (sin θ) |m| b0, where a0 and b0 are polynomials in cos θ. The only non-zero terms at the limits of the integrals are those for |m| = 1, in which case we have at the limits: a,θb,θ = a0b0, cot θ (ab),θ = 2a0b0, csc2 θ ab = a0b0, csc θ a,θb = cos θ a0b0, ab,θ,θ = 0. However these terms cancel in the sums so that the integrals vanish for all m. This completes the proof for the case p = 0. Now consider the case p = ±1 (n = −m ± 1), that is the projections on the sectoral spherical harmonics of degree one. We thus have to calculate dθ A±1 and dθ B±1 with A±1 ≡ sin2 θA(a ; b) , B±1 ≡ sin2 θB(a ; b) . We again write the integrands as total derivatives: A±1 = sin θ cos θ a,θ b,θ − sin θ cos θ +m2 (ab),θ − cos2 θ ∓ 2m a,θ b + sin2 θ a b,θ + 2m(m∓ 1) cot θ ab B±1 = −i (m∓ 1) sin θ a,θ b,θ +m sin θ a b,θ,θ − (m∓ 1) cos θ a,θ b −m cos θ a b,θ −m(1∓m) csc θ ab The sought integrals are thus given by dθ A±1 = −m2 (ab),θ − (1∓ 2m) a,θ b+ 2m(m∓ 1) cot θ ab dθ B±1 = i (m∓ 1) cos θ a,θ b+m cos θ a b,θ +m(1∓m) csc θ ab where we have dropped the terms containing at least one power of sin θ and replaced cos2 θ by its value at the limits. The remaining terms can be evaluated as in the case p = 0, but with a = (sin θ)|m| a0 and b = (sin θ)|m∓1| b0. All terms give zero at the limits of the integrals for all values of m. 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Introduction Fundamental equations of elasticity Three-dimensional elasticity theory Spherical shell Assumptions of the thin shell theory Strain-displacement equations Hooke's law Equilibrium equations Resolution Available methods Differential operators Transverse displacement Resolution of the equations of equilibrium Compatibility relation Tangential displacements Stresses Flexure equations and their properties Thin shell approximation Covariance Degree one Limit cases Membrane limit Euclidean limit Shell with constant thickness Displacements Comparison with the literature Breakdown of the third assumption of thin shell theory Conclusion Appendix Covariant, contravariant and normalized components Covariant derivatives Three-dimensional spherical geometry Two-dimensional spherical geometry Gradient, divergence, curl and Laplacian Rigid displacements Differential identities for the operators Oi No degree one in operators A and B

Planetary topography can either be modeled as a load supported by the lithosphere, or as a dynamical effect due to lithospheric flexure caused by mantle convection. In both cases the response of the lithosphere to external forces can be calculated with the theory of thin elastic plates or shells. On one-plate planets the spherical geometry of the lithospheric shell plays an important role in the flexure mechanism. So far the equations governing the deformations and stresses of a spherical shell have only been derived under the assumption of a shell of constant thickness. However local studies of gravity and topography data suggest large variations in the thickness of the lithosphere. In this article we obtain the scalar flexure equations governing the deformations of a thin spherical shell with variable thickness or variable Young's modulus. The resulting equations can be solved in succession, except for a system of two simultaneous equations, the solutions of which are the transverse deflection and an associated stress function. In order to include bottom loading generated by mantle convection, we extend the method of stress functions to include loads with a toroidal tangential component. We further show that toroidal tangential displacement always occurs if the shell thickness varies, even in the absence of toroidal loads. We finally prove that the degree-one harmonic components of the transverse deflection and of the toroidal tangential displacement are independent of the elastic properties of the shell and are associated with translational and rotational freedom. The flexure equations for a shell of variable thickness are useful not only for the prediction of the gravity signal in local admittance studies, but also for the construction of stress maps in tectonic analysis.

Introduction Terrestrial planets are flattened spheres only at first sight: close-up views reveal a rich topography with unique characteristics for each planet. Unity in diversity is found by studying the support mechanism for topographic deviations from the hydrostatic planetary shape. A simple mechanism, called isostasy, postulates mountains floating with iceberg-like roots in the higher density mantle. Another simple mech- anism assumes that mountains stand on a rigid membrane called the mechanical lithosphere. The two simple models predict very different gravity signals, from very weak in the first to very strong in the second. The truth lies in-between: an encompassing model views topographic structures as loads on an elastic shell with finite rigidity, called the elastic lithosphere (the elastic lithosphere is a subset of the mechanical lithosphere). The rigidity depends on the elastic properties of the rocks and on the apparent elastic thickness of the lithosphere. The latter parameter is the main objective of many studies, since its value can be related to the lithospheric composition and temperature. Another important application of the model of lithospheric flexure is the determination of stress maps, which can then be compared with the observed distribution of tectonic features. http://arxiv.org/abs/0704.1627v2 Besides the important assumption of elasticity, the model of lithospheric flexure is often simplified by two approximations. The first one states that the area to be analyzed is sufficiently small so that the curved lithospheric shell can be modeled as a flat plate. The second approximation states that the shell (or plate) is thin, which means that deformations are small and that the elastic thickness is small with respect to the wavelength of the load. On Earth, the model of lithospheric flexure has been very successful for the understanding of the topography of the oceanic plates, whereas the analysis of continental plates is fraught with difficulties due to the very old and complex structure of the continents. As far as we know, plate tectonics do not occur at the present time on other terrestrial planets, which are deemed one-plate planets [Solomon , 1978], although the term single-shell would fit better because of the curvature. A single shell can support loads of much larger extent and greater weight, the best example of which is the huge Tharsis volcanic formation covering a large portion of Mars. Indeed such a load cannot be supported by the bending moments present in a thin flat plate, whereas it can be supported by stresses tangent to the shell: the shell acts as a membrane. The first application of thin shell theory to a planet was done by Brotchie and Silvester [1969] for the lithosphere of the Earth (see also Brotchie [1971]). However the approximation of flat plate theory was seen to be sufficient when it was understood that the lithosphere of the Earth is broken into several plates [Tanimoto, 1998]. Brotchie and Silvester [1969] do not consider tangential loads and their flexure equation only includes dominant terms in derivatives; their equation is thus a special case of the equations of Kraus [1967] and Vlasov [1964] discussed below. The articles reviewed hereafter use the thin shell theory of Kraus or Vlasov unless mentioned otherwise. On the Moon, Solomon and Head [1979] used Brotchie’s equation to study displacement and stress in mare basins. Turcotte et al. [1981] estimated gravity-topography ratios for the mascons and discussed the type of stress supporting topographic loads. Arkani-Hamed [1998] modeled the support of mascons with Brotchie’s equation. Sugano and Heki [2004] and Crosby and McKenzie [2005] estimated the elastic thickness of the lithosphere from Lunar Prospector data. On Mars, the dominance of the Tharsis rise in the topography led to numerous applications of the theory of thin elastic shells to lithospheric flexure. Thurber and Toksöz [1978], Comer et al. [1985], Hall et al. [1986] and Janle and Jannsen [1986] used Brotchie’s equation to estimate the lithospheric thickness under Martian volcanoes. Turcotte et al. [1981] studied the transition between bending and membrane regimes. Willemann and Turcotte [1982] analyzed the lithospheric support of Tharsis. Sleep and Phillips [1985] analyzed the membrane stress distribution on the whole surface. Banerdt et al. [1992], Banerdt and Golombek [2000] and Phillips et al. [2001] used a model of lithospheric flexure including membrane support, bending stresses and tangent loads [Banerdt , 1986] in order to study the global stress distribution. Arkani-Hamed [2000] determined the elastic thickness beneath large volcanoes with Brotchie’s equation whereas Johnson et al. [2000] estimated the elastic thickness beneath the North Polar Cap. Using local admittance analysis with spatiospectral methods, McGovern et al. [2002] determined the elastic thickness at various locations [see also McGovern et al., 2004]. McKenzie et al. [2002] made local admittance analyses of line-of-sight grav- ity data both with flat plates and with spherical shell models. Turcotte et al. [2002] used the spherical shell formula for a one-dimensional wavelet analysis of the admittance in order to determine the average elastic thickness of the lithosphere. Zhong and Roberts [2003] and Lowry and Zhong [2003] studied the support of the Tharsis rise with an hybrid model including the flexure of a thin elastic shell as well as the internal loading of a thermal plume in the mantle. Belleguic et al. [2005] determined the elastic thickness and the density beneath large volcanoes. Searls et al. [2006] investigated the elastic thickness and the density beneath the Utopia and Hellas basins. Venus is considered as a one-plate planet but does not have giant volcanic or tectonic structures comparable to Tharsis. The spherical shell model has thus not been used as often for Venus as for Mars. Banerdt [1986] studied the global stress distribution. Janle and Jannsen [1988] and Johnson and Sandwell [1994] used Brotchie’s equation to estimate the lithospheric thickness in various locations. Sandwell et al. [1997] computed global strain trajectories for comparison with observed tectonics. Lawrence and Phillips [2003] inverted the admittance in order to estimate the elastic thickness and mantle density anomalies over two lowland regions and one volcanic rise. Anderson and Smrekar [2006] established a global map of the elastic thickness based on local admittance analysis. Mercury’s topography and gravity fields are not yet known well enough to warrant the application of a thin elastic shell model. We refer to Wieczorek [2007] for a review of recent results regarding the lithosphere of terrestrial planets. The elastic thickness of the lithosphere is not at all homogeneous over the surface of a planet. For example, McGovern et al. [2004] (for Mars) and Anderson and Smrekar [2006] (for Venus) find litho- spheric thickness variations of more than 100 km. The former study explains the variation in lithospheric thickness in terms of different epochs of loading, as the lithosphere is thickening with time. Other studies however inferred that spatial variations in lithospheric thickness on Mars are as important as temporal variations [Solomon and Head , 1982, 1990; Comer et al., 1985]. Thin spherical shell models have always been applied with a constant elastic thickness for the whole lithosphere. Local studies are done by win- dowing both the data (gravity and topography) and the model predictions for gravity [Simons et al., 1997; Wieczorek and Simons , 2005]. The assumptions behind these methods are that the elastic thick- ness is constant within the window and that the area outside the window can be neglected. The first assumption is of course true for a small enough window, but the size of the window is limited from below by the resolution of the data [Wieczorek and Simons , 2005]. Even if the first assumption were true, the second assumption is violated in two ways (unless the elastic thickness is spatially constant). First, the deformation of the shell within the window as well as the associated stress field are both modified if the elastic thickness is changed in the area outside the window. Second, the value of the predicted gravity field within the window depends on the shell deflection outside the window. These reasons make it interesting to develop a model of the lithospheric flexure for a spherical shell of variable thickness. Although a full inversion of the gravity and topography data is impractical with such a model because of the huge size of the parameter space, other applications are of high interest. For example a two-stage inversion can be considered: in the first stage a constant elastic thickness is assumed, and the resulting values are used in the second stage as a starting point for an inversion with variable elastic thickness (the parameter space can also be constrained with an a priori). Moreover this model can be used to produce synthetic data and thus allows us to check the validity of inversions assuming a constant elastic thickness. Finally, stress and strain fields can be computed for given variations of the elastic thickness, with the aim of comparing stress and strain maps with tectonic features. General equations governing the deformations of a thin elastic shell have been given by various authors [e.g. Love, 1944; Vlasov , 1964; Kraus, 1967]. However the possibility of variable shell thickness is only considered at the early stage where the strain-displacement relationships, Hooke’s law and the equilibrium equations are separately derived for the thin shell. The combination of these three sets of equations into a unique equation for the transverse deflection is made under the restriction of constant elastic thickness, ‘owing to the analytical complications which would otherwise arise’ [see Kraus, 1967, p. 199]. This article is dedicated to the derivation of the minimum set of equations governing the deformations of a thin elastic spherical shell with variable thickness. Using Kraus’ method of stress functions, we find that the transverse deflection is the solution of a simultaneous system of two differential equations of the fourth order. Contrary to the case of constant thickness, these equations cannot be combined due to the presence of products of derivatives of the thickness and derivatives of the deflection or of the stress function. In order to include bottom loading generated by mantle convection, we extend the method of stress functions to include toroidal tangential loads, which were not considered by Kraus [1967]. Non- toroidal loading is for example generated by tangential lithostatic forces whereas toroidal loading could be due to mantle flow generating drag at the base of the lithosphere (mantle flow also produces non- toroidal loading). With applications to tectonics in mind, we derive the equations relating the tangential displacements and the stresses to the transverse deflection and the stress functions. We further show that toroidal tangential displacement occurs even if there is no toroidal loading (unless the shell thickness is constant). Finally we prove three properties specific to the degree-one harmonic components: (1) the degree-one transverse deflection and the degree-one toroidal tangential displacement drop from the elasticity equations because they represent rigid displacements of the whole shell, (2) the transverse and tangential components of degree-one loads are related so that the shell does not accelerate, (3) degree- one loads can deform the shell and generate stresses. Though our aim is to introduce a variable shell thickness, the final equations are also valid for a variable Young’s modulus (Poisson’s ratio must be kept constant). Another way to take into account variations of the lithospheric thickness consists in treating the lithosphere as a three-dimensional spherical solid that is elastic [Métivier et al., 2006] or viscoelastic [e.g. Zhong et al., 2003; Latychev et al., 2005; Wang and Wu, 2006]. The resulting equations are exact (no thin shell approximation) and can be solved with finite element methods. The thin shell assumption is probably satisfied for known planetary lithospheres; in case of doubt, it is advisable to compare the results of thin shell theory with three-dimensional models assuming constant elastic thickness: static thick shell models [e.g. Banerdt et al., 1982; Janes and Melosh , 1990; Arkani-Hamed and Riendler , 2002] or time-dependent viscoelastic models [e.g. Zhong and Zuber , 2000; Zhong , 2002]. The advantage of thin shell equations is their two-dimensional character, making them much easier to program and quicker to solve on a computer. Solving faster either gives access to finer two-dimensional grids or allows to examine a larger parameter space. We choose to work within the formalism of differential calculus on curved surfaces, without which the final equations would be cumbersome. Actually the only tool used in this formalism is the covariant derivative, which can be seen by geophysicists as just a way of combining several terms into one ‘deriva- tive’. All necessary formulas are given in the Appendix. The possibility of writing a differential equation in terms of covariant derivatives (in a tensorial form) also provides a consistency check. The presence of derivatives that cannot be included into covariant derivatives is simply forbidden. This is not without meaning for differential equations of the fourth order including products of derivatives. In section 2, we show how to obtain the strain-displacement relationships, Hooke’s law and the equilibrium equations for a thin spherical shell. These equations are available in the literature for the general case of a thin shell [e.g. Kraus, 1967], but we derive them for the spherical case in a simpler way, starting directly with the metric for the spherical shell. We examine in detail the various approximations made to obtain the thin shell theory of flexure, refraining until the end from taking the ‘thin shell’ quantitative limit in order to ascertain its influence on the final equations. In section 3, we use the method of stress functions to obtain the flexure equations governing the displacements and the stresses. In section 4, we give the final form of the flexure equations in the thin shell approximation. We also study the covariance and the degree-one projection of the flexure equations. In section 5, we examine various limit cases in which the flexure equations take a simpler form: the membrane limit, the Euclidean limit and the limit of constant thickness. 2 Fundamental equations of elasticity 2.1 Three-dimensional elasticity theory Linear elasticity theory is based on three sets of equations. We directly state them in tensorial form for an isotropic material, since they are derived in Cartesian coordinates in many books [e.g. Ranalli , 1987; Synge and Schild , 1978]. Recall that, in tensorial notation, there is an implicit summation on indices that are repeated on the same side of an equation. The first set of equations includes strain-displacement relationships: ǫij = (ui,j + uj,i) , (1) where ǫij is the infinitesimal strain tensor and ui are the finite displacements. The ‘comma’ notation denotes the spatial derivative (see Appendix 7.2). The second set includes the constitutive equations of elasticity, or Hooke’s law, relating the strain tensor and the stress tensor σij : σij = λ ǫ δij + 2Gǫij , (2) where ǫ = ǫ11 + ǫ22 + ǫ33 and δij = 1 if i= j, otherwise it equals zero. The parameter λ is known as the first Lamé constant. The parameter G is known as the second Lamé constant, or the shear modulus, or the modulus of rigidity. Boundary conditions are given by σij nj = Ti , (3) with nj being the normal unit vector of the surface element and Ti being the surface force per unit area. The third set includes equations of motion which reduce to equilibrium equations for stresses if the problem is static: σij,j = 0 . (4) Body forces, such as gravity, are assumed to be absent. Both strain and stress tensors are symmetric: ǫij = ǫji and σij = σji. These three-dimensional equations do not yet have the right form for the description of the defor- mations of a two-dimensional spherical shell. Various methods have been used to generate appropriate equations. Love [1944] and Timoshenko and Woinowsky-Krieger [1964] derive strain-displacements and equilibrium equations directly on the surface of the sphere (the latter only for the special cases of no bending or axisymmetrical loading). Sokolnikoff [1956] derives strain-displacement equations in three- dimensional curvilinear coordinates using an arbitrary diagonal metric but states without proof the equi- librium equations in curvilinear coordinates. Kraus [1967] uses Sokolnikoff’s form of strain-displacement equations and derives equilibrium equations for an arbitrary two-dimensional surface using Hamilton’s principle (i.e. virtual displacements). Instead of directly deriving equations on the two-dimensional surface of the sphere, we will first obtain their form in three-dimensional curvilinear coordinates and then restrict them to the surface of the sphere. The first step can elegantly be done through the use of tensors [Synge and Schild , 1978]. Equations (1)-(4) are tensorial with respect to orthogonal transformations, but not with respect to other coordinate transforms (one reason being the presence of usual derivatives). In other words, they are only valid in Cartesian coordinates. In a three-dimensional Euclidean space, tensorial equations have a simplified form in Cartesian coordinates because supplementary terms that make them tensorial with respect to arbitrary coordinate transformations are zero. The missing terms can be reconstructed by using a set of rules, such as the replacement of usual derivatives by covariant derivatives and the substitution of tensorial contraction to sum on components. Correspondence rules lead to the following three sets of equations: ǫij = ui|j + uj|i , (5) σij = λ ǫ gij + 2Gǫij , (6) gjk σij|k = 0 , (7) where ǫ = gkl ǫkl. The notation ui|j denotes the covariant derivative of ui (see Appendix 7.2). The metric and its inverse are noted gij and g ij , respectively. Tensorial components cannot be expressed in a normalized basis (except for Cartesian coordinates) which is more common for physical interpretation (see Appendix 7.1). Covariant components in equations (5)-(7) are related to components defined in a normalized basis (written with a hat) by: gii ûi , ǫij = giigjj ǫ̂ij , σij = giigjj σ̂ij , where there is no implicit summation on repeated indices. In the next section we will introduce additional assumptions in order to restrict the equations to the two-dimensional surface of a spherical shell. 2.2 Spherical shell 2.2.1 Assumptions of the thin shell theory Suppose that the two first coordinates are the colatitude θ and longitude ϕ on the surface of the sphere, whereas the third coordinate ζ is radial. R is the shell radius. Assumptions of the thin shell theory are [see Kraus, 1967, chap. 2.2]: 1. The shell is thin (say less than one tenth of the radius of the sphere). 2. The deflections of the shell are small. 3. The transverse normal stress is negligible: σζζ = 0. 4. Normals to the reference surface of the shell remain normal to it and undergo no change of length during deformation: ǫθζ = ǫϕζ = ǫζζ = 0. The second assumption allows us to use linear equations to describe the deflections. The third and fourth assumptions are not fully consistent: we refer to Kraus [1967] for more details. We will relax them in the derivation of the equations for the deflection of a spherical shell. The crucial assumption is σζζ = 0 which is essential for the restriction of Hooke’s law to the two-dimensional shell. As we will see later, σζζ cannot be zero since it is related to the non-zero transverse load (besides the fact that it is incompatible with a vanishing transverse strain). What is absolutely necessary is that σζζ ≪ σii for i = (θ, ϕ). In section 5.3.3, we will show that this condition is satisfied if the wavelength of the load is much larger than the thickness of the shell. The reference surface is the middle surface of the shell. With the aim of integrating out the third coordinate, a coordinate system is chosen so that the radial coordinate ζ is zero on the reference surface. The metric is given by ds2 = (R+ ζ) dθ2 + sin2 θ dϕ2 + dζ2 . (8) Christoffel symbols necessary for the computation of the covariant derivatives are given in Appendix 7.3. 2.2.2 Strain-displacement equations With the metric (8), the strain-displacement equations (5) become ǫ̂θθ = (ûθ,θ + ûζ) , ǫ̂ϕϕ = (csc θ ûϕ,ϕ + cot θ ûθ + ûζ) , ǫ̂θϕ = (csc θ ûθ,ϕ − cot θ ûϕ + ûϕ,θ) , (9) ǫ̂ζζ = ûζ,ζ , ǫ̂θζ = ((R + ζ) ûθ,ζ − ûθ + ûζ,θ) , ǫ̂ϕζ = ((R + ζ) ûϕ,ζ − ûϕ + csc θ ûζ,ϕ) , where all quantities are given in a normalized basis. The fourth assumption of the thin shell theory implies that the displacements are linearly distributed across the thickness of the shell, with the transverse displacement being constant. Displacements can thus be expanded to first order in ζ: (ûθ, ûϕ, ûζ) = (vθ + ζβθ, vϕ + ζβϕ, w) . (10) The coefficients (vθ , vϕ, w, βθ, βϕ) are independent of ζ: (vθ, vϕ, w) represent the components of the displacement vector of a point on the reference surface, whereas (βθ, βϕ) represent the rotations of tangents to the reference surface oriented along the tangent axes. We determine βθ and βϕ by applying the fourth assumption of the thin shell theory and the expansion (10) to the last two strain-displacement equations (vθ − w,θ) , (vϕ − csc θ w,ϕ) . (11) The substitution of the expansion (10) into the fourth strain-displacement equation (ǫ̂ζζ = ûζ,ζ) leads to the condition ǫ̂ζζ = 0, as postulated by the thin shell theory. After the insertion of the expansion formulas (10) and (11), the first three strain-displacement equa- tions (9) become ǫ̂θθ = ǫ 1 + ζ/R κ0θ , ǫ̂ϕϕ = ǫ 1 + ζ/R κ0ϕ , (12) 2 ǫ̂θϕ = γ 1 + ζ/R The extensional strains ǫ0θ, ǫ ϕ and γ θϕ are defined by ǫ0θ = (vθ,θ + w) , ǫ0ϕ = (csc θ vϕ,ϕ + cot θ vθ + w) , (13) γ0θϕ = (vϕ,θ − cot θ vϕ + csc θ vθ,ϕ) . They represent the normal and shearing strains of the reference surface. The flexural strains κ0θ, κ ϕ and τ0 are given by κ0θ = − O1 w , κ0ϕ = − O2 w , (14) τ0 = − O3 w . They represent the changes in curvature and the torsion of the reference surface during deformation [Kraus, 1967]. The differential operators O1,2,3 are defined by + 1 , O2 = csc2 θ + cot θ + 1 , (15) O3 = csc θ − cot θ The zero upper index in (ǫ0θ, ǫ θϕ, κ 0) refers to the reference surface and appears in order to follow Kraus’ notation. In Love’s theory, one makes the approximation 1 ∼= 1R in equations (12). However this does not simplify the calculations when the shell is a sphere because it has only one radius of curvature (for a shell with two radii of curvature, this approximation is a great simplification). Our choice to keep the factor 1 leads to the same results as the theory of Flügge-Lur’e-Byrne, explained in Kraus [1967, chap. 3.3a] or Novozhilov [1964, p. 53], in which this factor is expanded up to second order. In any case the choice between the approximation or the expansion of this factor does not affect the equations (derived in the next section) relating the stress and moment resultants to the strains: they are the same is approximated to zeroth order, expanded to second order or fully kept. 2.2.3 Hooke’s law When the metric is diagonal and the basis is normalized, Hooke’s law (6) becomes σ̂ii = λ ǫ̂kk + 2G ǫ̂ii (i = j) , σ̂ij = 2G ǫ̂ij (i 6= j) . There is no implicit summation on repeated indices. The third assumption of the thin shell theory, σ̂ζζ = 0, can be used to eliminate ǫ̂ζζ from Hooke’s σ̂θθ = 1− ν2 (ǫ̂θθ + ν ǫ̂ϕϕ) , σ̂ϕϕ = 1− ν2 (ǫ̂ϕϕ + ν ǫ̂θθ) . Young’s modulus E and Poisson’s ratio ν are related to Lamé parameters by (1 + ν)(1 − 2ν) 2(1 + ν) In principle, the fourth assumption of the thin shell theory, ǫ̂θζ = ǫ̂ϕζ = 0, leads to σ̂θζ = σ̂ϕζ = 0 but non-vanishing values must be retained for purposes of equilibrium. Figure 1: Stress resultants and stress couples acting on a small element of the shell. The directions of the stress resultants (simple arrows) and the rotation sense of the stress couples (double arrows) correspond to positive components (tensile stress is positive). Loads (qθ, qϕ, q) act on the reference surface. The substitution of the expansion (12) into the thin shell approximation of Hooke’s law gives σ̂θθ = 1− ν2 ǫ0θ + νǫ 1 + ζ/R κ0θ + ν κ σ̂ϕϕ = 1− ν2 ǫ0ϕ + νǫ 1 + ζ/R κ0ϕ + ν κ , (16) σ̂θϕ = 2(1 + ν) γ0θϕ + 1 + ζ/R with ǫ0θ, ǫ θϕ, κ ϕ and τ 0 defined by equations (13) and (14). The expressions for σ̂θζ and σ̂ϕζ will not be needed. We now integrate the stress distributions across the thickness h of the shell (see Figure 1). The stress resultants and couples obtained in this way are defined per unit of arc length on the reference surface: ∫ h/2 σ̂ii (1 + ζ/R) dζ (i = θ, ϕ) , Nθϕ = Nϕθ = ∫ h/2 σ̂θϕ (1 + ζ/R) dζ , ∫ h/2 σ̂iζ (1 + ζ/R) dζ (i = θ, ϕ) , (17) ∫ h/2 σ̂ii (1 + ζ/R) ζ dζ (i = θ, ϕ) , Mθϕ = Mϕθ = ∫ h/2 σ̂θϕ (1 + ζ/R) ζ dζ . We evaluate these integrals with the expansion (16). The tangential stress resultants are Nθ = K ǫ0θ + ν ǫ Nϕ = K ǫ0ϕ + ν ǫ , (18) Nθϕ = K γ0θϕ . Explicit expressions for the transverse shearing stress resultants Qi are not needed since these quantities will be determined from the equilibrium equations. The moment resultants are Mθ = D κ0θ + ν κ ǫ0θ + ν ǫ Mϕ = D κ0ϕ + ν κ ǫ0ϕ + ν ǫ , (19) Mθϕ = D The extensional rigidity K and the bending rigidity D are defined by 1− ν2 , (20) 12(1− ν2) . (21) Their dimensionless ratio ξ is a large number, ξ = R2 , (22) the inverse of which will serve as an expansion parameter for thin shell theory. 2.2.4 Equilibrium equations With the metric (8), the components θ, ϕ and ζ of the equilibrium equations (7) respectively become (R+ ζ) (sin θ σ̂θθ),θ + σ̂θϕ,ϕ − cos θ σ̂ϕϕ + sin θ σ̂ζθ + sin θ (R+ ζ) = 0 , (R + ζ) (sin θ σ̂θϕ),θ + σ̂ϕϕ,ϕ + cos θ σ̂θϕ + sin θ σ̂ζϕ + sin θ (R+ ζ) = 0 , (R + ζ) (sin θ σ̂θζ),θ + σ̂ϕζ,ϕ − sin θ (σ̂θθ + σ̂ϕϕ) + sin θ (R+ ζ) = 0 , where the equations have been multiplied by sin θ (R + ζ), (R + ζ) and sin θ (R + ζ)2, respectively. The stress components are given in a normalized basis. The integration on ζ of these three equations in the range [−h/2, h/2] yields the equilibrium equations for the forces: (sin θ Nθ),θ +Nθϕ,ϕ − cos θ Nϕ + sin θ Qθ +R sin θ qθ = 0 , (23) (sin θNθϕ),θ +Nϕ,ϕ + cos θ Nθϕ + sin θ Qϕ +R sin θ qϕ = 0 , (24) (sin θ Qθ),θ +Qϕ,ϕ − sin θ (Nθ +Nϕ)−R sin θ q = 0 , (25) where qθ and qϕ are the components of the tangential load vector per unit area of the reference surface: (R+ ζ) = R2 qi (i = θ, ϕ) . We choose the convention that tensile stresses are positive (see Figure 1). The transverse load per unit area of the reference surface is noted q and is taken to be positive toward the center of the sphere: (R+ ζ) = −R2 q . (26) The first two equilibrium equations for the stresses can also be multiplied by ζ before the integration to yield the equilibrium equations for the moments: (sin θMθ),θ +Mθϕ,ϕ − cos θMϕ −R sin θ Qθ = 0 , (27) (sin θMθϕ),θ +Mϕ,ϕ + cos θMθϕ −R sin θ Qϕ = 0 . (28) We have neglected small terms in ζ (R+ ζ) where i = (θ, ϕ). A third equilibrium equation for the moments exists but has the form of an identity: Mθϕ =Mϕθ. 3 Resolution 3.1 Available methods At this stage the elastic theory for a thin spherical shell involves 17 equations: six strain-displacement relationships (13)-14), six stress-strain relations (18)-(19) making Hooke’s law, and five equilibrium equa- tions (23)-(23) and (27)-(28). There are 17 unknowns: six strain components (ǫ0θ, ǫ θϕ, κ three displacements (w, vθ , vϕ), three tangential stress resultants (Nθ, Nϕ, Nθϕ), two transverse shearing stress resultants (Qθ, Qϕ), and three moment resultants (Mθ,Mϕ,Mθϕ). The three equations (16) are also needed if the tangential stresses (σ̂θθ, σ̂θϕ, σ̂ϕϕ) are required. The quantities of primary interest to us are the transverse deflection and the tangential stresses (sometimes tangential strain is preferred, as in Sandwell et al. [1997] or Banerdt and Golombek [2000]). We thus want to find the minimum set of equations that must be solved to determine these quantities. We are aware of two methods of resolution [Novozhilov , 1964, p. 66]. In the first one, we insert the strain-displacement relationships into Hooke’s law, and substitute in turn Hooke’s law into the equilibrium equations. This method yields three simultaneous differential equations for the displacements. Once the displacements are known, it is possible to compute the strains and the stresses. The second method supplements the equilibrium equations with the equations of compatibility [Novozhilov , 1964, p. 27] that relate the partial derivatives of the strain components. It is then convenient to introduce the so-called stress functions [Kraus , 1967, p. 243], without direct physical interpretation, which serve to define the stress resultants without introducing the tangential displacements. Equations relating the transverse displacement and the stress functions are then found by applying the third equation of equilibrium and the third equation of compatibility (it is also possible to use all three equations of compatibility in order to directly solve for the stress and moment resultants). Once the transverse displacement and the stress functions are known, stresses can be computed. If the shell thickness is constant, the deformations of a thin spherical shell can be completely calcu- lated with both methods. If the shell thickness is variable, the three equations governing displacements, obtained with the first method, cannot be decoupled and are not easy to solve. Kraus’ method with stress functions leads to a system of three equations (relating the transverse displacement and the two stress functions), in which the first equation is decoupled and solved before the other two. This method thus provides a system of equations much easier to solve and will be chosen in this article. When solving the equations, one usually assumes from the beginning the large ξ limit, i.e. 1+ ξ ∼= ξ where ξ is defined by equation (22). We will only take this limit at the end of the resolution. This procedure will not complicate the computations, since we have to compute anyway many new terms because of the variable shell thickness. 3.2 Differential operators We will repeatedly encounter the operators Oi which intervene in the expressions (14) for the flexural strains (κ0θ, κ 0). Since we are looking for scalar equations, we need to find out how the operators Oi can be combined in order to yield scalar expressions, i.e. expressions that are invariant with respect to changes of coordinates on the sphere. The first thing is to relate the Oi to tensorial operators. Starting from the covariant derivatives on the sphere ∇i, we construct the following tensorial differential operators of the second degree in derivatives: Dij = ∇i∇j + gij , (29) where ∇i denotes the covariant derivative (see Appendix 7.2). These operators give zero when applied on spherical harmonics of degree one (considered as scalars): Dij Y1m = 0 (m = −1, 0, 1) . (30) This property can be explicitly checked on the spherical harmonics (109) with the metric and the formulas for the double covariant derivatives given in Appendix 7.4. In two-dimensional spherical coordinates (θ, ϕ), the three operators Oi defined by equations (15) are related to the operators Dij acting on a scalar function f through O1 f = Dθθ f , O2 f = csc2 θDϕϕ f , (31) O3 f = csc θDθϕ f . The operators O1,2,3f actually correspond to normalized Dijf , i.e. Dijf/( giigjj). The usual derivatives of the operators Oi satisfy the useful identities (112)-(113) which are proven in Appendix 7.7. These identities are the consequence of the path dependence of the parallel transport of vectors on the curved surface of the sphere. Invariant expressions are built by contracting all indices of the differential operators in their tensorial form. The indices can be contracted with the inverse metric gij or with the antisymmetric tensor εij (see Appendix 7.4), which should not be confused with the strain tensor ǫij . In the following, a and b are scalar functions on the sphere. At degree 2, the only non-zero contraction of the Dij is related to the Laplacian (104): ∆′a ≡ gij Dij a = (∆+ 2) a (32) = (O1 +O2) a = a,θ,θ + cot θ a,θ + csc 2 θ a,ϕ,ϕ + 2 a . At degree 4, a scalar expression symmetric in (a, b) is given by A(a ; b) ≡ [∆′a][∆′b]− [Dij a][Dij b] , = [∆ a][∆ b]− [∇i∇j a][∇i∇j b] + [∆ a] b+ a [∆ b] + 2 a b (33) = [O1 a][O2 b] + [O2 a][O1 b]− 2 [O3 a][O3 b] = (a,θ,θ + a) csc2 θ b,ϕ,ϕ + cot θ b,θ + b csc2 θ a,ϕ,ϕ + cot θ a,θ + a (b,θ,θ + b) −2 csc2 θ (a,θ,ϕ − cot θ a,ϕ) (b,θ,ϕ − cot θ b,ϕ) . where upper indices are raised with the inverse metric: Dij = gikgjlDkl. The action of an operator does not extend beyond the brackets enclosing it. If a is constant, A(a ; b) = a∆′b. It is useful to define an associated operator A0 that gives zero if its first argument is constant: A0(a ; b) = A(a ; b)− a [∆′b] . (34) A scalar expression of degree 4 antisymmetric in (a, b) is given by B1(a ; b) ≡ gij εkl [Dik a] [Djl b] = gij εkl [∇i∇k a] [∇j∇l b] (35) = [(O1 −O2) a] [O3b]− [O3a] [(O1 −O2) b] = csc θ a,θ,θ − csc2 θ a,ϕ,ϕ − cot θ a,θ (b,θ,ϕ − cot θ b,ϕ) − csc θ (a,θ,ϕ − cot θ a,ϕ) b,θ,θ − csc2 θ b,ϕ,ϕ − cot θ b,θ We will also need another operator of degree 4: B2(a ; b) ≡ εij [∇ia ] [∇j ∆′b] = csc θ a,θ [∆ ′b],ϕ − a,ϕ [∆ ′b],θ . (36) The sum of the operators B1 and B2 is noted B: B(a ; b) = B1(a ; b) + B2(a ; b) . (37) If either a or b is constant, B(a ; b) = 0. The operators A and B have an interesting property proven in Appendix 7.8: for arbitrary scalar functions a and b, A(a ; b) and B(a ; b) do not have a degree-one term in their spherical harmonic expansion. This is not true of A0, B1 and B2. 3.3 Transverse displacement 3.3.1 Resolution of the equations of equilibrium The first step consists in finding expressions for the moment resultants (Mθ,Mϕ,Mθϕ) in terms of the transverse displacement and the stress resultants. The extensional strains (ǫ0θ, ǫ θϕ) can be eliminated from the equations for stress and moment resultants (18)-(19). The flexural strains (κ0θ, κ 0) depend on the transverse displacement w through equations (14). We thus obtain Mθ = − (O1 + νO2)w + Mϕ = − (O2 + νO1)w + Nϕ , (38) Mθϕ = − (1− ν)O3 w + Nθϕ , where the parameter ξ is defined by equation (22). The second step consists in solving the equilibrium equations for moments in order to find the transverse shearing stress resultants (Qθ, Qϕ). We substitute expressions (38) into equations (27)-(28). Knowing that the stress resultants satisfy the equilibrium equations (23)-(24), we obtain new expressions for Qθ and Qϕ (identities (112)-(113) are helpful): Qθ = − (D∆′w),θ − (1− η)Rqθ (1− ν) (D,θ O2 − csc θD,ϕO3) w − (η,θNθ + csc θ η,ϕNθϕ) , (39) Qϕ = − csc θ (D∆′w),ϕ − (1− η)Rqϕ (1− ν) (−D,θ O3 + csc θD,ϕ O1) w − (η,θ Nθϕ + csc θ η,ϕNϕ) . (40) The operator ∆′ is defined by equation (32) and η is a parameter close to 1: 1 + ξ . (41) The third step consists in finding expressions for the stress resultants (Nθ, Nϕ, Nθϕ) in terms of stress functions by solving the first two equilibrium equations (23)-(24). Let us define the following linear combinations of the stress and moment resultants: (Pθ, Pϕ, Pθϕ) = Mθ, Nϕ + Mϕ, Nθϕ + . (42) We observe that these linear combinations satisfy simplified equations of equilibrium: (sin θ Pθ),θ + Pθϕ,ϕ − cos θ Pϕ +R sin θ qθ = 0 , (sin θ Pθϕ),θ + Pϕ,ϕ + cos θ Pθϕ +R sin θ qϕ = 0 . (43) Comparing these equilibrium equations with the identities (112)-(113), we see that the homogeneous equations (i.e. equations (43) with a zero tangential load qθ = qϕ = 0) are always satisfied if (Pθ, Pϕ, Pθϕ) = (O2,O1,−O3)F , (44) where F is an auxiliary function called stress function. For the moment, this function is completely arbitrary apart from being scalar and differentiable. Particular solutions of the full equations (43) can be found if we express the tangential load qT = qθθ̂ + qϕϕ̂ in terms of the surface gradient of a scalar potential Ω (consoidal or poloidal component) and the surface curl of a vector potential V r̂ (toroidal component): qT = − ∇̄Ω + 1 ∇̄ × (V r̂) . (45) Surface operators are defined in Appendix 7.5, where the terms consoidal/poloidal are also discussed. The covariant components of qT are (qθ, sin θqϕ) and can be expressed as − 1RΩ,i + gjkεikV,j , which gives qθ = − Ω,θ + R sin θ V,ϕ , sin θ qϕ = − Ω,ϕ − sin θ V,θ . (46) If the tangential load is consoidal (V = 0), a particular solution of equations (43) is given by (Pθ, Pϕ, Pθϕ) = (1, 1, 0)Ω . (47) If the tangential load is toroidal (Ω = 0), a particular solution of equations (43) is given by (Pθ, Pϕ, Pθϕ) = (2O3,−2O3,O2 −O1)H , (48) where we have introduced a second stress function H which satisfies the constraint ∆′H = −V + V0 , (49) where V0 is a constant (identities (112)-(113) are useful). This equation allows us to determine the stress function H if the toroidal source V is known. The general solution of the equations (43) is given by the sum of the general solution (44) of the homogeneous equations and the two particular solutions (47)-(48) of the full equations: (Pθ, Pϕ, Pθϕ) = (O2F +Ω + 2O3H,O1F +Ω− 2O3H,−O3F + (O2 −O1)H) . (50) The stress resultants (Nθ, Nϕ, Nθϕ) can now be obtained from (Pθ, Pϕ, Pθϕ) by using equations (42) and (38): (Nθ, Nϕ, Nθϕ) = η (Pθ, Pϕ, Pθϕ) + η (O1 + νO2,O2 + νO1, (1− ν)O3)w , (51) which finally give Nθ = η O2 F + (∆′ − (1− ν)O2)w +Ω+ 2O3H Nϕ = η O1 F + (∆′ − (1− ν)O1)w +Ω− 2O3H , (52) Nθϕ = η −O3 F + (1 − ν) O3 w + (O2 −O1)H The fourth step consists in expressing the third equation of equilibrium (25) in terms of the transverse displacement w and the stress functions F and H . For this purpose, it is handy to express the transverse shearing stress resultants (Qθ, Qϕ) in terms of (w,F,H). We thus substitute Nϕ, Nθ, Nθϕ, given by equations (52), into the expressions for Qθ and Qϕ, given by equations (39)-(40): Qθ = − (ηD∆′w),θ + (ηD),θ O2 − csc θ (ηD),ϕ O3 − (η,θ O2 − csc θ η,ϕO3)F + Ω,θ − (ηΩ),θ − csc θ V,ϕ − 2 (η,θ O3 + csc θ η,ϕ O2)H + csc θ (η,ϕ ∆′H − η(∆′H),ϕ) , (53) Qϕ = − csc θ (ηD∆′w),ϕ + − (ηD),θ O3 + csc θ (ηD),ϕ O1 − (−η,θ O3 + csc θ η,ϕ O1)F + csc θ (Ω,ϕ − (ηΩ),ϕ) + V,θ +2 (η,θ O1 + csc θ η,ϕ O3)H − (η,θ ∆′H − η(∆′H),θ) . (54) The various terms present in Qθ and Qϕ can be classified into generic types according to their differential structure. Each generic type will contribute in a characteristic way to the third equation of equilibrium. It is worthwhile to compute the generic contribution of each type before using the full expressions of the stress resultants with their multiple terms. Identities (112)-(113) are helpful in this calculation. We thus write the third equation of equilibrium (25) as follows: I (Qθ ;Qϕ ; 0)− (Nθ +Nϕ)−Rq = 0 , (55) where the differential operator I is defined for arbitrary expressions (X,Y, Z) by I(X ;Y ;Z) = csc θ (sin θX),θ + csc θ Y,ϕ − cot θ Z . (56) This operator must be evaluated for the following generic types present in (Qθ, Qϕ): I (a,θ ; csc θ a,ϕ ; 0) = ∆a , I (− csc θ a,ϕ ; a,θ ; 0) = 0 , I (a,θ O2b− csc θ a,ϕ O3b ;−a,θ O3b+ csc θ a,ϕO1b ; 0) = A0(a ; b) , (57) I (a,θ O3b+ csc θ a,ϕ O2b ;−a,θ O1b− csc θ a,ϕO3b ; 0) = B1(a ; b) , I (− csc θ (a,ϕ∆′b − a[∆′b],ϕ) ; a,θ ∆′b− a[∆′b],θ ; 0) = 2B2(a ; b) , where a and b are scalar functions. The operators A0, B1 and B2 are defined in section 3.2. We now substitute (Qθ, Qϕ) and (Nϕ, Nθ) into the third equation of equilibrium (55) and use formulas (57). We thus obtain the first of the differential equations that relate w and the stress functions F and ∆′ (ηD∆′w)− (1 − ν)A(ηD ;w) +R3 A(η ;F ) + 2R3 B(η ;H) = −R4 q +R3 (∆Ω−∆′(ηΩ)) . (58) The operators ∆′, A and B are defined in section 3.2. 3.3.2 Compatibility relation A second equation relating the transverse displacement and the stress functions comes from the compat- ibility relation which is derived by eliminating (vθ, vϕ) from the strain-displacement equations (13): sin θ γ0θϕ,ϕ sin2 θ ǫ0ϕ,θ + ǫ0θ,ϕ,ϕ − sin θ cos θ ǫ0θ,θ + 2 sin2 θ ǫ0θ − sin2 θ ∆′w . (59) The strain components are related to the stress resultants through Hooke’s law (18). The substitution of equations (18) into the compatibility equation (59) gives ∆′ (α (Nθ +Nϕ))− ∆′w − (1 + ν)J (αNθ ;αNϕ ;αNθϕ) = 0 . (60) where α is the reciprocal of the extensional rigidity: α ≡ 1 K(1− ν2) . (61) For arbitrary expressions (X,Y, Z), J (X ;Y ;Z) is defined by J (X ;Y ;Z) = csc2 θ sin2 θX,θ + Y,ϕ,ϕ + 2 (sin θ Z,ϕ),θ − cot θ Y,θ + 2 Y . (62) As in the case of the third equation of equilibrium, it is practical to classify the terms present in (Nθ, Nϕ, Nθϕ) into generic types and evaluate separately their contribution to the equation of compati- bility. There are three types of terms for which we must evaluate the operator J (identities (112)-(115) are helpful in this calculation): J (a ; a ; 0) = ∆′a , J (aO2b ; aO1b ;−aO3b) = A(a ; b) , (63) J (2aO3b ;−2aO3b ; a (O2 −O1)b) = 2B(a ; b) , where a and b are scalar functions. The operators ∆′, A and B are defined in section 3.2. We now evaluate J in equation (60) with expressions (52) and formulas (63): J (αNθ ;αNϕ ;αNθϕ) = A(ηα ;F ) + ∆′ (ηαD∆′w)− 1 (1− ν)A(ηαD ;w) + ∆′ (ηαΩ) + 2B(ηα ;H) . (64) The term A(ηαD ;w) can be rewritten with the help of the following equality: 1− ν2 A(ηαD ;w) = ∆′w −A(η ;w) , (65) since (1 − ν2)ηαD/R2 = η/ξ and (η/ξ),i = −η,i. We finally substitute Nθ + Nϕ, given by equations (52), into the compatibility equation (60) and use expressions (64)-(65). We thus obtain the second of the differential equations that relate w and the stress functions F and H : ∆′ (ηα∆′F )− (1 + ν)A(ηα ;F )− A(η ;w) − 2(1 + ν)B(ηα ;H) = −(1− ν)∆′(ηαΩ) . (66) 3.4 Tangential displacements Assuming that the flexure equations (58) and (66) for the transverse displacement w and the stress functions (F,H) have been solved, we now show how to calculate the tangential displacements. In analogy with the decomposition of the tangential load in equations (45)-(46), the tangential displacement can be separated into consoidal and toroidal components: v = ∇̄S + ∇̄ × (T r̂) , (67) where S and T are the consoidal and toroidal scalars, respectively. The covariant components of v are (vθ, sin θ vϕ) and can be expressed as S,i + g jkεikT,j (see Appendix 7.5), which gives vθ = S,θ + csc θ T,ϕ , sin θ vϕ = S,ϕ − sin θ T,θ . (68) The strain-displacement equations (13) become ǫ0θ = ((O1 − 1)S +O3 T + w) , ǫ0ϕ = ((O2 − 1)S −O3 T + w) , (69) γ0θϕ = (2O3S + (O2 −O1)T ) . The stress resultants (Nθ, Nϕ, Nθϕ) given by equations (18) become (∆S + (1 + ν)w − (1− ν) ((O2 − 1)S −O3 T )) , (∆S + (1 + ν)w − (1− ν) ((O1 − 1)S +O3 T )) , (70) Nθϕ = (2O3 S + (O2 −O1)T ) . The toroidal potential T cancels in the sum Nθ +Nϕ: Nθ +Nϕ = (1 + ν) (∆S + 2w) . The consoidal displacement potential S can thus be related to (w,F,Ω) by using expressions (52) for the stress resultants: ∆S = Rηα (1− ν) (∆′F + 2Ω) + η ∆′w − 2w , (71) where α is defined by equation (61). It is more difficult to extract the toroidal displacement potential T . When the shell thickness is constant, decoupled equations for the displacements can be found by suitable differentiation and combi- nation of the three equilibrium equations (23)-(25) for the stress resultants. This method does not work if the shell thickness is variable because the resulting equations are coupled. The trick consists in relating the tangential displacements to (w,F,H) by the way of equations similar to the homogeneous part of the first two equilibrium equations, but with (Nθ, Nϕ, Nθϕ) replaced by (Nθ, Nϕ, Nθϕ), so that derivatives do not mix with derivatives of T . We will thus calculate the following expression in two different ways (from equations (52) and (70)): = csc2 θ (−X,ϕ + sin θ Y,θ) , where X and Y are defined by X = sin θ I Nθϕ ; Y = sin θ I Nθϕ ; Nϕ ;− The operator I is defined by equation (56). As before, it is easier to begin with the evaluation of the operator Z for generic contributions: Z(a ; a ; 0) = 0 , Z(aO2b ; aO1b ;−aO3b) = −B(a ; b) , Z(2aO3b ;−2aO3b ; a(O2 −O1)b) = 2A(a ; b)−∆′ (a∆′b) , On the one hand, the evaluation Z for (Nθ, Nϕ, Nθϕ) given by equations (70) gives −1− ν ∆∆′ T . On the other hand, the evaluation of Z for (Nθ, Nϕ, Nθϕ) given by equations (52) gives + (1− ν)B + 2RA The equality of the two previous formulas yields the sought equation for T : ∆∆′ T = 2R (1 + ν) (B (ηα ;F )− 2A (ηα ;H) + ∆′ (ηα∆′H)) + 2B (η ;w) . (72) We have used the relation (η/ξ),i = −η,i. The equation for T shows that toroidal displacement always occurs when the shell thickness varies. The right-hand side of equation (72) only vanishes when two conditions are met: (1) there is no toroidal source (so that the stress function H vanishes) and (2) the shell thickness is constant (so that the terms in B vanish). 3.5 Stresses Stresses can be computed from (w,F,H) and Ω by substituting equations (14), (18) and (52) into equa- tions (16): σ̂θθ = (O2 F +Ω+ 2O3H) + R(1− ν2) (∆′ − (1− ν)O2)w , σ̂ϕϕ = (O1 F +Ω− 2O3H) + R(1− ν2) (∆′ − (1− ν)O1)w , (73) σ̂θϕ = (−O3 F + (O2 −O1)H) + R(1 + ν) O3w . Stresses at the surface are obtained by setting ζ = h/2. 4 Flexure equations and their properties 4.1 Thin shell approximation The flexure equations derived in section 3 already include several assumptions of the thin shell theory, but not yet the first one stating that the shell is thin. Of course, the three other assumptions can be seen to be consequences of the first one [see Kraus, 1967, chap. 2.2], but we have not imposed in a quantitative way the thinness condition on the equations. We thus impose the limit of small h/R or, equivalently, the limit of large ξ (defined by equation (22)) on the flexure equations for (H,F,w, S, T ). This procedure amounts to expand η ≈ 1 − 1/ξ (neglecting terms in 1/ξ wherever appropriate) and to neglect the derivatives of η in equations (49), (58), (66), (71) and (72). We thus obtain the final flexure equations for the displacements (w, S, T ) and for the stress functions (F,H): ∆′H = −V + V0 , (74) ∆′ (D∆′w)− (1 − ν)A(D ;w) +R3∆′F = −R4 q − 2R3Ω+ R ∆Ω , (75) ∆′ (α∆′F )− (1 + ν)A(α ;F )− 1 ∆′w = −(1− ν)∆′(αΩ) + 2(1 + ν)B(α ;H) , (76) ∆S = Rα (1 − ν) (∆′F + 2Ω) + 1 ∆′w − 2w , (77) ∆∆′ T = 2R (1 + ν) (B (α ;F )− 2A (α ;H) + ∆′ (α∆′H)) . (78) Recall that the differential operators ∆′, A and B are defined by equations (32), (33) and (37). The potential pairs (Ω, V ) and (S, T ) are related to the tangential load and displacement by equations (45) and (67), respectively. In the second equation, the term R3∆Ω/ξ has been kept since it could be large if Ω has a short wavelength. For the same reason, the term ∆′w/ξ has been kept in the fourth equation. In the third and fifth equations, the terms depending on H belong to the right-hand sides since they can be considered as a source once H has been calculated from the first equation. The same can be said of the terms depending on w and F in the fourth and fifth equations: they are supposed to be known from the simultaneous resolution of the second and third equations. The difficulty in solving the equations thus lies with the two flexure equations for w and F which are linear with non-constant coefficients (all other equations are linear - in their unknowns - with constant coefficients). Once these two core equations have been solved, all other quantities are easily derived from them. The bending rigidity D, defined by equation (21), characterizes the bending regime: the shell locally bends in a similar way as a flat plate undergoing small deflections with negligible stretching. The pa- rameter α, defined by equation (61), is the reciprocal of the extensional rigidity K and characterizes the membrane regime of the shell in which bending moments are negligible and the load is mainly supported by internal stresses tangent to the shell. Since D and K are respectively proportional to the third and first power of the shell thickness, the membrane regime (in which the D-depending terms are neglected) is obtained in the limit of an extremely thin shell. This observation and the fact that such a shell, lacking rigidity, cannot support bending moments justify the use of the term ‘membrane’. The weight of the various terms in the equations depends on two competing factors: the magnitude of the coefficient multiplying the derivative and the number of derivatives. On the one hand, a coefficient containing D will be smaller than a coefficient containing α−1 since αD/R2 ∼ ξ−1 is a small number (see equation (22)). On the other hand, a large number of derivatives will increase the weight of the term if the derived function has a small wavelength. The transition between the membrane and the bending regimes thus depends on the wavelength of the load: if the load has a large wavelength (with respect to the shell radius), the flexure of the shell will be well described by equations without the terms depending on D (see section 5.1), whereas the flexure under loads of small wavelength is well described by equations keeping only the D-depending terms with the largest number of derivatives (see section 5.2). Stress functions are associated with membrane stretching and give negligible contributions in the bending regime. Formulas (52) show that F (respectively H) plays the role of potential for the stress resultants in the membrane regime when the load is transversal (respectively tangential toroidal). There is no stress function associated with the tangential consoidal component of the load because it is identical to Ω, the consoidal potential of the load. The variation of the shell thickness has two effects. First, it couples the spherical harmonic modes that are solutions to the equations for a shell of constant thickness. Second, the toroidal part remains intertwined with the transversal and consoidal parts, whereas it decouples if the thickness is constant. For example, the toroidal load is a source for the transverse deflection through the term B in the third flexure equation. Furthermore, the stress function F is a source for the toroidal displacement in the fifth flexure equation. For numerical computations, the flexure equations (74)-(78) obtained in the thin shell approximation (to which we can add the equations (73) for the stresses) are adequate. For this purpose, it is not useful to keep small terms in 1/ξ since the theory rests on assumptions only true for a thin shell. In the rest of the article, we will continue to work with the equations (49), (58), (66), (71) and (72) for more generality. 4.2 Covariance Because of their tensorial form, the flexure equations (74)-(78) are covariant; this is also true of the more general equations (49), (58), (66), (71) and (72). This property means that their form is valid in all systems of coordinates on the sphere, though the tensor components (the covariant derivatives of scalar functions, the metric and the antisymmetric tensor) will have a different expression in each system. The scalar functions will also have a different dependence on the coordinates in each system. The covariance of the final equations was expected. We indeed started with tensorial equations in section 2.1; their restriction to the sphere in principle respected the tensoriality with respect to changes of coordinates on the sphere. However the covariance of the two-dimensional theory was not made explicit until we obtained the final equations. This property is thus a strong constraint on the form of the solution and a check of its validity, though only necessary and not sufficient (other covariant terms may have been ignored). Another advantage of the covariant form is the facility to express the final equations in different systems of coordinates (even non-orthogonal ones) with the aim of solving them. For example, the fi- nite difference method in spherical coordinates (θ, ϕ) suffers from a very irregular grid and from pole singularities. These problems can be avoided with the ‘cubed sphere’ coordinate system [Ronchi et al., 1996]. Operators including covariant derivatives (here the Laplacian and the operators A and B) can ex- pressed in any system of coordinates whose metric is known. Christoffel symbols and tensorial differential operators can be computed with symbolic mathematical software. 4.3 Degree one Displacements of degree one require special consideration. All differential operators acting on (w,F,H) (that is ∆′, A and B) in the flexure equations (58) and (66) can be expressed in terms of the operators Dij = ∇i∇j+gij (see section 3.2). The Dij have the interesting property that they give zero when acting on spherical harmonics of degree one (see equation (30)). Therefore the degree-one terms in the spherical harmonic expansion of w vanish from the flexure equations. The magnitude of the transverse deflection of degree one neither depends on the load nor on the elastic properties of the spherical shell. More generally, the homogeneous (q = Ω = V = 0) flexure equations (49), (58) and (66) are satisfied by w and F being both of degree one. According to equations (71)-(72), the corresponding tangential displacement is constrained by ∆S = −2w and ∆∆′T = 0, so that S and T are also of degree one, with S = w. These conditions lead to vanishing strains (see equations (69)) which indicate a rigid displacement. The total displacement is then given by u = w r̂+ ∇̄w + ∇̄ × (T r̂) , where w and T are of degree one. In Appendix 7.6, we show that the first two terms represent a rigid translation whereas the last term represents a rigid rotation. As expected, stresses vanish for such displacements (see equation (73)). This freedom in translating or rotating the solution reflects the freedom in the choice of the reference frame (in practice the reference frame is centered at the center of the undeformed shell). The same freedom of translation is also found in the theory of deformations of a spherical, radially stratified, gravitating solid [e.g. Farrell , 1972; Greff-Lefftz and Legros , 1997; Blewitt , 2003]. What can we say about degree-one loading? Let us first examine what happens when the flexure equations are projected on the spherical harmonics of degree one. Since the operator ∆′ annihilates the degree one in any spherical harmonic expansion, terms of the form ∆′f vanish when they are projected on the spherical harmonics of degree one. Moreover, the operators A and B also vanish in this projection since they do not contain a degree-one term (see equations (116)-(117) of Appendix 7.8). Therefore the degree-one component of the flexure equation (66) is identically zero, whereas the degree-one component of the flexure equation (58) is q + ∇̄ · qT Yi = 0 (i = x, y, z) , (79) where dω = sin θ dθ dϕ and Yi are the real spherical harmonics of degree one. The integral is taken over the whole spherical surface. We have used the relation ∆Ω = −R ∇̄ ·qT derived from equations (45) and (106). The first term in the integrand of equation (79) is the projection on the Cartesian axes (x̂, ŷ, ẑ) of the vector field q r̂: (q Yx, q Yy, q Yz) = (q r̂ · x̂, q r̂ · ŷ, q r̂ · ẑ) , where we have used formulas (109) for the spherical harmonics. The second term in equation (79) can be rewritten with the identity (102) and Gauss’ theorem (107): ∇̄ · qT Yi = − dω qT · ∇̄Yi (i = x, y, z) , where the Yi are considered as scalars. Since qT is orthogonal to r̂, the integrand is the projection of qT on the Cartesian axes (x̂, ŷ, ẑ): qT · ∇̄Yx,qT · ∇̄Yy,qT · ∇̄Yz = (qT · x̂,qT · ŷ,qT · ẑ) , where we have used formulas (110) for the gradients of the spherical harmonics. Recalling that the transverse load q was defined positive towards the center of the sphere (see equation (26)), we define a total load vector q = −q r̂+qT . With the above results, we can rewrite the degree-one projection (79) as dω (q · x̂ ,q · ŷ ,q · ẑ) = (0, 0, 0) , (80) which means that the integral (over the whole spherical surface) of the projection on the coordinate axes of the total load vector vanishes. This result is the consequence of the static assumption in the equations of motion (7), since a non-zero sum of the external forces would accelerate the sphere. In practice, degree-one loads on planetary surfaces are essentially due to mass redistribution [Greff-Lefftz and Legros , 1997] and have a tangential consoidal component (for example the gravitational force is not directed to- ward the center of figure of the shell). If the shell thickness is variable, a non-zero Ω of degree one will induce degrees higher than one in w and in S. If the shell thickness is constant, the degree-one load drops from the flexure equations (58)-(66) and w is not affected. However, the degree-one Ω generates (assuming a constant shell thickness) a degree-one tangential displacement through equation (71), so that S 6= w. Whether the shell thickness is variable or not, a degree-one Ω thus generates a total displacement which is not only a translation but also a tangential deformation [Blewitt , 2003], in which case stresses do not vanish as shown by equations (73). 5 Limit cases 5.1 Membrane limit A shell is in a membrane state of stress if bending moments (Mθ,Mϕ,Mθϕ) can be neglected, in analogy with a membrane which cannot support bending moments. Equations (19) show that this is true if the bending rigidity vanishes: D = 0. Consistency with equation (22) imposes the limit of infinite ξ (or η = 1). With these approximations, the first flexure equation (58) for (w,F,H) becomes ∆′F = −Rq − 2Ω . (81) The second flexure equation (66) for (w,F,H) becomes ∆′w = ∆′ (α∆′F )− (1 + ν)A(α ;F )− 2(1 + ν)B(α ;H) + (1− ν)∆′(αΩ) . (82) If q is independent of w, F and w can be successively determined with spherical harmonic transforms from equations (81) and (82) (though the right-hand side of the latter equation must be computed with another method). If q has a linear dependence in w (such as when the sphere is filled with a fluid), w can be eliminated between equations (81) and (82), so that F and w can also be computed in succession (however the equation for F cannot be solved by a spherical harmonic transform). H is supposed to be known since equation (49) is not modified and can be solved with spherical harmonics. The stresses are obtained from equations (73) with the additional approximation of neglecting the term in ζ: (σ̂θθ, σ̂ϕϕ, σ̂θϕ) = (O2F +Ω+ 2O3H,O1F +Ω− 2O3H,−O3F + (O2 −O1)H) . Bending moments play a small role if the load has a large wavelength. In practice, the threshold at which bending moments become significant can be evaluated from the constant thickness equation for w (see section 5.3.1). One must compare the magnitudes of the terms in D and 1/α in the left-hand side of equation (88). Bending moments are negligible (i.e. the term in D) if the spherical harmonic degree ℓ of the transverse displacement w is such that ℓ . k , (83) where k = (12(1− ν2))1/4 ≈ 1.8 if we take ν = 1/4. This threshold is about 10 for a planet with a radius of 3400 km and a lithospheric thickness equal to 100 km. Flexure equations in the membrane limit of a shell with constant thickness have been used by Sleep and Phillips [1985] to study the lithospheric stress in the Tharsis region of the planet Mars. 5.2 Euclidean limit The equation for the deflection of a rectangular plate with variable thickness has been derived by Timoshenko and Woinowsky-Krieger [1964, p. 173]. We will check here that the Euclidean limit of our equations gives the same answer. Let us define the coordinates (x, y) = (Rϕ,R θ′), where θ′ = θ/2 − θ is the latitude. We work in a small latitude band around the equator (so that θ′ ≪ 1) and in the limit of large spherical radius R and large ξ (η = 1). Under this change of coordinate, each derivative introduces a factor R so that terms with the largest number of derivatives dominate. In particular, covariant derivatives can be approximated by usual derivatives. The surface Laplacian (104) can be approximated as follows: ∆ ≈ R2 ≡ R2∆e . We assume that there are no tangential loads (Ω = V = H = 0). The flexure equations (75)-(76) for the transverse displacement become: R4∆e (D∆ew)− (1 − ν)R4 Ae(D ;w) +R5∆eF = −R4 q , (84) R4 ∆e (α∆eF )− (1 + ν)R4 Ae(α ;F )−R ∆ew = 0 , (85) where the operator Ae is defined by Ae(a ; b) = (∆e a) (∆e b)− a,i,j b,i,j Equation (85) gives a relation between the magnitudes of F and w: O(R3 ∆eF ) ∼ O(w/α) . In the large R limit, equation (84) thus becomes ∆e (D∆ew) − (1− ν)Ae(D ;w) = −q , which is the equation derived by Timoshenko and Woinowsky-Krieger [1964]. This equation has been used by Stark et al. [2003], Kirby and Swain [2004] and Pérez-Gussinyé et al. [2004] for the local analysis of the lithosphere of the Earth. Its one-dimensional version, in which Ae vanishes, has been used by Sandwell [1984] to describe the flexure of the oceanic lithosphere on Earth and by Stewart and Watts [1997] to model the flexure at mountain ranges. 5.3 Shell with constant thickness 5.3.1 Displacements If the thickness of the shell is constant, the toroidal part of the tangential displacement decouples. The terms in B indeed drop from the flexure equations (58) and (66) so that the equations for (w,F ) depend only on (q,Ω): ηD∆′∆′w − (1− ν) ηD∆′w + ηR3 ∆′F = −R4 q +R3 ((1 − η)∆Ω− 2ηΩ) , (86) ∆′∆′F − (1 + ν)∆′F − 1 ∆′w = −(1− ν)∆′Ω , (87) where we used the property A(a ; b) = a∆′b valid for constant a. We eliminate F from these equations and obtain a sixth order equation relating w to (q,Ω): ηD∆∆′∆′w + ∆′w = −R4 (∆′ − 1− ν) q +R3 1 + ξ ∆′ − 1− ν ∆Ω . (88) The elimination of ∆′F between equations (77) and (86) gives an equation relating the consoidal displacement potential S to (w, q,Ω): ∆S = − 1 1 + ν 1 + ξ ∆∆′w − 2w − (1 − ν)R2α q + 1− ν 1 + ξ Rα∆Ω . Terms not including a Laplacian can be eliminated with equation (88), so that we obtain an explicit solution for S in terms of (w, q,Ω): 1 + ξ 1− ν2 (∆ + 1 + ν)∆′w + w +R2α q − Rα 1 + ξ (∆− ξ(1 + ν))Ω , (89) where the integration constant has been set to zero. Equations (49) and (72) give an equation for the toroidal displacement potential: ∆′ T = −2ηRα (1 + ν)V , (90) where the integration constant has been set to zero. We have assumed that ∆V 6= 0, otherwise we get ∆′T = 0. The differential equations given in this section can be solved with spherical harmonics so that the co- efficients of the spherical harmonic expansions of (w, S, T ) can be expressed in terms of the corresponding coefficients of the loads (q,Ω, V ) (see Kraus [1967], Turcotte et al. [1981], Banerdt [1986]). 5.3.2 Comparison with the literature We now compare our equations for a shell of constant thickness with those found in the literature. The formulas of Banerdt [1986] (taken from the work of Vlasov [1964]) are the most general: ∆3 + 4∆2 (∆ + 2)w = −R4 (∆ + 1− ν) q +R3 ∆− 1− ν ∆Ω , (91) (∆ + 2)χ = Dξ(1− ν) ∆V . (92) Banerdt’s notation is slightly different: his formulas are obtained with the substitutions ξ → ψ, Ω → RΩ and V → RV . The normal rotation χ is proportional to the radial component (in a normalized basis) of the curl of the tangential displacement: (∇× v)r̂ . The curl ∇× v is related to our surface curl (103) by ∇× v = ∇̄ × v + csc θ (sin θv̂ϕ),θ − v̂θ,ϕ With the formulas (68) and (104), we get ∇× v = −∆T r̂ so that equation (92) becomes ∆′ T = −2Rα(1 + ν)V . (93) We see that Banerdt’s equations (91) and (93) coincide with our equations (88) and (90) in the limit of large ξ (η = 1), with one exception: the bending term for w is written D(∆3 + 4∆2)w instead of D∆∆′∆′w = (∆3 + 4∆2 + 4∆)w. This error has propagated in many articles and is of consequence for the degree-one harmonic component, since it violates the static assumption and spoils the translation invariance discussed in section 4.3. The impact on higher degrees is negligible. Because of this mistake, many authors give a separate treatment to the first harmonic degree. Banerdt also gives formulas for the tangential displacements in terms of consoidal and toroidal scalars (A,B) corresponding to our scalars (S, T ): his formula (A10) is equivalent to our equation (89) in the limit of large ξ. If we ignore temperature effects, Kraus’ first equation for (w,F ) is equivalent to our equation (86) in the limit of large ξ, whereas his second equation for (w,F ) is equivalent to the combination eq.(87)+ 1+ν eq.(86) in the limit of large ξ [see Kraus , 1967, eq. 6.54h and 6.55d]. Note that the definition of Kraus’ stress function F [Kraus, 1967, p. 243] differs from ours: FKraus = F − k(1 − ν) with k = 1. This freedom of redefining F for arbitrary k remains as long as D is constant. The flexure equation for w, equation (88), is unaffected so that the solution for w is unchanged. In the final step, Kraus makes a mistake when combining the two equations for (w,F ) and thus obtains a flexure equation for w with the same error as in equation (91). Kraus does not include toroidal loading. The flexure equation of Turcotte et al. [1981] is taken from Kraus [1967] without the tangential loading and is the same as equation (91) with Ω = 0. The flexure equation of Brotchie and Silvester [1969] is given in our notation by D∆2w + w = −R4 q , (94) where q includes their term γw describing the response of the enclosed liquid. This equation can be obtained from our equation (88) as follows: keep only the derivatives of the highest order in each term, set Ω = 0, take the limit of large ξ (η = 1) and integrate. Brotchie and Silvester choose to work in the approximation of a shallow shell and with axisymmetrical loading, solving their equation in polar coordinates with Bessel-Kelvin functions. The reduction to fourth order in equation (94), the shallow shell approximation and the axisymmetrical assumption are not justified nowadays since the full equation (88) can be quickly solved with computer-generated spherical harmonics. The contraction due to a transverse load of degree 0, w = −R2α(1 − ν)q/2, is equivalent to the radial displacement computed by Love in the limit of a thin shell [Love, 1944, p.142]. However ad- ditional assumptions about the initial state of stress and the internal density changes are necessary [Willemann and Turcotte, 1982] so that the degree 0 is usually excluded from the analysis. 5.3.3 Breakdown of the third assumption of thin shell theory With the spherical harmonic solutions of the equations for a shell of constant thickness, it is possible to check the thin shell assumption stating that the transverse normal stress is negligible with respect to the tangential normal stress. The magnitude of the former can be estimated by the load q (see definition (26)) whereas the magnitude of the latter can be approximated with formulas (73) evaluated on the outer surface: (σ̂θθ + σ̂ϕϕ) |h ∆′F − Eh 4R2(1− ν) ∆′w . where we have assumed the absence of tangential loads (Ω = 0) and the limit of large ξ. We can relate σT to q by using the solution in spherical harmonics of equations (87) and (88). Since the thin shell assumption is expected to fail for a load of sufficiently small wavelength, we assume that the spherical harmonic degree ℓ is large. Assuming ℓ≫ 1, we obtain (∆′F )ℓm ≈ wℓm , wℓm ≈ − 1 + ℓ ξ(1−ν2) qℓm , where the spherical harmonic coefficients are indexed by their degree ℓ and their order m. If the shell is not in a membrane state of stress (see equation (83)), ℓ2 > 2R/h so that σT can be approximated by (σT )lm ≈ ξ(1 + ν) qlm . The thin shell assumption holds if q < σT , that is if 3(1 + ν) or λ > 3(1 + ν) h , (95) where λ is the load wavelength (λ ≈ 2πR/ℓ). We have 3(1 + ν) ≈ 1.9 and 2π/ 3(1 + ν) ≈ 3.2 if we take ν = 1/4. This condition on λ is consistent with the transition zone between the thin and thick shell responses analyzed in Janes and Melosh [1990] and Zhong and Zuber [2000], but does not coincide with the constraint given in Willemann and Turcotte [1982], which is ℓ < 2π R/h (this last condition looks more like the threshold (83) for the membrane regime). Though the stress distribution is affected, the limit (95) on the degree ℓ is not important for the displacements, since they tend to zero at small wavelengths. Therefore the theory does not break down at short wavelength if one is interested in the computation of the gravity field associated to the transverse deflection of the lithosphere. 6 Conclusion The principal results of this article are the five flexure equations (74)-(78) governing the three displace- ments of the thin spherical shell and the two auxiliary stress functions. Stresses are derived quantities which can be obtained from equations (73). The shell thickness and Young’s modulus can vary, but Poisson’s ratio must be constant. The loads acting on the shell can be of any type since we extend the method of stress functions to include not only transverse and consoidal tangential loads, but also toroidal tangential loads. The flexure equations can be solved one after the other, except the two equations (75)-(76) for the transverse deflection w and the stress function F , which must be simultaneously solved. Tangential loading is usually neglected when solving for the deflection because of its small effect. In that case, it is sufficient to solve the two equations (75)-(76) with Ω = H = 0: ∆′ (D∆′w)− (1− ν)A(D ;w) +R3∆′F = −R4 q , ∆′ (α∆′F )− (1 + ν)A(α ;F )− 1 ∆′w = 0 . However tangential loading must be taken into account when computing stress fields [Banerdt , 1986]. In the long-wavelength limit (i.e. membrane regime), all equations can be solved one after the other because it is possible to solve for F before solving for the transverse deflection. If a small part of the shell is considered, the flexure equations reduce to the equations governing the deflection of a flat plate with variable thickness. If the shell thickness is constant, the flexure equations reduce to equations available in the literature which can be completely solved with spherical harmonics. Our rigorous treatment of the thin shell approximation has clarified the effect of the shell thickness on the flexure equations. We emphasize the need to use the correct form for the equations (without the common mistake in the differential operator acting on w) in order to have the correct properties for the degree-one deflection and degree-one load. We have also obtained two general properties of the flexure equations. First we have shown that there is always a toroidal component in the tangential displacement if the shell thickness is variable. Second we have proven that the degree-one harmonic components of the transverse deflection and of the toroidal component of the tangential displacement do not depend on the elastic properties of the shell. This property reflects the freedom under translations and rotations of the reference frame. Besides we have shown that degree-one loads are constrained by the static assumption but can deform the shell and generate stresses. This article was dedicated to the theoretical treatment of the flexure of a thin elastic shell with variable thickness. While the special case of constant thickness admits an analytical solution in terms of spherical harmonics, the general flexure equations must be solved with numerical methods such as finite differences, finite elements or pseudospectral methods. In a forthcoming paper, we will give a practical method of solution and discuss applications to real cases. Acknowledgments M. Beuthe is supported by a PRODEX grant of the Belgian Science Federal Policy. The author thanks Tim Van Hoolst for his help and Jeanne De Jaegher for useful comments. Special thanks are due to Patrick Wu for his constructive criticisms which helped to improve the manuscript. 7 Appendix 7.1 Covariant, contravariant and normalized components Tensors can be defined by their transformation law under changes of coordinates. The two types of tensor components, namely covariant and contravariant components, transform in a reciprocal way under changes of coordinates. Tensor components cannot be expressed in a normalized basis: the space must have a coordinate vector basis (for contravariant components) and a dual basis (for covariant components) which are not normalized. The only exception is a flat space with Cartesian coordinates, where covariant, contravariant and normalized components are identical. Since the metric is the scalar product of the elements of the coordinate vector basis, the covariant components are related to components defined in a normalized basis (written with a hat) by gii ûi , whereas the relation for contravariant components is ûi . The normalized Cartesian basis (x̂, ŷ, ẑ) is related to the normalized basis for spherical coordinates (r̂, θ̂, ϕ̂) by x̂ = cos θ cosϕ θ̂ − sinϕ ϕ̂+ sin θ cosϕ r̂ , ŷ = cos θ sinϕ θ̂ + cosϕ ϕ̂+ sin θ sinϕ r̂ , (96) ẑ = − sin θ θ̂ + cos θ r̂ . 7.2 Covariant derivatives Usual derivatives are indicated by a ‘comma’: vi,j = Covariant derivatives (defined below) are indicated by a ‘bar’ or by the operator ∇i: vi|j = ∇j vi . The former notation emphasizes the tensorial character of the covariant derivative since the covariant derivative adds a covariant index to the vector. The latter notation is more adapted when we are interested by the properties of the operator. The covariant derivative of a scalar function f is equal to the usual derivative, f|i = f,i, and is itself a covariant vector: f|i = vi. Covariant derivatives on covariant and contravariant vector components are defined by vi|j = vi,j − Γkij vk , (97) vi |j = v ,j + Γ k , (98) where the summation on repeated indices is implicit. The symbols Γkij are the Christoffel symbols of the second kind [Synge and Schild , 1978]. Their expressions for the metrics used in this article are given in sections 7.3 and 7.4. Covariant differentiation of higher order tensors is explained in Synge and Schild [1978] but we only need the rule for a covariant tensor of second order: σij|k = σij,k − Γlik σlj − Γljk σil . If some of the indices of the tensor are contravariant, the rule is changed according to equation (98). The covariant derivatives of the metric and of the inverse metric are zero: gij|k = 0 and g 7.3 Three-dimensional spherical geometry The geometry of a thin spherical shell of average radius R can be described with coordinates θ, ϕ and ζ, respectively representing the colatitude, longitude and radial coordinates. The radial coordinate ζ is zero on the reference surface (i.e. the sphere of radius R) of the shell. The non-zero components of the metric are given by gθθ = (R + ζ) gϕϕ = (R + ζ) 2 sin2 θ , gζζ = 1 . The non-zero Christoffel symbols are given by θθ = −(R+ ζ) , Γζϕϕ = −(R+ ζ) sin2 θ , Γθζθ = Γ θζ = Γ ζϕ = Γ Γθϕϕ = − sin θ cos θ , ϕθ = Γ θϕ = cot θ . 7.4 Two-dimensional spherical geometry If θ and ϕ respectively represent the colatitude and longitude coordinates, the non-zero components of the metric on the surface of the sphere are given by gθθ = 1 , gϕϕ = sin 2 θ . (99) The non-zero Christoffel symbols are given by Γθϕϕ = − sin θ cos θ , ϕθ = Γ θϕ = cot θ . The double covariant derivatives of a scalar function f are thus given by f|θ|θ = f,θ,θ , f|θ|ϕ = f|ϕ|θ = f,θ,ϕ − cot θ f,ϕ , f|ϕ|ϕ = f,ϕ,ϕ + sin θ cos θ f,θ . An antisymmetric tensor εij is defined by εij ≡ det gij ε̄ij , where ε̄ij is the antisymmetric symbol invariant under coordinate transformations: ε̄θϕ = −ε̄ϕθ = 1, ε̄θθ = ε̄ϕϕ = 0 (ε̄ij is usually called a tensor density; Synge and Schild [1978] call it a relative tensor of weight -1). The non-zero covariant components of εij are given for the metric of the spherical surface by εθϕ = −εϕθ = sin θ . The non-zero contravariant components, εij = gikgjlεkl, are given by εθϕ = −εϕθ = csc θ . The covariant derivative of the tensor εij is zero: εij|k = 0. 7.5 Gradient, divergence, curl and Laplacian Various differential operators on the surface of the sphere can be constructed with covariant derivatives. In this section, f and t are scalar functions defined on the sphere and v is a vector tangent to the sphere. Backus [1986] gives more details on surface operators and on Helmholtz’s theorem. As mentioned in Appendix 7.2, the covariant derivative of a scalar function f defined on the sphere is a covariant vector tangent to the sphere whose components are f,θ and f,ϕ. The surface gradient of f is the same vector with its components expressed in the normalized basis (θ̂, ϕ̂): ∇̄f = f,θ θ̂ + csc θ f,ϕ ϕ̂ . (100) The contraction of the covariant derivative with the components of a vector v yields a scalar: vi |i = v ,θ + cot θ v θ + vϕ,ϕ . The surface divergence is the corresponding operation on the vector with its components expressed in the normalized basis (θ̂, ϕ̂): ∇̄ · v = csc θ (sin θ v̂θ),θ + v̂ϕ,ϕ . (101) Since the result is a scalar, vi = ∇̄ · v. A useful identity is ∇̄ · (f v) = ∇̄f · v + f ∇̄ · v . (102) The contraction of the antisymmetric tensor εij with the covariant derivative of a scalar t yields the covariant components of a vector v: vi = g jk εik t,j . The components are given for the metric (99) by vθ = csc θ t,ϕ and vϕ = − sin θ t,θ. If t is considered as the radial component of the radial vector t = t r̂ (the covariant radial component is equal to the normalized one), vi are the non-zero covariant components of the three-dimensional curl of t, which is tangent to the sphere. This fact justifies the definition of the surface curl of t, which is equal to the vector v but with components given in the normalized basis (θ̂, ϕ̂): ∇̄ × t = csc θ t,ϕ θ̂ − t,θ ϕ̂ . (103) The contraction of the double covariant derivative acting on a scalar f defines the surface Laplacian: ∆f = gij f|i|j = f,θ,θ + cot θ f,θ + csc 2 θ f,ϕ,ϕ . (104) The surface Laplacian can also be seen as the composition of the surface divergence with the surface gradient: ∆f = ∇̄ · ∇̄f . According to Helmholtz’s theorem, a vector tangent to the sphere can be written as the sum of the surface gradient of a scalar f and the surface curl of a radial vector t r̂: v = ∇̄f + ∇̄ × (t r̂) . (105) While t is always called the toroidal scalar (or potential) for v, there is no standard terminology for f . Backus [1986] calls f the consoidal scalar for v. Some authors [e.g. Banerdt , 1986] call f the poloidal potential for v. The origin of this use lies in the theory of mantle convection, in which plate tectonics are assumed to be driven by mantle flow. Under the assumption of an incompressible mantle fluid, the velocity field of the fluid is solenoidal, i.e. its 3-dimensional divergence vanishes. In such a case, the velocity field can be decomposed into a poloidal part (∇×∇×(P r̂)) and a toroidal part (∇×(Q r̂)), where differential operators are 3-dimensional [Backus , 1986]. If the velocity field is tangent to the spherical surface, the poloidal component at the surface is also the consoidal component [Forte and Peltier , 1987]. However the fields for which we use Helmholtz’s theorem, i.e. the tangential surface load and the tangential surface displacement, do not belong to 3-dimensional solenoidal vector fields. We thus prefer to use the term ‘consoidal’. The surface divergence of v depends only on the consoidal scalar f : ∇̄ · v = ∆f . (106) The two-dimensional version of Gauss theorem is dω ∇̄ · v = 0 . (107) where dω = sin θ dθ dϕ and the integral is taken over the whole spherical surface. It can be proven with formula (101). 7.6 Rigid displacements At the surface of a sphere subjected to deformation, the displacement u of a point can be expressed with the help of Helmholtz’s theorem (105) in terms of three scalar functions (w, S, T ) depending on θ and ϕ: u = w r̂+ ∇̄S + ∇̄ × (T r̂) . (108) Strains (and stresses) vanish for rigid displacements. Equations (69) show that strains vanish when (w, S, T ) are of degree one, with S = w (recall that the operatorsOi annihilate the degree one). Assuming these conditions, we now show that utransl = w r̂ + ∇̄w represents a rigid translation whereas urot = ∇̄ × (t r̂) represents a rigid rotation of the sphere. We choose as basis the real spherical harmonics of degree one which form the components of the radial unit vector in Cartesian coordinates: (Yx, Yy, Yz) = (sin θ cosϕ, sin θ sinϕ, cos θ) = (x̂, ŷ, ẑ) · r̂ . (109) We need the surface gradient of the real spherical harmonics which can be computed with formulas (96) and (100): ∇̄Yx, ∇̄Yy, ∇̄Yz = (x̂− sin θ cosϕ r̂, ŷ − sin θ sinϕ r̂, ẑ− cos θ r̂) . (110) If the expansion of w in the degree-one basis is w = a Yx + b Yy + c Yz, then w r̂+ ∇̄w = a x̂+ b ŷ + c ẑ , so that utransl is indeed a rigid translation of the sphere. If the expansion of T in the degree-one basis is T = a′Yx + b ′Yy + c ′Yz, then ∇̄ × (T r̂) = a′ − sinϕ θ̂ − cos θ cosϕ ϕ̂ cosϕ θ̂ − cos θ sinϕ ϕ̂ + c′ sin θ ϕ̂ , so that urot includes a rigid rotation of the sphere, with (a ′, b′, c′) being the angles of rotation around the axes (x̂, ŷ, ẑ), respectively. Though urot seems to include a uniform radial expansion, one should recall that linearized strain-displacement equations are not valid for large displacements. Since strains vanish, the radial expansion is not physical and urot represents a pure rotation. Finite deformations are for example discussed in Love [1944][pp. 66-73] and Sokolnikoff [1956][pp. 29-33]. 7.7 Differential identities for the operators Oi The differential operators Oi defined by equations (15) satisfy differential identities useful when obtaining the flexure equations. They are special cases of differential identities valid in curved spaces. The presence of curvature makes the parallel transport of vectors path-dependent; this property quantifies the curvature of space and can be expressed as the lack of commutativity of the covariant derivatives of a vector v: vi|j|k − vi|k|j = Riljk vl , (111) where Riljk are the covariant components of the Riemann tensor. On the sphere, the Riemann tensor has only one independent component that is non-zero, Rθϕθϕ = − sin2 θ. Other components are related by the symmetries Rαβγδ = −Rβαγδ = −Rαβδγ = Rγδαβ. The substitution of f,i to vi in the commutation relation (111) provides two differential identities satisfied by double covariant derivatives acting on scalar functions: csc θ f|ϕ|ϕ − csc θ f|ϕ|θ,ϕ − cos θ f|θ|θ + sin θ f,θ = 0 , f|θ|θ,ϕ − f|ϕ|θ,θ − cot θ f|ϕ|θ + f,ϕ = 0 . The replacement in the above equations of the double covariant derivatives by the normalized dif- ferential operators (31) yields the following identities: (sin θO2f),θ − (O3f),ϕ − cos θO1f = 0 (I1) , (112) (sin θO3f),θ − (O1f),ϕ + cos θO3f = 0 (I2) . (113) These identities can also be directly checked with the definitions (15) of the operators Oi. The identities (I1)-(I2) can be differentiated to generate identities of higher order. A first useful identity is obtained from sin θ(I1),θ − (I2),ϕ = 0: csc2 θ sin2 θ (O2f),θ + (O1f),ϕ,ϕ − 2 sin θ (O3f),ϕ − cot θ (O1f),θ + 2O1f = ∆ ′f . (114) A second useful identity is obtained from (I1),ϕ + sin θ(I2),θ = 0: csc2 θ sin2 θ (O3f),θ − (O3f),ϕ,ϕ + sin θ ((O2 −O1) f),ϕ + cot θ (O3f),θ − 2O3f = 0 . (115) 7.8 No degree one in operators A and B We want to prove that the operators A and B defined by equations (33) and (37) do not have any degree-one term in their spherical harmonic expansion: dω A(a ; b)Y ∗1p = 0 (p = −1, 0, 1) , (116) dω B(a ; b)Y ∗1p = 0 (p = −1, 0, 1) , (117) where (a, b) are arbitrary scalar functions on the sphere, dω = sin θ dθ dϕ and the integral is taken over the whole spherical surface. This property is not a straightforward consequence of constructing A and B with Dij as a building block. Although A and B can be factored into terms without degree one (such as Dija or ∆′a), the product of the factors may contain degree-one terms in its spherical harmonic expansion. Without loss of generality, we can prove the above identities with the arguments (a, b) being spherical harmonics of given degree and order. The general result is then obtained by superposition. Let a and b be spherical harmonics of order m and n: a ∼ eimϕ and b ∼ einϕ (we will not use their harmonic degree in the proof). All derivatives with respect to ϕ in the operators A and B can then be replaced with the rules a,ϕ → ima and b,ϕ → inb. The integral over ϕ in equations (116)-(117) gives dϕ ei(m+n−p)ϕ = 2π δm+n−p,0 , so that the integral is zero unless n = p−m. First consider the case p = 0 (n = −m), that is the projection on the zonal spherical harmonic of degree one. We thus have to calculate dθ A0 and dθ B0 with A0 ≡ sin θ cos θA(a ; b) , B0 ≡ sin θ cos θ B(a ; b) . The trick consists in rewriting the integrands as total derivatives: cos2 θ a,θ b,θ + cot θ sin2 θ −m2 (ab),θ + sin2 θ +m2 csc2 θ cos 2θ B0 = −im cos θ a,θ b,θ + sin θ a b,θ − csc θ cos2 θ a,θ b+ cos θ a b,θ,θ The sought integrals are thus given by dθ A0 = a,θ b,θ −m2 cot θ (ab),θ +m2 csc2 θ ab dθ B0 = −im cos θ a,θ b,θ − csc θ a,θ b+ cos θ a b,θ,θ where we have dropped the terms containing at least one power of sin θ which vanish at the limits; we have also replaced cos2 θ and cos 2θ by their value at the limits. The remaining terms can be evaluated by recalling the dependence in sin θ of the spherical harmonics: a = (sin θ)|m| a0 and b = (sin θ) |m| b0, where a0 and b0 are polynomials in cos θ. The only non-zero terms at the limits of the integrals are those for |m| = 1, in which case we have at the limits: a,θb,θ = a0b0, cot θ (ab),θ = 2a0b0, csc2 θ ab = a0b0, csc θ a,θb = cos θ a0b0, ab,θ,θ = 0. However these terms cancel in the sums so that the integrals vanish for all m. This completes the proof for the case p = 0. Now consider the case p = ±1 (n = −m ± 1), that is the projections on the sectoral spherical harmonics of degree one. We thus have to calculate dθ A±1 and dθ B±1 with A±1 ≡ sin2 θA(a ; b) , B±1 ≡ sin2 θB(a ; b) . We again write the integrands as total derivatives: A±1 = sin θ cos θ a,θ b,θ − sin θ cos θ +m2 (ab),θ − cos2 θ ∓ 2m a,θ b + sin2 θ a b,θ + 2m(m∓ 1) cot θ ab B±1 = −i (m∓ 1) sin θ a,θ b,θ +m sin θ a b,θ,θ − (m∓ 1) cos θ a,θ b −m cos θ a b,θ −m(1∓m) csc θ ab The sought integrals are thus given by dθ A±1 = −m2 (ab),θ − (1∓ 2m) a,θ b+ 2m(m∓ 1) cot θ ab dθ B±1 = i (m∓ 1) cos θ a,θ b+m cos θ a b,θ +m(1∓m) csc θ ab where we have dropped the terms containing at least one power of sin θ and replaced cos2 θ by its value at the limits. The remaining terms can be evaluated as in the case p = 0, but with a = (sin θ)|m| a0 and b = (sin θ)|m∓1| b0. All terms give zero at the limits of the integrals for all values of m. 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Introduction Fundamental equations of elasticity Three-dimensional elasticity theory Spherical shell Assumptions of the thin shell theory Strain-displacement equations Hooke's law Equilibrium equations Resolution Available methods Differential operators Transverse displacement Resolution of the equations of equilibrium Compatibility relation Tangential displacements Stresses Flexure equations and their properties Thin shell approximation Covariance Degree one Limit cases Membrane limit Euclidean limit Shell with constant thickness Displacements Comparison with the literature Breakdown of the third assumption of thin shell theory Conclusion Appendix Covariant, contravariant and normalized components Covariant derivatives Three-dimensional spherical geometry Two-dimensional spherical geometry Gradient, divergence, curl and Laplacian Rigid displacements Differential identities for the operators Oi No degree one in operators A and B

arXiv:0704.1628v1 [cond-mat.mes-hall] 12 Apr 2007 Detection of single electron spin resonance in a double quantum dot F. H. L. Koppens,∗ C. Buizert, I. T. Vink, K.C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen Spin-dependent transport measurements through a double quantum dot are a valuable tool for detecting both the coherent evolution of the spin state of a single electron as well as the hybridization of two-electron spin states. In this paper, we discuss a model that describes the transport cycle in this regime, including the effects of an oscillating magnetic field (causing electron spin resonance) and the effective nuclear fields on the spin states in the two dots. We numerically calculate the current flow due to the induced spin flips via electron spin resonance and we study the detector efficiency for a range of parameters. The experimental data are compared with the model and we find a reasonable agreement. A. Introduction Recently, coherent spin rotations of a single electron were demonstrated in a double quantum dot device [1]. In this system, spin-flips of an electron in the dot were in- duced via an oscillating magnetic field (electron spin res- onance or ESR) and detected through a spin-dependent transition of the electron to another dot, which already contained one additional electron. This detection scheme is an extension of the proposal for ESR detection in a sin- gle quantum dot by Engel and Loss [2]. Briefly, the device can be operated (in a spin blockade regime [3]) such that the electron in the left dot can only move to the right dot if a spin flip in one of the two dots is induced via ESR. From the right dot, the electron exits to the right reser- voir and another electron enters the left dot from the left reservoir. A continuous repetition of this transition will result in a net current flow. Compared to the single dot detection scheme [2], us- ing the double-dot as the detector has two major advan- tages. First, the experiment can be performed at a lower static magnetic field and consequently with lower, tech- nically less demanding, excitation frequencies. Second, the spin detection is rather insensitive to unwanted oscil- lating electric fields, because the relevant dot levels can be positioned far from the Fermi energies of the leads. These electric fields are unavoidably generated together with the oscillating magnetic field as well. The drawback of the double-dot detector is that spin detection is based on the projection in the two-electron singlet-triplet basis, while the aim is to detect single spin rotations. However, this detection is still possible be- cause the electrons in the two dots experience different effective nuclear fields. This is due to the hyperfine inter- action of the electron spins with the (roughly 106) nuclear spins in the host semiconductor material of each quan- tum dot [4–11]. In order to provide more insight in this double-dot ESR detection scheme for single spin rota- tions, it is necessary to analyze the coherent evolution of the two-electron spin states together with the transitions ∗Electronic address: f.h.l.koppens@tudelft.nl; Kavli Institute of NanoScience Delft, P.O. Box 5046, 2600 GA Delft, The Netherlands in the transport cycle. In this paper, we discuss a model that describes the transport cycle in the spin blockade regime while includ- ing the coherent coupling between the two dots, and the influence of the static and oscillating magnetic field to- gether with the effective nuclear fields on the electron spin states. The aim is to understand how effectively single spin resonance will affect the measured quantity in the experiment, namely the current flow in the spin blockade regime. The organization of this paper is as follows. First, we will explain the transport cycle and the mechanism that causes spin blockade. Next, we will briefly discuss the static system Hamiltonian and the mixing of the two-electron spin states by the effective nuclear field. Then we add an oscillating magnetic field to this Hamiltonian, that forms -together with the double dot tunnelling processes- the basis of the rate equations that describe how the density matrix of the two-electron spin states evolves in time. The current flow, calculated from the steady state solution of the density operator, is then analyzed for different coherent coupling values, magnitudes of the oscillating magnetic field, in combina- tion with different effective nuclear fields in the two dots. This provides further insight in the optimal conditions for spin-flip detection with a double quantum dot. B. Spin blockade In the spin-blockade regime, the double-dot is tuned such that one electron always resides in the right dot, and a second electron can tunnel from the left reservoir through the left and right dots, to the right reservoir [3]. This current-carrying cycle can be described with the occupations (m, n) of the left and right dots: (1, 1) → (0, 2) → (0, 1) → (1, 1). When an electron enters the left dot and forms a double-dot singlet state S11 with the electron in the right dot (S = |↑↓〉 − |↓↑〉, normalization omitted for brevity), it is possible for the left electron to move to the right dot, because the right dot singlet state S02 is energetically accessible. Next, one electron tunnels from the right dot to the right lead and another electron can again tunnel into the left dot. If, however, the two electrons form a double-dot triplet state T11, the left electron cannot move to the right dot, as the right http://arxiv.org/abs/0704.1628v1 dot triplet state T02 is much higher in energy (due to the relatively large exchange splitting in a single dot). The electron can also not move back to the lead and therefore further current flow is blocked as soon as any of the (double-dot) triplet states is formed (Fig. 1a,b). Spin blockade only occurs if at least one of the eigen- states of the system Hamiltonian is a pure triplet state. If processes are present that induce transitions from all the three (1,1) triplet states to the (1,1) singlet state, spin blockade is lifted and a current will flow. As we will see below, the presence of the nuclear spins in the host semiconductor can give rise to these kind of transitions. This can be seen most easily by adding the effect of the hyperfine interaction to the system Hamiltonian. C. System Hamiltonian The system Hamiltonian is most conveniently writ- ten in the two-electron singlet-triplet basis with the quantization-axis in the z-direction. The basis states are S11, T 11, T 11, T 11 and S02. The subscript m,n denotes the dot occupancy. We exclude the T02 state from the model, because this state is energetically inaccessible and therefore does not play an important role in the trans- DLR>>t DLR=0 S - S11 02 S - S11 02 -10 -5 0 5 10 DLR/t b) a) FIG. 1: a) A schematic of the double dot and the electro- chemical potentials (energy relative to the (0,1) state) of the relevant two-electron spin states. For ∆LR > t, transitions from the S11 state to the S02 state are possible via inelastic relaxation with rate Γin. Spin blockade occurs when one of the T i11 states is occupied. b) Similar schematic for ∆LR = 0, where the singlet states are hybridized. Also in this case, spin blockade occurs when one of T i11 states is occupied. c) Energy levels as a function of detuning. At ∆LR = 0, the singlet states hybridize into bonding and anti-bonding states. The splitting between the triplets states corresponds to the Zeeman energy gµBBext. port cycle. Furthermore, we neglect the thermal energy kT in the description, which is justified when the bias over the two dots is much larger than kT . The system Hamiltonian is given by H0 = − ∆LR|S02〉〈S02|+ t |S11〉〈S02|+ |S02〉〈S11| − gµBBext |T+11〉〈T 11| − |T 11〉〈T , (1) where ∆LR is the energy difference between the |S11〉 and |S02〉 state (level detuning, see Fig.1a), t is the tun- nel coupling between the |S11〉 and |S02〉 states, Bext the external magnetic field in the z-direction and Sz L(R) the spin operator along z for the left (right) electron. The eigenstates of the Hamiltonian (1) for finite external field are shown in figure 1c. For ∆LR < t, the tunnel coupling t causes an anti-crossing of the |S11〉 and |S02〉 states. For ∆LR < 0, transport is blocked by Coulomb blockade (i.e. the final state |S02〉 is at a higher energy than the initial state S11). For ∆LR ≥ 0, transport will be blocked when one of the three triplet states becomes occupied (spin blockade). In Fig.1a and b, we distinguish two regimes: ∆LR > t where the (exchange) energy splitting between T 011 and S11 is negligibly small and transitions from S11 to S02 occur via inelastic relaxation with rate Γin and the energy. A different regime holds for ∆LR < t, where S11 is coherently coupled with S02 giving rise to a finite (ex- change) splitting between T 011 and the hybridized singlet states. We will return to this distinction in the discussion below. D. Singlet-triplet mixing by the nuclear spins The effect of the hyperfine interaction with the nuclear spins can be studied [12] by adding a static (frozen) ef- fective nuclear field BLN (B N ) at the left (right) dot to the system Hamiltonian: Hnucl = −gµB N · SL +BRN · SR = −gµB(BLN −BRN ) · (SL − SR)/2 −gµB(BLN +BRN ) · (SL + SR)/2. (2) For the sake of convenience, we separate the inhomoge- neous and homogeneous contribution, for reasons which we will discuss later. Considering the nuclear field as static is justified since the tunnel rates and electron spin dynamics are expected to be much faster than the dy- namics of the nuclear system [10, 13, 14]. Therefore, we will treat the Hamiltonian as time-independent. The effect of nuclear reorientation will be included later by ensemble averaging. We will show now that triplet states mix with the S11 state if the nuclear field is different in the two dots (in all three directions). This mixing will lift spin blockade, detectable as a finite current running through the dots for ∆LR ≥ 0. The effective nuclear field can be decomposed -30 -20 -10 0 10 20 30 Magnetic field (mT) Bext=0 B » Bext ND -10 -5 0 5 10 10 S02 a +b +g +kS T T T11 11 11 11 -10 -5 0 5 10 a +kS T11 11 DLR/t DLR/t FIG. 2: a) Observed current flow in the inelastic transport regime (gµB∆LR ≫ t) due to singlet-triplet mixing by the nuclei. b) Electrochemical potentials in the presence of Hnucl (t ∼ ∆BN ). Singlet and triplet eigenstates are denoted by red and blue lines respectively. Hybridized states (of sin- glet and triplet) are denoted by dotted purple lines. For gµBBext ≫ t, gµB∆BN , the split-off triplets (T and T− ) are hardly perturbed and current flow is blocked when they be- come occupied. Parameters: t = 0.2µeV, gµBBN,L=(0.1,0,- 0.1)µeV, gµBBN,R=(-0.1,-0.2,-0.2)µeV and gµBBext=2µeV. in a homogeneous and an inhomogeneous part (see right- hand side of (2)). The homogeneous part simply adds vectorially to the external field Bext, changing slightly the Zeeman splitting and preferred spin orientation of the triplet states. The inhomogeneous part ∆BN ≡ BLN − N on the other hand couples the triplet states to the singlet state, as can be seen readily by combining the spin operators in the following way SxL − SxR = |S11〉〈T−11| − |S11〉〈T 11| + h.c. i|S11〉〈T−11| − i|S11〉〈T 11| + h.c. SzL − SzR = |S11〉〈T 011|+ |T 011〉〈S11| . (3) The first two expressions reveal that the inhomogeneous field in the transverse plane ∆BxN , ∆B mixes the |T+11〉 and |T−11〉 states with the |S11〉. The longitudinal com- ponent ∆BzN mixes |T 011〉 with |S11〉 (third expression). The degree of mixing between two states will depend strongly on the energy difference between them [5]. In the case of gµBBext, t < gµB 〉, the three triplet states are close in energy to the |S11〉 state. Their intermixing will be strong, lifting spin blockade. For gµBBext ≫ t, gµB 〉 the |T+11〉 and |T 11〉 states are split off in energy by an amount of gµBBext. Con- sequently the perturbation of these states caused by the nuclei will be small. Although the |T 011〉 remains mixed with the |S11〉 state, the occupation of one of the two split-off triplet states can block the flow through the sys- The effect of nuclear mixing is shown in Fig. 2 [5]. The observed current flow through the system is typi- cally in the order of a few hundreds of fA (Fig. 2a). At zero field, where the mixing is strongest, the current flow is largest. Increasing the field gradually restores spin blockade. Fig. 2b shows the energy levels for zero and finite external field. The theoretical calculations of the nuclear-spin mediated current flow (obtained from a master equation approach) are discussed in references [12, 15]. E. Oscillating magnetic field and rate equations So far, we have seen that the occurrence of transitions between singlet and triplet spin states are detectable as a small current in the spin blockade regime. We will now discuss how this lifting of spin blockade can also be used to detect single spin rotations, induced via electron spin resonance. The basic idea is the following. The basic idea is the following. If the system is blocked in e.g. | ↑〉| ↑〉, and the driving field rotates e.g. the left spin, then transitions are induced to the state | ↓〉| ↑〉. This state contains a singlet component and therefore a probability for the electron to move to the right dot and right lead. Inducing single spin rotations can therefore lift spin blockade. However, together with the driving field, the spin tran- sitions are much more complicated due to the interplay of different processes: spin resonance of the two spins, inter- action with the nuclear fields, spin state hybridization by coherent dot coupling and inelastic transitions from the S(1,1) state to the S(0,2) state. In order to understand the interplay of these processes, we will first model the system with a time-dependent Hamiltonian and a den- sity matrix approach. Next, we will discuss the physical interpretation of the simulation results. The Hamiltonian now also contains a term with an os- cillating magnetic field in the x-direction with amplitude Hac(t) = gµBBac sin(ωτ) · (SxL + SxR). (4) We assume that Bac is equal in both dots, which is a reasonable approximation in the experiment (from simu- lations we find that the difference of Bac is 20% at most [1]). We assume Bext ≫ BN , Bac, which allows applica- tion of the rotating wave approximation [16]. Therefore, we will define B1 ≡ 12Bac, which is in the rotating frame the relevant driving field for the ESR process. In order to study the effect of ESR and the nuclear fields that are involved in the transport cycle, we will construct rate equations that include the unitary evo- lution of the spins in the dots governed by the time- dependent Hamiltonian. This approach is based on the model of reference [12], where the Hamiltonian contained only time-independent terms. Seven states are involved in the transport cycle, namely the three (1,1) triplets |T i11〉, the double and single dot singlet states |S11〉 and |S02〉 and the two (0,1) states | ↑01〉 and | ↓01〉, making the density operator a 7× 7 matrix. The rate equations based on the time-independent Hamiltonian are given in [12]. These are constructed from the term that gives the unitary evolution of the system governed by the Hamilto- nian (H = H0 +Hac) dρ̂k/dτ = − i~ 〈k|[H, ρ̂]|k〉, together with terms that account for incoherent tunnelling pro- cesses between the states. The rate equations for the diagonal elements are given by = − i 〈T+11|[H, ρ̂]|T ρ̂↑01 = − i 〈T−11|[H, ρ̂]|T ρ̂↓01 dρ̂T 0 〈T 011|[H, ρ̂]|T 011〉+ ρ̂↑01 + ρ̂↓01 dρ̂S11 〈S11|[H, ρ̂]|S11〉+ ρ̂↑01 + ρ̂↓01 − Γinρ̂S11 dρ̂S02 〈S02|[H, ρ̂]|S02〉+ Γinρ̂S11 − ΓRρ̂S02 dρ̂↑01 ρ̂S02 − ΓLρ̂↑01 dρ̂↓01 ρ̂S02 − ΓLρ̂↓01 (5) The rate equations for the off-diagonal elements are 0 2 4 6 8 10 Time ( s)m transport via S02 RF on Spin blockade (0, / ) FIG. 3: Time evolution of the diagonal elements of the density matrix for one particular nuclear configuration. Parameters: ~ω = gµB100mT, Bext =100 mT, B N,x,y,z =(0,0,2.2) mT, BRN,x,y,z =(0,0,0), B1 = 1.3 mT, ΓL = 73 MHz, ΓR = 73 MHz, ~Γin = gµBB N,z and ∆LR=200µeV, t=0.3 µeV. given by dρ̂jk = − i 〈j|[H, ρ̂]|k〉 − 1 Γj + Γk ρ̂jk (6) where the indices j, k ∈ T i11, S11, S02, ↑01, ↓01 label the states available to the system. The tunneling/projection rates Γj equal Γin and ΓR for the |S11〉 and |S02〉 states respectively, and equal zero for the other 5 states. The first term on the right-hand side describes the unitary evolution of the system, while the second term describes a loss of coherence due to the finite lifetime of the sin- glet states. This is the first source of decoherence in our model. The second one is the inhomogeneous broaden- ing due to the interaction with the nuclear system. We do not consider other sources of decoherence, as they are expected to occur on much larger timescales. Because we added a time-dependent term to the Hamiltonian (the oscillating field), we numerically calcu- late the time evolution of ρ̂(t), treating the Hamiltonian as stationary on the timescale ∆τ ≪ 2π/ω. To reduce the simulation time, we use the steady state solution ρ̂τ→∞ in the absence of the oscillating magnetic field as the ini- tial state ρ̂(τ = 0) for the time evolution. At τ = 0 the oscillating field is turned on and the system evolves to- wards a dynamic equilibrium on a timescale set by the inverse of the slowest tunnelling rate Γ. This new equi- librium distribution of populations is used to calculate the current flow, which is proportional to the occupation of the |S02〉 state (I = eΓRρ̂S02). An example of the time evolution of the density matrix elements is shown in Fig. 3. The figure clearly reveals that the blockade is lifted when the oscillating field is applied. This is visible as an increase of the occupation of the |S02〉 state. In order to simulate the measured current flow we have to consider the fact that the measurements are taken with a sampling rate of 1 Hz. As the timescale of the nuclear dynamics is believed to be much faster than 1 Hz [10, 13, 14], we expect each datapoint to be an in- tegration of the response over many configurations of the nuclei. The effect of the evolving nuclear system is included in the calculations by averaging the different values of the (calculated) current flow obtained for each frozen configuration. These configurations are randomly sampled from a gaussian distribution of nuclear fields in the left and right dot (similar as in [12]). Because the electron in the two dots interact with different nuclear spins, the isotropic gaussian distributions in the two dots are uncorrelated, such that 〉 and 〈B2N,x〉 = 〈B2N,y〉 = 〈B2N,z〉. For the sake of convenience we define 〉 and σN,z = -100 -50 0 50 100 B (mT)ext -g B =hm wB ext g B =hm wB ext Simulation: Inelastic transport ( )DLR>>t h )G m sin B N/(g B (mT)ext -g B =hm wB ext g B =hm wB ext -100 -50 0 50 100 Simulation: Resonant transport t/(gm sB N) ( )DLR=0 0 50-50 100-100 RF at 460 MHz P~-16dBm RF off -g B =hm wB ext g B =hm wB ext B (mT)ext Experimental data FIG. 4: a). Calculated average current flow in the inelastic transport regime. Parameters: ~ω = gµB100mT, Bext =100 mT, σN =2.2mT, B1 = 1.3mT, ΓL,R = 73 MHz, t=0.3 µeV and ∆LR = 200 µeV. Results are similar for any value for t, provided that ∆LR ≫ t. b) Calculated average current flow in the resonant transport regime at zero detuning for differ- ent values of t. Parameters: ~ω = gµB100mT, σN =2.2mT, B1 = 1.3mT, ΓL,R = 73 MHz, Γin = 0 and ∆LR = 0. Av- eraged over 400 nuclear configurations for t/(gµBσN ) > 0.5 and 60 configurations for t/(gµBσN) = 0.5. Simulation car- ried out for positive magnetic fields only; values shown for negative fields are equal to results obtained for positive field. c) Experimental data from Ref [1] with (curve offset by 100 fA for clarity) and without oscillating magnetic field. The fre- quency of the oscillating magnetic field is 460 MHz and the applied power is -16dBm. Simulation carried out for posi- tive magnetic fields only; values shown for negative fields are equal to results obtained for positive field. F. Simulation results and physical picture An example of the calculated (average) current flow as a function of Bext (Fig. 4a,b) shows a (split) peak around zero magnetic field and two satellite peaks for Bext = ±~ω/(gµB), where the spin resonance condition is satisfied. This (split) peak atBext = 0 is due to singlet- triplet mixing by the inhomogeneous nuclear field, and the splitting depends on the tunnel coupling, similar as the observations in [5]. The response from the induced spin flips via the driving field is visible for the both in- elastic and resonant transport regime, and the current flow has comparable magnitude to the peak at Bext = 0. The satellite peaks are also visible in the experimental data from [1] (also shown here in Fig. 4), although the shape and width of the satellite peaks are different, as we will discuss later. We want to stress that the ESR satellite peaks only appear when an inhomogeneous nuclear field is present in the simulations. In other words, for ∆BN = 0 and B1 equal in both dots, spin rotations are induced in both dots at the same time and at the same rate. Starting, for example, from the state |T+11〉 = | ↑↑〉 transitions are induced to the state | ↓↓〉 via the intermediate state | ↑ + ↓〉|↑ + ↓〉/ 2 = (|T+11〉+|T 11〉+2|T 011〉)/ 2. No mixing with the singlet state takes place (the evolution is in the triplet-subspace) and no current will therefore flow. The ESR sattelite peaks are visible for both resonant and inelastic transport regime (Figs. 4a,b). For the res- onant transport regime, we see that for t/σN < 5 the sattelite peak increases in height when increasing t, sim- ply because the coupling between the two singlet states increases. However, further increasing t reduces the sig- nal, and this is because the exchange splitting then plays a more important role. Namely, increasing the exchange splitting reduces the mixing between the T 011 state with the hybridized singlet state by the nuclear field gradi- ent. This mixing is a crucial element for detecting the induced rotations of one of the two electron spins. In the inelastic transport regime, this exchange splitting is negligibly small and therefore the height of the sattelite peak depends only on Γin and the driving field B1. A study of the height of the satellite peak as a function of B1 reveals a non-monotonous behaviour, which can be seen in Fig. 5a. The physical picture behind this behavior is most easily sketched by distinguishing three regimes: 1. For B1 < σN,z, for most of the nuclear configu- rations the spin in at most one of the two dots is on resonance, so spins are flipped in either the left or right dot. In that case transitions are in- duced from e.g. | ↑↑〉 to | ↑↓〉 = |S11〉 + |T 011〉 or |↓↑〉 = |S11〉 − |T 011〉. The resulting current flow ini- tially increases quadratically with B1, as one would normally expect (Fig. 5a). 2. For B1 ≫ σN,z, for most of the nuclear configu- rations two spins are rotated simultaneously due B1 (mT) 0 0.2 0.4 0.6 0.8 2 3 4 5 6 7 8 Experimental data Detuning ~360 eVm Detuning ~320 eVm Model sN,z=1.3mT 0.10.01 1 10 B1 N,z/ 2s FIG. 5: Height and width of the ESR satellite peak. a) Cir- cles: calculated ESR peak height as a function of driving am- plitude B1. Parameters: ~ω = gµB100mT, Bext =100 mT, σN =2.2mT, ΓL,R = 73 MHz, t=0.3 µeV, ~Γin = gµBσN and ∆LR = 200 µeV. Lines are the current measurements for 2 different values of ∆LR. The measurements show time- dependent (telegraph type) behavior. Therefore, the curves are obtained by repeating sweeps of B1 and then selecting the largest current value for each value of B1. b) Calculated width of the ESR satellite peaks as a function of B1. For small ESR power the peak is broadened by the random nu- clear fluctuations, at high powers it is broadened by B1. to power broadening of the Rabi resonance. The stronger B1, the more the transitions occur only in the triplet subspace (the driving field B1 that ro- tates two spins dominates the S−T0 mixing by the nuclear spins). As a result, the current decreases for increasing B1. 3. If B1 ∼ σN,z the situation is more complex because both processes (rotation of 2 spins simultaneously and transitions from T 011 to S11) are effective. We find that if both processes occur with comparable rates, the overall transition rate to the singlet state is highest. This is the reason why the current has a maximum at B1 ≈ σN,z (Fig. 5a). The experimental data of the ESR satellite peak height (normalized by the zero-field current flow) for two dif- ferent values of ∆LR are shown in Fig. 5a. In order to compare the experimental results with the model we have estimated the rate Γin from the measured current flow at Bext = 0 (we found similar values for both curves). The agreement of the experimental data with the model is reasonable, as it shows the expected quadratic increase with B1, as well as a comparable peak height. However, we see that variations of the level detuning ∆LR can re- sult in considerable differences of the measured ESR peak height. We have two possible explanations for the devi- ations of the experimental data with the model. First, we have found experimental signatures of dynamic nu- clear polarization when the ESR resonance condition was fulfilled. We expect that this is due to feedback of the electron transport on the nuclear spins (similar to that discussed in [11, 15, 17]), although the exact processes are not (yet) fully understood. Second, unwanted electric fields affect the electron tunnelling processes, but are not taken into account in the model. We expect that these electric fields will not change the location and width of the ESR sattelite peaks because this field does not couple the spin states. It is however possible that the height of the satellite peak is altered by the electric field because if can affect the coupling between the S(0,2) with the S(1,1) state. Finally, we discuss the width of the ESR satellite peak (Fig. 5b). If the inelastic tunnelling process between the dots (with rate Γin) and B1 are both smaller than σN,z, the ESR peak (obtained from simulations) is broad- ened by the statistical fluctuations of the effective nuclear field. For high B1, the width approaches asymptotically the line with slope 1 (see Fig. 5b). In this regime, the peak is broadened by the RF amplitude B1. In the ex- periment [1], the shape of the satellite peak was different (flat on top with sharp edges) than expected from the model. Furthermore, the FWHM was larger than ex- pected from just σN,z. We attribute this to feedback of the ESR-induced current flow on the nuclear spin bath. As a result, a clear FWHM increase with B1 could not be observed. It should be noted that in the simulation the central peak is broader than the satellite peaks. From study- ing the influence of various parameters in the model, we conclude that the greater width of the central peak is caused by the tranverse nuclear field fluctuations (BN,x and BN,y), which broaden the central peak but not the ESR satellite peaks. We conclude that the model discussed here qualita- tively agrees with the main features that were observed in the double dot transport measurements that aims at detecting (continuous wave) ESR of a single electron spin. The details of the ESR satellite peak height and width do not agree quantitatively with the model. We believe these deviations can be attributed to unwanted electric fields and feedback of the electron transport on the nu- clear spin polarization. Improving the understanding of these feedback mechanisms remains interesting for future investigation as it might point towards a direction to mit- igate the decoherence of the electron spin [12, 18]. Acknowledgments This study was supported by the Dutch Organization for Fundamental Research on Matter (FOM), the Nether- lands Organization for Scientific Research (NWO) and the Defense Advanced Research Projects Agency Quan- tum Information Science and Technology program. [1] F. Koppens, C. Buizert, K. Tielrooij, I. Vink, K. Nowack, T. Meunier, L. Kouwenhoven, and L. Vandersypen, Na- ture 442, 766 (2006). [2] H. A. Engel and D. Loss, Phys. Rev. Lett. 86, 4648 (2001). [3] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Sci- ence 297, 1313 (2002). [4] A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Nature 435, 925 (2005). [5] F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Han- son, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz, W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van- dersypen, Science 309, 1346 (2005). [6] P.-F. Braun,X. Marie, L. Lombez, B. Urbaszek, T. Amand, P.Renucci, V. K. Kalevich, K. V. Kavokin, O. Krebs, P. Voisin and Y. Masumoto, Phys. Rev. Lett. 94, 116601 (2005). [7] D. Gammon, Al. L. Efros, T. A. Kennedy, M. Rosen, D. S. Katzer, D. Park, S.W. Brown, V. L. Korenev, I. A. Merkulov, Phys. Rev. Lett. 86, 5176 (2001). [8] I.A. Merkulov, A.L. Efros, J. Rosen, Phys. Rev. B 65, 205309 (2002). [9] A. V. Khaetskii, D. Loss, L. Glazman, Phys. Rev. Lett. 88, 186802 (2002). [10] W. A. Coish and D. Loss, Phys. Rev. B 70, 195340 (2004). [11] K. Ono, S. Tarucha, Phys. Rev. Lett. 92, 256803 (2004). [12] O. N. Jouravlev and Y. V. Nazarov, Phys. Rev. Lett. 96, 176804 (2006). [13] R. de Sousa, S. Das Sarma, Phys. Rev. B 67, 033301 (2003). [14] D. Paget, G. Lampel, B. Sapoval, V.I. Safarov, Phys. Rev. B 15, 5780 (1977). [15] J. Inarrea, G. Platero, and A. H. MacDonald (2006), URL http://www.citebase.org/abstract?id= oai:arXiv.org:cond-mat/0609323. [16] C. Poole, Electron Spin Resonance, 2nd ed. (Wiley, New York, 1983). [17] M. S. Rudner and L. S. Levitov (2006), URL http://www.citebase.org/abstract?id=oai: arXiv.org:cond-mat/0609409. [18] D. Klauser, W. Coish, and D. Loss, Phys. Rev. B 73, 205302 (2006).

Spin-dependent transport measurements through a double quantum dot are a valuable tool for detecting both the coherent evolution of the spin state of a single electron as well as the hybridization of two-electron spin states. In this paper, we discuss a model that describes the transport cycle in this regime, including the effects of an oscillating magnetic field (causing electron spin resonance) and the effective nuclear fields on the spin states in the two dots. We numerically calculate the current flow due to the induced spin flips via electron spin resonance and we study the detector efficiency for a range of parameters. The experimental data are compared with the model and we find a reasonable agreement.

Introduction Recently, coherent spin rotations of a single electron were demonstrated in a double quantum dot device [1]. In this system, spin-flips of an electron in the dot were in- duced via an oscillating magnetic field (electron spin res- onance or ESR) and detected through a spin-dependent transition of the electron to another dot, which already contained one additional electron. This detection scheme is an extension of the proposal for ESR detection in a sin- gle quantum dot by Engel and Loss [2]. Briefly, the device can be operated (in a spin blockade regime [3]) such that the electron in the left dot can only move to the right dot if a spin flip in one of the two dots is induced via ESR. From the right dot, the electron exits to the right reser- voir and another electron enters the left dot from the left reservoir. A continuous repetition of this transition will result in a net current flow. Compared to the single dot detection scheme [2], us- ing the double-dot as the detector has two major advan- tages. First, the experiment can be performed at a lower static magnetic field and consequently with lower, tech- nically less demanding, excitation frequencies. Second, the spin detection is rather insensitive to unwanted oscil- lating electric fields, because the relevant dot levels can be positioned far from the Fermi energies of the leads. These electric fields are unavoidably generated together with the oscillating magnetic field as well. The drawback of the double-dot detector is that spin detection is based on the projection in the two-electron singlet-triplet basis, while the aim is to detect single spin rotations. However, this detection is still possible be- cause the electrons in the two dots experience different effective nuclear fields. This is due to the hyperfine inter- action of the electron spins with the (roughly 106) nuclear spins in the host semiconductor material of each quan- tum dot [4–11]. In order to provide more insight in this double-dot ESR detection scheme for single spin rota- tions, it is necessary to analyze the coherent evolution of the two-electron spin states together with the transitions ∗Electronic address: f.h.l.koppens@tudelft.nl; Kavli Institute of NanoScience Delft, P.O. Box 5046, 2600 GA Delft, The Netherlands in the transport cycle. In this paper, we discuss a model that describes the transport cycle in the spin blockade regime while includ- ing the coherent coupling between the two dots, and the influence of the static and oscillating magnetic field to- gether with the effective nuclear fields on the electron spin states. The aim is to understand how effectively single spin resonance will affect the measured quantity in the experiment, namely the current flow in the spin blockade regime. The organization of this paper is as follows. First, we will explain the transport cycle and the mechanism that causes spin blockade. Next, we will briefly discuss the static system Hamiltonian and the mixing of the two-electron spin states by the effective nuclear field. Then we add an oscillating magnetic field to this Hamiltonian, that forms -together with the double dot tunnelling processes- the basis of the rate equations that describe how the density matrix of the two-electron spin states evolves in time. The current flow, calculated from the steady state solution of the density operator, is then analyzed for different coherent coupling values, magnitudes of the oscillating magnetic field, in combina- tion with different effective nuclear fields in the two dots. This provides further insight in the optimal conditions for spin-flip detection with a double quantum dot. B. Spin blockade In the spin-blockade regime, the double-dot is tuned such that one electron always resides in the right dot, and a second electron can tunnel from the left reservoir through the left and right dots, to the right reservoir [3]. This current-carrying cycle can be described with the occupations (m, n) of the left and right dots: (1, 1) → (0, 2) → (0, 1) → (1, 1). When an electron enters the left dot and forms a double-dot singlet state S11 with the electron in the right dot (S = |↑↓〉 − |↓↑〉, normalization omitted for brevity), it is possible for the left electron to move to the right dot, because the right dot singlet state S02 is energetically accessible. Next, one electron tunnels from the right dot to the right lead and another electron can again tunnel into the left dot. If, however, the two electrons form a double-dot triplet state T11, the left electron cannot move to the right dot, as the right http://arxiv.org/abs/0704.1628v1 dot triplet state T02 is much higher in energy (due to the relatively large exchange splitting in a single dot). The electron can also not move back to the lead and therefore further current flow is blocked as soon as any of the (double-dot) triplet states is formed (Fig. 1a,b). Spin blockade only occurs if at least one of the eigen- states of the system Hamiltonian is a pure triplet state. If processes are present that induce transitions from all the three (1,1) triplet states to the (1,1) singlet state, spin blockade is lifted and a current will flow. As we will see below, the presence of the nuclear spins in the host semiconductor can give rise to these kind of transitions. This can be seen most easily by adding the effect of the hyperfine interaction to the system Hamiltonian. C. System Hamiltonian The system Hamiltonian is most conveniently writ- ten in the two-electron singlet-triplet basis with the quantization-axis in the z-direction. The basis states are S11, T 11, T 11, T 11 and S02. The subscript m,n denotes the dot occupancy. We exclude the T02 state from the model, because this state is energetically inaccessible and therefore does not play an important role in the trans- DLR>>t DLR=0 S - S11 02 S - S11 02 -10 -5 0 5 10 DLR/t b) a) FIG. 1: a) A schematic of the double dot and the electro- chemical potentials (energy relative to the (0,1) state) of the relevant two-electron spin states. For ∆LR > t, transitions from the S11 state to the S02 state are possible via inelastic relaxation with rate Γin. Spin blockade occurs when one of the T i11 states is occupied. b) Similar schematic for ∆LR = 0, where the singlet states are hybridized. Also in this case, spin blockade occurs when one of T i11 states is occupied. c) Energy levels as a function of detuning. At ∆LR = 0, the singlet states hybridize into bonding and anti-bonding states. The splitting between the triplets states corresponds to the Zeeman energy gµBBext. port cycle. Furthermore, we neglect the thermal energy kT in the description, which is justified when the bias over the two dots is much larger than kT . The system Hamiltonian is given by H0 = − ∆LR|S02〉〈S02|+ t |S11〉〈S02|+ |S02〉〈S11| − gµBBext |T+11〉〈T 11| − |T 11〉〈T , (1) where ∆LR is the energy difference between the |S11〉 and |S02〉 state (level detuning, see Fig.1a), t is the tun- nel coupling between the |S11〉 and |S02〉 states, Bext the external magnetic field in the z-direction and Sz L(R) the spin operator along z for the left (right) electron. The eigenstates of the Hamiltonian (1) for finite external field are shown in figure 1c. For ∆LR < t, the tunnel coupling t causes an anti-crossing of the |S11〉 and |S02〉 states. For ∆LR < 0, transport is blocked by Coulomb blockade (i.e. the final state |S02〉 is at a higher energy than the initial state S11). For ∆LR ≥ 0, transport will be blocked when one of the three triplet states becomes occupied (spin blockade). In Fig.1a and b, we distinguish two regimes: ∆LR > t where the (exchange) energy splitting between T 011 and S11 is negligibly small and transitions from S11 to S02 occur via inelastic relaxation with rate Γin and the energy. A different regime holds for ∆LR < t, where S11 is coherently coupled with S02 giving rise to a finite (ex- change) splitting between T 011 and the hybridized singlet states. We will return to this distinction in the discussion below. D. Singlet-triplet mixing by the nuclear spins The effect of the hyperfine interaction with the nuclear spins can be studied [12] by adding a static (frozen) ef- fective nuclear field BLN (B N ) at the left (right) dot to the system Hamiltonian: Hnucl = −gµB N · SL +BRN · SR = −gµB(BLN −BRN ) · (SL − SR)/2 −gµB(BLN +BRN ) · (SL + SR)/2. (2) For the sake of convenience, we separate the inhomoge- neous and homogeneous contribution, for reasons which we will discuss later. Considering the nuclear field as static is justified since the tunnel rates and electron spin dynamics are expected to be much faster than the dy- namics of the nuclear system [10, 13, 14]. Therefore, we will treat the Hamiltonian as time-independent. The effect of nuclear reorientation will be included later by ensemble averaging. We will show now that triplet states mix with the S11 state if the nuclear field is different in the two dots (in all three directions). This mixing will lift spin blockade, detectable as a finite current running through the dots for ∆LR ≥ 0. The effective nuclear field can be decomposed -30 -20 -10 0 10 20 30 Magnetic field (mT) Bext=0 B » Bext ND -10 -5 0 5 10 10 S02 a +b +g +kS T T T11 11 11 11 -10 -5 0 5 10 a +kS T11 11 DLR/t DLR/t FIG. 2: a) Observed current flow in the inelastic transport regime (gµB∆LR ≫ t) due to singlet-triplet mixing by the nuclei. b) Electrochemical potentials in the presence of Hnucl (t ∼ ∆BN ). Singlet and triplet eigenstates are denoted by red and blue lines respectively. Hybridized states (of sin- glet and triplet) are denoted by dotted purple lines. For gµBBext ≫ t, gµB∆BN , the split-off triplets (T and T− ) are hardly perturbed and current flow is blocked when they be- come occupied. Parameters: t = 0.2µeV, gµBBN,L=(0.1,0,- 0.1)µeV, gµBBN,R=(-0.1,-0.2,-0.2)µeV and gµBBext=2µeV. in a homogeneous and an inhomogeneous part (see right- hand side of (2)). The homogeneous part simply adds vectorially to the external field Bext, changing slightly the Zeeman splitting and preferred spin orientation of the triplet states. The inhomogeneous part ∆BN ≡ BLN − N on the other hand couples the triplet states to the singlet state, as can be seen readily by combining the spin operators in the following way SxL − SxR = |S11〉〈T−11| − |S11〉〈T 11| + h.c. i|S11〉〈T−11| − i|S11〉〈T 11| + h.c. SzL − SzR = |S11〉〈T 011|+ |T 011〉〈S11| . (3) The first two expressions reveal that the inhomogeneous field in the transverse plane ∆BxN , ∆B mixes the |T+11〉 and |T−11〉 states with the |S11〉. The longitudinal com- ponent ∆BzN mixes |T 011〉 with |S11〉 (third expression). The degree of mixing between two states will depend strongly on the energy difference between them [5]. In the case of gµBBext, t < gµB 〉, the three triplet states are close in energy to the |S11〉 state. Their intermixing will be strong, lifting spin blockade. For gµBBext ≫ t, gµB 〉 the |T+11〉 and |T 11〉 states are split off in energy by an amount of gµBBext. Con- sequently the perturbation of these states caused by the nuclei will be small. Although the |T 011〉 remains mixed with the |S11〉 state, the occupation of one of the two split-off triplet states can block the flow through the sys- The effect of nuclear mixing is shown in Fig. 2 [5]. The observed current flow through the system is typi- cally in the order of a few hundreds of fA (Fig. 2a). At zero field, where the mixing is strongest, the current flow is largest. Increasing the field gradually restores spin blockade. Fig. 2b shows the energy levels for zero and finite external field. The theoretical calculations of the nuclear-spin mediated current flow (obtained from a master equation approach) are discussed in references [12, 15]. E. Oscillating magnetic field and rate equations So far, we have seen that the occurrence of transitions between singlet and triplet spin states are detectable as a small current in the spin blockade regime. We will now discuss how this lifting of spin blockade can also be used to detect single spin rotations, induced via electron spin resonance. The basic idea is the following. The basic idea is the following. If the system is blocked in e.g. | ↑〉| ↑〉, and the driving field rotates e.g. the left spin, then transitions are induced to the state | ↓〉| ↑〉. This state contains a singlet component and therefore a probability for the electron to move to the right dot and right lead. Inducing single spin rotations can therefore lift spin blockade. However, together with the driving field, the spin tran- sitions are much more complicated due to the interplay of different processes: spin resonance of the two spins, inter- action with the nuclear fields, spin state hybridization by coherent dot coupling and inelastic transitions from the S(1,1) state to the S(0,2) state. In order to understand the interplay of these processes, we will first model the system with a time-dependent Hamiltonian and a den- sity matrix approach. Next, we will discuss the physical interpretation of the simulation results. The Hamiltonian now also contains a term with an os- cillating magnetic field in the x-direction with amplitude Hac(t) = gµBBac sin(ωτ) · (SxL + SxR). (4) We assume that Bac is equal in both dots, which is a reasonable approximation in the experiment (from simu- lations we find that the difference of Bac is 20% at most [1]). We assume Bext ≫ BN , Bac, which allows applica- tion of the rotating wave approximation [16]. Therefore, we will define B1 ≡ 12Bac, which is in the rotating frame the relevant driving field for the ESR process. In order to study the effect of ESR and the nuclear fields that are involved in the transport cycle, we will construct rate equations that include the unitary evo- lution of the spins in the dots governed by the time- dependent Hamiltonian. This approach is based on the model of reference [12], where the Hamiltonian contained only time-independent terms. Seven states are involved in the transport cycle, namely the three (1,1) triplets |T i11〉, the double and single dot singlet states |S11〉 and |S02〉 and the two (0,1) states | ↑01〉 and | ↓01〉, making the density operator a 7× 7 matrix. The rate equations based on the time-independent Hamiltonian are given in [12]. These are constructed from the term that gives the unitary evolution of the system governed by the Hamilto- nian (H = H0 +Hac) dρ̂k/dτ = − i~ 〈k|[H, ρ̂]|k〉, together with terms that account for incoherent tunnelling pro- cesses between the states. The rate equations for the diagonal elements are given by = − i 〈T+11|[H, ρ̂]|T ρ̂↑01 = − i 〈T−11|[H, ρ̂]|T ρ̂↓01 dρ̂T 0 〈T 011|[H, ρ̂]|T 011〉+ ρ̂↑01 + ρ̂↓01 dρ̂S11 〈S11|[H, ρ̂]|S11〉+ ρ̂↑01 + ρ̂↓01 − Γinρ̂S11 dρ̂S02 〈S02|[H, ρ̂]|S02〉+ Γinρ̂S11 − ΓRρ̂S02 dρ̂↑01 ρ̂S02 − ΓLρ̂↑01 dρ̂↓01 ρ̂S02 − ΓLρ̂↓01 (5) The rate equations for the off-diagonal elements are 0 2 4 6 8 10 Time ( s)m transport via S02 RF on Spin blockade (0, / ) FIG. 3: Time evolution of the diagonal elements of the density matrix for one particular nuclear configuration. Parameters: ~ω = gµB100mT, Bext =100 mT, B N,x,y,z =(0,0,2.2) mT, BRN,x,y,z =(0,0,0), B1 = 1.3 mT, ΓL = 73 MHz, ΓR = 73 MHz, ~Γin = gµBB N,z and ∆LR=200µeV, t=0.3 µeV. given by dρ̂jk = − i 〈j|[H, ρ̂]|k〉 − 1 Γj + Γk ρ̂jk (6) where the indices j, k ∈ T i11, S11, S02, ↑01, ↓01 label the states available to the system. The tunneling/projection rates Γj equal Γin and ΓR for the |S11〉 and |S02〉 states respectively, and equal zero for the other 5 states. The first term on the right-hand side describes the unitary evolution of the system, while the second term describes a loss of coherence due to the finite lifetime of the sin- glet states. This is the first source of decoherence in our model. The second one is the inhomogeneous broaden- ing due to the interaction with the nuclear system. We do not consider other sources of decoherence, as they are expected to occur on much larger timescales. Because we added a time-dependent term to the Hamiltonian (the oscillating field), we numerically calcu- late the time evolution of ρ̂(t), treating the Hamiltonian as stationary on the timescale ∆τ ≪ 2π/ω. To reduce the simulation time, we use the steady state solution ρ̂τ→∞ in the absence of the oscillating magnetic field as the ini- tial state ρ̂(τ = 0) for the time evolution. At τ = 0 the oscillating field is turned on and the system evolves to- wards a dynamic equilibrium on a timescale set by the inverse of the slowest tunnelling rate Γ. This new equi- librium distribution of populations is used to calculate the current flow, which is proportional to the occupation of the |S02〉 state (I = eΓRρ̂S02). An example of the time evolution of the density matrix elements is shown in Fig. 3. The figure clearly reveals that the blockade is lifted when the oscillating field is applied. This is visible as an increase of the occupation of the |S02〉 state. In order to simulate the measured current flow we have to consider the fact that the measurements are taken with a sampling rate of 1 Hz. As the timescale of the nuclear dynamics is believed to be much faster than 1 Hz [10, 13, 14], we expect each datapoint to be an in- tegration of the response over many configurations of the nuclei. The effect of the evolving nuclear system is included in the calculations by averaging the different values of the (calculated) current flow obtained for each frozen configuration. These configurations are randomly sampled from a gaussian distribution of nuclear fields in the left and right dot (similar as in [12]). Because the electron in the two dots interact with different nuclear spins, the isotropic gaussian distributions in the two dots are uncorrelated, such that 〉 and 〈B2N,x〉 = 〈B2N,y〉 = 〈B2N,z〉. For the sake of convenience we define 〉 and σN,z = -100 -50 0 50 100 B (mT)ext -g B =hm wB ext g B =hm wB ext Simulation: Inelastic transport ( )DLR>>t h )G m sin B N/(g B (mT)ext -g B =hm wB ext g B =hm wB ext -100 -50 0 50 100 Simulation: Resonant transport t/(gm sB N) ( )DLR=0 0 50-50 100-100 RF at 460 MHz P~-16dBm RF off -g B =hm wB ext g B =hm wB ext B (mT)ext Experimental data FIG. 4: a). Calculated average current flow in the inelastic transport regime. Parameters: ~ω = gµB100mT, Bext =100 mT, σN =2.2mT, B1 = 1.3mT, ΓL,R = 73 MHz, t=0.3 µeV and ∆LR = 200 µeV. Results are similar for any value for t, provided that ∆LR ≫ t. b) Calculated average current flow in the resonant transport regime at zero detuning for differ- ent values of t. Parameters: ~ω = gµB100mT, σN =2.2mT, B1 = 1.3mT, ΓL,R = 73 MHz, Γin = 0 and ∆LR = 0. Av- eraged over 400 nuclear configurations for t/(gµBσN ) > 0.5 and 60 configurations for t/(gµBσN) = 0.5. Simulation car- ried out for positive magnetic fields only; values shown for negative fields are equal to results obtained for positive field. c) Experimental data from Ref [1] with (curve offset by 100 fA for clarity) and without oscillating magnetic field. The fre- quency of the oscillating magnetic field is 460 MHz and the applied power is -16dBm. Simulation carried out for posi- tive magnetic fields only; values shown for negative fields are equal to results obtained for positive field. F. Simulation results and physical picture An example of the calculated (average) current flow as a function of Bext (Fig. 4a,b) shows a (split) peak around zero magnetic field and two satellite peaks for Bext = ±~ω/(gµB), where the spin resonance condition is satisfied. This (split) peak atBext = 0 is due to singlet- triplet mixing by the inhomogeneous nuclear field, and the splitting depends on the tunnel coupling, similar as the observations in [5]. The response from the induced spin flips via the driving field is visible for the both in- elastic and resonant transport regime, and the current flow has comparable magnitude to the peak at Bext = 0. The satellite peaks are also visible in the experimental data from [1] (also shown here in Fig. 4), although the shape and width of the satellite peaks are different, as we will discuss later. We want to stress that the ESR satellite peaks only appear when an inhomogeneous nuclear field is present in the simulations. In other words, for ∆BN = 0 and B1 equal in both dots, spin rotations are induced in both dots at the same time and at the same rate. Starting, for example, from the state |T+11〉 = | ↑↑〉 transitions are induced to the state | ↓↓〉 via the intermediate state | ↑ + ↓〉|↑ + ↓〉/ 2 = (|T+11〉+|T 11〉+2|T 011〉)/ 2. No mixing with the singlet state takes place (the evolution is in the triplet-subspace) and no current will therefore flow. The ESR sattelite peaks are visible for both resonant and inelastic transport regime (Figs. 4a,b). For the res- onant transport regime, we see that for t/σN < 5 the sattelite peak increases in height when increasing t, sim- ply because the coupling between the two singlet states increases. However, further increasing t reduces the sig- nal, and this is because the exchange splitting then plays a more important role. Namely, increasing the exchange splitting reduces the mixing between the T 011 state with the hybridized singlet state by the nuclear field gradi- ent. This mixing is a crucial element for detecting the induced rotations of one of the two electron spins. In the inelastic transport regime, this exchange splitting is negligibly small and therefore the height of the sattelite peak depends only on Γin and the driving field B1. A study of the height of the satellite peak as a function of B1 reveals a non-monotonous behaviour, which can be seen in Fig. 5a. The physical picture behind this behavior is most easily sketched by distinguishing three regimes: 1. For B1 < σN,z, for most of the nuclear configu- rations the spin in at most one of the two dots is on resonance, so spins are flipped in either the left or right dot. In that case transitions are in- duced from e.g. | ↑↑〉 to | ↑↓〉 = |S11〉 + |T 011〉 or |↓↑〉 = |S11〉 − |T 011〉. The resulting current flow ini- tially increases quadratically with B1, as one would normally expect (Fig. 5a). 2. For B1 ≫ σN,z, for most of the nuclear configu- rations two spins are rotated simultaneously due B1 (mT) 0 0.2 0.4 0.6 0.8 2 3 4 5 6 7 8 Experimental data Detuning ~360 eVm Detuning ~320 eVm Model sN,z=1.3mT 0.10.01 1 10 B1 N,z/ 2s FIG. 5: Height and width of the ESR satellite peak. a) Cir- cles: calculated ESR peak height as a function of driving am- plitude B1. Parameters: ~ω = gµB100mT, Bext =100 mT, σN =2.2mT, ΓL,R = 73 MHz, t=0.3 µeV, ~Γin = gµBσN and ∆LR = 200 µeV. Lines are the current measurements for 2 different values of ∆LR. The measurements show time- dependent (telegraph type) behavior. Therefore, the curves are obtained by repeating sweeps of B1 and then selecting the largest current value for each value of B1. b) Calculated width of the ESR satellite peaks as a function of B1. For small ESR power the peak is broadened by the random nu- clear fluctuations, at high powers it is broadened by B1. to power broadening of the Rabi resonance. The stronger B1, the more the transitions occur only in the triplet subspace (the driving field B1 that ro- tates two spins dominates the S−T0 mixing by the nuclear spins). As a result, the current decreases for increasing B1. 3. If B1 ∼ σN,z the situation is more complex because both processes (rotation of 2 spins simultaneously and transitions from T 011 to S11) are effective. We find that if both processes occur with comparable rates, the overall transition rate to the singlet state is highest. This is the reason why the current has a maximum at B1 ≈ σN,z (Fig. 5a). The experimental data of the ESR satellite peak height (normalized by the zero-field current flow) for two dif- ferent values of ∆LR are shown in Fig. 5a. In order to compare the experimental results with the model we have estimated the rate Γin from the measured current flow at Bext = 0 (we found similar values for both curves). The agreement of the experimental data with the model is reasonable, as it shows the expected quadratic increase with B1, as well as a comparable peak height. However, we see that variations of the level detuning ∆LR can re- sult in considerable differences of the measured ESR peak height. We have two possible explanations for the devi- ations of the experimental data with the model. First, we have found experimental signatures of dynamic nu- clear polarization when the ESR resonance condition was fulfilled. We expect that this is due to feedback of the electron transport on the nuclear spins (similar to that discussed in [11, 15, 17]), although the exact processes are not (yet) fully understood. Second, unwanted electric fields affect the electron tunnelling processes, but are not taken into account in the model. We expect that these electric fields will not change the location and width of the ESR sattelite peaks because this field does not couple the spin states. It is however possible that the height of the satellite peak is altered by the electric field because if can affect the coupling between the S(0,2) with the S(1,1) state. Finally, we discuss the width of the ESR satellite peak (Fig. 5b). If the inelastic tunnelling process between the dots (with rate Γin) and B1 are both smaller than σN,z, the ESR peak (obtained from simulations) is broad- ened by the statistical fluctuations of the effective nuclear field. For high B1, the width approaches asymptotically the line with slope 1 (see Fig. 5b). In this regime, the peak is broadened by the RF amplitude B1. In the ex- periment [1], the shape of the satellite peak was different (flat on top with sharp edges) than expected from the model. Furthermore, the FWHM was larger than ex- pected from just σN,z. We attribute this to feedback of the ESR-induced current flow on the nuclear spin bath. As a result, a clear FWHM increase with B1 could not be observed. It should be noted that in the simulation the central peak is broader than the satellite peaks. From study- ing the influence of various parameters in the model, we conclude that the greater width of the central peak is caused by the tranverse nuclear field fluctuations (BN,x and BN,y), which broaden the central peak but not the ESR satellite peaks. We conclude that the model discussed here qualita- tively agrees with the main features that were observed in the double dot transport measurements that aims at detecting (continuous wave) ESR of a single electron spin. The details of the ESR satellite peak height and width do not agree quantitatively with the model. We believe these deviations can be attributed to unwanted electric fields and feedback of the electron transport on the nu- clear spin polarization. Improving the understanding of these feedback mechanisms remains interesting for future investigation as it might point towards a direction to mit- igate the decoherence of the electron spin [12, 18]. Acknowledgments This study was supported by the Dutch Organization for Fundamental Research on Matter (FOM), the Nether- lands Organization for Scientific Research (NWO) and the Defense Advanced Research Projects Agency Quan- tum Information Science and Technology program. [1] F. Koppens, C. Buizert, K. Tielrooij, I. Vink, K. Nowack, T. Meunier, L. Kouwenhoven, and L. Vandersypen, Na- ture 442, 766 (2006). [2] H. A. Engel and D. Loss, Phys. Rev. Lett. 86, 4648 (2001). [3] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Sci- ence 297, 1313 (2002). [4] A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Nature 435, 925 (2005). [5] F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Han- son, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz, W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van- dersypen, Science 309, 1346 (2005). [6] P.-F. Braun,X. Marie, L. Lombez, B. Urbaszek, T. Amand, P.Renucci, V. K. Kalevich, K. V. Kavokin, O. Krebs, P. Voisin and Y. Masumoto, Phys. Rev. Lett. 94, 116601 (2005). [7] D. Gammon, Al. L. Efros, T. A. Kennedy, M. Rosen, D. S. Katzer, D. Park, S.W. Brown, V. L. Korenev, I. A. Merkulov, Phys. Rev. Lett. 86, 5176 (2001). [8] I.A. Merkulov, A.L. Efros, J. Rosen, Phys. Rev. B 65, 205309 (2002). [9] A. V. Khaetskii, D. Loss, L. Glazman, Phys. Rev. Lett. 88, 186802 (2002). [10] W. A. Coish and D. Loss, Phys. Rev. B 70, 195340 (2004). [11] K. Ono, S. Tarucha, Phys. Rev. Lett. 92, 256803 (2004). [12] O. N. Jouravlev and Y. V. Nazarov, Phys. Rev. Lett. 96, 176804 (2006). [13] R. de Sousa, S. Das Sarma, Phys. Rev. B 67, 033301 (2003). [14] D. Paget, G. Lampel, B. Sapoval, V.I. Safarov, Phys. Rev. B 15, 5780 (1977). [15] J. Inarrea, G. Platero, and A. H. MacDonald (2006), URL http://www.citebase.org/abstract?id= oai:arXiv.org:cond-mat/0609323. [16] C. Poole, Electron Spin Resonance, 2nd ed. (Wiley, New York, 1983). [17] M. S. Rudner and L. S. Levitov (2006), URL http://www.citebase.org/abstract?id=oai: arXiv.org:cond-mat/0609409. [18] D. Klauser, W. Coish, and D. Loss, Phys. Rev. B 73, 205302 (2006).

Microsoft Word - Maser Donor type semiconductor at low temperature as maser active medium Yuri Kornyushin Maître Jean Brunschvig Research Unit, Chalet Shalva, Randogne, CH-3975 In some semiconductors donor impurity atoms can attract additional electrons, forming negative donor impurity ions. Thus we have 3 energy levels for electrons: zero energy levels at the bottom of the conductivity band, negative energy levels of the bounded electrons of the negative donor impurity ions, and deeper negative energy levels of the outer electrons of the neutral donor impurity atoms. So the donor impurity atoms could serve as active centres for a maser. The maximum achievable relative population is 0.5. Typical wavelength of the generated oscillation is 0.14 mm; three level scheme could be realized at rather low temperatures, considerably lower than 6 K. Let us consider some donor impurity atom in a donor-type semiconductor having one more outer electron than the host atom of a semiconductor (e.g., the host silicon atom). In the simplest case of a non-degenerate standard conductivity band the equation of a motion of a superfluous electron is the same as that for the electron in a hydrogen atom [1]. The bond energy at that is as follows [1]: Ebd = (m0e )(m/m0 ), (1) where m0 is the free electron mass, m is the effective electron mass in semiconductor conductivity band, e is the electron charge, and  is the Planck constant divided by 2. Comparative to the bond energy in a hydrogen atom (see, e.g., [2]) the right-hand part of Eq. (1) contains additional factor (m/m0 ). At m = 0.1m0 and  = 12 [1] this factor, (m/m0 ) = 6.94410 Hydrogen atom can attract an additional electron, forming negative hydrogen ion [3]. The electron affinity of a free electron to a hydrogen atom is Eah = 0.754 eV [3]. The bond energy in a hydrogen atom, Ebh = (m0e ) = 13.598 eV [3]. Taking into account that in a donor semiconductor we have an additional factor (m/m0 ), we have the affinity of a conductivity band electron to a donor impurity atom Ead = 0.754(m/m0 ) eV and Ebd = 13.598(m/m0 ) eV. At m = 0.1m0 and  = 12 [1] we have Ead = 5.23610 eV = 6.076 K and Ebd = 9.44310 So at temperature considerably lower than Ead (about 6 K here) we have donor atoms acting as active maser centres. We have three energy levels of electrons: zero energy levels at the bottom of the conductivity band, negative energy levels of electron, forming negatively charged donor ions, and deeper negative energy levels of the outer electron of the neutral donor impurity atoms. So we can pump some outer electrons of some neutral donor impurity atoms to the conductivity band. At low enough temperature these electrons will form negative impurity ions with some other neutral donor impurity atoms, thus forming highly populated levels above the ground state level, Ebd. When high population of the upper levels is achieved, the frequency, Ebd  Ead = 8.9210 eV = 2.1410 (1/s) = 0.14 mm, could be generated or amplified. It is rather obvious that the maximum concentration of the negative donor impurity ions, which could be achieved, is 0.5nd (nd is the number of the donor impurity atoms per unit volume of a semiconductor). References 1. A. L. Efros. Semiconductors, in: Encyclopaedic Dictionary Solid State Physics, V. 2 (Kiev: Naukova Dumka, 1998), p. 91 (in Russian). 2. L. D. Landau and E. M. Lifshits, Quantum Mechanics (Oxford: Pergamon, 1986). 3. 1988 CRC Handbook of Chemistry and Physics, ed. R. C. Weast (Boca Raton, FL: CRC).

In some semiconductors donor impurity atoms can attract additional electrons, forming negative donor impurity ions. Thus we have 3 energy levels for electrons: zero energy levels at the bottom of the conductivity band, negative energy levels of the bounded electrons of the negative donor impurity ions, and deeper negative energy levels of the outer electrons of the neutral donor impurity atoms. So the donor impurity atoms could serve as active centres for a maser. The maximum achievable relative population is 0.5. Typical wavelength of the generated oscillation is 0.14 mm; three level scheme could be realized at rather low temperatures, considerably lower than 6 K.

Microsoft Word - Maser Donor type semiconductor at low temperature as maser active medium Yuri Kornyushin Maître Jean Brunschvig Research Unit, Chalet Shalva, Randogne, CH-3975 In some semiconductors donor impurity atoms can attract additional electrons, forming negative donor impurity ions. Thus we have 3 energy levels for electrons: zero energy levels at the bottom of the conductivity band, negative energy levels of the bounded electrons of the negative donor impurity ions, and deeper negative energy levels of the outer electrons of the neutral donor impurity atoms. So the donor impurity atoms could serve as active centres for a maser. The maximum achievable relative population is 0.5. Typical wavelength of the generated oscillation is 0.14 mm; three level scheme could be realized at rather low temperatures, considerably lower than 6 K. Let us consider some donor impurity atom in a donor-type semiconductor having one more outer electron than the host atom of a semiconductor (e.g., the host silicon atom). In the simplest case of a non-degenerate standard conductivity band the equation of a motion of a superfluous electron is the same as that for the electron in a hydrogen atom [1]. The bond energy at that is as follows [1]: Ebd = (m0e )(m/m0 ), (1) where m0 is the free electron mass, m is the effective electron mass in semiconductor conductivity band, e is the electron charge, and  is the Planck constant divided by 2. Comparative to the bond energy in a hydrogen atom (see, e.g., [2]) the right-hand part of Eq. (1) contains additional factor (m/m0 ). At m = 0.1m0 and  = 12 [1] this factor, (m/m0 ) = 6.94410 Hydrogen atom can attract an additional electron, forming negative hydrogen ion [3]. The electron affinity of a free electron to a hydrogen atom is Eah = 0.754 eV [3]. The bond energy in a hydrogen atom, Ebh = (m0e ) = 13.598 eV [3]. Taking into account that in a donor semiconductor we have an additional factor (m/m0 ), we have the affinity of a conductivity band electron to a donor impurity atom Ead = 0.754(m/m0 ) eV and Ebd = 13.598(m/m0 ) eV. At m = 0.1m0 and  = 12 [1] we have Ead = 5.23610 eV = 6.076 K and Ebd = 9.44310 So at temperature considerably lower than Ead (about 6 K here) we have donor atoms acting as active maser centres. We have three energy levels of electrons: zero energy levels at the bottom of the conductivity band, negative energy levels of electron, forming negatively charged donor ions, and deeper negative energy levels of the outer electron of the neutral donor impurity atoms. So we can pump some outer electrons of some neutral donor impurity atoms to the conductivity band. At low enough temperature these electrons will form negative impurity ions with some other neutral donor impurity atoms, thus forming highly populated levels above the ground state level, Ebd. When high population of the upper levels is achieved, the frequency, Ebd  Ead = 8.9210 eV = 2.1410 (1/s) = 0.14 mm, could be generated or amplified. It is rather obvious that the maximum concentration of the negative donor impurity ions, which could be achieved, is 0.5nd (nd is the number of the donor impurity atoms per unit volume of a semiconductor). References 1. A. L. Efros. Semiconductors, in: Encyclopaedic Dictionary Solid State Physics, V. 2 (Kiev: Naukova Dumka, 1998), p. 91 (in Russian). 2. L. D. Landau and E. M. Lifshits, Quantum Mechanics (Oxford: Pergamon, 1986). 3. 1988 CRC Handbook of Chemistry and Physics, ed. R. C. Weast (Boca Raton, FL: CRC).

Mon. Not. R. Astron. Soc. 000, 1–6 (2007) Printed 29 October 2018 (MN LATEX style file v2.2) Exciting the Magnetosphere of the Magnetar CXOU J164710.2-455216 in Westerlund 1 M. P. Muno,1 B. M. Gaensler,2,3 J. S. Clark,3,4 R. de Grijs,5 D. Pooley,6,7 I. R. Stevens,8 & S. F. Portegies Zwart9,10 1Space Radiation Laboratory, California Institute of Technology, Pasadena, CA 91125; mmuno@srl.caltech.edu 2School of Physics A29, The University of Sydney, NSW 2006, Australia 3Harvard-Smithsonian Center for Astrophysics, 60 Garden St. Cambridge, MA 02138 4Department of Physics & Astronomy, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK 5Department of Physics & Astronomy, The University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, U.K. 6Chandra Fellow 7Astronomy Department, University of California at Berkeley, 601 Campbell Hall, Berkeley, CA 94720, USA 8School of Physics & Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK 9Astronomical Institute ’Anton Pannekoek’ Kruislaan 403, 1098SJ Amsterdam, the Netherlands 10Section Computational Science Kruislaan 403, 1098SJ Amsterdam, the Netherlands Accepted 2007 March 15. Received 2007 February 13; in original form 2006 November 25 ABSTRACT We describe XMM-Newton observations taken 4.3 days prior to and 1.5 days subse- quent to two remarkable events that were detected with Swift on 2006 September 21 from the candidate magnetar CXOU J164710.2-455216: (1) a 20 ms burst with an en- ergy of 1037 erg (15–150 keV), and (2) a rapid spin-down (glitch) with ∆P/P ∼ −10−4. We find that the luminosity of the pulsar increased by a factor of 100 in the interval between observations, from 1×1033 to 1×1035 erg s−1 (0.5–8.0 keV), and that its spec- trum hardened. The pulsed count rate increased by a factor of 10 (0.5–8.0 keV), but the fractional rms amplitude of the pulses decreased from 65 to 11 per cent, and their profile changed from being single-peaked to exhibiting three peaks. Similar changes have been observed from other magnetars in response to outbursts, such as that of 1E 2259+586 in 2002 June. We suggest that a plastic deformation of the neutron star’s crust induced a very slight twist in the external magnetic field, which in turn generated currents in the magnetosphere that were the direct cause of the X-ray outburst. Key words: stars: neutron — pulsar: individual (CXOU J164710.2-455216) — X- rays: bursts — stars: magnetic fields 1 INTRODUCTION Young, isolated neutron stars come in a variety of mani- festations, including ordinary radio pulsars, compact cen- tral objects in supernova remnants, soft gamma repeaters (SGRs), and anomalous X-ray pulsars (AXPs). The latter two classes of source share long rotational periods (P=5–10 s), rapid spin-down rates (Ṗ&10−12 s s−1), X-ray luminosi- ties (LX&10 33 erg s−1) that exceed their spin-down power, and the frequent production of second-long soft gamma-ray bursts (Woods & Thompson 2006). These properties sug- gest that they are magnetars, neutron stars powered by the unwinding of extremely strong (B&1015 G) internal mag- netic fields (Thompson & Duncan 1995, 1996). The phenomenology associated with magnetars is thought to be driven by how the unwinding internal fields interact with the crusts of the neutron stars, which in turn determines the geometries of the external magnetic fields (Thompson & Duncan 1995, 1996; Thompson, Lyutikov, & Kulkarni 2002). In some cases, the crusts respond to the unwinding fields plastically, and the energy is gradu- ally deposited into the magnetospheres. This causes tran- sient ‘active periods,’ in which the persistent fluxes in- crease on timescales of weeks to years (Woods et al. 2004; Gotthelf et al. 2004). Fractures may also occur in the crust, which generate waves in the external fields, and in turn pro- duce sudden soft gamma-ray ‘bursts’ with energies up to 1041 erg (Göğüş et al. 2001; Gavriil, Kaspi, & Woods 2002). In the most extreme cases, instabilities can rearrange the entire external magnetic field, producing ‘giant flares’ with energies of 1044 −1046 erg (Hurley et al. 1999; Palmer et al. 2005; Hurley et al. 2005). Finally, changes in the coupling between the bulk of the crust and a superfluid component appear to change the crust’s angular momentum, as is sug- gested by both secular variations in the spin down rates c© 2007 RAS http://arxiv.org/abs/0704.1630v1 2 M. P. Muno et al. Figure 1. Images of the counts received by the EPIC-pn in the 0.5–10 keV band on 2006 September 16 (left) and 22 (right). The images are centred on the core of the star cluster Wester- lund 1 (α, δ = 251.h76792 –45.◦84972 [J2000]). In addition to the AXP CXOU J164710.2-455216, also visible in the images are three bright OB/WR stars. The blank strips in the image are gaps between the chips in the detector array. on time-scales of weeks (Gavriil & Kaspi 2004; Woods et al. 2006) or sudden, day-long episodes of spin-up (‘glitches’) or spin-down (Woods et al. 1999; Gavriil & Kaspi 2003; Dall’Osso et al. 2003; Kaspi et al. 2003; Woods et al. 2004). Unfortunately, the frequent, sensitive monitoring observa- tions that are required to identify transient active periods, to detect bursts, and to track the rotation of these pulsars have not always been available. Therefore, in many cases the causal connections between these phenomena have been unclear (e.g., Gavriil & Kaspi 2003; Woods et al. 2005). Here we report XMM-Newton observations of the 10.6 s X-ray pulsar, CXOU J164710.2-455216 (Muno et al. 2006), that bracketed a series of events that occurred near 2006 September 21. Near this time, Swift detected a soft gamma-ray burst (Krimm et al. 2006) and a glitch with ∆P/P ∼ −10−4 (Israel et al. 2007). These events con- firm our original hypothesis that this source is a magnetar (Muno et al. 2006). We find that during the interval between our two XMM-Newton observations, there were also dra- matic changes in the luminosity, spectrum, and pulse profile of CXOU J164710.2-455216. We compare these to changes observed during active periods from other magnetars, and discuss the implications for the interaction between the mag- netic fields and crusts of the these neutron stars. 2 OBSERVATIONS As part of the guest observer programme, XMM-Newton ob- served CXOU J164710.2-455216 for 46 ks starting on 2005 September 16 at 18:59:38 (UTC). Fortuitously, 4.3 days later, on 2006 September 21 at 01:34:53 (UTC), the Swift Burst Alert Telescope (BAT) detected a 20 ms burst from the direction of Westerlund 1 (Krimm et al. 2006), with an energy of 3×1037 erg (15–150 keV; for a distance D=5 kpc; Clark et al. 2005). In response, the director of XMM-Newton carried out an observation lasting 30 ks beginning 1.5 days later on 2006 September 22 at 12:40:27 (UTC). We analysed the XMM-Newton observations in order to study changes in the X-ray flux, spectrum, and pulse profile. We analysed the data taken with the European Photon Figure 2. Phase-averaged spectra of CXOU J164710.2-455216 taken on 2006 September 16 and 22 (top panels), in units of de- tector counts. Models for the spectra are shown with a solid line: a single absorbed blackbody on September 16, and two absorbed blackbodies on September 22. For the latter spectrum, the cool and hot blackbodies are indicated with the dotted and dashed lines, respectively. The bottom panels display the difference be- tween the data and the models, in units of the 1σ uncertainty on the data. There are systematic residuals at low energies in the September 22 spectrum, but these are not significant enough to affect the overall model. Imaging Camera (EPIC). For most of the timing and spec- tral analysis, we used data taken with 73.4 ms time resolu- tion using the pn array. The data from the MOS arrays were taken with 2.4 s time resolution, which was inadequate for studying the profile of this 10.6 s pulsar. Moreover, the data suffered from pile-up during the second observation, when the source was bright (see below). Therefore, we only used the MOS data to generate spectra for the first observation. We processed the observation data files using the stan- dard tools (epchain and emchain) from the Science Anal- ysis Software version 7.0. The events were filtered in the standard manner, and we adjusted the arrival times of the events to the Solar System barycentre. Images from the EPIC-pn data are displayed in Figure 1. Comparing the data from before and after the Swift burst, we find that CXOU J164710.2-455216 increased in count rate by a factor of 80 (0.5–8.0 keV). Next, we extracted pulse-phase-averaged spectra from within 15′′ of the location of CXOU J164710.2-455216 (α, δ = 251.h79250, –45.◦87136 [J2000]). Estimates of the back- ground were extracted from a 30′′ circular region that was located 1.′5 west of the source region. We obtained the detec- tor response and effective area using standard tools (rmfgen and arfgen). The EPIC-pn spectra are displayed in Figure 2. We modeled these spectra using XSPEC version 12.2.1. We first assumed that the spectra could be described as blackbody emission absorbed by interstellar gas and scat- tered by dust. This model was acceptable for the observa- tions before the burst on September 16 (χ2/ν = 59.4/67), but was inconsistent with the data from September 22 (χ2/ν = 2255/1136). For the later observation, we could model the spectrum with two continuum components, either the sum of two blackbodies, or a blackbody plus power law contin- uum. We assumed that the interstellar absorption column toward the source did not change between observations. The spectral parameters, fluxes, and luminosities for the above models are listed in Table 1. For completeness, we also list c© 2007 RAS, MNRAS 000, 1–6 Exciting a Magnetar’s Fields 3 Table 1. Spectral Models for CXOU J164710.2-455216 2005 2006 May–Jun Sep 16 Sep 22 Two Blackbodies NH (10 22 cm−2) 1.28 1.28 1.28(2) kT1 (keV) 0.60(1) 0.54(1) 0.67(1) Abb,1 (km 2) 0.09(1) 0.08(1) 3.62(2) kT2 (keV) . . . . . . 1.7(1) Abb,2 (km 2) . . . . . . 0.021(6) FX (10 −13 erg cm−2 s−1) 2.3 1.5 215.7 LX (10 33 erg s−1) 1.4 1.0 109.7 Blackbody Plus Power Law NH (10 22 cm−2) 1.44 1.44 1.44(1) kT1 (keV) 0.58(2) 0.52(1) 0.68(1) Abb,1 (km 2) 0.11(1) 0.11(1) 2.87(3) Γ . . . . . . 2.07(4) NΓ (10 −3 cm−2 s−1 keV−1) . . . . . . 3.7(9) FX (10 −13 erg cm−2 s−1) 2.3 1.5 214.1 LX (10 33 erg s−1) 1.5 1.1 130.3 The reduced chi-squared for both joint fits were 1423/1298. The interstellar absorption was assumed not to have changed over the course of these observations. To compute the area of the blackbody emission, we assumed D=5 kpc. NΓ is the photon flux density of the power law at 1 keV. Uncertainties are 1σ, for one degree of freedom. Fluxes are in the 0.5–8.0 keV band. parameters from models of the spectra taken with Chandra during 2005 May and June (Muno et al. 2006). For both models, we found that the luminosity was a factor of 100 higher (0.5–8.0 keV) 1.5 days after the burst than it was 4.3 days before the burst. The increase in flux was largely because the area of the ≈0.5 keV blackbody increased from 0.1 km2 before the burst to ≈3 km2 after the burst. It also resulted from the prominence of the hard component after the burst. Modeled as a kT=1.7 keV black- body, it produced 26 per cent of the observed flux on 2006 September 22 (18 per cent of the absorption-correction flux). Modeled as a Γ=2.07 power law, it produced 50 per cent of the observed flux (70 per cent of the intrinsic flux; 0.5-8.0 keV). If we add these components to our models for the spectra taken on 2006 September 16, we find that their frac- tional contribution to the observed flux was lower: <15 per cent for the blackbody, and <35 per cent for the power law. To identify pulsations from CXOU J164710.2-455216, we computed Fourier periodograms using the Rayleigh statistic. (A search for pulsations from other point sources in the field revealed no other pulsars.) This provided an initial estimate of the pulse period, which we then refined by computing pulse profiles from non-overlapping 5000 s intervals during each observation, measuring their phases by cross-correlating them with the average pulse profile from each observation, and modeling the differences be- tween the assumed and measured phases using first-order polynomials. The best-fitting periods were 10.61065(7) s and 10.61064(8) s for 2006 September 16 and 22, respec- tively. These values are within 1.5σ of the periods measured in 2005 May and June, 10.6112(4) s and 10.6107(1) s, re- spectively (Muno et al. 2006). The reference epochs of the pulse maxima for the two observations in 2006 September were 53994.786313(2) and 54000.526588(1) (MJD, Barycen- tre Dynamical Time). Monitoring observations taken with Figure 3. Pulse profiles of CXOU J164710.2-455216 taken on 2006 September 16 (top panels) and 2006 September 22 (bottom panels), and in three energy bands: 0.5–2.0 keV (left panels), 2.0– 3.5 keV (middle panels), and 3.5–7.0 keV (right panels). Two identical cycles are repeated in each panel. The dashed line in the top panel represents the background count rate. Swift reveal that a glitch with a fractional period change of ∆P/P ∼ −10−4 occurred between these two observations; a discussion of this result is presented in (Israel et al. 2007). We used these ephemerides to compute the pulse pro- files in the full band of 0.5–8.0 keV, and three sub-bands: 0.5–2.0 keV, 2.0–3.5 keV, and 3.5–7.0 keV. The root-mean- squared (rms) amplitudes of the pulsations in the full band (0.5–8.0 keV) increased from 0.02 count s−1 before the burst, to 0.29 count s−1 after the burst. At the same time, the frac- tional rms amplitudes declined from 64 per cent before the burst to 11 per cent after the burst. Moreover, the pulse pro- file changed dramatically after the burst, as can be seen in the profiles from the sub-bands displayed in Figure 3. Before the burst, the pulse at all energies was single peaked, and the differences in the pulse profile as a function of energy are not very pronounced. After the burst, the pulse in the full band displayed three distinct peaks, and a dependence on energy developed. Specifically, in the 3.5–7.0 keV band, the third peak was absent and the flux between the first two peaks (phases 0.1–0.3) was larger, so that the overall profile was more sinusoidal at high energies than at low. We examined whether phase-resolved spectroscopy could provide any insight into the origin of the pulses. Un- fortunately, CXOU J164710.2-455216 was too faint on 2006 September 16 to generate spectra for all but the peak of the pulse. We did examine phase-resolved spectra for 2006 September 22, but found no systematic trend relating the spectral parameters with the intensity as a function of phase. Finally, we searched for bursts by examining the time series of events recorded by the EPIC-pn. We found no ev- idence for bursts producing more than 4 counts within the 73.4 ms frame time, which placed an upper limit to their ob- served fluence of 3×10−11 erg cm−2 (for a Γ=1.8 power law; Krimm et al. 2006), or an energy of <2× 1035 erg (0.5–8.0 keV; D=5 kpc). c© 2007 RAS, MNRAS 000, 1–6 4 M. P. Muno et al. 3 DISCUSSION In the 5.8 days between our two XMM-Newton obser- vations of CXOU J164710.2-455216, a remarkable set of events occurred. First, the phase-averaged luminosity of CXOU J164710.2-455216 increased by a factor of ∼100, from LX = 1 × 10 33 to LX = 1 × 10 35 erg s−1 (0.5– 8.0 keV; Fig. 1; Campana & Israel 2006), and the spec- trum hardened (Table 1). Energetically, this is the most important feature of this active period. In the 1.5 days af- ter the burst, if we conservatively assume the persistent flux from CXOU J164710.2-455216 was constant, the to- tal energy released was ∼1040 erg (0.5–8.0 keV). Second, a 20 ms long burst with an energy of 3 × 1037 erg (15– 150 keV) was detected from this source with the BAT on board Swift (Krimm et al. 2006). Third, a glitch was ob- served in the spin period of the pulsar, with ∆P/P ∼ −10−4 (Israel et al. 2007). Fourth, the pulse profile changed from having a simple, single-peaked structure, to exhibiting three distinct peaks with pronounced energy dependence (Fig. 3). Similar changes in the fluxes, spectra, and timing properties of magnetars have been observed before, but the combina- tion observed from CXOU J164710.2-455216 is unique. It is common for the persistent luminosities of mag- netars to vary on time scales of weeks to years. The per- sistent luminosities from the SGRs 1900+14 (Woods et al. 2001) and 1806–20 (Woods et al. 2006) and the bright AXPs 1E 1048.1–5937 (Gavriil & Kaspi 2004; Tiengo et al. 2005) and 1E 2259+586 (Woods et al. 2004) have been observed to vary by factors of 2–3 around ∼1034 − 1035 erg s−1 (0.5–10 keV). The luminosities of SGR 1627– 41 (Kouveliotou et al. 2003) and the transient AXP XTE J1810–597 (Ibrahim et al. 2004; Gotthelf et al. 2004) have been observed to increase by factors of 100, from ∼1033 to ∼1035 erg s−1 (0.5–10 keV). The larger luminosities, ∼1035 erg s−1, appear to be a rough upper envelope for the persistent 0.5–8.0 keV fluxes of magnetars (not count- ing bursts and giant flares). Indeed, the active period from CXOU J164710.2-455216 also had LX ≈ 10 35 erg s−1 (0.5– 8.0 keV). This persistent flux is generally assumed to be produced because the unwinding internal fields induce grad- ual, plastic deformations in the crust and external magnetic fields, which in turn heats the surface or magnetosphere (Thompson & Duncan 1995, 1996). Therefore, the increase in the flux from CXOU J164710.2-455216 demonstrates that either the unwinding of the internal fields, or the response of the crust to that unwinding, is intermittent and can activate in .5 days. The active periods from magnetars are often accompa- nied by second-long bursts. These bursts are the hallmarks of SGRs, and during their active periods hundreds will oc- cur over the course of a year with energies of up to 1041 erg (2–60 keV; Göğüş et al. 2001).The bursts detected from AXPs have all been weaker, with peak energies of .1038 erg (2–60 keV). In the AXPs XTE J1810–597 (Woods et al. 2005) and 1E 1048.1–5937 (Gavriil, Kaspi, & Woods 2006), the bursts that have been detected are infrequent and rela- tively isolated. In 1E 2259+586 (Woods et al. 2004), a series of bursts were detected during an 11 ks observation that oc- curred within 7 days of the start of an active period in 2002 June. The burst detected from CXOU J164710.2-455216 re- sembles those from 1E 2259+586, in that it occurred very near the start of an active period. The energy of the burst (3×1037 erg; 15–150 keV) is trivial compared to that released as persistent flux (&1040 erg; 0.5–8.0 keV), so it is probably not a trigger, but a symptom of the active period. Under the magnetar model, the bursts that accompany the active pe- riods are caused by fractures that occur in the crust. These fractures inject into the magnetosphere currents that are unstable to to wave motion, which quickly generates hot, X- ray emitting plasma (Thompson & Duncan 1995, 1996). It is reasonable to expect that such fractures would be stronger and occur more frequently when the persistent flux is higher, because the crust is already under stress. Variations in the spin-down rates have been observed from several luminous (LX & 10 34 erg s−1; 0.5–8.0 keV) magnetars. Torque variations have been detected from 1E 1048.1–5937 (Gavriil & Kaspi 2004) and SGR 1806–20 (Woods et al. 2006), in association with their active periods. Sudden period changes have been seen in three cases. Two glitches have been detected from 1RXS J170849–400910 with ∆P/P ∼ −1 × 10−6 and −6 × 10−6 (Gavriil & Kaspi 2003; Dall’Osso et al. 2003). Neither were associated with active periods, but the monitoring observations were sparse, so one could have been missed (Dall’Osso et al. 2003). One glitch accompanied the 2002 June active period of 1E 2259+586 in which the spin period decreased by ∆P/P ∼ −10−6 (Kaspi et al. 2003; Woods et al. 2004). Finally, a dra- matic episode of spin-down occurred near the time of a 1044 erg (3–100 keV) giant flare from SGR 1900+14, with ∆P/P ∼ 10−4 (Woods et al. 1999). This is of comparable magnitude to the glitch from CXOU J164710.2-455216, al- beit of the opposite sign (Israel et al. 2007). The glitch appears to have been a major energetic component of the outburst from CXOU J164710.2-455216. Glitches are ascribed to sudden changes in the moments of inertia of the neutron stars that occur when crustal move- ments change how superfluid in the interior is coupled to the bulk of the crust (e.g., Dall’Osso et al. 2003; Kaspi et al. 2003). The change in rotational energy during the glitch, as- suming most of the star rotates as a solid body, is on order ∆Erot ∼ IΩ∆Ω, where I∼10 45 g cm2 is the moment of in- ertia of a neutron star with mass M=1.4 M⊙ and radius R=1 km. For CXOU J164710.2-455216 Ω=0.6 rad s−1 and ∆Ω=6×10−5 rad s−1, so ∆Erot∼10 40 erg. However, a larger input of energy into the stellar interior may be required to unpin the superfluid vortices and initiate the glitch, ∼1042 erg (e.g., Link & Epstein 1996; Thompson et al. 2000). In contrast, the radiative output of CXOU J164710.2-455216 in the first week of this active period was only ∼1040 erg (0.5– 8.0 keV). Whereas for the giant flare from SGR 1900+14 and the 2002 June active period from 1E 2259+586 it appeared that most of the energy was radiated away from the mag- netosphere (Thompson et al. 2000; Woods et al. 2004), for CXOU J164710.2-455216 most of the energy was probably input into the interior of the neutron star. The change in the pulse profile of CXOU J164710.2-455216 is also difficult to understand from an energetic standpoint. Changes in the qualitative shape of the pulse profiles (as opposed to changes in the pulsed fraction) have only been seen previously from three sources. For 1E 2259+586, the profile before the 2002 June burst exhibited two distinct peaks, whereas after the burst the phases between the peaks contained more flux, so that c© 2007 RAS, MNRAS 000, 1–6 Exciting a Magnetar’s Fields 5 part of the profile resembled a single plateau of emission (Woods et al. 2004). This change is minor compared to that from CXOU J164710.2-455216 in Figure 3. Large changes in the harmonic structure of the pulse profile have only been observed in response to the giant flares from SGRs. For SGR 1900+14 the profile had three peaks before the flare in 1998, and a single peak during and after (Woods et al. 2001). For SGR 1806–20, the opposite change occurred in 2004: it shifted from having a simple, single-pulsed profile to having multiple peaks (Woods et al. 2006). For the SGRs, the changes in the pulse profiles are thought to occur because the multipole structure of the ex- ternal magnetic fields are rearranged. This is reasonable, because the giant flares release a significant fraction of the energy in the external fields. For a dipole, this would be B2extR ∼ 1045 G, where we take Bext∼10 14 G, and R∼10 km (Woods et al. 1999; Hurley et al. 2005). However, for CXOU J164710.2-455216, and to a lesser degree for 1E 2259+586, it is unreasonable to suggest that active periods releasing only ∼1040 erg of X-rays resulted from a significant rearrangement of the exterior magnetic fields. Instead, we suggest that a change occurred in the dis- tribution of currents in the magnetosphere. We hypothesize that the emission in quiescence is thermal emission from the cooling neutron star, which emerges through a hot spot where the opacity of the highly-magnetized atmosphere is lowest (Heyl & Hernquist 1998). A single hot spot on the surface could explain the single-peaked, fully modulated (≈70 per cent rms) pulse in quiescence (Özel, Psaltiz, & Kaspi 2001). We suggest that the active period was initi- ated when a very small twist was imparted to the magnetic field by plastic motions of the crust. Currents formed to compensate for this twist, which heated the surface of the star and resonantly scattered the emission from its surface (Table 1). Both of these would contribute to creating the complex pulse profile (Thompson et al. 2002). If our sce- nario is correct, when this source returns to quiescence, the pulse should regain its single-peaked profile. 4 CONCLUSIONS We have examined the X-ray luminosity, spectrum, and pulse profile of CXOU J164710.2-455216 before and after an interval during which Swift detected a soft gamma-ray burst and a timing glitch from the source. The energy radi- ated from the exterior was too small to have resulted from a significant rearrangement of the external magnetic fields of CXOU J164710.2-455216. Instead, the dramatic change in the pulse profile indicates that the underlying emission mechanism changed. Before the burst, the X-ray emission was probably powered by the thermal energy of the star, whereas afterwards it was powered by currents in the mag- netosphere. Moreover, the glitch required an energy at least as large as the energy released as X-rays, &1040 erg, which suggests that much of the energy of this event was input into the interior of the neutron star. Future X-ray observa- tions of this source will reveal the duration and duty cycle of this active period, which would constrain the amount of energy input into the interior. This could help answer why the emission, which is thought to be produced as the inter- nal fields of magnetars unwind, can remain inactive for years and then suddenly turn on in a few days. ACKNOWLEDGMENTS We thank N. Schartel for providing the discretionary obser- vation, G. Israel for sharing the results of the Swift observa- tions, and the referee for helpful comments. MPM was sup- ported by the NASA XMM Guest Observer Facility; BMG by a Federation Fellowship from the Australian Research Council and an Alfred P. Sloan Research Fellowship; and SFPZ by the Royal Dutch Academy of Arts and Sciences. 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M. et al. 2004, ApJ, 605, 378 c© 2007 RAS, MNRAS 000, 1–6 http://arxiv.org/abs/astro-ph/0703684 6 M. P. Muno et al. Woods P. M., Kouveliotou C., Finger M. H., Göğüş E., Wilson C. A., Patel S. K., Hurley K., Swank J. H. 2006, ApJ, 654, 470 Woods P. M., Thompson C. 2006, in Compact Stellar X- ray Sources, eds. W. Lewin, M. van der Klis, Cambridge University Press, 547 c© 2007 RAS, MNRAS 000, 1–6 Introduction Observations Discussion Conclusions

We describe XMM-Newton observations taken 4.3 days prior to and 1.5 days subsequent to two remarkable events that were detected with Swift on 2006 September 21 from the candidate magnetar CXOU J164710.2-455216: (1) a 20 ms burst with an energy of 1e37 erg (15-150 keV), and (2) a rapid spin-down (glitch) with a fractionap period change of 1e-4. We find that the luminosity of the pulsar increased by a factor of 100 in the interval between observations, from 1e33 to 1e35 erg/s (0.5-8.0 keV), and that its spectrum hardened. The pulsed count rate increased by a factor of 10 (0.5-8.0 keV), but the fractional rms amplitude of the pulses decreased from 65 to 11 per cent, and their profile changed from being single-peaked to exhibiting three peaks. Similar changes have been observed from other magnetars in response to outbursts, such as that of 1E 2259+586 in 2002 June. We suggest that a plastic deformation of the neutron star's crust induced a very slight twist in the external magnetic field, which in turn generated currents in the magnetosphere that were the direct cause of the X-ray outburst.

Introduction Observations Discussion Conclusions

Further Evidence that the Redshifts of AGN Galaxies May Contain Intrinsic Components M.B. Bell1 ABSTRACT In the decreasing intrinsic redshift (DIR) model galaxies are assumed to be born as compact objects that have been ejected with large intrinsic redshift com- ponents, zi, out of the nuclei of mature AGN galaxies. As young AGN galaxies (quasars) they are initially several magnitudes sub-luminous to mature galaxies but their luminosity gradually increases over 108 yrs, as zi decreases and they evolve into mature AGN galaxies (BLLacs, Seyferts and radio galaxies). Evi- dence presented here that on a logz-mv plot the bright edge of the AGN galaxy distribution at z = 0.1 is unquestionably several magnitudes sub-luminous to the brightest radio galaxies is then strong support for this model and makes it likely that the high-redshift AGN galaxies (quasars) are also sub-luminous, having sim- ply been pushed above the radio galaxies on a logz-mv plot by the presence of a large intrinsic component in their redshifts. An increase in luminosity below z = 0.06 is also seen. It is associated in the DIR model with an increase in luminosity as the sources mature but, if real, is difficult to interpret in the cosmological redshift (CR) model since at this low redshift it is unlikely to be associated with a higher star formation rate or an increase in the material used to build galax- ies. Whether it might be possible in the CR model to explain these results by selection effects is also examined. Subject headings: galaxies: active - galaxies: distances and redshifts - galaxies: quasars: general 1. Introduction Because the belief that the redshift of quasars is cosmological has become so entrenched, and the consequences now of it being wrong are so enormous, astronomers are very reluctant 1Herzberg Institute of Astrophysics, National Research Council of Canada, 100 Sussex Drive, Ottawa, ON, Canada K1A 0R6; morley.bell@nrc.gc.ca http://arxiv.org/abs/0704.1631v2 – 2 – to consider other possibilities. However, there is increasing evidence that some galaxies may form around compact, seed objects ejected with a large intrinsic redshift component from the nuclei of mature active galaxies. In this model, as the intrinsic component decreases the compact objects evolve into mature active galaxies in a time frame of a few times 108 yrs (Arp 1997, 1998, 1999; Bell 2002a,b,c,d, 2004, 2006; Bell and McDiarmid 2006, 2007; Burbidge 1999; Galianni et al. 2005; Lopéz-Corredoira and Gutiérrez 2006). In the DIR model radio galaxies represent the end of the AGN galaxy evolutionary sequence, where most of the intrinsic redshift component has disappeared and their luminosity has peaked. Only then can these objects be detected to large cosmological distances and can it be seen that they are good standard candles. There is every reason to assume that at each stage of their evolution (at each zi value) they will also be good standard candles. In this paper AGN refers to the active nucleus and AGN galaxy refers to the nucleus plus host galaxy. It was recently demonstrated that the high redshift AGN galaxies detected to date appear to have a mean distance near 300 Mpc (Bell 2004), and therefore few beyond ∼ 500 Mpc will have been detected. However, in the DIR model it is assumed that this birthing process through compact object ejection has taken place at all cosmological epochs and that those galaxies that were born in the early universe still survive today, even though they will have almost certainly evolved beyond the mature AGN galaxy (radio galaxy) stage. Although they may no longer contain active nuclei, by this point in their evolution their redshifts will contain only a very small intrinsic redshift component. This remnant intrinsic redshift is observed to-day in common spiral galaxies (Tifft 1996, 1997; Bell and Comeau 2003; Bell, Comeau and Russell 2004), and the local Hubble constant is found to be Ho = 58 km s−1 Mpc−1 when the intrinsic components are removed (Bell and Comeau 2003; Bell, Comeau and Russell 2004). This value is smaller than the value (Ho = 72) obtained by Freedman et al. (2001) before removal of the intrinsic components. In most respects the DIR model is perfectly compatible with the standard Big Bang model of the Universe. It differs mainly in the way galaxies are born and the claim that in this model at least the radio galaxies pass through an initial short-lived AGN period (108 yrs) in which their redshifts contain an intrinsic component that quickly disappears. After that, as they evolve through the next 1010 years they can be used as they are today, to study cosmology. Although there is now a considerable amount of evidence supporting the DIR model, there are also some well-known arguments against this model that have been raised by those who support the CR model (e.g. the Lyman forest, lensing by intervening galaxies, etc.). An explanation of these arguments in the DIR model can be found in the Discussion section of a previous paper (Bell 2004). In the CR model the location of high-redshift AGN galaxies (quasars) on a logz-mv plot can be explained by the presence of a non-thermal component superimposed on their optical – 3 – luminosity. In the DIR model their location on this plot is explained by the presence of a non-cosmological redshift component superimposed on their redshift. This paper uses an updated logz-mv plot containing over 100,000 AGN galaxies to compare the most luminous radio galaxies and first-ranked cluster galaxies at each redshift to the high luminosity edge of the AGN galaxy distribution in an attempt to see which model (CR or DIR model) can best explain the data. In this paper the standard candle (constant luminosity) slope is used as a reference to make luminosity comparisons at a given redshift. This is shown as a dashed line in Fig 1 and a solid line in Fig 2. Luminosity increases to the left. 2. The Data A logz-mv plot for those radio sources with measured redshifts that were detected in the 1 Jy radio survey (Stickel et al. 1994) is presented in Fig 1. The quasars are plotted as filled circles and the radio galaxies as open squares. As discussed above, in the DIR model the radio galaxies are the objects that high-redshift quasars and other AGN galaxies evolve into when their intrinsic redshift component has largely disappeared. In Fig 1, first-ranked cluster galaxies (Sandage 1972a; Kristian et al. 1978) are indicated by the dashed line. The most luminous radio galaxies, like first-ranked cluster galaxies, are clearly good standard candles to large cosmological distances, and their redshifts must then be cosmological, as expected in both the CR and DIR models since any intrinsic redshift component will have almost completely disappeared. All the sources listed as quasars and active galaxies in the updated Véron-Cetty/Véron catalogue (Véron-Cetty and Véron 2006) (hereafter VCVcat) are plotted in Fig 2. Since the VCVcat is made up of AGN galaxies from many different surveys, there will undoubtedly be differences in the selection criteria involved. However, since AGN galaxies are easily distinguishable from other types of galaxies, the normally strict selection criteria are not required in this case to obtain a source sample that is made up almost entirely of AGN galaxies. In that sense the VCVcat is probably the most complete sample of AGN galaxies available to-day. Because the source distribution in the plot in Fig 2 is continuous, the sources listed as quasars and AGN are clearly the same, and there is therefore no reason to separate them into two different categories as was done arbitrarily in the VCVcat. This should not be too surprising since they have long been lumped together in unification models (Antonucci 1993). In Fig 2 the abrupt decrease in the number of sources for 0.5 < z < 3 and mv > 21 is explained by a faint magnitude cut-off near mv = 21m. It cannot affect the conclusions drawn here because at each redshift we are only comparing the bright, or high luminosity, edge of the source distribution (where the source density increases sharply when – 4 – moving from bright to faint). For example, in Fig 2, at z = 0.03, 0.06, 0.15 and 1, the high luminosity edge of the AGN galaxy distribution is at mv = 14, 15, 18, 17, respectively. However, some surveys have had other observer, or program-imposed limits applied that can also affect the bright edge of the source distribution and this is discussed in more detail in Section 3.1. The slope change in the high-redshift tail (z > 3) may be due to uncertainties in converting to visual magnitudes and/or to large k-dimming effects that have been unaccounted for. Whatever the cause, it will also not affect the arguments presented here that only apply to sources at lower redshifts. 3. Discussion In Fig 1, the large triangle shows where the quasars would be located in the DIR model if the intrinsic component in their redshifts could be removed. All must lie below the radio galaxies. In this plot there are no AGN galaxies below the radio galaxies, and it is therefore easy to conclude that quasars are at the distance implied by their redshifts and are therefore super-luminous to first ranked cluster galaxies at all epochs. This was the conclusion drawn by Sandage (1972b, see his Fig 4) from a plot similar to Fig 1. Sandage argued that since no quasars lie to the right (fainter) of the radio galaxy distribution, this can be understood if a quasar consists of a normal, strong radio galaxy with a non-thermal component superimposed on its optical luminosity. He concluded from this evidence that quasars redshifts are cosmological. In Fig 2 many of the high redshift quasars are also located above the radio galaxies, however, here most of the low- and intermediate-redshift AGN galaxies fall below the radio galaxy line. This is what is expected in the DIR model where AGN galaxies are born sub- luminous and reach their most luminous point when the intrinsic redshift component has disappeared. They must therefore all fall below the mature galaxy line. If those detected to date are all nearer than ∼ 500 Mpc (Bell 2004) most will also be located below the dashed line at z = 0.1 in Fig 2. This is what is seen in Fig 2 when the intrinsic component is small. The fact that low-redshift AGN galaxies are located below this line when the intrinsic component is too small to push them above it, suggests strongly that it is only the intrinsic component present in the high redshift sources that has pushed these sources above the radio galaxies. This argument is also supported by the shape of the plot in Fig 2, which starts out flat near z = 0.06, steepening gradually to z = 0.2 and then more rapidly to high redshifts. This conclusion is further supported by the fact that the zi ∼ 0 AGN galaxies (radio galaxies) are good standard candles, and there is therefore no reason to think that the other AGN galaxies will not be, for a given intrinsic redshift value. – 5 – Because almost all of the AGN galaxies are less luminous than the highest luminosity radio galaxies and first-ranked cluster galaxies at redshifts below z ∼ 0.3, the explanation proposed by Sandage (1972b) can no longer be valid. Quasars cannot be normal radio galaxies, or even Seyferts, with a non-thermal optical component superimposed. In fact, since the high luminosity edge of the AGN galaxy distribution in Fig 2 is ∼ 3 mag fainter than the high luminosity edge of the radio galaxies at z = 0.1, if quasars are sub-luminous galaxies brightened by a superimposed non-thermal optical component, at z = 2 this su- perimposed component would have to increase the optical luminosity of the source by up to ∼ 9 magnitudes. This could even get worse at higher redshifts when k-dimming effects are included, which would make the standard model involving a superimposed non-thermal nuclear component increasingly difficult to believe. In the CR model the peak in quasar activity (luminosity and number) near z = 2 is assumed to be associated with a period when the star formation rate was higher than at present, and because there was more raw material around to make galaxies. In Fig 2, not only does the high luminosity edge of the AGN galaxies get intrinsically much fainter towards low redshifts (moving further to the right relative to the standard candle slope), below z ∼ 0.3 this decrease in luminosity begins to slow down. Below z = 0.1 their luminosity begins to increase again, eventually approaching that of the brightest radio galaxies. How is this to be explained in the CR model when we can no longer use the argument that there is more raw material around? This is one of the questions that will need to be addressed if the CR model is to continue to be favored, since this increase is exactly what is predicted in the DIR model as the AGN galaxies mature into radio galaxies. One possible explanation in the CR model is discussed in the following section. 3.1. Selection Effects in the Data Although in a sample like VCVcat it is difficult to take into account all of the selection effects that might be active, since the Sloan Digital Sky Survey (SDSS) sources are likely to make up the largest single portion of the sample the target selection process in that survey is worth examining. First, the survey is sensitive to all redshifts lower than z = 5.8, and the overall completeness is expected to be over 90% (Richards et al. 2002). Extended sources were also targeted as low-redshift quasar candidates in order to investigate the evolution of AGN at the faint end of the luminosity function. During the color selection process no distinction was made between quasars and the less luminous Seyfert nuclei. Objects that had the colors of low-redshift AGN galaxies were targeted even if they were resolved. This policy was in contrast to some other quasar surveys that reject extended objects, thereby imposing – 6 – a lower limit to the redshift distribution of the survey (Richards et al. 2002). In addition to selecting normal quasars, the selection algorithm also makes it sensitive to atypical AGN such as broad absorption line quasars and heavily reddened quasars (Richards et al. 2002). In addition to the detection limit set by the sensitivity of the observing system the SDSS also contains two additional observer, or program-imposed, limits. One of these was a faint-edge limit at i∗ = 19.1m, and the other was a bright-edge cut-off at i∗ = 15m. The reasons why these limits were imposed can be found in Richards et al. (2002). Although color-selected quasar candidates below z = 3 were only targeted to a Galactic extinction- corrected i∗ magnitude of 19.1, as noted above, since we are only examining the bright edge of the logz-mv plot, this faint edge limit is not expected to have affected the results. However, the bright edge cut-off at i∗ = 15m could have affected the shape of the bright edge of the logz-mv plot and this needs to be examined more closely. In Fig 2, for 0.7 < z < 3 it is possible that the bright edge cut-off could have prevented the detection of some of the brighter sources, although if many were missed we might expect to see some evidence of a sharp cut-off along the bright edge similar to that seen at mv ∼ 21m. None is seen. Furthermore, since the bright edge of the distribution between z = 0.1 and z = 0.5 is at least 1 magnitude fainter than many sources detected at the higher redshifts it seems unlikely that the i∗ = 15 limit could have significantly affected the bright edge of the distribution in this redshift range. In fact, it is apparent from Fig 2 of Schneider et al. (2007) (which is a plot of the i magnitude of the 77,429 objects in the SDSS Fifth Data Release quasar catalogue versus redshift) that in the SDSS catalogue it is unlikely that many sources were missed at any redshift because of the cut-off at i = 15. It is also worth noting that the sources that lie outside the limits imposed in the SDSS have not been discarded. SDSS photometry for those objects brighter than i∗ = 15 is sufficiently accurate that they can be used in follow-up studies should the need arise. (Richards et al. 2002). In Fig 2 there is also an increase seen in the number of AGN galaxies as z increases. Such an increase is expected in the CR model where the redshift is distance related and where it would be due to the increasing volume of space sampled as z increases. This would then support the CR model. However, it needs to be kept in mind also that if a bright edge cut-off is affecting the shape of the bright edge of the source distribution, it would presumably also have created this increase in source number with redshift by preventing the detection of many more of the bright sources at low redshifts. In the DIR model, where the redshift of AGN galaxies is age related, the number density of sources as a function of cosmological redshift can only be determined after the intrinsic component is removed. – 7 – This paper examines the AGN galaxies listed in the VCVcat and draws conclusions based on that sample. It contains the quasars found in the SDSS that were available at the time the catalog was prepared, and approximately 11,000 Seyferts and BLLacs, but whether the current VCVcat contains many AGN galaxies found in the SDSS galaxy survey is unclear. Hao et al. (2005a,b) have pointed out that although the color selection technique used in the SDSS is very efficient, selecting AGN galaxies is a complex process and requires that the optical luminosity of the active nucleus be at least comparable to the luminosity of the host galaxy for the color to be distinctive. Thus the color selection systematically misses AGN galaxies with less luminous nuclei at low redshift. If mainly faint sources at redshifts below z = 0.08 were missed, it is conceivable that the bright edge currently visible near mv ∼ 14 might have been created by the selection process. In this case there might be no luminosity increase below z = 0.1, which would be more easily explained in the CR model. However, if the VCVcat does not contain many AGN galaxies found in the SDSS galaxy survey this would not be a problem here. Furthermore, in the DIR model, where the luminosity of the host galaxy is predicted to increase as it matures, presumably bright AGN galaxies as well as faint ones could be missed if the host galaxy has brightened significantly so as to swamp the nucleus. Also, in Fig 2, the bend in the distribution towards higher luminosities near z = 0.07 and mv = 18 does seem to point to a real increase in the luminosity at lower redshifts. However, if this sample is incomplete at low redshifts for a particular magnitude range, the conclusions drawn here may change when a more complete sample becomes available. Also, if AGN galaxies at vastly different redshifts are to be compared, as here, it is important that the optical magnitude of the entire galaxy be used and not simply that of the nucleus. It is the total magnitude that has invariably been used for high redshift quasars because of the difficulty of separating the nuclear and host contributions. Hao et al. (2005b) point out that, in attempting to obtain the luminosity function of the active nucleus, it is important that it not be contaminated by the host galaxy. Since the brightening predicted in the DIR model below z = 0.1 is due to the host galaxy maturing and increasing in luminosity, the contribution from the entire host galaxy must be included in the magnitudes used in the logz-mv plot if the brightening is to be detected. Although the luminosity of the nucleus may be adequate in determining the luminosity function of the active nucleus in the CR model, because of the complex process required to identify AGN galaxies (Hao et al. 2005b), obviously great care will be required in obtaining the magnitudes of low redshift AGN galaxies if they are to be used in logz-mv plots. – 8 – 4. Conclusion The most luminous radio galaxies and first-ranked cluster galaxies have been compared here to the high luminosity edge of the AGN galaxy distribution on a logz-mv plot. It is found that while the radio galaxies and cluster galaxies are good standard candles at all epochs, the luminosity of the AGN galaxies varies significantly from one epoch to another. Compared to the comparison galaxies the AGN galaxies are found to be super-luminous at high redshifts, but become sub-luminous as the redshift decreases. These new results show that below z = 0.3 the rate of luminosity decrease begins to slow down and below z = 0.1 the luminosity begins to increase again. Although their apparent super-luminous nature at high z can be explained by a higher star formation rate, and the fact that there might have been more raw material around to make galaxies at that epoch, a luminosity increase below z = 0.1 is more difficult to explain when these arguments are unlikely explanations. It is therefore concluded here that the evidence favors the argument that the high redshift AGN galaxies (quasars) that do lie above the mature galaxy line on a logz-mv plot have all been pushed there because of a large intrinsic component in their redshifts and not because they have a superimposed non-thermal component that is many magnitudes brighter than that seen in radio galaxies. All AGN galaxies then will be sub-luminous to mature galaxies, as predicted in the DIR model. For a given intrinsic redshift component, all are likely also to be good standard candles. Finally, if it turns out that many faint AGN galaxies at low redshifts have been missed in a particular magnitude range the conclusion that the bright edge of the logz-mv plot increases again in this redshift range may need to be re-evaluated. Such an effect might be introduced by the selection effect discussed by Hao et al. (2005a,b), but only if the VCVcat contains many of the SDSS AGN galaxies, as explained in Sec 3.1. I wish to thank two anonymous referees for suggestions on how this paper might be improved. I also thank Dr. D. McDiarmid for helpful comments. – 9 – REFERENCES Antonucci, R. 1993, ARA&A, 31, 473 Arp, H. 1997, A&A, 319, 33 Arp, H. 1998, ApJ, 496, 661 Arp, H. 1999, ApJ, 525, 596 Bell, M.B. 2002a, ApJ, 566, 705 Bell, M.B. 2002b, ApJ, 567, 801 Bell, M.B. 2002c, preprint astro-ph/0208320 Bell, M.B. 2002d, preprint astro-ph/0211091 Bell, M.B. and Comeau, S.P. 2003, preprint astro-ph/0305060 Bell, M.B. 2004, ApJ, 616, 738 Bell, M.B., Comeau, S.P. and Russell, D.G. 2004, astro-ph/0407591 Bell, M. B. 2006, astro-ph/0602242 Bell, M.B. and McDiarmid, D. 2006, ApJ, 648, 140 Bell, M.B. and McDiarmid, D. 2007, astro-ph/0701093 Burbidge, E.M. 1999, ApJ, 511, L9 Freedman, W.L., et al. 2001, ApJ, 553,47 Galianni, P., Burbidge, E.M., Arp, H., Junkkarinen, V., Burbidge, G., and Zibetti, S. 2005, ApJ, 620, 88 Hao, L., etal. 2005a, AJ, 129, 1783, (and arXiv:astro-ph/0501059v1) Hao, L., et al. 2005b, AJ, 129, 1795, (and arXiv:astro-ph/0501042v2) Kristian, J., Sandage, A., and Westphal, J.A. 1978, ApJ, 221, 383 Lopéz-Corredoira, M. and Gutiérrez, C.M. 2006, astro-ph/0609514 Richards, E.T. et al. 2002, AJ, 123, 2945 http://arxiv.org/abs/astro-ph/0208320 http://arxiv.org/abs/astro-ph/0211091 http://arxiv.org/abs/astro-ph/0305060 http://arxiv.org/abs/astro-ph/0407591 http://arxiv.org/abs/astro-ph/0602242 http://arxiv.org/abs/astro-ph/0701093 http://arxiv.org/abs/astro-ph/0501059 http://arxiv.org/abs/astro-ph/0501042 http://arxiv.org/abs/astro-ph/0609514 – 10 – Sandage, A. 1972a, ApJ, 178, 1 Sandage, A. 1972b, ApJ, 178, 25 Schneider et al. 2007, AJ, 134, 102 Stickel, M., Meisenheimer, K., and Kuhr, H. 1994, A&AS, 105, 211 Tifft, W.G. 1996, ApJ, 468, 491 Tifft, W.G. 1997, ApJ, 485, 465 Véron-Cetty, M.P. and Véron, P. 2006, A&A, 455, 773 This preprint was prepared with the AAS LATEX macros v5.2. – 11 – Fig. 1.— Plot of redshift versus optical magnitude for quasars (filled circles) and radio galaxies (open squares) from (Stickel et al. 1994). The dashed line represents brightest clus- ter galaxies from (Sandage 1972a; Kristian et al. 1978). See text for an explanation of the triangle. – 12 – Fig. 2.— Logz-mv plot of all 106,958 sources listed as quasars and active galaxies in the Véron-Cetty-Véron catalogue. The solid line indicates first-ranked clusters from Sandage (1972a); Kristian et al. (1978). The dashed line indicates the maximum distance for high- redshift AGN detected to date from (Bell 2004). In the DIR model any AGN that lie above this line have been pushed there by the presence of an intrinsic redshift component. Introduction The Data Discussion Selection Effects in the Data Conclusion

In the decreasing intrinsic redshift (DIR) model galaxies are assumed to be born as compact objects that have been ejected with large intrinsic redshift components, z_(i), out of the nuclei of mature AGN galaxies. As young AGN (quasars) they are initially several magnitudes sub-luminous to mature galaxies but their luminosity gradually increases over 10^8 yrs, as z_(i) decreases and they evolve into mature AGN (Seyferts and radio galaxies). Evidence presented here that low- and intermediate-redshift AGN are unquestionably sub-luminous to radio galaxies is then strong support for this model and makes it likely that the high-redshift AGN (quasars) are also sub-luminous, having simply been pushed above the radio galaxies on a logz-m_(v) plot by the presence of a large intrinsic component in their redshifts. An increase in luminosity below z = 0.06 is also seen. It is associated in the DIR model with an increase in luminosity as the sources mature but, if real, is difficult to interpret in the cosmological redshift (CR) model since at this low redshift it is unlikely to be associated with a higher star formation rate or an increase in the material used to build galaxies. Whether it might be possible in the CR model to explain these results by selection effects is also examined.

Introduction Because the belief that the redshift of quasars is cosmological has become so entrenched, and the consequences now of it being wrong are so enormous, astronomers are very reluctant 1Herzberg Institute of Astrophysics, National Research Council of Canada, 100 Sussex Drive, Ottawa, ON, Canada K1A 0R6; morley.bell@nrc.gc.ca http://arxiv.org/abs/0704.1631v2 – 2 – to consider other possibilities. However, there is increasing evidence that some galaxies may form around compact, seed objects ejected with a large intrinsic redshift component from the nuclei of mature active galaxies. In this model, as the intrinsic component decreases the compact objects evolve into mature active galaxies in a time frame of a few times 108 yrs (Arp 1997, 1998, 1999; Bell 2002a,b,c,d, 2004, 2006; Bell and McDiarmid 2006, 2007; Burbidge 1999; Galianni et al. 2005; Lopéz-Corredoira and Gutiérrez 2006). In the DIR model radio galaxies represent the end of the AGN galaxy evolutionary sequence, where most of the intrinsic redshift component has disappeared and their luminosity has peaked. Only then can these objects be detected to large cosmological distances and can it be seen that they are good standard candles. There is every reason to assume that at each stage of their evolution (at each zi value) they will also be good standard candles. In this paper AGN refers to the active nucleus and AGN galaxy refers to the nucleus plus host galaxy. It was recently demonstrated that the high redshift AGN galaxies detected to date appear to have a mean distance near 300 Mpc (Bell 2004), and therefore few beyond ∼ 500 Mpc will have been detected. However, in the DIR model it is assumed that this birthing process through compact object ejection has taken place at all cosmological epochs and that those galaxies that were born in the early universe still survive today, even though they will have almost certainly evolved beyond the mature AGN galaxy (radio galaxy) stage. Although they may no longer contain active nuclei, by this point in their evolution their redshifts will contain only a very small intrinsic redshift component. This remnant intrinsic redshift is observed to-day in common spiral galaxies (Tifft 1996, 1997; Bell and Comeau 2003; Bell, Comeau and Russell 2004), and the local Hubble constant is found to be Ho = 58 km s−1 Mpc−1 when the intrinsic components are removed (Bell and Comeau 2003; Bell, Comeau and Russell 2004). This value is smaller than the value (Ho = 72) obtained by Freedman et al. (2001) before removal of the intrinsic components. In most respects the DIR model is perfectly compatible with the standard Big Bang model of the Universe. It differs mainly in the way galaxies are born and the claim that in this model at least the radio galaxies pass through an initial short-lived AGN period (108 yrs) in which their redshifts contain an intrinsic component that quickly disappears. After that, as they evolve through the next 1010 years they can be used as they are today, to study cosmology. Although there is now a considerable amount of evidence supporting the DIR model, there are also some well-known arguments against this model that have been raised by those who support the CR model (e.g. the Lyman forest, lensing by intervening galaxies, etc.). An explanation of these arguments in the DIR model can be found in the Discussion section of a previous paper (Bell 2004). In the CR model the location of high-redshift AGN galaxies (quasars) on a logz-mv plot can be explained by the presence of a non-thermal component superimposed on their optical – 3 – luminosity. In the DIR model their location on this plot is explained by the presence of a non-cosmological redshift component superimposed on their redshift. This paper uses an updated logz-mv plot containing over 100,000 AGN galaxies to compare the most luminous radio galaxies and first-ranked cluster galaxies at each redshift to the high luminosity edge of the AGN galaxy distribution in an attempt to see which model (CR or DIR model) can best explain the data. In this paper the standard candle (constant luminosity) slope is used as a reference to make luminosity comparisons at a given redshift. This is shown as a dashed line in Fig 1 and a solid line in Fig 2. Luminosity increases to the left. 2. The Data A logz-mv plot for those radio sources with measured redshifts that were detected in the 1 Jy radio survey (Stickel et al. 1994) is presented in Fig 1. The quasars are plotted as filled circles and the radio galaxies as open squares. As discussed above, in the DIR model the radio galaxies are the objects that high-redshift quasars and other AGN galaxies evolve into when their intrinsic redshift component has largely disappeared. In Fig 1, first-ranked cluster galaxies (Sandage 1972a; Kristian et al. 1978) are indicated by the dashed line. The most luminous radio galaxies, like first-ranked cluster galaxies, are clearly good standard candles to large cosmological distances, and their redshifts must then be cosmological, as expected in both the CR and DIR models since any intrinsic redshift component will have almost completely disappeared. All the sources listed as quasars and active galaxies in the updated Véron-Cetty/Véron catalogue (Véron-Cetty and Véron 2006) (hereafter VCVcat) are plotted in Fig 2. Since the VCVcat is made up of AGN galaxies from many different surveys, there will undoubtedly be differences in the selection criteria involved. However, since AGN galaxies are easily distinguishable from other types of galaxies, the normally strict selection criteria are not required in this case to obtain a source sample that is made up almost entirely of AGN galaxies. In that sense the VCVcat is probably the most complete sample of AGN galaxies available to-day. Because the source distribution in the plot in Fig 2 is continuous, the sources listed as quasars and AGN are clearly the same, and there is therefore no reason to separate them into two different categories as was done arbitrarily in the VCVcat. This should not be too surprising since they have long been lumped together in unification models (Antonucci 1993). In Fig 2 the abrupt decrease in the number of sources for 0.5 < z < 3 and mv > 21 is explained by a faint magnitude cut-off near mv = 21m. It cannot affect the conclusions drawn here because at each redshift we are only comparing the bright, or high luminosity, edge of the source distribution (where the source density increases sharply when – 4 – moving from bright to faint). For example, in Fig 2, at z = 0.03, 0.06, 0.15 and 1, the high luminosity edge of the AGN galaxy distribution is at mv = 14, 15, 18, 17, respectively. However, some surveys have had other observer, or program-imposed limits applied that can also affect the bright edge of the source distribution and this is discussed in more detail in Section 3.1. The slope change in the high-redshift tail (z > 3) may be due to uncertainties in converting to visual magnitudes and/or to large k-dimming effects that have been unaccounted for. Whatever the cause, it will also not affect the arguments presented here that only apply to sources at lower redshifts. 3. Discussion In Fig 1, the large triangle shows where the quasars would be located in the DIR model if the intrinsic component in their redshifts could be removed. All must lie below the radio galaxies. In this plot there are no AGN galaxies below the radio galaxies, and it is therefore easy to conclude that quasars are at the distance implied by their redshifts and are therefore super-luminous to first ranked cluster galaxies at all epochs. This was the conclusion drawn by Sandage (1972b, see his Fig 4) from a plot similar to Fig 1. Sandage argued that since no quasars lie to the right (fainter) of the radio galaxy distribution, this can be understood if a quasar consists of a normal, strong radio galaxy with a non-thermal component superimposed on its optical luminosity. He concluded from this evidence that quasars redshifts are cosmological. In Fig 2 many of the high redshift quasars are also located above the radio galaxies, however, here most of the low- and intermediate-redshift AGN galaxies fall below the radio galaxy line. This is what is expected in the DIR model where AGN galaxies are born sub- luminous and reach their most luminous point when the intrinsic redshift component has disappeared. They must therefore all fall below the mature galaxy line. If those detected to date are all nearer than ∼ 500 Mpc (Bell 2004) most will also be located below the dashed line at z = 0.1 in Fig 2. This is what is seen in Fig 2 when the intrinsic component is small. The fact that low-redshift AGN galaxies are located below this line when the intrinsic component is too small to push them above it, suggests strongly that it is only the intrinsic component present in the high redshift sources that has pushed these sources above the radio galaxies. This argument is also supported by the shape of the plot in Fig 2, which starts out flat near z = 0.06, steepening gradually to z = 0.2 and then more rapidly to high redshifts. This conclusion is further supported by the fact that the zi ∼ 0 AGN galaxies (radio galaxies) are good standard candles, and there is therefore no reason to think that the other AGN galaxies will not be, for a given intrinsic redshift value. – 5 – Because almost all of the AGN galaxies are less luminous than the highest luminosity radio galaxies and first-ranked cluster galaxies at redshifts below z ∼ 0.3, the explanation proposed by Sandage (1972b) can no longer be valid. Quasars cannot be normal radio galaxies, or even Seyferts, with a non-thermal optical component superimposed. In fact, since the high luminosity edge of the AGN galaxy distribution in Fig 2 is ∼ 3 mag fainter than the high luminosity edge of the radio galaxies at z = 0.1, if quasars are sub-luminous galaxies brightened by a superimposed non-thermal optical component, at z = 2 this su- perimposed component would have to increase the optical luminosity of the source by up to ∼ 9 magnitudes. This could even get worse at higher redshifts when k-dimming effects are included, which would make the standard model involving a superimposed non-thermal nuclear component increasingly difficult to believe. In the CR model the peak in quasar activity (luminosity and number) near z = 2 is assumed to be associated with a period when the star formation rate was higher than at present, and because there was more raw material around to make galaxies. In Fig 2, not only does the high luminosity edge of the AGN galaxies get intrinsically much fainter towards low redshifts (moving further to the right relative to the standard candle slope), below z ∼ 0.3 this decrease in luminosity begins to slow down. Below z = 0.1 their luminosity begins to increase again, eventually approaching that of the brightest radio galaxies. How is this to be explained in the CR model when we can no longer use the argument that there is more raw material around? This is one of the questions that will need to be addressed if the CR model is to continue to be favored, since this increase is exactly what is predicted in the DIR model as the AGN galaxies mature into radio galaxies. One possible explanation in the CR model is discussed in the following section. 3.1. Selection Effects in the Data Although in a sample like VCVcat it is difficult to take into account all of the selection effects that might be active, since the Sloan Digital Sky Survey (SDSS) sources are likely to make up the largest single portion of the sample the target selection process in that survey is worth examining. First, the survey is sensitive to all redshifts lower than z = 5.8, and the overall completeness is expected to be over 90% (Richards et al. 2002). Extended sources were also targeted as low-redshift quasar candidates in order to investigate the evolution of AGN at the faint end of the luminosity function. During the color selection process no distinction was made between quasars and the less luminous Seyfert nuclei. Objects that had the colors of low-redshift AGN galaxies were targeted even if they were resolved. This policy was in contrast to some other quasar surveys that reject extended objects, thereby imposing – 6 – a lower limit to the redshift distribution of the survey (Richards et al. 2002). In addition to selecting normal quasars, the selection algorithm also makes it sensitive to atypical AGN such as broad absorption line quasars and heavily reddened quasars (Richards et al. 2002). In addition to the detection limit set by the sensitivity of the observing system the SDSS also contains two additional observer, or program-imposed, limits. One of these was a faint-edge limit at i∗ = 19.1m, and the other was a bright-edge cut-off at i∗ = 15m. The reasons why these limits were imposed can be found in Richards et al. (2002). Although color-selected quasar candidates below z = 3 were only targeted to a Galactic extinction- corrected i∗ magnitude of 19.1, as noted above, since we are only examining the bright edge of the logz-mv plot, this faint edge limit is not expected to have affected the results. However, the bright edge cut-off at i∗ = 15m could have affected the shape of the bright edge of the logz-mv plot and this needs to be examined more closely. In Fig 2, for 0.7 < z < 3 it is possible that the bright edge cut-off could have prevented the detection of some of the brighter sources, although if many were missed we might expect to see some evidence of a sharp cut-off along the bright edge similar to that seen at mv ∼ 21m. None is seen. Furthermore, since the bright edge of the distribution between z = 0.1 and z = 0.5 is at least 1 magnitude fainter than many sources detected at the higher redshifts it seems unlikely that the i∗ = 15 limit could have significantly affected the bright edge of the distribution in this redshift range. In fact, it is apparent from Fig 2 of Schneider et al. (2007) (which is a plot of the i magnitude of the 77,429 objects in the SDSS Fifth Data Release quasar catalogue versus redshift) that in the SDSS catalogue it is unlikely that many sources were missed at any redshift because of the cut-off at i = 15. It is also worth noting that the sources that lie outside the limits imposed in the SDSS have not been discarded. SDSS photometry for those objects brighter than i∗ = 15 is sufficiently accurate that they can be used in follow-up studies should the need arise. (Richards et al. 2002). In Fig 2 there is also an increase seen in the number of AGN galaxies as z increases. Such an increase is expected in the CR model where the redshift is distance related and where it would be due to the increasing volume of space sampled as z increases. This would then support the CR model. However, it needs to be kept in mind also that if a bright edge cut-off is affecting the shape of the bright edge of the source distribution, it would presumably also have created this increase in source number with redshift by preventing the detection of many more of the bright sources at low redshifts. In the DIR model, where the redshift of AGN galaxies is age related, the number density of sources as a function of cosmological redshift can only be determined after the intrinsic component is removed. – 7 – This paper examines the AGN galaxies listed in the VCVcat and draws conclusions based on that sample. It contains the quasars found in the SDSS that were available at the time the catalog was prepared, and approximately 11,000 Seyferts and BLLacs, but whether the current VCVcat contains many AGN galaxies found in the SDSS galaxy survey is unclear. Hao et al. (2005a,b) have pointed out that although the color selection technique used in the SDSS is very efficient, selecting AGN galaxies is a complex process and requires that the optical luminosity of the active nucleus be at least comparable to the luminosity of the host galaxy for the color to be distinctive. Thus the color selection systematically misses AGN galaxies with less luminous nuclei at low redshift. If mainly faint sources at redshifts below z = 0.08 were missed, it is conceivable that the bright edge currently visible near mv ∼ 14 might have been created by the selection process. In this case there might be no luminosity increase below z = 0.1, which would be more easily explained in the CR model. However, if the VCVcat does not contain many AGN galaxies found in the SDSS galaxy survey this would not be a problem here. Furthermore, in the DIR model, where the luminosity of the host galaxy is predicted to increase as it matures, presumably bright AGN galaxies as well as faint ones could be missed if the host galaxy has brightened significantly so as to swamp the nucleus. Also, in Fig 2, the bend in the distribution towards higher luminosities near z = 0.07 and mv = 18 does seem to point to a real increase in the luminosity at lower redshifts. However, if this sample is incomplete at low redshifts for a particular magnitude range, the conclusions drawn here may change when a more complete sample becomes available. Also, if AGN galaxies at vastly different redshifts are to be compared, as here, it is important that the optical magnitude of the entire galaxy be used and not simply that of the nucleus. It is the total magnitude that has invariably been used for high redshift quasars because of the difficulty of separating the nuclear and host contributions. Hao et al. (2005b) point out that, in attempting to obtain the luminosity function of the active nucleus, it is important that it not be contaminated by the host galaxy. Since the brightening predicted in the DIR model below z = 0.1 is due to the host galaxy maturing and increasing in luminosity, the contribution from the entire host galaxy must be included in the magnitudes used in the logz-mv plot if the brightening is to be detected. Although the luminosity of the nucleus may be adequate in determining the luminosity function of the active nucleus in the CR model, because of the complex process required to identify AGN galaxies (Hao et al. 2005b), obviously great care will be required in obtaining the magnitudes of low redshift AGN galaxies if they are to be used in logz-mv plots. – 8 – 4. Conclusion The most luminous radio galaxies and first-ranked cluster galaxies have been compared here to the high luminosity edge of the AGN galaxy distribution on a logz-mv plot. It is found that while the radio galaxies and cluster galaxies are good standard candles at all epochs, the luminosity of the AGN galaxies varies significantly from one epoch to another. Compared to the comparison galaxies the AGN galaxies are found to be super-luminous at high redshifts, but become sub-luminous as the redshift decreases. These new results show that below z = 0.3 the rate of luminosity decrease begins to slow down and below z = 0.1 the luminosity begins to increase again. Although their apparent super-luminous nature at high z can be explained by a higher star formation rate, and the fact that there might have been more raw material around to make galaxies at that epoch, a luminosity increase below z = 0.1 is more difficult to explain when these arguments are unlikely explanations. It is therefore concluded here that the evidence favors the argument that the high redshift AGN galaxies (quasars) that do lie above the mature galaxy line on a logz-mv plot have all been pushed there because of a large intrinsic component in their redshifts and not because they have a superimposed non-thermal component that is many magnitudes brighter than that seen in radio galaxies. All AGN galaxies then will be sub-luminous to mature galaxies, as predicted in the DIR model. For a given intrinsic redshift component, all are likely also to be good standard candles. Finally, if it turns out that many faint AGN galaxies at low redshifts have been missed in a particular magnitude range the conclusion that the bright edge of the logz-mv plot increases again in this redshift range may need to be re-evaluated. Such an effect might be introduced by the selection effect discussed by Hao et al. (2005a,b), but only if the VCVcat contains many of the SDSS AGN galaxies, as explained in Sec 3.1. I wish to thank two anonymous referees for suggestions on how this paper might be improved. I also thank Dr. D. McDiarmid for helpful comments. – 9 – REFERENCES Antonucci, R. 1993, ARA&A, 31, 473 Arp, H. 1997, A&A, 319, 33 Arp, H. 1998, ApJ, 496, 661 Arp, H. 1999, ApJ, 525, 596 Bell, M.B. 2002a, ApJ, 566, 705 Bell, M.B. 2002b, ApJ, 567, 801 Bell, M.B. 2002c, preprint astro-ph/0208320 Bell, M.B. 2002d, preprint astro-ph/0211091 Bell, M.B. and Comeau, S.P. 2003, preprint astro-ph/0305060 Bell, M.B. 2004, ApJ, 616, 738 Bell, M.B., Comeau, S.P. and Russell, D.G. 2004, astro-ph/0407591 Bell, M. 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The dashed line represents brightest clus- ter galaxies from (Sandage 1972a; Kristian et al. 1978). See text for an explanation of the triangle. – 12 – Fig. 2.— Logz-mv plot of all 106,958 sources listed as quasars and active galaxies in the Véron-Cetty-Véron catalogue. The solid line indicates first-ranked clusters from Sandage (1972a); Kristian et al. (1978). The dashed line indicates the maximum distance for high- redshift AGN detected to date from (Bell 2004). In the DIR model any AGN that lie above this line have been pushed there by the presence of an intrinsic redshift component. Introduction The Data Discussion Selection Effects in the Data Conclusion

SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Abstract. We compute the scattering amplitude for Schrödinger operators at a critical energy level, corresponding to the maximum point of the potential. We follow [30], using Isozaki-Kitada’s representation formula for the scattering amplitude, together with results from [5] in order to analyze the contribution of trapped trajectories. Contents 1. Introduction 2 2. Assumptions and main results 4 3. Proof of the main resolvent estimate 10 4. Representation of the Scattering Amplitude 16 5. Computations before the critical point 19 5.1. Computation of u− in the incoming region 19 5.2. Computation of u− along γ 6. Computation of u− at the critical point 23 6.1. Study of the transport equations for the phases 25 6.2. Taylor expansions of ϕ+ and ϕ 6.3. Asymptotics near the critical point for the trajectories 34 6.4. Computation of the ϕkj ’s 36 7. Computations after the critical point 43 7.1. Stationary phase expansion in the outgoing region 43 7.2. Behaviour of the phase function Φ 47 Date: April 12, 2007. 2000 Mathematics Subject Classification. 81U20,35P25,35B38,35C20. Key words and phrases. Scattering amplitude, critical energy, Schrödinger equation. Acknowledgments: We would like to thank Johannes Sjöstrand for helpful discussions during the prepa- ration of this paper. The first author also thanks Victor Ivrii for supporting visits to Université Paris Sud, Orsay, and the Department of Mathematics at Orsay for the extended hospitality. http://arxiv.org/abs/0704.1632v2 2 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND 7.3. Integration with respect to time 48 Appendix A. Proof of Proposition 2.5 53 Appendix B. A lower bound for the resolvent 53 Appendix C. Lagrangian manifolds which are transverse to Λ± 55 Appendix D. Asymptotic behaviour of certain integrals 57 References 60 1. Introduction We study the semiclassical behavior of scattering amplitude at energy E > 0 for Schrö- dinger operators (1.1) P (x, hD) = −h ∆ + V (x) where V is a real valued C∞ function on Rn, which vanishes at infinity. We shall suppose here that E is close to a critical energy level E0 for P , which corresponds to a non-degenerate global maximum of the potential. Here, we address the case where this maximum is unique. Let us recall that, if V (x) = O(〈x〉−ρ) for some ρ > (n + 1)/2, then for any ω 6= θ ∈ Sn−1 and E > 0, the problem P (x, hD)u = Eu, u(x, h) = ei 2Ex·ω/h +A(ω, θ,E, h)e 2E|x|/h |x|(n−1)/2 + o(|x|(1−n)/2) as x→ +∞, x|x| = θ, has a unique solution. The scattering amplitude at energy E for the incoming direction ω and the outgoing direction θ is the real number A(ω, θ,E, h). For potentials that are not decaying that fast at infinity, it is not that easy to write down a stationary formula for the scattering amplitude: If V (x) = O(〈x〉−ρ) for some ρ > 1, one can define the scattering matrix at energy E using wave operators (see Section 4 below). Then, writing (1.2) S(E, h) = Id− 2iπT (E, h), one can see that T (E, h) is a compact operator on L2(Sn−1), which kernel T (ω, θ,E, h) is smooth out of the diagonal in Sn−1×Sn−1. Then, the scattering amplitude is given for θ 6= ω, (1.3) A(ω, θ,E, h) = c(E))h(n−1)/2T (ω, θ,E, h), where (1.4) c(E) = −2π(2E)− 4 (2π) 2 e−i (n−3)π We proceed here as in [30], where D. Robert and H. Tamura have studied the semiclassical behavior of the scattering amplitude for short range potentials at a non-trapping energy E . SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 3 An energy E is said to be non-trapping when K(E), the trapped set K(E) at energy E, is empty. This trapped set is defined as (1.5) K(E) = (x, ξ) ∈ p−1(E), exp(tHp)(x, ξ) 6→ ∞ as t→ ±∞ where Hp is the Hamiltonian vector field associated to the principal symbol p(x, ξ) = V (x) of the operator P . Notice that the scattering amplitude has been first studied, in the semiclassical regime, by B. Vainberg [32] and Y. Protas [27] in the case of compactly supported potential, and for non-trapping energies, where they obtained the same type of result. Under the non-trapping assumption, and some other non-degeneracy condition (in fact our assumption (A4) below), D. Robert and H. Tamura have shown that the scattering amplitude has an asymptotic expansion with respect to h. The non-degeneracy assumption implies in particular that there is a finite number N∞ of classical trajectories for the Hamiltonian p, with asymptotic direction ω for t→ −∞ and asymptotic direction θ as t→ +∞. Robert and Tamura’s result is the following asymptotic expansion for the scattering amplitude: (1.6) A(ω, θ,E, h) = iS∞j /h aj,m(ω, θ,E)h m +O(h∞), h→ 0, where S∞j is the classical action along the corresponding trajectory. Also, they have computed the first term in this expansion, showing that it can be given in terms of quantities attached to the corresponding classical trajectory only. There are also some few works concerning the scattering amplitude when the non-trapping assumption is not fulfilled. In his paper [24], L. Michel has shown that, if there is no trapped trajectory with incoming direction ω and outgoing direction θ (see the discussion after (2.6) below), and if there is a complex neighborhood of E of size ∼ hN for some N ∈ N possibly large, which is free of resonances, then A(ω, θ,E, h) is still given by Robert and Tamura’s formula. The potential is also supposed to be analytic in a sector out of a compact set, and the assumption on the existence of a resonance free domain around E amounts to an estimate on boundary value of the meromorphic extension of the truncated resolvent of the for (1.7) ‖χ(P − (E ± i0))−1χ‖ = O(h−N ), χ ∈ C∞0 (Rn). Of course, these assumptions allow the existence of a non-empty trapped set. In [2] and [3], the first author has shown that at non-trapping energies or in L. Michel’s setting, the scattering amplitude is an h-Fourier Integral Operator associated to a natural scattering relation. These results imply that the scattering amplitude admits an asymptotic expansion even without the non-degeneracy assumption, and in the sense of oscillatory inte- grals. In particular, the expansion (1.6) is recovered under the non-degeneracy assumption and as an oscillatory integral. In [21], A. Lahmar-Benbernou and A. Martinez have computed the scattering amplitude at energy E ∼ E0, in the case where the trapped set K(E0) consists in one single point corresponding to a local minimum of the potential (a well in the island situation). In that case, the estimate (1.7) is not true, and their result is obtained through a construction of the resonant states. In the present work, we compute the scattering amplitude at energy E ∼ E0 in the case where the trapped set K(E0) corresponds to the unique global maximum of the potential. 4 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND The one-dimensional case has been studied in [28, 14, 15], with specific techniques, and we consider here the general n > 1 dimensional case. Notice that J. Sjöstrand in [31], and P. Briet, J.-M. Combes and P. Duclos in [7, 8] have described the resonances close to E0 in the case where V is analytic in a sector around R From their result, it follows that Michel’s assumption on the existence of a not too small resonance-free neighborhood of E0 is satisfied. However, we show below (see Proposition 2.5) that for any ω ∈ Sn−1, there is at least one half-trapped trajectory with incoming direction ω, so that L. Michel’s result never applies here. Here, we do not assume analyticity for V . We compute the contributions to the scattering amplitude arising from the classical trajectories reaching the unstable equilibrium point, which corresponds to the top of the potential barrier. At the quantum level, tunnel effect occurs, which permits the particle to pass through this point. Our computation here relies heavily on [5], where a precise description of this phenomena has been obtained. In a forthcoming paper, we shall show that in this case also, the scattering amplitude is an h-Fourier Integral Operator. This paper is organized in the following way. In Section 2, we describe our assumptions, and state our main results: a resolvent estimate, and the asymptotic expansion of the scattering amplitude in the semiclassical regime. Section 3 is devoted to the proof of the resolvent estimate, from which we deduce in Section 4 estimates similar to those in [30]. In that section, we also recall briefly the representation formula for the scattering amplitude proved by Isozaki and Kitada, and introduce notations from [30]. The computation of the asymptotic expansion of the scattering amplitude is conducted in sections 5, 6 and 7, following the classical trajectories. Eventually, we have put in four appendices the proofs of some side results or technicalities. 2. Assumptions and main results We suppose that the potential V satisfies the following assumptions (A1) V is a C∞ function on Rn, and, for some ρ > 1, ∂αV (x) = O(〈x〉−ρ−|α|). (A2) V has a non-degenerate maximum point at x = 0, with E0 = V (0) > 0 and ∇2V (0) = . . .  , 0 < λ1 ≤ λ2 ≤ . . . ≤ λn. (A3) The trapped set at energy E0 is K(E0) = {(0, 0)}. Notice that the assumptions (A1)–(A3) imply that V has an absolute global maximum at x = 0. Indeed, if L = {x 6= 0; V (x) ≥ E0} was non empty, the geodesic, for the Agmon distance (E0 − V (x))1/2+ dx, between 0 and L would be the projection of a trapped bicharacteristic (see [1, Theorem 3.7.7]). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 5 As in D. Robert and H. Tamura in [30], one of the key ingredient for the study of the scattering amplitude is a suitable estimate for the resolvent. Using the ideas in [5, Section 4], we have obtained the following result, that we think to be of independent interest. Theorem 2.1. Suppose assumptions (A1), (A2) and (A3) hold, and let α > 1 be a fixed real number. We have (2.1) ‖P − (E ± i0))−1‖α,−α . h−1| lnh|, uniformly for |E − E0| ≤ δ, with δ > 0 small enough. Here ‖Q‖α,β denotes the norm of the bounded operator Q from L2(〈x〉α dx) to L2(〈x〉β dx). Moreover, we prove in the Appendix B that our estimate is not far from optimal. Indeed, we have the Proposition 2.2. Under the assumptions (A1) and (A2), we have (2.2) ‖(P − E0 ± i0)−1‖α,−α & h−1 | lnh|. We would like to mention that in the case of a closed hyperbolic orbit, the same upper bound has been obtained by N. Burq [9] in the analytic category, and in a recent paper [11] by H. Christianson in the C∞ setting. As a matter of fact, in the present setting, S. Nakamura has proved in [26] an O(h−2) bound for the resolvent. Nakamura’s estimate would be sufficient for our proof of Theorem 2.6, but it is not sharp enough for the computation of the total scattering cross section along the lines of D. Robert and H. Tamura in [29]. In that paper, the proof relies on a bound O(h−1) for the resolvent, but it is easy to see that an estimate like O(h−1−ε) for any small enough ε > 0 is sufficient. If we denote (2.3) σ(ω,E0, h) = |A(ω, θ,E, h)|2dθ, the total scattering cross-section, and following D. Robert and H. Tamura’s work, our resolvent estimates gives the Theorem 2.3. Suppose assumptions (A1), (A2) and (A3) hold, and that ρ > n+1 , n ≥ 2. If |E − E0| < δ for some δ > 0 small enough, then (2.4) σ(ω,E, h) = 4 2−1(2E)−1/2h−1 V (y + sω)ds dy +O(h−(n−1)/(ρ−1)). Now we state our assumptions concerning the classical trajectories associated with the Hamiltonian p, that is curves t 7→ γ(t, x, ξ) = exp(tHp)(x, ξ) for some initial data (x, ξ) ∈ T ∗Rn. Let us recall that, thanks to the decay of V at infinity, for given α ∈ Sn−1 and z ∈ α⊥ ∼ Rn−1 (the impact plane), there is a unique bicharacteristic curve (2.5) γ±(t, z, α,E) = (x±(t, z, α,E), ξ±(t, z, α,E)) such that (2.6) |x±(t, z, α,E) − 2Eαt− z| = 0, |ξ±(t, z, α,E) − 2Eα| = 0. 6 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND We shall denote by Λ−ω the set of points in T n lying on trajectories going to infinity with direction ω as t → −∞, and Λ+θ the set of those which lie on trajectories going to infinity with direction θ as t→ +∞: (2.7) Λ−ω = γ−(t, z, ω,E) ∈ T ∗Rn, z ∈ ω⊥, t ∈ R Λ+θ = γ+(t, z, θ, E) ∈ T ∗Rn, z ∈ θ⊥, t ∈ R We shall see that Λ−ω and Λ θ are in fact Lagrangian submanifolds of T Under the assumptions (A1), (A2) and (A3) there are only two possible behaviors for x±(t, z, α,E) as t→ ∓∞: either it escapes to ∞, or it goes to 0. First we state our assumptions for the first kind of trajectories. For these, we also have, for some ξ∞(z, ω,E), ξ−(t, z, ω) = ξ∞(z, ω,E), and we shall say that the trajectory γ−(t, z, ω,E) has initial direction ω and final direction θ = ξ∞(z, ω,E)/2 E. As in [30] we shall suppose that there is only a finite number of trajectories with initial direction ω and final direction θ. This assumption can be given in terms of the angular density (2.8) σ̂(z) = |det(ξ∞(z, ω,E), ∂z1ξ∞(z, ω,E), . . . , ∂zn−1ξ∞(z, ω,E))|. Definition 2.4. The outgoing direction θ ∈ Sn−1 is called regular for the incoming direction ω ∈ Sn−1, or ω-regular, if θ 6= ω and, for all z′ ∈ ω⊥ with ξ∞(z′, ω,E) = 2 Eθ, the map ω⊥ ∋ z 7→ ξ∞(z, ω,E) ∈ Sn−1 is non-degenerate at z′, i.e. σ̂(z′) 6= 0. We fix the incoming direction ω ∈ Sn−1, and we assume that (A4) the direction θ ∈ Sn−1 is ω-regular. Then, one can show that Λ−ω ∩ Λ+θ is a finite set of Hamiltonian trajectories (γ∞j )1≤j≤N∞ , γ∞j (t) = γ ∞(t, z∞j ) = (x j (t), ξ j (t)), with transverse intersection along each of these curves. We turn to trapped trajectories. Let us notice that the linearization Fp at (0, 0) of the Hamilton vector field Hp has eigenvalues −λn, . . . ,−λ1, λ1, . . . , λn. Thus (0, 0) is a hyper- bolic fixed point for Hp, and the stable/unstable manifold Theorem gives the existence of a stable incoming Lagrangian manifold Λ− and a stable outgoing Lagrangian manifold Λ+ characterized by (2.9) Λ± = {(x, ξ) ∈ T ∗Rn, exp(tHp)(x, ξ) → 0 as t→ ∓∞} . In this paper, we shall describe the contribution to the scattering amplitude of the trapped trajectories, that is those going from infinity to the fixed point (0, 0). We have proved in Appendix A the following result, which shows that there are always such trajectories. Proposition 2.5. For every ω, θ ∈ Sn−1, we have (2.10) Λ−ω ∩ Λ− 6= ∅ and Λ+θ ∩ Λ+ 6= ∅. We suppose that (A5) Λ−ω and Λ− (resp. Λ and Λ+) intersect transversally. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 7 Under this assumption, Λ−ω ∩ Λ− and Λ+θ ∩ Λ+ are finite sets of bicharacteristic curves. We denote them, respectively, (2.11) γ−k : t 7→ γ −(t, z−k ) = (x k (t), ξ −(t)), 1 ≤ k ≤ N−, (2.12) γ+ℓ : t 7→ γ +(t, z+ℓ ) = (x +(t), ξ+(t)), 1 ≤ ℓ ≤ N+. Here, the z− (resp. the z+ ) belong to ω⊥ (resp. θ⊥) and determine the corresponding curve by (2.6). We recall from [18, Section 3] (see also [5, Section 5]), that each integral curve γ±(t) = (x±(t), ξ±(t)) ∈ Λ± satisfies, in the sense of expandible functions (see Definition 6.1 below), (2.13) γ±(t) ∼ γ±j (t)e ±µj t, as t→ ∓∞, where µ1 = λ1 < µ2 < . . . is the strictly increasing sequence of linear combinations over N of the λj’s. Here, the functions γ j : R → R2n are polynomials, that we write (2.14) γ±j (t) = M ′j∑ γ±j,mt Considering the base space projection of these trajectories, we denote (2.15) x±(t) ∼ g±j (t)e ±µj t, as t→ ∓∞, g±j (t) = M ′j∑ g±j,mt Let us denote ̂ the (only) integer such that µb = 2λ1. We prove in Proposition 6.11 below that if j < ̂, then M ′j = 0, or more precisely, that g j (t) = g j is a constant vector in Ker(Fp∓λj). We also have M ′ ≤ 1, and g− can be computed in terms of g−1 . In this paper, concerning the incoming trajectories, we shall assume that, (A6) For each k ∈ {1, . . . , N−}, g−1 (z ) 6= 0. Finally, we state our assumptions for the outgoing trajectories γ+ℓ ⊂ Λ+ ∩Λθ+. First of all, it is easy to see, using Hartman’s linearization theorem, that there exists always a m ∈ N such that g+m(z ℓ ) 6= 0. We denote (2.16) ℓℓℓ = ℓℓℓ(ℓ) = min{m, g+m(z+ℓ ) 6= 0} the smallest of these m’s. We know that µℓℓℓ is one of the λj’s, and that M ℓℓℓ = 0. In [5], we have been able to describe the branching process between an incoming curve γ− ⊂ Λ− and an outgoing curve γ+ ⊂ Λ+ provided 〈g−1 |g 1 〉 6= 0 (see the definition for Λ̃+(ρ−) before [5, Theorem 2.6]). Here, for the computation of the scattering amplitude, we can relax a lot this assumption, and analyze the branching in other cases that we describe now. Let us denote, for a given pair of paths (γ−(z−k ), γ +(z+ℓ )) in (Λ ω ∩ Λ−)× (Λ+θ ∩ Λ+), (2.17) M2(k, ℓ) = − j∈I1(2λ1) α,β∈I2(λ1) βV (0) (g−1 (z αV (0) (g+1 (z 8 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND M1(k, ℓ) =− α∈I2(λ1) αV (0) (g−1 (z −))α(g+ (z+))j + (g (z−))j(g α,β∈I2(λ1) (g−1 (z (g+1 (z Cα,β,(2.18) where Cα,β =− ∂α+βV (0) + j∈I1\I1(2λ1) λ2j(4λ 1 − λ2j) ∂α+γV (0)∂β+γV (0) γ,δ∈I2(λ1) γ+δ=α+β (γ + δ)! γ! δ! γV (0)∂j∂ δV (0).(2.19) Here, we have set I1 = {1, . . . , n}, 1j = (δij)i=1,...,n ∈ Nn and (2.20) Im(µ) = {β ∈ Nn, β = 1k1 + · · ·+ 1km with λk1 = · · · = λkm = µ}, the set of multi-indices β of length |β| = m with each index of its non-vanishing components in the set {j ∈ N, λj = µ}. We also denote Im ⊂ Nn the set of multi-indices of length m. We will suppose that (A7) For each pair of paths (γ−(z−k ), γ +(z+ℓ )), k ∈ {1, . . . , N−}, ℓ ∈ {1, . . . , N+}, one of the three following cases occurs: (a) The set m < ̂, 〈g−m(z−k )|g+m(z ℓ )〉 6= 0 is not empty. Then we denote k = min m < ̂, 〈g−m(z−k )|g )〉 6= 0 (b) For all m < ̂, we have 〈g−m(z−k )|g+m(z ℓ )〉 = 0, and M2(k, ℓ) 6= 0. (c) For all m < ̂, we have 〈g−m(z−k )|g+m(z ℓ )〉 = 0, M2(k, ℓ) = 0 and M1(k, ℓ) 6= 0. As one could expect (see [30], [28] or [15]), action integrals appear in our formula for the scattering amplitude. We shall denote S∞j = (|ξ∞j (t)|2 − 2E0)dt, j ∈ {1, . . . , N∞},(2.21) S−k = |ξ−k (t)| 2 − 2E01t<0 dt, k ∈ {1, . . . , N−},(2.22) (t)|2 − 2E01t>0 dt, ℓ ∈ {1, . . . , N+},(2.23) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 9 and ν∞j , ν ℓ , ν k the Maslov indexes of the curves γ j , γ ℓ , γ k respectively. Let also D−k = limt→+∞ ∣∣∣ det ∂x−(t, z, ω,E0) ∂(t, z) ∣∣∣ e−(Σλj−2λ1)t,(2.24) D+ℓ = limt→−∞ ∣∣∣ det ∂x+(t, z, ω,E0) ∂(t, z) ∣∣∣ e(Σλj−2λℓℓℓ)t,(2.25) be the Maslov determinants for γ− , and γ+ respectively. We show below that 0 < D− +∞. Eventually we set (2.26) Σ(E, h) = − iE − E0 Then, the main result of this paper is the Theorem 2.6. Suppose assumptions (A1) to (A7) hold, and that E ∈ R is such that E −E0 = O(h). Then A(ω, θ,E, h) = Aregj (ω, θ,E, h) + Asingk,ℓ (ω, θ,E, h) +O(h ∞),(2.27) where (2.28) Aregj (ω, θ,E, h) = e iS∞j /h j,m(ω, θ,E)h j,0 (ω, θ,E) = −iν∞j π/2 σ̂(zj)1/2 Moreover we have • In case (a) Asingk,ℓ (ω, θ,E, h) = e k,ℓ,m(ω, θ,E, lnh)h (Σ(E)+bµm)/µk−1/2,(2.29) where the a k,ℓ,m (ω, θ,E, ln h) are polynomials with respect to lnh, and k,ℓ,0(ω, θ,E, ln h) = π1−n/2 ei(nπ/4−π/2) )−1/2 (Σ(E) (2λ1λℓℓℓ) × e−iν π/2e−iν π/2(D− )−1/2 × |g−1 (z k )| |g ℓℓℓ (z (z−k ) (z+ℓ ) 〉)−Σ(E)/µk .(2.30) • In case (b) (2.31) Asingk,ℓ (ω, θ,E, h) = e k,ℓ (ω, θ,E) hΣ(E)/2λ1−1/2 | lnh|Σ(E)/λ1 (1 + o(1)), 10 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND where (ω, θ,E) = π1−n/2 ei(nπ/4−π/2) )−1/2 (Σ(E) (2λ1λℓℓℓ) 3/2(2λ1) Σ(E)/λ1−1 × e−iν π/2e−iν π/2(D−k D × |g−1 (z )| |g+ − iM2(k, ℓ) )−Σ(E)/2λ1 .(2.32) • In case (c) (2.33) Asingk,ℓ (ω, θ,E, h) = e k,ℓ (ω, θ,E) hΣ(E)/2λ1−1/2 | ln h|Σ(E)/2λ1 (1 + o(1)), where k,ℓ (ω, θ,E) = π1−n/2 ei(nπ/4−π/2) )−1/2 (Σ(E) (2λ1λℓℓℓ) 3/2(2λ1) Σ(E)/2λ1−1 × e−iν π/2e−iν π/2(D−k D × |g−1 (z k )| |g ℓℓℓ (z − iM1(k, ℓ) )−Σ(E)/2λ1 .(2.34) Here, the µ̂j are the linear combinations over N of the λk’s and λk − λ1’s, and the function z 7→ z−Σ(E)/µk is defined on C\]−∞, 0] and real positive on ]0,+∞[. Of course the assumption that 〈g−1 |g 1 〉 6= 0 (a subcase of (a)) is generic. Without the assumption (A4), the regular part Areg of the scattering amplitude has an integral rep- resentation as in [3]. When the assumption (A7) is not fulfilled, that is when the terms corresponding to the µj with j ≤ ̂ do not contribute, we don’t know if the scattering amplitude can be given only in terms of the g±1 ’s and of the derivatives of the potential. 3. Proof of the main resolvent estimate Here we prove Theorem 2.1 using Mourre’s Theory. We start with the construction of an escape function close to the stationary point (0, 0) in the spirit of [10] and [5]. Since Λ+ and Λ− are Lagrangian manifolds, one can choose local symplectic coordinates (y, η) such that (3.1) p(x, ξ) = B(y, η)y · η, where (y, η) 7→ B(y, η) is a C∞ mapping from a neighborhood of (0,0) in T ∗Rn to the space Mn(R) of n× n matrices with real entries, such that, (3.2) B(0, 0) = . . . We denote U a unitary Fourier Integral Operator (FIO) microlocally defined in a neighborhood of (0, 0), which canonical transformation is the map (x, ξ) 7→ (y, η), and we set (3.3) P̂ = UPU∗. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 11 Here the FIO U∗ is the adjoint of U , and we have UU∗ = Id+O(h∞) and U∗U = Id+O(h∞) microlocally near (0, 0). Then P̂ is a pseudodifferential operator, with a real (modulo O(h∞)) symbol p̂(y, η) = j p̂j(y, η)h j , such that (3.4) p̂0 = B(y, η)y · η. We set B1 = Oph(b1), (3.5) b1(y, η) = χ̃2(y, η), where M > 1 will be fixed later and χ̃1 ≺ χ̃2 ∈ C∞0 (T ∗Rn) with χ̃1 = 1 near (0, 0). In what follows, we will assume that hM < 1. In particular, b1 ∈ S1/2(| ln h|). Here and in what follows, we use the usual notation for classes of symbols. For m an order function, a function a(x, ξ, h) ∈ C∞(T ∗Rn) belongs to Sδh(m) when (3.6) ∀α ∈ N2n, ∃Cα > 0, ∀h ∈]0, 1], |∂αx,ξa(x, ξ, h)| ≤ Cαh−δ|α|m(x, ξ). Let us also recall that, if a ∈ Sα(1) and b ∈ Sβ(1), with α, β < 1/2, we have (3.7) Oph(a),Oph(b) = Oph ih{b, a} + h3(1−α−β) Oph(r), with r ∈ Smin(α,β)(1): In particular the term of order 2 vanishes. Hence, we have here (3.8) [B1, P̂ ] = Oph ih{p̂0, b1} + | lnh|h3/2 Oph(rM ), with rM ∈ S1/2(1). The semi-norms of rM may depend on M . We have (3.9) {p̂0, b1} = c1 + c2, {p̂0, χ̃2}(3.10) p̂0, ln By + (∂ηB)y · η hM + y2 Bη + (∂yB)y · η hM + η2 χ̃2.(3.11) The symbols c1 ∈ S1/2(| lnh|), c2 ∈ S1/2(1) satisfy supp(c1) ⊂ supp(∇χ̃2). Let ϕ̃ ∈ C∞0 (T n) be a function such that ϕ̃ = 0 near (0, 0) and ϕ̃ = 1 near the support of ∇χ̃2. We Oph(c1) =Oph(ϕ̃)Oph(c1)Oph(ϕ̃) +O(h∞) ≥− C1h| ln h|Oph(ϕ̃)Oph(ϕ̃) +O(h∞) ≥− C1h| ln h|Oph(ϕ̃2) +O(h2| ln h|),(3.12) for some C1 > 0. On the other hand, using [5, (4.96)–(4.97)], we get (3.13) Oph(c2) ≥ εM−1 Oph(χ̃1) +O(M−2), 12 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND for some ε > 0. With the notation A1 = U ∗B1U , the formulas (3.8), (3.9), (3.12) and (3.13) imply −i[A1, P ] =− iU∗[B1, P ]U ≥εhM−1U∗Oph(χ̃1)U − C1h| lnh|U∗ Oph(ϕ̃2)U +O(hM−2) +OM (h3/2| lnh|).(3.14) If κ is the canonical transformation associated to U , then χj = χ̃j ◦ κ, j = 1, 2 and ϕ = ϕ̃ ◦ κ are C∞0 (T ∗(Rn), [0, 1]) functions which satisfy χ1 = 1 near (0, 0) and ϕ = 0 near (0, 0). Using Egorov’s Theorem, (3.14) becomes (3.15) − i[A1, P ] ≥ εhM−1 Oph(χ1)− C1h| lnh|Oph(ϕ) +O(hM−2) +OM (h3/2| lnh|). Now, we build an escape function outside of supp(χ1) as in [22]. Let 1(0,0) ≺ χ0 ≺ χ1 ≺ χ2 ≺ χ3 ≺ χ4 ≺ χ5 be C∞0 (T ∗(Rn), [0, 1]) functions with ϕ ≺ χ4. We define a3 = g(ξ)(1−χ3(x, ξ))x ·ξ where g ∈ C∞0 (Rn) satisfies 1p−1([E0−δ,E0+δ]) ≺ g. Using [6, Lemma 3.1], we can find a bounded, C∞ function a2(x, ξ) such that (3.16) Hpa2 ≥ 0 for all (x, ξ) ∈ p−1([E0 − δ,E0 + δ]), 1 for all (x, ξ) ∈ supp(χ4 − χ0) ∩ p−1([E0 − δ,E0 + δ]), and we set A2 = Oph(a2χ5). We denote (3.17) A = A1 + C2| lnh|A2 + | lnh|A3, where C2 > 1 will be fixed later. Now let ψ̃ ∈ C∞0 ([E0 − δ,E0 + δ], [0, 1]) with ψ̃ = 1 near E0. We recall that ψ̃(P ) is a classical pseudodifferential operator of class Ψ 0(〈ξ〉−∞) with principal symbol ψ̃(p). Then, from (3.15), we obtain −iψ̃(P )[A,P ]ψ̃(P ) ≥εhM−1ψ̃(P )Oph(χ1)ψ̃(P )− C1h| ln h|ψ̃(P )Oph(ϕ)ψ̃(P ) + C2h| lnh|Oph ψ̃2(p)(χ4 − χ0) + C2h| ln h|Oph ψ̃2(p)a2Hpχ5 + h| ln h|Oph ψ̃2(p)(ξ2 − x · ∇V )(1 − χ3) + h| ln h|Oph ψ̃2(p)x · ξHp(gχ3) +O(hM−2) +OM (h3/2| lnh|).(3.18) From (A1), we have x·∇V (x) → 0 as x→ ∞. In particular, if χ3 is equal to 1 in a sufficiently large zone, we have (3.19) ψ̃2(p)(ξ2 − x · ∇V )(1− χ3) ≥ E0ψ̃2(p)(1− χ3). If C2 > 0 is large enough, the G̊arding inequality implies (3.20) C2 Oph ψ̃2(p)(χ4 − χ0) −C1 Oph ψ̃2(p)ϕ ψ̃2(p)x · ξHp(gχ3) ≥ Oph ψ̃2(p)(χ4 − χ0) +O(h). As in [22], we take χ5(x) = χ̃5(µx) with µ small and χ̃5 ∈ C∞0 ([E0 − δ,E0 + δ], [0, 1]). Since a2 is bounded, we get (3.21) ∣∣C2ψ̃2(p)a2Hpχ5 ∣∣ ≤ µC2‖a2‖L∞‖Hpχ̃5‖L∞ . µ. Therefore, if µ is small enough, (3.19) implies (3.22) Oph ψ̃2(p)(ξ2−x ·∇V )(1−χ3) +C2 Oph ψ̃2(p)a2Hpχ5 ψ̃2(p)(1−χ3) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 13 Then (3.18), (3.20), (3.22) and the G̊arding inequality give −iψ̃(P )[A,P ]ψ̃(P ) ≥εhM−1 Oph ψ̃2(p)χ1 + h| ln h|Oph ψ̃2(p)(χ4 − χ0) h| ln h|Oph ψ̃2(p)(1 − χ3) +O(hM−2) +OM (h3/2| lnh|) ≥εhM−1 Oph ψ̃2(p) +O(hM−2) +OM (h3/2| ln h|).(3.23) Choosing M large enough and 1E0 ≺ ψ ≺ ψ̃, we have proved the Lemma 3.1. Let M be large enough and ψ ∈ C∞0 ([E0− δ,E0+ δ]), δ > 0 small enough, with ψ = 1 near E0. Then, we have (3.24) − iψ(P )[A,P ]ψ(P ) ≥ εh−1ψ2(P ). Moreover (3.25) [A,P ] = O(h| ln h|). From the properties of the support of the χj, we have [[P,A], A] =[[P,A1], A1] + C2| lnh|[[P,A1], A2] + C2| ln h|[[P,A2], A1] + C22 | lnh|2[[P,A2], A2] + C2| lnh|2[[P,A2], A3] + C2| ln h|2[[P,A3], A2] + | lnh|2[[P,A3], A3] +O(h∞).(3.26) We also know that P ∈ Ψ0(〈ξ〉2), A2 ∈ Ψ0(〈ξ〉−∞) and A3 ∈ Ψ0(〈x〉〈ξ〉−∞). Then, we can show that all the terms in (3.26) with j, k = 2, 3 satisfy (3.27) [[P,Aj ], Ak] ∈ Ψ0(h2). On the other hand, (3.28) [[P,A1], A2] = U ∗[[P̂ , B1], UA2U ∗]U +O(h∞), with UA2U ∗ ∈ Ψ0(1). From (3.8) – (3.11), we have [P̂ , B1] ∈ Ψ1/2(h| ln h|) and then (3.29) [[P,A1], A2] = O(h3/2| lnh|). The term [[P,A2], A1] gives the same type of contribution. It remains to study (3.30) [[P,A1], A1] = U ∗[[P̂ , B1], B1]U +O(h∞). Let χ̃3 ∈ C∞0 (T ∗Rn), [0, 1]) with χ̃2 ≺ χ̃3 and (3.31) f = χ̃3(y, η) ∈ S1/2(| ln h|). Then, with a remainder rM ∈ S1/2(1) which differs from line to line, i[P̂ , B1] =hOph f{χ̃2, p̂0}+ c2 − h3/2| ln h|Oph(rM ) =hOph(f)Oph({χ̃2, p̂0}) + hOph(c2) + h3/2| lnh|Oph(rM ).(3.32) 14 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND In particular, since [P̂ , B1] ∈ Ψ1/2(h| ln h|), c2 ∈ S1/2(1) and f ∈ S1/2(| lnh|), [[P̂ , B1], B1] =[[P̂ , B1],Oph(fχ̃2)] =− ih[Oph(f)Oph({χ̃2, p̂0}),Oph(fχ̃2)]− ih[Oph(c2),Oph(fχ̃2)] +O(h3/2| ln h|2) =− ih[Oph(f)Oph({χ̃2, p̂0}),Oph(f)Oph(χ̃2)] +O(h| ln h|) =− ihOph(f)[Oph({χ̃2, p̂0}),Oph(f)]Oph(χ̃2) − ih[Oph(f),Oph(f)]Oph({χ̃2, p̂0})Oph(χ̃2) − ihOph(f)Oph(f)[Oph({χ̃2, p̂0}),Oph(χ̃2)] − ihOph(f)[Oph(f),Oph(χ̃2)]Oph({χ̃2, p̂0}) +O(h| ln h|) =O(h| ln h|).(3.33) From (3.26), (3.27), (3.29) and (3.33), we get (3.34) [[P,A], A] = O(h| lnh|). As a matter of fact, using [5], one can show that [[P,A], A] = O(h). Now we can use the following proposition which is an adaptation of the limiting absorption principle of Mourre [25] (see also [12, Theorem 4.9], [19, Proposition 2.1] and [4, Theorem 7.4.1]). Proposition 3.2. Let (P,D(P )) and (A,D(A)) be self-adjoint operators on a separable Hilbert space H. Assume the following assumptions: i) P is of class C2(A). Recall that P is of class Cr(A) if there exists z ∈ C \ σ(P ) such (3.35) R ∋ t→ eitA(P − z)−1e−itA, is Cr for the strong topology of L(H). ii) The form [P,A] defined on D(A) ∩D(P ) extends to a bounded operator on H and (3.36) ‖[P,A]‖ . β. iii) The form [[P,A],A] defined on D(A) extends to a bounded operator on H and (3.37) ‖[[P,A],A]‖ . γ. iv) There exist a compact interval I ⊂ R and g ∈ C∞0 (R) with 1I ≺ g such that (3.38) ig(P )[P,A]g(P ) & γg2(P ). v) β2 . γ . 1. Then, for all α > 1/2, limε→0〈A〉−α(P − E ± iε)−1〈A〉−α exists and (3.39) ∥∥〈A〉−α(P − E ± i0)−1〈A〉−α ∥∥ . γ−1, uniformly for E ∈ I. Remark 3.3. From Theorem 6.2.10 of [4], we have the following useful characterization of the regularity C2(A). Assume that (ii) and (iv) hold. Then, P is of class C2(A) if and only if, for some z ∈ C \ σ(P ), the set {u ∈ D(A); (P − z)−1u ∈ D(A) and (P − z)−1u ∈ D(A)} is a core for A. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 15 Proof. The proof follows the work of Hislop and Nakamura [19]. For ε > 0, we define M2 = ig(P )[P,A]g(P ) and Gε(z) = (P − iεM2 − z)−1 which is analytic for Re z ∈ I and Im z > 0. Following [12, Lemma 4.14] with (3.35)), we get (3.40) ‖g(P )Gε(z)ϕ‖ . (εγ)−1/2|(ϕ,Gε(z)ϕ)|1/2, (3.41) ‖(1− g(P ))Gε(z)‖ . 1 + εβ‖Gε(z)‖, and then (3.42) ‖Gε(z)‖ . (εγ)−1, for ε < ε0 with ε0 small enough, but independent on β, γ. As in [19], let Dε = (1 + |A|)−α(1 + ε|A|)α−1 for α ∈]1/2, 1] and Fε(z) = DεGε(z)Dε. Of course, from (3.42), (3.43) ‖Fε(z)‖ . (εγ)−1, and (3.40) and (3.41) with ϕ = Dεψ give (3.44) ‖Gε(z)Dε‖ . 1 + (εγ)−1/2‖Fε‖1/2. The derivative of Fε(z) is given by (see [12, Lemma 4.15]) (3.45) ∂εFε(z) = iDεGεM 2GεDε = Q0 +Q1 +Q2 +Q3, Q0 =(α− 1)|A|(1 + |A|)−α(1 + ε|A|)α−2Gε(z)Dε + (α− 1)DεGε(z)|A|(1 + |A|)−α(1 + ε|A|)α−2(3.46) Q1 =DεGε(1− g(P ))[P,A](1 − g(P ))GεDε(3.47) Q2 =DεGε(1− g(P ))[P,A]g(P )GεDε +DεGεg(P )[P,A](1 − g(P ))GεDε(3.48) Q3 =−DεGε[P,A]GεDε.(3.49) From (3.44), we obtain (3.50) ‖Q0‖ . εα−1 1 + (εγ)−1/2‖Fε‖1/2 and from (3.36), v) of Proposition 3.2, (3.41) and (3.42), we get (3.51) ‖Q1‖ . γ−1. Using in addition (3.44), we obtain (3.52) ‖Q2‖ . 1 + (εγ)−1/2‖Fε‖1/2. Now we write Q3 = Q4 +Q5 with Q4 = −DεGε[P − iεM2 − z,A]GεDε(3.53) Q5 = −iεDεGε[M2,A]GεDε.(3.54) For Q4, we have the estimate (3.55) ‖Q4‖ . εα−1 1 + (εγ)−1/2‖Fε‖1/2 On the other hand, (3.36), (3.37) and v) imply (3.56) ‖[M2,A]‖ . γ. 16 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Then (3.44) gives (3.57) ‖Q5‖ . 1 + ‖Fε‖. Using the estimates on the Qj, we get (3.58) ‖∂εFε‖ . εα−1 γ−1 + (εγ)−1/2‖Fε‖1/2 + ‖Fε‖ Using (3.43) and integrating (3.37) N times with respect to ε, we get (3.59) ‖Fε‖ . γ−1 1 + ε2α(1−2 −N )−1), so that, for N large enough, (3.60) lim sup ‖〈A〉−α(P − E ± iδ)−1〈A〉−α‖ . γ−1. Using, as in [19], that z 7→ F0(z) is Hölder continuous, we prove the existence of the limit limIm z→0 F0(z) for Re z ∈ I and the proposition follows from (3.60). � From Lemma 3.1 and (3.34), we can apply Proposition 3.2 with A = A/| ln h|, β = h and γ = h/| ln h|. Therefore we have the estimate (3.61) ∥∥〈A〉−α(P − E ± i0)−1〈A〉−α ∥∥ . h−1| ln h|, for E ∈ [E0 − δ,E0 + δ]. As usual, we have (3.62) ‖〈x〉−α〈A〉α‖ = O(1), for α ≥ 0. Indeed, (3.62) is clear for α ∈ 2N, and the general case follows by complex interpolation. Then, (3.61) and (3.26) imply Theorem 2.1. 4. Representation of the Scattering Amplitude As in [30], our starting point for the computation of the scattering amplitude is the rep- resentation given by Isozaki and Kitada in [20]. We recall briefly their formula, that they obtained writing parametrices for the wave operators W± as Fourier Integral Operators, tak- ing advantage of the well-known intertwining property W±P = P0W±, P = P0 + V . The wave operators are defined by (4.1) W± = s− lim eitP/he−itP0/h, where the limit exist thanks to the short-range assumption (A1). The scattering operator is by definition S = (W+)∗W−, and the scattering matrix S(E, h) is then given by the decompostion of S with respect to the spectral measure of P0 = −h2∆. Now we recall briefly the discussion in [30, Section 1,2] (see also [3]), and we start with some notations. If Ω is an open subset of T ∗Rn , we denote by Am(Ω) the class of symbols a such that (x, ξ) 7→ a(x, ξ, h) belongs to C∞(Ω) and (4.2) ∣∣∣∂αx ∂ ξ a(x, ξ) ∣∣∣ ≤ Cαβ〈x〉m−|α|〈ξ〉−L, for all L > 0, (x, ξ) ∈ Ω, (α, β) ∈ Nd × Nd. We also denote by (4.3) Γ±(R, d, σ) = (x, ξ) ∈ Rn × Rn : |x| > R, 1 < |ξ| < d,± cos(x, ξ) > ±σ SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 17 with R > 1, d > 1, σ ∈ (−1, 1), and cos(x, ξ) = 〈x,ξ〉|x| |ξ| , the outgoing and incoming subsets of T ∗Rn, respectively. Eventually, for α > 1 , we denote the bounded operator F0(E, h) : L2α(R n) → L2(Sn−1) given by (4.4) (F0(E, h)f) (ω) = (2πh)− 2 (2E) 2E〈ω,x〉f(x)dx,E > 0. Isozaki and Kitada have constructed phase functions Φ± and symbols a± and b± such that, for some R0 >> 0, 1 < d4 < d3 < d2 < d1 < d0, and 0 < σ4 < σ3 < σ2 < σ1 < σ0 < 1: i) Φ± ∈ C∞(T ∗Rn) solve the eikonal equation (4.5) |∇xΦ±(x, ξ)|2 + V (x) = in (x, ξ) ∈ Γ±(R0, d0,±σ0), respectively. ii) (x, ξ) 7→ Φ±(x, ξ) − x · ξ ∈ A0 (Γ±(R0, d0,±σ0)) . iii) For all (x, ξ) ∈ T ∗Rn (4.6) ∂xj∂ξk (x, ξ) − δjk ∣∣∣∣ < ε(R0), where δjk is the Kronecker delta and ε(R0) → 0 as R0 → +∞. iv) a± ∼ j=0 h ja±j , where a±j ∈ A−j(Γ±(3R0, d1,∓σ1)), supp a±j ⊂ Γ±(3R0, d1,∓σ1), a±j solve (4.7) 〈∇xΦ±|∇xa±0〉+ (∆xΦ±) a±0 = 0 (4.8) 〈∇xΦ±|∇xa±j〉+ (∆xΦ±) a±j = ∆xa±j−1, j ≥ 1, with the conditions at infinity (4.9) a±0 → 1, a±j → 0, j ≥ 1, as |x| → ∞. in Γ±(2R0, d2,∓σ2), and solve (4.7) and (4.8) in Γ±(4R0, d1,∓σ2). v) b± ∼ j=0 h jb±j, where b±j ∈ A−j(Γ±(5R0, d3,±σ4), supp b±j ⊂ Γ±(5R0, d3,±σ4), b±j solve (4.7) and (4.8) with the conditions at infinity (4.9) in Γ±(6R0, d4,±σ3), and solve (4.7) and (4.8) in Γ±(6R0, d3,±σ3). For a symbol c and a phase function ϕ, we denote by Ih(c, ϕ) the oscillatory integral (4.10) Ih(c, ϕ) = (2πh)n (ϕ(x,ξ)−〈y,ξ〉)c(x, ξ)dξ and we set (4.11) K±a(h) = P (h)Ih(a±,Φ±)− Ih(a±,Φ±)P0(h), K±b(h) = P (h)Ih(b±,Φ±)− Ih(b±,Φ±)P0(h). The operator T (E, h) for E ∈] 1 [ is then given by (see [20, Theorem 3.3]) (4.12) T (E, h) = T+1(E, h) + T−1(E, h) − T2(E, h), 18 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND where (4.13) T±1(E, h) = F0(E, h)Ih(a±,Φ±)∗K±b(h)F∗0 (E, h) (4.14) T2(E, h) = F0(E, h)K∗+a(h)R(E + i0, h) (K+b(h) +K−b(h))F∗0 (E, h), where we denote from now on R(E ± i0, h) = (P − (E ± i0))−1. Writing explicitly their kernel, it is easy to see, by a non-stationary phase argument, that the operators T±1 are O(h∞) when θ 6= ω. Therefore we have (4.15) A(ω, θ,E, h) = −c(E)h(n−1)/2T2(ω, θ,E, h) +O(h∞), where c(E) is given in (1.4). As in [30], we shall use our resolvent estimate (Theorem 2.1) in a particular form. It was noticed by L. Michel in [24, Proposition 3.1] that, in the present trapping case, the following proposition follows easily from the corresponding one in the non-trapping setting. Indeed, if ϕ is a compactly supported smooth function, it is clear that P̃ = −h2∆+ (1− ϕ(x/R))V (x) satisfies the non-trapping assumption for R large enough, thanks to the decay of V at ∞. Writing [30, Lemma 2.3] for P̃ , one gets the Proposition 4.1. Let ω± ∈ A0 has support in Γ±(R, d, σ±) for R > R0. For E ∈ [E0 − δ,E0 + δ], we have (i) For any α > 1/2 and M > 1, then, for any ε > 0, (4.16) ‖R(E ± i0, h)ω±(x, hDx)‖−α+M,−α = O(h−3−ε). (ii) If σ+ > σ−, then for any α≫ 1, (4.17) ‖ω∓(x, hDx)R(E ± i0, h)ω±(x, hDx)‖−α+δ,−α = O(h∞). (iii) If ω(x, ξ) ∈ A0 has support in |x| < (9/10)R, then for any α≫ 1 (4.18) ‖ω(x, hDx)R(E ± i0, h)ω±(x, hDx)‖−α+δ,−α = O(h∞). Then we can follow line by line the discussion after Lemma 2.1 of D. Robert and H. Tamura, and we obtain (see Equations 2.2-2.4 there): (4.19) A(ω, θ,E, h) = c(E)h−(n+1)/2〈R(E + i0, h)g−eiψ−/h, g+eiψ+/h〉+O(h∞), where (4.20) g± = e −iψ±/h[χ±, P ]a±(x, h)e iψ±/h, (4.21) ψ+(x) = Φ+(x, 2Eθ), ψ−(x) = Φ−(x, 2Eω). Moreover the functions χ± are C n) functions such that χ± = 1 on some ball B(0, R±), with support in B(0, R± + 1). Eventually, we shall need the following version of Egorov’s Theorem, which is also used in Robert and Tamura’s paper. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 19 Proposition 4.2 ([30, Proposition 3.1]). Let ω(x, ξ) ∈ A0 be of compact support. Assume that, for some fixed t ∈ R, ωt is a function in A0 which vanishes in a small neighborhood of {(x, ξ); (x, ξ) = exp(tHp)(y, η), (y, η) ∈ suppω}. ‖Oph(ωt)e−itP/hOph(ω)‖−α,α = O(h∞), for any α ≫ 1. Moreover, the order relation is uniform in t when t ranges over a compact interval of R. In the three next sections, we prove Theorem 2.6 using (4.19). We set (4.22) u− = u − = R(E + i0, h)g−eiψ−/h, and our proof consists in the computation of u− in different region of the phase space, following the classical trajectories γ∞j , or γ k and γ ℓ . It is important to notice that we have (P−E)u− = 0 out of the support of g−. 5. Computations before the critical point 5.1. Computation of u− in the incoming region. We start with the computation of u− in an incoming region which contains the micro- support of g−. Notice that, thanks to Theorem 2.1, 〈x〉−αu−(x) is a semiclassical family of distributions for α > 1/2. Lemma 5.1. Let P be a Schrödinger operator as in (1.1) satisfying only (A1). Suppose that I is a compact interval of ]0,+∞[, and d > 0 is such that I ⊂] 1 [. Suppose also that 0 < σ+ < 1, R is large enough and K ⊂ T ∗Rn is a compact subset of {|x| > R} ∩ p−1(I). Then there exists T0 > 0 such that, if ρ ∈ K and t > T0, (5.1) exp(tHp)(ρ) ∈ Γ+(R/2, d, σ+) ∪ (B(0, R/2) × Rn). Proof. Let δ > 0. From the construction of C. Gérard and J. Sjöstrand [17], there exists a function G(x, ξ) ∈ C∞(R2n) such that, (HpG)(x, ξ) ≥ 0 for all (x, ξ) ∈ p−1(] [),(5.2) (HpG)(x, ξ) > 2E(1− δ) for |x| > R0 and p(x, ξ) = E ∈] [,(5.3) G(x, ξ) = x · ξ for |x| > R0.(5.4) Let ρ ∈ K, and γ(t) = (x(t), ξ(t)) = exp(tHp)(ρ) be the corresponding Hamiltonian curve. We distinguish between 2 cases: 1) For all t > 0, we have |x(t)| > R0. Then G(γ(t)) > 2E(1 − δ)t+G(ρ) and, for t > T1 with T1 large enough, (5.5) G(γ(t)) > 2 sup x∈B(0,R0) p(x,ξ)∈I G(x, ξ). 20 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND By continuity, there exists a neighborhood U of γ such that, for all γ̃ ∈ U , we have (5.6) G(γ̃(T1)) > sup x∈B(0,R0) p(x,ξ)∈I G(x, ξ). Since G is non-decreasing on γ̃(t), we have |x̃(t)| > R0 for all t > T1, and then (5.7) G(γ̃(t)) > 2E(1 − δ)(t − T1) +G(γ̃(T1)) > 2E(1 − δ)t − C. On the other hand, by uniformly finite propagation, we have |x̃(t)| < 2E(1+ δ)t+C. From (5.7), we get |x̃(t)| > 1 t− C for all γ̃ ∈ U , and then |ξ̃(t)| = 2E + ot→∞(1). In particular, the previous estimates gives (5.8) |x(t)| > R/2, (5.9) cos x̃, ξ̃ (t) > 2E(1− δ)t− C 2E(1 + δ)t+ C)( 2E + ot→∞(1)) 1 + δ + ot→∞(1) > 1− 3δ, for t > T0 with T0 large enough but independent on γ̃ ∈ U . Thus, for t > T0 and γ̃ ∈ U , we (5.10) γ̃(t) ∈ Γ+(R/2, d, σ+), with σ+ = 1− 3δ. 2) There exist T2 > 0 such that |x(T2)| = R0. Then there exists V a neighborhood of γ such that, for all γ̃ ∈ V, we have |x̃(T2)| < 2R0. Let t > T2. a) If |x̃(t)| ≤ R/2, then γ̃(t) ∈ B(0, R/2) × Rn. b) Assume now |x̃(t)| > R/2. Denote by T3 (> T2) the last time (before t) such that |x̃(T3)| = 2R0. Then G(γ̃(t)) >2E(1 − δ)(t− T3) +G(γ̃(T3))(5.11) >2E(1 − δ)(t− T3)− C,(5.12) where C depend only on R0. On the other hand, the have |x̃(t)| < 2E(1 + δ)(t − T3) + C (where the constant C depend only on R0). Then, (5.13) t− T3 > |x̃(t)|√ 2E(1 + δ) 2E(1 + δ) (5.14) |ξ̃(t)| = 2E + oR→∞(1), x̃, ξ̃ (t) > 2E(1− δ)|x̃(t)| |x̃(t)|( 2E(1 + δ))( 2E + oR→∞(1)) +O(R−1) 1 + δ + oR→∞(1) > 1− 2δ + oR→∞(1).(5.15) So, if R is large enough, γ̃(t) ∈ Γ+(R/2, d, σ+), σ+ = 1− 3δ. Then a) and b) imply that, for all γ̃ ∈ V and t > T0 := T2, we have (5.16) γ̃(t) ∈ Γ+(R/2, d, σ+) ∪ (B(0, R/2) × Rn). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 21 The lemma follows from (5.10), (5.16) and a compactness argument. � Recall that the microsupport of g−(x)e iψ−(x)/h ∈ C∞0 (Rn) is contained in Γ−(R−, d1, σ1). Let ω−(x, ξ) ∈ A0 with ω− = 1 near Γ−(R−/2, d1, σ1) and supp(ω−) ⊂ Γ−(R−/3, d0, σ0). Using the identity (5.17) u− = e−it(P−E)/h(g−e iψ−/h)dt+R(E + i0, h)e−iT (P−E)/h(g−eiψ−/h), and Proposition 4.1, Proposition 4.2 and Lemma 5.1, we get (5.18) Oph(ω−)u− = Oph(ω−) e−it(P−E)/h(g−e iψ−/h)dt+O(h∞), for some T > 0 large enough. In particular, (5.19) MS(Oph(ω−)u−) ⊂ Λ−ω ∩ (B(0, R− + 1)× Rn). 5.2. Computation of u− along γ Now we want to compute u− microlocally along a trajectory γ k . We recall that γ k is a bicharacteristic curve (x−k (t), ξ k (t)) such that (x k (t), ξ k (t)) → (0, 0) as t → +∞, and such that, as t→ −∞, (5.20) |x−k (t)− 2E0ωt− z−k | → 0, 2E0ω| → 0. If R− is large enough, a− solves (4.7) and (4.8) microlocally near γ k ∩ MS(g−eiψ−/h). In particular, microlocally near γ−k ∩ Γ−(R−/2, d1, σ1) ∩ (B(0, R−)× Rn), u− is given by (5.18) e−it(P−E)/h([χ−, P ]a−e iψ−/h)dt+O(h∞) e−it(P−E)/h(χ−(P − E)a−eiψ−/h)dt e−it(P−E)/h((P − E)χ−a−eiψ−/h)dt+O(h∞) (P − E)e−it(P−E)/h(χ−a−eiψ−/h)dt+O(h∞) =(P − E)R(E + i0, h)a−eiψ−/h +O(h∞) iψ−/h +O(h∞).(5.21) Now, using (5.21), and the fact that u− is a semiclassical distribution satisfying (5.22) (P − E)u− = 0, we can compute u− microlocally near γ ∩ B(0, R−) using Maslov’s theory (see [23] for more details). Moreover, it is proved in Proposition C.1 (see also [5, Lemma 5.8]) that the Lagrangian manifold Λ−ω has a nice projection with respect to x in a neighborhood of γ close to (0, 0). Then, in such a neighborhood, u− is given by (5.23) u−(x) = a−(x, h)e π/2eiψ−(x)/h, 22 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND where ν−k denotes the Maslov index of γ k , and ψ− satisfies the usual eikonal equation (5.24) p(x,∇ψ−) = E0. Here, to the contrary of (4.21), we have written E = E0 + zh with z = O(1), and we choose to work with z in the amplitudes instead of the phases. As usual, we have (5.25) ∂t(ψ−(x (t))) = ∇ψ−(x−k (t)) · ∂tx (t) = ∇ψ−(x−k (t)) · ξ (t) = |ξ− (t)|2, so that (5.26) ψ−(x (t)) = ψ−(x (s)) + (u)|2du We also have ψ−(x k (s)) = ( 2E0ωs+ z k ) · 2E0ω + o(1) as s→ −∞, and then (5.27) ψ−(x k (t)) = 2E0s+ |ξ−k (u)| 2du+ o(1), s→ −∞. We have obtained in particular that (5.28) ψ−(x k (t)) = |ξ−k (u)| 2−2E01u<0 du = |ξ−k (u)| 2−V (x−k (u))+E0 sgn(u) du. We turn to the computation of the symbol. The function a−(x, h) ∼ k=0 a−,k(x)h satisfies the usual transport equations: (5.29) ∇ψ− · ∇a−,0 + (∆ψ− − 2iz)a−,0 = 0, ∇ψ− · ∇a−,k + (∆ψ− − 2iz)a−,k = i ∆a−,k−1, k ≥ 1, In particular, we get for the principal symbol (5.30) ∂t(a−,0(x (t))) = ∇a−,0(x−k (t)) · ξ (t) = ∇a−,0(x−k (t)) · ∇ψ−(x (t)), so that, (5.31) ∂t(a−,0(x k (t))) = − ∆ψ−(x k (t))− 2iz a−,0(x k (t)) and then (5.32) a−,0(x k (t)) = a−,0(x k (s)) exp ∆ψ−(x−(u)) du + i(t− s)z On the other hand, from [30, Lemma 4.3], based on Maslov theory, we have (5.33) a−,0(x k (t)) = (2E0) 1/4D−k (t) −1/2eitz , where (5.34) D− (t) = ∣∣ det ∂x−(t, z, ω,E0) ∂(t, z) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 23 6. Computation of u− at the critical point Now we use the results of [5] to get a representation of u− in a whole neighborhood of the critical point. Indeed we saw already that (P − E)u− = 0 out of the support of g−, in particular in a neighborhood of the critical point. First, we need to recall some terminology of [18] and [5]. We recall from Section 2 that (µj)j≥0 is the strictly growing sequence of linear combinations over N of the λj’s. Let u(t, x) be a function defined on [0,+∞[×U , U ⊂ Rm. Definition 6.1. We say that u : [0,+∞[×U → R, a smooth function, is expandible, if, for any N ∈ N, ε > 0, α, β ∈ N1+m, (6.1) ∂αt ∂ u(t, x)− uj(t, x)e −µj t e−(µN+1−ε)t for a sequence of (uj)j smooth functions, which are polynomials in t. We shall write u(t, x) ∼ uj(t, x)e −µj t, when (6.1) holds. We say that f(t, x) = Õ(e−µt) if for all α, β ∈ N1+m and ε > 0 we have (6.2) ∂αt ∂ xf(t, x) = O(e−(µ−ε)t). Definition 6.2. We say that u(t, x, h), a smooth function, is of class SA,B if, for any ε > 0, α, β ∈ N1+m, (6.3) ∂αt ∂ xu(t, x, h) = O hAe−(B−ε)t Let S∞,B = A SA,B. We say that u(t, x, h) is a classical expandible function of order (A,B), if, for any K ∈ N, (6.4) u(t, x, h) − uk(t, x)h k ∈ SK+1,B, for a sequence of (uk)k expandible functions. We shall write u(t, x, h) ∼ uk(t, x)h in that case. Since the intersection between Λ−ω and Λ− is transverse along the trajectories γ k ), and since g−1 (z k ) 6= 0, Theorem 2.1 and Theorem 5.4 of [5] implies that one can write, microlocally near (0, 0), (6.5) u− = ∫ N−∑ αk(t, x, h)eiϕ k(t,x)/hdt, 24 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND where the αk(t, x, h)’s are classical expandible functions in S0,2ReΣ(E): (6.6) αk(t, x, h) ∼ αkm(t, x)h αkm(t, x) ∼ αkm,j(t, x)e −2(Σ(E)+µj )t, and where the αkm,j(t, x)’s are polynomial with respect to t. We recall from (2.26) that, for E = E0 + hz, (6.7) Σ(E) = − iz. Following line by line Section 6 of [5], we obtain (see [5, (6.26)]) αk0,0(0) = e iπ/4(2λ1) 3/2e−iν π/2|g(γ−k )|(D −1/2(2E0) 1/4.(6.8) Notice that from (5.32) and Proposition C.1, we have 0 < D− < +∞. From [5, Section 5], we recall that the phases ϕk(t, x) satisfies the eikonal equation (6.9) ∂tϕ k + p(x,∇xϕk) = E0, and that they have the asymptotic expansion (6.10) ϕk(t, x) ∼ ϕkj,m(x)t me−µjt, with Mkj < +∞. In the following, we denote (6.11) ϕkj (t, x) = ϕkj,m(x)t and the first ϕkj ’s are of the form ϕk0(t, x) =ϕ+(x) + ck(6.12) ϕk1(t, x) =− 2λ1g−(z−k ) · x+O(x 2),(6.13) where ck ∈ R is the constant depending on k given by (6.14) ck = “ψ−(0)” = lim k (t)) = S thanks to (5.28) (see also [5, Lemma 5.10]). Moreover ϕ+ is the generating function of the outgoing stable Lagrangian manifold Λ+ with ϕ+(0) = 0. We have (6.15) ϕ+(x) = x2j +O(x3). The fact that ϕk1(t, x) does not depend on t and the expression (6.13) follows also from Corollary 6.6 and (6.109). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 25 6.1. Study of the transport equations for the phases. Now, we examine the equations satisfied by the functions ϕkj (t, x), defined in (6.10), for the integers j ≤ ̂ (recall that ̂ is defined by µb = 2λ1). For clearer notations, we omit the superscript k until further notice. Let us recall that the function ϕ(t, x) satisfies the eikonal equation (6.9), which implies (see (6.10)) (6.16) e−µjtϕj,m(x)(−µjtm+mtm−1)+ ∇ϕj,m(x)tme−µj t +V (x) ∼ E0, and then e−µjtϕj,m(x)(−µjtm +mtm−1) + ∇ϕj,m∇ϕe,em(x)e−(µj+µe)ttm+ em +V (x) ∼ E0.(6.17) When µj < 2λ1, the double product of the previous formula provides a term of the form e if and only if µj = 0 or µe = 0. In particular, the term in e −µjt in (6.17) gives (6.18) ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+(x) · ∇ϕj,m(x)tm = 0. When µj = 2λ1, one gets also a term in e −2λ1t for µj = µe = λ1 and then ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+(x) · ∇ϕj,m(x)tm tm+ em∇ϕ1,m(x)∇ϕ1, em(x) = 0.(6.19) We denote (6.20) L = ∇ϕ+(x) · ∇ the vector field that appears in (6.18) and (6.19). We set also L0 = j λjxj∂j its linear part at x = 0, and we begin with the study of the solution of (6.21) (L− µ)f = g, with µ ∈ R and f , g ∈ C∞(Rn). First of all, we show that it is sufficient to solve (6.21) for formal series. Proposition 6.3. Let g ∈ C∞(Rn) and g0 the the Taylor expansion of g at 0. For each formal series f0 such that (L−µ)f0 = g0, there exists one and only one function f ∈ C∞(Rn) defined near 0 such that f = f0 +O(x∞) and (6.22) (L− µ)f = g, near 0. 26 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Proof. Let f̃0 be a C ∞ function having f0 has Taylor expansion at 0. With the notation f = f̃0 + r, the problem (6.22) is equivalent to find r = O(x∞) with (6.23) (L− µ)r = g − (L− µ)f̃0 = r̃, where r̃ ∈ C∞ has g0 − (L− µ)f0 = 0 as Taylor expansion at 0. Let y(t, x) be the solution of (6.24) ∂ty(t, x) = ∇ϕ+(y(t, x)), y(0, x) = x. Thus, (6.23) is equivalent to (6.25) r(x) = e−µsr̃(y(s, x))ds + e−µtr(y(t, x)). Since r(x), r̃(x) = O(x∞) and y(s, x) = O(eλ1t|x|) for t < 0, the functions e−µtr(y(t, x)), e−µtr̃(y(t, x)) are O(eNt) as t→ −∞ for all N > 0. Then (6.26) r(x) = e−µsr̃(y(s, x))ds, and r(x) = O(x∞). The uniqueness follows and it is enough to prove that r given by (6.26) is C∞. We have (6.27) ∂t(∇xy) = (∇2xϕ+(y))(∇xy), and since ∇2xϕ+ is bounded, there exists C > 0 such that (6.28) |∇xy(t, x)| . e−Ct, has t → −∞. Then, e−µs(∇r̃)(y(s, x))(∂jy(t, x)) = O(eNt) as t → −∞ for all N > 0 and ∂jr(x) = −µs(∇r̃)(y(s, x))(∂jy(t, x))ds. The derivatives of order greater than 1 can be treated the same way. � We denote (6.29) Lµ = L− µ : CJxK → CJxK, where we use the standard notation CJxK for formal series, and CpJxK for formal series of degree ≥ p. We notice that (6.30) Lµx α = (L0 − µ)xα + C|α|+1JxK = (λ · α− µ)xα +C|α|+1JxK. Recall that Iℓ(µ) has been defined in (2.20). The number of elements in Iℓ(µ) will be denoted (6.31) nℓ(µ) = #Iℓ(µ). One has for example n2(µ) = n1(µ)(n1(µ)+1) Proposition 6.4. Suppose µ ∈]0, 2λ1[. With the above notations, one has KerLµ⊕ ImLµ = CJxK. More precisely: i) The kernel of Lµ has dimension n1(µ), and one can find a basis (Ej1 , . . . , Ejn1(µ) KerLµ such that Ej(x) = xj + C2JxK, j ∈ I1(µ). ii) A formal series F = F0 + Fjxj + C2JxK belongs to ImLµ if and only if Fj = 0 for all j ∈ I1(µ). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 27 Remark 6.5. Thanks to Propostion 6.3, the same result is true for germs of C∞ functions at 0. Notice that when µ 6= µj for all j, Lµ is invertible. Proof. For a given F = α Fαx α ∈ CJxK, we look for solutions E = α ∈ CJxK to the equation (6.32) Lµ The calculus of the term of order x0 in (6.32) leads to the equation (6.33) E0 = − With this value for E0, (6.32) becomes, using again (6.30), (6.34) |α|=1 (λ · α− µ)Eαxα = |α|=1 α + C2JxK. We have two cases: If α /∈ I1(µ), one should have (6.35) Eα = λ · α− µ. If α ∈ I1(µ), the formula (6.34) becomes Fα = 0. In that case, the corresponding Eα can be chosen arbitrarily. Now suppose that the Eα are fixed for any |α| ≤ n− 1 (with n ≥ 2), and such that (6.36) Lµ |α|≤n−1 α + CnJxK. We can write (6.32) as (6.37) Lµ |α|=n α − Lµ |α|≤n−1 + Cn+1JxK, or, using again (6.30), (6.38) |α|=n (λ · α− µ)Eαxα = |α|≤n α − Lµ |α|≤n−1 +Cn+1JxK. Since |α| ≥ 2, one has λ · α ≥ 2λ1 > µ, so that (6.38) determines by induction all the Eα’s for |α| = n in a unique way. � Corollary 6.6. If j < ̂, the function ϕj(t, x) does not depend on t, i.e. we have Mj = 0. Proof. Suppose that Mj ≥ 1, then (6.18) gives the system (6.39) (L− µj)ϕj,Mj = 0, (L− µj)ϕj,Mj−1 = −Mjϕj,Mj , with ϕj,Mj 6= 0. But this would imply that ϕj,Mj ∈ KerLµ ∩ ImLµ, a contradiction. � 28 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND As a consequence, for j < ̂, the equation (6.18) on ϕj reduces to (6.40) (L− µj)ϕj,0 = 0, and, from Proposition 6.4, we get that (6.41) ϕj(t, x) = ϕj,0(x) = k∈I1(µ) dj,kxk +O(x2). Now we pass to the case j = ̂, and we study (6.19). First of all, we have seen that ϕ1 does not depend on t, so that this equation can be written (6.42) ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+ · ∇ϕj,m(x)tm + ∣∣∇ϕ1(x) ∣∣2 = 0. As for the study of (6.18), we begin with that of (6.21), now in the case where µ = 2λ1. We denote Ψ : Rn1(2λ1) −→ Rn2(λ1) the linear map given by (6.43) Ψ(Eβ1 , . . . , Eβn1(2λ1) β∈I1(2λ1) ∂α(L− µ)xβ α∈I2(λ1) and we set (6.44) n(Ψ) = dimKerΨ. Recalling that L = ∇ϕ+(x) · ∇, we see that (6.45) Ψ(Eβ1 , . . . , Eβn1(2λ1) β∈I1(2λ1) ∂α∂βϕ+(0) α∈I2(λ1) More generally, for any |α| = 2, we denote (6.46) Ψα((Eβ)β∈I1(2λ1)) = β∈I1(2λ1) ∂α∂βϕ+(0) Then, at the level of formal series, we have the Proposition 6.7. Suppose µ = 2λ1. Then i) KerLµ has dimension n2(λ1) + n(Ψ). ii) A formal series F = α Fαx α belongs to ImLµ if and only if ∀α ∈ I1(2λ1), Fα = 0,(6.47) |β|=1 β /∈I1(2λ1) ∂β∂αϕ+(0) 2λ1 − λ · β α∈I2(λ1) ∈ ImΨ.(6.48) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 29 iii) If F ∈ ImLµ, any formal series E = α with LµE = F satisfies F0,(6.49) λ · α− 2λ1 Fα, for α ∈ I1 \ I1(2λ1),(6.50) β∈I1(2λ1) |β|=1 β /∈I1(2λ1) ∂β∂αϕ+(0) 2λ1 − λ · β α∈I2(λ1) .(6.51) Moreover for α ∈ I2 \ I2(λ1), one has (6.52) Eα = λ · α− 2λ1 Fα −Ψα((Eβ)β∈I1(2λ1)) + |β|=1 β /∈I1(2λ1) 2λ1 − λ · β ∂α+βϕ+(0) Last, E is completely determined by F and a choice of the Eα for |α| ≤ 2 such that (6.49)– (6.52) are satisfied. iv) KerLµ ∩ Im(Lµ)2 = {0}. Proof. For a given F = α Fαx α we look for a E = α such that L2λ1E = F . First of all, we must have (6.53) E0 = − When this is true, we get (6.54) |α|=1 Eα(L0 − 2λ1)xα = |α|=1 Fα(L− 2λ1)xα + C2JxK, and we obtain as necessary condition that Fα = 0 for any α ∈ I1(2λ1). So far, the Eα for α ∈ I1(2λ1) can be chosen arbitrarily, and we must have (6.55) Eα = λ · α− 2λ1 , α ∈ I2 \ I1(2λ1). We suppose that (6.53) and (6.55) hold. Then we should have (6.56) |α|=2 Eα(L0−2λ1)xα = |α|=2 |α|=1 α/∈I1(2λ1) |α|=1 Eα(L−2λ1)xα +C3JxK. Notice that the second term in the R.H.S of (6.56) belongs to C2JxK thanks to (6.55). Again, we have to cases: • When α ∈ I2(λ1), the corresponding Eα can be chosen arbitrarily, but one must have |β|=1 ∂α(L− 2λ1)xβ |x=0(6.57) =Ψα((Eβ)β∈I1(2λ1)) + |β|=1 β /∈I1(2λ1) ∂α+βϕ+(0) ,(6.58) 30 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND and this, with (6.55), gives (6.51). • When |α| = 2, α /∈ I2(λ1), one obtains λ · α− 2λ1 |β|=1 ∂α(L− 2λ1)xβ λ · α− 2λ1 Fα −Ψα((Eβ)β∈I1(2λ1))− |β|=1 β /∈I1(2λ1) ∂α+βϕ+(0) ,(6.59) and this, with (6.55), gives (6.52). Now suppose that (6.53), (6.55), (6.57) and (6.59) hold, and that we have chosen a value for the free variables Eα for α ∈ I1(2λ1)∪I2(λ1). Thanks to the fact that λ ·α 6= 2λ1 for any α ∈ Nn with |α| = 3, we see as in the proof of Propostion 6.4, that the equation (6.54) has a unique solution, and the points (i), (ii) and (iii) follows easily. We prove the last point of the proposition, and we suppose that (6.60) E = α ∈ KerLµ ∩ Im(Lµ)2. First, we have E ∈ KerLµ ∩ ImLµ. Thus, E0 = 0 by (6.49), Eα = 0 for α ∈ I1(2λ1) by (6.47), and Eα = 0 for α ∈ I1 \ I1(2λ1) by (6.50). Last, since LµE = 0, we also have Eα = 0 for α ∈ I2 \ I2(λ1), and finally, (6.61) E = α∈I2(λ1) α + C3JxK. Moreover, one can write E = LµG for some G ∈ ImLµ. Since E0 = 0, we must have G0 = 0. Since G ∈ ImLµ, by (6.47), we have Gα = 0 for α ∈ I1(2λ1). Finally, since Eα = 0 for |α| = 1, α /∈ I1(2λ1), the same is true for the corresponding Gα, and (6.62) G = |α|≥2 Then, since Lµx α = 0+C3[x] for α ∈ I2(λ1), we obtain Eα = 0 for α ∈ I2(λ1). As above, we then get that, for |α| ≥ 3, Eα = 0, and this ends the proof. � Corollary 6.8. We always have Mb ≤ 2. If, in addition, λk 6= 2λ1 for all k ∈ {1, . . . , n}, then Mb ≤ 1. Proof. Suppose that Mb ≥ 3. Then (6.42) gives (L− µb)ϕb,Mb = 0(6.63) (L− µb)ϕb,Mb −1 = −Mbϕb,Mb(6.64) (L− µb)ϕb,Mb −2 = −(Mb − 1)ϕb,Mb −1,(6.65) with ϕb,Mb 6= 0. Notice that we have used the fact that Mb − 2 > 0 in (6.65). But this gives ϕb,Mb ∈ Ker(L − µb) and (L − µb)2ϕb,Mb−2 = Mb(Mb − 1)ϕb,Mb , so that ϕb,Mb ∈ Im(L− µb)2. This contradicts point (iv) of Proposition 6.7. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 31 Now we suppose that λk 6= 2λ1 for all k ∈ {1, . . . n}, that is I1(2λ1) = ∅, and that Mb = 2. Then (6.42) gives (L− µb)ϕb,Mb = 0(6.66) (L− µb)ϕb,Mb −1 = −Mb ϕb,Mb(6.67) with ϕb,Mb 6= 0. Therefore we have ϕb,Mb ∈ KerLµb ∩ ImLµb , and we get the same conclusion as in (6.61): ϕb,Mb(x) = O(x2). Then, we write (6.68) ϕb,Mb = (L− µb)g, and we see, as in (6.62), that g = O(x2), here because I1(2λ1) = ∅. Finally, we conclude also that ϕb,Mb = 0, a contradiction. � 6.2. Taylor expansions of ϕ+ and ϕ Now we compute the Taylor expansions of the leading terms with respect to t, of the phase functions ϕ(t, x) = ϕk(t, x). Lemma 6.9. The smooth function ϕ+(x) = x2j +O(x3) satisfies (6.69) ∂αϕ+(0) = − λ · α∂ αV (0), for |α| = 3, and (6.70) ∂αϕ+(0) = − 2(λ · α) β,γ∈I2 α=β+γ β! γ! βV (0) λj + λ · β γV (0) λj + λ · γ λ · α∂ αV (0), for |α| = 4, where α, β, γ ∈ Nn are multi-indices. Proof. The smooth function x 7→ ϕ+(x) is defined in a neighborhood of 0, and it is charac- terized (up to a constant: we have chosen ϕ+(0) = 0) by (6.71) p(x,∇ϕ+(x)) = |∇ϕ+(x)|2 + V (x) = 0 ∇ϕ+(x) = (λjxj)j=1,...,n +O(x The Taylor expansion of ϕ+ at x = 0 is (6.72) ϕ+(x) = x2j + |α|=3,4 ∂αϕ+(0) xα +O(x5), and we have (6.73) ∂jϕ+(x) = λjxj + |α|=3,4 ∂αϕ+(0) xα−1j +O(x4). 32 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Therefore |∇ϕ+(x)|2 = j + 2 |α|=3 )∂αϕ+(0) xα + 2 |α|=4 )∂αϕ+(0) |α|=3 ∂αϕ+(0) xα−1j +O(x5).(6.74) Let us compute further the last term in (6.74): |α|=3 ∂αϕ+(0) xα−1j |β|,|γ|=3 ∂βϕ+(0) ∂γϕ+(0) xβ+γ−21j |α|=4 α=β+γ |β|,|γ|=2 βϕ+(0) γϕ+(0) ·(6.75) Writing the Taylor expansion of V at x = 0 as (6.76) V (x) = x2j + |α|=3,4 ∂αV (0) xα +O(x5), and using the eikonal equation (6.71), we obtain first, for any α ∈ Nn with |α| = 3, (6.77) ∂αϕ+(0) = − λ · α∂ αV (0). Then, (6.74) and (6.75) give (6.78) ∂αϕ+(0) = − λ · α∂ αV (0) − 1 2(λ · α) β,γ∈I2 α=β+γ βV (0) λj + λ · β γV (0) λj + λ · γ for |α| = 4. � Now we pass to the function ϕ1. This function is a solution, in a neighborhood of x = 0, of the transport equation (6.79) Lϕ1(x) = λ1ϕ1(x), where L is given in (6.20). Lemma 6.10. The C∞ function ϕ1(x) = −2λ1g−1 (z ) · x+O(x2) satisfies (6.80) ∂αϕ1(0) = 2λ1α! (λ1 − λ · α)(λ1 + λ · α) αV (0) g−1 (z SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 33 for |α| = 2, and ∂αϕ1(0) =− λ1 − λ · α k∈I1(λ1),j∈I1 β,γ∈I2 α+1j=β+γ βV (0) λj + λ · β γV (0) (λ1 − λ · γ)(λ1 + λ · γ) g−1 (z (λ1 − λ · α)(λ1 + λ · α) k∈I1,j∈I1(λ1) β,γ∈I2 1j+α=β+γ (α+ 1j)! βV (0) λk + λ · β γV (0) λk + λ · γ g−1 (z (λ1 − λ · α)(λ1 + λ · α) j∈I1(λ1) αV (0) g−1 (z .(6.81) for |α| = 3. Proof. We write (6.82) ϕ1(x) = ajxj + |α|=2,3 α +O(x4), and Lemma 6.9 together with (6.73) give all the coefficients in the expansion (6.83) ∇ϕ+(x) = λjxj + |α|=2,3 Aj,αx α +O(x4) j=1,...,n In fact, we have (6.84) Aj,α = ∂α+1jϕ+(0) and aα = ∂αϕ1(0) We get Lϕ1(x) = ∂jϕ+(x)∂jϕ1(x) ajλjxj + |α|=2 αjλjaα + ajAj,α |α|=3 αjλjaαx |β|=|γ|=2 Aj,βγjaγx β+γ−1j + |α|=3 ajAj,αx +O(x4) ajλjxj + |α|=2 λ · α aα + Aj,αaj |α|=3 λ · α aα + α=β+γ−1j |β|,|γ|=2 Aj,βγjaγ + ajAj,α xα +O(x4).(6.85) 34 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Thus, (6.79) gives, for all α ∈ Nn with |α| = 2, (6.86) aα = λ1 − λ · α Aj,αaj , and, for all α ∈ Nn with |α| = 3, (6.87) aα = λ1 − λ · α β,γ∈I2 α+1j=β+γ γjAj,βaγ + ajAj,α Then, the lemma follows from (6.84). � 6.3. Asymptotics near the critical point for the trajectories. The informations obtained so far are not sufficient for the computation of the ϕj ’s. We shall obtain here some more knowledge by studying the behaviour of the incoming trajectory γ−(t) as t → +∞. We recall from [18, Section 3] (see also [5, Section 5]), that the curve γ−(t) = (x−(t), ξ−(t)) ∈ Λ− ∩ Λ−ω satisfy, in the sense of expandible functions, (6.88) γ−(t) = M ′j∑ γ−j,mt me−µjt, Notice that we continue to omit the subscript k for γ−k = (x k , ξ k ), z k , . . . Writing also (6.89) x−(t) ∼ g−j,m(t, z−)e −µj t, g−j (z −, t) = M ′j∑ g−j,m(z −)tm, for some integers M ′j, we know that g −) = g−1,0(z −) 6= 0. Since ξ−(t) = ∂tx−(t), we have (6.90) ξ−(t) ∼ M ′j∑ g−j,m(z −)(−µjtm +mtm−1)e−µjt. Proposition 6.11. If j < ̂, then M ′j = 0. We also have M ≤ 1, and M ′ = 0 when I1(2λ1) 6= ∅. Moreover (6.91) (g− |α|=2 ∂α+βV (0) (g−1 (z −))α for β ∈ I1(2λ1), 0 for β /∈ I1(2λ1). and, for |β| = 1, β /∈ I1(2λ1), (6.92) (g− (2λ1 + λ · β)(2λ1 − λ · β) |α|=2 ∂α+βV (0) (g−1 (z −))α. Proof. First of all, since ∂tγ −(t) = Hp(γ −(t)), we can write (6.93) ∂tγ −(t) = Fp(γ −(t)) +O(t2M ′1e−2λ1t), SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 35 where (6.94) Fp = d(0,0)Hp = , Λ2 = diag(λ21, . . . , λ We obtain (6.95) 1≤j<b M ′j∑ (Fp + µj)γ 1≤j<b M ′j∑ γ−j,mmt m−1e−µj t. Now suppose j < ̂ and M ′j ≥ 1. We get, for this j, for some γ j,M ′j 6= 0, (6.96) (Fp + µj)γ j,M ′j (Fp + µj)γ j,M ′j−1 =M ′jγ j,M ′j so that Ker(Fp + µj) ∩ Im(Fp + µj) 6= {0}. Since Fp is a diagonizable matrix, this can easily be seen to be a contradiction. Now we pass to the study of M ′ . So far we have obtained that (6.97) γ−(t) = 1≤j<b γ−j e −µjt + tme−2λ1t +O(tCe−µb+1t), and we can write (6.98) Hp(x, ξ) = |α|=2 ∂α∇V (0) xα +O(x3)  . Thus we have (6.99) Hp(γ −(t)) = Fp γ−j e −µjt + tme−2λ1t + e−2λ1tA(γ−1 ) +O(e −(2λ1+ε)t), where, noticing that µj + µj′ = 2λ1 if and only if j = j ′ = 1, (6.100) A(γ−1 ) = |α|=2 ∂α∇V (0) (g−1 )  . For the terms of order e−2λ1t, we have, since ∂tγ −(t) = Hp(γ −(t)), (6.101) (Fp + 2λ1) mtm−1 −A(γ−1 ). Thus, if we suppose that M ′ ≥ 2, we obtain (6.102) (Fp + 2λ1)γ b,M ′ (Fp + 2λ1)γ b,M ′ b,M ′ 36 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Then again we have γ− b,M ′ ∈ Ker(Fp + 2λ1) ∩ Im(Fp + 2λ1), a contradiction. Eventually, if λj 6= 2λ1 for all j, then Ker(Fp + 2λ1) = {0}. Therefore, if we suppose that b = 1, we see that γb,1 6= 0 satisfies the first equation in (6.102) and we get a contradiction. Now we compute γ− (t) = γ− t+ γ− . We have (6.103) (Fp + 2λ1)γ (Fp + 2λ1)γ −A(γ−1 ), and we see that γ− = Πγ− = ΠA(γ−1 ), where Π is the projection on the eigenspace of Fp associated to −2λ1. We denote by ej = (δi,j ⊗ 0)i=1,...,n and εj = (0 ⊗ δi,j)i=1,...,n for j = 1, . . . , n, so that (e1, . . . en, ε1, . . . , εn) is the canonical basis of R 2n = T(0,0)T n. Then it is easy to check that , for all j, v±j = ej±λj1εj is an eigenvector of Fp for the eigenvalue ±λj . In the basis {e1, ε1, . . . , en, εn} the projector Π is block diagonal and, if Kj = Vect(ej , εj), we (6.104) Π|Kj 1/2 −1/4λ1 −λ1 1/2 for j ∈ I1(2λ1), 0 for j /∈ I1(2λ1). Therefore, we obtain (6.105) (g− |α|=2 ∂β∂αV (0) (g−1 (z −))α for β ∈ I1(2λ1), 0 for β /∈ I1(2λ1). Now suppose that k /∈ I1(2λ1). Then the second equality in (6.103) restricted to Kk gives (6.106) 2λ1 1 λ2k 2λ1 Πkγb,0 = −ΠkA(γ−1 ), where Πk denotes the projection onto Kk. Solving this system, one gets (6.107) (g− 4λ21 − λ2k ΠxΠkA(γ and, together with (6.100), this ends the proof of Proposition 6.11. � 6.4. Computation of the ϕkj ’s. Here we compute the ϕkj ’s for j ≤ ̂. We still omit the superscript k. From [5], we know that ξ−(t) = ∇xϕ t, x−(t) , so that, using (6.41), ξ−(t) =∇ϕ+(x−(t)) +∇ϕ1(x−(t))e−λ1t + 2≤j<b ∇ϕj(0)e−µj t +∇ϕb,2(0)t2e−2λ1t +∇ϕb,1(0)te−2λ1t +∇ϕb,0(0)e−2λ1t + Õ(e−µb+1t).(6.108) Since ϕ+ = −ϕ− and ξ− ∈ Λ−, we have ∇ϕ+(x−(t)) = −ξ−(t), and we obtain first, by (6.90), (6.109) ∇ϕj(0) = −2µjg−j (z for 1 ≤ j < ̂. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 37 Now we study ϕb(t, x) = ϕb,0(x) + tϕb,1(x) + t 2ϕb,2(x) when I1(2λ1) 6= ∅. It follows from (6.108) that we have (6.110) − 4λ1g−b,1(z −) = ∇ϕb,1(0), − 4λ1g−b,0(z −) + 2g− (z−) = ∇ϕb,0(0) +∇2ϕ1(0)g−1 (z On the other hand, we have seen that, by (6.19), the functions ϕb,2, ϕb,1 and ϕb,0 satisfy (6.111) (L− 2λ1)ϕb,2 = 0, (L− 2λ1)ϕb,1 = −2ϕb,2, (L− 2λ1)ϕb,2 = −ϕb,1 − |∇ϕ1(0)|2. In particular ϕb,2 ∈ Ker(L− 2λ1) ∩ Im(L− 2λ1) so that (see (6.61)), (6.112) ϕb,2(x) = α∈I2(λ1) c2,αx α +O(x3). Going back to (6.108), we notice that we obtain now ξ−(t) =∇ϕ+(x−(t)) +∇ϕ1(x−(t))e−λ1t + 2≤j<b ∇ϕj(0)e−µj t ∇ϕb,1(0)te−2λ1t +∇ϕb,0(0)e−2λ1t + Õ(e−µb+1t),(6.113) and this equality is consistent with Proposition 6.11. Then, (6.49) and (6.50) give (6.114) ϕb,1(x) = α∈I1(2λ1) c1,αx |α|=2 c1,αx α +O(x3), and, by (6.51), we have (6.115) Ψ((c1,β)β∈I1(2λ1)) = (−2c2,α)α∈I2(λ1). By (6.52), we also have for |α| = 2, α /∈ I2(λ1), (6.116) c1,α = 2λ1 − λ · α β∈I1(2λ1) ∂α+βϕ+(0) c1,β . The function ϕb,0(x) = |α|≤2 c0,αx α +O(x3) satisfies (see (6.42)) (6.117) (L− 2λ1)ϕb,0 = −ϕb,1 − ∣∣∇ϕ1(x) First of all, the compatibility condition (6.47) gives (6.118) ∀α ∈ I1(2λ1), c1,α = −∇ϕ1(0) · ∂α∇ϕ1(0), so that in particular, by (6.115), the function ϕb,2 is known up to O(x3) terms: (6.119) ∀α ∈ I2(λ1), c2,α = β∈I1(2λ1) ∂α+βϕ+(0) ∇ϕ1(0) · ∂β∇ϕ1(0), 38 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND (6.120) ∀α /∈ I2(λ1), |α| = 2, c1,α = − 2λ1 − λ · α β∈I1(2λ1) ∂α+βϕ+(0) ∇ϕ1(0) · ∂β∇ϕ1(0). Now (6.49) and (6.50) give (6.121) c0,0 = ϕb,0(0) = |∇ϕ1(0)|2, (6.122) ∀α /∈ I1(2λ1), |α| = 1, c0,α = 2λ1 − λ · α ∇ϕ1(0) · ∂α∇ϕ1(0). From the other compatibility condition (6.48), we know that c1,α + ∇ϕ1(0) · ∂α∇ϕ1(0) + β,γ∈I1(λ1) β+γ=α ∂β∇ϕ1(0) · ∂γ∇ϕ1(0) |β|=1 β/∈I1(2λ1) ∂α+βϕ+(0) ∇ϕ1(0) · ∂β∇ϕ1(0) 2λ1 − λ · β α∈I2(λ1) ∈ ImΨ,(6.123) and, from (6.51), we obtain a relation between the (c0,β)β∈I1(2λ1) and the (c1,α)α∈I2(λ1), namely ∀α ∈ I2(λ1), c1,α =− ∂α∇ϕ1(0) · ∇ϕ1(0)− β,γ∈I1(λ1) β+γ=α ∂β∇ϕ1(0) · ∂γ∇ϕ1(0) β∈I1(2λ1) ∂α+βϕ+(0) c0,β − |β|=1 β/∈I1(2λ1) ∂α+βϕ+(0) ∇ϕ1(0) · ∂β∇ϕ1(0) 2λ1 − λ · β ·(6.124) Using the second equation in (6.110), we obtain, for |β| = 1, (6.125) c0,β = −4λ1(g−b,0(z −))β + 2(g− (z−))β − ∂β∇ϕ1(0) · g−1 (z At this point, we have computed the functions ϕb,1(x) and ϕb,2(x) up to O(x3), in terms of derivatives of ϕ+ and ϕ1, and of the g (z−). We shall now use the expressions we have obtained in Section 6.2 and in Section 6.3 to give these functions in terms of g−1 and of derivatives of V only. First of all, by (6.112), (6.119), Lemma (6.9) and Lemma (6.10), we obtain ϕb,2(x) =− γ∈I1(2λ1) α,β∈I2(λ1) ∂β+γV (0) (g−1 (z ∂α+γV (0) +O(x3).(6.126) Then we have (6.127) ϕb,1(x) = −4λ1g−b,1(z −) · x+ α∈I2(λ1) c1,αx |α|=2 α/∈I2(λ1) c1,αx α +O(x3), SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 39 where the c1,α are given by (6.124) and (6.125) for α ∈ I2(λ1), and by (6.120) for α /∈ I2(λ1). • For |α| = 2, α /∈ I2(λ1), we obtain by (6.116), Lemma 6.9 and Lemma 6.10, c1,α = (2λ1 + λ · α)(2λ1 − λ · α) β∈I1(2λ1) ∂α+βV (0) (λ1 + λj)(3λ1 + λj) β∇V (0) · g−1 (z−)(g −))j .(6.128) Since (g−1 (z −))j = 0 but for j ∈ I1(λ1), we get, changing notations a bit, (6.129) c1,α = (2λ1 + λ · α)(2λ1 − λ · α) γ∈I1(2λ1) β∈I2(λ1) ∂α+γV (0) ∂β+γV (0) (g−1 (z −))β. • Now we compute c1,α for α ∈ I2(λ1). For the last term in the R.H.S. of (6.124), we obtain |β|=1 β /∈I1(2λ1) ∂α+βϕ+(0) ∇ϕ1(0) · ∂β∇ϕ1(0) 2λ1 − λ · β γ∈I1\I1(2λ1) β∈I2(λ1) (2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2 ∂α+γV (0) ∂β+γV (0) (g−1 (z −))β .(6.130) Using (6.91) and (6.125), we have also β∈I1(2λ1) ∂α+βϕ+(0) c0,β = γ∈I1(2λ1) ∂α+γV (0) (z−))γ + γ∈I1(2λ1) β∈I2(λ1) ∂α+γV (0) ∂β+γV (0) (g−1 (z −))β .(6.131) 40 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND We pass to the computation of − 1 ∂α∇ϕ1(0) · ∇ϕ1(0) for α ∈ I2(λ1). We obtain ∂α∇ϕ1(0) · ∇ϕ1(0) = − β∈I2(λ1) ∂α+βV (0) (g−1 (z j,p,k=1 β,γ∈I2 β+γ=α+1p+1j ((α+ 1p)j + 1)(αp + 1) (λk + λ · β)(λk + λ · γ) ∂β+1kV (0) ∂γ+1kV (0) (g−1 (z −))j(g + 2λ1 j,p,k=1 β,γ∈I2 β+γ=α+1p+1j (αp + 1)γj (λ1 − λ · γ)(λ1 + λ · γ)(λj + λ · β) β+1jV (0) ∂γ+1kV (0) (g−1 (z −))k(g = I + II + III. (6.132) Writing δ = 1j + 1p, we get (6.133) II = −1 β,γ,δ∈I2 β+γ=α+δ (α+ δ)! (λk + λ · β)(λk + λ · γ) ∂β+1kV (0) ∂γ+1kV (0) (g−1 (z α! δ! Since δ ∈ I2(λ1) (otherwise (g−1 (z−))δ = 0), we have β, γ ∈ I2(λ1) and, changing notations a (6.134) II = −1 β∈I2(λ1) (α+ β)! γ,δ∈I2(λ1) γ+δ=α+β (2λ1 + λj)2 γV (0) δV (0) (g−1 (z In the last term III, we can suppose that γ = 1j+1q for some q ∈ {1, . . . , n}. Then γj = γ! and, writing β = 1a + 1b we have III = λ1 j,k,p=1 (αp + 1)(g −))k(g a,b,q∈I1 1a+1b+1q=α+1p (αp + 1) (λ1 − λj − λq)(λ1 + λj + λq)(λj + λa + λb) ∂j,a,bV (0)∂j,q,kV (0).(6.135) Since α ∈ I2(λ1) and 1p ∈ I1(λ1) (otherwise (g−1 (z−))p = 0), we have 1a, 1b, 1q ∈ I1(λ1) so that we can write (6.136) III = − j,k,p=1 (αp + 1) λj(2λ1 + λj)2 (g−1 (z −))k(g a,b,q∈I1 1a+1b+1q=α+1p ∂j,a,bV (0)∂j,q,kV (0). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 41 Now it is easy to check, noticing that (α+ 1p)k ∈ {1, 2, 3, 4} and examining each case, that (6.137) a,b,q∈I1 1a+1b+1q=α+1p ∂j,a,bV (0)∂j,q,kV (0) = (α+ 1p)k a,b,c,d∈I1 1a+1b+1c+1d=α+1p+1k ∂j,a,bV (0)∂j,c,dV (0). Therefore, we have III = −1 j,k,p=1 (α + 1p + 1k)! λj(2λ1 + λj)2 (g−1 (z −))k(g a,b,c,d∈I1 1a+1b+1c+1d=α+1p+1k ∂j,a,bV (0)∂j,c,dV (0).(6.138) Eventually, setting β = 1p + 1k, γ = 1a + 1b and δ = 1c + 1d, we get (6.139) III = − β∈I2(λ1) (α+ β)! γ,δ∈I2(λ1) γ+δ=α+β λj(2λ1 + λj)2 γV (0) δV (0) (g−1 (z We are left with the computation of β,γ∈I1(λ1) β+γ=α ∂β∇ϕ1(0) · ∂γ∇ϕ1(0) = − β,γ∈I1(λ1) β+γ=α βϕ1(0) · ∂j∂γϕ1(0) λ2j(2λ1 + λj) β,γ∈I1(λ1) β+γ=α k,ℓ=1 ∂j∂k∂ βV (0)(g−1 (z −))k∂j∂ℓ∂ γV (0)(g−1 (z −))ℓ.(6.140) At this point, we notice that α∈I2(λ1) β,γ∈I1(λ1) β+γ=α ∂β∇ϕ1(0) · ∂γ∇ϕ1(0)xα λ2j (2λ1 + λj) β,γ∈I1(λ1) α∈I2(λ1) β+γ=α k,ℓ=1 ∂j∂k∂ βV (0)(g−1 (z −))k∂j∂ℓ∂ γV (0)(g−1 (z −))ℓ x λ2j (2λ1 + λj) α,β∈I2(λ1) (α+ β)! γ,δ∈I2(λ1) γ+δ=α+β γV (0) δV (0) (g−1 (z α,β∈I2(λ1) αV (0) βV (0) xα(g−1 (z }(6.141) 42 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND From (6.124), (6.130), (6.131) (6.139), and (6.141), we finally obtain that α∈I2(λ1) c1,αx γ∈I1\I1(2λ1) α,β∈I2(λ1) (2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2 ∂α+γV (0) ∂β+γV (0) (g−1 (z −))βxα γ∈I1(2λ1) α∈I2(λ1) ∂α+γV (0) (z−))γxα + γ∈I1(2λ1) α,β∈I2(λ1) ∂α+γV (0) ∂β+γV (0) (g−1 (z −))βxα α,β∈I2(λ1) ∂α+βV (0) (g−1 (z −))βxα α,β∈I2(λ1) (α+ β)! γ,δ∈I2 γ+δ=α+β (2λ1 + λj)2 γV (0) δV (0) (g−1 (z α,β∈I2(λ1) (α+ β)! γ,δ∈I2(λ1) γ+δ=α+β λj(2λ1 + λj)2 γV (0) δV (0) (g−1 (z α,β∈I2(λ1) (α + β)! γ,δ∈I2(λ1) γ+δ=α+β λ2j(2λ1 + λj) γV (0) δV (0) (g−1 (z α,β∈I2(λ1) λ2j(2λ1 + λj) αV (0) βV (0) xα(g−1 (z −))β , (6.142) or, more simply, α∈I2(λ1) c1,αx α = − γ∈I1(2λ1) α∈I2(λ1) ∂α+γV (0) (z−))γxα + α,β∈I2(λ1) (g−1 (z γ∈I1\I1(2λ1) (2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2 ∂α+γV (0)∂β+γV (0) γ∈I1(2λ1) ∂α+γV (0)∂β+γV (0)− ∂α+βV (0) − (α+ β)! γ,δ∈I2 γ+δ=α+β γV (0) δV (0) λ2j (2λ1 + λj) αV (0)∂j∂ βV (0) .(6.143) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 43 7. Computations after the critical point 7.1. Stationary phase expansion in the outgoing region. Now we compute the scattering amplitude starting from (4.19). First of all, we change the cut-off function χ+ so that the support of the right hand side of the scalar product in (4.19) is close to (0, 0). ℓχ+ = 0 supp(∇χ+) χ+ = 1 ℓeχ+ = 0 supp(∇eχ+) eχ+ = 1 Figure 1. The support of χ+ and χ̃+ in T Using Maslov’s theory, we construct a function v+ which coincides with a+(x, h)e iψ+(x)/h out of a small neighborhood of ∩ (B(0, R+ +1)×Rn) and such that v+ is a solution of (P−E)v+ = 0 microlocally near . Let χ̃+(x, ξ) ∈ C∞(T ∗Rn) such that χ̃+(x, ξ) = χ+(x) out of a small enough neighborhood of ∩ (B(0, R++1)×Rn). In particular, (P −E)v+ is microlocally 0 near the support of χ+ − χ̃+. So, we have 〈u−, [χ+, P ]v+〉 =〈u−, [Op(χ̃+), P ]v+〉+ 〈u−, (χ+ −Op(χ̃+))(P − E)v+〉 − 〈(P − E)u−, (χ+ −Op(χ̃+))v+〉 =〈u−, [Op(χ̃+), P ]v+〉+O(h∞)− 〈g−eiψ−/h, (χ+ −Op(χ̃+))v+〉 =〈u−, [Op(χ̃+), P ]v+〉+O(h∞),(7.1) since the microsupport of g−e iψ−/h and χ+− χ̃+ are disjoint. Thus, the scattering amplitude is given by (7.2) A(ω, θ,E, h) = c(E)h−(n+1)/2〈u−, [Op(χ̃+), P ]v+〉+O(h∞). Now we will prove that, modulo O(h∞), the only contribution to the scattering amplitude in (7.2) comes from the values of the functions u− and v+ microlocally on the trajectories γ and γ∞j . From (5.18), the fact that u− = O(h−C) and (P −E)u− = 0 microlocally out of the microsupport of g−e −iψ−/h, and the usual propagation of singularities theorem, we get (7.3) MS(u−) ⊂ Λ−ω ∪ Λ+. Moreover, we have (7.4) MS(v+) ⊂ Λ+θ . 44 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Now, let f∞j (resp. f ℓ ) be C n) functions with support in a small enough neighborhood of γ∞j (resp. γ ℓ ∩MS(v+)) such that f∞j = 1 (resp. f k = 1) in a neighborhood of γ j (resp. γ+ℓ ∩MS(v+)). In particular, we assume that all these functions have disjoint support. Since u− and v+ have disjoint microsupport out of the support of the f j and the f ℓ , we have A(ω, θ,E, h) =c(E)h−(n+1)/2 〈Op(f∞j )u−,Op(f∞j )[Op(χ̃+), P ]v+〉 + c(E)h−(n+1)/2 〈Op(f+ℓ )u−,Op(f ℓ )[Op(χ̃+), P ]v+〉+O(h =Areg +Asing.(7.5) Concerning the terms which contain f∞j , Areg, we are exactly in the same setting as in [30, Section 4]. The computation there gives (7.6) Areg = j,m(ω, θ,E)h iS∞j /h +O(h∞). Now we compute Asing. Proceeding as in Section 5.2 for u−, one can show that v+ can be written as (7.7) v+(x) = a+(x, h)e π/2eiψ+(x)/h, microlocally near any ρ ∈ γ+ close enough to (0, 0). Here ν+ is the Maslov index of γ+ . The phase ψ+ and the classical symbol a+ satisfy the usual eikonal and transport equations. In particular, as in (5.28) and (5.33), we have (7.8) ℓ (t)) = − |ξ+ℓ (u)| 2 − 2E01u>0 du = − |ξ+ℓ (u)| 2 − V (x+ℓ (u)) −E0 sgn(u) du, and a+(x, h) ∼ m a+,m(x)h m with (7.9) a+,0(x ℓ (t)) = (2E0) 1/4(D+ℓ (t)) −1/2eitz, where (7.10) D+ℓ (t) = ∣∣ det ∂x+(t, z, θ, E0) ∂(t, z) We can chose χ̃+ so that the microsupport of the symbol of Op(f ℓ )[Op(χ̃+), P ] is contained in a vicinity of such a point ρ ∈ γ+ℓ (see Figure 1). Then, microlocally near ρ, we have (7.11) Op(f+ )[Op(χ̃+), P ]v+ = ã+(x, h)e π/2eiψ+(x)/h, (7.12) ã+(x, h) = ã+,m(x)h (7.13) ã+,0(x) = −i{χ̃+, p}(x,∇ψ+(x))a+,0(x). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 45 From [5, Section 5], the Lagrangian manifold {(x,∇xϕk(t, x)); ∂tϕk(t, x) = 0}, coincides with Λ−ω . In particular, since MS(v+) ⊂ Λ+θ and since there is no curve γ∞(z∞j ) sufficiently closed to the critical point, the finite times in (6.5) give a contribution O(h∞) to the scattering amplitude (4.19). In view of the equations (6.5), (6.12) and (7.11), the principal contribution of Asing will come from the intersection of the manifolds Λ+θ and Λ+. Recall that, from (A5), the manifolds Λ+θ and Λ+ intersect transversally along γ In particular, to compute Asing, we can apply the method of stationary phase in the directions that are transverse to γ+ . For each ℓ, after a linear and orthonormal change of variables, we can assume that g+ ) is collinear to the xℓℓℓ–direction, and that V (x) satisfies (A2). We denote Hℓxℓℓℓ = {y = (y1, . . . , yn) ∈ R n; yℓℓℓ = xℓℓℓ} the hyperplane orthogonal to (0, . . . , 0, xℓℓℓ, 0, . . . , 0). We shall compute Asing in the case where there is only one incoming curve γ− and one outgoing curve γ+ . In the general case, Asing is simply given by the sum over k and ℓ of such contributions. Using (4.19), (6.5) and (7.11), we can write Asing =c(E)h −(n+1)/2 k(t,x)−ψ+(x))/hαk(t, x, h)ã+(x, h)e π/2dt dx c(E)h−(n+1)/2√ y∈Hxℓℓℓ k(t,x)−ψ+(x))/hαk(t, x, h)ã+(x, h)e π/2dt dy dxℓℓℓ.(7.14) Let Φ(y) = ϕk(t, xℓℓℓ, y) − ψ+(xℓℓℓ, y) be the phase function in (7.14). From (6.10)–(6.13), we can write (7.15) Φ(y) = S−k + (ϕ+ − ψ+)(xℓℓℓ, y) + ψ̃(t, xℓℓℓ, y), where ψ̃ = O(e−λ1t) is an expandible function. Since the manifolds Λ+ and Λ+ intersect transversally along γ+ , the phase function y → (ϕ+−ψ+)(xℓℓℓ, y) has a non degenerate critical point yℓ(xℓℓℓ) ∈ Hℓxℓℓℓ ∩ Πxγ , and xℓℓℓ 7→ yℓ(xℓℓℓ) is C∞ for xℓℓℓ 6= 0. Then, from the implicit function theorem, the function Φ has a unique critical point yℓ(t, xℓℓℓ) ∈ Hℓxℓℓℓ for t large enough depending on xℓℓℓ. The function (t, xℓℓℓ) 7→ yℓ(t, xℓℓℓ) is expandible and we have (7.16) yℓ(t, xℓℓℓ) = y ℓ(xℓℓℓ)−Hess(ϕ+ − ψ+)−1 yℓ(xℓℓℓ) yℓ(xℓℓℓ) e−µ1t + Õ e−µ2t As a consequence, Φ yℓ(t, xℓℓℓ) is also expandible. Since ϕ+ and ψ+ satisfy the same eikonal equation, we get (see (5.25)) (7.17) ∂t(ϕ+ − ψ+)(x+ℓ (t)) = |ξ (t)|2 − |ξ+ (t)|2 = 0. 46 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Thus, (ϕ+ − ψ+)(yℓ(xℓℓℓ)) does not depend of xℓℓℓ and is equal to (ϕ+ − ψ+)(yℓ(xℓℓℓ)) = lim (ϕ+ − ψ+)(x+ℓ (t)) |ξ+ℓ (s)| 2 − 2E01s>0 ds (s)|2 − V (x+ (s))− E0 sgn(s) ds ,(7.18) where we have used (7.8). Therefore, the phase function Φ at the critical point yℓ(t, xℓℓℓ) is equal to yℓ(t, xℓℓℓ) µm≤2λ1 t, yℓ(xℓℓℓ) e−µmt Hess(ϕ+ − ψ+)−1 yℓ(xℓℓℓ) yℓ(xℓℓℓ) · ∇ϕ1 yℓ(xℓℓℓ) e−2µ1t + Õ(e−eµt),(7.19) where µ̃ is the first of the µj’s such that µj > 2λ1. Using the method of the stationary phase for the integration with respect to y ∈ Hℓxℓℓℓ in (7.14), we get (7.20) Asing = c(E)h −(n+1)/2 (2πh)(n−1)/2 eiΦ(y ℓ(t,xℓℓℓ))/hf ℓ(t, xℓℓℓ, h) dt dxℓℓℓ +O(h∞). TheO(h∞) term follows from the fact that the error term stemming from the stationary phase method can be integrated with respect to time t, since αk ∈ S0,2ReΣ(E), with ReΣ(E) > 0 (see the beginning of Section 6). The symbol f ℓ(t, xℓℓℓ, h) is a classical expandible function of order S1,2ReΣ(E) in the sense of Definition 6.2: (7.21) f ℓ(t, xℓℓℓ, h) ∼ f ℓm(t, xℓℓℓ, lnh)h where the f ℓm are polynomials with respect to lnh and (7.22) f ℓ0(t, xℓℓℓ, lnh) = α t, yℓ(t, xℓℓℓ) ã+,0 yℓ(t, xℓℓℓ) π/2 e i sgnΦ′′ |Hℓxℓℓℓ (yℓ(t,xℓℓℓ))π/4 ∣∣detΦ′′|Hℓxℓℓℓ yℓ(t, xℓℓℓ) )∣∣1/2 Using Proposition C.1, we compute the Hessian of Φ, and we get yℓ(xℓℓℓ) =diag(−λ1, . . . ,−λℓℓℓ−1, λℓℓℓ,−λℓℓℓ+1, . . . ,−λn) + o(1), yℓ(xℓℓℓ) =diag(λ1, . . . , λn) + o(1). Then, for xℓℓℓ small enough and t large enough depending on xℓℓℓ, we have ∣∣detΦ′′|Hℓxℓℓℓ yℓ(t, xℓℓℓ) )∣∣1/2 = j 6=ℓℓℓ 2λj + o(1),(7.23) sgnΦ′′|Hℓxℓℓℓ yℓ(t, xℓℓℓ) = n− 1,(7.24) as xℓℓℓ goes to 0. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 47 7.2. Behaviour of the phase function Φ. Suppose that j ∈ N is such that j < ̂. From (6.40), we have (7.25) ϕkj (x (s0)) = e −µj(s−s0)ϕkj (x (s)). Combining (6.41) with (6.109), we get ϕkj (x ℓ (s0)) =e µjs0e−µjs − 2µj〈g−j (z k )|g ℓ 〉)e µjs +O(e2λ1s) =− 2µj g−j (z ∣∣g+j (z eµjs0 .(7.26) We suppose first that we are in the case (a) of assumption (A7). Then, (7.19) becomes (7.27) Φ yℓ(t, xℓℓℓ) − 2µk eµks(xℓℓℓ)e−µkt + Õ(e−µk+1t). Here s(xℓℓℓ) is such that x (s(xℓℓℓ)) = x ℓ(xℓℓℓ) and the Õ(e−µk+1t) is in fact expandible, uniformly with respect to xℓℓℓ when xℓℓℓ varies in a compact set avoiding 0. Suppose now that we are in the case (b) of assumption (A7). Of course, from (7.26), we have ϕj yℓ(xℓℓℓ) = 0 for all j < ̂. On the other hand, Corollary 6.8 and (6.111) imply (7.28) ϕk b,2(x ℓ (s0)) = e −2λ1(s−s0)ϕk b,2(x ℓ (s)). Combining this with (6.126), we get b,2(x (s0)) =e 2λ1s0e−2λ1s j∈I1(2λ1) α,β∈I2(λ1) ∂α+1jV (0) ∂β+1jV (0) g−1 (z g+1 (z e2λ1s +O(e3λ1s) j∈I1(2λ1) α,β∈I2(λ1) ∂α+1jV (0) ∂β+1jV (0) g−1 (z g+1 (z e2λ1s0 .(7.29) In particular, (7.19) becomes, in that case, yℓ(t, xℓℓℓ) =S−k − S j∈I1(2λ1) α,β∈I2(λ1) ∂α+1jV (0) ∂β+1jV (0) g−1 (z g+1 (z e2λ1s(xℓℓℓ) × t2e−2λ1t +O(te−2λ1t) =S−k + S ℓ +M2(k, ℓ)t 2e−2λ1t +O(te−2λ1t).(7.30) As in (7.27), the term O(te−2λ1t) is in fact expandible uniformly with respect to xℓℓℓ when xℓℓℓ varies in a compact set avoiding 0. Eventually, we suppose that we are in the case (c) of assumption (A7). Then we obtain from (7.26) and (7.29) that ϕj yℓ(xℓℓℓ) = 0 for all j < ̂ and ϕb,2 yℓ(xℓℓℓ) = 0. With the last identity in mind, Equation (6.111) on ϕk implies (7.31) ϕk b,1(x ℓ (s0)) = e −2λ1(s−s0)ϕk b,1(x ℓ (s)). 48 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND In order to compute ϕk b,1(x ℓ (s)), we put the expansion (2.15) for x ℓ (s) (with Proposition 6.11 in mind) into (6.127). The third term in (6.127) will be, at least, O(e(µ2+µ1)s) = o(e2λ1s). Thank to (6.91) and thanks to the fact that M2(k, ℓ) = 0, the first term in (6.127) will give no contribution of order se2λ1s and will be of the form (7.32) − 4λ1g−b,1(z −) · x+ℓ (s) = − α∈I2(λ1) αV (0) (g−1 (z −))α(g+ (z+))je 2λ1s + Õ(eµb+1s) It remains to study the contribution the second term in (6.127), as given in (6.143). As previously, the first term of the third line in (6.143) will give a term of order o(e2λ1s). The other terms will contribute to the order e2λ1s for α∈I2(λ1) αV (0) (z−))j(g +))α + α,β∈I2(λ1) (g−1 (z (g+1 (z − ∂α+βV (0) + j∈I1\I1(2λ1) λ2j (4λ 1 − λ2j ) ∂α+γV (0)∂β+γV (0) γ,δ∈I2(λ1) γ+δ=α+β (γ + δ)! γ! δ! γV (0)∂j∂ δV (0) .(7.33) Thus, combining (7.32) and (7.33), the discussion above leads to b,1(x ℓ (s0)) =e 2λ1s0e−2λ1s M1(k, ℓ)e2λ1s + o(e2λ1s) =M1(k, ℓ)e2λ1s0 .(7.34) In particular, (7.19) becomes, in that case, yℓ(t, xℓℓℓ) =S−k + S ℓ +M1(k, ℓ)e 2λ1s(xℓℓℓ)te−2λ1t +O(e−2λ1t).(7.35) As above, the O(e−2λ1t) is expandible uniformly with respect to the variable xℓℓℓ when xℓℓℓ varies in a compact set avoiding 0. 7.3. Integration with respect to time. Now we perform the integration with respect to time t in (7.20). We follow the ideas of [18, Section 5] and [5, Section 6]. Since yℓ(t, xℓℓℓ) is expandible (see (7.16)), and since Φ is C outside of xℓℓℓ = 0, the symbol f ℓ(t, xℓℓℓ, h) is expandible. We compute only the contribution of the principal symbol (with respect to h) of f ℓ, since the other terms can be treated the same way, and the remainder term will give a contribution O(h∞) to the scattering amplitude. In other word, we compute (7.36) Asing0 = c(E)h−(n+1)/2√ (2πh)(n−1)/2h eiΦ(y ℓ(t,xℓℓℓ))/hf ℓ0(t, xℓℓℓ) dt dxℓℓℓ +O(h∞). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 49 First, we assume that we are in the case (a) of the assumption (A7). In that case, Ψ is given by (7.27). For xℓℓℓ fixed in a compact set outside from 0, we set yℓ(t, xℓℓℓ) − (S−k + S =− 2µk eµks(xℓℓℓ)e−µkt +R(t, xℓℓℓ),(7.37) and we perform the change of variable t→ τ in (7.36), and we assume for a moment that (7.38) Here R(t, xℓℓℓ) = Õ(e−µk+1t) is expandible. As in [18, Section 5] and [5, Section 6], we get e−t ∼ − 2µk (z−k ) (z+ℓ ) eµks(xℓℓℓ) )−1/µkτ1/µk τ bµj/µkbj(− ln τ, xℓℓℓ) (7.39) t ∼− 1 ln τ + − 2µk (z−k ) (z+ℓ ) eµks(xℓℓℓ) τ bµj/µkbj(− ln τ, xℓℓℓ)(7.40) τ bµj/µkbj(− ln τ, xℓℓℓ),(7.41) where the bj ’s change from line to line. These expansions are valid in the following sense: Definition 7.1. Let f(τ, y) be defined on ]0, ε[×U where U ⊂ Rm. We say that f = Ô(g(τ)) (resp. f = ô(g(τ))), where g(τ) is a non-negative function defined in ]0, ε[ if and only if for all α ∈ N and β ∈ Nm, (7.42) (τ∂τ ) α∂βy f(τ, y) = O(g(τ)), (resp. o(g(τ))) for all (τ, y) ∈]0, ε[×U . Thus, an expression like f ∼ j=1 τ bµj/µkfj(− ln τ, xℓℓℓ), where fj(− ln τ, xℓℓℓ) is a polynomial with respect to ln τ , like in (7.39)–(7.41), means that, for all J ∈ N, (7.43) f(τ, x)− τ bµj/µkfj(− ln τ, xℓℓℓ) = Ô(τ bµJ/µk). We shall say that such symbols f are called expandible near 0. Since f ℓ0(t, xℓℓℓ, h) is expandible (see Definition 6.1) with respect to t, this symbol is also expandible near 0 with respect to τ in the previous sense. In particular, we get (7.44) f̃ ℓ0(τ, xℓℓℓ) = −f ℓ0(t, xℓℓℓ)τ τ (Σ(E)+cµj)/µk f̃ ℓ0,j(− ln τ, xℓℓℓ), where the f̃ ℓ0,j’s are polynomials with respect to ln τ . The principal symbol f̃ 0,0 is independent on ln τ and we have (7.45) f̃ ℓ0,0(xℓℓℓ) = − 2µk eµks(xℓℓℓ) )−Σ(E)/µkf ℓ0,0(xℓℓℓ). In that case, (7.36) becomes (7.46) Asing0 = c(E)h−1/2 (2π)1−n/2 ∫∫ +∞ eiτ/hf̃ ℓ0(τ, xℓℓℓ) dxℓℓℓ +O(h∞). 50 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Note that f̃ ℓ0(τ, xℓℓℓ) has in fact a compact support with respect to τ . Now, using Lemma D.1, we can perform the integration with respect to t of each term in the right hand side of (7.44), modulo a term O(h∞) (see (D.3)–(D.4) in Lemma D.1). Then, we get (7.47) Asing0 = c(E)h−1/2 (2π)1−n/2 f̂j(ln h)h (Σ(E)+bµj )/µk , where f̂j(lnh) is a polynomial in respect to lnh. f̂0 does not depend on h and we have (7.48) f̂0 = Γ(Σ(E)/µk)(−i)−Σ(E)/µk f̃ ℓ0,0(xℓℓℓ) dxℓℓℓ. To finish the proof, it remains to perform the integration with respect to xℓℓℓ in (7.48). From (7.22) and (7.45), it becomes f̂0 =Γ(Σ(E)/µk) (z−k ) (z+ℓ ) eµks(xℓℓℓ) )−Σ(E)/µk × α0,0(yℓ xℓℓℓ) ã+,0 yℓ(xℓℓℓ) π/2 e i sgnΦ′′ |Hℓxℓℓℓ (yℓ(xℓℓℓ))π/4 ∣∣ detΦ′′|Hℓxℓℓℓ yℓ(xℓℓℓ) )∣∣1/2 dxℓℓℓ.(7.49) Now we make the change of variable xℓℓℓ 7→ s given by yℓ(xℓℓℓ) = x+ℓ (s) (then s(xℓℓℓ) = s). In particular, (7.50) dxℓℓℓ = ∂s(x ℓ,ℓℓℓ(s))ds = λℓℓℓ|g ℓℓℓ (z ℓ )|e λℓℓℓs(1 + o(1))ds, as s→ −∞. In this setting, we get (7.51) α0,0(x (s)) = α0,0(0)(1 + o(1)), as s→ −∞, where α0,0(0) is given in (6.8). We also have, from (7.9) and (7.13), (7.52) ã+,0(x ℓ (s)) = −i∂s χ̃+(γ ℓ (s)) (2E0) 1/4(D+ℓ ) −1/2eisz. Then, putting (7.23), (7.24), (7.50), (7.51) and (7.52) in (7.49), we obtain f̂0 =Γ(Σ(E)/µk) (z−k ) (z+ℓ ) 〉)−Σ(E)/µkα0,0(0)∂s χ̃+(γ ℓ (s)) i(n−1)π/4 j 6=ℓℓℓ λℓℓℓ|g+ℓℓℓ (z )|(2E0)1/4(D+ℓ ) −1/2eisze−Σ(E)seλℓℓℓs(1 + o(1)) ds i(n+1)π/4 j 6=ℓℓℓ )−1/2 λℓℓℓ|g+ℓℓℓ (z )|Γ(Σ(E)/µk) 〉)−Σ(E)/µk × e−iν π/2α0,0(0)(2E0) 1/4(D+ℓ ) χ̃+(γ ℓ (s)) (1 + o(1)) ds.(7.53) Here the o(1) does not depend on χ̃+. Now, we choose a family of cut-off functions (χ̃ +)j∈N such that the support of ∂t goes to −∞ as j → +∞. We also assume that SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 51 ℓ (t)) is non negative (see Figure 1). Then f̂0 =− ei(n+1)π/4 j 6=ℓℓℓ )−1/2 λℓℓℓΓ(Σ(E)/µk)e π/2eiπ/4(2λ1) 3/2e−iν × |g−1 (z )| |g+ 〉)−Σ(E)/µk(7.54) (2E0) 1/2(D− )−1/2 × (1 + o(1)).(7.55) as j → +∞. Since f̂0 is also independent of χ̃+, we obtain Theorem 2.6 from (7.47) and (7.48), in the case (a) and under the assumption (7.38). When 〈g− (z−k )|g (z+ℓ )〉 > 0, we set τ as the opposite of the R.H.S. of (7.37), and we obtain the result along the same lines (see Remark D.2). Now we assume that we are in the case (b) of the assumption (A7). In that case, the phase function Ψ is given by (7.30). For xℓℓℓ fixed in a compact set outside from 0, we set, mimicking (7.37), yℓ(t, xℓℓℓ) − (S−k + S =M2(k, ℓ)e2λ1s(xℓℓℓ)t2e−2λ1t +R(t, xℓℓℓ)(7.56) where R(t, xℓℓℓ) = O(te−2λ1t) is expandible with respect to t. As above, we assume that M2(k, ℓ) is positive (the other case can be studied the same way). Following (7.39), we want to write s := e−t as a function of τ . Since t 7→ τ(t) is expandible with respect to t, we have (7.57) τ = M2(k, ℓ)e2λ1s(xℓℓℓ)(ln s)2s2λ1(1 + r(s, xℓℓℓ)), where r(s, xℓℓℓ) = ô(1). In particular, ∂sτ > 0 for s positive small enough and then, for ε > 0 small enough, s 7→ τ(s) is invertible for 0 < s < ε. We denote s(τ) the inverse of this function. We look for s(τ) of the form (7.58) s(τ) = (2λ1) M2(k, ℓ)e2λ1s(xℓℓℓ) )1/2λ1 u(τ, xℓℓℓ) (− ln τ)1/λ1 where u(τ, xℓℓℓ) has to be determined. Using (7.57), the equation on u is τ =M2(k, ℓ)e2λ1s(xℓℓℓ)(ln s)2s2λ1(1 + r(s, xℓℓℓ)) =τu2λ1 (2λ1) −2M2(k, ℓ)e2λ1s(xℓℓℓ) + 2λ1 − 2ln(− ln τ) 1 + r (2λ1) M2(k, ℓ)e2λ1s(xℓℓℓ) )1/2λ1 u (− ln τ)1/λ1 , xℓℓℓ =τF (τ, u, xℓℓℓ),(7.59) where F = u2λ1(1 + r̃(τ, u, xℓℓℓ)) and r̃ = ô(1) for u close to 1 (here (u, xℓℓℓ) are the variables y in Definition 7.1). In other word, to find u, we have to solve F (t, u, xℓℓℓ) = 1. First we remark that u 7→ F (τ, u, xℓℓℓ) is real-valued and continuous. Since, for δ > 0 and τ small enough, F (τ, 1 − δ, xℓℓℓ) < 1 < (τ, 1 + δ, xℓℓℓ), there exists u ∈ [1 − δ, 1 + δ] such that F (τ, 1 + δ, xℓℓℓ) = 1. Thank to the discussion before (7.58), the function s(τ) is of the form (7.58) with u(τ, xℓℓℓ) ∈ [1− δ, 1 + δ], for τ small enough. 52 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND For τ > 0, the function F is C∞ and, since r̃ = ô(1), we have (7.60) ∂u F (τ, u, xℓℓℓ)− 1 (u(τ, xℓℓℓ)) = 2λ1u 2λ1−1(1 + oτ (1)) > λ1, for τ small enough. The notation oτ (1) means a term which goes to 0 as τ goes to 0. Here we have used the fact that u(τ, xℓℓℓ) is close to 1. In particular, the implicit function theorem implies that u(τ, xℓℓℓ) is C We write u = 1 + v(τ, xℓℓℓ) and we known that v ∈ C∞ and v = oτ (1). Differentiating the equality (7.61) 1 = F (τ, u(τ, xℓℓℓ), xℓℓℓ) = u(τ, xℓℓℓ) )2λ1( 1 + r̃(τ, u(τ, xℓℓℓ), xℓℓℓ) one can show that v = ô(1). Thus we have e−t =s(τ) = (2λ1) M2(k, ℓ)e2λ1s(xℓℓℓ) )1/2λ1 1 + r̂(τ, xℓℓℓ) (− ln τ)1/λ1 ,(7.62) t =− ln τ (1 + r̂(τ, xℓℓℓ)),(7.63) + r̂(τ, xℓℓℓ),(7.64) where r̂(τ, xℓℓℓ) = ô(1) change from line to line. Since f ℓ0(t, xℓℓℓ, h) is expandible with respect to t, we get, from (7.62)–(7.64), (7.65) f̃ ℓ0(τ, xℓℓℓ) = −f ℓ0(t, xℓℓℓ)τ = τΣ(E)/2λ1(− ln τ)−Σ(E)/λ1 f̃ ℓ0,0(xℓℓℓ) + r̂(τ, xℓℓℓ) where r̂ = ô(1) and (7.66) f̃ ℓ0,0(xℓℓℓ) = (2λ1) Σ(E)/λ1−1 M2(k, ℓ)e2λ1s(xℓℓℓ) )−Σ(E)/2λ1 f ℓ0,0(xℓℓℓ). In that case, (7.36) becomes (7.67) Asing0 = c(E)h−1/2 (2π)1−n/2 ∫∫ +∞ eiτ/hf̃ ℓ0(τ, xℓℓℓ) dxℓℓℓ +O(h∞). Note that f̃ ℓ0(τ, xℓℓℓ) has in fact a compact support with respect to τ . Now, using Lemma D.1, we can perform the integration with respect to t in (7.67), modulo an error term given by (D.3)–(D.4) in Lemma D.1. Then, we get Asing0 = c(E)h−1/2 (2π)1−n/2 )/hΓ(Σ(E)/2λ1)(−i)−Σ(E)/2λ1 × hΣ(E)/2λ1(− lnh)−Σ(E)/λ1 f̃ ℓ0,0(xℓℓℓ) dxℓℓℓ + o(1) ,(7.68) as h goes to 0. The rest of the proof follows that of (7.55). At last, the proof of Theorem 2.6 in the case (c) can be obtained along the same lines, and we omit it. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 53 Appendix A. Proof of Proposition 2.5 We prove that Λ+θ ∩Λ+ 6= ∅. From the assumption (A2), the Lagrangian manifold Λ+ can be described, near (0, 0) ∈ T ∗(Rn), as (A.1) Λ+ = {(x, ξ); x = ∇ϕ̃+(ξ)}, for |ξ| < 2ε, with ε > 0 small enough. For η ∈ Sn−1, let (x(t, η), ξ(t, η)) be the bicharacteristic curve with initial condition (ϕ̃(εη), εη). We have (A.2) Λ+ = {(x(t, η), ξ(t, η)); t ∈ R, η ∈ Sn−1} ∪ {(0, 0)}. The function ξ(t, η) is continuous on R× Sn−1. From the classical scattering theory (see [13, Section 1.3]), we know that this function ξ(t, η) converges uniformly to (A.3) ξ(∞, η) := lim ξ(t, η), as t→ +∞ and ξ(∞, η) ∈ 2E Sn−1. Then, the function (A.4) F (t, η) = 1−t , η) |ξ( t 1−t , η)| is well defined for 0 ≤ t ≤ 1 with the convention F (1, η) = ξ(∞, η)/ 2E. Here we used that |ξ(t, η)| 6= 0 for each t ∈ [0,+∞], η ∈ Sn−1. The previous properties of ξ(t, η) imply the continuity of F (t, η) on [0, 1] × Sn−1. From (A.2), to prove that Λ+θ ∩ Λ+ 6= ∅ for all θ ∈ Sn−1, it is enough (equivalent) to show the surjectivity of η → F (1, η). But if η → F (1, η) is not onto, then ImF (1, ·) ⊂ Sn−1 \ {a point}. And since Sn−1 \ {a point} is a contractible space, F (1, ·) is homotopic to a constant (A.5) f : Sn−1 → Sn−1. On the other hand, F : [0, 1]× Sn−1 −→ Sn−1 gives a homotopy between F (0, ·) = IdSn−1 and F (1, ·). In particular, we have (A.6) 1 = deg(F (0, ·)) = deg(F (1, ·)) = deg(f(·)) = 0, which is impossible (see [16, Section 23] for more details). Appendix B. A lower bound for the resolvent Let χ ∈ C∞(]0,+∞[) be a non-decreasing function such that (B.1) χ(x) = x for 0 < x < 1 2 for 2 < x, Let also ϕ ∈ C∞0 (R) an even function such that 0 ≤ ϕ ≤ 1, 1[−1,1] ≺ ϕ, and suppϕ ⊂ [−2, 2]. We set (B.2) u(x) = j/2hϕ |xj |1/2 uj(x), 54 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND where 0 < α < 2β will be fixed later on. The uj ’s are of course C ∞ functions, and we have (B.3) (P − E0)u = − ∆u(x)− x2ju(x) +O(x3u(x)). Lemma B.1. For any h small enough, we have hβn| lnh|n/2 . ‖u‖L2(Rn) . hβn| lnh|n/2,(B.4) ∥∥|x|3u(x) L2(Rn) . h3αhβn| lnh|n/2.(B.5) Proof. First of all, the second estimate follow easily from the first one: we have ∥∥|x|3u(x) ∥∥2 = |x|6|u(x)|2dx . h6α‖u‖2, since u vanishes if |x| > 2hα. Thanks to the fact that u is a product of n functions of one variable, it is enough to estimate dt = 2 ∫ 2hα We have dt ≤ I ≤ 2 ∫ 2hα dt+ 2 so that dt ≤ I ≤ 2 ∫ 2hα dt+ 2 4 dt. The first estimate follows from the fact that 2β − α > 0, once we have noticed that ∫ Ahα dt = h2β (2β − α)| ln h|+ α lnA On the other hand, we have ∆u(x)− x2ju(x) = j 6=k uj(xj) u′′k(xk)− x2kuk(xk) From Lemma B.1, we get ∥∥(P − E0)u ∥∥ .hβ(n−1)| lnh|(n−1)/2 sup 1≤k≤n ∥∥h2u′′k(t) + λ 2uk(t) ∥∥+ h3αhβn| lnh|n/2 h−β | lnh|−1/2 sup 1≤k≤n ∥∥h2u′′k(t) + λ 2uk(t) ∥∥+ h3α ‖u‖.(B.6) We also have (B.7) h2u′′k(t) + λkt 2uk(t) = e h2v′′h(t) + ihλk(2t∂t + 1)vh(t) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 55 where we have set vh(t) = ϕ |t|1/2 . Notice that the right hand side of (B.7) is an even function, so that we only have to consider t > 0. The point here, is that we have, for t > 0, (B.8) (2t∂t + 1) = − h   2 if 0 < t < O(1) if h < t < h2β , 0 if h2β < t. Therefore, we obtain ∥∥(2t∂t + 1)vh ∥∥2 =2 ∫ 2hα (2t∂t + 1) |t|1/2 ∫ 2hα |t|1/2 ∫ h2β ∫ 2hα |t|1/2 dt . h2β .(B.9) On the other hand, an easy computation gives, still for t > 0, v′′h(t) =h −2αϕ′′ 4t5/2 .(B.10) Computing the L2–norm of each of these terms as in Lemma B.1 and (B.9), we obtain (B.11) ‖h2v′′h‖ . h2+β−2α + h2+β−2α + h2−3β + h2−3β , and, eventually, from (B.6), (B.7), (B.9) and (B.11), ∥∥(P − E0)u h−β | lnh|−1/2 h1+β + h2+β−2α + h2−3β + h3α Therefore we obtain Proposition 2.2 if we can find α > 0 and β > 0 such that 2− 2α > 1, 2− 4β > 1, 3α > 1 and 2β > α, and one can check that α = 5/12 and β = 11/48 satisfies these four inequalities. Appendix C. Lagrangian manifolds which are transverse to Λ± Let Λ ⊂ p−1(E0) be a Lagrangian manifold such that Λ ∩ Λ− is transverse along a Hamil- tonian curve γ(t) = (x(t), ξ(t)). Then, where exists a 6= 0 and ν ∈ {1, . . . , n} such that (C.1) γ(t) = (a+O(e−εt))e−λν t, as t→ +∞. The vector a is an eigenvector of (C.2) V ′′(0) 0 for the eigenvalue λν . Thus, up to a linear change of variable in R n, we can always assume that Πxa is collinear to the xν–direction. The goal of this section is to prove the following geometric result. 56 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Proposition C.1. For t large enough, Λ projects nicely on Rnx near γ(t). In particular, there exists ψ ∈ C∞(Rn) defined near Πxγ, unique up to a constant, such that Λ = Λψ := {(x,∇ψ(x)); x ∈ Rn}. Moreover, we have (C.3) ψ′′(x(t)) =  . . . . . .  +O(e−εt), as t→ +∞. Remark C.2. The same result hold in the outgoing region: If γ = Λ ∩ Λ+ is transverse, Λ projects nicely on Rnx near γ(t), t → −∞. Then Λ = Λψ for some function ψ satisfying ψ′′(x(t)) = diag(−λ1, . . . ,−λν−1, λν ,−λν+1, . . . ,−λn) +O(eεt). Proof. We follow the proof of [18, Lemma 2.1]. There exist symplectic local coordinates (y, η) centered at (0, 0) such that Λ− (resp. Λ+) is given by y = 0 (resp. η = 0) and (ξj + λjxj) +O((x, ξ)2),(C.4) (ξj − λjxj) +O((x, ξ)2).(C.5) Then, p(x, ξ) = A(y, η)y · η with A0 := A(0, 0) = diag(λ1, . . . , λn). (C.6) A0 +O(e−λ1t) 0 O(e−λ1t) A0 +O(e−λ1t) We denote by U(t, s) the linear operator such that U(t, s)δ solves (C.6) with U(s, s) = Id. Since Λ∩Λ− = γ is transverse, there exists En−1(t0) ⊂ Tγ(t0)Λ, a vector space of dimension n − 1 disjoint from Tγ(t0)Λ−. For convenience, we set En(t0) = En−1(t0) ⊕ Rv for some v /∈ Tγ(t0)Λ + Tγ(t0)Λ−. Let E•(t) = U(t, t0)E•(t0). From [18, Lemma 2.1], there exists Bt = O(e−λ1t) such that En(t) is given by δη = Btδx. Now, if δ ∈ En−1(t), we have σ(Hp, δ) = 0 since En−1(t)⊕ RHp = Tγ(t)Λ and Λ is a Lagrangian manifold. From (C.1), we have (C.7) Hp(γ(t)) = γ̇(t) = −λν(ãeην +O(e−εt))e−λν t, where eην is the basis vector corresponding to ην and then (C.8) 0 = σ(eλν tHp, δ) = λν ãδyν +O(e−εt)|δ|. It follows that δ ∈ En−1(t) if and only if (δyν , δη) = B̃tδy′ with B̃t = O(e−εt). Using Tγ(t)Λ = En−1(t)⊕ RHp, we obtain that Tγ(t)Λ has a basis formed of vector fj(t) such that fj =eyj +O(e−εt) for j 6= ν(C.9) fν =eην +O(e−εt).(C.10) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 57 In the (x, ξ)-coordinates, Tγ(t)Λ has a basis formed of vector f̃j(t) of the form f̃j =eξj + λjexj +O(e−εt) for j 6= ν(C.11) f̃ν =eξν − λjexν +O(e−εt),(C.12) and the lemma follows. � Appendix D. Asymptotic behaviour of certain integrals Lemma D.1. Let α ∈ C, Reα > 0, β ∈ R and χ ∈ C∞0 (]−∞, 1/2[) be such that χ = 1 near 0. As λ goes to +∞, we have (D.1) eiλttα(− ln t)βχ(t) dt = Γ(α)(ln λ)β(−iλ)−α(1 + o(1)). Moreover, if β ∈ N, we get (D.2) eiλttα(− ln t)βχ(t) dt = (−iλ)−α (j)(α)(−1)j ln(−iλ) +O(λ−∞). Finally, if s(t) ∈ C∞(]0,+∞[) satisfies (D.3) |∂jt s(t)| = o tα−j(− ln t)β for all j ∈ N and t→ 0, then (D.4) eiλts(t)χ(t) (ln λ)βλ−α Here (−iλ)−α = eiαπ/2λ−α and ln(−iλ) = lnλ− iπ/2. Remark D.2. Notice that one obtains the behaviour of these quantities as λ → −∞ by taking the complex conjugate in these expressions. Proof. We begin with (D.2) and assume first that β = 0. Then, we can write eiλttαχ(t) = lim ei(λ+iε)ttαχ(t) = lim I1(α, ε) + I2(α, ε) ,(D.5) where I1(α, ε) = e−(ε−iλ)ttα ,(D.6) I2(α, ε) = ei(λ+iε)ttα(1− χ(t)) dt ·(D.7) It is clear that (D.8) I1(α, ε) = (ε− iλ)−αΓ(α), where z−α is well defined on C\]−∞, 0] and real positive on ]0,+∞[. In particular (D.9) lim I1(α, ε) = (−iλ)−αΓ(α). 58 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Concerning, I2(α, ε), we remark that r(t, α) = t α−1(1− χ(t)) is a symbol which satisfies (D.10) |∂jt ∂kαr(t, α)| . 〈t〉Reα−1−j〈ln t〉k, for all j, k ∈ N uniformly for t ∈ [0,+∞[ and α in a compact set of {Re z > 0}. Then, making integration by parts in (D.7), we obtain (D.11) I2(α, ε) = (ε− iλ)j e(iλ−ε)t∂jt r(t, α) dt, for all j ∈ N. Now, if j is large enough (j > Reα), ∂jt r(t, α) is integrable in time uniformly with respect to ε. In particular, for such j, (D.12) lim I2(α, ε) = e ijπ/2λ−j eiλt∂ t r(t, α) dt, and then (see (D.10) or Cauchy’s formula) ∂kα lim I2(α, ε) =e ijπ/2λ−j eiλt∂ αr(t, α) dt =O(λ−∞),(D.13) for all k ∈ N. Then we obtain (D.2) for β = 0. To obtain the result for β ∈ N, it is enough to see that eiλttα(ln t)βχ(t) eiλttαχ(t) (−iλ)−αΓ(α) + ∂βα lim I2(α, ε) =(−iλ)−α (j)(α) − ln(−iλ) +O(λ−∞),(D.14) from (D.13). Thus, (D.2) is proved. Let u be a function C∞(]0,+∞[) be such that (D.15) |∂jt u(t)| . tReα−j(− ln t)β, near 0. Let ϕ ∈ C∞(R) such that ϕ = 1 for t < 1 and ϕ = 0 for t > 2. For δ > 0, we have (D.16) eiλtu(t)χ(t) 1−ϕ(t/δ) = (−iλ)−N eiλt∂Nt u(t)χ(t) 1−ϕ(t/δ) for all N . If one of the derivatives falls on 1−ϕ(t/δ), the support of this contribution is inside [δ, 2δ]. Therefore, the corresponding term will be bounded by δReα−N−1(ln δ)β and will contribute like δReα−N (− ln δ)β to the integral. If ones of the derivatives falls on χ(t), the support of the integrand will be a compact set outside of 0 and then this function will be O(1). The contribution to the integral of such term will be like 1. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 59 If all the derivatives fall on u(t)t−1, we corresponding term will satisfies eiλt∂Nt u(t)t−1 1− ϕ(t/δ) dt =O(1) tReα−1−N (− ln t)β(1− χ(t))dt .(− ln δ)βδReα−N ,(D.17) for N large enough (N > Reα). From this 3 cases, we deduce (D.18) eiλtu(t)χ(t) 1− ϕ(t/δ) (− ln δ)βδα−Nλ−N Taking δ = (ελ)−1, we get (D.19) eiλtu(t)χ(t) 1− ϕ(t/δ) ε(lnλ)βλ−α as λ→ +∞. We now assume (D.3), and we want to prove (D.4). Since, for t small enough (D.20) tReα−1(− ln t)β . tReα(− ln t)β we get eiλts(t)χ(t)ϕ(t/δ) ∣∣∣ =oδ→0(1) tReα−1(− ln t)βdt =oδ→0(1)δ Re α(− ln δ)β .(D.21) Here oδ→0(1) stands for a term which goes to 0 as δ goes to 0. If δ = (ελ) −1, we get (D.22) eiλts(t)χ(t)ϕ(t/δ) ∣∣∣ = oλ→+∞(1)λ−α(ln λ)β, when λ → +∞ and ε fixed. Taking ε small enough in (D.19), and then λ large enough in (D.22), we get (D.4). We are left with (D.1). We need to compute (D.23) I = eiλttα(− ln t)βϕ(t/δ) dt Performing the change of variable s = λt, we get I =λ−α ∫ 2/ε eissα(ln λ− ln s)βϕ(εs) ds =(lnλ)βλ−α ∫ 2/ε eissα(1− ln s/ lnλ)βϕ(εs) ds .(D.24) 60 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND We remark that, in the previous equation, − ln s/ lnλ > − ln(2/ε)/ ln λ > −1/2 for λ large enough. Using (1 + u)β = 1 +O(|u|+ |u|max(1,β)) for u > −1/2, we get I =(lnλ)βλ−α ∫ 2/ε eissαϕ(εs) + (lnλ)βλ−α ∫ 2/ε sReαO ( | ln s| ( | ln s| )max(1,β)) ϕ(εs) =(lnλ)β eiλttα(− ln t)βϕ(t/δ) dt (lnλ)β−1λ−α .(D.25) Note that the Oε in (D.25) depends on ε. 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Anal. i Priložen. 11 (1977), no. 4, 6–18, 96. http://arxiv.org/abs/math/0602069 62 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Ivana Alexandrova, Department of Mathematics, East Carolina University, Greenville, NC 27858, USA E-mail address: alexandrovai@ecu.edu Jean-François Bony, Institut de Mathématiques de Bordeaux, (UMR CNRS 5251), Université de Bordeaux I, 33405 Talence, France E-mail address: bony@math.u-bordeaux1.fr Thierry Ramond, Mathématiques, Université Paris Sud, (UMR CNRS 8628), 91405 Orsay, France E-mail address: thierry.ramond@math.u-psud.fr 1. Introduction 2. Assumptions and main results 3. Proof of the main resolvent estimate 4. Representation of the Scattering Amplitude 5. Computations before the critical point 5.1. Computation of u- in the incoming region 5.2. Computation of u- along -k 6. Computation of u- at the critical point 6.1. Study of the transport equations for the phases 6.2. Taylor expansions of + and k1 6.3. Asymptotics near the critical point for the trajectories 6.4. Computation of the jk's 7. Computations after the critical point 7.1. Stationary phase expansion in the outgoing region 7.2. Behaviour of the phase function 7.3. Integration with respect to time Appendix A. Proof of Proposition 2.5 Appendix B. A lower bound for the resolvent Appendix C. Lagrangian manifolds which are transverse to Appendix D. Asymptotic behaviour of certain integrals References

We compute the scattering amplitude for Schr\"odinger operators at a critical energy level, corresponding to the maximum point of the potential. We follow the wrok of Robert and Tamura, '89, using Isozaki and Kitada's representation formula for the scattering amplitude, together with results from Bony, Fujiie, Ramond and Zerzeri '06 in order to analyze the contribution of trapped trajectories.

Introduction 2 2. Assumptions and main results 4 3. Proof of the main resolvent estimate 10 4. Representation of the Scattering Amplitude 16 5. Computations before the critical point 19 5.1. Computation of u− in the incoming region 19 5.2. Computation of u− along γ 6. Computation of u− at the critical point 23 6.1. Study of the transport equations for the phases 25 6.2. Taylor expansions of ϕ+ and ϕ 6.3. Asymptotics near the critical point for the trajectories 34 6.4. Computation of the ϕkj ’s 36 7. Computations after the critical point 43 7.1. Stationary phase expansion in the outgoing region 43 7.2. Behaviour of the phase function Φ 47 Date: April 12, 2007. 2000 Mathematics Subject Classification. 81U20,35P25,35B38,35C20. Key words and phrases. Scattering amplitude, critical energy, Schrödinger equation. Acknowledgments: We would like to thank Johannes Sjöstrand for helpful discussions during the prepa- ration of this paper. The first author also thanks Victor Ivrii for supporting visits to Université Paris Sud, Orsay, and the Department of Mathematics at Orsay for the extended hospitality. http://arxiv.org/abs/0704.1632v2 2 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND 7.3. Integration with respect to time 48 Appendix A. Proof of Proposition 2.5 53 Appendix B. A lower bound for the resolvent 53 Appendix C. Lagrangian manifolds which are transverse to Λ± 55 Appendix D. Asymptotic behaviour of certain integrals 57 References 60 1. Introduction We study the semiclassical behavior of scattering amplitude at energy E > 0 for Schrö- dinger operators (1.1) P (x, hD) = −h ∆ + V (x) where V is a real valued C∞ function on Rn, which vanishes at infinity. We shall suppose here that E is close to a critical energy level E0 for P , which corresponds to a non-degenerate global maximum of the potential. Here, we address the case where this maximum is unique. Let us recall that, if V (x) = O(〈x〉−ρ) for some ρ > (n + 1)/2, then for any ω 6= θ ∈ Sn−1 and E > 0, the problem P (x, hD)u = Eu, u(x, h) = ei 2Ex·ω/h +A(ω, θ,E, h)e 2E|x|/h |x|(n−1)/2 + o(|x|(1−n)/2) as x→ +∞, x|x| = θ, has a unique solution. The scattering amplitude at energy E for the incoming direction ω and the outgoing direction θ is the real number A(ω, θ,E, h). For potentials that are not decaying that fast at infinity, it is not that easy to write down a stationary formula for the scattering amplitude: If V (x) = O(〈x〉−ρ) for some ρ > 1, one can define the scattering matrix at energy E using wave operators (see Section 4 below). Then, writing (1.2) S(E, h) = Id− 2iπT (E, h), one can see that T (E, h) is a compact operator on L2(Sn−1), which kernel T (ω, θ,E, h) is smooth out of the diagonal in Sn−1×Sn−1. Then, the scattering amplitude is given for θ 6= ω, (1.3) A(ω, θ,E, h) = c(E))h(n−1)/2T (ω, θ,E, h), where (1.4) c(E) = −2π(2E)− 4 (2π) 2 e−i (n−3)π We proceed here as in [30], where D. Robert and H. Tamura have studied the semiclassical behavior of the scattering amplitude for short range potentials at a non-trapping energy E . SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 3 An energy E is said to be non-trapping when K(E), the trapped set K(E) at energy E, is empty. This trapped set is defined as (1.5) K(E) = (x, ξ) ∈ p−1(E), exp(tHp)(x, ξ) 6→ ∞ as t→ ±∞ where Hp is the Hamiltonian vector field associated to the principal symbol p(x, ξ) = V (x) of the operator P . Notice that the scattering amplitude has been first studied, in the semiclassical regime, by B. Vainberg [32] and Y. Protas [27] in the case of compactly supported potential, and for non-trapping energies, where they obtained the same type of result. Under the non-trapping assumption, and some other non-degeneracy condition (in fact our assumption (A4) below), D. Robert and H. Tamura have shown that the scattering amplitude has an asymptotic expansion with respect to h. The non-degeneracy assumption implies in particular that there is a finite number N∞ of classical trajectories for the Hamiltonian p, with asymptotic direction ω for t→ −∞ and asymptotic direction θ as t→ +∞. Robert and Tamura’s result is the following asymptotic expansion for the scattering amplitude: (1.6) A(ω, θ,E, h) = iS∞j /h aj,m(ω, θ,E)h m +O(h∞), h→ 0, where S∞j is the classical action along the corresponding trajectory. Also, they have computed the first term in this expansion, showing that it can be given in terms of quantities attached to the corresponding classical trajectory only. There are also some few works concerning the scattering amplitude when the non-trapping assumption is not fulfilled. In his paper [24], L. Michel has shown that, if there is no trapped trajectory with incoming direction ω and outgoing direction θ (see the discussion after (2.6) below), and if there is a complex neighborhood of E of size ∼ hN for some N ∈ N possibly large, which is free of resonances, then A(ω, θ,E, h) is still given by Robert and Tamura’s formula. The potential is also supposed to be analytic in a sector out of a compact set, and the assumption on the existence of a resonance free domain around E amounts to an estimate on boundary value of the meromorphic extension of the truncated resolvent of the for (1.7) ‖χ(P − (E ± i0))−1χ‖ = O(h−N ), χ ∈ C∞0 (Rn). Of course, these assumptions allow the existence of a non-empty trapped set. In [2] and [3], the first author has shown that at non-trapping energies or in L. Michel’s setting, the scattering amplitude is an h-Fourier Integral Operator associated to a natural scattering relation. These results imply that the scattering amplitude admits an asymptotic expansion even without the non-degeneracy assumption, and in the sense of oscillatory inte- grals. In particular, the expansion (1.6) is recovered under the non-degeneracy assumption and as an oscillatory integral. In [21], A. Lahmar-Benbernou and A. Martinez have computed the scattering amplitude at energy E ∼ E0, in the case where the trapped set K(E0) consists in one single point corresponding to a local minimum of the potential (a well in the island situation). In that case, the estimate (1.7) is not true, and their result is obtained through a construction of the resonant states. In the present work, we compute the scattering amplitude at energy E ∼ E0 in the case where the trapped set K(E0) corresponds to the unique global maximum of the potential. 4 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND The one-dimensional case has been studied in [28, 14, 15], with specific techniques, and we consider here the general n > 1 dimensional case. Notice that J. Sjöstrand in [31], and P. Briet, J.-M. Combes and P. Duclos in [7, 8] have described the resonances close to E0 in the case where V is analytic in a sector around R From their result, it follows that Michel’s assumption on the existence of a not too small resonance-free neighborhood of E0 is satisfied. However, we show below (see Proposition 2.5) that for any ω ∈ Sn−1, there is at least one half-trapped trajectory with incoming direction ω, so that L. Michel’s result never applies here. Here, we do not assume analyticity for V . We compute the contributions to the scattering amplitude arising from the classical trajectories reaching the unstable equilibrium point, which corresponds to the top of the potential barrier. At the quantum level, tunnel effect occurs, which permits the particle to pass through this point. Our computation here relies heavily on [5], where a precise description of this phenomena has been obtained. In a forthcoming paper, we shall show that in this case also, the scattering amplitude is an h-Fourier Integral Operator. This paper is organized in the following way. In Section 2, we describe our assumptions, and state our main results: a resolvent estimate, and the asymptotic expansion of the scattering amplitude in the semiclassical regime. Section 3 is devoted to the proof of the resolvent estimate, from which we deduce in Section 4 estimates similar to those in [30]. In that section, we also recall briefly the representation formula for the scattering amplitude proved by Isozaki and Kitada, and introduce notations from [30]. The computation of the asymptotic expansion of the scattering amplitude is conducted in sections 5, 6 and 7, following the classical trajectories. Eventually, we have put in four appendices the proofs of some side results or technicalities. 2. Assumptions and main results We suppose that the potential V satisfies the following assumptions (A1) V is a C∞ function on Rn, and, for some ρ > 1, ∂αV (x) = O(〈x〉−ρ−|α|). (A2) V has a non-degenerate maximum point at x = 0, with E0 = V (0) > 0 and ∇2V (0) = . . .  , 0 < λ1 ≤ λ2 ≤ . . . ≤ λn. (A3) The trapped set at energy E0 is K(E0) = {(0, 0)}. Notice that the assumptions (A1)–(A3) imply that V has an absolute global maximum at x = 0. Indeed, if L = {x 6= 0; V (x) ≥ E0} was non empty, the geodesic, for the Agmon distance (E0 − V (x))1/2+ dx, between 0 and L would be the projection of a trapped bicharacteristic (see [1, Theorem 3.7.7]). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 5 As in D. Robert and H. Tamura in [30], one of the key ingredient for the study of the scattering amplitude is a suitable estimate for the resolvent. Using the ideas in [5, Section 4], we have obtained the following result, that we think to be of independent interest. Theorem 2.1. Suppose assumptions (A1), (A2) and (A3) hold, and let α > 1 be a fixed real number. We have (2.1) ‖P − (E ± i0))−1‖α,−α . h−1| lnh|, uniformly for |E − E0| ≤ δ, with δ > 0 small enough. Here ‖Q‖α,β denotes the norm of the bounded operator Q from L2(〈x〉α dx) to L2(〈x〉β dx). Moreover, we prove in the Appendix B that our estimate is not far from optimal. Indeed, we have the Proposition 2.2. Under the assumptions (A1) and (A2), we have (2.2) ‖(P − E0 ± i0)−1‖α,−α & h−1 | lnh|. We would like to mention that in the case of a closed hyperbolic orbit, the same upper bound has been obtained by N. Burq [9] in the analytic category, and in a recent paper [11] by H. Christianson in the C∞ setting. As a matter of fact, in the present setting, S. Nakamura has proved in [26] an O(h−2) bound for the resolvent. Nakamura’s estimate would be sufficient for our proof of Theorem 2.6, but it is not sharp enough for the computation of the total scattering cross section along the lines of D. Robert and H. Tamura in [29]. In that paper, the proof relies on a bound O(h−1) for the resolvent, but it is easy to see that an estimate like O(h−1−ε) for any small enough ε > 0 is sufficient. If we denote (2.3) σ(ω,E0, h) = |A(ω, θ,E, h)|2dθ, the total scattering cross-section, and following D. Robert and H. Tamura’s work, our resolvent estimates gives the Theorem 2.3. Suppose assumptions (A1), (A2) and (A3) hold, and that ρ > n+1 , n ≥ 2. If |E − E0| < δ for some δ > 0 small enough, then (2.4) σ(ω,E, h) = 4 2−1(2E)−1/2h−1 V (y + sω)ds dy +O(h−(n−1)/(ρ−1)). Now we state our assumptions concerning the classical trajectories associated with the Hamiltonian p, that is curves t 7→ γ(t, x, ξ) = exp(tHp)(x, ξ) for some initial data (x, ξ) ∈ T ∗Rn. Let us recall that, thanks to the decay of V at infinity, for given α ∈ Sn−1 and z ∈ α⊥ ∼ Rn−1 (the impact plane), there is a unique bicharacteristic curve (2.5) γ±(t, z, α,E) = (x±(t, z, α,E), ξ±(t, z, α,E)) such that (2.6) |x±(t, z, α,E) − 2Eαt− z| = 0, |ξ±(t, z, α,E) − 2Eα| = 0. 6 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND We shall denote by Λ−ω the set of points in T n lying on trajectories going to infinity with direction ω as t → −∞, and Λ+θ the set of those which lie on trajectories going to infinity with direction θ as t→ +∞: (2.7) Λ−ω = γ−(t, z, ω,E) ∈ T ∗Rn, z ∈ ω⊥, t ∈ R Λ+θ = γ+(t, z, θ, E) ∈ T ∗Rn, z ∈ θ⊥, t ∈ R We shall see that Λ−ω and Λ θ are in fact Lagrangian submanifolds of T Under the assumptions (A1), (A2) and (A3) there are only two possible behaviors for x±(t, z, α,E) as t→ ∓∞: either it escapes to ∞, or it goes to 0. First we state our assumptions for the first kind of trajectories. For these, we also have, for some ξ∞(z, ω,E), ξ−(t, z, ω) = ξ∞(z, ω,E), and we shall say that the trajectory γ−(t, z, ω,E) has initial direction ω and final direction θ = ξ∞(z, ω,E)/2 E. As in [30] we shall suppose that there is only a finite number of trajectories with initial direction ω and final direction θ. This assumption can be given in terms of the angular density (2.8) σ̂(z) = |det(ξ∞(z, ω,E), ∂z1ξ∞(z, ω,E), . . . , ∂zn−1ξ∞(z, ω,E))|. Definition 2.4. The outgoing direction θ ∈ Sn−1 is called regular for the incoming direction ω ∈ Sn−1, or ω-regular, if θ 6= ω and, for all z′ ∈ ω⊥ with ξ∞(z′, ω,E) = 2 Eθ, the map ω⊥ ∋ z 7→ ξ∞(z, ω,E) ∈ Sn−1 is non-degenerate at z′, i.e. σ̂(z′) 6= 0. We fix the incoming direction ω ∈ Sn−1, and we assume that (A4) the direction θ ∈ Sn−1 is ω-regular. Then, one can show that Λ−ω ∩ Λ+θ is a finite set of Hamiltonian trajectories (γ∞j )1≤j≤N∞ , γ∞j (t) = γ ∞(t, z∞j ) = (x j (t), ξ j (t)), with transverse intersection along each of these curves. We turn to trapped trajectories. Let us notice that the linearization Fp at (0, 0) of the Hamilton vector field Hp has eigenvalues −λn, . . . ,−λ1, λ1, . . . , λn. Thus (0, 0) is a hyper- bolic fixed point for Hp, and the stable/unstable manifold Theorem gives the existence of a stable incoming Lagrangian manifold Λ− and a stable outgoing Lagrangian manifold Λ+ characterized by (2.9) Λ± = {(x, ξ) ∈ T ∗Rn, exp(tHp)(x, ξ) → 0 as t→ ∓∞} . In this paper, we shall describe the contribution to the scattering amplitude of the trapped trajectories, that is those going from infinity to the fixed point (0, 0). We have proved in Appendix A the following result, which shows that there are always such trajectories. Proposition 2.5. For every ω, θ ∈ Sn−1, we have (2.10) Λ−ω ∩ Λ− 6= ∅ and Λ+θ ∩ Λ+ 6= ∅. We suppose that (A5) Λ−ω and Λ− (resp. Λ and Λ+) intersect transversally. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 7 Under this assumption, Λ−ω ∩ Λ− and Λ+θ ∩ Λ+ are finite sets of bicharacteristic curves. We denote them, respectively, (2.11) γ−k : t 7→ γ −(t, z−k ) = (x k (t), ξ −(t)), 1 ≤ k ≤ N−, (2.12) γ+ℓ : t 7→ γ +(t, z+ℓ ) = (x +(t), ξ+(t)), 1 ≤ ℓ ≤ N+. Here, the z− (resp. the z+ ) belong to ω⊥ (resp. θ⊥) and determine the corresponding curve by (2.6). We recall from [18, Section 3] (see also [5, Section 5]), that each integral curve γ±(t) = (x±(t), ξ±(t)) ∈ Λ± satisfies, in the sense of expandible functions (see Definition 6.1 below), (2.13) γ±(t) ∼ γ±j (t)e ±µj t, as t→ ∓∞, where µ1 = λ1 < µ2 < . . . is the strictly increasing sequence of linear combinations over N of the λj’s. Here, the functions γ j : R → R2n are polynomials, that we write (2.14) γ±j (t) = M ′j∑ γ±j,mt Considering the base space projection of these trajectories, we denote (2.15) x±(t) ∼ g±j (t)e ±µj t, as t→ ∓∞, g±j (t) = M ′j∑ g±j,mt Let us denote ̂ the (only) integer such that µb = 2λ1. We prove in Proposition 6.11 below that if j < ̂, then M ′j = 0, or more precisely, that g j (t) = g j is a constant vector in Ker(Fp∓λj). We also have M ′ ≤ 1, and g− can be computed in terms of g−1 . In this paper, concerning the incoming trajectories, we shall assume that, (A6) For each k ∈ {1, . . . , N−}, g−1 (z ) 6= 0. Finally, we state our assumptions for the outgoing trajectories γ+ℓ ⊂ Λ+ ∩Λθ+. First of all, it is easy to see, using Hartman’s linearization theorem, that there exists always a m ∈ N such that g+m(z ℓ ) 6= 0. We denote (2.16) ℓℓℓ = ℓℓℓ(ℓ) = min{m, g+m(z+ℓ ) 6= 0} the smallest of these m’s. We know that µℓℓℓ is one of the λj’s, and that M ℓℓℓ = 0. In [5], we have been able to describe the branching process between an incoming curve γ− ⊂ Λ− and an outgoing curve γ+ ⊂ Λ+ provided 〈g−1 |g 1 〉 6= 0 (see the definition for Λ̃+(ρ−) before [5, Theorem 2.6]). Here, for the computation of the scattering amplitude, we can relax a lot this assumption, and analyze the branching in other cases that we describe now. Let us denote, for a given pair of paths (γ−(z−k ), γ +(z+ℓ )) in (Λ ω ∩ Λ−)× (Λ+θ ∩ Λ+), (2.17) M2(k, ℓ) = − j∈I1(2λ1) α,β∈I2(λ1) βV (0) (g−1 (z αV (0) (g+1 (z 8 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND M1(k, ℓ) =− α∈I2(λ1) αV (0) (g−1 (z −))α(g+ (z+))j + (g (z−))j(g α,β∈I2(λ1) (g−1 (z (g+1 (z Cα,β,(2.18) where Cα,β =− ∂α+βV (0) + j∈I1\I1(2λ1) λ2j(4λ 1 − λ2j) ∂α+γV (0)∂β+γV (0) γ,δ∈I2(λ1) γ+δ=α+β (γ + δ)! γ! δ! γV (0)∂j∂ δV (0).(2.19) Here, we have set I1 = {1, . . . , n}, 1j = (δij)i=1,...,n ∈ Nn and (2.20) Im(µ) = {β ∈ Nn, β = 1k1 + · · ·+ 1km with λk1 = · · · = λkm = µ}, the set of multi-indices β of length |β| = m with each index of its non-vanishing components in the set {j ∈ N, λj = µ}. We also denote Im ⊂ Nn the set of multi-indices of length m. We will suppose that (A7) For each pair of paths (γ−(z−k ), γ +(z+ℓ )), k ∈ {1, . . . , N−}, ℓ ∈ {1, . . . , N+}, one of the three following cases occurs: (a) The set m < ̂, 〈g−m(z−k )|g+m(z ℓ )〉 6= 0 is not empty. Then we denote k = min m < ̂, 〈g−m(z−k )|g )〉 6= 0 (b) For all m < ̂, we have 〈g−m(z−k )|g+m(z ℓ )〉 = 0, and M2(k, ℓ) 6= 0. (c) For all m < ̂, we have 〈g−m(z−k )|g+m(z ℓ )〉 = 0, M2(k, ℓ) = 0 and M1(k, ℓ) 6= 0. As one could expect (see [30], [28] or [15]), action integrals appear in our formula for the scattering amplitude. We shall denote S∞j = (|ξ∞j (t)|2 − 2E0)dt, j ∈ {1, . . . , N∞},(2.21) S−k = |ξ−k (t)| 2 − 2E01t<0 dt, k ∈ {1, . . . , N−},(2.22) (t)|2 − 2E01t>0 dt, ℓ ∈ {1, . . . , N+},(2.23) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 9 and ν∞j , ν ℓ , ν k the Maslov indexes of the curves γ j , γ ℓ , γ k respectively. Let also D−k = limt→+∞ ∣∣∣ det ∂x−(t, z, ω,E0) ∂(t, z) ∣∣∣ e−(Σλj−2λ1)t,(2.24) D+ℓ = limt→−∞ ∣∣∣ det ∂x+(t, z, ω,E0) ∂(t, z) ∣∣∣ e(Σλj−2λℓℓℓ)t,(2.25) be the Maslov determinants for γ− , and γ+ respectively. We show below that 0 < D− +∞. Eventually we set (2.26) Σ(E, h) = − iE − E0 Then, the main result of this paper is the Theorem 2.6. Suppose assumptions (A1) to (A7) hold, and that E ∈ R is such that E −E0 = O(h). Then A(ω, θ,E, h) = Aregj (ω, θ,E, h) + Asingk,ℓ (ω, θ,E, h) +O(h ∞),(2.27) where (2.28) Aregj (ω, θ,E, h) = e iS∞j /h j,m(ω, θ,E)h j,0 (ω, θ,E) = −iν∞j π/2 σ̂(zj)1/2 Moreover we have • In case (a) Asingk,ℓ (ω, θ,E, h) = e k,ℓ,m(ω, θ,E, lnh)h (Σ(E)+bµm)/µk−1/2,(2.29) where the a k,ℓ,m (ω, θ,E, ln h) are polynomials with respect to lnh, and k,ℓ,0(ω, θ,E, ln h) = π1−n/2 ei(nπ/4−π/2) )−1/2 (Σ(E) (2λ1λℓℓℓ) × e−iν π/2e−iν π/2(D− )−1/2 × |g−1 (z k )| |g ℓℓℓ (z (z−k ) (z+ℓ ) 〉)−Σ(E)/µk .(2.30) • In case (b) (2.31) Asingk,ℓ (ω, θ,E, h) = e k,ℓ (ω, θ,E) hΣ(E)/2λ1−1/2 | lnh|Σ(E)/λ1 (1 + o(1)), 10 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND where (ω, θ,E) = π1−n/2 ei(nπ/4−π/2) )−1/2 (Σ(E) (2λ1λℓℓℓ) 3/2(2λ1) Σ(E)/λ1−1 × e−iν π/2e−iν π/2(D−k D × |g−1 (z )| |g+ − iM2(k, ℓ) )−Σ(E)/2λ1 .(2.32) • In case (c) (2.33) Asingk,ℓ (ω, θ,E, h) = e k,ℓ (ω, θ,E) hΣ(E)/2λ1−1/2 | ln h|Σ(E)/2λ1 (1 + o(1)), where k,ℓ (ω, θ,E) = π1−n/2 ei(nπ/4−π/2) )−1/2 (Σ(E) (2λ1λℓℓℓ) 3/2(2λ1) Σ(E)/2λ1−1 × e−iν π/2e−iν π/2(D−k D × |g−1 (z k )| |g ℓℓℓ (z − iM1(k, ℓ) )−Σ(E)/2λ1 .(2.34) Here, the µ̂j are the linear combinations over N of the λk’s and λk − λ1’s, and the function z 7→ z−Σ(E)/µk is defined on C\]−∞, 0] and real positive on ]0,+∞[. Of course the assumption that 〈g−1 |g 1 〉 6= 0 (a subcase of (a)) is generic. Without the assumption (A4), the regular part Areg of the scattering amplitude has an integral rep- resentation as in [3]. When the assumption (A7) is not fulfilled, that is when the terms corresponding to the µj with j ≤ ̂ do not contribute, we don’t know if the scattering amplitude can be given only in terms of the g±1 ’s and of the derivatives of the potential. 3. Proof of the main resolvent estimate Here we prove Theorem 2.1 using Mourre’s Theory. We start with the construction of an escape function close to the stationary point (0, 0) in the spirit of [10] and [5]. Since Λ+ and Λ− are Lagrangian manifolds, one can choose local symplectic coordinates (y, η) such that (3.1) p(x, ξ) = B(y, η)y · η, where (y, η) 7→ B(y, η) is a C∞ mapping from a neighborhood of (0,0) in T ∗Rn to the space Mn(R) of n× n matrices with real entries, such that, (3.2) B(0, 0) = . . . We denote U a unitary Fourier Integral Operator (FIO) microlocally defined in a neighborhood of (0, 0), which canonical transformation is the map (x, ξ) 7→ (y, η), and we set (3.3) P̂ = UPU∗. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 11 Here the FIO U∗ is the adjoint of U , and we have UU∗ = Id+O(h∞) and U∗U = Id+O(h∞) microlocally near (0, 0). Then P̂ is a pseudodifferential operator, with a real (modulo O(h∞)) symbol p̂(y, η) = j p̂j(y, η)h j , such that (3.4) p̂0 = B(y, η)y · η. We set B1 = Oph(b1), (3.5) b1(y, η) = χ̃2(y, η), where M > 1 will be fixed later and χ̃1 ≺ χ̃2 ∈ C∞0 (T ∗Rn) with χ̃1 = 1 near (0, 0). In what follows, we will assume that hM < 1. In particular, b1 ∈ S1/2(| ln h|). Here and in what follows, we use the usual notation for classes of symbols. For m an order function, a function a(x, ξ, h) ∈ C∞(T ∗Rn) belongs to Sδh(m) when (3.6) ∀α ∈ N2n, ∃Cα > 0, ∀h ∈]0, 1], |∂αx,ξa(x, ξ, h)| ≤ Cαh−δ|α|m(x, ξ). Let us also recall that, if a ∈ Sα(1) and b ∈ Sβ(1), with α, β < 1/2, we have (3.7) Oph(a),Oph(b) = Oph ih{b, a} + h3(1−α−β) Oph(r), with r ∈ Smin(α,β)(1): In particular the term of order 2 vanishes. Hence, we have here (3.8) [B1, P̂ ] = Oph ih{p̂0, b1} + | lnh|h3/2 Oph(rM ), with rM ∈ S1/2(1). The semi-norms of rM may depend on M . We have (3.9) {p̂0, b1} = c1 + c2, {p̂0, χ̃2}(3.10) p̂0, ln By + (∂ηB)y · η hM + y2 Bη + (∂yB)y · η hM + η2 χ̃2.(3.11) The symbols c1 ∈ S1/2(| lnh|), c2 ∈ S1/2(1) satisfy supp(c1) ⊂ supp(∇χ̃2). Let ϕ̃ ∈ C∞0 (T n) be a function such that ϕ̃ = 0 near (0, 0) and ϕ̃ = 1 near the support of ∇χ̃2. We Oph(c1) =Oph(ϕ̃)Oph(c1)Oph(ϕ̃) +O(h∞) ≥− C1h| ln h|Oph(ϕ̃)Oph(ϕ̃) +O(h∞) ≥− C1h| ln h|Oph(ϕ̃2) +O(h2| ln h|),(3.12) for some C1 > 0. On the other hand, using [5, (4.96)–(4.97)], we get (3.13) Oph(c2) ≥ εM−1 Oph(χ̃1) +O(M−2), 12 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND for some ε > 0. With the notation A1 = U ∗B1U , the formulas (3.8), (3.9), (3.12) and (3.13) imply −i[A1, P ] =− iU∗[B1, P ]U ≥εhM−1U∗Oph(χ̃1)U − C1h| lnh|U∗ Oph(ϕ̃2)U +O(hM−2) +OM (h3/2| lnh|).(3.14) If κ is the canonical transformation associated to U , then χj = χ̃j ◦ κ, j = 1, 2 and ϕ = ϕ̃ ◦ κ are C∞0 (T ∗(Rn), [0, 1]) functions which satisfy χ1 = 1 near (0, 0) and ϕ = 0 near (0, 0). Using Egorov’s Theorem, (3.14) becomes (3.15) − i[A1, P ] ≥ εhM−1 Oph(χ1)− C1h| lnh|Oph(ϕ) +O(hM−2) +OM (h3/2| lnh|). Now, we build an escape function outside of supp(χ1) as in [22]. Let 1(0,0) ≺ χ0 ≺ χ1 ≺ χ2 ≺ χ3 ≺ χ4 ≺ χ5 be C∞0 (T ∗(Rn), [0, 1]) functions with ϕ ≺ χ4. We define a3 = g(ξ)(1−χ3(x, ξ))x ·ξ where g ∈ C∞0 (Rn) satisfies 1p−1([E0−δ,E0+δ]) ≺ g. Using [6, Lemma 3.1], we can find a bounded, C∞ function a2(x, ξ) such that (3.16) Hpa2 ≥ 0 for all (x, ξ) ∈ p−1([E0 − δ,E0 + δ]), 1 for all (x, ξ) ∈ supp(χ4 − χ0) ∩ p−1([E0 − δ,E0 + δ]), and we set A2 = Oph(a2χ5). We denote (3.17) A = A1 + C2| lnh|A2 + | lnh|A3, where C2 > 1 will be fixed later. Now let ψ̃ ∈ C∞0 ([E0 − δ,E0 + δ], [0, 1]) with ψ̃ = 1 near E0. We recall that ψ̃(P ) is a classical pseudodifferential operator of class Ψ 0(〈ξ〉−∞) with principal symbol ψ̃(p). Then, from (3.15), we obtain −iψ̃(P )[A,P ]ψ̃(P ) ≥εhM−1ψ̃(P )Oph(χ1)ψ̃(P )− C1h| ln h|ψ̃(P )Oph(ϕ)ψ̃(P ) + C2h| lnh|Oph ψ̃2(p)(χ4 − χ0) + C2h| ln h|Oph ψ̃2(p)a2Hpχ5 + h| ln h|Oph ψ̃2(p)(ξ2 − x · ∇V )(1 − χ3) + h| ln h|Oph ψ̃2(p)x · ξHp(gχ3) +O(hM−2) +OM (h3/2| lnh|).(3.18) From (A1), we have x·∇V (x) → 0 as x→ ∞. In particular, if χ3 is equal to 1 in a sufficiently large zone, we have (3.19) ψ̃2(p)(ξ2 − x · ∇V )(1− χ3) ≥ E0ψ̃2(p)(1− χ3). If C2 > 0 is large enough, the G̊arding inequality implies (3.20) C2 Oph ψ̃2(p)(χ4 − χ0) −C1 Oph ψ̃2(p)ϕ ψ̃2(p)x · ξHp(gχ3) ≥ Oph ψ̃2(p)(χ4 − χ0) +O(h). As in [22], we take χ5(x) = χ̃5(µx) with µ small and χ̃5 ∈ C∞0 ([E0 − δ,E0 + δ], [0, 1]). Since a2 is bounded, we get (3.21) ∣∣C2ψ̃2(p)a2Hpχ5 ∣∣ ≤ µC2‖a2‖L∞‖Hpχ̃5‖L∞ . µ. Therefore, if µ is small enough, (3.19) implies (3.22) Oph ψ̃2(p)(ξ2−x ·∇V )(1−χ3) +C2 Oph ψ̃2(p)a2Hpχ5 ψ̃2(p)(1−χ3) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 13 Then (3.18), (3.20), (3.22) and the G̊arding inequality give −iψ̃(P )[A,P ]ψ̃(P ) ≥εhM−1 Oph ψ̃2(p)χ1 + h| ln h|Oph ψ̃2(p)(χ4 − χ0) h| ln h|Oph ψ̃2(p)(1 − χ3) +O(hM−2) +OM (h3/2| lnh|) ≥εhM−1 Oph ψ̃2(p) +O(hM−2) +OM (h3/2| ln h|).(3.23) Choosing M large enough and 1E0 ≺ ψ ≺ ψ̃, we have proved the Lemma 3.1. Let M be large enough and ψ ∈ C∞0 ([E0− δ,E0+ δ]), δ > 0 small enough, with ψ = 1 near E0. Then, we have (3.24) − iψ(P )[A,P ]ψ(P ) ≥ εh−1ψ2(P ). Moreover (3.25) [A,P ] = O(h| ln h|). From the properties of the support of the χj, we have [[P,A], A] =[[P,A1], A1] + C2| lnh|[[P,A1], A2] + C2| ln h|[[P,A2], A1] + C22 | lnh|2[[P,A2], A2] + C2| lnh|2[[P,A2], A3] + C2| ln h|2[[P,A3], A2] + | lnh|2[[P,A3], A3] +O(h∞).(3.26) We also know that P ∈ Ψ0(〈ξ〉2), A2 ∈ Ψ0(〈ξ〉−∞) and A3 ∈ Ψ0(〈x〉〈ξ〉−∞). Then, we can show that all the terms in (3.26) with j, k = 2, 3 satisfy (3.27) [[P,Aj ], Ak] ∈ Ψ0(h2). On the other hand, (3.28) [[P,A1], A2] = U ∗[[P̂ , B1], UA2U ∗]U +O(h∞), with UA2U ∗ ∈ Ψ0(1). From (3.8) – (3.11), we have [P̂ , B1] ∈ Ψ1/2(h| ln h|) and then (3.29) [[P,A1], A2] = O(h3/2| lnh|). The term [[P,A2], A1] gives the same type of contribution. It remains to study (3.30) [[P,A1], A1] = U ∗[[P̂ , B1], B1]U +O(h∞). Let χ̃3 ∈ C∞0 (T ∗Rn), [0, 1]) with χ̃2 ≺ χ̃3 and (3.31) f = χ̃3(y, η) ∈ S1/2(| ln h|). Then, with a remainder rM ∈ S1/2(1) which differs from line to line, i[P̂ , B1] =hOph f{χ̃2, p̂0}+ c2 − h3/2| ln h|Oph(rM ) =hOph(f)Oph({χ̃2, p̂0}) + hOph(c2) + h3/2| lnh|Oph(rM ).(3.32) 14 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND In particular, since [P̂ , B1] ∈ Ψ1/2(h| ln h|), c2 ∈ S1/2(1) and f ∈ S1/2(| lnh|), [[P̂ , B1], B1] =[[P̂ , B1],Oph(fχ̃2)] =− ih[Oph(f)Oph({χ̃2, p̂0}),Oph(fχ̃2)]− ih[Oph(c2),Oph(fχ̃2)] +O(h3/2| ln h|2) =− ih[Oph(f)Oph({χ̃2, p̂0}),Oph(f)Oph(χ̃2)] +O(h| ln h|) =− ihOph(f)[Oph({χ̃2, p̂0}),Oph(f)]Oph(χ̃2) − ih[Oph(f),Oph(f)]Oph({χ̃2, p̂0})Oph(χ̃2) − ihOph(f)Oph(f)[Oph({χ̃2, p̂0}),Oph(χ̃2)] − ihOph(f)[Oph(f),Oph(χ̃2)]Oph({χ̃2, p̂0}) +O(h| ln h|) =O(h| ln h|).(3.33) From (3.26), (3.27), (3.29) and (3.33), we get (3.34) [[P,A], A] = O(h| lnh|). As a matter of fact, using [5], one can show that [[P,A], A] = O(h). Now we can use the following proposition which is an adaptation of the limiting absorption principle of Mourre [25] (see also [12, Theorem 4.9], [19, Proposition 2.1] and [4, Theorem 7.4.1]). Proposition 3.2. Let (P,D(P )) and (A,D(A)) be self-adjoint operators on a separable Hilbert space H. Assume the following assumptions: i) P is of class C2(A). Recall that P is of class Cr(A) if there exists z ∈ C \ σ(P ) such (3.35) R ∋ t→ eitA(P − z)−1e−itA, is Cr for the strong topology of L(H). ii) The form [P,A] defined on D(A) ∩D(P ) extends to a bounded operator on H and (3.36) ‖[P,A]‖ . β. iii) The form [[P,A],A] defined on D(A) extends to a bounded operator on H and (3.37) ‖[[P,A],A]‖ . γ. iv) There exist a compact interval I ⊂ R and g ∈ C∞0 (R) with 1I ≺ g such that (3.38) ig(P )[P,A]g(P ) & γg2(P ). v) β2 . γ . 1. Then, for all α > 1/2, limε→0〈A〉−α(P − E ± iε)−1〈A〉−α exists and (3.39) ∥∥〈A〉−α(P − E ± i0)−1〈A〉−α ∥∥ . γ−1, uniformly for E ∈ I. Remark 3.3. From Theorem 6.2.10 of [4], we have the following useful characterization of the regularity C2(A). Assume that (ii) and (iv) hold. Then, P is of class C2(A) if and only if, for some z ∈ C \ σ(P ), the set {u ∈ D(A); (P − z)−1u ∈ D(A) and (P − z)−1u ∈ D(A)} is a core for A. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 15 Proof. The proof follows the work of Hislop and Nakamura [19]. For ε > 0, we define M2 = ig(P )[P,A]g(P ) and Gε(z) = (P − iεM2 − z)−1 which is analytic for Re z ∈ I and Im z > 0. Following [12, Lemma 4.14] with (3.35)), we get (3.40) ‖g(P )Gε(z)ϕ‖ . (εγ)−1/2|(ϕ,Gε(z)ϕ)|1/2, (3.41) ‖(1− g(P ))Gε(z)‖ . 1 + εβ‖Gε(z)‖, and then (3.42) ‖Gε(z)‖ . (εγ)−1, for ε < ε0 with ε0 small enough, but independent on β, γ. As in [19], let Dε = (1 + |A|)−α(1 + ε|A|)α−1 for α ∈]1/2, 1] and Fε(z) = DεGε(z)Dε. Of course, from (3.42), (3.43) ‖Fε(z)‖ . (εγ)−1, and (3.40) and (3.41) with ϕ = Dεψ give (3.44) ‖Gε(z)Dε‖ . 1 + (εγ)−1/2‖Fε‖1/2. The derivative of Fε(z) is given by (see [12, Lemma 4.15]) (3.45) ∂εFε(z) = iDεGεM 2GεDε = Q0 +Q1 +Q2 +Q3, Q0 =(α− 1)|A|(1 + |A|)−α(1 + ε|A|)α−2Gε(z)Dε + (α− 1)DεGε(z)|A|(1 + |A|)−α(1 + ε|A|)α−2(3.46) Q1 =DεGε(1− g(P ))[P,A](1 − g(P ))GεDε(3.47) Q2 =DεGε(1− g(P ))[P,A]g(P )GεDε +DεGεg(P )[P,A](1 − g(P ))GεDε(3.48) Q3 =−DεGε[P,A]GεDε.(3.49) From (3.44), we obtain (3.50) ‖Q0‖ . εα−1 1 + (εγ)−1/2‖Fε‖1/2 and from (3.36), v) of Proposition 3.2, (3.41) and (3.42), we get (3.51) ‖Q1‖ . γ−1. Using in addition (3.44), we obtain (3.52) ‖Q2‖ . 1 + (εγ)−1/2‖Fε‖1/2. Now we write Q3 = Q4 +Q5 with Q4 = −DεGε[P − iεM2 − z,A]GεDε(3.53) Q5 = −iεDεGε[M2,A]GεDε.(3.54) For Q4, we have the estimate (3.55) ‖Q4‖ . εα−1 1 + (εγ)−1/2‖Fε‖1/2 On the other hand, (3.36), (3.37) and v) imply (3.56) ‖[M2,A]‖ . γ. 16 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Then (3.44) gives (3.57) ‖Q5‖ . 1 + ‖Fε‖. Using the estimates on the Qj, we get (3.58) ‖∂εFε‖ . εα−1 γ−1 + (εγ)−1/2‖Fε‖1/2 + ‖Fε‖ Using (3.43) and integrating (3.37) N times with respect to ε, we get (3.59) ‖Fε‖ . γ−1 1 + ε2α(1−2 −N )−1), so that, for N large enough, (3.60) lim sup ‖〈A〉−α(P − E ± iδ)−1〈A〉−α‖ . γ−1. Using, as in [19], that z 7→ F0(z) is Hölder continuous, we prove the existence of the limit limIm z→0 F0(z) for Re z ∈ I and the proposition follows from (3.60). � From Lemma 3.1 and (3.34), we can apply Proposition 3.2 with A = A/| ln h|, β = h and γ = h/| ln h|. Therefore we have the estimate (3.61) ∥∥〈A〉−α(P − E ± i0)−1〈A〉−α ∥∥ . h−1| ln h|, for E ∈ [E0 − δ,E0 + δ]. As usual, we have (3.62) ‖〈x〉−α〈A〉α‖ = O(1), for α ≥ 0. Indeed, (3.62) is clear for α ∈ 2N, and the general case follows by complex interpolation. Then, (3.61) and (3.26) imply Theorem 2.1. 4. Representation of the Scattering Amplitude As in [30], our starting point for the computation of the scattering amplitude is the rep- resentation given by Isozaki and Kitada in [20]. We recall briefly their formula, that they obtained writing parametrices for the wave operators W± as Fourier Integral Operators, tak- ing advantage of the well-known intertwining property W±P = P0W±, P = P0 + V . The wave operators are defined by (4.1) W± = s− lim eitP/he−itP0/h, where the limit exist thanks to the short-range assumption (A1). The scattering operator is by definition S = (W+)∗W−, and the scattering matrix S(E, h) is then given by the decompostion of S with respect to the spectral measure of P0 = −h2∆. Now we recall briefly the discussion in [30, Section 1,2] (see also [3]), and we start with some notations. If Ω is an open subset of T ∗Rn , we denote by Am(Ω) the class of symbols a such that (x, ξ) 7→ a(x, ξ, h) belongs to C∞(Ω) and (4.2) ∣∣∣∂αx ∂ ξ a(x, ξ) ∣∣∣ ≤ Cαβ〈x〉m−|α|〈ξ〉−L, for all L > 0, (x, ξ) ∈ Ω, (α, β) ∈ Nd × Nd. We also denote by (4.3) Γ±(R, d, σ) = (x, ξ) ∈ Rn × Rn : |x| > R, 1 < |ξ| < d,± cos(x, ξ) > ±σ SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 17 with R > 1, d > 1, σ ∈ (−1, 1), and cos(x, ξ) = 〈x,ξ〉|x| |ξ| , the outgoing and incoming subsets of T ∗Rn, respectively. Eventually, for α > 1 , we denote the bounded operator F0(E, h) : L2α(R n) → L2(Sn−1) given by (4.4) (F0(E, h)f) (ω) = (2πh)− 2 (2E) 2E〈ω,x〉f(x)dx,E > 0. Isozaki and Kitada have constructed phase functions Φ± and symbols a± and b± such that, for some R0 >> 0, 1 < d4 < d3 < d2 < d1 < d0, and 0 < σ4 < σ3 < σ2 < σ1 < σ0 < 1: i) Φ± ∈ C∞(T ∗Rn) solve the eikonal equation (4.5) |∇xΦ±(x, ξ)|2 + V (x) = in (x, ξ) ∈ Γ±(R0, d0,±σ0), respectively. ii) (x, ξ) 7→ Φ±(x, ξ) − x · ξ ∈ A0 (Γ±(R0, d0,±σ0)) . iii) For all (x, ξ) ∈ T ∗Rn (4.6) ∂xj∂ξk (x, ξ) − δjk ∣∣∣∣ < ε(R0), where δjk is the Kronecker delta and ε(R0) → 0 as R0 → +∞. iv) a± ∼ j=0 h ja±j , where a±j ∈ A−j(Γ±(3R0, d1,∓σ1)), supp a±j ⊂ Γ±(3R0, d1,∓σ1), a±j solve (4.7) 〈∇xΦ±|∇xa±0〉+ (∆xΦ±) a±0 = 0 (4.8) 〈∇xΦ±|∇xa±j〉+ (∆xΦ±) a±j = ∆xa±j−1, j ≥ 1, with the conditions at infinity (4.9) a±0 → 1, a±j → 0, j ≥ 1, as |x| → ∞. in Γ±(2R0, d2,∓σ2), and solve (4.7) and (4.8) in Γ±(4R0, d1,∓σ2). v) b± ∼ j=0 h jb±j, where b±j ∈ A−j(Γ±(5R0, d3,±σ4), supp b±j ⊂ Γ±(5R0, d3,±σ4), b±j solve (4.7) and (4.8) with the conditions at infinity (4.9) in Γ±(6R0, d4,±σ3), and solve (4.7) and (4.8) in Γ±(6R0, d3,±σ3). For a symbol c and a phase function ϕ, we denote by Ih(c, ϕ) the oscillatory integral (4.10) Ih(c, ϕ) = (2πh)n (ϕ(x,ξ)−〈y,ξ〉)c(x, ξ)dξ and we set (4.11) K±a(h) = P (h)Ih(a±,Φ±)− Ih(a±,Φ±)P0(h), K±b(h) = P (h)Ih(b±,Φ±)− Ih(b±,Φ±)P0(h). The operator T (E, h) for E ∈] 1 [ is then given by (see [20, Theorem 3.3]) (4.12) T (E, h) = T+1(E, h) + T−1(E, h) − T2(E, h), 18 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND where (4.13) T±1(E, h) = F0(E, h)Ih(a±,Φ±)∗K±b(h)F∗0 (E, h) (4.14) T2(E, h) = F0(E, h)K∗+a(h)R(E + i0, h) (K+b(h) +K−b(h))F∗0 (E, h), where we denote from now on R(E ± i0, h) = (P − (E ± i0))−1. Writing explicitly their kernel, it is easy to see, by a non-stationary phase argument, that the operators T±1 are O(h∞) when θ 6= ω. Therefore we have (4.15) A(ω, θ,E, h) = −c(E)h(n−1)/2T2(ω, θ,E, h) +O(h∞), where c(E) is given in (1.4). As in [30], we shall use our resolvent estimate (Theorem 2.1) in a particular form. It was noticed by L. Michel in [24, Proposition 3.1] that, in the present trapping case, the following proposition follows easily from the corresponding one in the non-trapping setting. Indeed, if ϕ is a compactly supported smooth function, it is clear that P̃ = −h2∆+ (1− ϕ(x/R))V (x) satisfies the non-trapping assumption for R large enough, thanks to the decay of V at ∞. Writing [30, Lemma 2.3] for P̃ , one gets the Proposition 4.1. Let ω± ∈ A0 has support in Γ±(R, d, σ±) for R > R0. For E ∈ [E0 − δ,E0 + δ], we have (i) For any α > 1/2 and M > 1, then, for any ε > 0, (4.16) ‖R(E ± i0, h)ω±(x, hDx)‖−α+M,−α = O(h−3−ε). (ii) If σ+ > σ−, then for any α≫ 1, (4.17) ‖ω∓(x, hDx)R(E ± i0, h)ω±(x, hDx)‖−α+δ,−α = O(h∞). (iii) If ω(x, ξ) ∈ A0 has support in |x| < (9/10)R, then for any α≫ 1 (4.18) ‖ω(x, hDx)R(E ± i0, h)ω±(x, hDx)‖−α+δ,−α = O(h∞). Then we can follow line by line the discussion after Lemma 2.1 of D. Robert and H. Tamura, and we obtain (see Equations 2.2-2.4 there): (4.19) A(ω, θ,E, h) = c(E)h−(n+1)/2〈R(E + i0, h)g−eiψ−/h, g+eiψ+/h〉+O(h∞), where (4.20) g± = e −iψ±/h[χ±, P ]a±(x, h)e iψ±/h, (4.21) ψ+(x) = Φ+(x, 2Eθ), ψ−(x) = Φ−(x, 2Eω). Moreover the functions χ± are C n) functions such that χ± = 1 on some ball B(0, R±), with support in B(0, R± + 1). Eventually, we shall need the following version of Egorov’s Theorem, which is also used in Robert and Tamura’s paper. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 19 Proposition 4.2 ([30, Proposition 3.1]). Let ω(x, ξ) ∈ A0 be of compact support. Assume that, for some fixed t ∈ R, ωt is a function in A0 which vanishes in a small neighborhood of {(x, ξ); (x, ξ) = exp(tHp)(y, η), (y, η) ∈ suppω}. ‖Oph(ωt)e−itP/hOph(ω)‖−α,α = O(h∞), for any α ≫ 1. Moreover, the order relation is uniform in t when t ranges over a compact interval of R. In the three next sections, we prove Theorem 2.6 using (4.19). We set (4.22) u− = u − = R(E + i0, h)g−eiψ−/h, and our proof consists in the computation of u− in different region of the phase space, following the classical trajectories γ∞j , or γ k and γ ℓ . It is important to notice that we have (P−E)u− = 0 out of the support of g−. 5. Computations before the critical point 5.1. Computation of u− in the incoming region. We start with the computation of u− in an incoming region which contains the micro- support of g−. Notice that, thanks to Theorem 2.1, 〈x〉−αu−(x) is a semiclassical family of distributions for α > 1/2. Lemma 5.1. Let P be a Schrödinger operator as in (1.1) satisfying only (A1). Suppose that I is a compact interval of ]0,+∞[, and d > 0 is such that I ⊂] 1 [. Suppose also that 0 < σ+ < 1, R is large enough and K ⊂ T ∗Rn is a compact subset of {|x| > R} ∩ p−1(I). Then there exists T0 > 0 such that, if ρ ∈ K and t > T0, (5.1) exp(tHp)(ρ) ∈ Γ+(R/2, d, σ+) ∪ (B(0, R/2) × Rn). Proof. Let δ > 0. From the construction of C. Gérard and J. Sjöstrand [17], there exists a function G(x, ξ) ∈ C∞(R2n) such that, (HpG)(x, ξ) ≥ 0 for all (x, ξ) ∈ p−1(] [),(5.2) (HpG)(x, ξ) > 2E(1− δ) for |x| > R0 and p(x, ξ) = E ∈] [,(5.3) G(x, ξ) = x · ξ for |x| > R0.(5.4) Let ρ ∈ K, and γ(t) = (x(t), ξ(t)) = exp(tHp)(ρ) be the corresponding Hamiltonian curve. We distinguish between 2 cases: 1) For all t > 0, we have |x(t)| > R0. Then G(γ(t)) > 2E(1 − δ)t+G(ρ) and, for t > T1 with T1 large enough, (5.5) G(γ(t)) > 2 sup x∈B(0,R0) p(x,ξ)∈I G(x, ξ). 20 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND By continuity, there exists a neighborhood U of γ such that, for all γ̃ ∈ U , we have (5.6) G(γ̃(T1)) > sup x∈B(0,R0) p(x,ξ)∈I G(x, ξ). Since G is non-decreasing on γ̃(t), we have |x̃(t)| > R0 for all t > T1, and then (5.7) G(γ̃(t)) > 2E(1 − δ)(t − T1) +G(γ̃(T1)) > 2E(1 − δ)t − C. On the other hand, by uniformly finite propagation, we have |x̃(t)| < 2E(1+ δ)t+C. From (5.7), we get |x̃(t)| > 1 t− C for all γ̃ ∈ U , and then |ξ̃(t)| = 2E + ot→∞(1). In particular, the previous estimates gives (5.8) |x(t)| > R/2, (5.9) cos x̃, ξ̃ (t) > 2E(1− δ)t− C 2E(1 + δ)t+ C)( 2E + ot→∞(1)) 1 + δ + ot→∞(1) > 1− 3δ, for t > T0 with T0 large enough but independent on γ̃ ∈ U . Thus, for t > T0 and γ̃ ∈ U , we (5.10) γ̃(t) ∈ Γ+(R/2, d, σ+), with σ+ = 1− 3δ. 2) There exist T2 > 0 such that |x(T2)| = R0. Then there exists V a neighborhood of γ such that, for all γ̃ ∈ V, we have |x̃(T2)| < 2R0. Let t > T2. a) If |x̃(t)| ≤ R/2, then γ̃(t) ∈ B(0, R/2) × Rn. b) Assume now |x̃(t)| > R/2. Denote by T3 (> T2) the last time (before t) such that |x̃(T3)| = 2R0. Then G(γ̃(t)) >2E(1 − δ)(t− T3) +G(γ̃(T3))(5.11) >2E(1 − δ)(t− T3)− C,(5.12) where C depend only on R0. On the other hand, the have |x̃(t)| < 2E(1 + δ)(t − T3) + C (where the constant C depend only on R0). Then, (5.13) t− T3 > |x̃(t)|√ 2E(1 + δ) 2E(1 + δ) (5.14) |ξ̃(t)| = 2E + oR→∞(1), x̃, ξ̃ (t) > 2E(1− δ)|x̃(t)| |x̃(t)|( 2E(1 + δ))( 2E + oR→∞(1)) +O(R−1) 1 + δ + oR→∞(1) > 1− 2δ + oR→∞(1).(5.15) So, if R is large enough, γ̃(t) ∈ Γ+(R/2, d, σ+), σ+ = 1− 3δ. Then a) and b) imply that, for all γ̃ ∈ V and t > T0 := T2, we have (5.16) γ̃(t) ∈ Γ+(R/2, d, σ+) ∪ (B(0, R/2) × Rn). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 21 The lemma follows from (5.10), (5.16) and a compactness argument. � Recall that the microsupport of g−(x)e iψ−(x)/h ∈ C∞0 (Rn) is contained in Γ−(R−, d1, σ1). Let ω−(x, ξ) ∈ A0 with ω− = 1 near Γ−(R−/2, d1, σ1) and supp(ω−) ⊂ Γ−(R−/3, d0, σ0). Using the identity (5.17) u− = e−it(P−E)/h(g−e iψ−/h)dt+R(E + i0, h)e−iT (P−E)/h(g−eiψ−/h), and Proposition 4.1, Proposition 4.2 and Lemma 5.1, we get (5.18) Oph(ω−)u− = Oph(ω−) e−it(P−E)/h(g−e iψ−/h)dt+O(h∞), for some T > 0 large enough. In particular, (5.19) MS(Oph(ω−)u−) ⊂ Λ−ω ∩ (B(0, R− + 1)× Rn). 5.2. Computation of u− along γ Now we want to compute u− microlocally along a trajectory γ k . We recall that γ k is a bicharacteristic curve (x−k (t), ξ k (t)) such that (x k (t), ξ k (t)) → (0, 0) as t → +∞, and such that, as t→ −∞, (5.20) |x−k (t)− 2E0ωt− z−k | → 0, 2E0ω| → 0. If R− is large enough, a− solves (4.7) and (4.8) microlocally near γ k ∩ MS(g−eiψ−/h). In particular, microlocally near γ−k ∩ Γ−(R−/2, d1, σ1) ∩ (B(0, R−)× Rn), u− is given by (5.18) e−it(P−E)/h([χ−, P ]a−e iψ−/h)dt+O(h∞) e−it(P−E)/h(χ−(P − E)a−eiψ−/h)dt e−it(P−E)/h((P − E)χ−a−eiψ−/h)dt+O(h∞) (P − E)e−it(P−E)/h(χ−a−eiψ−/h)dt+O(h∞) =(P − E)R(E + i0, h)a−eiψ−/h +O(h∞) iψ−/h +O(h∞).(5.21) Now, using (5.21), and the fact that u− is a semiclassical distribution satisfying (5.22) (P − E)u− = 0, we can compute u− microlocally near γ ∩ B(0, R−) using Maslov’s theory (see [23] for more details). Moreover, it is proved in Proposition C.1 (see also [5, Lemma 5.8]) that the Lagrangian manifold Λ−ω has a nice projection with respect to x in a neighborhood of γ close to (0, 0). Then, in such a neighborhood, u− is given by (5.23) u−(x) = a−(x, h)e π/2eiψ−(x)/h, 22 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND where ν−k denotes the Maslov index of γ k , and ψ− satisfies the usual eikonal equation (5.24) p(x,∇ψ−) = E0. Here, to the contrary of (4.21), we have written E = E0 + zh with z = O(1), and we choose to work with z in the amplitudes instead of the phases. As usual, we have (5.25) ∂t(ψ−(x (t))) = ∇ψ−(x−k (t)) · ∂tx (t) = ∇ψ−(x−k (t)) · ξ (t) = |ξ− (t)|2, so that (5.26) ψ−(x (t)) = ψ−(x (s)) + (u)|2du We also have ψ−(x k (s)) = ( 2E0ωs+ z k ) · 2E0ω + o(1) as s→ −∞, and then (5.27) ψ−(x k (t)) = 2E0s+ |ξ−k (u)| 2du+ o(1), s→ −∞. We have obtained in particular that (5.28) ψ−(x k (t)) = |ξ−k (u)| 2−2E01u<0 du = |ξ−k (u)| 2−V (x−k (u))+E0 sgn(u) du. We turn to the computation of the symbol. The function a−(x, h) ∼ k=0 a−,k(x)h satisfies the usual transport equations: (5.29) ∇ψ− · ∇a−,0 + (∆ψ− − 2iz)a−,0 = 0, ∇ψ− · ∇a−,k + (∆ψ− − 2iz)a−,k = i ∆a−,k−1, k ≥ 1, In particular, we get for the principal symbol (5.30) ∂t(a−,0(x (t))) = ∇a−,0(x−k (t)) · ξ (t) = ∇a−,0(x−k (t)) · ∇ψ−(x (t)), so that, (5.31) ∂t(a−,0(x k (t))) = − ∆ψ−(x k (t))− 2iz a−,0(x k (t)) and then (5.32) a−,0(x k (t)) = a−,0(x k (s)) exp ∆ψ−(x−(u)) du + i(t− s)z On the other hand, from [30, Lemma 4.3], based on Maslov theory, we have (5.33) a−,0(x k (t)) = (2E0) 1/4D−k (t) −1/2eitz , where (5.34) D− (t) = ∣∣ det ∂x−(t, z, ω,E0) ∂(t, z) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 23 6. Computation of u− at the critical point Now we use the results of [5] to get a representation of u− in a whole neighborhood of the critical point. Indeed we saw already that (P − E)u− = 0 out of the support of g−, in particular in a neighborhood of the critical point. First, we need to recall some terminology of [18] and [5]. We recall from Section 2 that (µj)j≥0 is the strictly growing sequence of linear combinations over N of the λj’s. Let u(t, x) be a function defined on [0,+∞[×U , U ⊂ Rm. Definition 6.1. We say that u : [0,+∞[×U → R, a smooth function, is expandible, if, for any N ∈ N, ε > 0, α, β ∈ N1+m, (6.1) ∂αt ∂ u(t, x)− uj(t, x)e −µj t e−(µN+1−ε)t for a sequence of (uj)j smooth functions, which are polynomials in t. We shall write u(t, x) ∼ uj(t, x)e −µj t, when (6.1) holds. We say that f(t, x) = Õ(e−µt) if for all α, β ∈ N1+m and ε > 0 we have (6.2) ∂αt ∂ xf(t, x) = O(e−(µ−ε)t). Definition 6.2. We say that u(t, x, h), a smooth function, is of class SA,B if, for any ε > 0, α, β ∈ N1+m, (6.3) ∂αt ∂ xu(t, x, h) = O hAe−(B−ε)t Let S∞,B = A SA,B. We say that u(t, x, h) is a classical expandible function of order (A,B), if, for any K ∈ N, (6.4) u(t, x, h) − uk(t, x)h k ∈ SK+1,B, for a sequence of (uk)k expandible functions. We shall write u(t, x, h) ∼ uk(t, x)h in that case. Since the intersection between Λ−ω and Λ− is transverse along the trajectories γ k ), and since g−1 (z k ) 6= 0, Theorem 2.1 and Theorem 5.4 of [5] implies that one can write, microlocally near (0, 0), (6.5) u− = ∫ N−∑ αk(t, x, h)eiϕ k(t,x)/hdt, 24 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND where the αk(t, x, h)’s are classical expandible functions in S0,2ReΣ(E): (6.6) αk(t, x, h) ∼ αkm(t, x)h αkm(t, x) ∼ αkm,j(t, x)e −2(Σ(E)+µj )t, and where the αkm,j(t, x)’s are polynomial with respect to t. We recall from (2.26) that, for E = E0 + hz, (6.7) Σ(E) = − iz. Following line by line Section 6 of [5], we obtain (see [5, (6.26)]) αk0,0(0) = e iπ/4(2λ1) 3/2e−iν π/2|g(γ−k )|(D −1/2(2E0) 1/4.(6.8) Notice that from (5.32) and Proposition C.1, we have 0 < D− < +∞. From [5, Section 5], we recall that the phases ϕk(t, x) satisfies the eikonal equation (6.9) ∂tϕ k + p(x,∇xϕk) = E0, and that they have the asymptotic expansion (6.10) ϕk(t, x) ∼ ϕkj,m(x)t me−µjt, with Mkj < +∞. In the following, we denote (6.11) ϕkj (t, x) = ϕkj,m(x)t and the first ϕkj ’s are of the form ϕk0(t, x) =ϕ+(x) + ck(6.12) ϕk1(t, x) =− 2λ1g−(z−k ) · x+O(x 2),(6.13) where ck ∈ R is the constant depending on k given by (6.14) ck = “ψ−(0)” = lim k (t)) = S thanks to (5.28) (see also [5, Lemma 5.10]). Moreover ϕ+ is the generating function of the outgoing stable Lagrangian manifold Λ+ with ϕ+(0) = 0. We have (6.15) ϕ+(x) = x2j +O(x3). The fact that ϕk1(t, x) does not depend on t and the expression (6.13) follows also from Corollary 6.6 and (6.109). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 25 6.1. Study of the transport equations for the phases. Now, we examine the equations satisfied by the functions ϕkj (t, x), defined in (6.10), for the integers j ≤ ̂ (recall that ̂ is defined by µb = 2λ1). For clearer notations, we omit the superscript k until further notice. Let us recall that the function ϕ(t, x) satisfies the eikonal equation (6.9), which implies (see (6.10)) (6.16) e−µjtϕj,m(x)(−µjtm+mtm−1)+ ∇ϕj,m(x)tme−µj t +V (x) ∼ E0, and then e−µjtϕj,m(x)(−µjtm +mtm−1) + ∇ϕj,m∇ϕe,em(x)e−(µj+µe)ttm+ em +V (x) ∼ E0.(6.17) When µj < 2λ1, the double product of the previous formula provides a term of the form e if and only if µj = 0 or µe = 0. In particular, the term in e −µjt in (6.17) gives (6.18) ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+(x) · ∇ϕj,m(x)tm = 0. When µj = 2λ1, one gets also a term in e −2λ1t for µj = µe = λ1 and then ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+(x) · ∇ϕj,m(x)tm tm+ em∇ϕ1,m(x)∇ϕ1, em(x) = 0.(6.19) We denote (6.20) L = ∇ϕ+(x) · ∇ the vector field that appears in (6.18) and (6.19). We set also L0 = j λjxj∂j its linear part at x = 0, and we begin with the study of the solution of (6.21) (L− µ)f = g, with µ ∈ R and f , g ∈ C∞(Rn). First of all, we show that it is sufficient to solve (6.21) for formal series. Proposition 6.3. Let g ∈ C∞(Rn) and g0 the the Taylor expansion of g at 0. For each formal series f0 such that (L−µ)f0 = g0, there exists one and only one function f ∈ C∞(Rn) defined near 0 such that f = f0 +O(x∞) and (6.22) (L− µ)f = g, near 0. 26 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Proof. Let f̃0 be a C ∞ function having f0 has Taylor expansion at 0. With the notation f = f̃0 + r, the problem (6.22) is equivalent to find r = O(x∞) with (6.23) (L− µ)r = g − (L− µ)f̃0 = r̃, where r̃ ∈ C∞ has g0 − (L− µ)f0 = 0 as Taylor expansion at 0. Let y(t, x) be the solution of (6.24) ∂ty(t, x) = ∇ϕ+(y(t, x)), y(0, x) = x. Thus, (6.23) is equivalent to (6.25) r(x) = e−µsr̃(y(s, x))ds + e−µtr(y(t, x)). Since r(x), r̃(x) = O(x∞) and y(s, x) = O(eλ1t|x|) for t < 0, the functions e−µtr(y(t, x)), e−µtr̃(y(t, x)) are O(eNt) as t→ −∞ for all N > 0. Then (6.26) r(x) = e−µsr̃(y(s, x))ds, and r(x) = O(x∞). The uniqueness follows and it is enough to prove that r given by (6.26) is C∞. We have (6.27) ∂t(∇xy) = (∇2xϕ+(y))(∇xy), and since ∇2xϕ+ is bounded, there exists C > 0 such that (6.28) |∇xy(t, x)| . e−Ct, has t → −∞. Then, e−µs(∇r̃)(y(s, x))(∂jy(t, x)) = O(eNt) as t → −∞ for all N > 0 and ∂jr(x) = −µs(∇r̃)(y(s, x))(∂jy(t, x))ds. The derivatives of order greater than 1 can be treated the same way. � We denote (6.29) Lµ = L− µ : CJxK → CJxK, where we use the standard notation CJxK for formal series, and CpJxK for formal series of degree ≥ p. We notice that (6.30) Lµx α = (L0 − µ)xα + C|α|+1JxK = (λ · α− µ)xα +C|α|+1JxK. Recall that Iℓ(µ) has been defined in (2.20). The number of elements in Iℓ(µ) will be denoted (6.31) nℓ(µ) = #Iℓ(µ). One has for example n2(µ) = n1(µ)(n1(µ)+1) Proposition 6.4. Suppose µ ∈]0, 2λ1[. With the above notations, one has KerLµ⊕ ImLµ = CJxK. More precisely: i) The kernel of Lµ has dimension n1(µ), and one can find a basis (Ej1 , . . . , Ejn1(µ) KerLµ such that Ej(x) = xj + C2JxK, j ∈ I1(µ). ii) A formal series F = F0 + Fjxj + C2JxK belongs to ImLµ if and only if Fj = 0 for all j ∈ I1(µ). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 27 Remark 6.5. Thanks to Propostion 6.3, the same result is true for germs of C∞ functions at 0. Notice that when µ 6= µj for all j, Lµ is invertible. Proof. For a given F = α Fαx α ∈ CJxK, we look for solutions E = α ∈ CJxK to the equation (6.32) Lµ The calculus of the term of order x0 in (6.32) leads to the equation (6.33) E0 = − With this value for E0, (6.32) becomes, using again (6.30), (6.34) |α|=1 (λ · α− µ)Eαxα = |α|=1 α + C2JxK. We have two cases: If α /∈ I1(µ), one should have (6.35) Eα = λ · α− µ. If α ∈ I1(µ), the formula (6.34) becomes Fα = 0. In that case, the corresponding Eα can be chosen arbitrarily. Now suppose that the Eα are fixed for any |α| ≤ n− 1 (with n ≥ 2), and such that (6.36) Lµ |α|≤n−1 α + CnJxK. We can write (6.32) as (6.37) Lµ |α|=n α − Lµ |α|≤n−1 + Cn+1JxK, or, using again (6.30), (6.38) |α|=n (λ · α− µ)Eαxα = |α|≤n α − Lµ |α|≤n−1 +Cn+1JxK. Since |α| ≥ 2, one has λ · α ≥ 2λ1 > µ, so that (6.38) determines by induction all the Eα’s for |α| = n in a unique way. � Corollary 6.6. If j < ̂, the function ϕj(t, x) does not depend on t, i.e. we have Mj = 0. Proof. Suppose that Mj ≥ 1, then (6.18) gives the system (6.39) (L− µj)ϕj,Mj = 0, (L− µj)ϕj,Mj−1 = −Mjϕj,Mj , with ϕj,Mj 6= 0. But this would imply that ϕj,Mj ∈ KerLµ ∩ ImLµ, a contradiction. � 28 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND As a consequence, for j < ̂, the equation (6.18) on ϕj reduces to (6.40) (L− µj)ϕj,0 = 0, and, from Proposition 6.4, we get that (6.41) ϕj(t, x) = ϕj,0(x) = k∈I1(µ) dj,kxk +O(x2). Now we pass to the case j = ̂, and we study (6.19). First of all, we have seen that ϕ1 does not depend on t, so that this equation can be written (6.42) ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+ · ∇ϕj,m(x)tm + ∣∣∇ϕ1(x) ∣∣2 = 0. As for the study of (6.18), we begin with that of (6.21), now in the case where µ = 2λ1. We denote Ψ : Rn1(2λ1) −→ Rn2(λ1) the linear map given by (6.43) Ψ(Eβ1 , . . . , Eβn1(2λ1) β∈I1(2λ1) ∂α(L− µ)xβ α∈I2(λ1) and we set (6.44) n(Ψ) = dimKerΨ. Recalling that L = ∇ϕ+(x) · ∇, we see that (6.45) Ψ(Eβ1 , . . . , Eβn1(2λ1) β∈I1(2λ1) ∂α∂βϕ+(0) α∈I2(λ1) More generally, for any |α| = 2, we denote (6.46) Ψα((Eβ)β∈I1(2λ1)) = β∈I1(2λ1) ∂α∂βϕ+(0) Then, at the level of formal series, we have the Proposition 6.7. Suppose µ = 2λ1. Then i) KerLµ has dimension n2(λ1) + n(Ψ). ii) A formal series F = α Fαx α belongs to ImLµ if and only if ∀α ∈ I1(2λ1), Fα = 0,(6.47) |β|=1 β /∈I1(2λ1) ∂β∂αϕ+(0) 2λ1 − λ · β α∈I2(λ1) ∈ ImΨ.(6.48) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 29 iii) If F ∈ ImLµ, any formal series E = α with LµE = F satisfies F0,(6.49) λ · α− 2λ1 Fα, for α ∈ I1 \ I1(2λ1),(6.50) β∈I1(2λ1) |β|=1 β /∈I1(2λ1) ∂β∂αϕ+(0) 2λ1 − λ · β α∈I2(λ1) .(6.51) Moreover for α ∈ I2 \ I2(λ1), one has (6.52) Eα = λ · α− 2λ1 Fα −Ψα((Eβ)β∈I1(2λ1)) + |β|=1 β /∈I1(2λ1) 2λ1 − λ · β ∂α+βϕ+(0) Last, E is completely determined by F and a choice of the Eα for |α| ≤ 2 such that (6.49)– (6.52) are satisfied. iv) KerLµ ∩ Im(Lµ)2 = {0}. Proof. For a given F = α Fαx α we look for a E = α such that L2λ1E = F . First of all, we must have (6.53) E0 = − When this is true, we get (6.54) |α|=1 Eα(L0 − 2λ1)xα = |α|=1 Fα(L− 2λ1)xα + C2JxK, and we obtain as necessary condition that Fα = 0 for any α ∈ I1(2λ1). So far, the Eα for α ∈ I1(2λ1) can be chosen arbitrarily, and we must have (6.55) Eα = λ · α− 2λ1 , α ∈ I2 \ I1(2λ1). We suppose that (6.53) and (6.55) hold. Then we should have (6.56) |α|=2 Eα(L0−2λ1)xα = |α|=2 |α|=1 α/∈I1(2λ1) |α|=1 Eα(L−2λ1)xα +C3JxK. Notice that the second term in the R.H.S of (6.56) belongs to C2JxK thanks to (6.55). Again, we have to cases: • When α ∈ I2(λ1), the corresponding Eα can be chosen arbitrarily, but one must have |β|=1 ∂α(L− 2λ1)xβ |x=0(6.57) =Ψα((Eβ)β∈I1(2λ1)) + |β|=1 β /∈I1(2λ1) ∂α+βϕ+(0) ,(6.58) 30 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND and this, with (6.55), gives (6.51). • When |α| = 2, α /∈ I2(λ1), one obtains λ · α− 2λ1 |β|=1 ∂α(L− 2λ1)xβ λ · α− 2λ1 Fα −Ψα((Eβ)β∈I1(2λ1))− |β|=1 β /∈I1(2λ1) ∂α+βϕ+(0) ,(6.59) and this, with (6.55), gives (6.52). Now suppose that (6.53), (6.55), (6.57) and (6.59) hold, and that we have chosen a value for the free variables Eα for α ∈ I1(2λ1)∪I2(λ1). Thanks to the fact that λ ·α 6= 2λ1 for any α ∈ Nn with |α| = 3, we see as in the proof of Propostion 6.4, that the equation (6.54) has a unique solution, and the points (i), (ii) and (iii) follows easily. We prove the last point of the proposition, and we suppose that (6.60) E = α ∈ KerLµ ∩ Im(Lµ)2. First, we have E ∈ KerLµ ∩ ImLµ. Thus, E0 = 0 by (6.49), Eα = 0 for α ∈ I1(2λ1) by (6.47), and Eα = 0 for α ∈ I1 \ I1(2λ1) by (6.50). Last, since LµE = 0, we also have Eα = 0 for α ∈ I2 \ I2(λ1), and finally, (6.61) E = α∈I2(λ1) α + C3JxK. Moreover, one can write E = LµG for some G ∈ ImLµ. Since E0 = 0, we must have G0 = 0. Since G ∈ ImLµ, by (6.47), we have Gα = 0 for α ∈ I1(2λ1). Finally, since Eα = 0 for |α| = 1, α /∈ I1(2λ1), the same is true for the corresponding Gα, and (6.62) G = |α|≥2 Then, since Lµx α = 0+C3[x] for α ∈ I2(λ1), we obtain Eα = 0 for α ∈ I2(λ1). As above, we then get that, for |α| ≥ 3, Eα = 0, and this ends the proof. � Corollary 6.8. We always have Mb ≤ 2. If, in addition, λk 6= 2λ1 for all k ∈ {1, . . . , n}, then Mb ≤ 1. Proof. Suppose that Mb ≥ 3. Then (6.42) gives (L− µb)ϕb,Mb = 0(6.63) (L− µb)ϕb,Mb −1 = −Mbϕb,Mb(6.64) (L− µb)ϕb,Mb −2 = −(Mb − 1)ϕb,Mb −1,(6.65) with ϕb,Mb 6= 0. Notice that we have used the fact that Mb − 2 > 0 in (6.65). But this gives ϕb,Mb ∈ Ker(L − µb) and (L − µb)2ϕb,Mb−2 = Mb(Mb − 1)ϕb,Mb , so that ϕb,Mb ∈ Im(L− µb)2. This contradicts point (iv) of Proposition 6.7. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 31 Now we suppose that λk 6= 2λ1 for all k ∈ {1, . . . n}, that is I1(2λ1) = ∅, and that Mb = 2. Then (6.42) gives (L− µb)ϕb,Mb = 0(6.66) (L− µb)ϕb,Mb −1 = −Mb ϕb,Mb(6.67) with ϕb,Mb 6= 0. Therefore we have ϕb,Mb ∈ KerLµb ∩ ImLµb , and we get the same conclusion as in (6.61): ϕb,Mb(x) = O(x2). Then, we write (6.68) ϕb,Mb = (L− µb)g, and we see, as in (6.62), that g = O(x2), here because I1(2λ1) = ∅. Finally, we conclude also that ϕb,Mb = 0, a contradiction. � 6.2. Taylor expansions of ϕ+ and ϕ Now we compute the Taylor expansions of the leading terms with respect to t, of the phase functions ϕ(t, x) = ϕk(t, x). Lemma 6.9. The smooth function ϕ+(x) = x2j +O(x3) satisfies (6.69) ∂αϕ+(0) = − λ · α∂ αV (0), for |α| = 3, and (6.70) ∂αϕ+(0) = − 2(λ · α) β,γ∈I2 α=β+γ β! γ! βV (0) λj + λ · β γV (0) λj + λ · γ λ · α∂ αV (0), for |α| = 4, where α, β, γ ∈ Nn are multi-indices. Proof. The smooth function x 7→ ϕ+(x) is defined in a neighborhood of 0, and it is charac- terized (up to a constant: we have chosen ϕ+(0) = 0) by (6.71) p(x,∇ϕ+(x)) = |∇ϕ+(x)|2 + V (x) = 0 ∇ϕ+(x) = (λjxj)j=1,...,n +O(x The Taylor expansion of ϕ+ at x = 0 is (6.72) ϕ+(x) = x2j + |α|=3,4 ∂αϕ+(0) xα +O(x5), and we have (6.73) ∂jϕ+(x) = λjxj + |α|=3,4 ∂αϕ+(0) xα−1j +O(x4). 32 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Therefore |∇ϕ+(x)|2 = j + 2 |α|=3 )∂αϕ+(0) xα + 2 |α|=4 )∂αϕ+(0) |α|=3 ∂αϕ+(0) xα−1j +O(x5).(6.74) Let us compute further the last term in (6.74): |α|=3 ∂αϕ+(0) xα−1j |β|,|γ|=3 ∂βϕ+(0) ∂γϕ+(0) xβ+γ−21j |α|=4 α=β+γ |β|,|γ|=2 βϕ+(0) γϕ+(0) ·(6.75) Writing the Taylor expansion of V at x = 0 as (6.76) V (x) = x2j + |α|=3,4 ∂αV (0) xα +O(x5), and using the eikonal equation (6.71), we obtain first, for any α ∈ Nn with |α| = 3, (6.77) ∂αϕ+(0) = − λ · α∂ αV (0). Then, (6.74) and (6.75) give (6.78) ∂αϕ+(0) = − λ · α∂ αV (0) − 1 2(λ · α) β,γ∈I2 α=β+γ βV (0) λj + λ · β γV (0) λj + λ · γ for |α| = 4. � Now we pass to the function ϕ1. This function is a solution, in a neighborhood of x = 0, of the transport equation (6.79) Lϕ1(x) = λ1ϕ1(x), where L is given in (6.20). Lemma 6.10. The C∞ function ϕ1(x) = −2λ1g−1 (z ) · x+O(x2) satisfies (6.80) ∂αϕ1(0) = 2λ1α! (λ1 − λ · α)(λ1 + λ · α) αV (0) g−1 (z SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 33 for |α| = 2, and ∂αϕ1(0) =− λ1 − λ · α k∈I1(λ1),j∈I1 β,γ∈I2 α+1j=β+γ βV (0) λj + λ · β γV (0) (λ1 − λ · γ)(λ1 + λ · γ) g−1 (z (λ1 − λ · α)(λ1 + λ · α) k∈I1,j∈I1(λ1) β,γ∈I2 1j+α=β+γ (α+ 1j)! βV (0) λk + λ · β γV (0) λk + λ · γ g−1 (z (λ1 − λ · α)(λ1 + λ · α) j∈I1(λ1) αV (0) g−1 (z .(6.81) for |α| = 3. Proof. We write (6.82) ϕ1(x) = ajxj + |α|=2,3 α +O(x4), and Lemma 6.9 together with (6.73) give all the coefficients in the expansion (6.83) ∇ϕ+(x) = λjxj + |α|=2,3 Aj,αx α +O(x4) j=1,...,n In fact, we have (6.84) Aj,α = ∂α+1jϕ+(0) and aα = ∂αϕ1(0) We get Lϕ1(x) = ∂jϕ+(x)∂jϕ1(x) ajλjxj + |α|=2 αjλjaα + ajAj,α |α|=3 αjλjaαx |β|=|γ|=2 Aj,βγjaγx β+γ−1j + |α|=3 ajAj,αx +O(x4) ajλjxj + |α|=2 λ · α aα + Aj,αaj |α|=3 λ · α aα + α=β+γ−1j |β|,|γ|=2 Aj,βγjaγ + ajAj,α xα +O(x4).(6.85) 34 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Thus, (6.79) gives, for all α ∈ Nn with |α| = 2, (6.86) aα = λ1 − λ · α Aj,αaj , and, for all α ∈ Nn with |α| = 3, (6.87) aα = λ1 − λ · α β,γ∈I2 α+1j=β+γ γjAj,βaγ + ajAj,α Then, the lemma follows from (6.84). � 6.3. Asymptotics near the critical point for the trajectories. The informations obtained so far are not sufficient for the computation of the ϕj ’s. We shall obtain here some more knowledge by studying the behaviour of the incoming trajectory γ−(t) as t → +∞. We recall from [18, Section 3] (see also [5, Section 5]), that the curve γ−(t) = (x−(t), ξ−(t)) ∈ Λ− ∩ Λ−ω satisfy, in the sense of expandible functions, (6.88) γ−(t) = M ′j∑ γ−j,mt me−µjt, Notice that we continue to omit the subscript k for γ−k = (x k , ξ k ), z k , . . . Writing also (6.89) x−(t) ∼ g−j,m(t, z−)e −µj t, g−j (z −, t) = M ′j∑ g−j,m(z −)tm, for some integers M ′j, we know that g −) = g−1,0(z −) 6= 0. Since ξ−(t) = ∂tx−(t), we have (6.90) ξ−(t) ∼ M ′j∑ g−j,m(z −)(−µjtm +mtm−1)e−µjt. Proposition 6.11. If j < ̂, then M ′j = 0. We also have M ≤ 1, and M ′ = 0 when I1(2λ1) 6= ∅. Moreover (6.91) (g− |α|=2 ∂α+βV (0) (g−1 (z −))α for β ∈ I1(2λ1), 0 for β /∈ I1(2λ1). and, for |β| = 1, β /∈ I1(2λ1), (6.92) (g− (2λ1 + λ · β)(2λ1 − λ · β) |α|=2 ∂α+βV (0) (g−1 (z −))α. Proof. First of all, since ∂tγ −(t) = Hp(γ −(t)), we can write (6.93) ∂tγ −(t) = Fp(γ −(t)) +O(t2M ′1e−2λ1t), SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 35 where (6.94) Fp = d(0,0)Hp = , Λ2 = diag(λ21, . . . , λ We obtain (6.95) 1≤j<b M ′j∑ (Fp + µj)γ 1≤j<b M ′j∑ γ−j,mmt m−1e−µj t. Now suppose j < ̂ and M ′j ≥ 1. We get, for this j, for some γ j,M ′j 6= 0, (6.96) (Fp + µj)γ j,M ′j (Fp + µj)γ j,M ′j−1 =M ′jγ j,M ′j so that Ker(Fp + µj) ∩ Im(Fp + µj) 6= {0}. Since Fp is a diagonizable matrix, this can easily be seen to be a contradiction. Now we pass to the study of M ′ . So far we have obtained that (6.97) γ−(t) = 1≤j<b γ−j e −µjt + tme−2λ1t +O(tCe−µb+1t), and we can write (6.98) Hp(x, ξ) = |α|=2 ∂α∇V (0) xα +O(x3)  . Thus we have (6.99) Hp(γ −(t)) = Fp γ−j e −µjt + tme−2λ1t + e−2λ1tA(γ−1 ) +O(e −(2λ1+ε)t), where, noticing that µj + µj′ = 2λ1 if and only if j = j ′ = 1, (6.100) A(γ−1 ) = |α|=2 ∂α∇V (0) (g−1 )  . For the terms of order e−2λ1t, we have, since ∂tγ −(t) = Hp(γ −(t)), (6.101) (Fp + 2λ1) mtm−1 −A(γ−1 ). Thus, if we suppose that M ′ ≥ 2, we obtain (6.102) (Fp + 2λ1)γ b,M ′ (Fp + 2λ1)γ b,M ′ b,M ′ 36 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Then again we have γ− b,M ′ ∈ Ker(Fp + 2λ1) ∩ Im(Fp + 2λ1), a contradiction. Eventually, if λj 6= 2λ1 for all j, then Ker(Fp + 2λ1) = {0}. Therefore, if we suppose that b = 1, we see that γb,1 6= 0 satisfies the first equation in (6.102) and we get a contradiction. Now we compute γ− (t) = γ− t+ γ− . We have (6.103) (Fp + 2λ1)γ (Fp + 2λ1)γ −A(γ−1 ), and we see that γ− = Πγ− = ΠA(γ−1 ), where Π is the projection on the eigenspace of Fp associated to −2λ1. We denote by ej = (δi,j ⊗ 0)i=1,...,n and εj = (0 ⊗ δi,j)i=1,...,n for j = 1, . . . , n, so that (e1, . . . en, ε1, . . . , εn) is the canonical basis of R 2n = T(0,0)T n. Then it is easy to check that , for all j, v±j = ej±λj1εj is an eigenvector of Fp for the eigenvalue ±λj . In the basis {e1, ε1, . . . , en, εn} the projector Π is block diagonal and, if Kj = Vect(ej , εj), we (6.104) Π|Kj 1/2 −1/4λ1 −λ1 1/2 for j ∈ I1(2λ1), 0 for j /∈ I1(2λ1). Therefore, we obtain (6.105) (g− |α|=2 ∂β∂αV (0) (g−1 (z −))α for β ∈ I1(2λ1), 0 for β /∈ I1(2λ1). Now suppose that k /∈ I1(2λ1). Then the second equality in (6.103) restricted to Kk gives (6.106) 2λ1 1 λ2k 2λ1 Πkγb,0 = −ΠkA(γ−1 ), where Πk denotes the projection onto Kk. Solving this system, one gets (6.107) (g− 4λ21 − λ2k ΠxΠkA(γ and, together with (6.100), this ends the proof of Proposition 6.11. � 6.4. Computation of the ϕkj ’s. Here we compute the ϕkj ’s for j ≤ ̂. We still omit the superscript k. From [5], we know that ξ−(t) = ∇xϕ t, x−(t) , so that, using (6.41), ξ−(t) =∇ϕ+(x−(t)) +∇ϕ1(x−(t))e−λ1t + 2≤j<b ∇ϕj(0)e−µj t +∇ϕb,2(0)t2e−2λ1t +∇ϕb,1(0)te−2λ1t +∇ϕb,0(0)e−2λ1t + Õ(e−µb+1t).(6.108) Since ϕ+ = −ϕ− and ξ− ∈ Λ−, we have ∇ϕ+(x−(t)) = −ξ−(t), and we obtain first, by (6.90), (6.109) ∇ϕj(0) = −2µjg−j (z for 1 ≤ j < ̂. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 37 Now we study ϕb(t, x) = ϕb,0(x) + tϕb,1(x) + t 2ϕb,2(x) when I1(2λ1) 6= ∅. It follows from (6.108) that we have (6.110) − 4λ1g−b,1(z −) = ∇ϕb,1(0), − 4λ1g−b,0(z −) + 2g− (z−) = ∇ϕb,0(0) +∇2ϕ1(0)g−1 (z On the other hand, we have seen that, by (6.19), the functions ϕb,2, ϕb,1 and ϕb,0 satisfy (6.111) (L− 2λ1)ϕb,2 = 0, (L− 2λ1)ϕb,1 = −2ϕb,2, (L− 2λ1)ϕb,2 = −ϕb,1 − |∇ϕ1(0)|2. In particular ϕb,2 ∈ Ker(L− 2λ1) ∩ Im(L− 2λ1) so that (see (6.61)), (6.112) ϕb,2(x) = α∈I2(λ1) c2,αx α +O(x3). Going back to (6.108), we notice that we obtain now ξ−(t) =∇ϕ+(x−(t)) +∇ϕ1(x−(t))e−λ1t + 2≤j<b ∇ϕj(0)e−µj t ∇ϕb,1(0)te−2λ1t +∇ϕb,0(0)e−2λ1t + Õ(e−µb+1t),(6.113) and this equality is consistent with Proposition 6.11. Then, (6.49) and (6.50) give (6.114) ϕb,1(x) = α∈I1(2λ1) c1,αx |α|=2 c1,αx α +O(x3), and, by (6.51), we have (6.115) Ψ((c1,β)β∈I1(2λ1)) = (−2c2,α)α∈I2(λ1). By (6.52), we also have for |α| = 2, α /∈ I2(λ1), (6.116) c1,α = 2λ1 − λ · α β∈I1(2λ1) ∂α+βϕ+(0) c1,β . The function ϕb,0(x) = |α|≤2 c0,αx α +O(x3) satisfies (see (6.42)) (6.117) (L− 2λ1)ϕb,0 = −ϕb,1 − ∣∣∇ϕ1(x) First of all, the compatibility condition (6.47) gives (6.118) ∀α ∈ I1(2λ1), c1,α = −∇ϕ1(0) · ∂α∇ϕ1(0), so that in particular, by (6.115), the function ϕb,2 is known up to O(x3) terms: (6.119) ∀α ∈ I2(λ1), c2,α = β∈I1(2λ1) ∂α+βϕ+(0) ∇ϕ1(0) · ∂β∇ϕ1(0), 38 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND (6.120) ∀α /∈ I2(λ1), |α| = 2, c1,α = − 2λ1 − λ · α β∈I1(2λ1) ∂α+βϕ+(0) ∇ϕ1(0) · ∂β∇ϕ1(0). Now (6.49) and (6.50) give (6.121) c0,0 = ϕb,0(0) = |∇ϕ1(0)|2, (6.122) ∀α /∈ I1(2λ1), |α| = 1, c0,α = 2λ1 − λ · α ∇ϕ1(0) · ∂α∇ϕ1(0). From the other compatibility condition (6.48), we know that c1,α + ∇ϕ1(0) · ∂α∇ϕ1(0) + β,γ∈I1(λ1) β+γ=α ∂β∇ϕ1(0) · ∂γ∇ϕ1(0) |β|=1 β/∈I1(2λ1) ∂α+βϕ+(0) ∇ϕ1(0) · ∂β∇ϕ1(0) 2λ1 − λ · β α∈I2(λ1) ∈ ImΨ,(6.123) and, from (6.51), we obtain a relation between the (c0,β)β∈I1(2λ1) and the (c1,α)α∈I2(λ1), namely ∀α ∈ I2(λ1), c1,α =− ∂α∇ϕ1(0) · ∇ϕ1(0)− β,γ∈I1(λ1) β+γ=α ∂β∇ϕ1(0) · ∂γ∇ϕ1(0) β∈I1(2λ1) ∂α+βϕ+(0) c0,β − |β|=1 β/∈I1(2λ1) ∂α+βϕ+(0) ∇ϕ1(0) · ∂β∇ϕ1(0) 2λ1 − λ · β ·(6.124) Using the second equation in (6.110), we obtain, for |β| = 1, (6.125) c0,β = −4λ1(g−b,0(z −))β + 2(g− (z−))β − ∂β∇ϕ1(0) · g−1 (z At this point, we have computed the functions ϕb,1(x) and ϕb,2(x) up to O(x3), in terms of derivatives of ϕ+ and ϕ1, and of the g (z−). We shall now use the expressions we have obtained in Section 6.2 and in Section 6.3 to give these functions in terms of g−1 and of derivatives of V only. First of all, by (6.112), (6.119), Lemma (6.9) and Lemma (6.10), we obtain ϕb,2(x) =− γ∈I1(2λ1) α,β∈I2(λ1) ∂β+γV (0) (g−1 (z ∂α+γV (0) +O(x3).(6.126) Then we have (6.127) ϕb,1(x) = −4λ1g−b,1(z −) · x+ α∈I2(λ1) c1,αx |α|=2 α/∈I2(λ1) c1,αx α +O(x3), SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 39 where the c1,α are given by (6.124) and (6.125) for α ∈ I2(λ1), and by (6.120) for α /∈ I2(λ1). • For |α| = 2, α /∈ I2(λ1), we obtain by (6.116), Lemma 6.9 and Lemma 6.10, c1,α = (2λ1 + λ · α)(2λ1 − λ · α) β∈I1(2λ1) ∂α+βV (0) (λ1 + λj)(3λ1 + λj) β∇V (0) · g−1 (z−)(g −))j .(6.128) Since (g−1 (z −))j = 0 but for j ∈ I1(λ1), we get, changing notations a bit, (6.129) c1,α = (2λ1 + λ · α)(2λ1 − λ · α) γ∈I1(2λ1) β∈I2(λ1) ∂α+γV (0) ∂β+γV (0) (g−1 (z −))β. • Now we compute c1,α for α ∈ I2(λ1). For the last term in the R.H.S. of (6.124), we obtain |β|=1 β /∈I1(2λ1) ∂α+βϕ+(0) ∇ϕ1(0) · ∂β∇ϕ1(0) 2λ1 − λ · β γ∈I1\I1(2λ1) β∈I2(λ1) (2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2 ∂α+γV (0) ∂β+γV (0) (g−1 (z −))β .(6.130) Using (6.91) and (6.125), we have also β∈I1(2λ1) ∂α+βϕ+(0) c0,β = γ∈I1(2λ1) ∂α+γV (0) (z−))γ + γ∈I1(2λ1) β∈I2(λ1) ∂α+γV (0) ∂β+γV (0) (g−1 (z −))β .(6.131) 40 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND We pass to the computation of − 1 ∂α∇ϕ1(0) · ∇ϕ1(0) for α ∈ I2(λ1). We obtain ∂α∇ϕ1(0) · ∇ϕ1(0) = − β∈I2(λ1) ∂α+βV (0) (g−1 (z j,p,k=1 β,γ∈I2 β+γ=α+1p+1j ((α+ 1p)j + 1)(αp + 1) (λk + λ · β)(λk + λ · γ) ∂β+1kV (0) ∂γ+1kV (0) (g−1 (z −))j(g + 2λ1 j,p,k=1 β,γ∈I2 β+γ=α+1p+1j (αp + 1)γj (λ1 − λ · γ)(λ1 + λ · γ)(λj + λ · β) β+1jV (0) ∂γ+1kV (0) (g−1 (z −))k(g = I + II + III. (6.132) Writing δ = 1j + 1p, we get (6.133) II = −1 β,γ,δ∈I2 β+γ=α+δ (α+ δ)! (λk + λ · β)(λk + λ · γ) ∂β+1kV (0) ∂γ+1kV (0) (g−1 (z α! δ! Since δ ∈ I2(λ1) (otherwise (g−1 (z−))δ = 0), we have β, γ ∈ I2(λ1) and, changing notations a (6.134) II = −1 β∈I2(λ1) (α+ β)! γ,δ∈I2(λ1) γ+δ=α+β (2λ1 + λj)2 γV (0) δV (0) (g−1 (z In the last term III, we can suppose that γ = 1j+1q for some q ∈ {1, . . . , n}. Then γj = γ! and, writing β = 1a + 1b we have III = λ1 j,k,p=1 (αp + 1)(g −))k(g a,b,q∈I1 1a+1b+1q=α+1p (αp + 1) (λ1 − λj − λq)(λ1 + λj + λq)(λj + λa + λb) ∂j,a,bV (0)∂j,q,kV (0).(6.135) Since α ∈ I2(λ1) and 1p ∈ I1(λ1) (otherwise (g−1 (z−))p = 0), we have 1a, 1b, 1q ∈ I1(λ1) so that we can write (6.136) III = − j,k,p=1 (αp + 1) λj(2λ1 + λj)2 (g−1 (z −))k(g a,b,q∈I1 1a+1b+1q=α+1p ∂j,a,bV (0)∂j,q,kV (0). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 41 Now it is easy to check, noticing that (α+ 1p)k ∈ {1, 2, 3, 4} and examining each case, that (6.137) a,b,q∈I1 1a+1b+1q=α+1p ∂j,a,bV (0)∂j,q,kV (0) = (α+ 1p)k a,b,c,d∈I1 1a+1b+1c+1d=α+1p+1k ∂j,a,bV (0)∂j,c,dV (0). Therefore, we have III = −1 j,k,p=1 (α + 1p + 1k)! λj(2λ1 + λj)2 (g−1 (z −))k(g a,b,c,d∈I1 1a+1b+1c+1d=α+1p+1k ∂j,a,bV (0)∂j,c,dV (0).(6.138) Eventually, setting β = 1p + 1k, γ = 1a + 1b and δ = 1c + 1d, we get (6.139) III = − β∈I2(λ1) (α+ β)! γ,δ∈I2(λ1) γ+δ=α+β λj(2λ1 + λj)2 γV (0) δV (0) (g−1 (z We are left with the computation of β,γ∈I1(λ1) β+γ=α ∂β∇ϕ1(0) · ∂γ∇ϕ1(0) = − β,γ∈I1(λ1) β+γ=α βϕ1(0) · ∂j∂γϕ1(0) λ2j(2λ1 + λj) β,γ∈I1(λ1) β+γ=α k,ℓ=1 ∂j∂k∂ βV (0)(g−1 (z −))k∂j∂ℓ∂ γV (0)(g−1 (z −))ℓ.(6.140) At this point, we notice that α∈I2(λ1) β,γ∈I1(λ1) β+γ=α ∂β∇ϕ1(0) · ∂γ∇ϕ1(0)xα λ2j (2λ1 + λj) β,γ∈I1(λ1) α∈I2(λ1) β+γ=α k,ℓ=1 ∂j∂k∂ βV (0)(g−1 (z −))k∂j∂ℓ∂ γV (0)(g−1 (z −))ℓ x λ2j (2λ1 + λj) α,β∈I2(λ1) (α+ β)! γ,δ∈I2(λ1) γ+δ=α+β γV (0) δV (0) (g−1 (z α,β∈I2(λ1) αV (0) βV (0) xα(g−1 (z }(6.141) 42 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND From (6.124), (6.130), (6.131) (6.139), and (6.141), we finally obtain that α∈I2(λ1) c1,αx γ∈I1\I1(2λ1) α,β∈I2(λ1) (2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2 ∂α+γV (0) ∂β+γV (0) (g−1 (z −))βxα γ∈I1(2λ1) α∈I2(λ1) ∂α+γV (0) (z−))γxα + γ∈I1(2λ1) α,β∈I2(λ1) ∂α+γV (0) ∂β+γV (0) (g−1 (z −))βxα α,β∈I2(λ1) ∂α+βV (0) (g−1 (z −))βxα α,β∈I2(λ1) (α+ β)! γ,δ∈I2 γ+δ=α+β (2λ1 + λj)2 γV (0) δV (0) (g−1 (z α,β∈I2(λ1) (α+ β)! γ,δ∈I2(λ1) γ+δ=α+β λj(2λ1 + λj)2 γV (0) δV (0) (g−1 (z α,β∈I2(λ1) (α + β)! γ,δ∈I2(λ1) γ+δ=α+β λ2j(2λ1 + λj) γV (0) δV (0) (g−1 (z α,β∈I2(λ1) λ2j(2λ1 + λj) αV (0) βV (0) xα(g−1 (z −))β , (6.142) or, more simply, α∈I2(λ1) c1,αx α = − γ∈I1(2λ1) α∈I2(λ1) ∂α+γV (0) (z−))γxα + α,β∈I2(λ1) (g−1 (z γ∈I1\I1(2λ1) (2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2 ∂α+γV (0)∂β+γV (0) γ∈I1(2λ1) ∂α+γV (0)∂β+γV (0)− ∂α+βV (0) − (α+ β)! γ,δ∈I2 γ+δ=α+β γV (0) δV (0) λ2j (2λ1 + λj) αV (0)∂j∂ βV (0) .(6.143) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 43 7. Computations after the critical point 7.1. Stationary phase expansion in the outgoing region. Now we compute the scattering amplitude starting from (4.19). First of all, we change the cut-off function χ+ so that the support of the right hand side of the scalar product in (4.19) is close to (0, 0). ℓχ+ = 0 supp(∇χ+) χ+ = 1 ℓeχ+ = 0 supp(∇eχ+) eχ+ = 1 Figure 1. The support of χ+ and χ̃+ in T Using Maslov’s theory, we construct a function v+ which coincides with a+(x, h)e iψ+(x)/h out of a small neighborhood of ∩ (B(0, R+ +1)×Rn) and such that v+ is a solution of (P−E)v+ = 0 microlocally near . Let χ̃+(x, ξ) ∈ C∞(T ∗Rn) such that χ̃+(x, ξ) = χ+(x) out of a small enough neighborhood of ∩ (B(0, R++1)×Rn). In particular, (P −E)v+ is microlocally 0 near the support of χ+ − χ̃+. So, we have 〈u−, [χ+, P ]v+〉 =〈u−, [Op(χ̃+), P ]v+〉+ 〈u−, (χ+ −Op(χ̃+))(P − E)v+〉 − 〈(P − E)u−, (χ+ −Op(χ̃+))v+〉 =〈u−, [Op(χ̃+), P ]v+〉+O(h∞)− 〈g−eiψ−/h, (χ+ −Op(χ̃+))v+〉 =〈u−, [Op(χ̃+), P ]v+〉+O(h∞),(7.1) since the microsupport of g−e iψ−/h and χ+− χ̃+ are disjoint. Thus, the scattering amplitude is given by (7.2) A(ω, θ,E, h) = c(E)h−(n+1)/2〈u−, [Op(χ̃+), P ]v+〉+O(h∞). Now we will prove that, modulo O(h∞), the only contribution to the scattering amplitude in (7.2) comes from the values of the functions u− and v+ microlocally on the trajectories γ and γ∞j . From (5.18), the fact that u− = O(h−C) and (P −E)u− = 0 microlocally out of the microsupport of g−e −iψ−/h, and the usual propagation of singularities theorem, we get (7.3) MS(u−) ⊂ Λ−ω ∪ Λ+. Moreover, we have (7.4) MS(v+) ⊂ Λ+θ . 44 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Now, let f∞j (resp. f ℓ ) be C n) functions with support in a small enough neighborhood of γ∞j (resp. γ ℓ ∩MS(v+)) such that f∞j = 1 (resp. f k = 1) in a neighborhood of γ j (resp. γ+ℓ ∩MS(v+)). In particular, we assume that all these functions have disjoint support. Since u− and v+ have disjoint microsupport out of the support of the f j and the f ℓ , we have A(ω, θ,E, h) =c(E)h−(n+1)/2 〈Op(f∞j )u−,Op(f∞j )[Op(χ̃+), P ]v+〉 + c(E)h−(n+1)/2 〈Op(f+ℓ )u−,Op(f ℓ )[Op(χ̃+), P ]v+〉+O(h =Areg +Asing.(7.5) Concerning the terms which contain f∞j , Areg, we are exactly in the same setting as in [30, Section 4]. The computation there gives (7.6) Areg = j,m(ω, θ,E)h iS∞j /h +O(h∞). Now we compute Asing. Proceeding as in Section 5.2 for u−, one can show that v+ can be written as (7.7) v+(x) = a+(x, h)e π/2eiψ+(x)/h, microlocally near any ρ ∈ γ+ close enough to (0, 0). Here ν+ is the Maslov index of γ+ . The phase ψ+ and the classical symbol a+ satisfy the usual eikonal and transport equations. In particular, as in (5.28) and (5.33), we have (7.8) ℓ (t)) = − |ξ+ℓ (u)| 2 − 2E01u>0 du = − |ξ+ℓ (u)| 2 − V (x+ℓ (u)) −E0 sgn(u) du, and a+(x, h) ∼ m a+,m(x)h m with (7.9) a+,0(x ℓ (t)) = (2E0) 1/4(D+ℓ (t)) −1/2eitz, where (7.10) D+ℓ (t) = ∣∣ det ∂x+(t, z, θ, E0) ∂(t, z) We can chose χ̃+ so that the microsupport of the symbol of Op(f ℓ )[Op(χ̃+), P ] is contained in a vicinity of such a point ρ ∈ γ+ℓ (see Figure 1). Then, microlocally near ρ, we have (7.11) Op(f+ )[Op(χ̃+), P ]v+ = ã+(x, h)e π/2eiψ+(x)/h, (7.12) ã+(x, h) = ã+,m(x)h (7.13) ã+,0(x) = −i{χ̃+, p}(x,∇ψ+(x))a+,0(x). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 45 From [5, Section 5], the Lagrangian manifold {(x,∇xϕk(t, x)); ∂tϕk(t, x) = 0}, coincides with Λ−ω . In particular, since MS(v+) ⊂ Λ+θ and since there is no curve γ∞(z∞j ) sufficiently closed to the critical point, the finite times in (6.5) give a contribution O(h∞) to the scattering amplitude (4.19). In view of the equations (6.5), (6.12) and (7.11), the principal contribution of Asing will come from the intersection of the manifolds Λ+θ and Λ+. Recall that, from (A5), the manifolds Λ+θ and Λ+ intersect transversally along γ In particular, to compute Asing, we can apply the method of stationary phase in the directions that are transverse to γ+ . For each ℓ, after a linear and orthonormal change of variables, we can assume that g+ ) is collinear to the xℓℓℓ–direction, and that V (x) satisfies (A2). We denote Hℓxℓℓℓ = {y = (y1, . . . , yn) ∈ R n; yℓℓℓ = xℓℓℓ} the hyperplane orthogonal to (0, . . . , 0, xℓℓℓ, 0, . . . , 0). We shall compute Asing in the case where there is only one incoming curve γ− and one outgoing curve γ+ . In the general case, Asing is simply given by the sum over k and ℓ of such contributions. Using (4.19), (6.5) and (7.11), we can write Asing =c(E)h −(n+1)/2 k(t,x)−ψ+(x))/hαk(t, x, h)ã+(x, h)e π/2dt dx c(E)h−(n+1)/2√ y∈Hxℓℓℓ k(t,x)−ψ+(x))/hαk(t, x, h)ã+(x, h)e π/2dt dy dxℓℓℓ.(7.14) Let Φ(y) = ϕk(t, xℓℓℓ, y) − ψ+(xℓℓℓ, y) be the phase function in (7.14). From (6.10)–(6.13), we can write (7.15) Φ(y) = S−k + (ϕ+ − ψ+)(xℓℓℓ, y) + ψ̃(t, xℓℓℓ, y), where ψ̃ = O(e−λ1t) is an expandible function. Since the manifolds Λ+ and Λ+ intersect transversally along γ+ , the phase function y → (ϕ+−ψ+)(xℓℓℓ, y) has a non degenerate critical point yℓ(xℓℓℓ) ∈ Hℓxℓℓℓ ∩ Πxγ , and xℓℓℓ 7→ yℓ(xℓℓℓ) is C∞ for xℓℓℓ 6= 0. Then, from the implicit function theorem, the function Φ has a unique critical point yℓ(t, xℓℓℓ) ∈ Hℓxℓℓℓ for t large enough depending on xℓℓℓ. The function (t, xℓℓℓ) 7→ yℓ(t, xℓℓℓ) is expandible and we have (7.16) yℓ(t, xℓℓℓ) = y ℓ(xℓℓℓ)−Hess(ϕ+ − ψ+)−1 yℓ(xℓℓℓ) yℓ(xℓℓℓ) e−µ1t + Õ e−µ2t As a consequence, Φ yℓ(t, xℓℓℓ) is also expandible. Since ϕ+ and ψ+ satisfy the same eikonal equation, we get (see (5.25)) (7.17) ∂t(ϕ+ − ψ+)(x+ℓ (t)) = |ξ (t)|2 − |ξ+ (t)|2 = 0. 46 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Thus, (ϕ+ − ψ+)(yℓ(xℓℓℓ)) does not depend of xℓℓℓ and is equal to (ϕ+ − ψ+)(yℓ(xℓℓℓ)) = lim (ϕ+ − ψ+)(x+ℓ (t)) |ξ+ℓ (s)| 2 − 2E01s>0 ds (s)|2 − V (x+ (s))− E0 sgn(s) ds ,(7.18) where we have used (7.8). Therefore, the phase function Φ at the critical point yℓ(t, xℓℓℓ) is equal to yℓ(t, xℓℓℓ) µm≤2λ1 t, yℓ(xℓℓℓ) e−µmt Hess(ϕ+ − ψ+)−1 yℓ(xℓℓℓ) yℓ(xℓℓℓ) · ∇ϕ1 yℓ(xℓℓℓ) e−2µ1t + Õ(e−eµt),(7.19) where µ̃ is the first of the µj’s such that µj > 2λ1. Using the method of the stationary phase for the integration with respect to y ∈ Hℓxℓℓℓ in (7.14), we get (7.20) Asing = c(E)h −(n+1)/2 (2πh)(n−1)/2 eiΦ(y ℓ(t,xℓℓℓ))/hf ℓ(t, xℓℓℓ, h) dt dxℓℓℓ +O(h∞). TheO(h∞) term follows from the fact that the error term stemming from the stationary phase method can be integrated with respect to time t, since αk ∈ S0,2ReΣ(E), with ReΣ(E) > 0 (see the beginning of Section 6). The symbol f ℓ(t, xℓℓℓ, h) is a classical expandible function of order S1,2ReΣ(E) in the sense of Definition 6.2: (7.21) f ℓ(t, xℓℓℓ, h) ∼ f ℓm(t, xℓℓℓ, lnh)h where the f ℓm are polynomials with respect to lnh and (7.22) f ℓ0(t, xℓℓℓ, lnh) = α t, yℓ(t, xℓℓℓ) ã+,0 yℓ(t, xℓℓℓ) π/2 e i sgnΦ′′ |Hℓxℓℓℓ (yℓ(t,xℓℓℓ))π/4 ∣∣detΦ′′|Hℓxℓℓℓ yℓ(t, xℓℓℓ) )∣∣1/2 Using Proposition C.1, we compute the Hessian of Φ, and we get yℓ(xℓℓℓ) =diag(−λ1, . . . ,−λℓℓℓ−1, λℓℓℓ,−λℓℓℓ+1, . . . ,−λn) + o(1), yℓ(xℓℓℓ) =diag(λ1, . . . , λn) + o(1). Then, for xℓℓℓ small enough and t large enough depending on xℓℓℓ, we have ∣∣detΦ′′|Hℓxℓℓℓ yℓ(t, xℓℓℓ) )∣∣1/2 = j 6=ℓℓℓ 2λj + o(1),(7.23) sgnΦ′′|Hℓxℓℓℓ yℓ(t, xℓℓℓ) = n− 1,(7.24) as xℓℓℓ goes to 0. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 47 7.2. Behaviour of the phase function Φ. Suppose that j ∈ N is such that j < ̂. From (6.40), we have (7.25) ϕkj (x (s0)) = e −µj(s−s0)ϕkj (x (s)). Combining (6.41) with (6.109), we get ϕkj (x ℓ (s0)) =e µjs0e−µjs − 2µj〈g−j (z k )|g ℓ 〉)e µjs +O(e2λ1s) =− 2µj g−j (z ∣∣g+j (z eµjs0 .(7.26) We suppose first that we are in the case (a) of assumption (A7). Then, (7.19) becomes (7.27) Φ yℓ(t, xℓℓℓ) − 2µk eµks(xℓℓℓ)e−µkt + Õ(e−µk+1t). Here s(xℓℓℓ) is such that x (s(xℓℓℓ)) = x ℓ(xℓℓℓ) and the Õ(e−µk+1t) is in fact expandible, uniformly with respect to xℓℓℓ when xℓℓℓ varies in a compact set avoiding 0. Suppose now that we are in the case (b) of assumption (A7). Of course, from (7.26), we have ϕj yℓ(xℓℓℓ) = 0 for all j < ̂. On the other hand, Corollary 6.8 and (6.111) imply (7.28) ϕk b,2(x ℓ (s0)) = e −2λ1(s−s0)ϕk b,2(x ℓ (s)). Combining this with (6.126), we get b,2(x (s0)) =e 2λ1s0e−2λ1s j∈I1(2λ1) α,β∈I2(λ1) ∂α+1jV (0) ∂β+1jV (0) g−1 (z g+1 (z e2λ1s +O(e3λ1s) j∈I1(2λ1) α,β∈I2(λ1) ∂α+1jV (0) ∂β+1jV (0) g−1 (z g+1 (z e2λ1s0 .(7.29) In particular, (7.19) becomes, in that case, yℓ(t, xℓℓℓ) =S−k − S j∈I1(2λ1) α,β∈I2(λ1) ∂α+1jV (0) ∂β+1jV (0) g−1 (z g+1 (z e2λ1s(xℓℓℓ) × t2e−2λ1t +O(te−2λ1t) =S−k + S ℓ +M2(k, ℓ)t 2e−2λ1t +O(te−2λ1t).(7.30) As in (7.27), the term O(te−2λ1t) is in fact expandible uniformly with respect to xℓℓℓ when xℓℓℓ varies in a compact set avoiding 0. Eventually, we suppose that we are in the case (c) of assumption (A7). Then we obtain from (7.26) and (7.29) that ϕj yℓ(xℓℓℓ) = 0 for all j < ̂ and ϕb,2 yℓ(xℓℓℓ) = 0. With the last identity in mind, Equation (6.111) on ϕk implies (7.31) ϕk b,1(x ℓ (s0)) = e −2λ1(s−s0)ϕk b,1(x ℓ (s)). 48 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND In order to compute ϕk b,1(x ℓ (s)), we put the expansion (2.15) for x ℓ (s) (with Proposition 6.11 in mind) into (6.127). The third term in (6.127) will be, at least, O(e(µ2+µ1)s) = o(e2λ1s). Thank to (6.91) and thanks to the fact that M2(k, ℓ) = 0, the first term in (6.127) will give no contribution of order se2λ1s and will be of the form (7.32) − 4λ1g−b,1(z −) · x+ℓ (s) = − α∈I2(λ1) αV (0) (g−1 (z −))α(g+ (z+))je 2λ1s + Õ(eµb+1s) It remains to study the contribution the second term in (6.127), as given in (6.143). As previously, the first term of the third line in (6.143) will give a term of order o(e2λ1s). The other terms will contribute to the order e2λ1s for α∈I2(λ1) αV (0) (z−))j(g +))α + α,β∈I2(λ1) (g−1 (z (g+1 (z − ∂α+βV (0) + j∈I1\I1(2λ1) λ2j (4λ 1 − λ2j ) ∂α+γV (0)∂β+γV (0) γ,δ∈I2(λ1) γ+δ=α+β (γ + δ)! γ! δ! γV (0)∂j∂ δV (0) .(7.33) Thus, combining (7.32) and (7.33), the discussion above leads to b,1(x ℓ (s0)) =e 2λ1s0e−2λ1s M1(k, ℓ)e2λ1s + o(e2λ1s) =M1(k, ℓ)e2λ1s0 .(7.34) In particular, (7.19) becomes, in that case, yℓ(t, xℓℓℓ) =S−k + S ℓ +M1(k, ℓ)e 2λ1s(xℓℓℓ)te−2λ1t +O(e−2λ1t).(7.35) As above, the O(e−2λ1t) is expandible uniformly with respect to the variable xℓℓℓ when xℓℓℓ varies in a compact set avoiding 0. 7.3. Integration with respect to time. Now we perform the integration with respect to time t in (7.20). We follow the ideas of [18, Section 5] and [5, Section 6]. Since yℓ(t, xℓℓℓ) is expandible (see (7.16)), and since Φ is C outside of xℓℓℓ = 0, the symbol f ℓ(t, xℓℓℓ, h) is expandible. We compute only the contribution of the principal symbol (with respect to h) of f ℓ, since the other terms can be treated the same way, and the remainder term will give a contribution O(h∞) to the scattering amplitude. In other word, we compute (7.36) Asing0 = c(E)h−(n+1)/2√ (2πh)(n−1)/2h eiΦ(y ℓ(t,xℓℓℓ))/hf ℓ0(t, xℓℓℓ) dt dxℓℓℓ +O(h∞). SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 49 First, we assume that we are in the case (a) of the assumption (A7). In that case, Ψ is given by (7.27). For xℓℓℓ fixed in a compact set outside from 0, we set yℓ(t, xℓℓℓ) − (S−k + S =− 2µk eµks(xℓℓℓ)e−µkt +R(t, xℓℓℓ),(7.37) and we perform the change of variable t→ τ in (7.36), and we assume for a moment that (7.38) Here R(t, xℓℓℓ) = Õ(e−µk+1t) is expandible. As in [18, Section 5] and [5, Section 6], we get e−t ∼ − 2µk (z−k ) (z+ℓ ) eµks(xℓℓℓ) )−1/µkτ1/µk τ bµj/µkbj(− ln τ, xℓℓℓ) (7.39) t ∼− 1 ln τ + − 2µk (z−k ) (z+ℓ ) eµks(xℓℓℓ) τ bµj/µkbj(− ln τ, xℓℓℓ)(7.40) τ bµj/µkbj(− ln τ, xℓℓℓ),(7.41) where the bj ’s change from line to line. These expansions are valid in the following sense: Definition 7.1. Let f(τ, y) be defined on ]0, ε[×U where U ⊂ Rm. We say that f = Ô(g(τ)) (resp. f = ô(g(τ))), where g(τ) is a non-negative function defined in ]0, ε[ if and only if for all α ∈ N and β ∈ Nm, (7.42) (τ∂τ ) α∂βy f(τ, y) = O(g(τ)), (resp. o(g(τ))) for all (τ, y) ∈]0, ε[×U . Thus, an expression like f ∼ j=1 τ bµj/µkfj(− ln τ, xℓℓℓ), where fj(− ln τ, xℓℓℓ) is a polynomial with respect to ln τ , like in (7.39)–(7.41), means that, for all J ∈ N, (7.43) f(τ, x)− τ bµj/µkfj(− ln τ, xℓℓℓ) = Ô(τ bµJ/µk). We shall say that such symbols f are called expandible near 0. Since f ℓ0(t, xℓℓℓ, h) is expandible (see Definition 6.1) with respect to t, this symbol is also expandible near 0 with respect to τ in the previous sense. In particular, we get (7.44) f̃ ℓ0(τ, xℓℓℓ) = −f ℓ0(t, xℓℓℓ)τ τ (Σ(E)+cµj)/µk f̃ ℓ0,j(− ln τ, xℓℓℓ), where the f̃ ℓ0,j’s are polynomials with respect to ln τ . The principal symbol f̃ 0,0 is independent on ln τ and we have (7.45) f̃ ℓ0,0(xℓℓℓ) = − 2µk eµks(xℓℓℓ) )−Σ(E)/µkf ℓ0,0(xℓℓℓ). In that case, (7.36) becomes (7.46) Asing0 = c(E)h−1/2 (2π)1−n/2 ∫∫ +∞ eiτ/hf̃ ℓ0(τ, xℓℓℓ) dxℓℓℓ +O(h∞). 50 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Note that f̃ ℓ0(τ, xℓℓℓ) has in fact a compact support with respect to τ . Now, using Lemma D.1, we can perform the integration with respect to t of each term in the right hand side of (7.44), modulo a term O(h∞) (see (D.3)–(D.4) in Lemma D.1). Then, we get (7.47) Asing0 = c(E)h−1/2 (2π)1−n/2 f̂j(ln h)h (Σ(E)+bµj )/µk , where f̂j(lnh) is a polynomial in respect to lnh. f̂0 does not depend on h and we have (7.48) f̂0 = Γ(Σ(E)/µk)(−i)−Σ(E)/µk f̃ ℓ0,0(xℓℓℓ) dxℓℓℓ. To finish the proof, it remains to perform the integration with respect to xℓℓℓ in (7.48). From (7.22) and (7.45), it becomes f̂0 =Γ(Σ(E)/µk) (z−k ) (z+ℓ ) eµks(xℓℓℓ) )−Σ(E)/µk × α0,0(yℓ xℓℓℓ) ã+,0 yℓ(xℓℓℓ) π/2 e i sgnΦ′′ |Hℓxℓℓℓ (yℓ(xℓℓℓ))π/4 ∣∣ detΦ′′|Hℓxℓℓℓ yℓ(xℓℓℓ) )∣∣1/2 dxℓℓℓ.(7.49) Now we make the change of variable xℓℓℓ 7→ s given by yℓ(xℓℓℓ) = x+ℓ (s) (then s(xℓℓℓ) = s). In particular, (7.50) dxℓℓℓ = ∂s(x ℓ,ℓℓℓ(s))ds = λℓℓℓ|g ℓℓℓ (z ℓ )|e λℓℓℓs(1 + o(1))ds, as s→ −∞. In this setting, we get (7.51) α0,0(x (s)) = α0,0(0)(1 + o(1)), as s→ −∞, where α0,0(0) is given in (6.8). We also have, from (7.9) and (7.13), (7.52) ã+,0(x ℓ (s)) = −i∂s χ̃+(γ ℓ (s)) (2E0) 1/4(D+ℓ ) −1/2eisz. Then, putting (7.23), (7.24), (7.50), (7.51) and (7.52) in (7.49), we obtain f̂0 =Γ(Σ(E)/µk) (z−k ) (z+ℓ ) 〉)−Σ(E)/µkα0,0(0)∂s χ̃+(γ ℓ (s)) i(n−1)π/4 j 6=ℓℓℓ λℓℓℓ|g+ℓℓℓ (z )|(2E0)1/4(D+ℓ ) −1/2eisze−Σ(E)seλℓℓℓs(1 + o(1)) ds i(n+1)π/4 j 6=ℓℓℓ )−1/2 λℓℓℓ|g+ℓℓℓ (z )|Γ(Σ(E)/µk) 〉)−Σ(E)/µk × e−iν π/2α0,0(0)(2E0) 1/4(D+ℓ ) χ̃+(γ ℓ (s)) (1 + o(1)) ds.(7.53) Here the o(1) does not depend on χ̃+. Now, we choose a family of cut-off functions (χ̃ +)j∈N such that the support of ∂t goes to −∞ as j → +∞. We also assume that SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 51 ℓ (t)) is non negative (see Figure 1). Then f̂0 =− ei(n+1)π/4 j 6=ℓℓℓ )−1/2 λℓℓℓΓ(Σ(E)/µk)e π/2eiπ/4(2λ1) 3/2e−iν × |g−1 (z )| |g+ 〉)−Σ(E)/µk(7.54) (2E0) 1/2(D− )−1/2 × (1 + o(1)).(7.55) as j → +∞. Since f̂0 is also independent of χ̃+, we obtain Theorem 2.6 from (7.47) and (7.48), in the case (a) and under the assumption (7.38). When 〈g− (z−k )|g (z+ℓ )〉 > 0, we set τ as the opposite of the R.H.S. of (7.37), and we obtain the result along the same lines (see Remark D.2). Now we assume that we are in the case (b) of the assumption (A7). In that case, the phase function Ψ is given by (7.30). For xℓℓℓ fixed in a compact set outside from 0, we set, mimicking (7.37), yℓ(t, xℓℓℓ) − (S−k + S =M2(k, ℓ)e2λ1s(xℓℓℓ)t2e−2λ1t +R(t, xℓℓℓ)(7.56) where R(t, xℓℓℓ) = O(te−2λ1t) is expandible with respect to t. As above, we assume that M2(k, ℓ) is positive (the other case can be studied the same way). Following (7.39), we want to write s := e−t as a function of τ . Since t 7→ τ(t) is expandible with respect to t, we have (7.57) τ = M2(k, ℓ)e2λ1s(xℓℓℓ)(ln s)2s2λ1(1 + r(s, xℓℓℓ)), where r(s, xℓℓℓ) = ô(1). In particular, ∂sτ > 0 for s positive small enough and then, for ε > 0 small enough, s 7→ τ(s) is invertible for 0 < s < ε. We denote s(τ) the inverse of this function. We look for s(τ) of the form (7.58) s(τ) = (2λ1) M2(k, ℓ)e2λ1s(xℓℓℓ) )1/2λ1 u(τ, xℓℓℓ) (− ln τ)1/λ1 where u(τ, xℓℓℓ) has to be determined. Using (7.57), the equation on u is τ =M2(k, ℓ)e2λ1s(xℓℓℓ)(ln s)2s2λ1(1 + r(s, xℓℓℓ)) =τu2λ1 (2λ1) −2M2(k, ℓ)e2λ1s(xℓℓℓ) + 2λ1 − 2ln(− ln τ) 1 + r (2λ1) M2(k, ℓ)e2λ1s(xℓℓℓ) )1/2λ1 u (− ln τ)1/λ1 , xℓℓℓ =τF (τ, u, xℓℓℓ),(7.59) where F = u2λ1(1 + r̃(τ, u, xℓℓℓ)) and r̃ = ô(1) for u close to 1 (here (u, xℓℓℓ) are the variables y in Definition 7.1). In other word, to find u, we have to solve F (t, u, xℓℓℓ) = 1. First we remark that u 7→ F (τ, u, xℓℓℓ) is real-valued and continuous. Since, for δ > 0 and τ small enough, F (τ, 1 − δ, xℓℓℓ) < 1 < (τ, 1 + δ, xℓℓℓ), there exists u ∈ [1 − δ, 1 + δ] such that F (τ, 1 + δ, xℓℓℓ) = 1. Thank to the discussion before (7.58), the function s(τ) is of the form (7.58) with u(τ, xℓℓℓ) ∈ [1− δ, 1 + δ], for τ small enough. 52 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND For τ > 0, the function F is C∞ and, since r̃ = ô(1), we have (7.60) ∂u F (τ, u, xℓℓℓ)− 1 (u(τ, xℓℓℓ)) = 2λ1u 2λ1−1(1 + oτ (1)) > λ1, for τ small enough. The notation oτ (1) means a term which goes to 0 as τ goes to 0. Here we have used the fact that u(τ, xℓℓℓ) is close to 1. In particular, the implicit function theorem implies that u(τ, xℓℓℓ) is C We write u = 1 + v(τ, xℓℓℓ) and we known that v ∈ C∞ and v = oτ (1). Differentiating the equality (7.61) 1 = F (τ, u(τ, xℓℓℓ), xℓℓℓ) = u(τ, xℓℓℓ) )2λ1( 1 + r̃(τ, u(τ, xℓℓℓ), xℓℓℓ) one can show that v = ô(1). Thus we have e−t =s(τ) = (2λ1) M2(k, ℓ)e2λ1s(xℓℓℓ) )1/2λ1 1 + r̂(τ, xℓℓℓ) (− ln τ)1/λ1 ,(7.62) t =− ln τ (1 + r̂(τ, xℓℓℓ)),(7.63) + r̂(τ, xℓℓℓ),(7.64) where r̂(τ, xℓℓℓ) = ô(1) change from line to line. Since f ℓ0(t, xℓℓℓ, h) is expandible with respect to t, we get, from (7.62)–(7.64), (7.65) f̃ ℓ0(τ, xℓℓℓ) = −f ℓ0(t, xℓℓℓ)τ = τΣ(E)/2λ1(− ln τ)−Σ(E)/λ1 f̃ ℓ0,0(xℓℓℓ) + r̂(τ, xℓℓℓ) where r̂ = ô(1) and (7.66) f̃ ℓ0,0(xℓℓℓ) = (2λ1) Σ(E)/λ1−1 M2(k, ℓ)e2λ1s(xℓℓℓ) )−Σ(E)/2λ1 f ℓ0,0(xℓℓℓ). In that case, (7.36) becomes (7.67) Asing0 = c(E)h−1/2 (2π)1−n/2 ∫∫ +∞ eiτ/hf̃ ℓ0(τ, xℓℓℓ) dxℓℓℓ +O(h∞). Note that f̃ ℓ0(τ, xℓℓℓ) has in fact a compact support with respect to τ . Now, using Lemma D.1, we can perform the integration with respect to t in (7.67), modulo an error term given by (D.3)–(D.4) in Lemma D.1. Then, we get Asing0 = c(E)h−1/2 (2π)1−n/2 )/hΓ(Σ(E)/2λ1)(−i)−Σ(E)/2λ1 × hΣ(E)/2λ1(− lnh)−Σ(E)/λ1 f̃ ℓ0,0(xℓℓℓ) dxℓℓℓ + o(1) ,(7.68) as h goes to 0. The rest of the proof follows that of (7.55). At last, the proof of Theorem 2.6 in the case (c) can be obtained along the same lines, and we omit it. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 53 Appendix A. Proof of Proposition 2.5 We prove that Λ+θ ∩Λ+ 6= ∅. From the assumption (A2), the Lagrangian manifold Λ+ can be described, near (0, 0) ∈ T ∗(Rn), as (A.1) Λ+ = {(x, ξ); x = ∇ϕ̃+(ξ)}, for |ξ| < 2ε, with ε > 0 small enough. For η ∈ Sn−1, let (x(t, η), ξ(t, η)) be the bicharacteristic curve with initial condition (ϕ̃(εη), εη). We have (A.2) Λ+ = {(x(t, η), ξ(t, η)); t ∈ R, η ∈ Sn−1} ∪ {(0, 0)}. The function ξ(t, η) is continuous on R× Sn−1. From the classical scattering theory (see [13, Section 1.3]), we know that this function ξ(t, η) converges uniformly to (A.3) ξ(∞, η) := lim ξ(t, η), as t→ +∞ and ξ(∞, η) ∈ 2E Sn−1. Then, the function (A.4) F (t, η) = 1−t , η) |ξ( t 1−t , η)| is well defined for 0 ≤ t ≤ 1 with the convention F (1, η) = ξ(∞, η)/ 2E. Here we used that |ξ(t, η)| 6= 0 for each t ∈ [0,+∞], η ∈ Sn−1. The previous properties of ξ(t, η) imply the continuity of F (t, η) on [0, 1] × Sn−1. From (A.2), to prove that Λ+θ ∩ Λ+ 6= ∅ for all θ ∈ Sn−1, it is enough (equivalent) to show the surjectivity of η → F (1, η). But if η → F (1, η) is not onto, then ImF (1, ·) ⊂ Sn−1 \ {a point}. And since Sn−1 \ {a point} is a contractible space, F (1, ·) is homotopic to a constant (A.5) f : Sn−1 → Sn−1. On the other hand, F : [0, 1]× Sn−1 −→ Sn−1 gives a homotopy between F (0, ·) = IdSn−1 and F (1, ·). In particular, we have (A.6) 1 = deg(F (0, ·)) = deg(F (1, ·)) = deg(f(·)) = 0, which is impossible (see [16, Section 23] for more details). Appendix B. A lower bound for the resolvent Let χ ∈ C∞(]0,+∞[) be a non-decreasing function such that (B.1) χ(x) = x for 0 < x < 1 2 for 2 < x, Let also ϕ ∈ C∞0 (R) an even function such that 0 ≤ ϕ ≤ 1, 1[−1,1] ≺ ϕ, and suppϕ ⊂ [−2, 2]. We set (B.2) u(x) = j/2hϕ |xj |1/2 uj(x), 54 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND where 0 < α < 2β will be fixed later on. The uj ’s are of course C ∞ functions, and we have (B.3) (P − E0)u = − ∆u(x)− x2ju(x) +O(x3u(x)). Lemma B.1. For any h small enough, we have hβn| lnh|n/2 . ‖u‖L2(Rn) . hβn| lnh|n/2,(B.4) ∥∥|x|3u(x) L2(Rn) . h3αhβn| lnh|n/2.(B.5) Proof. First of all, the second estimate follow easily from the first one: we have ∥∥|x|3u(x) ∥∥2 = |x|6|u(x)|2dx . h6α‖u‖2, since u vanishes if |x| > 2hα. Thanks to the fact that u is a product of n functions of one variable, it is enough to estimate dt = 2 ∫ 2hα We have dt ≤ I ≤ 2 ∫ 2hα dt+ 2 so that dt ≤ I ≤ 2 ∫ 2hα dt+ 2 4 dt. The first estimate follows from the fact that 2β − α > 0, once we have noticed that ∫ Ahα dt = h2β (2β − α)| ln h|+ α lnA On the other hand, we have ∆u(x)− x2ju(x) = j 6=k uj(xj) u′′k(xk)− x2kuk(xk) From Lemma B.1, we get ∥∥(P − E0)u ∥∥ .hβ(n−1)| lnh|(n−1)/2 sup 1≤k≤n ∥∥h2u′′k(t) + λ 2uk(t) ∥∥+ h3αhβn| lnh|n/2 h−β | lnh|−1/2 sup 1≤k≤n ∥∥h2u′′k(t) + λ 2uk(t) ∥∥+ h3α ‖u‖.(B.6) We also have (B.7) h2u′′k(t) + λkt 2uk(t) = e h2v′′h(t) + ihλk(2t∂t + 1)vh(t) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 55 where we have set vh(t) = ϕ |t|1/2 . Notice that the right hand side of (B.7) is an even function, so that we only have to consider t > 0. The point here, is that we have, for t > 0, (B.8) (2t∂t + 1) = − h   2 if 0 < t < O(1) if h < t < h2β , 0 if h2β < t. Therefore, we obtain ∥∥(2t∂t + 1)vh ∥∥2 =2 ∫ 2hα (2t∂t + 1) |t|1/2 ∫ 2hα |t|1/2 ∫ h2β ∫ 2hα |t|1/2 dt . h2β .(B.9) On the other hand, an easy computation gives, still for t > 0, v′′h(t) =h −2αϕ′′ 4t5/2 .(B.10) Computing the L2–norm of each of these terms as in Lemma B.1 and (B.9), we obtain (B.11) ‖h2v′′h‖ . h2+β−2α + h2+β−2α + h2−3β + h2−3β , and, eventually, from (B.6), (B.7), (B.9) and (B.11), ∥∥(P − E0)u h−β | lnh|−1/2 h1+β + h2+β−2α + h2−3β + h3α Therefore we obtain Proposition 2.2 if we can find α > 0 and β > 0 such that 2− 2α > 1, 2− 4β > 1, 3α > 1 and 2β > α, and one can check that α = 5/12 and β = 11/48 satisfies these four inequalities. Appendix C. Lagrangian manifolds which are transverse to Λ± Let Λ ⊂ p−1(E0) be a Lagrangian manifold such that Λ ∩ Λ− is transverse along a Hamil- tonian curve γ(t) = (x(t), ξ(t)). Then, where exists a 6= 0 and ν ∈ {1, . . . , n} such that (C.1) γ(t) = (a+O(e−εt))e−λν t, as t→ +∞. The vector a is an eigenvector of (C.2) V ′′(0) 0 for the eigenvalue λν . Thus, up to a linear change of variable in R n, we can always assume that Πxa is collinear to the xν–direction. The goal of this section is to prove the following geometric result. 56 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Proposition C.1. For t large enough, Λ projects nicely on Rnx near γ(t). In particular, there exists ψ ∈ C∞(Rn) defined near Πxγ, unique up to a constant, such that Λ = Λψ := {(x,∇ψ(x)); x ∈ Rn}. Moreover, we have (C.3) ψ′′(x(t)) =  . . . . . .  +O(e−εt), as t→ +∞. Remark C.2. The same result hold in the outgoing region: If γ = Λ ∩ Λ+ is transverse, Λ projects nicely on Rnx near γ(t), t → −∞. Then Λ = Λψ for some function ψ satisfying ψ′′(x(t)) = diag(−λ1, . . . ,−λν−1, λν ,−λν+1, . . . ,−λn) +O(eεt). Proof. We follow the proof of [18, Lemma 2.1]. There exist symplectic local coordinates (y, η) centered at (0, 0) such that Λ− (resp. Λ+) is given by y = 0 (resp. η = 0) and (ξj + λjxj) +O((x, ξ)2),(C.4) (ξj − λjxj) +O((x, ξ)2).(C.5) Then, p(x, ξ) = A(y, η)y · η with A0 := A(0, 0) = diag(λ1, . . . , λn). (C.6) A0 +O(e−λ1t) 0 O(e−λ1t) A0 +O(e−λ1t) We denote by U(t, s) the linear operator such that U(t, s)δ solves (C.6) with U(s, s) = Id. Since Λ∩Λ− = γ is transverse, there exists En−1(t0) ⊂ Tγ(t0)Λ, a vector space of dimension n − 1 disjoint from Tγ(t0)Λ−. For convenience, we set En(t0) = En−1(t0) ⊕ Rv for some v /∈ Tγ(t0)Λ + Tγ(t0)Λ−. Let E•(t) = U(t, t0)E•(t0). From [18, Lemma 2.1], there exists Bt = O(e−λ1t) such that En(t) is given by δη = Btδx. Now, if δ ∈ En−1(t), we have σ(Hp, δ) = 0 since En−1(t)⊕ RHp = Tγ(t)Λ and Λ is a Lagrangian manifold. From (C.1), we have (C.7) Hp(γ(t)) = γ̇(t) = −λν(ãeην +O(e−εt))e−λν t, where eην is the basis vector corresponding to ην and then (C.8) 0 = σ(eλν tHp, δ) = λν ãδyν +O(e−εt)|δ|. It follows that δ ∈ En−1(t) if and only if (δyν , δη) = B̃tδy′ with B̃t = O(e−εt). Using Tγ(t)Λ = En−1(t)⊕ RHp, we obtain that Tγ(t)Λ has a basis formed of vector fj(t) such that fj =eyj +O(e−εt) for j 6= ν(C.9) fν =eην +O(e−εt).(C.10) SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 57 In the (x, ξ)-coordinates, Tγ(t)Λ has a basis formed of vector f̃j(t) of the form f̃j =eξj + λjexj +O(e−εt) for j 6= ν(C.11) f̃ν =eξν − λjexν +O(e−εt),(C.12) and the lemma follows. � Appendix D. Asymptotic behaviour of certain integrals Lemma D.1. Let α ∈ C, Reα > 0, β ∈ R and χ ∈ C∞0 (]−∞, 1/2[) be such that χ = 1 near 0. As λ goes to +∞, we have (D.1) eiλttα(− ln t)βχ(t) dt = Γ(α)(ln λ)β(−iλ)−α(1 + o(1)). Moreover, if β ∈ N, we get (D.2) eiλttα(− ln t)βχ(t) dt = (−iλ)−α (j)(α)(−1)j ln(−iλ) +O(λ−∞). Finally, if s(t) ∈ C∞(]0,+∞[) satisfies (D.3) |∂jt s(t)| = o tα−j(− ln t)β for all j ∈ N and t→ 0, then (D.4) eiλts(t)χ(t) (ln λ)βλ−α Here (−iλ)−α = eiαπ/2λ−α and ln(−iλ) = lnλ− iπ/2. Remark D.2. Notice that one obtains the behaviour of these quantities as λ → −∞ by taking the complex conjugate in these expressions. Proof. We begin with (D.2) and assume first that β = 0. Then, we can write eiλttαχ(t) = lim ei(λ+iε)ttαχ(t) = lim I1(α, ε) + I2(α, ε) ,(D.5) where I1(α, ε) = e−(ε−iλ)ttα ,(D.6) I2(α, ε) = ei(λ+iε)ttα(1− χ(t)) dt ·(D.7) It is clear that (D.8) I1(α, ε) = (ε− iλ)−αΓ(α), where z−α is well defined on C\]−∞, 0] and real positive on ]0,+∞[. In particular (D.9) lim I1(α, ε) = (−iλ)−αΓ(α). 58 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Concerning, I2(α, ε), we remark that r(t, α) = t α−1(1− χ(t)) is a symbol which satisfies (D.10) |∂jt ∂kαr(t, α)| . 〈t〉Reα−1−j〈ln t〉k, for all j, k ∈ N uniformly for t ∈ [0,+∞[ and α in a compact set of {Re z > 0}. Then, making integration by parts in (D.7), we obtain (D.11) I2(α, ε) = (ε− iλ)j e(iλ−ε)t∂jt r(t, α) dt, for all j ∈ N. Now, if j is large enough (j > Reα), ∂jt r(t, α) is integrable in time uniformly with respect to ε. In particular, for such j, (D.12) lim I2(α, ε) = e ijπ/2λ−j eiλt∂ t r(t, α) dt, and then (see (D.10) or Cauchy’s formula) ∂kα lim I2(α, ε) =e ijπ/2λ−j eiλt∂ αr(t, α) dt =O(λ−∞),(D.13) for all k ∈ N. Then we obtain (D.2) for β = 0. To obtain the result for β ∈ N, it is enough to see that eiλttα(ln t)βχ(t) eiλttαχ(t) (−iλ)−αΓ(α) + ∂βα lim I2(α, ε) =(−iλ)−α (j)(α) − ln(−iλ) +O(λ−∞),(D.14) from (D.13). Thus, (D.2) is proved. Let u be a function C∞(]0,+∞[) be such that (D.15) |∂jt u(t)| . tReα−j(− ln t)β, near 0. Let ϕ ∈ C∞(R) such that ϕ = 1 for t < 1 and ϕ = 0 for t > 2. For δ > 0, we have (D.16) eiλtu(t)χ(t) 1−ϕ(t/δ) = (−iλ)−N eiλt∂Nt u(t)χ(t) 1−ϕ(t/δ) for all N . If one of the derivatives falls on 1−ϕ(t/δ), the support of this contribution is inside [δ, 2δ]. Therefore, the corresponding term will be bounded by δReα−N−1(ln δ)β and will contribute like δReα−N (− ln δ)β to the integral. If ones of the derivatives falls on χ(t), the support of the integrand will be a compact set outside of 0 and then this function will be O(1). The contribution to the integral of such term will be like 1. SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 59 If all the derivatives fall on u(t)t−1, we corresponding term will satisfies eiλt∂Nt u(t)t−1 1− ϕ(t/δ) dt =O(1) tReα−1−N (− ln t)β(1− χ(t))dt .(− ln δ)βδReα−N ,(D.17) for N large enough (N > Reα). From this 3 cases, we deduce (D.18) eiλtu(t)χ(t) 1− ϕ(t/δ) (− ln δ)βδα−Nλ−N Taking δ = (ελ)−1, we get (D.19) eiλtu(t)χ(t) 1− ϕ(t/δ) ε(lnλ)βλ−α as λ→ +∞. We now assume (D.3), and we want to prove (D.4). Since, for t small enough (D.20) tReα−1(− ln t)β . tReα(− ln t)β we get eiλts(t)χ(t)ϕ(t/δ) ∣∣∣ =oδ→0(1) tReα−1(− ln t)βdt =oδ→0(1)δ Re α(− ln δ)β .(D.21) Here oδ→0(1) stands for a term which goes to 0 as δ goes to 0. If δ = (ελ) −1, we get (D.22) eiλts(t)χ(t)ϕ(t/δ) ∣∣∣ = oλ→+∞(1)λ−α(ln λ)β, when λ → +∞ and ε fixed. Taking ε small enough in (D.19), and then λ large enough in (D.22), we get (D.4). We are left with (D.1). We need to compute (D.23) I = eiλttα(− ln t)βϕ(t/δ) dt Performing the change of variable s = λt, we get I =λ−α ∫ 2/ε eissα(ln λ− ln s)βϕ(εs) ds =(lnλ)βλ−α ∫ 2/ε eissα(1− ln s/ lnλ)βϕ(εs) ds .(D.24) 60 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND We remark that, in the previous equation, − ln s/ lnλ > − ln(2/ε)/ ln λ > −1/2 for λ large enough. Using (1 + u)β = 1 +O(|u|+ |u|max(1,β)) for u > −1/2, we get I =(lnλ)βλ−α ∫ 2/ε eissαϕ(εs) + (lnλ)βλ−α ∫ 2/ε sReαO ( | ln s| ( | ln s| )max(1,β)) ϕ(εs) =(lnλ)β eiλttα(− ln t)βϕ(t/δ) dt (lnλ)β−1λ−α .(D.25) Note that the Oε in (D.25) depends on ε. 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Anal. i Priložen. 11 (1977), no. 4, 6–18, 96. http://arxiv.org/abs/math/0602069 62 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND Ivana Alexandrova, Department of Mathematics, East Carolina University, Greenville, NC 27858, USA E-mail address: alexandrovai@ecu.edu Jean-François Bony, Institut de Mathématiques de Bordeaux, (UMR CNRS 5251), Université de Bordeaux I, 33405 Talence, France E-mail address: bony@math.u-bordeaux1.fr Thierry Ramond, Mathématiques, Université Paris Sud, (UMR CNRS 8628), 91405 Orsay, France E-mail address: thierry.ramond@math.u-psud.fr 1. Introduction 2. Assumptions and main results 3. Proof of the main resolvent estimate 4. Representation of the Scattering Amplitude 5. Computations before the critical point 5.1. Computation of u- in the incoming region 5.2. Computation of u- along -k 6. Computation of u- at the critical point 6.1. Study of the transport equations for the phases 6.2. Taylor expansions of + and k1 6.3. Asymptotics near the critical point for the trajectories 6.4. Computation of the jk's 7. Computations after the critical point 7.1. Stationary phase expansion in the outgoing region 7.2. Behaviour of the phase function 7.3. Integration with respect to time Appendix A. Proof of Proposition 2.5 Appendix B. A lower bound for the resolvent Appendix C. Lagrangian manifolds which are transverse to Appendix D. Asymptotic behaviour of certain integrals References